Dualities and Representations of Lie Superalgebras 0821891189, 978-0-8218-9118-6

Citation preview

Dualities and Representations of Lie Superalgebras 3HUN *EN #HENG 7EIQIANG 7ANG

'RADUATE 3TUDIES IN -ATHEMATICS 6OLUME 

!MERICAN -ATHEMATICAL 3OCIETY

Dualities and Representations of Lie Superalgebras

Dualities and Representations of Lie Superalgebras

Shun-Jen Cheng Weiqiang Wang

Graduate Studies in Mathematics Volume 144

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 17B10, 17B20.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-144

Library of Congress Cataloging-in-Publication Data Cheng, Shun-Jen, 1963– Dualities and representations of Lie superalgebras / Shun-Jen Cheng, Weiqiang Wang. pages cm. — (Graduate studies in mathematics ; volume 144) Includes bibliographical references and index. ISBN 978-0-8218-9118-6 (alk. paper) 1. Lie superalgebras. 2. Duality theory (Mathematics) I. Wang, Weiqiang, 1970– II. Title. QA252.3.C44 2013 512.482—dc23 2012031989

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

17 16 15 14 13 12

To Mei-Hui, Xiaohui, Isabelle, and our parents

Contents

Preface Chapter 1. Lie superalgebra ABC

xiii 1

§1.1. Lie superalgebras: Definitions and examples 1.1.1. Basic definitions 1.1.2. The general and special linear Lie superalgebras 1.1.3. The ortho-symplectic Lie superalgebras 1.1.4. The queer Lie superalgebras 1.1.5. The periplectic and exceptional Lie superalgebras 1.1.6. The Cartan series 1.1.7. The classification theorem

1 2 4 6 8 9 10 12

§1.2. Structures of classical Lie superalgebras 1.2.1. A basic structure theorem 1.2.2. Invariant bilinear forms for gl and osp 1.2.3. Root system and Weyl group for gl(m|n) 1.2.4. Root system and Weyl group for spo(2m|2n + 1) 1.2.5. Root system and Weyl group for spo(2m|2n) 1.2.6. Root system and odd invariant form for q(n)

13 13 16 16 17 17 18

§1.3. Non-conjugate positive systems and odd reflections 1.3.1. Positive systems and fundamental systems 1.3.2. Positive and fundamental systems for gl(m|n) 1.3.3. Positive and fundamental systems for spo(2m|2n + 1) 1.3.4. Positive and fundamental systems for spo(2m|2n) 1.3.5. Conjugacy classes of fundamental systems

19 19 21 22 23 25

§1.4. Odd and real reflections 1.4.1. A fundamental lemma

26 26 vii

Contents

viii

1.4.2. 1.4.3. 1.4.4. 1.4.5.

Odd reflections Real reflections Reflections and fundamental systems Examples

27 28 28 30

§1.5. Highest weight theory 1.5.1. The Poincar´e-Birkhoff-Witt (PBW) Theorem 1.5.2. Representations of solvable Lie superalgebras 1.5.3. Highest weight theory for basic Lie superalgebras 1.5.4. Highest weight theory for q(n)

31 31 32 33 35

§1.6. Exercises

37

Notes

40

Chapter 2. Finite-dimensional modules

43

§2.1. Classification of finite-dimensional simple modules 2.1.1. Finite-dimensional simple modules of gl(m|n) 2.1.2. Finite-dimensional simple modules of spo(2m|2) 2.1.3. A virtual character formula 2.1.4. Finite-dimensional simple modules of spo(2m|2n + 1) 2.1.5. Finite-dimensional simple modules of spo(2m|2n) 2.1.6. Finite-dimensional simple modules of q(n)

43 43 45 45 47 50 53

§2.2. Harish-Chandra homomorphism and linkage 2.2.1. Supersymmetrization 2.2.2. Central characters 2.2.3. Harish-Chandra homomorphism for basic Lie superalgebras 2.2.4. Invariant polynomials for gl and osp 2.2.5. Image of Harish-Chandra homomorphism for gl and osp 2.2.6. Linkage for gl and osp 2.2.7. Typical finite-dimensional irreducible characters

55 55 56 57 59 62 65 68

§2.3. Harish-Chandra homomorphism and linkage for q(n) 2.3.1. Central characters for q(n) 2.3.2. Harish-Chandra homomorphism for q(n) 2.3.3. Linkage for q(n) 2.3.4. Typical finite-dimensional characters of q(n)

69 70 70 74 76

§2.4. Extremal weights of finite-dimensional simple modules 2.4.1. Extremal weights for gl(m|n) 2.4.2. Extremal weights for spo(2m|2n + 1) 2.4.3. Extremal weights for spo(2m|2n)

77 77 80 82

§2.5. Exercises

85

Notes

89

Chapter 3. Schur duality

91

Contents

§3.1. Generalities for associative superalgebras 3.1.1. Classification of simple superalgebras 3.1.2. Wedderburn Theorem and Schur’s Lemma 3.1.3. Double centralizer property for superalgebras 3.1.4. Split conjugacy classes in a finite supergroup

ix

91 92 94 95 96

§3.2. Schur-Sergeev duality of type A 3.2.1. Schur-Sergeev duality, I 3.2.2. Schur-Sergeev duality, II 3.2.3. The character formula 3.2.4. The classical Schur duality 3.2.5. Degree of atypicality of λ 3.2.6. Category of polynomial modules

98 98 100 104 105 106 108

§3.3. Representation theory of the algebra Hn 3.3.1. A double cover 3.3.2. Split conjugacy classes in Bn 3.3.3. A ring structure on R− 3.3.4. The characteristic map 3.3.5. The basic spin module 3.3.6. The irreducible characters

109 110 111 114 116 118 119

§3.4. Schur-Sergeev duality for q(n) 3.4.1. A double centralizer property 3.4.2. The Sergeev duality 3.4.3. The irreducible character formula

121 121 123 125

§3.5. Exercises

125

Notes

128

Chapter 4. Classical invariant theory

131

§4.1. FFT for the general linear Lie group 4.1.1. General invariant theory 4.1.2. Tensor and multilinear FFT for GL(V ) 4.1.3. Formulation of the polynomial FFT for GL(V ) 4.1.4. Polarization and restitution

131 132 133 134 135

§4.2. Polynomial FFT for classical groups 4.2.1. A reduction theorem of Weyl 4.2.2. The symplectic and orthogonal groups 4.2.3. Formulation of the polynomial FFT 4.2.4. From basic to general polynomial FFT 4.2.5. The basic case

137 137 139 140 141 142

§4.3. Tensor and supersymmetric FFT for classical groups 4.3.1. Tensor FFT for classical groups 4.3.2. From tensor FFT to supersymmetric FFT

145 145 147

Contents

x

§4.4. Exercises

149

Notes

150

Chapter 5. Howe duality

151

§5.1. Weyl-Clifford algebra and classical Lie superalgebras 5.1.1. Weyl-Clifford algebra 5.1.2. A filtration on Weyl-Clifford algebra 5.1.3. Relation to classical Lie superalgebras 5.1.4. A general duality theorem 5.1.5. A duality for Weyl-Clifford algebras

152 152 154 155 157 159

§5.2. 5.2.1. 5.2.2. 5.2.3. 5.2.4.

Howe duality for type A and type Q Howe dual pair (GL(k), gl(m|n)) (GL(k), gl(m|n))-Howe duality Formulas for highest weight vectors (q(m), q(n))-Howe duality

160 160 162 164 166

§5.3. 5.3.1. 5.3.2. 5.3.3. 5.3.4. 5.3.5.

Howe duality for symplectic and orthogonal groups Howe dual pair (Sp(V ), osp(2m|2n)) (Sp(V ), osp(2m|2n))-Howe duality Irreducible modules of O(V ) Howe dual pair (O(k), spo(2m|2n)) (O(V ), spo(2m|2n))-Howe duality

169 170 172 175 177 178

§5.4. 5.4.1. 5.4.2. 5.4.3. 5.4.4. 5.4.5.

Howe duality for infinite-dimensional Lie algebras Lie algebras a∞ , c∞ , and d∞ The fermionic Fock space (GL(), a∞ )-Howe duality (Sp(k), c∞ )-Howe duality (O(k), d∞ )-Howe duality

180 180 183 184 187 190

§5.5. Character formula for Lie superalgebras 5.5.1. Characters for modules of Lie algebras c∞ and d∞ 5.5.2. Characters of oscillator osp(2m|2n)-modules 5.5.3. Characters for oscillator spo(2m|2n)-modules

192 192 193 195

§5.6. Exercises

197

Notes

201

Chapter 6. Super duality §6.1. Lie superalgebras of classical types 6.1.1. Head, tail, and master diagrams 6.1.2. The index sets 6.1.3. Infinite-rank Lie superalgebras 6.1.4. The case of m = 0 6.1.5. Finite-dimensional Lie superalgebras

205 206 206 208 208 211 213

Contents

6.1.6. §6.2. 6.2.1. 6.2.2. 6.2.3. 6.2.4. 6.2.5. §6.3. 6.3.1. 6.3.2. 6.3.3. 6.3.4. §6.4. 6.4.1. 6.4.2. 6.4.3. 6.4.4. 6.4.5. §6.5. 6.5.1. 6.5.2. 6.5.3. 6.5.4. 6.5.5. 6.5.6. §6.6. Notes

xi

Central extensions The module categories Category of polynomial modules revisited Parabolic subalgebras and dominant weights  The categories O, O, and O n The categories On , On , and O Truncation functors The irreducible character formulas Two sequences of Borel subalgebras of  g Odd reflections and highest weight modules The functors T and T Character formulas Kostant homology and KLV polynomials Homology and cohomology of Lie superalgebras Kostant u− -homology and u-cohomology Comparison of Kostant homology groups Kazhdan-Lusztig-Vogan (KLV) polynomials Stability of KLV polynomials Super duality as an equivalence of categories Extensions a` la Baer-Yoneda  Relating extensions in O, O, and O f f f Categories O , O , and O Lifting highest weight modules Super duality and strategy of proof The proof of super duality Exercises

Appendix A. Symmetric functions §A.1. The ring Λ and Schur functions A.1.1. The ring Λ A.1.2. Schur functions A.1.3. Skew Schur functions A.1.4. The Frobenius characteristic map §A.2. Supersymmetric polynomials A.2.1. The ring of supersymmetric polynomials A.2.2. Super Schur functions §A.3. The ring Γ and Schur Q-functions A.3.1. The ring Γ A.3.2. Schur Q-functions A.3.3. Inner product on Γ

213 214 215 217 218 220 221 222 223 225 228 231 232 232 235 236 239 240 241 241 243 247 247 248 250 255 258 261 261 261 265 268 270 271 271 273 275 275 277 278

Contents

xii

A.3.4. A.3.5.

A characterization of Γ Relating Λ and Γ

280 281

§A.4. The Boson-Fermion correspondence A.4.1. The Maya diagrams A.4.2. Partitions A.4.3. Fermions and fermionic Fock space A.4.4. Charge and energy A.4.5. From Bosons to Fermions A.4.6. Fermions and Schur functions A.4.7. Jacobi triple product identity

282 282 282 284 286 287 289 289

Notes

290

Bibliography

291

Index

299

Preface

Lie algebras, Lie groups, and their representation theories are parts of the mathematical language describing symmetries, and they have played a central role in modern mathematics. An early motivation of studying Lie superalgebras as a generalization of Lie algebras came from supersymmetry in mathematical physics. Ever since a Cartan-Killing type classification of finite-dimensional complex Lie superalgebras was obtained by Kac [60] in 1977, the theory of Lie superalgebras has established itself as a prominent subject in modern mathematics. An independent classification of the finite-dimensional complex simple Lie superalgebras whose even subalgebras are reductive (called simple Lie superalgebras of classical type) was given by Scheunert, Nahm, and Rittenberg in [106]. The goal of this book is a systematic account of the structure and representation theory of finite-dimensional complex Lie superalgebras of classical type. The book intends to serve as a rigorous introduction to representation theory of Lie superalgebras on one hand, and, on the other hand, it covers a new approach developed in the past few years toward understanding the Bernstein-Gelfand-Gelfand (BGG) category for classical Lie superalgebras. In spite of much interest in representations of Lie superalgebras stimulated by mathematical physics, these basic topics have not been treated in depth in book form before. The reason seems to be that the representation theory of Lie superalgebras is dramatically different from that of complex semisimple Lie algebras, and a systematic, yet accessible, approach toward the basic problem of finding irreducible characters for Lie superalgebras was not available in a great generality until very recently. We are aware that there is an enormous literature with numerous partial results for Lie superalgebras, and it is not our intention to make this book an encyclopedia. Rather, we treat in depth the representation theory of the three most important classes of Lie superalgebras, namely, the general linear Lie superalgebras gl(m|n),

xiii

Preface

xiv

the ortho-symplectic Lie superalgebras osp(m|2n), and the queer Lie superalgebras q(n). To a large extent, representations of sl(m|n) can be understood via gl(m|n). The lecture notes [32] by the authors can be considered as a prototype for this book. The presentation in this book is organized around three dualities with a unifying theme of determining irreducible characters: Schur duality,

Howe duality, and super duality.

The new book of Musson [90] treats in detail the ring theoretical aspects of the universal enveloping algebras of Lie superalgebras as well as the basic structures of simple Lie superalgebras. There are two superalgebra generalizations of Schur duality. The first one, due to Sergeev [110] and independently Berele-Regev [7], is an interplay between the general linear Lie superalgebra gl(m|n) and the symmetric group, which incorporates the trivial and sign modules in a unified framework. The irreducible polynomial characters of gl(m|n) arising this way are given by the super Schur polynomials. The second one, called Sergeev duality, is an interplay between the queer Lie superalgebra q(n) and a twisted hyperoctahedral group algebra. The Schur Q-functions and related combinatorics of shifted tableaux appear naturally in the description of the irreducible polynomial characters of q(n). It has been observed that much of the classical invariant theory for the polynomial algebra has a parallel theory for the exterior algebra as well. The First Fundamental Theorem (FFT) for both polynomial invariants and skew invariants for classical groups admits natural reformulation and extension in the theory of Howe’s reductive dual pairs [51, 52]. Lie superalgebras allow an elegant and uniform treatment of Howe duality on the polynomial and exterior algebras (cf. Cheng-Wang [29]). For the general linear Lie groups, Schur duality, Howe duality, and FFT are equivalent. Unlike Schur duality, Howe duality treats classical Lie groups and (super)algebras beyond type A equally well. The Howe dualities allow us to determine the character formulas for the irreducible modules appearing in the dualities. The third duality, super duality, has a completely different flavor. It views the representation theories of Lie superalgebras and Lie algebras as two sides of the same coin, and it is an unexpected and rather powerful approach developed in the past few years by the authors and their collaborators, culminating in Cheng-LamWang [24]. The super duality approach allows one to overcome in a conceptual way various major obstacles in super representation theory via an equivalence of module categories of Lie algebras and Lie superalgebras. Schur, Howe, and super dualities provide approaches to the irreducible character problem in increasing generality and sophistication. Schur and Howe dualities only offer a solution to the irreducible character problem for modules in some semisimple subcategories. On the other hand, super duality provides a conceptual solution to the long-standing irreducible character problem in fairly general BGG

Preface

xv

categories (including all finite-dimensional modules) over classical Lie superalgebras in terms of the usual Kazhdan-Lusztig polynomials of classical Lie algebras. Totally different and independent approaches to the irreducible character problem of finite-dimensional gl(m|n)-modules have been developed by Serganova [107] and Brundan [11]. Also Brundan’s conjecture on irreducible characters of gl(m|n) in the full BGG category O has recently been proved in [26]. Super duality again plays a crucial role in the proof. However, this latest approach to the full BGG category O is beyond the scope of this book. The book is largely self-contained and should be accessible to graduate students and non-experts as well. Besides assuming basic knowledge of entry-level graduate algebra (and some familiarity with basic homological algebra in the final Chapter 6), the other prerequisite is a one-semester course in the theory of finitedimensional semisimple Lie algebras. For example, either the book by Humphreys or the first half of the book by Carter on semisimple Lie algebras is sufficient. Some familiarity with symmetric functions and representations of symmetric groups can be sometimes useful, and Appendix A provides a quick summary for our purpose. It is possible that super experts may also benefit from the book, as several “folklore” results are rigorously proved and occasionally corrected in great detail here, sometimes with new proofs. The proofs of some of these results can be at times rather difficult to trace or read in the literature (and not merely because they might be in a different language). Here is a broad outline of the book chapter by chapter. Each chapter ends with exercises and historical notes. Though we have tried to attribute the main results accurately and fairly, we apologize beforehand for any unintended omissions and mistakes. Chapter 1 starts by defining various classes of Lie superalgebras. For the basic Lie superalgebras, we introduce the invariant bilinear forms, root systems, fundamental systems, and Weyl groups. Positive systems and fundamental systems for basic Lie superalgebras are not conjugate under the Weyl group, and the notion of odd reflections is introduced to relate non-conjugate positive systems. The PBW theorem for the universal enveloping algebra of a Lie superalgebra is formulated, and highest weight theory for basic Lie superalgebras and q(n) is developed. In Chapter 2, we focus on Lie superalgebras of types gl, osp and q. We classify their finite-dimensional simple modules using odd reflection techniques. We then formulate and establish precisely the images of the respective Harish-Chandra homomorphisms and linkage principles. We end with a Young diagrammatic description of the extremal weights in the simple polynomial gl(m|n)-modules and finite-dimensional simple osp(m|2n)-modules. It takes considerably more effort to formulate and prove these results for Lie superalgebras than for semisimple Lie algebras because of the existence of non-conjugate Borel subalgebras and the limited role of Weyl groups for Lie superalgebras.

xvi

Preface

Schur duality for Lie superalgebras is developed in Chapter 3. We start with some results on the structure of associative superalgebras including the super variants of the Wedderburn theorem, Schur’s lemma, and the double centralizer property. The Schur-Sergeev duality for gl(m|n) is proved, and it provides a classification of irreducible polynomial gl(m|n)-modules. As a consequence, the characters of the simple polynomial gl(m|n)-modules are given by the super Schur polynomials. On the algebraic combinatorial level, there is a natural super generalization of the notion of semistandard tableau, which is a hybrid of the traditional version and its conjugate counterpart. The Schur-Sergeev duality for q(n) requires understanding the representation theory of a twisted hyperoctahedral group algebra, which we develop from scratch. The characters of the simple polynomial q(n)-modules are given by the Schur Q-polynomials up to some 2-powers. In Chapter 4, we give a quick introduction to classical invariant theory, which serves as a preparation for Howe duality in the next chapter. We describe several versions of the FFT for the classical groups, i.e., a tensor algebra version, a polynomial algebra version, and a supersymmetric algebra version. Howe duality is the main topic of Chapter 5. Like Schur duality, Howe duality involves commuting actions of a classical Lie group G and a classical superalgebra g on a supersymmetric algebra. The precise relation between the classical Lie superalgebras and Weyl-Clifford algebras WC is established. According to the FFT for classical invariant theory in Chapter 4 when applied to the G-action on the associated graded algebra gr WC, the basic invariants generating (gr WC)G turn out to form the associated graded space for a Lie superalgebra g . From this it follows that the algebra of G-invariants WCG is generated by g . Multiplicity-free decompositions for various (G, g )-Howe dualities are obtained explicitly. Character formulas for the irreducible g -modules appearing in (G, g )-Howe duality are then obtained via a comparison with Howe duality involving classical groups G and infinite-dimensional Lie algebras, which we develop in detail. Finally in Chapter 6, we develop a super duality approach to obtain a complete and conceptual solution of the irreducible character problem in certain parabolic Bernstein-Gelfand-Gelfand categories for general linear and ortho-symplectic Lie superalgebras. This chapter is technically more sophisticated than the earlier chapters. Super duality is an equivalence of categories between parabolic categories for Lie superalgebras and their Lie algebra counterparts at an infinite-rank limit, and it matches the corresponding parabolic Verma modules, irreducible modules, Kostant u-homology groups, and Kazhdan-Lusztig-Vogan polynomials. Truncation functors are introduced to relate the BGG categories for infinite-rank and finite-rank Lie superalgebras. In this way, we obtain a solution a` la Kazhdan-Lusztig of the irreducible character problem in the corresponding parabolic BGG categories for finite-dimensional basic Lie superalgebras.

Preface

xvii

There is an appendix in the book. In Appendix A, we have included a fairly self-contained treatment of some elementary aspects of symmetric function theory, including Schur functions, supersymmetric functions and Schur Q-functions. The celebrated boson-fermion correspondence serves as a prominent example relating superalgebras to mathematical physics and algebraic combinatorics. The Fock space therein is used in setting up the Howe duality for infinite-dimensional Lie algebras in Chapter 5. For a one-semester introductory course on Lie superalgebras, we recommend two plausible ways of using this book. A first approach uses Chapters 1, 2, 3, with possible supplements from Chapter 5 and Appendix A. A second approach uses Chapters 1, 3, 5 with possible supplements from Chapter 4 and Appendix A. It is also possible to use this book for a course on the interaction between representations of Lie superalgebras and algebraic combinatorics. The more advanced Chapter 6 can be used in a research seminar. Acknowledgment. The book project started with the lecture notes [32] of the authors, which were an expanded written account of a series of lectures delivered by the second-named author in the summer school at East China Normal University, Shanghai, in July 2009. In a graduate course at the University of Virginia in Spring 2010, the second-named author lectured on what became a large portion of Chapters 3, 4, and 5 of the book. The materials in Chapter 3 on Schur duality have been used by the second-named author in the winter school in Taipei in December 2010. Part of the materials in the first three chapters have also been used by the first-named author in a lecture series in Shanghai in March 2011, and then in a lecture series by both authors in a workshop in Tehran in May 2011. We thank the participants in all these occasions for their helpful suggestions and feedback, and we especially thank Constance Baltera, Jae-Hoon Kwon, Li Luo, Jinkui Wan, and Youjie Wang for their corrections. We are grateful to Ngau Lam for his collaboration which has changed our way of thinking about the subject of Lie superalgebras. The first-named author gratefully acknowledges the support from the National Science Council, Taiwan, and the second-named author gratefully acknowledges the continuing support of the National Science Foundation, USA.

Shun-Jen Cheng Weiqiang Wang

Chapter 1

Lie superalgebra ABC

We start by introducing the basic notions and definitions in the theory of Lie superalgebras, such as basic and queer Lie superalgebras, Cartan and Borel subalgebras, root systems, positive and fundamental systems. We formulate the main structure results for the basic Lie superalgebras and the queer Lie superalgebras. We describe in detail the structures of Lie superalgebras of type gl, osp and q. A distinguishing feature for Lie superalgebras is that Borel subalgebras, positive systems, or fundamental systems of a simple finite-dimensional Lie superalgebra may not be conjugate under the action of the corresponding Weyl group; rather, they are shown to be related to each other by real and odd reflections. A highest weight theory is developed for Lie superalgebras. We describe how fundamental systems are related and how highest weights are transformed by an odd reflection.

1.1. Lie superalgebras: Definitions and examples Throughout this book we will work over the field C of complex numbers. Let ¯ 1} ¯ Z2 = {0, denote the group of two elements and let Sn denote the symmetric group in n letters. In this section, we introduce many examples of Lie superalgebras. The examples, most relevant to this book, of the general linear and ortho-symplectic Lie superalgebras are introduced first. Other series of finite-dimensional simple Lie 1

1. Lie superalgebra ABC

2

superalgebras of classical type, namely, the queer and the periplectic Lie superalgebras, along with the three exceptional ones, are then described. The finitedimensional Cartan type Lie superalgebras are then realized explicitly as subalgebras of the Lie superalgebra of polynomial vector fields on a purely odd dimensional superspace. The section ends with Kac’s classification theorem of the finite-dimensional simple Lie superalgebras over C, which we state without proof. 1.1.1. Basic definitions. A vector superspace V is a vector space endowed with a Z2 -gradation: V = V0¯ ⊕V1¯ . The dimension of the vector superspace V is the tuple dimV = (dimV0¯ | dimV1¯ ) or sometimes dimV = dimV0¯ + dimV1¯ (which should be clear from the context). The superdimension of V is defined to be sdimV := dimV0¯ − dimV1¯ . We denote the superspace with even subspace Cm and odd subspace Cn by Cm|n . It has dimension (m|n). The parity of a homogeneous element a ∈ Vi is denoted by |a| = i, i ∈ Z2 . An element in V0¯ is called even, while an element in V1¯ is called odd. A subspace of a vector superspace V = V0¯ ⊕ V1¯ is a vector superspace W = W0¯ ⊕ W1¯ ⊆ V with compatible Z2 -gradation, i.e., Wi ⊆ Vi , for i ∈ Z2 . Let V be a superspace. Throughout the book, when we write |v| for an element v ∈ V , we will always implicitly assume that v is a homogeneous element and automatically extend the relevant formulas by linearity (whenever applicable). Also, note that if V and W are superspaces, then the space of linear transformations from V to W is naturally a vector superspace. In particular, the space of endomorphisms of V , denoted by End(V ), is a vector superspace. When V = Cm|n we write I = Im|n = IV for the identity matrix on V . There is a parity reversing functor Π on the category of vector superspaces. For a vector superspace V = V0¯ ⊕V1¯ , we let Π(V ) = Π(V )0¯ ⊕ Π(V )1¯ ,

Π(V )i = Vi+1¯ , ∀i ∈ Z2 .

Clearly, Π2 = I. Definition 1.1. A superalgebra A, sometimes also called a Z2 -graded algebra, is a vector superspace A = A0¯ ⊕ A1¯ equipped with a bilinear multiplication satisfying Ai A j ⊆ Ai+ j , for i, j ∈ Z2 . A module M over a superalgebra A is always understood in the Z2 -graded sense, that is M = M0¯ ⊕ M1¯ such that Ai M j ⊆ Mi+ j , for i, j ∈ Z2 . Subalgebras and ideals of superalgebras are also understood in the Z2 -graded sense. A superalgebra that has no nontrivial ideal is called simple. A homomorphism between A-modules M and N is a linear map f : M → N satisfying that f (am) = a f (m), for all a ∈ A, m ∈ M. A homomorphism f : M → N is of degree |f| ∈ Z2 if f (Mi ) ⊆ Mi+| f | for i ∈ Z2 . A homomorphism between modules M and N of a superalgebra A is sometimes understood in the literature as a linear map f : M → N of parity | f | ∈ Z2 which satisfies () f (am) = (−1)|a|·| f | a f (m), for homogeneous a ∈ A, m ∈ M. Let us call

1.1. Lie superalgebras: Definitions and examples

3

such a map a -homomorphism. These two definitions can be converted to each other as follows. Given a homomorphism f : M → N of degree | f | in the sense of Definition 1.1, we define f † : M → N by the formula (1.1)

f † (x) := (−1)| f |·|x| f (x).

Then f † is a -homomorphism. Conversely, (1.1) also converts a -homomorphism into a homomorphism as in Definition 1.1. Now we come to the definition of the main object of our study. Definition 1.2. A Lie superalgebra is a superalgebra g = g0¯ ⊕ g1¯ with bilinear multiplication [·, ·] satisfying the following two axioms: for homogeneous elements a, b, c ∈ g, (1) Skew-supersymmetry: [a, b] = −(−1)|a|·|b| [b, a]. (2) Super Jacobi identity: [a, [b, c]] = [[a, b], c] + (−1)|a|·|b|[b, [a, c]]. A bilinear form (·, ·) : g × g → C on a Lie superalgebra g is called invariant if ([a, b], c) = (a, [b, c]), for all a, b, c ∈ g. For a Lie superalgebra g = g0¯ ⊕ g1¯ , the even part g0¯ is a Lie algebra. Hence, if g1¯ = 0, then g is just a usual Lie algebra. A Lie superalgebra g with purely odd part, i.e., g0¯ = 0, has to be abelian, i.e., [g, g] = 0. Definition 1.3. Let g and g be Lie superalgebras. A homomorphism of Lie superalgebras is an even linear map f : g → g satisfying f ([a, b]) = [ f (a), f (b)],

a, b ∈ g.

Example 1.4. (1) Let A = A0¯ ⊕ A1¯ be an associative superalgebra. We can make A into a Lie superalgebra by letting [a, b] := ab − (−1)|a|·|b| ba, for homogeneous a, b ∈ A and extending [·, ·] by bilinearity. (2) Let g be a Lie superalgebra. Then End(g) is an associative superalgebra, and hence it carries a structure of a Lie superalgebra by (1). We define the adjoint map ad : g → End(g) by ad (a)(b) := [a, b],

a, b ∈ g.

Then ad is a homomorphism of Lie superalgebras due to the super Jacobi identity. The resulting action of g on itself is called the adjoint action. (3) Let A = A0¯ ⊕ A1¯ be a superalgebra. An endomorphism D ∈ End(A)s , for s ∈ Z2 , is called a derivation of degree s if it satisfies that D(ab) = D(a)b + (−1)s|a| aD(b),

a, b ∈ A.

1. Lie superalgebra ABC

4

Denote by Der(A)s the space of derivations on A of degree s. One verifies that the superspace of derivations of A, Der(A) = Der(A)0¯ ⊕ Der(A)1¯ , is a subalgebra of the Lie superalgebra (End(A), [·, ·]). In the case when g is a Lie superalgebra we have adg ∈ Der(g), for all g ∈ g, by the super Jacobi identity. Indeed, such derivations are called inner derivations. The inner derivations form an ideal in Der(g). (4) Let g = g0¯ ⊕ g1¯ be a superspace such that g0¯ = Cz is one-dimensional. Suppose that we have a symmetric bilinear form B(·, ·) on g1¯ . We can make g into a Lie superalgebra by letting z commute with g and declaring [v, w] := B(v, w)z,

v, w ∈ g1¯ .

The special cases when B(·, ·) is zero and when B(·, ·) is non-degenerate, respectively, are of particular interest. Indeed, their corresponding universal enveloping superalgebras (see Section 1.5.1) are isomorphic to the exterior and Clifford superalgebras of Section 1.1.6 and Definition 3.33, respectively. For a Lie superalgebra g = g0¯ ⊕ g1¯ , the restriction of the adjoint homomorphism ad |g0¯ : g0¯ → End(g1¯ ) is a homomorphism of Lie algebras. That is, g1¯ is a g0¯ -module under the adjoint action. Remark 1.5. To a Lie superalgebra g = g0¯ ⊕ g1¯ we associate the following data: (1) A Lie algebra g0¯ . (2) A g0¯ -module g1¯ induced by the adjoint action. (3) A g0¯ -homomorphism S2 (g1¯ ) → g0¯ induced by the Lie bracket. (4) The condition coming from Definition 1.2(2) with a, b, c ∈ g1¯ . Conversely, the above data determine a Lie superalgebra structure on g0¯ ⊕ g1¯ . 1.1.2. The general and special linear Lie superalgebras. Let V = V0¯ ⊕ V1¯ be a vector superspace so that End(V ) is an associative superalgebra. As in Example 1.4(1), End(V ), equipped with the supercommutator, forms a Lie superalgebra called the general linear Lie superalgebra and denoted by gl(V ). When V = Cm|n we also write gl(m|n) for gl(V ). Choose ordered bases for V0¯ and V1¯ that combine to a homogeneous ordered basis for V . We will make it a convention to parameterize such a basis by the set (1.2)

I(m|n) = {1, . . . , m; 1, . . . , n}

with total order (1.3)

1 < . . . < m < 0 < 1 < . . . < n.

Here 0 is inserted for notational convenience later on. The elementary matrices are accordingly denoted by Ei j , with i, j ∈ I(m|n). With respect to such an ordered

1.1. Lie superalgebras: Definitions and examples

5

basis, End(V ) and gl(V ) can be realized as (m + n) × (m + n) complex matrices of the block form   a b (1.4) g= , c d where a, b, c, and d are respectively m × m, m × n, n × m, and n × n matrices. The even subalgebra gl(V )0¯ consists of matrices of the form (1.4) with b = c = 0, while the odd subspace gl(V )1¯ consists of those with a = d = 0. In particular, gl(V )0¯ ∼ = gl(m) ⊕ gl(n), and as a gl(V )0¯ -module, gl(V )1¯ is self-dual and is isomorphic to (Cm ⊗ Cn∗ ) ⊕ (Cm∗ ⊗ Cn ). Here and below, Cn∗ denotes the dual space of Cn . Remark 1.6. Let Π be the parity reversing functor defined in Section 1.1.1. We have an isomorphism of Lie superalgebras from gl(V ) to gl(ΠV ) by sending T to ΠT Π−1 . When dimV = (m|n), we obtain an isomorphism of Lie superalgebras gl(m|n) ∼ = gl(n|m). For each element g ∈ gl(m|n) of the form (1.4) we define the supertrace as str(g) := tr(a) − tr(d), where tr(x) denotes the trace of the square matrix x. One checks that str([g, g ]) = 0,

for g, g ∈ gl(m|n).

Thus, the subspace sl(m|n) := {g ∈ gl(m|n) | str(g) = 0} is a subalgebra of gl(m|n), and it is called the special linear Lie superalgebra. One verifies directly that [gl(m|n), gl(m|n)] = sl(m|n). Furthermore, sl(m|n) ∼ = sl(n|m), and when m = n and m + n ≥ 2, sl(m|n) is simple. When m = n, sl(m|m) contains a nontrivial center generated by the identity matrix Im|m . For m ≥ 2, sl(m|m)/CIm|m is simple. Example 1.7. Let g = gl(1|1) and consider the following basis for g:         0 1 0 0 1 0 0 0 e= , f= , E1. , E11 = . ¯ 1¯ = 0 0 1 0 0 0 0 1 Set h := E1, ¯ 1¯ + E11 = I1|1 . Then h is central, [e, f ] = h, and sl(1|1) has a basis {e, h, f }. Let I = I0¯ I1¯ be a parametrization of a homogeneous basis of the superspace V = V0¯ ⊕ V1¯ , where I0¯ and I1¯ parameterize the corresponding bases of V0¯ and V1¯ , respectively. For an element i in I we define  0, if i ∈ I0¯ , |i| := 1, if i ∈ I1¯ . For example, for the parametrization I(m|n) with ordering (1.3), we have |i| = 0 for i < 0, and |i| = 1 for i > 0. Choosing a total ordering of the homogeneous

1. Lie superalgebra ABC

6

basis we may identify gl(V ) with the space of |I| × |I| matrices. For such a matrix A = ∑i, j∈I ai j Ei j , ai j ∈ C, we define the supertranspose of A to be (1.5)

Ast :=

∑ (−1)| j|(|i|+| j|)ai j E ji .

i, j∈I

We define the Chevalley automorphism τ : gl(V ) → gl(V ) by the formula (1.6)

τ(A) := −Ast .

It is straightforward to check that τ is an automorphism of Lie superalgebras. We note that τ restricts to an automorphism of sl(V ). Also, for m, n both nonzero, τ has order 4 and hence is, in general, not an involution. 1.1.3. The ortho-symplectic Lie superalgebras. Definition 1.8. Let V = V0¯ ⊕V1¯ be a vector superspace. A bilinear form B(·, ·) : V ×V −→ V is called even (respectively, odd), if B(Vi ,V j ) = 0 unless i + j = 0¯ (respectively, ¯ An even bilinear form B is said to be supersymmetric if B|V¯ ×V¯ is i + j = 1). 0 0 symmetric and B|V1¯ ×V1¯ is skew-symmetric, and it is called skew-supersymmetric if B|V0¯ ×V0¯ is skew-symmetric and B|V1¯ ×V1¯ is symmetric. Let B be a non-degenerate even supersymmetric bilinear form on a vector superspace V = V0¯ ⊕V1¯ . It follows that dimV1¯ is necessarily even. For s ∈ Z2 , let osp(V )s := {g ∈ gl(V )s | B(g(x), y) = −(−1)s·|x| B(x, g(y)), ∀x, y ∈ V }, osp(V ) := osp(V )0¯ ⊕ osp(V )1¯ . One checks that osp(V ) is a Lie superalgebra, called the ortho-symplectic Lie superalgebra; that is, osp(V ) is the subalgebra of gl(V ) that preserves a nondegenerate supersymmetric bilinear form. Its even subalgebra is isomorphic to so(V0¯ ) ⊕ sp(V1¯ ), a direct sum of the orthogonal Lie algebra on V0¯ and the symplectic Lie algebra on V1¯ . When V = C|2m , we write osp(|2m) for osp(V ). Note that when  (respectively, m) is zero, the ortho-symplectic Lie superalgebra reduces to the classical Lie algebra sp(2m) (respectively, so()). Similarly, we define the Lie superalgebra spo(V ) as the subalgebra of gl(V ) that preserves a non-degenerate skew-supersymmetric bilinear form on V (here dimV0¯ has to be even). When V = C2m| , we write spo(2m|) for spo(V ). Remark 1.9. A non-degenerate supersymmetric bilinear form B on V induces a non-degenerate skew-supersymmetric bilinear form BΠ on Π(V ), defined by   BΠ Π(v), Π(v ) := (−1)|v| B(v, v ), for v, v ∈ V. The restriction of the isomorphism gl(V ) ∼ = gl(ΠV ) in Remark 1.6 gives rise to a Lie superalgebra isomorphism between osp(V ) (with respect to B) and spo(ΠV ) (with respect to BΠ ); see Exercise 1.3. It follows that osp(|2m) ∼ = spo(2m|).

1.1. Lie superalgebras: Definitions and examples

7

We now give an explicit matrix realization of the ortho-symplectic Lie superalgebra. To this end, we first observe that the supertranspose (1.5) of a matrix in the block form (1.4) is equal to  st  t  a b ct a , = c d −bt d t where xt denotes the usual transpose of a matrix x. Define the (2m + 2n + 1) × (2m + 2n + 1) matrix in the (m|m|n|n|1)-block form ⎛ ⎞ 0 Im 0 0 0 ⎜−Im 0 0 0 0⎟ ⎜ ⎟ ⎟. 0 0 0 I 0 J2m|2n+1 := ⎜ (1.7) n ⎜ ⎟ ⎝ 0 0 In 0 0⎠ 0 0 0 0 1 Let J2m|2n denote the (2m + 2n) × (2m + 2n) matrix obtained from J2m|2n+1 by deleting the last row and column. For  = 2n or 2n + 1, by definition spo(2m|) is the subalgebra of gl(2m|) that preserves the bilinear form on C2m| with matrix J2m| relative to the standard basis of C2m| , and hence spo(2m|) = {g ∈ gl(2m|) | gst J2m| + J2m| g = 0}. By a direct computation, spo(2m|2n + 1) consists of the (2m + 2n + 1) × (2m + 2n + 1) matrices of the following (m|m|n|n|1)-block form ⎛ ⎞ d e yt1 xt1 zt1 ⎜ f −d t −yt −xt −zt ⎟ ⎜ ⎟ a b −vt ⎟ (1.8) ⎜ ⎜ x x1 ⎟ , b, c skew-symmetric, e, f symmetric. t t ⎝ y y1 c −a −u ⎠ z z1 u v 0 ∼ C2m ⊗ C2n+1 (which is self-dual) as a module over Note that spo(2m|2n + 1)1¯ = spo(2m|2n + 1)0¯ ∼ = sp(2m) ⊕ so(2n + 1). The Lie superalgebra spo(2m|2n) consists of matrices (1.8) with the last row and column removed. Note that spo(2m|2n)1¯ ∼ = C2m ⊗ C2n (which is self-dual) as a module over spo(2m|2n)0¯ ∼ = sp(2m) ⊕ so(2n). Here and below, the rows and columns of the matrices J2m| and (1.8) (or its modification) are indexed by the finite set I(2m|). Proposition 1.10. The automorphism τ in (1.6) restricts to an automorphism of spo(2m|). Proof. Take an element g ∈ spo(2m|). Thus, J2m| g + gst J2m| = 0, and hence (1.9)

gst Jst2m| + Jst2m| (gst )st = 0.

1. Lie superalgebra ABC

8

Observe that Jst2m| = (J2m| )−1 . So if we multiply (1.9) on the left and on the right by J2m| , we obtain the identity J2m| gst + (gst )st J2m| = 0. This implies that J2m| τ(g) + τ(g)st J2m| = 0, and so τ(g) ∈ spo(2m|).



1.1.4. The queer Lie superalgebras. Let V = V0¯ ⊕V1¯ be a vector superspace with dimV0¯ = dimV1¯ . Choose P ∈ End(V )1¯ such that P2 = In|n . The subspace q(V ) = {T ∈ End(V ) | [T, P] = 0} is a subalgebra of gl(V ) called the queer Lie superalgebra. Different choices of P give rise to isomorphic queer Lie superalgebras. If V = Cn|n , then q(V ) is also denoted by q(n). To give an explicit matrix realization of q(n), let us take P to be the 2n × 2n matrix   √ 0 In P := −1 (1.10) . −In 0 Then, for g ∈ gl(n|n) of the form (1.4), we have g ∈ q(n) if and only if gP − (−1)|g| Pg = 0, and in turn, if and only if   a b (1.11) g= , b a ∼ gl(n) as Lie where a, b are arbitrary complex n × n matrices. Thus we have q(n)0¯ = ∼ algebras, and q(n) ¯ = gl(n) as the adjoint q(n) ¯ -module. A linear basis for q(n) 1

0

consists of the following elements: (1.12)

Ei j := Ei¯ j¯ + Ei j , E i j := Ei j¯ + Ei¯ j ,

1 ≤ i, j ≤ n.

The derived superalgebra [q(n), q(n)] consists of matrices of the form (1.11), with a ∈ gl(n) and b ∈ sl(n), and so it contains a one-dimensional center generated by the identity matrix In|n . The quotient superalgebra [q(n), q(n)]/CIn|n has even part isomorphic to sl(n) and odd part isomorphic to the adjoint module, and one can show that it is simple for n ≥ 3. For n = 2, the odd part of the quotient Lie superalgebra is an abelian ideal, since the adjoint module of sl(2) does not appear in the symmetric square of the adjoint module. Remark 1.11. Consider the subspace  q(n) of gl(n|n) consisting of elements that commute with P. That is,  q(n) := {g ∈ gl(n|n)|gP − Pg = 0}. In matrix form,  q(n) consists of the following n|n-block matrices:   a b (1.13) , −b a

1.1. Lie superalgebras: Definitions and examples

9

where a and b are arbitrary n × n matrices. One checks that  q(n) is closed under  the Lie bracket and hence is a subalgebra of gl(n|n). Indeed q(n) is isomorphic to q(n), since the map τ in (1.6) sends q(n) to  q(n), and vice versa. Thus, (1.13) gives another realization of the queer Lie superalgebra. 1.1.5. The periplectic and exceptional Lie superalgebras. Let us describe more examples of Lie superalgebras. The periplectic Lie superalgebras. Let V = V0¯ ⊕ V1¯ be a vector superspace with dimV0¯ = dimV1¯ . Let C(·, ·) be a non-degenerate odd symmetric bilinear form on V . One checks that the subspace of gl(V ) preserving C is closed under the Lie bracket and hence is a Lie subalgebra of gl(V ). This superalgebra is called the periplectic Lie superalgebra and will be denoted by p(V ). Different choices of C give rise to isomorphic periplectic Lie superalgebras. In the case V = Cn|n , p(V ) is also denoted by p(n). To write down an explicit matrix realization of p(n) as a subalgebra of gl(n|n), let us take the 2n × 2n matrix   0 In P := (1.14) , In 0 which determines an odd symmetric bilinear form C on Cn|n . Then, g ∈ p(n) if and only if gst P + Pg = 0. It follows that    a b (1.15) p(n) = , where b is symmetric and c is skew-symmetric . c −at We have (1.16)

p(n)0¯ ∼ = gl(n),

p(n)1¯ ∼ = S2 (Cn ) ⊕ ∧2 (Cn∗ ).

For n ≥ 3, the  derivedsuperalgebra [p(n), p(n)] is simple, and it consists of matrices a b of the form with tr(a) = 0, bt = b, and ct = −c. c −at Remark 1.12. One checks that the Lie subalgebra  p(n) of gl(n|n) preserving the non-degenerate odd skew-symmetric  bilinear  form corresponding to P in (1.10) a b consists of matrices of the form with bt = −b and ct = c. Similar to c −at Remark 1.11, the map τ in (1.6) restricts to an isomorphism between p(n) and  p(n). The exceptional Lie superalgebra D(2|1, α). We take three copies of the Lie algebra sl(2) denoted by gi (i = 1, 2, 3), and we associate to each gi a copy of the standard sl(2)-module Vi . Clearly, as gi -modules, we have an isomorphism S2 (Vi ) ∼ = gi . By Schur’s

Lemma we may associate a nonzero scalar αi ∈ C to any such isomorphism. Now consider g = g0¯ ⊕ g1¯ , where g0¯ ∼ = g1 ⊕ g2 ⊕ g3 , and g1¯ is the irreducible g0¯ -module

10

1. Lie superalgebra ABC

V1 ⊗ V2 ⊗ V3 . We can associate three nonzero complex numbers αi , i = 1, 2, 3, to any surjective g0¯ -homomorphism from S2 (g1¯ ) to g0¯ . Thus, our vector superspace g satisfies Conditions (1)–(3) of Remark 1.5. It is easy to see that Condition (4) of Remark 1.5 is equivalent to ∑3i=1 αi = 0. Thus, we obtain a Lie superalgebra g(α1 , α2 , α3 ) depending on three nonzero parameters αi , i = 1, 2, 3. For σ ∈ S3 , we have g(α1 , α2 , α3 ) ∼ = g(ασ(1) , ασ(2) , ασ(3) ). Also, g(α1 , α2 , α3 ) ∼ = g(λα1 , λα2 , λα3 ), for any nonzero λ ∈ C. Thus, we have a one-parameter family of Lie superalgebras D(2|1, α) := g(α, 1, −1 − α) that are simple for α = 0, −1. We have g(α, 1, −1 − α) ∼ = g(1, α, −1 − α) ∼ = g(1, α−1 , −α−1 − 1), and also ∼ g(α, 1, −1 − α) = g(−1 − α, 1, α), which imply D(2|1, α) ∼ = D(2|1, α−1 ) ∼ = D(2|1, −1 − α). The maps α → α−1 and α → (−1 − α) generate an action of S3 on C \ {0, −1}. We have D(2|1, α) ∼ = D(2|1, β) if and only if β ∈ S3 · α. This gives additional isomorphisms D(2|1, α) ∼ = D(2|1, −(1 + α)−1 α) ∼ = D(2|1, −1 − α−1 ) ∼ = D(2|1, −(1 + α)−1 ). Thus, any orbit of (C \ {0, −1}) /S3 consists of six points, except for the orbit corresponding to the three points α = 1, −2, − 12 , and the orbit corresponding to the  1 two points α = − 2 ± − 34 . Finally, note that D(2|1, 1) ∼ = osp(4|2). The exceptional Lie superalgebra F(3|1). There is a simple Lie superalgebra F(3|1) with F(3|1)0¯ ∼ = sl(2) ⊕ so(7). The odd part, as an F(3|1)0¯ -module, is isomorphic to the tensor product of the standard sl(2)-module and the simple so(7)-spin module. Hence, dim F(3|1) = (24|16). In the literature, F(3|1) is often denoted by F(4), which could be confused with the simple Lie algebra of type F4 . The exceptional Lie superalgebra G(3). There is a simple Lie superalgebra G(3) with G(3)0¯ ∼ = sl(2) ⊕ G2 . The odd part as a G(3)0¯ -module is isomorphic to the tensor product of the standard sl(2)-module and the fundamental 7-dimensional G2 -module. Hence, dim G(3) = (17|14). 1.1.6. The Cartan series. In this subsection, we describe the Cartan series of finite-dimensional simple Lie superalgebras without proof. This part is included for the sake of presenting a complete classification of finite-dimensional simple Lie superalgebras in Theorem 1.13 and will not be used elsewhere in the book. Lie superalgebra W (n). Let ∧(n) be the exterior algebra in n indeterminates ξ1 , ξ2 , . . . , ξn . We have ξi ξ j = −ξ j ξi , for all i, j, and in particular, ξ2i = 0, for all i. ¯ for all i, the algebra ∧(n) becomes a superalgebra which we also Setting |ξi | = 1, refer to as an exterior superalgebra. By general construction in Example 1.4, we have a finite-dimensional Lie superalgebra of derivations on ∧(n), which will be denoted by W (n).

1.1. Lie superalgebras: Definitions and examples

For i = 1, . . . , n, the derivation mined by

∂ ∂ξi

11

: ∧(n) → ∧(n) of degree 1¯ is uniquely deter-

∂ (ξ j ) = δi j , ∂ξi

j = 1, . . . , n.

Given an element f = ( f1 , f 2 , . . . , f n ) ∈ ∧(n)n , with | fi | = | f j |, for all i, j, the linear map D f : ∧(n) → ∧(n) of the form n

D f = ∑ fi i=1

∂ ∂ξi

is a derivation of ∧(n). Furthermore, all homogeneous derivations of ∧(n) are of this form, since a derivation is determined by its values at ξi for all i. Therefore, sending f → D f defines a linear isomorphism from ∧(n)n to W (n), and so W (n) has dimension 2n n. Setting degξi = 1 and deg ∂ξ∂ i = −1, for all i, gives rise to a Z-gradation on W (n), called the principal gradation. We have W (n) =

n−1 

W (n) j .

j=−1

The Z-gradation is compatible with the super structure on W (n); that is, W (n)s = ⊕ j≡s mod 2W (n) j , for s ∈ Z2 . The 0th degree component W (n)0 is a Lie algebra isomorphic to gl(n), and each W (n) j is a gl(n)-module isomorphic to ∧ j+1 (Cn )⊗Cn∗ . In particular, when n = 2, we have W (2)0 ∼ = gl(2), W (2)−1 ∼ = C2∗ and W (2)1 ∼ = C2 as gl(2)-modules. Indeed, we have isomorphisms of Lie superalgebras W (2) ∼ = osp(2|2) ∼ = sl(2|1) (see Exercises 1.4 and 1.5). The Lie superalgebra W (n) is simple, for n ≥ 2. Moreover, W (n) contains the following three series of simple Lie superalgebras as subalgebras that we shall describe. Lie superalgebra S(n). The first series is the super-analogue of the Lie algebra of divergence-free vector fields given by  n  n ∂ ∂ S(n) := ∑ f j ∈ W (n) | ∑ ( f j) = 0 . j=1 ∂ξ j j=1 ∂ξ j The Lie superalgebra S(n) is a Z-graded subalgebra of W (n) and we have S(n) = j=−1 S(n) j . The Lie algebra S(n)0 is isomorphic to sl(n), and the jth degree component S(n) j is isomorphic to the top irreducible summand of the sl(n)-module ∧ j+1 (Cn ) ⊗ Cn∗ . The Lie superalgebra S(n) is simple, for n ≥ 3.

n−2

 Lie superalgebra S(n). Let n be even so that ω = 1 + ξ1 ξ2 · · · ξn is a Z2 homogeneous invertible element in ∧(n). Consider the subspace of W (n) given

1. Lie superalgebra ABC

12

by  := S(n)



n

∂

n

∂

∑ f j ∂ξ j ∈ W (n) | ∑ ∂ξ j (ω f j ) = 0

j=1

 .

j=1

 is a subalgebra of the Lie superalgebra W (n). The Lie It can be shown that S(n)  is no longer Z-graded, as the defining condition is not homosuperalgebra S(n)  geneous. However, S(n) inherits a natural filtration from the filtration on W (n) induced by the principal gradation. The associated graded Lie superalgebra of  is isomorphic to S(n). Explicitly, S(n)  is the following direct sum of vector S(n) spaces inside W (n): (1.17)

 = S(n)

n−2 

 j, S(n)

j=−1

 −1 is spanned by {(1 − ξ1 ξ2 · · · ξn ) ∂ |i = 1, . . . , n}, and S(n)  j = S(n) j , where S(n) ∂ξi  is simple. Note that S(2)  ∼ for j = 0, . . . , n−2. For n ≥ 2 and n even, S(n) = spo(2|1). Lie superalgebra H(n). As in the classical setting, W (n) contains a subalgebra H(n) as defined below, which is a super-analogue of the Lie algebra of Hamiltonian vector fields. For f , g ∈ ∧(n), we define the Poisson bracket by ∂ f ∂g . j=1 ∂ξ j ∂ξ j n

{ f , g} := (−1)| f | ∑

The Poisson bracket makes ∧(n) into a Lie superalgebra, which we will denote by   H(n). Now putting deg f := k − 2, for f ∈ ∧(n)k , H(n) becomes a Z-graded Lie  superalgebra. The superalgebra H(n) is not simple, as it has center C1. However,  the derived superalgebra of H(n)/C1, which we denote by H(n), is simple, for n−3 n ≥ 4. Moreover, H(n) = j=−1 H(n) j is a graded Lie superalgebra. The 0th degree component is a Lie algebra isomorphic to so(n). As an so(n)-module we have H(n) j ∼ = ∧ j+2 (Cn ), for −1 ≤ j ≤ n − 3. Finally, H(n) can be viewed as a ∂f ∂ subalgebra of W (n), since the assignment f → (−1)| f | ∑nj=1 ∂ξ , for f ∈ ∧(n), j ∂ξ j  gives rise to an embedding of Lie superalgebras from H(n)/C1 into W (n). 1.1.7. The classification theorem. The following theorem of Kac [60] gives a classification of finite-dimensional complex simple Lie superalgebras. Note the following isomorphisms of Lie superalgebras (see Exercises 1.5, 1.4, and 1.7): osp(2|2) ∼ = sl(2|1) ∼ = W (2), sl(2|2)/CI2|2 ∼ = H(4), [p(3), p(3)] ∼ = S(3). Theorem 1.13. The following is a complete list of pairwise non-isomorphic finitedimensional simple Lie superalgebras over C. (1) A finite-dimensional simple Lie algebra in the Killing-Cartan list.

1.2. Structures of classical Lie superalgebras

13

(2) sl(m|n), for m > n ≥ 1 (excluding (m, n) = (2, 1)); sl(m|m)/CIm|m , for m ≥ 3; spo(2m|n), for m, n ≥ 1.   (3) D(2|1, α), for α ∈ C \ {0, ±1, −2, − 12 } /S3 ; F(3|1); and G(3). (4) [p(n), p(n)] and [q(n), q(n)]/CIn|n , for n ≥ 3.  (5) W (n), for n ≥ 3; S(n), for n ≥ 4; S(2n), for n ≥ 2; and H(n), for n ≥ 4. A Lie superalgebra g = g0¯ ⊕ g1¯ in Theorem 1.13(1)-(4) has the property that g0¯ is a reductive Lie algebra and the adjoint g0¯ -module g1¯ is semisimple. To distinguish between such a Lie superalgebra from one in the Cartan series of Theorem 1.13(5), we introduce the following terminology. Definition 1.14. A Lie superalgebra g = g0¯ ⊕ g1¯ in Theorem 1.13(2)-(4) is called classical. A classical Lie superalgebra in Theorem 1.13(2)-(3) is called basic. The Lie superalgebra gl(m|n) for m, n ≥ 1 is also declared to be basic. Remark 1.15. The basic Lie superalgebras admit non-degenerate even supersymmetric bilinear forms, and this property characterizes the simple basic Lie superalgebras among all simple Lie superalgebras in the list of Theorem 1.13. Remark 1.16. Let us comment on the simplicity of the Lie superalgebras in Theorem 1.13. It is not difficult to check directly the simplicity of the sl series, spo(2m|2), and those in (4). A Lie superalgebra g in the remaining cases in (2) and (3), with the exception of spo(2m|2), satisfies the properties that the adjoint g0¯ -module g1¯ is irreducible and faithful, and [g1¯ , g1¯ ] = g0¯ . A Lie superalgebra g with such properties can be easily shown to be simple (see Exercise 1.9). 

For the Z-graded Cartan type Lie superalgebras g = j≥−1 g j in (5), we first note that they are all transitive (i.e., [g−1 , x] = 0 implies that x = 0, for x ∈ g j and j ≥ 0) and irreducible (i.e., the g0 -module g−1 is irreducible). Furthermore, we have [g−1 , g1 ] = g0 , [g0 , g1 ] = g1 , and g j is generated by g1 (i.e., g j = [g j−1 , g1 ]), for j ≥ 2. A Lie superalgebra g with these properties is simple (see Exercise 1.10).  can be verified with a bit of extra work using the Finally, the simplicity of S(n) explicit realization given in (1.17).

1.2. Structures of classical Lie superalgebras In this section, Cartan subalgebras, root systems, Weyl groups, and invariant bilinear forms for basic Lie superalgebras are introduced and described in detail for type gl, osp and q. A structure theorem is formulated for the basic and type q Lie superalgebras. 1.2.1. A basic structure theorem. In this subsection, we shall assume that g = g0¯ ⊕ g1¯ is a basic Lie superalgebra (see Definition 1.14).

1. Lie superalgebra ABC

14

A Cartan subalgebra h of g is defined to be a Cartan subalgebra of the even subalgebra g0¯ . Since every inner automorphism of g0¯ extends to one of Lie superalgebra g and Cartan subalgebras of g0¯ are conjugate under inner automorphisms, we conclude that the Cartan subalgebras of g are conjugate under inner automorphisms. Let h be a Cartan subalgebra of g. For α ∈ h∗ , let gα = {g ∈ g | [h, g] = α(h)g, ∀h ∈ h}. The root system for g is defined to be Φ = {α ∈ h∗ | gα = 0, α = 0}. Define the sets of even and odd roots, respectively, to be Φ0¯ := {α ∈ Φ|gα ∩ g0¯ = 0},

Φ1¯ := {α ∈ Φ|gα ∩ g1¯ = 0}.

Definition 1.17. For a basic or queer Lie superalgebra g = g0¯ ⊕g1¯ , the Weyl group W of g is defined to be the Weyl group of the reductive Lie algebra g0¯ . As we shall see, the Weyl groups play a less vital though still important role in determining central characters and the linkage principle for Lie superalgebras that is somewhat different from the theory of semisimple Lie algebras. The following theorem shows that the structures of the basic Lie superalgebras are similar to those of semisimple Lie algebras. Theorem 1.18. Let g be a basic Lie superalgebra with a Cartan subalgebra h. (1) We have a root space decomposition of g with respect to h: g = h⊕



gα ,

and g0 = h.

α∈Φ

(2) dim gα = 1, for α ∈ Φ. (Now fix some nonzero eα ∈ gα .) (3) [gα , gβ ] ⊆ gα+β , for α, β, α + β ∈ Φ. (4) Φ, Φ0¯ and Φ1¯ are invariant under the action of the Weyl group W on h∗ . (5) There exists a non-degenerate even invariant supersymmetric bilinear form (·, ·) on g. (6) (gα , gβ ) = 0 unless α = −β ∈ Φ. (7) The restriction of the bilinear form (·, ·) on h × h is non-degenerate and W -invariant. (8) [eα , e−α ] = (eα , e−α )hα , where hα is the coroot determined by (hα , h) = α(h) for h ∈ h. (9) Φ = −Φ, Φ0¯ = −Φ0¯ , and Φ1¯ = −Φ1¯ . (10) Let α ∈ Φ. Then, kα ∈ Φ for some integer k = ±1 if and only if α is an odd root such that (α, α) = 0; in this case, we must have k = ±2.

1.2. Structures of classical Lie superalgebras

15

Proof. For g = gl(m|n) or osp(2m|), we will describe explicitly the root systems and trace forms below in this section, and so the theorem in these cases follows by inspection. The theorem for the other infinite series basic Lie superalgebras sl(m|n) follows by some easy modification of the gl(m|n) case. Part (4) follows by the invariance, under the action of the Weyl group W , of the sets of weights for the adjoint g0¯ -modules g0¯ and g1¯ , respectively. Since the remaining three exceptional Lie superalgebras D(2|1, α), G(3) and F(3|1) will not be studied in detail in the book, we will be sketchy. Most parts of the theorem, with the exception of (5), again follow by inspection from the constructions of these superalgebras and standard arguments as for simple Lie algebras (with the help of (5)). As the exceptional Lie superalgebras are simple, any nonzero invariant supersymmetric bilinear form must be non-degenerate. For G(3) and F(3|1), the Killing form is such a nonzero even form. A direct and ad hoc construction of a nonzero invariant bilinear form on D(2|1, α) is possible.  It follows by Theorem 1.18(1) that h is self-normalizing in g (and h is abelian), justifying the terminology of a Cartan subalgebra. Since h ⊆ g0¯ and dim gα = 1 for each α ∈ Φ by Theorem 1.18(2), there exists i ∈ Z2 such that gα ⊆ gi . Hence Φ is a disjoint union of Φ0¯ and Φ1¯ , and we have Φi = {α ∈ Φ | gα ⊆ gi },

i ∈ Z2 .

Remark 1.19. One uniform approach to establish Theorem 1.18 is as follows (see [60, Proposition 2.5.3]). One follows the standard construction of contragredient (i.e., Kac-Moody) (super)algebras (see [18, Chapter 14]) to show that any Lie superalgebra in Theorem 1.18 is a Kac-Moody superalgebra (or rather the quotient by its possibly nontrivial center) associated to some generalized Cartan matrix with Z2 -grading. These (quotients of) Kac-Moody superalgebras carry non-degenerate even invariant supersymmetric bilinear forms by a standard Kac-Moody type argument (see [18, Chapter 16]). A root α ∈ Φ is called isotropic if (α, α) = 0. An isotropic root is necessarily an odd root. Denote the set of isotropic odd roots by (1.18)

¯ 1¯ := {α ∈ Φ1¯ | (α, α) = 0} Φ = {α ∈ Φ1¯ | 2α ∈ Φ}.

The second equation above follows by Theorem 1.18(10). Moreover, we have 1 e2α = [eα , eα ] = 0, 2

¯ 1¯ . for α ∈ Φ

We also introduce the following set of roots (1.19)

¯ 0¯ = {α ∈ Φ0¯ | α/2 ∈ Φ}. Φ

1. Lie superalgebra ABC

16

1.2.2. Invariant bilinear forms for gl and osp. In contrast to the semisimple Lie algebras, the Killing form for a basic Lie superalgebra may be zero, and even when it is nonzero, it may not be positive definite on the real vector space spanned by Φ. In this subsection, we give a down-to-earth description of an invariant nondegenerate even supersymmetric bilinear form for Lie superalgebras of type gl and osp. The supertrace str on the general linear Lie superalgebra gives rise to a nondegenerate supersymmetric bilinear form (·, ·) : gl(m|n) × gl(m|n) → C,

(a, b) = str(ab),

where ab denotes the matrix multiplication. It is straightforward to check that this form is invariant. Restricting to the Cartan subalgebra h of diagonal matrices, we obtain a non-degenerate symmetric bilinear form on h: ⎧ ⎨ 1 if 1 ≤ i = j ≤ m, (Eii , E j j ) = −1 if 1 ≤ i = j ≤ n, ⎩ 0 if i = j, where i, j ∈ I(m|n). We recall here that I(m|n) is defined in (1.2). Denote by {δi , ε j }i, j the basis of h∗ dual to {Ei i , E j j }i, j , where 1 ≤ i ≤ m and 1 ≤ j ≤ n. Using the bilinear form (·, ·) we can identify δi with (Ei,i , ·) and ε j with −(E j j , ·). When it is convenient we also use the notation (1.20)

εi := δi ,

for 1 ≤ i ≤ m.

The form (·, ·) on h induces a non-degenerate bilinear form on h∗ , which will be denoted by (·, ·) as well. Then, for i, j ∈ I(m|n), we have ⎧ ⎨ 1 if 1 ≤ i = j ≤ m, (εi , ε j ) = (1.21) −1 if 1 ≤ i = j ≤ n, ⎩ 0 if i = j. Such a bilinear form on gl(2n|) restricts to a non-degenerate invariant supersymmetric bilinear form on the subalgebra spo(2n|), which will also be denoted by (·, ·). The further restriction to a Cartan subalgebra of spo(2n|) remains nondegenerate. This allows us to identify a Cartan subalgebra h with its dual h∗ , and one also obtains a bilinear form on h∗ . 1.2.3. Root system and Weyl group for gl(m|n). Let g = gl(m|n) and h be the Cartan subalgebra of diagonal matrices. Its root system Φ = Φ0¯ ∪ Φ1¯ is given by Φ0¯ = {εi − ε j | i = j ∈ I(m|n), i, j > 0 or i, j < 0}, Φ1¯ = {±(εi − ε j ) | i, j ∈ I(m|n), i < 0 < j}. Observe that Ei j is a root vector corresponding to the root εi −ε j , for i = j ∈ I(m|n).

1.2. Structures of classical Lie superalgebras

17

The Weyl group of gl(m|n), which is by definition the Weyl group of the even subalgebra g0¯ = gl(m) ⊕ gl(n), is isomorphic to Sm × Sn , where we recall that Sn denotes the symmetric group of n letters. 1.2.4. Root system and Weyl group for spo(2m|2n + 1). Now we describe the root system for the ortho-symplectic Lie superalgebra spo(2m|2n + 1), which is defined in matrix form (1.8) in Section 1.1.3. Recall that the rows and columns of the matrices are indexed by I(2m|2n + 1). The subalgebra h of g of diagonal matrices has a basis given by Hi := Ei,i − Em+i,m+i ,

1 ≤ i ≤ m,

H j := E j j − En+ j,n+ j ,

1 ≤ j ≤ n,

and it is the standard Cartan subalgebra for spo(2m|2n + 1). Let {δi , ε j | 1 ≤ i ≤ m, 1 ≤ j ≤ n} be the corresponding dual basis in h∗ . With respect to h, the root system Φ = Φ0¯ ∪ Φ1¯ for spo(2m|2n + 1) is {±δi ± δ j , ±2δ p , ±εk ± εl , ±εq } ∪ {±δ p ± εq , ±δ p }, where 1 ≤ i < j ≤ m, 1 ≤ k < l ≤ n, 1 ≤ p ≤ m, 1 ≤ q ≤ n. A root vector is a nonzero vector in gα for α ∈ Φ and will be denoted by eα . The root vectors for spo(2m|2n + 1) can be chosen explicitly as follows in (1.22)(1.29) (1 ≤ i = j ≤ m, 1 ≤ k = l ≤ n): (1.22)

eεk = E2n+1,k+n − Ek,2n+1 ,

e−εk = E2n+1,k − Ek+n,2n+1 ,

(1.23)

e2δi = Ei,i+m ,

(1.24)

eδi +δ j = Ei, j+m + E j,i+m ,

e−δi −δ j = E j+m,i + Ei+m, j ,

(1.25)

eδi −δ j = Ei, j − E j+m,i+m ,

eεk −εl = Ekl − El+n,k+n ,

(1.26)

eεk +εl = Ek,l+n − El,k+n ,

e−εk −εl = Ek+n,l − El+n,k ,

(1.27)

eδi +εk = Ek,i+m + Ei,k+n ,

e−δi −εk = Ek+n,i − Ei+m,k ,

(1.28)

eδi −εk = Ek+n,i+m + Ei,k ,

e−δi +εk = Ek,i − Ei+m,k+n ,

(1.29)

eδi = E2n+1,i+m + Ei,2n+1 ,

e−δi = E2n+1,i − Ei+m,2n+1 .

e−2δi = Ei+m,i ,

The Weyl group of spo(2m|2n + 1), which is by definition the Weyl group of n g0¯ = sp(2m) ⊕ so(2n + 1), is isomorphic to (Zm 2  Sm ) × (Z2  Sn ). 1.2.5. Root system and Weyl group for spo(2m|2n). Let g = spo(2m|2n). The abelian Lie subalgebra h spanned by {Hi , H j |1 ≤ i ≤ m, 1 ≤ j ≤ n} is a Cartan subalgebra for spo(2m|2n). Again, when it is convenient, we will use the notation εi¯ = δi , for 1 ≤ i ≤ m. With respect to h, the root system Φ = Φ0¯ ∪ Φ1¯ is given by {±δi ± δ j , ±2δ p , ±εk ± εl } ∪ {±δ p ± εq }, where 1 ≤ i < j ≤ m, 1 ≤ k < l ≤ n, 1 ≤ p ≤ m, 1 ≤ q ≤ n.

1. Lie superalgebra ABC

18

The root vectors for spo(2m|2n) are given by (1.23)–(1.28). The Weyl group of spo(2m|2n), which is by definition the Weyl group of the even subalgebra g0¯ = sp(2m) ⊕ so(2n), is an index 2 subgroup of (Zm 2  Sm ) × (Zn2  Sn ), with only an even number of signs in Zn2 permitted. 1.2.6. Root system and odd invariant form for q(n). In this subsection let g be the queer Lie superalgebra q(n) (see Section 1.1.4). Since the case of q(n) is different from the basic Lie superalgebra case, we will describe altogether its Cartan subalgebras, root systems, positive systems, Borel subalgebras, and invariant bilinear form.   a b Recall that q(n) can be realized as matrices in the n|n block form b a indexed by I(n|n). The subalgebra consisting of matrices with a, b being diagonal (which we refer to as “block diagonal matrices”) will be called the standard Cartan subalgebra. We define a Cartan subalgebra h to be any subalgebra conjugate to the standard Cartan subalgebra by the adjoint action of some element in the group GL(n) associated to q(n)0¯ . Note that h is self-normalizing in g and h is nilpotent, justifying the terminology of a Cartain subalgebra. However, h = h0¯ ⊕ h1¯ is not abelian, since [h0¯ , h] = 0 and [h1¯ , h1¯ ] = h0¯ . Now fix h = h0¯ ⊕ h1¯ to be the standard Cartan subalgebra. The vectors (1.30)

Hi := Ei¯i¯ + Eii ,

i = 1, . . . , n

form a basis for h0¯ , while the vectors (1.31)

H i := Ei¯i + Ei i¯,

i = 1, . . . , n

form a basis for h1¯ . We let {εi | i = 1, . . . , n} denote the basis in h∗0¯ dual to {Hi | i = 1, . . . , n}. With respect to h0¯ , we have the root space decomposition  g = h ⊕ α∈Φ gα with root system Φ = Φ0¯ ∪ Φ1¯ , where Φ0¯ and Φ1¯ are understood as distinct isomorphic copies of the root system {εi − ε j | 1 ≤ i = j ≤ n} for gl(n). We have dimC gα = 1, for each α ∈ Φ, and gα ⊆ gi , for α ∈ Φi and i ∈ Z2 . The Weyl group of q(n) is identified with the symmetric group Sn .   a b The matrices with a, b being upper triangular (which we refer to as b a “block upper triangular matrices”) form a solvable subalgebra b, which will be called the standard Borel subalgebra of g. The positive system corresponding to the Borel subalgebra b is Φ+ = Φ+ ∪ Φ+ , where Φ+ = Φ+ = {εi − ε j | 1 ≤ i < j ≤ 0¯ 1¯ 0¯ 1¯ − + + − n}. Let Φ = −Φ and so Φ = Φ ∪ Φ . Let n+ =



α∈Φ+

gα ,

n− =



gα .

α∈Φ−

Then, we have b = h ⊕ n+ , and we have a triangular decomposition g = n− ⊕ h ⊕ n+ .

1.3. Non-conjugate positive systems and odd reflections

19

To be compatible with the definition of Weyl vector ρ in (1.35) for basic Lie superalgebras later on, it makes sense to set ρ = 0 for q(n). Any subalgebra of g that is conjugate to b by GL(n) will be referred to as a Borel subalgebra of g.   a b For g = in q(n), the odd trace is defined to be b a (1.32)

otr(g) := tr(b).

Using this, we obtain an odd non-degenerate invariant symmetric bilinear form (·, ·) on q(n) defined by (1.33)

(g, g ) = otr(gg ),

g, g ∈ q(n).

Here “odd” is understood in the sense of Definition 1.8.

1.3. Non-conjugate positive systems and odd reflections In this section, positive systems, fundamental systems, and Dynkin diagrams for basic Lie superalgebras are defined and classified, along with Borel subalgebras. In contrast to semisimple Lie algebras, the fundamental systems for a Lie superalgebra may not be conjugate under the Weyl group action. 1.3.1. Positive systems and fundamental systems. Let Φ be a root system for a basic Lie superalgebra g with a given Cartan subalgebra h, and let E be the real vector space spanned by Φ. We have E ⊗R C = h∗ , for g = gl(m|n). For g = gl(m|n) the space E ⊗R C is a subspace of h∗ of codimension one. A total ordering ≥ on E below is always assumed to be compatible with the real vector space structure; that is, v ≥ w and v ≥ w imply that v + v ≥ w + w , −w ≥ −v, and cv ≥ cw for c ∈ R and c > 0. A positive system Φ+ is a subset of Φ consisting precisely of all those roots α ∈ Φ satisfying α > 0 for some total ordering of E. Given a positive system Φ+ , we define the fundamental system Π ⊂ Φ+ to be the set of α ∈ Φ+ which cannot be written as a sum of two roots in Φ+ . We refer to elements in Φ+ as positive roots and elements in Π as simple roots. Similarly, we denote by Φ− the − + − corresponding set of negative roots. Set Φ+ i = Φ ∩ Φi and Φi = Φ ∩ Φi , for + i ∈ Z2 . By Theorem 1.18(9), we have Φ− = −Φ+ and Φ− i = −Φi , for i ∈ Z2 . Then, we have Φ+ = Φ+ ∪ Φ+ . 0¯ 1¯ ¯ 1¯ from (1.18). Associated to a positive system Φ+ , we let Recall Φ (1.34)

¯+ ¯ 1¯ ∩ Φ+ . Φ := Φ 1¯

1. Lie superalgebra ABC

20

Proposition 1.20. Let g be a basic Lie superalgebra with a Cartan subalgebra h. There is a one-to-one correspondence between the set of positive systems for (g, h) and the set of fundamental systems for (g, h). The Weyl group of g acts naturally on the set of the positive systems (respectively, fundamental systems). Proof. It follows by definition that the fundamental system exists and is unique for a given positive system. A positive root, if not simple, can be written as a sum of two positive roots. Continuing this way, any positive root is a Z+ -linear combination of simple roots, and hence a positive system is determined by its fundamental system. By Theorem 1.18, Φ = −Φ and Φ is W -invariant. Then W acts naturally on the set of positive systems, and then on the set of fundamental systems by the above correspondence.  We define n+ =



gα ,

n− =



gα .

α∈Φ−

α∈Φ+

Then n± are ad h-stable nilpotent subalgebras of g and we obtain a triangular decomposition g = n− ⊕ h ⊕ n+ . The solvable subalgebra b = h ⊕ n+ is called a Borel subalgebra of g (corresponding to Φ+ ). We have b = b0¯ ⊕ b1¯ , where bi = b ∩ gi for i ∈ Z2 . Remark 1.21. The rank one subalgebra corresponding to an isotropic simple root is isomorphic to sl(1|1), which is solvable. Therefore, if we enlarge a Borel subalgebra by adding the root space corresponding to a negative isotropic simple root, then the resulting subalgebra is still solvable. Thus, a Borel subalgebra is not a maximal solvable subalgebra for Lie superalgebras in general. Given a positive system Φ+ = Φ+ ∪ Φ+ , the Weyl vector ρ is defined by 0¯ 1¯ ρ = ρ0¯ − ρ1¯ ,

(1.35) where ρ0¯ =

1 ∑+ α, 2 α∈Φ 0¯

ρ1¯ =

1 ∑+ β. 2 β∈Φ 1¯

Denote (1.36)

1m|n = (δ1 + . . . + δm ) − (ε1 + . . . + εn ).

The following lemma is proved by a direct computation. Lemma 1.22. We have the following formulas for the Weyl vector ρ for the standard positive system Φ+ : n 1 (1) ρ = ∑m i=1 (m − i + 1)δi − ∑ j=1 jε j − 2 (m + n + 1)1m|n , for g = gl(m|n).

1.3. Non-conjugate positive systems and odd reflections

21

1 n 1 (2) ρ = ∑m i=1 (m − n − i + 2 )δi + ∑ j=1 (n − j + 2 )ε j , for g = spo(2m|2n + 1). n (3) ρ = ∑m i=1 (m − n − i + 1)δi + ∑ j=1 (n − j)ε j , for g = spo(2m|2n).

We shall next describe completely the positive and fundamental systems for Lie superalgebras of type gl and osp case-by-case. 1.3.2. Positive and fundamental systems for gl(m|n). Recall the root system Φ for gl(m|n) and the standard Cartan subalgebra h described in Section 1.2.3. The subalgebra of upper triangular matrices is the standard Borel subalgebra of g that contains h, and the corresponding standard positive system of Φ is given by {εi − ε j | i, j ∈ I(m|n), i < j}. Bearing in mind εi = δi , the standard fundamental system for gl(m|n) is {δi − δi+1 , ε j − ε j+1 , δm − ε1 | 1 ≤ i ≤ m − 1, 1 ≤ j ≤ n − 1}, with the corresponding simple root vectors ei := Ei,i+1 , for i ∈ I(m − 1|n − 1), and em := Em,1 . The simple coroots are h j := E j j − E j+1, j+1 , for j ∈ I(m − 1|n − 1), and hm := Em,m + E11 . Denote fi := Ei+1,i , for i ∈ I(m − 1|n − 1), and fm := E1,m . (Here we have slightly abused notation to let i+1 mean ι + 1, for i = ¯ι with 1 ≤ ι ≤ m−1.) Then {ei , hi , f i | i ∈ I(m|n − 1)} is a set of Chevalley generators for sl(m|n). Note that (δi − δi+1 , δi − δi+1 ) = 2,

1 ≤ i ≤ m − 1,

(δm − ε1 , δm − ε1 ) = 0, (ε j − ε j+1 , ε j − ε j+1 ) = −2,

1 ≤ j ≤ n − 1.

Thus δm − ε1 is an isotropic simple root. Following the usual convention for Lie algebras, we draw the corresponding standard Dynkin diagram with its fundamental system attached:

(1.37)

 δ1 − δ2

 δ2 − δ3

···

 δm − ε1



ε1 − ε2

···



εn−2 − εn−1



εn−1 − εn

Here, as usual, we denote by  an even simple root α such that 12 α is not a root.  Following Kac’s notation, denotes an odd isotropic simple root. Let us classify all possible positive systems for gl(m|n), keeping in mind εi¯ = δi , for 1 ≤ i ≤ m. If we ignore the parity of roots for the moment, the root system of gl(m|n) is the same as the root system for gl(m + n). Hence, by definition, their positive systems (respectively, fundamental systems) are exactly described in the same way, and so there are (m + n)! of them in total. It follows from the wellknown classification for gl(m + n) that a fundamental system for gl(m|n) consists of (m + n − 1) roots εi1 − εi2 , εi2 − εi3 , . . . , εim+n−1 − εim+n , where {i1 , i2 , . . . , im+n } = I(m|n). Then we restore the parity of the simple roots in a fundamental system for gl(m|n). The corresponding Dynkin diagram is of the form

1. Lie superalgebra ABC

22



(1.38)



εi1 − εi2

εi2 − εi3



where is either  or is even or odd.







···



···

εik − εik+1

 εim+n−1 − εim+n

, depending on whether the corresponding simple root

Example 1.23. For n = m, there exists a fundamental system consisting of only odd roots {δ1 − ε1 , ε1 − δ2 , δ2 − ε2 , . . . , δm − εm }, whose corresponding Dynkin diagram is 



δ1 − ε1

ε1 − δ2





···

δk − εk



···

εk − δk+1



εm−1 − δm

δm − εm

The εδ-sequence for a fundamental system Π as in (1.38) is obtained by switching the ordered sequence εi1 εi2 . . . εim+n for Π to the εδ-notation via the identification εi¯ = δi and then dropping the indices. Clearly, an εδ-sequence has m δ’s and n ε’s. In general, there exist positive systems for Φ that are not conjugate to each other under the action of the Weyl group, in contrast to the semisimple Lie algebra case. Example 1.24. (1) The standard Borel subalgebra of gl(m|n) corresponds to the sequence δ · · · δ ε · · · ε while the Borel opposite to the standard one m

n

corresponds to ε · · · ε δ · · · δ. n

m

(2) The three W -conjugacy classes of fundamental systems for gl(1|2) correspond to the three sequences δεε, εδε, εεδ, respectively. 1.3.3. Positive and fundamental systems for spo(2m|2n + 1). Now we describe the positive/fundamental systems and Dynkin diagrams for spo(2m|2n + 1) whose root system is described in Section 1.2.4. The standard positive system Φ+ = Φ+ ∪ Φ+ corresponding to the standard Borel subalgebra for spo(2m|2n + 1) is 0¯ 1¯ {δi ± δ j , 2δ p , εk ± εl , εq } ∪ {δ p ± εq , δ p }, where 1 ≤ i < j ≤ m, 1 ≤ k < l ≤ n, 1 ≤ p ≤ m, 1 ≤ q ≤ n. The fundamental system Π of Φ+ contains one odd simple root δm − ε1 , and it is given by Π = {δi − δi+1 , δm − ε1 , εk − εk+1 , εn | 1 ≤ i ≤ m − 1, 1 ≤ k ≤ n − 1}. The corresponding standard Dynkin diagram for spo(2m|2n + 1) is

(1.39)

 δ1 − δ2

 δ2 − δ3

···

 δm − ε1



ε1 − ε2

···

=⇒

εn−1 − εn

εn

1.3. Non-conjugate positive systems and odd reflections

23

Another often used positive system in Φ for spo(2m|2n + 1) is given by {δi ± δ j , 2δ p , εk ± εl , εq } ∪ {εq ± δ p , δ p }, where 1 ≤ i < j ≤ m, 1 ≤ k < l ≤ n, 1 ≤ p ≤ m, 1 ≤ q ≤ n, with the following fundamental system {εk − εk+1 , εn − δ1 , δi − δi+1 , δm |1 ≤ i ≤ m − 1, 1 ≤ k ≤ n − 1}. The corresponding Dynkin diagram is (1.40)



ε1 − ε2



ε2 − ε3

···

 εn − δ1

 δ1 − δ2

where we follow Kac’s convention and use root, as (δm , δm ) = 1.

···

=⇒ δm−1 − δm

δm

to denote a non-isotropic odd simple

Now let us classify all the possible fundamental systems for spo(2m|2n + 1) with given Cartan subalgebra h, keeping in mind εi¯ = δi . Note that 2ε p ∈ Φ+ if and only if ε p ∈ Φ+ , and that ±2ε p are never in any fundamental system by definition. Hence, for the sake of classification of positive systems and fundamental systems ˜ := Φ \ {±2ε p | 1 ≤ p ≤ m}. Ignoring in Φ, it suffices to consider the subset Φ ˜ may be identified with the root system of the classical the parity of the roots, Φ Lie algebra so(2m + 2n + 1), whose fundamental systems are completely known and acted upon simply transitively by the Weyl group of so(2m + 2n + 1) (which is ∼ = Z2m+n  Sm+n ). The number of W -conjugacy classes   of fundamental systems for spo(2m|2n + 1) is |W (so(2m + 2n + 1))|/|W | = m+n m . The εδ-sequence associated to a fundamental system (or a Dynkin diagram) for spo(2m|2n + 1) is defined as for gl(m|n), starting from the type A end of the Dynkin diagram (to fix the ambiguity). For example, the εδ-sequence associated to the standard Dynkin diagram above is m δ’s followed by n ε’s. Example 1.25. Let g = osp(1|2). Then its even subalgebra g0¯ is isomorphic to sl(2) = Ce, h, f , and as a g0¯ -module g1¯ is isomorphic to the 2-dimensional natural sl(2)-module CE + CF. The simple root consists of a (unique) odd non-isotropic root δ1 so that 2δ1 is an even root. The Dynkin diagram is . The root vectors E and F associated to the odd roots δ1 and −δ1 can be chosen such that [E, E] = 2e, [F, F] = −2 f , [E, F] = h. 1.3.4. Positive and fundamental systems for spo(2m|2n). Now we consider g = spo(2m|2n), whose root system is described in Section 1.2.5. The standard positive system Φ+ = Φ+ ∪ Φ+ in Φ corresponding to the standard Borel subalgebra 0¯ 1¯ is {δi ± δ j , 2δ p , εk ± εl } ∪ {δ p ± εq },

1. Lie superalgebra ABC

24

where 1 ≤ i < j ≤ m, 1 ≤ k < l ≤ n, 1 ≤ p ≤ m, 1 ≤ q ≤ n, with its fundamental system being Π = {δi − δi+1 , δm − ε1 , εk − εk+1 , εn−1 + εn | 1 ≤ i ≤ m − 1, 1 ≤ k ≤ n − 1}. The corresponding standard Dynkin diagram is

(1.41)

 δ1 − δ2



···

δ2 − δ3

 δm − ε1



ε1 − ε2

···



εn−1 − εn

@

εn−1 + εn



εn−2 − εn−1@

Another often used positive system in Φ is given by {δi ± δ j , 2δ p , εk ± εl } ∪ {εq ± δ p }, where 1 ≤ i < j ≤ m, 1 ≤ k < l ≤ n, 1 ≤ p ≤ m, 1 ≤ q ≤ n, with its fundamental system being {εk − εk+1 , εn − δ1 , δi − δi+1 , 2δm | 1 ≤ i ≤ m − 1, 1 ≤ k ≤ n − 1}. The corresponding Dynkin diagram is (1.42)



ε1 − ε2



ε2 − ε3

···

 εn − δ1

 δ1 − δ2

···

⇐= δm−1 − δm

2δm

As we have already observed above, there are (at least) two Dynkin diagrams for spo(2m|2n) of different shapes. The classification of all fundamental systems for the root system Φ of spo(2m|2n) is divided into 2 cases below. As usual, we keep in mind εi¯ = δi . (1) First, we classify the fundamental systems Π in Φ that do not contain any long root (i.e., a root of the form ±2ε p ). With parity ignored, the subset ˜ := Φ \ {±2ε p | 1 ≤ p ≤ m} may be identified with the root system of the clasΦ sical Lie algebra so(2m + 2n), whose fundamental systems are completely known, and they are acted upon simply transitively by the Weyl group of so(2m + 2n) (which is an index 2 subgroup of Z2m+n  Sm+n ). Observe that the positive system Φ+ for Φ corresponding to Π is completely determined by the positive system ˜ ∩ Φ+ for Φ ˜ (and vice versa). We conclude that the fundamental systems for Φ ˜ Φ that do not contain any long root are exactly the fundamental systems for Φ. ˜ However, not every fundamental system of Φ gives rise to a fundamental system of Φ. To be precise, a fundamental system of Φ cannot contain a pair of roots of the form {±εi ± ε p , ±εi ∓ ε p }, i = p and 1 ≤ p ≤ m. It follows that there are n of m+n · |W (so(2m + 2n))| such fundamental systems of Φ, and hence the number m+n−1 n W -conjugacy classes for g in this case is m+n · |W (so(2m + 2n))|/|W | = . m (2) Next, we classify the fundamental systems Π that contain some long root.  := Φ ∪ {±2ε j , 1 ≤ j ≤ n}. With the parity ignored, Φ  In this case, we consider Φ may be identified with the root system of sp(2m + 2n), whose fundamental systems

1.3. Non-conjugate positive systems and odd reflections

25

are completely described. Observe that the positive system Φ+ for Φ that corre + of Φ  that contains sponds to Π is completely determined by the positive system Φ + Φ . We conclude that the fundamental systems for Φ that contain some long root  whose long root is of the form ±2ε p . It are exactly the fundamental systems for Φ follows that the number of W -conjugacy of such fundamental systems for   classes m g is m+n . · |W (sp(2m + 2n))|/|W | = 2 m+n−1 n The εδ-sequence for spo(2m|2n) associated to a positive system is now defined as follows. We can first obtain an ordered sequence of ε’s and δ’s just as for spo(2m|2n + 1). If this sequence has an ε as its last member, then it is the εδ-sequence. If the last member in this sequence is a δ, then the εδ-sequence is obtained from this sequence by attaching a sign to the last ε. Example 1.26. For spo(4|4), the εδ-sequences εδδε, εδεδ, and εδ(−ε)δ are distinct. 1.3.5. Conjugacy classes of fundamental systems. We now describe the classification of the W -conjugacy classes of fundamental systems for these Lie superalgebras via the εδ-sequences. Proposition 1.27. Let Φ be the root system and let W be the Weyl group for a Lie superalgebra g of type gl or osp. Then the W -conjugacy classes of fundamental systems in Φ are in one-to-one m+n correspondence with the associated εδ-sequences. In particular, there are m W -conjugacy classes of fundamental systems for   m+n−1 gl(m|n) and spo(2m|2n+1), while there are m+n W -conjugacy classes m + n of fundamental systems for spo(2m|2n). Proof. By the case-by-case classification of fundamental systems in Φ and definition of εδ-sequences, we clearly have a well-defined map Θ : {fundamental systems in Φ}/W −→ {εδ-sequences for g}. As we can easily construct a fundamental system Π for a given εδ-sequence, Θ is surjective. To show Θ is a bijection, it remains to show that the two finite sets have the same cardinalities. The number of W -conjugacy classes of fundamental systems equals |W  |/|W |, where W  denotes the Weyl group of gl(m + n) when g = gl(m|n) and W  denotes the Weyl group  so(2m + 2n + 1) when g = spo(2m|2n + 1). In either case,  of |W  |/|W | = m+n is simply an m . On the other hand, in either case, an εδ-sequence   ordered arrangement of m δ’s and n ε’s. Thus the total number is also m+n m . In the case of spo(2m|2n), when the last slot of an εδ-sequence is an ε, the (m + n − 1) slots can be filled with m δ’s and (n − 1) ε’s. Thus we obtain previous m+n−1 different εδ-sequences this way. When the last slot in an εδ-sequence is m a δ, then the previous (m + n − 1) slots are filled with (m − 1) δ’s and n ε’s, with the last ε having either a positive or negative sign. This way we obtain additional

1. Lie superalgebra ABC

26

      distinct εδ-sequences. Now m+n−1 + 2 m+n−1 is exactly the number 2 m+n−1 n m n of W -conjugacy classes of fundamental systems for spo(2m|2n) that we have computed earlier.  The fundamental and positive systems for the exceptional Lie superalgebras G(3), F(3|1), D(2|1, α) are also listed completely by Kac [60] (with one missing for F(3|1); see [130, 5.1]). From the classification of the fundamental and positive systems we immediately obtain the following proposition and lemma. Proposition 1.28. Let g be a basic Lie superalgebra, excluding sl(n|n)/CIn|n and gl(m|n). Let h be a Cartan subalgebra of g. Then any fundamental system for the root system of (g, h) forms a basis in h∗ . (For gl(m|n), any fundamental system is linearly independent in h∗ .) Lemma 1.29. Let g be a basic Lie superalgebra. Let Π be the fundamental system in a positive system Φ+ , and let α be an isotropic odd root in Φ+ . Then there exists w ∈ W such that w(α) ∈ Π.

1.4. Odd and real reflections In this section, beside real reflections associated to even roots as for semisimple Lie algebras, we introduce odd reflections associated to isotropic odd simple roots. Both real and odd reflections permute the fundamental systems of a root system. 1.4.1. A fundamental lemma. We have seen that the fundamental systems of a root system Φ are not always W -conjugate due to the existence of odd roots. Recall a root α ∈ Φ is isotropic if (α, α) = 0, and an isotropic root must be odd. The following lemma plays a fundamental role in the representation theory of Lie superalgebras. Lemma 1.30. Let g be a basic Lie superalgebra and let Π be a fundamental system of a positive system Φ+ . Let α be an odd isotropic simple root. Then, (1.43)

+ Φ+ α := {−α} ∪ Φ \{α}

is a new positive system whose corresponding fundamental system Πα is given by (1.44) Πα = {β ∈ Π | (β, α) = 0, β = α} ∪ {β + α | β ∈ Π, (β, α) = 0} ∪ {−α}. Proof. For g = gl(m|n), spo(2m|2n), or spo(2m|2n + 1), which we are mostly concerned about in this book, we have already obtained complete descriptions of all fundamental systems. For most Π’s, the set {β ∈ Π | (β, α) = 0} has cardinality at most 2 and consists of roots of the form ±εi ±ε j for i = j ∈ I(m|n). By inspection we see that Πα is a fundamental system. There are a few extra cases when the root α corresponds to a node in the corresponding Dynkin diagram which is either

1.4. Odd and real reflections

27

connected to a long simple root, or connected to a short non-isotropic root, or is a branching node, or is one of the (short) end nodes of a branching node, as follows: 

···

 α

···

⇐=





=⇒



···

α@  @

α



···



@  @ α 

Here the vertical dashed lines indicate an edge when it connects two ’s and no    edge otherwise, means either  or , and means either or . It can be checked directly in these cases that Πα is also a fundamental system. + + Take β ∈ Φ+ α ∩ Φ . Since Π is a fundamental system for Φ , by Proposition 1.28 we have a unique expression β = ∑γ∈Π\{α} mγ γ + mα α for mγ ∈ Z+ and mα ∈ Z+ . It follows by definition of Πα that β can be expressed as a linear combination of Πα of the form β = ∑κ∈Πα \{−α} mκ κ + mα (−α) for some suitable integer mα . By choice of β we have mκ > 0 for some κ, and hence mα ∈ Z+ , since Πα is a fundamental system. This shows that Φ+ α is the positive system corresponding to Πα .

For an exceptional Lie superalgebra g = D(2|1, α), G(3), or F(3|1), which will not be studied in any detail in this book, our proof shall be rather sketchy. A conceptual approach (see [66]) would be to follow Remark 1.19 to regard g as a Kac-Moody superalgebra with Chevalley generators ei , f i , hi for i ∈ I associated to a Cartan matrix A (and a fundamental system Π = {αi | i ∈ I}), where I is Z2 graded. For an isotropic odd root α ∈ Π, one can construct a new set of Chevalley generators ei , f i , hi associated to Πα , which gives rise to a new Cartan matrix A . By standard machinery of the Kac-Moody theory, the Kac-Moody superalgebra associated to A and A coincide with g. From this it follows that Πα is a fundamental system for g. The same argument as above shows that Φα is the positive system associated to the fundamental system Πα . This uniform approach is applicable to all basic Lie superalgebras.  1.4.2. Odd reflections. Let b = h ⊕ n+ , where n+ corresponds to the positive system Φ+ . Then the new Borel subalgebra corresponding to Φ+ α in (1.44) for an isotropic odd simple root α is given by bα := h ⊕

(1.45)



gβ .

β∈Φ+ α

Observe that b0¯ = bα0¯ by Lemma 1.30. The process of obtaining Πα (respectively, α + Φ+ α or b ) from Π (respectively, Φ or b) will be referred to as odd reflection (with respect to α) and will be denoted by rα . We shall write (1.46)

rα (Π) = Πα ,

Note that r−α rα = 1.

rα (Φ+ ) = Φ+ α,

rα (b) = bα .

1. Lie superalgebra ABC

28

Remark 1.31. In contrast to a real reflection associated to an even root (see below), an odd reflection rα with respect to an isotropic odd root α may not be extended to a linear transformation on h∗ which sends a simple root in Π to a simple root in Πα . For example, for spo(2m|2n) with a fundamental system Π corresponding to the first Dynkin diagram below, take α = δm − εn . The odd reflection transforms Π to Πα corresponding to the second Dynkin diagram below (with the · · · portion unchanged). A plausible transformation would have to fix those δi , ε j (for i < m, j < n) and interchange εn and δm , but this transformation would not send δm +εn to 2δm . 

···



εn−1 − δm@

@

δm + εn

··· δm − εn



εn−1 − εn



⇐=

εn − δm

2δm

1.4.3. Real reflections. By Theorem 1.18, a basic Lie superalgebra g admits an even non-degenerate supersymmetric bilinear form (·, ·), which restricts to a nondegenerate form (·, ·) on h and on h∗ . For an even root α (which is automatically non-isotropic), we define the real reflection rα as a linear map on h∗ given by rα (x) = x − 2

(x, α) α, (α, α)

for x ∈ h∗ .

In particular, rα preserves Φ, Φ0¯ , and Φ1¯ , respectively. The group generated by real reflections rα , for α ∈ Φ0¯ , is precisely the Weyl group W of g0¯ (and hence the Weyl group of g by Definition 1.17). ¯ 0¯ as defined in For an even simple root α, we must have α/2 ∈ Φ, that is, α ∈ Φ + (1.19). In this case, (1.46) can be understood with Φα as in (1.43), bα as in (1.45), and Πα as the image of Π under rα . For an odd root α such that 2α ∈ Φ, we define the reflection rα as the real reflection r2α associated to 2α. In this case, it is understood that Πα = rα (Π), and (1.47)

+ Φ+ α = {−α, −2α} ∪ Φ \{α, 2α}

is the new positive system associated to Πα . 1.4.4. Reflections and fundamental systems. Proposition 1.32. For two fundamental systems Π and  Π of a basic Lie superalgebra g, there exists a sequence consisting of real and odd reflections r1 , r2 , . . . , rk such that rk . . . r2 r1 (Π) =  Π. Proof. Denote by Φ+ and  Φ+ the positive systems associated to Π and  Π respectively. We prove the corollary by induction on |Φ+ ∩  Φ− |. If |Φ+ ∩  Φ− | = 0, then Π =  Π. Assume now |Φ+ ∩  Φ− | > 0. Then Π =  Π, and we pick α ∈ Π ∩  Φ− = 0/

1.4. Odd and real reflections

29

and apply the real or odd simple reflection rα . Observe that Φ+ α is the positive system with fundamental system Πα = rα (Π), regardless of the parity of the simple root α. Note that  − +  − |Φ+ α ∩ Φ | < |Φ ∩ Φ |. By the inductive assumption, there exists a sequence of real and odd reflections r2 , . . . , rk such that rk . . . r2 (Πα ) =  Π. Hence rk . . . r2 rα (Π) =  Π.  Proposition 1.33. Let g be a basic Lie superalgebra. Let Φ+ be a positive system with Π as its fundamental system and ρ as its associated Weyl vector. Then, (ρ, β) = 1 2 (β, β) for every simple root β ∈ Π. Proof. Let α ∈ Π. Let Πα and Φ+ α be respectively the fundamental and positive systems obtained by a real or odd reflection rα defined above. Let ρα denote the + Weyl vector for the positive system Φ+ α . Recall Φα is given in (1.43) for α odd or ¯ 0¯ , and in (1.47) otherwise. We compute that for α ∈ Φ  ¯ 0¯ or non-isotropic odd, ρ − α, for α ∈ Φ ρα = ρ + α, for α isotropic odd. Claim. Assuming that the proposition holds for a fundamental system Π, it holds for the new fundamental system Πα for α ∈ Π. Let us prove the claim. First assume α ∈ Π is even or non-isotropic odd. In this case, ρα = rα (ρ), Πα = rα (Π), and so we have for each β ∈ Π that 1 1 (ρα , rα (β)) = (ρ, β) = (β, β) = (rα (β), rα (β)). 2 2 Now assume α ∈ Π is isotropic odd. We will check case-by-case, recalling the definition of Πα from (1.44). If (β, α) = 0 for β ∈ Π, then β ∈ Πα . By the assumption of the claim, we have 1 (ρα , β) = (ρ + α, β) = (ρ, β) = (β, β). 2 If (β, α) = 0 for β ∈ Π, then β + α ∈ Πα . By the assumption of the claim, (ρ, α) = 1 2 (α, α) = 0, and hence 1 1 (ρα , β + α) = (ρ + α, β + α) = (ρ + α, β) = (β, β) + (α, β) = (β + α, β + α). 2 2 This completes the proof of the claim. By the claim and Proposition 1.32, it suffices to prove the proposition for just one fundamental system, e.g., the standard fundamental system for type gl and osp. Assume that β is an even simple root or a non-isotropic odd simple root. In either case, one checks that rβ (ρ) = ρ − β. Since (·, ·) is W -invariant, we have   (ρ, β) = rβ (ρ), rβ (β) = (ρ − β, −β) = −(ρ, β) + (β, β). It follows that (ρ, β) = 12 (β, β) just as for semisimple Lie algebras.

1. Lie superalgebra ABC

30

For isotropic odd simple root β, we do a case-by-case inspection. For type gl or osp, the equation (ρ, β) = 12 (β, β) follows directly from Lemma 1.22. For the three exceptional cases, it can be checked directly by using a fundamental system with exactly one isotropic simple root and the detailed description of root systems (cf. Kac [60]). We skip the details. 

1.4.5. Examples. Below we illustrate the notion of odd reflections by examples. Example 1.34. Associated with gl(1|2), we have Φ0¯ = {±(ε1 − ε2 )} and Φ1¯ = {±(δ1 − ε1 ), ±(δ1 − ε2 )}. There are 6 fundamental systems that are related by real and odd reflections as follows. There are three conjugacy classes of Borel subalgebras corresponding to the three columns below, and each vertical pair corresponds to such a conjugacy class.  δ1 − ε1

 δ1 − ε2



rδ1 −ε1



ε1 − ε2



rε1 −ε2



rδ1 −ε2





ε1 − δ1

δ1 − ε2







ε2 − ε1

ε2 − δ1

rδ1 −ε2



ε1 − ε2

rε1 −ε2

rδ1 −ε1





ε2 − δ1

 

ε2 − ε1

δ1 − ε1





rε1 −ε2



ε1 − δ1

Example 1.35. Let g = gl(4|2). The following sequence of fundamental systems Π0 = Πst , Π1 , Π2 , Π3 , Π4 (in descending order) is obtained from the standard fundamental system Πst by applying consecutively the sequences of odd reflections with respect to the odd roots δ4 − ε1 , δ3 − ε1 , δ2 − ε1 , and δ1 − ε1 .  δ1 − δ2

 δ1 − δ2

 ε1 − δ1







δ2 − δ3 δ3 − δ4 δ4 − ε1







δ2 − ε1 ε1 − δ3 δ3 − δ4







δ1 − δ2 δ2 − δ3 δ3 − δ4



ε1 − ε2

 δ4 − ε2



rδ4 −ε1 XX X X rδ3 −ε1

  rδ2 −ε1 XX X X rδ1 −ε1

 

 δ1 − δ2

 δ1 − ε1







δ2 − δ3 δ3 − ε1 ε1 − δ4







ε1 − δ2 δ2 − δ3 δ3 − δ4

 δ4 − ε2

 δ4 − ε2

δ4 − ε2

If we continue to apply to Π4 consecutively the sequence of odd reflections with respect to the odd roots δ4 −ε2 , δ3 −ε2 , δ2 −ε2 , and δ1 −ε2 , we obtain the following fundamental systems Π4 , Π5 , Π6 , Π7 , Π8 :

1.5. Highest weight theory  ε1 − δ1

 ε1 − δ1



ε1 − ε2







δ1 − δ2 δ2 − δ3 δ3 − δ4







δ1 − δ2 δ2 − ε2 ε2 − δ3



31  δ4 − ε2

rδ4 −ε2 XX X X rδ3 −ε2

 

 δ3 − δ4



 



ε1 − δ1

rδ2 −ε2 XX X X rδ1 −ε2





 ε1 − δ1







δ1 − δ2 δ2 − δ3 δ3 − ε2







 ε2 − δ4



δ1 − ε2 ε2 − δ2 δ2 − δ3 δ3 − δ4

ε2 − δ1 δ1 − δ2 δ2 − δ3 δ3 − δ4

1.5. Highest weight theory In this section, we formulate the Poincar´e-Birkhoff-Witt Theorem for Lie superalgebras. Finite-dimensional irreducible modules over certain solvable Lie superalgebras, including all Borel subalgebras, are classified. This is then used to develop a highest weight theory of the basic and queer Lie superalgebras. 1.5.1. The Poincar´e-Birkhoff-Witt (PBW) Theorem. Let g = g0¯ ⊕ g1¯ be a Lie superalgebra. A universal enveloping algebra of g is an associative superalgebra with unity U(g) together with a homomorphism of Lie superalgebras ι : g → U(g) characterized by the following universal property. Given an associative superalgebra A and a homomorphism of Lie superalgebras ϕ : g → A, there exists a unique homomorphism of associative superalgebras ψ : U(g) → A such that the following diagram commutes: ι

/ U(g) BB BB ∃ψ ϕ BBB ! 

gB B

A

In particular, this implies that representations of g are representations of U(g) and vice versa. It follows by a standard tensor algebra construction that a universal enveloping algebra exists, and it is unique up to isomorphism by the defining universal property. Theorem 1.36 (Poincar´e-Birkhoff-Witt Theorem). Let {x1 , x2 , . . . , x p } be a basis for g0¯ and let {y1 , y2 , . . . , yq } be a basis for g1¯ . Then the set  r1 r2  r s x1 x2 . . . x pp ys11 ys22 . . . yqq | r1 , r2 , . . . , r p ∈ Z+ , s1 , s2 , . . . , sq ∈ {0, 1} is a basis for U(g). The proof for Theorem 1.36 is a straightforward super generalization of the Lie algebra case and will be omitted (see [102, Theorem 2.1] for detail).

1. Lie superalgebra ABC

32

The superalgebra U(g) carries a filtered algebra structure by letting Uk (g) = span{x1r1 x2r2 . . . x pp ys11 ys22 . . . yqq | ∑ ri + ∑ s j ≤ k}, s

r

i

k ∈ Z+ .

j

Its associated graded algebra is isomorphic to S(g0¯ ) ⊗ ∧(g1¯ ). Given a Lie subalgebra l of a finite-dimensional Lie superalgebra g and an l-module V , we define the induced module as IndglV = U(g) ⊗U(l) V. By the PBW Theorem, if V is finite-dimensional, then so is Indgg0¯ V . 1.5.2. Representations of solvable Lie superalgebras. Just as for Lie algebras, a finite-dimensional Lie superalgebra g = g0¯ ⊕ g1¯ is called solvable if g(n) = 0 for some n ≥ 1, where we define inductively g(n) = [g(n−1) , g(n−1) ] and g(0) = g. Let g = g0¯ ⊕ g1¯ be a finite-dimensional solvable Lie superalgebra such that [g1¯ , g1¯ ] ⊆ [g0¯ , g0¯ ]. Given λ ∈ g∗0¯ with λ([g0¯ , g0¯ ]) = 0, we define a one-dimensional g-module Cλ = Cvλ by xvλ = λ(x)vλ , yvλ = 0,

for x ∈ g0¯ ,

for y ∈ g1¯ .

There is a canonical linear isomorphism (g0¯ /[g0¯ , g0¯ ])∗ ∼ = {λ ∈ g∗0¯ | λ([g0¯ , g0¯ ]) = 0}. Lemma 1.37. Let g = g0¯ ⊕ g1¯ be a finite-dimensional solvable Lie superalgebra such that [g1¯ , g1¯ ] ⊆ [g0¯ , g0¯ ]. Then every finite-dimensional irreducible g-module is one-dimensional. A complete list of finite-dimensional irreducible g-modules is given by Cλ , for λ ∈ (g0¯ /[g0¯ , g0¯ ])∗ . Proof. We have already shown that any such Cλ is a g-module. Clearly every one-dimensional g-module is isomorphic to Cλ , for some λ ∈ (g0¯ /[g0¯ , g0¯ ])∗ . So it suffices to show that every finite-dimensional irreducible g-module V is one dimensional. (n)

The even subalgebra g0¯ is solvable since g is solvable and g0¯ ⊆ g(n) for every n. By applying Lie’s theorem to g0¯ , V contains a nonzero g0¯ -invariant vector vλ , where λ ∈ g∗0¯ with λ([g0¯ , g0¯ ]) = 0. Here, by a g0¯ -invariant vector we mean that xvλ = λ(x)vλ , ∀x ∈ g0¯ . Now Frobenius reciprocity says that Homg (Indgg0¯ Cvλ ,V ) ∼ = Homg0¯ (Cvλ ,V ), and thus, by the irreducibility of V , there exists a g-epimorphism from Indgg0¯ Cvλ ∼ = ∧(g1¯ ) ⊗ Cvλ onto V . Thus, to prove that V is one dimensional, it suffices to prove that Indgg0¯ Cvλ has a composition series with one-dimensional composition factors. We shall construct such a composition series explicitly.

1.5. Highest weight theory

33

Set dim g1¯ = n. By Lie’s theorem, the g0¯ -module g1¯ has an ordered basis {y j | j = 1, . . . , n} such that for all 1 ≤ i ≤ n we have n

(1.48)

ad g0¯ (yi ) ⊆ ∑ Cy j ,

ad [g0¯ , g0¯ ](yi ) ⊆

j=i

n

∑

Cy j .

j=i+1

Let B1 = {1, y1 }, and define inductively Bk = {Bk−1 , Bk−1 yk }, for 2 ≤ k ≤ n, where the set Bk−1 yk denotes the ordered set of elements obtained by multiplying all elements in Bk−1 on the right by yk . Then Bn vλ is an ordered basis for Indgg0¯ Cvλ n of cardinality 2n . Denote this ordered basis by {v1 , · · · , v2n }. Set Vi := ⊕2j=i Cv j . Thanks to (1.48), we have a filtration of g0¯ -modules V = V1 ⊇ V2 ⊇ V3 ⊇ · · · ⊇ · · ·V2n ⊇ 0. Clearly we have, for 1 ≤ k,  ≤ n and i1 < . . . < i ≤ n,  [yk , yi1 ] yi2 . . . yi vλ − yi1 yk yi2 . . . yi vλ , (1.49) y k y i1 y i2 . . . y i v λ = 1 2 [yk , yi1 ] yi2 . . . yi vλ ,

if k = i1 , if k = i1 .

Using the assumption that [g1¯ , g1¯ ] ⊆ [g0¯ , g0¯ ], it follows from (1.48) and (1.49) that every yk leaves Vi invariant, and hence each Vi is a g-module. Thus, the above filtration is a composition series with one-dimensional composition factors.  Example 1.38. The Lie superalgebra g = Cz ⊕ g1¯ in Example 1.4(4) associated to a non-degenerated symmetric bilinear form B on a nonzero space g1¯ is solvable. But an irreducible module of g with the central element z acting as a nonzero scalar has dimension more than one. So the condition [g1¯ , g1¯ ] ⊆ [g0¯ , g0¯ ] in Lemma 1.37 cannot be dropped. 1.5.3. Highest weight theory for basic Lie superalgebras. Let g be a basic Lie superalgebra. Let h be the standard Cartan subalgebra and let Φ be the root system. Let b = h ⊕ n+ be a Borel subalgebra of g = b ⊕ n− and let Φ+ be the associated positive system. The condition of Lemma 1.37 is satisfied for the solvable Lie superalgebra b, since we have b1¯ = n+ , and 1¯ [b1¯ , b1¯ ] = [n+ , n+ ] ⊆ n+ = [h, n+ ] ⊆ [b0¯ , b0¯ ]. 1¯ 1¯ 0¯ 0¯ Let V be a finite-dimensional irreducible representation of g. Then by Lemma 1.37 V contains a one-dimensional b-module that is of the form Cλ = Cvλ , for λ ∈ h∗ ∼ = (b/[b, b])∗ . That is, hvλ = λ(h)vλ (h ∈ h),

xvλ = 0 (x ∈ n+ ).

By the PBW theorem and the irreducibility of V , we obtain V = U(n− )vλ and thus a weight space decomposition (1.50)

V=



µ∈h∗

Vµ ,

1. Lie superalgebra ABC

34

where the µ-weight space Vµ is given by Vµ := {v ∈ V | hv = µ(h)v, ∀h ∈ h}. By (1.50), Vµ = 0 unless λ − µ is a Z+ -linear combination of positive roots. The weight λ is called the b-highest weight (and sometime called an extremal weight) of V , the space Cvλ is called the b-highest weight space, and the vector vλ is called a b-highest weight vector for V . When no confusion arises, we will simply say highest weight by dropping b. Hence, we have established the following. Proposition 1.39. Let g be a basic Lie superalgebra with a Borel subalgebra b. Then every finite-dimensional irreducible g-module is a b-highest weight module. We shall denote the highest weight irreducible module of highest weight λ by L(λ), L(g, λ), or L(g, b, λ), depending on whether b and g are clear from the context. Recall from Section 1.4 the notations Πα and bα associated to an isotropic odd simple root α. Denote by ·, · : h∗ × h → C the standard bilinear pairing. Denote by hα the corresponding coroot for α, and denote by eα and f α the root vectors of roots α and −α so that [eα , f α ] = hα . Lemma 1.40. Let L be a (not necessarily finite-dimensional) simple g-module and let v be a b-highest weight vector of L of b-highest weight λ. Let α be an isotropic odd simple root. (1) If λ, hα  = 0, then L is a g-module of bα -highest weight λ and v is a bα -highest weight vector. (2) If λ, hα  = 0, then L is a g-module of bα -highest weight (λ − α) and fα v is a bα -highest weight vector. Proof. We first observe three simple identities: (i) eα f α v = [eα , f α ]v = hα v = λ, hα v. (ii) eβ f α v = [eβ , f α ]v = 0 for any β ∈ Φ+ ∩ Φ+ α , since either β − α is not a root + + or it belongs to Φ ∩ Φα . (iii) fα2 v = 0, since α is an isotropic odd root and so f α2 = 0. Now, we consider two cases separately. (1) Assume that λ, hα  = 0. Then we must have fα v = 0, for otherwise fα v would be a b-singular vector in the simple g-module L by (i) and (ii). Also eβ v = 0, α for β ∈ Φ+ ∩ Φ+ α . Thus Lemma 1.30 implies that v is a b -highest weight vector of weight λ in the g-module L. (2) Assume that λ, hα  = 0. Then (i) above implies that f α v is nonzero. Now it follows by (ii), (iii), and Lemma 1.30 that fα v is a bα -highest weight vector of weight λ − α in L. 

1.5. Highest weight theory

35

Example 1.41. Let g = gl(4|2). Denote the sequence of Borel subalgebras corresponding to the fundamental systems Πi in Example 1.35 as bi , for 0 ≤ i ≤ 8, with bst = b0 . Consider the finite-dimensional irreducible gl(4|2)-module L(λ), where λ = a1 δ1 + a2 δ2 + a3 δ3 + a4 δ4 + b1 ε1 + b2 ε2 with a1 ≥ a2 ≥ a3 ≥ a4 ≥ 2 and b1 ≥ b2 ≥ 0. We identify λ = (a1 , a2 , a3 , a4 |b1 , b2 ). The bi -extremal weights, denoted by λi for 0 ≤ i ≤ 4, are computed as follows: λ0 = λ = (a1 , a2 , a3 , a4 |b1 , b2 ), λ1 = (a1 , a2 , a3 , a4 − 1|b1 + 1, b2 ), λ2 = (a1 , a2 , a3 − 1, a4 − 1|b1 + 2, b2 ), λ3 = (a1 , a2 − 1, a3 − 1, a4 − 1|b1 + 3, b2 ), λ4 = (a1 − 1, a2 − 1, a3 − 1, a4 − 1|b1 + 4, b2 ). If we continue to consecutively apply to λ4 the sequence of odd reflections with respect to the odd roots δ4 −ε2 , δ3 −ε2 , δ2 −ε2 , and δ1 −ε2 , we obtain the following bi -extremal weights λi , for 5 ≤ i ≤ 8: λ5 = (a1 − 1, a2 − 1, a3 − 1, a4 − 2|b1 + 4, b2 + 1), λ6 = (a1 − 1, a2 − 1, a3 − 2, a4 − 2|b1 + 4, b2 + 2), λ7 = (a1 − 1, a2 − 2, a3 − 2, a4 − 2|b1 + 4, b2 + 3), λ8 = (a1 − 2, a2 − 2, a3 − 2, a4 − 2|b1 + 4, b2 + 4). We shall see in Section 2.4 that all these weights λi afford very simple visualization in terms of Young diagrams. 1.5.4. Highest weight theory for q(n). Let g = q(n) be the queer Lie superalgebra. We recall several subalgebras of g from Section 1.2.6. Let h be the standard Cartan subalgebra consisting of block diagonal matrices and let b be the standard Borel subalgebra of block upper triangular matrices of g. We have b = h ⊕ n+ and g = b ⊕ n− . The Cartan subalgebra h = h0¯ ⊕ h1¯ is a solvable but nonabelian Lie superalgebra, since [h1¯ , h1¯ ] = h0¯ and [h0¯ , h0¯ + h1¯ ] = 0. For λ ∈ h∗0¯ , define a symmetric bilinear form ·, ·λ on h1¯ by v, wλ := λ([v, w]),

for v, w ∈ h1¯ .

Denote by Rad·, ·λ the radical of the form ·, ·λ . Then ·, ·λ descends to a nondegenerate symmetric bilinear form on h1¯ /Rad·, ·λ , and it gives rise to a superalgebra Cλ as follows. Choosing an orthonormal basis {ei } of ∈ h1¯ /Rad·, ·λ with respect to the form ·, ·λ, the associative superalgebra Cλ is generated by {ei } subject to the relations (3.23), and hence is isomorphic to the Clifford superalgebra Ck , where k = dim h1¯ /Rad·, ·λ (see Definition 3.33). By definition we have an isomorphism of associative superalgebras (1.51) Cλ ∼ = U(h)/Iλ ,

1. Lie superalgebra ABC

36

where Iλ denotes the ideal of U(h) generated by Rad·, ·λ and a − λ(a) for a ∈ h0¯ . Let h1¯ ⊆ h1¯ be a maximal isotropic subspace with respect to ·, ·λ , and define the Lie subalgebra h := h0¯ ⊕ h1¯ . The one-dimensional h0¯ -module Cvλ , defined by hvλ = λ(h)vλ , extends to an h -module by letting h1¯ .vλ = 0. Define the induced h-module Wλ := Indhh Cvλ . Recall the well-known fact that a Clifford superalgebra admits a unique irreλ (see, for example, Exercise 3.11 in Chapter 3). ducible (Z2 -graded) module W λ (viewed as an Lemma 1.42. For λ ∈ h∗0¯ , the h-module Wλ is isomorphic to W h-module via the pullback through (1.51)) and is irreducible. Furthermore, every finite-dimensional irreducible h-module is isomorphic to Wλ , for some λ ∈ h∗0¯ . Lemma 1.42 shows that the h-module Wλ is independent of a choice of a maximal isotropic subspace h1¯ . Proof. The action of U(h) on Wλ descends to an action of U(h)/Iλ , and via (1.51) λ of the Clifford superalgebra we identify Wλ with the unique irreducible module W Cλ . Hence, the h-module Wλ is irreducible. Suppose we are given an irreducible h-module U. Then it contains an h0¯ weight vector vλ of weight λ ∈ h∗0¯ . Recall that h = h0¯ ⊕ h1¯ , and consider the   h -submodule π,U(h )vλ of U such that π([h1¯ , h1¯ ]) = π([h0¯ , h0¯ ]) = 0. By applying Lemma 1.37 to Lie superalgebra π(h ), there exists a one-dimensional h submodule Cvλ of U(h )vλ ⊆ U. By Frobenius reciprocity we have a surjective h-homomorphism from Wλ onto U. Since Wλ is irreducible, we have U ∼ = Wλ .  Let V be a finite-dimensional irreducible g-module. Pick an irreducible hmodule Wλ in V , where λ ∈ h∗0¯ can be taken to be maximal in the partial order induced by the positive system Φ+ by the finite dimensionality of V . By definition, Wλ is h0¯ -semisimple of weight λ. For any α ∈ Φ+ with associated even root vector eα and odd root vector eα in n+ , the space CeαWλ + CeαWλ is an h-module that is h0¯ -semisimple of weight λ + α. If CeαWλ + CeαWλ = 0 for some α ∈ Φ+ , then it contains an isomorphic copy of Wλ+α as an h-submodule, contradicting the maximal weight assumption of λ. Hence, we have n+Wλ = 0. By irreducibility of V we must have U(n− )Wλ = V , which gives rise to a weight space decomposition  of V = µ∈h∗¯ Vµ . The space Wλ = Vλ is the highest weight space of V , and it 0 completely determines the irreducible module V . We denote V by L(g, λ), or L(λ), if g is evident from the context. Summarizing, we have proved the following. Proposition 1.43. Let g = q(n). Any finite-dimensional irreducible g-module is a highest weight module.

1.6. Exercises

37

Let (λ) be the number of nonzero parts in a composition λ (generalizing the notation for length of a partition λ). We set  0, if (λ) is even, (1.52) δ(λ) = 1, if (λ) is odd. Now for λ ∈ h∗ , recalling the notation Hi from (1.30), we identify λ with the composition λ = (λ(H1 ), . . . , λ(Hn )), and hence, (λ) is equal to the dimension of the space h1¯ /Rad·|·λ. We remark that the highest weight space Wλ of L(λ) has dimension 2((λ)+δ(λ))/2 . Note that the Clifford superalgebra Cλ admits an odd automorphism if and only if (λ) is odd. Hence, the h-module Wλ , or equivalently the irreducible Cλ -module Wλ , has an odd automorphism if and only if (λ) is an odd integer. An automorphism of the irreducible g-module L(λ) clearly induces an h-module automorphism of its highest weight space. Conversely, any h-module automorphism on Wλ induces an automorphism of the g-module IndgbWλ . Since an automorphism preserves the maximal submodule, it induces an automorphism of the unique irreducible quotient g-module. We have proved the following. Lemma 1.44. Let g = q(n) be the queer Lie superalgebra with a Cartan subalgebra h. Let L(λ) be the irreducible g-module of highest weight λ ∈ h∗0¯ . Then, dim Endg (L(λ)) = 2δ(λ) . This suggests that Schur’s Lemma requires modification for superalgebras. This will be discussed in depth in Chapter 3, Lemma 3.4.

1.6. Exercises Exercises 1.13, 1.14, 1.15, 1.16, and 1.24 below indicate that various classical theorems in the theory of Lie algebras fail for Lie superalgebras. Exercise 1.1. Let ϕ be an automorphism of Lie superalgebra gl(m|n), and let J ∈ gl(m|n)0¯ . Define g(ϕ, J) := {g ∈ gl(m|n) | Jg − ϕ(g)J = 0}. Prove that g(ϕ, J) is a subalgebra of gl(m|n). Exercise 1.2. Let J2m| be as defined in 1.1.3, and let J2m| be the matrix obtained from J2m| by substituting Im with −Im . Prove: (1) There exists an automorphism ϕ of Lie superalgebra gl(2m|) given by ϕ(g) = −gst . (2) g(ϕ, J2m| ) = spo(2m|). (3) g(ϕ3 , J2m| ) = g(ϕ, J2m| ). Exercise 1.3. Prove that Πgst Π−1 = (ΠgΠ−1 )st , for g ∈ gl(m|n); conclude that osp(m|n) ∼ = spo(n|m), for n even (as stated in Remark 1.9). 3

1. Lie superalgebra ABC

38

Exercise 1.4. The space ∧(2) is naturally a W (2)-module with submodule C1. Show that the action of W (2) on ∧(2)/C1 is faithful, and thus induces, by identifying ∧(2)/C1 with C1|2 , an isomorphism of Lie superalgebras W (2) ∼ = sl(1|2). Exercise 1.5. Prove that the following linear map ψ : spo(2|2) → sl(1|2) is an isomorphism of Lie superalgebras: ⎛ ⎞ √ ⎞ √ ⎛ d e y1 x1 2y 2y 2a 1 ⎜ f −d −y x ⎟ ψ √ ⎜ ⎟ −→ ⎝ 2x1 d + a e ⎠. ⎝−x x1 −a 0 ⎠ √ 2x f −d + a y y1 0 a Exercise 1.6. Suppose that g = g−1 ⊕ g0 ⊕ g1 is a Z-graded Lie superalgebra such that g0¯ = g0 , g1¯ = g1 ⊕ g−1 , and [g1 , g−1 ] = g0 . Assume further that g0 is a simple Lie algebra and g±1 are irreducible g0 -modules such that Homg0 (g0 , g1 ⊗g−1 ) ∼ = C. Prove that these data determine the Lie superalgebra structure on g uniquely. Exercise 1.7. Prove: (1) The simple Lie superalgebras S(3) and [p(3), p(3)] are isomorphic. (Hint: use Exercise 1.6.) (2) The Lie superalgebra H(4) and sl(2|2)/CI2|2 are isomorphic. Exercise 1.8. Let g = g0¯ ⊕ g1¯ be a finite-dimensional simple Lie superalgebra with g1¯ = 0. Prove: (1) [g0¯ , g1¯ ] = g1¯ . (2) [g1¯ , g1¯ ] = g0¯ . (3) The g0¯ -module g1¯ is faithful. Exercise 1.9. Let g = g0¯ ⊕ g1¯ be a Lie superalgebra such that [g1¯ , g1¯ ] = g0¯ and the adjoint g0¯ -module g1¯ is faithful and irreducible. Prove that g is simple. Exercise 1.10. Let g =

n

j=−1 g j

be a Z-graded Lie superalgebra satisfying

(1) [g−1 , x] = 0 implies that x = 0, for all x ∈ g j with j ≥ 0; (2) The adjoint g0 -module g−1 is irreducible; (3) [g−1 , g1 ] = g0 , [g0 , g1 ] = g1 , and g j+1 = [g j , g1 ] for j ≥ 1. Prove that g is simple. Exercise 1.11. Let g = sl(2|2)/CI2|2 . Prove that dim gα = 2, for α ∈ Φ1¯ . Exercise 1.12. Let g be a Lie superalgebra. Prove that g ⊗ ∧(n) is a Lie superalgebra with Lie bracket defined by [a ⊗ λ, b ⊗ µ] = (−1)|λ||b| [a, b] ⊗ λµ,

a, b ∈ g; λ, µ ∈ ∧(n).

1.6. Exercises

39

Exercise 1.13. Let g be a finite-dimensional simple Lie superalgebra. The Lie superalgebra W (n) acts naturally on g ⊗ ∧(n) so that the semidirect product G = (g ⊗ ∧ (n))  W (n) is a Lie superalgebra. Prove that G is semisimple (which by definition means that G has no nontrivial solvable ideal.) Exercise 1.14. Find a filtration of subalgebras for G in Exercise 1.13 of the form 0 = G0  G1  G2  · · ·  Gk−1  Gk = G, for some k, such that Gi−1 is an ideal in Gi wtih Gi /Gi−1 simple, for all i = 1, . . . , k. Prove that G is not a direct sum of simple Lie superalgebras. Exercise 1.15. Let G be constructed as in Exercise 1.13 from a not necessarily simple Lie superalgebra g. Suppose that V is a faithful irreducible representation of g. Prove that, as a representation of G, V ⊗ ∧(n) is faithful and irreducible. Exercise 1.16. Continuing Exercise 1.15, assume that g is the one-dimensional abelian Lie algebra and V is a nontrivial irreducible representation of g. Prove that G is not reductive, i.e., G is not a direct sum of a semisimple Lie superalgebra and a one-dimensional even subalgebra. (It is known that a finite-dimensional Lie algebra possessing a finite-dimensional faithful representation is reductive.) Exercise 1.17. Let g be a simple Lie superalgebra whose Killing form is nondegenerate. Prove that every derivation of g is inner. Exercise 1.18. Prove that the Killing forms on the Lie superalgebras sl(k|) with k = , k +  ≥ 2, and osp(m|2n) with m − 2n = 2, m + 2n ≥ 2 are non-degenerate. Conclude that the derivations of these Lie superalgebras are all inner. Exercise 1.19. Let g = g0¯ ⊕ g1¯ be a finite-dimensional simple Lie superalgebra. Suppose that g0¯ is a semisimple Lie algebra and let Der(g) = ad(g) ⊕ W be a decomposition of ad(g0¯ )-modules. Prove: (1) W is a trivial ad(g0¯ )-module. (2) For D ∈ W , the map D : g → g is an ad(g0¯ )-homomorphism vanishing on g0¯ . Exercise 1.20. Use Exercise 1.19(2) to prove the following: (1) Every derivation of osp(m|2n) is inner. (2) The space Der(g)/ad(g) for sl(m + 1|m + 1)/CIm+1|m+1 , [p(m), p(m)], and [q(m), q(m)]/CIm|m, for m ≥ 2, are all one-dimensional. (3) The space Der(g)/ad(g) for sl(2|2)/CI2|2 is three-dimensional. 

Exercise 1.21. Let g = j≥−1 g j be the simple Z-graded Lie superalgebra W (m), S(m), or H(m + 1), for m ≥ 3, with principal gradation. Let b0 be a Borel subalge  bra of g0 . Set g≥0 = j≥0 g j , g>0 = j>0 g j , and b = b0 ⊕ g>0 . Prove: (1) b is a solvable Lie superalgebra satisfying [b1¯ , b1¯ ] ⊆ [b0¯ , b0¯ ].

1. Lie superalgebra ABC

40

(2) A finite-dimensional irreducible representation of g≥0 is an irreducible representation of g0 on which the subalgebra g>0 acts trivially. (3) Any finite-dimensional irreducible module of g is a quotient of Indgg≥0 V , where V is a finite-dimensional irreducible representation of g≥0 . Conclude that, up to isomorphism, finite-dimensional irreducible g0 -modules are in one-to-one correspondence with finite-dimensional irreducible g-modules. Exercise 1.22. Let g = [p(2), p(2)]. Prove: (1) [g, g]  g, and hence g is not simple. (2) g is semisimple. (3) Any nontrivial finite-dimensional irreducible g-module is a direct sum of two copies of the same irreducible g0¯ -module. Exercise 1.23. It is known that every finite-dimensional Lie algebra has a finitedimensional faithful representation. Prove that every finite-dimensional Lie superalgebra has a finite-dimensional faithful representation. Exercise 1.24. Prove that the exact sequence of Lie superalgebras 0 −→ CIm|m → sl(m|m) −→ sl(m|m)/CIm|m −→ 0 is non-split, though sl(m|m)/CIm|m is simple, for m ≥ 2. (Hence Levi’s theorem fails for Lie superalgebras.) Exercise 1.25. Prove that the bilinear form (1.33) induced from the odd trace form (1.32) is a symmetric non-degenerate invariant form on q(n).

Notes Section 1.1. The classification of finite-dimensional simple complex Lie superalgebras was first announced by Kac in [59]. The detailed proof of the classification, along with many other fundamental results on Lie superalgebras, appeared in Kac [60] two years later. An independent proof of the classification of the finitedimensional complex simple Lie superalgebras whose even subalgebras are reductive was given in the two papers by Scheunert, Nahm, and Rittenberg in [106] around the same time. Section 1.2. The structure theory of root systems, root space decompositions, and invariant bilinear forms of basic Lie superalgebras presented here is fairly standard. More details can be found in the standard references (see Kac [60, 62] and Scheunert [105]). Section 1.3. The concept of odd reflections was introduced by Leites, Saveliev, and Serganova [78] to relate non-conjugate Borel subalgebras, fundamental and positive systems. The list of conjugacy classes of fundamental systems under the Weyl group action for the basic Lie algebras was given by Kac [60]. We introduce

Notes

41

a notion of εδ-sequences (which appeared in Cheng-Wang [32]) to facilitate the parametrization of the conjugacy classes of fundamental systems for Lie superalgebras of type gl and osp. For definitions of Borel subalgebras for general Lie superalgebras, see [90, 95]. Section 1.4. Lemma 1.30 on odd reflections appeared for more general KacMoody Lie superalgebras in Kac-Wakimoto [66, Lemma 1.2], and it is sometimes attributed to Serganova’s 1988 thesis (cf., e.g., [108]). Section 1.5. A detailed proof of the PBW Theorem for Lie superalgebras, Theorem 1.36, can be found in Milnor-Moore [85, Theorem 6.20] and Ross [102, Theorem 2.1]. As in the classical theory, the first step to develop a highest weight theory for the basic and queer Lie superalgebras is to study representations of Borel subalgebras. Lemma 1.37 on solvable Lie superalgebras appeared in Kac [60, Proposition 5.2.4]. The proof given here is different. Lemma 1.40 on the change of the extremal weights of an irreducible module under an odd reflection (which appeared in Penkov-Serganova [94, Lemma 1]) plays a fundamental role in representation theory of Lie superalgebras developed in the book. Exercises 1.8, 1.9, and 1.17–1.20 are taken from [60] and [105], while Exercise 1.21 was implicit in [60]. Exercise 1.16 was inspired by similar examples for Lie algebras in prime characteristic, which we learned from A. Premet.

Chapter 2

Finite-dimensional modules

In this chapter, we mainly work on Lie superalgebras of type gl, osp, and q. The finite-dimensional irreducible modules for these Lie superalgebras are classified in terms of highest weights. In contrast to semisimple Lie algebras, there is more work to do to achieve such classifications for Lie superalgebras. We describe the images of the Harish-Chandra homomorphisms in terms of (super)symmetric functions and formulate the linkage principle on composition factors of a Verma module. As an application, we present a Weyl-type character formula for the socalled typical finite-dimensional irreducible modules. The chapter concludes with a study of extremal weights, i.e., highest weights with respect to (not necessarily conjugate) Borel subalgebras, of various finite-dimensional irreducible modules.

2.1. Classification of finite-dimensional simple modules In this section, the highest weights of finite-dimensional irreducible modules over Lie superalgebras of type gl, osp, and q are determined.

2.1.1. Finite-dimensional simple modules of gl(m|n). Let g be the Lie superalgebra gl(m|n) and let h be the Cartan subalgebra of diagonal matrices spanned by the basis elements {Eii |i ∈ I(m|n)}. Let n+ (respectively, n− ) be the subalgebra of strictly upper (respectively, strictly lower) triangular matrices of gl(m|n) so that Φ+ corresponds to (1.37). Then we have the standard triangular decomposition g = n − ⊕ h ⊕ n+ . 43

2. Finite-dimensional modules

44

The even subalgebra admits a compatible triangular decomposition g0¯ = n− ⊕ h ⊕ n+ , 0¯ 0¯ where n± = g0¯ ∩ n± . Let b = h ⊕ n+ and b0¯ = h ⊕ n+ . 0¯ 0¯

Moreover, the Lie superalgebra g admits a Z-gradation

(2.1)

g = g−1 ⊕ g0 ⊕ g1 ,

where g−1 (respectively, g1 ) is spanned by all Ei j with i, j ∈ I(m|n) such that i > 0 > j (respectively, i < 0 < j). Note that g−1 and g1 are abelian Lie superalgebras, and that the Z-degree zero subspace g0 coincides with the Z2 -degree zero subspace g0¯ . For λ ∈ h∗ , let L0 (λ) be the simple g0¯ -module of highest weight λ ∈ h∗ (relative to the Borel b0¯ ). Then L0 (λ) may be extended trivially to a g0¯ ⊕ g1 -module due to (2.1). Define the Kac module over g by K(λ) = Indgg0 ⊕g1 L0 (λ), which, as a vector space, can be identified by the PBW Theorem 1.36 with (2.2)

K(λ) = ∧(g−1 ) ⊗ L0 (λ).

Proposition 2.1. Let g = gl(m|n). There exists a surjective g-module homomorphism (unique up to a scalar multiple) K(λ)  L(λ). Moreover, the following are equivalent: (1) L(λ) is finite dimensional. (2) L0 (λ) is finite dimensional. (3) K(λ) is finite dimensional. Proof. The existence of a surjective homomorphism K(λ)  L(λ) follows by the embedding of g0¯ -modules L0 (λ) → L(λ) and Frobenius reciprocity. (1) ⇒ (2). Note that L0 (λ) is an irreducible direct summand of L(λ) regarded as a g0¯ -module. (2) ⇒ (3). Follows from (2.2). (3) ⇒ (1). Follows from the surjectivity of the map K(λ)  L(λ).



According to Proposition 1.39, every finite-dimensional simple g-module is a highest weight module L(λ), for some λ ∈ h∗ . Moreover, L(λ) ∼  L(µ), if λ = µ. = By Proposition 2.1, the classification of finite-dimensional simple g-modules is the same as the classification of finite-dimensional simple modules of g0¯ = gl(m) ⊕ gl(n), which is well known. Hence, we have established the following. Proposition 2.2. A complete list of pairwise non-isomorphic finite-dimensional n ∗ simple gl(m|n)-modules are L(λ), for λ = ∑m i=1 λi δi + ∑ j=1 υ j ε j ∈ h satisfying λi − λi+1 ∈ Z+ and υ j − υ j+1 ∈ Z+ for all possible i, j.

2.1. Classification of finite-dimensional simple modules

45

Remark 2.3. Note that g0¯ is a Levi subalgebra of g = gl(m|n) (corresponding to the removal of the odd simple root from the standard Dynkin diagram of g), and hence the Kac module K(λ) is a distinguished parabolic Verma module relative to the standard Borel subalgebra. Since K(λ) is a highest weight module, it is indecomposable. However, for m ≥ 1 and n ≥ 1, K(λ) is reducible for suitable λ, e.g., when λ(hα ) = 0 for the odd simple root α by Lemma 1.40. Hence, the category of finite-dimensional gl(m|n)-modules is not semisimple when m ≥ 1 and n ≥ 1. 2.1.2. Finite-dimensional simple modules of spo(2m|2). The classification of finite-dimensional simple modules for g = spo(2m|2) is carried out similarly to the gl(m|n) case. Let Φ+ be the positive system of the root system Φ of g corresponding to the Dynkin diagram (1.42). Associated to Φ+ we have the triangular decomposition g = n− ⊕ h ⊕ n+ , compatible with that of its even subalgebra g0¯ = n− ⊕ h ⊕ n+ , where n± = g0¯ ∩ n± . In addition, g = spo(2m|2) admits the 0¯ 0¯ 0¯ following two favorable properties that are shared by gl(m|n) but not by other spo superalgebras: first, g admits a Z-graded Lie superalgebra structure of the form g = g−1 ⊕ g0 ⊕ g1 , where g0 = g0¯ , g1¯ = g−1 ⊕ g1 , and the set of roots for g1 is given by Φ+ = {ε1 ± δi , 1 ≤ i ≤ m}. Secondly, g0¯ is a Levi subalgebra of g corresponding 1¯ to the removal of the odd simple root from the Dynkin diagram (1.42). Hence we define the Kac module K(λ) of g as before: K(λ) = Indgg0 ⊕g1 L0 (λ), where L0 (λ) denotes the simple g0¯ -module of highest weight λ relative to Φ+ . 0¯ Proposition 2.1 remains valid in the current setting, and it implies that the classification of finite-dimensional simple g-modules is the same as the classification of finite-dimensional simple modules of g0¯ ∼ = sp(2m) ⊕ C. Proposition 2.4. A complete list of pairwise non-isomorphic finite-dimensional ∗ simple spo(2m|2)-modules are L(λ), for λ = ∑m i=1 λi δi +υ1 ε1 ∈ h , such that υ1 ∈ C and (λ1 , . . . , λm ) is a partition. 2.1.3. A virtual character formula. Let g be a Lie superalgebra of type gl or osp. With respect to a positive system Φ+ = Φ+ ∪ Φ+ (see Sections 1.3.2, 1.3.3, and 0¯ 1¯ 1.3.4), we have the triangular decomposition g = n− ⊕ h ⊕ n+ . Letting b = n+ ⊕ h, we define the Verma module Δ(λ) = Indgb Cvλ associated to a weight λ ∈ h∗ . Here, as usual, Cvλ stands for the one-dimensional b-module on which h acts by the character λ, and n+ acts trivially. Then Δ(λ) admits a unique simple quotient g-module, which is the irreducible g-module L(λ) of highest weight λ. A weight λ ∈ h∗ is called dominant integral with respect to Φ+ or sim0¯ + + ply Φ0¯ -dominant integral if λ, hα  ∈ Z+ , for all α ∈ Φ0¯ . A difference of two

2. Finite-dimensional modules

46

characters of finite-dimensional g-modules is called a virtual character of finitedimensional g-modules. Also recall the Weyl vector ρ = ρ0¯ − ρ1¯ from (1.35). We shall need the following proposition (see Santos [104, Proposition 5.7]) to determine if certain weights could be highest weights of finite-dimensional g-modules. Proposition 2.5. Let g be of type gl or osp. For any Φ+ -dominant integral weight 0¯ λ ∈ h∗ , the expression ∏β∈Φ+¯ (eβ/2 + e−β/2 ) 1

(2.3)

∑ (−1)(w)ew(λ+ρ)

∏α∈Φ+¯ (eα/2 − e−α/2 ) w∈W 0

is a virtual character of finite-dimensional g-modules. Proof. Following the proof of [104, Proposition 5.7], we shall make use of the so-called Bernstein functor L0 [104, (12)] from the category of h-semisimple gmodules to the category of g0¯ -semisimple g-modules. Actually, when restricting to the category of h-semisimple g0¯ -modules, the functor L0 is just the classical Bernstein functor, which is a special case of the functor P defined in Knapp and Vogan [72, (2.8)]. As we shall only need some standard properties of the Bernstein functor, which can be found in [72, 104], we will not recall here the precise definition, which is rather involved. An immediate consequence of the definition of the functor L0 is that every g0¯ -composition factor of L0 (V ) is also a g0¯ -composition factor of V . Hence L0 takes a finitely generated h-semisimple g0¯ -module in the BGG category to a finitedimensional g0¯ -module. Denote by Li the ith derived functor of L0 . Then, each Li (V ) is a finite-dimensional g-module for any Verma g-module V (which lies in the BGG category of g0¯ -modules). Now it is a classical result (cf. [72, Corollary 4.160]) that the resulting Euler characteristic when applying L0 to a g0¯ -Verma module is given by the Weyl character formula, i.e., ∞

∑ (−1)i ch Li (Indb Cλ ) = g0¯

0¯

i=0

∑w∈W (−1)(w) ew(λ+ρ0¯ ) , ∏α∈Φ+¯ (eα/2 − e−α/2 ) 0

g

and Li (Indb0¯¯ Cλ ) = 0 for i  0. This can be viewed as an algebraic version of the 0 Borel-Weil-Bott theorem. Given a short exact sequence of finite-dimensional h-semisimple g0¯ -modules 0 −→ M −→ E −→ N −→ 0, it follows by the Euler-Poincar´e principle that ∞

∞

∞

i=0

i=0

i=0

∑ (−1)i ch Li (M) + ∑ (−1)i ch Li (N) = ∑ (−1)ich Li (E).

2.1. Classification of finite-dimensional simple modules

47

As a g0¯ -module, Δ(λ) and Indb0¯¯ Cλ ⊗ Λ(n− ) have finite g0¯ -Verma flags, where 1¯ 0 a Verma flag is a filtration whose sections are Verma modules. Since these two g0¯ modules also have identical characters, their g0¯ -Verma flags have the same Verma module multiplicities. Now the character of the b0¯ -module Cλ ⊗ Λ(n− ) equals 1¯ λ −β β/2 −β/2 e ∏β∈Φ+¯ (1 + e ). Noting the W -invariance of ∏β∈Φ+¯ (e + e ), we have g

1

1

∞

∞

i=0

i=0

∑ (−1)i ch Li (Δ(λ)) = ∑ (−1)ich Li =

=

 g0¯  Indb ¯ Cλ ⊗ Λ(n− ) 1¯ 0

  ∑w∈W (−1)(w) w eλ+ρ0¯ ∏β∈Φ+¯ (1 + e−β ) 1

∏α∈Φ+¯ (eα/2 − e−α/2 ) 0   ∑w∈W (−1)(w) w eλ+ρ0¯ −ρ1¯ ∏β∈Φ+¯ (eβ/2 + e−β/2 ) 1

∏α∈Φ+¯ (eα/2 − e−α/2 ) 0

=

∏β∈Φ+¯ (eβ/2 + e−β/2 ) 1

∑ (−1)(w)ew(λ+ρ).

∏α∈Φ+¯ (eα/2 − e−α/2 ) w∈W 0

Since the left-hand side is a virtual character of finite-dimensional g-modules, so is the right-hand side. This proves the proposition.  Remark 2.6. Upon changing the sign + in the numerator of (2.3) to −, the resulting expression ∏β∈Φ+¯ (eβ/2 − e−β/2 ) 1

∏

α∈Φ+ 0¯

∑ (−1)(w)ew(λ+ρ)

(eα/2 − e−α/2 ) w∈W

is a virtual supercharacter (see Section 2.2.4) of finite-dimensional g-modules. 2.1.4. Finite-dimensional simple modules of spo(2m|2n + 1). We start with a general remark for a Lie superalgebra g of type osp. By Proposition 1.39, every finite-dimensional irreducible g-module is necessarily of the form L(λ), for some λ ∈ h∗ . Moreover, we have L(λ) ∼ = L(µ) if and only if λ = µ. However, the classification of finite-dimensional simple g-modules is nontrivial, partly because the even subalgebra g0¯ of g is not a Levi subalgebra. Lemma 2.7. Let g be a basic Lie superalgebra. A necessary condition for the finite dimensionality of the g-module L(λ) is that λ is Φ+ -dominant integral. 0¯ Proof. The proof follows by noting that λ is a highest weight with respect to Φ+ 0¯ for L(λ), regarded as a g0¯ -module.  In this subsection we consider g = spo(2m|2n + 1) and let Φ+ be the standard positive system (1.39) for g. The next lemma is a reformulation of the well-known dominance integral (DI) conditions for Lie algebras sp(2m) and so(2n + 1).

2. Finite-dimensional modules

48

Lemma 2.8. Let g = spo(2m|2n + 1) with g0¯ = sp(2m) ⊕ so(2n + 1). A weight + n λ = ∑m i=1 λi δi + ∑ j=1 υ j ε j is Φ0¯ -dominant integral if and only if the following two conditions are satisfied: (DI.b-1) λ1 ≥ . . . ≥ λm with all λi ∈ Z+ . (DI.b-2) υ1 ≥ . . . ≥ υn , with either (i) all υ j ∈ Z+ or (ii) all υ j ∈ 12 + Z+ . For g = spo(2m|2n + 1), a Φ+ -dominant integral weight is called an integer 0¯ weight if it satisfies (DI.b-1), and it is called a half-integer weight if it satisfies (DI.b-2). The L(λ)’s for integer and half-integer weights λ turn out to be quite different, and they are analyzed separately. n ∗ Proposition 2.9. Let g = spo(2m|2n + 1) and let λ = ∑m i=1 λi δi + ∑ j=1 υ j ε j ∈ h + be a half-integer Φ0¯ -dominant integral weight. Then the highest weight g-module L(λ) is finite dimensional if and only if λm ≥ n.

Proof. Take a half-integer Φ+ -dominant integral weight λ such that λm ≥ n. We 0¯ will look for the highest weight appearing in the virtual character (2.3) of finitedimensional g-modules in Proposition 2.5. Recalling the Weyl vector ρ = ρ0¯ − ρ1¯ from (1.35), we compute that ρ1¯ =

2n + 1 m ∑ δi , 2 i=1

ρ0¯ = mδ1 + (m − 1)δ2 + . . . + δm + (n − 1/2)ε1 + (n − 3/2)ε2 + . . . + (1/2)εn . Since the Weyl group of sp(2m) is ∼ = Zm 2  Sm and it consists of the signed permutations among δ1 , . . . , δm , we conclude by the assumption λm ≥ n that the weight λ + ρ remains Φ+ -dominant and regular. Thus the highest weight occurring in the 0¯ virtual character of finite-dimensional g-modules (2.3) is (λ + ρ) + ρ1¯ − ρ0¯ = λ, and so L(λ) must be finite dimensional by Proposition 2.5. Conversely, suppose that the module L(λ) is finite dimensional for a halfinteger Φ+ -dominant integral weight λ. We apply the following sequence of mn 0¯ odd reflections to the standard fundamental and positive systems of g in (1.39). We first apply the n odd reflections with respect to δm − ε1 , δm − ε2 , . . . , δm − εn , then the n odd reflections with respect to δm−1 − ε1 , . . . , δm−1 − εn etc., until we finally apply the n odd reflections with respect to δ1 − ε1 , δ1 − ε2 , . . . , δ1 − εn . After a computation similar to Example 1.35, the resulting fundamental system is that of the diagram (1.40). By the half-integer weight assumption on λ, the value of λ at any isotropic odd coroot belongs to 12 + Z and hence is nonzero. By Lemma 1.40, after an odd reflection with respect to an isotropic odd root α, the new highest weight is λ − α, which again takes value in 12 + Z at any odd isotropic coroot. Continuing this way and using a computation very similar to Example 1.41, we see that the new highest weight corresponding to the above Dynkin diagram is equal to

2.1. Classification of finite-dimensional simple modules

49

n λ − n ∑m i=1 δi + m ∑ j=1 ε j . Now this weight must take non-negative integer value at h2δm by the finite dimensionality of L(λ), from which we conclude that λm ≥ n. 

We next determine for which integer Φ+ -dominant integral weight λ the g0¯ module L(λ) is finite dimensional. Recall µ denotes the conjugate partition of a partition µ. The following definition plays a fundamental role in the book. Definition 2.10. A partition µ = (µ1 , µ2 , . . .) is called an (m|n)-hook partition (or simply a hook partition when m, n are implicitly understood) if µm+1 ≤ n. Equivalently, µ is an (m|n)-hook partition if µn+1 ≤ m. Diagrammatically, the hook condition means that the (m + 1, n + 1)-box in the Young diagram is missing; it can be visualized as follows: n

µ m

Given an (m|n)-hook partition µ, we denote by µ+ = (µm+1 , µm+2 , . . .) and write its conjugate, which is necessarily of length ≤ n, as ν = (µ+ ) = (ν1 , . . . , νn ). We define the weights (2.4)

µ = µ1 δ1 + . . . + µm δm + ν1 ε1 + . . . + νn−1 εn−1 + νn εn , µ− = µ1 δ1 + . . . + µm δm + ν1 ε1 + . . . + νn−1 εn−1 − νn εn .

(µ− is only used for spo(2m|2n) below.) It is sometimes convenient to identify µ and µ− with the following tuples of integers: µ = (µ1 , . . . , µm ; ν1 , . . . , νn ), µ− = (µ1 , . . . , µm ; ν1 , . . . , −νn ). n ∗ Theorem 2.11. Let g = spo(2m|2n + 1), and let λ = ∑m i=1 λi δi + ∑ j=1 υ j ε j ∈ h be a Φ+ -dominant integral weight. 0¯

(1) Suppose that λ is a half-integer weight. Then L(λ) is finite dimensional if and only if λm ≥ n. (2) Suppose that λ is an integer weight. Then L(λ) is finite dimensional if and only if λ is of the form µ , where µ is an (m|n)-hook partition. Proof. Part (1) is Proposition 2.9. So it remains to prove (2).

50

2. Finite-dimensional modules

Let µ be an (m|n)-hook partition. For an integer M such that M ≥ max{m, (µ)}, consider the Lie superalgebra spo(2M|2n + 1) acting on its standard representation C2M|2n+1 . It has highest weight δ1 . The module ∧k (C2M|2n+1 ) contains a highest weight vector of highest weight ∑kj=1 δ j , for k ≤ M. Write the conjugate partition µ = (ν1 , ν2 , . . .). Then the spo(2M|2n + 1)-module ∧ν1 (C2M|2n+1 ) ⊗ ∧ν1 (C2M|2n+1 ) ⊗ · · · contains the irreducible spo(2M|2n + 1)-module V of highest weight µ as a quotient, and hence V is finite dimensional. Now, we use a sequence of odd reflections to change the standard Borel subalgebra of spo(2M|2n + 1) into a Borel subalgebra that contains as a subalgebra the standard Borel subalgebra of spo(2m|2n + 1). (Here spo(2m|2n + 1) is regarded as a subalgebra of spo(2M|2n + 1).) A sequence of odd reflections that we may use for this purpose is the following: We first apply the n odd reflections with respect to δM − ε1 , δM − ε2 , . . . , δM − εn , then the n odd reflections with respect to δM−1 − ε1 , . . . , δM−1 − εn etc., until we finally apply the n odd reflections with respect to δm+1 −ε1 , δm+1 −ε2 , . . . , δm+1 −εn . Thus, we have applied a total of (M −m)n odd reflections. The subdiagram consisting of the first (m + n − 1) vertices of the new Dynkin diagram is of type gl(m|n). Similar to Example 1.41, one shows that the highest weight of V with respect to the new Borel subalgebra is µ . Hence by restriction we conclude that the spo(2m|2n + 1)-module L(µ ) is finite dimensional, proving the “if” direction of (2). By Lemma 2.8, an integer Φ+ -dominant integral highest weight for a simple 0¯ finite-dimensional g-module is necessarily of the form µ = µ1 δ1 + . . . + µm δm + ν1 ε1 + . . . + νn εn , where (µ1 , . . . , µm ) and (ν1 , . . . , νn ) are partitions. To prove the remaining condition ν1 ≤ µm , it suffices to prove ν1 ≤ µ1 in the case of m = 1 by noting that spo(2|2n + 1) is a subalgebra of spo(2m|2n + 1). Let V be a finitedimensional simple spo(2|2n + 1)-module of highest weight µ. Via the sequence of odd reflections with respect to δ1 − ε1 , δ1 − ε2 , . . . , δ1 − εn , we change the standard Borel subalgebra to a Borel subalgebra with an odd non-isotropic simple root δ1 . Now if µ1 ≥ n, then there is no condition on µ and we are done. Suppose now that µ1 = k < n; we shall prove by contradiction by assuming that νk+1 ≥ 1 (and so ν1 > µ1 ). This implies that νi ≥ 1, for all 1 ≤ i ≤ k, and hence, when applying the odd reflection with respect to δ1 − εi , the new highest weight is obtained from the old one by subtracting δ1 − εi , for 1 ≤ i ≤ k. This also remains true when applying the odd reflection with respect to δ1 −εk+1 . Hence the resulting new highest weight has negative δ1 -coefficient. But this implies that the new highest weight evaluated at the coroot h2δ1 must be negative, contradicting the finite dimensionality of the module V . Hence µ must satisfy the hook condition, as claimed.  2.1.5. Finite-dimensional simple modules of spo(2m|2n). Let n ≥ 2. Let g = spo(2m|2n) with g0¯ = sp(2m) ⊕ so(2n). In this subsection Φ+ is the standard positive system (1.41) for g. The next lemma is the well-known dominance integral conditions for classical Lie algebras sp(2m) and so(2n).

2.1. Classification of finite-dimensional simple modules

51

Lemma 2.12. Let g = spo(2m|2n). Assume n ≥ 2. A weight λ = ∑m i=1 λi δi + + n ∗ ∑ j=1 υ j ε j ∈ h is Φ0¯ -dominant integral if and only if the following two conditions are satisfied: (DI.d-1) λ1 ≥ . . . ≥ λm with all λi ∈ Z+ . (DI.d-2) υ1 ≥ . . . ≥ υn−1 ≥ |υn |, with either (i) all υ j ∈ Z or (ii) all υ j ∈ 12 + Z. For g = spo(2m|2n), a Φ+ -dominant integral weight λ ∈ h∗ is called an integer 0¯ weight if it satisfies (DI.d-1), and it is called a half-integer weight if it satisfies (DI.d-2). The L(λ)’s for integer and half-integer weights λ again are quite different. n Proposition 2.13. Let g = spo(2m|2n) with n ≥ 2. Let λ = ∑m i=1 λi δi + ∑ j=1 υ j ε j be a half-integer Φ+ -dominant integral weight. Then the highest weight g-module 0¯ L(λ) is finite dimensional if and only if λm ≥ n.

Proof. Take a half-integer Φ+ -dominant integral weight λ with λm ≥ n. We com0¯ m pute that ρ1 = n ∑i=1 δi , and so µ := λ − ρ1 is a Φ+ -dominant integral weight by 0¯ Lemma 2.12. By virtue of the classical Weyl character formula, the virtual character of finite-dimensional g-modules (2.3) in Proposition 2.5 can be rewritten as chL0 (µ) ∏β∈Φ+¯ (eβ/2 + e−β/2 ). The highest weight in this expression is clearly 1 µ + ρ1 = λ, and we conclude by Proposition 2.5 that L(λ) is finite dimensional. Now let λ be a half-integer Φ+ -dominant integral weight such that L(λ) is finite 0¯ dimensional. We apply exactly the same sequence of mn odd reflections as in the proof of Proposition 2.9 now to the standard fundamental system for spo(2m|2n) in (1.41) in Section 1.3.4. The resulting fundamental system is that of the diagram in (1.42). Bearing in mind that λ is half-integer, as in the proof of Proposition 2.9, we can show that the highest weight with respect to this new Borel subalgebra is n λ − n ∑m i=1 δi + m ∑ j=1 ε j . Since this weight must take non-negative integer value at h2δm to ensure that L(λ) is finite dimensional, we conclude that λm ≥ n.  n Theorem 2.14. Let g = spo(2m|2n) with n ≥ 2. Let λ = ∑m i=1 λi δi + ∑ j=1 υ j ε j be + a Φ0¯ -dominant integral weight.

(1) Suppose that λ is a half-integer weight. Then L(λ) is finite dimensional if and only if λm ≥ n. (2) Suppose that λ is an integer weight. Then L(λ) is finite dimensional if and only if λ is of the form µ or µ− , where µ is an (m|n)-hook partition. Proof. Part (1) is Proposition 2.13. Since the proof for (2) is very similar to the proof for Theorem 2.11(2), we shall only give a sketch below. We recall from (1.41) that the standard Dynkin diagram of g = spo(2m|2n) possesses a nontrivial diagram involution, which induces an involution of the Lie superalgebra g. Twisting the module structure with this involution interchanges the irreducible modules L(µ ) and L(µ− ).

52

2. Finite-dimensional modules

Given an (m|n)-hook partition µ, we choose M ≥ max{m, (µ)} and consider the Lie superalgebra spo(2M|2n). Then, arguing as in the proof of Theorem 2.11, the irreducible spo(2M|2n)-module of highest weight µ is finite dimensional. We apply the same sequence of odd reflections as in that proof to change the standard Borel subalgebra of spo(2M|2n) to a Borel subalgebra that is compatible with the standard Borel subalgebra of spo(2m|2n). Again, the subdiagram consisting of the first (m + n − 1) vertices of the new Dynkin diagram is of type gl(m|n), and it is easy to see that the new highest weight equals µ . By restriction to spo(2m|2n), we see that L(µ ) is finite dimensional, and so L(µ− ) is also finite dimensional by applying the Dynkin diagram involution. Conversely, let µ = µ1 δ1 + . . . + µm δm + ν1 ε1 + . . . + νn εn be an integer Φ+ 0¯ dominant integral weight. By Lemma 2.12, (µ1 , . . . , µm ) and (ν1 , . . . , |νn |) are partitions. To prove that ν1 ≤ µm , it suffices to prove ν1 ≤ µ1 in the case that m = 1. Using the sequence of odd reflections with respect to δ1 − ε1 , δ1 − ε2 , . . . , δ1 − εn , we change the standard Borel subalgebra to a Borel subalgebra with an even simple root 2δ1 . Now the same argument as in the proof of Theorem 2.11(2) shows that if µ does not satisfy the hook condition, then µ must take a negative value at h2δ1 , contradicting the finite dimensionality of L(µ).  Example 2.15. Let ξ, ξ¯ be two odd indeterminates, and let x = {x1 , · · · , xm }, x¯ = {x¯1 , · · · , x¯m } be 2m even indeterminates with m > 0. We identify the span of these indeterminates with C2|2m so that the exterior algebra ∧ (C2|2m ) := ∧(C2 ) ⊗ S(C2m ) is identified with the polynomial superalgebra generated by these indeterminates, ¯ x, x]. which we denote by C[ξ, ξ, ¯ We denote by Π the fundamental system of osp(2|2m) given by {ε1 − δ1 , δ1 − δ2 , . . . , δn−1 − δn , 2δn } with Dynkin diagram as in (1.42) with n = 1. Let eα and f α , for α ∈ Π, denote the Chevalley generators. ¯ x, x]: Define the following first-order differential operators on C[ξ, ξ, ¯ ∂ ∂ ∂ ∂ + x¯1 ¯ , f˘ε1 −δ1 = x1 − ξ¯ , ∂x1 ∂ξ ∂x¯1 ∂ξ ∂ ∂ ∂ ∂ e˘δi −δi+1 = xi − x¯i+1 , f˘δi −δi+1 = xi+1 − x¯i , ∂xi+1 ∂x¯i ∂xi ∂x¯i+1 ∂ ∂ e˘2δm = xm , f˘2δm = x¯m . ∂x¯m ∂xm

e˘ε1 −δ1 = ξ

i = 1, · · · , m − 1,

The map eα → e˘α and f α → f˘α , for α ∈ Π, defines a representation of osp(2|2m) ¯ x, x]. on C[ξ, ξ, ¯ Indeed, this gives an explicit realization of the irreducible representations L(ε1 + kδ1 ) ∼ = ∧ k+1 (C2|2m ), for all k ∈ Z+ (see Exercise 2.12). Example 2.16. In order to construct the symmetric tensors of the natural osp(2|2m)module we consider x, x, ¯ two even, and ξ = {ξ1 , · · · , ξm }, ξ¯ = {ξ¯ 1 , · · · , ξ¯ m }, 2m odd

2.1. Classification of finite-dimensional simple modules

53

indeterminates. We identify the span of these indeterminates with C2|2m so that the supersymmetric algebra S(C2|2m ) := S(C2 ) ⊗ ∧(C2m ) ¯ We continue to use the notation from Example 2.15 is identified with C[x, x, ¯ ξ, ξ]. ¯ and define the following first-order differential operators on C[x, x, ¯ ξ, ξ]: eˆε1 −δ1 = x eˆδi −δi+1 = ξi

∂ ∂ + ξ¯ 1 , ∂ξ1 ∂x¯

∂ ∂ − ξ¯ i+1 ¯ , ∂ξi+1 ∂ξ i ∂ eˆ2δm = ξm ¯ , ∂ξ m

∂ ∂ fˆε1 −δ1 = ξ1 − x¯ ¯ , ∂x ∂ξ 1 ∂ ∂ fˆδi −δi+1 = ξi+1 − ξ¯ i ¯ , ∂ξi ∂ξi+1 ∂ fˆ2δm = ξ¯ m . ∂ξm

The map eα → eˆα and f α → fˆα , for α ∈ Π, defines a representation of osp(2|2m) ¯ Just like for the orthogonal Lie algebra, the symmetric tensors on C[x, x, ¯ ξ, ξ]. are not irreducible. The computation of the composition factors of the symmetric tensor here is more involved than in the case of orthogonal Lie algebras (see Exercises 2.13–2.16). 2.1.6. Finite-dimensional simple modules of q(n). Now suppose that g is the queer Lie superalgebra q(n). Let h be the standard Cartan subalgebra of block diagonal matrices and let b = h ⊕ n+ be the standard Borel subalgebra of block upper triangular matrices; cf. Section 1.2.6. Recall that {Hi = Ei,i + Eii | 1 ≤ i ≤ n} forms the standard basis for h0¯ , with dual basis {εi | 1 ≤ i ≤ n} for h∗0¯ . We have a triangular decomposition g = n− ⊕ h ⊕ n+ . Recall from Lemma 1.42 that Wλ is a finite-dimensional irreducible h-module, for λ ∈ h∗0¯ . Regarding Wλ as a b-module by letting n+ act trivially, we define the Verma module of g to be Δ(λ) = IndgbWλ . The highest weight space of Δ(λ) can be naturally identified with Wλ and has dimension 2((λ)+δ(λ))/2 . Any nonzero vector in the highest weight space Wλ generates the whole g-module Δ(λ). Following standard arguments for semisimple Lie algebras, any g-submodule of Δ(λ) has a weight space decomposition, and then the Verma module Δ(λ) contains a unique maximal proper submodule J(λ) (which is the sum of all proper submodules). Hence, Δ(λ) has a unique irreducible quotient g-module L(λ) = Δ(λ)/J(λ). Assume that the g-module L(λ) is finite dimensional, and write λ = ∑ni=1 λi εi . Since g0¯ ∼ = gl(n), the g0¯ -dominance integral condition imposes a necessary condition on λ: λi − λi+1 ∈ Z+ , for all 1 ≤ i ≤ n − 1. However, this condition on λ is not sufficient for L(λ) to be finite dimensional, as we see in the following lemma.

2. Finite-dimensional modules

54

Lemma 2.17. Let g = q(2) and λ = mε1 + mε2 ∈ h∗0¯ with m = 0. Then L(λ) is infinite dimensional. Proof. Recall the notations Ei j , E i j , etc. from Sections 1.1.4 and 1.2.6. Then E12 , E21 , and H1 − H2 = E11 − E22 form a standard sl(2)-triple s. Take a highest weight vector vλ in L(λ). For λ = mε1 + mε2 , we have E12 vλ = 0 = (H1 − H2 )vλ . To prove the lemma, it suffices to show that E21 vλ = 0 in L(λ). Indeed, by the sl(2) representation theory, E21 vλ = 0 implies that the s-submodule generated by vλ must be the infinite-dimensional Verma module of zero highest weight. To that end, recalling H i = Eii + Eii , we compute that E 12 E21 vλ = E21 E 12 vλ + (H 1 − H 2 )vλ = (H 1 − H 2 )vλ , and (H 1 − H 2 )2 vλ = (E11 + E22 )vλ = 2mvλ = 0. Hence, we have (H 1 − H 2 )vλ = 0 and E21 vλ = 0. This proves the lemma.  Theorem 2.18. The highest weight irreducible q(n)-module L(λ) is finite dimensional if and only if λ = ∑ni=1 λi εi satisfies the conditions (i)-(ii): (i) λi − λi+1 ∈ Z+ ; (ii) λi = λi+1 implies that λi = 0,

for 1 ≤ i ≤ n − 1.

Proof. For L(λ) to be finite dimensional, λ = ∑ni=1 λi εi must satisfy the usual dominance integral condition for q(n)0¯ ∼ = gl(n), which is Condition (i). Now λ also must satisfy Condition (ii) by applying Lemma 2.17 to various standard subalgebras of q(n) that are isomorphic to q(2). This proves the “only if” part. Assume that λ satisfies Conditions (i)-(ii), and assume in addition that λi ∈ Z. This implies that λi = 0, for all i, and λ j > λ j+1 , for j = 1, . . . , n − 1. We want + to prove that L(λ) is finite dimensional. Write gε = n− ε + hε + nε for the standard triangular decomposition of gε , for ε ∈ Z2 . The Verma g-module Δ(λ), regarded as a g0¯ -module, has a g0¯ -Verma flag, and we have chΔ(λ) = 2

(n+1)/2!

  g ch Indh0¯ +n+ Cλ ⊗ ∧(n− ) , 1¯ 0¯

0¯

where r! for a real number r denotes the largest integer no greater than r. As in the proof of Proposition 2.5, we apply the Bernstein functor L0 and its derived functor Li (with g0¯ = gl(n)) to Δ(λ) and compute the corresponding Euler characteristic

2.2. Harish-Chandra homomorphism and linkage

55

(recall Φ1¯ and Φ0¯ are identical copies of the root system for gl(n)):   g ∑ (−1)ichLi (Δ(λ)) =2 (n+1)/2! ∑ (−1)i chLi Indh0¯ +n+ Cλ ⊗ ∧(n−1¯ ) i≥0

i≥0

=2

(n+1)/2!



0¯

0¯

∑w∈W (−1)(w) w eλ+ρ0¯ ∏α∈Φ+¯ (1 + e−α )



1

∏α∈Φ+¯ (eα/2 − e−α/2 ) 0

=2

−α ) + (1 + e ∏ (n+1)/2! α∈Φ ¯ 1

∑ (−1)(w)ew(λ).

∏α∈Φ+¯ (1 − e−α ) w∈W 0

Since the highest weight appearing in this last expression is clearly λ and the Euler characteristic is a virtual character of finite-dimensional g-modules (see the proof of Proposition 2.5), L(λ) is a finite-dimensional module. Now suppose that λ satisfies Conditions (i)-(ii) of the theorem and in addition that λi ∈ Z for each i. We write λ = (λ1 , . . . , λk , 0, . . . , 0, λl , . . . , λn ) such that λ1 > · · · > λk > 0 > λl > · · · > λn . Denote λ+ = (λ1 , . . . , λk , 0, . . ., 0) and λ− = (0, . . . , 0, λl , . . . , λn ). By Theorem 3.49 on Sergeev duality in Chapter 3, L(µ) is finite dimensional for µ = ∑i µi εi corresponding to a strict partition (µ1 , . . . , µn ). Then, the dual module L(µ)∗ ∼ = L(µ∗ ) is finite dimensional, where we have denoted n ∗ µ = − ∑i=1 µn+1−i εi (see Exercise 2.18). It follows that both L(λ+ ) and L(λ− ) are finite dimensional. Hence, as a quotient of the finite-dimensional g-module L(λ+ ) ⊗ L(λ− ), L(λ) must be finite dimensional. 

2.2. Harish-Chandra homomorphism and linkage In this section, we study the center of the universal enveloping algebra U(g) when g is a basic Lie superalgebra, in particular of type gl and osp. The image of the Harish-Chandra homomorphism is described with the help of supersymmetric functions (see Appendix A). Then, we provide a characterization of when two central characters are equal and hence obtain a linkage principle for composition factors of a Verma module. The finite-dimensional typical irreducible representations of g are classified, and a Weyl-type character formula for these modules is obtained. 2.2.1. Supersymmetrization. Let g be a finite-dimensional Lie superalgebra. Denote by Z(g) = Z(g)0¯ ⊕ Z(g)1¯ the center of the enveloping superalgebra U(g), where Z(g)i = {z ∈ U(g)i | za = (−1)ik az, ∀a ∈ gk for k ∈ Z2 },

i ∈ Z2 .

Recall S(g) denotes the symmetric superalgebra of g. As a straightforward super generalization of a standard construction for Lie algebras (cf. Carter [18, Proposition 11.4]), we may define a supersymmetrization map γ : S(g)−→U(g)

2. Finite-dimensional modules

56

and show that it is a g-module isomorphism. When restricting to the g-invariants, we have the following. Proposition 2.19. The supersymmetrization map γ : S(g)→U(g) induces a linear isomorphism γ : S(g)g → Z(g). Given a finite-dimensional representation (π,V ) of a Lie superalgebra g, we denote its supercharacter by schV. That is, schV =

∑ sdim(Vµ)eµ,

µ∈h∗

where we recall from Section 1.1.1 that sdimW = dimW0¯ − dimW1¯ stands for the superdimension of a vector superspace W = W0¯ ⊕ W1¯ . If we regard e as the Euler number, a formal expansion of eµ gives us ∞ 1  [schV ](h) = ∑ str π(h)i , ∀h ∈ g. i=0 i! A straightforward super generalization  of the arguments for Lie algebras (see Carter [18, pp. 212-3]) shows that x → str π(x)i defines a g-invariant polynomial function on g. From this we conclude the following. Proposition 2.20. Let g be a basic Lie superalgebra with Cartan subalgebra h. Let (π,V ) be a finite-dimensional g-module. Then the polynomial functions str π(h)i on h, for i > 0, can be identified with the homogeneous components of the supercharacter of π, and they arise as restrictions of g-invariant polynomial functions on g. 2.2.2. Central characters. In this subsection, we let g be a basic Lie superalgebra, and we let h be its Cartan subalgebra. Recall that g is equipped with an even non-degenerate invariant supersymmetric bilinear form (see Section 1.2.2). Furthermore, the bilinear form (·, ·) on g allows us to identify g with its dual g∗ and h with h∗ so that we have corresponding isomorphisms S(g)g ∼ = S(g∗ )g and S(h) ∼ = S(h∗ ). Via such identifications, the ∗ ∗ restriction map S(g ) → S(h ) and the induced map η : S(g) −→ S(h)

(2.5)

are homomorphisms of algebras. Let g = n− ⊕h⊕n+ be a triangular decomposition. By the PBW Theorem 1.36, we have U(g) = U(h) ⊕ (n−U(g) +U(g)n+ ), and we denote by φ : U(g) → U(h) the projection associated to this direct sum decomposition. Since [h, n± ] ⊆ n± , by considering ad h(z) = 0 for any element z ∈ Z(g), we conclude that z affords a unique expression of the form (2.6)

+ z = hz + ∑ n − i hi ni , i

± ± for hz , hi ∈ U(h), n± i ∈ n U(n ).

2.2. Harish-Chandra homomorphism and linkage

57

Hence Z(g) consists of only even elements. The restriction of φ to Z(g), denoted again by φ, defines an algebra homomorphism from Z(g) to U(h) that sends z to hz . As usual, we freely identify S(h) with the algebra of polynomial functions on h∗ . Extending the standard pairing ·, · : h∗ × h → C, we can make sense of λ, p for λ ∈ h∗ and p ∈ S(h) by regarding it as the value at λ of the polynomial function p on h∗ . For λ ∈ h∗ , define a linear map (2.7)

χλ : Z(g) −→ C χλ (z) =λ, φ(z),

∀z ∈ Z(g).

Lemma 2.21. Let g be a basic Lie superalgebra. An element z ∈ Z(g) acts as the scalar χλ (z) on any highest weight g-module V (λ) of highest weight λ. Proof. Our proof here is phrased so that it makes sense for q(n) later on as well. Let z ∈ Z(g). The highest weight space V (λ)λ is an irreducible h-module, which is one-dimensional for basic g (but may fail to be one-dimensional in the q(n) case). The even element z commutes with h, and hence by Schur’s Lemma, z acts as a scalar ξ(z) on V (λ)λ . Now let vλ be a highest weight vector of V (λ), and write an arbitrary vector u ∈ V (λ) as u = xvλ for x ∈ U(h + n− ). Since z is central, we have zu = xzvλ = ξ(z)u. It follows by (2.6) that   + zvλ = hz + ∑ n− i hi ni vλ = λ, hz vλ = χλ (z)vλ . i

Hence ξ(z) = χλ (z). The lemma is proved.



Therefore, χλ : Z(g) → C in (2.7) defines a one-dimensional representation of Z(g), which will be called the central character associated to a highest weight λ. 2.2.3. Harish-Chandra homomorphism for basic Lie superalgebras. In this subsection, we assume that g is a basic Lie superalgebra. Let Φ+ be a positive system in the root system Φ for g. Recall the Weyl vector ρ from (1.35). Regarding S(h) as the algebra of polynomial functions on h∗ , we define an automorphism τ of the algebra S(h) by τ( f )(λ) := f (λ − ρ),

∀ f ∈ S(h) and λ ∈ h∗ .

We then define the Harish-Chandra homomorphism to be the composition hc = τφ : Z(g) −→ S(h). The non-degenerate invariant bilinear form (·, ·) on h induces a W -equivariant algebra isomorphism θ : S(h) → S(h∗ ) and hence an isomorphism θ : S(h)W → S(h∗ )W . By composition we obtain an algebra homomorphism hc∗ = θhc : Z(g) −→ S(h∗ ). Proposition 2.22. Let g be a basic Lie superalgebra. We have

2. Finite-dimensional modules

58

(1) χλ = χw(λ+ρ)−ρ , for all w ∈ W and λ ∈ h∗ . (2) hc(Z(g)) ⊆ S(h)W and hc∗ (Z(g)) ⊆ S(h∗ )W . Proof. The character of the Verma module Δ(µ) of highest weight µ ∈ h∗ is given by chΔ(µ) = eµ+ρ D−1 , where D :=

∏β∈Φ+¯ (eβ/2 − e−β/2 ) 0

∏α∈Φ+¯ (eα/2 + e−α/2 )

.

1

Since the character of a module is equal to the sum of the characters of its composition factors, we have (2.8)

chΔ(λ) = ∑ bµλ chL(µ), µ

where bµλ ∈ Z+ and bλλ = 1. Since Δ(λ) is a highest weight module, the µ’s in the summation can be assumed to satisfy λ − µ ∈ ∑α∈Φ+ Z+ α and also χλ = χµ by Lemma 2.21. We choose a total order ≥ on the set λ − ∑α Z+ α with the property that ν ≥ µ if ν − µ ∈ ∑α∈Φ+ Z+ α. Then the ordered sets {chΔ(µ)}µ,≥ and {chL(µ)}µ,≥ are two ordered bases for the same space. Now (2.8) says that the matrix expressing {chΔ(µ)}µ,≥ in terms of {chL(µ)}µ,≥ is upper triangular with 1 along the diagonal, and hence we have chL(λ) = ∑ aµλ chΔ(µ), µ

where aµλ ∈ Z with aλλ = 1, and (2.9)

aµλ = 0 unless λ − µ ∈

∑

α∈Φ+

Z+ α and χλ = χµ .

Thus (2.10)

DchL(λ) = ∑ aµλ eµ+ρ . µ

Assume for now that λ ∈ h∗ is chosen so that the irreducible g-module L(λ) is finite dimensional. Then L(λ) is a semisimple g0¯ -module, and so chL(λ) is W invariant. On the other hand, D is W -anti-invariant by Theorem 1.18(4). Thus the right-hand side of (2.10) is W -anti-invariant, and hence can be written as (2.11)

∑ aµλ ∑ (−1)(w)ew(µ+ρ) ,

µ∈X

w∈W

where X consists of Φ+ -dominant integral weights such that aµλ = 0. We compute 0¯ that aw(λ+ρ)−ρ,λ = ±aλλ = ±1, and hence by (2.9) we have χλ = χw(λ+ρ)−ρ , for all w ∈ W.

Now Part (1) of the proposition as a polynomial identity on λ ∈ h∗ follows from the following claim. Claim. S := {λ ∈ h∗ | L(λ) is finite dimensional} is Zariski dense in h∗ .

2.2. Harish-Chandra homomorphism and linkage

59

Let us verify the claim. Given a Φ+ -dominant integral weight µ ∈ h∗ , the 0¯ induced module Indgg0¯ L0 (µ) is finite dimensional and has a highest weight µ + 2ρ1¯ . It is clear that the set of such weights µ+2ρ1¯ associated to all Φ+ -dominant integral 0¯ µ ∈ h∗ (which is a subset of S) is Zariski dense in h∗ . Hence the claim follows. By (1), we have that λ + ρ, hc(z) = w(λ + ρ), hc(z), for all w ∈ W and z ∈ Z(g). Thus hc(z) ∈ S(h)W , and equivalently, hc∗ (Z(g)) ⊆ S(h∗ )W by definition of hc∗ . This proves (2).  By Proposition 2.22, we have (2.12)

hc∗ = θhc : Z(g) −→ S(h∗ )W ,

hc : Z(g) −→ S(h)W ,

and we further define a homomorphism (see (2.5) for η) (2.13)

η∗ = θη : S(g)g −→ S(h∗ ).

In the remainder of this section, we shall strengthen Proposition 2.22 for g of type gl and osp by describing hc∗ (Z(g)) precisely. 2.2.4. Invariant polynomials for gl and osp. Suppose that g is of type gl or osp. The goal of this subsection is to establish Proposition 2.23 which partially describes the image Im(η∗ ) via supersymmetric polynomials; see Appendix A.2.1. (The word “partially” can and will be removed subsequently.) The description of this image plays a key role in the study of the Harish-Chandra homomorphism later on. ¯+ Associated to a positive system Φ+ , we have the subset Φ of positive isotropic 1¯ odd roots defined in (1.34). Introduce the following element P :=

(2.14)

∏

α ∈ S(h∗ ).

¯+ α∈Φ 1¯

Below we shall treat g = gl(m|n), spo(2m|2n + 1), and spo(2m|2n) case-by-case. Invariant polynomials for g = gl(m|n). Recall that the Weyl group W for g = gl(m|n) is W = Sm × Sn , which is by definition the same as the Weyl group of its even subalgebra gl(m) ⊕ gl(n),. Applying Proposition 2.20 to the natural representation (π,V ) of g = gl(m|n) and recalling η∗ from (2.13), we see that (2.15)

m

n

i=1

j=1

σkm,n := ∑ δki − ∑ εkj ∈ Im(η∗ ),

∀k ∈ Z+ .

Here and further, we shall identify S(h∗ ) = C[δ, ε], where we set δ = {δ1 , . . . , δm },

ε = {ε1 , . . . , εn }.

Hence an element of S(h∗ )W is precisely a polynomial that is symmetric in δ and symmetric in ε. Recalling from A.2.1 the definition of supersymmetric polynomials, we introduce the following subalgebra of S(h∗ )W : (2.16)

S(h∗ )W sup = {supersymmetric polynomials in variables δ, ε}.

2. Finite-dimensional modules

60

By (A.33), the polynomials {σkm,n | k ∈ Z+ } generate the algebra S(h∗ )W sup . Thus ∗ W we have S(h )sup ⊆ Im(η∗ ). The element P defined in (2.14), up to a possible sign that depends on the choice of positive systems for gl(m|n), can be computed to be

∏

P=

(δi − ε j ).

1≤i≤m,1≤ j≤n ∗ W It follows by (A.30) that f P ∈ S(h∗ )W sup for any f ∈ S(h ) .

Invariant polynomials for g = spo(2m|2n + 1). Recall the Weyl group W for g = spo(2m|2n + 1), which is by definition the same as for sp(2m) ⊕ so(2n + 1), is n WB := (Zm 2  Sm ) × (Z2  Sn ),

(2.17)

and hence an element f ∈ S(h∗ )W = C[δ, ε]W is exactly a polynomial that is symmetric among δ2i ’s and symmetric among ε2j ’s. We introduce the following subalgebra of S(h∗ )W : 2 2 (2.18) S(h∗ )W sup = {supersymmetric polynomials in δi , ε j , 1 ≤ i ≤ m, 1 ≤ j ≤ n}.

Applying Proposition 2.20 to the natural representation of spo(2m|2n + 1), we have that m

n

i=1

j=1

2k 2k σ2k m,n = ∑ δi − ∑ ε j ∈ Im(η∗ ),

∀k ∈ Z+ .

2k ∗ W Hence we have S(h∗ )W sup ⊆ Im(η∗ ), since σm,n for k ≥ 0 generate S(h )sup by (A.33). ¯+ In this case, we calculate that Φ ¯ = {δi ± ε j | 1 ≤ i ≤ m, 1 ≤ j ≤ n} for the 1

standard positive system Φ+ , and hence, P in (2.14) is given by P=

∏

1≤i≤m,1≤ j≤n

(δi − ε j )(δi + ε j ) =

∏

(δ2i − ε2j ).

1≤i≤m,1≤ j≤n

(For other positive systems, P might differ from the above formula by an irrelevant ∗ W sign.) By (A.30), we have that f P ∈ S(h∗ )W sup for any f ∈ S(h ) . Invariant polynomials for g = spo(2m|2n). We regard g = spo(2m|2n) as a subalgebra of spo(2m|2n + 1) with the same Cartan subalgebra h. We continue to denote by WB the Weyl group of spo(2m|2n + 1) and then identify the Weyl group W of g = spo(2m|2n) with a subgroup of WB of index 2. By the same consideration as for g = spo(2m|2n + 1) above using Proposition 2.20, we conclude B that the image Im(η∗ ) contains S(h∗ )W sup , which consists of all the supersymmetric polynomials in δ2i , ε2j , 1 ≤ i ≤ m, 1 ≤ j ≤ n.

2.2. Harish-Chandra homomorphism and linkage

61

Noting now that Φ+ = {δi ± ε j | 1 ≤ i ≤ m, 1 ≤ j ≤ n} for the standard positive 1¯ + system Φ , we set D1¯ : =

∏ (eα/2 − e−α/2 )

α∈Φ+ 1¯

= ∏(eδi /2−ε j /2 − e−δi /2+ε j /2 )(eδi /2+ε j /2 − e−δi /2−ε j /2 ). i, j

For a homogeneous polynomial g ∈ Z[δ, ε]WB , the expression n

Q := ∏ (eε j − e−ε j ) · g(eδ1 − e−δ1 , . . . , eδm − e−δm ; eε1 − e−ε1 , . . . , eεn − e−εn ) j=1

has ε1 ε2 · · · εn g(δ1 , . . . , δm ; ε1 , . . . , εn ) as its homogeneous term of lowest degree. Also observe that D1¯ has P as its lowest degree term, where the element P in (2.14) associated to the standard positive system Φ+ of spo(2m|2n) is given by P=

∏

(δ2i − ε2j ).

1≤i≤m,1≤ j≤n

(For other positive systems, P might differ by a sign from the above formula.) Since Q is an integral polynomial in e±δi and e±ε j that is W -invariant, Q is actually a virtual g0¯ -character. By Remark 2.6, D1¯ Q is a virtual supercharacter of finitedimensional g-modules. Hence every homogeneous term of D1¯ Q lies in Im(η∗ ) by Proposition 2.20, and in particular so does the lowest degree term of D1¯ Q, i.e., ε1 ε2 · · · εn Pg ∈ Im(η∗ ). Introduce the following subspace S(h∗ )W sup (which can be easily shown to be a subalgebra) of S(h∗ )W = C[δ, ε]W : (2.19)

∗ WB S(h∗ )W sup = S(h )sup



C[δ, ε]WB ε1 ε2 · · · εn P.

Summarizing the above discussions, we have shown that S(h∗ )W sup ⊆ Im(η∗ ). It is a standard fact about the classical Weyl group W that any element f ∈ S(h∗ )W = C[δ, ε]W can be written as f = h + ε1 ε2 · · · εn g, ∗ W where h, g ∈ C[δ, ε]WB . Thus, we have hP ∈ S(h∗ )W sup and ε1 ε2 · · · εn gP ∈ S(h )sup , and hence, f P ∈ S(h∗ )W sup .

Summarizing all three cases, we have established the following. Proposition 2.23. Let g be either gl(m|n), spo(2m|2n + 1), or spo(2m|2n), and let W be its Weyl group. Then, we have (1) P ∈ S(h∗ )W . ∗ W (2) f P ∈ S(h∗ )W sup , for all f ∈ S(h ) .

(3) S(h∗ )W sup ⊆ Im(η∗ ).

2. Finite-dimensional modules

62

2.2.5. Image of Harish-Chandra homomorphism for gl and osp. Let g be a basic Lie superalgebra for now. We have a non-degenerate invariant bilinear form (·, ·) on g, h, and h∗ . We recall here that (α, α) = 0, for any isotropic odd root α. The non-degenerate form (·, ·) induces an isomorphism h ∼ = h∗ . Also recall the central characters χλ from (2.7). Lemma 2.24. Let g be a basic Lie superalgebra. Let α be an isotropic odd root such that (λ + ρ, α) = 0. Then we have χλ = χλ+tα , for all t ∈ C. Proof. Fix a positive system Φ+ . We may assume that the isotropic odd root α ∈ Φ+ (otherwise, use −α to replace α). First suppose that α is an isotropic simple root in Φ+ . Then (ρ, α) = 12 (α, α) = 0, and hence (λ, α) = 0 by assumption. Let vλ be a highest weight vector in the Verma module Δ(λ). Then, eα f α vλ = [eα , f α ]vλ = hα vλ = (λ, α)vλ = 0. Also, we have eγ f α vλ = 0 for simple roots γ = α by weight considerations. Hence, fα vλ is a singular vector in Δ(λ). This gives rise to a nontrivial g-module homomorphism Δ(λ − α) → Δ(λ), whence χλ−α = χλ . Now suppose α is any isotropic root in Φ+ . By Lemma 1.29, there exists w ∈ W such that β = w(α) is simple. Then (α, α) = (β, β) = 0 by the W -invariance of (·, ·). By Proposition 2.22, we have χλ−α = χw(λ−α+ρ)−ρ = χµ , where we have denoted µ := w(λ + ρ) − ρ − β and used w(α) = β. Now by the W -invariance of (·, ·) again, we compute that (µ + ρ, β) = (w(λ + ρ), w(α)) − (β, β) = (λ + ρ, α) − 0 = 0. Thus, we can apply the special case established in the preceding paragraph to the weight µ and simple root β to obtain that χλ−α = χµ = χµ+β = χw(λ+ρ)−ρ = χλ , where we have used Proposition 2.22 again in the last equality. This implies that χλ = χλ−tα , for any t ∈ Z+ and any isotropic odd root α with (λ + ρ, α) = 0. Since Z+ is Zariski dense in C, we conclude that the polynomial identity χλ = χλ+tα holds for all t ∈ C.  Now we specialize to the type gl and osp. Proposition 2.25. Let g be either gl(m|n), spo(2m|2n + 1), or spo(2m|2n). Then ∗ W we have hc∗ (Z(g)) ⊆ S(h∗ )W sup , where S(h )sup is given in (2.16), (2.18), and (2.19). Proof. Let us proceed case-by-case.

2.2. Harish-Chandra homomorphism and linkage

63

First let us consider the case of g = gl(m|n). Assume that λ + ρ = ∑m i=1 ai δi + n ∑ j=1 b j ε j satisfies (λ + ρ, α) = 0 for some isotropic root α = δi − ε j , which means that ai = −b j . Let z ∈ Z(g). Then by Lemma 2.24 we have λ + ρ, hc(z) = χλ (z) = χλ−tα (z) = λ − tα + ρ, hc(z),

∀t ∈ C.

This implies that the specialization hc∗ (z)|δi =−ε j =t is independent of t, where we recall from (2.12) that hc∗ = θhc. In addition, we have hc∗ (z) ∈ S(h∗ )W by Proposition 2.22, whence hc∗ (z) ∈ S(h∗ )W sup by the definition (2.16).

The case for g = spo(2m|2n + 1) is analogous. Let λ ∈ h∗ be such that (λ + ρ, α) = 0 for some isotropic root α = δi ± ε j . We show by the same argument as above that the specialization hc∗ (z)|δi =∓ε j =t is independent of t. In addition, it follows by Proposition 2.22 that hc∗ (z) ∈ S(h∗ )W for W = WB (see (2.17)), which means that hc∗ (z) is symmetric in δ2i for all i and symmetric in ε2j for all j. Then we conclude that hc∗ (z) ∈ S(h∗ )W sup by the definition (2.18).

Now suppose that g = spo(2m|2n). Let W be the Weyl group of g, which is regarded as a subgroup of WB of index 2 as usual. Given z ∈ Z(g), we can write hc∗ (z) ∈ S(h∗ )W as hc∗ (z) = f + ε1 · · · εn g,

for f , g ∈ S(h∗ )WB = C[δ2 , ε2 ]Sm ×Sn .

Since χλ = χλ−tα , for any λ and any isotropic root α = δi − ε j with (λ + ρ, α) = 0, the specialization hc∗ (z)|δi=ε j =t = f 1 + g1 is independent of t, where we denote f1 := f |δi =ε j =t ,

g1 := tε1 · · · ε j−1 ε j+1 · · · εn g|δi =ε j =t .

As a polynomial of t, f 1 has even degree while g1 has odd degree. Thus both f1 and g1 must be independent of t. Hence, recalling (2.18), we must have that B f ∈ S(h∗ )W sup ,

g|δi =ε j =t = 0.

Since the polynomial g is symmetric in δ21 , . . . , δ2m and symmetric in ε21 , . . . , ε2n , we conclude that g is divisible by (δ2i − ε2j ), for all i and j, and hence divisible by P = ∏i, j (δ2i − ε2j ). Thus we have g = g0 P for some g0 ∈ S(h∗ )WB . The inclusion hc∗ (Z(g)) ⊆ S(h∗ )W sup now follows by the definition (2.19).



Theorem 2.26. Let g = gl(m|n), spo(2m|2n + 1), or spo(2m|2n), and let W be its Weyl group. Then we have hc∗ (Z(g)) = Im(η∗ ) = S(h∗ )W sup , where S(h∗ )W sup is defined in (2.16), (2.18), and (2.19), respectively. Proof. By Propositions 2.23 and 2.25, we have a map hc∗ : Z(g) → Im(η∗ ) and that hc∗ (Z(g)) ⊆ S(h∗ )W sup ⊆ Im(η∗ ). So it suffices to show that hc∗ (Z(g)) = Im(η∗ )

2. Finite-dimensional modules

64

in order to complete the proof. The commutative diagram (where gr denotes the associated graded) (2.20)

gr γ−1 ∼ H =

/ S(g)g HH HH HH η∗ HH gr hc∗ $ 

gr Z(g)

S(h∗ )

implies that gr hc∗ : gr Z(g) → Im(η∗ ) is surjective. Now the surjectivity of the map hc∗ : Z(g) → Im(η∗ ) follows by a standard filtered algebra argument. Indeed, consider hc∗ (Z(g)) and Im(η∗ ) as filtered spaces: C∼ = hc∗ (Z(g))0 ⊆ hc∗ (Z(g))1 ⊆ · · · ,

C∼ = Im(η∗ )0 ⊆ Im(η∗ )1 ⊆ · · · .

We shall prove by induction on k that hc∗ (Z(g))k = Im(η∗ )k , for all k. Clearly we have hc∗ (Z(g))0 = Im(η∗ )0 . Now assume that hc∗ (Z(g))k−1 = Im(η∗ )k−1 . Let us take a = η∗ (x) ∈ Im(η∗ )k for x ∈ S(g)g . Then a = gr (hc∗ γ)(x), and hc∗ γ(x) − a ∈ Im(η∗ )k−1 = hc∗ (Z(g))k−1 . Therefore, we conclude that a ∈ hc∗ (Z(g))k .



∗ W Define the subalgebra S(h)W sup of S(h) to be the preimage of S(h )sup under ∗ the isomorphism θ : S(h) → S(h ). Various maps in this section can now be put together in the following diagram, which only becomes commutative when we pass from Z(g) to its associated graded (note that the other algebras are already graded).

(2.21)

γ−1 ∼ =

/ S(g)g t t tt η hc∗ ttη t  ztt ∼ ∗  = S(h∗ )W o S(h)W

Z(g)

sup

θ

sup

Remark 2.27. The surjective homomorphism hc : Z(g) → S(h)W sup is actually injective as well, and hence all the homomorphisms in the diagram (2.21) are isomorphisms. Actually Sergeev [111, Corollary 1.1] showed that η : S(g)g → S(h) is injective, and the injectivity of hc follows from this by a filtered algebra argument. Except for its intrinsic value, the injectivity does not seem to play any crucial role in the representation theory of g. The following corollary is immediate from Proposition 2.23 and Theorem 2.26, and will be used later on. Corollary 2.28. Let g be one of the Lie superalgebras gl(m|n), spo(2m|2n + 1), or spo(2m|2n). Then for any f ∈ S(h)W , we have f P ∈ hc∗ (Z(g)).

2.2. Harish-Chandra homomorphism and linkage

65

2.2.6. Linkage for gl and osp. In this subsection, we let g denote the Lie superalgebra gl(m|n), spo(2m|2n), or spo(2m|2n + 1). Using the description of the images of Harish-Chandra homomorphisms, we obtain a necessary and sufficient condition for two central characters of g to be equal. Let h be its standard Cartan subalgebra of diagonal matrices. Let {δi , ε j |1 ≤ i ≤ m, 1 ≤ j ≤ n} be the standard basis of h∗ . We fix a positive system Φ+ of the ¯+ root system Φ for g. Recall the subset Φ ⊆ Φ+ defined in (1.34). 1¯ Definition 2.29. The degree of atypicality of an element λ ∈ h∗ , denoted by #λ, is ¯+ the maximum number of mutually orthogonal roots α ∈ Φ such that (λ + ρ, α) = 1¯ ∗ + 0. An element λ ∈ h is said to be typical (relative to Φ ) if #λ = 0 and is atypical otherwise. We define a relation ∼ on h∗ by declaring λ ∼ µ,

for λ, µ ∈ h∗ ,

if there exist mutually orthogonal isotropic odd roots α1 , α2 , . . . , α , complex numbers c1 , c2 , . . . , c , and an element w ∈ W satisfying that    µ + ρ = w λ + ρ − ∑ ca αa , (λ + ρ, α j ) = 0, j = 1, . . . , . (2.22) a=1

The weights λ and µ are said to be linked if λ ∼ µ. It follows from Theorem 2.30 below that linkage is an equivalence relation. Theorem 2.30. Let g be gl(m|n), spo(2m|2n), or spo(2m|2n + 1), and let h be its Cartan subalgebra. Let λ, µ ∈ h∗ . Then λ is linked to µ if and only if χλ = χµ . Proof. Assume first that λ ∼ µ, i.e., (2.22), holds. Since the αi ’s are orthogonal to one another, the second condition in (2.22) implies that we can repeatedly apply Lemma 2.24 to conclude that χλ = χλ−c1 α1 = χλ−c1 α1 −c2 α2 = . . . = χλ−∑

a=1 ca αa

.

Since µ + ρ = w(λ + ρ − ∑a=1 ca αa ) by (2.22) again, it follows by Proposition 2.22 that χλ−∑ ca αa = χµ , whence χλ = χµ . a=1

n Now assume that χλ = χµ . Let us write λ = ∑m i=1 λi δi + ∑ j=1 ν j ε j and µ = m n ∑i=1 µi δi + ∑ j=1 η j ε j . Recall from Theorem 2.26 that, for g = gl(m|n), the polynomials σkm,n given in (2.15) lie in hc∗ (Z(g)), for all k ∈ Z+ , while in the case of g = spo(2m|2n) or spo(2m|2n + 1), the polynomials σkm,n lie in hc∗ (Z(g)), for all even k ∈ Z+ . Let zkm,n be an element in Z(g) with hc∗ (zkm,n ) = σkm,n , where k is assumed to be even for type spo. Then for g in all three cases, m

n

i=1

j=1

χλ (zkm,n ) = ∑ (λ + ρ, δi )k − ∑ (λ + ρ, ε j )k . We now proceed case-by-case.

2. Finite-dimensional modules

66

(1) First, suppose that g = gl(m|n). Then, for all k ≥ 0, we have χλ (zkm,n ) = χµ (zkm,n ). This implies that, for all k ≥ 0, m

n

m

n

i=1

j=1

i=1

j=1

∑ (λ + ρ, δi)k − ∑ (λ + ρ, ε j )k = ∑ (µ + ρ, δi )k − ∑ (µ + ρ, ε j )k .

Using exponential generating functions in a formal indeterminate t, this is equivalent to m

(2.23)

n

m

n

∑ e(λ+ρ,δ )t − ∑ e(λ+ρ,ε )t = ∑ e(µ+ρ,δ )t − ∑ e(µ+ρ,ε )t . i

j

i=1

i

j=1

i=1

j

j=1

Pick a maximal number of pairs (ia , ja ) such that (λ + ρ, δia ) = (λ + ρ, ε ja ) and all ia and all ja are distinct, for 1 ≤ a ≤ . Similarly, pick a maximal number of pairs (ib , jb ) such that (µ + ρ, δib ) = (µ + ρ, ε jb ) and all ib and all jb are distinct, for 1 ≤ b ≤ r. Note that the functions eat are linearly independent for distinct a. In the identity obtained from (2.23) by canceling the terms corresponding to all the pairs (ia , ja ) and (ib , jb ), the survived (λ + ρ, δi )’s must match bijectively with the survived (µ + ρ, δi )’s, while the (λ + ρ, ε j )’s and (µ + ρ, ε j )’s must match bijectively. So  = r. Now we can find a pair of permutations w = (σ1 , σ2 ) ∈ Sm × Sn that extends the above bijections such that w−1 (µ + ρ) − λ − ρ is a linear combination of these mutually orthogonal isotropic odd roots δia − ε ja , and hence λ ∼ µ. (2) Now suppose that g = spo(2m|2n + 1). Then, for all even k ≥ 0, we have χλ (zkm,n ) = χµ (zkm,n ). This is equivalent to the following generating function identity: m

(2.24)

n

m

n

∑ e(λ+ρ,δi) t − ∑ e(λ+ρ,ε j ) t = ∑ e(µ+ρ,δi) t − ∑ e(µ+ρ,ε j ) t .

i=1

2

2

j=1

2

i=1

2

j=1

Pick a maximal number of pairs (ia , ja ) such that (λ + ρ, δia ) = sa (λ + ρ, ε ja ) for some sign sa ∈ {±} and all ia and all ja are distinct, for 1 ≤ a ≤ . Similarly, pick a maximal number of pairs (ib , jb ) such that (µ + ρ, δib )2 = (µ + ρ, ε jb )2 and all ib and all jb are distinct, for 1 ≤ b ≤ r. In the identity obtained from (2.24) by canceling the terms corresponding to all the pairs (ia , ja ) and (ib , jb ), the survived (λ + ρ, δi )’s must match bijectively with the survived (µ + ρ, δi )’s up to signs, while the (λ+ρ, ε j )’s and (µ+ρ, ε j )’s must match bijectively up to signs. So  = r. Then, there exists a pair of signed permutations w = (w1 , w2 ) ∈ W = WB = (Zm 2  Sm ) × (Zn2  Sn ) that extends the above bijections such that (2.25)

w−1 (µ + ρ) − (λ + ρ) =



∑ ca (δi

a=1

This implies that λ ∼ µ.

a

− sa ε ja ),

ca ∈ C.

2.2. Harish-Chandra homomorphism and linkage

67

(3) Finally, consider the case when g = spo(2m|2n). For all even k ≥ 0, we have χλ (zkm,n ) = χµ (zkm,n ). This is equivalent again to the generating function identity (2.24). Following the case of spo(2m|2n + 1), we make a similar choice of pairs (ia , ja ), for 1 ≤ a ≤ , and (ib , jb ), for 1 ≤ b ≤ , which leads to a choice of w = (w1 , w2 ) ∈ WB and a sequence of isotropic odd roots satisfying (2.25). But we are not done yet, as the Weyl group W of spo(2m|2n) is not WB , but a subgroup of WB of index 2. We will finish the job by considering two cases separately depending on whether λ is typical or atypical. Assume first that λ is atypical (and hence so is µ), and we have  ≥ 1. If w = (w1 , w2 ) ∈ W , we are done. Otherwise, let τ1 ∈ WB be the element changing the sign for δi1 while fixing all ε j and δi (i = i1 ), and let σ1 ∈ WB be the element changing the sign for ε j1 while fixing all δi and ε j ( j = j1 ). Note that τ1 ∈ W and σ1 w−1 ∈ W , and τ21 = σ21 = 1. By the definition of the pair (i1 , j1 ) and using (2.25), we have τ1 σ1 w−1 (µ + ρ) − (λ + ρ)     = τ1 σ1 w−1 (µ + ρ) − (λ + ρ) + τ1 σ1 (λ + ρ) − (λ + ρ)    = − 2(λ + ρ, δi1 ) + c1 (δi1 − s1 ε j1 ) + ∑ ca (δia − sa ε ja ). a=2

Hence we have obtained a set of mutually orthogonal isotropic odd roots, a corresponding set of complex numbers, and the Weyl group element wσ1 τ1 ∈ W satisfying (2.22). Thus, λ ∼ µ. By the description of the center Z(g) in Theorem 2.26 and (2.19), we have additional elements in Z(g) to use. In particular, we have ! " ! " (2.26) ε1 ε2 · · · εn P, θ−1 (λ + ρ) = ε1 ε2 · · · εn P, θ−1 (µ + ρ) . Assume now that λ is typical   (and so is µ), and  hence  = 0. Then the(m + n)tuple (λ + ρ, δi )2 , (λ + ρ, ε j )2 i, j coincides with (µ + ρ, δi )2 , (µ + ρ, ε j )2 i, j , up to a permutation in Sm × Sn . Recall P = ∏i, j (δ2i − ε2j ). Note that P, θ−1 (λ + ρ) = P, θ−1 (µ + ρ) = 0, since λ and µ are typical. By canceling this nonzero factor in (2.26), we obtain that ! " ! " (2.27) ε1 ε2 · · · εn , θ−1 (λ + ρ) = ε1 ε2 · · · εn , θ−1 (µ + ρ) . Note that in this case (2.22) reads that µ + ρ = w(λ + ρ), where we recall w = (w1 , w2 ). If (εi , λ + ρ) = 0 for all i, we conclude from (2.27) that w2 changes the signs for an even number of ε j ’s, and hence w = (w1 , w2 ) lies in the Weyl group W . If (εi , λ + ρ) = 0 for some i and w ∈ W , then w = (w1 , w2 σ) ∈ W satisfies that µ + ρ = w (λ + ρ), where σ denotes the sign change at εi . Hence λ ∼ µ in this case as well. The theorem is proved.



2. Finite-dimensional modules

68

Theorem 2.30 has the following implication, which will be referred to as the linkage principle for Lie superalgebra g of type gl and osp. Proposition 2.31. Let g be a Lie superalgebra of type gl and osp. For a composition factor L(µ) in a Verma module Δ(λ) of g, µ must satisfy the following conditions: λ − µ ∈ Z+ Φ+ and µ ∼ λ. Corollary 2.32. If χλ = χµ , then the degrees of atypicality for λ and µ coincide. Proof. This can be read off from the proof of Theorem 2.30.



Corollary 2.33. A finite-dimensional spo(2m|1)-module is completely reducible. Proof. Recall from Theorem 2.11 that the irreducible spo(2m|1)-module L(λ) is finite dimensional if and only if the sequence (λ1 , . . . , λm ) associated to the highest weight λ = ∑m i=1 λi δi is a partition. Let λ and µ be highest weights for two irreducible finite-dimensional spo(2m|1)-modules. Then λ and µ are Φ+ -dominant 0¯ integral, and hence λ ∼ µ if and only if λ = µ. By Theorem 2.30 this implies that χλ = χµ if and only if λ = µ, and hence every finite-dimensional spo(2m|1)-module is completely reducible.  2.2.7. Typical finite-dimensional irreducible characters. In this subsection we assume that g is of type gl or osp. As an application of the linkage principle, we obtain a character formula for the typical finite-dimensional irreducible g-modules. We fix a positive system Φ+ of g and denote by g = n− ⊕ h ⊕ n+ the associated ¯+ triangular decomposition. Recall the subset Φ ⊆ Φ+ defined in (1.34). Clearly, 1¯ ∗ ¯+ λ ∈ h is typical (cf. Definition 2.29) if and only if (λ + ρ, α) = 0, for all α ∈ Φ . 1¯   −1 Alternatively, λ is typical if and only if λ + ρ, P  = 0, where P := θ (P) ∈ S(h) in terms of P in (2.14). The following is a partial converse of Proposition 2.22 and follows immediately from Theorem 2.30. Below we shall provide a second proof based on the weaker Corollary 2.28. Lemma 2.34. Let g be of type gl or osp. Let λ, µ ∈ h∗ with λ typical. Suppose that χλ = χµ . Then, there exists w ∈ W such that w(λ + ρ) = µ + ρ, and µ is also typical. Proof. Suppose that there is no w ∈ W such that w(λ + ρ) = µ + ρ. Then we have / We can choose an element f ∈ S(h) that takes the value W (λ + ρ) ∩W (µ + ρ) = 0. 1 on the finite set W (λ + ρ) and 0 on the finite set W (µ + ρ). Averaging over W if necessary, we may also assume that f ∈ S(h)W . Now by Corollary 2.28 we have θ( f )P ∈ hc∗ (Z(g)). Thus there exists z ∈ Z(g) so that θ( f )P = hc∗ (z) and equivalently f P = hc(z). Hence, χλ (z) = λ + ρ, hc(z) = λ + ρ, f P  = λ + ρ, f λ + ρ, P  = λ + ρ, P  = 0. Similarly, we have χµ (z) = µ + ρ, hc(z) = µ + ρ, f P  = µ + ρ, f µ + ρ, P  = 0.

2.3. Harish-Chandra homomorphism and linkage for q(n)

69

This contradicts χλ = χµ . Hence there exists w ∈ W such that w(λ + ρ) = µ + ρ. ¯ 1¯ = Φ ¯+ ¯+ It follows by definition that Φ ∪ −Φ is W -invariant. Hence, for any 1¯ 1¯ + ¯ α ∈ Φ ¯ , we have 1

(µ + ρ, α) = (w(λ + ρ), α) = (λ + ρ, w−1 α) = 0. 

Therefore, µ is typical.

Recall that the finite-dimensional irreducible g-modules have been classified in terms of highest weights in Section 2.1. Theorem 2.35. Let g be of type gl or osp. Let λ ∈ h∗ be a typical weight such that L(λ) is a finite-dimensional irreducible representation of g. Then chL(λ) =

∏α∈Φ+¯ (1 + e−α ) 1

∏

β∈Φ+ 0¯

∑ (−1)(w)ew(λ+ρ)−ρ.

(1 − e−β ) w∈W

Proof. Recall from the proof of Proposition 2.22 that (2.28)

DchL(λ) =

∑ aµλ ∑ (−1)(w)ew(µ+ρ),

µ∈X

w∈W

where X consists of Φ+ -dominant weights such that aµλ = 0. 0¯ Since λ is typical and χλ = χµ for all µ ∈ X, it follows by Lemma 2.34 that X = {λ}. Recalling that aλλ = 1, we can rewrite (2.28) as DchL(λ) =

∑ (−1)(w)ew(λ+ρ),

w∈W

from which the theorem follows.



Example 2.36. Let g = gl(1|1). Then λ = aδ+bε is typical if and only if a+b = 0. Example 2.37. The character formula in Theorem 2.35 applies to all irreducible finite-dimensional spo(2m|1)-modules, as their highest weights are always typical.

2.3. Harish-Chandra homomorphism and linkage for q(n) This section is the counterpart of Section 2.2 for the queer Lie superalgebra q(n). The image of the Harish-Chandra homomorphism for q(n) is described with the help of symmetric functions such as Schur Q-functions (see Appendix A). Then, we obtain a linkage principle on composition factors of a Verma module of q(n). The finite-dimensional typical irreducible representations of q(n) are classified, and a Weyl-type character formula for these modules is obtained.

2. Finite-dimensional modules

70

2.3.1. Central characters for q(n). In this section, let g = q(n) be the queer Lie superalgebra, and let h = h0¯ + h1¯ be its (nonabelian) Cartan subalgebra. As this subsection is parallel to that of the basic Lie superalgebra case in Section 2.2.2, we will be brief and only point out the differences from therein. Recall that g is equipped with a non-degenerate odd invariant supersymmetric bilinear form (·, ·), which allows us to identify g with its dual g∗ , h with h∗ , and h0¯ with h∗1¯ . Via such identifications, the restriction map S(g∗ ) → S(h∗ ) and the induced map η : S(g) → S(h) are homomorphisms of algebras. Let g = n− ⊕ h ⊕ n+ be a triangular decomposition. As before, we have a projection φ : U(g) → U(h), and any element z ∈ Z(g) affords a unique expression of the form (2.6). Moreover, it follows from ad h(z) = 0 that hz ∈ U(h0¯ ). Hence Z(g) consists of only even elements. The restriction of φ to Z(g), also denoted by φ, defines an algebra homomorphism from Z(g) to U(h0¯ ), which sends z to hz . For λ ∈ h∗0¯ , define a linear map χλ : Z(g) → C by χλ (z) = λ, φ(z). The proof of Lemma 2.21 also works for the following lemma. Lemma 2.38. Let g = q(n). An element z ∈ Z(g) acts as the scalar χλ (z) on any highest weight g-module V (λ) of highest weight λ. Therefore χλ : Z(g) → C defines a one-dimensional Z(g)-module, which will be called the central character of g = q(n) associated to the highest weight λ. 2.3.2. Harish-Chandra homomorphism for q(n). Let g = n− ⊕ h ⊕ n+ be the standard triangular decomposition. Bearing in mind that ρ = 0 for q(n), we call hc = φ : Z(g) → S(h0¯ ) the Harish-Chandra homomorphism for q(n). Recall the elements Hi , H i , Ei j , E i j in q(n) etc. from Sections 1.1.4 and 1.2.6. A singular vector in the Verma module Δ(λ) of q(n) is a nonzero vector v such that Ei j v = E i j v = 0, for 1 ≤ i < j ≤ n. Lemma 2.39. Let g = q(n). Let λ ∈ h∗0¯ be such that λi − λi+1 = k ∈ N for some 1 ≤ i ≤ n − 1. Then k+1 u := E i,i+1 Ei+1,i vλ is a singular vector in the Verma module Δ(λ) of weight si (λ) = λ − k(εi − εi+1 ). Proof. We have the following identities in Δ(λ), for λ ∈ h∗0¯ and m ≥ 0: (2.29)

m−1 m m (H i − H i+1 )Ei+1,i vλ = Ei+1,i (H i − H i+1 )vλ − 2mE i+1,i Ei+1,i vλ ,

(2.30)

m+1 m m−1 E i,i+1 Ei+1,i vλ = (m + 1)Ei+1,i (H i − H i+1 )vλ − m(m + 1)E i+1,i Ei+1,i vλ .

Indeed, (2.29) can be directly verified by induction on m, and then (2.30) follows by induction on m using (2.29).

2.3. Harish-Chandra homomorphism and linkage for q(n)

71

It follows by (2.30) for m = k that u = 0. Now note that k+1 E i,i+1 u = (E i,i+1 )2 Ei+1,i vλ = 0, k+1 Ei,i+1 u = E i,i+1 Ei,i+1 Ei+1,i vλ = 0,

by a standard sl(2)-calculation. By weight considerations in Δ(λ), we must have E j, j+1 u = Ej, j+1 u = 0,

j = i.

Thus, we conclude that u is a singular vector in Δ(λ), and the weight of u is clearly equal to si (λ) = λ − k(εi − εi+1 ).  Recall that the Weyl group W for g = q(n) is the symmetric group Sn . Proposition 2.40. Let g = q(n). We have χλ = χw(λ) , for all w ∈ W and λ ∈ h∗0¯ . Also, we have hc(Z(g)) ⊆ S(h0¯ )W . Proof. By Lemma 2.39, we have a nonzero homomorphism Δ(si (λ)) → Δ(λ), for λ ∈ h∗0¯ with λi − λi+1 ∈ N, which implies by Lemma 2.21 that χλ = χsi (λ) . Hence, χλ = χw(λ) , for all w ∈ W and for all integral weights λ = ∑ni=1 λi εi satisfying λ1 > . . . > λn . Since the set of such weights is Zariski dense in h∗0¯ , it follows that χλ = χw(λ) , for all w ∈ W and λ ∈ h∗0¯ . That is, λ, hc(z) = w(λ), hc(z), for all z ∈ Z(g), w ∈ W and λ ∈ h∗0¯ . Hence, we conclude that hc(Z(g)) ⊆ S(h0¯ )W , thanks to the W -invariance of (·, ·).  Remark 2.41. One can imitate the proof of Proposition 2.22 to give an alternative proof of Proposition 2.40, bypassing Lemma 2.39. On the other hand, a second proof of Proposition 2.22 can be given in the spirit of the proof of Proposition 2.40 with some extra work involving odd reflections, as the Weyl group W is not generated by the even simple reflections. The odd non-degenerate invariant bilinear form (·, ·) on h induces an (even) isomorphism h0¯ ∼ = Πh∗1¯ , where Π denotes the parity reversing functor from Section 1.1.1. This in turn induces (even) isomorphisms θ : S(h0¯ ) → S(Πh∗1¯ ) and θ : S(h0¯ )W → S(Πh∗1¯ )W . We further define hc∗ = θhc : Z(g) → S(Πh∗1¯ )W and η∗ = θη : S(g)g → S(Πh∗1¯ ). Let ε1 , ε2 , . . . , εn ∈ Πh∗1¯ be determined by

εi (H j ) = δi j . That is, θ(Hi ) = εi , for each i. Let (π,V ) be the natural representation of g = q(n) on the vector superspace Cn|n . We recall the odd trace operator otr : g → C from (1.32). A straightforward super generalization of the arguments  for Lie algebras as 2k−1 given in Carter [18, pp. 212-213] shows that x → otr π(x) , for k ∈ N, defines a   g-invariant homogeneous polynomial on g of degree k (note that otr π(x)2k = 0 by definition of the odd trace). Restricting to h1¯ we obtain a homogeneous polynomial

2. Finite-dimensional modules

72

on Πh1¯ of degree 2k − 1. Then a direct calculation shows that, for x ∈ h1¯ and k ≥ 1,  2k−1 otr π(x) equals the value at x of the polynomial n

p2k−1 (ε) = ∑ εi2k−1 ,

k ∈ N.

i=1

Recall that a ring Γn of symmetric functions in n variables ε1 , ε2 , . . . , εn is defined in Appendix A.3. By (A.53), Γn has a basis that consists of Schur Q-functions Qλ (ε1 , ε2 , . . . , εn ), for λ ∈ SP, such that (λ) ≤ n (where SP denotes the set of strict partitions). By (A.45), the algebra Γn,C := C ⊗Z Γn is generated by the odd degree power sums p2k−1 (ε) for k ≥ 1. Summarizing the above discussions, we have established the following. Proposition 2.42. Let g = q(n). We have Γn,C ⊆ Im(η∗ ). We define (2.31)

P :=

∏

(εi + ε j ) ∈ S(Πh∗1¯ ).

1≤i< j≤n

The following lemma is a reformulation of (A.59) in Appendix A. Lemma 2.43. Let f ∈ S(Πh∗1¯ )W . Then we have f P ∈ Im(η∗ ). Lemma 2.44. Assume that a weight λ ∈ h∗0¯ satisfies that λi = −λi+1 = k ∈ C, for some 1 ≤ i ≤ n − 1. (1) If k = 0, then u := Ei+1,i vλ for any highest weight vector vλ is a singular vector in Δ(λ) of weight λ − εi + εi+1 . (2) If k = 0, then there exists a highest weight vector vλ in the Verma module Δ(λ) such that (H i − H i+1 )vλ = 0 and (H i + H i+1 )vλ = 0. Moreover, u := Ei+1,i (H i − H i+1 )vλ − 2kE i+1,i vλ is a singular vector in Δ(λ) of weight λ − εi + εi+1 . Proof. (1) Assume that k = 0. Note that H i and H i+1 act trivially on the highest weight space Wλ of Δ(λ), and hence H i vλ = H i+1 vλ = 0 for any highest weight vector vλ . Then, Ei,i+1 u = (Hi − Hi+1 )vλ = 0, and E i,i+1 u = (H i − H i+1 )vλ = 0. Also it follows from weight consideration that Ej, j+1 u = E j, j+1 u = 0 for j = i. Hence, u is a singular vector. (2) Assume that k = 0. We again identify the highest weight space of the Verma module Δ(λ) as the h-module Wλ (see Section 2.1.6 and also see Lemma 1.42). We compute that λ([H i ± H i+1 , H i ± H i+1 ]) = 2λ(Hi + Hi+1 ) = λi + λi+1 = 0.

2.3. Harish-Chandra homomorphism and linkage for q(n)

73

Hence, by Lemma 1.42, there exists a highest weight vector vλ ∈ Wλ such that wλ := (H i − H i+1 )vλ = 0 and (H i + H i+1 )vλ = 0. It follows that (H i + H i+1 )wλ = λ(Hi − Hi+1 )vλ = 2kvλ , (H i − H i+1 )wλ = 0. We compute that Ei,i+1 Ei+1,i wλ = λ(Hi − Hi+1 )wλ = 2kwλ , E i,i+1 Ei+1,i wλ = (H i − H i+1 )wλ = 0, Ei,i+1 E i+1,i vλ = (H i − H i+1 )vλ = wλ , E i,i+1 E i+1,i vλ = (H i + H i+1 )vλ = 0. It follows that Ei,i+1 u = E i,i+1 u = 0. Also we have by weight consideration that Ej, j+1 u = E j, j+1 u = 0 for j = i. Hence, u is a singular vector in Δ(λ) whose weight is clearly equal to λ − εi + εi+1 .  Lemma 2.45. Let λ = ∑ni=1 λi εi ∈ h∗0¯ . Assume that λi = −λ j for some 1 ≤ i = j ≤ n. Then, χλ = χλ−tα for α = εi − ε j and any t ∈ C. Proof. We first claim that χλ = χλ−εi +ε j . By Proposition 2.40 we are reduced to proving the claim when j = i + 1, and the claim in this case follows by Lemmas 2.38 and 2.44. By a repeated application of the claim, we have χλ = χλ−tα for any t ∈ Z+ . Since Z+ is Zariski dense in C, the lemma follows.  The following theorem is the q(n)-counterpart of Theorem 2.26. Theorem 2.46. Let g = q(n). Then hc∗ (Z(g)) = Im(η∗ ) = Γn,C . Proof. Let λ = ∑ni=1 λi εi be an integral weight. Let z ∈ Z(g). By Proposition 2.40, we know that hc∗ (z) ∈ S(Πh∗1¯ )W . By unraveling the definition of χλ , we obtain by Lemma 2.45 that hc∗ (z)|εi =−ε j =t is independent of t. Then by the characterization of Γn given in (A.60), we have hc∗ (z) ∈ Γn,C . Together with Proposition 2.42, we have shown that hc∗ (Z(g)) ⊆ Γn,C ⊆ Im(η∗ ). It remains to show that hc∗ (Z(g)) = Im(η∗ ). This follows by the corresponding identity on the associated graded and then a standard filtered algebra argument, exactly as in the proof of Theorem 2.26. We leave the details to the interested reader.  We summarize various maps for the q(n) case in the following diagram, which only becomes commutative when we pass from Z(g) to its associated graded (note

2. Finite-dimensional modules

74

that the other algebras are graded). γ−1 ∼ =

/ S(g)g t t t η ttt hc∗ t t η ∗  ytt ∼  = S(Πh∗ )W o S(h ¯ )W

Z(g)

(2.32)

1¯

θ

0

Remark 2.47. Actually hc and hc∗ are injective (see Sergeev [111]), and so we have an isomorphism of algebras hc∗ : Z(g) → Γn,C . The injectivity of hc∗ does not play any role in the representation theory of q(n) in the book. 2.3.3. Linkage for q(n). For a root α of the form εi − ε j , define α∨ := Hi + H j ∈ h0¯ . We define a relation on h∗0¯ for λ, µ ∈ h∗0¯ ,

λ ∼ µ,

if there exist a collection of roots αi , complex numbers ci , for 1 ≤ i ≤ , and an element w ∈ W such that   # $ # $  µ = w λ − ∑ ca αa , (2.33) αi , α∨j = 0, λ, α∨j = 0, 1 ≤ i, j ≤ . a=1

If λ ∼ µ, we say that λ and µ are linked. The following theorem is a q(n)counterpart of Theorem 2.30. It also implies that linkage for q(n) is an equivalence relation. Theorem 2.48. Let h = h0¯ + h1¯ be a Cartan subalgebra of q(n), and let λ, µ ∈ h∗0¯ . Then λ is linked to µ if and only if χλ = χµ . Proof. Assume first that λ ∼ µ. By a repeated application of Lemma 2.45, the second and third conditions in (2.33) for λ ∼ µ imply that χλ = χλ−c1 α1 = χλ−c1 α1 −c2 α2 = . . . = χλ−∑

a=1 ca αa

.

Since µ = w(λ − ∑a=1 ca αa ) by (2.33) again, it follows by Proposition 2.40 that χλ−∑ ca αa = χµ , whence χλ = χµ . a=1

Now assume that χλ = χµ . Write λ = ∑ni=1 λi εi and µ = ∑ni=1 µi εi . Let z2k+1 ∈ Z(g) such that hc(z2k+1 ) = p2k+1 , the (2k + 1)st power sum in H1 , . . . , Hn . From χλ (z2k+1 ) = χµ (z2k+1 ) we obtain that λ, hc(z2k+1 ) = µ, hc(z2k+1 ). Hence we have n

∑ λi2k+1 =

i=1

n

, ∑ µ2k+1 j

j=1

∀k ∈ Z+ ,

2.3. Harish-Chandra homomorphism and linkage for q(n)

75

which is equivalent to the following generating function identity in an indeterminate t: n

∑ sinh(λit) =

i=1

n

∑ sinh(µ jt).

j=1

Here we recall that the function sinh(t) =

et − e−t t 2k+1 =∑ 2 k≥0 (2k + 1)!

satisfies sinh(−at) = − sinh(at) for any scalar a ∈ C. Pick a maximal number of pairs (ia , ja ) such that λia = −λ ja , with 1 ≤ ia < ja ≤ n and all ia , ja are distinct, for 1 ≤ a ≤ . Denote Iλ = {1 ≤ i ≤ n | i = ia , i = ja , ∀a}. Similarly, pick a maximal number of pairs (ib , jb ) such that µib = −µ jb , with 1 ≤ ib < jb ≤ n and all ib , jb are distinct, for 1 ≤ b ≤ r, and define Iµ accordingly. It follows by definition that (2.34)

∑ sinh(λit) = ∑ sinh(µ jt).

i∈Iλ

j∈Iµ

Let c be the maximum of the set {|λi |, |µ j | | i ∈ Iλ , j ∈ Iµ }. Furthermore, assume without loss of generality that for some s we have c = |λs | and λs appears in the set {λi |i ∈ Iλ } with multiplicity . The identity (2.34) which is valid for all t implies that there exists µr with r ∈ Iµ such that µr = λs and also µr must appear in {µ j | j ∈ Iµ } with multiplicity  as well. Canceling the corresponding hyperbolic sine functions from both sides of (2.34) we can proceed similarly as before and conclude that there is a bijection between Iλ and Iµ such that the corresponding λi and µ j coincide. Now we can find a permutation w ∈ Sn that extends the bijection between Iλ and Iµ such that w−1 µ − λ is a linear combination of the roots εia − ε ja . Thus, λ ∼ µ.  Theorem 2.48 has the following implications. For a composition factor L(µ) in a Verma module Δ(λ) of q(n), µ must satisfy the following conditions: λ − µ ∈ Z+ Φ+ , and µ ∼ λ. This will be referred to as the linkage principle for q(n). Definition 2.49. Let g = q(n) and let h be its standard Cartan subalgebra. The degree of atypicality of a weight λ ∈ h∗0¯ , denoted by #λ, is the maximum number of pairs (ia , ja ) with all ia , ja distinct such that λ, Hia + H ja  = 0. A weight λ ∈ h∗0¯ is called typical if #λ = 0, and is atypical otherwise. We have the following corollary from the proof of Theorem 2.48. Corollary 2.50. If χλ = χµ for λ, µ ∈ h∗0¯ , then the degrees of atypicality for λ and µ coincide.

2. Finite-dimensional modules

76

2.3.4. Typical finite-dimensional characters of q(n). Note that λ ∈ h∗0¯ is typical (cf. Definition 2.49) if and only if

∏

λ, Hi + H j  = 0.

1≤i< j≤n

Recall that the finite-dimensional irreducible q(n)-modules are classified in terms of highest weights in Theorem 2.18. The following is the q(n)-counterpart of Lemma 2.34. Lemma 2.51. Let g = q(n) and λ, µ ∈ h∗0¯ with λ typical. Suppose that χλ = χµ . Then, there exists w ∈ W such that µ = w(λ), and µ is typical. Proof. Recalling P from (2.31), we have εi + ε j = θ(Hi + H j ), and   P=θ ∏ (Hi + H j ) . 1≤i< j≤n

Clearly P is W -invariant. By Lemma 2.43 and Theorem 2.46, we have θ( f )P ∈ hc∗ (Z(g)), for f ∈ S(h0¯ )W . Also recall ρ = 0. With these ingredients in place, the argument in the proof of Lemma 2.34 goes through in this case as well. We leave the details to the interested reader.  The following is a q(n)-analogue of Theorem 2.35. Theorem 2.52. Let g = q(n) and let λ ∈ h∗0¯ be a typical weight such that the irreducible q(n)-module L(λ) is finite dimensional. Then chL(λ) = 2

(λ)+δ(λ) 2

∏β∈Φ+¯ (1 + e−β ) 1

∑ (−1)(w)ew(λ).

∏α∈Φ+¯ (1 − e−α ) w∈W 0

Proof. We follow the strategy of the proof of Theorem 2.35. The character of the Verma module Δ(λ) is given by chΔ(λ) = 2

(λ)+δ(λ) 2

eλ

∏β∈Φ+¯ (1 + e−β ) 1

∏α∈Φ+¯ (1 − e−α )

.

0

Observe that the highest weight spaces of L(µ) and Δ(µ), for every µ ∈ W λ, have (λ)+δ(λ) dimension 2 2 , thanks to (µ) = (λ). Imitating the proof of Proposition 2.22, we obtain that (2.35)

∏α∈Φ+¯ (1 − e−α ) 0

∏

β∈Φ+ 1¯

(1 + e−β )

chL(λ) =

∑ aµλ ∑ (−1)(w)ew(µ),

µ∈X

w∈W

where X consists of Φ+ -dominant integral weights such that aµλ = 0. 0¯

2.4. Extremal weights of finite-dimensional simple modules

77

Since λ is typical and χλ = χµ for all µ ∈ X, it follows by Lemma 2.51 that X = {λ}. Recalling that aλλ = 2 ∏α∈Φ+¯ (1 − e−α ) 0

∏β∈Φ+¯ (1 + e−β ) 1

(λ)+δ(λ) 2

, we can rewrite (2.35) as

chL(λ) = 2

(λ)+δ(λ) 2

∑ (−1)(w)ew(λ+ρ),

w∈W

from which the theorem follows.



Remark 2.53. The assumption on λ in Theorem 2.52 forces λi > λi+1 for all i, and so (λ) = n or (λ) = n − 1.

2.4. Extremal weights of finite-dimensional simple modules In this section, we study the extremal weights of a finite-dimensional irreducible module over a Lie superalgebra of type gl and osp. A main complication arises from the existence of non-conjugate Borel subalgebras for Lie superalgebras. For the Lie superalgebra of type gl the extremal weights for irreducible polynomial representations are determined. For the Lie superalgebras of type osp the extremal weights of finite-dimensional irreducible representations of integer weights are determined. In all cases, the answers are given in terms of the hook diagrams. 2.4.1. Extremal weights for gl(m|n). We start with some general remarks for any basic Lie superalgebra g. Let L be a finite-dimensional g-module. Then given any Borel subalgebra b of g associated to a positive system Φ+ , there exists a unique weight λb for L such that λb +α is not a weight for L for any α ∈ Φ+ , and the weight space Lλb is one-dimensional (cf. Proposition 1.39). The weight λb is called the b-extremal weight for the g-module L. Two Borel subalgebras b and b of g are in general not conjugate. Thus, the b-extremal weight and the b -extremal weight of L may in general not be conjugate by the Weyl group W , in contrast to the semisimple Lie algebra setting. In this subsection we let g = gl(m|n). Let h be its standard Cartan subalgebra with standard Borel subalgebra bst . For an (m|n)-hook partition λ, we may regard λ as an element in h∗ as in (2.4). We shall determine the b-extremal weight of L(g, bst , λ ) for any Borel subalgebra b. The class of gl(m|n)-modules L(g, bst , λ ), for all (m|n)-hook partitions λ, are called the polynomial modules of gl(m|n), and they will feature significantly in Chapter 3 on Schur duality. Recall the weights δi and ε j from Section 1.2.3. Let b be a Borel subalgebra of g and let Φ+ be its positive system. Assume that the εδ-sequence from Section 1.3 associated to b is δd1 εe1 δd2 εe2 · · · δdr εer where the exponents denote the corresponding multiplicities (all di , ei are positive except possibly d1 = 0 or er = 0). There exist a permutation s of {1, . . . , m} and a permutation t of {1, . . . , n} such that the δ’s and ε’s appearing in the εδ-sequence from left to right are δs(1) , δs(2) , . . . , δs(m) , and εt(1) , εt(2) , . . . , εt(n) , respectively.

2. Finite-dimensional modules

78

Define u

(2.36)

du :=

∑ da

u

and

eu :=

a=1

∑ ea

a=1

for u = 1, . . . , r, and let d0 = e0 = 0. Note dr = m, er = n. Define the b-Frobenius coordinates (pi |q j ) of an (m|n)-hook partition λ as follows. For 1 ≤ i ≤ m, 1 ≤ j ≤ n, let (2.37)

pi = max{λi − eu , 0},

if du < i ≤ du+1 for some 0 ≤ u ≤ r − 1,

q j = max{λj − du+1 , 0},

if eu < j ≤ eu+1 for some 0 ≤ u ≤ r − 1.

Associated to an (m|n)-hook Young diagram λ and a Borel subalgebra b, we define a weight λb ∈ h∗ in terms of the b-Frobenius coordinates (pi |qi ) to be m

n

i=1

j=1

λb := ∑ pi δs(i) + ∑ q j εt( j) . It is elementary to read off the b-Frobenius coordinates of λ from the Young diagram of λ in general, as illustrated by the next example. Example 2.54. Let b be the Borel subalgebra of gl(5|4) associated to the following fundamental system: δ2 − δ3



 δ3 − ε3

ε3 − ε2





δ1 − δ4



ε2 − δ1

 δ4 − ε4

ε4 − ε1



 ε1 − δ5

Consider the (5|4)-hook diagram λ = (14, 11, 8, 8, 7, 4, 3, 2). The b-Frobenius coordinates for λ are: p1 = 14, p2 = 11, p3 = p4 = 6, p5 = 3;

q1 = q2 = 6, q3 = 3, q4 = 2.

These are read off from the Young diagram of λ by following the εδ sequence δδεεδδεεδ (with δ for rows and ε for columns of λ) as follows: n=4 ← p1 → ← p2 → ← p3 → ← p4 → ↑ ↑ ↑ ↑ m = 5 q1 q2 q3 q4 ↓ ↓ ↓ ↓

← p5 →

2.4. Extremal weights of finite-dimensional simple modules

79

Then we obtain that λ b = p1 δ 2 + p2 δ 3 + p3 δ 1 + p4 δ 4 + p5 δ 5 + q1 ε 3 + q2 ε 2 + q3 ε 4 + q4 ε 1 = 6δ1 + 14δ2 + 11δ3 + 6δ4 + 3δ5 + 2ε1 + 6ε2 + 6ε3 + 3ε4 . Theorem 2.55. Let λ be an (m|n)-hook partition. Let b be an arbitrary Borel subalgebra of gl(m|n). Then, the b-extremal weight of the simple gl(m|n)-module L(g, bst , λ ) is λb . Proof. Set V = L(g, bst , λ ). One distinguished feature for V is that all weights ν of V are polynomial in the sense that ν(Eii ) ∈ Z+ for all i ∈ I(m|n); see Chapter 3, Theorem 3.11. The theorem holds for the standard Borel subalgebra bst , which corresponds to st the sequence of m δ’s followed by n ε’s, thanks to λb = λ . By Corollary 1.32, bst can be converted to b by a finite number of real and odd reflections. Thus, we are led to consider two Borel subalgebras b1 and b2 , where b2 is obtained from b1 by applying a simple reflection corresponding to a simple root α. Let µ be the b2 -extremal weight for V . To complete the proof of the theorem, we will show that µ = λb2 , assuming the theorem holds for b1 . If α is an even root, then rα (λb1 ) = λb2 by definition, and the validity of the theorem for b1 implies its validity for b2 . Now assume that α = ±(δi − ε j ) is an odd simple root. The corresponding coroot is hα = ±(Ei¯,i¯ + E j j ). If λb1 (hα ) = 0, then we must have λb1 (Ei¯,i¯) = λb1 (E j j ) = 0 (as each has to be nonnegative), and λb1 = λb2 . Then by Lemma 1.40 we have µ = λb1 = λb2 . Suppose that λb1 (hα ) = 0 and α = δi − ε j . Diagrammatically we can represent λb1 (Ei¯,i¯) = a and λb1 (E j j ) = b by Diagram (ii) below, and this forces that a > 0. By Lemma 1.40, we have µ = λb1 − δi + ε j , and hence µ(E j j ) = b + 1 and µ(Ei¯,i¯) = a − 1, which are represented by Diagram (i) below. The remaining parts of µ are the same as those for λb1 and hence we conclude that µ = λb2 , as claimed. Suppose that λb1 (hα ) = 0 and α = ε j − δi . Diagrammatically we can represent λb1 (E j j ) = b + 1 and λb1 (Ei¯,i¯) = a − 1 by Diagram (i) below, and this forces that b ≥ 0 and a ≥ 1. Lemma 1.40 implies that µ = λb1 − ε j + δi and hence µ(Ei¯,i¯) = a and µ(E j j ) = b can be represented by Diagram (ii) below. Hence µ = λb2 . a−1

a

b+1 b

(i) This completes the proof of the theorem.

(ii) 

2. Finite-dimensional modules

80

Example 2.56. Let us describe the highest weights in Theorem 2.55 with respect to the three Borel subalgebras of special interest. st

(1) As seen above, λb = λ . (2) If we take the opposite Borel subalgebra bop corresponding to a sequence of n ε’s followed by m δ’s, then λb = λ1 ε1 + . . . + λn εn + max{λ1 − n, 0}δ1 + . . . + max{λm − n, 0}δm . op

(3) In the case when |m − n| ≤ 1, we may take a Borel subalgebra bo whose simple roots are all odd (or equivalently, the corresponding εδ-sequence is alternating between ε and δ): 





···





o

In this case, Theorem 2.55 states that the coefficients of δ and ε in λb are given by the modified Frobenius coordinates (pi |qi )i≥1 of the partition λ (respectively, λ ), when the first simple root is of the form δ − ε (respectively, ε − δ). Here by modified Frobenius coordinates we mean pi = max{λi − i + 1, 0},

qi = max{λi − i, 0}

so that ∑i (pi + qi ) = |λ|. “Modified” here refers to a shift by 1 from the pi coordinates defined in [83, Chapter 1, Page 3]. The modified Frobenius coordinates in this case can be read off from the Young diagram by alternatively reading off (and deleting thereafter) the number of boxes in rows and columns. As an illustration, if λ = (7, 5, 4, 3, 1), then we have (p1 , p2 , p3 |q1 , q2 , q3 ) = (7, 4, 2|4, 2, 1). 7 4 2 4

2

1

2.4.2. Extremal weights for spo(2m|2n + 1). Let us denote the weights of the natural spo(2m|2n + 1)-module C2m|2n+1 by ±δi , 0, ±ε j for 1 ≤ i ≤ m, 1 ≤ j ≤ n. Recall that the standard Borel subalgebra bst of spo(2m|2n + 1) is associated to the fundamental system (1.39) which we recall:  δ1 − δ2

 δ2 − δ3

···

 δm−1 − δm

δm − ε1





ε1 − ε2

···

=⇒

εn−1 − εn

εn

2.4. Extremal weights of finite-dimensional simple modules

81

Let b be a Borel subalgebra. As explained in Section 1.3.3, a Dynkin diagram for spo(2m|2n + 1) always has a type A end while the other end is a short (even or odd) root. For example, the above Dynkin diagram has its type A end labeled by the simple root δ1 − δ2 . Starting from the type A end, the simple roots for b of spo(2m|2n + 1) give rise to an εδ-sequence δd1 εe1 δd2 εe2 · · · δdr εer and sequences of ±1’s: (ξi )1≤i≤m ∪ (η j )1≤ j≤n (all the di and e j are positive except possibly d1 = 0 or er = 0). Furthermore, the Dynkin diagram contains a short odd root if and only if er = 0. Recall the definitions of du and eu from (2.36) and those of pi and q j from (2.37). Similar to the type A case there exist a permutation s of {1, . . . , m} and a permutation t of {1, . . . , n}, so that the simple roots for b are given by ξi δs(i) − ξi+1 δs(i+1) , 1 ≤ i ≤ m, i ∈ {du |u = 1, . . . , r}; η j εt( j) − η j+1 εt( j+1) , 1 ≤ j ≤ n, j ∈ {eu |u = 1, . . . , r}; ξdu δs(du ) − η1+eu−1 εt(1+eu−1 ) , for 1 ≤ u ≤ r if er > 0 (or 1 ≤ u < r if er = 0); ηeu εt(eu ) − ξ1+du δs(1+du ) , u = 1, . . . , r − 1; ηer εt(er ) ,

(or ξdr δs(dr ) if er = 0).

if er > 0

By Theorem 2.11, a complete list of finite-dimensional irreducible modules of spo(2m|2n + 1) of integer weights are the highest weight g-modules L(g, bst , λ ) of bst -highest weights λ in (2.4), for some (m|n)-hook partition λ. Example 2.57. Consider the following Dynkin diagram of spo(10|9) with simple roots attached: δ2 + δ3





−ε3 − ε2

−δ3 + ε3



 ε2 − δ1

δ1 − δ4



 δ4 − ε4

ε4 − ε1





=⇒ y

ε1 + δ5

−δ5

We read off a signed sequence with indices δ2 (−δ3 )(−ε3 )ε2 δ1 δ4 ε4 ε1 (−δ5 ). In particular, we obtain a sequence δδεεδδεεδ by ignoring the signs and indices. In this case, d1 = d2 = 2, d3 = 1, and e1 = e2 = 2. Furthermore, the sequences (ξi )1≤i≤5 and (η j )1≤ j≤4 are (1, −1, 1, 1, −1) and (−1, 1, 1, 1), respectively. Theorem 2.58. Let λ be an (m|n)-hook partition. Let b be a Borel subalgebra of spo(2m|2n + 1) and retain the above notation. Then, the b-highest weight of the simple spo(2m|2n + 1)-module L(g, bst , λ ) is m

n

i=1

j=1

λb := ∑ ξi pi δs(i) + ∑ η j q j εt( j) . Proof. The arguments here are similar to the ones given in the proof of Theorem 2.55 for gl(m|n), and so we will be sketchy.

2. Finite-dimensional modules

82

We first note that the theorem holds for the standard Borel subalgebra bst , which corresponds to the sequence of m δ’s followed by n ε’s with all signs ξi st and η j being positive, i.e., λb = λ . All Borels are linked to the standard Borel by a sequence of even and odd simple reflections. Let b and b be two Borel subalgebras related by a simple reflection with respect to a simple root α, and assume the theorem holds for b . If  α is even, then clearly rα (λb ) = λb , and the theorem holds for b. In the case when α is odd, we have four cases to consider, by setting α = ±δi ± ε j . Each case is verified by the same type of argument as in the proof of Theorem 2.55.  Example 2.59. With respect to the Borel b of spo(10|9) as in Example 2.57, the b-extremal weight of L(g, bst , λ ) for λ as given in Example 2.54 equals λ b = p1 δ 2 − p2 δ 3 + p3 δ 1 + p4 δ 4 − p5 δ 5 − q1 ε 3 + q2 ε 2 + q3 ε 4 + q4 ε 1 = 6δ1 + 14δ2 − 11δ3 + 6δ4 − 3δ5 + 2ε1 + 6ε2 − 6ε3 + 3ε4 . Corollary 2.60. Every finite-dimensional irreducible spo(2m|2n + 1)-module of integer highest weight is self-dual. Proof. By a standard fact in highest weight theory, the dual module L(g, bst , λ )∗ has bst -highest weight equal to the opposite of the bst -lowest weight of L(g, bst , λ ). Denote by bop the opposite Borel to the standard one bst . It follows by Theorem 2.58 that the bop -extremal weight (i.e., the bst -lowest weight) of the module L(g, bst , λ ) is −λ . This proves the corollary.  Consider the following Dynkin diagram of spo(2m|2n + 1) with simple roots attached, and let us denote its associated Borel subalgebra by bsd : ⇐=

−ε1

ε1 − ε2



ε2 − ε3

···

 εn − δm

 δm − δm−1



···

 δ2 − δ1

We have the following immediate corollary of Theorem 2.11 and Theorem 2.58. Corollary 2.61. An irreducible spo(2m|2n + 1)-module of integer highest weight with respect to the Borel subalgebra bsd is finite dimensional if and only if the bsd -highest weight is of the form (2.38)

m

n

i=1

j=1

λb = − ∑ λi δi − ∑ max{λj − m, 0} εn− j+1 , sd

where λ = (λ1 , λ2 , . . .) is an (m|n)-hook partition. 2.4.3. Extremal weights for spo(2m|2n). Let n ≥ 2. Let us denote the weights of the natural spo(2m|2n)-module C2m|2n by ±δi , ±ε j for 1 ≤ i ≤ m, 1 ≤ j ≤ n. The standard Borel subalgebra bst of spo(2m|2n) is the one associated to the Dynkin diagram (1.41), which we recall for the reader’s convenience:

2.4. Extremal weights of finite-dimensional simple modules





δ1 − δ2

···

δ2 − δ3





δm − ε1





ε1 − ε2

δm−1 − δm

83

···

εn−1 − εn



εn−2 − εn−1@ @



εn−1 + εn

According to Section 1.3.4, there are two types of Dynkin diagrams and corresponding Borel subalgebras for spo(2m|2n): (i) Diagrams of |-shape, i.e., Dynkin diagrams with a long simple root ±2δi .

(ii) Diagrams of -shape, i.e., Dynkin diagrams with no long simple root.

We will follow the notation for spo(2m|2n + 1) in Section 2.4.2 for fundamental systems in terms of signed εδ-sequences, so we have permutations s,t, and signs ξi , η j . We fix the ambiguity on the choice of the last sign ηn associated to a Borel b of -shape by demanding the total number of negative signs among η j (1 ≤ j ≤ n) to be always even. Let λ be an (m|n)-hook partition, and let the b-Frobenius coordinates (pi |q j ) be as defined in Section 2.4.1. Introduce the following weights: λb := λb− :=

m

n

∑ ξi pi δs(i) + ∑ η j q j εt( j),

i=1

j=1

m

n−1

i=1

j=1

∑ ξi pi δs(i) + ∑ η j q j εt( j) − ηn qn εt(n).

The weight λb− will only be used for Borel b of -shape. Note that λb = λ and st we shall denote λ− := λb− . st

Given a Borel b of |-shape, we define s(b) to be the sign of ∏nj=1 η j . Recall from Theorem 2.14 that a finite-dimensional irreducible spo(2m|2n)module of integer highest weight with respect to the standard Borel subalgebra is of the form L(g, bst , λ ) or L(g, bst , λ− ), where λ is an (m|n)-hook partition. Theorem 2.62. Let n ≥ 2, and let λ be an (m|n)-hook partition. (1) Assume b is of -shape. Then, (i) λb is the b-extremal weight for the module L(g, bst , λ ). (ii) λb− is the b-extremal weight for the module L(g, bst , λ− ). (2) Assume b is of |-shape. Then, (i) λb is the b-extremal weight for L(g, bst , λ ) if s(b) = +. (ii) λb is the b-extremal weight for L(g, bst , λ− ) if s(b) = −. Proof. Let b and b be Borel subalgebras. Suppose that they are related by b = w(b), for some Weyl group element w. Note that the set of weights for L(g, bst , λ )   is W -invariant. By definition we observe that λb = w(λb ) and λb− = w(λb− ), and hence the validity of the theorem for b implies its validity for b . Thus it suffices to

2. Finite-dimensional modules

84

prove the theorem for a Borel subalgebra b with even roots {δi ± δ j |i ≤ j} ∪ {εk ± εl |k < l}. We shall make this assumption for the remainder of the proof.

Suppose that the Dynkin diagram of b is of -shape. Thus the corresponding εδ-sequence ends with an ε, i.e., it is of the form · · · δ · · · ε · · · ε.

The fundamental system of b can be brought to the fundamental system corresponding to bst by applying a sequence of odd reflections corresponding to odd isotropic root of the form εi − δ j or δ j − εi . As Part (1) of the theorem is true for the standard Borel bst , exactly the same argument as in the proof of Theorem 2.55 proves (1). Now suppose that b is of |-shape. We first consider the Borel subalgebras b1 and b2 corresponding to the two fundamental systems below, respectively: 

···

ε1 − ε2



ε1 − ε2

εn−1 − εn



 εn − δ1

···

εn−1 + εn







···



−εn − δ1 δ1 − δ2

⇐= δm−1 − δm

δ1 − δ2

···

2δm

⇐=

δm−1 − δm

2δm

Note the sign sequence (η1 , . . . , ηn ) associated to the εi ’s is (1, . . . , 1, 1) for b1 and (1, . . ., 1, −1) for b2 . So we have s(b1 ) = + and s(b2 ) = −. First, using sequences of odd reflections, we can transform the standard fundamental system to the fundamental systems associated with b1 and b2 . We list below two such sequences consisting of mn odd reflections that will take bst to b1 and b2 , respectively. {δm − ε1 , . . . , δ1 − ε1 ; δm − ε2 , . . . , δ1 − ε2 ; . . . ; δm − εn , . . . , δ1 − εn }, {δm − ε1 , . . . , δ1 − ε1 ; δm − ε2 , . . . , δ1 − ε2 ; . . . ; δm + εn , . . . , δ1 + εn }. We note that the above two sequences are identical except for the last n reflections st (where + signs replace − signs). Starting from λb± for the standard Borel bst and repeatedly using Lemma 1.30 at each step, it is straightforward to show that Part (2) of the theorem is true for b1 and b2 . Now if s(b) = + (respectively, s(b) = −) then b can be brought to b1 (respectively b2 ) by a sequence of odd reflections corresponding to odd isotropic roots of the form δ j − εi or εi − δ j . Now the same type of argument as in the proof of Theorem 2.55 proves (2).  Corollary 2.63. Let n ≥ 0 be even. Then every finite-dimensional irreducible spo(2m|2n)-module of integral highest weight is self-dual. Proof. The argument here is similar to the proof of Corollary 2.60. For n = 0, the corollary follows by the well-known fact that the longest element in the Weyl group

2.5. Exercises

85

for sp(2m) is −1. Now assume n ≥ 2 is even. Denote by bop the opposite Borel to the standard one bst . It follows by Theorem 2.62 that the bop -extremal weight of the module L(g, bst , λ ) (and respectively, L(g, bst , λ− )) is −λ (and respectively, −λ− ). The corollary follows.  Remark 2.64. The remaining b-extremal weights for the modules L(g, bst , λ ) when s(b) = − or for the modules L(g, bst , λ− ) when s(b) = + do not seem to afford a uniform simple answer. Consider the following Dynkin diagram of spo(2m|2n) whose Borel subalgebra is denoted by bsd . ε1 − ε2

−ε1 − ε2



@ @

ε2 − ε3



···



εn−1 − εn

εn − δm



 δm − δm−1

···

 δ2 − δ1

We record the following corollary of Theorems 2.14 and 2.62. Corollary 2.65. Let n ≥ 2. An irreducible spo(2m|2n)-module of integral highest weight with respect to the Borel subalgebra bsd is finite dimensional if and only if the highest weight is of the form (2.39)

m

n−1

i=1

j=1

− ∑ λi δi − ∑ max{λj − m, 0} εn− j+1 ± max{λn − m, 0} ε1 ,

where λ = (λ1 , λ2 , . . .) is an (m|n)-hook partition.

2.5. Exercises Exercise 2.1. Let m ≥ 2. Prove: (1) A highest weight irreducible representation L(λ) of gl(m|m) remains an irreducible representation when restricted to sl(m|m). (2) Every highest weight irreducible representation of sl(m|m) extends to a representation of gl(m|m). Hints: For (1) if v were an sl(m|m)-singular weight vector in L(λ) and we can write v = ∑i vi , where the vi ’s are gl(m|m)-weight vectors of distinct weights. But then each vi would be gl(m|m)-singular. Use (1) to prove (2). Exercise 2.2. We use the notation for osp(1|2) from Example 1.25. Prove: (1) The identities [E, F 2n ] = −nF 2n−1 and [E, F 2n+1 ] = F 2n (h − n) hold in U(osp(1|2)), for all n ∈ Z+ . (2) A complete list of inequivalent irreducible finite-dimensional representations of osp(1|2) is given by {L(nδ1 ) | n ∈ Z+ }.

2. Finite-dimensional modules

86

(3) The character of L(nδ1 ) is given by e(n+ 2 )δ1 − e−(n+ 2 )δ1 1

ch L(nδ1 ) =

1

e 2 δ 1 − e− 2 δ 1 1

1

.

Exercise 2.3. Let g be a finite-dimensional simple Lie superalgebra equipped with an even supersymmetric invariant bilinear form (·, ·). Let {ui |1 ≤ i ≤ k} be a homogeneous basis for g and let {ui |1 ≤ i ≤ k} be its dual basis with respect to (·, ·) so that (ui , u j ) = δi j . Prove: (1) The Casimir operator Ω := ∑ki=1 (−1)|ui | ui ui ∈ U(g) is independent of the choice of the basis {ui |1 ≤ i ≤ k}. (2) Ω ∈ Z(g) and Ω acts on a highest weight g-module of highest weight λ as the scalar (λ + 2ρ, λ). Exercise 2.4. Let g = osp(1|2). Prove that the Casimir operator Ω acts as different scalars on inequivalent finite-dimensional irreducible g-modules and that every finite-dimensional g-module is completely reducible. Exercise 2.5. Let g = gl(1|1). Prove: (1) The Verma g-module Δ(aδ1 + bε1 ) is reducible if and only if a = −b. (2) There exists a non-split short exact sequence of g-modules: 0 −→ L((a − 1)δ1 − (a − 1)ε1 ) −→ Δ(aδ1 − aε1 ) −→ L(aδ1 − aε1 ) −→ 0. (Hence the infinitely many simple g-modules L(aδ1 −aε1 ) for all a in a congruence class C/Z belong to one block.) Exercise 2.6. Let g = sl(1|2). Prove that the Casimir operator acts trivially on the polynomial module L(λ), for λ = (k + 1)δ1 + (k − 1)ε1 , k ∈ N. Exercise 2.7. Let g be a basic Lie superalgebra and let b be a Borel subalgebra with Weyl vector ρb . Let b be a Borel subalgebra with b0¯ = b0¯ . Suppose that b is obtained from b by a sequence of odd reflections with respect to the odd roots α1 , α2 , . . . , αk . Set λ0 = λb and define inductively for i = 1, . . . , k  λi−1 , if (λi−1 , αi ) = 0, λi := λi−1 − α, if (λi−1 , αi ) = 0. 

Set λb = λk . Prove: 

(1) λb is well-defined, i.e., it is independent of the sequence of odd reflections. ¯ 1¯ , if and only if (λb + ρb , α) = 0, ∀α ∈ Φ ¯ 1¯ (here (2) (λb + ρb , α) = 0, ∀α ∈ Φ ¯ we recall the notation Φ1¯ from (1.18)).

2.5. Exercises

87

Exercise 2.8. Let g = gl(m|n) and let b be its standard Borel subalgebra with positive system Φ+ . Let b be another Borel subalgebra of g with positive system  Φ+ . Assume that b = b and −Φ+ =  Φ+ . Give explicitly a sequence of mn odd 0¯ 1¯ 0¯ 1¯ reflections that transforms b to b . Exercise 2.9. Suppose that two positive systems of a basic Lie superalgebra are related by an odd reflection α and they contain identical even positive roots. Prove that the respective Harish-Chandra homomorphisms are identical. (Hints: It suffices to prove that the images of the Harish-Chandra homomorphisms evaluated at typical weights coincide, and note that the Weyl vectors corresponding to the two positive systems differ by α.) Exercise 2.10. Let g be a finite-dimensional reductive Lie algebra. Let V (λ) be a highest weight module of g of highest weight λ, and let E be a finite-dimensional representation of g. Denote the set of weights of E by PE . Show that if L(µ) is a composition factor of V (λ) ⊗ E then µ + ρ ∈ W (λ + ρ + PE ). Exercise 2.11. Let g = gl(m|n) or osp(2|2m). Let g = g−1 ⊕ g ⊕ g1 be the Zgradation as in (2.1). For an integral weight λ let K(λ) denote the (not necessarily finite-dimensional) Kac module. Prove: (1) If there exists a singular vector of weight µ in K(λ), then µ + ρ is of the form w(λ − ∑α kα α + ρ), w ∈ W , kα ∈ Z+ , and α ∈ Φ+ ∩ g1 . (Hint: Use Exercise 2.10.) (2) If there exists a singular vector of weight µ in K(λ) and λ is typical, then µ + ρ = w(λ + ρ), for some w ∈ W . (3) K(λ) is irreducible, for λ typical. Exercise 2.12. Let g = osp(2|2m) with m > 0 and g0¯ = so(2) ⊕ sp(2m). Prove: (1) The map eα → e˘α and f α → f˘α , for α ∈ Π, given in Example 2.15, defines ¯ x, x]. a representation of osp(2|2m) on C[ξ, ξ, ¯ 2|2m 0 ∼ (2) As a g0¯ -module, we have C = L (ε1 ) ⊕ L0 (−ε1 ) ⊕ L0 (δ1 ), with high¯ and x1 , respectively. est weight vectors ξ, ξ, (3) For k ≥ 2, the g0¯ -module ∧ k (C2|2m ) decomposes as L0 ((k − 1)δ1 + ε1 ) ⊕ L0 ((k − 1)δ1 − ε1 ) ⊕ L0 (kδ1 ) ⊕ L0 ((k − 2)δ1 ), ¯ k−1 , xk , and xk−2 ξξ, ¯ respectively. with highest weight vectors ξx1k−1 , ξx 1 1 1 (4) The osp(2|2m)-module ∧ k (C2|2m ) is irreducible for all k ≥ 1. (Hint: Use (4) and the formulas for e˘ε1 −δ1 and f˘ε1 −δ1 .) In Exercises 2.13–2.16 below, g = osp(2|2m) with m > 0 and g0¯ = so(2) ⊕ sp(2m), and we follow the notation of Example 2.16. Exercise 2.13. Prove:

2. Finite-dimensional modules

88

(1) The map eα → eˆα and f α → fˆα , for α ∈ Π, given in Example 2.16, defines a representation of osp(2|2n). (2) The Laplace operator Δ=

m ∂ ∂ ∂ ∂ : Sk (C2|2m ) → Sk−2 (C2|2m ) −∑ ∂x ∂x¯ i=1 ∂ξi ∂ξ¯ i

is surjective and commutes with the action of osp(2|2m), for k ≥ 2. Exercise 2.14. Let k ≥ 2m + 1. Prove that dimSk (C2|2m ) − dimSk−2 (C2|2m ) = 22m+1 . ¯ Exercise 2.15. Let Ξ := ∑m i=1 ξi ξi and k ≥ 2m. Set   m i k−m Γ := ∑ (−1) x¯k−m−i xm−i Ξ ∈ Sk (C2|2m ). i i=0 Prove: (1) Δ(xk ) = 0 and eˆα xk = 0, for α ∈ Π. (2) Γ = 0, Δ(Γ) = 0, and eˆα Γ = 0, for α ∈ Π. Exercise 2.16. Let k ≥ 2m + 1. Prove: (1) kε1 and (2m − k)ε1 are typical weights for osp(2|2m). (2) We have an isomorphism of osp(2|2m)-modules: ker Δ ∼ = L(kε1 ) ⊕ L ((2m − k)ε1 ) . What are the highest weights of these two irreducible modules with respect to the following new fundamental system? 



δ1 − δ2

 δ2 − δ3

···



δm−1 − δ@ m

@

δm − ε1

δm + ε1

Exercise 2.17. Let λ = λ1 ε1 + λ2 ε2 with λ1 > λ2 , λ1 = −λ2 , and λ1 , λ2 ∈ Z. Write α = ε1 − ε2 . Prove: (1) There is a short exact sequence of q(2)-modules: 0 −→ Δ(λ2 ε1 + λ1 ε2 ) −→ Δ(λ) −→ L(λ) −→ 0. (2) We have the following decomposition of L(λ) as gl(2)-modules:  2L0 (λ), if λ1 − λ2 = 1, L(λ) ∼ = 2L0 (λ) ⊕ 2L0 (λ − α), if λ1 − λ2 > 1. Exercise 2.18. Let L(µ) be a finite-dimensional irreducible q(n)-module of highest weight µ = ∑ni=1 µi εi . Prove that L(µ∗ ) ∼ = L(µ)∗ , where µ∗ := − ∑ni=1 µn+1−i εi . (Hint: + If ν is a Φ0¯ -dominant integral weight of L(µ), then w0 ν−w0 µ ∈ ∑α∈Φ+¯ Z+ α, where 0 w0 is the longest element in W .)

Notes

89

Notes Section 2.1. A systematic study of the representation theory of finite-dimensional (simple) Lie superalgebras was initiated by Kac [62]. The classification of finitedimensional irreducible representations of the Lie superalgebras of types gl and spo (Propositions 2.2 and 2.2; Theorems 2.11 and 2.14) appeared first in Kac [60], where the finite dimensionality condition was formulated in terms of Dynkin labels instead of hook partitions used in this book. The proof using odd reflections in the most challenging spo type given here follows Shu-Wang [115] for the cases of integer weights, and the proof using odd reflections and the Bernstein functor (see Santos [104]) for the cases of half-integer weights here is new. Yet another closely related approach to the classification is provided in Azam-Yamane-Yousofzadeh [4] by means of Weyl groupoids. Theorem 2.18 on the classification of finitedimensional irreducible modules of q(n) was formulated by Penkov [93, Theorem 4], who proved the “only if” direction. We are not aware of a written proof for the “if” direction in the literature. Our proof uses the Bernstein functor again. Section 2.2. The Harish-Chandra homomorphisms for Lie superalgebras were first studied by Kac [62], who formulated results in terms of fractional fields. Proposition 2.22 and Lemma 2.24 (and their proofs) are taken from [61] and [62], respectively. The analogue of Chevalley theorem on invariant polynomials for simple Lie superalgebras has been established by Sergeev [111], which can be recovered as a part of Theorem 2.26 for type gl and osp. Our approach to calculating the images of the Harish-Chandra homomorphisms in this book is different and new, and it uses the connection to the theory of supersymmetric functions for types gl and spo and the theory of Schur Q-functions for type q. There is another approach developed by Gorelik [48] following Kac [63] on the centers of the universal enveloping algebras (also see Alldridge [1]). A variant of Theorem 2.30 on linkage principle for gl and osp was originally stated as a conjecture in Kac [62, §6]. The notion of typicality for basic Lie superalgebras is due to Kac [62]. Corollary 2.28 (which is due to Kac [62, Theorem 2] by different arguments) was used by Kac to derive the irreducible character formula for finite-dimensional typical modules of basic classical Lie superalgebas (see Theorem 2.35). Also see Gorelik [47]. Section 2.3. This section is the counterpart to Section 2.2 for the queer Lie superalgebra q(n). Theorem 2.46 on the image of the Harish-Chandra homomorphism for the queer Lie superalgebra was formulated without proof in a short announcement of Sergeev [109]. We fill in the details of a proof by presenting several explicit formulas of singular vectors in Verma modules. The notion of typicality for q(n) was formulated by Penkov [93]. Theorem 2.52 on the typical character formula for q(n) is due to Penkov [93, Theorem 2]. Theorem 2.48 on linkage principle for q(n) or its variant has been regarded as folklore after [109].

90

2. Finite-dimensional modules

Section 2.4. As we shall see in Chapter 3, the irreducible polynomial representations of gl(m|n) were classified by Sergeev [110] (and independently later by Berele and Regev [7]). The results on extremal weights of the irreducible polynomial representations for gl and finite-dimensional irreducible integer weight modules for spo were due to Ngau Lam and the authors (for the spo case see [24] and for the gl case see [32]). The special case for type gl when a fundamental system consists of only odd simple roots goes back to [30]. The irreducible character problem for finite-dimensional modules over Lie superalgebras has been a challenging one. For gl(m|n), there have been complete solutions by completely different approaches due to Serganova [107], Brundan [11], Brundan-Stroppel [15], and Cheng-Lam [23] (also see Cheng-Wang-Zhang [34]). There is also a combinatorial dimension formula by Su-Zhang [120]. For q(n), there has been complete solutions due to Penkov-Serganova [96] and Brundan [12]. There were numerous partial results in the literature over the years; see for example [8, 33, 123, 124, 96, 104, 110, 133]. Chapter 6 of this book offers a complete solution for a wider class of modules (including all finite-dimensional ones) of gl(m|n) and osp(k|2m). Exercise 2.12 implies that the exterior powers of the natural osp(2|2m)-module are irreducible. For general osp(k|2m) a similar argument as outlined there can be used to show the irreducibility of every exterior power of the natural module. As in the orthogonal Lie algebra case, (super)symmetric tensors are not irreducible. In contrast to the orthogonal Lie algebra, the determination of the composition factors in the symmetric powers is significantly more involved as seen in Exercises 2.13– 2.16, which are taken from [33]. In [33] it is shown that the kernel of the Laplacian in the case of osp(2|2m) can either have one, two, or three composition factors. The proof outlined in Exercise 2.11 for the irreducibility of Kac modules of a typical highest weight is taken from [80]. A different argument of this fact can be found in [62].

Chapter 3

Schur duality

Schur duality, which was popularized in Weyl’s book The Classical Groups, is an interplay between representations of the general linear Lie group/algebra and representations of the symmetric group. On the combinatorial level, it explains their mutual connections to Schur functions and Young tableaux. In this chapter, we describe Sergeev’s superalgebra generalization of Schur duality. The first generalization is a duality between the general linear Lie superalgebra and the symmetric group. This allows us to classify the irreducible polynomial representations of the general linear Lie superalgebra and obtain their character formula in terms of supertableaux and also in terms of super Schur functions. The second generalization is a duality between the queer Lie superalgebra and a twisted (or spin) group algebra Hd of the hyperoctahedral group. The representation theory of the algebra Hd is systematically developed, and its connection with symmetric functions such as Schur Q-functions is explained in detail. The irreducible polynomial representations of the queer Lie superalgebra are classified in terms of strict partitions, and their characters are shown to be the Schur Q-functions up to some 2-powers.

3.1. Generalities for associative superalgebras In this section, we classify the finite-dimensional simple associative superalgebras over C. We also formulate the superalgebra generalizations of Schur’s Lemma, the double centralizer property, and Wedderburn’s Theorem. By studying the center of a finite group superalgebra, we obtain a numerical identity relating the number of simple supermodules to the number of split conjugacy classes. 91

92

3. Schur duality

In this section, a superalgebra is always understood to be a finite-dimensional associative superalgebra with unity 1, and a module is always understood to be finite-dimensional. 3.1.1. Classification of simple superalgebras. Let V be a vector superspace with even and odd subspaces of equal dimension. Given an odd automorphism P of V of order 2, we define the following subalgebra of the endomorphism superalgebra End(V ): Q(V ) = {x ∈ End(V ) | [x, P] = 0}. In the case when V = Cn|n and P is the linear transformation in the block matrix form (1.10), we write Q(V ) as Q(n), which consists of 2n × 2n matrices of the form:   a b , b a where a and b are arbitrary n × n matrices, for n ≥ 1. Note that we have a superalgebra isomorphism Q(V ) ∼ = Q(n) by properly choosing coordinates in V , whenever dimV = (n|n). The matrix superalgebra M(m|n) is the algebra of matrices in the m|n-block form   a b , c d whose even subspace consists of the matrices with b = 0 and c = 0, and whose odd subspace consists of the matrices with a = 0 and d = 0. The matrix superalgebra M(m|n) is isomorphic to the endomorphism superalgebra of Cm|n . Recall that by an ideal I and a module M of a superalgebra A, we always mean that I and M are Z2 -graded, i.e., I = (I ∩ A0¯ ) ⊕ (I ∩ A1¯ ), and M = M0¯ ⊕ M1¯ such that Ai M j ⊆ Mi+ j , for i, j ∈ Z2 . For a superalgebra A and an A-module M, we shall denote by |A| and |M| the underlying (i.e., ungraded) algebra and module. Ideals and modules of |A| are understood in the usual (i.e., ungraded) sense. We shall denote by A-mod the category of (finite-dimensional) modules of the superalgebra A (with morphisms of degree one allowed). The underlying even subcategory A-mod0¯ , which consists of the same objects and only even morphisms of A-mod, is an abelian category. Recall the parity reversing functor Π from Section 1.1.1. We define the Grothendieck group [A-mod] of the category A-mod to be the Z-module generated by all objects in A-mod subject to the following two relations: (i) [M] = [L] + [N]; (ii) [ΠM] = −[M], for all L, M, N in A-mod satisfying a short exact sequence 0 → L → M → N → 0 with even morphisms. Denote by Z(|A|) the center of |A|. Clearly, Z(|A|) = Z(|A|)0 ⊕ Z(|A|)1 , where Z(|A|)i = Z(|A|) ∩ Ai for i ∈ Z2 . Recall a superalgebra A is simple if A has no nontrivial ideal. Theorem 3.1. Let A be a finite-dimensional simple associative superalgebra.

3.1. Generalities for associative superalgebras

93

(1) If Z(|A|)1 = 0, then A is isomorphic to the matrix superalgebra M(m|n) for some m and n. (2) If Z(|A|)1 = 0, then A is isomorphic to Q(n) for some n. We shall need the following lemma. Let pi : A → Ai be the projection for i ∈ Z2 . Lemma 3.2. Let A be a simple superalgebra. If J is a proper ideal of |A|, then the induced maps pi |J : J → Ai are isomorphisms of vector spaces and, moreover, J ∩ Ai = 0 for i ∈ Z2 . Proof. We claim that if I is a nonzero ideal of A0¯ , then (3.1)

I + A1¯ IA1¯ = A0¯ ,

and A1¯ I + IA1¯ = A1¯ . Indeed, the Z2 -graded subspace (I + A1¯ IA1¯ ) + (A1¯ I + IA1¯ ) of A is closed under left and right multiplications by elements of A0¯ and A1¯ ; hence it is an ideal of A. Now the claim follows from the simplicity of A. For a proper ideal J of |A|, J ∩ A0¯ and p0¯ (J) are ideals of A0¯ , and J ∩ A0¯ ⊆ p0¯ (J). Actually, we must have J ∩ A0¯  p0¯ (J), for otherwise J would be an ideal of A, contradicting the simplicity of A. By inspection, A1¯ (J ∩ A0¯ )A1¯ ⊆ J ∩ A0¯ and A1¯ p0¯ (J)A1¯ ⊆ p0¯ (J). It follows by (3.1) that (3.2)

J ∩ A0¯ = 0,

p0¯ (J) = A0¯ .

We have A21¯ = A0¯ , for otherwise A21¯ + A1¯ would be a proper ideal of A, which contradicts with the simplicity of A. Hence, we have (3.3)

J ∩ A1¯ = 0,

p1¯ (J) = A1¯ ,

by the following computations (which use (3.2) and that 1 ∈ A0¯ ): J ∩ A1¯ = A0¯ (J ∩ A1¯ ) = A21¯ (J ∩ A1¯ ) ⊆ A1¯ (J ∩ A0¯ ) = 0, p1¯ (J) ⊇ A1¯ p0¯ (J) = A1¯ A0¯ = A1¯ . Now the lemma follows from (3.2) and (3.3).



Proof of Theorem 3.1. We claim that Z(|A|)0¯ = C. To see this, let z ∈ Z(|A|)0¯ . The map z : A → A defined by left multiplication by z has an eigenvalue c ∈ C, and the kernel of z − c is a nonzero 2-sided ideal of A. The simplicity of A implies that z − c = 0, and hence z = c · 1. (1) Assume Z(|A|)1 = 0. We claim that |A| is a simple algebra (and so |A| is a matrix algebra by a classical structure theorem of simple algebras). Indeed, assume |A| is a non-simple algebra with a proper ideal J. Denote w = p−1 (1) ∈ J 0¯ and u = p1¯ (w) ∈ A1¯ , where u = 0 by Lemma 3.2. Then, w = 1 + u. For any homogeneous element x ∈ Ai for some i ∈ Z2 , we have xw = x+xu and wx = x+ux, and so pi (xw) = x = pi (wx). By Lemma 3.2, xw = wx and so xu = ux; that is, u ∈ Z(|A|)1 . It follows by assumption that u = 0, which is a contradiction.

3. Schur duality

94

Consider the automorphism σ : |A| → |A| given by σ(ai ) = (−1)i ai for ai ∈ Ai and i ∈ Z2 . There exists u ∈ |A| such that σ(a) = uau−1 for all a ∈ |A|, since every automorphism of a matrix algebra is inner. We have u ∈ A0¯ because σ(u) = u. It follows from σ2 = 1 that u2 ∈ Z(|A|)0¯ = C. We can choose u such that u2 = 1 and then, by a change of basis, take u to be a diagonal matrix with m 1’s followed by n (−1)’s in the main diagonal. Hence, we conclude that A ∼ = M(m|n). (2) For a nonzero element w ∈ Z(|A|)1 , we have w2 ∈ Z(|A|)0 = C. We must have w2 = 0, for otherwise the annihilator of w is a proper ideal of the superalgebra A, contradicting the simplicity of A. Then we may assume w2 = 1. We have Z(|A|)1 = (Z(|A|)1 w)w ⊆ Z(|A|)0 w = Cw ⊆ Z(|A|)1 , and hence, Z(|A|) = C + Cw. Moreover, A1¯ = (A1¯ w)w ⊆ A0¯ w ⊆ A1¯ , i.e., A1¯ = A0¯ w. The algebra A0¯ must be simple, for otherwise a proper ideal I of A0¯ would give rise to a proper ideal I + Iw of A (which is again a contradiction). Then A0¯ is a matrix algebra M(n) for some n, and the superalgebra A = M(n) ⊕ M(n)w is isomorphic to Q(n).  3.1.2. Wedderburn Theorem and Schur’s Lemma. The basic results of finitedimensional semisimple (unital associative) algebras over C admit natural super generalizations. A module M of a superalgebra A is called semisimple if every A-submodule of M is a direct summand of M. The following are straightforward generalizations of the classical structure results on semisimple modules and algebras, and they can be proved in the same way as in the non-super setting. Theorem 3.3 (Super Wedderburn Theorem). The following statements are equivalent for a superalgebra A: (1) Every A-module is semisimple. (2) The left regular module A is a direct sum of minimal left ideals. (3) A is a direct sum of simple superalgebras. A superalgebra A is called semisimple if it satisfies one of the three equivalent conditions (1)–(3) in Theorem 3.3. By Theorems 3.1 and 3.3, a finite-dimensional semisimple superalgebra A is isomorphic to a direct sum of simple superalgebras as follows: (3.4)

A∼ =

m  i=1

M(ri |si ) ⊕

q 

Q(n j ),

j=1

where m = m(A) and q = q(A) are invariants of A. Observe that Cr|s is a simple module (unique up to isomorphism) of the simple superalgebra M(r|s), and Cn|n is a simple module (unique up to isomorphism) of the simple superalgebra Q(n). A simple A-module V is annihilated by all but one of the summands in (3.4). We say V is of type M if this summand is of the form M(ri |si ) and of type Q if this summand

3.1. Generalities for associative superalgebras

95

is of the form Q(n j ). These two types of simple modules are distinguished by a super analogue of Schur’s Lemma. Lemma 3.4 (Super Schur’s Lemma). If M and L are simple modules over a superalgebra A, then ⎧ = L is of type M, ⎨ 1 if M ∼ 2 if M ∼ dim HomA (M, L) = = L is of type Q, ⎩ 0 if M ∼ = L. Proof. In the case when M ∼ = L is of type M, we can assume that A ∼ = M(r|s) for r|s some r, s, and that M = L = C . Then it is straightforward to check (as in the non-super case) that HomM(r|s) (Cr|s , Cr|s ) = CI. In the case when M ∼ = L is of type Q, we can assume that A ∼ = Q(n) for some n, and that M = L = Cn|n . Then one checks that HomQ(n) (Cn|n , Cn|n ) = CI + CP, where P is the linear transformation in the block matrix form (1.14). The case where M ∼  L is clear.  =

3.1.3. Double centralizer property for superalgebras. Given two associative superalgebras A and B, the tensor product A ⊗ B is naturally a superalgebra, with multiplication defined by (3.5)



(a ⊗ b)(a ⊗ b ) = (−1)|b|·|a | (aa ) ⊗ (bb )

(a, a ∈ A, b, b ∈ B).

Note that we have a superalgebra isomorphism (see Exercise 3.10) Q(m) ⊗ Q(n) ∼ = M(mn|mn). Hence, as a Q(m) ⊗ Q(n)-module, the tensor product Cm|m ⊗ Cn|n is a direct sum of two isomorphic copies of a simple module (which is ∼ = Cmn|mn ), and we have HomQ(n) (Cn|n , Cmn|mn ) ∼ = Cm|m as a Q(m)-module. Let A and B be semisimple superalgebras. Let M be a simple A-module of type Q and let N be a simple B-module of type Q. Then, the A ⊗ B-module M ⊗ N is a direct sum of two isomorphic copies of a simple module of type M, denoted by 2−1 M ⊗ N. Moreover, HomB (N, 2−1 M ⊗ N) is naturally an A-module, which is isomorphic to the A-module M. Proposition 3.5. Suppose that W is a finite-dimensional vector superspace, and B is a semisimple subalgebra of End(W ). Let A = EndB (W ). Then, EndA (W ) = B. As an A ⊗ B-module, W is multiplicity-free, i.e., W∼ = ∑ 2−δi Ui ⊗Vi , i

where δi ∈ {0, 1}, {Ui } are pairwise non-isomorphic simple A-modules, and {Vi } are pairwise non-isomorphic simple B-modules. Moreover, Ui and Vi for a given i are of the same type, and they are of type M if and only if δi = 0.

3. Schur duality

96

Proof. Assume that Va are all the pairwise non-isomorphic simple B-modules of type M, and Vb are all the pairwise non-isomorphic simple B-modules of type Q. Then the hom-spaces Ua := HomB (Va ,W ) and Ub := HomB (Vb ,W ) are naturally A-modules. By the super Schur’s Lemma, we know dimUb = (kb |kb ) for some kb . By the semisimplicity assumption on B, we have an isomorphism of B-modules W∼ =

 a

Ua ⊗Va ⊕



2−1Ub ⊗Vb .

b

By applying the super Schur’s Lemma, we obtain the following superalgebra isomorphisms: A = EndB (W ) ∼ =



EndB (Ua ⊗Va ) ⊕

a

∼ =

 

EndB (2−1Ub ⊗Vb )

b

End(Ua ) ⊗ IVa ⊕

a

∼ =





(End(2−1Ub ) ⊗ Q(1)) ⊗ IVb

b

End(Ua ) ⊗ IVa ⊕



a

Q(Ub ) ⊗ IVb .

b

Hence, A is a semisimple superalgebra, Ua are all the pairwise non-isomorphic simple A-modules of type M, and Ub are all the pairwise non-isomorphic simple A-modules of type Q. Since A is now a semisimple superalgebra, we can reverse the roles of A and B in the above computation and obtain the following superalgebra isomorphisms: EndA (W ) ∼ =

 a

The proposition is proved.

IUa ⊗ End(Va ) ⊕



IUb ⊗ Q(Vb ) ∼ = B.

b



3.1.4. Split conjugacy classes in a finite supergroup. For a finite group G, the group algebra CG is a semisimple algebra. It follows by using the Wedderburn Theorem and comparing two different bases for the center of CG that the number of simple G-modules coincides with the number of conjugacy classes of G. Now assume that the finite group G contains a subgroup G0 of index 2. We call elements in G0 even and elements in G\G0 odd. Then the group algebra of G is naturally a superalgebra and we shall denote it by C[G, G0 ] = CG0 ⊕ C[G\G0 ] to make clear its Z2 -grading and its dependence on G0 . Just as for the usual group algebras, it is standard to show that C[G, G0 ] is a semisimple superalgebra. Since elements in a given conjugacy class of G share the same parity (i.e., Z2 -grading), it makes sense to talk about even and odd conjugacy classes. Proposition 3.6. (1) The number of simple C[G, G0 ]-modules coincides with the number of even conjugacy classes of G. (2) The number of simple C[G, G0 ]-modules of type Q coincides with the number of odd conjugacy classes of G.

3.1. Generalities for associative superalgebras

97

Proof. Set A = C[G, G0 ]. By (3.4), A can be decomposed as a direct sum of simple superalgebras: A∼ =

m 

M(ri |si ) ⊕

i=1

q 

Q(n j ).

j=1

The center of the algebra |(M(r|s)| is ∼ the center of the algebra |Q(n)| = CI,  and  0 In is of dimension (1|1), spanned by I2n and . Hence, dim Z(|A|)0¯ = m + q In 0 and dim Z(|A|)1¯ = q. On the other hand, note that |A| is the usual group algebra CG. Hence the class sums for the even (respectively, odd) conjugacy classes of G form a basis of Z(|A|)0¯ (respectively, Z(|A|)1¯ ). The proposition is proved.   that contains a subgroup G 0 of index 2, let us further For a finite group G  contains a central element z that is even and of order 2. Setting assume that G  G = G/{1, z}, we have a short exact sequence of groups θ  −→ 1 −→ {1, z} −→ G G −→ 1.

(3.6)

Let C be a conjugacy class of G. Depending on whether an element x in θ−1 (C)  or it splits into is conjugate to z x, θ−1 (C) is either a single conjugacy class of G  two conjugacy classes of G (see Exercise 3.2). In the latter case, C is called a split conjugacy class, and either conjugacy class in θ−1 (C) will also be called split. An element x ∈ G is called split if the conjugacy class of x is split. If we denote θ−1 (x) = {x, ˜ zx}, ˜ then x is split if and only if x˜ is not conjugate to zx. ˜ − G 0 ] by the ideal generated Denote by CG the quotient superalgebra of C[G,  by z + 1. Now z acts as ±I on any simple G-module by Schur’s Lemma. Thus, we G 0 ]-modules have an isomorphism of left C[G, G 0 ] ∼ C[G, = C[G, G0 ]

(3.7)



CG− ,

according to the eigenvalues of z. Since z is central, (3.7) is an isomorphism of superalgebras as well. Hence, CG− is naturally a semisimple superalgebra. A spin G 0 ]-module M means a Z2 -graded module over the superalgebra C[G, G 0 ] on C[G,  which z acts by −I, and we will sometimes simply refer to it as a spin G-module   (with Z2 -grading implicitly assumed). A spin C[G, G0 ]-module is then naturally identified as a CG− -module.  n (nontrivial for n ≥ 4) of the symmetric group Sn Example 3.7. A double cover S was constructed by Schur (see [57]). The subgroup of Sn of index 2 in this case is  the alternating group An . The quotient algebra CS− n = CSn /z + 1 by the ideal generated by (z + 1) is call the spin symmetric group algebra. The algebra CS− n is an algebra generated by t1 ,t2 , . . . ,tn−1 subject to the relations: ti2 = 1,

titi+1ti = ti+1titi+1 ,

tit j = −t j ti ,

|i − j| > 1.

3. Schur duality

98

 n can be obtained from the above formulas by keeping the (A presentation for S first two relations and replacing the third one by tit j = zt j ti .) The algebra CS− n is a superalgebra with each ti being odd, for 1 ≤ i ≤ n − 1.  will be presented in Section 3.3.1. Another major example of a double cover G G 0 ]-modules equals the Proposition 3.8. (1) The number of simple spin C[G, number of even split conjugacy classes of G. G 0 ]-modules of type Q equals the number (2) The number of simple spin C[G, of odd split conjugacy classes of G. G 0 ], and further denote by m¯ (respectively, q) Proof. Let us denote by A = C[G, ¯ the number of simple spin A-modules of type M (respectively, type Q). The central element z acts on Z(|A|) by multiplication with eigenvalues ±1, and the (−1)eigenspace Z− (|A|) can be naturally identified with Z(|CG− |); see (3.7). Hence, G 0 ] as a direct sum of simples, we can using (3.7) and a decomposition of C[G, show by a similar argument as for Proposition 3.6 that (3.8)

dim Z− (|A|)0¯ = m¯ + q, ¯

dim Z− (|A|)1¯ = q. ¯

 as follows: D1 , zD1 , . . . , D , zD , D+1 , . . . , Dk , List all conjugacy classes of G where Di ∩ zDi = 0/ for 1 ≤ i ≤  and zDi = Di for  + 1 ≤ i ≤ k. The corresponding class sums, d1 , zd1 , . . . , d , zd , d+1 , . . . , dk , form a basis for Z(|A|). It follows that a basis for Z− (|A|) is {di − zdi | 1 ≤ i ≤ }, and thus, dim Z− (|A|) = , which is the number of split conjugacy classes of G. The proposition follows by a division of the split conjugacy classes by parity and a comparison with (3.8). 

3.2. Schur-Sergeev duality of type A In this section, we formulate the Schur-Sergeev duality as a double centralizer property for the commuting actions of the Lie superalgebra gl(V ) and the symmetric group Sd on the tensor space V ⊗d , where V = Cm|n . We obtain a multiplicityfree decomposition of V ⊗d as a U(gl(V )) ⊗ CSd -module. The character formula for the irreducible gl(V )-modules arising this way is given in terms of super Schur polynomials. We then provide a Young diagrammatic interpretation of the degree of atypicality of the weight λ . We further show that the category of polynomial gl(m|n)-modules is semisimple. 3.2.1. Schur-Sergeev duality, I. Let g = gl(m|n) and let V = Cm|n be the natural g-module. Then V ⊗d is naturally a g-module by letting Φd (g).(v1 ⊗ v2 ⊗ . . . ⊗ vd ) =g.v1 ⊗ . . . ⊗ vd + (−1)|g|·|v1 | v1 ⊗ g.v2 ⊗ . . . ⊗ vd + . . . + (−1)|g|·(|v1 |+...+|vd−1 |) v1 ⊗ v2 ⊗ . . . ⊗ g.vd , where g ∈ g and vi ∈ V is assumed to be Z2 -homogeneous for all i.

3.2. Schur-Sergeev duality of type A

99

On the other hand, let Ψd ((i, i + 1)).v1 ⊗ . . . ⊗ vi ⊗ vi+1 ⊗ . . . ⊗ vd (3.9)

= (−1)|vi |·|vi+1 | v1 ⊗ . . . ⊗ vi+1 ⊗ vi ⊗ . . . ⊗ vd ,

1 ≤ i ≤ d − 1,

where (i, j) denotes a transposition in Sd and vi , vi+1 are Z2 -homogeneous. Lemma 3.9. The formula (3.9) defines a left action of the symmetric group Sd on V ⊗d . The actions of (gl(m|n), Φd ) and (Sd , Ψd ) on V ⊗d commute with each other. Proof. It is straightforward to check that the formula (3.9) satisfies the Coxeter relations for Sd (see Exercise 3.5), and hence it defines a left action of Sd . The verification of the commuting action boils down to the case below when d = 2 and i = 1 (where we let s = (1, 2) ∈ S2 , g ∈ gl(m|n), and v1 , v2 ∈ V ): Ψd (s)Φd (g)(v1 ⊗ v2 ) = Ψd (s)(g.v1 ⊗ v2 + (−1)|g|·|v1 | v1 ⊗ g.v2 ) = (−1)(|g|+|v1 |)|v2 | v2 ⊗ g.v1 + (−1)|g|·|v1 |+|v1 |(|g|+|v2 |) g.v2 ⊗ v1 , Φd (g)Ψd (s)(v1 ⊗ v2 ) = (−1)|v1 |·|v2 | (g.v2 ⊗ v1 + (−1)|g|·|v2 | v2 ⊗ g.v1 ). Clearly, Ψd (s)Φd (g) = Φd (g)Ψd (s).



We are going to formulate a superalgebra analogue of Schur duality. Theorem 3.10 (Schur-Sergeev duality, Part I). Let g = gl(m|n). The images of Φd and Ψd , Φd (U(g)) and Ψd (CSd ), satisfy the double centralizer property, i.e., Φd (U(g)) =EndSd (V ⊗d ), EndU(g) (V ⊗d ) = Ψd (CSd ). Proof. By Lemma 3.9, we have Φd (U(g)) ⊆ EndSd (V ⊗d ). We shall proceed to prove that Φd (U(g)) ⊇ EndSd (V ⊗d ).

We have a natural isomorphism of vector superspaces End(V )⊗d ∼ = End(V ⊗d ), which is then made Sd -equivariant, as both superspaces admit natural Sd -actions. This isomorphism allows us to identify EndSd (V ⊗d ) ≡ Symd (End(V )), the space of Sd -invariants in End(V )⊗d . Denote by Yk , 1 ≤ k ≤ d, the C-span of the super-symmetrization ϖ(x1 , . . . , xk ) :=

∑

σ∈Sd

σ.(x1 ⊗ . . . ⊗ xk ⊗ I ⊗d−k ),

for all xi ∈ End(V ). Note that Yd = Symd (End(V )) ≡ EndSd (V ⊗d ). Let x˜ = Φd (x) = ∑di=1 I ⊗i−1 ⊗ x ⊗ I ⊗d−i , for x ∈ g = End(V ), and denote by Xk , 1 ≤ k ≤ d, the C-span of x˜1 . . . x˜k for all xi ∈ End(V ). Claim. We have Yk ⊆ Xk for 1 ≤ k ≤ d.

3. Schur duality

100

Assuming the claim is true (in particular for k = d), then the theorem follows thanks to EndSd (V ⊗d ) = Yd ⊆ Xd ⊆ Φd (U(g)). We will prove the claim by induction on k. The case k = 1 holds, thanks to ϖ(x) = (d − 1)!x. ˜ Assuming that Yk−1 ⊆ Xk−1 , we have ϖ(x1 , . . . , xk−1 ) · x˜k ∈ Xk .

(3.10) On the other hand, we have ϖ(x1 , . . . , xk−1 ) · x˜k =

∑

σ∈Sd

σ.(x1 ⊗ . . . ⊗ xk−1 ⊗ I ⊗d−k+1 ) · x˜k

d

=

∑ ∑

j=1 σ∈Sd

  σ. (x1 ⊗ . . . ⊗ xk−1 ⊗ I ⊗d−k+1 ) · (I ⊗ j−1 ⊗ xk ⊗ I ⊗d− j ) ,

which can be written as a sum A1 + A2 , where k−1

A1 =

∑ ϖ(x1, . . . , x j xk , . . . , xk−1 ) ∈ Yk−1 ,

j=1

and d

A2 =

∑ ∑

j=k σ∈Sd

σ.(x1 ⊗ . . . ⊗ xk−1 ⊗ I ⊗ j−k ⊗ xk ⊗ I ⊗d− j )

= (d − k + 1)ϖ(x1 , . . . , xk−1 , xk ). Note that A1 ∈ Xk , since Yk−1 ⊆ Xk−1 ⊆ Xk . Hence (3.10) implies that A2 ∈ Xk , and so, Yk ⊆ Xk . This proves the claim. Hence, Φd (U(g)) = EndSd (V ⊗d ) = EndB (V ⊗d ), for B := Ψd (CSd ). Note that B is a semisimple algebra, and so the assumption of Proposition 3.5 is satisfied. Therefore, we have EndU(g) (V ⊗d ) = Ψd (CSd ).  3.2.2. Schur-Sergeev duality, II. Recall from Definition 2.10 that an (m|n)-hook partition µ = (µ1 , µ2 , . . .) is a partition with µm+1 ≤ n. Given an (m|n)-hook partition µ, we denote by µ+ = (µm+1 , µm+2 , . . .) and write its conjugate partition, which is necessarily of length ≤ n, as ν = (µ+ ) = (ν1 , . . . , νn ). We recall the weight defined in (2.4): µ = µ 1 δ 1 + . . . + µ m δ m + ν 1 ε 1 + . . . + ν n ε n . Denote by Pd (m|n) the set of all (m|n)-hook partitions of size d and let P(m|n) = ∪d≥0 Pd (m|n) = {λ | λ is a partition with λm+1 ≤ n}.

3.2. Schur-Sergeev duality of type A

101

In particular, Pd (m) = Pd (m|0) is the set of partitions of d of length at most m. Accordingly, we let Pd = {λ | λ is a partition of d}. We denote by L(λ ) for λ ∈ P(m|n) the simple g-module of highest weight λ with respect to the standard Borel subalgebra. For a partition λ of d, we denote by Sλ the Specht module of Sd . It is well known that {Sλ | λ ∈ Pd } is a complete list of simple Sd -modules. For example, S(d) is the trivial representation 1d , and d S(1 ) = sgnd is the sign representation. Theorem 3.11 (Schur-Sergeev duality, Part II). As a U(gl(m|n)) ⊗ CSd -module, we have  (Cm|n )⊗d ∼ L(λ ) ⊗ Sλ . = λ∈Pd (m|n)

We will refer to a U(gl(m|n)) ⊗ CSd -module as a (gl(m|n), Sd )-module. Similar conventions also apply to similar setups later on. Proof. The proof of this theorem will occupy most of Section 3.2.2. Let V = Cm|n with standard basis {ei |i ∈ I(m|n)}, and let W = V ⊗d . Note that Ψd (CSd ) is a semisimple algebra and all its simple modules are of type M. By Proposition 3.5 and Theorem 3.10, we have a multiplicity-free decomposition of the (gl(m|n), Sd )module W of the following form: W ≡ V ⊗d ∼ =



L[λ] ⊗ Sλ ,

λ∈Pd (m|n)

where L[λ] is some simple gl(m|n)-module associated to λ, whose highest weight (with respect to the standard Borel) is to be determined. Also to be determined is the index set Pd (m|n) = {λ & d | L[λ] = 0}. First we need to prepare some notation. Let CP(m|n) be the set of pairs ν|µ of compositions ν = (ν1 , . . . , νm ) and µ = (µ1 , . . . , µn ), and let CPd (m|n) = {ν|µ ∈ CP(m|n) | ∑ νi + ∑ µ j = d}. i

j

We have the following weight space decomposition with respect to the Cartan subalgebra of diagonal matrices h ⊂ gl(m|n): (3.11)

W=



Wν|µ ,

ν|µ∈CPd (m|n)

where Wν|µ has a linear basis ei1 ⊗ . . .⊗ eid , with the indices satisfying the following equality of multisets: {i1 , . . . , id } = {1, . . . , 1, . . . , m, . . . , m; 1, . . . , 1, . . . , n, . . . , n}.             ν1

νm

µ1

µn

3. Schur duality

102

Introduce the following Young subgroup of Sd : Sν|µ = Sν1 × . . . × Sνm × Sµ1 × . . . × Sµn . ⊗µ

⊗µ

1 n m 1 The span of the vector eν|µ := e⊗ν ⊗ . . . ⊗ e⊗ν can be identim ⊗ e1 ⊗ . . . ⊗ en 1 fied with the one-dimensional Sν|µ -module 1ν ⊗ sgnµ . Here 1ν is the trivial Sν module and sgnµ is the Sµ -module sgnµ1 ⊗ · · ·⊗ sgnµn . Since the orbit Sd eν|µ spans d Wν|µ , we have a surjective Sd -homomorphism from IndS Sν|µ (1ν ⊗sgnµ ) onto Wν|µ by Frobenius reciprocity. By counting the dimensions we have an Sd -isomorphism:

(3.12)

S Wν|µ ∼ = IndSdν|µ (1ν ⊗ sgnµ ).

Let us denote the decomposition of the Sd -module Wν|µ into irreducibles by Wν|µ ∼ =



Kλ,ν|µ Sλ ,

for Kλ,ν|µ ∈ Z+ .

λ

Let λ be a partition that is identified with its Young diagram. Recall I(m|n) is totally ordered by (1.3). A supertableau T of shape λ, or a super λ-tableau T , is an assignment of an element in I(m|n) to each box of the Young diagram λ satisfying the following conditions: (HT1) The numbers are weakly increasing along each row and column. (HT2) The numbers from {1, . . . , m} are strictly increasing along each column. (HT3) The numbers from {1, . . . , n} are strictly increasing along each row. Such a supertableau T is said to have content ν|µ ∈ CP(m|n) if i¯ (1 ≤ i ≤ m) appears νi times and j (1 ≤ j ≤ n) appears µ j times. Denote by HT(λ, ν|µ) the set of super λ-tableaux of content ν|µ. In the case n = 0, a supertableau becomes a usual (semistandard) tableau. Lemma 3.12. We have Kλ,ν|µ = #HT(λ, ν|µ). d ∼ Proof. Recall that IndS Sν|µ (1ν ⊗ sgnµ ) =



λ Kλ,ν|µ S

λ.

/ and we prove the formula by induction on the length First assume that µ = 0, r = (ν). A (semistandard) tableau T of shape λ and content ν gives rise to a sequence of partitions 0/ = λ0 ⊂ λ1 ⊂ . . . ⊂ λr = λ such that λi has the shape given by the parts of T with entries ≤ i, and λi /λi−1 has νi boxes for each i. This sets up a bijection between HT(λ, ν) and the set of such sequences of partitions. Denote  Sd d1 = d − νr and  ν = (ν1 , . . . , νr−1 ). We have IndS1 1d1 ∼ = ρ&d1 Kρ,ν Sρ , where ν Kρ,ν = #HT(ρ,  ν) by the induction hypothesis. Now the induction step is simply the following representation theoretic version of Pieri’s formula (A.22) (which can be seen using the Frobenius characteristic map and (A.26)): for a partition ρ & d1 , d IndS Sd

×Sνr (S 1

ρ

⊗ 1νr ) ∼ =

 λ

Sλ ,

3.2. Schur-Sergeev duality of type A

103

where λ is such that λ/ρ is a horizontal strip of νr boxes (that is, λ/ρ is a skew diagram of size νr whose columns all have length at most one). / as the initial step, we complete Then using the above special case (for µ = 0) the proof in the general case by induction on the length of µ, in which the induction step is exactly the conjugated Pieri’s formula.  Lemma 3.13. Let λ ∈ Pd and ν|µ ∈ CPd (m|n). Then Kλ,ν|µ = 0 unless λ ∈ Pd (m|n). Proof. By the identity Kλ,ν|µ = #HT(λ, ν|µ), it suffices to prove that if a super λ-tableau T of content ν|µ exists, then λm+1 ≤ n. By applying the supertableau condition (HT2) to the first column of T , we see that the first entry k ∈ I(m|n) in row (m + 1) satisfies k > 0. Applying the supertableau condition (HT3) to the (m + 1)st row, we conclude that λm+1 ≤ n.  Recall that Pd (m|n) = {λ & d | L[λ] = 0}. It follows by Lemma 3.13 that Pd (m|n) ⊂ Pd (m|n). On the other hand, given λ ∈ Pd (m|n), clearly a super λtableau exists; e.g., we can fill in the numbers 1, . . . , m on the first m rows of λ row-by-row downward, and then for the remaining subdiagram of λ, we fill in the numbers 1, . . . , n column-by-column from left to right. This distinguished super λ-tableau will be denoted by Tλ . Hence, we have proved that Pd (m|n) = Pd (m|n). 

[λ]

For a given λ ∈ Pd (m|n), we have L[λ] = ν|µ∈CPd (m|n) Lν|µ . Among all the contents of super λ-tableaux, the one for Tλ corresponds to a highest weight relative to the standard Borel subalgebra of g by the supertableau conditions (HT1-3). Hence we conclude that L[λ] = L(λ ), the simple g-module of highest weight λ . This completes the proof of Theorem 3.11.  Remark 3.14. (1) For n = 0, the Schur-Sergeev duality reduces to the usual Schur duality. If in addition d = 2, then (Cm )⊗2 = S2 (Cm )⊕∧2 (Cm ). This fits well with the well-known fact that, as gl(m)-modules, S2 (Cm ) and ∧2 (Cm ) are, respectively, irreducible of highest weights 2δ1 and δ1 + δ2 . (2) If d ≤ mn + m + n, then Pd (m|n) is the set of all partitions of d, and every simple Sd -module appears in the Schur-Sergeev duality decomposition. (3) For d = 2, the Schur-Sergeev duality reads that (Cm|n )⊗2 = S2 (Cm|n ) ⊕ ∧2 (Cm|n ), where we recall that S2 and ∧2 are understood in the super sense. In particular, as an ordinary vector space, S2 (Cm|n ) = S2 (Cm ) ⊕ (Cm ⊗ Cn ) ⊕ ∧2 (Cn ).

3. Schur duality

104

3.2.3. The character formula. Let x = {x1 , . . . , xm } and y = {y1 , . . . , yn } be two sets of independent variables. We shall compute the character of the gl(m|n)module L(λ ):

∑

E

Em¯ m¯ E11 chL(λ ) := tr |L(λ ) x1 1¯ 1¯ . . . xm y1 . . . yEn nn =

dim L(λ )ν|µ ∏ xiνi y j j . µ

ν|µ∈CP(m|n)

i, j

Recall that mν denotes the monomial symmetric function associated to a partition ν (cf. Appendix A), and that hsν denotes the super Schur function (A.37). Theorem 3.15. Let λ be an (m|n)-hook partition, i.e., a partition with λm+1 ≤ n. Let x = {x1 , . . . , xm } and y = {y1 , . . . , yn }. Then the following character formula holds: chL(λ ) = hsλ (x; y). Proof. It follows from Theorem 3.11 and (3.11) that   Wν|µ ∼ L(λ ) ⊗ Sλ . = ν|µ∈CPd (m|n)

λ∈Pd (m|n) E

Em¯ m¯ yE11 . . . yEnn and Let us apply simultaneously the trace operator tr |L(λ ) x1 1¯ 1¯ . . . xm n 1 the Frobenius characteristic map chF (see Appendix A.1.4) to both sides of the above isomorphism. Then, summing over all d ≥ 0 and using (3.12) and (A.26), we have that

∑

ν|µ∈CP(m|n)

mν (x)mµ (y)hν (z)eµ (z) =

∑

λ∈P(m|n)

chL(λ )sλ (z),

where z = {z1 , z2 , . . .} is infinite. By the identities in (A.11), we have

∑

ν|µ∈CP(m|n)

∞

m

n

1 + y j zk = ∑ hsλ (x; y)sλ (z). k=1 i=1 j=1 1 − xi zk λ∈P(m|n)

mν (x)mµ (y)hν (z)eµ (z) = ∏ ∏ ∏

Now the theorem follows by comparing the above two equations and noting the linear independence of sλ (z).  Given a super λ-tableau T of content ν|µ ∈ CP(m|n), we denote by νm 1 (x|y)T := x1ν1 x2ν2 · · · xm y1 · · · yµnn . µ

We have the following alternative character formula. Theorem 3.16. Let λ be an (m|n)-hook partition. Then, (3.13)

chL(λ ) = ∑(x|y)T , T

where the summation is taken over all super λ-tableaux T . Also, we have (3.14)

chL(λ ) = ∑ #HT(λ, ν|µ) mν (x)mµ (y), ν,µ

where the summation is over partitions ν and µ of length at most m and n, respectively, such that |ν| + |µ| = d.

3.2. Schur-Sergeev duality of type A

105

Proof. The formula (3.13) is simply a reformulation of Theorem 3.15 by the definition of super Schur functions in (A.37) and the combinatorial formula for (skew) Schur functions (A.19). Let us collect the same monomials together in the sum (3.13). Regarding L(λ ) as a module over gl(m) × gl(n), we observe that chL(λ ) is symmetric with respect to x1 , . . . , xm and symmetric with respect to y1 , . . . , yn . Hence, we must have #HT(λ,  ν| µ) = #HT(λ, ν|µ) if  ν and  µ are compositions obtained by rearranging the parts from ν and µ, respectively. Now (3.14) follows from this observation and (3.13).  Corollary 3.17. The following weight multiplicity formula for the module L(λ ) holds: dim L(λ )ν|µ = #HT(λ, ν|µ). Example 3.18. The standard basis vectors for S2 (Cm|n ) are ei ⊗ e j + (−1)|i|| j| e j ⊗ ei , where i, j ∈ I(m|n) satisfy i ≤ j < 0, i < 0 < j, or 0 < i < j (cf. Remark 3.14 (4)). They are in bijection with the supertableaux of shape λ = (2): i

j

This is compatible with the isomorphism S2 (Cm|n ) ∼ = L(gl(m|n), 2δ1 ). 3.2.4. The classical Schur duality. We sketch below a more standard argument for the standard Schur duality on W = (Cn )⊗d , which emphasizes the decomposition of W as a gl(n)-module instead of as an Sd -module. Given a composition (or a partition) µ of d of length ≤ n, we denote by Wµ the µ-weight space of the gl(n)-module. Clearly Wµ has a basis (3.15)

e i1 ⊗ . . . ⊗ e id ,

where {i1 , . . . , id } = {1, . . . , 1, . . . , n, . . ., n}.       µ1

As a (gl(n), Sd )-module,

W∼ =



µn

L(λ) ⊗U λ ,

λ

∼ W n+ (the space of highest weight vectors in W := Homgl(n) (L(λ),W ) = λ of weight λ). Only λ ∈ Pd (n) can be highest weights of gl(n)-modules that appear in the decomposition of (Cn )⊗d , and every such λ indeed appears as L(λ) is clearly   a summand of the submodule ∧λ1 (Cn ) ⊗ ∧λ2 (Cn ) ⊗ · · · of W . where U λ

Note that CPd (n) has two interpretations: one as the polynomial weights for gl(n) and the other as compositions of d in at most n parts. A remarkable fact is that the partial order on weights induced by the positive roots of gl(n) coincides with the dominance partial order ≥ on compositions.

3. Schur duality

106

Since W =



µ Wµ

=



µ,λ:λ≥µ L(λ)µ ⊗U

Wµ ∼ =

(3.16)



λ , we conclude that, as an S

d -module,

dim L(λ)µU λ .

λ≥µ

On the other hand, Sd acts on the basis (3.15) of Wµ transitively, and the stabilizer ⊗µ ⊗µ of the basis element e1 1 ⊗ . . . ⊗ en n is the Young subgroup Sµ . Therefore we have  ∼ IndSd 1µ = (3.17) Wµ = Kλµ Sλ , Sµ

λ≥µ

where Kλµ is the Kostka number that satisfies Kλλ = 1. It is well known that Kλµ is equal to the number of semistandard λ-tableaux of content µ. By the double centralizer property (see Proposition 3.5 and Theorem 3.10), U λ has to be an irreducible Sd -module for each λ. We compare the interpretations (3.16) and (3.17) of Wµ in the special case when µ is dominant (i.e., a partition). One by one downward along the dominance order starting with µ = (d), this provides the identification U µ = Sµ for every µ, and moreover, we obtain the well-known equality dim L(λ)µ = Kλµ . 3.2.5. Degree of atypicality of λ . In this section, we provide a Young diagrammatic interpretation of the degree of atypicality of the weight λ , for an (m|n)-hook partition λ. Up to a shift by − 12 (m + n + 1)1m|n (which is irrelevant in all applications), the Weyl vector ρ for the standard positive system of gl(m|n) from Lemma 1.22 can be written as m

n

i=1

j=1

ρ = ∑ (m − i + 1)δi − ∑ jε j .

(3.18)

We introduce an integer iλ , with 0 ≤ iλ ≤ min{m, n}, to stand for the smallest nonnegative integer i such that the (m − i, n − i)-th box belongs to the diagram λ. Example 3.19. Let λ = (7, 4, 2, 2, 1, 1) with m = 4 and n = 5. Then iλ is the number of (m − i, n − i) boxes that do not lie in the diagram of λ, for i = 0, 1, . . . , min{m, n}. Such boxes are marked with crosses in this example. So iλ = 2. n n − iλ m − iλ m

@ @ @ @

3.2. Schur-Sergeev duality of type A

107

Lemma 3.20. For 0 ≤ j ≤ iλ − 1, we have λm− j ≤ n − iλ ≤ λm−iλ . Proof. The lemma is clearly equivalent to the claim that λm−iλ +1 ≤ n − iλ ≤ λm−iλ . The latter is evident from the diagram in Example 3.19, as iλ is equal to the number of boxes of coordinates (m − i, n − i) that do not lie in the diagram of λ, for i = 0, 1, . . ., min{m, n}.  Recall the degree of atypicality #µ for a weight µ ∈ h∗ from Definition 2.29. Proposition 3.21. Let λ be an (m|n)-hook partition. Then we have iλ = #λ . Proof. We compute by (3.18) that (3.19)

m

n

i=1

j=1

λ + ρ = ∑ (λi + m − i + 1)δi − ∑ ( j − ν j )ε j .

We observe that the sequence {λ1 +m, λ2 +m−1, . . . , λm +1} is strictly decreasing, while the sequence {1 − ν1 , 2 − ν2 , . . . , n − νn } is strictly increasing. Suppose that iλ = 0. This is equivalent to saying that λm ≥ n, by Lemma 3.20. Thus λi + m − i + 1 ≥ λm + 1 > j − ν j , ∀1 ≤ j ≤ n, 1 ≤ i ≤ m. It follows by Definition 2.29 that #λ = 0. Now suppose that iλ > 0. Then, we have λm = n − j0 < n for some 0 < j0 ≤ n. This implies that νn = · · · = νn− j0 +1 = 0, and thus {n − j0 + 1 − νn− j0 +1 , . . . , n − νn } = {n − j0 + 1, . . . , n}. It follows by Lemma 3.20 that, for 0 ≤ j ≤ iλ − 1, n − j0 + 1 = λm + 1 ≤ λm− j + j + 1 ≤ n − iλ + j + 1 ≤ n. Thus, in the set {n − j0 + 1 − νn− j0 +1 , . . . , n − νn } = {n − j0 + 1, . . . , n}, there is a unique element that is equal to λm− j + j + 1, for 0 ≤ j ≤ iλ − 1. Hence, #λ ≥ iλ . Finally, for i = 1, . . . , m − iλ , Lemma 3.20 implies that λi + m − i + 1 ≥ λm−iλ + iλ + 1 ≥ (n − iλ ) + iλ + 1 = n + 1. Thus any such λi + m − i + 1, for i = 1, . . . , m − iλ , cannot be equal to an element of the form j − ν j , for j = 1, . . . , n, whence #λ = iλ .  In light of (3.19), we define Hλ = {λi + m − i + 1 | 1 ≤ i ≤ m}, Tλ = { j−ν j | 1 ≤ j ≤ n},

Iλ = Hλ ∩ Tλ .

The following corollary can be read off from the proof of Proposition 3.21.

3. Schur duality

108

Corollary 3.22. Let λ be an (m|n)-hook partition. Then Iλ consists of precisely the smallest #λ numbers in Hλ , and also, Iλ consists of precisely the largest #λ numbers in {1, 2, . . . , n} that are not in Tλ \ Iλ . Example 3.23. Let λ = (7, 4, 2, 2, 1, 1) with m = 4 and n = 5 as in Example 3.19. Then Hλ = {11, 7, 4, 3}, Tλ = {−1, 2, 3, 4, 5}, and Iλ = {3, 4}. This agrees with Corollary 3.22. 3.2.6. Category of polynomial modules. In this subsection g = gl(m|n) and L(ν) is the irreducible highest weight g-module of highest weight ν ∈ h∗ with respect to the standard Borel subalgebra. Recall from Section 2.2.2 the notion of central characters χν : Z(g) → C. Recall that, given a Lie (super)algebra G and G-modules M and N, the vector space Ext1G (M, N) classifies (up to equivalence) the short exact sequences of Gmodules of the form (3.20)

0 −→ N −→ E −→ M −→ 0.

Proposition 3.24. Let λ, µ be (m|n)-hook diagrams. Then, (1) χλ = χµ if and only if λ = µ.   (2) Ext1gl(m|n) L(λ ), L(µ ) = 0. Proof. (1) Suppose that χλ = χµ . Then by Theorem 2.30 we have Hλ \ Iλ = Hµ \ Iµ ,

Tλ \ Iλ = Tµ \ Iµ .

Thus, it suffices to show that λ can be reconstructed from the sets Hλ \ Iλ and Tλ \ Iλ . To that end, note that λ + ρ and λ can be recovered from the sets Hλ and Tλ , which in turn are determined by the three sets Iλ , Hλ \ Iλ and Tλ \ Iλ . But by Corollary 3.22 the set Iλ is determined from the set Tλ \ Iλ . This proves (1). (2) Consider a short exact sequence of the form (3.20) with M = L(λ ) and N = L(µ ). First assume λ = µ. It follows by (1) that there exists a central element z such that χλ (z) = χµ (z). Then, ker(z − χλ (z)) is a nonzero proper submodule of E, which must be isomorphic to M as it cannot be isomorphic to N. Hence the short exact sequence splits. Now assume λ = µ. Then by weight considerations, the two-dimensional µ -weight subspace of E has to be the highest weight space. Thus E contains two simple highest weight submodules of highest weight µ that must intersect trivially. Hence the short exact sequence splits in this case as well.  n Definition 3.25. A weight µ = ∑m i=1 ai δi + ∑ j=1 b j ε j is called a polynomial weight for gl(m|n) if all ai , b j are nonnegative integers. A gl(m|n)-module M is called a polynomial module if M is h-semisimple and every weight of M is a polynomial weight.

Proposition 3.26. An irreducible highest weight gl(m|n)-module M is a polynomial module if and only if M ∼ = L(µ ) for some (m|n)-hook partition µ.

3.3. Representation theory of the algebra Hn

109

Proof. Assume that M ∼ = L(µ ), for some (m|n)-hook partition µ. Denote by d = |µ|. Then, by Theorem 3.11, L(µ ) is a direct summand of (Cm|n )⊗d , which is clearly a polynomial module. Now assume that M is an irreducible polynomial gl(m|n)-module, which by n Definition 3.25(3) is isomorphic to L(λ) for some λ = ∑m i=1 λi δi + ∑ j=1 b j ε j , with all λi , b j ∈ Z+ . Since L(gl(m) ⊕ gl(n), λ) is a polynomial module, we must have λi ≥ λi+1 and b j ≥ b j+1 , for all possible i, j. We shall proceed to complete the proof by contradiction. Suppose that λ = µ , for any (m|n)-hook partition µ. Then we have λm = k − 1 and bk > 0, for some 1 ≤ k ≤ n. We apply the sequence of odd reflections corresponding to the odd roots δm − ε1 , . . . , δm − εk−1 to obtain a new Borel subalgebra  b from the standard one. By applying Lemma 1.40 repeatedly, we compute the  b-highest weight of M to be λ = λ−(k −1)δm +ε1 +. . . +εk−1 . Observe by a repeated use of Lemma 1.30 that δm − εk is a simple root of  b. Let wλ be a nonzero  b-highest weight vector of M. Then the vector e w is nonzero in M, and its weight λ − δm + εk has −1 −δm +εk

λ

as the coefficient for δm , thanks to λm = k − 1 and bk > 0. This contradicts the assumption that M is a polynomial module. 

Theorem 3.27. The category of polynomial gl(m|n)-modules is a semisimple tensor category. Proof. Let M be a polynomial gl(m|n)-module. For v ∈ M note that U(gl(m|n)0¯ )v is finite-dimensional. Thus, U(gl(m|n))v is also finite-dimensional, and hence it is a direct sum of irreducible polynomial modules by Propositions 3.24 and 3.26. It follows that M is a sum of irreducible polynomial modules, and hence it is a direct sum of irreducible polynomial modules. It remains to show that the tensor product of any two irreducible polynomial gl(m|n)-modules is a direct sum of irreducible polynomial gl(m|n)-modules. To that end, note that by Theorem 3.11 any irreducible polynomial module is a direct summand of (Cm|n )⊗d , for some d ≥ 0, and furthermore any such tensor power is a direct sum of irreducible polynomial modules. Now take L(λ ) ⊆ (Cm|n )⊗d and L(µ ) ⊆ (Cm|n )⊗l , for (m|n)-hook partitions λ and µ. Then L(λ ) ⊗ L(µ ) ⊆ (Cm|n )⊗k ⊗ (Cm|n )⊗l ∼ = (Cm|n )⊗d+l . Since (Cm|n )⊗d+l is a direct sum of irreducible polynomial modules, so is the submodule L(λ ) ⊗ L(µ ). 

3.3. Representation theory of the algebra Hn In this section, we develop systematically the representation theory of an algebra Hn , which is equivalent to the spin representation theory of a distinguished double cover Bn of the hyperoctahedral group Bn . We classify the split conjugacy classes in

3. Schur duality

110

Bn . We then define a characteristic map using the character table for the simple spin modules of Bn , analogous to the Frobenius characteristic map for the symmetric groups. The images of the irreducible spin characters of Bn under the characteristic map are shown to be Schur Q-functions (up to some 2-powers). 3.3.1. A double cover. Let Πn be the finite group generated by ai (i = 1, . . . , n) and the central element z subject to the relations (3.21)

a2i = 1,

z2 = 1,

ai a j = za j ai

(i = j).

The symmetric group Sn acts on Πn by σ(ai ) = aσ(i) , σ ∈ Sn . The semidirect product Bn := Πn  Sn admits a natural finite group structure and will be called the twisted hyperoctahedral group. Explicitly, the multiplication in Bn is given by (a, σ)(a , σ ) = (aσ(a ), σσ ), a, a ∈ Πn , σ, σ ∈ Sn . Since Πn /{1, z} ' Zn2 , the group Bn is a double cover of the hyperoctahedral group Bn := Zn2  Sn , and the order |Bn | is 2n+1 n!. That is, we have a short exact sequence of groups θn 1 −→ {1, z} −→ Bn −→ Bn −→ 1,

where θn sends each ai to the generator bi of the ith copy of Z2 in Bn . We define a Z2 -grading on the group Bn by setting the degree of each ai to ¯ Hence the group Bn fits into the be 1¯ and the degree of elements in Sn to be 0.  general setting of G in Section 3.1.4. The group Bn inherits a Z2 -grading from Bn via the homomorphism θn . This induces parity epimorphisms p : Bn → Z2 and p : Bn → Z2 . The conjugacy classes of the group Bn (a special case of a wreath product) can be described as follows, cf. Macdonald [83, I, Appendix B]. It is convenient to identify Z2 as {+, −} with + being the identity element. Given a cycle t = (i1 , . . . , im ), we call the set {i1 , . . . , im } the support of t, denoted by supp(t). The subgroup Zn2 of Bn consists of elements of the form bI := ∏i∈I bi for I ⊂ {1, . . . , n}. Each element bI σ ∈ Bn can be written as a product of the form (unique up to reordering) bI σ = (bI1 σ1 )(bI2 σ2 ) . . . (bIk σk ), where σ ∈ Sn is a product of disjoint cycles σ = σ1 . . . σk , and Ia ⊂ supp(σa ) for each 1 ≤ a ≤ k; bIa σa is called a signed cycle of bI σ. The cycle-product of each signed cycle bIa σa is defined to be the element ∏i∈Ia bi ∈ Z2 (which can be con− veniently thought of as a sign + or −). Let m+ i (respectively, mi ) be the number of i-cycles of bI σ with associated cycle-product being + (respectively, −). Then + − ρ+ = (imi )i≥1 and ρ− = (imi )i≥1 are partitions such that |ρ+ | + |ρ− | = n. The pair of partitions (ρ+ , ρ− ) will be called the type of the element bI σ.

3.3. Representation theory of the algebra Hn

111

The following is the basic fact on the conjugacy classes of Bn , cf. [83, I, Appendix B]. We leave the proof to the reader (Exercise 3.9). Lemma 3.28. Two elements of Bn are conjugate if and only if their types are the same. We shall denote by Cρ+ ,ρ− the conjugacy class of type (ρ+ , ρ− ). Note that if bI σ ∈ Cρ+ ,ρ− , then Cρ+ ,ρ− is even (respectively, odd) if |I| is even (respectively, odd). Example 3.29. Let τ = (1, 2, 3, 4)(5, 6, 7)(8, 9), σ = (1, 3, 8, 6)(2, 7, 9)(4, 5) ∈ S10 . It is straightforward to check that both x = ((+, +, +, −, +, +, +, −, +, −), τ) and y = ((+, −, −, −, +, −, −, −, +, −), σ) in B10 have the same type (ρ+ , ρ− ) = ((3), (4, 2, 1)). Thus, x is conjugate to y in B10 . 3.3.2. Split conjugacy classes in Bn . A partition λ = (λ1 , . . . , λ ) of length  is called strict if λ1 > λ2 > . . . > λ , and it is called odd if each part λi is odd. We denote by SPn the set of all strict partitions of n, and by OPn the set of all odd partitions of n. Moreover, we denote SP =



SPn ,

OP =

n≥0



OPn .

n≥0

Recall Pn denotes the set of all partitions of n. Let P+ = {λ ∈ Pn | (λ) is even}, n P− = {λ ∈ Pn | (λ) is odd}. n Given σ ∈ Sn of cycle type µ, we denote by d(σ) = n − (µ). For an (ordered) subset I = {i1 , i2 , . . . , im } of {1, . . . , n}, we denote aI = ai1 i2 ...im = ai1 ai2 . . . aim . / then aI aJ = z|I||J| aJ aI . Also we can It follows that p(aI ) ≡ |I| mod 2. If I ∩ J = 0, easily show by induction that (3.22)

ai1 i2 ...im = zd(s) as(i1 )s(i2 )...s(im )

for a permutation s such that s fixes the letters other than i1 , i2 , . . . , im . We can write a general element of Bn as zk aI s = zk (aI1 s1 ) . . . (aIq sq ), where s = s1 . . . sq is a cycle decomposition of s and I j ⊂ supp(s j ) for each j. We denote by J c the complement of a subset J ⊆ {1, . . . , n}.

3. Schur duality

112

Lemma 3.30. Let aI s = (aI1 s1 ) . . . (aIq sq ) be an element of Bn in its cycle decomposition. Let J ⊆ supp(s1 ) ∩ I1c . Then (aJ s1 )(aI s)(aJ s1 )−1 = zd(s1 )+|J||I| aI s. Proof. Observe that a2I = z(|I|−1)|I|/2 for any subset I. For k > 1 we have (aJ s1 )(aIk sk )(aJ s1 )−1 = z|J||Ik | aIk sk . Therefore it remains to see that (aJ s1 )(aI1 s1 )(aJ s1 )−1 = z(|J|−1)|J|/2 aJ as1 (I1 ∪J) s1 = z(|J|−1)|J|/2+d(s1) aJ a(I1 ∪J) s1 = z(|J|−1)|J|/2+d(s1)+|J||I1 | a2J aI1 s1 = z|J||I1 |+d(s1 ) aI1 s1 , where we have used the fact that supp(s1 ) ⊇ I1 ∪ J and (3.22).



Theorem 3.31. The conjugacy class Cρ+ ,ρ− in Bn splits if and only if / (1) for even Cρ+ ,ρ− , we have ρ+ ∈ OPn and ρ− = 0, (2) for odd Cρ+ ,ρ− , we have ρ+ = 0/ and ρ− ∈ SP− n. Proof. Assume that Cρ+ ,ρ− is an even conjugacy class such that ρ+ ∈ OP. Then θ−1 n (Cρ+ ,ρ− ) contains an element aI s with a signed cycle decomposition of the form aI s = s1 (aI2 s2 ) . . . (aIp s p ), where s1 = (1, 2, . . ., r) for r = 2k even, I1 = 0/ and |I| is even. Consider the element x = a12...r (1, 2, . . ., r) ∈ Bn . By Lemma 3.30 we have x(aI s)x−1 = z(2k−1)+2k·|I| aI s = zaI s. Therefore, if an even conjugacy class Cρ+ ,ρ− splits, then ρ+ ∈ OP. / Then ρ− Assume that Cρ+ ,ρ− is an even conjugacy class such that ρ− = 0. contains at least two parts, and θ−1 n (Cρ+ ,ρ− ) contains an element of the form aI s = (ai1 s1 )(ai2 s2 )(aI3 s3 ) . . . (aIp s p ), where i1 ∈ supp(s1 ), i2 ∈ supp(s2 ). Then (ai1 s1 )−1 aI s(ai1 s1 ) = (ai1 s1 )−1 (ai1 s1 ai2 s2 )(ai1 s1 )(aI3 s3 ) . . . (aIp s p ) = ai2 s2 (ai1 s1 )(aI3 s3 ) . . . (aIp s p ) = z(ai1 s1 )(ai2 s2 )(aI3 s3 ) . . . (aIp s p ) = zaI s. / Together with the Hence, if an even conjugacy class Cρ+ ,ρ− splits, then ρ− = 0. above, we have shown that an even split conjugacy class should satisfy (1).

3.3. Representation theory of the algebra Hn

113

/ Then, Now assume that Cρ+ ,ρ− is an odd conjugacy class such that ρ+ = 0. −1 θn (Cρ+ ,ρ− ) contains an element aI s with signed cycle decomposition of the form aI s = (s1 )(aI2 s2 ) . . . (aIq sq ), where I1 = 0/ and |I| is odd. Let J = supp(s1 ). Then, by Lemma 3.30, (aJ s1 )(aI s)(aJ s1 )−1 = z(|J|−1)+|I||J|aI s = zaI s, / since |I| is odd. Hence if Cρ+ ,ρ− is an odd split conjugacy class then ρ+ = 0. Next, assume that Cρ+ ,ρ− is an odd conjugacy class such that ρ− contains two identical parts. Then θ−1 n (Cρ+ ,ρ− ) contains an element of the form aI s = (ai1 (i1 , i2 , . . . , ik ))(a j1 ( j1 , j2 , . . . , jk )) . . . (aIq sq ). Consider the element t = (i1 , j1 ) . . . (ik , jk ). We have t(aI s)t −1 = at(I) s = a j1 ( j1 , . . . , jk )ai1 (i1 , . . . , ik ) . . . = zaI s. Hence, if an odd conjugacy class Cρ+ ,ρ− splits then ρ− ∈ SP. Together with the above, we have shown that an odd split conjugacy class satisfies (2). Assume that (1) holds. Then θ−1 n (Cρ+ ,0/ ) contains an element s = s1 . . . sq ∈ Sn with each si being an odd cycle. Suppose on the contrary the conjugacy class Cρ+ ,0/ does not split, that is, (aJ t)s(aJ t)−1 = zs for some element aJ t. Then (aJ t)s = zs(aJ t), and so zas(J) = aJ , which implies that supp(s) ⊆ J. On the other hand, as(J) = zd(s) aJ = aJ by (3.22), since s is a product of disjoint odd cycles. This contradiction implies that the conjugacy class Cρ+ ,0/ splits. − Now assume that we are given an odd conjugacy class C0,ρ / − with ρ strict as specified in (2). Thus θ−1 / − ) contains an element aI s = (ai1 s1 ) . . . (aiq sq ), n (C0,ρ where q is odd and ik ∈ supp(sk ). Suppose on the contrary that the conjugacy class −1 = z(a s) for some element a t. It C0,ρ I J / − does not split, that is, (aJ t)(aI s)(aJ t) r follows that t commutes with s and hence t = sr11 . . . sqq for 0 ≤ ri < order(si ), since the cycle type of s is a strict partition. Write aJ t = (aJ1 t1 ) . . . (aJq tq ) with tm = srmm . As in the proof of Lemma 3.30 we have

(aJ t)(ai1 s1 )(aJ t)−1 = (aJ1 t1 )(aJ2 t2 ) · · · (aJq tq )(ai1 s1 )((aJ2 t2 ) · · · (aJq tq ))−1 (aJ1 t1 )−1 = z|J|−|J1 | (aJ1 t1 )(ai1 s1 )(aJ1 t1 )−1 , which must equal ai1 s1 up to a power of z. Set (aJ1 t1 )(ai1 s1 )(aJ1 t1 )−1 = z∗ ai1 s1 where ∗ is 0 or 1. We claim that ∗ is always 0. Note that aJ1 at1 (i1 ) = z∗ ai1 as1 (J1 ) , and so J1 differs from s1 (J1 ) by one element. Without loss of generality, we let i1 = 1, s1 = (1, 2, . . ., k), J1 = {1, 2, . . . , r} with 0 ≤ r < k. Then we have a1 . . . ar · ar1 +1 = aJ1 at1 (i1 ) = z∗ ai1 as1 (J1 ) = z∗ a1 · a2 . . . ar+1 ,

3. Schur duality

114

which implies that r1 = r and ∗ = 0. Therefore, (aJ t)(ai1 s1 )(aJ t)−1 = z|J|−|J1 | ai1 s1 , and similarly we have (aJ t)(aI s)(aJ t)−1 = (aJ t)(aI1 s1 ) . . . (aIq sq )(aJ t)−1 = zq|J|−(|J1 |+...+|Jq |) aI s = z(q−1)|J| aI s = aI s, 

since q is odd. This is a contradiction.

−1  For α ∈ OPn we let C+ α be the split conjugacy class in Bn which lies in θn (Cα,0/ ) + and contains a permutation in Sn of cycle type α. Then zCα is the other conjugacy − class in θ−1 n (Cα,0/ ), which will be denoted by Cα . Recall from Appendix A that zα denotes the order of the centralizer of an element of type α in Sn . The order of the centralizer of an element of a given cycle type is known explicitly for Bn (and actually for any wreath product, cf. Macdonald [83, I, Appendix B]). The next lemma follows from this classical fact.

Lemma 3.32. Let α ∈ OPn . The order of the centralizer of an element in the 1+(α) z . Thus, the order of the conjugacy class C+ is  conjugacy class C+ α α of Bn is 2 α n−(α) z−1 . |C+ α | = n!2 α 3.3.3. A ring structure on R− . Let us introduce a basic example of superalgebras. Definition 3.33. The Clifford algebra Cn is the C-algebra generated by ci , for 1 ≤ i ≤ n, subject to relations (3.23)

c2i = 1,

ci c j = −c j ci (i = j).

¯ ∀i, Cn becomes a superalgebra also called the Clifford superalgeLetting |ci | = 1, bra. The symmetric group Sn acts as automorphisms on the algebra Cn naturally. We will refer to the semi-direct product Hn := Cn  CSn as the Hecke-Clifford algebra, where (3.24)

σci = cσ(i) σ,

∀σ ∈ Sn .

Note that the algebra Hn is naturally a superalgebra by letting each σ ∈ Sn be even and each ci be odd. Recall the group Πn from (3.21). The quotient algebra CΠn /z + 1 of the group algebra CΠn by the ideal generated by z + 1 is isomorphic to the Clifford superalgebra Cn with the identification a¯i = ci , 1 ≤ i ≤ n. Hence, we have an isomorphism of superalgebras (3.25) CBn /z + 1 ∼ = Hn . Recall our convention from Section 3.1.1 that a module of a superalgebra is always understood to be Z2 -graded. We shall denote by Hn -mod the category of modules of the superalgebra Hn (with morphisms of degree one allowed). Thanks

3.3. Representation theory of the algebra Hn

115

to the superalgebra isomorphism (3.25), Hn -mod is equivalent to the category of spin Bn -modules. (Recall from Section 3.1.4 that a spin Bn -module means a Z2 graded Bn -module M on which z ∈ Bn acts as −I.) We shall not distinguish these two isomorphic categories below, and the latter one has the advantage that one can apply the standard arguments from the theory of finite groups directly. Denote by R− n = [Hn -mod] the Grothendieck group of Hn -mod (cf. Section 3.1.1). As in the usual (ungraded) case, we may replace the isomorphism classes of modules by their characters and then regard R− n as the free abelian group with a basis consisting of the characters of the simple spin Bn -modules. Let −

R :=

∞ 

R− n,

R− Q

n=0

:=

∞ 

Q ⊗Z R− n,

n=0

− where it is understood that R− 0 = Z. We shall define a ring structure on R as follows. Denote by Bm,n the subgroup of Bm+n generated by Sm × Sn and Πm+n . Then  Bm,n can be identified with the quotient group Bm × Bn /{(1, 1), (z, z)}, where Bm × Bn denotes the product group in the super sense, i.e., elements from Bm and Bn supercommute with each other. −  ˆ Given ϕ ∈ R− m and ψ ∈ Rn , we define the spin character ϕ×ψ of Bm,n by letting

ˆ ϕ×ψ(x, y) = ϕ(x)ψ(y), where (x, y) is the image of (x, y) in Bm,n ∼ = (Bm × Bn )/{(1, 1), (z, z)}. We define a − product on R by B

ˆ ϕ · ψ = IndBm+n (ϕ×ψ), m,n

R− m,

R− n

where ϕ ∈ ψ∈ for all m, n, and Ind denotes the induced character. It follows from the properties of the induced characters that the multiplication on R− is commutative and associative. Remark 3.34. Equivalently, the multiplication in R− can be described as follows. Let Hm,n be the subalgebra of Hm+n generated by Cm+n and Sm × Sn . For M ∈ Hm -mod and N ∈ Hn -mod, M ⊗ N is naturally an Hm,n -module, and we define the product [M] · [N] = [Hm+n ⊗Hm,n (M ⊗ N)], and then extend it by Z-bilinearity. + For ϕ ∈ R− n , we shall write ϕα = ϕ(x) for x ∈ Cα , α ∈ OPn ; hence ϕ(y) = −ϕα − for y ∈ Cα . Given two partitions α, β, we let α ∪ β denote the partition obtained by collecting the parts of α and β together and rearranging them in descending order.

3. Schur duality

116

− Lemma 3.35. Let ϕ ∈ R− m , ψ ∈ Rn , and γ ∈ OPm+n . Then zγ (ϕ · ψ)γ = ∑ zα zβ ϕ α ψβ . α∈OPm ,β∈OPn α∪β=γ

Proof. It can be checked directly that  C+ γ ∩ Bm,n =



α,β∈OP,α∪β=γ

ˆ + C+ α ×Cβ .

It follows from this, Lemma 3.32, and the standard induced character formula for finite groups that (ϕ · ψ)γ =

|Bm+n |

ˆ ∑ (ϕ×ψ)(w)

|Bm,n | · |C+ γ | w∈C+ γ

21+(γ) zγ ∑ ϕαψβ |C+α | · |C+β | 2m+n+1 m!n! α,β∈OP,α∪β=γ zγ n−(β) −1 = zβ n!ϕα ψβ ∑ 2m−(α) z−1 α m!2 m+n−(γ) m!n!2 α,β∈OP,α∪β=γ zγ = ϕ α ψβ , ∑ z z α,β∈OP,α∪β=γ α β

=



where we have used (γ) = (α) + (β).

3.3.4. The characteristic map. Recall from (A.45) in Appendix A that the ring ΓQ := Q ⊗Z Γ has a basis given by the power-sum symmetric functions pµ for µ ∈ OP. Moreover, ΓQ is equipped with a bilinear form ·, · given in (A.55). We define the (spin) characteristic map ch : R− Q −→ ΓQ to be the linear map given by ch(ϕ) =

∑

α∈OPn

z−1 α ϕ α pα ,

for ϕ ∈ R− n , n ≥ 0.

Proposition 3.36. The characteristic map ch : R− Q → ΓQ is an algebra homomorphism. − Proof. For ϕ ∈ R− m , ψ ∈ Rn , we have by Lemma 3.35 that

ch(ϕ · ψ) =

∑

γ∈OP

=

z−1 γ (ϕ · ψ)γ pγ

∑

∑

γ α,β∈OP,α∪β=γ

where we have used pγ = pα pβ .

z−1 γ

zγ ϕα ψβ pγ = ch(ϕ)ch(ψ), zα zβ 

3.3. Representation theory of the algebra Hn

117

Denote by [M], [N] = dim HomBn (M, N) for spin Bn -modules M, N. This defines a bilinear form ·, · on R− n by Z-bilinearity. The HomBn here can be either understood as in the category of Bn -modules (with degree one morphisms allowed) or in the category of (non-graded) Bn -modules, and they give the same dimension. In light of super Schur’s Lemma 3.4, this is reduced to a straightforward verification in the case when M ∼ = N is simple of type Q. Then R− , and hence also R− Q , carry a symmetric bilinear form, still denoted by − − ·, ·, which is induced from the ones on R− n , for all n, such that Rn and Rm are orthogonal whenever n = m. Lemma 3.37. For ϕ, ψ ∈ R− n , we have

∑

ϕ, ψ =

α∈OPn

2−(α) z−1 α ϕ α ψα .

−1 ∈ C+ . Also note that ϕ(x) = 0 unless x is Proof. Note that x ∈ C+ α implies that x α even and split. By Lemma 3.32 and applying the standard formula for the bilinear form on characters of a finite group, we have

ϕ, ψ = = = =

1 |Bn |

∑ ϕ(x−1)ψ(x)

x∈Bn

1 2n+1 n!

∑

α∈OPn

− (|C+ α |ϕα ψα + |Cα |ϕα ψα )

1 ∑ 2n−(α) z−1 α n!ϕα ψα n 2 n! α∈OP n

∑

α∈OPn

2−(α) z−1 α ϕ α ψα . 

Proposition 3.38. The characteristic map ch : R− Q → ΓQ is an isometry, i.e., it preserves the bilinear forms ·, · on R− Q and ΓQ . Proof. This follows from a direct computation using Lemma 3.37 and (A.55) from Appendix A: for ϕ, ψ ∈ R− n, ch(ϕ), ch(ψ) =

∑

α,β∈OPn

=

∑

α∈OPn

−1 z−1 α zβ pα , pβ ϕα ψβ

−(α) z−2 zα ϕα ψα = ϕ, ψ. α 2



3. Schur duality

118

3.3.5. The basic spin module. The algebra Hn acts on the Clifford superalgebra Cn by the formulas ci .(ci1 ci2 . . .) = ci ci1 ci2 . . . ,

σ.(ci1 ci2 . . .) = cσ(i1 ) cσ(i2 ) . . . ,

for σ ∈ Sn . The Hn -module Cn is called the basic spin module of Bn . Let σ = σ1 . . . σ ∈ Sn be a product of disjoint cycles with the cycle length of σi being µi , for i = 1, . . . , . If I is a union of some of the supp(σi )’s, say I = supp(σi1 ) ∪ . . . ∪ supp(σis ), then σ(cI ) = (−1)µi1 +...+µis −s cI . If I is not such a union, then σ(cI ) is not a scalar multiple of cI . Lemma 3.39. The value of the character ξn of the basic spin Bn -module at the conjugacy class C+ α is given by ξnα = 2(α) ,

(3.26)

α ∈ OPn .

Proof. Let α = (α1 , α2 , . . .) ∈ OPn and (α) = . Let σ = σ1 . . . σ be an element in Sn of cycle type α. The elements cI := ∏i∈I ci (which are defined up to a nonessential sign) for I ⊂ {1, . . . , n} form a basis of the basic spin module Cn . Observe that σcI = cI if I is a union of a subset of the supports supp(σ p ) for 1 ≤ p ≤ (α); otherwise σcI is equal to ±cJ for some J = I. Hence the character value ξnα , which is the trace of σ on Cn , is equal to 2(α) .  Below we will freely use the statements in Appendix A.3 on Schur Q-functions Qλ . Recall the symmetric function qn defined by the generating function (A.42), which is 1 + xi t ∑ qn t n = ∏ 1 − x i t . i≥1 n≥0 Lemma 3.40. Let n ≥ 1. We have (1) ch(ξn ) = qn ; (2) ξn , ξn  = 2; (3) the basic spin Bn -module Cn is simple of type Q. Proof. (1) It follows by the definition of ch, Lemma 3.39, and (A.47) that ch(ξn ) =

∑

α∈OPn

2(α) z−1 α pα = qn .

(2) Note that pk (1, 0, 0, . . .) = 1 for each k ≥ 1, and hence pα (1, 0, 0, . . .) = 1. Also, it follows by definition that qn (1, 0, 0, . . .) = 2, for n ≥ 1. Thus, specializing the identity ∑α∈OPn 2(α) z−1 α pα = qn at (x1 , x2 , x3 , . . .) = (1, 0, 0, . . .), we obtain (α) −1 ∑α∈OPn 2 zα = 2. We compute, by Lemmas 3.37 and 3.39, that ξn , ξn  =

∑

α∈OPn

(α) 2 2−(α) z−1 ) = α (2

∑

α∈OPn

2(α) z−1 α = 2.

3.3. Representation theory of the algebra Hn

119

(3) As the Bn -module Cn is semisimple, to prove (3) it suffices to exhibit an odd automorphism of the Bn -module Cn by (2) and super Schur’s Lemma 3.4. Indeed, the right multiplication with the element √1n (c1 + . . . + cn ) provides such an automorphism of order 2.  Proposition 3.41. The characteristic map ch : R− Q → ΓQ is an isomorphism of graded vector spaces. Proof. Since ΓQ is generated by qr for r ≥ 1 and ch is an algebra homomorphism by Proposition 3.36, ch is surjective by Lemma 3.40. By Proposition 3.8 and Theon rem 3.31, the dimension of Q ⊗ R− n is |SPn |, which is the same as dim ΓQ , for each n. Hence ch is an isomorphism.  3.3.6. The irreducible characters. Using the algebra structure on R− , we define the elements ξλ for a strict partition λ by the following recursive relations: λ2

(3.27)

ξ(λ1 ,λ2 ) = ξλ1 ξλ2 + 2 ∑ (−1)i ξλ1 +i ξλ2 −i , i=1

(3.28)

ξλ =

k

∑ (−1) j ξ(λ ,λ ) ξ(λ ,...λ ...,λ ) , 1

j

2

ˆj

for k = (λ) even,

k

j=2

(3.29)

ξλ =

k

∑ (−1) j−1 ξλ ξ(λ ,...λ ...,λ ) , j

1

ˆj

k

for k = (λ) odd.

j=1

We emphasize that these are precisely the same recursive relations for the Schur Q-functions Qλ (see (A.51)). Lemma 3.42. We have ch(ξλ ) = Qλ and ξλ , ξµ  = 2(λ) δλµ , for λ, µ ∈ SP. Proof. By Lemma 3.40, we have ch(ξn ) = qn = Q(n) . The general case of the first identity follows since ch is a ring homomorphism, and, in addition, ξλ and Qλ are obtained from ξn ’s and qn ’s, respectively, by the same recursive relations. The second identity follows from the fact that ch is an isometry and the formula Qλ , Qµ  = 2(λ) δλµ from (A.57).  Recalling the notation δ(λ) from (1.52) we define ζλ := 2−

(λ)−δ(λ) 2

ξλ ,

for λ ∈ SPn .

Lemma 3.43. The element ζλ lies in R− n , for λ ∈ SPn . Proof. We proceed by induction on (λ). For (λ) = 1, it is clear. Since Cn is a simple spin Bn -module of type Q by Lemma 3.40, the induced module with character ξm ξn is a sum of two isomorphic copies of a genuine module (the two odd automorphisms of ξm and ξn give rise to an even automorphism of order 2); that is,

3. Schur duality

120

∈ R− . Hence, for (λ) = 2, 12 ξ(λ1 ,λ2 ) lies in R− by (3.27). The general case follows easily by induction using the recursive relations (3.28) and (3.29).  1 m n 2ξ ξ

Corollary 3.44. For strict partitions λ, µ, we have  1 for (λ) even λ λ ζ , ζ  = 2 for (λ) odd, ζλ , ζµ  = 0,

for λ = µ.

Corollary 3.45. For each λ ∈ SPn , we have Qλ = 2

(λ)−δ(λ) 2

∑

α∈OPn

λ z−1 α ζ α pα .

Also, for each α ∈ OPn , we have pα =

∑

2−

(λ)+δ(λ) −(α) 2

λ∈SPn

ζλα Qλ .

Proof. The first identity follows from Lemma 3.42 and the definitions of ζλ and ch. Write pα = ∑λ∈SPn aλα Qλ for some scalars aλα . Recall from (A.55) and (A.57) that Qλ , Qµ  = 2(λ) δλµ and pα , pβ  = 2−(α) zα δαβ . Then aλα = 2−(λ) pα , Qλ  = 2−(λ) 2

(λ)−δ(λ) 2

λ − z−1 α ζα pα , pα  = 2

(λ)+δ(λ) −(α) 2

where the second equality above uses the first identity of the corollary.

ζλα , 

Theorem 3.46. Let λ ∈ SPn and (λ) = . Then ζλ is the character of a simple spin Bn -module (which is to be denoted by Dλ ). Moreover, the degree of ζλ is equal to −δ(λ) λi − λ j n! 2n− 2 . λ1 ! . . . λ ! ∏ i< j λi + λ j Proof. Since ζλ ∈ R− by Lemma 3.43 and ζλ , ζλ  = 1 for (λ) even by Corollary 3.44, ζλ or −ζλ , for (λ) even, is a simple character of type M. By Corollary 3.44, these simple characters are distinct. A simple count using Proposition 3.8 and Theorem 3.31 implies that these are all simple characters of type M. Now since ζλ , ζλ  = 2 for (λ) odd by Corollary 3.44, we have two possibilities: (a) either ζλ or −ζλ is simple of type Q, or (b) ζλ is of the form ±ζµ ± ζν with both ζµ and ζν being simple of type M, which means both (µ) and (ν) are even. Case (b) cannot occur; otherwise, it would contradict the linear independence of ζλ for all λ ∈ SPn . Hence these must be all type Q simple characters by a counting argument using Proposition 3.8 and Theorem 3.31 again.

3.4. Schur-Sergeev duality for q(n)

121

To show that ζλ rather than −ζλ for each λ ∈ SPn is a character of a simple module, it suffices to know that ζλ(1n ) or ξλ(1n ) is positive. To that end, we claim that ξλ(1n ) = 2n

λi − λ j n! , λ1 ! . . . λ ! ∏ i< j λi + λ j

which is equivalent to the degree formula in the theorem. The claim can be proved by induction on (λ) using the relations (3.27)–(3.29). The initial case with (λ) = 1 is taken care of by Lemma 3.39, and the case with (λ) = 2 can be checked directly by using (3.27). As the induction is elementary though lengthy (see [57]), we will simply remark here that the sought-for identity by using (3.28) and (3.29) precisely corresponds to the Laplacian-type expansion of the classical Pfaffian identity (cf. Macdonald [83, III.8, Ex. 5]):   ti − t j ti − t j Pf = ∏ . ti + t j 1≤i, j≤2n 1≤i< j≤2n ti + t j We refer to [57, proof of Proposition 4.13] for detail.



3.4. Schur-Sergeev duality for q(n) In this section, we formulate a double centralizer property for the actions of the Lie superalgebra q(n) and the algebra Hd on the tensor space (Cn|n )⊗d . We obtain a multiplicity-free decomposition of (Cn|n )⊗d as a U(q(n)) ⊗ Hd -module. The characters of the simple q(n)-modules arising this way are shown to be Schur Qfunctions (up to some 2-powers). 3.4.1. A double centralizer property. Recall from Section 3.2 (by setting m = n) that we have a representation (Φd ,V ⊗d ) of gl(n|n), and hence of its subalgebra q(n); and we also have a representation (Ψd ,V ⊗d ) of the symmetric group Sd . Moreover, the actions of gl(n|n) and the symmetric group Sd on V ⊗d commute with each other. Note in addition that the Clifford superalgebra Cd acts on V ⊗d , and we denote this action also by Ψd : Ψd (ci ).(v1 ⊗ . . . ⊗ vd ) = (−1)(|v1 |+...+|vi−1 |) v1 ⊗ . . . ⊗ vi−1 ⊗ Pvi ⊗ . . . ⊗ vd , where P is given in (1.10), each vi ∈ V is assumed to be Z2 -homogeneous, and 1 ≤ i ≤ n. Lemma 3.47. Let V = Cn|n . The actions of Sd and Cd above give rise to a representation (Ψd ,V ⊗d ) of Hd . Moreover, the actions of q(n) and Hd on V ⊗d commute with each other.

3. Schur duality

122

Proof. To see that (3.24) holds, it suffices to check for σ = (i, i + 1), 1 ≤ i ≤ d − 1. We compute that Ψd ((i, i + 1))Ψd (ci ).(v1 ⊗ . . . ⊗ vd ) = Ψd ((i, i + 1))(−1)(|v1|+...+|vi−1 |) v1 ⊗ . . . ⊗ Pvi ⊗ vi+1 ⊗ . . . ⊗ vd = (−1)(|vi |+1)|vi+1 | (−1)(|v1 |+...+|vi−1 |) v1 ⊗ . . . ⊗ vi+1 ⊗ Pvi ⊗ . . . ⊗ vd , Ψd (ci+1 )Ψd ((i, i + 1)).(v1 ⊗ . . . ⊗ vd ) = Ψd (ci+1 )(−1)(|vi |·|vi+1 |) v1 ⊗ . . . ⊗ vi+1 ⊗ vi ⊗ . . . ⊗ vd = (−1)(|vi |·|vi+1 |) (−1)(|v1 |+...+|vi−1 |+|vi+1 |) v1 ⊗ . . . ⊗ vi+1 ⊗ Pvi ⊗ . . . ⊗ vd . Hence, we have Ψd ((i, i + 1))Ψd (ci ) = Ψd (ci+1 )Ψd ((i, i + 1)). This further implies that Ψd ((i, i + 1))Ψd (ci+1 ) = Ψd (ci )Ψd ((i, i + 1)). A similar calculation shows that Ψd (( j, j + 1))Ψd (ci ) = Ψd (ci )Ψd (( j, j + 1)) for j = i, i − 1. By the definitions of q(n) and of Ψd (ci ) via P, the action of q(n) commutes with the action of ci for 1 ≤ i ≤ d. Since gl(n|n) commutes with Sd , so does the subalgebra q(n) of gl(n|n). Hence, the action of q(n) commutes with the action of Hd on V ⊗d .  Theorem 3.48. The images Φd (U(q(n))) and Ψd (Hd ) satisfy the double centralizer property, i.e., Φd (U(q(n))) =EndHd (V ⊗d ), Endq(n) (V ⊗d ) = Ψd (Hd ). Proof. Let g = q(n). We denote by Q(V ) the associative subalgebra of endomorphisms on V which (super)commute with the linear operator P. By Lemma 3.47, we have (3.30)

Φd (U(g)) ⊆ EndHd (V ⊗d ).

We shall proceed to prove that Φd (U(g)) ⊇ EndHd (V ⊗d ). By examining the action of Cd on V ⊗d , we see that the natural isomorphism End(V )⊗d ∼ = End(V ⊗d ) allows us to identify EndCd (V ⊗d ) ≡ Q(V )⊗d . As Hd = Cd  Sd , this further leads to the identification EndHd (V ⊗d ) ≡ Symd (Q(V )), the space of Sd -invariants in Q(V )⊗d . Denote by Yk , 1 ≤ k ≤ d, the C-span of the super-symmetrization ϖ(x1 , . . . , xk ) :=

∑

σ∈Sd

σ.(x1 ⊗ . . . ⊗ xk ⊗ I ⊗d−k ),

for all xi ∈ Q(V ). Note that Yd = Symd (Q(V )) ≡ EndHd (V ⊗d ). Let x˜ = Φd (x) = ∑di=1 I ⊗i−1 ⊗ x ⊗ I ⊗d−i , for x ∈ Q(V ), and denote by Xk , 1 ≤ k ≤ d, the C-span of x˜1 . . . x˜k for all xi ∈ Q(V ).

3.4. Schur-Sergeev duality for q(n)

123

By the same argument as in the proof of Theorem 3.10 (Schur-Sergeev duality for gl(m|n)), we have Yk ⊆ Xk for 1 ≤ k ≤ d. This implies that (3.31)

EndHd (V ⊗d ) = Yd ⊆ Xd ⊆ Φd (U(g)).

Combining (3.31) with (3.30), we conclude that Φd (U(g)) = EndHd (V ⊗d ) = EndB (V ⊗d ), for B := Ψd (Hd ). Note that the spin group algebra Hd , and hence also B, are semisimple superalgebras, and so the assumption of Proposition 3.5 is satisfied. Therefore, we have EndU(g) (V ⊗d ) = Ψd (Hd ).  3.4.2. The Sergeev duality. Recall from (1.52) that, for a partition λ of d with length (λ),  0, if (λ) is even, δ(λ) = 1, if (λ) is odd. Recall further that Dλ stands for the simple Hd -module with character ζλ (see Theorem 3.46), and L(λ) for the simple q(n)-module with highest weight λ (see Section 2.1.6). Theorem 3.49. Let V = Cn|n . As a U(q(n)) ⊗ Hd -module, we have (3.32)



V ⊗d ∼ =

2−δ(λ) L(λ) ⊗ Dλ .

λ∈SPd ,(λ)≤n

Here, 2−1 has the same meaning as in Proposition 3.5. Proof. Let W = V ⊗d . Recall Dλ is of type M if and only if δ(λ) = 0. It follows from Proposition 3.5, Theorem 3.48, and the semisimplicity of the superalgebra Hd that we have a multiplicity-free decomposition of the (q(n), Hd )-module W : W∼ =



2−δ(λ) L[λ] ⊗ Dλ ,

λ∈Qd (n)

where L[λ] is some simple q(n)-module associated to λ whose highest weight (with respect to the standard Borel) is to be determined. Also to be determined is the index set Qd (n) = {λ ∈ SPd | L[λ] = 0}. We shall identify a weight µ = ∑ni=1 µi εi occuring in W with a composition µ = (µ1 , . . . , µn ) ∈ CPd (n). We have the following weight space decomposition: (3.33)



W=

Wµ ,

µ∈CPd (n)

where Wµ has a linear basis ei1 ⊗ . . . ⊗ eid , with the indices satisfying the following equality of multisets: {i1 , . . . , id } = {1, . . . , 1, , 1, . . . , 1, . . . , n, . . . , n, n, . . . , n}.       µ1

µn

3. Schur duality

124

We have an Hd -module isomorphism: H (3.34) Wµ ∼ = IndCSd µ 1µ = Hd ⊗CSµ 1µ . λµ for a composition µ and a strict partition λ defined via Recall the integers K the following symmetric function identity in (A.53) : λµ Qλ , (3.35) qµ = ∑ K λ∈SP,λ≥µ

λλ = 1. To complete the proof of the theorem, we shall need the following. where K Lemma 3.50. Let µ = (µ1 , . . . , µn ) be a composition of d. We have the following decomposition of Wµ as an Hd -module: 

Wµ ∼ =

2

(λ)−δ(λ) 2

λµ Dλ . K

λ∈SP,λ≥µ

λµ ∈ Z+ . In particular, K Proof. Decompose the Hd -module Wµ into irreducibles: (3.36)

Wµ ∼ =



K˜ λµ Dλ ,

for K˜ λµ ∈ Z+ .

λ∈SPd

Recall the characteristic map ch : R− → ΓQ from Section 3.3.4, where we have  denoted R− = n≥0 [Hn -mod]. By Lemmas 3.42 and 3.43, for λ ∈ SPd , (3.37)

ch([Dλ ]) = 2−

l(λ)−δ(λ) 2

Qλ .

It follows by the ring structure on R− and Remark 3.34 that, for any composition d µ = (µ1 , µ2 , . . .) of d, IndH CSµ 1µ is equal to the product of the basic spin characters ξµ1 · ξµ2 . . ., and, hence by Lemma 3.40,   d ch(Wµ ) = ch IndH (3.38) CSµ 1µ = qµ . Applying the characteristic map to both sides of (3.36), and using (3.37), (3.38), and (3.34), we obtain that qµ = ∑ 2− λ

(λ)−δ(λ) 2

K˜ λµ Qλ .

It follows by a comparison of this identity with (3.35) that K˜ λµ = 2

(λ)−δ(λ) 2

λµ . K



We return to the proof of Theorem 3.49. Since the simple q(n)-module L[λ] , for  [λ] λ ∈ Qd (n), has a weight space decomposition L[λ] = µ∈CPd (n),µ≤λ Lµ , we must have (λ) ≤ n and so λ ∈ SPd (n). Now let λ ∈ SPd with (λ) ≤ n. Clearly λ is a weight of L[λ] and, moreover, corresponds to a highest weight. Hence, we conclude that L[λ] = L(λ), the simple g-module of highest weight λ, and that Qd (n) = {λ ∈ SPd | (λ) ≤ n}. This completes the proof of Theorem 3.49. 

3.5. Exercises

125

3.4.3. The irreducible character formula. A character of a q(n)-module with weight space decomposition M = ⊕µ Mµ is by definition

∑

chM = tr |M x1H1 . . . xnHn =

µ

dim Mµ · x11 . . . xnµn .

µ=(µ1 ,...,µn )

Theorem 3.51. Let λ be a strict partition of length ≤ n. The character of the (λ)−δ(λ) simple q(n)-module L(λ) is given by chL(λ) = 2− 2 Qλ (x1 , . . . , xn ). H Proof. By (3.33) and (3.34), we have V ⊗d ∼ = ⊕µ∈CPd (n) IndSµd 1µ . Using (3.38) and

applying the characteristic map ch and the trace operator tr x1H1 . . . xnHn simultaneously, which we will denote by ch2 , we obtain that ch2 (V ⊗d ) =

∑

qµ (z)mµ (x),

µ∈Pd ,(µ)≤n

which can be written using (A.54) and (A.57) as ch2 (V ⊗d ) =

1 + xi z j = ∑ 2−(λ) Qλ (x1 , . . . , xn )Qλ (z), 1 − x z i j 1≤i≤n,1≤ j λ∈SP

∏

where z = {z1 , z2 , . . .} is infinite. On the other hand, by applying ch2 to (3.32) and using (3.37), we obtain that ch2 (V ⊗d ) =

∑

λ∈SPd ,(λ)≤n

2−δ(λ) chL(λ) · 2−

(λ)−δ(λ) 2

Qλ (z).

Now the theorem follows by comparing the above two identities and noting the linear independence of the Qλ (z)’s. 

3.5. Exercises Exercise 3.1. (1) Show that the ungraded algebra |Q(n)| for the superalgebra Q(n) is isomorphic to M(n) ⊕ M(n). (2) Assume that a semisimple superalgebra A has m (respectively, q) nonisomorphic simple modules of type M (respectively, of type Q). Show that the algebra |A| is semisimple and that the number of non-isomorphic simple |A|-modules is equal to m + 2q.  be a double cover of a finite group G with the projection Exercise 3.2. Let G  → G; see (3.6). Let C be a conjugacy class of G. Show that θ−1 (C) is either θ:G  or splits into two conjugacy classes of G.  a single conjugacy class of G Exercise 3.3. Define an involution α on a superalgebra A by letting α(ak ) = (−1)k ak , for ak ∈ Ak , k = 0, 1. Given an A-module N, we define another A-module N  with the same underlying vector superspace as N but with the action of A twisted by α. Show that N ∼ = N  as A-modules.

3. Schur duality

126

Exercise 3.4. Let A be a superalgebra and N an |A|-module. Regarding α in Exercise 3.3 as an involution on |A|, we have an |A|-module N  obtained from N by a twisted action via α. Prove: (1) If M is a simple A-module of type M, then |M| ∼ = |M| as |A|-modules and |M| is a simple |A|-module.

(2) If Q is a simple A-module of type Q, then there exists a simple |A|-module N such that Q ∼  N  as |A|-modules. = N ⊕ N  and N ∼ = Exercise 3.5. Prove that the formula (3.9) satisfies the Coxeter relations for the symmetric group Sd . Exercise 3.6. (1) Let λ be an (m|n)-hook partition with m = 0. Show that λ = λ , (λ ) ≤ n, and hsλ (0; y) = sλ (y), for y = {y1 , . . . , yn }. (2) Show that the Schur-Sergeev duality (Theorem 3.11) for m = 0 reduces to a version of (gl(n), Sd )-Schur duality twisted by the sign representa  tion of Sd , i.e., as a (gl(n), Sd )-module, (C0|n )⊗d ∼ = µ∈Pd (n) L(µ) ⊗ Sµ .  (Note here the well-known fact that Sµ ⊗ sgn ∼ = Sµ .) Exercise 3.7. For an (m|n)-hook partition λ, show that the weight λ is typical (i.e., the degree of atypicality #λ = 0) if and only if λm ≥ n. Exercise 3.8. Let λ be an (m|n)-hook partition such that λm ≥ n, and let the Young diagram λ be depicted below as a union of 3 regions: an m × n rectangle diagram and two subdiagrams µ, ν: n µ m ν

(1) Prove that the Kac gl(m|n)-module K(λ ) is irreducible. (2) Let x = {x1 , . . . , xm } and y = {y1 , . . . , yn }. Prove the following factorization identity: m

n

hsλ (x; y) = ∏ ∏ (xi + y j )sµ (x)sν (y). i=1 j=1

Exercise 3.9. Prove Lemma 3.28 which states that two elements of the hyperoctahedral group Bn are conjugate if and only if their types are the same. Exercise 3.10. Prove the following isomorphisms of superalgebras (see (3.5)):

3.5. Exercises

127

(1) Q(m) ⊗ Q(n) ∼ = M(mn|mn). (2) Q(m) ⊗ M(r|s) ∼ = Q(m(r + s)).

(3) M(m|n) ⊗ M(r|s) ∼ = M(mr + ns|ms + nr). Exercise 3.11. Prove the following isomorphisms for Clifford superalgebras (see Definition 3.33): (1) Cm ⊗ Cn ∼ = Cm+n . ∼ (2) C1 = Q(1), C2 ∼ = M(1|1). (3) C2m+1 ∼ = Q(2m ),

C2m ∼ = M(2m−1 |2m−1 ).

Hence, Cn is a simple superalgebra, and it is of type M if and only if n is even. Exercise 3.12. Recall the spin symmetric group algebra from Example 3.7. Prove that sending √ −1 ci → ci , 1 ≤ i ≤ n; t j → −2 s j (c j − c j+1 ), 1 ≤ j ≤ n − 1 defines a superalgebra isomorphism CS− n ⊗ Cn −→ Hn . Exercise 3.13. Recall a homomorphism ϕ : Λ −→ Γ from (A.62) in Appendix A. n For n ≥ 0, we define functors Ωn : Sn -mod −→ Hn -mod, Ωn (M) = IndH CSn M. This induces a functor Ω = ⊕n Ωn , which gives rise to a Z-linear map Ω : R −→ R− , by letting Ω([M]) = [Ωn (M)] for M ∈ Sn -mod. Prove that Ω : RQ → R− Q and ϕ : ΛQ −→ ΓQ are homomorphisms of (Hopf) algebras. Moreover, the following is a commutative diagram of (Hopf) algebras: Ω

(3.39)

RQ −−−−→ R− ⏐ ⏐Q ⏐ ⏐ F &∼ ch&∼ = ch = ϕ

ΛQ −−−−→ ΓQ Exercise 3.14. Show that the Hd -module Wµ defined in (3.33) and (3.34) is isomord phic to Ω(M µ ), where M µ = IndS Sµ 1µ is the permutation module of Sd . (This way, Lemma 3.50 fits well with (3.35) and the diagram (3.39) as well as Exercise 3.15 below.) Exercise 3.15.

(1) Prove using (A.53), (A.54), and (A.57) that, for λ ∈ SP, Qλ =

∑

λµ mµ . 2(λ) K

µ∈P,µ≤λ

(2) Prove that the µ-weight multiplicity of the simple q(n)-module L(λ), for λ ∈ SPd (n) and µ ∈ CPd (n), is given by dim L(λ)µ = 2

(λ)+δ(λ) 2

λµ . K

Exercise 3.16. For λ ∈ SPd , write Qλ (x) = ∑µ∈P 2(λ) gλµ sµ (x), for gλµ ∈ Q. Prove:

3. Schur duality

128

(1) dim HomHd (Dλ , Ω(Sµ )) = 2

(λ)+δ(λ) 2

gλµ .

(2) As a gl(n)-module, the q(n)-module L(λ) decomposes into 

L(λ) ∼ =

2

(λ)+δ(λ) 2

gλµ L0 (µ).

µ∈P,µ≤λ,(µ)≤n

Here L0 (µ) denotes the gl(n)-module of highest weight µ. (3) gλµ = 0 unless λ ≥ µ;

gλλ = 1.

(4) gλµ ∈ Z+ . Exercise 3.17. The Jucys-Murphy elements Jk (1 ≤ k ≤ n) in Hn are defined to be Jk = ∑1≤ j N, since the case k = N is trivial. Since P(V k ) is a complete reducible GL(k)-module, we may assume without loss of generality that L is an irreducible GL(k)-module. Denote by U(k) the subgroup of GL(k) consisting of upper triangular k × k-matrices with all diagonal entries being 1. Then P(V k )U(k) is the space of highest weight vectors in P(V k ). Claim. We have P(V k )U(k) ⊆ P(V N ). Note by the standard highest weight theory that LU(k) = 0. Granting the claim, we have LU(k) ⊆ P(V k )U(k) ⊆ P(V N ), and hence, L ∩ P(V N ) = 0. Then, it follows by the irreducibility of L that L = L ∩ P(V N )GL(k) . So it remains to prove the claim. Recall dimV = N, and introduce the following Zariski-open subset in V k : Z = {(v = (v1 , . . . , vk ) ∈ V k | v1 , . . . , vN are linearly independent in V }. Given v = (v1 , . . . , vk ) ∈ Z, we can find α1 , . . . , αk−1 ∈ C such that α1 v1 + . . . + αk−1 vk−1 + vk = 0.

4. Classical invariant theory

138

Hence, there exists u ∈ U(k) such that π(u)(v1 , . . . , vk−1 , vk ) = (v1 , . . . , vk−1 , 0). For example, the following element ⎛ ⎞−1 1 0 ··· 0 α1 ⎜0 1 · · · 0 α2 ⎟ ⎜ ⎟ ⎜ .. . . . . . . .. ⎟ ⎜ . . . . ⎟ u = ⎜. ⎟ ⎜ .. . . . . . . ⎟ ⎝. . . . αk−1 ⎠ 0 ··· ··· 0 1 will do the job. Let f ∈ P(V k )U(k) . Then, f (v) = f (v1 , . . . , vk−1 , 0), for v ∈ Z. By induction on k > N, we see that f (v) = f (v1 , . . . , vN , 0, . . ., 0), for v ∈ Z. Since the polynomial function f is determined by its restriction to the Zariski-open subset Z, f (v) = f (v1 , . . . , vN , 0, . . . , 0), for all v ∈ V . This completes the proof of the claim and hence of the theorem.  Theorem 4.11 (Weyl). Let G be an arbitrary subgroup of GL(V ). Let N = dimV and assume k ≥ N. Then, P(V k )G is generated by P(V N )G GL(k) . In particular, if a subspace S generates the algebra P(V N )G , then SGL(k) generates the algebra P(V k )G . Proof. Thanks to the commuting actions of G ⊆ GL(V ) and of GL(k) on P(V k ), we see that P(V k )G is a GL(k)-module. The first part of the theorem follows by specializing Theorem 4.10 to the case L = P(V k )G and noting that P(V N )G = P(V k )G ∩ P(V N ). Denote by A the subalgebra of P(V k ) generated by SGL(k) . Then, A is contained in the algebra P(V k )G , thanks to SGL(k) ⊆ P(V k )G . On the other hand, A ⊇ S, and so the algebra A contains the algebra P(V N )G that S generates. Since it follows by definition that A is GL(k)-stable, A contains P(V N )G GL(k) , and so we can apply the first part of the theorem to conclude that A contains P(V k )G . Hence, A = P(V k )G .  The following proposition will be useful later on. Proposition 4.12. Let G be a subgroup of GL(V ), and let N = dimV . Assume that S is a subspace of P(V N )G that generates the algebra P(V N )G . Then, for k ≤ N, the algebra of invariants P(V k )G is generated by the subset Sk that consists of the restrictions to V k of elements in S. Moreover, Sk = S ∩ P(V k ) if S is GL(N)-stable. Proof. The inclusion V k → V N and the projection V N → V k induce the restriction map res : P(V N ) → P(V k ) and the inclusion P(V k ) → P(V N ), respectively. Since either of the compositions res

P(V k ) → P(V N ) → P(V k ),

res

P(V k )G → P(V N )G → P(V k )G

4.2. Polynomial FFT for classical groups

139

is the identity map, res : P(V N )G → P(V k )G is surjective. Hence, P(V k )G is generated by the restrictions Sk . Now assume that S is GL(N)-stable. In particular, TN acts on S semisimply, where Tk denotes the diagonal subgroup of GL(k), for k ≤ N. Note that TN = Tk × TN−k . Both Sk and S ∩ P(V k ) can easily be seen to be equal to the subspace of S where TN−k acts trivially, and hence they must be equal.  4.2.2. The symplectic and orthogonal groups. Let V be a vector space equipped with a non-degenerate symmetric or skew-symmetric bilinear form ω : V ×V → C. We denote by G(V, ω) the subgroup of GL(V ) that preserves the form ω, that is, G(V, ω) = {g ∈ GL(V ) | ω(gv1 , gv2 ) = ω(v1 , v2 ), ∀v1 , v2 ∈ V }. The group G(V, ω) is called an orthogonal group and is denoted by O(V ) if ω is symmetric, and it is called a symplectic group and is denoted by Sp(V ) if ω is skew-symmetric. The Lie algebra of the group G(V, ω), called the orthogonal and symplectic Lie algebras respectively, is g(V, ω) = {x ∈ gl(V ) | ω(xv1 , v2 ) = −ω(v1 , xv2 ), ∀v1 , v2 ∈ V }. Let V = CN , which will mostly be viewed as the space of column vectors. A non-degenerate symmetric (respectively, skew-symmetric) bilinear form relative to the standard basis {e1 , . . . , eN } in CN is determined by its associated non-singular symmetric (respectively, skew-symmetric) matrix J. That is, ω(v1 , v2 ) = vt1 Jv2 , for v1 , v2 ∈ CN . Note that N is necessarily even in the skew-symmetric case. When a form ω on V = CN has its associated matrix J, we have G(V, ω) = {X ∈ GL(N) | X t JX = J}. Different choices of non-singular symmetric (respectively, skew-symmetric) N × N matrices J lead to isomorphic orthogonal (respectively, symplectic) groups in different matrix forms. Some useful choices for the non-singular symmetric matrices are the identity matrix IN , the following anti-diagonal matrix ⎛ ⎞ 0 ··· 0 1 ⎜0 · · · 1 0⎟ ⎜ ⎟ (4.6) JN := ⎜ . . . . ⎟ , . . . . ⎝. . . .⎠ 1 ··· 0 0 and



0 I I 0

⎞ 0 I 0 ⎝I 0 0⎠ (for N = 2 + 1). 0 0 1 ⎛

 (for N = 2),

When we choose ω to be the standard symmetric form on V = CN corresponding to the identity matrix IN , O(V ) is simply the group O(N) of all N × N orthogonal matrices X, i.e., X t X = IN .

4. Classical invariant theory

140

On the other hand, some common choices for non-singular skew-symmetric matrices with N = 2 used to define the symplectic Lie group and algebra are ⎛ ⎞ Ω 0 ··· 0   ⎜0 Ω ··· 0⎟ 0 I ⎜ ⎟ and ⎜ . . . , . . ... ⎟ −I 0 ⎝ .. .. ⎠ 0 0 ··· Ω   0 1 where Ω = . Another common choice of a non-singular skew-symmetric −1 0 matrix is   0 J (4.7) . −J 0 The choices of (4.6) and (4.7) play prominent roles in the later Sections 5.3.2 and 5.3.1, respectively. 4.2.3. Formulation of the polynomial FFT. Let V = CN be equipped with a symmetric (respectively, skew-symmetric) form ω+ (respectively, ω− ) whose associated matrix is J+ (respectively, J− ). It will be convenient to write G± (V ) for G(V, ω± ), as this allows us to treat the + and − cases uniformly. Given k ∈ N, we denote by SM+ k = {k × k complex symmetric matrices}, SM− k = {k × k complex skew-symmetric matrices}. We have a natural identification of G± (V )-modules between V k and the space MNk of N × k complex matrices (on which G± (V ) acts by left multiplication). Define maps τ+ : V k ≡ MNk −→ SM+ k,

τ+ (X) = X t J+ X,

τ− : V k ≡ MNk −→ SM− k,

τ− (X) = X t J− X.

It can be checked that τ± (X)t = ±τ± (X), so τ± are well-defined.

The map τ± is G± (V )-equivariant, where we let G± (V ) act on SM± k trivially. ± Indeed, for g ∈ G (V ), X ∈ MNk , we have τ± (gX) = (gX)t J± (gX) = X t gt J± gX = X t J± X = τ± (X). Hence, the pullback via τ± gives rise to an algebra homomorphism k G τ∗± : P(SM± k ) −→ P(V ) ±

± (V )

,

τ∗± ( f ) = f ◦ τ± ,

where P(V k )G (V ) denotes the algebra of G± (V )-invariant polynomials on V k . Indeed, it follows from the G± (V )-equivariance of τ± that the image of τ∗± lies in the G± (V )-invariant subalgebra of P(V k ).

4.2. Polynomial FFT for classical groups

141

Write X = (v1 , . . . , vk ) with each vi ∈ V = CN . It follows by definition that (X t J± X)i j = ω± (vi , v j ). Denote by the same notation the restriction to SM± k the matrix coefficients xi j on Mkk , for 1 ≤ i, j ≤ k. Introduce the G± (V )-invariant functions (i| j) on V k , for 1 ≤ i, j ≤ k, by letting (i| j)(v1 , . . . , vk ) := ω± (vi , v j ). Note that these functions are not independent in general, and (i| j) = ±( j|i) in the ± case, respectively. It follows by definitions that τ∗± (xi j )(v1 , . . . , vk ) = (i| j). Theorem 4.13 (Polynomial FFT for G± (V )). The homomorphism τ∗± : P(SM± k )→ ± (V ) ± (V ) k G k G P(V ) is surjective. Equivalently, P(V ) is generated by the functions (i| j), where 1 ≤ i ≤ j ≤ k in the + case and 1 ≤ i < j ≤ k in the − case. The proof of Theorem 4.13 will be given in the subsequent subsections. Remark 4.14. One can show that the image of τ± consists of all the matrices Z ∈ SM± k such that rank(Z) ≤ min{k, dimV }. In particular, if dimV ≥ k, then τ± is surjective. Thus, under the assumption that dimV ≥ k, τ∗± is injective and so τ∗± is actually an isomorphism. Equivalently, the algebra of G± (V )-invariant polynomials in V k is a free polynomial algebra generated by the k(k ± 1)/2 polynomials as listed in Theorem 4.13. Remark 4.15. The non-degenerate form ω± induces a canonical isomorphism of G± (V )-modules: V ∗ ∼ = V . Via this isomorphism, we can obtain from Theorem 4.13 a (seemingly more general) polynomial FFT for the algebra of G± (V )-invariant polynomials in (V ∗ )m ⊕V k . 4.2.4. From basic to general polynomial FFT. Let us now refer to the statement in the polynomial FFT for G± (V ) as formulated in Theorem 4.13 as FFT(k), for k ≥ 1. Set N = dimV as before. Assuming a basic case FFT(N) holds for now, let us complete the proof of FFT(k) for all k. Denote by Sk the subspace spanned by the functions (i| j), where 1 ≤ i, j ≤ k. FFT(N) =⇒ FFT(k) for k < N. (Indeed, the same argument below shows that FFT() =⇒ FFT(k) for all  ≥ k ≥ 1.) The subspace SN is GL(N)-stable. Moreover, SN ∩ P(V k ) is simply Sk . Now FFT(k) follows by applying FFT(N) and Proposition 4.12 for G = G± (V ) and S = SN .

4. Classical invariant theory

142

FFT(N) =⇒ FFT(k) for k > N. According to FFT(N), SN generates the algebra P(V N )G . Observe that SN GL(k) = Sk . Hence, by Weyl’s Theorem 4.11, the algebra P(V k )G is generated by Sk , whence FFT(k). This completes the proof of Theorem 4.13, modulo the proof of the basic case FFT(N). We shall prove the basic case FFT(N) as Theorem 4.16 in the next subsection. 4.2.5. The basic case. In this subsection, we shall prove the following basic case of the polynomial FFT for G± (V ) (that is, Theorem 4.13 for k = dimV ). Theorem 4.16. Let N = dimV . Then, the algebra P(V N )G functions (i| j), for 1 ≤ i, j ≤ N.

± (V )

is generated by the

Proof. We proceed by induction on N. We treat the cases of orthogonal and symplectic groups separately, though the overall strategy of the proof is the same. The orthogonal group case. Let {e1 , . . . , eN } be the standard basis of V = CN . Take (x, y) = ∑Ni=1 xi yi , where x = (x1 , . . . , xN ), y = (y1 , . . . , yN ) in CN , and so O(V ) = O(N). Denote V1 := CN−1 × {0} ⊆ V and denote by (·, ·)1 the bilinear form on V1 obtained by restriction from (·, ·) on V . For N = 1, O1 = {±1}, Theorem 4.16 is clear. Assume now that N > 1. We have an orthogonal decomposition V = V1 ⊕ CeN with respect to (·, ·). Let f ∈ P(V N )O(N) . Since −IN ∈ O(N), f (X) = f (−IN · X) = f (−X), f cannot contain a nonzero odd degree term. Without loss of generality, let us assume that f is homogeneous of even degree 2d. Below, we will consider the restriction of f to the Zariski open subset of vectors v = (v1 , . . . , vN ) of V N given by the inequality (vN , vN ) = 0. Let κ ∈ C such that κ2 = (vN , vN ). Since (vN , vN ) = (κeN , κeN ), there exists an orthogonal matrix g ∈ O(N) such that gvN = κeN . Set (4.8)

vi = κ−1 gvi ,

(i = 1, . . . , N − 1).

Then, f (v) = f (gv) = κ2d f (v1 , . . . , vN−1 , eN ) = (vN , vN )d f (v1 , . . . , vN−1 , eN ). Let (4.9)

vi = vi + ti eN

(i = 1, . . . , N − 1),

be the orthogonal decomposition in V = V1 ⊕ CeN for vi ∈ V1 . Using (4.8), we compute that (4.10)

ti = (vi , eN ) = (κ−1 gvi , κ−1 gvN ) = (vi , vN )/(vN , vN ).

4.2. Polynomial FFT for classical groups

143

i

N−1 Writing t I = t1i1 · · ·tN−1 for multi-indices I = (i1 , . . . , iN−1 ) and regarding ti as independent variables, we have a decomposition of the following form:

(4.11)

f (v1 , . . . , vN−1 , eN ) = ∑ fˆI (v1 , . . . , vN−1 )t I , I

where fˆI are polynomial functions uniquely determined by f . Noting from (4.9) and (4.10) that ti is invariant under the transformation vi → hvi for h ∈ O(N − 1), we conclude from (4.11) that fˆI is O(N − 1)-invariant for each I. By induction hypothesis, there exist polynomials ϕI in the functions (i| j), 1 ≤ i ≤ j ≤ N − 1, such that fˆI = ϕI when evaluated on (v1 , . . . , vN−1 ). Using (4.8), (4.9), and (4.10), we compute that (4.12)

(vi , vj ) = (vi , vj ) − (ti eN ,t j eN ) =

(vi , v j )(vN , vN ) − (vi , vN )(v j , vN ) . (vN , vN )2

Let us recall that we have been considering the restriction of f to the open subset defined by (vN , vN ) = 0. Plugging (4.12) into fˆI = ϕI , we have f (v) = (vN , vN )−p F1 for some polynomial F1 in (vi , v j ), i ≤ j ≤ N, and some positive integer p whenever (vN , vN ) = 0. We will be done if we can show that the polynomial function (vN , vN ) p divides F1 . Indeed, the same type of argument above allows us to conclude that f (v) = (v1 , v1 )−q F2 for some polynomial F2 in (vi , v j ), i ≤ j ≤ N, and some positive integer q whenever (v1 , v1 ) = 0. In this way, we obtain a polynomial equation (v1 , v1 )q F1 = (vN , vN ) p F2 , which holds on a Zariski open subset of V N given by the inequalities (v1 , v1 ) = 0 and (vN , vN ) = 0, and hence the polynomial equation must hold on V N . This implies (vN , vN ) p divides F1 , since the polynomials (vN , vN ) and (v1 , v1 ) are relatively prime. The symplectic group case. Let V = CN be equipped with the standard basis {e1 , . . . , eN }, where N = 2 is even. For definiteness, let us take the symplectic group Sp(N) = Sp(V ) defined via the symplectic form ω on V : 

ω(x, y) = ∑ (x2i−1 y2i − x2i y2i−1 ), i=1

where x = (x1 , . . . , xN ), y = (y1 , . . . , yN ) in CN . We have an orthogonal decomposition V = V1 ⊕W with respect to the symplectic form ω, where V1 := CN−2 × {0} ⊆ V,

W := CeN−1 ⊕ CeN .

We shall denote by ω1 the bilinear form on V1 induced from ω on V and regard Sp(N − 2) = Sp(V1 ) as a natural subgroup of Sp(N) in this way. Let f ∈ P(V N )Sp(N) . Since −IN ∈ Sp(N), we may assume without loss of generality that f is homogeneous of even degree 2d, just as for the O(N) case

4. Classical invariant theory

144

above. Below, we will consider the restriction of f to the Zariski open subset of vectors v = (v1 , . . . , vN ) of V N given by the inequality ω(vN−1 , vN ) = 0. Choose κ ∈ C such that κ2 = ω(vN−1 , vN ). Since ω(eN−1 , eN ) = 1 and so ω(vN−1 , vN ) = ω(κeN−1 , κeN ), it is standard to show that there exists g ∈ Sp(N) such that gvN−1 = κeN−1 , gvN = κeN . Set vi = κ−1 gvi ,

(4.13)

(i = 1, . . . , N − 2).

Then, f (v) = f (gv) = κ2d f (v1 , . . . , vN−2 , eN−1 , eN ) = ω(vN−1 , vN )d f (v1 , . . . , vN−2 , eN−1 , eN ). When N = 2, we obtain a polynomial equation f (v1 , v2 ) = f (e1 , e2 )ω(v1 , v2 )d , which holds on a Zariski open set ω(v1 , v2 ) = 0 and hence holds everywhere. This proves the theorem for N = 2. Now assume N > 2. Let vi = vi + si eN−1 + ti eN

(4.14)

(i = 1, . . . , N − 2),

be the orthogonal decomposition in V = V1 ⊕ W for vi ∈ V1 . It follows by (4.13) that  si = ω(vi , eN ) = ω(κ−1 gvi , κ−1 gvN ) = ω(vi , vN )/ω(vN−1 , vN ), (4.15) ti = −ω(vi , eN−1 ) = −ω(vi , vN−1 )/ω(vN−1 , vN ). j

j

i

N−2 i1 N−2 Write sJ t I = s11 · · · sN−2 t1 · · ·tN−2 associated to the multi-indices I = (i1 , . . . , iN−2 ) and J = ( j1 , . . . , jN−2 ). Regarding si and ti as independent variables, we have a decomposition of the following form: (4.16) f (v1 , . . . , vN−2 , eN−1 , eN ) = ∑ fˆIJ (v1 , . . . , vN−2 )sJ t I ,

I,J

where fˆIJ are polynomial functions uniquely determined by f . Noting from (4.15) that si ,ti are invariant under the transformation vi → gvi for g ∈ Sp(N − 2), we conclude from (4.16) that fˆIJ is Sp(N − 2)-invariant for every I, J. By inductive assumption, there exist polynomials ϕIJ in the functions (i| j), 1 ≤ i ≤ j ≤ N − 2, such that fˆIJ = ϕIJ when evaluated at (v1 , . . . , vN−2 ). Using (4.13), (4.14), and (4.15), we compute that ω(vi , vj ) = ω(vi , vj ) + ti s j − sit j (4.17)

=

ω(vi , v j )ω(vN−1 , vN ) − ω(vi , vN−1 )ω(v j , vN ) + ω(vi , vN )ω(v j , vN−1 ) . ω(vN−1 , vN )2

Recall that we have been considering the restriction of f to the open subset defined by ω(vN−1 , vN ) = 0. Plugging (4.17) into fˆIJ = ϕIJ , we have f (v) = ω(vN−1 , vN )−p F

4.3. Tensor and supersymmetric FFT for classical groups

145

for some polynomial F in ω(vi , v j )i≤ j≤N and some positive integer p whenever ω(vN−1 , vN ) = 0. Repeating the same trick used in the orthogonal group case, we see that the polynomial function ω(vN−1 , vN ) p divides F. 

This completes the proof of the theorem.

4.3. Tensor and supersymmetric FFT for classical groups In this section, we will use the polynomial FFT for classical groups to derive the corresponding tensor FFT. Finally, the supersymmetric algebra version of FFT is derived from the tensor FFT. 4.3.1. Tensor FFT for classical groups. Let G = G(V, ω), where V is a vector space equipped with a non-degenerate symmetric or skew-symmetric bilinear form ω. Since V ∼ = V ∗ as a G-module, it suffices to consider the G-invariants on the tensor space V ⊗m rather than a mixed tensor space V ⊗m ⊗ (V ∗ )⊗n , for m, n ≥ 0. Observe that −IV ∈ G acts by the scalar multiple by (−1)m on V ⊗m . Hence, (V ⊗m )G = 0 unless m = 2k is even. Via the canonical identification of G-modules: V ⊗V ∼ = V ∗ ⊗V ∼ = End(V ), we obtain a natural G-module isomorphism: ∼

T : V ⊗2k −→ End(V ⊗k ), which maps u = v1 ⊗ v2 ⊗ . . . ⊗ v2k to T (u) defined as follows: T (u)(x1 ⊗ . . . ⊗ xk ) := ω(x1 , v2 )ω(x2 , v4 ) . . . ω(xk , v2k )v1 ⊗ v3 ⊗ . . . ⊗ v2k−1 . Clearly the identity IV ⊗k is G-invariant, and hence so is T −1 (IV ⊗k ). We describe T −1 (IV ⊗k ) explicitly as follows. Take a basis { f i } for V and its dual basis { f i } for V with respect to ω, i.e., ω( fi , f j ) = δi j . Define N

θ = ∑ fi ⊗ f i , i=1

θk = θ ⊗ . . . ⊗ θ . k

Then T −1 (IV ⊗k ) = θk . Recall from Section 3.2 the following commuting actions: Φ2k

Ψ2k

GL(V )  V ⊗2k  S2k . Since G is a subgroup of GL(V ) that commutes with S2k , θσ := Ψ2k (σ)(θk ) is G-invariant for any σ ∈ S2k .

4. Classical invariant theory

146

Theorem 4.17 (Tensor FFT for G(V, ω)). Let G = G(V, ω), where ω is a nondegenerate symmetric or skew-symmetric bilinear form on V . Then (V ⊗m )G = 0 for m odd. Moreover, (V ⊗2k )G has a spanning set {θσ | σ ∈ S2k }. The tensor FFT for G(V, ω) affords an equivalent dual reformulation on the G-invariants of (V ∗⊗m ) ∼ = (V ⊗m )∗ . We let (i| j) be the function such that (4.18)

(i| j)(v1, . . . , vm ) = ω(vi , v j ).

Denote by θ∗k = (1|2)(3|4) . . .(2k − 1|2k) the G-invariant function on (V ⊗m ) that corresponds to the identity under the iso∼ morphism T ∗ : V ∗⊗2k −→ End(V ⊗k ). Denote by Ψ∗2k the natural action of S2k on (V ∗⊗2k ), which clearly commutes with the natural action of G. Theorem 4.18 (Tensor FFT for G(V, ω), dual version). Let G = G(V, ω), where ω is non-degenerate symmetric or skew-symmetric. Then (V ∗⊗m )G = 0 for m odd. Moreover, (V ∗⊗2k )G has a spanning set {Ψ∗2k (σ)(θ∗k ) | σ ∈ S2k }. Theorems 4.17 and 4.18 are equivalent, and we shall only present a detailed proof of the tensor FFT as formulated in Theorem 4.18. The strategy of the proof can be outlined as follows. First, we introduce an auxiliary polarization variable X ∈ End V to reformulate this tensor FFT problem as one involving the basic case of the polynomial FFT as in Theorem 4.16 (note End V ∼ = V N for N = dimV ). Then, by applying Theorem 4.16, we transfer the problem at hand to an FFT problem for GL(V ), for which Theorem 4.2 applies. Proof of Theorem 4.18. Since −IV ∈ G acts on V ⊗m by a multiple of (−1)m , we have (V ⊗m )G = 0, for m odd. Now let m = 2k be even. We shall define an injective linear map Φ : V ∗⊗m −→ Pm (End V ) ⊗V ∗⊗m ,

λ → Φλ ,

where Φλ is defined as a function on End V × V ⊗m , which is of polynomial degree m on End V and linear on V ⊗m , by letting Φλ (X, w) = λ, X ⊗m w, for X ∈ End V, w ∈ V ⊗m . The injectivity follows since λ is recovered as Φλ (IV , −). The space Pm (End V ) ⊗V ∗⊗m carries commuting actions of the groups G and GL(V ) as follows: g. f (X, w) = f (g−1 X, w) for g ∈ G; h. f (X, w) = f (Xh, h−1 w) for h ∈ GL(V ). Note that Φ is a G-equivariant map, i.e., g.Φλ = Φg.λ , for g ∈ G. Also note that Φλ is GL(V )-invariant, i.e., h.Φλ = Φλ , for h ∈ GL(V ). Hence, we obtain by restriction of Φ the following injective linear map (which can actually be shown to be an isomorphism, though this stronger statement is not needed below) GL(V )  Φ : (V ∗⊗m )G −→ (Pm (End V ))G ⊗V ∗⊗m .

4.3. Tensor and supersymmetric FFT for classical groups

147

For notational convenience below, let us fix V = CN (regarded as a space of column vectors), and ω(x, y) = xt Jy, for x, y ∈ V . In this way, we identify End V as the space MNN of N × N matrices. The action of G on End V relevant to Φ above gives rise to a G-equivariant isomorphism End V ∼ = V N . When applying the basic case of the polynomial FFT (Theorem 4.16), we obtain, for λ ∈ (V ∗⊗m )G , that Φλ (X, w) = Fλ (X t JX, w), where Fλ is a polynomial function on SMN± ×V ⊗2k , which is of polynomial degree k on SMN± and linear on V ⊗2k (here we recall m = 2k). We have a natural isomorphism SMN+ ∼ = S2 (V ∗ ) in the case of symmetric ω, and −∼ 2 ∗ SMN = ∧ (V ) in the case of skew-symmetric ω. Recall that the action of GL(V ) ∼ MNN relevant to Φ above is given by left multiplication on MNN . on End V =

This action induces an action of GL(V ) on SMN± , which corresponds precisely to the GL(V )-action on S2 (V ∗ ) (respectively, ∧2 (V ∗ )) coming from the GL(V )module structure on V ∗ . Hence, we have a GL(V )-invariant polynomial function Fλ : S2 (V ∗ ) ×V ⊗2k → C in the case of symmetric ω (or Fλ : ∧2 (V ∗ ) ×V ⊗2k → C in the case of skew-symmetric ω), which is linear on V ⊗2k and of polynomial degree k on S2 (V ∗ ) (or ∧2 (V ∗ )). Thus, by the tensor FFT for GL(V ) (see Theorem 4.2), Fλ can be written as a linear combination of the functions   j j (M, w) → ∑ ui1 Mi j v1 · · · ∑ uik Mi j vk , i, j

i, j

where w is the tensor product of u1 , v1 , . . . , uk , vk ∈ CN in some order, ui1 ’s are the coordinates of u1 , and so on. Evaluating Fλ at M = J, we deduce that λ = Fλ (J, −) is a linear combination of the (i| j)’s.  4.3.2. From tensor FFT to supersymmetric FFT. Let W = W0¯ ⊕W1¯ be a finitedimensional vector superspace. Recall the supersymmetric algebra of W : S(W ) = S(W0¯ ) ⊗ ∧(W1¯ ). Let G = GL(V ). We set (4.19)

W0¯ = V m ⊕V ∗n ,

W1¯ = V k ⊕V ∗ .

As a GL(V )-module, we have a decomposition S(W ) = ⊕r≥0 Sr (W ) by the total degree, where Sr (W ) as a GL(V )-module is a direct sum of the following tensor spaces (which are also GL(V )-submodules) (4.20)

Sαβγδ :=Sα1 (V ) ⊗ . . . ⊗ Sαm (V ) ⊗ Sβ1 (V ∗ ) ⊗ . . . ⊗ Sβn (V ∗ ) ⊗ ∧γ1 (V ) ⊗ . . . ⊗ ∧γk (V ) ⊗ ∧δ1 (V ∗ ) ⊗ . . . ⊗ ∧δ (V ∗ ),

where the summation is taken over all nonnegative integers αa , βb , γc , and δd such that |α| + |β| + |γ| + |δ| = r. Here we have denoted α = (α1 , . . . , αm ), |α| = ∑a αa , similarly for β, |β|, and so on.

4. Classical invariant theory

148

We now fix α, β, γ, δ, and want to describe a spanning set for (Sαβγδ )GL(V ) . To that end, we regard the space (or the GL(V )-module) Sαβγδ as a subspace of the following mixed tensor space (or GL(V )-module) (4.21)

T αβγδ :=V ⊗α1 ⊗ . . . ⊗V ⊗αm ⊗ (V ∗ )⊗β1 ⊗ . . . ⊗ (V ∗ )⊗βn ⊗V ⊗γ1 ⊗ . . . ⊗V ⊗γk ⊗ (V ∗ )⊗δ1 ⊗ . . . ⊗ (V ∗ )⊗δ

(which is isomorphic to V ⊗(|α|+|γ|) ⊗ (V ∗ )⊗(|β|+|δ|) ). Hence, the space (Sαβγδ )GL(V ) can be viewed as a subspace of (T αβγδ )GL(V ) . By the tensor FFT for GL(V ) (Theorem 4.2), we will assume below that |α| + |γ| = |β| + |δ| = N for some N, for otherwise (T αβγδ )GL(V ) = 0. By the same tensor FFT again, the GL(V )-invariants of T αβγδ are spanned by Θσ for σ ∈ SN , and each Θσ as defined in (4.3) is a suitable tensor product of N degree-two invariants that arise from various pairings between V and V ∗ such that each copy of V and V ∗ in T αβγδ is used exactly in one such pairing. The subspace (Sαβγδ )GL(V ) is obtained from the space (T αβγδ )GL(V ) by a (par− tial) supersymmetrization. Applying the Sα ×Sβ ×S− γ ×Sδ -supersymmetrization operator ϖαβγδ to the Θσ ’s gives us a spanning set {ϖαβγδ (Θσ )} for (Sαβγδ )GL(V ) , where Sα = Sα1 × . . . × Sαm and by supersymmetrization we mean the symmetrization over the subgroup Sα × Sβ and the anti-symmetrization (denoted by the minus sign superscripts above) over the subgroup Sγ × Sδ . Denote the degreetwo GL(V )-invariants of S(W ) = S(V m ⊕V ∗n ) ⊗ ∧(V k ⊕V ∗ ), which are straightforward variants of the contractions  j|i∗  introduced in (4.4), by (4.22)

a|b∗ ||,

a|||d ∗ ,

|b∗ |c|,

||c|d ∗ ,

where 1 ≤ a ≤ m, 1 ≤ b ≤ n, 1 ≤ c ≤ k, 1 ≤ d ≤ . Here in the notation of (4.22) and similar ones below, we have used a vertical line | to indicate the location of the chosen pairs of V and V ∗ in S(V m ⊕V ∗n ⊕V k ⊕V ∗ ). From the tensor product form of Θσ we derive that ϖαβγδ (Θσ ) ∈ (Sαβγδ )GL(V ) is the corresponding product of N invariants among (4.22). The cases for G = O(V ) or Sp(V ) can be pursued in an analogous manner. Set (4.23)

W0¯ = V m ,

W1¯ = V k .

We do not need V ∗ here thanks to V ∗ ∼ = V as a G-module. We have a similar decomposition of the G-module S(W ) = ⊕α,γ Sαγ , where Sαγ is modified from (4.20) with β and δ being zero. The space Sαγ is a subspace (and also a G-submodule) of the tensor space T αγ , where T αγ is modified from (4.21) with β and δ being zero. Hence, the space (Sαγ )G can be viewed as a subspace of (T αγ )G . By the tensor FFT (Theorem 4.17), we may assume below that |α| + |γ| is even, say, equal to 2N. Then by the same FFT again, the G-invariants of T αγ are spanned by θσ for σ ∈ S2N , and each θσ is a suitable tensor product of N degree two invariants that arise from pairings V ⊗V so that each V in T αγ is used exactly in one such pairing.

4.4. Exercises

149

The subspace (Sαγ )G is obtained from the space (T αγ )G by a (partial) supersymmetrization. Hence, applying the Sα × S− γ -supersymmetrization operator ϖαγ to the θσ ’s gives us a spanning set {ϖαγ (θσ )} for (Sαγ )G . Denote the degree-two Ginvariants in Sαγ , which are straightforward variants of (i| j) introduced in (4.18), by (4.24)

(a|c),

(a; a |),

(|c; c ),

where 1 ≤ a ≤ a ≤ m and 1 ≤ c ≤ c ≤ k. Note in addition that (a; a|) = 0 for G = Sp(V ) and (|c; c) = 0 for G = O(V ). Hence, we derive from the tensor product form of θσ that ϖαγ (θσ ) is a product of N invariants among (a|c), (a; a|), (|c; c ). Summarizing, we have established the following. Theorem 4.19 (Supersymmetric FFT). Let G = GL(V ), O(V ), or Sp(V ). Let W be as given in (4.19) for GL(V ) and in (4.23) for O(V ) or Sp(V ), respectively. The algebra of G-invariants in the supersymmetric algebra S(W ) is generated by its quadratic invariants given in (4.22) for GL(V ) and in (4.24) for O(V ) or Sp(V ), respectively. Remark 4.20. Theorem 4.19 has two most important specializations: (1) When W1¯ = 0, the supersymmetric algebra on W reduces to the symmetric algebra S(W0¯ ). The corresponding FFT admits an equivalent reformulation as a polynomial FFT that describes the G-invariants in a polynomial algebra on a direct sum of copies of V (and V ∗ ). (2) When W0¯ = 0, the supersymmetric algebra on W reduces to the exterior algebra ∧(W1¯ ). The G-invariants in this case are known as skew invariants.

4.4. Exercises Exercise 4.1. Prove that the image of τ : Hom(Ck ,V ) ⊕ Hom(V, Cm ) → Mmk ,

τ(x, y) = yx,

consists of all the matrices Z ∈ Mmk such that rank(Z) ≤ min{m, k, dimV }. Exercise 4.2. Consider the cyclic diagonal subgroup Cn = {diag (ξ, ξ−1 ) | ξn = 1} of SL(2) and identify the symmetric algebra S(C2 ) with C[x, y]. Find a presentation in terms of generators and relations for the algebra of invariants C[x, y]Cn . Exercise 4.3. Let V be a complex vector space of dimension n. For A ∈ End(V ), consider its characteristic polynomial n

det(tI − A) = t n + ∑ (−1)i ei (A)t n−i . i=1

Also introduce the polynomial function tri : End(V ) → C, A → tr Ai , for i ≥ 1. Prove that:

4. Classical invariant theory

150

(1) The ei and tri , for 1 ≤ i ≤ n, are invariant polynomials on End(V ). (2) The algebra of invariants for the conjugation action of GL(V ) on End(V ) is isomorphic to a polynomial algebra generated by e1 , . . . , en . (3) The algebra of invariants for the conjugation action of GL(V ) on End(V ) is isomorphic to a polynomial algebra generated by tr1 , . . . , trn . (Hint: Establish connection to symmetric polynomials in n variables and note that the set of diagonalizable n × n matrices is Zariski dense in Mnn .) Exercise 4.4. Let U and V be two finite-dimensional simple GL(n)-modules, and assume that U ∼ = V as SL(n)-modules. Then U ∼ = V ⊗ detr for some r ∈ Z as GL(n)-modules, where det : GL(n) → C is the determinant GL(n)-module.

Notes Section 4.1. The materials here are mostly classical and standard. Weyl’s book [131] on invariant theory was influential, where the terminology of First Fundamental Theorem (FFT) for invariants was introduced. See Howe [51, 52]; also see the notes of Kraft-Procesi [74]. Theorem 4.1 on finite generation of an invariant subalgebra of the supersymmetric algebra is somewhat novel, as it is usually formulated for a polynomial algebra only. Section 4.2. The results here are classical. Our proof of Weyl’s theorem follows Kraft-Procesi [74]. Weyl’s theorem allows us to reduce the proof of the polynomial FFT for classical groups in general to a basic case. The basic case is then proved by induction on dimensions, and our presentation here does not differ much from Goodman-Wallach [46]. In [46], the general polynomial FFT was derived using the tensor FFT instead of Weyl’s theorem, while in Weyl [131], the Cappelli identity was used essentially in the proof of the polynomial FFT. Section 4.3. The tensor FFT for classical groups can be found in Weyl [131]. The quick proof of the tensor FFT for the orthogonal groups here is due to AtiyahBott-Patodi [3], and it works for the symplectic groups similarly (also see a presentation in Goodman-Wallach [46]). The supersymmetric FFT for classical groups, which is a common generalization of the polynomial FFT and skew FFT, is taken from Howe [51]. The supersymmetric FFT will be used in the next chapter on Howe duality for Lie superalgebras. The exercises are pretty standard.

Chapter 5

Howe duality

The goal of this chapter is to explain Howe duality for classical Lie groups, Lie algebras, and Lie superalgebras, and applications to irreducible character formulas for Lie superalgebras. Like Schur duality, Howe duality involves commuting actions of a pair of Lie group G and Lie superalgebra g on a supersymmetric algebra S(U), and S(U) is naturally an irreducible module over a Weyl-Clifford algebra WC. In the main examples, G is one of the three classical groups GL(V ), Sp(V ), or O(V ), g is a classical Lie algebra or superalgebra, and U is a direct sum of copies of the natural G-module and its dual. According to the FFT for classical invariant theory in Chapter 4, when applied to the G-action on the associated graded gr WC, the basic invariants generating (gr WC)G turn out to form the associated graded space for a Lie (super)algebra g . From this it follows that the algebra of G-invariants WCG is generated by g . The most important cases of S(U) are the two cases of a symmetric algebra and of an exterior algebra, corresponding respectively to U1¯ = 0 and U0¯ = 0. In these two cases, the Howe dualities only involve Lie groups and Lie algebras, and they may be regarded as Lie theoretic reformulations of the classical polynomial invariant and skew invariant theories, respectively, treated in Chapter 4. Howe duality provides a realization of an important class of irreducible modules, called oscillator modules, of the Lie (super)algebra g . The irreducible Gmodules and irreducible oscillator g -modules appearing in the decomposition of S(U) are paired in “duality”, which is much stronger than a simple bijection. Character formulas for the irreducible oscillator g -modules are then obtained via a comparison with Howe duality involving classical groups G and infinite-dimensional Lie algebras, which we develop in detail.

151

5. Howe duality

152

5.1. Weyl-Clifford algebra and classical Lie superalgebras In this section, we introduce a mixture of Weyl algebra and Clifford algebra uniformly in the framework of superalgebras. We formulate the basic properties of this Weyl-Clifford algebra and its connections to classical Lie algebras and superalgebras. We then establish a general duality theorem that works particularly well in the setting of Weyl-Clifford algebra. 5.1.1. Weyl-Clifford algebra. Recall that for a superspace U = U0¯ ⊕ U1¯ , we denote the supersymmetric algebra by S(U) = S(U0¯ ) ⊗ ∧(U1¯ ). In a first round of reading of the constructions below, the reader is advised to understand first the two special cases with U1¯ = 0 and then U0¯ = 0, respectively. The superspace S(U) is spanned by monomials of the form u1 . . . u p w1 . . . wq , for u1 , . . . , u p ∈ U0¯ and w1 , . . . , wq ∈ U1¯ . Note that ui u j = u j ui , ui wk = wk ui , and wk w = −w wk . The space S(U) is Z×Z2 -graded by letting the degree of a nonzero element u1 . . . u p w1 . . . wq be (p + q, q), ¯ where q¯ denotes q modulo 2. For x ∈ U0¯ (respectively, x ∈ U1¯ ), we define the left multiplication operator of ¯ (respectively, (1, 1)) ¯ degree (1, 0) Mx : S(U) −→ S(U),

y → xy.

We regard U0¯∗ ⊆ U ∗ by extending u∗ ∈ U0¯∗ to an element in U ∗ by setting u∗ , w = 0 for all w ∈ U1¯ . Similarly we have U1¯∗ ⊆ U ∗ . In this way, we identify U ∗ ≡ U0¯∗ ⊕U1¯∗ as a vector superspace. For u∗ ∈ U0¯∗ , we define the even derivation ¯ and for w∗ ∈ U ¯∗ , we define the odd derivation Dw∗ of degree Du∗ of degree (−1, 0), 1 ¯ as follows: (−1, 1) Du∗ : S(U) −→S(U), ∗

Dw∗ : S(U) −→ S(U), p

u u1 . . . u p w1 . . . wq −→ ∑ u∗, ui u1 . . . uˆi . . . u p w1 . . . wq ,

D

D

∗

w u1 . . . u p w1 . . . wq −→

i=1 q

∑ (−1)k−1w∗ , wk u1 . . . u p w1 . . . wˆ k . . . wq ,

k=1

where ui ∈ U0¯ and w j ∈ U1¯ . Here and below uˆi means the omittance of ui , and so on. The linear operators Mu and Du∗ , for u ∈ U0¯ , u∗ ∈ U0¯∗ , in the superalgebra End(S(U)) defined above are even, while Mw , Dw∗ ∈ End(S(U)), for w ∈ U1¯ , w∗ ∈ U1¯∗ , are odd. We shall denote by [·, ·] the supercommutator among linear operators. Lemma 5.1. For x∗ , y∗ ∈ U ∗ and x, y ∈ U, we have [Dx∗ , Dy∗ ] = [Mx , My ] = 0,

[Dx∗ , My ] = x∗ , y1.

5.1. Weyl-Clifford algebra and classical Lie superalgebras

153

Let us reformulate. Set

U = U ⊕U ∗ . Then U is a superspace U = U 0¯ ⊕U U 1¯ with U 0¯ = U0¯ ⊕U0¯∗ ,

(5.1)

U 1¯ = U1¯ ⊕U1¯∗ .

Let ·, · be the even skew-supersymmetric bilinear form on the superspace U defined as follows: for u1 , u2 ∈ U0¯ , u∗1 , u∗2 ∈ U0¯∗ , w1 , w2 ∈ U1¯ , and w∗1 , w∗2 ∈ U1¯∗ , set u1 + u∗1 , u2 + u∗2  = u∗1 , u2  − u∗2 , u1 , w1 + w∗1 , w2 + w∗2  = w∗1 , w2  + w∗2 , w1 , u1 + u∗1 , w1 + w∗1  = 0. Define a linear map ι : U −→ End(S(U)) by letting

ι(x) = Mx , ι(x∗ ) = Dx∗ , for x ∈ U, x∗ ∈ U ∗ . It is straightforward to check that Lemma 5.1 affords the following equivalent reformulation. Lemma 5.2. We have [ι(a), ι(b)] = a, b 1,

(5.2)

for a, b ∈ U .

The subalgebra in End(S(U)) generated by ι(U U ) will be denoted by WC(U U) and referred to as the Weyl-Clifford (super)algebra. When U1¯ = 0, WC(U U ) reduces to the usual Weyl algebra W(U U 0¯ ). Choose a basis x1 , . . . , xm for the m∗ , so that S(U ) is identified with dimensional space U0¯ , with dual basis x1∗ , . . . , xm 0¯ C[x1 , . . . , xm ]. The operators Mxi can then be identified with the multiplication operators xi , while the derivations Dx∗i are identified with the differential operators ∂ U 0¯ ) is identified with the algebra of polynomial differential op∂xi ≡ ∂i . Hence, W(U erators on C[x1 , . . . , xm ]. When U0¯ = 0 and dimU1¯ = n, WC(U U ) is isomorphic to the Clifford superalgebra C2n (see Definition 3.33), which we also denote by C(U U 1¯ ). In the same spirit as above, let η1 , . . . , ηn be a basis for U1¯ , and let η∗1 , . . . , η∗n be its dual basis, so that ∧(U1¯ ) ≡ ∧(η1 , . . . , ηn ), the exterior superalgebra generated by η1 , . . . , ηn . The multiplication operators Mηi correspond to left multiplications by ηi and shall be denoted accordingly by ηi . The derivations Dη∗i correspond to the odd differential operators ∂η∂ i ≡ ði . Hence, C(U U 1¯ ) can be regarded as the superalgebra of differential operators on ∧(η1 , . . . , ηn ). For general U, there is an algebra isomorphism WC(U U) ∼ U ¯ ) ⊗ C(U U ¯) = W(U 0

1

and an identification S(U) ≡ C[x, η] := C[x1 , . . . , xm ] ⊗ ∧(η1 , . . . , ηn ). The superalgebra WC(U U ) is then identified with the superalgebra of polynomial differential operators on the superspace U, that is, the subalgebra of End(S(U))

5. Howe duality

154

generated by the multiplication operators xi , η j and the (super)derivations ∂i , ð j , for 1 ≤ i ≤ m, 1 ≤ j ≤ n. 5.1.2. A filtration on Weyl-Clifford algebra. The linear map ι|U : U → WC(U U) induces an injective algebra homomorphism κ : S(U) → WC(U U ), and similarly, ι|U ∗ : U ∗ → WC(U U ) induces an injective algebra homomorphism κ∗ : S(U ∗ ) → WC(U U ). Proposition 5.3. The map m : S(U)⊗S(U ∗ ) → WC(U U ) given by x⊗y → κ(x)κ∗ (y) is a linear isomorphism. Moreover, the algebra WC(U U ) is isomorphic to the algebra generated by ι(U U ) subject to the relation (5.2). Proof. The two statements in the proposition are equivalent. It follows by definition and (5.2) that the map m is surjective. The injectivity of m is equivalent to the claim that the “monomials” of an ordered basis for U and U ∗ form a basis of WC(U U ), and the claim can then be established by a straightforward inductive argument. Alternatively, the proposition for WC(U U ) reduces to the well-known counterparts for W(U U 0¯ ) and C(U U 1¯ ), thanks to WC(U U) ∼ U 0¯ ) ⊗ C(U U 1¯ ) (note that = W(U W(U U 0¯ ) and C(U U 1¯ ) commute).  We continue to identify WC(U U ) with the superalgebra of polynomial differential operators generated by the multiplication operators xi , η j and the (super)derivations ∂i , ð j , for 1 ≤ i ≤ m, 1 ≤ j ≤ n. By Proposition 5.3, WC(U U) α1 α β γ δ m n α α m has a linear basis {x η ∂ ð | α, γ ∈ Z+ , β, δ ∈ Z2 }, where x = x1 · · · xm for α = (α1 , . . . , αm ) and ηβ , ∂γ , ðδ are defined similarly. Define the degree deg xα ηβ ∂γ ðδ = |α| + |β| + |γ| + |δ|, where |α| = α1 + . . . + αm and |β|, |γ|, |δ| are similarly defined. The Weyl-Clifford algebra WC(U U ) carries a natural filtration {WC(U U )k }k≥0 α β γ δ by letting WC(U U )k be spanned by the elements x η ∂ ð of degree no greater than k. Clearly, WC(U U )k WC(U U ) ⊆ WC(U U )k+ , and so WC(U U ) is a filtered algebra. Actually, W(U U 0¯ ) and C(U U 1¯ ) are filtered subalgebras of the filtered algebra WC(U U ), and WC(U U ) = W(U U 0¯ ) ⊗ C(U U 1¯ ) as a tensor product of filtered algebras. As usual, the associated graded algebra for WC(U U ) is defined to be gr WC(U U) =

∞ 

WC(U U )k /WC(U U )k−1 .

k=0

For T = ∑α,β,γ,δ cαβγδ xα ηβ ∂γ ðδ with cαβγδ ∈ C of filtered degree j, i.e., T ∈ WC(U U ) j \WC(U U ) j−1 , we define the symbol or the Weyl symbol of T to be σ(T ) =

∑

|α|+|β|+|γ|+|δ|= j

cα,β,γ,δ xα ηβ yγ ξδ ∈ S j (U U ),

5.1. Weyl-Clifford algebra and classical Lie superalgebras

155

where we denote by yi and ξi the generators in S(U U ) that correspond to ∂i and ði , respectively. We define a symbol map, for k ≥ 0, σ : WC(U U )k →

(5.3)

k 

S j (U U)

j=0

by sending T to σ(T ). Proposition 5.4. Let G be a classical subgroup of GL(U0¯ ) × GL(U1¯ ). The symbol  map σ is a G-module isomorphism from WC(U U )k to kj=0 S j (U U ) for each k. As a G-module, the associated graded algebra gr WC(U U ) is isomorphic to the supersymmetric algebra S(U U ). Proof. The G-module WC(U U )k is semisimple and hence isomorphic to the assok ciated graded j=0 WC(U U ) j /WC(U U ) j−1 . For a fixed j, the linear span of {xα ηβ ∂γ ðδ | |α| + |β| + |γ| + |δ| = j} as a G-module is isomorphic to WC(U U ) j /WC(U U ) j−1 , which in turn is isomorphic j to S (U U ). The proposition follows.  5.1.3. Relation to classical Lie superalgebras. There is an intimate relation between the Weyl-Clifford algebra and classical Lie algebras/superalgebras, which we will now formulate. Introduce G = G0¯ ⊕ G1¯ , where G0¯ = S2 (ι(U U 0¯ )) ⊕ ∧2 (ι(U U 1¯ )),

G1¯ = ι(U U 0¯ ) ⊗ ι(U U 1¯ ).

Proposition 5.5. (1) The subspace G of WC(U U ) is closed under the super∼ commutator and as Lie superalgebras G = spo(U U ). In particular, G0¯ is a Lie algebra isomorphic to sp(U U 0¯ ) ⊕ so(U U 1¯ ). (2) Under the adjoint action, ι(U U ) is isomorphic to the natural representation ∼ of G = spo(U U ). Proof. We identify S2 (ι(U U 0¯ )) with the space of anti-commutators [ι(U U 0¯ ), ι(U U 0¯ )]+ , that is, the linear span of [a, b]+ = ab + ba, for a, b ∈ ι(U U 0¯ ); note by (5.2) that [a, b] = ab − ba ∈ C. Similarly, we identify ∧2 (ι(U U 1¯ )) with [ι(U U 1¯ ), ι(U U 1¯ )], the linear span of [c, d] = cd − dc, for c, d ∈ ι(U U 1¯ ); note by (5.2) that [c, d]+ ∈ C. We claim that S2 (ι(U U 0¯ )) is closed under the commutator, and moreover, we have S2 (ι(U U 0¯ )) ∼ U 0¯ ) as Lie algebras. To see this, let x1 , . . . , xm be a basis for = sp(U U0¯ . Regard xi and ∂i as elements in WC(U U ) as before. We may identify S2 (ι(U U 0¯ )) with the space spanned by elements of the form (5.4)

xi ∂ j + ∂ j xi ,

xi x j ,

∂i ∂ j ,

1 ≤ i, j ≤ m.

By a direct computation via this spanning set, one checks that S2 (ι(U U 0¯ )) is closed  under the commutator. Using that [b, a1 ] = b, a1  by (5.2), one further computes

5. Howe duality

156

that, for a, b, a1 ∈ ι(U U 0¯ ),  [a, b]+ , a1 = a[b, a1 ] + [b, a1 ]a + [a, a1 ]b + b[a, a1 ] (5.5) = 2b, a1  a + 2a, a1  b. Hence, ι(U U 0¯ ) is a representation of the Lie algebra S2 (ι(U U 0¯ )) under the adjoint action. With respect to the ordered basis {x1 , . . . , xm , ∂1 , . . . , ∂m } for ι(U U 0¯ ), we can write down the matrices of (5.4), the span of which is precisely given by the 2m × 2m matrices of the m|m-block form   A B , B = Bt ,C = Ct . C −At This shows that S2 (ι(U U 0¯ )) ∼ U 0¯ ) and that ι(U U 0¯ ) is its natural representation. = sp(U 2 Similarly, ∧ (ι(U U ¯ )) is closed under the Lie bracket, ∧2 (ι(U U ¯ )) ∼ U ¯ ) as = so(U 1

1

1

Lie algebras, and ι(U U 1¯ ) under the adjoint action of ∧2 (ι(U U 1¯ )) is its natural representation. Since the case for ∧2 (ι(U U 1¯ )) is completely parallel to the case for S2 (ι(U U 0¯ )) above, we shall omit the details except recording the following identity analogous to (5.5):  (5.6) [c, d], c1 = 2d, c1  c − 2c, c1  d, ∀c, d, c1 ∈ ι(U U 1¯ ). On the other hand, by a direct computation, we have that, for a, b ∈ U 0¯ and c, d ∈ U 1¯ , 1 [a ⊗ c, b ⊗ d]+ = ([a, b] ⊗ [c, d] + [a, b]+ ⊗ [c, d]+ ). 2 Here we recall [a, b] ∈ C and [c, d]+ ∈ C. This implies that [G1¯ , G1¯ ]+ ⊆ G0¯ . Hence G is closed under the super-commutator and so is a Lie subalgebra of WC(U U ). We further observe that G preserves the bilinear form ·, · on U and hence is a subalgebra of spo(U U ). Indeed, it follows by using (5.5) twice that, for a, b, a1 , b1 ∈ ι(U U 0¯ ), ! " " !  (5.7) [a, b]+ , a1 , b1 = − a1 , [a, b]+ , b1 . Similarly, it follows by (5.6) that, for c, d, c1 , d1 ∈ ι(U U 1¯ ), ! " " !  (5.8) [c, d], c1 , d1 = − c1 , [c, d], d1 . The remaining identity for the G-invariance of ·, · can be verified similarly (see Exercise 5.1). Since G0¯ = sp(U U 0¯ ) ⊕ so(U U 1¯ ) and G1¯ ∼ = U 0¯ ⊗ U 1¯ , it follows that ∼ G = spo(U U ). Now (5.5) and (5.6) together with [[a, b]+ , c] = [[c, d], a] = 0 show that ι(U U ) is indeed the natural representation of G under the adjoint action.  Proposition 5.6. The adjoint action of the Lie superalgebra G on WC(U U ) preserves the filtration on WC(U U ). The action of G0¯ can be lifted to an action by automorphisms of the group Sp(U U 0¯ ) × O(U U 1¯ ) on WC(U U ).

5.1. Weyl-Clifford algebra and classical Lie superalgebras

157

Proof. The first statement follows from Proposition 5.5(2) and the definition of the filtration on WC(U U ). Clearly the adjoint action of G0¯ on ι(U U ) lifts to an action of the group Sp(U U 0¯ ) × O(U U 1¯ ). As each finite-dimensional filtered subspace WC(U U )k for k ≥ 1 is preserved by the adjoint action of G0¯ , the G0¯ -module WC(U U) is semisimple and hence isomorphic to S(U U ) by a standard induction argument. The second statement follows.  5.1.4. A general duality theorem. In this subsection, we establish a duality theorem in a general setting, which will be used subsequently. Let G be one of the classical Lie groups GL(V ), Sp(V ), or O(V ). Assume (ρ, L) is a rational representation of G of countable dimension, which in particular means that it is a direct sum of finite-dimensional simple G-modules. Assume that a subalgebra R ⊆ End(L) satisfies: (i) R acts irreducibly on L. (ii) For g ∈ G, T ∈ R, we have ρ(g)T ρ(g)−1 ∈ R. That is, R is a G-module by conjugation. (iii) As a G-module, R is a direct sum of finite-dimensional irreducible Gmodules. As we shall see, in the main applications developed in this chapter, the algebra R is taken to be the Weyl-Clifford algebra on a sum of copies of the natural G-module V , and Conditions (i)-(iii) are easily verified. Define the algebra of G-invariants in R to be RG = {T ∈ R | ρ(g)T = T ρ(g), ∀g ∈ G}. We write the G-module L as a direct sum of finite-dimensional simple G-modules   L(λ) of the isomorphism class λ as L ∼ L(λ) ⊗ M λ , where G(ρ) denotes = λ∈G(ρ)  the set of isomorphism classes of simple G-modules appearing in (ρ, L), and M λ := HomG (L(λ), L) denotes the multiplicity space of λ. RG ,

Then M λ is naturally a left RG -module by left multiplication. Indeed, for T ∈ f ∈ M λ = HomG (L(λ), L), v ∈ L(λ), g ∈ G, we have T f (gv) = T ρ(g) f (v) = ρ(g)T f (v), Mλ .

that is, T f ∈ Hence, we have a decomposition of the CG ⊗ RG -module L RG )-module): (which we shall simply refer to as a (G,R (5.9)

L∼ =



L(λ) ⊗ M λ .

 λ∈G(ρ)

We shall need the following variant of the Jacobson density theorem, a proof of which can be found in [46, Corollary 4.1.6].

5. Howe duality

158

Proposition 5.7. Let L be a vector space of countable dimension. Assume that a subalgebra R of End(L) acts on L irreducibly. Then, for any finite-dimensional subspace X of L, we have R|X = Hom(X, L). Theorem 5.8. Assume that (ρ, L) is a representation of a classical group G of countable dimension, which is a direct sum of finite-dimensional simple G-modules. Further assume that G, R, L satisfy Conditions (i)-(iii). Then the M λ ’s, for λ ∈  G(ρ), are pairwise non-isomorphic irreducible RG -modules. Proof. We shall need the following G-invariant version of Proposition 5.7. Claim. Let X ⊆ L be a finite-dimensional G-invariant subspace. Then RG |X = HomG (X, L). Indeed, by G-invariance of X, Hom(X, L) is a G-module, and the restriction res map R −→ Hom(X, L) given by r → r|X is a G-homomorphism. Since R and Hom(X, L) are completely reducible G-modules, they contain RG and HomG (X, L) as direct summands of trivial submodules. We shall denote by π the respective Gequivariant projection maps. Hence we have the following commutative diagram: (5.10)

R res



Hom(X, L)

π

/ RG res

π

 / HomG (X, L)

Now, suppose that T ∈ HomG (X, L). Regarding T ∈ Hom(X, L), we may apply Proposition 5.7 to find an element r ∈ R such that r|X = T . Then we have T = π(T ) = π(r|X ) = π(r)|X , proving the claim, since π(r) ∈ RG .  Recall M λ = HomG (L(λ), L) by definition, for λ ∈ G(ρ). Take two nonzero −1 elements S, T ∈ HomG (L(λ), L). Then ψ = ST : T (L(λ)) → S(L(λ)) is an isomorphism of irreducible G-modules. It follows by the claim above that there exists u ∈ RG that restricts to ψ : T (L(λ)) → L. Hence, uT : L(λ) → S(L(λ)) is an isomorphism of irreducible G-modules. By Schur’s Lemma, we must have S = cuT for some scalar c, and this proves the irreducibility of the RG -module HomG (L(λ), L).  It remains to show that, for distinct λ, µ ∈ G(ρ), an RG -module homomorphism ϕ : HomG (L(λ), L) → HomG (L(µ), L) must be zero. To this end, we take an arbitrary element T ∈ HomG (L(λ), L). Set S := ϕ(T ), U := T (L(λ)) ⊕ S(L(µ)), and let p : U → S(L(µ)) ⊆ L be the natural projection, which is clearly G-equivariant. By the above claim, there exists r ∈ RG such that p = r|U . Then, it follows from pT = 0 that rT = 0 and then, 0 = ϕ(rT ) = rϕ(T ) = rS = pS = S. Hence we conclude that ϕ = 0 as desired.



5.1. Weyl-Clifford algebra and classical Lie superalgebras

159

We will call a decomposition of the (G, RG )-module L as in (5.9) strongly multiplicity-free, in light of Theorem 5.8. 5.1.5. A duality for Weyl-Clifford algebras. Let U = U0¯ ⊕ U1¯ be a vector superspace, and let G be a classical Lie subgroup of GL(U0¯ ) × GL(U1¯ ). In all the examples that give rise to concrete Howe dualities in the next sections, U0¯ and U1¯ are actually direct sums of the natural G-module. The induced G-module (ρ, S(U)) is semisimple. Also, the G-module WC(U U ) ⊆ End(S(U)), with G-action given by conjugation g.T = ρ(g)T ρ(g−1 ), for g ∈ G and T ∈ WC(U U ), is semisimple. Lemma 5.9. As a module over the Weyl-Clifford algebra WC(U U ), S(U) is irreducible. Proof. Identify S(U) with C[x, η], and identify WC(U U ) with the superalgebra of polynomial differential operators on U as in Section 5.1.2. Then, given any nonzero element f in C[x, η], we can find a suitable constant-coefficient differential operator D ∈ WC(U U ) such that D. f = 1. Now applying suitable multiplication operators in WC(U U ) to 1, we obtain all the monomials xα ηβ that span C[x, η]. The lemma follows.  Letting L = S(U) and R = WC(U U ), the triple (G, L, R) satisfies Conditions (i)– (iii) of Section 5.1.4. Hence Theorem 5.8 is applicable in light of Lemma 5.9 and gives us the following. Theorem 5.10. Retain the notations above. As a (G, WC(U U )G )-module, S(U) is strongly multiplicity-free, i.e.,  S(U) ∼ L(λ) ⊗ M λ . =  λ∈G(ρ)

Here the M λ ’s are pairwise non-isomorphic irreducible WC(U U )G -modules, while the L(λ)’s are pairwise non-isomorphic irreducible finite-dimensional G-modules. Recall by Theorem 4.1 that the algebra S(U U )G is finitely generated. Also recall from (5.3) the symbol map σ : WC(U U ) → S(U U ). Proposition 5.11. Let {S1 , . . . , Sr } be a set of generators of the algebra S(U U )G , where G is a classical subgroup of GL(U0¯ ) × GL(U1¯ ). Suppose T j ∈ WC(U U )G is chosen so that σ(T j ) = S j , for j = 1, . . . , r. Then {T1 , . . . , Tr } generate the algebra WC(U U )G . Proof. It follows from Proposition 5.4 that the symbol map σ restricts to an isok j morphism from WC(U U )G U )G , for each k ≥ 0. j=0 S (U k to Let a ∈ WC(U U )G k . We shall show that a is generated by T1 , . . . , Tr by induction on k. For k = 0, the claim is trivial. Suppose that k ≥ 1. By assumption σ(a) = f (S1 , S2 , . . . , Sr ), for some polynomial f . Now consider the element

5. Howe duality

160



k−1 j f (T1 , T2 , . . . , Tr ) ∈ WC(U U )G U ), and j=0 S (U k . We note that σ(a − f (T1 , . . . , Tr )) ∈ G hence a − f (T1 , . . . , Tr ) ∈ WC(U U )k−1 . By induction hypothesis, a − f (T1 , . . . , Tr ) is generated by T1 , . . . , Tr , and hence so is a. 

Proposition 5.12. Suppose that the generators T1 , . . . , Tr for the algebra WC(U U )G r  G can be chosen such that g := i=1 CTi is a Lie subalgebra of WC(U U ) under the G λ super commutator. Then the WC(U U ) -modules M ’s (see Theorem 5.10) are pairwise non-isomorphic irreducible g -modules, and we have the following (G, g )module decomposition: S(U) ∼ =



L(λ) ⊗ M λ .

 λ∈G(ρ)

Proof. By assumption, the action of g on S(U) extends to an action of the universal enveloping superalgebra U(g ), so we obtain an algebra homomorphism ρ : U(g ) → End(S(U)). The assumption that T1 , . . . , Tr generate the algebra WC(U U )G implies that ρ (U(g )) = WC(U U )G . The proposition now follows from Theorem 5.10.  We shall refer to (G, g ) in Proposition 5.12 as a Howe dual pair. In the next sections, by specializing to the case when G is a classical group and U0¯ and U1¯ are direct sums of the natural G-module, we shall be able to make an explicit choice of the generators T1 , . . . , Tr so that Proposition 5.12 can be applied. Note that S(U) = S(U0¯ ) when U1¯ = 0, and that S(U) = ∧(U1¯ ) when U0¯ = 0. In either special case, the g appearing in Proposition 5.12 will be a Lie algebra.

5.2. Howe duality for type A and type Q In this section, we first formulate Howe duality for the classical group GL(V ) and give a precise multiplicity-free decomposition as predicted by Proposition 5.12. We find explicit formulas for the joint highest weight vectors in the above decomposition, which will also be useful in Section 5.3. In addition, we formulate another Howe duality between a pair of queer Lie superalgebras and provide an explicit multiplicity-free decomposition. 5.2.1. Howe dual pair (GL(k), gl(m|n)). Take V = Ck , and hence identify GL(V ) with GL(k) and gl(V ) with gl(k). Given m, n ≥ 0, we consider the superspace U = U0¯ ⊕U1¯ with U0¯ = V m and U1¯ = V n , and identify naturally U ≡ V ⊗ Cm|n . We let e1 , . . . , ek denote the standard basis for the natural gl(k)-module Ck , and we let ei , i ∈ I(m|n), denote the standard basis for the natural module Cm|n of the general linear Lie superalgebra gl(m|n). We denote, for 1 ≤ r ≤ k, 1 ≤ a ≤ m, 1 ≤ b ≤ n, (5.11)

xra := er ⊗ ea¯ ,

ηrb := er ⊗ eb .

5.2. Howe duality for type A and type Q

161

Then the set {xra , ηrb } is a basis for U. We will denote by C[x, η] the polynomial superalgebra generated by the elements in (5.11), and from now on further identify S(U) ≡ S(V ⊗ Cm|n ) ≡ C[x, η].

(5.12)

We introduce the following first-order differential operators E pq =

(5.13)

m

∂

n

∂

∑ x p j ∂xq j + ∑ η ps ∂ηqs ,

j=1

1 ≤ p, q ≤ k.

s=1

We also introduce the following first-order differential operators Ei¯ j¯ =

k

∑ x pi

p=1

(5.14) Est =

k

∑ η ps

p=1

k ∂ ∂ , Ei¯s = ∑ x pi , ∂x p j ∂η ps p=1 k ∂ ∂ , Esi¯ = ∑ η ps , ∂η pt ∂x pi p=1

1 ≤ i, j ≤ m, 1 ≤ s,t ≤ n.

The natural commuting actions of gl(k) and gl(m|n) on V ⊗ Cm|n induce commuting actions on C[x, η]. The following lemma is standard and can be verified by a direct computation (see Exercise 5.2(1)). Lemma 5.13. The commuting actions of gl(k) and gl(m|n) on S(Ck ⊗ Cm|n ) = C[x, η] are realized by the formulas (5.13) and (5.14), respectively. We shall denote by g the linear span of the elements in (5.14). Indeed, the elements in (5.14) form a linear basis for g . Let U = U ⊕ U ∗ as before. In light of (5.12), we identify WC(U U ) with the superalgebra of polynomial differential operators on C[x, η], and so the elements in (5.14) may be regarded as elements in WC(U U ). Theorem 5.14. The g , which forms an isomorphic copy of gl(m|n), generates the algebra of GL(V )-invariants in WC(U U ). Moreover, as a (GL(k), gl(m|n))-module, k m|n S(C ⊗ C ) is strongly multiplicity-free. Proof. Set G = GL(k). Recall from Proposition 5.4 that the associated graded of the filtered algebra WC(U U ) is S(U U ), and that the action of gl(k) on S(U U ) lifts to an action of G. Let {y pi } denote the basis in U0¯∗ dual to the basis {x pi } for U0¯ , and let {ξ pt } denote the basis in U1¯∗ dual to the basis {η pt } for U1¯ . By the supersymmetric First Fundamental Theorem for GL(V ) (see Theorem 4.19), the  G algebra S(U U )G = S(U0¯ ⊕U0¯∗ ) ⊗ ∧(U1¯ ⊕U1¯∗ ) is generated by k

zi¯ j¯ =

∑ x pi y p j ,

k

zts =

p=1

(5.15)

k

zi¯t =

∑ x pi ξ pt ,

p=1

∑ η pt ξ ps ,

p=1 k

zt i¯ =

∑ η pt y pi ,

p=1

1 ≤ i, j ≤ m, 1 ≤ s,t ≤ n.

5. Howe duality

162

Observe that the symbol map σ in Section 5.1.2 sends each Eab in (5.14) to zab for a, b ∈ I(m|n), and hence by Proposition 5.11, the Eab ’s form a set of generators for WC(U U )G . Now the theorem follows from Proposition 5.12, since the Eab ’s generate the Lie superalgebra gl(m|n) by Lemma 5.13.  Remark 5.15. When n = 0, S(U) = S(U0¯ ) becomes the symmetric algebra on U0¯ = Ck ⊗Cm . When m = 0, S(U) = ∧(U1¯ ) is the exterior algebra on U1¯ = Ck ⊗Cn . In these two important specializations of Theorem 5.14, g becomes a Lie algebra, namely gl(m) and gl(n), respectively. 5.2.2. (GL(k), gl(m|n))-Howe duality. We will first find the multiplicity-free decomposition of the (GL(V ), gl(m))-module S(Ck ⊗ Cm ) explicitly. To do that, we will take advantage of Schur duality which was established in Chapter 3. As the formulation below will involve Lie algebra gl(k) for various k, we will add the index k to denote by Lk (λ) the irreducible gl(k)-module of highest weight λ. For dominant integral weight λ, the gl(k)-module Lk (λ) lifts to a GL(k)-module, which will be denoted by the same notation. Let us consider the natural action of gl(k) × gl(m) on Ck ⊗ Cm and its induced action on the dth symmetric tensor Sd (Ck ⊗ Cm ), for d ≥ 0. As usual, we identify a polynomial weight for gl(k) with a partition of length at most k. Recall Pd denotes the set of partitions of d. Theorem 5.16 ((GL(k), gl(m))-Howe duality). As a (GL(k), gl(m))-module, we have  Sd (Ck ⊗ Cm ) ∼ Lk (λ) ⊗ Lm (λ). = λ∈Pd (λ)≤min{k,m}

Proof. We shall show that Schur duality implies Howe duality. By Schur duality (see Theorem 3.11), we have   (Ck )⊗d ∼ (5.16) Lk (λ) ⊗ Sλ , (Cm )⊗d ∼ Lm (µ) ⊗ Sµ . = = λ∈Pd (λ)≤k

µ∈Pd (µ)≤m

Using (5.16) and denoting by ΔSd the diagonal subgroup in Sd × Sd , we have Sd  Sd (Ck ⊗ Cm ) ∼ = (Ck ⊗ Cm )⊗d ΔSd  ∼ = (Ck )⊗d ⊗ (Cm )⊗d ΔSd   ∼ Lk (λ) ⊗ Lm (µ) ⊗ Sλ ⊗ Sµ = λ,µ∈Pd (λ)≤k,(µ)≤m

∼ =



λ∈Pd (λ)≤min{k,m}

Lk (λ) ⊗ Lm (λ).

5.2. Howe duality for type A and type Q

163

In the last step, we have used the well-known fact that Sλ ∼ = (Sλ )∗ , and hence by Schur’s lemma (Sλ ⊗ Sµ )ΔSd ∼ = HomS (Sλ , Sµ ) ∼ = δλ,µ C. d

This completes the proof of the theorem.



Remark 5.17. It is possible to derive Schur duality from the (GL(k), gl(m))-Howe duality by making use of the well-known fact that the zero-weight subspace of the GL(d)-module Ld (λ) for a partition λ of d is naturally an Sd -module, which is isomorphic to the Specht module Sλ . Also, it is possible to derive the FFT for GL(k) from Howe duality. Hence the FFT for the general linear group, Schur duality, and Howe duality of type A are essentially all equivalent. Now let us consider the induced action of gl(k) × gl(m) on the dth exterior tensor space ∧d (Ck ⊗ Cm ), for 0 ≤ d ≤ km. Recall that λ denotes the conjugate partition of a partition λ. We have the following skew version of Howe duality. Theorem 5.18 (Skew (GL(k), gl(m))-Howe duality). As a (GL(k), gl(m))-module, we have  ∧d (Ck ⊗ Cm ) ∼ Lk (λ) ⊗ Lm (λ ). = λ∈Pd (λ)≤k,(λ )≤m

Proof. We shall use Schur duality to derive the skew Howe duality. For an Sd module M, let us denote by M Sd ,sgn the submodule of M that transforms by the sign character. Using (5.16), we have Sd ,sgn  ∧d (Ck ⊗ Cm ) ∼ = (Ck ⊗ Cm )⊗d ΔSd ,sgn  ∼ = (Ck )⊗d ⊗ (Cm )⊗d ΔSd ,sgn   ∼ Lk (λ) ⊗ Lm (µ) ⊗ Sλ ⊗ Sµ = λ,µ∈Pd (λ)≤k,(µ)≤m

∼ =



Lk (λ) ⊗ Lm (λ ).

λ∈Pd (λ)≤k,(λ )≤m  In the last step, we have used the isomorphism Sµ ⊗ sgn ∼ = Sµ from (A.29), and hence  ΔSd ∼ (Sλ ⊗ Sµ )ΔSd ,sgn ∼ = Sλ ⊗ Sµ ⊗ sgn = HomSd (Sλ , Sµ ⊗ sgn) ∼ = δλ,µ C.

The theorem is proved.



5. Howe duality

164

The natural action of gl(k) × gl(m|n) on Ck ⊗ Cm|n induces a natural action on the supersymmetric algebra S(Ck ⊗ Cm|n ). Recall from (2.4) the weight λ associated to an (m|n)-hook partition λ. The following theorem, which is a generalization of both Theorems 5.16 and 5.18, provides an explicit form of the strongly multiplicity-free decomposition in Theorem 5.14. Theorem 5.19 ((GL(k), gl(m|n))-Howe duality). As a (GL(k), gl(m|n))-module, we have  Sd (Ck ⊗ Cm|n ) ∼ Lk (λ) ⊗ Lm|n (λ ). = λ∈Pd (λ)≤k,λm+1 ≤n

The proof of Theorem 5.19 is completely analogous to the one for Theorem 5.16, now using the Schur-Sergeev duality (Theorem 3.11) which we recall: as (gl(m|n), Sd )-modules, (Cm|n )⊗d ∼ = ⊕λ∈Pd ,λm+1 ≤n L(λ ) ⊗ Sλ . We leave this as an exercise (see Exercise 5.3). Recall that the character of the irreducible gl(k)-module Lk (λ) is given by the Schur function sλ , while the character of the irreducible gl(m|n)-module Lm|n (λ ) is given by the super Schur function hsλ (x; y) (see Theorem 3.15). Now by computing the characters of both sides of the isomorphism in Theorem 5.19 and summing over all d ≥ 0, we immediately recover the super Cauchy identity (A.41) in Appendix A: ∏ j,k (1 + y j zk ) = ∑ hsλ (x; y)sλ (z). ∏i,k (1 − xi zk ) λ∈P By setting the variables y j (respectively, xi ) to zero and noting that hsλ (x; 0) = sλ (x) (respectively, hsλ (0; y) = sλ (y)), we further recover the classical Cauchy identities as given in (A.11), which correspond to the Howe duality in Theorem 5.16 and the skew-Howe duality in Theorem 5.18. Remark 5.20. When we derive the (GL(k), gl(m|n))-Howe duality from the SchurSergeev duality, we could have replaced GL(k) by the Lie algebra gl(k) without any change. Moreover, using Schur-Sergeev duality in the same fashion, one can derive a more general (gl(k|), gl(m|n))-Howe duality on S(Ck| ⊗ Cm|n ). 5.2.3. Formulas for highest weight vectors. In this subsection, we shall find explicit formulas for the joint highest weight vectors in the (GL(k), gl(m|n))-Howe duality decomposition of Theorem 5.19. These formulas will play a key role in the subsequent section on Howe duality for symplectic and orthogonal groups. Recalling the notation from (5.13) and (5.14) that the simple root vectors of the standard fundamental systems for gl(k) and gl(m|n) are respectively (1 ≤ i ≤ k − 1, 1 ≤ s ≤ m − 1, 1 ≤ t ≤ n − 1): (5.17)

Ei,i+1 =

m

∑ xi j

j=1

∂ ∂xi+1, j

n

+ ∑ ηis s=1

∂ ∂ηi+1,s

,

5.2. Howe duality for type A and type Q

165

(5.18) Es,s+1 = ¯

k

∂

∑ x ps ∂x p,s+1 ,

Et,t+1 =

p=1

k

∂

∑ η pt ∂η p,t+1 ,

Em,1 =

p=1

k

∂

∑ x pm ∂η p1 .

p=1

Let λ be an (m|n)-hook partition of length at most k. We are looking for the joint highest weight vector with respect to the standard Borel subalgebra of gl(k) × gl(m|n), or equivalently the vector annihilated by (5.17) and (5.18), in the summand Lk (λ) ⊗ Lm|n (λ ) in the decomposition of C[x, η]. Such a vector is unique up to a scalar multiple, thanks to the multiplicity-free decomposition in Theorem 5.19. For 1 ≤  ≤ min{k, m}, define ⎛ x11 x12 · · · ⎜x21 x22 · · · ⎜ (5.19) ♦ := det ⎜ . .. .. ⎝ .. . . x1 x2 · · ·

⎞ x1 x2 ⎟ ⎟ .. ⎟ . . ⎠ x

Note that ♦ is annihilated by all the operators in (5.17) and (5.18), and hence is a joint highest weight vector (see Exercise 5.4(1)). The column determinant of an  ×  matrix A = (ai j ) with possibly noncommuting entries is defined to be cdet A =

∑ (−1)(σ)aσ(1)1 aσ(2)2 · · · aσ().

σ∈S

In the case when m < k, we introduce the following determinant: ⎞ ⎛ x11 x12 · · · x1m η1t · · · η1t ⎜x21 x22 · · · x2m η2t · · · η2t ⎟ ⎟ ⎜ (5.20) ♦t, := cdet ⎜ . .. . . .. .. .. .. ⎟ , . ⎝ . . . . . . . ⎠ x1 x2 · · · xm ηt · · · ηt for m <  ≤ k and 1 ≤ t ≤ n, where the last ( − m) columns are filled with the same vector. For notational convenience, we set ♦t, := ♦ , for all t and 1 ≤  ≤ m. Remark 5.21. The element ♦t, is always nonzero (Exercise 5.4). It reduces to (5.19) when  ≤ m, and, up to a scalar multiple, reduces to η1t · · · ηt when m = 0. For a partition µ with µ1 ≤ k and 1 ≤ t1 ,t2 , . . . ,tµ1 ≤ n, we define (5.21)

♦t1 ,t2 ,...,tµ1 ,µ := ♦t1 ,µ1 ♦t2 ,µ2 · · · ♦tµ1 ,µµ . 1

Now let r be fixed by the conditions λr > m and λr+1 ≤ m for an (m|n)-hook partition λ. Note that we must have 0 ≤ r ≤ n. Denote by λ≤r the subdiagram of the Young diagram λ which consists of the first r columns of λ, i.e., the columns of length greater than m.

5. Howe duality

166

Lemma 5.22. The vector ♦1,2,...,r,λ≤r is a joint highest weight vector for the gl(k) × gl(m|n)-module Lk (λ≤r ) ⊗ Lm|n (λ≤r ) in the decomposition of C[x, η].

Proof. Set µ = λ≤r and ♦ = ♦1,2,...,r,µ in this proof. By Remark 5.21 and a direct computation using (5.13), (5.14), and (5.17), we can verify the following: (1) ♦ is nonzero. (2) ♦ has weight (µ, µ ) with respect to the action of gl(k) × gl(m|n). (3) Each factor ♦i,µi in ♦ is annihilated by the operators in (5.17). By the strongly multiplicity-free decomposition of Theorem 5.19 (recalling S(Ck ⊗ Cm|n ) ≡ C[x, η]), we may identify Lm|n (µ ) ≡ C[x, η]nµ1 , the space of highest weight vectors in the gl(k)-module C[x, η] of highest weight µ (here n1 denotes the nilradical of the standard Borel for gl(k)). It follows by (3) that ♦ is annihilated by the operators in (5.17), and hence ♦ ∈ C[x, η]nµ1 by (1)–(3). Since ♦ has weight µ with respect to the action of gl(m|n) by (2) again, it must be the unique (up to a scalar multiple) joint highest weight vector in C[x, η] of weight (µ, µ ) under the action of gl(k) × gl(m|n).  Theorem 5.23. Let λ be a partition such that (λ) ≤ k and λm+1 ≤ n. Then, a joint highest weight vector of weight (λ, λ ) in the gl(k) × gl(m|n)-module C[x, η] is given by ♦1,2,...,λ1 ,λ = ♦1,λ1 ♦2,λ2 · · · ♦λ1 ,λλ . 1

Proof. We have ♦1,2,...,λ1 ,λ = ♦1,2,...,r,λ≤r ♦λr+1 · · · ♦λλ . 1

Thus, it is a product of a nonzero element in C[x, η] and a nonzero element in C[x], and so it is nonzero. Also we see that it is a product of the joint highest weight vectors ♦1,2,...,r,λ≤r (by Lemma 5.22) and ♦ ,  = λr+1 , . . . , λλ1 , and hence is also a joint highest weight vector. Also it is straightforward to verify by (5.13) and (5.14) that ♦1,2,...,λ1 ,λ has the correct weight (λ, λ ).  Remark 5.24. When n = 0, the highest weight vector in Theorem 5.23 does not involve the odd generators ηi j . When m = 0, the vector becomes simply a monomial in the anti-commuting variables ηi j . 5.2.4. (q(m), q(n))-Howe duality. In this subsection, we obtain a (q(m), q(n))Howe duality using the (q(n), Hd )-Sergeev duality in Theorem 3.49, in a similar fashion as in Section 5.2.2. Recall that, by definition, q(m) consists of X ∈ gl(m|m) super-commuting with given in (1.10), i.e., q(m) consists of the m|m-block matrices of the form  P A B . Recall from Remark 1.11 that we have another nonstandard realization B A

5.2. Howe duality for type A and type Q

167

of the Lie superalgebra q(m) as  q(m) = {X ∈gl(m|m)  | XP − PX = 0}, which conA B sists of the m|m-block matrices of the form . We let q(m) act on Cm|m −B A as the nonstandard isomorphic copy  q(m), while we let q(n) act on Cn|n by linear maps in the standard way, i.e. linear maps super-commuting with P. This induces an action of q(m) × q(n) on Cm|m ⊗ Cn|n . Define the linear map PΔ : Cm|m ⊗ Cn|n → Cm|m ⊗ Cn|n by PΔ (v ⊗ w) := Pv ⊗ Pw,

∀v ∈ Cm|m , w ∈ Cn|n .

Lemma 5.25. The actions of PΔ and q(m) × q(n) on Cm|m ⊗ Cn|n commute with each other. Proof. For Z2 -homogeneous X ∈ q(m), Y ∈ q(n), v ∈ Cm|m , and w ∈ Cn|n , we have XPΔ (v ⊗ w) = X(Pv ⊗ Pw) = (XPv) ⊗ (Pw) = PXv ⊗ Pw = PΔ X(v ⊗ w), Y PΔ (v ⊗ w) = Y (Pv ⊗ Pw) = (−1)|Y |+|Y ||v| Pv ⊗Y Pw = (−1)|Y ||v| Pv ⊗ PY w = (−1)|Y ||v| PΔ (v ⊗Y w) = PΔY (v ⊗ w). 

The lemma follows.

Recall that the finite group Πd is generated by a1 , . . . , ad , z with relations (3.21), and this gives rise to the group Bd = Πd  Sd . The natural action of Πd (respectively, Bd ) on (Cn|n )⊗d factors through an action of the Clifford superalgebra Cd by Lemma 3.47 (respectively, the superalgebra Hd ), where ai acts as I ⊗i−1 ⊗P⊗I ⊗d−i with P given in (1.10). Hence the diagonal subgroups ΔΠd ⊆ Πd × Πd and ΔBd ⊆ Bd × Bd act on the tensor product (Cm|m )⊗d ⊗ (Cn|n )⊗d ∼ = (Cm|m ⊗ Cn|n )⊗d . In this subsection, it is more convenient to talk about the groups Πd and Bd than about the algebras Cd and Hd . Lemma 5.26. We have a natural identification:  Δ(Bd ) ∼ (Cm|m ⊗ Cn|n )⊗d = Sd (Cmn|mn ), 

where (·)Δ(Bd ) denotes the subspace of Δ(Bd )-invariants. Δ

Proof. Consider first the case when d = 1. We observe that (Cm|m ⊗ Cn|n )P consists of elements of the form vm ⊗ vn + P(vm ) ⊗ P(vn ) and vm ⊗ P(vn ) + P(vm ) ⊗ vn , Δ where vm ∈ Cm|0 and vn ∈ Cn|0 . Hence (Cm|m ⊗ Cn|n )P is isomorphic to Cmn|mn .

5. Howe duality

168

Since ΔΠd is an even subgroup of Δ(Bd ) generated by the d copies of PΔ ’s, we have  Δ(Bd )   ΔSd m|m ⊗d n|n ⊗d m|m ⊗d n|n ⊗d ΔΠd ∼ (C ) ⊗ (C ) = (C ) ⊗ (C ) ΔSd  Δ ∼ = ((Cm|m ⊗ Cn|n )P )⊗d ∼ = ((Cmn|mn )⊗d )Sd ∼ = Sd (Cmn|mn ). 

This proves the lemma.

By Sergeev duality in Theorem 3.49, there is a commuting action of q(n) and Bd on (Cn|n )⊗d . Hence there is a natural q(m) × q(n)-action on ((Cm|m ⊗  Cn|n )⊗d )Δ(Bd ) , and by the identification in Lemma 5.26, on Sd (Cmn|mn ) as well. We shall write down this action on Sd (Cmn|mn ) explicitly in terms of differential operators. Let x pi and ξ pi , for 1 ≤ p ≤ m and 1 ≤ i ≤ n, denote the standard even and odd coordinates of Cmn|mn , respectively. We may then identify S(Cmn|mn ) with the polynomial superalgebra generated by the x pi and ξ pi , for 1 ≤ p ≤ m and 1 ≤ i ≤ n. We introduce the following first-order differential operators:  pq = E

j=1 n

(5.22) E

pq

n

=

∂

∂

∂

∂

∂

∂

∂

∂

∑ (x p j ∂xq j + ξ p j ∂ξq j ), ∑ (x p j ∂ξq j − ξ p j ∂xq j ),

1 ≤ p, q ≤ m.

j=1

i j = E (5.23) Ei j =

m

∑ (x pi ∂x p j + ξ pi ∂ξ p j ),

p=1 m

∑ (x pi ∂ξ p j + ξ pi ∂x p j ),

1 ≤ i, j ≤ n.

p=1

The following lemma is proved by a direct computation (see Exercise 5.2(2)).  pq and E pq in (5.22), for 1 ≤ p, q ≤ m, form a copy of Lemma 5.27. The operators E i j and Ei j in (5.23), for 1 ≤ i, j ≤ n, form a copy of q(n). Furthermore, q(m), while E they define commuting actions of q(m) and q(n) on S(Cmn|mn ). Recall from (1.52) that, for a partition λ of d with length (λ),  0, if (λ) is even, δ(λ) = 1, if (λ) is odd. Furthermore, recall that SPd denotes the set of strict partitions of d, and Ln (λ) stands for the simple q(n)-module with highest weight λ, for λ ∈ SPd of length not exceeding n (see Section 2.1.6).

5.3. Howe duality for symplectic and orthogonal groups

169

Theorem 5.28 ((q(m), q(n))-Howe duality). As a (q(m), q(n))-module, Sd (Cmn|mn ) admits the following strongly multiplicity-free decomposition: 

Sd (Cmn|mn ) ∼ =

2−δ(λ) Lm (λ) ⊗ Ln (λ).

λ∈SPd (λ)≤min{m,n}

Proof. By Sergeev duality (Theorem 3.49), we have as q(m) × Bd modules (Cm|m )⊗d ∼ =



2−δ(λ) Lm (λ) ⊗ Dλ ,

λ∈SPd (λ)≤m

where we recall that Dλ denotes an irreducible Bd -module (or equivalently, an irreducible Hd -module). Therefore, combined with Lemma 5.26, this gives us  (5.24) Sd (Cmn|mn ) ∼ = (((Cm|m )⊗d ) ⊗ ((Cn|n )⊗d ))Δ(Bd )



∼ =



2−δ(λ) 2−δ(µ) (Lm (λ) ⊗ Dλ ⊗ Ln (µ) ⊗ Dµ )Δ(Bd )

λ,µ∈SPd (λ)≤m,(µ)≤n



∼ =



2−δ(λ) 2−δ(µ) (Lm (λ) ⊗ Ln (µ)) ⊗ (Dλ ⊗ Dµ )Δ(Bd ) .

λ,µ∈SPd (λ)≤m,(µ)≤n

The Bd -module Dλ is self-dual, i.e., (Dλ )∗ ∼ = Dλ , since by Theorem 3.46 and λ (3.27)–(3.29) all the character values of D are real. Recalling Dλ is of type M if and only if δ(λ) = 0, we conclude by super Schur’s Lemma 3.4 that 

dim(Dλ ⊗ Dµ )Δ(Bd ) = dim HomBd (Dλ , Dµ ) = 2δ(λ) δλµ . This together with (5.24) completes the proof of the theorem.



Recall that the character of the irreducible q(n)-module Ln (λ) is given by the Schur Q-function Qλ up to a 2-power by Theorem 3.51. Now by calculating the characters of both sides of the isomorphism in Theorem 5.28 and summing over all d ≥ 0, we recover the following Cauchy identity (A.57) in Appendix A: 1 + xi y j

∏ 1 − xi y j = ∑ i, j

λ∈SP

2−(λ) Qλ (x)Qλ (y).

5.3. Howe duality for symplectic and orthogonal groups In this section, we use the First Fundamental Theorem (FFT) for classical Lie groups in Chapter 4 to construct the Howe dual pairs (Sp(V ), osp(2m|2n)) and (O(V ), spo(2m|2n)). We also obtain precise multiplicity-free decompositions of these Howe dualities for symplectic and orthogonal groups. The case of orthogonal groups, which are not connected, requires some extra work.

5. Howe duality

170

5.3.1. Howe dual pair (Sp(V ), osp(2m|2n)). We take V = Ck , where k = 2 is assumed to be even in this subsection. We consider the superspace U = U0¯ ⊕ U1¯ with U0¯ = V m and U1¯ = V n , and identify naturally U ≡ V ⊗ Cm|n as before. We continue to identify the supersymmetric algebra of U with C[x, η] as in (5.12). Recall the skew-symmetric matrix (4.7). We let Sp(k) = G(V, ω− ) be the symplectic subgroup of GL(k), consisting of elements preserving the non-degenerate skew-symmetric bilinear form ω− on V = Ck corresponding to the matrix (4.7), and let sp(k) ⊆ gl(k) be its Lie algebra. In order to write down explicitly the Lie algebra sp(k), we introduce the following notation. For an  ×  matrix A = (ai j ), we denote by A = J At J the transpose of A with respect to the “opposite” diagonal: ⎛

(5.25)

a−1, a, ⎜a,−1 a−1,−1 ⎜ ⎜a,−2 a−1,−2 ⎜  A =⎜ . .. ⎜ .. . ⎜ ⎝ a,2 a−1,2 a,1 a−1,1

⎞ a2, a1, a2,−1 a1,−1 ⎟ ⎟ a2,−2 a1,−2 ⎟ ⎟ .. .. ⎟ . . . ⎟ ⎟ a2,2 a1,2 ⎠ a2,1 a1,1

··· ··· ··· .. . ··· ···

It is then straightforward to check that sp(k) consists precisely of the following k × k matrices in |-block form:  (5.26)

  A B | D = −A , B = B,C = C . C D

sp(k) =

The standard Borel of sp(k) consists of the k × k upper triangular matrices of the form (5.26) and so is compatible with the standard Borel of gl(k). Furthermore, the Cartan subalgebra of sp(k) is spanned by the basis vectors Eii − Ek+1−i,k+1−i , for 1 ≤ i ≤ k/2. The action of the Lie algebra sp(k) as the subalgebra of gl(k) on C[x, η] given in (5.13) lifts to an action of the Lie group Sp(V ) = Sp(k). On the other hand, the action ρ of gl(m|n) on C[x, η] with ρ(Eab ) = Eab , for a, b ∈ I(m|n), is modified from (5.14) by a shift of scalars on the diagonal matrices:

Ei¯ j¯ =

k

∑ x pi

p=1

(5.27) Est =

k

∂ k + δi j , ∂x p j 2

Ei¯s =

∂ k − δst , ∂η pt 2

Esi¯ =

∑ η ps

p=1

k

∂

∑ x pi ∂η ps ,

1 ≤ i, j ≤ m,

p=1 k

∂

∑ η ps ∂x pi ,

p=1

1 ≤ s,t ≤ n.

5.3. Howe duality for symplectic and orthogonal groups

171

Introduce the following additional operators: Ii¯ j¯ =

k 2

∑



 x pi xk+1−p, j − xk+1−p,i x p j ,

p=1

Ii¯s =

k 2

∑

  x pi ηk+1−p,s − xk+1−p,i η ps ,

p=1

Ist =

k 2

∑

  η ps ηk+1−p,t − ηk+1−p,s η pt ,

p=1

(5.28) Δi¯ j¯ =

 ∂ ∂ ∂  ∂ , − ∂xk+1−p,i ∂x p j p=1 ∂x pi ∂xk+1−p, j

Δi¯s =

 ∂ ∂ ∂  ∂ , − ∂xk+1−p,i ∂η ps p=1 ∂x pi ∂ηk+1−p,s

1 ≤ i < j ≤ m,

Δst =

 ∂ ∂ ∂  ∂ , − ∂ηk+1−p,s ∂η pt p=1 ∂η ps ∂ηk+1−p,t

1 ≤ s ≤ t ≤ n.

k 2

∑ k 2

∑ k 2

∑

Let us denote by g the linear span of the elements in (5.27) and (5.28). Clearly, the elements in (5.27) and (5.28) form a linear basis for g . Lemma 5.29. (1) Under the super-commutator, g forms a Lie superalgebra, which is isomorphic to osp(2m|2n). (2) The actions of Sp(k) and osp(2m|2n) on S(Ck ⊗ Cm|n ) commute. Proof. (1) With U = Ck ⊗ Cm|n and U = U ⊕ U ∗ , we identify WC(U U ) with the superalgebra of differential operators on S(U). The operators Ei¯ j¯ and Est from (5.27) can be written as   ∂ ∂ 1 k  ∂ 1 k  ∂ Ei¯ j¯ = ∑ x pi + x pi , Est = ∑ η ps − η ps , 2 p=1 ∂x p j ∂x p j 2 p=1 ∂η pt ∂η pt so that g can be regarded as a subspace inside G ∼ U ) as in Proposition 5.5. = spo(U A direct computation using elements in (5.27) and (5.28) shows that g forms a Lie superalgebra under the super-commutator. Moreover, a further direct computation shows that g ∼ = osp(2m|2n), with identification of root vectors eα , for a root α, as follows: eδi −δ j = Ei¯ j¯,

eδi −εs = Ei¯s ,

eεs −εt = Est ,

eεs −δi = Esi¯,

eδi +δ j = Ii¯ j¯,

eδi +εs = Ii¯s ,

eεs +εt = Ist ,

e−δi −δ j = Δi¯ j¯,

1 ≤ i = j ≤ m, 1 ≤ s = t ≤ n.

5. Howe duality

172

e−εs −εt = Δst ,

e−δi −εs = Δi¯s ,

1 ≤ i < j ≤ m, 1 ≤ s ≤ t ≤ n.

We omit the details. (2) Recall that e1 , . . . , ek denote the standard basis for Ck . It follows by definition of ω− that ω− (e p , ek+1−p ) = −ω− (ek+1−p , e p ) = 1, k/2

for 1 ≤ p ≤ k/2, and hence ∑ p=1 (e p ⊗ ek+1−p − ek+1−p ⊗ e p ) ∈ Ck ⊗ Ck is Sp(k)invariant. Thus, Ii¯ j¯, Ii¯s , Ist defined above are Sp(k)-invariant in WC(U U ), where we recall the definition of the basis (5.11) for U and we have regarded ∂x∂pi and ∂ ∂η ps

as elements in U ∗ . Similarly, Δi¯ j¯, Δi¯s , Δst defined above are Sp(k)-invariant in WC(U U ). On the other hand, the elements in (5.27) are GL(k)-invariant (and hence Sp(k)-invariant) in WC(U U ). Hence, for g ∈ G, x ∈ g and z ∈ C[x, η], we have gx.z = xg.z, whence (2).



Theorem 5.30 ((Sp(k), osp(2m|2n))-Howe duality, I). Sp(k) and osp(2m|2n) form a Howe dual pair on S(Ck ⊗ Cm|n ), and the (Sp(k), osp(2m|2n))-module S(Ck ⊗ Cm|n ) is strongly multiplicity-free. Proof. Set G = Sp(k). The action of sp(k) on S(Ck ⊗Cm|n ) clearly lifts to an action of G. By Lemma 5.29, the actions of G and osp(2m|2n) on S(U) = S(Ck ⊗ Cm|n ) commute. Let U = U ⊕ U ∗ . The associated graded for WC(U U ) is S(U U ), and the action of sp(k) on S(U U ) clearly lifts to an action of G. Let {y pi } denote the basis in U0¯∗ dual to the basis {x pi } for U0¯ , and let {ξ pt } denote the basis in U1¯∗ dual to the basis {η pt } for U1¯ . By the supersymmetric First Fundamental Theorem (FFT) for G (see  G Theorem 4.19), a generating set T for the algebra S(U U ) = S(U0¯ ⊕U0¯∗ ) ⊗ ∧(U1¯ ⊕  G U1¯∗ ) consists of zi¯ j¯, zst , zi¯s , zsi¯ in (5.15), Ii¯ j¯, Ii¯s , Ist , and k 2

∑





y pi yk+1−p, j − yk+1−p,i y p j ,

p=1

k 2

∑



 y pi ξk+1−p,s − yk+1−p,i ξ ps ,

p=1

  ξ , ξ − ξ ξ ps pt k+1−p,t k+1−p,s ∑ k 2

p=1

for 1 ≤ i, j ≤ m, 1 ≤ s,t ≤ n. Observe that the symbol map σ sends the basis elements (5.27) and (5.28) for g to the above elements in T for S(U U )G . Now the theorem follows from Proposition 5.12 and Lemma 5.29.  5.3.2. (Sp(V ), osp(2m|2n))-Howe duality. Denote by u+ (respectively, u− ) the subalgebra of osp(2m|2n) spanned by the Δ (respectively, I) operators in (5.28). The irreducible highest weight module L(osp(2m|2n), µ) below is understood to be relative to the Borel subalgebra of osp(2m|2n) corresponding to the fundamental system specified in the following Dynkin diagram:

5.3. Howe duality for symplectic and orthogonal groups 

δ1 − δ2

@ @



−δ1 − δ2



···

δ2 − δ3







···

δm − ε1

173



εn−1 − εn

The root vectors associated to the simple roots −δ1 − δ2 , δi − δi+1 (1 ≤ i ≤ m − 1), δm − ε1 , εt − εt+1 (1 ≤ t ≤ n − 1), in the notations of (5.27) and (5.28) are, respectively, (5.29)

Δ1¯ 2¯ ,

Ei¯,i+1 (1 ≤ i ≤ m − 1),

Em1 ,

Et,t+1 (1 ≤ t ≤ n − 1).

Then gl(m|n) is a Levi subalgebra of osp(2m|2n) corresponding to the removal of the simple root −δ1 − δ2 . We have a triangular decomposition (5.30)

osp(2m|2n) = u− ⊕ gl(m|n) ⊕ u+ .

In particular, the Lie superalgebras gl(m|n) and osp(2m|2n) share the same Cartan subalgebra. An element f ∈ C[x, η] is called harmonic if f is annihilated by the + subalgebra u+ . The space of harmonics C[x, η]u will be denoted by Sp H. Recall that P(m|n) denotes the set of all (m|n)-hook partitions λ (i.e., λm+1 ≤ n), and also recall from (2.4) that, for an (m|n)-hook partition λ, λ is a weight for gl(m|n). Then λ + 1m|n can be regarded as a weight for the Lie superalgebras gl(m|n) and osp(2m|2n), where we recall the definition of 1m|n from (1.36). Theorem 5.31 ((Sp(k), osp(2m|2n))-Howe duality, II). Let k = 2. We have the following decomposition as a (Sp(k), osp(2m|2n))-module:    S(Ck ⊗ Cm|n ) = L(Sp(k), λ) ⊗ L osp(2m|2n), λ + 1m|n . λ∈P(m|n),(λ)≤ +

Proof. Observing that M u is a gl(m|n)-module for any osp(2m|2n)-module M by the triangular decomposition (5.30), we may regard the space of harmonics Sp H as an (Sp(k), gl(m|n))-module. Furthermore, since each graded subspace of S(Ck ⊗ Cm|n ) as a gl(m|n)-module is a submodule of a tensor power of Cm|n + and hence is completely reducible by Theorem 3.11, L(osp(2m|2n), µ)u is also a completely reducible gl(m|n)-module for any irreducible osp(2m|2n)-module L(osp(2m|2n), µ) appearing in S(Ck ⊗ Cm|n ). It follows by the irreducibility of + L(osp(2m|2n), µ) that L(osp(2m|2n), µ)u is an irreducible gl(m|n)-module, and a highest weight vector of highest weight µ in L(osp(2m|2n), µ) remains a highest + weight vector in L(osp(2m|2n), µ)u . Therefore, we must have (5.31)

+ L(osp(2m|2n), µ)u ∼ = Lm|n (µ).

By Theorem 5.30, the (Sp(k), osp(2m|2n))-module S(Ck ⊗ Cm|n ) is strongly multiplicity-free, and hence by (5.31), the (Sp(k), gl(m|n))-module Sp H is also strongly multiplicity-free. To establish the explicit form of decomposition as stated

5. Howe duality

174

in the theorem, it suffices (and is indeed equivalent) to establish the following explicit decomposition of Sp H as an (Sp(k), gl(m|n))-module: (5.32)

Sp

H∼ =



L(Sp(k), λ) ⊗ Lm|n (λ + 1m|n ).

λ∈P(m|n),(λ)≤

We first consider the limit case n = ∞ with the space of harmonics denoted by Sp H∞ , where the condition λ ∈ P(m|n) reduces to λ ∈ P. Set ♦ = ♦1,2,...,λ1 ,λ . Observe by Theorem 5.23 that the vector ♦ associated to any partition λ with (λ) ≤  is a joint sp(k) × gl(m|n)-highest weight vector and ♦ is a polynomial independent of the variables x pi and η pt for p ≥ +1. Each Δ-operator in (5.28) is a second-order differential operator whose summands always involve differentiation with respect to one of these variables for p ≥  + 1. Thus, ♦ is annihilated by u+ , and hence we have ♦ ∈ Sp H∞ . One also easily checks that the vector ♦ has weight (λ, λ + 1m|n ) under the transformations of sp(k) × osp(2m|2n). Therefore, all the summands on the right-hand side of (5.32) indeed occur in Sp H∞ (with multiplicity one), and all irreducible representations of Sp(k) occur. We conclude that (5.32) must hold (when n = ∞). Now consider the finite n case. We may regard S(Cd ⊗ Cm|n ) ⊆ S(Cd ⊗ Cm|∞ ) with compatible actions of osp(2m|2n) ⊆ osp(2m|2∞). We claim the following equality for spaces of joint highest weight vectors holds: (Sp H)n1 ×n(n) = (Sp H∞ )n1 ×n(∞) ∩ C[x, η]. Here n1 (respectively, n(n)) denotes the nilradical of the standard Borel for sp(k) (respectively, the nilradical generated by (5.29) for osp(2m|2n)). Note by definition that (Sp H)n1 ×n(n) ⊇ (Sp H∞ )n1 ×n(∞) ∩ C[x, η]. Since each summand of the operators in (5.27) and the Δ-operators in (5.28) that lie in osp(2m|2∞)\osp(2m|2n) must involve ∂η∂ps for s > n, we conclude that (Sp H)n1 ×n(n) ⊆ (Sp H∞ )n1 ×n(∞) ∩ C[x, η], and hence the claim is proved. Observe from the explicit formulas of the joint highest vectors in Sp H∞ (see Theorem 5.23) that precisely those vectors corresponding to (m|n)-hook partitions will not involve the variables η ps for s > n, and hence they lie in (Sp H∞ )n1 ×n(∞) ∩ C[x, η]. This together with the claim above completes the proof of the theorem.  The irreducible osp(2m|2n)-modules appearing in the (Sp(k), osp(2m|2n))Howe duality decomposition above (for varied k) are called the oscillator modules of osp(2m|2n). In general the osp(2m|2n)-oscillator modules are infinite dimensional, as their highest weights evaluated at the simple root −δ1 − δ2 is −λ1 − λ2 − k < 0. Specializing n = 0 in Theorem 5.31 gives us the following.

5.3. Howe duality for symplectic and orthogonal groups

175

Corollary 5.32 ((Sp(k), so(2m))-Howe duality). Let k = 2. We have the following decomposition of (Sp(k), so(2m))-modules:   m  S(Ck ⊗ Cm ) ∼ L(Sp(k), λ) ⊗ L so(2m), ∑ (λi + )δi . = λ∈P,(λ)≤min{,m}

i=1

We now specialize m = 0 in Theorem 5.31. The irreducible highest weight sp(2n)-modules in Corollary 5.33 below are relative to the Borel subalgebra of sp(2n) corresponding to the fundamental system specified in the following Dynkin diagram: =⇒ −2ε1

ε1 − ε2



···





εn−1 − εn

Corollary 5.33 ((Sp(k), sp(2n))-Howe duality). Let k = 2. We have the following decomposition of (Sp(k), sp(2n))-modules:   n  ∧(Ck ⊗ Cn ) ∼ L(Sp(k), λ) ⊗ L sp(2n), ∑ (λi − )εi . = λ∈P (λ)≤,(λ )≤n

i=1

5.3.3. Irreducible modules of O(V ). Take V = Ck . Let O(k) = G(V, ω+ ) be the orthogonal subgroup of GL(k) consisting of elements preserving the non-degenerate symmetric bilinear form ω+ on V = Ck associated to the k × k matrix Jk in (4.6), and let so(k) ⊆ gl(k) be its Lie algebra. Let SO(V ) = O(V ) ∩ SL(V) denote the special orthogonal group. The Lie algebras so(k), for k = 2 and k = 2 + 1 with  ∈ Z+ , consist precisely of the following k × k matrices in |- and |1|-block forms, respectively (recall A from (5.25)):    A B    so(2) = | D = −A , B = −B,C = −C , (5.33) C D ⎧⎛ ⎫ ⎞ ⎨ A x B ⎬ so(2 + 1) = ⎝ y 0 z ⎠ | D = −A , B = −B,C = −C , (5.34) ⎩ ⎭ C w D where, in addition, the  × 1 matrices x = (xi1 ) and w = (wi1 ) and the 1 ×  matrices y = (y1i ) and z = (z1i ) are related by z1i = −x+1−i,1 and y1i = −w+1−i,1 , for all 1 ≤ i ≤ . The standard Borel of so(k), for k even and odd, is the subalgebra of upper triangular matrices in so(k), and hence is compatible with the standard Borel of gl(k). The Cartan subalgebra of so(k) is spanned by the basis vectors Eii − Ek+1−i,k+1−i , for 1 ≤ i ≤ , where k = 2 or 2 + 1. The following fact is well known (cf. [46]). Lemma 5.34. A finite-dimensional simple so(k)-module L(so(k), λ) of highest weight λ = ∑i=1 λi εi lifts to an SO(k)-module if and only if

5. Howe duality

176

(1) (λ1 , . . . , λ ) is a partition, for odd k = 2 + 1, (2) (λ1 , . . . , λ−1 , |λ |) is a partition, for even k = 2. Next we will describe the classification of simple O(k)-modules and give a parametrization in terms of partitions. Let det denote the one-dimensional determinant module of O(k). First consider the case when k = 2 + 1 is odd. In this case, −I ∈ O(k)\SO(k) and O(k) is the direct product SO(k)×Z2 . Associated to a partition λ with (λ) ≤ , we let L(O(k), λ) stand for the irreducible O(k)-module that restricts to the irreducible SO(k)-module L(SO(k), λ) and on which the element −I acts trivially. We shall denote the irreducible O(k)-module L(O(k), λ) ⊗det by L(O(k), λ), where the Young diagram of the partition λ is obtained from λ by replacing the first column λ1 by k − λ1 . This way we have obtained a complete parametrization of the simple O(k)-modules L(O(k), µ) in terms of partitions µ with µ1 + µ2 ≤ k = 2 + 1. Now consider the case when k = 2 is even. Proposition 5.35. Let k = 2 be even. A simple O(k)-module L is exactly one of the following (where all λi ∈ Z+ ): (1) L is a direct sum of two irreducible SO(k)-modules of highest weights (λ1 , · · · , λ−1 , λ ) and (λ1 , · · · , λ−1 , −λ ), where λ > 0. In this case, L∼ = L ⊗ det. (2) For λ = (λ1 , · · · , λ−1 , 0), there are exactly two inequivalent irreducible O(k)-modules that restrict to the irreducible SO(k)-module of highest weight λ. If one of these modules is L, then the other one is L ⊗ det. We shall denote the simple O(k)-module in Proposition 5.35(1) by L(O(k), λ). Let τ ∈ O(k) \ SO(k) denote the element that switches the basis vector e with e+1 and fixes all other standard basis vectors ei ’s of Ck . We declare L(O(k), λ) to be the O(k)-module in Proposition 5.35(2) on which τ acts on an SO(k)-highest weight vector trivially. Note that τ transforms an SO(k)-highest weight vector in the O(k)-module L(O(k), λ) ⊗ det by −1. We shall denote the simple O(k)-module  where as before the Young diagram of the partition L(O(k), λ) ⊗ det by L(O(k), λ), λ is obtained from λ by replacing the first column λ by k − λ . In this way, we 1 1 have obtained a complete parametrization of the simple O(k)-modules L(O(k), µ) in terms of partitions µ with µ1 + µ2 ≤ k = 2. Summarizing the above discussion for k odd and even, we have obtained the following uniform description. Proposition 5.36. A complete set of pairwise non-isomorphic simple O(k)-modules consists of L(O(k), µ), where µ runs over all partitions of length no greater than k with µ1 + µ2 ≤ k.

5.3. Howe duality for symplectic and orthogonal groups

177

5.3.4. Howe dual pair (O(k), spo(2m|2n)). We consider the superspace U = U0¯ ⊕ U1¯ with U0¯ = V m and U1¯ = V n , and identify naturally U ≡ V ⊗ Cm|n as before. We continue the identification S(U) ≡ C[x, η] as in (5.12). The action of the Lie algebra so(k) as a subalgebra of gl(k) on C[x, η] given in (5.13) lifts to an action of Lie group O(V ) = O(k). On the other hand, recall the action of gl(m|n) on C[x, η] given by (5.27). Introduce the following additional operators on C[x, η]: Ji¯ j¯ =

k

∑ x pi xk+1−p, j ,

∇i¯ j¯ =

p=1

(5.35)

Ji¯s =

k

∑ x pi ηk+1−p,s , k

∑ η ps ηk+1−p,t ,

p=1

∂

∂

∂

∂

∂

∂

∑ ∂x pi ∂xk+1−p, j ,

p=1

∇i¯s =

p=1

Jst =

k

k

∑ ∂x pi ∂ηk+1−p,s ,

1 ≤ i ≤ j ≤ m,

p=1

∇st =

k

∑ ∂η ps ∂ηk+1−p,t ,

1 ≤ s < t ≤ n.

p=1

Let us denote by g the linear span of the elements in (5.27) and (5.35). Note that the elements in (5.27) and (5.35) form a linear basis for g . Lemma 5.37. (1) Under the super-commutator, g forms a Lie superalgebra, which is isomorphic to spo(2m|2n). (2) The actions of O(k) and spo(2m|2n) on S(Ck ⊗ Cm|n ) commute. Proof. (1) Letting U = Ck ⊗ Cm|n and U = U ⊕U ∗ , we identify WC(U U ) with the  superalgebra of differential operators on S(U). We can then regard g as a subspace inside G ∼ U ) as in Proposition 5.5. In a completely analogous fashion as = spo(U in the proof of Lemma 5.29, we can show that g ∼ = spo(2m|2n) with an explicit identification of root vectors with elements in (5.27) and (5.35). We leave it to the reader to fill in the details (Exercise 5.9). (2) Recall the standard basis e1 , . . . , ek for Ck . By definition of ω+ , we have ω+ (e p , ek+1−p ) = 1, for 1 ≤ p ≤ k, and hence ∑kp=1 e p ⊗ ek+1−p ∈ Ck ⊗ Ck is O(k)-invariant. It follows that Ji¯ j¯, Ji¯s , Jst defined above are O(k)-invariant in WC(U U ). Dually, the elements ∇i¯ j¯, ∇i¯s , ∇st are also O(k)-invariant in WC(U U ). On the other hand, the elements in (5.27) are clearly O(k)-invariant in WC(U U ). Hence (2) follows.  Theorem 5.38 ((O(k), spo(2m|2n))-Howe duality, I). O(k) and spo(2m|2n) form a Howe dual pair on S(Ck ⊗Cm|n ), and the (O(k), spo(2m|2n))-module S(Ck ⊗Cm|n ) is strongly multiplicity-free. Proof. Set G = O(k). The action of so(k) on S(Ck ⊗ Cm|n ) lifts to an action of G. By Lemma 5.37, the actions of G and spo(2m|2n) on S(U) = S(Ck ⊗ Cm|n ) commute.

5. Howe duality

178

Let U = U ⊕U ∗ . The associated graded for WC(U U ) is S(U U ). Thus, the action of so(k) on WC(U U ) lifts to an action of G. Let {y pi } denote the basis in U0¯∗ dual to the basis {x pi } for U0¯ , and let {ξ pt } denote the basis in U1¯∗ dual to the basis {η pt } for U1¯ . By the supersymmetric First Fundamental Theorem for G (Theorem 4.19),  G a generating set T for the algebra S(U U )G = S(U0¯ ⊕U0¯∗ ) ⊗ ∧(U1¯ ⊕U1¯∗ ) consists of zi¯ j¯, zst , zi¯s , zsi¯ in (5.15), Ji¯ j¯, Ji¯s , Jst , and k

k

k

p=1

p=1

p=1

∑ y pi yk+1−p, j , ∑ y pi ξk+1−p,s , ∑ ξ ps ξk+1−p,t ,

for 1 ≤ i, j ≤ m, 1 ≤ s,t ≤ n. The symbol map σ sends the basis elements (5.27) and (5.35) for g to the above elements in T, and hence the theorem follows from Proposition 5.12 and Lemma 5.37.  5.3.5. (O(V ), spo(2m|2n))-Howe duality. Denote by u+ (respectively, u− ) the subalgebra of spo(2m|2n) spanned by the ∇ (respectively, J) operators in (5.35). The irreducible highest weight module L(spo(2m|2n), µ) below is understood to be relative to the Borel subalgebra of spo(2m|2n) corresponding to the fundamental system specified in the following Dynkin diagram: =⇒ −2δ1

δ1 − δ2



···

δ2 − δ3

 δm − ε1

···



ε1 − ε2



εn−1 − εn

The root vectors associated to the simple roots −2δ1 , δi − δi+1 (1 ≤ i ≤ m − 1), δm − ε1 , εt − εt+1 (1 ≤ t ≤ n − 1), in the notations of (5.27) and (5.35), are respectively (5.36)

∇1¯ 1¯ ,

Ei¯,i+1 (1 ≤ i ≤ m − 1),

Em1 ¯ ,

Et,t+1 (1 ≤ t ≤ n − 1).

Then gl(m|n) is a Levi subalgebra of spo(2m|2n) corresponding to the removal of the simple root −2δ1 . We have a triangular decomposition (5.37)

spo(2m|2n) = u− ⊕ gl(m|n) ⊕ u+ .

In particular, the Lie superalgebras gl(m|n) and spo(2m|2n) share the same Cartan subalgebra. For λ ∈ P(m|n), λ + 2k 1m|n is a weight for gl(m|n), and hence can be regarded as a weight for spo(2m|2n) as well. Theorem 5.39 ((O(k), spo(2m|2n))-Howe duality, II). We have the following decomposition of (O(k), spo(2m|2n))-modules:    k S(Ck ⊗ Cm|n ) = L(O(k), λ) ⊗ L spo(2m|2n), λ + 1m|n . 2 λ∈P(m|n) λ1 +λ2 ≤k

+

+

Proof. Denote the space of harmonics C[x, η]u by O H. Then M u is a gl(m|n)module for any spo(2m|2n)-module M by the triangular decomposition (5.37), so

5.3. Howe duality for symplectic and orthogonal groups

179

that O H is an (O(k), gl(m|n))-module. Similar to the proof of Theorem 5.31, it is enough to establish the following decomposition of O H as an (O(k), gl(m|n))module:    k O (5.38) H∼ L(O(k), λ) ⊗ Lm|n λ + 1m|n . = 2 λ∈P(m|n) λ1 +λ2 ≤k

The validity of (5.38) for finite n follows from the case when n = ∞ in a completely parallel way as in the proof of Theorem 5.31. Hence it remains to establish (5.38) in the limit case n = ∞, where the condition λ ∈ P(m|n) reduces to λ ∈ P. Assume now n = ∞ in the remainder of the proof. Set ♦ = ♦1,2,...,λ1 ,λ , for λ ∈ P with λ1 + λ2 ≤ k. We claim that ♦ ∈ O H, i.e, ♦ is annihilated by u+ . By Theorem 5.23 ♦ is a joint so(k) × gl(m|n)-highest weight vector. To establish the claim, it suffices to show that ♦ is annihilated by the simple root vector ∇1¯ 1¯ , since ♦ is annihilated by the remaining simple root vectors in (5.36) that lie in gl(m|n). ∂ To show that ∇1¯ 1¯ (♦) = 0, we recall that ∇1¯ 1¯ = ∑kp=1 ∂x∂p1 ∂xk+1−p,1 and recall from (5.21) that ♦ = ♦1,λ1 · · · ♦λ1 ,λλ . We observe that each summand in the expan1

sion of the double differentiation ∇1¯ 1¯ (♦) is either of the form (i)

∂♦i,λ

i

∂♦ j,λ

j

∂x p1 ∂xk+1−p,1

···

∂2 ♦i,λ

i for i = j, or of the form (ii) ∂x p1 ∂xk+1−p,1 · · · . Since λ1 + λ2 ≤ k by assumption, we   have λi + λ j ≤ k for i = j. Note by definitions (5.19) and (5.20) that the x pq ’s appearing in ♦i,λi satisfy the constraints p ≤ λi and the xrs ’s appearing in ♦ j,λj satisfy r ≤ λj , and hence p + r ≤ λi + λj ≤ k. Thus, the derivative in (i) must be zero. Expanding the determinant ♦i,λi along the first column, we obtain ♦i,λi = ∑ p x p1 A p , where A p is an expression not containing any xq1 . Therefore, the derivative in (ii) must also be zero, and hence ∇1¯ 1¯ (♦) = 0.

When k = 2, recall the element τ ∈ O(k) \ SO(k) defined in Section 5.3.3. When k = 2 + 1, we let τ be the linear map on Ck that fixes e p , for all p =  + 1, and that sends e+1 to −e+1 . By examining how τ transforms the so(k)-highest weight vector ♦, for λ1 + λ2 ≤ k, one concludes that the O(k)-module generated by ♦ is L(O(k), λ). Another direct computation implies that ♦ has spo(2m|2n)-weight λ + 2k 1m|n . Thus, all irreducible representations of O(k) occur in O H, and all the summands on the right-hand side of (5.38) occur in the space of harmonics O H. Therefore, (5.38) holds in the case when n = ∞, and the theorem is proved.  The simple spo(2m|2n)-modules appearing in the above (O(k), spo(2m|2n))Howe duality decomposition (for varied k) will be called the oscillator modules of spo(2m|2n). The spo(2m|2n)-oscillator modules are, in general, infinite dimensional, as their highest weights evaluated at the simple root −2δ1 is −2λ1 − k < 0. Specializing n = 0 in Theorem 5.39 gives us the following.

5. Howe duality

180

Corollary 5.40 ((O(k), sp(2m))-Howe duality). We have the following decomposition of (O(k), sp(2m))-modules:  m  k  S(Ck ⊗ Cm ) ∼ L(O(k), λ) ⊗ L sp(2m), ∑ (λi + )δi . = 2   i=1 (λ)≤m,λ1 +λ2 ≤k

We now specialize m = 0 in Theorem 5.39. The irreducible highest weight so(2n)-modules in Corollary 5.41 below are relative to the Borel subalgebra of so(2n) corresponding to the fundamental system specified in the following Dynkin diagram: ε1 − ε2



@ @



−ε1 − ε2



ε2 − ε3

···





εn−1 − εn

Corollary 5.41 ((O(k), so(2n))-Howe duality). We have the following decomposition of (O(k), so(2n))-modules:  n  k  ∧(Ck ⊗ Cn ) ∼ L(O(k), λ) ⊗ L so(2n), ∑ (λi − )εi . = 2    i=1 (λ )≤n,λ1 +λ2 ≤k

5.4. Howe duality for infinite-dimensional Lie algebras In this section, we formulate Howe dualities between classical Lie groups and infinite-dimensional Lie algebras in fermionic Fock spaces. The results of this section will be applied in Section 5.5 to derive character formulas for the irreducible oscillator modules of the Lie superalgebras arising in the Howe duality decompositions in Section 5.3. 5.4.1. Lie algebras a∞ , c∞ , and d∞ . We define the infinite-dimensional Lie alge ∞ and its Lie subalgebras c∞ and d∞ of type C and D. bras a∞ ≡ gl  ∞ . Denote by gl∞ the Lie algebra of all matrices The Lie algebra a∞ ≡ gl (ai j )i, j∈Z with ai j = 0 for all but finitely many i’s and j’s. Let the degree of the elementary matrix Ei j be j − i. This defines the Z-principal gradation on gl∞ =   k∈Z gl∞,k . Denote by a∞ ≡ gl∞ = gl∞ ⊕ CK the central extension associated to the following 2-cocycle τ: (5.39)

τ(A, B) = Tr ([J, A]B) ,

where J = ∑ j≤0 Eii . Denoting by [·, ·] the Lie bracket on gl∞ , we introduce a bracket [·, ·] on the central extension a∞ = gl∞ ⊕ CK by [A + aK, B + bK] := [A, B] + τ(A, B)K,

A, B ∈ gl∞ , a, b ∈ C.

5.4. Howe duality for infinite-dimensional Lie algebras

181

By definition, a 2-cocycle τ satisfies the following conditions: for A, B,C ∈ gl∞ , (i) τ(A, B) = −τ(B, A);

(ii) τ(A, [B,C]) + τ(B, [C, A]) + τ(C, [A, B]) = 0.

The 2-cocyle conditions on τ ensure that the bracket [·, ·] is skew-symmetric and satisfies the Jacobi identity. More explicitly, we have the following commutation relation on a∞ : for i, j, m, n ∈ Z,   [Ei j , Emn ] = δ jm Ein − δin Em j + δ jm δin θ{i ≤ 0} − θ{ j ≤ 0} K, where the Boolean characteristic function is defined to be  1, if the statement P is true, (5.40) θ{P} = 0, if the statement P is false. Remark 5.42. The Lie algebra a∞ can be “completed” in the following sense (see  ∞ the completed Lie algebra of matrices (ai j )i, j∈Z with ai j = 0 [37]). Denote by gl  ∞ and thus we for |i − j|  0. The formula for τ in (5.39) also makes sense for gl  obtain a central extension of gl∞ (cf. [65, 87]). While τ is a 2-coboundary on gl∞ ,  ∞ is leading to a trivial central extension a∞ as we have defined in this book, τ on gl  no longer a 2-coboundary, and it gives a nontrivial central extension of gl∞ . Similar  ∞ below in this section. remarks apply to the classical subalgebras of gl∞ and gl The completed Lie algebras and their central extensions are important because of their relations to various subalgebras such as Heisenberg, Virasoro, and affine Lie algebras, but they play no particular role in this book. Actually in Chapter 6, the “uncompleted” variants are preferred as we need to compare these with finite rank Lie algebras. The main results in this section make sense when formulated for these completed Lie algebras. We extend the Z-gradation of the Lie algebra gl∞ to a∞ by putting the degree of K to be 0. In particular, we have a triangular decomposition a∞ = a∞+ ⊕ a∞0 ⊕ a∞− , where a∞± =



gl∞,± j ,

a∞0 = gl∞,0 ⊕ CK.

j∈N

Denote by εi for i ∈ Z the element in (a∞0 )∗ determined by εi (E j j ) = δi j and εi (K) = 0, for j ∈ Z. Then the root system of a∞ is {εi − ε j | i, j ∈ Z, i = j} with a fundamental system {εi − εi+1 | i ∈ Z}. The corresponding set of simple coroots is given by {Hia := Eii − Ei+1,i+1 + δi,0 K | i ∈ Z}. Let Λaj , for j ∈ Z, be the jth fundamental weight for a∞ , i.e., Λaj (Hia ) = δi j , for all i ∈ Z, and Λaj (K) = 1. A direct computation shows that  a Λ0 − ∑0k= j+1 εk , for j < 0, a Λj = (5.41) j Λa0 + ∑k=1 εk , for j ≥ 1.

5. Howe duality

182

Let L(a∞ , Λ) denote the irreducible highest weight a∞ -module of highest weight Λ ∈ (a∞0 )∗ with respect to the standard Borel subalgebra a∞+ ⊕ a∞0 . The level of Λ is given by the scalar Λ(K) = ∑i∈Z Λ(Hia ). The Lie algebra c∞ . Now consider the natural gl∞ -module V0 with basis {vi | i ∈ Z} such that Ei j vk = δ jk vi . Let C be the following skew-symmetric bilinear form on V0 : C(vi , v j ) = (−1)i δi,1− j , ∀i, j ∈ Z. Denote by c∞ the Lie subalgebra of gl∞ that preserves the bilinear form C: c∞ = {a ∈ gl∞ | C(a(u), v) +C(u, a(v)) = 0, ∀u, v ∈ V0 } = {(ai j )i, j∈Z ∈ gl∞ | ai j = −(−1)i+ j a1− j,1−i }. Denote by c∞ = c∞ ⊕ CK the central extension of c∞ associated to the 2-cocycle (5.39) restricted to c∞ . Then c∞ inherits from a∞ a natural triangular decomposition: c∞ = c∞+ ⊕ c∞0 ⊕ c∞− , where c∞± = c∞ ∩ a∞± , c∞0 = c∞ ∩ a∞0 . Let {εi | i ∈ N} be the basis in (c∞0 )∗ dual to the basis {Ei,i − E−i+1,−i+1 | i ∈ N} in c∞0 . The Dynkin diagram for c∞ with a standard simple system is given as follows: (5.42)

=⇒

β× := −2ε1 ε1 − ε2



ε2 − ε3

···





εi − εi+1



···

A set of simple coroots {Hic , i ≥ 0} for c∞ is Hic = Eii + E−i,−i − Ei+1,i+1 − E1−i,1−i , H0c

i ∈ N,

= E0,0 − E1,1 + K.

Then we denote by Λcj ∈ (c∞0 )∗ , for j ∈ Z+ , the jth fundamental weight of c∞ , i.e., Λcj (Hic ) = δi j for all i ∈ Z+ . Note that Λcj (K) = 1 for all j. A direct computation shows that j

Λcj =

(5.43)

∑ εk + Λc0 ,

for j ≥ 1.

k=1

Denote by L(c∞ , Λ) the irreducible highest weight module of c∞ of highest weight Λ ∈ (c∞0 )∗ , whose level is given by Λ(K) = ∑i≥0 Λ(Hic ). The Lie algebra d∞ . We denote by d∞ the Lie subalgebra of gl∞ preserving the following symmetric bilinear form D on V0 : D(vi , v j ) = δi,1− j ,

∀i, j ∈ Z.

Namely, we have d∞ = {a ∈ gl∞ | D(a(u), v) + D(u, a(v)) = 0, ∀u, v ∈ V0 } = {(ai j )i, j∈Z ∈ gl∞ | ai j = −a1− j,1−i }.

5.4. Howe duality for infinite-dimensional Lie algebras

183

Denote by d∞ = d∞ ⊕ CK the central extension associated to the 2-cocycle (5.39)  ∞: restricted to d∞ . Then d∞ has a triangular decomposition induced from gl d∞ = d∞+ ⊕ d∞0 ⊕ d∞− , where d∞± = d∞ ∩ a∞± and d∞0 = d∞ ∩ a∞0 . The Cartan subalgebra d∞0 of d∞ is the same as c∞0 (both as subalgebras of a∞ ), and we let ε j ∈ (d∞0 )∗ be defined in the same way as in the case of c∞ . The Dynkin diagram with a standard fundamental system for d∞ is given as follows: β× := −ε1 − ε2 

@ @



(5.44) ε1 − ε2



ε2 − ε3



ε3 − ε4

···





εi − εi+1



···

A set of simple coroots {Hid | i ≥ 0} for d∞ is given by Hid = Eii + E−i,−i − Ei+1,i+1 − E1−i,1−i H0d

(i ∈ N),

= E0,0 + E−1,−1 − E2,2 − E1,1 + 2K.

Denote by Λdj ∈ (d∞0 )∗ , for j ∈ Z+ , the jth fundamental weight of d∞ , i.e, Λdj (Hid ) = δi j for i ∈ Z+ . Denote by L(d∞ , Λ) the irreducible highest weight d∞ -module of highest weight Λ, whose level is Λ(K) = 12 Λ(H0d ) + 12 Λ(H1d ) + ∑i≥2 Λ(Hid ). 5.4.2. The fermionic Fock space. Recall from (A.68) that F denotes the fermionic Fock space generated by a pair of fermions ψ± (z), whose components ψ± r , r ∈ 1 + Z, satisfy the Clifford commutation relation (A.66). 2 We shall denote by F the fermionic Fock space of  pairs of fermions ψ±,p (z) =

∑

ψr±,p z−r− 2 , 1

1 ≤ p ≤ .

r∈ 21 +Z

 be the Clifford algebra generated by ψr±,p , where 1 ≤ More precisely, we let C p ≤  and r ∈ 12 + Z, with anti-commutation relations given by [ψr+,p , ψ−,q s ]+ = δ p,q δr,−s I,

−,p −,q [ψr+,p , ψ+,q s ]+ = [ψr , ψs ]+ = 0,

 -module generated by for all r, s ∈ 12 + Z, 1 ≤ p, q ≤ . Then F is the simple C ±,p the vacuum vector |0, which satisfies the condition ψr |0 = 0 for all r > 0 and 1 ≤ p ≤ . Here I is identified with the identity operator on F . Introduce a neutral fermionic field φ(z) =

∑

r∈ 12 +Z

φr z−r− 2 1

5. Howe duality

184

whose components satisfy the following anti-commutation relation: [φr , φs ]+ = δr,−s I,

r, s ∈

1 + Z. 2

1

+ 2 the Clifford algebra generated by φr and ψr±,p subject to the We denote by C ±,p additional anti-commutation relation [φr , ψs ]+ = 0, where 1 ≤ p ≤ . By the Fock 1 1 + 2 -module generated by the vacuum vector |0, space F+ 2 we mean the simple C ±,p

which satisfies the condition φr |0 = ψr |0 = 0 for all r > 0 and 1 ≤ p ≤ . 1

 and C + 2 are naturally filtered algebras by letting the degree The algebras C ±,p of each ψr and φr be 1 and the degree of I be 0. The associated graded algebras are exterior algebras. We introduce natural 12 Z+ -gradations (called the principal 1 gradation) on F and F+ 2 by the eigenvalues of the degree operator d on F and 1 F+ 2 defined by * + ±,p ±,p d|0 = 0, d, ψ−r = rψ−r , [d, φ−r ] = rφ−r , ∀r, p. 1

Every graded subspace of F and F+ 2 with respect to the principal gradation is finite dimensional. Introduce the normal ordered product (denoted by : :) :ψr+,p ψ−,q s :

 = 

(5.45) :ψr−,p ψ+,q s :

=

−,q

+,p

+,q

−,p

−ψs ψr , if s = −r < 0, −,q ψr+,p ψs , otherwise, −ψs ψr , if s = −r < 0, −,p +,q ψr ψs , otherwise. −,q

−,q

Also let :ψr+,p ψs : = ψr+,p ψs , :ψr−,p ψs : = ψr−,p ψs , :ψr±,p φs : = ψr±,p φs , for all p, q, r, s. +,q

+,q

5.4.3. (GL(), a∞ )-Howe duality. Let

∑

E∗i j zi−1 w− j =



∑ :ψ+,p(z)ψ−,p(w):.

p=1

i, j∈Z

Equivalently, we have E∗i j =

(5.46)



:. ∑ :ψ+,p−i ψ−,p j−

p=1

1 2

1 2

Consider the following generating functions

∑ e∗pq (n)z−n−1 = :ψ+,p(z)ψ−,q(z):,

n∈Z

p, q = 1, . . . , .

q

q

5.4. Howe duality for infinite-dimensional Lie algebras

185

Such generating functions are “vertex operators” in the theory of vertex algebras, pq but we will avoid such terminology in this book. We will only need e∗ (0) below, pq pq and hence introduce a short-hand notation E∗ ≡ e∗ (0). It follows that

∑

E∗pq =

(5.47)

:ψ−r ψ−,q r :, +,p

p, q = 1, . . . , .

r∈ 21 +Z

Lemma 5.43. (1) Sending Ei j to E∗i j , for i, j ∈ Z, defines a representation of the Lie algebra a∞ on F of level . pq

(2) Sending E pq to E∗ in (5.47), for 1 ≤ p, q ≤ , defines an action of the Lie algebra gl(), which lifts to an action of GL() on F . (3) GL() and a∞ form a Howe dual pair on F . Proof. (1) We need to check that

  [E∗i j , E∗mn ] = δ jm E∗in − δin E∗m j + δ jm δin θ{i ≤ 0} − θ{ j ≤ 0} I.

Let us check in detail the case when m = j and n = i. Since clearly [E∗ii , E∗ii ] = 0, we can assume that i = j. It follows from (5.46) that +  * −,p +,p −,p [E∗i j , E∗ji ] = ∑ ψ+,p ψ ψ 1 1 ,ψ1 1 −i j− − j i− 2

p=1 

=

∑





∑





= =

∑



2

−,p

+,p

+,p

+,p

2

2

2

−,p

−,p

+,p

2

−,p

ψ 1 −i ψi− 1 − ψ 1 − j ψ j− 1 2

p=1

−,p

2

+,p

−,p

ψ 1 −i ψ j− 1 ψ 1 − j ψi− 1 − ψ 1 − j ψi− 1 ψ 1 −i ψ j− 1 2

p=1

=

+,p

2

2

2

2

2



2



2

   +,p −,p +,p −,p :ψ 1 −i ψi− 1 : − :ψ 1 − j ψ j− 1 : + θ{i ≤ 0} − θ{ j ≤ 0} I

2 2 p=1  E∗ii − E∗j j + θ{i ≤

2

2

 0} − θ{ j ≤ 0} I.

The remaining cases are similar. pq

uq

pv (2) We need to check that [E∗ , Euv ∗ ] = δqu E∗ − δ pv E∗ for 1 ≤ p, q, u, v ≤ . Indeed, it follows by (5.47) that  +,p −,q −,v [E∗pq , Euv :ψ−r ψr :, :ψ+,u −r ψr : ∗ ]= ∑ r∈ 21 +Z

=

∑

 +,p +,u −,q  pv uq δqu :ψ−r ψ−,v r : − δvp :ψ−r ψr : = δqu E∗ − δ pv E∗ .

r∈ 21 +Z pq

As the operators E∗ have degree zero in the principal gradation, the action of gl() preserves each (principally) graded subspace of F , which is finite dimensional and of integral weight. Hence the action of gl() lifts to an action of the Lie group GL() on F .

5. Howe duality

186

(3) It follows by (5.46) and (5.47) that [E∗pq , E∗i j ] =

*

∑

r∈ 21 +Z

+  +,p −,q −,u :ψ−r ψr :, ∑ :ψ+,u ψ 1 1: −i j− 2

u=1

2

* + * + +,q −,q +,p −,q +,p −,q +,p −,p = :ψ 1 −i ψi− 1 :, :ψ 1 −i ψ j− 1 : + :ψ 1 − j ψ j− 1 :, :ψ 1 −i ψ j− 1 : 2

=

2

2

2

−,q −,q :ψ+,p ψ j− 1 : − :ψ+,p ψ j− 1 : 1 1 −i −i 2 2 2 2

2

2

2

2

= 0.

Hence the actions of GL() and a∞ on F commute. The statement on the Howe dual pair can be proved by a similar argument as for Theorem 5.14, and we rephrase briefly the argument as follows. Set G = GL(), U = C ⊗ C∞ ⊕ C∗ ⊗ C∞ , where C∞ is a vector space with a basis {wr | r ∈ − 12 − Z+ }. The dual basis in C∞∗ is denoted by {w−r | r ∈ − 12 − Z+ }. We take a basis {v+,p | 1 ≤ p ≤ } for C , and a dual basis {v−,p | 1 ≤ p ≤ } for C∗ .  can be naturally identified with WC(U The Clifford algebra C U ) for the purely odd ±,p

space U = U ⊕U ∗ by setting ψr ≡ v±,p ⊗ wr , so that its associated graded can be identified with ∧(U U ). Observe that the images under the symbol map σ (defined in 5.1.2) of E∗i j in (5.46) form a set of generators for ∧(U U )G by the supersymmetric First Fundamental Theorem of invariant theory for GL(V ) (see Theorem 4.19). Now the claim on the Howe dual pair follows from Proposition 5.12.  Recall the well-known fact that a simple gl()-module L(gl(), λ) of highest weight λ = ∑i=1 λi δi (with respect to the standard Borel subalgebra spanned by E pq for p ≤ q) lifts to a simple GL()-module L(GL(), λ) if and only if (λ1 , . . . , λ ) is a generalized partition in the sense that all λi ∈ Z and λ1 ≥ . . . ≥ λ . By abuse of notation, we shall identify a weight λ for gl() with a generalized partition of the form λ = (λ1 , . . . , λ ). For a generalized partition λ = (λ1 , . . . , λ ) we define, for i ∈ Z,  |{ j | λ j ≥ i}|, for i ≥ 1, λi := −|{ j | λ j < i}|, for i ≤ 0. Theorem 5.44 ((GL(), a∞ )-Howe duality). As a (GL(), a∞ )-module, we have (5.48)

F ∼ =



L(GL(), λ) ⊗ L(a∞ , Λa (λ)),

λ

where the summation is over all generalized partitions of the form λ = (λ1 , . . . , λ ), and Λa (λ) := Λaλ1 + . . . + Λaλ = Λa0 + ∑i∈Z λi εi . Proof. By Lemma 5.43 and Proposition 5.12, we have a strongly multiplicity-free decomposition of the (GL(), a∞ )-module F . To obtain the explicit decomposition as given in the theorem, we will exhibit an explicit formula for a joint highest

5.4. Howe duality for infinite-dimensional Lie algebras

187

weight vector associated to each λ; see Proposition 5.45 below. The theorem follows as we observe by Proposition 5.45 that every simple GL()-module appears in the decomposition (5.48) of F .  For 1 ≤ p ≤  and m ≥ 1, we denote +,p +,p +,p +,p Ξm := ψ−m+ 1 ···ψ 3 ψ 1 , − − 2

−,p Ξm

:= ±,p

We make the convention that Ξ0

−,p ψ−m+ 1 2

2

2

−,p −,p · · · ψ− 3 ψ 1 . −2 2

= 1.

Proposition 5.45. Given a generalized partition of the form λ = (λ1 , . . . , λ ), let i, j be such that λ1 ≥ · · · ≥ λi > λi+1 = · · · = λ j−1 = 0 > λ j ≥ · · · ≥ λ . Then the joint highest weight vector in F associated to λ with respect to the standard Borel for gl() × a∞ is −, j

−, j+1

+,2 +,i −, vaλ = Ξ+,1 λ1 Ξλ2 · · · Ξλi · Ξ−λ j Ξ−λ j+1 · · · Ξ−λ |0,

(5.49)

whose weights with respect to gl() and a∞ are λ and Λa (λ), respectively. Proof. The vector vaλ in (5.49) is indeed a highest weight vector for gl() and a∞ , respectively, since applying any positive root vector in either gl() or a∞ to vaλ is either manifestly zero or gives rise to two identical ψ∗∗ ∗ in the resulting monomial, whence also zero (see Exercise 5.10). Another direct calculation shows that the highest weight of vaλ for gl() is (λ1 , . . . , λ ). −,p

We recall that E∗mm = ∑p=1 : ψ 1 −m ψm− 1 :. When m ≥ 1 and n ≥ 1, one easily 2 2 computes +,p

−,q

[E∗mm , ψ−n+ 1 ] = δm,n ψ−n+ 1 , +,q

[E∗mm , ψ−n+ 1 ] = 0.

+,q

2

2

2

For m ≤ 0 and n ≥ 1 we have −,q

−,q

[E∗mm , ψ−n+ 1 ] = −δ−m+1,n ψ−n+ 1 , 2

2

[E∗mm , ψ−n+ 1 ] = 0. +,q

2

This implies that the weight of the vector vaλ with respect to the action of a∞ equals Λa0 + ∑i λi εi , which is also equal to Λaλ1 + . . . + Λaλ by (5.41).  5.4.4. (Sp(k), c∞ )-Howe duality. Let k = 2. It follows by (5.46) that

∑ (E∗i j − (−1)i+ j E∗1− j,1−i )zi−1w− j

i, j∈Z

= ∑p=1 (:ψ+,p (z)ψ−,p (w): + :ψ+,p (−w)ψ−,p (−z):) .

Equivalently, we have (5.50)

E∗i j − (−1)i+ j E∗1− j,1−i =



∑

p=1



 +,p −,p +,p −,p :ψ 1 −i ψ j− 1 : − (−1)i+ j :ψ j− 1 ψ 1 −i : . 2

2

2

2

5. Howe duality

188

Let p, q = 1, . . . , . Consider the following generating functions

∑ e˜pq (n)z−n−1 = :ψ−,p (z)ψ−,q(−z):,

n∈Z

∑ e˜∗∗pq (n)z−n−1 = :ψ+,p (z)ψ+,q(−z):.

n∈Z

We introduce the following short-hand notation: E˜ pq ≡ e˜ pq (0),

pq pq E˜ ∗∗ ≡ e˜∗∗ (0).

It follows that E˜ pq = (5.51)

−,p

∑

(−1)−r− 2 :ψ−r ψ−,q r :,

∑

(−1)−r− 2 :ψ−r ψ+,q r :.

1

r∈ 21 +Z pq E˜ ∗∗ =

1

+,p

r∈ 21 +Z pq qp Note that E˜ pq = E˜ qp and E˜ ∗∗ = E˜ ∗∗ .

Lemma 5.46. Let k = 2. (1) The formula (5.50) defines an action of Lie algebra c∞ on F of level . pq pq (2) The operators E˜ pq and E˜ ∗∗ in (5.51) together with E∗ in (5.47), for 1 ≤ p, q ≤ , define an action of the Lie algebra sp(k) on F , which lifts to an action of Sp(k). (3) Sp(k) and c∞ form a Howe dual pair on F . Proof. (1) This follows from Lemma 5.43(1), (5.50), and the definition of c∞ . (2) Let us denote a standard basis for V = C ⊕ C∗ by v±,p , for p = 1, . . . , , and denote a standard basis for C∞ by wr with r ∈ (− 12 −Z+ ). Letting U = V ⊗C∞ , ±,p we naturally identify F with ∧(U) by setting ψr ≡ v±,p ⊗ wr . Note that V is a symplectic space with a natural pairing, and so is U. Hence, we have a natural action of sp(V ) = sp(k) on F . The formula (5.51) is simply a precise way of writing down this action in terms of coordinates. Indeed, comparing these formulas with (5.26) we can see that the matrices there with B = C = 0 correspond to the pq pq E∗ ’s, while the ones with A = C = 0 correspond to the E˜ ∗∗ ’s, and the ones with A = B = 0 correspond to the E˜ pq ’s. Alternatively, it can be verified by a direct computation that the operators in (5.51) generate sp(k). Note that the action of sp(k) preserves the principal gradation of F and every graded subspace of F is finite dimensional. Thus, the action of sp(k) lifts to that of Sp(k). pq

(3) As part of the proof of Lemma 5.43, we see that E∗ commutes with the action of c∞ . We will now check that [E˜ uv , c∞ ] = 0, and leave the similar verification that [E˜ uv ∗∗ , c∞ ] = 0 to the reader. Indeed, we have that  uv ∗ E˜ ,Ei j − (−1)i+ j E∗1− j,1−i

5.4. Howe duality for infinite-dimensional Lie algebras * =

∑



189

−,p

+,p

2

2

−,p

−,v i+ j (−1)−r− 2 :ψ−,u :ψ j− 1 ψ 1 −i :) −r ψr :, ∑ (:ψ 1 −i ψ j− 1 : − (−1) 1

+,p 2

p=1

r∈ 21 +Z

+

2

* i −,u −,v j −,v −,u = (−1)i ψ−,v ψ−,u ψi− 1 + (−1) j ψ−,u ψ−,v ψ j− 1 , 1 1 + (−1) ψ 1 1 1 + (−1) ψ 1 −i −i − j i− j− 2 2 2 2 2 2 2−j 2 + +,u −,u +,v −,v i+ j +,u −,u i+ j +,v −,v ψ 1 −i ψ j− 1 + ψ 1 −i ψ j− 1 − (−1) ψ j− 1 ψ 1 −i − (−1) ψ j− 1 ψ 1 −i 2

2

2

2

2

2

2

2

=(−1)i ψ−,v ψ−,u + (−1)i ψ−,u ψ−,v − (−1)i ψ−,u ψ−,v − (−1)i ψ−,v ψ−,u = 0. 1 1 1 1 −i j− 1 −i j− 1 −i j− 1 −i j− 1 2

2

2

2

Hence the actions of Sp(k) and c∞ on

F

2

2

2

2

commute.

The Howe dual pair claim follows by the same type of argument as the one given in the proof of Lemma 5.43(3) using now the First Fundamental Theorem of invariant theory for Sp(k) (see Theorem 4.19).  Theorem 5.47 ((Sp(k), c∞ )-Howe duality). As an (Sp(k), c∞ )-module, we have F ∼ =



L(Sp(k), λ) ⊗ L(c∞ , Λc (λ)),

λ∈P()

where Λc (λ) := Λc0 + ∑i≥1 λi εi = Λcλ1 + . . . + Λcλ . The joint highest weight vector in F associated to λ with respect to the standard Borel for sp(k) × c∞ is vcλ := +,2 +, Ξ+,1 λ1 Ξλ2 · · · Ξλ |0. Proof. We observe that vcλ has weights λ and Λc0 + ∑i≥1 λi εi with respect to the actions of sp(k) and c∞ , respectively, the latter of which equals Λcλ1 + . . . + Λcλ by pq +,p (5.43). Also E˜ ∗∗ annihilates vcλ , since in both expressions only ψ∗ ’s are involved pq and E˜ ∗∗ |0 = 0. Since vcλ is known to be a joint (gl(), a∞ )-highest weight vector by Proposition 5.45, we conclude that vcλ is a joint (sp(k), c∞ )-highest weight vector. By Lemma 5.46 and Proposition 5.12, F is strongly multiplicity-free as an (Sp(k), c∞ )-module. Since vcλ is a nonzero vector in F of weight λ under the action of sp(k), for every λ ∈ P(), we also observe that all finite-dimensional Sp(k)modules appear in the decomposition of F . So the multiplicity-free (Sp(k), c∞ )module decomposition of F follows.  Let z1 , . . . , z , y1 , y2 , . . . be indeterminates. The character of the Sp(k)-module pp F is the trace of the operator ∏p=1 zEp∗ on F , while the character of the c∞ E∗ −E∗1−i,1−i

module F is the trace of the operator ∏i∈N yi ii pp E∗ii −E∗1−i,1−i  ∏i∈N yi ∏ p=1 zEp∗

on F . Computing the trace

of on both sides of the isomorphism in Theorem 5.47, we obtain the following character identity: ∞

(5.52)



∏ ∏ (1 + yi z p)(1 + yi z−1 p )= ∑ i=1 p=1

λ∈P()

ch L(Sp(k), λ) ch L(c∞ , Λc (λ)), E∗ −E∗1−i,1−i

where by definition ch L(c∞ , Λc (λ)) = Tr|L(c∞ ,Λc (λ)) ∏i∈N yi ii

.

5. Howe duality

190

5.4.5. (O(k), d∞ )-Howe duality. Let k = 2. It follows by (5.46) that

∑



 E∗i, j − E∗1− j,1−i zi−1 w− j =





∑

 :ψ+,p (z)ψ−,p (w): − :ψ+,p (w)ψ−,p (z): .

p=1

i, j∈Z

Equivalently, we have the following operators on F : E∗i, j − E∗1− j,1−i =

(5.53)



∑



 −,p +,p −,p :ψ+,p ψ : . 1 1 : − :ψ 1 ψ1 j− −i j− −i 2

p=1

2

2

2

Let p, q = 1, . . . , . We introduce the following generating functions ∑n∈Z e pq (n)z−n−1 = :ψ−,p (z)ψ−,q (z):, pq ∑n∈Z e∗∗ (n)z−n−1 = :ψ+,p (z)ψ+,q (z):. We will adopt the following short-hand notation: Eˇ pq ≡ e pq (0),

pq pq Eˇ ∗∗ ≡ e∗∗ (0).

−,p

pq Eˇ ∗∗ =

It follows that Eˇ pq =

(5.54)

∑

:ψ−r ψ−,q r :,

r∈ 21 +Z

∑

+,p

:ψ−r ψ+,q r :.

r∈ 21 +Z

pq qp Note that Eˇ pq = −Eˇ qp and Eˇ ∗∗ = −Eˇ ∗∗ .

Lemma 5.48. (1) The formula (5.53) defines an action of the Lie algebra d∞  on F of level . pq pq (2) The operators Eˇ pq , Eˇ ∗∗ in (5.54) together with E∗ in (5.47), for 1 ≤ p, q ≤  , define an action of so(2) on F , which lifts to an action of O(2). (3) O(2) and d∞ form a Howe dual pair on F . Proof. The proof is completely analogous to that of Lemma 5.46 for type C. For example, (2) can be easily obtained by comparing the formulas in (5.54) and (5.47) with the one in (5.33), similar to the type C case. We leave the details to the reader (Exercise 5.13).  1

Let k = 2 + 1. We introduce the following operators on F+ 2 , for i, j ∈ Z: E∗i j − E∗1− j,1−i =

(5.55)

  +,p −,p +,p −,p p p :ψ ψ : − :ψ ψ : + :φ 1 −i φ j− 1 :. ∑ 1 −i j− 1 j− 1 1 −i 

p=1

2

2

2

2

2

Equivalently, we have   ∑ E∗i j − E∗1− j,1−i zi−1 w− j i, j∈Z

= ∑p=1 (:ψ+,p (z)ψ−,p (w): − :ψ+,p (w)ψ−,p (z):) + :φ(z)φ(w):.

2

5.4. Howe duality for infinite-dimensional Lie algebras

191

We introduce the following additional generating functions:

∑ e p(n)z−n−1 = :ψ−,p(z)φ(z):,

n∈Z

∑ e∗p(n)z−n−1 = :ψ+,p(z)φ(z):,

p = 1, . . . , .

n∈Z

p p Introducing the short-hand notation Eˇ ∗ = e∗ (0) and Eˇ p = e p (0), we have the following formulas:

(5.56)

Eˇ p =

∑

−,p

:ψ−r φr :,

r∈ 21 +Z

Eˇ ∗p =

∑

+,p

:ψ−r φr :.

r∈ 21 +Z

The following is a counterpart for k = 2 + 1 of Lemma 5.48. Lemma 5.49. (1) The formula (5.55) defines an action of the Lie algebra d∞ 1 + 2 on F . pq pq p (2) The operators Eˇ ∗∗ , Eˇ pq in (5.54), Eˇ ∗ , Eˇ p in (5.56), together with E∗ in (5.47), for 1 ≤ p, q ≤ , define an action of the Lie algebra so(2 + 1) on 1 F+ 2 , which lifts to an action of O(2 + 1). 1

(3) O(2 + 1) and d∞ form a Howe dual pair when acting on F+ 2 . Proof. The proof is analogous to that of Lemma 5.46 in type C and will be left to the reader (Exercise 5.13).  Recall the partition parametrization of simple O(k)-modules from Proposition 5.36. The next theorem treats even and odd k uniformly. Theorem 5.50 ((O(k), d∞ )-Howe duality). As an (O(k), d∞ )-module we have    k F2 ∼ L(O(k), λ) ⊗ L d∞ , Λd (λ) , = λ∈P,λ1 +λ2 ≤k

where Λd (λ) := kΛd0 + ∑i≥1 λi εi . Proof. By Lemmas 5.48 and 5.49, we have a strongly multiplicity-free decompok sition of the (O(k), d∞ )-module F 2 . The proof is completed by finding explicit joint highest weight vectors with prescribed weights and noting that every simple k O(k)-module appears in the decomposition of F 2 , analogous to the proofs of Theorems 5.44 and 5.47. We will refer the reader to [129, Theorem 3.2(2)] and [129, Theorem 4.1(2)] for the precise forms of these joint highest weight vectors for k even and odd, respectively.  E∗ −E∗

pp

Computing the trace of ∏i∈N yi ii 1−i,1−i ∏p=1 zEp∗ on both sides of the identity in Theorem 5.50, we obtain the following character identity (recall the Boolean

5. Howe duality

192

characteristic function (5.40)): ∞

(5.57)

∞



∏(1+yi )θ{k is odd} · ∏ ∏ (1 + yi z p )(1 + yiz−1 p ) i=1

i=1 p=1

=

∑

λ∈P,λ1 +λ2 ≤k

ch L(O(k), λ) ch L(d∞ , Λd (λ)), E∗ −E∗1−i,1−i

where ch L(d∞ , Λd (λ)) = Tr|L(d∞ ,Λd (λ)) ∏i∈N yi ii

.

5.5. Character formula for Lie superalgebras In this section, we obtain character formulas for the irreducible oscillator modules of the Lie superalgebras of type osp constructed in Section 5.3 as a simple application of the Howe dualities established in Section 5.4. 5.5.1. Characters for modules of Lie algebras c∞ and d∞ . For x ∈ {c, d}, let x∞0 be the Cartan subalgebra, and let W be the Weyl group of the Lie algebras x∞ defined in Section 5.4. Let l be the Levi subalgebra of c∞ (respectively of d∞ ) corresponding to the removal of the simple root β× in the Dynkin diagram (5.42) (respectively in the Dynkin diagram (5.44)), and let W0 denote the Weyl group of l. Also, let Φ+ and Φ+ l denote the sets of positive roots of x∞ and l, respectively, corresponding to the respective Dynkin diagrams. Let u+ and u− be, respectively, the nilradical and opposite nilradical associated to l so that we have x∞ = u− ⊕ l ⊕ u+ . Let Wr0 be the set of the minimal length representatives of the right cosets W0 \W of length r for x∞ . It is well known that the Weyl group W can be written as  W = W0W 0 with W 0 = r≥0 Wr0 , and we have W 0 = {w ∈ W | w(−Φ+ ) ∩ Φ+ ⊆ Φ+ \ Φ+ l }.

(5.58)

For µ ∈ (x∞0 )∗ and w ∈ W , we set w ◦ µ := w(µ + ρx ) − ρx , where ρx ∈ (x∞0

)∗

is determined by ρx , Hix  = 1, for every simple coroot Hix .

Let λ be as in Theorem 5.47 in the case of c∞ and as in Theorem 5.50 in the case of d∞ . Since Λx (λ), H xj  ∈ Z+ , for every j, it follows by (5.58) that w ◦ Λx (λ), Hix  ∈ Z+ , for w ∈ W 0 and Hix ∈ l. In our setting, noting that W0 is the group of permutations of N that fix all but finitely many numbers, we may find explicitly a partition λxw = λw = ((λw )1 , (λw )2 , . . .) such that w ◦ Λx (λ) can be written as  kΛx + ∑ j>0 (λw ) j ε j , x w ◦ Λ (λ) = k 0x 2 Λ0 + ∑ j>0 (λw ) j ε j ,

if x = d, if x = c.

5.5. Character formula for Lie superalgebras

193

Recall that sµ denotes the Schur function associated to a partition µ (see, e.g., A.1). Let  ∏1≤i≤ j (1 − yi y j ), for x = c, Dx := ∏1≤i< j (1 − yi y j ), for x = d. Note that Dx = ∏α∈Φ+ \Φ+ (1 − e−α ), where we recall that Φ+ l is the set of the l positive roots in l. Proposition 5.51. Let λ be as in Theorem 5.47 in the case of c∞ and as in Theorem 5.50 in the case of d∞ . We have the following character formula: ch L(x∞ , Λx (λ)) =

1 ∞ ∑ (−1)r ∑ 0 sλxw (y1, y2, . . .), Dx r=0 w∈W

for x = c, d.

r

Proof. For any such λ, set Λ = Λx (λ). The module L(x∞ , Λ) is integrable and hence affords the Weyl-Kac character formula (see e.g. [64]) (5.59)

ch L(x∞ , Λ) =

eσ(Λ+ρx )−ρx , −α ) + (1 − e

∑ (−1)(σ) ∏α∈Φ

σ∈W

where we recall that Φ+ denotes the set of positive roots. Let ρl ∈ (x∞0 )∗ be the element determined by ρl (H j ) = 1, for all j ∈ N, and ρl (H0x ) = 0. Then τ(ρx −ρl ) = ρx − ρl , for τ ∈ W0 . We now rewrite (5.59) as follows: ch L(x∞ , Λ) =

∑ ∑

(−1)(τ) (−1)(w)

w∈W 0 τ∈W0

eτw(Λ+ρx )−ρx ∏α∈Φ+ (1 − e−α )

τw(Λ+ρx )−τ(ρx )+τ(ρx )−ρx (−1)(w) (τ) e (−1) Dx ∏α∈Φ+l (1 − e−α ) w∈W 0 τ∈W0   τ w(Λ+ρx )−ρx +τ(ρl )−ρl (w) (−1) e = ∑ ∑ (−1)(τ) x D ∏α∈Φ+ (1 − e−α ) w∈W 0 τ∈W0

=

∑ ∑

l

=

∑

(−1)(w)

w∈W 0 ∞

=

Dx

eτ(w◦Λ+ρl )−ρl

∑ (−1)(τ) ∏

τ∈W0

(1 − e α∈Φ+ l

−α )

1 ∑ (−1)r ∑ 0 sλw (y1, y2, . . .). Dx r=0 w∈W r

In the last identity we have used the fact that l is of type A+∞ , and so the irreducible l-character of integrable highest weight w ◦ Λ is simply the Schur function sλw .  5.5.2. Characters of oscillator osp(2m|2n)-modules. Recall that hsλ denotes the super Schur function defined in Appendix (A.37). By the character of a module of pp k+1−p.k+1−p Sp(k) or of osp(2m|2n), we mean the trace of the operator ∏1≤p≤ zEp −E E

E

or ∏1≤i≤m ∏1≤ j≤n xi i¯i¯ y j j j , respectively. We have the following character formula

5. Howe duality

194

for the irreducible oscillator osp(2m|2n)-modules appearing in the Howe duality decomposition in Theorem 5.31. Theorem 5.52. Let x = {x1 , . . . , xm } and y = {y1 , . . . , yn }. For λ ∈ P() with λm+1 ≤ n, we have the following character formula: ch L(osp(2m|2n), λ + 1m|n ) r   ∏1≤i≤m (1 + xi ys ) · ∑∞ r=0 (−1) ∑w∈Wr0 hsλcw (y; x) x1 · · · xm  1≤s≤n = · . y1 · · · yn ∏1≤i< j≤m ∏1≤s≤t≤n (1 − xi x j )(1 − ys yt ) E

E

jj n i¯i¯ Proof. Computing the trace of the operator ∏p=1 zEp −E ∏m i=1 ∏ j=1 xi y j on both sides of the isomorphism in Theorem 5.31, we obtain that   x1 · · · xm   m n (1 + y j z−1 p )(1 + y j z p ) ∏ ∏ ∏ −1 y1 · · · yn p=1 i=1 j=1 (1 − xi z p )(1 − xi z p ) (5.60) = ∑ ch L(Sp(2), λ) ch L(osp(2m|2n), λ + 1m|n ). pp

k+1−p,k+1−p

λ∈P() λm+1 ≤n

Next, by formal algebraic manipulations starting from (5.52), we shall obtain a new identity with the same left-hand side as (5.60). Replacing ch L(c∞ , Λc (λ)) in (5.52) by the expression in Proposition 5.51, we obtain an identity of symmetric functions in variables y1 , y2 , . . .. We replace yn+i by xi , for all i ≥ 1, and get the following identity: ∞

n



−1 ∏ ∏ ∏ (1 + yi z p)(1 + yi z−1 p )(1 + x j z p )(1 + x j z p ) = ∑

×

ch L(Sp(k), λ)

λ∈P()

j=1 i=1 p=1

∞ 1 (−1)r ∑ sλcw (y1 , . . . , yn ; x1 , . . .). ∑ 1≤s≤t≤n 1≤i≤ j (1 − xi ys )(1 − xi x j )(1 − ys yt ) r=0 w∈W 0

∏ ∏

r

Now we apply the standard involution ωx (see (A.7)) to both sides of this new identity on the ring of symmetric functions in the variables x1 , x2 , . . .. In the process we use the identities (A.7), (A.24), and (A.38):   ωx ∏(1 − ys xi )−1 = ∏(1 + ys xi ), ωx



i≥1

∏ (1 − xi x j )−1

1≤i≤ j



i≥1



= 

∏ (1 − xi x j )−1 ,

1≤i< j

ωx sµ (y, x) = hsµ (y; x). Finally, in the resulting identity we set x j = 0 for j ≥ m + 1 and multiply both sides   m by xy11···x . In this way we obtain the following identity which shares the same ···yn

5.5. Character formula for Lie superalgebras

left-hand side as (5.60):   x1 · · · xm   m n (1 + y j z−1 p )(1 + y j z p ) = ∏ ∏ ∏ −1 y1 · · · yn p=1 i=1 j=1 (1 − xi z p )(1 − xi z p ) ×

195

∑

ch L(Sp(2), λ)

λ∈P() λm+1 ≤n r   ∏ (1 + xi ys ) ∑∞ r=0 (−1) ∑w∈Wr0 hsλcw (y; x) x1 · · · xm  1≤i≤m 1≤s≤n

y1 · · · yn

∏1≤i< j≤m ∏1≤s≤t≤n (1 − xi x j )(1 − ys yt )

.

The theorem follows by comparing this identity with (5.60) and noting the linear independence of the characters ch L(Sp(2), λ).  5.5.3. Characters for oscillator spo(2m|2n)-modules. The goal of this subsection is to find a character formula for the irreducible oscillator spo(2m|2n)-modules, which appear in the (O(k), spo(2m|2n))-Howe duality (Theorem 5.39). First, let k = 2 + 1 be odd and let λ be a partition with λ1 + λ2 ≤ k. Following the notations of Section 5.3.3 we denote by λ the partition obtained from λ by replacing the first column by k − λ1 . Since the restrictions to SO(k) of the modules L(O(k), λ) and L(O(k), λ) are isomorphic, the usual character will not distinguish them. Recall −I ∈ O(k)\SO(k). So we define the (enhanced) character ch M of an pp k+1−p,k+1−p O(k)-module M to be the trace of the operator ε−I ∏p=1 zEp −E , where ε is an additional formal variable such that ε2 = 1. The character of an spo(2m|2n)Ei¯i¯ E j j n module is defined as usual to be the trace of the operator ∏m i=1 ∏ j=1 xi y j . Theorem 5.53. Let k = 2 + 1. Let x = {x1 , . . . , xm } and y = {y1 , . . . , yn }. For λ ∈ P(m|n) with λ1 + λ2 ≤ k, we have the following character formula:   k ch L spo(2m|2n), λ + 1m|n 2 r   2k ∏1≤i≤m (1 + xi ys ) · ∑∞ r=0 (−1) ∑w∈Wr0 hsλdw (y; x) x1 · · · x m 1≤s≤n = · . y1 · · · yn ∏1≤i≤ j≤m ∏1≤s . 

Since ε|λ| = ε|λ|+1 , we conclude that the (enhanced) characters ch L(O(k), λ), where λ1 + λ2 ≤ k, are linearly independent. E

E

¯¯ jj Computing the trace of the operator ε−I ∏p=1 zEp −E ∏i, j xi ii y j on both sides of the isomorphism in Theorem 5.39, we obtain the following character pp

k+1−p,k+1−p

5. Howe duality

196

identity: 

x1 · · · xm y1 · · · yn

 2k

(5.61)

m



n

∏ ∏∏

(1 + εy j z−1 p )(1 + εy j z p )(1 + εy j )

(1 − εxi z−1 p )(1 − εxi z p )(1 − εxi )   k = ∑ ch L(O(k), λ) ch L spo(2m|2n), λ + 1m|n . 2 λ +λ ≤k

p=1 i=1 j=1

1

2

λm+1 ≤n

Analogous to the proof of Theorem 5.52, by means of algebraic manipulations starting from (5.57), we shall obtain a new identity with the same left-hand side as (5.61). Replacing ch L(d∞ , Λd (λ)) in (5.57) by the expression in Proposition 5.51 and taking into account the eigenvalue of the element −I, we obtain an identity of symmetric functions in the variables y1 , y2 , . . .. We replace yn+i by xi , for all i ≥ 1, and then apply the involution ωx on the ring of symmetric functions in the variables x1 , x2 , . . .. Finally, we set x j = 0 for j ≥ m + 1 and multiply both sides of   2k m the resulting identity by xy11···x . In this way we obtain ···yn 

x1 · · · x m y1 · · · yn =

 2k

∑

λ1 +λ2 ≤k λm+1 ≤n

(1 + εy j z−1 p )(1 + εy j z p )(1 + εy j )



∏ ∏ (1 − εx z−1 p )(1 − εx z

i p )(1 − εxi )

i

p=1 i, j

ch L(O(k), λ) 

x1 · · · xm × y1 · · · yn

r  2k ∏1≤i≤m (1 + xi ys ) ∑∞ r=0 (−1) ∑w∈Wr0 hsλdw (y; x) 1≤s≤n

∏1≤i≤ j≤m ∏1≤s n, we may regard λ as a weight in h∗n in a natural  way. Similarly, for λ ∈ P+ with + λ j = 0 for j > n, we regard λ as a weight in ∗ hn . Finally, for λ ∈ P+ with θ(+ λ) j = 0 for j ∈ 12 N with j > n, we regard λθ as a ∗ weight in  h∗n . These weights λ, λ , λθ in h∗n , hn , and  h∗n , respectively, with the above constraints, will be called dominant weights. The subsets of dominant weights in ∗ + h∗n , hn , and  h∗n will be denoted by Pn+ , Pn , and Pn+ , respectively. Corresponding to a fixed Y0 ⊆ Π(k), the Levi and parabolic subalgebras of the finite-rank Lie superalgebra gn are ln = l ∩ g n ,

pn = p ∩ g n ,

respectively. This allows us to define the corresponding parabolic Verma and irreducible gn -modules Δn (µ) and Ln (µ) with highest weight µ ∈ Pn+ . The corresponding category of gn -modules is denoted by On , which is defined as in Definition 6.6, now with h, l, P+ , and Π therein replaced by hn , ln , Pn+ , and Πn , respectively. The statements in the previous paragraph admit obvious counterparts for the Lie superalgebras gn and  gn as well. We introduce the self-explanatory notation +  n l,  n (ν), L n (ν), O p for  gn , where υ ∈ Pn and Δn (υ), Ln (υ), On , l, p for gn , and Δ ν ∈ Pn+ . 

6.2.5. Truncation functors. Let n < k ≤ ∞. For M ∈ Ok , we write M = γ Mγ , where γ ∈ ∑−1 i=−m Cεi + ∑0< j≤k Z+ ε j + CΛ0 , according to its weight space decomposition. The truncation functor trkn : Ok → On is defined by trkn (M) =

 ν

Mν ,

where the summation is over ν ∈ ∑−1 i=−m Cεi + ∑0< j≤n Z+ ε j +CΛ0 . When it is clear from the context we shall also write trn instead of trkn . Analogously, truncation k → O  n are defined. These functors are obviously functors trkn : Ok → On and trkn : O k exact. (The notation trn used earlier in Section 6.2.1 corresponds to the extreme case here when k = 0.) Recall from Sections 6.1.3 and 6.1.6 the notation E j and Ej for basis elements of the Cartan subalgebras. Proposition 6.9. Let n < k ≤ ∞ and X = L, Δ.

6. Super duality

222 

Xn (µ), if µ, E j  = 0, ∀ j > n, 0, otherwise.    X n (µ), if µ, E j  = 0, ∀ j > n, + (2) For µ ∈ Pk we have trn X k (µ) = 0, otherwise.    Xn (µ), if µ, E j  = 0, ∀ j > n, (3) For µ ∈ Pk+ we have trn Xk (µ) = 0, otherwise. (1) For µ ∈

Pk+

  we have trn Xk (µ) =

Proof. We shall only show (1), as similar arguments prove (2) and (3). Since trkn ◦ trlk = trln , it suffices to show (1) for k = ∞. First suppose that µ, En+1  > 0. Then every weight ν of L(l, µ) also satisfies ν, En+1  > 0 by the choice of fundamental systems of g and hence of l. Recall from the proof of Proposition 6.7 that as an l-module u− decomposes into a direct sum of irreducibles with highest weights in P+ . Thus, every weight υ in Δ(µ) = S(u− ) ⊗ L(l, µ) must also satisfy υ, En+1  > 0. Therefore, trn (Δ(µ)) = 0, and hence trn (L(µ)) = 0. Now suppose that µ, En+1  = 0. Since µ ∈ Pk+ , this implies that µ, E j  = 0 for all j > n. Let l and p denote the standard Levi and parabolic subalgebras of g corresponding to the removal of the vertex βn of the Dynkin diagram of g. Then l ∼ = gn ⊕ gl(W ), where W is the subspace of V0+ spanned by the vectors vi , i > n. Consider the parabolic Verma module Indgp Ln (µ), where Ln (µ) extends to an l -module by the trivial action of gl(W ) and is then extended to a p -module in a trivial way. Clearly, L(µ) is the unique irreducible quotient of Indgp Ln (µ). Since trn (Indgp Ln (µ)) = Ln (µ) and trn (L(µ)) is a nonzero gn -module (as it contains a nonzero vector of weight µ), we conclude that trn (L(µ)) = Ln (µ). To complete the proof, we observe that Δ(µ) ∼ = S(u− ) ⊗ L(l, µ), and that the gn -module trn (Δ(µ)) has highest weight µ with character equal to ch Δn (µ). From these we conclude that trn (Δ(µ)) = Δn (µ). 

6.3. The irreducible character formulas  → O and T : O  → O are introduced, and they In this section, two functors T : O are shown to send irreducible and parabolic Verma modules to irreducible and parabolic Verma modules, respectively. The main ingredients for studying these functors are two sequences of odd reflections on the standard Borel subalgebra of  g, leading to new Borel subalgebras which are “approximately compatible” with the standard Borel subalgebras of g and g. As a consequence, the solution of the irreducible character problem in O via the classical Kazhdan-Lusztig theory also  provides a complete solution to the irreducible character problem for O and O.

6.3. The irreducible character formulas

223

6.3.1. Two sequences of Borel subalgebras of  g. Let us recall briefly the basics of odd reflections from Chapter 1, Sections 1.3 and 1.5, which are applied now to the Lie superalgebra  g. Under the odd reflection with respect to an isotropic odd   , the resulting new fundamental system simple root α in a fundamental system Π  α is given by (1.44) as follows: Π  α = {β ∈ Π   | (β, α) = 0, β = α} ∪ {β + α | β ∈ Π   , (β, α) = 0} ∪ {−α}. Π   by b and the resulting new Let us denote the Borel subalgebra corresponding to Π α Borel subalgebra by b . According to Lemma 1.40, an irreducible  g-module of b highest weight λ is also a bα -highest weight module with bα -highest weight equal to either λ or λ − α, depending on whether (λ, α) = 0. It is instructive to review Examples 1.35 and 1.41 to see how the highest weights change under a sequence of odd reflections, as they offer in a simpler setting a similar pattern to what we shall see below. Recall from (6.2) the odd roots αr and the even roots βr , for r ∈ 12 N. Recall from Section 6.1 that the standard Dynkin diagram associated to  g is given by (6.17)

  

k α×





α1/2

α1

···







αn−1/2

αn

αn+1/2

···

Fix n ∈ N. We shall specify a sequence of n(n+1) odd reflections, which will trans2 form the standard diagram (6.17) into a new diagram with a new fundamental system of the form (6.19) below. The ordered sequence of n(n+1) odd roots we use 2 are: ε1/2 − ε1 , ε3/2 − ε2 , ε5/2 − ε3 ,

(6.18)

ε1/2 − ε2 ,

ε3/2 − ε3 ,

ε1/2 − ε3 ,

··· ,··· ,··· ,··· , εn−1/2 − εn ,

εn−3/2 − εn ,

...,

ε3/2 − εn ,

ε1/2 − εn .

Let us go into the details. First, applying the odd reflection corresponding to α1/2 = ε1/2 − ε1 , we obtain the following diagram:  

k  β×

 ε1 − ε1/2



β1/2









α3/2

α2

α5/2

α3

···

Next, applying to it consecutively the two odd reflections corresponding to α3/2 = ε3/2 − ε2 and α1/2 + α1 + α3/2 = ε1/2 − ε2 , we obtain the following two diagrams:  

k  β×









ε1 − ε1/2 ε1/2 − ε2 −α3/2 ε3/2 − ε5/2





α5/2

α3

···

6. Super duality

224  

k 



β×

β1







ε2 − ε1/2 β1/2

β3/2





α5/2

α3

···

Now, to the last diagram above we apply consecutively the three odd reflections corresponding to α5/2 = ε5/2 − ε3 , α3/2 + α2 + α5/2 = ε3/2 − ε3 , and α1/2 + α1 + α3/2 + α2 + α5/2 = ε1/2 − ε3 and obtain the following three diagrams:  

k 



 

k 



 

k 





β1

β2

β×

β1

β×

β1

β×





ε2 − ε1/2 β1/2







ε3/2 − ε3

−α5/2





ε2 − ε1/2 ε1/2 − ε3 ε3 − ε3/2

 ε3 − ε1/2



β1/2



β5/2









β3/2

β3/2

β5/2

β5/2

 α7/2

 α7/2

 α7/2

···

···

···

We continue this way until we finally apply the n odd reflections corresponding to αn−1/2 , αn−3/2 + αn−1 + αn−1/2 , . . ., ∑2n−1 i=1 αi/2 , which equal εn−1/2 − εn , εn−3/2 − εn , . . . , ε1/2 − εn , respectively. The resulting new Borel subalgebra for  g in the end c  will be denoted by b (n), and the corresponding positive system will be denoted  c (n). The corresponding fundamental system, denoted by Π  c (n), is listed in by Φ   Tn the following Dynkin diagram (recall the tail diagram

from Section 6.1.1): (6.19)

    

k Tn ε − ε n

1/2



β1/2

···



βn−1/2





αn+1/2

αn+1

··· 

The crucial point here is that the subdiagram to the left of the first in (6.19) is the Dynkin diagram of gn , and the precise detail of the remaining part of (6.19) is not needed. On the other hand, starting with the standard Dynkin diagram (6.17) of  g , we n(n+1) may apply a different sequence of 2 odd reflections associated to the following ordered sequence of n(n+1) odd roots: 2 ε1 − ε3/2 , ε2 − ε5/2 , ε1 − ε5/2 , ε3 − ε7/2 ,

(6.20)

ε2 − ε7/2 ,

ε1 − ε7/2 ,

··· ,··· ,··· ,··· , εn − εn+1/2 ,

εn−1 − εn+1/2 ,

...,

ε2 − εn+1/2 ,

ε1 − εn+1/2 .

Since the procedure here is completely parallel to the previous one for the sequence (6.18), we skip the details. The resulting new Borel subalgebra for  g in the end s  will be denoted by b (n), and the corresponding positive system will be denoted by

6.3. The irreducible character formulas

225

 s (n). The corresponding fundamental system, denoted by Π  s (n), is listed in the Φ   following Dynkin diagram (recall the tail diagram

Tn from Section 6.1.1): (6.21)

     Tn+1

k

ε −ε n+1/2

 β1

1

···

 βn





αn+1

αn+3/2

···

Again the crucial fact here is that the subdiagram to the left of the odd simple root εn+1/2 − ε1 in (6.21) is the Dynkin diagram of gn+1 . +

Recall that Φ+ and Φ denote the standard positive systems of g and g, respectively. Also, b and b denote the corresponding standard Borel subalgebras of g and g, respectively. By definition, g and g are naturally subalgebras of  g. +  c (n), and b =   s (n), Proposition 6.10. We have Φ+ ⊆ Φ bc (n) ∩ g. Also, Φ ⊆ Φ s  and b = b (n) ∩ g.

 c (n) and Π ⊆ Φ  s (n), where Π and Π are the Proof. It suffices to check that Π ⊆ Φ standard fundamental systems of g and g, respectively. These inclusions can then be observed directly from (6.19) and (6.21).  6.3.2. Odd reflections and highest weight modules. Recall the standard Levi subalgebra l of  g with nilradical  u and opposite nilradical  u− from Section 6.2.2. Lemma 6.11. The sequences of odd reflections (6.18) and (6.20) leave the set of roots of  u and the set of roots of  u− invariant. Proof. The isotropic odd roots used in the sequences of odd reflections (6.18) and (6.20) are of the form αi + αi+1/2 + . . . + α j for 0 < i < j; hence, they are all roots of l. Since  u and  u− are both invariant under the action of l, the lemma follows by applying Lemma 1.30.  We denote by  bc (n) and  bs (n) the Borel subalgebras of the Levi subalgebra l l l  c (n) ∩ ∑  Zα and Π  s (n) ∩ ∑  Zα, corresponding to the fundamental systems Π α∈Y α∈Y respectively (here we recall Y from (6.13)). By Lemma 6.11 we have, for all n ∈ N, (6.22)

 bc (n) =  bcl (n) +  u,

 bs (n) =  bsl (n) +  u.

∗ + Below we will regard P+ ⊆  h∗ and P ⊆  h∗ , via h∗ ⊆  h∗ and h ⊆  h∗ .

Proposition 6.12. Let λ = (dΛ0 , − λ, + λ) ∈ P+ , where + λ = (+ λ1 , + λ2 , . . .) is a partition. Let n ∈ N. (1) Suppose that (+ λ) ≤ n. Then the highest weight of L(l, λθ ) with respect to the Borel subalgebra  bc (n) is λ. l

6. Super duality

226

(2) Suppose that + λ1 ≤ n. Then the highest weight of L(l, λθ ) with respect to the Borel subalgebra  bs (n) is λ . l

Proof. The proof of (2) using the sequence (6.20) is completely parallel to the proof of (1) which uses (6.18). We shall only prove (1).    We observe that the odd  reflections (6.18) only affect the tail diagram

and T leave the head diagram k unchanged, and the summand −1

∑

λ|k := dΛ0 +

(6.23)

λi εi

i=−m

of λθ in (6.16) is unchanged by these odd reflections. We will show more generally by induction on k that, after applying the first k(k + 1)/2 odd reflections of the sequence (6.18), the highest weight of L(l, λθ ) with respect to the Borel subalgebra  bc (k) becomes l

k

(6.24)

k



λ[k] =λ|k + ∑ + λi εi + ∑ + λi − kεi−1/2 i=1

+

∑

i=1

 + λ j −

j + 1ε j−1/2 +

j≥k+1

∑

+ λ j − jε j ,

j≥k+1

where q := max{q, 0} for q ∈ Z. Part (1) follows as a special case of (6.24) for k = n. Note that λ[0] = λθ = λ|k + ∑ j≥1 λj − j + 1ε j−1/2 + ∑ j≥1 λ j − jε j . Suppose that k = 1. If (+ λ) < 1, then + λ = 0 and λθ = λ|k . So in particular λθ , E1/2 + E1  = 0, and thus by Lemma 1.40, the  bc (1)-highest weight is λ[1] = λθ . l If (+ λ) ≥ 1, then λθ , E1/2 + E1  > 0, as we recall that 

λθ = λ|k + + λ1 ε1/2 + (+ λ1 − 1)ε1 + · · · . Hence, by Lemma 1.40, the  bc (1)-highest weight after the odd reflection with rel spect to ε1/2 − ε1 is 

λ[1] = λ|k + + λ1 ε1 + (+ λ1 − 1)ε1/2 + · · · , proving (6.24) in the case k = 1. Now by the induction hypothesis, suppose that λ[k] is the new highest weight after applying the first k(k + 1)/2 odd reflections of the sequence (6.18). If (+ λ) ≤ k, then λ[k] = λ|k + ∑ki=1 + λi εi . Therefore, for 1 ≤ i ≤ k + 1, we have λ[k] , Ei−1/2 + Ek+1  = 0, and by Lemma 1.40, the odd reflections with respect to εi−1/2 − εk+1 leave λ[k] unchanged. Thus we have λ[k+1] = λ[k] , and in this case we are done. Now assume that (+ λ) ≥ k + 1. Let s = + λk+1 . We further separate this into two cases (i) and (ii) below.

6.3. The irreducible character formulas

227 

(i) Suppose that + λk+1 ≥ k + 1. Then + λk+1 ≥ k + 1, and hence we can rewrite k

k

i=1

i=1





λ[k] = λ|k + ∑ + λi εi + ∑ (+ λi − k)εi−1/2 + (+ λk+1 − k)εk+1/2 

∑

+ ( λk+1 − k − 1)εk+1 + +

∑

+ λ j − j + 1ε j−1/2 +

j≥k+2

+ λ j − jε j .

j≥k+2

Now we apply the next (k + 1) odd reflections in the sequence (6.18) consecutively. As λ[k] , Ek+1/2 + Ek+1  > 0, by Lemma 1.40 we calculate the new weight after the odd reflection with respect to εk+1/2 − εk+1 to be k

k

i=1

i=1





λ[k,1] = λ|k + ∑ + λi εi + ∑ (+ λi − k)εi−1/2 + (+ λk+1 − k − 1)εk+1/2 + ( λk+1 − k)εk+1 + +

∑



+ λ j − j + 1ε j−1/2 +

j≥k+2

∑

+ λ j − jε j .

j≥k+2

Now λ[k,1] , Ek−1/2 + Ek+1  > 0, so by Lemma 1.40 we calculate the new weight after the odd reflection with respect to εk−1/2 − εk+1 to be k

k−1





λ[k,2] =λ|k + ∑ + λi εi + ∑ (+ λi − k)εi−1/2 + (+ λk − k − 1)εk−1/2 i=1 i=1 +  + ( λk+1 − k − 1)εk+1/2 + (+ λk+1 − k + 1)εk+1  + + λ j − j + 1ε j−1/2 + + λ j − jε j . j≥k+2 j≥k+2

∑

∑

Continuing this way, after a total of (k + 1) odd reflections we end up with the weight k+1

k+1

i=1

i=1



λ[k,k+1] =λ|k + ∑ + λi εi + ∑ (+ λi − k − 1)εi−1/2 +

∑

 + λ j −

j + 1ε j−1/2 +

j≥k+2

∑

+ λ j − jε j ,

j≥k+2

which is exactly λ[k+1] . The induction step is completed in this case. 

(ii) Now suppose that + λk+1 = s < k + 1. Since + λ is a partition, the weight (6.24) becomes k

s

i=1

i=1



λ[k] = λ|k + ∑ + λi εi + ∑ (+ λi − k)εi−1/2 , 

where + λi − k > 0, for i ≤ s. It follows by Lemma 1.40 that odd reflections with respect to εk+1/2 − εk+1 , · · · , εs+1/2 − εk+1 leave λ[k] unchanged, while odd reflections with respect to εs−1/2 − εk+1 , · · · , ε1/2 − εk+1 do affect λ[k] . In a similar way,

6. Super duality

228

the new weight after these (k + 1) odd reflections is calculated to be k+1

s

i=1

i=1



λ[k,k+1] = λ|k + ∑ + λi εi + ∑ (+ λi − k − 1)εi−1/2 , which is again equal to λ[k+1] . The induction step is completed.



 be a Proposition 6.13. Let n ∈ N and λ = (dΛ0 , − λ, + λ) ∈ P+ . Let V (λθ ) ∈ O θ highest weight  g-module of highest weight λ with respect to the standard Borel subalgebra  b. The following statements hold. (1) Suppose that (+ λ) ≤ n. Then V (λθ ) is a highest weight module of highest weight λ with respect to the Borel subalgebra  bc (n). (2) Suppose that + λ1 ≤ n. Then V (λθ ) is a highest weight module of highest weight λ with respect to the Borel subalgebra  bs (n).  θ ) = U( Proof. By definition, Δ(λ g) ⊗U(p) L(l, λθ ), and hence its  g-quotient V (λθ ) contains a unique copy of L(l, λθ ) that is annihilated by  u. By Proposition 6.12(1), L(l, λθ ) has  bc (n)-highest weight λ, and let us denote by v a corresponding  bc (n)l

l

highest weight vector. Thus, by (6.22), v is also a  bc (n)-singular vector in V (λθ ) of weight λ. The vector v clearly generates the l-module L(l, λθ ) and hence the  g-module V (λθ ), proving (1). The proof of (2) based on Proposition 6.12(2) is similar and hence omitted.  6.3.3. The functors T and T . By definition, g and g are naturally subalgebras of  g, l and l are subalgebras of l, while h and h are subalgebras of  h. Also, we have ∗ inclusions of the restricted duals h∗ ⊆  h∗ and h ⊆  h∗ .  =  ∗ M γ , we form the following subGiven a semisimple  h-module M γ∈h

 spaces of M: (6.25)

 := T (M)



γ , M

and

 := T (M)

 ∗

γ∈h∗

γ . M

γ∈h

 is an h-submodule of M,  and T (M)  is an h-submodule of M.  One Note that T (M)  is also an l-module, then T (M)  is an l-submodule of M  and T (M)  checks that if M    is an l-submodule of M. Furthermore, if M is a  g-module, then T (M) is a g and T (M)  is a g-submodule of M.  submodule of M  give rise to the natural projections The direct sum decompositions in M  −−−−→ T (M)  TM : M

and

 −−−−→ T (M)  T M : M

→N  is an  that are h- and h-module homomorphisms, respectively. If f : M h homomorphism, then the following maps induced by restrictions of f ,  −−−−→ T (N)  T [ f] : T (M)

and

 −−−−→ T (N),  T [ f] : T (M)

6.3. The irreducible character formulas

229

→N  is a are also h- and h-module homomorphisms, respectively. Also if f : M    g-module homomorphism, then TM and T [ f ] (respectively, T M and T [ f ]) are gmodule (respectively, g-module) homomorphisms. It follows that T and T define exact functors from the category of  h-semisimple  g-modules to the category of hsemisimple g-modules and the category of h-semisimple g-modules, respectively. Also, we have the following commutative diagrams: 

(6.26)

 −−−f−→ M ⏐ ⏐T & M

 N ⏐ ⏐T & N



f]  −−T−[−  T (M) → T (N)

Lemma 6.14. For λ ∈ P+ , we have   T L(l, λθ ) = L(l, λ),



 −−−f−→ M ⏐ ⏐T & M

 N ⏐ ⏐T & N



f]  −−T−[−  T (M) → T (N)

  T L(l, λθ ) = L(l, λ ).

Proof. We prove the first formula using a comparison of the characters. The proof of the second formula is similar and will be omitted. Let λ = (dΛ0 , − λ, + λ) ∈ P+ . To be consistent with the notation of λ|k in (6.23), it is convenient to make the convention that k contains the central element K, and let L(l ∩ k, λ|k ) denote the irreducible l ∩ k-module of highest weight λ|k . It follows from the discussion in Section 6.2.1 that (6.27)

ch L(l, λθ ) = ch L(l ∩ k, λ|k ) hs+ λ (x1/2 , x3/2 , . . . ; x1 , x2 , . . .),

where xr := eεr for r ∈ 12 N. Note that l ∩ k = l ∩ k. Now L(l, λθ ) is completely reducible as an l-module, since a polynomial module of gl(V0+ ) = gl(∞|∞) is completely reducible when restricted to gl(V0+ ) = gl(∞). On the character level, applying T to L(l, λθ ) corresponds to setting x1/2 , x3/2 , x5/2 , . . . in the character formula (6.27) to zero. Thus,   by (A.37), T L(l, λθ ) is an l-module with character ch L(l ∩ k, λ|k ) s+ λ (x1 , x2 , . . .), which is precisely the character of L(l, λ). This proves the first formula.   to O, and T is an exact functor Proposition 6.15. T is an exact functor from O  to O. from O  then T (M)  ∈ O,  ∈O Proof. In light of Proposition 6.14, it suffices to show that if M    and T (M) ∈ O. We shall only show that T (M) ∈ O, as the argument for T (M) ∈ O is analogous. First consider the case when the head diagram is non-degenerate, i.e., m > 0  Then there exists ζθ , ζθ , . . . , ζθ , with ζ1 , . . . , ζr ∈  ∈ O. (see Section 6.1.1). Let M r 1 2  is bounded by some ζθ . Ignoring dΛ0 recall that we P+ , such that any weight of M i

6. Super duality

230

have ζ j = (− ζ j , + ζ j ), where + ζ j is a partition with |+ ζ j | = k j . For each ζ j , let Pj be the following finite subset of h∗ : Pj := {(− ζ j , µ) | µ ∈ P with |µ| = k j }.  and P(M) := rj=1 Pj . Set M := T (M) Claim. Given any weight ν of M, there exists γ ∈ P(M) such that γ − ν ∈ Z+ Π.  we It suffices to prove the claim for ν ∈ P+ . Since ν is also a weight of M,  for some i. Thus have ζθi − ν ∈ Z+ Π, ζθi − ν = pα× + − κ + + κ, where p ∈ Z+ , − κ ∈ ∑α∈Π(k) Z+ α, and + κ ∈ ∑β∈Π(T)\α Z+ β. This implies that + ν  × is a partition of size ki + p, and hence there exists γ ∈ P(M) such that + ν is obtained from the partition + γ by adding p boxes to it. For every such a box, we record the row number in which it was added in the multiset J with |J| = p. Then we have ν = γ − − κ − ∑ (ε−1 − ε j ), j∈J

and hence γ − ν ∈ Z+ Π. Thus, we conclude that M ∈ O. The limit case m = 0 now can be proved case-by-case by slight modification of the argument above (see Exercise 6.1).   is a highest weight  Proposition 6.16. Let λ ∈ P+ . If V (λθ ) ∈ O g-module of highest θ θ θ   weight λ , then T (V (λ )) and T (V (λ )) are highest weight g- and g-modules of highest weights λ and λ , respectively. Proof. We will only prove the statement for T (V (λθ )), as the case of T (V (λθ )) is analogous. By Proposition 6.13, for n > (+ λ), V (λθ ) is a  bc (n)-highest weight module of highest weight λ, and thus a nonzero vector v in V (λθ ) of weight λ (unique up to a scalar multiple and independent of n) is a  bc (n)-highest weight vector of V (λθ ). θ Evidently we have v ∈ T (V (λ )), and by Proposition 6.10, v is a b-singular vector in T (V (λθ )). The g-module T (V (λθ )), regarded as an l-module, is completely reducible by Proposition 6.7 and Lemma 6.14. To complete the proof, it suffices to show that every vector w ∈ T (V (λθ )) of weight µ = (dΛ0 , − µ, + µ) ∈ P+ lies in U(n− )v, where n− denotes the opposite nilradical of b. To that end, we choose n such that n > max{(+ λ), (+ µ)}. Then we have w ∈ U( nc (n)− )v, where  nc (n)− denotes the opposite nilradical of  bc (n). Now the + + condition n > max{( λ), ( µ)} implies that (6.28)

λ−µ =

−1

∑

i=−m

n−1

ai ε i + ∑ b j ε j , j=1

ai , b j ∈ Z.

6.3. The irreducible character formulas

231

(We emphasize that only ε j with integer indices j appear in (6.28).) But λ − µ is  c (n), i.e., also a Z+ -linear combination of simple roots from Π λ−µ =

∑

 c (n) α∈Π

aα α,

with all aα ∈ Z+ and finitely many aα > 0. We have the following. Claim. λ − µ is a Z+ -linear combination of Πn = Π(k) {β−1 , β1 , . . . , βn−1 }. The claim then implies that w ∈ U(n− )v, and the proposition follows. It remains to prove the claim. Assume on the contrary that there were some  c (n) \ Πn = {εn − ε1/2 , β1/2 , . . . , βn−1/2 , αn+1/2 , αn+1 , . . .} with aα = 0. If α∈Π aεn −ε 1 = 0, then we must also have aα = 0, for α = β1/2 , . . . , βn−1/2 , αn+1/2 by the 2

constraint (6.28). Similarly, if aβi−1/2 = 0 for some 1 ≤ i ≤ n, then we must also have aα = 0, for α = βi+1/2 , . . . , βn−1/2 , αn+1/2 as well. In all cases, λ − µ, Er  = 0, for some r ≥ n, contradicting (6.28).  6.3.4. Character formulas. The following theorem is the first main result of this chapter. Theorem 6.17. Let λ ∈ P+ . We have      θ ) = Δ(λ), T L(λ  θ ) = L(λ); T Δ(λ      θ ) = Δ(λ ), T L(λ  θ ) = L(λ ). T Δ(λ Proof. We will prove only the statements involving T . The statements involving T can be proved in an analogous way.  θ ) = U( Let us write Δ(λ) = U(u− ) ⊗C L(l, λ) and Δ(λ u− ) ⊗C L(l, λθ ). We observe that all the weights in U(u− ), L(l, λ), U( u− ), and L(l, λθ ) are of the form u− )) = ∑ j0 br εr with br ∈ Z+ (possibly modulo dΛ0 ). Since T (U(      θ ) = ch Δ(λ). Since T Δ(λ  θ) U(u− ), it follows by Lemma 6.14 that ch T Δ(λ isa highest  weight module of highest weight λ by Proposition 6.16, we have θ  T Δ(λ ) = Δ(λ).  = L(λ  θ ). Suppose that M := T (M)  is reducible. Since by ProposiSet M tion 6.16 the module M is a highest weight g-module, it must have a b-singular vector, say w of weight µ ∈ P+ , inside M that is not a highest weight vector. We  is a  choose n > max{(+ λ), (+ µ)}. By Proposition 6.13, the  g-module M bc (n)highest weight module of highest weight λ, say with a highest weight vector vλ . By Proposition 6.16, vλ is a b-highest weight vector of M, and hence w ∈ U(a)vλ , where a is the subalgebra of n− generated by the root vectors corresponding to the roots in −Π = −Π(k) {−β−1 , −β1 , . . . , −βk }, for some k. Now choose q > max{n, k + 1}. Note that vλ is also a  bc (q)-highest weight  of weight λ. Since w is b-singular, it is annihilated by vector of the  g-module M

6. Super duality

232

the root vectors corresponding to the roots in Π(k) {β−1 , β1 , β2 , . . .}. But w is  c (q) complealso annihilated by the root vectors corresponding to the roots in Π mentary to the subset Π = Π(k) {β−1 , β1 , β2 , . . . , βq−1 }, since these root vectors  commute with a and w ∈ U(a)vλ . It follows that w is a  bc (q)-singular vector in M,  contradicting the irreducibility of M.  It is standard to write (6.29)

ch L(λ) =

∑

aµλ ch Δ(µ),

µ∈P+

for λ ∈ P+ and aµλ ∈ Z. For x = a, b, c, d, the triangular transition matrix (aµλ ) in Theorem 6.18 or its inverse is known in the Kazhdan-Lusztig theory, since the Kazhdan-Lusztig polynomials determine the composition multiplicities of parabolic Verma modules in the parabolic BGG category O of g-modules; see, e.g., Soergel [116, p. 455]. The infinite rank of g here does not cause any difficulty in light of Proposition 6.9. We have the following character formulas for g and  g. Theorem 6.18. Let λ ∈ P+ . Then (1) ch L(λ ) = ∑µ∈P+ aµλ ch Δ(µ ),  θ ).  θ ) = ∑µ∈P+ aµλ ch Δ(µ (2) ch L(λ Proof. This follows immediately from Theorem 6.17 and (6.29).



Remark 6.19. Theorem 6.18 together with Proposition 6.9 provide a complete solution a` la Kazhdan-Lusztig to the irreducible character problem in the category On for the finite-rank basic Lie superalgebras of type gl and osp.

6.4. Kostant homology and KLV polynomials In this section, we formulate and study the Kostant homology groups of the Lie superalgebras  u− , u− , and u¯ − , with coefficients in modules belonging to the re We show that the functors T : O  → O and spective categories O, O, and O.  → O match perfectly the corresponding Kostant homology groups (and also T :O Kostant cohomology groups). Such matchings are then interpreted as equalities of  the Kazhdan-Lusztig-Vogan polynomials for the categories O, O, and O. Some basic facts on Lie algebra homology and cohomology needed in this section can be found in the book of Kumar [76]. 6.4.1. Homology and cohomology of Lie superalgebras. Let L = L0¯ ⊕ L1¯ be a vector superspace, and let T(L) be the tensor superalgebra of L. Then T(L) = ∞ ⊗n is a Z-graded associative superalgebra. Recall that the exterior supern=0 L algebra of L is the quotient superalgebra ∧ (L) := T(L)/J, where J is the homogeneous two-sided ideal of T(L) generated by elements of the form x ⊗ y +

6.4. Kostant homology and KLV polynomials

233

(−1)|x||y| y ⊗ x, where x and y are Z2 -homogeneous elements of L. Then ∧ (L) = ∞ n n=0 ∧ L is also a Z-graded associative superalgebra. For elements x1 , x2 , . . . , xk in L, the image of the element x1 ⊗ x2 ⊗ · · · ⊗ xk under the canonical quotient map from L⊗k to ∧ k (L) will be denoted by x1 x2 · · · xk . We emphasize that the exterior algebra defined and used in this subsection is always understood in the super sense. Now let L = L0¯ ⊕ L1¯ be a Lie superalgebra of countable dimension. For an L-module V we define Cn (L,V ) = ∧ n (L) ⊗V, Define the boundary operator d :=

C• (L,V ) =





Cn (L,V ).

n≥0

n dn

: C• (L,V ) → C• (L,V ) by

dn (x1 x2 · · · xn ⊗ v) := (6.30)

∑

(−1)s+t+|xs | ∑i=1 |xi |+|xt | ∑ j=1 |x j |+|xs ||xt | [xs , xt ]x1 · · · xs · · · xt · · · xn ⊗ v

1≤s n, we have the following ln -module isomorphisms.   − (1) trkn ∧ (u− k ) = ∧ (un ).   (2) trkn ∧ (u− ) ⊗ L (λ) = ∧ (u− k n ) ⊗ Ln (λ). k We now employ the same method as in Section 6.4.3 to study the effect of the truncation functors trkn on the corresponding u− k -homology groups with coefficients − in the module Lk (λ), for λ ∈ Pk+ . Let d (n) : ∧ (u− n ) ⊗ Ln (λ) → ∧ (un ) ⊗ Ln (λ) denote the corresponding boundary operator of the chains of the finite-dimensional Lie

6.5. Super duality as an equivalence of categories

241

(k) superalgebra u− = d (n) , n with coefficients in Ln (λ). It is evident that d |∧ (u− n )⊗Ln (λ) and furthermore we have d (n) trkn = trkn d (k) . The proof of Theorem 6.27 can be adapted to prove the following.

Theorem 6.33. Let λ ∈ Pk+ and i ∈ Z+ . We have    Hi (u− n , Ln (λ)) , k − trn Hi (uk , Lk (λ)) = 0,

if λ ∈ Pn+ , otherwise.

For λ, µ ∈ Pn+ , the expression ∞   (n) µλ (q) := ∑ (−q)−i dim Homln L(ln , µ), Hi (u− n , Ln (λ)) i=0

equals the parabolic Kazhdan-Lusztig polynomial of the Lie algebra gn via Vogan’s homological interpretation. Theorem 6.33 and (6.37) imply that the KazhdanLusztig polynomials for finite-rank semisimple or reductive Lie algebras of classical type admit a remarkable stability, i.e., we have (6.38)

(n)

µλ (q) = µλ (q),

λ, µ ∈ Pn+ .

A similar argument leads to analogous stability of the KLV polynomials in  n , which coincide for n  0 with  θ θ (q) and    (q), recategories On and O µ λ µλ spectively. Theorem 6.31 then further allows us to suitably identify these KLV polynomials for finite-rank Lie (super)algebras gn , gn , and  gn .

6.5. Super duality as an equivalence of categories In this section, we establish super duality, which is formulated as an equivalence of the two module categories O and O. More precisely, we shall show that the  → O and T : O  → O are equivalences of categories. In the case functors T : O when x = a, b, c, d, the category O, as a variant of the BGG category for classical Lie algebras, is well understood, and super duality leads to new insight on the module category O for Lie superalgebras. Some basic facts on category theory and homological algebra needed in this section can be found in the book of Mitchell [86]. 6.5.1. Extensions a` la Baer-Yoneda. In this subsection, we shall review the BaerYoneda extension Ext1 in a general abelian category C. This is needed later on for  since there may not be enough projective objects in these the categories O, O, O, categories. A reader can also take for granted the exact sequence (6.42) and safely skip this subsection. We shall omit most of the proofs, and refer to Mitchell [86, Chapter VII] for details. For two objects A,C ∈ C, let E, E  be short exact sequences of the form (6.39)

i

j

E : 0 −→ A −→ B −→ C −→ 0,

6. Super duality

242

i

j

E  : 0 −→ A −→ B −→ C −→ 0. We say that E ∼ E  if there exists f : B → B such that i = f ◦i and j  ◦ f = j. By the 5-lemma f is an isomorphism, and so ∼ is indeed an equivalence relation. The set of extensions of degree 1 of C by A is by definition the set of equivalence classes of exact sequences of the form (6.39) with respect to the relation ∼, and will be denoted by Ext1C (C, A). We shall write E ∈ Ext1C (C, A) for the class represented by the exact sequence E by abuse of notation. Denote by E0 the equivalence class corresponding to the short exact sequence E0 : 0 −→ A−→A ⊕C−→C −→ 0. Given an object C ∈ C, a morphism γ : C → C, and an element E ∈ Ext1C (C, A), we have the following diagram: C ⏐ ⏐γ & E:

0 −−−−→ A −−−−→ B −−−−→ C −−−−→ 0

Then, there exists a unique element E  ∈ Ext1C (C , A) such that the following diagram is commutative: E :

0 −−−−→ A −−−−→ / / /

B −−−−→ ⏐ ⏐β &

C −−−−→ 0 ⏐ ⏐γ &

E:

0 −−−−→ A −−−−→ B −−−−→ C −−−−→ 0

Indeed, B is simply the pullback of γ and the morphism B → C. The morphism A → B is then uniquely determined by the commutativity of the left square and the exactness of the first row of the diagram. We denote E  ∈ Ext1 (C , A) by Eγ. E· Letting γ vary, we obtain a map E· : HomC (C ,C) → Ext1C (C , A) defined by γ → Eγ. On the other hand, letting E vary, one shows that a morphism γ : C → C induces a well-defined map ·γA : Ext1C (C, A) → Ext1C (C , A). That is, for a fixed object A in C, Ext1C (·, A) is a contravariant functor. With this notation ·γA = Ext1C (γ, A) we shall write ·γA = ·γ when A is clear from the context. Dually, suppose we are given an object A ∈ C, a morphism α : A → A , and an element E ∈ Ext1C (C, A), so that we have the following diagram: E:

0 −−−−→ A −−−−→ B −−−−→ C −−−−→ 0 ⏐ ⏐α & A

6.5. Super duality as an equivalence of categories

243

We obtain uniquely an element E  ∈ Ext1C (C, A) such that the following diagram is commutative: E : 0 −−−−→ A −−−−→ B −−−−→ C −−−−→ 0 ⏐ ⏐ / ⏐α ⏐β / & & / E :

0 −−−−→ A −−−−→ B −−−−→ C −−−−→ 0

The resulting element E  is denoted by αE. Fixing E we thus obtain a map ·E : ·E HomC (A, A ) → Ext1C (C, A ) given by α → αE. On the other hand, by letting E vary, we can show that a fixed morphism α : A → A induces a well-defined map 1 1 1  C α· : ExtC (C, A) → ExtC (C, A ) so that ExtC (C, ·) is a covariant functor. With this 1 notation we have C α· = ExtC (C, α). Thus, Ext1C (·, ·) gives a bi-functor on C. We shall write C α· = α·, when C is clear from the context. For an extension E, with morphisms α and γ as above, we have (αE)γ = α(Eγ). We define an abelian group structure on Ext1C (C, A). For A ∈ C, we define a morphism Δ = ΔA : A → A ⊕ A, Δ(a) := (a, a), and a morphism ∇ = ∇A : A ⊕ A → A by ∇(a, a ) := a + a . Given two elements E, E  ∈ Ext1C (C, A) we let E ⊕ E  be the obvious element in Ext1C (C ⊕C, A ⊕ A). To define the group operation we consider ∇A (E ⊕ E  )ΔC ,

(6.40)

which is an exact sequence of the form (6.39). Define E + E  to be the element in Ext1C (C, A) corresponding to (6.40). It can be shown that the operation + : Ext1C (C, A) × Ext1C (C, A) → Ext1C (C, A) is well-defined, abelian, and associative. Furthermore, it can be shown that the equivalence class of the split exact sequence E0 is an additive identity and that (−1A )E is the additive inverse of E. For A,C ∈ C, it is convenient to set Ext0C (C, A) := HomC (C, A).

Let X be an object in C. Applying the functor HomC (X, ·) to the exact sequence (6.39) gives us an exact sequence (6.41)

0 → HomC (X, A) → HomC (X, B) → HomC (X,C).

Now for each γ ∈ HomC (X,C), we obtain Eγ ∈ Ext1C (X, A). It can be shown that the map E· extends (6.41) to the following exact sequence: (6.42)

0 −→HomC (X, A) −→ HomC (X, B) −→ HomC (X,C) E·

i·

j·

−→Ext1C (X, A) −→ Ext1C (X, B) −→ Ext1C (X,C).  Now we return to the categories O, O, 6.5.2. Relating extensions in O, O, and O.   and let E ∈ Ext1 (C,  C ∈ O  A).  and O, of g-, g-, and  g-modules, respectively. Let A,  O  of the form Then we have by Section 6.5.1 an exact sequence in O E :

 −→ B −→ C −→ 0. 0 −→ A

6. Super duality

244

Applying the exact functors T and T , we obtain exact sequences in O and O, respectively. It can be easily seen that, in this way, the functors T and T define   T (A)   A)  to Ext1 T (C), natural homomorphisms of abelian groups from Ext1 (C, O O   1    and ExtO T (C), T (A) , respectively. The respective elements are denoted by T [E]  and T [E]. Recall the Lie superalgebra g with parabolic subalgebra p, Levi subalgebra l and nilradical u from Section 6.2.2. We have the following relative Koszul resolution for the trivial p-module (see Knapp-Vogan [72, II.7]): (6.43)

δ

δ

ε

k 1 · · · −→Ck −→ Ck−1 −→ · · · −→ C0 −→ C −→ 0.

Here Ck := U(p) ⊗U(l) ∧ k (p/l) is a p-module with p acting on the left of the first factor, and ε is the augmentation map from U(p) to C. The p-homomorphism δk is given by δk (a ⊗ x1 x2 · · · xk ) :=

∑

(−1)s+t a ⊗ [xs , xt ]x1 · · · xs · · · xt · · · xk

1≤s 0, and δ0 := ε ⊗ 1: (6.45)

δ

δ

δ

0 k 1 · · · −→ Dk −→ Dk−1 −→ · · · −→ D0 −→ L(l, λ) −→ 0.

The above construction admits straightforward generalizations for the Lie superalgebras g and  g.  and V ∈ O. We have, for i = 0, 1, Theorem 6.34. Let λ ∈ P+ , V ∈ O, V ∈ O,   Homl L(l, λ), H i (u,V ) ∼ = ExtiO (Δ(λ),V ),    θ ), V  ), Homl L(l, λθ ), H i ( u, V ) ∼ = ExtiO (Δ(λ   Homl L(l, λ ), H i (u,V ) ∼ = ExtiO (Δ(λ ),V ). Proof. We shall only prove the first isomorphism for l for i = 0, 1, and the remaining two isomorphisms can be proved in an analogous fashion. Let V ∈ O. Recall the (unrestricted) cohomology group Hi (u,V ) from Remark 6.20 in contrast to the (restricted) cohomology group H i (u,V ) used mostly in this chapter. Let Hi (u,V )ss be the maximal h-semisimple submodule of Hi (u,V ). We have the following

6.5. Super duality as an equivalence of categories

245

Claim. Hi (u,V )ss ∼ = H i (u,V ). To see this, observe by definition that we have a natural injective map from Hi (u,V )ss . Since the coboundary operator ∂i is an h-homomorphism i ∧ (u),V ) is a direct product of finite-dimensional h-weight spaces, Ker ∂i and Hom(∧ and Im ∂i are also direct products of finite-dimensional h-weight spaces. Now given x ∈ Ker ∂i representing a cohomology class in Hi (u,V )ss , there exists an element y ∈ Im ∂i−1 such that x − y ∈ Ci (u,V ) ∩ Ker ∂i . We have then ι(x − y) ∈ x + Im ∂i−1 , and hence ι is also a surjection. This proves the claim. ι H i (u,V ) →

Now since L(l, λ) is h-semisimple and Hi (u,V ) is a direct product of finitedimensional h-weight spaces, it follows by the claim that     Homl L(l, λ), Hi (u,V ) ∼ = Homl L(l, λ), H i (u,V ) .

(6.46)

This allows us to freely switch between H i and Hi below. Unraveling the definition, we have H 0 (u,V ) = V u , the u-invariants of V . Thus, Homl (L(l, λ), H 0 (u,V )) ∼ = Homl (L(l, λ),V u ) ∼ = Homp (L(l, λ),V ) ∼ = HomO (Δ(λ),V ), ∼ is due to Frobenius reciprocity. This proves the isomorphism where the last “=” for l with i = 0 in the theorem.   It remains to show that Homl L(l, λ), H 1 (u,V ) ∼ = Ext1O (Δ(λ),V ). We shall construct the isomorphism explicitly.   First, take f¯ ∈ Homl L(l, λ), H 1 (u,V ) . Recall H 1 (u,V ) = Ker d1 /Im d0 , with Ker d1 ⊆ Hom(u,V ). Let f ∈ Homl (L(l, λ), Hom(u,V )) ∼ = Homl (u ⊗ L(l, λ),V ) be a representative that, by Frobenius reciprocity, is regarded as an element in   Homp Indpl (u ⊗ L(l, λ)) ,V . Applying the exact functor U(g) ⊗U(p) - to (6.45), we obtain an exact sequence   ∂2 ∂1 U(g) ⊗U(l) ∧ 2 (p/l) ⊗ L(l, λ) −→U(g) ⊗U(l) (p/l ⊗ L(l, λ)) −→ ∂

0 U(g) ⊗U(l) L(l, λ) −→ Δ(λ) −→ 0,

where ∂i denotes the differentials induced from the differentials δi . This gives rise to the following diagram ∂

U(g) ⊗U(l) (p/l ⊗ L(l, λ)) −−−1−→ U(g) ⊗U(l) L(l, λ) −−−−→ Δ(λ) −−−−→ 0 ⏐ ⏐f & V

6. Super duality

246

Taking the pushout E f of f and ∂1 , we obtain the following commutative diagram U(g)

U(g)

IndU(l) (p/l ⊗ L(l, λ)) −−−−→ IndU(l) L(l, λ) −−−−→ Δ(λ) −−−−→ 0 ⏐ ⏐ / ⏐f ⏐ / & & / V

α

−−−−→

Ef

−−−−→ Δ(λ) −−−−→ 0

Since f represents a cocycle, we have f ◦∂2 = 0. This implies that α is an injection. Now since V and Δ(λ) lie in O, so does E f , and hence we obtain an element in Ext1O (Δ(λ),V ). One verifies in a straightforward manner that if f is a coboundary, then E f ∼ = V ⊕ Δ(λ), and hence the map f¯ → E f¯ is well-defined. Conversely, suppose that we have an extension of the form 0 −→ V −→ E −→ Δ(λ) −→ 0. Note that E ∈ O, and hence there exists an l-submodule of E that is mapped isomorphically to L(l, λ) ⊆ Δ(λ). We denote this l-module also by L(l, λ) and note that uL(l, λ) ⊆ V . This allows us to define an element fE ∈ Homl (u ⊗ L(l, λ),V ) ∼ = Homl (L(l, λ), Hom(u,V )) by fE (u ⊗ w) = uw. Now fE corresponds to a cocycle in Homl (L(l, λ), Hom(u,V )), because V is a submodule of the u-module E. Also, one sees that if E is the trivial extension, then f E must be a coboundary. Thus, sending E to f¯E gives us a well-defined map Ext1O (Δ(λ),V ) → Homl (L(l, λ), H1 (u,V )) ∼ = Homl (L(l, λ), H 1 (u,V )); see (6.46). It can be  checked1 that these  two1 maps are inverses to each other. Hence, we have Homl L(l, λ), H (u,V ) ∼  = ExtO (Δ(λ),V ).  It makes sense to apply T and T to the short exact sequences (i.e., Ext1 ) in O. O

Corollary 6.35. Let λ, µ ∈ P+ . We have, for i = 0, 1,       θ ), L(µ  θ ) = ExtiO Δ(λ), L(µ) , T ExtiO Δ(λ       θ ), L(µ  θ ) = Exti Δ(λ ), L(µ ) . T ExtiO Δ(λ O Proof. We shall only prove the identity for T , as the proof for T is analogous. By Theorem 6.34, for i = 0, 1, we have     ExtiO Δ(λ), L(µ) ∼ = Homl L(l, λ), H i (u, L(µ)) , (6.47)      θ ), L(µ  θ) ∼  θ )) . ExtiO Δ(λ u, L(µ = Homl L(l, λθ ), H i (

   θ ))) ∼ Recall that T L(l, λθ ) = L(l, λ) by Lemma 6.14, and that T (H n ( u, L(λ = H n (u, L(λ)) by Corollary 6.28. Now the identity for T in the corollary follows by applying the functor T to (6.47) and the naturality of T . 

6.5. Super duality as an equivalence of categories

247

f  f . The following proposition is standard and well 6.5.3. Categories O f , O , and O known. For a proof the reader may consult Kumar [76, Lemma 2.1.10]. Recall Π denotes the standard fundamental system of g.

Proposition 6.36. Let M ∈ O. Then there exists a (possibly infinite) increasing filtration of g-modules 0 = M0 ⊆ M1 ⊆ M2 ⊆ · · · ⊆ Mi−1 ⊆ Mi ⊆ · · ·

(6.48) such that (1) M =



i≥0 Mi ;

(2) Mi /Mi−1 is a highest weight module of highest weight νi with νi ∈ P+ , for i ≥ 1; (3) the condition νi − ν j ∈ ∑α∈Π Z+ α implies that i < j; (4) for any weight µ of M, there exists an r ∈ N such that (M/Mr )µ = 0.   ∈ O. Similar statements hold for M ∈ O and M Let O f denote the full subcategory of O consisting of finitely generated U(g)f  f are defined in a similar fashion. modules. The categories O and O Corollary 6.37. Let M ∈ O. Then M ∈ O f if and only if there exists a finite increasing filtration 0 = M0 ⊆ M1 ⊆ M2 ⊆ · · · ⊆ Mk = M of g-modules such that Mi /Mi−1 is a highest weight module of highest weight νi with νi ∈ P+ , for 1 ≤ i ≤ k. Similar   ∈ O. statements hold for M ∈ O and M Proof. If M is a g-module with a finite filtration of the form (6.48), then clearly M is finitely generated. Conversely, suppose that u1 , u2 , . . . , un are vectors in M of weights ν1 , ν2 , . . . , νn generating M over U(g). By Proposition 6.36(4), M has a (possibly infinite) filtration of the form (6.48) such that there exists r ∈ N with Mr containing all vectors in M of weights ν1 , ν2 , . . . , νn . Thus ui ∈ Mr , for 1 ≤ i ≤ n, and hence Mr = M. The proofs for  g and g are analogous and omitted.  6.5.4. Lifting highest weight modules. The following proposition is the converse of Proposition 6.16. Proposition 6.38. (1) Suppose that V (λ) is a highest weight g-module of highest weight λ ∈ P+ . Then there is a highest weight  g-module V (λθ ) of highest weight λθ such that T (V (λθ )) = V (λ). (2) Suppose that U(λ ) is a highest weight g-module of highest weight λ  θ ) of highest with λ ∈ P+ . Then there is a highest weight  g-module U(λ θ θ   weight λ such that T (U(λ )) = U(λ ).

6. Super duality

248

Proof. We shall only prove (1), as (2) is similar. Let W be the kernel of the surjective g-homomorphism from Δ(λ) to V (λ).  θ )) = Δ(λ). By the exactness of the functor T , Now Theorem 6.17 says that T (Δ(λ  θ ) such that T (W  of Δ(λ  ) = W. it suffices to prove that W lifts to a submodule W We recall the standard Borel subalgebra b and standard fundamental system Π for g. Recall the sequence of Borel subalgebras  bc (n) and corresponding fun c (n) from Section 6.3.1, and that Πn = Π(k) Π(Tn ) from damental systems Π Section 6.1.1. We make the following. Claim. Let vµ be a b-singular vector of weight µ ∈ P+ in a  g-subquotient of θ c   Δ(λ ). Then vµ is a b (n)-singular vector, for n  0. This claim can be seen as follows. There exists n  0 such that λ − µ is a non-negative sum of even simple roots lying in Πn , and such that every weight in  θ ) is of the form λ − ν, where ν is a non-negative sum of simple roots in Π  c (n). Δ(λ c  (n) \ Πn , then µ + α cannot be a weight in Therefore, if α is a simple root in Π θ  Δ(λ ). This proves the claim. There is an increasing filtration of g-modules for W , 0 = W0 ⊆ W1 ⊆ W2 ⊆ · · · , satisfying the properties of Proposition 6.36. For each i > 0, let wi be a weight vector of weight νi in Wi such that wi +Wi−1 is a nonzero b-highest weight vector  θ ) = µ∈P+ L(l, µθ )m(µ) if and only if of Wi /Wi−1 . Now by Theorem 6.17, Δ(λ   θ ), there is a highest weight Δ(λ) = µ∈P+ L(l, µ)m(µ). Then, regarding Δ(λ) ⊆ Δ(λ ∼ L(l, νθ ) with respect to the Borel subalgebra i of the l-module U(l)wi = vector w i   θ ) generated by w i be the  1 , w 2 , . . . , w i b ∩l, for each i. Let W g-submodule of Δ(λ  for i ≥ 1, and set W0 = 0. i ) = Wi , for all i. The case i = 0 is We will prove by induction on i that T (W i−1 ) = Wi−1 . By construction, wi + W i−1 is a btrivial. Assume that i ≥ 1 and T (W    singular vector in Wi /Wi−1 . It follows by the claim above that wi + Wi−1 is a  bc (n)singular vector of weight νi for n  0. By (6.22), we have  b= b ∩l +  u⊆ b ∩l + c    i above, w i + Wi−1 is a b-highest weight vector b (n). Thus, by the construction of w −   i /W i−1 , and thus of Wi /Wi−1 . Now U(u )L(l, νi ) = Wi /Wi−1 and U( u− )L(l, νθi ) = W i /W i−1 ) = Wi /Wi−1 . This together with the inductive assumption implies that T (W  T (Wi ) = Wi .  = i≥1 W i , we obtain that T (W  ) = W. Finally, setting W



6.5.5. Super duality and strategy of proof. Recall the functors T and T from Section 6.3.3. The following is a main result of this chapter.  → O is an equivalence of categories. Theorem 6.39. (1) T : O  → O is an equivalence of categories. (2) T : O (3) The categories O and O are equivalent.

6.5. Super duality as an equivalence of categories

249

The equivalence of categories in Theorem 6.39(3) is called super duality. Define an equivalence relation ∼ on  h∗ by letting µ ∼ ν if and only if µ − ν lies  in the root lattice ZΠ of  g. For each such equivalence class [µ], fix a representative o ∗ o  ¯ For ε = 0, ¯ 1, ¯ set [µ] ∈ h and declare [µ] to have Z2 -grading 0.   ! "  h∗ε = µ ∈  µ − [µ]o , Er ≡ ε (mod 2) , for x = a, b, c, d, h∗ | ∑ r∈1/2+Z+

  m ! " ! "  h∗ε = µ ∈  h∗ | ∑ µ − [µ]o , E−i + ∑ µ − [µ]o , Er ≡ ε (mod 2) , for x = b• . i=1

r∈N

  is a semisimple  Recall that V ∈ O h-module with V = γ∈h∗ Vγ . Then V gives rise to a Z2 -graded  g-module V  :

(6.49)

V  := V0¯



V1¯ ,

Vε :=



Vµ

¯ 1), ¯ (ε = 0,

µ∈ h∗ε

which is compatible with the Z2 -grading on  g. ¯ ¯  0 and O  f ,0 to be the full subcategories of O  and O  f , respectively, We define O consisting of objects equipped with Z2 -gradation given by (6.49). Note that the  0¯ and O  f ,0¯ are of degree 0.  V  ¯ It is clear that for a given V ∈ O, morphisms in O  defined in (6.49) is isomorphic to V in O. It then follows by definition that the  and O  0¯ are equivalent. Similarly, O  f and O  f ,0¯ are equivalent two categories O categories. ¯

¯

0¯

f ,0¯

We can now analogously define O0 , O f ,0 , O , and O to be the respective f full subcategories of O, O f , O, and O consisting of objects equipped with Z2 0¯ f ,0¯ f ¯ ¯ gradation (6.49). Similarly, O0 ∼ = O, O ∼ = O, and also O f ,0 ∼ = Of , O ∼ =O . ¯

0 ¯  it suffices to prove Theorem 6.39(1) for  0¯ ∼ Since O0 ∼ = O, O ∼ = O, and O = O, ¯ 0  0¯ → O0¯ and T : O  0¯ → O . In order to keep notation simple, we will from T :O  O f , and O  f to denote the respective now on drop the superscript 0¯ and use O, O, 0¯  0¯ f ,0¯  f ,0¯ . Henceforth, when we write Δ(λ f,  θ ), L(λ  θ) ∈ O categories O , O , O , and O

λ ∈ P+ , we will mean the corresponding modules equipped with the Z2 -gradation (6.49). Similar convention applies to Δ(λ ) and L(λ ).

In the remainder of this subsection, we outline the strategy of the proof of Theorem 6.39. The detailed proof will be carried out in the next subsection. We shall restrict our proof for T entirely, since the arguments for the functors T and T are completely parallel. First, we study the relation between the morphisms  and O under T . In particular, we show in Proposition 6.44 in the categories O that T induces an isomorphism between morphisms of the two categories. In conjunction with Theorem 6.34 that relates extensions with homology groups, this isomorphism is then used to show that T induces an isomorphism between the first

6. Super duality

250

extension groups of the two categories. These, together with well-known facts from homological algebra, imply that T is an equivalence of categories. 6.5.6. The proof of super duality. We shall freely use the following short-hand  Similar conventions  and M = T (M),  for M  ∈ O. notation below. Set M = T (M)  In addition, set V (λ) = T (V (λθ )) , M   , N  ∈ O. apply to the images of T and T of M  θ and V (λ ) = T (V (λ )), for a highest weight  g-module V (λθ ) of highest weight λθ with λ ∈ P+ (see Proposition 6.16 for a justification of notation).  and let  ∈ O, Lemma 6.40. Let N E :



i   −→    −→ 0 0 −→ M M−→ M  be an exact sequence of  g-modules in O.

(6.50)

(1) The exact sequence (6.50) induces the following commutative diagram with exact rows. (We will use subscripts to distinguish various maps induced by T .)          ,N  N  −−→ Hom  M  ,N  −−→ Hom  M,  0 −−→ HomO M O O ⏐ ⏐ ⏐ ⏐T    ⏐T   ⏐T    & M ,N & M,N & M ,N        0 −−→ HomO M , N −−→ HomO M, N −−→ HomO M  , N         ·E  ,N  N  −−→ Ext1 M  ,N  −−→ Ext1 M,  −−→ Ext1 M   O ⏐ O⏐ O ⏐ ⏐T 1 ⏐T 1 ⏐T 1 & M  ,N & M, N & M  ,N        ·T (E) −−−→ Ext1O M  , N −−→ Ext1O M, N −−→ Ext1O M  , N .

(2) The analogous statement holds replacing T by T in (1), M by M, etc. Proof. By (6.42), the rows are exact, and it remains to show that the following diagram is commutative:       ·E  ,N  ,N  −−−  HomO M −→ Ext1 M O ⏐ ⏐ ⏐T    ⏐T 1 (6.51) & M ,N & M  ,N     ·T (E)  HomO M  , N −−−−→ Ext1O M  , N .        ,N  ,N  . Then fE ∈ Ext1 M  is the bottom exact row of Let f ∈ HomO M  O the following commutative diagram:

(6.52)

0 −−−i−→ 0 −−−−→ M ⏐ ⏐ &f

 −−−−→ M ⏐ ⏐ &

  −−−−→ 0 M / / /

 −−−−→ F −−−−→ M   −−−−→ 0. 0 −−−−→ N

6.5. Super duality as an equivalence of categories

251

 which is the pushout of f and i. Applying Here we identify fE with the module F, the functor T to (6.52) gives us a commutative diagram with exact rows: TM  ,M [i]

0 −−−−→ M  −−−−→ ⏐ ⏐T [ f] & M  ,N

M ⏐ ⏐ &

−−−−→ M  −−−−→ 0 / / /

 −−−−→ M  −−−−→ 0. 0 −−−−→ N −−−−→ T (F)  ≡ T 1  [ fE]  is the pushout of T    [ f] and T    [i]. ThereWe conclude that T (F)  ,N  M ,N M ,M M fore, we obtain that  ≡ T (F)  ≡ T    [ f]T [E],  TM1  ,N [ fE] M ,N 

i.e., the diagram (6.51) is commutative.  Then  N  ∈ O. Lemma 6.41. Let M,  N)  → HomO (M, N) is an injection. (1) T : HomO (M,  N)  → Hom (M, N) is an injection. (2) T : Hom  (M, O

O

 : L(l, µθ ) → L(l, µθ ), with µ ∈ P+ , Proof. By Lemma 6.14, any l-isomorphism ϕ ] : L(l, µ) → L(l, µ) and T [ϕ ] : L(l, µ ) → L(l, µ ). The induces isomorphisms T [ϕ lemma follows.  Lemma 6.42. Let V (λθ ) be a highest weight  g-module of highest weight λθ with +  Then  ∈ O. λ ∈ P , and let N  −→ HomO (V (λ), N) is an isomorphism. (1) T : HomO (V (λθ ), N)  −→ Hom (V (λ ), N) is an isomorphism. (2) T : Hom  (V (λθ ), N) O

O

Proof. Consider the commutative diagram with exact rows  θ ) −−−−→ V (λθ ) −−−−→ 0  −−−−→ Δ(λ 0 −−−−→ M ⏐ ⏐ ⏐ ⏐T ⏐T θ ⏐T θ & M & Δ(λ ) & V (λ ) 0 −−−−→ M −−−−→ Δ(λ) −−−−→ V (λ) −−−−→ 0. This gives rise to the following commutative diagram with exact rows:        θ ), N  N   −−−→ Hom  Δ(λ  −−−→ Hom  M, 0 −−−→ HomO V (λθ ), N O O ⏐ ⏐ ⏐ ⏐T θ  ⏐T θ  ⏐T   & V (λ ),N & Δ(λ ),N & M,N       0 −−−→ HomO V (λ), N −−−→ HomO Δ(λ), N −−−→ HomO M, N . Corollary 6.28(3) and Theorem 6.34 imply that TΔ(λ  θ ),N  is an isomorphism. By Lemma 6.41, TM, N  is an injection. Now a standard diagram chasing implies that TV (λθ ),N is an isomorphism. 

6. Super duality

252

Lemma 6.43. Let V (λθ ) be a highest weight  g-module of highest weight λθ with  Then  ∈ O. λ ∈ P+ , and let N  → Ext1 (V (λ), N) is an injection. (1) T : Ext1 (V (λθ ), N) O (2) T :

O  → Ext1 (V (λθ ), N) O

Ext1O (V (λ ), N) is an injection.

Proof. Let (6.53)



f  −→ E −→ 0 −→ N V (λθ ) −→ 0

be an exact sequence of  g-modules. Suppose that (6.53) gives rise to a split exT [ f]

act sequence of g-modules 0 → N → E → V (λ) → 0. Thus there exists ψ ∈ ∈ HomO (V (λ), E) such that T [ f] ◦ ψ = 1V (λ) . By Lemma 6.42, there exists ψ θ  such that T [ψ  ] = ψ. Thus T [ f◦ ψ  ] = T [ f] ◦ T [ψ  ] = 1V (λ) . By HomO (V (λ ), E)  = 1  θ , and hence (6.53) splits. Lemma 6.42 again, we have f◦ ψ  V (λ )

 Then  N  ∈ O. Proposition 6.44. Let M,  N)  −→ HomO (M, N) is an isomorphism. (1) T : HomO (M,  N)  −→ Hom (M, N) is an isomorphism. (2) T : HomO (M, O  f . We proceed by induction on the length of ∈O Proof. First we assume that M  If M  is a highest weight module, then it is true by Lemma 6.42. a filtration of M.   2 ⊂ · · · ⊂ M k = M  be an increasing filtration of  Let 0 = M0 ⊂ M1 ⊂ M g-modules  i /M i−1 is a highest weight module of highest weight νθ with νi ∈ P+ , such that M i  i /M i−1 and Zi := T (Zi ). for 1 ≤ i ≤ k. Let Zi := M Consider the following commutative diagram with an exact top row of  gmodules and an exact bottom row of g-modules, for i ≥ 1.

(6.54)

ιi i−1 −−− 0 −−−−→ M −→ ⏐ ⏐T  & Mi−1

i −−−−→ M ⏐ ⏐T  & Mi

Zi −−−−→ 0 ⏐ ⏐T & Zi

ι

0 −−−−→ Mi−1 −−−i−→ Mi −−−−→ Zi −−−−→ 0. The sequence (6.54) induces the following commutative diagram with exact rows.     i , N  −−−−→ Hom  M  0 −−−−→ HomO Zi , N O ⏐ ⏐ ⏐T  ⏐T   & Zi ,N & M,N     0 −−−−→ HomO Zi , N −−−−→ HomO Mi , N

6.5. Super duality as an equivalence of categories

253

    ·ιi i−1 , N  −−−−→ Ext1 Zi , N  −−−− → HomO M  O⏐ ⏐ ⏐T   ⏐T 1 & Mi−1 ,N & Zi ,N     ·ιi −−−−→ HomO Mi−1 , N −−−−→ Ext1O Zi , N . The map TMi−1 ,N is an isomorphism by induction. The map TZ1 ,N is an injection i by Lemma 6.43. Also TZi ,N is an isomorphism by Lemma 6.42. Now a standard diagram chasing implies that TMi ,N is an isomorphism.  By Proposition 6.36, we may  ∈ O. Now we consider the general case of M 0 ⊂ M 1 ⊂ M 2 ⊂ · · · such that choose an increasing filtration of  g-modules 0 = M  θ     i≥0 Mi = M and Mi /Mi−1 is a highest weight module of highest weight νi with i }i is lim M  ∼  νi ∈ P+ , for i ≥ 1. Then the direct limit of {M −→ i = M and the inverse ∼   ∼   limit is ← lim lim − HomO (Mi , N) = HomO (M, N). Similarly we have − → Mi = M and ∼ lim ←− HomO (Mi , N) = HomO (M, N). Furthermore, we have the following commutative diagram (where ϕ = ← lim −TMi ,N ):     ∼ =  N  −−− i , N  HomO M, M −→ ← lim Hom  O − ⏐ ⏐ ⏐T   ⏐ϕ & M,N &     ∼ = HomO M, N −−−−→ ← lim − HomO Mi , N . Since ϕ is an isomorphism, so is TM,  N .  Then  N  ∈ O. Lemma 6.45. Let M,  N)  → Ext1 (M, N) is an injection. (1) T : Ext1 (M, O (2) T :

O  N)  Ext1O (M,

→ Ext1O (M, N) is an injection.

Proof. The proof is virtually identical to the proof of Lemma 6.43, where we use Proposition 6.44 in place of Lemma 6.42.  Proposition 6.46. Let V (λθ ) be a highest weight  g-module of highest weight λθ  Then  ∈ O. with λ ∈ P+ , and let N  → Ext1 (V (λ), N) is an isomorphism. (1) T : Ext1O (V (λθ ), N) O

 → Ext1 (V (λ ), N) is an isomorphism. (2) T : Ext1 (V (λθ ), N) O O

Proof. Consider the following commutative diagram with exact rows:

(6.55)

 θ ) −−−π−→ V (λθ ) −−−−→ 0  −−−−→ Δ(λ 0 −−−−→ M ⏐ ⏐ ⏐ ⏐T ⏐T θ ⏐T θ & M & Δ(λ ) & V (λ ) π

0 −−−−→ M −−−−→ Δ(λ) −−−−→ V (λ) −−−−→ 0.

6. Super duality

254

The sequence (6.55) induces the following commutative diagram with exact rows:        θ ), N  N  −−−−→ Ext1 V (λθ ), N  −−−−→ Hom  M,  HomO Δ(λ  O O ⏐ ⏐ ⏐ ⏐T θ  ⏐T   ⏐T 1 & Δ(λ ),N & M,N & V (λθ ),N       HomO Δ(λ), N −−−−→ HomO M, N −−−−→ Ext1O V (λ), N     · π  θ ), N  N   −−−−→ Ext1 M, −−−−→ Ext1O Δ(λ  O ⏐ ⏐ ⏐T 1 ⏐T 1 & Δ(λ & M, N  θ ),N      ·π −−−−→ Ext1O Δ(λ), N −−−−→ Ext1O M, N . 1 is an injection by Lemma 6.45. The map T 1 The map TM, N   θ ),N  is an isomorphism Δ(λ by Corollary 6.35. Also TΔ(λ  θ ),N N  are isomorphisms by Proposition 6.44.  and TM, Now a standard diagram chasing implies that TV1(λθ ),N is an isomorphism. 

Proof of Theorem 6.39. Let C, C be abelian categories. Recall from Mitchell [86, II.4] that by definition a full and faithful functor F : C → C is an equivalence of categories if it satisfies the representative property that for every M  ∈ C there exists M ∈ C with F(M) ∼ = M . Proposition 6.44 implies that the functor T is full and faithful. Now for every M ∈ O, there is a filtration of g-modules for M, 0 = M0 ⊂ M1 ⊂ M2 ⊂ · · · , with Mi ∈ O satisfying the properties of Proposition 6.36. The filtration {Mi }i of M  f such that T (M i }i with M i ∈ O i ) ∼ lifts to a filtration {M = Mi by induction using    ∼ Propositions 6.38 and 6.46. Set M := ∪i≥0 Mi . It follows that T (M) = M.   ∈ O. To do that we proceed as in the proof of the We need to prove that M claim in Proposition 6.15. Suppose that ζ1 , . . . , ζr ∈ P+ such that every weight of M is bounded by ζi , for some i. For j = 1, . . . , r, define Pj and P(M) as in  = M, it suffices to prove that if ν ∈ P+ and Proposition 6.15. Now since T (M)  Let ζi − ν ∈ Z+ Π for some i, then there exists γ ∈ P(M) such that γθ − νθ ∈ Z+ Π. + | ζi | = ki . We write ζi − ν = − κ + p(ε−1 − ε1 ) + + κ, where p ∈ Z+ , − κ ∈ ∑α∈Π(k) Z+ α, and + κ ∈ ∑β∈Π(T)\β× Z+ β. Then + ν is a partition of size ki + p. Hence, there exists γ ∈ P(M) such that the partition + ν is obtained from + γ by adding p boxes. We record the sub-indices of the boxes in θ(+ γ) \ θ(+ ν) in the multiset J θ . Then we have γθ − νθ = − κ + ∑s∈J θ (ε−1 − εs ),  This implies that all the weights of M  are bounded above and hence γθ −νθ ∈ Z+ Π. θ   by the finite set {γ |γ ∈ P(M)}, and hence M ∈ O. We conclude that the functor T satisfies the representative property. Hence,  T : O → O is an equivalence of categories. This proves (1).

6.6. Exercises

255

The proof of (2) is entirely parallel to (1), while (3) follows from (1) and (2) (see, e.g., [86, II.10]).  Remark 6.47. Super duality helps to provide a categorical explanation and new proofs for results obtained in earlier chapters by other means: the irreducible polynomial character formula for gl(m|n) in terms of super Schur functions via SchurSergeev duality in Chapter 3, and also the irreducible character formulas for the oscillator modules of Lie superalgebras via Howe duality in Chapter 5.

6.6. Exercises Exercise 6.1. Prove Proposition 6.15 in the limiting case m = 0. Exercise 6.2. Prove the following identities of symmetric functions:   (1) ω ∏1≤i≤ j (1 − xi x j )−1 = ∏1≤i< j (1 − xi x j )−1 .   (2) ω ∏1≤i≤ j (1 + xi x j ) = ∏1≤i< j (1 + xi x j ). (Hint: Use T ( u− ) = u− and T ( u− ) = u− .) In Exercises 6.3 and 6.4 we shall use the following notation: Let Λ (respectively Λ) denote the ring of symmetric functions in the variables x1 , x2 , . . . (respectively x1/2 , x3/2 , . . .). Identifying Λ with Λ, the involution of symmetric functions is identified with . ∞ ∞ 1 ω ∏(1 + xit) = ∏ , (1 − x i− 1 t) i=1 i=1 2

for an indeterminate t. Exercise 6.3. Let Ωk be the character ring of the BGG category of (k + CK) denote modules. For an indeterminate e we let xr = eεr , for r ∈ 12 Z+ , and let Ωk ⊗Λ the topological completion of Ωk ⊗Λ with respect to the degree filtration of Λ. Let  Prove:  ∈ O. M ∈ O, M ∈ O, and M  and ch M ∈ Ωk ⊗Λ.  (1) ch M ∈ Ωk ⊗Λ    = ch T (M).  (2) ω ch T (M)     for all i ∈ Z+ . (3) ω ch Hi (u− , T (M)) = ch Hi (u− , T (M)),  and let λ, µ, ν ∈ P+ . Prove:  and N  be two objects in O, Exercise 6.4. Let M   ⊗N  is an object in O. (1) The tensor product M  ⊗ N)  = T (M)  ⊗ T (N),  and T (M  ⊗ N)  = T (M)  ⊗ T (N).  (2) T (M    ⊗ N)  = ch T (M  ⊗ N).  (3) ω ch T (M

6. Super duality

256

(4) L(λ) appears as a composition factor in L(µ) ⊗ L(ν) with the same multiplicity as L(λ ) appears as a composition factor in L(µ ) ⊗ L(ν ). Exercises 6.5 and 6.6 show that a finite-dimensional simple module over the general linear Lie superalgebra may not have a BGG resolution in terms of Verma modules, but may have a resolution in terms of Kac modules. In the notation of this chapter we set m = 1 and x = a. Let b2 be the standard Borel subalgebra of the Lie superalgebra g2 = gl(1|2) with Cartan subalgebra h2 . By highest weight we always mean with respect to b2 . ∗

Exercise 6.5. Let M(ε−1 ) be the Verma module of highest weight ε−1 ∈ h2 with highest weight vector vε−1 . Let L2 (ε−1 ) be its unique irreducible quotient. Prove: (1) If N ⊆ M(ε−1 ) is the g2 -submodule generated by E2,1 vε−1 , then N contains all nontrivial singular weight vectors of M(ε−1 ). (2) The g2 -module M(ε−1 )/N is reducible. (3) There is no exact sequence of g2 -modules of the form 

M(µi ) −→ M(ε−1 ) −→ L2 (ε−1 ) −→ 0,

∗

µi ∈ h 2 .

i

Exercise 6.6. Let wλ denote a highest weight vector in the g2 -Kac module Δ2 (λ), + for λ ∈ P2 . Prove: (1) The only nontrivial singular vectors of Δ2 (ε−1 ) are scalar multiples of E1,−1 E2,−1 wε−1 . (2) For k ∈ N the unique maximal submodule of Δ2 (−kε−1 + kε1 + ε2 ) is generated by the singular vector E1,−1 w−kε−1 +kε1 +ε2 . Furthermore, it is isomorphic to L2 (−(k + 1)ε−1 + (k + 1)ε1 + ε2 ). (3) For every k ∈ N we have an exact sequence of g2 -modules of the form Δ2 (−(k + 1)ε−1 + (k + 1)ε1 + ε2 ) −→ Δ2 (−kε−1 + kε1 + ε2 ) −→ · · · · · · −→ Δ2 (−ε−1 + ε1 + ε2 ) −→ Δ2 (ε−1 ) −→ L2 (ε−1 ) −→ 0. Conclude that L2 (ε−1 ) can be resolved in terms of Kac modules. (4) Recalling D from Section 2.2.3 and W ∼ = S2 we have the following identity: . λ+ρ 1 e chL2 (ε−1 ) = ∑ (−1)(w)w 1 + e−ε−1 +ε2 . D w∈W Exercise 6.7. Let g = gx . Define (6.56)

P++ := {λ ∈ P+ | dim Ln (λ) < ∞, for n  0}.

∞ + ++ if and only if (1) Let x = a. Prove that λ = ∑−1 i=−m λi εi + ∑ j=1 λ j ε j ∈ P (λ−m , . . . , λ−1 , + λ1 , + λ2 , . . .) is a partition.

6.6. Exercises

257

∞ + ++ (2) Let d ∈ Z and x = c. Prove that λ = dΛ0 + ∑−1 i=−m λi εi + ∑ j=1 λ j ε j ∈ P if and only if d ∈ −Z+ , 0 ≥ λ−m , and (λ−m −d, . . . , λ−1 −d, + λ1 , + λ2 , . . .) is a partition (see (6.14)).

(3) For x = b, b• , d and d ∈ 12 Z, determine the set P++ as in (2). Exercise 6.8. Let g = gx , where x = a, b, c, d. Let l and u− denote the Levi subalgebra and opposite nilradical, respectively. For λ in P++ as in (6.56) we have the following parabolic BGG resolution [6, 79, 100] (recall W 0 from Section 5.5): (i) There exists an exact sequence of g-modules of the form · · · −→





Δ(w ◦ λ) −→

w∈Wr0

Δ(w ◦ λ) −→ · · ·

0 w∈Wr−1

· · · −→



Δ(w ◦ λ) −→ Δ(λ) −→ L(λ) −→ 0.

w∈W10

(ii) As l-modules we have Hi (u− , L(λ)) =



w∈Wi0 L(l, w ◦ λ),

for all i ∈ Z+ .

Prove: (1) The KLV polynomials µλ (q) are monomials, for µ ∈ P+ . (2) Every g-module L(λ ) has a BGG resolution in terms of Δ(µ ), µ ∈ P+ . (3) The irreducible gl(m|n)-tensor modules have resolutions in terms of Kac modules. Furthermore, if ν is such a highest weight and µ ∈ Pn+ , then the KLV polynomial µν (q), if nonzero, is a monomial of q. (4) Let d ∈ Z+ and assume that (λ) ≤ min{n, d}. The irreducible highest weight osp(1|2n)-module of highest weight ∑ni=1 (λi + d)εi with respect to the fundamental system {−ε1 , ε1 − ε2 , . . . , εn−1 − εn } has a resolution in terms of parabolic Verma modules with Levi obtained by removing −ε1 . Also, similarly to (2), the corresponding nonzero KLV polynomials are monomials of q. Exercise 6.9. Let g = ga with Y0 = Π(k). Let λ ∈ P+ with λi ∈ Z. Suppose that for any λ j − j with j < 0 there exists an i > 0 such that λ j − j = λi − i. Prove that Δ(λ) is irreducible. (Hint: Use Exercise 2.11 and super duality.) Exercise 6.10. Let g = ga and let λ, µ ∈ P+ . Prove: (1) There exists k ∈ N such that for n ≥ k the multiplicity with which Ln (µ ) appears in a composition series of Δn (λ ) is independent of n. (2) The parabolic Verma module Δ(λ ) has a finite composition series. (3) The parabolic Verma module Δ(λ) has a finite composition series. Exercise 6.11. Let L be a finite-dimensional irreducible osp(k|2n)-module, for k = 2m or k = 2m + 1. Prove that there exist gx with x = b, d and λ ∈ P+ such that the gn -module Ln (λ ) is isomorphic to L.

6. Super duality

258

Exercise 6.12. Recall a∞ from Section 5.4.1 and let l be its Levi subalgebra obtained by removing the simple root ε0 − ε1 with opposite nilradical u− . Then l ∼ = a≤0 ⊕ a>0 ⊕ CK, where a≤0 (respectively a>0 ) consists of those matrices (ai j ) ∈ a∞ with ai j = 0 unless i, j ∈ −Z+ (respectively i, j ∈ N). For d ∈ Z let Cd be the parabolic BGG category of a∞ -modules on which K acts as the scalar d and that, as l-modules, decompose into direct sums of irreducibles of the form L(a≤0 , ν) ⊗ L(a>0 , µ), where ν = ∑ j≤0 ν j ε j and µ = ∑ j≥1 µ j ε j , with (−ν0 , −ν−1 , . . .) ∈ P and (µ1 , µ2 , . . .) ∈ P. Denote the set of such weights ν + µ by P+ . Prove: (1) The categories Cd and C−d are equivalent. (2) For a generalized partition of length d of the form λ = (λ1 ≥ λ2 ≥ . . . ≥ λi > 0 = . . . = 0 > λ j+1 ≥ . . . ≥ λd ), let Λ(λ) := −dΛ0 + ∑ik=1 λi εi + ∑dk= j+1 λk εk−d ∈ P+ . Then the KLV polynomial µΛ(λ) is a monomial for every µ ∈ P+ . Exercise 6.13. Let g = ga with Y0 = Π(k). Let d, m, n ∈ N and ν, µ ∈ P with n (ν) ≤ m, (µ) ≤ n, and (ν) + (µ) ≤ d. Set ξ := ∑m i=1 (−d − νi )ε−i + ∑ j=1 µ j ε j . Prove: (1) For any η ∈ Pn+ the KLV polynomial ηξ for the finite-dimensional Lie algebra gn is a monomial. (2) The gn -module Ln (ξ) has a resolution in terms of Δn (η) with η ∈ P+ . (Hint: Use Exercise 6.12.)

Notes This chapter is an exposition on the formulation and proof of super duality, which is an equivalence of the categories O and O, in Cheng-Lam-Wang [24], which in turn was built on Cheng-Lam [23] for type A. The super duality conjecture for type A in the maximal parabolic case was first formulated in Cheng-Wang-Zhang [34] and more generally in Cheng-Wang [31], motivated by and partially based on Brundan [11]. There is a different approach by Brundan-Stroppel [15] in the special case of the super duality conjecture formulated in [34]. Section 6.1. The presentation of the materials in this section follows closely Cheng-Lam-Wang [24]. Section 6.2. The truncation functors and their basic properties regarding Verma and irreducible modules (Proposition 6.9) have counterparts in the algebraic group setting, which was developed by Donkin [40]. In the super duality context, the truncation functors first appeared in Cheng-Wang-Zhang [34], and then were generalized in [31, 23, 24].

Notes

259

Section 6.3. The odd reflection approach leading to Theorems 6.17 and 6.18 was developed in [23, 24]. It is remarkable that, when combined with Proposition 6.9, this solves completely the irreducible character problem for modules in the category On , which include all finite-dimensional simple modules of type gl and osp. Totally different approaches and solutions for the finite-dimensional irreducible character problem have been developed in Serganova [107], Brundan [11], and Brundan-Stroppel [15] for gl(m|n), and in Gruson-Serganova [49] for osp. For recent works, see also [17, 82, 120, 121]. Section 6.4. We follow Tanaka [122] for a formula of the (co)boundary operator for Lie superalgebras (see also Iohara-Koga [55]). The results on the identi are fication of Kazhdan-Lusztig-Vogan polynomials in the categories O, O, and O due to [23, 24]. The study of u-cohomology groups was initiated in the fundamental paper of Kostant [73]. We have followed Vogan’s homological interpretation of the Kazhdan-Lusztig polynomials [125, Conjecture 3.4] (also see Serganova [107] in the setting of the finite-dimensional module category for gl(m|n)), as the usual approach via Hecke algebras is not applicable for Lie superalgebras. The Kazhdan-Lusztig conjectures [70, 71] for the BGG category of a semisimple Lie algebra were proved in Beilinson-Bernstein [5] and Brylinski-Kashiwara [16]. The polynomials µλ (q) for the category O coincide with the usual parabolic KazhdanLusztig polynomials [39]. Theorem 6.24, which describes an explicit connection between u− -homology and u-cohomology groups, is based on [81, Section 4]. The stability of Kazhdan-Lusztig polynomials (6.38) even for classical Lie algebras has not been formulated in the literature until we needed it in the super duality framework. Section 6.5. The formulation of super duality (Theorem 6.39) follows ChengLam-Wang [24] (also see [34, 31, 23] for earlier formulation). The proof here using Ext1 only is somewhat different from the original one, and it is based on Theorem 6.34, which is adapted from Rocha-Caridi and Wallach [101, §7, Theorem  0¯ and its variants for type A was first 2]. The definition of the module subcategory O given in [23, Section 2.5], and in general in [24, Section 5.2]. The tilting modules, formulated earlier by Ringel and Donkin in different contexts, can be shown to exist in the categories On , On , On (cf. Brundan [11], Cheng → O and Wang-Zhang [34] and Cheng-Wang [28] for type A). The functors T : O  → O can be shown to respect tilting modules (see Cheng-Lam-Wang [25]). T :O We do not treat tilting modules in the book, and refer the reader to the original papers for details. We also refer to [26] for a proof of the Brundan-Kazhdan-Lusztig conjecture for the full BGG category O of gl(m|n)-modules. The results in Exercises 6.3(3) and 6.4(4) were taken from [24] (various special cases appeared earlier in [34, 23]), while Exercise 6.5 is taken from [20]. An

260

6. Super duality

analogous character formula as in Exercise 6.6(4) holds for finite-dimensional irreducible modules over the Lie superalgebras gl(m|n) and osp(2|2n) in the case when the highest weights have atypicality degree equal to 1, and it appeared for the first time in [8] (see also [123, 124]). The resolutions in Exercise 6.8(2) were first constructed in [20], while the KLV polynomials in this particular setting were first computed in [35], both without using super duality. The observation in Exercise 6.9 was first made in [34], while Exercise 6.10 is taken from [23] (a special case of (2) goes back to [34]). We are not aware of a direct proof for Exercise 6.10(3) without using super duality. The KLV polynomials in Exercise 6.12 were computed in [19] for the first time. Exercise 6.13(1) and (2) are classical results that appeared already in [41] and [42], respectively, stated in a different form (see [53]). They may now be regarded as direct applications of super duality.

Appendix A

Symmetric functions

A.1. The ring Λ and Schur functions A.1.1. The ring Λ. A partition λ = (λ1 , λ2 , . . . , λ ) is a sequence of decreasing nonnegative integers λ1 ≥ λ2 ≥ . . . ≥ λ ≥ 0. Each λi is called a part of λ, and the number of nonzero parts of λ is called the length of λ, which is denoted by (λ). The sum of all its nonzero parts is called the size of λ, and is denoted by |λ|. We sometimes use the notation λ = (1m1 2m2 . . .) to denote a partition λ, where mi is the number of parts of λ that are equal to i for i ≥ 1. We often identify a partition λ with its associated Young diagram, and denote by λ the conjugate partition whose Young diagram is the transpose of the Young diagram of λ. Let P denote the set of all partitions and let Pn denote the set of all partitions of n, for n ≥ 0. For λ, µ ∈ P, we write λ ⊇ µ if λi ≥ µi for all i ≥ 1. The dominance order on P, denoted by ≥, is the partial order defined by λ≥µ

⇔

|λ| = |µ| and λ1 + . . . + λi ≥ µ1 + . . . + µi , for all i ≥ 1.

Consider the Z-module of formal power series Z[[x]] in the infinite set of indeterminates x := {x1 , x2 , x3 , . . .}. For α = (α1 , α2 , . . . , αk ) ∈ Zk+ , we write xα := x1α1 x2α2 · · · xkαk ∈ Z[[x]]. The symmetric group Sn acts on Z[[x]] by permuting the indeterminates, i.e., if σ ∈ Sn and f ∈ C[[x]], then σ f (x1 , x2 , . . .) := f (xσ(1) , xσ(2) , . . .), where by definition σ(i) = i, for all i > n. Since the action of Sn+1 is compatible with that of Sn , we obtain an action of the direct limit, denoted by S∞ , on Z[[x]]. 261

A. Symmetric functions

262

The monomial symmetric function associated to a partition λ is mλ (x) = mλ :=

∑

σxλ ∈ Z[[x]],

σ∈S∞ /Sλ

where Sλ is the stabilizer subgroup of the monomial xλ in S∞ . Definition A.1. The ring of symmetric functions Λ is the Z-span of the elements {mλ | λ ∈ P} in Z[[x]]. We remark that Λ is closed under multiplication and hence is indeed a ring. Also note that Λ is naturally graded by degree. We denote the Z-submodule of  degree n by Λn so that Λ = n≥0 Λn . Note that Λn is a free Z-module of rank equal to the number of partitions of n. For a field F (which is often taken to be Q or C), we denote the base change by ΛF = F ⊗Z Λ and ΛnF = F ⊗Z Λn . There are several other bases for Λ that we use freely in this book.

Definition A.2. Let n ∈ N. We define pn := m(n) = ∑ xin , i≥1

∑

en := m(1n ) =

x i1 x i2 · · · x in ,

i1 n to 0, and Λ is the inverse limit of Λn . For a partition λ of length no greater than n, we let aλ :=

∑

σ∈Sn

λn λ1 λ2 sgn(σ)xσ(1) xσ(2) · · · xσ(n) ,

where sgn(σ) denotes the sign of the permutation σ. Then aλ is skew-symmetric in the sense that τaλ = sgn(τ)aλ ,

∀τ ∈ Sn .

Let ρ = (n − 1, n − 2, . . . , 1, 0), so that we have aλ+ρ =

∑

σ∈Sn

  λ +n− j λn λ1 +n−1 λ2 +n−2 sgn(σ)xσ(1) xσ(2) · · · xσ(n) = det xi j

1≤i, j≤n

.

In particular, aρ is the Vandermonde determinant. Since aλ+ρ = 0 when xi = x j for i = j, it follows that aλ+ρ is divisible by xi − x j , for all i = j. Therefore the expression aλ+ρ sλ := aρ is a symmetric polynomial in x1 , x2 , . . . , xn , called the Schur polynomial associated to λ. Since sλ (x1 , . . . , xn , 0) = sλ (x1 , . . . , xn ), the inverse limit, denoted by sλ ∈ Λ, exists and is called the Schur function associated to λ. Let An denote the Z-module of skew-symmetric polynomials in x1 , x2 , . . . , xn . Then the aλ+ρ ’s, as λ runs over partitions with (λ) ≤ n, form a Z-basis of An . Since every element in An is divisible by aρ , the map Λn → An given by multiplication by aρ is a Z-module isomorphism. Thus, the sλ ’s, as λ runs over all partitions with (λ) ≤ n, form a Z-basis for Λn . Therefore, we conclude that the set of Schur functions associated with partitions of size k is a basis for Λk , for each k ≥ 0. (A.10) For a partition λ we have   sλ = det hλi −i+ j 1≤i, j≤n , (λ) ≤ n,   sλ = det eλi −i+ j , (λ ) ≤ n. 1≤i, j≤n

In particular, we have ω(sλ ) = sλ . (k)

Proof. Let er denote the rth elementary symmetric polynomial in the variables x1 , x2 , . . . , xk , . . . , xn , i.e., with xk deleted. Set   (k) M := (−1)n−i en−i . 1≤i,k≤n

A. Symmetric functions

266

Let α = (α1 , . . . , αn ) ∈ Zn+ . Let   Aα = xαj i ,

Hα := (hαi −n+ j )1≤i, j≤n ,

1≤i, j≤n

and let E (k) (t) =

n−1

∑ er

r=0

t = ∏(1 + xit).

(k) r

i=k

Then H(t)E (k) (−t) =

1 . 1 − xk t

This implies that n

∑ hα −n+ j (−1)n− j en− j = xkα . (k)

i

i

j=1

This equation can be recast in a matrix form as Aα = Hα M. Taking the determinant of both sides with α = λ + ρ, we obtain aλ+ρ = det Hλ+ρ det M. Setting λ = 0/ and noting det Hρ = 1, we obtain that aρ = det M, and so aλ+ρ = det Hλ+ρ aρ , which is an equivalent form of the first identity in (A.10) we wish to prove.   Now the matrices H = (hi− j )0≤i, j≤N and E = (−1)i− j ei− j 0≤i, j≤N are inverses of each other by (A.5), each with determinant equal to 1. This gives a relationship between the minors of H and the cofactors of E t . Exploiting this relationship, one can prove the following identity:     det hλi −i+ j 1≤i, j≤n = det eλ i −i+ j 1≤i, j≤n , for n ≥ max{(λ), (λ )}. The second identity in (A.10) now follows by the first identity in (A.10) and the identity above. The formula ω(sλ ) = sλ follows by applying ω to the first identity and using the second identity with λ and λ switched.  Take two sets of independent variables x = {x1 , x2 , . . .} and y = {y1 , y2 , . . .}. Applying (A.8) to the set of variables {xi y j } and setting t = 1, we obtain that 1

∏ (1 − xi y j ) = ∑ z−1 λ pλ (x)pλ (y). i, j

λ

(A.11) The following identities hold: 1 ∏ 1 − xi y j = ∑ hλ (x)mλ(y), i, j λ∈P

∏(1 + xi y j ) = ∑ eλ (x)mλ(y), i, j

λ∈P

A.1. The ring Λ and Schur functions

267

1

∏ 1 − xi y j = ∑ sλ (x)sλ(y), λ∈P

i, j

∏(1 + xi y j ) = ∑ sλ (x)sλ (y). 

λ∈P

i, j

(The last two are known as Cauchy identities.) Proof. We compute 1 ∏ 1 − xi y j = ∏ H(y j ) = ∏ i, j j j

-

.

∑ hr (x)yrj

r≥0

= ∑ hα (x)yα1 1 yα2 2 · · · , α

where the last sum is over all compositions α = (α1 , α2 , . . .). Hence it equals ∑λ hλ (x)mλ (y), proving the first identity in (A.11). The second identity follows by applying ω in the x variables to the first identity and using (A.10). Assume that the number of variables in x and y are both equal to n. The general by letting n go to infinity, as usual. Consider the n × n matrix  case is−1proved  (1 − xi y j ) 1≤i, j≤n . If we multiply the ith row of this matrix by ∏nj=1 (1 − xi y j ), then we obtain a matrix whose (i, k)th entry is equal to

∏(1 − xi yr ) =

r=k

n

∑ (−1)n− j en− j (y)xi (k)

n− j

.

j=1

That is, it is the (i, k)th entry of the matrix Aρ (x)t M(y), where Aρ and M are as in the proof of (A.10). This implies that   1 det (1 − xi y j )−1 1≤i, j≤n = aρ (x)aρ (y) ∏ (A.12) . 1 − xi y j i, j Next, we compute that ∞   det (1 − xi y j )−1 1≤i, j≤n = det( ∑ xik ykj )1≤i, j≤n = ∑

∑

α σ∈Sn

k=0

n

i sgn(σ) ∏ xiαi yασ(i) ,

i=1

where the first summation is over compositions α. Thus, we can write   det (1 − xi y j )−1 1≤i, j≤n = ∑ xα aα (y) = ∑ ∑ (−1)(τ) xτ(λ+ρ) aλ+ρ (y) α

(A.13)

λ τ∈Sn

= ∑ aλ+ρ (x)aλ+ρ (y), λ

where the summation over λ is over partitions of length no greater than n. The first Cauchy identity follows now by combining (A.12) and (A.13). The second Cauchy identity follows by applying ω in the y variables to the first Cauchy identity and using (A.10).  We define a Z-valued bilinear form (·, ·) on Λ by declaring (A.14)

(hλ , mµ ) = δλµ .

A. Symmetric functions

268

This extends to a Q-valued bilinear form on ΛQ . (A.15) Let {uλ | λ ∈ P} and {vλ | λ ∈ P} be two bases for ΛQ . Then the following conditions are equivalent. (1) (uλ , vµ ) = δλµ . (2) ∑λ uλ (x)vλ (y) = ∏i, j (1−x1 i y j ) . Proof. We write uλ = ∑ aλν hν , ν

vµ = ∑ bµη mη , η

aλν , bµη ∈ Q.

Then (1) is equivalent to ∑ν aλν bµν = δλµ , which is equivalent to ∑λ aλν bλη = δνη . Now (2) is equivalent to ∑λ ∑ν,η aλν bλ,η hν (x)mη (y) = ∑λ hλ (x)mλ (y) by using (A.11). This is equivalent to ∑λ aλν bλη = δνη .  It follows from (A.15) and (A.11) that (A.16)

(sλ , sµ ) = δλµ .

A.1.3. Skew Schur functions. For partitions µ and ν, we write the product sµ sν as (A.17)

sµ sν = ∑ cλµν sλ , λ

cλµν ∈ Z.

The cλµν associated with a triple of partitions are the Littlewood-Richardson coefficients, and they allow us to define the skew Schur function associated to partitions λ, µ by the formula sλ/µ := ∑ cλµν sν . ν

This definition is equivalent to the following identity: (A.18)

(sµ f , sλ ) = ( f , sλ/µ ),

∀ f ∈ Λ.

This can be seen by checking for f = sν . It follows from the definitions that sλ/0/ = sλ , and by (A.10) that ω(sλ ) = sλ . Since ω is a ring isomorphism, we obtain by the definition of the Littlewood Richardson coefficients above that cλµ ν = cλµν . It now follows from the definition of the skew Schur functions that ω(sλ/µ ) = sλ /µ . Let µ ⊆ λ be partitions which we may regard as Young diagrams. The skew diagram λ/µ is obtained from λ by removing the sub-diagram µ. A tableau of shape λ/µ is a filling of the boxes of λ/µ with elements from the set N. A tableau T of λ/µ is called semistandard if the entries in T are strictly increasing from top to

A.1. The ring Λ and Schur functions

269

bottom along each column and weakly increasing from left to right along each row. Given such a semistandard tableau T with entries t ∈ T , we define xT := ∏t∈T xt . The skew Schur function sλ/µ admits the following combinatorial description, a proof of which can be found in [83, (5.12)]. (A.19) For partitions µ ⊆ λ, we have sλ/µ = ∑ xT , T

where the summation is over all semistandard tableaux of shape λ/µ. The following is an easy consequence of (A.19). (A.20) Let x = {x1 , x2 , . . .} and y = {y1 , y2 , . . .} be two independent sets of variables. We have

∑ sµ (x)sλ/µ(y).

sλ (x, y) =

µ⊆λ

Let µ ⊆ λ be partitions. Recall that the weight of a semistandard tableau T of shape λ/µ is (1m1 2m2 · · · ), where mi is the number of times i appears in T . Denote by Kλ−µ,ν the number of semistandard tableaux of shape λ/µ and weight ν. Then (A.19) implies that sλ/µ =

(A.21)

∑ Kλ−µ,ν mν .

ν∈P

It follows from (A.18) and (A.21) that sµ hν = ∑ Kλ−µ,ν sλ . λ

Taking ν = (r) to be the one-part partition, we obtain Pieri’s formula: sµ hr = ∑ s λ ,

(A.22)

λ

where the summation is over partitions λ such that λ/µ is a skew diagram of size r whose columns all have length at most one. Recall that a partition is called even if all of its parts are even. We have the following two symmetric function identities ([83, I.5, Example 5]):

(A.23)

∑

sµ =

∑

sν =

µ even ν even

∏ (1 − xi x j )−1 ,

1≤i≤ j

∏ (1 − xi x j )−1 .

1≤i< j

Since the involution ω interchanges sλ with sλ , we conclude from these identities that (see also Exercise 6.2)   ω ∏ (1 − xi x j )−1 = ∏ (1 − xi x j )−1 . (A.24) 1≤i≤ j

1≤i< j

A. Symmetric functions

270

A.1.4. The Frobenius characteristic map. We shall freely use the notations from Section 3.2.2. Let Rn denote the Grothendieck group of the category of Sn -modules, which has [Sλ ] for λ ∈ Pn as a Z-basis. Let R=

∞ 

Rn .

n=0

Then R is a graded algebra with multiplication given by S

f g = IndSm+n ( f ⊗ g), m ×Sn

for f ∈ Rm , g ∈ Rn .

The Frobenius characteristic map chF : R → Λ is the Z-linear map such that, for n ≥ 0, (A.25)

chF (χ) =

∑ z−1 µ χ µ pµ ,

χ ∈ Rn ,

µ∈Pn

where χµ denotes the character value of χ at a permutation of cycle type µ. Denote by 1λ and sgnλ the trivial and sign representations or characters of the Young subgroup Sλ , respectively. The following basic properties of chF are well known. (A.26) The characteristic map chF is an isomorphism of graded rings. Moreover, for n ≥ 0 and λ ∈ Pn , we have that   n chF IndS = hλ , Sλ 1λ   n chF IndS = eλ , Sλ sgnλ chF ([Sλ ]) = sλ . Proof. One can use the induced character formula to prove that chF is a ring homomorphism. By (A.8) and (A.9) we have, respectively, chF (1n ) = hn and chF (sgnn ) = en . This immediately implies the first two identities since chF is a homomorphism. It is known (e.g., as a special case of Lemma 3.12) that (A.27)

n ∼ IndS Sµ 1n =



Kλµ Sλ ,

λ≥µ

where the Kostka number Kλµ is equal to the number of semistandard λ-tableaux of content µ. Note that Kµµ = 1. Also from (A.21) we conclude that sλ =

∑ Kλµ mµ .

µ≤λ

Using the relations of dual bases (A.14) and (A.16), this identity admits the following dual version: (A.28)

hµ =

∑ Kλµ sλ .

λ≥µ

A.2. Supersymmetric polynomials

271

It follows by comparing (A.27) and (A.28) and using the first identity in (A.26) that chF ([Sλ ]) = sλ . Since [Sλ ] and sλ for λ ∈ P form Z-bases for R and Λ respectively, we conclude that chF is an isomorphism.  (A.29) For µ ∈ Pn , we have an Sn -module isomorphism: Sµ ⊗ sgnn ∼ = Sµ . 

Proof. Let ψ denote the character of Sµ . By (A.7) and (A.25), we have that chF ([Sµ ⊗ sgnn ]) =

∑ (−1)n−(ν) z−1 ν ψ ν pν

ν∈Pn

=ω



∑

ν∈Pn

z−1 ν ψ ν pν

 

= ω(sµ ) = sµ = chF ([Sµ ]). Now the isomorphism follows from (A.26) that chF is an isomorphism.



A.2. Supersymmetric polynomials A.2.1. The ring of supersymmetric polynomials. Let x = {x1 , x2 , . . . , xm } and y = {y1 , y2 , , . . . , yn } be independent indeterminates. A polynomial f in Z[x, y] is called supersymmetric if the following conditions are satisfied: (1) f is symmetric in x1 , . . . , xm . (2) f is symmetric in y1 , . . . , yn . (3) The polynomial obtained from f by setting xm = yn = t is independent of t. Denote by Λ(m|n) the ring of supersymmetric polynomials in Z[x, y]. For a field F, we denote the base change by Λ(m|n),F = F⊗Z Λ(m|n) . We call an element f ∈ Z[x, y] satisfying (1) and (2) above doubly symmetric. For r ≥ 1 set m

n

i=1

j=1

σrm,n := ∑ xir − ∑ yrj . Then

σrm,n

is supersymmetric. Let m

n

m,n := ∏ ∏ (xi − y j ). i=1 j=1

Note that m,n is supersymmetric. Also, if g is doubly symmetric, then gm,n is supersymmetric. (A.30) Let q be doubly symmetric. Then qm,n is a polynomial in {σrm,n | r ≥ 1}. Proof. We proceed by induction on n (for arbitrary m), with the case n = 0 being clear.

A. Symmetric functions

272

Let n ≥ 1. We may assume, without loss of generality, that q is homogeneous of degree k ≥ 0, and we proceed by induction on k. Consider qm,n |yn =0 = q|yn =0 x1 · · · xm m,n−1 . By the induction hypothesis on n, there exists a polynomial g in {σ1m,n−1 , . . . , σrm,n−1 } such that qm,n |yn =0 = g(σ1m,n−1 , . . . , σrm,n−1 ). Note that g(σ1m,n , . . . , σrm,n )|xm =yn =t = qm,n |xm =yn =0 = 0. Thus xm − yn divides g(σ1m,n , . . . , σrm,n ). Since g(σ1m,n , . . . , σrm,n ) is doubly symmetric, it follows that g(σ1m,n , . . . , σrm,n ) = q∗ m,n ,

(A.31)

for some doubly symmetric polynomial q∗ . Now q∗ m,n |yn =0 = g(σ1m,n−1 , . . . , σrm,n−1 ) = qm,n |yn =0 . Hence yn divides q∗ − q. Since q∗ − q is symmetric in y1 , . . . , yn , we conclude that there exists a doubly symmetric polynomial f such that qm,n = q∗ m,n + y1 · · · yn f m,n .

(A.32)

Note that f has degree k − n if k ≥ n and f = 0 if k < n. In the case when k < n we have f = 0. Hence we are done by (A.31). Suppose that k ≥ n. Write f as a polynomial in terms of σum,0 and σv0,n , for u, v ≥ 1. Replace every σum,0 in this very polynomial by σum+1,0 and denote by f ∗ the resulting homogeneous doubly symmetric polynomial in Z[x1 , . . . , xm+1 , y1 , . . . , yn ]. Since f ∗ has degree k − n < k, we have by the induction hypothesis on k that there exists some polynomial h such that f ∗ m+1,n = h(σ1m+1,n , . . . , σsm+1,n ). This implies that y1 · · · yn f m,n = (−1)n f ∗ m+1,n |xm+1 =0 = (−1)n h(σ1m,n , . . . , σsm,n ), and hence (A.30) follows from (A.32).



(A.33) The algebra Λ(m|n),Q of supersymmetric polynomials is generated by σrm,n for all r ≥ 1. Proof. We proceed by induction on n to show that every supersymmetric polynomial p ∈ Q[x, y] is a polynomial in {σrm,n | r ≥ 1}. The case when n = 0 is clear.

A.2. Supersymmetric polynomials

273

Let n ≥ 1 and p ∈ Q[x, y] be a supersymmetric polynomial. Then, the polynomial p|xm =yn =0 in x1 , . . . , xm−1 and y1 , . . . , yn−1 is supersymmetric. By the induction hypothesis on n, there exists a polynomial f such that p|xm =yn =0 = f (σ1m−1,n−1 , . . . , σrm−1,n−1 ). Set f ∗ := f (σ1m,n , . . . , σrm,n ). Then (p − f ∗ )|xm =yn =t = 0, and hence xm − yn divides p − f ∗ . By double symmetry m,n divides p − f ∗ , and hence there exists a doubly symmetric polynomial q such that p = f ∗ + qm,n . By (A.30), qm,n is generated by {σrm,n | r ≥ 1}, and hence so is p.



We similarly define the ring of supersymmetric functions Λ(m|∞) and prove the following n = ∞ version of (A.33). (A.34) The algebra Λ(m|∞),Q of supersymmetric functions is generated by σrm,∞ for all r ≥ 1. Proof. Let f ∈ Λ(m|∞),Q . Assume that f is homogeneous of degree k. Then f(n) := f |yn+1 =yn+2 =···=0 is supersymmetric and hence, by (A.33), f (n) = g(σ1m,n , . . . , σrm,n ). Consider the element f − g(σ1m,∞ , . . . , σrm,∞ ) ∈ Λ(m|∞),Q . The expression f − g(σ1m,∞ , . . . , σrm,∞ ) is homogeneous of degree k and contains no monomials made up of only x1 , . . . , xm , y1 , . . . , yn . It follows by the symmetry of the y variables that f − g(σ1m,∞ , . . . , σrm,∞ ) = 0.   (m|n) to be the subring of the ring of douA.2.2. Super Schur functions. Define Λ bly symmetric polynomials over Z in the two sets of variables {x1 , . . . , xm } and {y1 , . . . , yn } consisting of polynomials f such that f |xm =t=−yn is independent of t.  m|n if and only if f (x, −y) is supersymmetric; hence, we have Evidently f (x, y) ∈ Λ an isomorphism of rings (A.35)

∼ =  ρy : Λ(m|n) −→ Λ (m|n) ,

 (m|∞) accordingly, and it is given by ρy ( f (x, y)) = f (x, −y). We define the ring Λ isomorphic to Λ(m|∞) . Regard Λ as the ring of symmetric functions in x1 , x2 , . . ., and identify the variables xm+i = yi , for i ≥ 1. By applying the involution ω on the y variables, denoted by ωy , to the set of infinite variables {y1 , y2 , . . .}, we obtain a ring homomorphism  (m|∞) . Indeed, by (A.7), we have ωy : Λ → Λ (A.36)

r  (m|∞) . ωy (pr ) = x1r + . . . + xm + (−1)r−1 (yr1 + yr2 + . . .) ∈ Λ

A. Symmetric functions

274

Since the pr ’s, for r ≥ 1, are algebraically independent generators of ΛQ by (A.2)  (m|∞),Q , and so ωy (Λ) ⊆ Λ  (m|∞),Q ∩Z[x, y] = and (A.6), we conclude that ωy (ΛQ ) ⊆ Λ r −1  (m|∞) . Now (A.34) and the fact that ρ−1 Λ y (ωy (pr )) = σm,∞ , r ≥ 1 , imply that ρy ◦  (m|∞),Q ωy : ΛQ → Λ(m|∞),Q is a ring isomorphism. This implies that ωy : ΛQ → Λ  is an isomorphism. The inverse isomorphism ω−1 y : Λ(m|∞),Q → ΛQ , which is again  given by the involution ω on the y variables, clearly satisfies that ω−1 y (Λ(m|∞) ) ⊆ Λ.  (m|∞) is a ring isomorphism. Hence, ωy : Λ → Λ Define the super Schur function hsλ associated to a partition λ by (A.37)

hsλ (x; y) =

∑ sµ(x)sλ /µ (y). 



µ⊆λ

The definition of hsλ (x; y) makes sense for x and y being finite and infinite. By (A.20), we have for y infinite   (A.38) ωy sλ (x, y) = hsλ (x; y).  (m|∞) is an isomorphism of rings. In addition, the set (A.39) The map ωy : Λ → Λ  (m|∞) . {hsλ (x1 , . . . , xm ; y1 , y2 , . . .) | λ ∈ P} is a Z-basis for Λ  (m|∞) → Λ  (m|n) obtained by setting the variables yi = 0, for i > n, is The map Λ a ring epimorphism. Furthermore, it sends the super Schur function corresponding to an (m|n)-hook partition to the respective super Schur polynomial corresponding to the same (m|n)-hook partition. The other super Schur functions are sent to zero. Evidently, the super Schur polynomials corresponding to (m|n)-hook partitions are linearly independent. Thus we have the following. (A.40) The set {hsλ }, as λ runs over all (m|n)-hook partitions, forms a Z-basis for  (m|n) . Λ The Cauchy identities in (A.11) admit the following super generalization. (A.41) Let x = {x1 , x2 , . . .}, y = {y1 , y2 , . . .}, z = {z1 , z2 , . . .} be three sets of (possibly infinite) indeterminates. The following identity holds: ∏ j,k (1 + y j zk ) = ∑ hsλ (x; y)sλ (z). ∏i,k (1 − xi zk ) λ∈P Proof. It suffices to prove the claim in the case when all three sets of indeterminates are infinite. A variant of the Cauchy identity in (A.11) can be written as 1

1

∏ (1 − xi zk ) ∏ (1 − y j zk ) = ∑ sλ (x, y)sλ(z). i,k

j,k

λ∈P

Now the claim follows by applying the involution ωy on the y variables to both sides of the above equation and using (A.38). 

A.3. The ring Γ and Schur Q-functions

275

A.3. The ring Γ and Schur Q-functions A.3.1. The ring Γ. Let x = {x1 , x2 , . . .}. Define a family of symmetric functions qr = qr (x), r ≥ 0, via the generating function 1 + txi Q(t) := ∑ qr (x)t r = ∏ (A.42) . i 1 − txi r≥0 Note that q0 (x) = 1, and that Q(t) satisfies the relation (A.43)

Q(t)Q(−t) = 1,

which is equivalent to the identities:

∑

(−1)r qr qs = 0,

n ≥ 1.

r+s=n

These identities are vacuous for n odd. When n = 2m is even, we have m−1

(A.44)

q2m =

1

∑ (−1)r−1qr q2m−r − 2 (−1)m q2m .

r=1

Let Γ be the Z-subring of Λ generated by the qr ’s: Γ = Z[q1 , q2 , q3 , . . .].



The ring Γ is graded by the degree of functions: Γ = n≥0 Γn , where Γn = Γ ∩ Λn . We set ΓQ = Q ⊗Z Γ, ΓC = C ⊗Z Γ. For a partition µ = (µ1 , . . . , µ ), we denote 

qµ = ∏ qµi . i=1

We shall denote by OP and SP the sets of odd and strict partitions, respectively. (A.45) The ring ΓQ enjoys the following remarkable properties: (1) ΓQ is a polynomial algebra with polynomial generators p2r−1 for r ≥ 1. (2) ΓQ is a polynomial algebra with polynomial generators q2r−1 for r ≥ 1. (1 ) {pµ | µ ∈ OP} forms a linear basis for ΓQ . (2 ) {qµ | µ ∈ OP} forms a linear basis for ΓQ . Proof. Recall the generating function for the power-sums: P(t) = ∑r≥1 prt r−1 . We have Q (t) d = ln Q(t) = P(t) + P(−t) = 2 ∑ p2r+1t 2r . Q(t) dt r≥0 It follows that Q (t) = 2Q(t) ∑r≥0 p2r+1t 2r , from which we deduce that rqr = 2(p1 qr−1 + p3 qr−3 + . . .), with the last term on the right-hand side being pr−1 q1 for r even and pr for r odd.

A. Symmetric functions

276

By using induction on r, we conclude that (i) each qr is expressible as a polynomial in terms of ps ’s with odd s; (ii) each pr with odd r is expressible as a polynomial in terms of qs ’s, which can be further restricted to odd s (note that each qr can be written as a polynomial of qs with odd s by applying (A.44) and induction on r). So ΓQ = Q[p1 , p3 , . . .] = Q[q1 , q3 , . . .]. Since the pr ’s (for r odd) are algebraically independent by (A.6), we have proved (1). (2) follows from (1) by the above equation and dimension counting. Clearly, (1 ) is equivalent to (1), while (2 ) is equivalent to (2).



(A.46) {qµ | µ ∈ SP} forms a Z-basis for Γ. Moreover, for any partition λ, we have qλ =

∑

aµλ qµ ,

µ∈SP,µ≥λ

for some aµλ ∈ Z. Proof. We claim that qλ lies in the Z-span of {qµ | µ ∈ SPn , µ ≥ λ}, for each partition λ of a given n. This can be seen by induction downward on the dominance order on λ as follows. The initial step for λ = (n) is clear, as (n) is strict. For nonstrict λ, we have λi = λ j = m for some i < j and some m > 0. Applying (A.44) to rewrite qλ easily provides the inductive step. According to a formula of Euler, |SPn | = |OPn |. Now (A.46) follows from Part (2) of (A.45) and the above claim. We record here a short proof of Euler’s formula:

∑ |SPn|qn =

n≥0

=

∏(1 + qr ) =

r≥1

∏r≥1 (1 + qr )(1 − qr ) ∏r≥1 (1 − qr )

1 ∏r≥1 (1 − q2r ) = = ∑ |OPn |qn . r ∏r≥1 (1 − q ) ∏r≥1,odd (1 − qr ) n≥0 

(A.47) We have qn = ∑α∈OPn 2(α) z−1 α pα . Proof. As seen in the proof of (A.45) above, we have . 1 Q(t) = exp 2 ∑ p2r−1t 2r−1 . r≥1 2r − 1 Now (A.47) follows by comparing the coefficients of t n on both sides.



A.3. The ring Γ and Schur Q-functions

277

A.3.2. Schur Q-functions. We shall define the Schur Q-functions Qλ , for λ ∈ SP. Let Q(n) = qn , n ≥ 1. Consider the generating function Q(t1 ,t2 ) := (Q(t1 )Q(t2 ) − 1)

t1 − t2 . t1 + t2

By (A.43), Q(t1 ,t2 ) is a power series in t1 and t2 , and we write Q(t1 ,t2 ) =

∑ Q(r,s)t1rt2s .

r,s≥0

The following can be checked easily from the definition by noting Q(t1 ,t2 ) = −Q(t2 ,t1 ). (A.48) We have Q(r,s) = −Q(s,r) , Q(r,0) = qr . In addition, s

Q(r,s) = qr qs + 2 ∑ (−1)i qr+i qs−i ,

r > s.

i=1

We recall the following classical facts (see Wikipedia). Any 2n × 2n skewsymmetric matrix A = (ai j ) satisfies det(A) = Pf(A)2 , where Pf(A) is the Pfaffian of A given by (A.49)

Pf(A) = ∑ sgn(σ)aσ(1)σ(2) . . . aσ(2n−1)σ(2n) , σ

summed over σ ∈ S2n such that σ(2i − 1) < σ(2i) and σ(2i − 1) < σ(2i + 1) for all admissible i. Equivalently, Pf(A) can be defined as the above sum over the whole group S2n divided by 2n n!. Definition A.4. The Schur Q-function Qλ , for λ ∈ SP with (λ) ≤ 2n, is defined to be the Pfaffian of the 2n × 2n skew-symmetric matrix (Q(λi ,λ j ) )1≤i, j≤2n . Example A.5. For indeterminates t1 , . . . ,t2n , let A = (ai j ) be the 2n × 2n skewt −t symmetric matrix with ai j = tii +t jj . Then, Pf(A) =

ti − t j . 1≤i< j≤2n ti + t j

∏

λ2n Remark A.6. Equivalently, Qλ is the coefficient of t1λ1 . . .t2n in

Q(t1 , . . . ,t2n ) := Pf (Q(ti ,t j )). It can be shown that Q(t1 , . . . ,t2n ) is alternating in the sense that Q(tσ(1) , . . . ,tσ(2n) ) = sgn(σ)Q(t1 , . . . ,t2n ), It follows that Qλ = 0 unless all parts of λ are distinct.

∀σ ∈ S2n .

A. Symmetric functions

278

(A.50) For λ ∈ SP with (λ) ≤ n, Qλ is equal to the coefficient of t1λ1 t2λ2 . . . in Q(t1 ) . . . Q(tn ) where it is understood that

ti −t j ti +t j

ti − t j , 1≤i< j≤n ti + t j

∏

= 1 + 2 ∑r≥1 (−1)r (ti−1t j )r .

The statement of (A.50) can be found in [83, (8.8), pp. 253]. The following recursive relations for Qλ are obtained directly by the Pfaffian definition of Qλ and the Laplacian expansion of Pfaffians (see Wikipedia). (A.51) For a strict partition λ = (λ1 , . . . , λm ), m

Qλ =

∑ (−1) j Q(λ ,λ ) Q(λ ,...,λˆ ,...,λ 1

j

2

j=2 m

Qλ =

j

∑ (−1) j−1 Qλ Q(λ ,...,λˆ ,...,λ j

1

j=1

j

m)

m)

,

,

for m even, for m odd.

(A.52) For λ ∈ SPn , we have

∑

Qλ = qλ +

aλµ qµ .

µ∈SPn ,µ>λ

It follows from the recursive relation (A.51) that Qλ is equal to qλ plus a linear combination of qν for ν > λ (not necessarily strict). Now (A.52) follows since each qν is expressible as a linear combination of qµ with strict partitions µ ≥ ν by (A.44) (this fact has been used in the induction step in the proof of (A.46)). As an immediately corollary of (A.46) and (A.52), we have (A.53) {Qλ | λ ∈ SP} is a Z-basis for Γ. Moreover, for any partition µ, we have qµ =

∑

λ∈SP,λ≥µ

λµ Qλ , K

λµ ∈ Z and K µµ = 1. (Clearly, one can further assume that µ is a composiwhere K tion instead of a partition in the statement.) λµ ≥ 0 Indeed from representation-theoretical consideration it can be seen that K (cf. Chapter 3, Lemma 3.50) . A.3.3. Inner product on Γ. Let x = {x1 , x2 , . . .} and y = {y1 , y2 , . . .} be two independent sets of variables. (A.54) We have 1 + xi y j

∏ 1 − xi y j = ∑

α∈OP

i, j

=

2(α) z−1 α pα (x)pα (y)

∑ mµ (x)qµ(y).

µ∈P

A.3. The ring Γ and Schur Q-functions

279

Proof. Let xy = {xi y j | i, j = 1, 2, . . .}. It follows by (A.42) that 1 + xi y j

∏ 1 − xi y j = ∑ qn(xy) = ∑ i, j

α∈OP

n≥0

2(α) z−1 α pα (xy).

Now the first identity follows since pα (xy) = pα (x)pα (y). We further compute that ∞ 1 + xi y j = ∏ 1 − xi y j ∏ ∑ qri (y)xiri i, j i ri =0

∑ mµ (x)qµ(y).

=

µ∈P



The second identity is proved. We define an inner product ·, · on ΓQ by letting pα , pβ  = 2−(α) zα δαβ .

(A.55)

(A.56) Let {uλ }, {vλ } be dual bases for ΓQ with respect to ·, ·. The following are equivalent: (i) (ii)

uλ , vµ  = δλµ

∀λ, µ; 1 + xi y j ∑ uλ (x)vλ(y) = ∏ 1 − xi y j . i, j λ

Such an equivalence can be established by the same standard argument as (A.15), now based on (A.54). (A.57) We have Qλ , Qµ  = 2(λ) δλµ ,

λ, µ ∈ SP.

Equivalently, we have 1 + xi y j

∏ 1 − xi y j = ∑ i, j

λ∈SP

2−(λ) Qλ (x)Qλ (y).

A rather nontrivial direct proof of (A.57) can be found in J´ozefiak [57]. Alternatively, one first works on the generality of Hall-Littlewood symmetric functions Qλ (x;t) for all partitions λ, and then (A.57) can be obtained as a specialization at t = −1. For details we refer to [83]. From the Hall-Littlewood approach, the Schur Q-function, for a strict partition λ with (λ) =  and m ≥ , is equal to  xi + x j  Qλ (x1 , . . . , xm ) = 2 (A.58) ∑ w x1λ1 · · · xλ ∏ ∏ xi − x j , 1≤i≤ i< j≤m w∈S /S m

m−

where the symmetric group Sm acts by permuting the variables x1 , . . . , xm and Sm− is the subgroup acting on x+1 , . . . , xm .

A. Symmetric functions

280

A.3.4. A characterization of Γ. Denote by Γm the counterpart of Γ in m variables, which consists of symmetric polynomials obtained by setting all but the first m variables to be zero in the functions in Γ. We define  :=

∏

(xi + x j ).

1≤i< j≤m

(A.59) Let f be any symmetric polynomial in m variables. Then f  lies in Γm . Proof. Let λ be a partition with (λ) ≤ m and let ρ = (m − 1, m − 2, . . . , 1, 0) so that λ + ρ is a strict partition. Then Qλ+ρ (x1 , . . . , xm ) ∈ Γm . Now by (A.58) we see that the Schur Q-polynomial Qλ+ρ (x1 , . . . , xm ), up to a 2-power, is equal to  xi + x j  λ1 +m−1 λm w x · · · x ∑ ∏ m 1 1≤i< j≤m xi − x j w∈Sm (xi + x j ) λm sgn(w)w(x1λ1 +m−1 · · · xm ) ∑ (x − x ) j w∈Sm 1≤i< j≤m i aλ+ρ = ∏ (xi + x j ) = sλ (x1, . . . , xm). aρ 1≤i< j≤m

=

∏

Since the Schur polynomials sλ (x1 , . . . , xm ) form a basis for the space Λm , this proves the lemma.  We have the following characterization of Γ. (A.60) Let g ∈ Λ. Then g ∈ Γ if and only if it satisfies the cancelation property that g|xi =−x j =t is independent of t for some i = j. Proof. Clearly, the power sums pr for r odd satisfy the above cancelation property, and so does any element in Γ. To prove the converse, it suffices to do so in the setting Γm of m variables. We shall show that if g ∈ Λm is a symmetric polynomial in x1 , . . . , xm such that g|xi =−x j =t is independent of t, then g can be written as a polynomial in the odd power sums {p2k+1 | k ∈ Z+ }. Observe that g|xi =−x j =t is independent of t for some particular choice of i, j with i = j if and only if it is independent of t for all (nonidentical) pairs i, j. We first note by (A.59) that q ∈ Γm for any symmetric polynomial q. We proceed by induction on m. If m = 0, 1, then the cancelation property is vacuous. Assume that m ≥ 2. Let p2k+1 denote the (2k + 1)st power-sum in the variables x1 , . . . , xm−2 . Then the polynomial g|xm−1 =−xm =0 is symmetric in x1 , . . . , xm−2 , and it satisfies the cancelation property. By the induction hypothesis on m, there exists a polynomial f such that g|xm−1 =−xm =0 = f (p1 , . . . , p2r+1 ).

A.3. The ring Γ and Schur Q-functions

281

Set f ∗ := f (p1 , . . . , p2r+1 ). Then (g − f ∗ )|xm−1 =−xm =t = 0, and hence xm−1 + xm divides g − f ∗ . Since g − f ∗ is symmetric in x1 , . . . , xm ,  divides g − f ∗ , and thus there exists a symmetric polynomial q such that g = f ∗ + q. By (A.59), q is generated by the odd power sums in the variables x1 , . . . , xm , and hence so is g.  A.3.5. Relating Λ and Γ. There is an intimate connection between the rings Λ and Γ. Recall a homomorphism ϕ (cf. [83, III, §8, Example 10]) defined by

(A.61)

ϕ : Λ −→ Γ,  for r odd, 2pr , ϕ(pr ) = 0, otherwise,

where pr denotes the rth power sum. One checks by definition that   ϕ H(t) = Q(t). This can be reformulated as follows. (A.62) We have ϕ(hn ) = qn (∀n ≥ 0),

ϕ(hµ ) = qµ , (∀µ ∈ P).

The homomorphism ϕ admits a categorification involving representations of the symmetric group; see Exercise 3.13. Suppose that λ is a strict partition of n. Let λ∗ be the associated shifted diagram that is obtained from the ordinary Young diagram by shifting the kth row to the right by k − 1 squares, for each k. Denoting (λ) = , we define the double partition λ to be λ = (λ1 , . . . , λ |λ1 − 1, λ2 − 1, . . . , λ − 1) in Frobenius notation. Example A.7. Let λ = (4, 3, 1). The corresponding shifted diagram λ∗ and double diagram λ are λ =

λ∗ =

We have the following basic property that relates Schur and Q-Schur functions. (see [83, III, §8, 10]). (A.63)

ϕ(sλ ) = 2−(λ) Q2λ ,

∀λ ∈ SPn .

A. Symmetric functions

282

A.4. The Boson-Fermion correspondence A.4.1. The Maya diagrams. The Maya diagrams are by definition the functions f : 12 + Z → {±} such that f (n) = + and f (−n) = −, for n  0. In other words, a Maya diagram is an assignment of the signs ± to the unit intervals on the real line associated to the lattice Z with fixed asymptotic values ± at infinity. These intervals are naturally labeled by 1/2 + Z, and we shall identify them. A Maya diagram f can be reconstructed by knowing the subset m of 12 + Z which is the preimage of + for f , which is simply an increasing sequence m = {m j } j≥1 in 12 + Z, such that m1 < m2 < m3 < . . . , and m j = m j−1 + 1 for j  0. In this appendix, we shall identify Maya diagrams with such sequences m. Example A.8. Consider the following Maya diagrams: ma :

mb :

mc :

··· 

···  ··· 

−



−

−3

−

 

−3

−

−2

−

−3

−



 

−2

−

−1

+

−2

+



 

−1

+

0

+

−1

−



  0

+

1

+

0

+



  1

+

+



+

 2

+



+

 3



+



+

···

4

+

3

2

–

 3

2

1

+





+



+

···

4

–



+



+

···

4

The Maya diagram mb is obtained from ma by changing signs from − to + at intervals − 32 and − 12 . The diagram mc is obtained from ma by changing signs from − to + at intervals − 52 , − 12 and from + to − at 32 , 72 . In terms of infinite sequences, we have ma = 12 + Z+ , mb = (− 32 , − 12 , 12 , 32 , 52 , . . .), and mc = (− 52 , − 12 , 12 , 52 , 92 , 11 2 , . . .). A.4.2. Partitions. To each Maya diagram m = {m j } j≥1 , we assign an integer, called the charge of m,  1 m = lim j − m j − j→∞ 2 and a partition (A.64)

λm = (1/2 − m1 − m , 3/2 − m2 − m , 5/2 − m3 − m , . . .).

Recall that P denotes the set of all partitions. The following can be viewed as the Boson-Fermion correspondence at the combinatorial level. (A.65) The map m → (m , λm ) is a bijection between the set of Maya diagrams and the set Z × P of “charged partitions”.

A.4. The Boson-Fermion correspondence

283

Proof. We will use primarily the example of mc in Example A.8 to illustrate how to visualize the Young diagram associated to λmc . Let us draw a new diagram following the signs on a Maya diagram from far left to far right as follows. We move to the southeast direction by one unit for each unit interval with a minus sign on the Maya diagram of mc , and move to the northeast by one unit for each unit interval with a plus sign on. As a result, we obtain the zigzag path in the first diagram below. The path is labeled by the unit intervals on the Maya diagram. Extending the two line segments coming from minus infinity and going to plus infinity, respectively, a finite bounded region emerges. This region can be further partitioned by solid lines as in the second diagram, and from which we read off the partition λmc = (3, 2, 2, 1) starting from the southwest row. It is easy to identify the partition obtained in this way for a Maya diagram m with the partition λm defined in (A.64). The vertical dashed line on the first diagram indicates that the intersection of the two infinite line segments corresponds to the coordinate 0 separating the two unit intervals − 12 and 12 . From this we read off the charge mc = 0. @ @ @ .. @ . − 52 − 32 − 12 @ @ 7 −2 @ @ @ @ @ @ @

1 2

3

@2 @

5 2

7

@2 @

··

·

··

·

9 2

0

@ @ @ @ @ .. @ @ @ . @ @ @ @ 7 7 @ −2 @ @ 2 @ @ @ 5 − 52@ @ @ 2 @ @ 3 − 32 @ @ 2 @ 1 − 12 @ 2

9 2

Observe that uniformly shifting every sign on a Maya diagram m one unit interval to the left gives rise to a Maya diagram whose charge is m + 1. For example, the mb can be obtained from ma by such a uniform shift to the left twice, and so mb = ma + 2. Such a shift clearly corresponds to a uniform shift of the labels on the corresponding zigzag path in the first diagram. Hence, it suffices to show that the set of Maya diagrams with charge 0 is in bijection with the set of partitions via m → λm . This bijection follows since we can easily reconstruct the labeled zigzag path (and hence the Maya diagram) from a given partition λ by first using a dashed

A. Symmetric functions

284

diagonal line on the upside-down Young diagram for λ (whose first southwest row has λ1 boxes) to locate the coordinate 0 point.  A.4.3. Fermions and fermionic Fock space. Let F be the vector space with a linear basis given by the symbols |m parameterized by the Maya diagrams m. We will call F the fermionic Fock space. Note that different Maya diagrams are related by changing signs at finitely many intervals, and let us consider changing signs to a Maya diagram one interval 1 − at a time. Accordingly, we define the linear operators ψ+ n , ψn on F, for n ∈ 2 + Z, as follows:   i , mi+1 , . . ., if mi = −n for some i, (−1)i−1 | . . . , mi−1 , m − ψn |m = 0, otherwise,  i (−1) | . . . , mi , n, mi+1 , . . ., if mi < n < mi+1 for some i, ψ+ n |m = 0, otherwise,  i denotes omission of the term mi . That is, ψ− where, as usual, m n corresponds to + the sign change at −n if the initial sign at −n is +, and ψn corresponds to the sign change at n if the initial sign at n is −. Remark A.9. Let V be a vector space with a standard basis vi , i ∈ 12 + Z, and let {v∗i } denote its dual basis. It is instructive to regard each vector |m as a semiinfinite wedge vm := vm1 ∧ vm2 ∧ vm3 ∧ · · · . The signs in the above definition of ψ± n ∗ can be explained naturally by identifying ψ− n as the contraction operator v−n and ψ+ n as taking the exterior product with vn . In this way, the Maya diagram model is canonically identified with a semi-infinite wedge model of the fermionic Fock space F. It is possible to use the subset m := f −1 (−) of 12 + Z, which is the complementary subset of m = f −1 (+), to identify with a Maya diagram f . Then m would be naturally identified with a semi-infinite wedge that goes to −∞ (instead of ∞). We will not adopt this convention in this book. We shall refer to | 12 , 32 , 52 , . . . as the vacuum vector of F and denote it by |0. Let us quote Dirac from his book “The Theory of Quantum Mechanics”: The perfect vacuum is a region where all the states of positive energy are unoccupied and all those of negative energy are occupied. We took the liberty in switching “positive” with “negative” in this exposition. Our convention is to consider the operators ψ± n as odd operators and the commutators among them to be anti-commutators: [A, B]+ = AB + BA. (A.66) The following anti-commutation relations hold for all m, n ∈ 12 + Z: − [ψ+ n , ψm ]+ = δn,−m I, 2 − 2 In particular, (ψ+ n ) = (ψn ) = 0.

+ [ψ+ n , ψm ]+ = 0,

− [ψ− n , ψm ]+ = 0.

A.4. The Boson-Fermion correspondence

285

Proof. This follows by a direct verification using the definitions of ψ± n . It corresponds to the fact that the processes of changing signs at two (possibly identical) intervals are interchangeable.  One also verifies directly the following. − (A.67) We have ψ+ n |0 = ψn |0 = 0, for n > 0.

 the Clifford (super)algebra generated by ψ± , for n ∈ 1 +Z, subject Denote by C n 2 to the relations in (A.66). (A.68) A linear basis for F can be given by (A.69)

+ + − − − ψ+ −p1 ψ−p2 . . . ψ−pr ψ−q1 ψ−q2 . . . ψ−qs |0,

for p1 > p2 > . . . > pr > 0 and q1 > q2 > . . . > qs > 0 with r, s ≥ 0. The basis element (A.69) is equal to m (up to a sign), where m is obtained from the set 1 2 + Z+ by deleting q1 , . . . , qs and adding −p1 , . . . , −pr .  generated The Fock space F is an irreducible module of the Clifford algebra C by the vacuum vector |0.

Proof. The identification of the vector (A.69) with m up to a sign follows from the definitions of ψ± n , which correspond to the additions and removals of the + signs. The irreducibility follows from the fact that one Maya diagram can be transformed into another by changing the signs one interval at a time.  − − + c Example A.10. We have |mc  = −ψ+ −5/2 ψ−1/2 ψ−7/2 ψ−3/2 |0, for m from Example A.8.

We define the half-integral Frobenius coordinates for a partition λ as follows. Draw the Young diagram for λ in the English convention, and draw a diagonal line from the vertex (0, 0) that divides the Young diagram into two halves. We then record the lengths of rows of the upper half and the columns of the lower half as (p1 , p2 , . . . , pr |q1 , q2 , . . . , qr ), where each pi , qi lie in 12 + Z+ , and p1 > p2 > . . . > pr > 0,

q1 > q2 > . . . > qr > 0.

The sequence (p1 − 12 , p2 − 12 , . . . , pr − 12 |q1 − 12 , . . . , qr − 12 ) is Frobenius’ original notation for the partition λ (see Macdonald [83, p. 3]). For example, the modified Frobenius coordinates for the partition (3, 2, 2, 1) are ( 52 , 12 | 72 , 32 ). It turns out that the indices pi , q j in (A.69) (at least when r = s) have natural interpretations in terms of partitions. (A.70) Let m be a Maya diagram of charge 0. Write (p1 , p2 , . . . , pr |q1 , q2 , . . . , qr ) for the modified Frobenius coordinates for the partition λm . Then, up to a sign, − − − + + |m is equal to ψ+ −p1 ψ−p2 . . . ψ−pr ψ−q1 ψ−q2 . . . ψ−qr |0. r2

The sign in (A.70) can be computed to be (−1)q1 +...+qr − 2 .

A. Symmetric functions

286

Proof. The proof follows from two observations that work for any general Maya diagram of charge 0, and we will illustrate below using our running example mc from Example A.8: (i) In the diagrams appearing in the proof of (A.65), the two 2 corresponding to the + sign to the left of the dashed vertical line are − 52 , − 12 , and the opposites of the 2 labels corresponding to the − sign to the right of the dashed vertical line are − 32 , − 72 ; they are exactly the indices of the ψ± s in the formula − − + |mc  = −ψ+ −5/2 ψ−1/2 ψ−7/2 ψ−3/2 |0 by (A.68). (ii) The modified Frobenius coor-

dinates for the partition (3, 2, 2, 1) are ( 52 , 12 | 72 , 32 ), and they precisely correspond to the 2 + 2 labels specified in (i).  We form generating functions (called fermionic vertex operators) ψ± (z) in a formal variable z by letting (A.71)

ψ+ (z) =

∑

−n− 2 ψ+ , nz

ψ− (z) =

1

n∈ 21 +Z

∑

−n− 2 ψ− . nz 1

n∈ 12 +Z

A.4.4. Charge and energy. We define the energy operator L0 as a linear operator on F that diagonalizes the basis elements (A.69) with eigenvalues p1 + . . . + pr + q1 + . . . + qs . In addition, we define a charge operator α0 as a linear operator on F that diagonalizes the basis elements (A.69) with eigenvalues r − s. The element (A.69) will be said to have energy p1 + . . . + pr + q1 + . . . + qs and charge r − s. We have the following charge decomposition: F=



F() ,

∈Z

where F() is spanned by the basis vectors (A.69) of charge . We shall see that the notion of charge here matches the notion of charge m for a Maya diagram m. (A.72) For each  ∈ Z, F() has a basis |m parameterized by the Maya diagrams m = {m j } j≥1 that satisfy m = . Proof. By definition, ψ− n of charge −1 corresponds to the removal of a term from a Maya diagram m = (m1 , m2 , . . .), which decreases the integer m by 1 . On the other hand, the operator ψ+ n of charge 1 corresponds to the creation of a new sequence (i.e. Maya diagram) by the insertion of a term into a sequence m, which increases the integer m by 1. This means that the two notions of charge are identical up to a possible constant shift. This constant must be zero once we observe that |0 = |ma  is clearly in F(0) and also ma = 0 by definition.  For  ∈ Z, we let (A.73)

⎧ + + + ⎪ ⎨ ψ 12 − . . . ψ− 32 ψ− 12 |0, |0, | = ⎪ ⎩ ψ−1 . . . ψ− 3 ψ− 1 |0, − − + 2

2

2

if  > 0 if  = 0 if  < 0.

A.4. The Boson-Fermion correspondence

287

In particular, |1 = ψ+ |0 and |-1 = ψ− |0. The following is easily verified. −1 −1 2

2

(A.74) For each  ∈ Z, the element | ∈ F() has the minimal energy, which is 2 /2, among all vectors (A.69) of charge . A.4.5. From Bosons to Fermions. The Heisenberg algebra Heis is the Lie algebra generated by ak (k ∈ Z) and a cental element c with the commutation relation: (A.75)

m, n ∈ Z.

[am , an ] = mδm,−n c,

In particular, a0 is central. The Heisenberg algebra Heis acts irreducibly on a polynomial algebra B() := C[p1 , p2 , . . .] in infinitely many variables p1 , p2 , . . ., by letting a−k (k > 0) act as the multiplication operator by kpk , ak (k > 0) act as the differential operator ∂p∂ k , c act as the identity I, and a0 act as I, for  ∈ C.

The Heis-module B() is irreducible, since any nonzero polynomial in B() can be transformed to a nonzero constant polynomial by a suitable sequence of differential operators and then produce arbitrary monomials by the multiplication operators. The Heis-module B() is a highest weight module in the sense that an .1 = 0 for n > 0, and any highest weight Heis-module generated by a highest weight vector v such that c.v = v and a0 .v = v is isomorphic to B() . Similarly to (5.45), we introduce the normal ordered product  + if n = −m < 0, −ψ− − n ψm , :ψ+ ψ : = m n + − ψm ψn , otherwise. Define the bosonic vertex operator α(z) = ∑k∈Z αk z−k−1 by letting (A.76)

α(z) =

∑ αk z−k−1 =

:ψ+ (z)ψ− (z):,

k∈Z

which is equivalent to defining componentwise (A.77)

αk =

∑

− :ψ+ n ψk−n :,

k ∈ Z.

n∈ 12 +Z

Note that α0 =

∑ ψ+−n ψ−n − ∑ ψ−n ψ+−n

n>0

n 0 and m of charge , ψ+ −p ψ−q |m = ±|m  if it is nonzero, where m is obtained from m by replacing a term q appearing in m by a new term −p, and the charge of m remains . Recalling the definition (A.64) of the partition λm , we see that |λm | = |λm | + p + q. This implies that the energy of m is equal to |λm | up to a constant shift that depends on the charge . The constant is determined to be / Hence, the energy of the element |m 2 /2 by recalling (A.74) and noting λ| = 0. is |λm | + 2m /2. From this it follows that the q-dimension (or graded dimension) of F() is given by 2

q2 tr|F() q = ∞ . ∏k=1 (1 − qk ) L0

On the other hand, note that the energy operator L0 satisfies (and is indeed ± characterized by) the following properties: (i) L0 |0 = 0; (ii) [L0 , ψ± −n ] = nψ−n , for 1 all n ∈ 2 + Z. It follows from (A.77) that [L0 , α−k ] = kα−k ,

(A.80)

k ∈ Z.

Note that αk | = 0, for k > 0, and α0 | = |. Hence, the irreducible Heissubmodule of F() generated by | is isomorphic to B() and has q-dimension 2

2 equal to ∏∞ q(1−qk ) , the same as k=1 Heis-module F() is irreducible.

the graded dimension of F() . It follows that the 

1 ± Remark A.11. From the proof, we have [α0 , ψ± n ] = ±ψn for n ∈ 2 + Z. This together with α0 |0 = 0 implies that α0 here can be identified with the charge operator defined earlier.

The other half of the Boson-Fermion correspondence allows one to reconstruct the fermions ψ± (z) in terms of the boson α(z). We will formulate the statement but skip its proof. Denote by S : F() → F(+1) the shift operator that sends | to | + 1 and commutes with the action of αk for all k = 0. Then ±

±1

ψ (z) = S : exp(±



α(z)dz):

= S±1 z±α0 e∓ ∑ j0

e

z− j j

αj

.

A.4. The Boson-Fermion correspondence

289

A.4.6. Fermions and Schur functions. By the Boson-Fermion correspondence (A.78), we have an isomorphism F(0) ∼ = B(0) . On the other hand, we can naturally (0) identify B with the ring of symmetric functions Λ by identifying α−λ1 . . . α−λ |0 with the power-sum symmetric functions pλ , for all partitions λ = (λ1 , . . . , λ ). Let us denote by Θ : F(0) → Λ the composition of the above two isomorphisms. Recall that the Schur functions sλ form a linear basis for Λ, and recall from (A.64) that a partition λm is associated to a Maya diagram m. (A.81) The linear isomorphism Θ : F(0) → Λ sends a Maya diagram m of charge 0 to the Schur function sλm . A proof of this can be found in [65]. The following supporting examples can be verified via a direct computation by repeatedly using (A.77); the elements in terms of ψ± ’s can be converted to the elements in terms of Maya diagrams via (A.70). Example A.12. We have 1 2 ψ− |0 (α + α−2 )|0 = ψ+ − 32 − 12 2 −1 1 2 ψ− |0. (α − α−2 )|0 = −ψ+ − 12 − 32 2 −1 In addition, we have   1 3 1 1 ψ− |0 α + α−2 α−1 + α−3 |0 = ψ+ − 52 − 12 6 −1 2 3   1 3 1 1 ψ− |0 α−1 − α−2 α−1 + α−3 |0 = ψ+ − 12 − 52 6 2 3 1 3 ψ− |0. (α − α−3 )|0 = −ψ+ − 32 − 32 3 −1 A.4.7. Jacobi triple product identity. (A.82) Let q, y be formal variables. The Jacobi triple product identity holds: ∞

∏(1 − qk )(1 + qk−

1 2

y)(1 + qk− 2 y−1 ) =

k=1

1

∑q

2 2

y .

∈Z

Proof. We compute the trace of the operator qL0 yα0 on F in two ways. First, by the linear basis (A.68) of F and knowing each fermionic operator ψ± − contributes  to the energy and ±1 to the charge, we have tr|F qL0 yα0 =

∏

n∈ 12 +Z+

(1 + qn y)(1 + qn y−1 ).

A. Symmetric functions

290

On the other hand, F() for each  ∈ Z is identified with the bosonic Fock space B() by (A.78). Using (A.74), we compute that 2

L0 α0

tr|F q y

=

∑ tr|B

()

∈Z

L0 α0

q y

q 2 y =∑ ∞ . k ∈Z ∏k=1 (1 − q )

The Jacobi triple product identity follows now by equating the two formulas for the trace and clearing the denominator. 

Notes Section A.1. The materials on symmetric functions and the Frobenius characteristic map are fairly standard, and they can be found in Macdonald [83] in possibly different order. Section A.2. The characterization of supersymmetric polynomials in A.2.1 appeared in Stembridge [117]. The results on super Schur functions and super Cauchy identity in A.2.2 can be found in [7, 110]. Section A.3. The materials can be found in Macdonald [83] and in J´ozefiak [57]. The characterization of the ring Γ in A.3.4 appeared in Pragacz [98, Theorem 2.11], and is used in Section 2.3.2. Section A.4. The materials on Boson-Fermion correspondence are standard. The use of Maya diagrams follows Miwa-Jimbo-Date [87]. Additional applications of Boson-Fermion correspondence, most notably to soliton equations, can be found in Kac-Raina [65] and Miwa-Jimbo-Date [87].

Bibliography

[1] A. Alldridge, The Harish-Chandra isomorphism for reductive symmetric superpairs, Transform. Groups, DOI: 10.1007/S00031-012-9200-y. [2] F. Aribaud, Une nouvelle d´emonstration d’un th´eor`eme de R. Bott et B. Kostant, Bull. Soc. Math. France 95 (1967), 205–242. [3] M. Atiyah, R. Bott, and V.K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330. [4] S. Azam, H. Yamane, and M. Yousofzadeh, Classification of Finite Dimensional Irreducible Representations of Generalized Quantum Groups via Weyl Groupoids, preprint, arXiv:1105.0160. [5] A. Beilinson and J. Bernstein, Localisation de g-modules, C.R. Acad. Sci. Paris Ser. I Math. 292 (1981), 15–18. [6] I. Bernstein, I. Gelfand and S. Gelfand: Differential operators on the base affine space and a study of g-modules. Lie groups and their representations (Proc. Summer School, Bolyai Janos Math. Soc., Budapest, 1971), pp. 21–64. Halsted, New York, 1975. [7] A. Berele and A. Regev, Hook Young Diagrams with Applications to Combinatorics and to Representations of Lie Superalgebras, Adv. Math. 64 (1987), 118–175. [8] I.N. Bernstein and D.A. Leites, A formula for the characters of the irreducible finitedimensional representations of Lie superalgebras of series gl and sl (Russian), C. R. Acad. Bulgare Sci. 33 (1980), 1049–1051. [9] B. Boe, J. Kujawa, and D. Nakano, Cohomology and support varieties for Lie superalgebras, Trans. Amer. Math. Soc. 362 (2010), 6551–6590. [10] A. Brini, A. Palareti, and A. Teolis, Gordan-Capelli series in superalgebras, Proc. Natl. Acad. Sci. USA 85 (1988), 1330–1333. [11] J. Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n), J. Amer. Math. Soc. 16 (2003), 185–231. [12] J. Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra q(n), Adv. Math. 182 (2004), 28–77. [13] J. Brundan and A. Kleshchev, Projective representations of symmetric groups via Sergeev duality, Math. Z. 239 (2002), 27–68.

291

292

Bibliography

[14] J. Brundan and J. Kujawa, A new proof of the Mullineux conjecture, J. Algebraic Combin. 18 (2003), 13–39. [15] J. Brundan and C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebras IV: the general linear supergroup, J. Eur. Math. Soc. 14 (2012), 373–419. [16] J.L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387–410. [17] B. Cao and L. Luo, Generalized Verma Modules and Character Formulae for osp(3|2m), preprint, arXiv:1001.3986. [18] R. Carter, Lie Algebras of Finite and Affine Type. Cambridge Studies in Advanced Mathematics 96. Cambridge University Press, Cambridge, 2005. xviii+632 pp. [19] S.-J. Cheng and J.-H. Kwon, Howe duality and Kostant homology formula for infinitedimensional Lie superalgebras, Int. Math. Res. Not. 2008, Art. ID rnn 085, 52 pp. [20] S.-J. Cheng, J.-H. Kwon, and N. Lam, A BGG-type resolution for tensor modules over general linear superalgebra, Lett. Math. Phys. 84 (2008), 75–87. [21] S.-J. Cheng, J.-H. Kwon, and W. Wang, Kostant homology formulas for oscillator modules of Lie superalgebras, Adv. Math. 224 (2010), 1548–1588. [22] S.-J. Cheng and N. Lam, Infinite-dimensional Lie superalgebras and hook Schur functions, Commun. Math. Phys. 238 (2003), 95–118. [23] S.-J. Cheng and N. Lam, Irreducible characters of general linear superalgebra and super duality, Commun. Math. Phys. 280 (2010), 645–672. [24] S.-J. Cheng, N. Lam and W. Wang, Super duality and irreducible characters of orthosymplectic Lie superalgebras, Invent. Math. 183 (2011), 189–224. [25] S.-J. Cheng, N. Lam and W. Wang, Super duality for general linear Lie superalgebras and applications, arXiv:1109.0667, Proc. Symp. Pure Math. (to appear). [26] S.-J. Cheng, N. Lam and W. Wang, Brundan-Kazhdan-Lusztig conjecture for general linear Lie superalgebras, arXiv:1203.0092. [27] S.-J. Cheng, N. Lam, and R.B. Zhang, Character formula for infinite dimensional unitarizable modules of the general linear superalgebra, J. Algebra 273 (2004), 780–805. [28] S.-J. Cheng and W. Wang, Remarks on Schur-Howe-Sergeev duality, Lett. Math. Phys. 52 (2000), 143–153. [29] S.-J. Cheng and W. Wang, Howe duality for Lie superalgebras, Compositio Math. 128 (2001), 55–94. [30] S.-J. Cheng and W. Wang, Lie subalgebras of differential operators on the super circle, Publ. Res. Inst. Math. Sci. 39 (2003), 545–600. [31] S.-J. Cheng and W. Wang, Brundan-Kazhdan-Lusztig and Super Duality Conjectures, Publ. Res. Inst. Math. Sci. 44 (2008), 1219–1272. [32] S.-J. Cheng and W. Wang, Dualities for Lie superalgebras, Lie Theory and Representation Theory, 1–46, Surveys of Modern Mathematics 2, International Press and Higher Education Press, 2012. [33] S.-J. Cheng, W. Wang, and R.B. Zhang, A Fock space approach to representation theory of osp(2|2n), Transform. Groups 12 (2007), 209–225. [34] S.-J. Cheng, W. Wang, and R.B. Zhang, Super duality and Kazhdan-Lusztig polynomials, Trans. Amer. Math. Soc. 360 (2008), 5883–5924. [35] S.-J. Cheng and R.B. Zhang, Analogue of Kostant’s u-homology formula for general linear superalgebras, Int. Math. Res. Not. 2004, 31–53. [36] S.-J. Cheng and R.B. Zhang, Howe duality and combinatorial character formula for orthosymplectic Lie superalgebras, Adv. Math. 182 (2004), 124–172.

Bibliography

293

[37] E. Date, M. Jimbo, M. Kashiwara, and T. Miwa: Transformation groups for soliton equations. III. Operator approach to the Kadomtsev-Petviashvili equation, J. Phys. Soc. Japan 50 (1981), 3806–3812. Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP-type, Phys. D 4 (1981/82), 343–365. [38] M. Davidson, E. Enright, and R. Stanke, Differential Operators and Highest Weight Representations, Mem. Amer. Math. Soc. 94 (1991), no. 455. [39] V. Deodhar, On some geometric aspects of Bruhat orderings II: the parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra 111 (1987), 483–506. [40] S. Donkin, On tilting modules for algebraic groups, Math. Z. 212 (1993), 39–60. [41] T. Enright, Analogues of Kostant’s u-cohomology formulas for unitary highest weight modules, J. Reine Angew. Math. 392 (1988), 27–36. [42] T. Enright and J. Willenbring, Hilbert series, Howe duality, and branching rules for classical groups, Ann. of Math. 159 (2004), 337–375. [43] L. Frappat, A. Sciarrino, and P. Sorba, Dictionary on Lie algebras and superalgebras. Academic Press, Inc., San Diego, CA, 2000. [44] I. Frenkel, Representations of affine Lie algebras, Hecke modular forms and Kortweg-de Vries type equations, Lect. Notes Math. 933 (1982), 71–110. [45] J. Germoni, Indecomposable representations of osp(3, 2), D(2, 1; α) and G(3). Colloquium on Homology and Representation Theory (Spanish) (Vaquerias, 1998). Bol. Acad. Nac. Cienc. (Cordoba) 65 (2000), 147–163. [46] R. Goodman and N. Wallach, Representations and invariants of the classical groups. Encyclopedia of Mathematics and its Applications, 68. Cambridge University Press, Cambridge, 1998. [47] M. Gorelik, Strongly typical representations of the basic classical Lie superalgebras, J. Amer. Math. Soc. 15 (2002), 167–184. [48] M. Gorelik, The Kac construction of the centre of U(g) for Lie superalgebras, J. Nonlinear Math. Phys. 11 (2004), 325–349. [49] C. Gruson and V. Serganova, Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras, Proc. Lond. Math. Soc. 101 (2010), 852–892. [50] D. Hill, J. Kujawa, and J. Sussan, Degenerate affine Hecke-Clifford algebras and type Q Lie superalgebras, Math. Z. 268 (2011), 1091–1158. [51] R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), 539– 570. [52] R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur Lectures, Israel Math. Conf. Proc. 8, Tel Aviv (1992), 1–182. [53] P.-Y. Huang, N. Lam, and T.-M. To, Super duality and homology of unitarizable modules of Lie algebras, Publ. Res. Inst. Math. Sci. 48 (2012), 45–63. [54] J. E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Graduate Studies in Mathematics, 94. American Mathematical Society, Providence, RI, 2008. [55] K. Iohara and Y. Koga, Second homology of Lie superalgebras, Math. Nachr. 278 (2005), 1041–1053. [56] T. J´ozefiak, Semisimple superalgebras, In: Algebra–Some Current Trends (Varna, 1986), pp. 96–113, Lect. Notes in Math. 1352, Springer-Verlag, Berlin-New York, 1988. [57] T. J´ozefiak, Characters of projective representations of symmetric groups, Expo. Math. 7 (1989), 193–247.

294

Bibliography

[58] T. J´ozefiak, A class of projective representations of hyperoctahedral groups and Schur Qfunctions, Topics in Algebra, Banach Center Publ., 26, Part 2, PWN-Polish Scientific Publishers, Warsaw (1990), 317–326. [59] V. Kac, Classification of simple Lie superalgebras (in Russian), Funkcional. Anal. i Priloˇzen 9 (1975), 91–92. [60] V. Kac, Lie superalgebras, Adv. Math. 26 (1977), 8–96. [61] V. Kac, Characters of typical representations of classical Lie superalgebras, Comm. Algebra 5 (1977), 889–897. [62] V. Kac, Representations of classical Lie superalgebras. Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977), pp. 597–626, Lecture Notes in Math. 676, Springer, Berlin, 1978. [63] V. Kac, Laplace operators of infinite-dimensional Lie algebras and theta functions, Proc. Nat. Acad. Sci. USA 81 (1984), 645–647. [64] V. Kac, Infinite dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990. [65] V. Kac and A. Raina, Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, Advanced Series in Mathematical Physics 2. World Scientific Publishing Co., Inc., Teaneck, NJ, 1987. [66] V. Kac and M. Wakimoto, Integrable highest weight modules over affine superalgebras and Appell’s function, Commun. Math. Phys. 215 (2001), 631–682. [67] V. Kac, W. Wang, and C. Yan, Quasifinite representations of classical Lie subalgebras of W1+∞ , Adv. Math. 139 (1998), 56–140. [68] S.-J. Kang and J.-H. Kwon, Graded Lie superalgebras, supertrace formula, and orbit Lie superalgebras, Proc. London Math. Soc. 81 (2000), 675–724. [69] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), 1–47. [70] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184. [71] D. Kazhdan and G. Lusztig, Schubert varieties and Poincar´e duality. Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 185–203, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980. [72] A. Knapp and D. Vogan, Cohomological Induction and Unitary Representations, Princeton Mathematical Series, 45. Princeton University Press, Princeton, NJ, 1995. [73] B. Kostant, Lie Algebra Cohomology and the Generalized Borel-Weil Theorem, Ann. Math. 74 (1961), 329–387. [74] H. Kraft and C. Procesi, Classical Invariant Theory, A Primer. 1996, 128 pp. Available at http://www.math.unibas.ch/∼kraft/Papers/KP-Primer.pdf. [75] S. Kudla, Seesaw reductive pairs, in Automorphic forms in several variables, Taniguchi Symposium, Katata 1983, Birkh¨auser, Boston, 244–268. [76] S. Kumar, Kac-Moody groups, their flag varieties and representation theory. Progress in Mathematics, 204. Birkhauser Boston, Inc., Boston, MA, 2002. [77] N. Lam and R.B. Zhang, Quasi-finite modules for Lie superalgebras of infinite rank, Trans. Amer. Math. Soc. 358 (2006), 403–439. [78] D. Leites, M. Saveliev, and V. Serganova, Embedding of osp(N/2) and the associated nonlinear supersymmetric equations. Group theoretical methods in physics, Vol. I (Yurmala, 1985), 255–297, VNU Sci. Press, Utrecht, 1986.

Bibliography

295

[79] J. Lepowsky: A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra 49 (1977), 496–511. [80] J.-A. Lin, Categories of gl(m|n) with typical central characters, Master’s degree thesis, National Taiwan University, 2009. [81] L. Liu, Kostant’s Formula for Kac-Moody Lie Algebras, J. Algebra 149 (1992), 155–178. [82] L. Luo, Character Formulae for Ortho-symplectic Lie Superalgebras osp(n|2), J. Algebra 353 (2012), 31–61. [83] I. G. Macdonald, Symmetric functions and Hall polynomials, Second Edition, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. [84] Yu. Manin, Gauge field theory and complex geometry, Grundlehren der mathematischen Wissenschaften 289, Second Edition, Springer-Verlag, Berlin, 1997. [85] J. Milnor and J. Moore, On the Structure of Hopf Algebras, Ann. Math. 81 (1965), 211–264. [86] B. Mitchell, Theory of categories, Pure and Applied Mathematics XVII, Academic Press, New York-London, 1965. [87] T. Miwa, M. Jimbo, and E. Date, Solitons. Differential equations, symmetries and infinitedimensional algebras, Cambridge Tracts in Mathematics 135. Cambridge University Press, Cambridge, 2000. [88] E. Moens and J. van der Jeugt, A determinantal formula for supersymmetric Schur polynomials, J. Algebraic Combin. 17 (2003), 283–307. [89] I. Musson, Lie superalgebras, Clifford algebras, induced modules and nilpotent orbits, Adv. Math. 207 (2006), 39–72. [90] I. Musson, Lie superalgebras and enveloping algebras, Graduate Studies in Mathematics, 131. American Mathematical Society, Providence, RI, 2012. ´ [91] M. Nazarov, Capelli identities for Lie superalgebras, Ann. Sci. Ecole Norm. Sup. 30 (1997), 847–872. [92] M. Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv. Math. 127 (1997), 190–257. [93] I. Penkov, Characters of typical irreducible finite-dimensional q(n)-modules, Funct. Anal. App. 20 (1986), 30–37. [94] I. Penkov and V. Serganova, Cohomology of G/P for classical complex Lie supergroups G and characters of some atypical G-modules, Ann. Inst. Fourier 39 (1989), 845–873. [95] I. Penkov and V. Serganova, Generic irreducible representations of finite-dimensional Lie superalgebras, Internat. J. Math. 5 (1994), 389–419. [96] I. Penkov and V. Serganova, Characters of finite-dimensional irreducible q(n)-modules, Lett. Math. Phys. 40 (1997), 147–158. [97] N. Popescu, Abelian categories with applications to rings and modules. London Mathematical Society Monographs 3, Academic Press, London-New York, 1973. [98] P. Pragacz, Algebro-geometric applications of Schur S- and Q-polynomials. Topics in invariant theory (Paris, 1989/1990), 130–191, Lecture Notes in Math. 1478, Springer, Berlin, 1991. [99] E. W. Read, The α-regular classes of the generalized symmetric groups, Glasgow Math. J. 17 (1976), 144–150. [100] A. Rocha-Caridi, Splitting Criteria for g-Modules Induced from a Parabolic and the Bernstein-Gelfand-Gelfand Resolution of a Finite Dimensional Irreducible g-Module, Trans. Amer. Math. Soc. 262 (1980) 335–366.

296

Bibliography

[101] A. Rocha-Caridi and N. Wallach, Projective modules over graded Lie algebras I, Math. Z. 180 (1982), 151–177. [102] L. Ross, Representations of graded Lie algebras, Trans. Amer. Math. Soc. 120 (1965), 17– 23. [103] B. Sagan, Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley, J. Combin. Theory Ser. A 45 (1981), 62–103. [104] J. Santos, Foncteurs de Zuckermann pour les superalg´ebres de Lie, J. Lie Theory 9 (1999), 69–112. [105] M. Scheunert, The Theory of Lie Superalgebras, Lect. Notes in Math. 716, Springer, Berlin, 1979. [106] M. Scheunert, W. Nahm, and V. Rittenberg, Classification of all simple graded Lie algebras whose Lie algebra is reductive I, II. Construction of the exceptional algebras, J. Math. Phys. 17 (1976), 1626–1639, 1640–1644. [107] V. Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra gl(m|n), Selecta Math. (N.S.) 2 (1996), 607–651. [108] V. Serganova, Kac-Moody superalgebras and integrability, In: Developments and trends in infinite-dimensional Lie theory, 169–218, Progr. Math. 288, Birkh¨auser, 2011. [109] A. Sergeev, The centre of enveloping algebra for Lie superalgebra Q(n, C), Lett. Math. Phys. 7 (1983), 177–179. [110] A. Sergeev, The tensor algebra of the identity representation as a module over the Lie superalgebras gl(n, m) and Q(n), Math. USSR Sbornik 51 (1985), 419–427. [111] A. Sergeev, The invariant polynomials of simple Lie superalgebras, Represent. Theory 3 (1999), 250–280 (electronic). [112] A. Sergeev, The Howe duality and the projective representation of symmetric groups, Represent. Theory 3 (1999), 416–434. [113] A. Sergeev, An Analog of the Classical Invariant Theory, I, II, Michigan J. Math. 49 (2001), 113–146, 147–168. [114] A. Sergeev and A. Veselov, Grothendieck rings of basic classical Lie superalgebras, Ann. of Math. 173 (2011), 663-703. [115] B. Shu and W. Wang, Modular representations of the ortho-symplectic supergroups, Proc. London Math. Soc. 96 (2008), 251–271. [116] W. Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represent. Theory (electronic) 2 (1998), 432–448. [117] J. Stembridge, A characterization of supersymmetric polynomials, J. Algebra 95 (1985), 439–444. [118] J. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), 87–134. [119] Y. Su, Composition factors of Kac modules for the general linear Lie superalgebras, Math. Z. 252 (2006), 731–754. [120] Y. Su and R.B. Zhang, Character and dimension formulae for general linear superalgebra, Adv. Math. 211 (2007), 1–33. [121] Y. Su and R.B. Zhang, Generalised Verma modules for the orthosymplectic Lie superalgebra osp(k|2), J. Algebra 357 (2012), 94–115. [122] J. Tanaka, On homology and cohomology of Lie superalgebras with coefficients in their finite-dimensional representations, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), 51–53. [123] J. Van der Jeugt, Character formulae for Lie superalgebra C(n), Comm. Algebra 19 (1991), 199–222.

Bibliography

297

[124] J. Van der Jeugt, J.W.B. Hughes, R. C. King, and J. Thierry-Mieg, Character formulas for irreducible modules of the Lie superalgebras sl(m/n), J. Math. Phys. 31 (1990), 2278– 2304. [125] D. Vogan, Irreducible characters of semisimple Lie Groups II: The Kazhdan-Lusztig Conjectures, Duke Math. J. 46 (1979), 805–859. [126] C.T.C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213 (1964), 187–199. [127] J. Wan and W. Wang, Lectures on spin representation theory of symmetric groups, Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), 91–164. [128] J. Wan and W. Wang, Spin Kostka polynomials, J. Algebraic Combin., DOI: 10.1007/s10801-012-0362-4. [129] W. Wang, Duality in infinite dimensional Fock representations, Commun. Contem. Math. 1 (1999), 155–199. [130] W. Wang and L. Zhao, Representations of Lie superalgebras in prime characteristic I, Proc. London Math. Soc. 99 (2009), 145–167. [131] H. Weyl, The classical groups. Their invariants and representations. Fifteenth printing. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997. [132] M. Yamaguchi, A duality of the twisted group algebra of the symmetric group and a Lie superalgebra, J. Alg. 222 (1999), 301–327. [133] Y. Zou, Categories of finite-dimensional weight modules over type I classical Lie superalgebras, J. Algebra 180 (1996), 459–482.

Index

εδ-sequence, 22, 23 adjoint action, 3 adjoint map, 3 algebra Clifford, 114, 183, 285 exterior, 10, 52 Hecke-Clifford, 114 Heisenberg, 287 supersymmetric, 53 univeral enveloping, 31 Weyl, 153 Weyl-Clifford, 153 atypical, 75 bilinear form even, 6 invariant, 3 odd, 6 skew-supersymmetric, 6 supersymmetric, 6 Boolean characteristic function, 181 boson fermion correspondence, 287 boundary, 233 category O, 219  219 O, O, 218 polynomial modules, 108 Cauchy identity, 267 central character, 57 character formula, 232 characteristic map, 116 Frobenius, 270 charge, 286

Chevalley automorphism, 6 Chevalley generators, 21 coboundary, 234 cocycle, 214 cohomology, 234 Kostant, 235 restricted, 234 unrestricted, 244 column determinant, 165 conjugacy class split, 97 contraction, 134 coroot, 14 cycle signed, 110 type of, 110 support of, 110 degree atypicality, 65, 75, 106, 107 even, 2 odd, 2 derivation, 3 diagram Dynkin, 21, 22, 24, 206, 207, 211 head, 207 master, 207 Maya, 282 charge of, 282 skew, 268 tail, 206 dimension, 2 duality super, 249 energy, 286

299

Index

300

extension, 108, 242 Baer-Yoneda, 241 central, 180, 181, 213 fermionic Fock space, 284 FFT, 132 multilinear GL(V ), 134 polynomial GL(V ), 135 O(V ), Sp(V ), 141 supersymmetric O(V ), Sp(V ), 149 tensor GL(V ), 133 O(V ), Sp(V ), 146 First Fundamental Theorem, 132 Fock space fermionic, 183 Frobenius coordinates, 78, 281 half-integral, 285 modified, 80, 215 functor T , 228 T , 228 duality, 235 parity reversing, 2, 71, 92 truncation, 221, 240 fundamental system, 19 group orthogonal, 139 symplectic, 139 harmonic, 173 highest weight, 34 highest weight vector, 34 homology, 233 Kostant, 235 homomorphism Harish-Chandra, 57 Lie superalgebra, 3 module, 2 Howe dual pair, 160 ideal, 2 identity matrix, 2 Jacobi triple product identity, 289 Kac module, 44 Kazhdan-Lusztig-Vogan polynomial, 239 level, 182 Lie algebra orthogonal, 139 symplectic, 139

Lie superalgebra, 3  g, g, g, 214 basic, 13 Cartan type H(n), 12 S(n), 11 W (n), 10  H(n), 12  S(n), 11 classical, 13 exceptional D(2|1, α), 9 F(3|1), 10 G(3), 10 general linear, 4 ortho-symplectic, 6 periplectic, 9 queer, 8 solvable, 32 special linear, 5 type a, 209 type b• , 209 type b, 210 type c, 209 type d, 210 linkage principle gl,osp, 68 q, 75 Littlewood-Richardson coefficient, 268 module Kac, 44, 45 oscillator osp(2m|2n), 174 spo(2m|2n), 179 parabolic Verma, 218 polynomial, 108, 215, 217 spin, 97 basic, 118 type M, 94 type Q, 94 multiplicity-free strongly, 159 nilradical, 217 opposite, 217 normal ordered product, 184 odd reflection, 27 odd trace, 19 operator boundary, 233 charge, 286 coboundary, 234 energy, 286 parity

Index

in supergroup, 110 in superspace, 2 parity reversing functor, 2 partition, 215, 261, 282 size, 261 conjugate, 261 dominance order, 261 even, 199, 269 generalized, 186 hook, 49, 215 length, 261 odd, 111, 275 part, 261 strict, 111, 275 Pfaffian, 277 Pieri’s formula, 269 Poincar´e’s lemma, 199 Poincar´e-Birkhoff-Witt Theorem, 31 polarization, 135 polynomial doubly symmetric, 271 Kazhdan-Lusztig-Vogan, 239 Schur, 265 supersymmetric, 271 positive system, 19 standard, 21, 23 pullback, 242 pushout, 251 reflection odd, 27, 225 real, 28 restitution, 135 restricted dual, 233 ring of symmetric functions, 262 root even, 14 isotropic, 15 odd, 14 simple, 19 root system, 14 gl(m|n), 16 q(n), 18 spo(2m|2n), 17 spo(2m|2n + 1), 17 Schur function, 265 skew, 268 super, 274 Schur polynomial, 265 Schur’s lemma, 95 spherical harmonics, 199 strongly multiplicity-free, 159 subalgebra, 2 Borel, 19, 20  bc (n), 224  bs (n), 224

301

standard, 18, 21, 22, 215, 217 Cartan, 14 standard, 17, 18, 215 Levi, 221 standard, 217 parabolic, 217, 221 subspace, 2 super duality, 249 superalgebra, 2 Clifford, 114, 153, 285 exterior, 10 semisimple, 94 simple, 2 Weyl-Clifford, 153 supercharacter, 56 superdimension, 2 supergroup, 96 superspace, 2 supertableau, 102 content of, 102 supertrace, 5 supertranspose, 6 symbol, 154 symbol map, 155, 159 symmetric function complete, 262 elementary, 262 involution ω, 264 monomial, 262 power sum, 262 ring of, 262 system fundamental, 19 positive, 19 root, 14 tableau semistandard, 268 skew, 268 trace odd, 19 typical, 65, 75 univeral enveloping algebra, 31 vector highest weight, 34 singular, 70 vacuum, 284 virtual character, 46 Wedderburn’s theorem, 94 weight dominant, 218, 221 dominant integral, 45 extremal, 77 half-integer, 48

302

highest, 34 integer, 48 linked gl,osp, 65 q, 74 polynomial, 108 Weyl group, 14 Weyl symbol, 154 Weyl vector, 20

Index

This book gives a systematic account of the structure and representation theory of finite-dimensional complex Lie superalgebras of classical type and serves as a good introduction to representation theory of Lie superalgebras. Several folklore results are rigorously proved (and occasionally corrected in detail), sometimes with new proofs.Three important dualities are presented in the book, with the unifying theme of determining irreducible characters of Lie superalgebras. In order of increasing sophistication, they are Schur duality, Howe duality, and super duality. The combinatorics of symmetric functions is developed as needed in connections to Harish-Chandra homomorphism as well as irreducible characters for Lie superalgebras. Schur-Sergeev duality for the queer Lie superalgebra is presented from scratch with complete detail. Howe duality for Lie superalgebras is presented in book form for the first time. Super duality is a new approach developed in the past few years toward understanding the Bernstein-Gelfand-Gelfand category of modules for classical Lie superalgebras. Super duality relates the representation theory of classical Lie superalgebras directly to the representation theory of classical Lie algebras and thus gives a solution to the irreducible character problem of Lie superalgebras via the Kazhdan-Lusztig polynomials of classical Lie algebras.

For additional information and updates on this book, visit WWWAMSORGBOOKPAGESGSM 

GSM/144

AMS on the Web w w w. a m s . o r g www.ams.org