Positivity in Lie Theory: Open Problems [Reprint 2011 ed.] 3110161125, 9783110161120

Table of contents :
Jordan algebras and conformal geometry
Asymptotic problems – from control systems to semigroups
Exponential function of Lie groups
Quelques problèmes d’analyse sur les espaces symétriques ordonnés
Tube domains in Stein symmetric spaces
Invariant cones in real representations
Universal objects in Lie semigroup theory
Introduction to total positivity
An introduction to the embedding problem for probabilities on locally compact groups
Compression semigroups in semisimple Lie groups: a direct approach
Discrete series and analyticity
Some open problems in representation theory related to complex geometry
Boundary values of holomorphic functions and some spectral problems for unitary repesentations
Open problems in harmonic analysis on causal symmetric spaces
Linear algebraic monoids
List of contributors
Index

Citation preview

de Gruyter Expositions in Mathematics 26

Editors

Ο. H. Kegel, Albert-Ludwigs-Universität, Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, The University of Melbourne, Parkville R.O.Wells, Jr., Rice University, Houston

de Gruyter Expositions in Mathematics 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

The Analytical and Topological Theory of Semigroups, Κ. H. Hofmann, J. D. Lawson, J. S. Pym (Eds.) Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues The Stefan Problem, A. M. Meirmanov Finite Soluble Groups, K. Doerk, T. O. Hawkes The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin Contact Geometry and Linear Differential Equations, V. E. Nazaikinskii, V. E. Shatalov, B. Yu. Stemin Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, Μ. V. Zaicev Nilpotent Groups and their Automorphisms, Ε. I. Khukhro Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon, G. Zhao Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. A. Nazarov, B. A. Plamenevsky Subgroup Lattices of Groups, R. Schmidt Orthogonal Decompositions and Integral Lattices, A. I. Kostrikin, PH. Tiep The Adjunction Theory of Complex Projective Varieties, M. C. Beltrametti, A. J. Sommese The Restricted 3-Body Problem: Plane Periodic Orbits, A. D. Bruno Unitary Representation Theory of Exponential Lie Groups, H. Leptin, J. Ludwig Blow-up in Quasilinear Parabolic Equations, A.A. Samarskii, V.A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov Semigroups in Algebra, Geometry and Analysis, Κ. H. Hofmann, J. D. Lawson, Ε. B. Vinberg (Eds.) Compact Projective Planes, H. Salzmann, D. Betten, Τ. Grundhöf er, Η. Hühl, R. Löwen, M. Stroppel An Introduction to Lorentz Surfaces, Τ. Weinstein Lectures in Real Geometry, F. Broglia (Ed.) Evolution Equations and Lagrangian Coordinates, A. M. Meirmanov, V. V. Pukhnachov, S. I. Shmarev Character Theory of Finite Groups, B. Huppert

Positivity in Lie Theory: Open Problems Editors

Joachim Hilgert Jimmie D. Lawson Karl-Hermann Neeb Ernest B.Vinberg

W DE _G Walter de Gruyter · Berlin · New York 1998

Editors Joachim Hilgert Institut für Mathematik Technische Universität Clausthal 38678 Clausthal-Zellerfeld Germany

Jimmie D. Lawson Dept. of Mathematics Louisiana State University Baton Rouge, LA 70803 USA

Karl-Hermann Neeb Mathematisches Institut Universität ErlangenNürnberg 91054 Erlangen Germany

Ernest Β. Vinberg Chair of Algebra Dept. of Mechanics & Mathematics Moscow State University 119899 Moscow, Russia

1991 Mathematics Subject Classification: 17Bxx, 17Cxx, 20Gxx, 20Mxx, 22Exx, 32Exx, 4 3 - X X , 5 2 - X X , 60Bxx, 9 3 - X X

Keywords: Causality, compression semigroups, conformal geometry, control systems, discrete series representations, exponential function, harmonic analysis, holomorphic representations, invariant cones, Jordan algebras, Lie algebras, Lie semigroups, linear monoids, one-parameter groups of probability measures, reductive groups, reproducing kernels, singular representations, spherical functions, Stein manifolds, symmetric spaces, total positivity, tube domains

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Positivity in Lie theory : open problems / editors, J. Hilgert . .. [et al.]. p. cm. — (De Gruyter expositions in mathematics, ISSN 0938-6572 ; 26) Includes index. ISBN 3-11-016112-5 (alk. paper) 1. Lie groups. I. Hilgert, Joachim. II. Series. QA387.P68 1998 512'.55-dc21 98-6433 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication

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Positivity in Lie theory : open problems / ed. J. Hilgert ... - Berlin ; New York : de Gruyter, 1998 (De Gruyter expositions in mathematics ; 26) ISBN 3-11-016112-5

© Copyright 1998 by Walter de Gruyter G m b H & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Typeset using the authors' T ß X files: I. Zimmermann, Freiburg Printing: Arthur Collignon G m b H , Berlin. Binding: Lüderitz & Bauer G m b H , Berlin. Cover design: Thomas Bonnie, Hamburg.

Preface

In December 1996 the Mathematical Research Institute at Oberwolfach hosted a conference on Positivity in Lie Theory in which many of the main contributors to this rapidly developing area discussed recent results and interesting open problems. A characteristic feature of the field is that notions of positivity in Lie theory occur in quite diverse settings, are motivated by a wide variety of problems and applications, and are approached from quite varying mathematical viewpoints. While this diversity is attractive for the specialists who work on these problems, it is often difficult for newcomers to the field to see the relation between the various aspects and to pick the right problems. The participants of the Oberwolfach meeting decided to put together a collection of problems with commentary that would serve as an invitation and a guide to the field. The result of this joint effort is the present book. In each chapter the reader is introduced to a specific open problem or circle of problems that the author considers important for further development. The level of presentation is chosen in such a way that a graduate student with a sound knowledge of basic Lie theory should be able to grasp the gist of the problem. The main definitions are explained, preliminary results are quoted, and the relevant references are listed. Moreover, the authors often tried to formulate smaller problems that might serve as guidelines to the main problems. To make sure that this book does not loose its value too quickly if certain problems get solved, a website has been set up under http://lie.math.tu-clausthal.de/~Hilgert/Problembook where known solutions will be announced and preprints can be made available. The scope of the field. We briefly describe the scope of the field we call "Positivity in Lie Theory". In chronogeometry one considers causal structures which consist of a continuous choice of one half of the double cone of timelike vectors on a Lorentzian manifold. The chosen cone is the positive or forward light cone. Then a central problem of chronogeometry is to decide which pairs of points can be connected by causal curves, i.e., curves with derivatives in the forward light cone. In geometric control theory one models the states of a system by the points of a differentiable manifold and the controls by a set of (control) vector fields on the manifold. A typical problem for the theory is the problem of controllability, where one wants to describe the set of states of a system reachable in positive time from some fixed state with a given set of controls. Such questions lead to a manifold of states and a set of (control) vector fields on the manifold. Many important features of the control vector fields are encoded in the convex cones spanned by the vector fields in the tangent spaces. In this way one again arrives at the concept of a general cone field or "causal structure" on a manifold and the associated causal order.

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If a group acts differentiably on a manifold one asks for invariant causal structures and considers the subset of the group leaving invariant the causal future. In this way one finds the "future" semigroup, which can be viewed as a "dynamical polysystem," a control theoretic analog of the one-parameter semigroups describing a dynamical system. Thus semigroups play an important role in positivity questions related to causal and control structures. Following the basic strategy of Lie theory one studies causal structures via their infinitesimal objects. This leads to the study of convex cones with invariance conditions in Lie algebras or more general representation spaces. It turns out that invariant cones in Lie algebras also play an important role in representation theory where they show up in dissipativity conditions on the spectrum of the operators representing the Lie algebra. This connection also leads to complex analysis via the possibility of finding holomorphic extensions of unitary representations and at the same time yields tools to study certain Stein domains. In the study of general homogeneous spaces another manifestation of positivity appears in the form of "compression" semigroups associated to distinguished open subsets associated with various classes of homogeneous manifolds. Here to any subset of the manifold one considers the semigroup of transformations carrying this set into itself. Important examples that occur naturally in questions of control theory, harmonic analysis and complex geometry can be found in flag manifolds. Important classes of compression semigroups arise in complex analysis where one studies actions of real groups G on complex manifolds Μ by biholomorphic mappings. In many important cases, such as the action of the automorphism group of a bounded symmetric domain, the action of the real group extends to a holomorphic action of a certain complex semigroup S by integrating the corresponding flows also for positive imaginary time. The holomorphic action of the complex semigroup S and its holomorphic representation theory now can be used to study the complex geometry of Μ with respect to the action of G. This correspondence ties up complex geometry and representation theory in a quite subtle way. The classical positivity concepts for matrices have been generalized in various ways. In harmonic analysis and representation theory one considers positive definite functions and kernels. On the other hand the notion of a totally positive matrix has been generalized from GL(n, M) to arbitrary real reductive groups. Finally a generalization of a toric variety, the Zariski closures of a linear algebraic group, which again turns out to be a semigroup, has been considered. Semigroups of this type are of a quite different character, however, than those associated with an infinitesimal cone of generators. These examples show that the link between the topics treated under the heading of "positivity" in Lie theory is the presence of orderings at the level of manifolds, semigroups at the level of groups, and cones at the level of vector spaces and Lie algebras.

Preface

vii

Recent developments. The problems described in this book primarily reflect recent developments. The intensive work during the eighties on the structure of subsemigroups of Lie groups as well as semigroup closures of linear algebraic groups laid the foundation for more specialized and application-oriented research conducted in recent years. New results on structure theory of Lie semigroups and causal spaces are now usually motivated by and obtained in the context of either control theory, harmonic analysis or representation theory. Harmonic analysis. A major undertaking at the moment is the study of various types of Hardy spaces associated to symmetric spaces with invariant causal structures. It has its origins in the "Gelfand-Gindikin Program" launched in 1977 with the goal of developing a new approach to the Plancherel Theorem for semisimple Lie groups by grouping irreducible representations together into geometrically constructed "chunks." The way to construct these geometric pieces is to consider G-invariant domains in the complexification of G and boundary values of holomorphic functions (or more general objects). Hardy spaces for causal symmetric spaces live on (curved) tube domains modelled on invariant cones, and yield the so called holomorphic discrete series. They are a first successful attempt to give content to the Gelfand-Gindikin Program for semisimple symmetric spaces. It is an open problem (even in the group case) to construct cohomological Hardy spaces for the "non-convex" tube domains. New light is shed on this problem through the use of Jordan algebras. It opens the way to compare the "curved" Hardy spaces with classical families of Hardy spaces on "linear" tube domains. The contributions of W. BERTRAM, J. FARAUT, S. GlNDIKIN and G. 0LAFSSON all deal with different aspects of this circle of questions. Whereas FARAUT and 0LAFSSON concentrate more on the harmonic analysis of causal symmetric spaces in which convex cones play a central role, W. BERTRAM explains the setting in which one can transcend the causal situation using Jordan algebra techniques involving also non-convex cones and generalized conformal structures. GlNDIKIN exposits his philosophy of generalized conformal structures in the context of complex analysis, in particular, Stein spaces. Another very active field is the study of spherical functions on ordered symmetric spaces. Originally motivated by the study of integral operators in scattering theory respecting causality, it has developed into a mature theory parallel to Harish-Chandra's treatment of Riemannian symmetric spaces. Very interesting new phenomena which have no parallel in the Riemannian setting are emerging in the case of non-reductive causal symmetric spaces. These aspects are also touched upon in the articles of FARAUT a n d 0LAFSSON.

Representation theory. A central question in the representation theory of semisimple groups is the construction of the inner products for Harish-Chandra modules, known to be unitarizable. Positivity methods are a powerful tool to attack this question for the class of highest weight representations. In the special case of representations with scalar lowest ÄT-type they already yield a construction of the Hilbert space structure based on Kirillov's orbit method. It is to be expected that they also give new insights in

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the general case. Problems related to the construction of Hilbert spaces carrying singular unitary representations are contained in the chapters written by V. F. MOLCHANOV a n d Y u . NERETIN.

Representation theory of Lie semigroups plays an important role in the study of Hardy spaces, spherical functions and unitarizable highest weight representations, but these examples do not exhaust its potential. A key feature of the holomorphic representation theory of complex semigroups is that it can be viewed as a technique extending the method of Fourier expansion - a central tool in the realm of compact groups - by more general desintegration methods available for representations of complex semigroups which are based on highest weight representations. So far these techniques are well developed for representations by bounded operators on Hilbert spaces. The contribution of K.-H. NEEB describes a circle of ideas aiming at an extension of this theory to certain unbounded representations on more general Banach spaces. A n important step in this direction is a better understanding of spaces of holomorphic functions on complex semigroups and in particular a description of a good convolution algebra of holomorphic functions. The articles of S. GLNDIKIN and Yu. NERETIN deal with the converse problem of using complex analytic methods to obtain interesting representations of real groups. These two aspects of the theory illustrate quite well that there are many bridges between complex analysis and representation theory that should be explored in the future. These bridges are most clearly visible for group actions on complexifications of causal symmetric spaces which often serve as model cases. Convex cones in Lie algebras which are invariant under the adjoint action have played an important role in the development of Lie semigroup theory and its applications to representation theory and harmonic analysis. Later one also considered convex cones in the tangent space of a symmetric space invariant under the isotropy representation of the stabilizer group. In both these cases one has a fairly complete picture. Much less is known for the general case of convex cones left invariant by an arbitrary linear group action. J. HILGERT and K.-H. NEEB give an overview what is known in general and present a series of suggestions for attacking the problem of classifying invariant cones in real representations other than the adjoint representation. Control theory. Concepts and methods from Lie theory play an important role in various aspects of geometric control theory. Typically in this approach to control theory one models the states of a system as points on a smooth manifold and the controls of the system as vector fields on the manifold. This set of controls may be viewed as a "dynamical polysystem," and in studying such questions as reachability and controllability, one is interested in the "forward semigroup" generated by finite compositions of positive time phase transitions from the flows corresponding to the given vector fields. This semigroup is in general infinite dimensional, but for a number of interesting problems arising from mechanics or other problems demonstrating a high degree of symmetry, the semigroup in question is both finite dimensional and an infinitesimally generated subsemigroup of a Lie group. In this case the problem at hand can often be pulled back to an equivalent problem on the Lie group, and

Preface

ix

the methods of Lie group and Lie semigroup theory can be applied. The most typical class of problems arising in this context concerns the so-called bilinear control systems, which can be interpreted as control systems on matrix groups in which the vector fields in question are right-invariant. These systems and related problems are discussed in t h e c o n t r i b u t i o n of F. COLONIUS, W. KLIEMANN and L. SAN MARTIN.

For studying reachable sets, investigators have found the notion of the Lie saturate of a set of vector fields useful (where the Lie saturate is the largest collection of vector fields having the same closed reachable sets). These Lie saturates are closely connected to Lie wedges, the appropriate tangent objects of Lie semigroups in the Lie algebra in the finite dimensional setting. The invariant cone fields on coset spaces of Lie groups, which are a prevalent aspect of positivity in Lie groups, are an important special case of the typical setting postulated in the non-smooth approach to control theory. The Pontryagin Maximum Principle has sharpened and simplified versions for the setting of Lie groups, which makes sharper results available and also sometimes facilitates calculations. Conversely the concepts and methods of geometric control theory have proved to be useful tools in the development of the Lie theory of semigroups. We mention some important examples of this interaction between geometric control theory and Lie semigroup theory. Questions of controllability on a Lie group are closely related to a knowledge of the maximal subsemigroups of the Lie group, since a subset of the Lie algebra of a group G viewed as a dynamical polysystem of right invariant vector fields on G is controllable if and only if it is not contained in the Lie tangent wedge (the appropriate tangential object in the Lie algebra) of a maximal subsemigroup. Substantial progress has been made in understanding the maximal subsemigroups, although they have only been characterized in the solvable case. In the case of semisimple Lie groups with finite center the maximal subsemigroups arise as compression semigroups of certain open domains in the flag manifolds of the Lie group. These open domains can be typically located in a "big cell" associated to a Bruhat decomposition of the Lie group and hence pulled back to an open domain in one of the nilpotent algebras associated with the Bruhat decomposition. The article of D. MlTTENHUBER considers the construction of compression semigroups from such a viewpoint. The asymptotic behavior of a constant linear dynamical system is determined by the spectrum of the coefficient matrix. As one moves to the study of analogous problems in time-varying and control systems, generalizations of these ideas are needed. Recent developments and open problems in this direction are discussed in the article of F. COLONIUS, W. KLIEMANN, and L. SAN ΜΑΚΉΝ. Instead of o n e - p a r a m e t e r

semigroups, one now has multi-parameter semigroups to consider. Varying notions of spectrum and the theory of Lyapunov exponents replace the classical matrix spectrum. The eigenspaces are replaced by control sets, maximal sets on which the semigroup (almost) acts transitively. Various other aspects. A classical problem in probability theory is the embedding of a probability measure on a locally compact group into a one-parameter semigroup of probability measures. As is explained in M. McCRUDDEN's contribution the last

χ

Preface

years have seen impressive progress on this problem, but the problem is also a source of rather subtle Lie theoretic questions. A related question is the identification of those elements in a connected Lie group which lie on a one-parameter group. These are precisely the elements in the image of the exponential mapping. Major results and open problems related to the image of the exponential map are presented by D. ©OKOVIC and Κ. H. HOFMANN.

The theory of algebraic semigroups has been developed since 1980. It weds the theories of abstract semigroups and algebraic groups and contributes to both of them. Reductive algebraic semigroups have been studied most extensively. Analogues of Cartan subgroups, the Weyl group, the Bruhat decomposition, and the Tits building have been introduced for them. Recently a complete classification of reductive algebraic semigroups was obtained. At the same time some important questions about their structure and representations remain to be answered. Making use of reductive algebraic semigroups, the so-called wonderful completion of a semisimple algebraic group was obtained in a most natural way. The asymptotic semigroup of a semisimple algebraic group was defined. In a sense it reflects the behavior of the group at infinity in a manner analogous to the way its Lie algebra reflects its infinitesimal behavior. The semigroup analogue of the Weyl group gives rise to a generalization of the Hecke algebra, an interesting combinatorial object. The article of M. PUTCHA and L. RENNER discusses some open problems in this area. One of the earliest examples of a Lie semigroup was the semigroup of totally positive matrices in GL(n, K). In his contribution G. LUSZTIG gives a generalization of total positivity from GL(n, R) to arbitrary real reductive groups; the resulting elements again form a Lie semigroup. An unexpected connection arising in the treatment is the use of canonical bases and perverse sheaves in the study of these semigroups. The contribution explains this connection and points out a number of problems arising in this context. In spite of the extensive work on foundational questions there remain a number of important questions unanswered. J. D. LAWSON's article describes the problems arising from attempts to find the right framework for Lie semigroups which are not necessarily subsemigroups of Lie groups and the efforts to understand semigroups which are freely generated by a local Lie semigroup.

Acknowlegdments. We thank the Mathematische Forschungsinstitut Oberwolfach for providing the logistics for the conference on which the content of this book draws. To the editors of the series de Gruyter Expositions in Mathematics and to the mathematics editor of the de Gruyter Verlag, MANFRED KARBE, we express our gratitude for publishing this collection in the usual superb form. The editors are grateful to KARL HEINRICH HOFMANN to whom the field owes so much and who is always willing to share his experience.

Table of contents

Jordan algebras and conformal geometry, by Wolfgang Bertram

1

Asymptotic problems - from control systems to semigroups, by Fritz Colonius, Wolfgang Kliemann, and Luiz A. B. San Martin

21

Exponential function of Lie groups, by Dragomir -Dokovic and Karl H. Hofmann

45

Quelques problemes d'analyse sur les espaces symetriques ordonnes, by Jacques Faraut

71

Tube domains in Stein symmetric spaces, by Simon Gindikin

81

Invariant cones in real representations, by Joachim Hilgert and Karl-Hermann Neeb

99

Universal objects in Lie semigroup theory, by Jimmie D. Lawson

121

Introduction to total positivity, by George Lusztig

133

An introduction to the embedding problem for probabilities on locally compact groups, by Mick McCrudden

147

Compression semigroups in semisimple Lie groups: a direct approach, by Dirk Mittenhuber

165

Discrete series and analyticity, by Vladimir F. Molchanov

185

Some open problems in representation theory related to complex geometry, by Karl-Hermann Neeb

195

Boundary values of holomorphic functions and some spectral problems for unitary repesentations, by Yurii A. Neretin

221

Open problems in harmonic analysis on causal symmetric spaces, by Gestur Ölafsson

249

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Table of contents

Linear algebraic monoids, by Lex Renner and Mohan Putcha

271

List of contributors

283

Index

285

Jordan algebras and conformal geometry Wolfgang Bertram

0. Introduction 0.1. Jordan theory and "Positivity in Lie theory." The present note is an introduction to some open problems related to the interplay of Lie and Jordan theory. Jordan algebras were invented in the thirties as a concept in the foundations of quantum mechanics; this attempt was not quite satisfactory, and Jordan algebras became interesting again later by the work of Koecher, Vinberg and others who remarked the geometric significance of Jordan algebras. In fact, "positive" Jordan algebras correspond to symmetric cones and "positive" Jordan triple systems to Hermitian symmetric spaces. A glance at other contributions to this volume will show that these geometric objects are indeed of basic importance for many topics related to "positivity in Lie theory," especially in harmonic analysis and representation theory. Therefore one would like to have a deeper understanding of the role Jordan theory plays in these areas. This leads automatically to the bigger class of "all" Jordan algebras and "all" Jordan triple systems, and to the question what their geometric counterpart is. Our approach to this question is guided by the model of the "positive" objects - with basic references [FK94] for symmetric cones and [Lo77], [Sa80] for Hermitian symmetric spaces. 0.2. The "Jordan functor." Lie algebras generalize the anti-symmetrized product [X, F] = Χ ο Υ — Υ ο X on the matrix-space M(n, K) over a field K. In a similar way, Jordan algebras generalize the symmetrized product XY = ^ (Χ ο y + y ο X) on M(n, K). Lie algebras correspond to geometric objects, namely to Lie groups·, this Lie fiinctor is well-understood and of fundamental importance. Therefore the following question is natural: Problem A. Is there a Jordan functor, what is the geometric object corresponding to Jordan algebras? Such a correspondence is in fact known (by a result of Koecher and Vinberg) for the "positive objects" in the category of Jordan algebras, namely for the so-called Euclidean or formally-real Jordan algebras: they correspond bijectively to symmetric cones which is a certain class of open, convex cones. We recall the basic facts about the Koecher-Vinberg correspondence in Section 1.0. In Chapter 1 we construct the "Jordan functor" and give at the same time a selfcontained introduction into basic Jordan theory. Our approach is geometric: we start with symmetric spaces which appear as open orbits of a linear group action in a vector

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space; a basic example is given by the general linear group, considered as a symmetric space, which is open in the space of all matrices. By elementary linear algebra, one associates to such an object a (commutative, non-associative) algebra-structure. The algebras we obtain are known as Lie triple algebras·, they form a class of algebras strictly bigger than the class of Jordan algebras. Therefore Problem A takes the following more specific form: Problem B. What geometric condition corresponds to the algebraic condition that a Lie triple algebra be a Jordan algebra? This condition defines the geometric object corresponding to Jordan algebras. We give a preliminary answer to this problem (Th. 1.2.2). The answer is quite satisfactory in that it allows to build Jordan theory only on simple geometric notions, but it is not satisfactory in that it does not yet really separate geometry and algebra - here the problem remains open. 0.3. "Generalized conformal structures." In Chapter 2 we recall the classical theorem of Liouville on conformal mappings of Euclidean 3-space and its generalization to (semi-simple) Jordan algebras ([Be96a]). This gives an occasion to discuss notions of "generalized conformal structure" and to stress analogies of the Liouville theorem with the fundamental theorem of projective geometry. Already Hermann Weyl ([Wey23]) discussed these two classical theorems in a common context. Problem C. How can the Liouville theorem for Jordan algebras and the fundamental theorem of projective geometry be understood in a common framework? Such a general theorem would have a great variety of different geometric aspects, too wide to be covered by a single concept of "generalized conformal structure" proposed so far. We believe that a good understanding of the Jordan-functor would help to find the appropriate notion of "conformal structure" and to understand the deeper nature of the Liouville theorem. 0.4. The "Jordan triple functor" and Makarevic spaces. Lie triple systems generalize Lie algebras with the triple product [[Χ, Κ], Z]. In a similar way, Jordan triple systems (JTS) generalize Jordan algebras with the triple product T(x,y, ζ) = x(yz) ~ y(xz) + (xy)z', this triple product is natural, as we show in Chapter 1. The geometric objects corresponding to Lie triple systems are symmetric spaces·, the Lie triple product is simply the curvature tensor of a symmetric space, cf. [Lo69]. We ask what geometric objects correspond to Jordan triple systems. Again, for "positive objects" in the category of JTS the answer is known (due to Koecher): the positive Hermitian JTS correspond bijectively to Hermitian symmetric spaces and real positive JTS to their real forms (cf. [Lo77]). In Chapter 3 we indicate how to generalize this correspondence for arbitrary JTS (see [Be97]): the appropriate generalization of Hermitian symmetric spaces is the

Jordan algebras and conformal geometry

3

notion of twisted complex symmetric spaces; they correspond bijectively to Hermitian Jordan triple systems, and general Jordan triple systems correspond to real forms of twisted complex symmetric spaces. Put in another way: Jordan triple systems correspond bijectively to symmetric spaces with a twisted complexification. Such objects have been classified (though defined in a different way) by B. O. Makarevic ([Ma73]), and therefore we call them Makarevic spaces. 0.5. The "Jordan-Lie functor." Plain enough, we have a forgetful-functor from the class of Makarevic spaces into the class of symmetric spaces just by forgetting the twisted complexification. We call it the geometric Jordan-Lie functor. There are two fundamental problems: Problem D (Existence problem). How far is this functor from being surjective; what is its image? In other words, which symmetric spaces admit a twisted complexification? Problem Ε (Uniqueness problem). How far is the functor from being injective? In other words, how many twisted complexifications does a symmetric space have, if at all? Both problems remain open. Moreover, a closer look at the list of Makarevic [Ma73] reveals a very intriguing fact, namely: the Jordan-Lie functor seems to be very close to being surjective and injective! In other words: most symmetric spaces are Makarevic, and most of them admit exactly one twisted complexification. In fact, the number of twisted complexifications for the examples we know of is either 0, 1, 2 or 3 - this may be taken as an expression of a sort of equilibrium between the Jordan side and the Lie side. Both problems can be put in a purely algebraic form by introducing the infinitesimal version of the geometric Jordan-Lie functor which associates to a JTS Τ the LTS RT(X, Y)Z := T{X, Υ, Ζ) - T(Y, X, Z); we call it the algebraic Jordan-Lie functor JTS —> LTS. Again, it is an empirical fact (already remarked by E. Neher, cf. [N85]) that this functor is very close to being injective and surjective; the fundamental problem is how this can be understood. 0.6. Extension problem. There is a third fundamental problem, closely related to the existence problem. In a somewhat vague formulation: What is the relation between Jordan algebras and Jordan triple systems? More geometrically: How are the geometric problems of conformal structures (Section 0.3) and twisted complexifications (Section 0.5) related? We distinguish between Makarevic spaces of the first kind and of the second kind; the former are closely related to Jordan algebras, generalizing tube type Hermitian symmetric spaces which can very well be investigated by Jordan algebra methods, cf. [FK94]. In a similar way, all Makarevic spaces of the first kind have a nice structure theory based on Jordan algebras ([Be96c]). Classification sug-

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gests that spaces of the second (i.e., not of the first) kind always have an imbedding as a sort of "subspace" into a space of the first kind. A typical example is given by the group SO(2n + 1) which is of the second kind; it can be imbedded into the group GL(n + 1, R) which is of the first kind. Problem F. Does every Makarevic space of the second kind have an extension to a Makarevic space of the first kind? Again, the problem can be put in a purely algebraic way and then takes the form of an extension problem for Jordan triple systems (Section 3.2).

1. Prehomogeneous symmetric spaces 1.0. The Koecher-Vinberg correspondence. A symmetric cone is an open convex cone Ω in a Euclidean vector space V = Mn which is self-dual (i.e., equal to its dual cone Ω* = {x G V | (χ|Ω) > 0}) and homogeneous under the group G(Q) := {g e GL(V) I g-Ω, = Ω}. By a result of Koecher and Vinberg (cf. [FK94], Th. III.3.1), such cones correspond bijectively to Euclidean Jordan algebras: a Jordan algebra is a commutative algebra satisfying the identity (J2)

x2(xy)=x(x2y).

It is called Euclidean if the trace-form (*|_y) := tr L(xy) (where L (α) is the operator of left multiplication by a) is positive definite. Such a Jordan algebra has a unit element e. The connected component of e of the set of invertible elements is then a symmetric cone, and every symmetric cone is obtained in this way. When establishing this result, one makes the following important observation: the open orbit Ω = G(Q) • e is a symmetric space. We are thus lead to the concept of a prehomogeneous symmetric space: Definition 1.0.1. Let G be a closed subgroup of the general linear group GL(V) of a complex or real vector space V. If there exists a point e e V such that the orbit Ω := G · e is open in V, then we say that (G, V, e) is a prehomogeneous vector space (with base point). A homogeneous space G/H is called a symmetric space if Η is open in the fixed point group Ga of a non-trivial involution (automorphism of order 2) σ : G ->· G. Putting these two concepts together, we say that (G, σ, V, e) is a prehomogeneous symmetric space if (G, V, e) is a prehomogeneous space such that the open orbit G · e = G/H is a symmetric space with associated involution σ. Example 1.0.2. Let V be a complex or real vector space equipped with a nondegenerate symmetric bilinear form b and G C GL(V) be the subgroup generated by the group of isometries 0(b) and the non-zero multiples of the identity. Choose a base point e such that b(e, e) = 1. It is a fact of elementary geometry that the orbit G · e

Jordan algebras and conformal geometry

5

is open in V (if b is positive definite, G · e is the complement of the origin, and if b has signature (1,3), then the connected component of e of G · e is open the Lorentz cone). Let J be the orthogonal reflection with respect to Ke. Then the stabilizer Ge is essentially the fixed point group of the involution gXidy JgJ λ - 1 idv, (g 6 0(b)), and therefore G · e is a symmetric space. Example 1.0.3. Let V = M(n, IF), the space of η χ η-matrices over the field F of real, complex or quaternionic numbers. The group G = GL(n,F) χ GL(n,F) acts on V by (g, h) · X = gXh~l. The G-orbit of the identity matrix e = I is equal to GL(n, F) and therefore open in M(n, F). It is symmetric with respect to the automorphism (g, h) i-»· (h, g). Similarly, the subspaces of V of symmetric, resp. Hermitian matrices are seen to be prehomogeneous symmetric spaces.

1.1. The algebra associated to a prehomogeneous symmetric space. If (G, V, e) is a prehomogeneous vector space, the projection κ :G->V,gh-^g-e therefore the differential ic := Dic(e) : g

is open, and

V,

dt t= 0 is surjective. Here g c gt(^) denotes the Lie algebra of G c GL(V); it is identified with the tangent space TeG. The kernel of κ is f), the Lie algebra of the stabilizer Η of the base point e. If q is a complementary subspace of f) in q, then /c|q is a bijection whose inverse we denote by L : V —»• q. Then the action of g on V restricted to q yields a bilinear map (in other words, an algebra structure) Ae : V χ V

V,

(*, y) i-* xy :=

L(x)y,

and L(x) is the operator of left translation by χ in this algebra. The base point e becomes a right unit for this algebra: L(x)e = K(L(X)) = Χ for all x, whence xy = L(x)L(y)e. If we have [q, q] C f), then*)* — yx = [L(jc), L(y)] e e i) · e = 0, and the product will be commutative. Now, if (G, σ, V, e) is a prehomogeneous symmetric space, then we have the canonical decomposition q = t) θ q into ±l-eigenspaces of the differential σ = Dea of the involution σ. By definition of a symmetric space, the +l-eigenspace f) is just the Lie algebra of the stabilizer group H, and the — 1-eigenspace q will be used in the construction just described. The algebra structure Ae : V ® V —V thus obtained will be called the algebra structure associated to the prehomogeneous symmetric space (G, σ, V, e). It is commutative since [q, q] c f). This implies that e is also a left unit, i.e., L(e) = idy. Hence exp(XL(e)) = ek idv € G, and we thus see that all non-zero scalars belong to G in the complex case, and all positive scalars belong to G in the real case. Hence the open orbit Ω is a (in general non-convex) cone. Lemma 1.1.1. The map L : V -> q is H- and i)-equivariant. In other words, Η is a subgroup of the group Aut(V, Ae) of automorphisms of the algebra structure Ae, and f) is a subalgebra of the Lie algebra Der(V, Ae) of derivations of Ae.

6

W. Bertram

Proof. For h e H, h · e = e, and hence ic(h ο Χ ο h~l) = h(X · e) = h(k(X)), thus ic is //-equivariant. By derivation, it follows that ic is also [)-equivariant, and the same is true for L = /c _1 . Let us recall some basic notions related to algebras in order to explain the second statement. We identify Ae : V V —• V and L : V -»· q c Hom(V, V) under the canonical isomorphism Hom(V V, V) = Hom(V, Hom(V, V)). Recall from elementary representation theory that GL( V) acts in a natural way on these spaces such that they are both isomorphic to V* ® V* ® V as GL(V)-modules. The automorphism group Aut(V, Ae) of Ae is the stabilizer of Ae in GL(V); it is the group of invertible linear maps commuting with Ae, or (equivalentely) commuting with L. Its Lie algebra is the Lie algebra Der (V, Ae) := [X e gl(^) I XAe = 0} of derivations of Ae. Here XA = XoA —Ao(X(8)id -|-id is the natural action ofgliV) onHom(V(8) V, V). The condition X · A = 0 is equivalent to the rule X(uv) = (Xu)v + u(Xv), where uv = A(u ® υ), and to [X, L{v)] = L(Xv) for all u, ν € V. • The lemma implies that [L(V), L(V)] = [q, q] C f) C Der(V, Ae). Putting X = [L(x), L(y)] 6 Der(V, Ae), the condition [X, L(a)] = L{Xa) reads [[!(*), L(y)], L(a)] = L(x(ya) -

y(xa)),

or (LTA) x(y(ab)) - y(x(ab)) + b(y(xa)) - b(x(ya)) + a{x(yb)) - a(y(xb)) = 0 for all x, y,a,b e V\this identity is equivalent to the condition that [L(V), L(V)] C Der(V, Ae). A commutative algebra satisfying (LTA) has been called a Lie triple algebra by H. Petersson, see [Pe67]. It is known that (LTA) is satisfied in any Jordan algebra, but it does not imply the Jordan identity as show counter-examples given in [Pe67]; see also Exercise 1.2.6. However, a semi-simple Lie triple algebra is a Jordan algebra; this has first been observed by Vinberg [Vi60], cf. also Remark 1.2.8. Exercise 1.1.2. Derive explicit formulas for the product associated to Examples 1.0.2 and 1.0.3 and verify that these are indeed Jordan products. Exercise 1.1.3. Define morphisms of prehomogeneous symmetric spaces and show that the association of an algebra structure to such a space is functorial. Problem 1.1.4. Given a unital Lie triple algebra A, construct a prehomogenous symmetric space having this algebra as associated algebra. (On the infinitesimal level, take g to be the Lie algebra generated by L(V); on the global level, it may be necessary to pass to a universal covering.) When is this construction functorial? 1.2. Characterization of Jordan algebras. We denote by Alg(V) := Hom(V (8) V, V) = Hom(V, Hom(V, V))

Jordan algebras and conformal geometry

7

the space of algebra structures on V and by A l g i ( V ) : = H o m ( S 2 V , V) the space of commutative algebra structures. Since the L i e triple algebra Ae associated to a prehomogeneous symmetric space is invariant under the stabilizer Η of the base point e, we may transport it by the group G to any point of Ω = G · e and obtain thus a

G-invariant algebrafieldon Ω.

Identifying all tangent spaces of Ω canonically with

V , this field can be described as a G-equivariant map A :Ω

g-e^g-Ae

Alg,(V),

(It is easily verified that A(x)

= goAeo

g

g~l).

for χ = g · e e Ω is the algebra structure associated to

the prehomogeneous symmetric space ( G , cg οσ ocj1, by

V,

where cg is conjugation

in G . )

L e m m a 1.2.1.

The map A is homogeneous of degree

—1,

and so is the geodesic

symmetry ae

g · e i-> a{g) · e.

: Ω —*• Ω ,

The map λ = A ο ae : Ω

Alg,(V),

g • e

a(g)

• Ae

is homogeneous of degree 1. Proof We Xidy e G. Α(λχ) for all

χ

have seen in Section 1.1 that G contains the (positive) scalars. Let

g =

Then

= A(g-x) = gG Ω. Using that tf(Xidv)

A ( x ) = λ idy OA(JC) Ο ( λ - 1 i d v ® λ _ 1 i d y ) = λ - 1 Α (Χ)

a(L(e)) = —L(e),

we see that

= a ( e x p ( l o g ( X ) L ( e ) ) ) = e x p ( - log(X)L(