Unitary Representations and Coadjoint Orbits of Low-Dimensional Nilpotent Lie Groups

Table of contents :
Contents
Preface
Guide to the Reader
Chapter 1. Groups of Dimensions 3, 4, and 5
G_3
G_4
G_{5, 1}
G_{5, 2}
G_{5, 3}
G_{5, 4}
G_{5, 5}
G_{5, 6}
Chapter 2. Groups of Dimension 6
G_{6, 1}
G_{6, 2}
G_{6, 3}
G_{6, 4}
G_{6, 5}
G_{6, 6}
G_{6, 7}
G_{6, 8}
G_{6, 9}
G_{6, 10}
G_{6, 11}
G_{6, 12}
G_{6, 13}
G_{6, 14}
G_{6, 15}
G_{6, 16}
G_{6, 17}
G_{6, 18}
G_{6, 19}
G_{6, 20}
G_{6, 21}
G_{6, 22}
G_{6, 23}
G_{6, 24}
Appendix
References

Citation preview

UNITARY REPRESENTATIONS AND COADJOINT ORBITS OF LOW-DIMENSIONAL NILPOTENT LIE GROUPS

by

Ole A. Nielsen

Queen's Papers in Pure and Applied Mathematics - No.

Queen's University Kingston, Ontario Canada

198 3

63

Copyright

1983

This -book, or parts thereof, may not be reproduced in any form without written permission fro~ the author.

CONTENTS

PREFACE GUI DE TO THE REA DER ·•

V

• viii

CHAPTER 1

' 1

CHAPTER 2

21

APPENDIX

114

REFERENCES.

117

.PREFACE

In studying nilpotent Lie algebras and groups the author has on va~ious occasions found it useful to examine low-dimensional examples.

The useful data con-

cerning a given nilpotent Lie grotip and its Lie algebra was, roughly speaking, that which enters into the construction of the irreducible unitary represeritations of the group using Kirillov's method of coadjoint orbits in the dual of the Lie algebra. relevant data for

a number

After calculating the

of examples the author

decided that it might be useful to proceed more system' atically and to make this data available to other

mathematicians.

The present volume is the outcome of

this project. The nilpotent Lie algebras of dimension 5 or less were computed by J.

Dixmier in 1958 [1], and he

even found, for each of these algebras, a corresponding connected and simply-connected Lie group and all of its irreducible unitary representations.

But as these cal-

culations were done before the advent of Kirillov theory they did not include the coadjoint orbits, for example, and so these algebras and groups are inclµded here as Chapter 1. -

V

-

The 6-dimensional nilpotent Lie algebras have · t·1mes, by K.A. Umlauf in been computed at least four

1891 [6], by

v.v.

Morozov in 1958 ['3], by M. Vergne in

1966 [7], and by T. Skjelbred and T. Sund in 1977 [5]. To the author's knowledge, however, the corresponding connected and simply-connected Lie groups, let alone their irreducible unitary representations, have not been computed.

The main part of the work which resulted in

the present volume consisted in taking each of the known 6-dimensional nilpotent Lie algbras and calculating a corresponding connected and simply-connected Lie group and its coadjoint orbits, irreducible unitary representations, and related data. of this data.

Chapter 2 contains all

The isomorphisms which exist between the

6-dimensional nilpotent Lie algebras listed in Chapter 2 and those computed by Morozov [3], Skjelbred and Sund [5], Umlauf [6], and Vergne [7] are indicated in an appendix. The list contained in Chapter 2 as well as those in

[5] and [7] consists of mutually non-isomorphic algebras none of which are isomorphic to products of lowerdimensional algebras; this is not the case for the lists contained in [3] and [6], however (cf. the appendix).

- vi -

It is a pleasure to thank a number of people and one organization for the valuable help they provided during the preparation of. this volume:

Barbara Wyslouzil

for computing the coadjoint orbits in dimension 6 during the spring of 1980; Sally Cockburn for carrying out many of the calculations needed for Chapter 2 and the Appendix during the summer of 1982; the Canadian NSERC for providing the funds (through both an operating grant and the Summer Undergraduate Award Program) which made it possible to hire Barbara Wyslouzil and Sally Cockburn; T.W. Hawkins, Jr.,

A.T. Lau, and T. Sund for providing

the author with copies of

[6], [3],

and

[5],

respectively;

and, finally, Marge Lambert for typing this volume with her usual skill.

Ole A. Nielsen

Kingston, Ontario November, 1982

- vii -

GUIDE TO THE READER

The data which has been computed .for each of the examples under consideration is presented in eight sections, as follows: (1)

defined by giving a Jordan-Holder basis

OJ- ,

OJ,, ,

n = dim

where

The Lie algebra is also given a

isomorphic to

Here a Lie group

OJ, G

x 1 , ... ,Xn

11

[ Xj, Xk] , 1

for

is given.

G

< j ~2+(!;3+x2!;1)f3+(!;4-x5s1)f4

+ (!; 5 + x4';1 )f 5

- 6 ( 6)

f

= 1;1 f 1 + •.• ·+ I; 5f 5

(a)

~1

Io :

~(f)

=

'JRX1

d=4; (b)

(7)

j 1 =2,

II'

~l = 0

.•

OJ-(f) =

·OJ,

f=I;· 1 f 1

j

2 =3,

j 3 =4,

j

4 =s

+··· +!; . 5f 5 r; 2

=···=s 5 =o

can be realized on

/-ro2) 12~

by

[nf(x1 , ... ,x 5 )~](s,t)

= [exp2ni(xl (b)

';,l =O .

II'

nf

(8)

-xzs-x4t)s1J~(s-x3,t-x5)

is the character given by

w=giER:s 1 ;f=o} 8 ( 0, ... , 0) =

J t rn S W

( 8 )s ~ dA.l ( Sl)

f

1 1

GS,2 ( l).

OJ =

o/5, 2 ==

JR.Xl

+ .. • +

JRX 5

-

(2)

G=Gs,2=

7 -

:Rs~

( xl' ... 'x 5 )( Y1' ••• 'y 5)

=,= (

xl + y 1 + xsy 3' x2 + y 2 + x 5Y 4' x3 +y3,x4 +y4,x5 +y5)

(xl, • • • 'X 5)-1 = (-xl + X3X 5' ""."X2 + X4X5' -X3' -X4' -x5) (3

r

b

X .=--

ox1 ·

1

X

=x

3

X =x

4

X

(4)

5

b

b

-+-

5 QXl

.

ox3

o b ox +ox-

5

4

2

=L bxs

exp(a 1 x1

+a 5x5 )= (a 1 +~a 3 a 5 ,a 2 +;a 4 a 5 ,a 3 ,a 4 ,a 5 )

+···

log(x1 , •.. ,x 5 ) = (x1 - ~x 3 x 5 )x1 + (x2 + x3X3 (5)

-½x4 x 5 )x2

+ x4X4 + x5X5

Ad(x1 , ••. ,x 5 )(a 1 x1 + • ·· +a 5x5 )

=

(al+ x5a3 - x3a 5)Xl + (a2 + x5a4 - x4a 5)X2 + a3X3 + a4X4 + a 5X5

1~

.

.

Ad ( xl' • • • ' x 5) ( ~ l f 1 + .. • + ; Sf 5)

= ; 1 f 1 + I; 2 f 2 + ( I; 3 ... x Sr; 1) f 3 + (!; 4 - x 5S 2 ) f 4 + (!; 5

+xl;2 +x3f;l)f 5

- 8 ( 6)

f=I; (a)

1.

+···+!; f

5 5

e; 1 ~o: ~(f)= :RXl + JRX2+ JR.(1;2X3-1;1X4) d=2;

j 1 =3,

·. ·

P4 (tl' t2) = (b)

1; 1 =O,

OJ-( f) = d=2;

(c)

=5

s2 ( 1;21;3) ½ tl + e;4 - ½ .

e; 2 /o: JR.XI + JRX2 + JRX3 j 1 =4,

j 2 =5

s1 =1; 2 =0 OJ(f)=

( 7)

j 2

OJ,

f=I; f +···+t; f 1 1 5 5

(a)

e: 1 ,/=o,

e:- 3 =~~ 5 =0 2

L (JR)

can be realized on

by

[rrf(x 1 , ... ,x 5 )cp](t) = [exp2rri((x 1 -- x 3 t g 1 + (x2 - x4t)e;2 +x41;4)]cp(t - xs) (b)

e; 1 =o, nf

s2 t=o, s4 =s 5 =o 2

L (JR)

can be realized on

by

[ n f ( x 1 , ... , x S) cp J (t )

= [ exp2ni ( (,x 2 (c)

X

4t

g

2

+ x 3 ; 3 )] i:p( t -

e;l =;2=0

nf

is the character given by

X

5)

-

(8)

w=={(e; 1 ,i; 2 ,e; 4 )E

eco, . . . , o > = J

W

:R. 3 :

t rrr r:

f

9 -

e; 1 fo} +r: f

~1 1 ~2 2

+r: f

~4 4

(

e >I e; 1 I d11. 3 ( e; 1 , e; 2 , s

4

>

GS,3 (1)

01 7

= fJ1,, = ~ 5,3

JRX

1

+ ••• +

JR.X

5

[XS' x4 J = x2 (2)

G= G

5, 3

= lRS

.

· 1 2 ( xl' • · · ' x 5 )( y l ' · · · ' y 5 ) = ( xl + y l + x 4 y 3 + x 5Y 2 + Zx 5Y 4'

x2 + y 2 + X: 5Y 4' x3 + y 3' x 4 + y 4' x 5 + y 5) (x1,•••,x5)

-1

· 1

2

= (--xl +x2x5+X3X4 -zX4X5,-X2 +x4X5' -x3,-x4,-x5)

(3)

X =x

.3

( t - x 5 ) ( d) . !;

= !; 2 = !; 3 = 0

l

rrt'

is the character given by rrf (x1 , ••• ,

(8)

x 5 ) = exp2rri(x 4 1; 4 + x 5i; 5 )

. . 3 . W={(!;l,e;2,!;5)E :R: l;l;fo} e(o, ••• ,o)=J trrr!; f

.

W

1 1

:RXl

+... +

+1; f

+1; f

2 2 .

5 5

(e)l1;1ld°A.3(1;1,1;2,1;5)

. .

GS, 5 (l )

l J, = OJ'5, 5 =

]RX 5

[X5,X4]=X3

( 2)

G= G

( Xl'

=

= :JR. 5

5, 5 0

•

(

•

'

X

5 )( yl'

0

•

•

'

y 5)

1 2 · 1 3 1 2 xl +yl +x5y2 +2x5Y3 +~5Y4,x2 +y2 +x5Y3 +,2X5Y4, X3 +y 3 +

X

5Y 4' X 4 + y 4' X 5 + y 5)

- 15 -

3 l 2 4 x 5' -x2 + x 3 x 5 - 2x 4 x 5'

-x3 + x4x 5' -x4' -x 5) (3)

X

1

=L ()XI

X = 2

_b_+g__

X

5 oxl

bx2

X 1 2 g__+x L+_o_ · 3 = ? 5 b :x:1 5 bx 2 b x 3 1 3 b

X4

= 6X 5

C)

1 2

c>

xi + 2x 5 b x2 + X 5

_c>- + L C) X 3

C) X 4

C) .

X5=bx 5

( 4)

exp (a 1 x1

+ · · · + a 5x5 )

1 1 2 1 3 1 1 2 = ( a 1 + 2a 2 a 5 + 6a 3 a 5 + 2 4 a 4 a 5' a 2 + 2a 3 a 5 + 6a 4 a 5' 1

a 3 + 2a 4 a 5' a 4' a 5) 1 1 2 log(xl'. •. ,x5) = (xl - 2x2x5 + 12x3x5)Xl 1

1

2

1

+ (x2 -2x3x5 + 12x4x5)X2 + (x3 - 2x4x5)X3 + x4X4 + x5X5 (5)

Ad(x1 , ••• ,x 5 )(a 1 x1 +··· +a 5x 5 ) 1 2 1 3 = (al+ x5a2 - x2a 5 +2x5a3 + 6x5a4 )XI 1 2 + (a2 + x5a3 - x3a 5 +2x5a4}X2 + (a3 + x5a4..,. x4a 5)X3

+ a4X4 +a5X5

- 16 (6 )

f =

(a)

s1 f 1 + · · · + s 5f 5 ~ 1 ,/=o: o;}(f)= :RXl + JR(!;2X2-1;1X3)+ JR(E;3X2-!;1X4)

(b)

s1 =O, s2 ,/=o OJ( f ) =

JR.Xl + JRX2 + JR ( e; 3 X3 j 1

d=Z;

:

=3,

j 2

S2 X4 )

=5

P4(t1,tzl~z~z t~+(~4~2:D (c)

;1

=s 2 =0, s3 ,/=o:

0}( f) = JR.Xl + JRX2 + JRX3

( d)

S1

=!;

2

{Jj(f) =

(7 )

r

=

(a,)

=!;,.,=O ,j

°r

sif i + . . . + ssf s s1 ,/=o, s2 =s 5 =o nf

can be realized on

1 2 (JR)

by

- 17 [nf(x1,···,xs)cp}(t)

·.

.

1

= [exp2ni((x1 -x 2 t+ 2x 3 t

2

1 3 - 6 x 4 t )r; 1

+ (x3 - x4t)!;3 +x4f;4)]cp(t - xs) (b)

i; 1

1hfo,

=o,

s 3 =s 5 =o

can b e rea 1 ize d on

"f

L 2 t-ro·) \..l.l'-

by

[nf(x1,···,xs)cp](t)

= [exp2rri( (x 2 - x 3 t + ;"'4 t 2 g 2 + x 4 i; 4 )]cp(t - x ) . 5 (c)

s1 =s 2 =o, s3 fo, s4 =s 5 =o "f

can be realized on

L 2 (:m.)

by

[nf(x1 , ••. ,x 5 )cp](t)

= [exp2ni((x3 - x4t)s3)]cp(t - xs) (d)

s1 =s 2 =r;3 =o nf

(8)

is the character given by

3 w={(!; 1 ,r; 3 ,s 4 )E :m.:

e