On the Fourier transform of a compact semisimple Lie ...

J. Austral. Math. Soc. (Series A) 56 (1994), 64-116

ON THE FOURIER TRANSFORM OF A COMPACT SEMISIMPLE LIE GROUP N. J. WILDBERGER

(27 May 1991)

Communicated by I. Raeburn

Abstract We develop a concrete Fourier transform on a compact Lie group by means of a symbol calculus, or *-product, on each integral co-adjoint orbit. These *-products are constructed by means of a moment map defined for each irreducible representation. We derive integral formulae for these algebra structures and discuss the relationship between two naturally occurring inner products on them. A global Kirillov-type character is obtained for each irreducible representation. The case of SU(2) is treated in some detail, where some interesting connections with classical spherical trigonometry are obtained. 1991 Mathematics subject classification (Amer. Math. Soc): 22 E 46, 58 F 06.

Introduction Classical harmonic analysis on the real line R and the torus T is largely the study of the Fourier transform. This is a map from functions on the group G to functions on the dual G, the set of unitary characters of G. It is given by the formula

JG

where is a function on G and x € G. Of course this is a formal definition whose meaning must be investigated for (p belonging to various spaces of functions (or © 1994 Australian Mathematical Society 0263-6115/94 $A2.00 + 0.00

64

[2]

Fourier transforms of Lie groups

65 A,

distributions) on G. The point here is that for these groups, the dual G is itself a topological group and the Fourier transform defined by (*) is a very explicit and computable object. The resulting theory has a well-known generalisation to locally compact abelian groups. For non-abelian groups, the situation is considerably more complicated. We will restrict our attention here to real Lie groups. Then the proper definition of the dual G is the set of equivalence classes of irreducible unitary representations. One then defines the Fourier transform for G by generalising (*). That is for a function on G and p e G one defines

(**)

4>\P)=

I(p(g)p(g)dg. JG Ai

This is then an operator-valued function of G, with the operators A(p) acting on (generally) different Hilbert spaces for different p e G. As such it also really depends on our choice of representation in each equivalence class of G. It is also clear that a necessary preliminary to any systematic study of the Fourier transform is a description of the dual G. This turns out to be a deep and difficult problem. For example, the dual of G = GL(n, R) has only recently been classified (see Vogan [21]), and the situation for general reductive groups is not yet understood, although progress has been made (see for example, Knapp and Speh [10]). Even with the dual G in hand however, the abstract nature of (**) is an obstacle to further study. It may not be possible to even formulate analogues of classical results from the abelian theory in any natural way. The purpose of this paper is to describe an alternate framework for the definition and study of the Fourier transform for G a compact connected semisimple Lie group. In effect we construct a new, geometric Fourier transform F for such a group and then show that it incorporates the abstract definition of (**). We may motivate this construction by going back to the Fourier transform for R or T and rewriting (*) as

(***)

• g*. Now if Q is the unit sphere in V and e : Q ->• PV the natural map, we define O = 4>' o e and call it the moment map of p. Our definition is actually more direct but equivalent. We study the image of $ and prove various functorial properties of it. These results are analogues of the functorial relationships between representations and co-adjoint orbits first discovered by Kirillov [9] for nilpotent groups. We show that the extremal set of Im is a single G-orbit Gp and that ~l(0p) = Jtp is a single G-orbit in £2. Furthermore 4> : Jtp —>• Op is an S1 bundle, which we call the canonical bundle of p (Proposition 4.9). In Theorem 4.11 we show that Jtp is an effective orbit. The association of the orbit Cp to each p e G is injective and is closely related to classical descriptions of G. If we fix a choice of positive roots, then 6P and Jtp can be considered to be the orbits of the highest weight and highest weight vector respectively. It also turns out that the canonical bundle of p is the same as that constructed in geometric quantization—this fact was communicated to us in conversation by I. Frenkel, to whom we are grateful. In Section 5 we use to transfer structure from Jtp to Gp. We show how the canonical 2-form on £?p and the connection 1-form on Jtp are related to the complex structure of V. We define the space of functions Ap on Gp. In Section 6 the main definitions concerning the Fourier kernel e and the Fourier transform F are introduced. The main results here have already been described. In Section 7 we turn in some detail to the case of G = SU(2). Here the

[6]


69

integral orbits are spheres in g* of radii I = 0, 1/2, 1, 3/2, We compute the Fourier kernel e of G in Theorem 7.5 and discover that it is closely related to the Cayley transform for G. At this point we remark that in the thesis Wildberger [22] we have constructed a similar Fourier transform theory for nilpotent Lie groups using entirely different methods (see also Arnal and Cortet [2]). For the group of real upper triangular matrices we found that the Fourier kernel also involved the Cayley transform. This phenomenon has been investigated for a wide class of nilpotent groups in Howe, Ratcliff and Wildberger [7] where it is shown that the role of the Cayley transform is closely related to the theory of the oscillator representation of the symplectic group. We also remark that our Fourier transform theory coincides in the case of SU(2) with that introduced by Sherman [15] in his work on Fourier analysis on spheres (see also Helgason [6]). In Theorems 7.12 and 7.13 we give formulae for the triple-kernels of the *-products on the integral spheres. These are related in a surprising way to certain classical identities of spherical trigonometry, and shed new light on the *-products of Moreno and Ortega-Navarro [14] as well as on Sherman's work. More generally, our theory provides an explicit construction of a *-product on any integral co-adjoint orbit of a compact semisimple Lie group. It would be of interest to compare this with the general construction of •-products on symplectic manifolds studied by Lichnerowicz [13], Gutt [5] and others. This paper is meant to be largely self-contained, the major requirement of the reader being a familiarity with elementary representation theory, for which references are Varadarajan [18] and Helgason [6]. This paper has existed in preprint form for some years now. A different version of our geometric Fourier transform construction has since appeared in Figueroa, Gracia-Bondia and Vaxilly [4] (see also Varilly and Gracia-Bondia [19], Arnal, Cahen and Gutt [1]). Additional work on the relationship between representations and moment maps has appeared in Wildberger [23,24] and Arnal and Ludwig [3].

Section 1 Let G be a compact Lie group and (V, p) an irreducible unitary representation of G. The Hermitian form on V will be denoted by (, ), and will be taken to be linear in the first variable and conjugate linear in the second. Let dim V = n. Let ./#be a non-zero orbit in V, so that there exists v ^ 0 in V with M =

70

N. J. Wildberger

[7]

{g • v\g e G}. Here the use of p is suppressed, so that g • v = p(g)(v). The orbit ^ c a r r i e s a G-invariant measure. LEMMA 1.1. The G-invariant measure dm on j^may for any v &V,

(1.1)

be normalised so that

v— I {v,m)mdm.

PROOF. Consider the operator A e End V defined by A(v) = / {v, m)mdm for v € V. Then for any g e G, A(g • v) == fI (8 (g•• v,v,m)mdm

L

= I (v, g =

-m)mdm

I (v, m)g • mdm JM

= g-A(v). Thus A commutes with all g e G and so by Schur's Lemma is a multiple of the identity. By an appropriate normalisation of dm we may choose the multiple to bel. Fix the measure dm on ^ # s o that (1.1) holds. Introduce L 2 (^#, dm) with inner product (, ) 2 . Thus for fa, 2 6 L2{M, dm), 2(m)dm.

For v e V, let t; be the function on Jtdefined by v(m) — {v, m). Let V be the space of all v, v e V, and let ~ : V -*• V be the map that sends v to v. LEMMA 1 . 2 . " : V - > V C l?(Jt, dm) is an isometry.

[8]


71

PROOF. Since (V, p) is irreducible, v = 0 if and only if v = 0, so ~ is an isomorphism. If vuv2 e V, then by Lemma 1.1,

{v\,v2)=

I

=

(vum)(m,v2)dm

(t>l, t>2>2-

DEFINITION 1.3. For T e End V, define KT : Jt x J( -+ C, the kernel of T with respect to ^ , by the formula

KT(m',m) = {T -m',m) for m', m e JK. LEMMA 1.4. For TX,T2& End V andm", m € Jt, KT2Tl(m",m)=

I

KTl(m",m')KT2(m',m)dm'.

PROOF. For T e End V, let f be the integral operator on L2(^, kernel KT. That is, for e L2{J%, dm) and m e ^#, f(/)(m)=

I

KT{m',m)(t){m')dm'.

Then for v e V and m e Jt,

fv(m) = [ (T • m\ m)(v, m') dm' =

(T-v,m)

Here we have used Lemma 1.1. Now let V x = {€ L2(J?,dm)\{„ is a function Ba e W0) such that for all wx, w2 e W wx x w2{m) = I I Ba(m,

mi,m2)wi(ml)w2(m2)dm1dm2-

Ja Ja

LEMMA 2.6. Suppose as before that {eu ..., ep} is an orthonormal basis of W with respect to {, )„ and that for 1 < /, j < p, e, x ej = k=\

Then a triple-kernel for the algebra (W, x) with respect to { , )a is given by Ba(m,mum2)

= ij.k

Furthermore it is unique (so does not depend on the choice of basis). PROOF. Let wi,w2

P

e W with wi = £]f=1 Oi-e,-, w2 = 5Zf=i ^ W with respect to { , )2 w the function N € W given by PROPOSITION

(2)

N(m',m) = $m',m)\2. PROOF.

We must show that N(m', m) e W(2) and that for all w € W and

rj(w)(m) = I

N(m',m)w(m')dm.

JM

Now for fixed m' G ^ t h e operator Tm/, € End V defined by Tm,•) = 7] e End V, i = 1,2. Let = r2Ti and let Kt be the kernel of 7] with respect to J(. By Proposition 1.16

= I {Tim,m)((n + l){mi,m){m,m2) -

{mum2))dm

for/ = 1,2. Thus K3(m",m) = • (T2m2,m2) ((« + ${m',m2) (m2,m) — {m',m)) dmxdm2dm! wx(mx)w2(rn2)((n +

=

l)2(m",mx)(mx,m2}(rn2,m)

JMJM

- (« + l)(m",mx)(mi,m) - (n + I)(m",m2){m2,m) + {m", m))dmxdm2. Thus wx x w2(m) = K3(m,m) =

wx(.mx)w2(rn2)B2(m,mx,

where B2{m,mum2)

= (n + \)2{m,mx){mum2){rn2,m)

- (n +

\)\{m,ml)\2

-(n + l)\(m,m2)\2 + \. In contrast, the triple-kernel B{ has a simple form even in the general case. THEOREM 3.11. Let JCbe an effective orbit with \J%\ = 1. m,nti,m2 e J(, Bi(m,mum2)

=

{m,mx){mx,m2)(m2,m).

Then for

88

N. J. Wildberger PROOF.

[25]

Let TuT2e End V. Then for v eV,

(T2TiV,v)—

I {Tiv, /n)(r 2 /n, v) dm

-11

~l(Tl)muml)(v,m1){ml,m)dml • /

— I I

(v~l(T2)m2,m2)(m,m2)(m2,v}dm2dm

(r)~1(Ti)m1,mi)(r)~l(T2)m2,m2)(v,mi) • (mx, m2)(m2, v) dm\dm2-

Thus for w\, w2 e W, wlxw2(m)=

I

v-\wx)(ml)rj-\w2){m2)(m,ml)(mum2)(m2,m)dmldm2.

But for w, w' € W, (f)~\w),w')2 = (w,w')u so u>i x w2(m) —If

u>l(m1)w2(m2)(m,nii)(m1,m2)(m2,m)dmldm2

for all wu w2 e W. Thus Bl(m,mum2)

= (m, mx)(mx, ni2}(m2, m).

Section 4 Let G be a compact Lie group and (V, p) a finite-dimensional unitary representation of G (not necessarily irreducible). Let Q = [v e V\(v, v) = 1} be the unit sphere, and suppose dim V — n. Let g be the Lie algebra of G and g* its dual. We will continue to suppress the notation p, so that for X € g and v & V, p(X)(v) will be denoted simply by X • v.

[26]


89

D E F I N I T I O N 4 . 1 . : £2 - > g*

v ^

(v)(X)

=

-{X-v,v)

i

for v € £2, X e g, is the moment map of (V, p). LEMMA 4.2. w a G-map between Q and g*, where G acts on g* via the co-adjoint action Ad*. PROOF. For g e G,

I

j(g-l-(X-(g.v)),v)

=

-{Adg~l(X)-v,v)

= i

= (v)(Adg-l(X)) = Ad*g(^>(v)). Let Im ! and 4>2 respectively. Let V = Vi © V2 a«2 respectively. Let V — V\ ® V2 and p = P\® fa the representation of G x G = G' given by p(g\, g2){v\ v2) = (gi • Vi) ® (gi • v2), g\, g2 G G and V\,v2 & V, with moment map . Then Im c f)* x g* satisfies x Im2 C Im c conv (Im^!) x conv (Im2) where conv (5) means the convex hull of S. PROOF. Let £2i, Q2 and f2 be the unit spheres of V\, V2 and V respectively. Thenforanyu! e Qu v2 € Q2, v = V\®v2 e Q so that for any (Xu X2) e g x g ,

X2) = r ^ C Z i • W,) ® V2 + U! 0 (X 2 • V2), Vi ® U2> '

, v2) + (uj, Ui)(X2 • v2, v2))

Thus Im i x Im $2 C Im O. Now let {vu ..., vn] be an orthonormal basis of Vi. Then if v € S2, we may write v = YH=i vk ® wk where iyt e V2 with 2 = 1. Then n

ir + Vic (X2 • Wi), Vi

so that n

• i)* be the projection dual to the inclusion i : t) ->• g. IfH '• & —*• b* is the moment map of(V, p\H) then H = p o . PROOF. Obvious from the definitions. In general Im cj> need not be convex. For example, if G = SU(2) and (V, p), is the standard two-dimensional representation of G, then Im $ is easily seen to be a single G-orbit in g*, namely a 2-sphere. However we have the following. LEMMA 4.6. Let T be a torus with Lie algebra t. Let (V, p) be a finite dimensional representation ofT with weights ik\, ..., ikn G it*. Then Im is the convex hull o / ( l i A , ) C t ' . / n particular ImO is convex. PROOF. Write V = Vi © . . . 0 Vn with V, the one-dimensional representation with weight ikj. That is, for X € t and Vj € Vj, X • vj = ikj(X)vj. Now if ; is the moment map of V, and if vj e Qj, the unit sphere in Vj, we have U j j ) Thus Im 7 = kj, and the result follows from Lemma 4.3 by induction. Now let T C G be a maximal torus with Lie algebra t. Choose a G-invariant positive-definite form (, ) on g. Let /A-i,..., ikn e it* be the weights of (V, p) restricted to T and let D c t* be the convex hull of {ku ..., kn}. Then if p : g* —>• t* is the projection, Lemmas 4.5 and 4.6 imply immediately that p(Im ) = D. Denote by Ext (D) those points of D which are not contained in any open line segment in D. Clearly Ext (D) is a subset of {ku . . . , kn}. It will be useful to sometimes identify g with g* via (, ) and thus to consider t* c g*. LEMMA 4.7.

If f e ImO c g* and p(f) = ke Ext(D), then f = k.

PROOF. Write f = k + n with \x e g*, / i ( t ) = 0. Let 6 be the co-adjoint orbit through / and let X • f e TG{ with X e g. Then dp(X • / ) is a tangent

92

N. J. Wildberger

[29]

vector to t* at p(f) = A.. Now since p(0) C D, any curve in ^through / with tangent vector X • f is sent by p to a curve in D through X € Ext (D). But any such curve must be singular since D is a solid convex polyhedron in t* with a finite number of vertices and A, is such a vertex. Thus p(X • f) = 0, so that for all Z € t, X e g, X • f(Z) = 0. Now a basic fact about Lie algebras of compact groups is that they are reductive. This implies that g = t© [t, g], an orthogonal direct sum with respect to the form ( , ). Thus since we can rewrite the above as (A. + fi)([Z, X]) — 0, we get /x([Z, X]) = 0 for all Z e t and l e g s o that /x = 0 and / = A.. LEMMA 4.8. Let X e Ext(D). Then v € ft is a weight vector for T of weight iX if and only if(v)) = X so that by the previous lemma, (u) = A.. On the other hand let v e ftwith3>(u) = X. If {u 1 ; ..., vn) is an orthonormal basis of weight vectors with Vj of weight iXj then we can write v = YH=i aiv' with J21=i l«/l2 = 1. Then if Z e t, { Z )

w=i

Y\aj\2x\(Z). But if (v) = A. then X = Yl"=i \aj l%- N o w s i n c e ^ e E x t ( ° ) a11 t n e ai be zero except for one and so v is actually a weight vector of weight iX. We remark more generally that if F C D is a face of D and

then the set of v e ft for which p( 0 are integers not all zero. Then A(p) contains a unique highest weight /Xi with the property that /xi > fi for any other weight fi € A(p). It is the extension to t) of the functional ik\ on t, for some A-i € Ext (D). For each /x e A(p), we denote the weight space of /x by VM so that

For ui, v2 e V, define 7;,,^ e End V by

For /x, /x' € A(yo) define f/MiM' c End V to be the span of all TVtV> with v e V^, u' e V^'. Then clearly End V - 0 M , M ' eA (p)^,M'The spaces C / ^ can be partially ordered by setting U^,^ > £/,,,< if /x > ?j or if /x = ?? and /x' > rj'. Let ^ be the representation of G on End V by conjugation so that for g € G, T e End V and u e V, J r ( « ) ( r ) u = g • (T • (g-1 • v))

while if X e g then TT(X)(7> = X • T • v -

T • X • v.

F o r a € A + , v e V^, u' 6 V^-, iw e V, jr(Xa)Tu,A*>)

= X* • (w, v')v - (Xa • w, v')v = (w, v')Xa • v — (w, Ya • v')v = TXa.v,v-{w) -

Thus (4.1) Similarly so that

Tt{Xa)U^

TvJ

C tV*./.' © ^/

[32]


95

Now let U = {T eEndV\{Tm,m)=O V m s i , ) . We must show U = 0. Clearly U is invariant under n{G) since jftp is a G-orbit, so it is also n(g ) invariant. For any T e U, we may write T as

where 5MM' e f/^,M'. Consider the set of all spaces U^^' for which there exists a T € U with non-zero component 5^M< in f/M,M-. We will show this set to be empty. If not, let Un^ be a maximal element, so that there exists T € U with 1

— "1.1 ^

/ . "/i H,n'€A(p)

where 0 ^ S,,,,- e £/„,,- and SMiM- e t/MiM' for all /i, /LI' e A(p). We claim that r\ = /;' = \i\. For suppose first that fit > rf. Write "1.1' ~ 7 , •*i«;.u>,'

where [w[,..., w's] is a basis of V,' and HI, e V,, with say w\ # 0. Then since (V, p) is irreducible, we can find a e A + such that Xa • w\ ^ 0. Thus

has a nonzero component in [/,,+„, ^ since {iy'p . . . , u;^} is linearly independent. Combining this with the fact that none of the elements 7r(Xa)5/ii/t can have a component in Un+a^ (by the maximality of t/,^ and (4.1)), wefindthat n{Xa)T has a non-zero component in Un+a
l

g'ftg'- )\

•

= fi are arbitrary elements of 6t and

m, = g • mt, m2 = g' • mt with (/w,) = / • . COROLLARY

(a) (b) (c)

7.8. Ifmum2

e Jtt and (/«,) = / e Ot, i - 1,2, /Ae/i

| (mi, m 2 )| 2 = 1 if and only if fx = f2; | (m,, m2) |2 = 0 i/and on/y jy / , = -f2; |(m,, m 2 )| 2 = (1/2)2* if and only if (/,, / 2 ) = 0.

PROOF. These statements follow immediately from the previous proposition and the fact that for yi, y2 e Gt

-i2(X) e At, then

X * (X) -

(X2-(2ir)2)

d

2£

dX