Representations of Semisimple Lie Groups. III. Characters

Representations of Semisimple Lie Groups. III. CharactersAuthor(s): Harish-ChandraSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 37, No. 6 (Jun. 15, 1951), pp. 366-369Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/88201 .

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366 MATHEMATICS: HARISH-CHANDRA PROC. N. A. S.

REPRESENTATIONS OF SEMISIMPLE LIE GRO UPS. III. CHARACTERS

BY HARISH-CHANDRA

DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY

Communicated by P. A. Smith, April 23, 1951

We shall adhere strictly to the notation of the preceding note.' Making use of an unpublished result of Chevalley one can prove the following theorem.

THEOREM 1. Let Xr be a quasisimple irreducible representation of G on a

Hilbert space >. Then there exists an integer N such that

dim MS < N(d(S))'

for any i) e 2. Moreover if 3) { {0} for some 2 e ?,2 then it can be shown that we

may take N equal to the order of the Weyl group W. Let xr be as above and let Cc (G) denote the class of all complex-valued

functions on G which are indefinitely differentiable everywhere and which vanish outside a compact set. Let A be a bounded operator on ~. We

say that A has a trace if for every complete orthonormal set2 (41, '2, ..., fn,, ...) in ! the series . (g4/, A4Z) converges to a finite number inde-

i>l

pendent of the choice of this orthonormal set. We denote this number by spA. Let A* be the adjoint of A. We say that A is of the Hilbert- Schmidt class if AA* has a trace.

THEOREM 2. Let 7r be a quasisimple irreducible representation of G on

a Hilbert space 9. Then for any f e Cc'(G) the operator' fGf(x)r(x) dx has a trace. Put

T,(f) = sp(fGf (x)x(x) dx)

and for any a e G let afa denote the function afa(x) =a f(a-xa) (x e G). Then

T, is a distribution in the sense of L. Schwartz4 and

T.(afa) = Tr(f) (f e CC,(G), a e G).

We shall call the distribution T, the character of the representation 7r.

THEOREM 3. Let 7rl and 7r2 be quasisimple irreducible representations of G on two Hilbert spaces. If T,, = cT,,(c C) then -Tx and T2 are infinitesi-

mally equivalent. Conversely if 7r1 and r2 are infinitesimally equivalenl T = T,.

Since for irreducible unitary representations infinitesimal equivalence is the same as ordinary equivalence, such a representation is completely determined within equivalence by its character.



VOL. 37, 1951 MA THEMA TICS: HARISH-CHA NDRA 367

THEOREM 4. Let mr be a quasisimple. irreducible representation of G and let f be any measurable function on G such that f vanishes outside a compact set and fG cf(x)l2 dx < oo. Then the operator fGf(x)r(x) dx is of the

Hilbert-Schmidt class.

For any X e go and f e Cc6 (G) put

(*Xf)(x) {df(exp( tX) )} dt I=o'

Then the mapping X -- *X is a representation of go on Cc' (G) which can

be extended to a representation b -> *b (b e 3) of 3. Let <p be the anti-

automorphism of *3 such that 50(X) = -X (X e g). If T is any dis-

tribution on G we define bT (b e 3) as follows:

bT(f) = T(*(p(b))f) (f Cc'(G)).

Moreover for any function f on G we denote by yf, f, and yfz (y, z, e G) the

functions

yf(z) = f(y-lx), fz(x) = f(xz), yfz(x) == f(y-lxz) (z G).

Let Z denote the center of G and let 7r be a quasisimple irreducible representation of G on a Hilbert space. Let x be the infinitesimal character

of 7r and let X be the homomorphism of Z into C such that 7r(a) =- (a)7r(l)

(a e Z). Then if T, is the character of mr it is easily seen that

zT, = x(z)rT (z e3)

T,r(yf) = T,((f), T,(fa) a= (a)T,(f) (f C,c(G), y eG, a E Z).

Now put5 MI1 MZ and let x -> x* denote the adjoint representation of G. Put (x)Y* = yxy-'(x, y e G) and let Cc(G) denote the class of all

continuous functions on G which vanish outside a compact set. Let a

and g be continuous functions5 on A+ and M1, respectively. Put6

T(f) = ff((n-lh-lm-l)u*)a(h)g(m) dm dh dn du* (f e C,(G))

Here dm, dh, dn, du* are the left invariant Haar measures on MI, A+, N, K*, respectively, and the integral extends over K* X M1 X A+ X N. It

can be shown that T(f) = T(yfy)(y e G). Let a be an irreducible representation of M1 on a finite-dimensional Hilbert space U. Let 8 be the

equivalence class of the representation -o of M defined by a. Let 40o ' 0 te an element in U belonging to the highest weight X of -o and let 7 be

the homomorphism of Z into C such that a(a) = r(a)a(l) (a e Z). Put

g(m) = (ano, a(m-l)4o) (m E Mi) and a(h) = e-(v+2p)(log h) where v is a linear

function on D1 and log h is the unique element in t)0 such that exp(log h) h. If we regard T as a distribution it is easily seen that

zT = xA(z)T (z T 3)



368 MATHEMATICS: HARiSH-CHANDRA PROC. N. A. S.

where A(11, + H2)-= v(Hi) -+ XB(II2) (Hi e br, H2 e )?). Moreover

T(fa) = a (a)T(f) (a E Z, f e Cc(G))

and it is easy to check that

T(f) = ffn((n --h-m-')"*)e-(" +2)(Iog h) (m_-) dm dh dn du* (f E Cc(G)) where

(m) a jo0i(

l2 spo(m)

and d(6) is the degree of o. Let A_ be the analytic subgroup of M corre-

sponding to btj,. Put A = A+A- and Al. = AZ. Let V be the set of all elements in G which can be written in the form xyx-~ with x e G and

y e A,N. We shall say that an element y e V is regular if y = xhx-~ for some x c G and h e A1 and y* has exactly5 1 eigen-values equal to 1. Let V0 be the set of all elements in V which are regular. Then V0 is open in G and V is the closure of V0. Let Wo be the subgroup of the Weyl group W consisting of those elenments s e W for which there exists an x E G such that sHi = x*II for all H e tf. It is easily seen that every s E Wo leaves both btl and ba invariant. Put

A-() H (ea(tt) - e- a(H)), A+(H) II (eam - e-^(H))

a e P_ a f P+4 (H e i)

and define e(s) - 4 1 (s 6 Wo) in such a way that

A(sHI) a (s)A-.(H)

for all H e t)i. In particular if P_ is empty e(s) = 1 for all s e Wo. Con- sider the function OA, on Vo defined as follows:

S E (s)e' (A+p")(H1+H) a Q Wo

0A. ,(y) = ,/(7) -?"~r)'J+(H q ) (y Vo) A-(H1)I A+(IIi + H)1 ( )

where y = x(y exp(IlH + H ))x-1 for soume x c G, - e Z, Hr e fD, andH2 E },. It can be shown that in spite of the ambiguity in the choice of 7, H1 and

H2, 0 A , is well defined on V0. We extend OA, 7 on G by defining it to be zero outside V0. Then it can be proved that6

T(f) = fGf(x) O,,, (x) dx (f e CG(G))

provided the Haar measure dx on G is suitably normalized. Let si = 1

S2, . . , Sr be a maximal set of distinct elements in the Weyl group W with the following properties:

(I) Let A, = s,(A + p) - p 1 < i < r. Then Ai + p s(Aj + p) if i j (1 < i, j< r) andse Wo.

(II) For each i (1 < i < r) there exists a i, e w such that Ai coincides on



VOL. 37, 1951 MlATIHEMATICS: Z. NEHARI 369

tf) with the highest weight5 Xs, of a8. Moreover if ac e 8 and 7 e M fn Z

a(y) = (7y)(l1).

Put Oi = OAi, a1 < i <_ r and Ti(f) = fGf(x)0i(x) dx (f e Cc(G)). Then we see that the distributions T, 1 < i < r are solutions of the equations,

zI7 = xA(z)T (z e3)

T(yfy) = T(f), T(f,) = v(a)T(f) (f E Cc(G), y G, a Z).

We have seen above that if mr is any quasisimple irreducible representation of G such that xA is the infinitesimal character of 7r and r(a) = a(a)r(l) (a 6 Z), then its character T, is also a solution of the above equations. Hence one might hope that in most cases TV would be a linear combination of Ti 1 < i < r.

In conclusion I should like to thank Professor C. Chevalley for his help and advice on several questions connected with the results of this note.

"Representations of semisimple Lie groups. II," PROC. NATL. ACAD. Sci., 37, 362

(1951), quoted hereafter as RII. 2 Since 7r is irreducible it follows easily that ~ is separable. 3 Here dx denotes the left invariant Haar measure on G. 4 Schwartz, L., Theorie des distributions, Hermann, Paris, 1950. 5 See RII for the meaning of the various symbols. 6 Compare with Gelfand I. M., and Naimark M. A., Izvestiya Akad. Nauk SSR, Ser.

Mat., 1947, vol. 11, pp. 411-504; Mat. Sbornik N. S., 1947, vol. 21 (63), pp. 405-434; Doklady Akad. Nauk SSSR (N. S.), 1948, vol. 61, pp. 9-11.

ON THE NUMERICAL COMPUTATION OF MAPPING FUNCTIONS B Y ORTHOGONALIZA TION

BY ZEEV NEHARI

WASHINGTON UNIVERSITY

Communicated by J. L. Walsh, April 23, 1951

By means of a technique introduced by Bergman (reference 1, and literature quoted there), it is possible to reduce the computation of a

large number of conformal mapping functions and harmonic domain functions to the orthonormalization of given complete sets of functions with respect to suitably defined inner products.-"3 While the convergence of the expansions obtained in this way is assured, the general method gives no indication as to the quality of the convergence. No estimate for the error committed in replacing the infinite expansions by finite partial sums is given and this, of course, represents a serious drawback as far as the

practical application of the Bergman method is concerned.

VOL. 37, 1951 MlATIHEMATICS: Z. NEHARI 369

tf) with the highest weight5 Xs, of a8. Moreover if ac e 8 and 7 e M fn Z

a(y) = (7y)(l1).

Put Oi = OAi, a1 < i <_ r and Ti(f) = fGf(x)0i(x) dx (f e Cc(G)). Then we see that the distributions T, 1 < i < r are solutions of the equations,

zI7 = xA(z)T (z e3)

T(yfy) = T(f), T(f,) = v(a)T(f) (f E Cc(G), y G, a Z).

We have seen above that if mr is any quasisimple irreducible representation of G such that xA is the infinitesimal character of 7r and r(a) = a(a)r(l) (a 6 Z), then its character T, is also a solution of the above equations. Hence one might hope that in most cases TV would be a linear combination of Ti 1 < i < r.

In conclusion I should like to thank Professor C. Chevalley for his help and advice on several questions connected with the results of this note.

"Representations of semisimple Lie groups. II," PROC. NATL. ACAD. Sci., 37, 362

(1951), quoted hereafter as RII. 2 Since 7r is irreducible it follows easily that ~ is separable. 3 Here dx denotes the left invariant Haar measure on G. 4 Schwartz, L., Theorie des distributions, Hermann, Paris, 1950. 5 See RII for the meaning of the various symbols. 6 Compare with Gelfand I. M., and Naimark M. A., Izvestiya Akad. Nauk SSR, Ser.

Mat., 1947, vol. 11, pp. 411-504; Mat. Sbornik N. S., 1947, vol. 21 (63), pp. 405-434; Doklady Akad. Nauk SSSR (N. S.), 1948, vol. 61, pp. 9-11.

ON THE NUMERICAL COMPUTATION OF MAPPING FUNCTIONS B Y ORTHOGONALIZA TION

BY ZEEV NEHARI

WASHINGTON UNIVERSITY

Communicated by J. L. Walsh, April 23, 1951

By means of a technique introduced by Bergman (reference 1, and literature quoted there), it is possible to reduce the computation of a

large number of conformal mapping functions and harmonic domain functions to the orthonormalization of given complete sets of functions with respect to suitably defined inner products.-"3 While the convergence of the expansions obtained in this way is assured, the general method gives no indication as to the quality of the convergence. No estimate for the error committed in replacing the infinite expansions by finite partial sums is given and this, of course, represents a serious drawback as far as the

practical application of the Bergman method is concerned.



Representations of Semisimple Lie Groups. III. Characters

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