Representations of Semisimple Lie Groups. IIAuthor(s): Harish-ChandraSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 37, No. 6 (Jun. 15, 1951), pp. 362-365Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/88200 .

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362 MATHEMATICS: HARISH-CHANDRA PRoc. N. A. S.

and at least one of kl, k3, k6, k7 is not divisible by p-say k6. . We write

Al'A k3AkA6A77 - E, (9)

and have Etr = 1, so that the period of E is pS where s < r.

Since p does not divide k6, we have the existence of two integers x, y such that

xk6 + ypl6 = 1. (10)

Raise (9) to the x power, and

A m' = 1 (11)

to the y power, then multiply; we find, with attention to (10),

AlkXA3AkA3XA A7k = EX; (12)

therefore

As = Akx A3-klaxA 7-E k7xx (13)

This expresses A6 as a power-product of other unbarred A's together with an element E whose period ?pr. But pr divides h6, which by (7) is less than pm6; therefore pr < pm. Hence the period of E is less than

that of A6.

Consequently, also E, as well as AI, A3, A7, was unbarred at tlie time we

came to A in our regular procedure-this precisely because of our agreement to arrange the elements of G in non-ascending order of the periods. But then

A6 should be barred, according to our rule of barring and leaving unbarred.

This contradiction establishes the independence of S, and so brings us

to our goal of proving the theorem which is the subject of this note.

REPRESENTATIONS OF SEMISIMPLE LIE GROUPS. HI

BY HARISH-CHANDRA

DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY

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

The object of this note is to announce some further results on repre- sentations of a connected semisimple Lie group on a Hilbert space. We

shall omit all proofs and assume that the reader is familiar with the con-

tents of an earlier note' (quoted henceforward as RI). Let R and C denote the fields of real and complex numbers, respectively,

and let G be a connected, simply connected, semisimple Lie group and

o3 its Lie algebra over R. Let x -> Ad(x)(x e G) denote the adjoint repre- sentation of G and let K be the complete inverse image in G of some maxi-

362 MATHEMATICS: HARISH-CHANDRA PRoc. N. A. S.

and at least one of kl, k3, k6, k7 is not divisible by p-say k6. . We write

Al'A k3AkA6A77 - E, (9)

and have Etr = 1, so that the period of E is pS where s < r.

Since p does not divide k6, we have the existence of two integers x, y such that

xk6 + ypl6 = 1. (10)

Raise (9) to the x power, and

A m' = 1 (11)

to the y power, then multiply; we find, with attention to (10),

AlkXA3AkA3XA A7k = EX; (12)

therefore

As = Akx A3-klaxA 7-E k7xx (13)

This expresses A6 as a power-product of other unbarred A's together with an element E whose period ?pr. But pr divides h6, which by (7) is less than pm6; therefore pr < pm. Hence the period of E is less than

that of A6.

Consequently, also E, as well as AI, A3, A7, was unbarred at tlie time we

came to A in our regular procedure-this precisely because of our agreement to arrange the elements of G in non-ascending order of the periods. But then

A6 should be barred, according to our rule of barring and leaving unbarred.

This contradiction establishes the independence of S, and so brings us

to our goal of proving the theorem which is the subject of this note.

REPRESENTATIONS OF SEMISIMPLE LIE GROUPS. HI

BY HARISH-CHANDRA

DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY

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

The object of this note is to announce some further results on repre- sentations of a connected semisimple Lie group on a Hilbert space. We

shall omit all proofs and assume that the reader is familiar with the con-

tents of an earlier note' (quoted henceforward as RI). Let R and C denote the fields of real and complex numbers, respectively,

and let G be a connected, simply connected, semisimple Lie group and

o3 its Lie algebra over R. Let x -> Ad(x)(x e G) denote the adjoint repre- sentation of G and let K be the complete inverse image in G of some maxi-

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VOL. 37, 1951 MA/1THEMA TICS: HARISH-CHANDRA 363

mal compact subgroup of Ad(G). Then K is a closed connected subgroup of G. Let Vo be the Lie algebra of K and let X --, adX(X E go) denote the

adjoint representation of go. Put B(X, Y) = sp(adXadY)(X, Y c go). Let $o be the set of all elements Y e go such that B(X, Y) = 0 for all X e 0o. Then go = Vo + $0-, So n $0 = {0 } and there exists an automorphism 0 of go over R such that 0(X + Y) - X - Y for X E 0o, Y e o. Let tb)0 be a maximal abelian subspace of 3o. We extend ),3 to a maximal abelian

subalgebra /o of go. Then lbo = b)o + bI-0 where Teo = Do n So. Let g be the complexification of go and let $3, 3, ), f)j, t) be the subspaces of

g spanned by o3, So, )o, fbo, fjo respectively, over C. We extend the bilinear form B and the automorphism 0 on g by linearity over C. Choose bases (t1i, ..., Hp) and (Hp+i, ..., H,), respectively, for ),0 and v/-1 -Io over R. Let a be the space of all linear functions on ). Given X e we can find a unique element Hx e b such that X(H) = B(H, Hx) for all H e

). We shall say that X is real if Hx e )o, + V-/ lieo- Moreover if X is real and Hx = E cH1(ci E R) we say that X > 0 if X 5 O and cj > 0 where

1 < i < I

j is the least index (1 < j K 1) such that cj 5 O. We know that ) is a Cartan subalgebra of g and O) = t. For every root a of g with respect to 1 let X, ?- 0 denote an element in g such that [H, X]a = a(H)X((H e t). Let P be the set of all roots a > 0. For any X e ~ let OX denote the linear function given by OX(H) = X(6H)( H e b). Then if a is a root Oa is also a root. Let P+ be the set of all a E P such that Oa < 0 and P the remaining set of positive roots. Iwasawa2 has shown that Xa, X - e V for a e P. Put

9t= Z CX,a = E t + E CX, + E CX_ . ae P+ a e P a e P

Then 9 is a nilpotent subalgebra of g and 9) is a subalgebra of V. Put 90 =a 9 n go, 9)o0 = 9J n V0. Then go = o -1= o ,p + 9o where the sum is direct2 and [9), 91] c 91, [93, lp3] = {o}. Moreover V and 9) are re- ductive algebras, i.e., they are the direct sums of their centers and their derived algebras which are semisimple. Let 6 be the center of V. Put So = ( n go and So' = [S0, So]. Let D, K', A+, M and N, respectively,

be the analytic subgroups of G corresponding to o, o', , tB), to and 9o0. Then M is closed and the mappings (y, u) -- -yu(y E D, u e K') and (v, h, n) -> vhn(v e K, h F A+, n e N) are analytic isomorphisms of DXK' with K and K X A+ X N with G, respectively.

Let 2, f2' and w, respectively, be the set of all equivalence classes of finite-dimensional irreducible representations of K, K' and M. Then if e Q2 and a e S it is easily seen that o(K') is irreducible and ao(M) is

fully reducible. We denote by T' the equivalence class of the irreducible

representation of K' defined by o. Also if a E c w e say that a < Z if 6 occurs in the reduction of a(M). Let a e 8 e co. Then a defines a repre-

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364 MA TiHEMA TICS: IHARISH-CHA NDRA PROc. N. A. S.

sentation /3 of 9)o atnd therefore of 9Jt. Let X anld A be any two weights of 3 with respect to lv. We extelnd X and A on t1 by defining them to be zero on )l,. Then it is easily seen that X -- is a real linear function on t. Hence /3 has a weight X such that X - u > 0 for every other weight Au.

Clearly X depends on 5 alone and. we shall call it the highest weight of 8.

Let !3 be the universal enveloping algebra of q and let 3 be the center

of 33. Let C[x] denote the ring of all commutative polynomials in I

independent variables xl, ..., x, with coefficients in C. We denote by / the isomorphic mapping of C[x] into Q3 given by (xlm"x2"'. . .xim,) =

Hilm...Hi".lm For any z e 3 there exists a unique element3 Xx(z) E C[x] such that z - -/(xr(z)) E 1X,. A being any linear function on f) we

e P

denote by XA(z) the value of the polynomial X,(z) at xi = A(H)[I i < 1.

Then the mapping XA:z -- XA(z)(z e 3) is a homomorphism of 3 into C and

conversely every homomorphism of 3 into C is of the form' XA for some

A e 8. Let p = 1/2 E a and let W be the Weyl group of g with respect to acP

T. Then if A,s A2 , A , XA, = XA if and only if3 s(Aj + p) = A2 + p for

some s C W. We denote by 2Q the set of all 2 e S2 for which there exists a finite-

dimensional representation 7r of G such that 2' occurs in the reduction of

7r(K').

THEOREM 1. Let T be a quasisimplel representation of G on a Banach

space with the infinitesimal character4 x. Suppose 2) is an element in OF

such that 3) occurs" in yr. Then there exists a linear function A on b and a

5 E w such that 8 < S and x =% XA and A(IJ) = Xa,(H)(IH E lg), where XS is

the highest weight of 8. Conversely suppose we are given a linear function

A on ) and S C 2 such that A coincides on be with the highest weight Xa of some 6 < ) (8 e co). Then there exists a quasisimple irreducible representation r of G on a Hilbert space with the infinitesimal character XA such that S occurs

in -r.

Remarks.-It seems likely that the first part of the above theorem is

actually true for all , E S2 and not merely for ,) e QF, but so far it has not

been possible to prove this. Notice that if G is a complex semisimple

group 0Q = QF and so in this case the theorem holds without any restriction

on 3).

Let 7rm and -r2 be two quasisimple representations of G on the Hilbert

spaces 51i and .2, respectively. For any 2) e Q let 1., Z denote the set

of all elements in ,2 which transform' under ri(K) according to 3(i =

1, 2). Put i0 =a ~i, s. Then we get a representation ri? of Q3 on

S?U such that

mr?0(X)k = Lim t {(7r(expt X) 1 - V} (1 e C o, X e Qo, t e R). t-*0 t

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VOL. 37, 1951 MATHEMATICS: HARISH-CHANDRA 365

We say that nil and 72m are infinitesimally equivalent if the representations t1? and 7r2? are algebraically equivalent, i.e., if there exists an isomorphism a of 01? onto '2? such that r2?(b) a = a,'?(b)ta(b 6 3, ' E h1?). Clearly if 7rl and 7r2 are equivalent they are also infinitesimally equivalent. Con-

versely it can be shown that if 7rl and 7r2 are both unitary, their infinitesimal

equivalence implies their equivalence in the usual sense.

Let r be a quasisimple irreducible representation of G on a Hilbert space. We know (see Theoremn 3 of RI) that dim ) < oo for all 21 U 2. More-

over, we may assume without loss of generality that the subspaces , are

all mutually orthogonal for distinct ). Let E:, denote the orthogonal

projection on p on SZ). Put

ozrr(x) = sp(EXr(x)ET) (x e G).

Then we can restate Theorem 7 of RI in a slightly improved formn as

follows.

THEOREM 2. Let 7ri, 7r2 be irreducible quasisimple representations of G

on two Hilbert spaces. Suppose thatfor some 3) e 2 and c e C, (~p` = Cp~,rt2 7 0. Then rl and 7r2 are infinitesimally equivalent. Conversely if 7rl and mr2 are infinitesimally equivalent (p'a = (p72 for all ) E 2.

We have seen above that every element x 6 G can be written uniquely in the form x = vhn(v c K, h e A+, n N). For any v e K and x e G let

v, and H(x, v) denote the unique elements in K and t0,, respectively, such that xv = v (exp II(x, v))n for some n e N. Moreover let r(v) denote the element in So such that v = (exp r(v))u for some u e K'. Put r(x, v) =

F(vx) - r(v). Let Z be the center of G and let z -> z* denote the adjoint representation of G. Then K : Z and K* is compact. It is easily seen that (va)x = vxa, H(x, va) = H(x, v), r(x, va) = rP(x, v)(a e Z). Hence we may write H(x, v) = H(x, v*), r(x, v) = r(x, v*). Let dv* denote the Haar measure on K* such that fK* dv* = 1. For any S E Q let He denote the linear function on ( such that

o(exp r) - e"(r)" (l) (r e o0)

for a e 2i. Also let d(3)) denote the degree of o. THEOREM 3. Let 7r be a quasisimple irreducible representation of G on

a Hilbert space S and let 2) be an element in Q such that d(3) = 1 and 2) occurs5 in 7r. Then dim ,D a = 1 and there exists a linear function A on b such that XA is the infinitesimal4 character of 7r and

f Z (X) = K* e"Px(r(X' v*))eA(H(x, v*)) dv* (x e G).

1Harish-Chandra, PROC. NATL. ACAD. SCI., 37, 170-173 (1951). 2 Iwasawa, K., Ann. Math., 50, 507-558 (1949). 3 Harish-Chandra, Trans. Am. Math. Soc., 70, 28-96 (1951). 4 The infinitesimal character of 7r was called simlply the character of 7r in RI. ' This means that 21 occurs in the reduction of 7r(K).

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