Proc. Natl. Acad. Sci. USA Vol. 75, No. 3, pp. 1063-1065, March 1978 Mathematics

On the algebraic construction and classification of Harish-Chandra modules (Lie algebras/semisimple Lie groups/representation theory) THOMAS J. ENRIGHT Department of Mathematics, University of California, San Diego, La Jolla, California 92093 Communicated by E. M. Stein, December 22, 1977

ABSTRACT Let G be a connected real semisimple Lie the set of all equivalence classes of Harish-Chandra modules group. In this article a functor is defined which assigns to each and Ajir(g,K) its subset of classes of irreducible modules. irreducible finite dimensional representation of a Cartan It follows from the early work of Harish-Chandra (1) and subgroup of G a Harish-Chandra module for G. This functor is described by an explicit construction of modules and a sufficient more recent variations of it (16, 17) that there is a natural bi- condition is given for the image module to be irreducible. In the jection between infinitesimal equivalence classes of admissible case when G is a linear group, this functor is used to exhibit all representations of G and Aji,(g,K). irreducible Harish-Chandra modules of G. Let 0H,1H, . .. ,rH be a complete set of 0-stable represen- tatives of the G-conjugacy classes of Cartan subgroups of G. For 1. Introduction 0 < i < r, let W1 = W(G,'H) equal the Weyl group of P and let 'iH denote the equivalence classes of irreducible finite di- The representation theory of compact (reductive) connected mensional representations of 'H. For any a e 'iH, let L, denote Lie groups can be given a fully satisfactory treatment through the representation space of a. either geometric or analytic techniques on the Lie group or by The work of Harish-Chandra, Trombi, Langlands, and algebraic means at the level of the Lie algebra. For noncompact Knapp and Zuckerman suggests that it is very fruitful to ex- connected real semisimple Lie groups, although the represen- amine the following (vaguely formulated) questions: tation theory of the group and the (K-finite) representation Question A: Is there a natural map r from o6<.i'H/We into theory of the Lie algebra are equivalent (1), nearly all of the A(g,K)? definitive results on the classification of irreducible represen- Question B: To what extent does an answer to Question A tations of the group (2-6) are based on analytic techniques, lead to a determination of .Aj(g,K)? In particular, is it true that especially the character theory and harmonic analysis of Har- if an element of 'Osr IH/W1 is in "general position", the cor- ish-Chandra. However, as notable exceptions to this we must responding element (under r) of AY(g,K) lies in Aj,(g,K), and mention the work of Kostant (7) (see also refs. 8-10) on the every element of 4Aj,(gK) in "general position" can be so ob- spherical principal series and the work of Vogan (11). tained? It is only recently that new techniques have been discovered in the infinitesimal approach to representation theory that 3. Methods of construction: Completions appear to have succeeded in making it fully effective in the We briefly describe the three methods of constructing g-mo- problem of constructing and classifying irreducible represen- dules that are necessary for our method. tations of the group. For example (refs. 12-15 and unpublished) (a) Principal series for split groups. Let M be a connected it has been possible to treat the discrete series, and more gen- real Lie group with (real) Lie algebra m. Let 0 be the Cartan erally the fundamental series, by this method. In this an- involution of m and let m = f 9 i be the Cartan decomposition. nouncement we shall describe some of the results that enable Assume that m has a Cartan subalgebra a with a c 4. Let 4' us to obtain canonically "almost all" and noncanonically all denote the roots for (m,a) and let 4'+ be a positive system for 4'. irreducible representations by the infinitesimal method. Set n = Yas+rMa, n- = saeo+rna, and b = a eD n. Then m = 2. Notation n CD a a n. Let K, A, and N denote the analytic subgroups of M that correspond to t, a, and n and let D equal the centralizer Let G be a connected real semisimple Lie group with finite of A in K. Then H = DA is a direct product and is the Cartan center and Lie algebra go. Let K be a maximal compact subgroup with Lie algebra a. note Lie K. 0 denotes subgroup of G and let to the algebra of If Let a be an (finite dimensional) irreducible representation the corresponding Cartan involution, then let go = fo to be of H and let L., denote the associated H-module (representation a Cartan decomposition of go. Let g (resp. f,g) denote the space). Extend L,7 to an N module by letting N act trivially. Let complexification of go (resp. fofo). For any Lie algebra a, let X(0) be the space of all measurable functions from M to L, such U(a) denote the universal enveloping algebra of a. that fRab) = a(b-1)f(a) for all a e M and b e D A N. X(a) is Definition 1: By a Harish-Chandra module for G we mean an M module. Let X7 denote the K-finite vectors in X(U7). Then a g-module A such that (i) A is finitely generated; (ii) A is a Xo is an m-module, which we call the principal series m-mo- module for K; and (iii)-for any finite dimensional irreducible dule induced by a. Let p = 1/2 ca6 o+a and let Up denote the f-module E, dim Homr(E,A) is finite. The integer dim one-dimensional H-module defined by letting D act trivially Homr(E,A) is called the multiplicity ofE in A. Let A(g,K) be and A act by a e(l~g(a),x), a e A. The unitary principal series m-modules are those for which a 0 Up` is a unitary represen- The costs of publication of this article were defrayed in part by the tation of H. payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate (b) For a subalgebra m of g and an m-module L, with U(g) this fact. regarded as a right U(m) module, U(g) Ou(m)L becomes a left 1063 Downloaded by guest on September 27, 2021 1064 Mathematics: Enright Proc. Natl. Acad. Sci. USA 75 (1978) g-module and is called the g-module induced by L from m to Q n A(mjt), set Q = Qm u Q'. We assume Q is chosen so that g (see ref. 18). n = IaeQ'ga. Since Q is 0-stable there is a unique positive sys- (c) Completions: We now introduce the notion of the com- tem Qf of A(f,tf) such that if f3 e Qr, then ,3 = alt1 for some a pletion of a g-module. It appears to be new and is a refinement e Q. Similarly Qm is 0-stable and determines a positive system of ideas occurring in the work done jointly with Varadarajan Qm,t of A(m n ,tt). Let t denote the unique element of the (12) and Wallach (13) (and unpublished). Weyl group W(tt) such that t Qf = Qmf u - (Qt\Qm,t). Set Let a be a subalgebra of g with basis HX,Y forming a stan- nf = Qf dard triple in g (i.e., [HX] = 2X, [HY] = -2Y, [X,Y] = H). Let Choose a factorization t = si... sn of minimal length such L be a g-module such that (i) L = $nL(n) where the sum is over that si is the reflection corresponding to the simple root 'yi of integers and L(n) = fveL I H v = navI and (ii) the action of X Q . For 1 < i < n, choose standard triplets Hi, Xi, Y1 such that is locally nilpotent [i.e., for each v E L, Xn v = 0 for some n = Xi e fy, and Y1 fea.E Define inductively a chain of g-modules n(v) > 1] and set Lo equal to the g-submodule for L of U(a)- as follows: Let Wo equal the g-module U, defined in Step 2. For finite vectors. For any g-module A, let Ax = la e A X-a = 1 < i < n, define'Wi to be the completion of Wi-1 with respect 0J. to the triple Hi, Xi, Yj. PROPOSITION 1. There exists a g-module L that contains Step 4 (the canonical quotient). Let Wn be the last element the quotient module L1 = LLLo and such that (i) L = L(n), in the chain constructed in Step 3. Wn contains a unique min- (ii) X is locally nilpotent on L, (iii) Y is injective on L and lo- imal submodule Z such that Wn/Z is U(f)-finite. Set W. = cally nilpotent on L/L1, and (iv) for any non-negative integer Wn/Z. n yn+1 gives a bijection of LX(n) onto LlX( 'n - 2). Moreover, THEOREM 1. The map af W, gives a map from Oreal form of m and both m and mlo are 0-stable. The restriction of a to the connected Let v and , be linear functionals on b whose real parts are component of 1 in 'H determines a unique element in b* by the dominant with respect to a positive system P of A(g,b). Assume identity u(exp X) = ex(X), X E b n g. Set A = (1/2) (A + OX) and also that 1, is integral and let 0/V+M denote the functor introduced v = (1/2) (X - OX); then A = At + P. If A(m,) denotes the roots by G. Zuckerman (19). Let A and B be elements of AiA,(gK). of the pair (n,b), let Pm be a positive system for A(M,). Extend Then we say A is {-related to B if for some v,,t as above ,,A'r+ P,, to a positive system P for A(g,b), the roots of (by), such that B has a Jordan-Holder series with all irreducible factors iso- P Pm u P', Pm P'"= XP = P", and Re (a,A) < O for all morphic to A. a e P (Re denotes the real part). Set n = ZaeP' gQa and p = m Let a e i'H, 0 < i < r. Let notation be as in Section 4. Assume n; then p is a 0-stable parabolic subalgebra of g with nilradical that the principal series m-module X'T is (when restricted to n. When elements of P satisfy the above inequality, we say that [m,m]) a class one principal series module for [m,m]. b n go is P, n, and p are aligned with a. Let a = bp= 1/2 1aepa and a split Cartan subalgebra of mo = m n go. Let m = f1 f3 CDn@ assume that a corresponds to an element 4a of 'H of degree be the complexification of an Iwasawa decomposition of mo one. with F1 = f n [mm] and ni = Z2aePmga with P as in Section 4. We now describe in several steps the construction of a map Let wr denote the projection of U(m)fi into U(g) determined by = from OS r 'f into A(g,K). the decomposition U(m) U(b) @ (f1U(m) + U(m)ni) (20, 21). Step 1 (principal series for split subgroups). Let M'equal the ir maps U(m)fi onto U(b))Wm, where Wm is the Weyl group for centralizer of fN in G; then M is connected with Lie algebra mo the pair (my) and the superscript denotes the invariants in U(b) 'H is an Iwasawa Cartan subgroup of M. Let = or@0 , and under the affine action of W.. The action of U(m)tl on the from Section 3 a let A+ = -Pm and let XI" denote the principal fl-fixed vector in X'T is given by x (r(x),X), x E U(m)t'. series m-module induced by M. Let p be the 0-stable parabolic subalgebra aligned with a Step 2 (induction). Make XG' a p-module by letting n act given in Section 4, p = m C n. Set n- = 2a'spg-a; then g = n- trivially and then induce from p to g (see Section 3b). We obtain * m@ n and U(g) = U(m) $d (n-U(g) + U(g)n). Let np denote the g-module U, = U(g) @u(p)X6'. the projection of U(g) onto U(m) given by this decomposition Step 3 (the chain of completions). Let t be a 0-stable funda- and note that qp maps U(g)f into U(m)fi. mental Cartan subalgebra of g which we may and do assume THEOREM 6. Assume G is linear. Let A be an element of is contained in m. Since b is central in mIN, a tf = t ri f. Let Q AjfA(g,K). Then A is {p-related to an element B in Jirr(gK) be a 0-stable positive system for the roots A(g,t); and if Q. = with the following properties: Downloaded by guest on September 27, 2021 Mathematics: Enright Proc. Natl. Acad. Sci. USA 75 (1978) 1065 (a) Let Pf denote a positive system of roots for A(ft), t X 5. ;T1rmbi, P. C. (1977) Am. J. Math. 99, 57-75. Cartan subalgebra of t, and set nk = jfpPi. Let t denote a 6. Zelobenko, D. P. (1970) Func. Anal. Appl. 4j 163-165. weight of Bnr whose norm is minimal. Then t is unique and the 7. Kostant, B. (1975) in Summer School ofBolyai Jfanos Math. Soc., t weight space, B'f[4], in Bnf is one dimensional. ed. Gel'fand, I. M. (Halsted Press, Division of John Wiley and Sons, New York), pp. 21-64. (b) For some 0 < i and for some a and as the i,