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We determine the reducibility and number of components of any representation of a quasi-split unitary group which is parabolically induced from a discrete series representation. The R-groups are computed explicitly, in terms of reducibility for maximal parabolics. This gives a description of the elliptic representations.

We prove that the restriction to (2, R) of the underlying HarishChandra module of a Discrete Series representation of a noncompact simple Lie group has no sl(2)-irreducible subrepresentations unless it is holomorphic. We also give a characterization of the holomorphic Discrete Series. Let G be a connected real simple Lie group. Fix a maximal compact subgroup K of G. Let g (t) denote the complex Lie algebra of G (respectively, K). Hence we have the Cartan decomposition g = t + p. Thus, p is Kinvariant under the restriction of the adjoint representation of G to K and p . Let T C K be a maximal torus. Henceforth we assume that T is a Cartan subgroup of G. Let t be the complex Lie algebra of T. Under these hypotheses Harish-Chandra in [HC2] has given a classification of the square-integrable representations of G in terms of the regular characters of T. Let (D(g, t) denote the root system of the pair (g, t) . A root of the pair (g, t) is called compact (noncompact) if its root space is contained in t (respectively, p). We now fix a noncompact root y . Then the root vectors of y span a copy by of sl(2, C) . Moreover by is invariant under the conjugation of g with respect to the Lie algebra of G. Let s,, be the real form of by associated to the conjugation of g with respect the Lie algebra of G restricted to by . Let Hy be the connected subgroup of G corresponding to sy I Let (7r, V) be the Harish-Chandra module underlying a square-integrable, irreducible representation of G. We then have Proposition 1. If the restriction of (7r, V) to by contains an irreducible bysubmodule, then G/K is a Hermitian symmetric space and (7r, V) corresponds to a holomorphic square-integrable representation. A corollary to Proposition 1 is Corollary. If G is as above, G/K is not Hermitian, and H is a semisimple subgroup of G such that its symmetric space is Hermitian, then (7r, V) restricted to [ has no [-holomorphic irreducible subrepresentations. Received by the editors April 29, 1991 and, in revised form, July 11, 1991. 1991 Mathematics Subject Classification. Primary 22E47.

Introduction. The representation theory of GL(2, K), for K a nonarchimedean local field, is somewhat more complicated than the corresponding theory for GL(2, R). In particular, the formulas for the characters of discrete series representation of GL(2, R) have a concise closed form, while the published character formulas for supercuspidal representations of GL(2, K) occupy several pages [10], [14]. The purpose of this paper is to give a concise formula for the character of certain infinite dimensional representations of GL(2, K). The representations considered are admissible in the sense that the stabilizer of a vector in the representation space is open in GL(2, K), and the subspace stabilized by an open compact subgroup of GL(2, K) is finite dimensional. Irreducible admissible representations ir of GL(2, K) satisfy Schur's lemma, so that the center K* of GL(2, K) acts by multiplication by a quasicharacter ,. There exists a locally constant character function ch , defined on a dense open subgroup of GL(2, K), which determines the isomorphism class of ir [7, Section 7]. Jacquet and Langlands [7, Section 2] introduced factors E(10 X) attached to the twist of ir by quasicharacters X of K* which describe the action of -r in a specific model. For each separable quadratic extension L of K, there exists an irreducible admissible representation BCL/K(1r) of GL(2, L), the base change of ir to L [9, Section 2]. This representation of GL(2, L) enters into the expression for the character. The main result of this paper is the following formula. Recall that conjugacy classes of nonsplit Cartan subgroups are parameterized by quadratic separable field extensions L of K, via embeddings of L* in GL(2, K).

We consider the problem of finding all the “embeddings” of a discrete series representation in the principal series in the case of a simple real Lie group G of real rank one. More precisely, we solve the problem when G is Spin ⁡ ( 2 n , 1 ) , SU ( n , 1 ) , SP ( n , 1 ) or F 4 ( n ⩾ 2 ) \operatorname {Spin} (2n,\,1),{\text {SU}}(n,\,1),\,{\text {SP}}(n,\,1)\,{\text {or}}\,{F_4}\,(n\, \geqslant \,2) . The problem is reduced to considering only discrete series representations with trivial infinitesimal character, by means of tensoring with finite dimensional representations. Various other techniques are employed.

In the theory of irreducible representations of a compact Lie group, the formula for the multiplicity of a weight and the so-called theorem of the highest weight are among the most important results. At least conjecturally, both of these statements have analogues for the discrete series of representations of a semisimple Lie group. Let G be a connected, semisimple Lie group, K c G a maximal compact subgroup, and suppose that rk K = rk G. Exactly in this situation, G has a non-empty discrete series [8]. Blattner's conjecture predicts how a given discrete series representation should break up under the action of K; precise statements can be found in [10], [15], [16]. Formally, the conjectured multiplicity formula looks just like the formula for the multiplicity of a weight. Partial results toward the conjecture have been proved in [10], [15]. More recently, the full conjecture was established for those linear groups G, whose quotient G/K admits a Hermitian symmetric structure [16]. As this paper was being completed, H. Hecht and I succeeded in proving Blattner's conjecture for all linear groups, by extending the arguments of [16]. According to Blattner's conjecture, any particular discrete series representation 7t contains a distinguished irreducible K-module V, with multiplicity one; moreover, w contains no irreducible K-module with a highest weight which is lower, in the appropriate sense, than that of V,:. For "most" discrete series representations, it was known that these two properties characterize at, up to infinitesimal equivalence, among all irreducible representations of G [10], [15]. In this paper, I shall give an infinitesimal characterization, by lowest K-type, for all discrete series representations. The result, which is stated as Theorem (1.3) below, closely resembles the theorem of the highest weight. I shall also draw a number of conclusions from it. The methods of this paper have some further, less immediate consequences, which will be taken up elsewhere. For the remainder of the introduction, I assume that G is a linear group.