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Reducibility of Generalized Principal Series Representations REDUCIBILITY OF GENERALIZED PRINCIPAL SERIES REPRESENTATIONS BY BIRGIT SPEH and DAVID A. VOGAN, JR(1) Mathematische Institut Massachusetts Institute der Universitlit Bonn of Technology, Cambridge Bonn, West Germany Massachusetts, U.S.A. 1. Introduction Let G be a connected semisimple matrix group, and P___ G a cuspidal parabolic~sub- group. Fix a Langlands decomposition P = MAN of P, with N the unipotent radical and A a vector group. Let ~ be a discrete series re- presentation of M, and v a (non-unitary) character of A. We call the induced representation ~(P, ~ | v) = Ind~ (6 | v | 1) (normalized induction) a generalized principal series representation. When v is unitary, these are the representations occurring in Harish-Chandra's Plancherel formula for G; and for general v they may be expected to play something of the same role in harmonic analysis on G as complex characters do in R n. Langlands has shown that any irreducible admissible representation of G can be realized canonically as a subquotient of a generalized principal series representation (Theorem 2.9 below). For these reasons and others (some of which will be discussed below) one would like to understand the reducibility of these representa- tions, and it is this question which motivates the results of this paper. We prove THEOREM 1.1. (Theorems 6.15 and 6.19). Let g(P, 5| be a generalized principal series representation. Fix a compact Caftan subgroup T + o/M (which exists because M has a discrete series). Let ~)= t++a, g (') Support by an AMS Research Fellowship. 228 B. SPEll AlgD D. A. VOGAN, JR denote the complexilied Lie algebras o1 T+A and G, respectively. Let ). e (~+)* denote the Harish-Chandra parameter o1 some constituent o1 the representation 5I Mo (the iden- tity component ot M). Set y:: (~, r)e ~*; here we write v E a*/or the dillerential ol v. Let 0 be the automorphism ol ~ which is 1 on t + and - 1 on a; then 0 preserves the root system A ol ~ in g. Then z(P, 5 | v) is reducible only i I there is a root o~E A such that n = 2<a, ~,>/<~, ~> e Z; and either (a) <~, r> >0, <0~, ~'> <0, and a4 --Oa, or (b) g = -Ocz, and a parity condition (relating the parity ol n and the action ol 5 on the disconnected part ol M) is satisfied. Suppose /urther that <fl, ~) ~=0 /or any fl E A. Then these conditions are also sul/icient /or ~z(P, ~| to be reducible. The parity condition is stated precisely in Proposition 6.1. The simplest kind of direct application of this theorem is the analysis of so-called complementary series representations. Whenever v is a unitary character of A, :~(P, 5| is a unitary representation. But :~(P, ~t | v) can also be given a unitary structure for certain other values of v; it is these representations which are called complementary series, and they have been studied by many people. The following theorem is well known, and we will not give a proof. It is included only to illustrate the applicability of Theorem 1.1 to the study of unitary representations. THEOREM 1.2. Suppose P=MAN is a parabolic subgroup o/ G, ~EATI is a unitary series representation, and dim A = 1. Assume that there is an element x E G normalizing M and A,/ixing 5 (in its action on I~l) and acting by a~a -1 on A. Fix a non-trivial real.valued character v E A; and/or t ER, write tv /or the character whose differential is t times that o/v. Let t o = sup {t El{ I~z(P, (~ | t, v) is irreducible/or all t 1 with It 1 [ <t}. Then whenever It[ <to, every irreducible composition [actor o/:~(P, 5 | is unitarizable. REDUCIBILITY OF GENERALIZED PRINCIPAL SERIES REPRESEI~TATIONS 229 The point is that Theorem 1.1 gives a lower bound on t o when 6 is a discrete series. It is far from best possible, however, and the converse of Theorem 1.2 is not true; so this result (and its various simple generalizations) are very far from the final word on com- plementary series. The techniques of this paper are actually directed at the more general problem of determining all of the irreducible composition factors of generalized principal series re- presentations, and their multiplicities. (We will call this the composition series problem.) This is of interest for several reasons. First, it is equivalent to determining the distribution characters of all irreducible representations of G, a problem which is entertaining in its own right. Next, it would allow one to determine the reducibility of any representations induced from parabolic subgroups of G (and not merely those induced from discrete series). For technical reasons this general reducibility problem is not easy to approach directly; but results about it give more complementary series representations, because of results like Theorem 1.2. Unfortunately our results about the composition series problem are very weak. The first, described in Section 3, relies on the theory of integral intertwining oper- ators. Corollary 3.15 provides a partial reduction of the composition series problem to the case when dim A = l; in particular, Theorem 1.1 is completely reduced to that case. Next, we study the Lie algebra cohomology of generalized principal series representations. When the parameter v is not too large, this leads to a reduction of the composition series problem to a proper subgroup (Theorem 4.23). Section 5 contains a series of technical results refining Zuckerman's "periodicity" ([21]). This leads easily to the "only if" part of Theorem 1.1. Section 6 is devoted to locating certain specific composition factors in generalized principal series representations, and thus to finding sufficient conditions for reducibility. All of the ideas described above appear as reduction techniques. We begin with two-well-known types of reducibility--the Schmid embeddings of discrete series into generalized principal series, and the embeddings of finite- dimensional representations into principal series--and do everything possible to complicate them. The main result is Theorem 6.9. For the benefit of casual readers, here is a guide to understanding the theorems of this paper. We regard a generalized principal series representation as parametrized (roughly) by the Cartan subalgebra ~) and weight ~ defined in Theorem 1.1. This is made precise in 2.3-2.6. Accordingly, we write g(y) for such a representation. By a theorem of Langlands, ~(~) has a canonical irreducible subquotient ~(y) (roughly), and in this way irreducible representations are also parametrized by weights of Cartan subalgebras. This is made precise in 2.8-2.9. The main result of Section 3 is Theorem 3.14, which reduces the question of reducibility 230 B. SPEH AND D. A. VOGAN, JR of ~(~) to the case when dim A = 1. The notation is explained between 3.2 and 3.4, and between 3.12 and 3.13. For the reader already familiar with the factorization of inter- twining operators, all of the results of Section 3 should be obvious consequences of Lemma 3.13. The main result of Section 4 is Theorem 4.23, whose statement is self-contained. The proof consists of a series of tricks, of which the only serious one is Proposition 4.21. A more conceptual explanation of the results can be given in terms of recent unpublished work of Zuckerman; but this does not simplify the proofs significantly. Section 5 consists of technical results on tensor products of finite dimensional re- presentations and irreducible admissible representations. The major new results are Theo- rem 5.15 (for which notation is defined at 5.1) and Theorem 5.20 (notation after 5.5). We also include a complete account of Schmid's theory of coherent continuation (after 5.2- after 5.5), and a formulation of the Hecht-Schmid character identities for disconnected groups (Proposition 5.14; notation after 5.6, and 5.12-5.14). Proposition 5.22 (due to Schmid) describes one kind of reducibility for generalized principal series. Section 6 begins by constructing more reducibility (Theorem 6.9). This leads to the precise forms of Theorem 1.1 (Theorems 6.15 and 6.19). Theorem 6.16 is a technical result about tensor products with finite dimensional representations; it can be interpreted as a calculation of the Borho-Jantzen-Duflo T-invariant of a Harish-Chandra module, in terms of the Langlands classification. Theorem 6.18 states that any irreducible has a unique irreducible pre-image under Zuckerman's ~-functor (Definition 5.1). In conjunction with Corollary 5.12, this reduces the composition series problem to the case of regular infinite- simal character. Section 7 contains the proof of Theorem 6.9 for split groups of rank 2, which are not particularly amenable to our reduction techniques. The questions considered in this paper have been studied by so many people that it is nearly impossible to assign credit accurately. We have indicated those results which we know are not original, but even then it has not always been possible to give a reference. Eearlier work may be found in [2], [7], [10], [11], and the references listed there. 2. Notation and the Langlands classification It will be convenient for inductive purposes to have at our disposal a slightly more general class of groups than that considered in the introduction. Let G be a Lie group, with Lie algebra go and identity component Go; put g = (g0)c. Notation such as H, H 0, ~)0, and I) will be used analogously. Let Gc be the connected adjoint group of 6, let g~=[g0, go], REDUCIBILITY OF GENERALIZED PRINCIPAL SERIES REPRESENTATIONS 231 and let G~ be the connected subgroup of G with Lie algebra ~oi.
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