INTRODUCTION This Introduction provides historical background and motivation for cohomological induction and gives an overview of the five main theorems. The section of Notes at the end of the book points to expositions where more detail can be found, and it gives references for the results that are cited. The Introduction is not logically necessary for the remainder of the book, and it occasionally uses mathematics that is not otherwise a prerequisite for the book. The first part of the Introduction tells the sense in which representation theory of a semisimple Lie group G with finite center reduces to the study of “(g, K ) modules,” and it describes the early constructions of infinite-dimensional group representations. One of the constructions is from complex analysis and produces representations in spaces of Dolbeault cohomology sections over a complex homogeneous space of G. This construction is expected to lead often to irreducible unitary representations. Passage to Taylor coefficients leads to an algebraic analog of this construction and to the definition of the left-exact Zuckerman functor . Cohomological induction involves more than the Zuckerman functor and its derived functors; it involves also the passage from a parabolic subalgebra of g to g itself. The original construction of Zuckerman’s does not lend itself naturally to the introduction of invariant Hermitian forms, and for this reason a right-exact version of the Zuckerman functor, known as the Bernstein functor , is introduced. The definition of depends on introduction of a “Hecke algebra” R(g, K ), and is then given as a change of rings. 1. Origins of Algebraic Representation Theory Harish-Chandra’s first work in representation theory used particular representations of specific noncompact groups to address problems in mathematical physics. In the late 1940s, long after Elie´ Cartan and Hermann Weyl had completed their development of the representation theory of compact connected Lie groups, Harish-Chandra turned his attention to compact groups and reworked the Cartan-Weyl theory in his own way. Introducing what are now known as Verma modules, he gave a uniform, completely algebraic construction of the irreducible repre- sentations of such groups. (Chevalley independently gave a different such algebraic construction. See the Notes for details.) Motivated by a question in Mautner [1950] of whether connected semisimple Lie groups are of “type I,” and perhaps emboldened by the success with finite-dimensional representations, Harish-Chandra began an algebraic treatment of the infinite-dimensional representations of noncompact groups, concentrating largely on connected real semisimple 3 4 INTRODUCTION groups. Although he initially allowed arbitrary connected semisimple groups, he eventually imposed the hypothesis that the groups have finite center, and we shall concentrate on that case. Let G be such a group. Harish-Chandra worked with representations of G on a complex Ba- nach space V, with the continuity property that the action G × V → V is continuous. His early goal was to strip away any need for real or complex analysis with such representations and to handle them purely in terms of algebra. If π is a continuous representation of G on the Banach space V, then the norms of the operators π(x) are uniformly bounded for x in compact neighborhoods of 1, and we can average π by L1 functions of compact support: π( f )v = f (x)π(x)v dx for v ∈ V and f compactly G supported in L1(G). Here dx is a Haar measure on G and is two-sided invariant because G is semisimple. The integral may be interpreted either as a vector-valued “Bochner integral” or as an ordinary Lebesgue integral for the value of every continuous linear functional ·, l on π( f )v, namely π( f )v, l= ( )π( )v, G f x x l dx. We say that v ∈ V is a C∞ vector if x → π(x)v is a C∞ function from G into V.Garding˚ observed that the subspace C∞(V ) of C∞ vectors is dense in V. In fact, if v is in V and if fn ≥ 0 is a sequence of compactly supported C∞ functions on G of integral 1 and with support shrinking ∞ to {1}, then π( fn)v is a C vector for each n and π( fn)v → v. Let g0 be the Lie algebra of G. We can make g0 act on the space of C∞ vectors by the definition d ∞ π(X)v = π(exp tX)v| = for X ∈ g ,v∈ C (V ), dt t 0 0 and one can check that π becomes a representation of g0: ∞ π([X, Y ]) = π(X)π(Y ) − π(Y )π(X) on C (V ). As a consequence if g = g0 ⊗R C denotes the complexification of g0 and if U(g) denotes the universal enveloping algebra of g, then π extends ∞ uniquely from a linear map π : g0 → EndC(C (V )) to a complex-linear ∞ algebra homomorphism π : U(g) → EndC(C (V )) sending 1 to 1. In this way C∞(V ) becomes a U(g) module. Under this construction a group representation leads to a U(g) module in such a way that closed G invariant subspaces W yield U(g) submodules 1. ORIGINS OF ALGEBRAIC REPRESENTATION THEORY 5 C∞(W). This correspondence, however, is inadequate as a reduction to algebra of the analytic aspects of representation theory. For one thing, simple examples show that the closure of a U(g) submodule of C∞(V ) need not be G invariant. For another, U(g) has countable vector- space dimension while C∞(V ) typically has uncountable dimension; thus C∞(V ) is usually not close to being irreducible (i.e., simple as a U(g) module) even if V is irreducible. Harish-Chandra rectified the first problem by using the U(g) invariant subspace Cω(V ) ⊆ C∞(V ) of analytic vectors, the subspace of v’s for which x → π(x)v is real analytic on G. It is not hard to check that the closure in V of a U(g) invariant subspace of Cω(V ) is G invariant. It is still true that Cω(V ) is dense in V, but this fact is much more difficult to prove than its C∞ analog and we state it as a theorem. Theorem 0.1 (Harish-Chandra). If π is a continuous representation of G on a Banach space V, then the subspace Cω(V ) of analytic vectors is dense in V. To deal with the problem that C∞(V ) and even Cω(V ) are too large, Harish-Chandra made use of a maximal compact subgroup K of the semisimple group G. We say that v ∈ V is K finite if π(K )v spans a finite-dimensional space. The subspace of K finite vectors breaks into a (possibly infinite) direct sum of finite-dimensional subspaces on which K operates irreducibly. The subspace VK of K finite vectors is dense in V, by an averaging argument similar to the proof that C∞(V ) is dense. ω ω Actually even the subspace C (V )K of K finite vectors in C (V ) is dense and is a direct sum of finite-dimensional subspaces on which K operates irreducibly. ω ∞ It is a simple matter to show that C (V )K and C (V )K are U(g) sub- ∞ ∞ ω modules of C (V ). Thus C (V )K and C (V )K are both U(g) modules and representation spaces for K , and the U(g) and K structures evi- dently satisfy certain compatibility conditions. Following terminology ∞ introduced by Lepowsky [1973], we call C (V )K with its U(g) and K structures the underlying (g, K ) module of V. (See Chapter I for the precise definition of (g, K ) module.) For consistency of terminology, we often refer to the representation of π on V as “the representation V.” ω Even the correspondence V → C (V )K is not one-one. For example, V might be the closure in a suitable norm of a G invariant space of func- tions on a homogeneous space of G. If the L2 norm is used, one version of V results, while if an L2 norm on the function and its first partial derivatives is used, another version of V results. Especially because of Theorem 0.6 below, it is customary to define away this problem. We say 6 INTRODUCTION ∞ ∞ that V1 and V2 are infinitesimally equivalent if C (V1)K and C (V2)K are equivalent algebraically—i.e., if there is a C linear isomorphism of ∞ ∞ C (V1)K onto C (V2)K respecting the U(g) and K actions. The reduction to algebra works best for representations that are irreducible or almost irreducible. Theorems 0.2 and 0.3 below prepare the setting. We say that V or its underlying (g, K ) module is quasisimple if the center Z(g) of U(g) operates as scalars in the (g, K ) module. Theorem 0.2 should be regarded as a version of Schur’s Lemma. Theorem 0.2 (Segal, Mautner). If V is an irreducible unitary repre- sentation of G on a Hilbert space, then V is quasisimple. If τ is an irreducible finite-dimensional representation of K , let Vτ be the sum of all K invariant subspaces of VK for which the K action under π is equivalent with τ. We say that V is admissible if each Vτ ω is finite-dimensional. From the denseness of C (V )K in V given in ω Theorem 0.1, it follows that C (V )τ is dense in Vτ . Consequently if Vτ ω ∞ is finite-dimensional, then C (V )τ = C (V )τ = Vτ .