Transcending classical invariant theory

Abstract

Let Sp2, (R) = Sp2, = Sp be the real symplectic group of rank n. Let Sp denote the two-fold cover of Sp. There is a unitary representation, constructed by Shale [Sh] and Weil [WA], of Sp that is of considerable interest in the theory of automorphic forms. We denote this representation by co and call it the oscillator representation. The purpose of this paper is to establish about C? a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co. We begin by formulating the result. Let E be a reductive subgroup of Sp, and let E denote the inverse image of E in Sp. Denote by R(E) the set of infinitesimal equivalence classes of continuous irreducible admissible representations of E on locally convex spaces. Let co' be the smooth representation of Sp associated to co. Let co be realized on a Hilbert space % and let the subspace of smooth vectors, on which space co' is defined, be written . Denote by R9(E , co) the set of elements of (E) which are realized as quotients by co' (E)-invariant closed subspaces of t'g . Consider a reductive dual pair (G, G') C Sp [H2]. It is not hard to show that G and G' commute with one another. Hence, we may regard co'I G. G' as a representation of Gx G' . It is well known [F] that R (Gx G') S9 '(G) xR9(G') . The identification associates to p E R(G) and p' E i(G') the tensor product p X p'. (We note that p X p' is not defined as a topological vector space; nevertheless, the infinitesimal equivalence class of p op' is well defined.) Select p X p' E (G x G') . Suppose that, in fact, p? p' E 9(G_ G', c) . Then clearly p E R (G, c) and p' E ,9(G', c) . Hence, ( G' , co) defines the graph of a correspondence between certain subsets of R9(G, cv) and ?J(G', co) . In fact, the situation is quite precise.