Transcending Classical Invariant Theory
نویسنده
چکیده
Let SP2n(R) = SP2n = 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 wand call it the oscillator representation. The purpose of this paper is to establish about w a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using w. We begin by formulating the result. Let E be a reductive subgroup of Sp, and let £ denote the inverse image of E in Sp. Denote by .9l (£) the set of infinitesimal equivalence classes of continuous irreducible admissible representations of £ on locally convex spaces. Let WOO be the smooth representation of sp associated to w. Let w be realized on a Hilbert space 'Y, and let the subspace of smooth vectors, on which space WOO is defined, be written 'Yoo . Denote by .9l(£, w) the set of elements of .9l(£) which are realized as quotients by woo(£)-invariant closed subspaces of 'Yoo . Consider a reductive dual pair (G, G') ~ Sp [H2]. It is not hard to show that -, 00 -, G and G commute with one another. Hence, we may regard wiG· G as a representation of Gx G' . It is well known [F] that .9l(GxG') ~ .9l(G) x.9l(G'). The identification associates to p E .9l(G) and p' E .9l(G') the tensor product p ® p'. (We note that p ® p' is not defined as a topological vector space; nevertheless, the infinitesimal equivalence class of P®P' is well defined.) Select , -, , -, P ® P E.9l (G x G ). Suppose that, in fact, p ® P E.9l (G· G ,w). Then clearly , -, -, P E .9l(G, w) and p E .9l(G ,w). Hence, .9l(G· G ,w) defines the graph of a correspondence between certain subsets of .9l (G , w) and .9l (G' , w). In fact, the situation is quite precise.
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