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ROGER HOWE 'Y oo .Denote by .9l(£,w) the set of elements of .9l(£)which are realized as quotients by woo(£)-invariant closed subspaces of 'Y oo .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 aThe 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 , --, , --,--, , 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.Theorem 1.The set .9l (G .G' , w) is the graph oj a bijection between (all oJ) .9l(G, w) and (all of) .9l(G' ,w).Moreover, an element p® p' oj .9l(G.G' ,w) occurs as a quotient oj WOO in a unique way.
Roger Howe (Sun,) studied this question.