CHAPTER I. REDUCTIVE DUAL PAIRS AND THE HOWE CORRESPONDENCE One of the basic strategies behind the proof of Theorem 0.3 of the introduction is to use the machinery of Howe's papers [Hoi] and [Ho2]. Since these papers are unpublished, we collect in this chapter the notation and general results from these papers that we will need. In Chapter II we will give the more detailed analysis that will be required in the proofs of our main results. Thus, for example and in the notation of the introduction, it is shown in [Hoi] that O(Xk) = L(Xk + p) is a simple highest weight module, and hence that J(k) is a primitive ideal. But it needs the more explicit computations of Chapter II to find A & and the associated variety of J(k). 1. Reductive Dual Pairs. 1.1. Let U~ be a complex vector space, equipped with a symplectic form , ~ . Write T for the group Sp(U~, , ~) and suppose that G and G' are two reductive subgroups of T. Then (G,Gl) is called a reductive dual pair if G and G' are mutual commutators in T. The pair is called a classical reductive dual pair if G and G' are also classical Lie groups. Set sp(U~) = Lie(T). In general, closed subgroups of F will be denoted by capital roman letters while the corresponding Lie subalgebras of sp(U~) will be denoted by the same letter in lower case German script. Thus, write Q = Lie G and g' = LieG'. Note that Q is the centraliser of #' in sp(U~), and vice-versa. The classical reductive dual pairs (G, G') in T have been classified in [Hoi] and all arise from the following construction. One may write U~ = E ® V , where E and V are equipped with bilinear forms such that , ~ may be deduced from these forms. Then G and G' are appropriate subgroups of GL(E) and GL(V), respectively. The precise subgroups of GL(E) and GL(V) that occur will be described in Chapter II, but that level of detail is unnecessary for the development of this chapter. The action of a group H on a space V will always be written as h v for h £ H and v £ V. 1.2. In the cases that interest us, U~ admits a polarisation U~ = U @U* such that G' C GL(U). Thus, as a subgroup of T, G' will act on U* via the contragradient representation (g u*)(u) = w*(^_1 u), for u G U, u* £ U* and g £ G'. We will always identify the ring of regular functions 0(U*) with the symmetric algebra S(U) via , ~ . Thus u £ U corresponds to u, ~ £ 0(J7*). This in turn identifies V(U*) = V(0(U*)) with V(S(U)) and hence with S(U)*S(U*) as left 5(17)-modules. 10
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