8 T. LEVASSEUR and J. T. STAFFORD INDEX OF NOTATION Throughout this paper we will use the Lie algebra notation from [Bl] while [Di] and [Ja] will form the basic references for enveloping algebras. The following notation will be assumed without comment. Unless otherwise stated, g will denote a semi-simple, complex Lie algebra, with a Cartan subalgebra \) and corresponding triangular decompo- sition Q = n~®f)©n+ . The Weyl group of fl will be denoted by W and the weight vector in Q corresponding to a root a £ I)* will be written Xa . Write p for the half sum of the positive roots and let M{\ + p) denote the Verma module with highest weight A and unique irreducible factor L(\+p). Write J(\ + p) for the annihilator ann^/(fl)(L(A-f p)). Unless otherwise specified, all tensor products and vector spaces will be over C while all algebras will be C-algebras. Given a vector space V , denote Homc(V, C) by V* . For more specialised notation the reader may refer to the following index. Many pieces of notation are given in Sections (II, §2), (II, §3) and (II, §4) for Cases A, B and C respectively. In order to combine these cases, we will use the notation (II, m.3), for example, to denote (II, 2.3), (II, 3.3) and (II, 4.3) in Cases A, B and C respectively. (0.1 (0.2 (0.3 (1,1-1 (1,1 2 (1,1.5 (1,3.1 (11,1.2 ( I I , m . l (II, m.3 (11,4.3 (II, m.4 (11,3.4 (11,4.4 (11,5.1 (11,5.2 (11,6.1 (III, 1.2 (III, 1.3 (111,1.6 V{R), Vm(R), V{X) Mp,q(C) J(k), sufficiently small £/~, T = Sp(U~), , ~ , classical reductive dual pair (G,G') U~ = U®U*, grnV(U*) = Qn/Qn-U Sn(U), Sn(U), metaplectic representation w sp(i j \ g (i ' i} , P+ Ajt, J(k) (see also (II, m.6)) V, E, F, k, p, q, n Xk, Xk, /(m), g.X h, Jk M, m, r+, t" , I(y) Symn(C) AUn(C) ^+ k, i(i) , Killing form K V( Jk) = Ok R = rKU(B)), C(p), V(X)p DerA Vk

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