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