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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