I. REDUCTIVE DUAL PAIRS

11

Here an element u* G U* is viewed as a derivation on S(U) by the rule v i—• u*(v) for

v € S(U). As will be seen shortly, this naturally gives rise to an action of T (and hence

of G')on V(U*) and 0(U*).

First, however, we need some more notation. Filter T(U*) by total degree; thus set

ft0 = C, Oi =(C/ef7*)eft

0

and for m 2 let 0

m

= ftm_ifti . Write grn(V(U*)) =

n

n

/ O

n

_ i (where ft_i = 0 ) . Note that

flrrP(IH = @grnV(U*) S S(£7~).

n0

Similarly for a vector space W, write 5

m

(W ) = {/ € S(W) : de#/ m} and Sm(W) =

5 '

m

( W ) / 5

m

- i ( W ) . In particular,

gr2V(U*) £ S2(!7~).

Moreover, the bracket [P, Q] = P Q — Q P on T(U*) induces the structure of a Lie

algebra, denoted by sp, on gr2V(U*) and there is an isomorphism UJ : sp(£7~) — » sp .

In fact we may even identify sp(U~) with a Lie subalgebra of T(U*). For, the set of

anticommutators ab -f ba of elements a, 6 £ 51(£7~) = [7 © 17* forms a Lie subalgebra

a inside V(U*). The projection from T(U*) onto D(U*)/£li induces an isomorphism

between a and s p . Thus we will henceforth identify sp = S2(U~) with a and regard

w a s a map from sp(U~) into T(U*). This is called the metaplectic representation of

sp(J7~). (All of this is described in [Hoi, Theorems 4 and 5], but it is straightforward

to write down u(5p(U~)), as we will do in the next section.)

1.3. Now consider the actions of the Lie groups. First, the adjoint action of

w(sp(U~)) on V(U*) integrates to give an action of T on V(U*) as algebra auto-

morphisms. Equivalently,

t-p = [W(0,P] for£e«p(0, Pev(u*)

and the map u is T-equivariant, where T is given the adjoint action on U(sp(U~)).

By [Hoi, Theorem 5], this action of T on 2(£7*) is just the natural extension of the

given action of r on U~ = S 1 ^ ) .

We will need to view the action of G' on G(U*) and V(U*) in a number of different

ways and we should emphasise that they are all the same. First, G' acts on T(U*) by

restricting the T -action. Of course, this is the natural extension of the action of G' on

U~ = U @U* obtained from the inclusion G1 C GL(U). Secondly, the contragradient

representation of G' on U* gives an action of G' on 0(11*) by (g-0)(u*) — 6(g~1 -u*),

for geG',0e (D(U*) and u E U* . When 0(17*) and 5(17) are identified by means of