12

T. LEVASSEUR and J. T. STAFFORD

, ~ this becomes the natural action of G' on S(U) extending that on U . Finally,

one also has the abstract action of G' on T(U*) given by

(g-P)*9 = g(P* GT 1 • 9)) for g G G', P G V{U*) and 9 G 0(U*). (1.3.1)

Let us check, for example, that this is the same action as the one we began with. Thus,

let P eU* (identified with derivations on 5(17)) and 9 G U. Then

(g-P)*e = g-(P*(g-1'0)) = g • (POT 1 • 9)) = Pig-1-9),

where the final equality comes from the fact that P{g~l -9) G C. Thus the actions do

indeed coincide.

1.4. Set V(U*)G' = {P G V(U*) : g- P = P for all g G G'} , and define 0{U*)G'

similarly. By universality, the Lie algebra homomorphism UJ defined in (1.2) extends to

a ring homomorphism UJ : U(sp(U~)) — T(U*). The starting point of our investigation

is:

T H E O R E M . [Hoi,Theorem 7] V(U*)G = W(U(Q)) .

REMARKS . With respect to the notation in [Hoi], we have interchanged the roles of

G and G', as it will be g rather than #' that will be our main interest.

We will not give a proof of this theorem here, since that would involve a fair amount

of extra notation and invariant theory. However, the idea behind the proof is fairly easy.

The isomorphism S(U~) = grV(U*) is G'-equivariant and so, by classical invariant

theory, (grV(U*)) is generated by the degree 2 invariants. Moreover, under the

identification gr2V(U*) = sp(U~) given in (1.2), the degree 2 invariants are simply Q.

The theorem follows from these observations by means of a straightforward induction.

1.5. The inclusion 0(U*)G C 0(U*) induces a homomorphism

V?: V(U*f - • V(0(U*)G'),

and hence, by Theorem 1.4, a homomorphism \j = (puj from U(Q) to T(0(U*)G ) . This

raises the natural question as to the image and kernel of ip, and much of this paper

is devoted to answering it. For, it will be the case that U* is the variety X of (0.2)

of the introduction, while G' is the group GL(k), O(k) or Sp(2k) that acts upon it.

Similarly, Q is the Lie algebra defined in (0.3).