4

T. LEVASSEUR and J. T. STAFFORD

Some remarks on this theorem are in order. First, note that it covers all the in-

teresting values of k. For, the ring A is non-regular if and only if k n. Moreover,

for k n the group Z/2Z acts trivially on A and so this case is already covered by

the earlier results concerning 0(k) invariants. If k n, then Theorem 0.3 shows that

R = V{Xk) = V{AZ/2Z) and so the above theorem implies that V(A)ZI2Z = D(A Z / 2 Z ) .

The case k = n is rather curious. For, (0.4) now implies that R ^ T(Xk) and one

can even show that V(Xk) cannot be finitely generated as an R-module (see (IV, Re-

mark 1.5)). Equivalently, V(AZ/2Z) is infinitely generated as a module over the subring

V{A)Z/2Z .

It is natural to ask whether one can extend these results to cover the rings of invari-

ants under the action of SL(k). Unfortunately the methods of this paper will not apply

to this case since there is no obvious enveloping algebra that can be used to generate all,

or most of, V(G(X)SL^).

0.7. There is an alternative way of viewing Theorems 0.3 and 0.6. Given a group

K acting on a variety y then this induces a natural action of K on V{y). Moreover,

one always has a map

v-.v(y)K

—

v(0(y)K)

obtained by restriction of differential operators. Now, Howe in [Hoi] actually provides

a map uo from U(Q) onto V(X)G . The map i/ of (0.3) is then simply z/ = ipco. Thus

an equivalent formulation of Theorem 0.3 is that p is surjective if (and only if) k is

sufficiently small. Similarly, in the set-up described in (0.6) one can show that

p':V(X)s°W —• V{0(X)s°W)

is surjective if and only if k n. Consequently, at least when k is small enough,

these results may be regarded as a non-commutative analogue of one of the basic results

of classical invariant theory - that Q(X)G = O(Xk). However, in contrast to the

commutative case, p will not be injective (see (IV, Lemma 1.7)).

0.8. We next give a brief outline of the proofs of the main results and of the

organisation of the paper. As was remarked earlier, one of the basic ideas behind the

proof of Theorem 0.3 is to use Howe's work on reductive dual pairs from [Hoi]. Since

Howe's paper has not been published, we give a brief survey of the relevant material in

Chapter I. In Chapter II, we then give the more detailed computations that we will need

concerning this material. In particular, in Chapter II we prove the results mentioned in

(0.5). We remark that many of these results, at least for Cases A and B, can be found