purposes of this introduction we merely note that 0{X)GL^ = O(Xk), where Xk is
the variety of p X q matrices of rank k.
(CASE B) Given n 1, let G' = 0(k) = 0(k, C) act on X = M*,n(C) by g^ =
for g G 0(k) and £ G X. In this case 0(X)°^ ^ 0(Xk), where A7* is the variety of
symmetric n x n matrices of rank k (see [DP] or (II, Theorem 3.3)).
(CASE C) Given n 1, let G = Sp(2k) = S(2fc,C) act on X = M2Jfc,n(C) by
g-Z = for g e Sp(2k) and £ G A . Then 0(AT)s^2* ^ 0(ArJb) where, now, ^
f c
defined to be the variety of all antisymmetric n x n matrices of rank 2k (see [DP]
or (II, Theorem 4.3)).
One may, of course, also consider the rings of invariants under the action of SO(k)
and 5L(fc), but we defer comment on these cases until later in the introduction.
0.3. It follows immediately from (0.2) that O(Xk) is a polynomial ring if (and only
if) k q in Case A, k n in Case B and 2k n 1 in Case C. Thus in these cases
V{Xk) is nothing more than the Weyl algebra,
Am(C) = C[xi,...,xm, d/dxu...,d/dxm]
of an appropriate index m. The aim of this paper is to study T(Xk) for the remaining
values of k. We therefore define k to be sufficiently small if 1 k q 1 in Case A,
1 A: n 1 in Case B and 2 2k n 2 in Case C. We remark that, with a little
work, one can prove that k is sufficiently small if and only if Xk is singular.
The method we use to study V(Xk) is Howe's notion of a classical reductive dual
pair in a symplectic group [Hoi , Ho2] (see Chapter I for the basic definitions and
results). This provides, by means of the metaplectic representation, a natural map
^ U($) V(Xk). Here g is the Lie algebra 9 = gl(p + q) in Case A, fl = sp(2n) in
Case B and Q = so(2n) in Case C, while U(&) denotes the enveloping algebra of Q.
We can now state the main result of this paper.
T H E O R E M . Suppose that k is sufficiently small, and that Xk is defined as in (0.2).
Then J(k) = kerfy) is a completely prime, maximal ideal of U(Q) . Moreover, ip induces
an isomorphism
U(8)/m -=* V(Xk).
This result is obtained by combining the main results of Chapters III and IV (see
(III, Theorem 1.10) and (IV, Theorem 1.3), respectively).
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