6

T. LEVASSEUR and J. T. STAFFORD

sentations of the classical groups in question. For example, the algebra T(A) introduced

in (0.6) is naturally a Harish-Chandra bimodule for the pair (sp(2n), Sp(2n)) and, as

such, decomposes as a direct sum T(A) = R@V- , where R = U($)/J(k) and D_ are

irreducible Harish-Chandra bimodules. At least when k is odd, with k n, these two

modules can be viewed as concrete realisations (in terms of differential operators) of the

two unipotent representations attached to the orbit O* in [BV2, Theorem III]. For more

details see (V, §5).

Again in relation to unipotent representations, we remark that the orbits O* con-

sidered here are complex analogues of the orbits G' • //* of [Ad2, pp. 144-5]. Similarly,

the representations 7r(/i) and TT(/Z)^ introduced in [Ad2, ibid] and [Adl , Definition 4.6]

correspond in the orthogonal case to the Harish-Chandra bimodules R and V- .

0.12. The results of this paper may also be viewed as part of a programme that

aims to attach a completely prime, primitive ideal J to some of the nilpotent orbits O

in g and to give the structure of a commutative ring to certain highest weight modules

1(A). In this paper, J = J(k) , 0 = 0 * and L(A) = L(Xk + p) = 0{Xk). The

results we have obtained also continue the idea, begun in [LS] and [LSS] ,of realizing

the corresponding primitive factor domain U(Q)/J as an algebra (or subalgebra) of the

ring of differential operators on some irreducible component of O f i n + . By (0.5.3), Xk

is such a component. The results of [LSS] for Lie algebras of type An , Cn and Dn

are just the case k = 1 of Theorem 0.3. However, [LSS] also proves a corresponding

result for Lie algebras of type Bn, E$ and E7, and it would be interesting to know

if Theorem 0.3 could be extended to cover these cases. Recently, Goncharov has also

asserted (without proof) that ip is surjective in the case k = 1 (see [Go]).

0.13. Finally, consider V(Z), where Z is an irreducible affine algebraic variety.

As noted in (0.1) for the cubic cone, it can happen that T(Z) has almost no pleasant

properties. Nevertheless we have now found a number of examples where this algebra

has a nice structure; for example, when:

0{Z) = 0(X)G' is as in (0.2); see Corollary 0.4,

O(Z) = 0{X)so^ see Theorem 0.6,

O(Z) = 0(Cn)G for G finite; see [Lei],

Z is a quadratic cone in C n ; see [LSS].

It would be interesting to know for what other varieties Z the ring of differential

operators is pleasant. In each of the cases mentioned above, Z has rational singularities,

but unfortunately this condition is insufficient by itself to ensure that V(Z) is pleasant.