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.
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