in the literature, notably in [KV]. We have included them here, both for the reader's
convenience and because most of the details will be needed elsewhere in the paper.
0.9. The main step in the proof of Theorem 0.3 is given in Chapter III. This proves
that (i) the rings R = tp(U(Q)) and V = V{Xk) have the same full quotient ring Q(R)
and (ii) V is a finitely generated R-module. This is, essentially, equivalent to showing
that V is the ring of I -finite vectors L{L{Xk + p), L(Xk + p)). To prove this, we adopt
the approach used in [LSS]. Thus, the key point is to prove that GKdimjiD/R
GKdim R 2 ,where GKdim stands for Gelfand-Kirillov dimension. This is done by
computing the dimensions of certain associated varieties. One may then apply Gabber's
Lemma [Le2] to conclude that V is a finitely generated R-module. (In the present
situation, Gabber's Lemma implies that there exists a unique, maximal, finitely generated
submodule M of Q(R) satisfying GKdimM/R GKdimR 2.) Since [Le2] will not
be published, we include a proof of this result, in its full generality, in an appendix to
this paper. We remark that, after this research was completed, Joseph found another
completely different method of proving that V = L(L(Xk + /), L(Xk + p)) (see [Jo3]).
The equality V = L(L(Xk + p), L(Xk + p)) is remarkably stable under translation.
Indeed, let E be a finite dimensional U(Q) -module, and N any direct summand of the
E7(fl)-module E ®c L(Xk + p). Then L(iV, N) £ VU X~\N), the ring of "twisted
differential operators" on N, regarded as a U(t~)-module. Here r~ is a certain abelian
sub-Lie algebra of n~ , related to A'jt. See (III, §3) for the full details.
0.10. In order to complete the proof of Theorem 0.3 it suffices to show that J{k)
is a maximal ideal of U(Q). This is proved in Chapter IV and follows from the fact
that, since A* is known and J(k) = ann L(Xk 4- p) (see (0.5)), one can use the results of
Barbash-Vogan, Joseph, et al to determine whether J(k) is maximal.
Finally, the results on SO(k) invariants are proved in Chapter V. For any k it is
not difficult to show, in the notation of (0.6), that
R = ^(J7(«p(2n))) C V(A)Z/2Z C V = V{Xk).
Thus, when k n, Theorem 0.3 forces R = V(A)2/22 = V, which is a major part
of Theorem 0.6. However, if k n then V is not even finitely generated as an R-
module. Instead, some explicit calculations are needed to prove that R = D(A) Z / 2 Z (see
(V, Theorem 3.11)).
0.11. The results obtained in this paper have some connection with unipotent repre-
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