On the Zassenhaus varieties of finite W -algebras in prime characteristic

Let Z(W) be the center of the finite W-algebra W(g, e) associated with g=Lie(G) and a nilpotent element e € g for a connected reductive algebraic group G over an algebraically closed field k of prime characteristic p under the standard hypotheses (H1)-(H3) (see [8, section 6 3]). In this paper, we first demonstrate that our previous results in [20] on the structure and geometric properties of Z(W) for p≥0 are still true under the present weakened restriction on p. Then we study the Zassenhaus variety Z of W(g, e), which is by definition the maximal spectrum Specm(Z(W)) of Z(W). On basis of the structure properties of Z(W), we describe Z via a good transverse slice S and show that Z is birationally equivalent to S, thereby a rational affine scheme. In the special case when e=0, we reobtain one of the main results of [26] on the rationality of the Zassenhaus varieites for reductive Lie algebras in prime characteristic.