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 for a connected reductive algebraic group G over an algebraically closed field x1D55C 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 p0 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.