On the Zassenhaus varieties of finite <i>W</i> -algebras in prime characteristic

Abstract Let $Z(\mathcal{W})$ be the center of the finite W -algebra $\mathcal{W}({\mathfrak{g}},e)$ associated with $\mathfrak{g}=\text{Lie}(G)$ and a nilpotent element $e\in\mathfrak{g}$ for a connected reductive algebraic group G over an algebraically closed field ${\unicode{x1D55C}}$ 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 $Z({\mathcal{W}})$ for $p\gg0$ are still true under the present weakened restriction on p . Then we study the Zassenhaus variety $\mathscr{Z}$ of $\mathcal{W}({\mathfrak{g}},e)$ , which is by definition the maximal spectrum $\text{Specm}(Z({\mathcal{W}}))$ of $Z({\mathcal{W}})$ . On basis of the structure properties of $Z({\mathcal{W}})$ , we describe $\mathscr{Z}$ via a good transverse slice ${\mathcal{S}}$ and show that $\mathscr{Z}$ is birationally equivalent to ${\mathcal{S}}$ , 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.