The Zassenhaus variety of a reductive Lie algebra in positive characteristic

Abstract

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p > 0 . Let Z be the centre of the universal enveloping algebra U = U ( g ) of g . Its maximal spectrum is called the Zassenhaus variety of g . We show that, under certain mild assumptions on G, the field of fractions Frac ( Z ) of Z is G-equivariantly isomorphic to the function field of the dual space g ∗ with twisted G-action. In particular Frac ( Z ) is rational. This confirms a conjecture of J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac ( Z ) , a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand–Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G 2 .