The Zassenhaus variety of a reductive Lie algebra in positive characteristic

TL;DR

This paper proves that the Zassenhaus variety's field of fractions is rational and that its center is a unique factorization domain, confirming two conjectures and impacting the understanding of the Gelfand-Kirillov conjecture for certain Lie algebras.

Contribution

It establishes the rationality of the Zassenhaus variety's function field and the unique factorization property of its center under mild assumptions, confirming conjectures by Alev, Braun, and Hajarnavis.

Findings

Frac(Z) is G-equivariantly isomorphic to the function field of g*

Z is a unique factorization domain

Confirmed conjectures of Alev, Braun, and Hajarnavis

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 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.