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 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 G2.

Modular Lie algebras and the Gelfand–Kirillov conjecture

“…The rule g ⋆ f := φ −1 (g(φ(f ))) defines a rational action of G k on S(g k ) (1) = k[(g (1) k ) * ] ∼ = k[(g * k ) (1) )]. In [37], the induced action of G k on (g * k ) (1) is called the the star action. By construction, it has the property that (23) g Fr ⋆ χ = g · χ ∀ g ∈ G k , χ ∈ (g * k ) (1) .…”

“…In [37], Tange introduced a principal open subset Z rs of Z and showed that it is isomorphic to a principal open subset of g * k contained in (g * k ) rs . In order to explain his construction in detail we need a more explicit description of the variety Z.…”

“…In this section we investigate the modular situation under the assumption that p ≫ 0. We are going to apply recent results of Rudolf Tange [37] on the Zassenhaus variety of g k to show that if the Gelfand-Kirillov conjecture folds for g k , then the field k(g * k ) = Frac S(g k ) is purely transcendental over its subfield k(g * k ) G k = Frac S(g k ) G k . To explain Tange's results in detail we need a geometric description of the Zassenhaus variety of g k .…”

Factorisation properties of group scheme actions

“…We have U Gr = Z Gr , where Z = U G 1 = U g is the centre of U . By [13,Thm. 2] Z is a UFD and by [13, 1.4] Z has a G-stable filtration A 0 ⊆ A 1 ⊆ A 2 ⊆ · · · with A 0 = k such that gr(Z) is a domain.…”

“…We finish the paper with some generalisations of results in [12] and [13]. For results on generators of algebras of infinitesimal invariant we refer to [5].…”