Modular Lie algebras and the Gelfand-Kirillov conjecture

Let g be a finite dimensional simple Lie algebra over an algebraically closed field K of characteristic 0. Let g(Z) be a Chevalley Z-form of g and g(k) = g(Z) circle times(Z) k, where k is the algebraic closure of F(p). Let G(k) be a simple, simply connected algebraic k-group with Lie(G(k)) = g(k). In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U(g(k)) to show that if the Gelfand-Kirillov conjecture (from 1966) holds for g, then for all p >> 0 the field of rational functions k(g(k)) is purely transcendental over its subfield k(g(k))(Gk). Very recently, it was proved by Colliot-Thelene, Kunyavskii, Popov, and Reichstein that the field of rational functions K(g) is not purely transcendental over its subfield K(g)(g) if g is of type B(n), n >= 3, D(n), n >= 4, E(6), E(7), E(8) or F(4). We prove a modular version of this result (valid for p >> 0) and use it to show that, in characteristic 0, the Gelfand-Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if g is of type B(n), n >= 3, D(n), n >= 4, E(6), E(7), E(8) or F(4), then the Lie field of g is more complicated than expected.