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 gℤ be a Chevalley ℤ-form of g and gK = gℤ ⊗ℤ K, where K is the algebraic closure of Fp. Let Gk be a simple, simply connected algebraic K-group with Lie(Gk)= gk. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U(gk) to show that if the Gelfand-Kirillov conjecture (from 1966) holds for g, then for all p≫0 the field of rational functions K(gK) is purely transcendental over its subfield. Very recently, it was proved by Colliot-Thélène, Kunyavskiǐ, 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 Bn, n≥3, Dn, n≥4, E6, E7, E8 or F4. 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 Bn, n≥3, Dn, n≥4, E6, E7, E8 or F4, then the Lie field of g is more complicated than expected. © 2010 Springer-Verlag.