Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

Abstract

AbstractLetkbe a field of characteristic zero, letGbe a connected reductive algebraic group overkand let 𝔤 be its Lie algebra. Letk(G), respectively,k(𝔤), be the field ofk-rational functions onG, respectively, 𝔤. The conjugation action of Gon itself induces the adjoint action of Gon 𝔤. We investigate the question whether or not the field extensionsk(G)/k(G)Gandk(𝔤)/k(𝔤)Gare purely transcendental. We show that the answer is the same fork(G)/k(G)Gandk(𝔤)/k(𝔤)G, and reduce the problem to the case whereGis simple. For simple groups we show that the answer is positive ifGis split of typeAnorCn, and negative for groups of other types, except possiblyG2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of Gon itself.