IS THE FUNCTION FIELD OF A REDUCTIVE LIE ALGEBRA PURELY.pdf

IS THE FUNCTION FIELD OF A REDUCTIVE LIE ALGEBRA PURELY TRANSCENDENTAL OVER THE FIELD OF INVARIANTS FOR THE ADJOINT ACTION? ´ ` ˘ JEAN-LOUIS COLLIOT-THELENE, BORIS KUNYAVSKII, VLADIMIR L. POPOV, AND ZINOVY REICHSTEIN Valentinu Evgenьeviqu Voskresenskomu, kollege i uqitel, s uvaжeniem i blagodarnostь Abstract. Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k-rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)G and k(g)/k(g)G are purely transcendental. We show that the answer is the same for k(G)/k(G)G and k(g)/k(g)G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A or C , and negative for groups of other types, except possibly n n G . A key ingredient in the proof of the negative result is a recent formula for the unramified 2 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 G on itself. Contents