IS THE FUNCTION FIELD OF A REDUCTIVE LIE ALGEBRA PURELY TRANSCENDENTAL OVER THE FIELD OF INVARIANTS FOR THE ADJOINT ACTION?

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 An or Cn, and negative for groups of other types, except possibly G2. 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 G on itself.