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

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_n or C_n, and negative for groups of other types, except possibly G_2. 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.