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.