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 $\mathfrak{g}$ be its Lie algebra. Let $k(G)$, respectively, $k(\mathfrak{g})$, be the field of $k$-rational functions on $G$, respectively, $\mathfrak{g}$. The conjugation action of $G$ on itself induces the adjoint action of $G$ on $\mathfrak{g}$. We investigate the question whether or not the field extensions $k(G)\mathrm{/}k(G{)}^G$ and $k(\mathfrak{g})\mathrm{/}k(\mathfrak{g}{)}^G$ are purely transcendental. We show that the answer is the same for $k(G)\mathrm{/}k(G{)}^G$ and $k(\mathfrak{g})\mathrm{/}k(\mathfrak{g}{)}^G$, 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 ${\mathsf{A}}_n$ or ${\sf C}n$,andnegativeforgroupsofothertypes,exceptpossibly${\sf G}{2}$. 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. The results and methods of this paper have played an important part in recent A. Premet’s negative solution (arxiv:0907.2500) of the Gelfand–Kirillov conjecture for finite-dimensional simple Lie algebras of every type, other than ${\mathsf{A}}_n$, ${\mathsf{C}}_n$, and ${\mathsf{G}}_2$.