Let $\mathfrak{g}=\mbox{Lie}(G)$ be the Lie algebra of a simple algebraic group $G$ over an algebraically closed field of characteristic $0$. Let $e$ be a nilpotent element of $\mathfrak{g}$ and let $\mathfrak{g}_e=\mbox{Lie}(G_e)$ where $G_e$ stands for the stabiliser of $e$ in $G$. For $\mathfrak{g}$ classical, we give an explicit combinatorial formula for the codimension of $[\mathfrak{g}_e,\mathfrak{g}_e]$ in $\mathfrak{g}_e$ and use it to determine those $e\in \mathfrak{g}$ for which the largest commutative quotient $U(\mathfrak{g},e)^{\mbox{ab}}$ of the finite $W$-algebra $U(\mathfrak{g},e)$ is isomorphic to a polynomial algebra. It turns out that this happens if and only if $e$ lies in a unique sheet of $\mathfrak{g}$. The nilpotent elements with this property are called non-singular in the paper. Confirming a recent conjecture of Izosimov, we prove that a nilpotent element $e\in \mathfrak{g}$ is non-singular if and only if the maximal dimension of the geometric quotients $\mathcal{S}/G$, where $\mathcal{S}$ is a sheet of $\mathfrak{g}$ containing $e$, coincides with the codimension of $[\mathfrak{g}_e,\mathfrak{g}_e]$ in $\mathfrak{g}_e$ and describe all non-singular nilpotent elements in terms of partitions. We also show that for any nilpotent element $e$ in a classical Lie algebra $\mathfrak{g}$ the closed subset of Specm  $U(\mathfrak{g},e)^{\mbox{ab}}$ consisting of all points fixed by the natural action of the component group of $G_e$ is isomorphic to an affine space. Analogues of these results for exceptional Lie algebras are also obtained and applications to the theory of primitive ideals are given.

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.