Quantisation of Slodowy slices (аfter Gan-Ginzburg and Premet)

Let $\mathfrak g$ be a complex semisimple Lie algebra and $G$ the adjoint group of $\mathfrak g$. For any $\mathfrak{sl}_2$-triple $(e,h,f)$ in $\mathfrak g$ the affine space $S=(e+\mathrm{Ker}\, \mathrm{ad}(f))$ is said to be the Slodowy slice to the nilpotent orbit $Ge$. Using the Killing form on $\mathfrak g$ one can identify $S$ with $\mathfrak z(e)^*$, where $\mathfrak z(e)$ is the centraliser of $e$ in $\mathfrak g$. But $S$ has another Poisson structure (different from that of $\mathfrak z(e)^*$) obtained via Hamiltonian reduction. This Poisson structure has several nice properties, for example, the Poisson centre of $\mathbb C[S]$ is a polynomial algebra. In this talk, we will construct a quantisation of $S$.