Centers and Azumaya loci of finite $W$-algebras
In this paper, we study the center $Z$ of the finite $W$-algebra $\mathcal{T}(\mathfrak{g},e)$ over an algebraically closed field $\mathds{k}$ of characteristic $p>0$. We obtain an analogue of Veldkamp's theorem on the center, and then decide the structure of the reduced center of the associated reduced $W$-algebra $\mathcal{T}_\chi(\mathfrak{g},e)$ via that of the reduced enveloping algebra $U_\chi(\mathfrak{g})$. For the maximal spectrum $\text{Specm}(Z)$ of the center $Z$, we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $\mathcal{T}(\mathfrak{g},e)$.