Centers and Azumaya loci of finite $W$-algebras

TL;DR

This paper investigates the structure of the center of finite W-algebras associated with semi-simple Lie algebras over fields of large characteristic, establishing a Veldkamp-type theorem and characterizing the Azumaya locus.

Contribution

It provides an analogue of Veldkamp's theorem for the center of finite W-algebras and describes the Azumaya locus as the smooth locus, linking algebraic and geometric properties.

Findings

Established an analogue of Veldkamp's theorem for the center.

Proved the Azumaya locus coincides with the smooth locus.

Connected the Azumaya locus to irreducible representations of maximal dimension.

Abstract

In this paper, we study the center $Z$ of the finite $W$-algebra $\mathcal{T}(\mathfrak{g},e)$ associated with a semi-simple Lie algebra $\mathfrak{g}$ over an algebraically closed field $\mathds{k}$ of characteristic $p\gg0$, and an arbitrarily given nilpotent element $e\in\mathfrak{g}$. We obtain an analogue of Veldkamp's theorem on the center. For the maximal spectrum $\text{Specm}(Z)$, we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $\mathcal{T}(\mathfrak{g},e)$.