A Localization Theorem for Finite W-algebras

TL;DR

The paper develops a Beilinson–Bernstein–style localization framework for finite $W$-algebras by realizing $U(\mathfrak g,e)$ as a quantization obtained via Hamiltonian reduction of $D_h$-type algebras on a resolution $\tilde S_e$ of the Slodowy slice. It proves an anti-dominant localization theorem for the quantized setting, and then leverages $\mathbb{C}^*$-equivariant Hamiltonian reduction to produce a quantization $D_h(\lambda,\chi)$ of $\tilde S_e$ whose modules correspond to finitely generated modules over $U(\mathfrak g,e)_\lambda$; this yields a Skryabin-type equivalence between $M_l$-twisted $U(\mathfrak g)_\lambda$-modules and modules over the finite $W$-algebra. The approach integrates Beilinson–Bernstein localization, equivariant and asymptotic differential operator techniques, and Hamiltonian reduction to connect Slodowy slice geometry with representation theory of $W$-algebras, providing both conceptual insight and concrete category equivalences with potential affine generalizations.

Abstract

Following the work of Beilinson-Bernstein and Kashiwara-Rouquier, we give a geometric interpretation of certain categories of modules over the finite W-algebra. As an application we reprove the Skryabin equivalence.

Theorems & Definitions (52)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Corollary 2.6
• Definition 3.1
• Definition 3.2
• Definition 3.3