Associated varieties of modules over Kac-Moody algebras and $C_2$-cofiniteness of W-algebras

TL;DR

This paper explores the relationship between associated varieties of modules over Kac-Moody and affine W-algebras, proves a conjecture on singular supports, and establishes C_2-cofiniteness for many simple W-algebras, advancing understanding of their geometric and algebraic properties.

Contribution

It establishes the link between associated varieties of modules over Kac-Moody and affine W-algebras, proves the Feigin-Frenkel conjecture, and shows C_2-cofiniteness for numerous W-algebras.

Findings

Associated varieties of G-integrable admissible representations are irreducible G-invariant subvarieties of the nullcone.

Confirmed the Feigin-Frenkel conjecture on singular supports.

Proved C_2-cofiniteness for all minimal series principal W-algebras and recent exceptional W-algebras.

Abstract

First, we establish the relation between the associated varieties of modules over Kac-Moody algebras \hat{g} and those over affine W-algebras. Second, we prove the Feigin-Frenkel conjecture on the singular supports of G-integrable admissible representations. In fact we show that the associated variates of G-integrable admissible representations are irreducible G-invariant subvarieties of the nullcone of g, by determining them explicitly. Third, we prove the C_2-cofiniteness of a large number of simple W-algebras, including all minimal series principal W-algebras and the exceptional W-algebras recently discovered by Kac-Wakimoto.