Fusion and convolution: applications to affine Kac-Moody algebras at the critical level

TL;DR

This paper investigates the action of convolution functors on modules over affine Kac-Moody algebras at the critical level, revealing eigen-module structures through fusion products and geometric methods.

Contribution

It demonstrates that modules at the critical level are eigen-modules under convolution functors using fusion products and geometric techniques.

Findings

Modules are eigen-modules with respect to convolution functors.

Fusion product techniques are effective in analyzing module actions.

The approach links algebraic and geometric methods in representation theory.

Abstract

Let g be a semi-simple Lie algebra, and let g^ be the corresponding affine Kac-Moody algebra. Consider the category of g^-modules at the critical level, on which the action of the Iwahori subalgebra integrates to algebraic action of the Iwahori subgroup I. We study the action on this category of the convolution functors with the "central" sheaves on the affine flag scheme G((t))/I. We show that each object of our category is an "eigen-module" with respect to these functors. In order to prove this, we use the fusion product of modules over the affine Kac-Moody algebra.