Irregular Wakimoto modules and the Casimir connection

TL;DR

This paper explores non-highest weight modules over affine Kac-Moody algebras at non-critical levels, revealing their structure via free field realization and connecting differential operators to the Casimir connection.

Contribution

It introduces a new class of modules called irregular Wakimoto modules and links their endomorphism rings to differential operators and the Casimir connection.

Findings

Embedded rings of differential operators in module endomorphisms

Identified the Casimir connection as an action on coinvariants

Provided a new perspective on non-highest weight modules

Abstract

We study some non-highest weight modules over an affine Kac-Moody algebra at non-critical level. Roughly speaking, these modules are non-commutative localizations of some non-highest weight "vacuum" modules. Using free field realization, we embed some rings of differential operators in endomorphism rings of our modules. These rings of differential operators act on a localization of the space of coinvariants of any module over the Kac-Moody algebra with respect to a certain level subalgebra. In a particular case this action is identified with the Casimir connection.