Irregular Wakimoto modules and the Casimir connection

We study some non-highest weight modules over an affine Kac–Moody algebra ^(\mathfrak g)]{\hat{\mathfrak g}} at the 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 ^(\mathfrak g)]{\hat{\mathfrak g}}-module with respect to a certain level subalgebra. In a particular case this action is identified with the Casimir connection.