A Conjectural Extension of the Kazhdan–Lusztig Equivalence

Abstract

A theorem of Kazhdan and Lusztig establishes an equivalence between the category of $G(\mathcal{O})$-integrable representations of the Kac--Moody algebra $\widehat{\mathfrak{g}}\_{-\kappa}$ at a negative level $-\kappa$ and the category $\operatorname{Rep}\_q(G)$ of (algebraic) representations of the “big” (a.k.a. Lusztig's) quantum group. In this paper we propose a conjecture that describes the category of Iwahori-integrable Kac–Moody modules. The corresponding object on the quantum group side, denoted $\operatorname{Rep}^{\operatorname{mxd}}\_q(G)$, involves Lusztig's version of the quantum group for the Borel and the De Concini–Kac version for the negative Borel.