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 ${\hat{\mathfrak{g}}}_{-\kappa }$ at a negative level $-\kappa$ and the category ${\mathrm{Rep}}_q(G)$ 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 ${\mathrm{Rep}}_q^{\mathrm{mxd}}(G)$, involves Lusztig's version of the quantum group for the Borel and the De Concini–Kac version for the negative Borel.