A conjectural extension of the Kazhdan-Lusztig equivalence

TL;DR

This paper proposes a conjecture extending the Kazhdan-Lusztig equivalence to Iwahori-integrable Kac-Moody modules, involving a mixed quantum group structure combining Lusztig's and De Concini-Kac's versions.

Contribution

It introduces a conjectural framework linking Iwahori-integrable Kac-Moody modules with a novel mixed quantum group category, extending known equivalences.

Findings

Conjecture relating Iwahori modules to mixed quantum group representations.

Proposed category involves Lusztig's and De Concini-Kac's quantum groups.

Extension of Kazhdan-Lusztig equivalence to new module categories.

Abstract

A theorem of Kazhdan and Lusztig establishes an equivalence between the category of G(CO)-integrable representations of the Kac-Moody algebra \hat{g}_{-\kappa} at a negative level -\kappa and the category \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 Rep^{mxd}_q(G), involves Lusztig's version of the quantum group for the Borel and the De Concini-Kac version for the negative Borel.