A Conjectural Extension of the Kazhdan–Lusztig Equivalence

  • Dennis Gaitsgory

    Harvard University, Cambridge, USA
A Conjectural Extension of the Kazhdan–Lusztig Equivalence cover
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Abstract

A theorem of Kazhdan and Lusztig establishes an equivalence between the category of G(O)G(\mathcal{O})-integrable representations of the Kac--Moody algebra g^κ\widehat{\mathfrak{g}}_{-\kappa} at a negative level κ-\kappa and the category Repq(G)\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 Repqmxd(G)\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.

Cite this article

Dennis Gaitsgory, A Conjectural Extension of the Kazhdan–Lusztig Equivalence. Publ. Res. Inst. Math. Sci. 57 (2021), no. 3/4, pp. 1227–1376

DOI 10.4171/PRIMS/57-3-14