Twisted Whittaker model and factorizable sheaves

TL;DR

This paper explores a geometric approach to understanding quantum group representations related to the Langlands dual group by extending the geometric Satake equivalence to a quantum setting using the affine Grassmannian.

Contribution

It introduces a novel geometric framework connecting quantum group representations to the affine Grassmannian, building on the classical Satake equivalence.

Findings

Establishes a first step towards a geometric realization of quantum group representations.

Proposes a twisted Whittaker model related to factorizable sheaves.

Lays groundwork for future geometric and categorical constructions in quantum Langlands theory.

Abstract

Let G be a reductive group. The geometric Satake equivalence realized the category of representations of the Langlands dual group ^LG in terms of spherical perverse sheaves (or D-modules) on the affine Grassmannian Gr_G=G((t))/G[[t]] of the original group G. In the present paper we perform a first step in realizing the category of representations of the quantum group corresponding to ^LG in terms of the geometry of Gr_G. The idea of the construction belongs to Jacob Lurie.