Perverse sheaves on affine flags and Langlands dual group

TL;DR

This paper extends the geometric Satake isomorphism to derived categories of l-adic sheaves on affine flag varieties, linking them to structures in the Iwahori Hecke algebra and Langlands dual group representations.

Contribution

It provides a new geometric description of derived categories related to affine flag varieties, connecting them to algebraic structures in the Iwahori Hecke algebra and Langlands dual group.

Findings

Established a geometric correspondence for derived categories on affine flag varieties.

Linked categories of sheaves to maximal commutative subalgebras in the Iwahori Hecke algebra.

Connected geometric categories to Iwahori-invariant Whittaker functions.

Abstract

The geometric Satake isomorphism is an equivalence between the categories of spherical perverse sheaves on affine Grassmanian and the category of representations of the Langlands dual group. We provide a similar description for derived categories of l-adic sheaves on an affine flag variety which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra; of the anti-spherical module over this algebra; and of the space of Iwahori-invariant Whitakker functions.