Perverse sheaves on twisted affine flag varieties and Langlands duality

TL;DR

This paper characterizes Iwahori-Whittaker equivariant perverse sheaves on affine flag varieties for tamely ramified groups using Langlands dual data, extending prior split group results.

Contribution

It extends the theory of Wakimoto sheaves and establishes tilting properties for non-split groups, advancing geometric representation theory.

Findings

Extended Wakimoto sheaves to tamely ramified groups

Proved convolution exactness and filtration properties

Established tilting properties for Iwahori-Whittaker averaging

Abstract

We provide a description of Iwahori-Whittaker equivariant perverse sheaves on affine flag varieties associated to tamely ramified reductive groups, in terms of Langlands dual data. This extends the work of Arkhipov-Bezrukavnikov from the case of split reductive groups. To achieve this, we first extend the theory of Wakimoto sheaves to our context and prove convolution exact central objects admit a filtration by such. We then establish the tilting property of the Iwahori-Whittaker averaging of certain central objects arising from the geometric Satake equivalence, which enables us to address the absence of an appropriate analogue of Gaitsgory's central functor for non-split groups.