An Iwahori-Whittaker model for the Satake category

TL;DR

This paper establishes a new geometric model for the Satake category of a reductive group using Iwahori-Whittaker sheaves, confirming a conjecture about tilting objects in this setting.

Contribution

It introduces an Iwahori-Whittaker model for the Satake category and proves a conjecture on tilting objects within this framework.

Findings

The Satake category can be described via Iwahori-Whittaker perverse sheaves.

Confirmation of the Juteau-Mautner-Williamson conjecture on tilting objects.

Extension of geometric Satake equivalence to new coefficient rings.

Abstract

In this paper we prove, for G a connected reductive algebraic group satisfying a technical assumption, that the Satake category of G (with coefficients in a finite field, a finite extension of Q_l, or the ring of integers of such a field) can be described via Iwahori-Whittaker perverse sheaves on the affine Grassmannian. As an application, we confirm a conjecture of Juteau-Mautner-Williamson describing the tilting objects in the Satake category.