The geometric Satake correspondence has played a central role in the geometric Langlands program. It is indispensable for the construction of global Langlands correspondences over globa function fields (by Vincent Lafforgue) and for the geometrization of the local Langlands correspondence over local fields (by Laurent Fargues and Peter Scholze). It is a categorical lifting of the classical Satake isomorphism for groups over locally compact nonarchimedean local fields.

Although there are now versions of the geometric Satake correspondence over mixed characteristic local fields, and there are now even motivic and diamond versions, in this course we will concentrate on the original fields of definition, the Laurent series fields over an algebraically closed field, and we work only with suitable $\ell$-adic coefficients. We will extend the standard treatments for split groups to cover quasi-split groups defined over power series rings. The goal is to give a streamlined and mostly self-contained treatment of the geometric Satake correspondence in this level of generality. The results we present were originally proved by Xinwen Zhu and Timo Richarz, but we hope to take a more direct path, not relying on nearby cycles to link the ramified geometric Satake correspondence to the unramified case, but rather proving the general case all at once, effectively treating quasi-split groups following the approach Timo Richarz used for split groups in his dissertation (with appropriate modifications and a few corrections).

Target Audience:

Graduate students

Teaching Language:

English

Registration

Prerequisite

Some knowledge of abstract algebraic geometry and the fundamental results on algebraic groups over local fields will be assumed. Notions such as perverse sheaves and Tannakian duality will be quickly reviewed as necessary.

Reference

The main references will be papers of Timo Richarz and Xinwen Zhu, but we will depart from these sources at some key points.

About the speaker

Thomas Haines is a Professor at University of Maryland. His research interests including Shimura Varieties, Flag varieties and Grassmannians for groups and loop groups, Representations of p-adic groups, Langlands program.

Homepage：

https://www.math.umd.edu/~tjh/