Simon Riche: The geometric Satake equivalence and some applications


In this course we will start by reviewing the statement of the geometric Satake equivalence, and give an overview of its proof for general coefficients. We will then explain the "Iwahori-Whittaker model" for this category, and explain how it can be used to study the representation theory of reductive algebraic groups over fields of positive characteristic. Finally we will explain the construction of Gaitsgory's central functor, and discuss its use in the construction of the Arkhipov-Bezrukavnikov equivalence. Prerequisites: We will assume that the structure theory of reductive algebraic groups over algebraically closed fields is known, as well as the basic theory of constructible complexes on algebraic varieties and of the perverse t-structure. References for the first subject include the classical books of Borel, Humphreys and Springer; for the second subject we recommend the book "Perverse Sheaves and Applications to Representation Theory" by P. N. Achar. (Shorter overviews of these theories can also easily be found online.)

Bibliography:

(1) Historical references:

S. Arkhipov and R. Bezrukavnikov, Perverse sheaves on affine flags and Langlands dual group, Israel J. Math. 170 (2009), 135-183.

R. Bezrukavnikov, D. Gaitsgory, I. Mirkovic, S. Riche, and L. Rider, An Iwahori-Whittaker model for the Satake category, J. Ec. polytech. Math. 6 (2019), 707-735.

D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math. 144 (2001), 253-280.

I. Mirkovic and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. 166 (2007), 95-143.

S. Riche and G. Williamson, Smith-Treumann theory and the linkage principle, Publ. Math. Inst. Hautes Etudes Sci. 136 (2022), 225-292.

(2) More detailed accounts of some of these constructions:

P. Achar and S. Riche, Central sheaves on affine flag varieties, preliminary version of a book.

P. Baumann and S. Riche, Notes on the geometric Satake equivalence, in Relative aspects in representation theory, Langlands functoriality and automorphic forms, 1-134, Lecture Notes in Math., 2221, CIRM Jean-Morlet Ser., Springer, Cham, 2018.

T. Richarz, Basics on affine Grassmannians, notes.