MATH 600D: Automorphic forms Seminar

When : Fridays 1:00-3:00
Where : MATH 126
What : Towards geometric Satake isomorphism

Overview : Satake isomorphism identifies a Hecke algebra with an algebra of symmetric polynomials (this just one approximate version of the statement). Versions of Satake isomorphism arise in representation theory, Langlands programme, combinatorics (in the study of Macdonald and Kazhdan-Lusztig polynomials). Its geometrization (due to Drinfeld, Mirkovic-Vilonen and others) plays a role in geometric representation theory and algebraic geometry.

Initial topic list from the organisation meeting :

Borel-Weil-Bott Theorem.
Perverse sheaves.
Whittaker functions, Casselman-Shalika formula
Bruhat decomposition, affine Grassmannians
Geometric Satake isomorphism

Talk schedule :

January 17th : Borel-Weil-Bott Theorem. By Thomas.
March 6th : Affine Grassmannians. By Nick. [Notes]

References :

B. Gross Notes On the Satake isomorphism.

T. Haines, R.Kottwitz, and A.Prasad notes on Iwahori-Hecke algebras.

I.G.Macdonald article on spherical functions of p-adic Chevalley group.

B. Casselman notes on Macdonald's book on p-adic spherical functions.

B. Casselman article on Symmetric powers and the Satake transform.

B. Casselman notes on Symmetric power decomposition for GL2.

X.Zhu article on geometric Satake, categorical traces, and arithmetic of Shimura varieties.

M. Kamgarpour notes on local geometric langlands.

T.Richarz notes on affine Grassmannians.

Notes from S.Riche talk on geometric Satake taken during CIRM summer school in buildings and affine Grassmannians. Part1. Part2.

P.Baumann and S.Riche notes on the geometric Satake equivalence.

I.Mirkovic and K.Vilonen notes on geometric Langlands duality and representations of algebraic groups over commutative rings.

V.Ginzburg notes on perverse sheaves on a loop group and Langlands' duality.

J.L.Taylor book Several Complex Variables with COnnections to Algebraic Geometry and Lie Groups.