**When**: Thursdays 2:00-3:30**Where**: MATH 126**What**: Satake isomorphism, classical and geometric

**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** :

- Spherical representations of a reductive group over a local field; intertwining operators.
- Representation theory of complex Lie groups : highest weight, Weyl character formula.
- Iwahori-Hecke algebra, Bernstein presentation
- Macdonald formula
- Whittaker functions, Casselman-Shalika formula
- Bruhat decomposition, affine Grassmannians
- Geometric Satake isomorphism

**Talk schedule** :

- September 19th : Reductive groups over local fields. Spherical representations and first version of the Satake isomorphism. By Thomas. [Notes]
- October 1st : Representation theory of complex Lie groups, Weyl character formula. By Nicholas.[Notes]
- October 10th : Intertwining operators. Part 1. By Thomas. [Notes for both parts]
- October 17th : Intertwining operators. Part 2. By Thomas.
- October 24th : No talk.
- October 31st : Weyl character formula continued by Nicholas [Notes]. Macdonald's formula by Ed.[Notes]
- November 7th : Macdonald's formula continued, by Ed. [Notes]
- November 21st : Satake, Weyl character formula, MacDonald summary, by Julia.[Notes]
- November 28th : Satake, Weyl character formula, MacDonald summary, by Julia. [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.