# Motives Seminar-WS 2013/14

## Meeting time and place:

We meet on Tuesdays,14-16 Uhr (c.t.), in WSC-S-U-3.01

## Program

This semester we will hold a seminar on Hecke algebras. Here is an preliminary outline:

**1. Definitions and examples** (based on [Bump], see also [Rouq]). 2-3 lectures

ai) Convolution Hecke algebra for `(G,K)`-definitions for `G` a `p`-adic Chevalley group or finite group, relation with representation theory of `G` (Bump, sections 1,2, 3, main results, theorem 1, lemma 1, theorem 2)

aii) Coxeter groups and Iwahori Hecke algebras: definition of Coxeter groups, examples of Weyl group of a finite root system, definition of the Iwahori Hecke algebra of a Coxeter group (Bump, sections 4, 5, 6, main results: prop. 14, theorem 6, prop 23, theorem 7, theorem 8 (no proof needed), examples for material in section 5,6).

bi) Finite field Iwahori Hecke algebras. Main result: the convolution Hecke algebra for `G( F_{q})` agrees with the Iwahori Hecke algebra

`H`for the Weyl group of

_{q}(W(G))`G`(here

`G`is a simple classical algebraic group associated to some finite irreducible Dynkin diagram). (Bump section 8, theorem 11).

bii) Affine Weyl groups: definition, extended Dynkin diagram, affine Weyl group of a

`p`-adic group (Bump sections 10, 11, 13, theorem 13, prop. 45, theorem 14, examples from section 11, theorem 15)

c) Iwahori Hecke algebra of

`p`-adic groups: the convolution Hecke algebra of

`G(`with respect to the Iwahori subgroup

**Q**_{p})`J`is isomorphic to the Iwahori Hecke algebra of the affine Weyl group (Bump section 15, theorem 21).

**2. Geometric representation theory ** (based on [Ginz]). 4 lectures

a) generalities on Borel-Moore homology and intersection cohomology,

b) convolution algebras and the main idea of geometric representation theory,

c) Springer construction of the Weyl group and its representations,

d) Springer construction of the enveloping algebra of `sl(n)` and its representations.

**3. Kahzdan-Lusztig conjectures.** ([Springer], [KL]) 3 lectures

a) KL polynomials

b) Riemann-Hilbert correspondence

c) Kahzdan-Lusztig conjectures

**4. Equivariant K-theory and the representation theory of Hecke algebras ** ([CG]-Chap. 7, [Vass]) 4 lectures.

a) Basics of

`K`-theory and equivariant

`K`-theory.

b) geometric construction of affine Hecke algebra,

c) epimorphism from quantized loop algebra onto affine Hecke algebra,

d) variations: Nil-Hecke algebras.

## Bibliography

[Borel] A. Borel, Algebraic `D`-modules, Perspectives in Mathematics, vol. 2, Academic Press, Inc., Harcourt Brace Jovanovic, 1987,

[Bryl] J-L. Brylinski, (Co)-homologie d'intersection and faisceaux perverses, seminaire Bourbaki, 1981-82, exp. 585.

[Bump] D.Bump, Hecke algebras, course notes from his web page at Stanford

[CG] Chriss N., Ginzburg V. Representation theory and complex geometry (Birkhauser, 1997)

[Ginz] V. Ginzburg, Geometric methods in representation theory of Hecke algebras and quantum groups (notes by V. Baranovsky), arxiv 9802004,

[KL] D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras. Invent. Math. 53 (1979), no. 2, 165–184.

[Springer] T. Springer, Quelques applications d'homologie d'intersection, seminaire Bourbaki, 1981-82, exp. 589,

[Vass] E. Vasserot, Affine quantum groups and equivariant `K`-theory, Transformation Groups, 1998,

[Rouq] R. Rouquier, Weyl groups, affine Weyl groups and reflection groups.

## Schedule

### Part 1: Definitions and examples

Lecture 1-10.22,29-1(a): "Convolution Hecke algebras, Coxeter groups and Iwahori Hecke algebras"-Brad Drew

Here are Lecture Notes for Lecture 1.

Lecture 2-11.05-1(b): "Finite field Iwahori Hecke algebras, affine Weyl groups"-Utsav Choudhury

Lecture 3-11.12-1(c): "The Iwahori Hecke algebra of `p`-adic groups"-Utsav Choudhury

### Part 2-Geometric representation theory for finite Weyl grouops and affine Weyl groups

Lecture 4-11.19-2(a): "Borel-Moore homology and intersection cohomology"-Girja Tripathi

Lecture 5-11.26-2(b): "Convolution algebras and geometric representation theory"-Giuseppe Ancona

Here are Lecture Notes for Lecture 5.

Lecture 6-12.3-2(c): "Springer construction of the Weyl group and its representations"-Ting-Yu Lee

Lecture 7-12.10-4(a) "Basics of `K`-theory and equivariant `K`-theory" Jerzy Weyman

Lecture 8-12.17-4(b) "Geometric construction of affine Hecke algebra" Jerzy Weyman

### Part 3-Kahzdan-Lusztig conjectures

Lectures 9 and 10-14, 21.01.2014-3(a) "Kahzdan-Lusztig polynomials", 3(b) "Riemann-Hilbert correspondence", 3(c) "Kahzdan-Lusztig conjectures"-Ivan Barrientos

### Part 4-Geometric representation theory for the enveloping algebra

Lecture 11-28.01-2(d): "Springer construction of the enveloping algebra of sl(n) and its representations"-Marc Levine

Lecture 12-04.02-4(c) "Epimorphism from quantized loop algebra onto the affine Hecke algebra"-Marc Levine

Lecture 13-???-4(d) "Variations: Nil-Hecke algebras"