Hecke algebras in Number Theory and Categorification

Abstracts

A. Ellis: Odd symmetric functions and odd categorified quantum sl(2)

We introduce odd analogues of the symmetric functions, the nilHecke algebra, and the cohomology of Grassmannians.  These algebras are used in constructing an odd categorifi.cation of quantum sl(2) and, conjecturally, odd Khovanov homology.  By work of Hill and Wang, there is a relation to the categorification of Kac-Moody superalgebras as well.  Joint with Mikhail Khovanov and Aaron Lauda. 

S. Cautis: Hecke algebras and quantum groups

This talk is almost entirely a survey. I will discuss how Hecke algebras show up in the context of quantum groups (both classical and categorified) and conversely how quantum groups appear naturally in the study of Hecke algebras.

K. Koziol: Towards a Langlands correspondence for Hecke modules of SL_n in characteristic p

In this talk, we show how to realize the pro-p-Iwahori-Hecke algebra of SL_n as a subalgebra of the pro-p-Iwahori-Hecke algebra of GL_n.  Using the interplay between these two algebras, we deduce two main results: one on an equivalence of categories between Hecke modules and mod-p representations of SL_n, and another on a numerical Langlands correspondence between "packets" of Hecke modules and mod-p projective Galois representations.

M.-F. Vignéras: p-Iwahori Hecke algebras of reductive p-adic groups.

The modules of these algebras are related to the representations of the groups in characteristic p, and to the local Langlands correspondence in characteristic p.

T. Haines: The stable Bernstein center and its role in number theory

I will explain the stable Bernstein center of a p-adic group, its relations with the local Langlands correspondence and the usual Bernstein center, its geometric construction, and its applications to Hasse-Weil zeta functions of Shimura varieties. I will also discuss some unconditional results in the context of parahoric and pro-p Iwahori Hecke algebras.

M. Harris: Arithmetic Hecke algebras and representations of the absolute Galois group of Q.

M. Khovanov: Categorification of the Hecke algebra and its applications.

This is an overview talk introducing the audience to a categorification of the Hecke algebra via the Soergel bimodules and its application to link homology.

M. Patnaik: Eisenstein Series and their Constant Terms on Loop Groups

We will explain several related constructions of Eisenstein series on loop groups.  Using some tools from the theory of Hecke algebras for p-adic groups, we can then compute the constant terms of these Eisenstein series.  We shall also try and explain the connection with the (conjectural) extension of the Langlands-Shahidi method to this setting, a method which aims to relate the analytic properties of these loop Eisenstein series to L-functions on finite-dimensional groups.

This work is joint in parts with: A. Braverman, H. Garland, D. Kazdhan, and S.D. Miller.

Y. Qi: Categorification of some small quantum groups

We propose an algebraic approach to categorify some small quantum groups at prime roots of unity, using p-differential graded algebras. The talk will focus on the example of the sl(2) case.

Organizers

Rachel Ollivier (Columbia University)

Joshua Sussan (CUNY Medgar Evers)