University of Sydney Algebra Seminar

Syu Kato (Kyoto University)

Friday 5th November, 12.05-12.55pm, Carslaw 175

Exotic representation theory of affine Hecke algebras

Affine Hecke algebras are ubiquitous in representation theory, like the study of representation theory of p-adic groups and the study of special orthogonal polynomials. Actually, a common classification of simple modules of one-parameter affine Hecke algebras is called the Deligne-Langlands-Lusztig conjecture ("DLL" for short) and can be seen as a part of the local Langlands correspondence.

In 2005, we found a certain deformation of this picture, yielding a classification of simple modules of multi-parameter affine Hecke algebras of classical types. This "deformation" was achieved by replacing the nilpotent cones of Lie algebras with some Hilbert nilcones which we propose to call "exotic nilpotent cones". Compared with the "DLL" picture, our picture has two advantages:

1. its geometric structure looks simpler (and was even more simplified by the work of Achar-Henderson);
2. it captures the parameter deformation in a essential way.

In this talk, I present our main construction and how it differs from the "DLL" picture. Based on the time remaining, I might present how 1) and 2) played a role in representation theory.