Algebraic group

Pure mathematics

Flag (linear algebra)

Mathematics

Centralizer and normalizer

Crystallography

Group theory

Current algebra

Algebra over a field

Mathematical analysis

Langlands dual group

Reductive group

Chemistry

Unipotent

Center (category theory)

Weyl group

Affine Lie algebra

Affine representation

Conjecture

Conjugacy class

Affine transformation

Algebraic number

Advanced studies in pure mathematics

This note is devoted to Lusztig's bijection between unipotent conjugacy classes in a simple complex algebraic group and 2-sided cells in the affine Weyl group of the Langlands dual group; and also to the description of the reductive quotient of the centralizer of the unipotent element in terms of convolution of perverse sheaves on affine flag variety of the dual group conjectured by Lusztig in [L4]. Our main tool is a recent construction by Gaitsgory (based on an idea of Beilinson and Kottwitz), the so-called sheaf-theoretic construction of the center of an affine Hecke algebra (see [Ga]). We show how this remarkable construction provides a geometric interpretation of the bijection, and allows to prove the conjecture.

On tensor categories attached to cells in affine Weyl groups