On tensor categories attached to cells in affine Weyl groups. (English) Zbl 1078.20044

Bezrukavnikov, Roman

Shoji, Toshiaki (ed.) et al., Representation theory of algebraic groups and quantum groups. Papers from the conference held as the 10th International Research Institute of the Mathematical Society of Japan (MSJ-IRI) at Sophia University, Tokyo, Japan, August 1–10, 2001. Tokyo: Mathematical Society of Japan (ISBN 4-931469-25-6/hbk). Advanced Studies in Pure Mathematics 40, 69-90 (2004).

This paper is devoted to Lusztig’s bijection between unipotent conjugacy classes in a simple complex algebraic group and two-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 the affine flag variety of the dual group conjectured by G. Lusztig [in Adv. Math. 129, No. 1, 85-98 (1997; Zbl 0884.20026)]. His main tool is a recent construction by D. Gaitsgory, the so-called sheaf-theoretic construction of the center of an affine Hecke algebra [see Invent. Math. 144, No. 2, 253-280 (2001; Zbl 1072.14055)]. The author shows how this remarkable construction provides a geometric interpretation of the bijection, and allows to prove the conjecture.
For the entire collection see [Zbl 1050.20001].

Reviewer: Chen Chengdong (Shanghai)

MSC:

20G05	Representation theory for linear algebraic groups
20F55	Reflection and Coxeter groups (group-theoretic aspects)
20C08	Hecke algebras and their representations
14M15	Grassmannians, Schubert varieties, flag manifolds
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)

Keywords:

tensor categories; two-sided cells; affine Weyl groups; unipotent conjugacy classes; simple complex algebraic groups