Advanced Studies in Pure Mathematics
- Adv. Stud. Pure Math.
- Representation Theory of Algebraic Groups and Quantum Groups, T. Shoji, M. Kashiwara, N. Kawanaka, G. Lusztig and K. Shinoda, eds. (Tokyo: Mathematical Society of Japan, 2004), 69 - 90
On tensor categories attached to cells in affine Weyl groups
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.
Received: 13 February 2002
Revised: 12 November 2002
First available in Project Euclid: 3 January 2019
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Bezrukavnikov, Roman. On tensor categories attached to cells in affine Weyl groups. Representation Theory of Algebraic Groups and Quantum Groups, 69--90, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/04010069. https://projecteuclid.org/euclid.aspm/1546542382