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On tensor categories attached to cells in affine Weyl groups. II. (English) Zbl 1078.20045
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, 101-119 (2004).
[For part I cf. the first author, ibid. 69-90 (2004; see the preceding review Zbl 1078.20044).]
Let $$W$$ be an affine Weyl group, and let $$\widehat W$$ be an extended affine Weyl group. Let $$\mathcal H$$ (respectively $$\widehat{\mathcal H}$$) be the corresponding Hecke algebra. G. Lusztig defined an asymptotic version of the Hecke algebra, the ring $$J$$ [see J. Algebra 109, 536-548 (1987; Zbl 0625.20032)]. By definition the ring $$J$$ is a direct sum $$J=\bigoplus_cJ_c$$ where summation is over the set of two-sided cells in the affine Weyl group. He proposed a conjecture describing the rings $$J_c$$ in terms of convolution algebras. This conjecture was verified in many cases. In this paper, the authors give a more conceptual proof of all previously known results. The authors prove a weak version of Lusztig’s conjecture on explicit description of the asymptotic affine Hecke algebra in terms of convolution algebras.
For the entire collection see [Zbl 1050.20001].

MSC:
 20G05 Representation theory for linear algebraic groups 20C08 Hecke algebras and their representations 20F55 Reflection and Coxeter groups (group-theoretic aspects)
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