On Tensor Categories Attached to Cells in Affine Weyl Groups Ii Roman Bezrukavnikov and Viktor Ostrik

Let R be a root system. Let W be the corresponding affine Weyl group, and let Ŵ be an extended affine Weyl group. Let H (respectively Ĥ) be the corresponding Hecke algebras. George Lusztig defined an asymptotic version of the Hecke algebra, the ring J , see [10]. By definition the ring J is a direct sum J = ⊕ c Jc where summation is over the set of two-sided cells in the affine Weyl group. Further, G. Lusztig proved that the set of two-sided cells in W is bijective to the set of unipotent conjugacy classes in an algebraic group over C with root system R, see [10] IV. Moreover, he proposed a Conjecture describing rings Jc in terms of convolution algebras, see [10] IV, 10.5 (a), (b). This Conjecture was verified in many cases by Nanhua Xi, see [16, 17, 18]. In this note we give a more conceptual proof of all previously known results. Our proof also works in some new cases. In general, we prove a statement (see Theorem 4 below) which is weaker than Lusztig’s Conjecture. The proof relies on many results of G.Lusztig in [10]. Our new essential tool is the theory of central sheaves on affine flag manifold due to A. Beilinson, D. Gaitsgory, R. Kottwitz, see [6]. One of us used this theory to prove a part of Lusztig’s Conjecture, see [4]. We would like to thank George Lusztig for useful conversations.