Let G be a connected reductive group and e be a nilpotent element in the Lie algebra of G. Let F_e be a maximal reductive subgroup of the centralizer of e in G. Let W be the extended affine Weyl group associated to G, and c the two-sided cell of W corresponding to e. Lusztig conjectured that the based ring J_c of c should be isomorphic to a F_e-equivarant K-group on the square of a finite set Y_e (G simply-connected).  Bezrukavnikov and Ostrik proved a weaker form of this conjecture in 2004, by introducing an additional centrally extended structure. We prove that when e lies in the minimal nilpotent orbit, this centrally extended structure is trivial, and hence Lusztig’s conjecure holds for the second lowest cell. As a corollary, we confirm a conjecture posed by Jianyi Shi in 2011 that the the number of left cell in the second lowest two-sided cell is half cardinality of the finite Weyl group. This is joint work with Qianfan Zhou.

谢迅，北京理工大学副教授。研究领域：李理论, 主要研究Hecke代数的Kazhdan-Lusztig基及其相关问题。代表性成果发表在著名综合期刊IMRN、Adv. Math.，以及代数专业杂志J. Algebra、Representation theory等。主持国家青年基金与国家面上项目各一项。