The based ring the lowest generalized two-sided cell of an extended affine Weyl group

2014/03/13 by Xun Xie, Xie, Xun
Mathematics · #20C08 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1403.3213

openalex publication_date 2014/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let c₀ be the lowest generalized two-sided cell of an extended affine Weyl group W. We determine the structure of the based ring of c₀. For this we show that certain conjectures of Lusztig on generalized cells (called P1-P15) hold for c₀. As an application, we use the structure of the based ring to study certain simple modules of Hecke algebras of W with unequal parameters, namely those attached to c₀. Also we give a set of prime ideals \mathfrakp of the center Z of the generic affine Hecke algebra H such that the reduced affine Hecke algebra k_\mathfrakpH is simple over k_\mathfrakp, where k_\mathfrakp=\rmFrac(Z/\mathfrakp) is the residue field of Z at \mathfrakp. In particular, we show that the algebra H⊗_Z\rmFrac(Z) is a split simple algebra over the field \rmFrac(Z).

Citations

• A factorization theorem for affine Kazhdan-Lusztig basis elements

Related

• The based ring of the lowest two-sided cell of an affine Weyl group, III
• The Based Rings of Two-sided cells in an Affine Weyl group of type B₃, IV
• The based rings of two-sided cells in an affine Weyl group of type B₃, II
• The Based Rings of Two-sided cells in an Affine Weyl group of type B₃, III
• The based rings of two-sided cells in an affine weyl group of type B₃, I