2015/01/26 by Jie Du, Du, Jie, Brian Parshall +4
Mathematics · #16G99 #20G43 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:16G99 #msc:20G43
paper · pdf · doi:10.48550/arxiv.1501.06481
This is a revised version. The paper will appear in Pacific J. Math., in volume dedicated to Robert Steinberg
openalex publication_date 2015/01/26 · arxiv created 2015/09/26 · arxiv updated 2015/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The (Iwahori-)Hecke algebra in the title is a q-deformation \sH of the group algebra of a finite Weyl group W. The algebra \sH has a natural enlargement to an endomorphism algebra \sA=\End_\sH(\sT) where \sT is a q-permutation module. In type An (i.e., W≅ \mathfrak Sn+1), the algebra \sA is a q-Schur algebra which is quasi-hereditary and plays an important role in the modular representation of the finite groups of Lie type. In other types, \sA is not always quasi-hereditary, but the authors conjectured 20 year ago that \sT can be enlarged to an \sH-module \sT+ so that \sA+=\End_\sH(\sT+) is at least standardly stratified, a weaker condition than being quasi-hereditary, but with "strata" corresponding to Kazhdan-Lusztig two-sided cells. The main result of this paper is a "local" version of this conjecture in the equal parameter case, viewing \sH as defined over \mathbb Z[t,t-1], with the localization at a prime ideal generated by a cyclotomic polynomial Φ2e(t), e\not=2. The proof uses the theory of rational Cherednik algebras (also known as RDAHAs) over similar localizations of \mathbb C[t,t-1]. In future paper, the authors expect to apply these results to prove global versions of the conjecture, at least in the equal parameter case with bad primes excluded.