Extending Hecke endomorphism algebras

Jie Du, Brian J. Parshall and Leonard L. Scott

Vol. 279 (2015), No. 1-2, 229–254

DOI: 10.2140/pjm.2015.279.229

Abstract

The (Iwahori–)Hecke algebra in the title is a $q$-deformation $\mathcal{\mathscr{H}}$ of the group algebra of a finite Weyl group $W$. The algebra $\mathcal{\mathscr{H}}$ has a natural enlargement to an endomorphism algebra $\mathcal{A}={End}_{\mathcal{\mathscr{H}}}\left(\mathcal{T}\right)$ where $\mathcal{T}$ is a $q$-permutation module. In type $A_n$ (i.e., $W\cong {\mathfrak{S}}_{n+1}$), the algebra $\mathcal{A}$ 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, $\mathcal{A}$ is not always quasi-hereditary, but the authors conjectured 20 year ago that $\mathcal{T}$ can be enlarged to an $\mathcal{\mathscr{H}}$-module ${\mathcal{T}}^{+}$ so that ${\mathcal{A}}^{+}={End}_{\mathcal{\mathscr{H}}}\left({\mathcal{T}}^{+}\right)$ 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 $\mathcal{\mathscr{H}}$ as defined over $\mathbb{Z}\left[t,t^{-1}\right]$, with the localization at a prime ideal generated by a cyclotomic polynomial ${\Phi }_{2e}\left(t\right)$, $e\ne 2$. The proof uses the theory of rational Cherednik algebras (also known as RDAHAs) over similar localizations of $ℂ\left[t,t^{-1}\right]$. In future papers, the authors hope to prove global versions of the conjecture, maintaining these localizations.

Keywords

Hecke algebra, Cherednik, root of unity, stratified, quasi-hereditary

Mathematical Subject Classification 2010

Primary: 20C08, 20C33

Secondary: 16S50, 16S80

Milestones

Received: 15 June 2015

Revised: 25 September 2015

Accepted: 25 September 2015

Published: 21 December 2015