Extending Hecke endomorphism algebras at roots of unity

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 $A_n$ (i.e., $W\cong {\mathfrak S}_{n+1}$), 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 $\Phi_{2e}(t)$, $e\not=2$. 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.