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The (Iwahori-)Hecke algebra in the title is a $q$-deformation $\sH$ of thegroup algebra of a finite Weyl group $W$. The algebra $\sH$ has a naturalenlargement to an endomorphism algebra $\sA=\End_\sH(\sT)$ where $\sT$ is a$q$-permutation module. In type $A_n$ (i.e., $W\cong {\mathfrak S}_{n+1}$), thealgebra $\sA$ is a $q$-Schur algebra which is quasi-hereditary and plays animportant role in the modular representation of the finite groups of Lie type.In other types, $\sA$ is not always quasi-hereditary, but the authorsconjectured 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 weakercondition than being quasi-hereditary, but with "strata" corresponding toKazhdan-Lusztig two-sided cells.  The main result of this paper is a "local" version of this conjecture in theequal 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 Cherednikalgebras (also known as RDAHAs) over similar localizations of ${\mathbbC}[t,t^{-1}]$. In future paper, the authors expect to apply these results toprove global versions of the conjecture, at least in the equal parameter casewith bad primes excluded.

Extending Hecke endomorphism algebras at roots of unity