  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.
