  Let $G$ be a finite group of Lie type. In studying the cross-characteristic
representation theory of $G$, the (specialized) Hecke algebra
$H=\End_G(\ind_B^G1_B)$ has played a important role. In particular, when
$G=GL_n(\mathbb F_q)$ is a finite general linear group, this approach led to
the Dipper-James theory of $q$-Schur algebras $A$. These algebras can be
constructed over $\sZ:=\mathbb Z[t,t^{-1}]$ as the $q$-analog (with $q=t^2$) of
an endomorphism algebra larger than $H$, involving parabolic subgroups. The
algebra $A$ is quasi-hereditary over $\sZ$. An analogous algebra, still denoted
$A$, can always be constructed in other types. However, these algebras have so
far been less useful than in the $GL_n$ case, in part because they are not
generally quasi-hereditary.
  Several years ago, reformulating a 1998 conjecture, the authors proposed (for
all types) the existence of a $\sZ$-algebra $A^+$ having a stratified derived
module category, with strata constructed via Kazhdan-Lusztig cell theory. The
algebra $A$ is recovered as $A=eA^+e$ for an idempotent $e\in A^+$. A main goal
of this monograph is to prove this conjecture completely. The proof involves
several new homological techniques using exact categories. Following the proof,
we show that $A^+$ does become quasi-hereditary after the inversion of the bad
primes. Some first applications of the result -- e.g., to decomposition
matrices -- are presented, together with several open problems.
