Forced gradings in integral quasi-hereditary algebras with applications to quantum groups

Let $\sO$ be a discrete valuation ring with fraction field $K$ and residue field $k$. A quasi-hereditary algebra $\wA$ over $\sO$ provides a bridge between the representation theory of the quasi-hereditary algebra $\wA_K:=K\otimes \wA$ over the field $K$ and the quasi-hereditary algebra $A_k:=k\otimes_\sO\wA$ over $k$. In one important example, $\wA_K$--mod is a full subcategory of the category of modules for a quantum enveloping algebra while $\wA_k$--mod is a full subcategory of the category of modules for a reductive group in positive characteristic. This paper considers first the question of when the positively graded algebra $\gr \wA:= \bigoplus_{n\geq 0}(\wA\cap\rad^n\wA_K)/(\wA\cap\rad^{n+1}\wA_K)$ is quasi-hereditary. A main result gives sufficient conditions that $\gr\wA$ be quasi-hereditary. The main requirement is that each graded module $\gr\wDelta(\lambda)$ arising from a $\wA$-standard (Weyl) module $\wDelta(\lambda)$ have an irreducible head. An additional hypothesis requires that the graded algebra $\gr \wA_K$ be quasi-hereditary, a property recently proved by us to hold in some important cases involving quantum enveloping algebras. In the case where $\wA$ arises from regular dominant weights for a quantum enveloping algebra at a primitive $p$th root of unity for a prime $p>2h-2$ (where $h$ is the Coxeter number), a second main result shows that $\gr\wA$ is quasi-hereditary. The proof depends on previous work of the authors, including a continuation of the methods there involving tightly graded subalgebras, and a development of a quantum deformation theory over $\sO$, worthy of attention in its own right, extending the work of Andersen-Jantzen-Soergel. As we point out, this work provides an essential step in our work on $p$-filtrations of Weyl modules for reductive algebraic groups over fields of positive characteristic.