Bounding Ext for modules for algebraic groups, finite groups, and quantum groups

Given a finite root system $\Phi$, we show that there is an integer $c=c(\Phi)$ such that $\dim\Ext_G^1(L,L')<c$, for any reductive algebraic group $G$ with root system $\Phi$ and any irreducible rational $G$-modules $L,L'$. There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for $\Ext^n$, for any integer $n\geq 0$, using a constant depending only on $n$ and the root system. Weaker versions of this are proved in the algebraic and finite group cases, sufficient to give similar results for algebraic and generic cohomology. The results both use, and have consequences for, Kazhdan-Lusztig polynomials. An appendix proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.