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

TL;DR

This paper establishes uniform bounds on the dimensions of Ext groups for modules over algebraic groups, finite groups, and quantum groups, linking these bounds to the root system and providing new insights into cohomological properties.

Contribution

It proves the existence of uniform bounds on Ext dimensions depending only on the root system for various classes of groups, extending to quantum groups and connecting with Kazhdan-Lusztig polynomials.

Findings

Bound on Ext^1 dimensions for algebraic groups depending on root system

Bound on Ext^n for quantum groups depending on root system and n

Results have implications for algebraic and generic cohomology

Abstract

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.