Bounding Ext for Modules for Algebraic Groups, Finite Groups and Quantum Groups

Given a finite root system Φ, we show there is an integer c = c(Φ) such that dimExtG(L,L ) < c, for any reductive algebraic groupG with root system Φ and any irreducible rational G-modules L,L. We also prove that there 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, we are able to obtain a similar result for Ext, for any integer n ≥ 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 for large characteristic. Our results both use, and have consequences for, Kazhdan-Lusztig polynomials. We also introduce the new tool of a shifted standard (or costandard) module filtration for an object in the derived category of a highest weight category.