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

Abstract

Given a finite root system Φ , we show that there is an integer c = c ( Φ ) such that dim Ext G 1 ( L , L ′ ) < c , for any reductive algebraic group G with root system Φ 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 ⩾ 0 , using a constant depending only on n and the root system. When L is the trivial module, the same result is proved in the algebraic group case, thus giving similar bounded properties, independent of characteristic, for algebraic and generic cohomology. (A similar result holds for any choice of L = L ( λ ) , even allowing λ to vary, provided the p -adic expansion of lambda is limited to a fixed number of terms.) In particular, because of the interpretation of generic cohomology as a limit for underlying families of finite groups, the same boundedness properties hold asymptotically for finite groups of Lie type. The results both use, and have consequences for, Kazhdan–Lusztig polynomials. Appendix A proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.