Leonard Scott (Charlottesville), Semisimple series for q-Weyl and q-Specht modules, and some related integral representation theory of affine Lie algebras and quantum groups.

Abstract:

This talk will be divided into two related sessions. Both take as their starting point weaker results I intend to discuss at the DMV conference in Koeln. However, the techniques required to go further are independent and quite different, and will occupy my discussions here in Stuttgart. All results discussed are part of joint work with Brian Parshall.

The first topic is filtrations of q-Schur algebras Weyl modules (q-Weyl modules) with semisimple and explicitly computable sections. (Multiplicities are given by coefficients for inverse Kazhdan-Lusztig polynomials). The result for Koeln will take us as far as large roots of unity, but I will discuss here how to parlay that into results for all roots of unity. The methods involve passing to affine Kac-Moody Lie algebras. In part, we will make use of deformation results of Peter Fiebig, which remind me, at least, of constructions of orders in finite group rings, though they are carried out for affine Lie algebras. Because we are able to handle all roots of unity, we can pass by a Schur functor to q-Specht modules, and obtain explicit semisimple series for them. This is all in characteristic 0, though there are some implications for characteristic p>0, if you are willing to assume the James conjecture holds, as well as a suitable bipartite conjecture (on Ext1 quivers).

The second topic of the talk concerns filtrations on integral versions of certain quotients of quantum enveloping algebras. I will review the notions of integral quasi-hereditary algebras and integral versions of generalized q-Schur algebras. Filtrations will be introduced which lead to natural integral graded algebras, modifications of the integral generalized q-Schur algebras. In some cases, the reductions mod p of these integral graded algebras are both Kozul and quasi-hereditary, and enjoy the strong parity properties of a "Kazhdan-Lusztig theory" in their homological algebra. If we just stick to algebras over fields, then the subject matter here is something that will be discussed in Koeln, with more restrictions on primes, and on the highest weights of composition factors. But the extension to the integral case both broadens and deepens this theory. For instance, filtration sections, that could only be viewed as semisimple for small weights, become "p-filtrations", involving tensors of restricted irreducible modules with twisted Weyl modules, when larger weights are considered. It seems necessary to come down from the integral case in characteristic 0 to achieve such results.