Macdonald polynomials and decomposition numbers for finite unitary groups

(work in progress with R. Rouquier) I will present a computational (yet conjectural) method to determine some decomposition matrices for finite groups of Lie type. These matrices encode how ordinary representations decompose when they are reduced to a field with positive characteristic ℓℓ. There is an algorithm to compute them for GL(n,q)GL(n,q) when ℓℓ is large enough, but finding these matrices for other groups of Lie type is a very challenging problem.

In this talk I will focus on the finite general unitary group GU(n,q)GU(n,q). I will first explain how one can produce a "natural" self-equivalence in the case of GL(n,q)GL(n,q) coming from the topology of the Hilbert scheme of the complex plane. The combinatorial part of this equivalence is related to Macdonald's theory of symmetric functions and gives (q,t)(q,t)-decomposition numbers. The evidence suggests that the case of finite unitary groups is obtained by taking a suitable square root of that equivalence, which encodes the relation between GU(n,q)GU(n,q) and GL(n,−q)GL(n,−q).