Unipotent Brauer Character Values of GL(n, \mathbb{F}_q ) and the Forgotten Basis of the Hall Algebra

We give a formula for the values of irreducible unipotent p-modular Brauer characters of at unipotent elements, where p is a prime not dividing q, in terms of (unknown!) weight multiplicities of quantum GL _n and certain generic polynomials S _,(q). These polynomials arise as entries of the transition matrix between the renormalized Hall-Littlewood symmetric functions and the forgotten symmetric functions. We also provide an alternative combinatorial algorithm working in the Hall algebra for computing S _,(q).