Un(q) acting on flags and supercharacters

Let U=Un(q) be the group of lower unitriangular n×n-matrices with entries in the field Fq with q elements for some prime power q and n∈N. We investigate the restriction to U of the permutation action of GLn(q) on flags in the natural GLn(q)-module Fqn. Applying our results to the special case of flags of length two we obtain a complete decomposition of the permutation representation of GLn(q) on the cosets of maximal parabolic subgroups into irreducible CU-modules.