$U_n(q)$ acting on flags and supercharacters

Let $U=U_n(q)$ be the group of lower unitriangular $n\times n$-matrices with entries in the field ${\mathbb{F}}_q$ with $q$ elements for some prime power $q$ and $n\in \mathbb{N}$. We investigate the restriction to $U$ of the permutation action of $G{L}_n(q)$ on flags in the natural $G{L}_n(q)$-module ${\mathbb{F}}_q^n$. Applying our results to the special case of flags of length two we obtain a complete decomposition of the permutation representation of $G{L}_n(q)$ on the cosets of maximal parabolic subgroups into irreducible $\mathbb{C}U$-modules.