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Let $U=U_n(q)$ be the group of lower unitriangular $n \times n$-matrices withentries 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 permutationaction of $GL_n(q)$ on flags in the natural $GL_n(q)$-module $\mathbb F_q^n$.Applying our results to the special case of flags of length two we obtain acomplete decomposition of the permutation representation of $GL_n(q)$ on thecosets of maximal parabolic subgroups into irreducible $\mathbb C U$-modules.

$U_n(q)$ acting on flags and supercharacters