Let $V$ be an $n$-dimensional vector space over a finite field $F$ (of order $q$). Denote by $\mathrm{AGL}(V)$ the group of invertible affine transformations of $V$; so $\mathrm{AGL}(V)$ consists of all mappings $x\mapsto Tx+v$ where $T\in GL(V)$ and $v\in V$.

Clearly, $\mathrm{AGL}(V)$ acts on the poset of all affine subspaces of $V$ where an affine subspace is, of course, a coset of a linear subspace. Let $P$ be the subposet of proper affine subspaces and let $\Delta(P)$ be the order complex of $P$. Then $\mathrm{AGL}(V)$ acts simplicially on $\Delta(P)$ and hence acts on the reduced homology of $\Delta(P)$.

Now $P$ is, it seems to me, the proper part of a geometric lattice. If we declare a subset of $V$ to be independent if its elements are affinely independent, then I believe this gives a matroid whose corresponding lattice of flats is the set of affine subspaces (and I guess the empty set). So $P$ should be the proper part of this and hence its reduced homology should be concentrated in the top dimension, which I guess is $n-1$ if I count correctly. Does anybody know any literature describing the decomposition of $\widetilde{H}_{n-1}(\Delta(P),\mathbb C)$ into irreducible $\mathbb C\mathrm{AGL}(V)$-modules? Is it irreducible?

Of course, the easy case is when dimension $n$ of $V$ is $1$. Then $AGL(V)$ consists of all invertible $ax+b$-maps and the proper affine subspaces are points. So the reduced homology in dimension $0$ is the augmentation submodule of the permutation module of $F\rtimes F^\times$ acting on $F$. This action is doubly transitive, and so the augmentation submodule is irreducible of degree $q-1$ (and is the unique irreducible representation of degree greater than $1$).