On Lusztig-Dupont homology of flag complexes

Let $V$ be an $n$-dimensional vector space over the finite field of order $q$. The spherical building $X_V$ associated with $GL(V)$ is the order complex of the nontrivial linear subspaces of $V$. Let $\mathfrak{g}$ be the local coefficient system on $X_V$, whose value on the simplex $\sigma=[V_0 \subset \cdots \subset V_p] \in X_V$isgivenby$\mathfrak{g}(\sigma)=V_0$. Following the work of Lusztig and Dupont, we study the homology module $D^k(V)={\tilde{H}}_{n-k-1}(X_V;\mathfrak{g})$. Our results include a construction of an explicit basis of $D^1(V)$, and the following twisted analogue of a result of Smith and Yoshiara: For any $1\le k\le n-1$, the minimal support size of a non-zero $(n-k-1)$-cycle in the twisted homology ${\tilde{H}}_{n-k-1}(X_V;{\wedge }^k\mathfrak{g})$ is $\frac{(n-k+2)!}{2}$.