On Lusztig-Dupont homology of flag complexes

Let V be an n-dimensional vector space over the finite field F _q . The spherical building X _V associated with GL(V) is the order complex of the nontrivial linear subspaces of V. Let g be the local coefficient system on X _V , whose value on the simplex σ=[V ₀ ⊂⋯⊂V _p ]∈X _V is given by g(σ)=V ₀ . The homology module D ¹ (V)=H˜ _n−2 (X _V ;g) plays a key role in Lusztig's seminal work on the discrete series representations of GL(V). Here, some further properties of g and its exterior powers are established. These include a construction of an explicit basis of D ¹ (V), a computation of the dimension of D ^k (V)=H˜ _n−k−1 (X _V ;∧ ^k g), and the following twisted analogue of a result of Smith and Yoshiara: For any 1≤k≤n−1, the minimal support size of a non-zero (n−k−1)-cycle in the twisted homology H˜ _n−k−1 (X _V ;∧ ^k g) is ([Formula presented].