Resolutions for principal series representations of p-adic GL(n)

Let F be a nonarchimedean locally compact field with residue characteristic p and G (F) the group of F-rational points of a connected reductive group. In [12], Schneider and Stuhler realize, in a functorial way, any smooth complex finitely generated representation of G (F) as the 0-homology of a certain coefficient system on the semisimple building of G. It is known that this method does not apply in general for smooth mod p representations of G (F), even when G = GL2. However, we prove that a principal series representation of GLn (F) over a field with arbitrary characteristic can be realized as the 0-homology of the corresponding coefficient system as defined in [12].