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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. Following Schneider and Stuhler, one can 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 semi-simple building of G(F). It is known that this method does not apply in general for smooth mod p representations of G(F), even when G= GL(2). However, we prove that a principal series representation of GL(n,F) over a field with arbitrary characteristic can be realized as the 0-homology of the corresponding coefficient system.