Abstract

For a locally compact totally disconnected topological group G one can define a ‘smooth’ representation. This is just a representation with an extra continuity condition. The category of all such representations is abelian, Noetherian and has enough projectives. In particular, one can study its projective dimension. In this talk we explain how to put a bound on the projective dimension of this category and moreover we show how to explicitly construct a projective resolution for each smooth G-module. The construction is inspired by a Theorem of Bernstein, who shows how this is done in the case of reductive p-adic groups. We generalise his approach to the case of an arbitrary locally compact totally disconnected group. However, the resolutions which Bernstein constructs are not of finitely generated projective modules. In the second part of the talk, following the work of Peter Schneider and Ulrich Stuhler for reductive algebraic groups, we explain how to construct finitely generated resolutions by passing to a category of G-equivariant objects, more precisely – a category of cosheaves on a simplicial complex on which G acts. Throughout we give examples of all our constructions for SL_n(Q_p) and GL_n(Q_p).