| The two standard procedures for constructing representations of a reductive p-adic group G are: parabolic induction from a Levi subgroup; and compact induction from a compact, open subgroup. Parabolic induction, along with its adjoint Jacquet restriction, underlie the theory of the Bernstein center. On the other hand, the representations of the compact open subgroups of G are organized by chamber homology, which is a kind of equivariant homology for the Bruhat-Tits building. This dissertation studies the action of the parabolic/Jacquet functors and the Bernstein center on chamber homology.;We define an action of the Jacquet functors on chamber homology, and on the Hochschild homology of the Hecke algebra of G. Explicit descriptions of these actions are given: for general G, in the case of Jacquet restriction; and for G = SL2, in the case of parabolic induction. Our computations extend earlier results of van Dijk, Nistor, and Dat.;We conjecture (for general G) and prove (for SL 2) that a formula of Clozel, relating the Jacquet functors in degree zero with a certain geometric partition of G, continues to hold for the action of the Jacquet functors on higher homology. Our result for G = SL2 implies that the idempotents in the Bernstein center act on higher homology as diagonal operators, with respect to the decomposition of G into its compact and non-compact parts; this extends an earlier result of Dat in degree zero.;We also construct a canonical-up-to-homotopy chain complex which computes the Bernstein components of chamber homology. The problem of computing these components was first raised by Baum, Higson and Plymen in their paper of 2000, and our result provides a new perspective on the conjectures made in that paper. |
