Iwasawa Algebras and Parabolic Induction of p-adic Banach Representations

Author

Jeremiah Roberts, Southern Illinois University CarbondaleFollow

Date of Award

5-1-2024

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Ban, Dubravka

Abstract

Let G be a reductive group, and P a parabolic subgroup. Let L ⊆ K be finiteextensions of Qp and let G = G(L), P = P(L). In this thesis, we define the Iwasawa algebra K[[G]] and prove that it is isomorphic to the convolution algebra of compactly supported distributions on G. We show that under Schneider-Teitelbaum duality the func- tor of parabolic induction on the side of the admissible representations corresponds to the functor K[[G]] ⊗K[[P ]] − on the side of the K[[G]]-modules.This has important applications in the theory of admissible representations of G on p-adicBanach spaces. In particular, we prove the parabolic induction of an admissible represen- tation is again admissible, and prove Frobenius reciprocity for admissible representations.