On Continuous Principal Series Representations of reductive $p$-adic groups on $p$-adic Banach Spaces

Author

Venkata Sai Manisha Varahagiri, Southern Illinois University CarbondaleFollow

Date of Award

8-1-2023

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Ban, Dubravka

Abstract

This thesis focuses on $p$-adic Banach space representations, with a specific emphasis on continuous principal series representations. The motivation for this research stems from the desire to investigate continuous principal series representations in the context of general connected reductive groups.The objective is to broaden existing knowledge by extending the results obtained for continuous principal series representations of split-connected reductive $p$-adic groups to include any arbitrary connected reductive $p$-adic groups. Building upon the work of D. Ban and J. Hundley in 2016 on split-connected reductive groups, we follow a similar approach but encompass a wider class of reductive groups. The structural aspects of Bruhat-Tits building theory are utilized to analyze connected reductive $p$-adic groups.We investigate continuous principal series representations, which are defined for connected reductive groups. These representations are explored both on the connected reductive group $G$ and its maximal compact open subgroup $G_0$. One of the primary results of this research is the duality theorem of continuous principal series representations which establishes an isomorphism between the continuous dual of principal series of $G_0$, denoted by ${\text{Ind}_{P_0}^{G_0}}(\chi^{-1})$, and $\mathcal{M}^{(\chi)}$, where $\mathcal{M}^{(\chi)}$ is an Iwasawa module. Consequently, this result shows that ${\text{Ind}_{P_0}^{G_0}}(\chi^{-1})$ is an admissible banach space representation. Furthermore, we present a projective limit realization for the Iwasawa module $M_0^{(\chi)}= \mathcal{O}_K[[G_0]] {\otimes}_{\mathcal{O}_K[[P_0]]} \mathcal{O}_K$.