Intertwining maps between 𝑝-adic principal series of 𝑝-adic groups

Ban, Dubravka; Hundley, Joseph

doi:10.1090/ert/571

Intertwining maps between $p$-adic principal series of $p$-adic groups
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by Dubravka Ban and Joseph Hundley

Represent. Theory 25 (2021), 975-993

DOI: https://doi.org/10.1090/ert/571

Published electronically: December 1, 2021

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Abstract:

In this paper we study $p$-adic principal series representation of a $p$-adic group $G$ as a module over the maximal compact subgroup $G_0$. We show that there are no non-trivial $G_0$-intertwining maps between principal series representations attached to characters whose restrictions to the torus of $G_0$ are distinct, and there are no non-scalar endomorphisms of a fixed principal series representation. This is surprising when compared with another result which we prove: that a principal series representation may contain infinitely many closed $G_0$-invariant subspaces. As for the proof, we work mainly in the setting of Iwasawa modules, and deduce results about $G_0$-representations by duality.

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