Locally unitary principal series representations of ${\rm GL}_{d+1}(F)$

TL;DR

This paper studies tamely ramified principal series representations of ${ m GL}_{d+1}(F)$ over local fields, showing that locally unitary cases admit well-structured integral models with explicit modular reductions.

Contribution

It demonstrates that locally unitary principal series representations have explicitly constructible integral structures and modular reductions, enhancing understanding of their algebraic properties.

Findings

Existence of well-organized integral structures for locally unitary representations

Explicit description of modular reductions of these integral structures

Clarification of the structure of ${ m GL}_{d+1}(F)$ representations in the tamely ramified case

Abstract

For a local field $F$ we consider tamely ramified principal series representations $V$ of $G={\rm GL}_{d+1}(F)$ with coefficients in a finite extension $K$ of ${\mathbb Q}_p$. Let $I_0$ be a pro-$p$-Iwahori subgroup in $G$, let ${\mathcal H}(G,I_0)$ denote the corresponding pro-$p$-Iwahori Hecke algebra. If $V$ is locally unitary, i.e. if the ${\mathcal H}(G,I_0)$-module $V^{I_0}$ admits an integral structure, then such an integral structure can be chosen in a particularly well organized manner, in particular its modular reduction can be made completely explicit.