Globally analytic principal series representation and Langlands base change

Jishnu Ray

Vol. 307 (2020), No. 2, 455–490

DOI: 10.2140/pjm.2020.307.455

Abstract

S. Orlik and M. Strauch have studied locally analytic principal series representation for general $p$-adic reductive groups generalizing an earlier work of P. Schneider for $GL\left(2\right)$ and related the condition of irreducibility of such locally analytic representation with that of a suitable Verma module. We take the case of $GL\left(n\right)$ and study the globally analytic principal series representation under the action of the pro-$p$ Iwahori subgroup of $GL\left(n,{\mathbb{Z}}_p\right)$, following the notion of globally analytic representations introduced by M. Emerton. Furthermore, we relate the condition of irreducibility of our globally analytic principal series to that of a Verma module. Finally, using the Steinberg tensor product theorem, we construct the Langlands base change of our globally analytic principal series to a finite unramified extension of ${\mathbb{Q}}_p$.

Keywords

rigid analytic representations, Tate algebra, affinoids, rigid analytic geometry, $p$-adic Langlands, $p$-adic representations, $p$-adic Lie groups

Mathematical Subject Classification 2010

Primary: 20G25, 22E50

Secondary: 20G05

Milestones

Received: 22 September 2018

Revised: 27 February 2020

Accepted: 5 March 2020

Published: 4 September 2020