#### Vol. 307, No. 2, 2020

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Globally analytic principal series representation and Langlands base change

### Jishnu Ray

Vol. 307 (2020), No. 2, 455–490
##### 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 $GL\left(n,{ℤ}_{p}\right)$, 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 ${ℚ}_{p}$.

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##### 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
##### Authors
 Jishnu Ray Department of Mathematics The University of British Columbia Vancouver BC Canada