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 ${\mathbb{Q}}_p$.

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Mathematical Subject Classification 2010

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