Globally analytic principal series representation and Langlands base change

TL;DR

This paper studies globally analytic principal series representations of GL(n) over p-adic fields, linking their irreducibility to Verma modules and constructing Langlands base change to unramified extensions, extending prior work for GL(2).

Contribution

It extends the analysis of locally and globally analytic principal series representations to GL(n), establishing irreducibility criteria and constructing Langlands base change in higher dimensions.

Findings

Irreducibility of globally analytic principal series linked to Verma modules.

Constructed Langlands base change for GL(n) over unramified extensions.

Generalized earlier results from GL(2) to higher-dimensional GL(n).

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 $GL(2)$ and related the condition of irreducibility of such locally analytic representation with that of a suitable Verma module. In this article, we take the case of $GL(n)$ and study the globally analytic principal series representation under the action of the pro-$p$ Iwahori subgroup of $GL(n,\mathbb{Z}_p)$, 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 Steinberg tensor product theorem, we construct Langlands base change of our globally analytic principal series to a finite unramified extension of $\mathbb{Q}_p$, generalizing an earlier work of Clozel for $GL(2)$.