# Globally Analytic *p*-adic Representations of the Pro–*p* Iwahori Subgroup of *GL*(2) and Base Change, II: A Steinberg Tensor Product Theorem

## Abstract

In this paper, which is a sequel to Clozel (Globally analytic p-adic representations of the pro-p Iwahori subgroup of GL(2) and base change, I: Iwasawa algebras and a base change map, to appear in Bull. Iran Math Soc, [4]), we exploit the base change map for globally analytic distributions constructed there, relating distributions on the pro-p Iwahori subgroup of *GL*(2) over \(\mathbb {Q}_p\) and those on the pro-p Iwahori subgroup of *GL*(2, *L*) where *L* is an unramified extension of \(\mathbb {Q}_p\). This is used to obtain a functor, the ‘Steinberg tensor product’, relating globally analytic *p*-adic representations of these two groups. We are led to extend the theory, sketched by Emerton (Locally analytic vectors in representations of locally p-adic analytic groups, [6]), of these globally analytic representations. In the last section we show that this functor exhibits, for principal series, Langlands’ base change (at least for the restrictions of these representations to the pro-p Iwahori subgroups.)

## Keywords

11R23 11F70 14G22## References

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