Schneider-Teitelbaum duality for locally profinite groups

Tomoki Mihara

doi:10.29252/cgasa.14.1.119

Abstract

We define monoidal structures on several categories of linear topological modules over the valuation ring of a local field, and study module theory with respect to the monoidal structures. We extend the notion of the Iwasawa algebra to a locally profinite group as a monoid with respect to one of the monoidal structure, which does not necessarily form a topological algebra. This is one of the main reasons why we need monoidal structures. We extend Schneider--Teitelbaum duality to duality applicable to a locally profinite group through the module theory over the generalised Iwasawa algebra, and give a criterion of the irreducibility of a unitary Banach representation.

Highlights

Let k denote a non-Archimedean local field, and Ok ⊂ k the valuation ring of k
We show that every Banach k-vector space and every compact linear topological Ok-module are CG
cg and •0 ×cg •1 : (C cg) contains both of the categories of Banach k-vector spaces and compact Hausdorff flat linear topological Ok-modules, which play the roles of the foundation in Schneider–Teitelbaum duality

Summary

Introduction

Let k denote a non-Archimedean local field, and Ok ⊂ k the valuation ring of k. One topic is to give monoidal structures on several categories of linear topological Ok-modules. We show that every Banach k-vector space and every compact linear topological Ok-module are CG. C cg contains both of the categories of Banach k-vector spaces and compact Hausdorff flat linear topological Ok-modules, which play the roles of the foundation in Schneider–Teitelbaum duality (cf [13] Theorem 2.3). The other topic is to define a generalised Iwasawa algebra Ok[[G]] associated to a locally profinite group G, and to extend Schneider–Teitelbaum duality, which is applicable to a profinite group, to duality applicable to G by using module theory over Ok[[G]].

Preliminaries

Monoidal structures

Modules over Iwasawa algebras

Generalised Schneider-Teitelbaum duality

Applications