Types et contragr\'edientes

TL;DR

This paper investigates the behavior of contragredient representations of p-adic reductive groups, establishing conditions under which their properties mirror those over complex fields, with applications to the theory of types for GL(n).

Contribution

It provides a general framework for understanding contragredient representations over algebraically closed fields beyond complex numbers, with specific applications to GL(n) and its inner forms.

Findings

Conditions for contragredient behavior equivalence

Application to types in GL(n) and inner forms

Extension of classical results to algebraically closed fields

Abstract

Let G be a p-adic reductive group, and R an algebraically closed field. Let us consider a smooth representation of G on an R-vector space V. Fix an open compact subgroup K of G and a smooth irreducible representation of K on a finite-dimensional R-vector space W. The space of K-homomorphisms from W to V is a right module over the intertwining algebra H(G,K,W). We examine how those constructions behave when we pass to the contragredient representations of V and W, and we give conditions under which the behaviour is the same as in the case of complex representations. We take an abstract viewpoint and use only general properties of G. In the last section, we apply this to the theory of types for the group GL(n) and its inner forms over a non-Archimedean local field.