Ordinary representations of G(Q_p) and fundamental algebraic representations

TL;DR

This paper constructs a new class of unitary Banach space representations of G(Q_p) associated with ordinary G-hat representations, inspired by p-adic Langlands ideas, and explores their occurrence in cohomology for G=GL_n.

Contribution

It introduces a novel construction of ordinary representations of G(Q_p) from G-hat representations, extending the p-adic Langlands program framework.

Findings

Pi(rho)^ord is shown to occur in the rho-part of cohomology for G=GL_n under certain conditions.

A weaker version of the cohomological occurrence result is proven in the p-adic case.

The construction generalizes the ordinary part of tensor products of fundamental algebraic representations.

Abstract

Let G be a split connected reductive algebraic group over Q_p such that both G and its dual group G-hat have connected centres. Motivated by a hypothetical p-adic Langlands correspondence for G(Q_p) we associate to an n-dimensional ordinary (i.e. Borel valued) representation rho : Gal(Q_p-bar/Q_p) to G-hat(E) a unitary Banach space representation Pi(rho)^ord of G(Q_p) over E that is built out of principal series representations. (Here, E is a finite extension of Q_p.) Our construction is inspired by the "ordinary part" of the tensor product of all fundamental algebraic representations of G. There is an analogous construction over a finite extension of F_p. In the latter case, when G=GL_n we show under suitable hypotheses that Pi(rho)^ord occurs in the rho-part of the cohomology of a compact unitary group. We also prove a weaker version of this result in the p-adic case.