Ordinary representations of G ( Q p ) and fundamental algebraic representations

Abstract

Let $G$ be a split connected reductive algebraic group over ${\mathbb{Q}}_p$ such that both $G$ and its dual group $\hat{G}$ have connected centers. Motivated by a hypothetical $p$-adic Langlands correspondence for $G\left({\mathbb{Q}}_p\right)$, we associate to an $n$-dimensional ordinary (i.e., Borel-valued) representation $\rho :Gal({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)\to \hat{G}\left(E\right)$ a unitary Banach space representation $\Pi (\rho {)}^{\mathrm{ord}}$ of $G\left({\mathbb{Q}}_p\right)$ over $E$ that is built out of principal series representations. (Here, $E$ is a finite extension of ${\mathbb{Q}}_p$.) 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 ${\mathbb{F}}_p$. When $G=G{L}_n$, we show under suitable hypotheses that $\Pi (\rho {)}^{\mathrm{ord}}$ occurs in the $\rho$-part of the cohomology of a compact unitary group.