Whittaker functionals and contragredient in characteristic not $p$

TL;DR

This paper establishes a dimension equality for certain Hom spaces of smooth representations over fields of arbitrary characteristic, leading to new insights into Whittaker models and multiplicity one results in non-Archimedean harmonic analysis.

Contribution

It proves a dimension equality for Hom spaces involving contragredients over fields of arbitrary characteristic, extending classical results to positive characteristic settings.

Findings

Dimension equality for Hom spaces with finite dimension

Application to Whittaker models and multiplicity one theorems

Generalization of Prasad's conjecture over arbitrary fields

Abstract

Let $R$ be an algebraically closed field and $\ell$ be its characteristic. Let $G$ be a locally profinite group having a compact open subgroup of invertible pro-order in $R$. Take $N$ a closed subgroup of $G$ exhausted by compact subgroups of invertible pro-orders in $R$ and fix a smooth character $\theta$ of $N$. For $\pi$ an irreducible smooth $R$-representation of $G$ whose matrix coefficients are compactly supported modulo the center (we call it $Z$-compact), we show that the dimensions $\mathrm{Hom}_{N}(\pi,\theta)$ and $\mathrm{Hom}_{N}(\pi^\vee,\theta^{-1})$ are equal provided one of the two is finite. We derive a few applications from this result. First, we prove that any $G$-intertwiner from $\pi$ to $\mathrm{Ind}_N^G(\theta)$ has image in $\mathrm{ind}_{ZN}^G(\omega_\pi \theta)$, where $\omega_\pi$ is the central character of $\pi$, and the Whittaker space of $\pi$ agrees with that of its Whittaker periods. Second, it applies to quasi-split groups over non Archimedean local fields of residual characteristic $p \neq \ell$ and where $N$ is the unipotent radical of a Borel subgroup of $G$ together with a generic character $\theta$. Our equality of dimensions turns out to be a good replacement for Rodier's crucial use of complex conjugation in the proof of Whittaker multiplicity at most one for cuspidal representations. Then by a lifting argument, we recover Rodier's generalization of the Gelfand-Kazhdan property for $R$-valued $(\theta^{-1}\otimes \theta)$-equivariant distributions on $G$. This latter fact, together with Rodier's heridity property, which is valid in our context, leads to the multiplicity at most one of Whittaker functionals over $R$. We also give other applications, including a generalization over $R$ of a result for complex representations proved by Chang Yang and initially conjectured by Dipendra Prasad.