A matrix Paley-Wiener theorem for non-connected $p$-adic reductive groups

TL;DR

This paper extends the Paley-Wiener theorem to non-connected p-adic reductive groups, characterizing the Fourier transform image for functions on such groups with abelian component quotients.

Contribution

It provides necessary and sufficient conditions for the existence of functions matching prescribed Fourier transforms on a class of induced representations of non-connected p-adic groups.

Findings

Characterization of the Fourier transform image for non-connected p-adic groups.

Conditions for the existence of functions with prescribed Fourier transforms.

Uniqueness of such functions in the specified setting.

Abstract

Let $F$ be a local non archimedian field of characteristic $0$, and $G$ a non-connected reductive group over $F$. We denote $G^0$ the connected component of the identity and assume the quotient $G/G^0$ is abelian. For $f$ a locally constant compactly supported function on $G$ and $\pi$ a complex smooth representation of $G$, we define the Fourier transform of $f$ evaluated at $\pi$ to be $\pi(f) = \int_{G} f(g) \pi(g) \, dg$, which is an endomorphism of the underlying vector space of $\pi$. We give a description of the image of this Fourier transform map : given, for every $\pi$ in a certain family of induced representations of $G$, an endomorphism $\varphi(\pi)$ of the underlying vector space, we provide necessary and sufficient conditions under which there exists a function $f$ (necessarily unique) such that $\pi(f) = \varphi(\pi)$ for all $\pi$ in the family.