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

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 $π$ a complex smooth representation of $G$, we define the Fourier transform of $f$ evaluated at $π$ to be $π(f) = \int_{G} f(g) π(g) \, dg$, which is an endomorphism of the underlying vector space of $π$. We give a description of the image of this Fourier transform map : given, for every $π$ in a certain family of induced representations of $G$, an endomorphism $φ(π)$ of the underlying vector space, we provide necessary and sufficient conditions under which there exists a function $f$ (necessarily unique) such that $π(f) = φ(π)$ for all $π$ in the family.