A twisted invariant Paley-Wiener theorem for real reductive groups

Abstract

Let $G^{+}$ be the group of real points of a possibly disconnected linear reductive algebraic group defined over $\mathbb{R}$ which is generated by the real points of a connected component $G^{\text{'}}$. Let $K$ be a maximal compact subgroup of the group of real points of the identity component of this algebraic group. We characterize the space of maps $\pi \mapsto \mathrm{tr}(\pi (f))$, where $\pi$ is an irreducible tempered representation of $G^{+}$ and $f$ varies over the space of smooth, compactly supported functions on $G^{\text{'}}$ which are left and right $K$-finite. This work is motivated by applications to the twisted Arthur-Selberg trace formula