Fourier Transform from the Symmetric Square Representation of PGL2 and SL2

Abstract

Abstract Let $G$ be a connected reductive group over $\overline{{\mathbb{F}}}_{q}$ and let $\rho ^{\vee }:G^{\vee }\rightarrow \mathrm{GL}_{n}$ be an algebraic representation of the dual group $G^{\vee }$. Assuming that $G$ and $\rho ^{\vee }$ are defined over ${\mathbb{F}}_{q}$, Braverman and Kazhdan defined an operator on the space ${{\mathcal{C}}}(G({\mathbb{F}}_{q}))$ of complex valued functions on $G({\mathbb{F}}_{q})$. In this paper we are interested in the case where $G$ is either $\mathrm{SL}_{2}$ or $\mathrm{PGL}_{2}$ and $\rho ^{\vee }$ is the symmetric square representation of $G^{\vee }$. We construct a natural $G\times G$-equivariant embedding $G\hookrightarrow{{\mathcal{G}}}={{\mathcal{G}}}_{\rho }$ and an involutive operator (Fourier transform) ${{\mathcal{F}}}^{{{\mathcal{G}}}}$ on the space of functions ${{\mathcal{C}}}({{\mathcal{G}}}({\mathbb{F}}_{q}))$ that extends Braverman-Kazhdan’s operator.