Fourier transform from the symmetric square representation of $PG{L}_2$ and $S{L}_2$

Let $G$ be a connected reductive group over ${\bar{\mathbb{F}}}_q$ and let ${\rho }^{\vee }:G^{\vee }\to G{L}_n$ 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 $S{L}_2$ or $PG{L}_2$ 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.