  Let $G$ be a connected reductive group over $\overline{\mathbb{F}}_q$ and let
$\rho^\vee:G^\vee\rightarrow GL_n$ 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 $SL_2$ or $PGL_2$ 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.
