Fourier transform from the symmetric square representation of $PGL_2$ and $SL_2$

TL;DR

This paper constructs a Fourier transform operator extending Braverman-Kazhdan's work for specific groups, namely $SL_2$ and $PGL_2$, using the symmetric square representation of their dual groups.

Contribution

It introduces a natural embedding and an involutive Fourier transform operator for the case of $SL_2$ and $PGL_2$, extending existing operators in the Braverman-Kazhdan framework.

Findings

Constructed a $G imes G$-equivariant embedding of $G$ into $ ext{Lie algebra}$

Defined an involutive Fourier transform operator on functions over the extended group

Extended Braverman-Kazhdan's operator to specific cases of $SL_2$ and $PGL_2$

Abstract

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.