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Fourier transform from the symmetric square representation of $PGL 2$ and $SL 2$ by Laumon, Gérard; Letellier, Emmanuel is a scholarly article available to read on EtoBox.

What is Fourier transform from the symmetric square representation of $PGL 2$ and $SL 2$ about?

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.

Author

Laumon, Gérard; Letellier, Emmanuel

Published

2023

Language

EN