Non-Abelian Fourier Transforms and Normalized Intertwining Operators for General Parabolics over Finite Fields, and the Kloosterman Fourier Transform for Quadric Cones

TL;DR

This paper extends the theory of Fourier transforms over finite fields to general parabolic subgroups of reductive groups, introducing new intertwining operators and establishing a finite-field analogue of the quadric Fourier transform.

Contribution

It constructs and analyzes intertwining operators for parabolic basic affine spaces, generalizing previous work, and connects these to classical Fourier transforms and Kloosterman sums over finite fields.

Findings

Transform reduces to classical Fourier transform for certain parabolics in SL_n.

Transforms are given by Fourier transforms on quadric cones involving Kloosterman sums.

Proves Fourier inversion for the quadric Fourier transform over finite fields.

Abstract

Let $G$ be a (split) reductive group over $\mathbb{F}_q$, and let $M$ be a standard Levi subgroup of $G$. Consider $P$ and $P'$ parabolics in $G$, containing $M$, with Levi factor $M$. we let $U = R_u(P)$ (resp., $U' = R_u(P')$) denote the unipotent radical; and we denote by $\overline{G/U}$ (resp., $\overline{G/U'}$) the affinization of the corresponding homogeneous space. Extending the work of Braverman-Kazhdan ( arXiv:9809.112 , arXiv:0206.119 ) to general parabolic basic affine (or ``paraspherical") space, we propose a construction for certain intertwining operators $F_{P',P}: S(\overline{G/U}(\mathbb{F}_q), \mathbb{C}) \to S(\overline{G/U'}(\mathbb{F}_q), \mathbb{C})$ for a suitable function spaces $S$, defined via kernels analogous to those appearing (loc. cit.). We then study the extent to which these intertwiners are normalized. We show that, for opposite $(n-1)+1$ parabolics of $\textrm{SL}_n$, our transform reduces to the classical linear Fourier transform; and that, for opposite unipotents in $\textrm{SL}_3$ or opposite Siegel parabolics in $\textrm{Sp}_4$, our transforms are given by a Fourier transform on a quadric cone (with kernel coming from a Kloosterman sum). We prove Fourier inversion for this transform on (a natural subclass of) functions on the quadric cone, establishing a finite-field analogue of the quadric Fourier transform of Gurevich-Kazhdan arXiv:2304.13993 and Getz-Hsu-Leslie arXiv:2103.10261.