Parabolic Kazhdan-Laumon and the Kloosterman Fourier Transform for Quadric Cones

Abstract

Let $G$ be a split reductive group over $F_q$, and let $M$ be a standard Levi subgroup of $G$. Let $Par_M(G)$ denote the set of parabolic subgroups of $G$ with Levi factor $M$. For $P$ and $P'$ in $Par_M(G)$, we let $U=R_u(P)$ and $U'=R_u(P')$ denote the unipotent radicals, and we denote by $\overline{G/U}$ and $\overline{G/U'}$ the affinizations of the corresponding homogeneous spaces. Extending the work of Kazhdan-Laumon and Braverman-Kazhdan (arXiv:math/9809112, arXiv:math/0206119) to general parabolic basic affine, or paraspherical, spaces, we propose a construction for certain intertwining operators $F_{P',P}: S(\overline{G/U}(F_q),C) \to S(\overline{G/U'}(F_q),C)$ for suitable function spaces $S$, defined via kernels analogous to those appearing in those works. We then study the extent to which these intertwiners are normalized. We show that, for opposite $(n-1)+1$ parabolics of $SL_n$, our transform reduces to the classical linear Fourier transform, and that, for opposite unipotents in $SL_3$ or opposite Siegel parabolics in $Sp_4$, 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, Getz-Hsu-Leslie, and Kobayashi-Mano (arXiv:2304.13993, arXiv:2103.10261, arXiv:0712.1769).