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

Let $G$ be a (split) reductive group over ${\mathbb{F}}_q$, and let $M$ be a standard Levi subgroup of $G$. Consider $P$ and $P^{\mathrm{\prime }}$ parabolics in $G$, containing $M$, with Levi factor $M$. we let $U=R_u(P)$ (resp., $U^{\mathrm{\prime }}=R_u(P^{\mathrm{\prime }})$) denote the unipotent radical; and we denote by $\overline{G\mathrm{/}U}$ (resp., $\overline{G\mathrm{/}{U}^{\mathrm{\prime }}}$) 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 $F_{P^{\mathrm{\prime }},P}:S(\overline{G\mathrm{/}U}({\mathbb{F}}_q),\mathbb{C})\to S(\overline{G\mathrm{/}{U}^{\mathrm{\prime }}}({\mathbb{F}}_q),\mathbb{C})$ 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 ${\text{SL}}_n$, our transform reduces to the classical linear Fourier transform; and that, for opposite unipotents in ${\text{SL}}_3$ or opposite Siegel parabolics in ${\text{Sp}}_4$, 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.