Let G be a (split) reductive group over \({\mathbb {F}}_q\) , and let M be a standard Levi subgroup of G. Let \(\mathcal {P}\) denote the set of parabolic subgroups of G with Levi factor M. For P and \(P'\) in \(\mathcal {P}\) , 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 Kazhdan-Laumon [28] and Braverman-Kazhdan [6, 7] to general parabolic basic affine (or “paraspherical") space, we propose a construction for certain intertwining operators \(\mathcal {F}_{P',P}: \mathcal {S}(\overline{G/U}({\mathbb {F}}_q), {\mathbb {C}}) \rightarrow \mathcal {S}(\overline{G/U'}({\mathbb {F}}_q), {\mathbb {C}})\) for suitable function spaces \(\mathcal {S}\) , defined via kernels analogous to those appearing in 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, Getz-Hsu-Leslie, and Kobayashi-Mano [18, 23, 29].
