Non-Abelian Fourier Transforms and Normalized Intertwining Operators over Finite Fields for General Parabolic Subgroups

Let $G$ be a (split) reductive group over $F_q$, $T$ a maximal torus, $T' \subset T$ a subtorus, and $M = Z_G(T')$ -- a reductive subgroup of $G$. Let $P$ denote a parabolic subgroup of $G$ with Levi factor $M$; let $U = R_u(P)$ (resp., $U^{op} = R_u(P^{op})$); let $\overline{G/U}$ (resp., $\overline{G/U^{op}}$) denote the affinizations of the quasi-affine homogeneous spaces $G/U$ (resp., $G/U^{op}$). We propose a construction for a non-abelian Fourier transform $\mathcal{S}(\overline{G/U}(F_q), C) \to \mathcal{S}(\overline{G/Uop}(F_q), C)$, where $\mathcal{S}$ denotes a certain restricted subspace of functions on $\overline{G/U}(F_q)$. We conjecture that this transform is involutive (i.e., satisfies the analogue of Fourier inversion) and provide evidence for this conjecture in concrete low-rank cases. Along the way, we will prove a finite-field Fourier transform for affine quadratic cones, and construct the complete set of finite-field normalized intertwining operators for the standard Borels in $SL_3$.