@article{THESIS,
      recid = {12420},
      author = {Slipper, Aaron Arthur Gordon},
      title = {Non-Abelian Fourier Transforms and Normalized Intertwining  Operators over Finite Fields for General Parabolic  Subgroups},
      publisher = {University of Chicago},
      school = {Ph.D.},
      address = {2024-06},
      number = {THESIS},
      pages = {78},
      abstract = {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 $SL_3$.},
      url = {http://knowledge.uchicago.edu/record/12420},
      doi = {https://doi.org/10.6082/uchicago.12420},
}