Some questions about representations of p-adic groups
Dipendra Prasad
TL;DR
The notes present a framework of open questions linking p-adic and finite-field representation theory, focusing on which $G(F)$-representations contain finite-field constituents such as the Steinberg representation or Weyl-group–parametrized modules, and how these relate to enhanced Langlands parameters. They develop and probe degenerate Whittaker models for $\mathrm{GL}_n(D)$, explore twisted Jacquet modules for $\mathrm{GL}_{2n}(\mathbb{F}_q)$ with explicit formulas, and propose p-adic analogues and generalizations, including connections to the Jacquet–Langlands correspondence and Langlands parameters. The discussion extends to central covering groups, showing how to lift supercuspidals from $G$ to covers and asserting multiplicity-one for Whittaker models in the covering setting, while raising questions about splitting conditions and a possible complete classification. Finally, it suggests a Fourier-analytic program on $\mathrm{M}_n(\mathbb{F}_q)$ to relate orbit-function transforms to $\mathrm{GL}_n(\mathbb{F}_q)$ characters and to general Lie algebras, aiming to illuminate structural parallels across finite fields and local fields.
Abstract
Some question about representations of $p$-adic groups are discussed.