Construction of tame supercuspidal representations in arbitrary residue characteristic
Jessica Fintzen, David Schwein
TL;DR
The paper tackles the problem of constructing tame supercuspidal representations of $G(F)$ in arbitrary residue characteristic, including $p=2$, by relaxing Yu's genericity condition and adopting a uniform input framework. It advances the theory by developing a robust Heisenberg–Weil apparatus that remains valid in characteristic two, using Clifford theory to manage extensions from $K^-$ to $K$ and to $ ilde{K}$, and thereby producing irreducible, supercuspidal representations through compact induction from an open, compact-mod-center subgroup. The key contributions include a uniform input akin to Yu's, the removal of the $GE2$ constraint, a detailed treatment of $p=2$-specific phenomena (including the linearization of projective Weil representations on suitable subgroups), and a proof of irreducibility and supercuspidality for $q>3$, thereby recovering Yu’s construction as a special case while broadening its applicability. This work broadens the landscape of explicit supercuspidal representations for general tame reductive $F$-groups and enhances the toolbox for harmonic analysis on $p$-adic groups, with potential implications for the local Langlands program in even residual characteristics.
Abstract
Let F be a nonarchimedean local field whose residue field has at least four elements. Let G be a connected reductive group over F that splits over a tamely ramified field extension of F. We provide a construction of supercuspidal representations of G(F) via compact induction that contains, among others, all the supercuspidal representations constructed by Yu in 2001, but that also works in residual characteristic two. The input for our construction is described uniformly for all residual characteristics and is analogous to Yu's input except that we do not require our input to satisfy the second genericity condition (GE2) that Yu imposes.