Lifting semisimple characters of $p$-adic types from fixed-point subgroups

TL;DR

The paper addresses the problem of lifting semisimple characters from fixed-point subgroups G^{[Γ]} to the ambient p-adic group G, where Γ is a finite group of automorphisms with order prime to p. It provides an explicit, constructive method to lift the truncated Kim–Yu data (character-datum) parametrizing semisimple characters for G^{[Γ]} to a Γ-stable character-datum for G, ensuring that the Γ-fixed point recovers the original data via refactorization. The core technical contributions include a lifting theorem for a single quasicharacter and an iterative procedure to lift an entire character-datum while preserving compatibility, together with a refined treatment of restriction and refactorization. The results connect to broader constructions such as Howe factorizations and G-factorizations, and have potential implications for base change and functoriality in the Langlands program through explicit control of semisimple characters across fixed-point subgroups.

Abstract

Given a $p$-adic group $G=\mathbf{G}(F)$ and a finite group $Γ\subset\mathrm{Aut}_F(\mathbf{G})$ such that the fixed-point subgroup $\mathbf{G}^Γ$ is reductive, we show that every semisimple character (in the sense of Bushnell and Kutzko) of a type for $G^Γ= \mathbf{G}^Γ(F)$ arises as the restriction of a semisimple character of a type for $G$. We achieve this by explicitly lifting the truncated Kim--Yu datum (or character-datum) that parametrizes the semisimple character for $G^Γ$ to a character-datum that parametrizes a semisimple character for $G$. Our proof, which is of independent interest, uses state-of-the-art techniques and, as a special case, defines a lift of a Howe factorization of a character of a maximal torus of $G^Γ$.

Figures (6)

• Figure 1.1: Roadmap for proving Theorem \ref{['th:lift']}.
• Figure 3.1: Illustration of how part of the character-datum collapses when taking the $\Gamma$-fixed point as per Theorem \ref{['th:restriction']}. Here, ${\mathbf{G}}^{i,[\Gamma]} = {\mathbf{G}}^{i+1,[\Gamma]} = \dots = {\mathbf{G}}^{i+k,[\Gamma]}$.
• Figure 4.1: Lift (top) produced by Theorem \ref{['T:liftone']}, given the $H$-generic quasicharacter $\xi$ of $\mathbf{H}'$ of depth $t=r_n$ (bottom). The $\Gamma$-fixed point of the lift is $(\mathbf{H}',t,\varphi\xi)$.
• Figure 4.2: Naive application of Theorem \ref{['T:liftone']} to the single-quasicharacter character-data $({\mathbf{H}}^{i}\subseteq {\mathbf{H}}^{i+1}, y, t_i, {\xi}_{i})$, $0\leq i\leq n$.
• Figure 4.3: Illustration of Step $j$, for $j=n, n-1,n-2,\dots,1,0$, in the proof of Theorem \ref{['th:lift']}. The vertical arrows illustrate the application of Theorem \ref{['T:liftone']} to $({\mathbf{H}}^{j}\subseteq ({\mathbf{G}}^{j+1,0})^{[\Gamma]}, y, t_j, {\varphi}_{j+1}^{-1}{\xi}_{j})$ with $s=t_{j-1}$ (setting $\varphi_{n+1}={\bf 1}$ and $t_{-1} = 0$), which makes the correction factor ${\varphi}_{j}$ appear. If $j \geq 1$, the inverse of this correction factor then gets folded into ${\xi}_{j-1}$ in preparation for Step $j-1$, as illustrated by the dashed arrow.

Theorems & Definitions (42)

• Theorem : Theorem \ref{['th:lift']}
• Lemma 2.1
• proof
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Remark 2.7
• Lemma 2.8