Lifting representations of finite reductive groups II: Explicit conorms

TL;DR

This paper provides explicit descriptions of a map relating stable conjugacy classes in dual groups of reductive groups, facilitating the understanding of representation liftings, including cases matching Shintani lifting.

Contribution

The authors explicitly describe the map $ abla^{st}$ in terms of simpler cases and construct explicit morphisms, clarifying the lifting of representations for reductive groups.

Findings

Compatibility of the map with isogenies and Weil restriction.

Expression of the map as a composition of simpler maps.

Construction of explicit morphisms matching the stable conjugacy class lifting.

Abstract

Let $k$ be a field, $\tilde{G}$ a connected reductive $k$-group, and $\Gamma$ a finite group. In a previous work, the authors defined what it means for a connected reductive $k$-group $G$ to be "parascopic" for $(\tilde{G},\Gamma)$. Roughly, this is a simultaneous generalization of several settings. For example, $\Gamma$ could act on $\tilde{G}$, and $G$ could be the connected part of the group of $\Gamma$-fixed points in $\tilde{G}$. Or $G$ could be an endoscopic group, a pseudo-Levi subgroup, or an isogenous image of $\tilde{G}$. If $G$ is such a group, and both $\tilde{G}$ and $G$ are $k$-quasisplit, then we constructed a map $\hat{\mathcal{N}}^{\text{st}}$ from the set of stable semisimple conjugacy classes in the dual $G^\wedge(k)$ to the set of such classes in $\tilde{G}^\wedge(k)$. When $k$ is finite, this implies a lifting from packets of representations of $G(k)$ to those of $\tilde{G}(k)$. In order to understand such a lifting better, here we describe two ways in which $\hat{\mathcal{N}}^{\text{st}}$ can be made more explicit. First, we can express our map in the general case in terms of simpler cases. We do so by showing that $\hat{\mathcal{N}}^{\text{st}}$ is compatible with isogenies and with Weil restriction, and also by expressing it as a composition of simpler maps. Second, in many cases we can construct an explicit $k$-morphism $\hat N \colon G^\wedge \longrightarrow \tilde{G}^\wedge$ that agrees with $\hat{\mathcal{N}}^{\text{st}}$. As a consequence, our lifting of representations is seen to coincide with Shintani lifting in some important cases.