Abstract

We classify to some extent the pairs of group morphisms

\mathrm{\Gamma }\to \text{Spin}(7)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma \rightarrow \textrm{Spin}(7)$$\end{document}

which are element-conjugate but not globally conjugate. As an application, we study the case where

\mathrm{\Gamma }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document}

is the Weil group of a p-adic local field, which is relevant to the recent approach to the local Langlands correspondence for

{\text{G}}_2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{G}_2$$\end{document}

and

{\text{PGSp}}_6

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{PGSp}_6$$\end{document}

in Gan and Savin (Forum Math Pi 11:e28, 2023). As a second application, we improve some result in Kret and Shin (J Eur Math Soc 25(1):75–152, 2023) about

{\text{GSpin}}_7

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{GSpin}_7$$\end{document}

-valued Galois representations.

Keywords

Spin groups / Local-global conjugacy / Langlands correspondence / 22C05 / 20G15 / 20G41 / 11F80 / 11R39

References

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Funding

CNRS(ANR-19-CE40-0015-02 (COLOSS))

Singapore Government MOE(R-146-000-320-114)

RIGHTS & PERMISSIONS

Peking University