---
address: 'BICMR, Peking University, No. 5 Yiheyuan Road, Haidian District, Beijing 100871, China.'
author:
- Jun Yu
date: June 2018
title: Acceptable compact Lie groups
---

[**Mathematics Subject Classification (2010).**]{} 22C05, 22E15.

[**Keywords.**]{} Acceptable group, strongly controlling fusion.

Introduction
============

Acceptability of a group is defined by Michael Larsen in [@Larsen1]. The motivation of studying acceptability comes from its connection with multiplicity one question in the automorphic form theory ([@Arthur]). Recently it is also found its connection with Langlands correspondence through Lafforgue’s operators ([@Lafforgue]).

This paper contributes to the goal of classifying acceptable compact Lie groups. The major results in this paper is summarized with the following theorem.

\[T1\] Let $G$ be a connected compact semisimple Lie group. For $G$ to be acceptable it is necessary and sufficient that $G$ is isomorphic to a direct product of the following groups,

-   $\SU(n)$ ($n\geq 1$);

-   $\Sp(n)$ ($n\geq 1$);

-   $\SO(2n+1)$ ($n\geq 1$);

-   $\G_2$;

-   $\SO(4)$.

This paper contains two main aspects. In the first aspect, we study the notion of strongly controlling fusion (“SCF" for short) defined by Robert Griess ([@Griess]). We give new proofs for “SCF" pairs shown by Griess in [@Griess], and show new “SCF" pairs 