Character Estimates of Adjoint Simple Lie Groups

TL;DR

This paper establishes that for adjoint simple Lie groups, the identity element can be approximated by products of elements from any nontrivial conjugacy class, and character images of the group are contained in specific disks.

Contribution

It proves the existence of a uniform number of products needed to approximate the identity and character bounds for adjoint simple Lie groups, extending understanding of their algebraic and analytical properties.

Findings

Existence of N such that e is in the interior of C^n for all n ≥ N

Every normalized character maps into a disk of radius less than 1 tangent to the unit disk

Results hold uniformly for all nontrivial conjugacy classes in adjoint simple Lie groups

Abstract

Call a compact, connected, simple Lie group $G$ {\emph{adjoint simple}} if it has trivial center. Let $C\subset G$ be a nontrivial conjugacy class, $e\in G$ the identity element of $G$. We prove the existence of an $N\in\mathbb{N}$, depending on $G$ but not $C$, such that $e$ lies in the interior of $C^n$ for all $n\geq N$. We then prove that a disk $D\subset\mathbb{C}$ of radius less than 1, contained in the unit disk $D_1$ and tangent to $D_1$ at $z=1$, contains the image of every normalized character $\chi(e)^{-1}\chi$ of $G$.