Thomae's function on a Lie group

TL;DR

This paper establishes an inequality relating automorphism order and fixed-point subalgebra dimension in simple Lie algebras, characterizes cases of equality, and explores applications to characters, graded algebras, and Swan conductors.

Contribution

It introduces a novel inequality and characterization for automorphisms of simple Lie algebras, linking automorphism order to fixed-point subalgebra dimension.

Findings

Derived an inequality relating automorphism order to fixed-point subalgebra dimension.

Characterized automorphisms where the inequality becomes equality.

Applied results to characters of zero weight spaces, graded Lie algebras, and Swan conductors.

Abstract

Let $\mathfrak g$ be a simple complex Lie algebra of finite dimension. This paper gives an inequality relating the order of an automorphism of $\mathfrak g$ to the dimension of its fixed-point subalgebra, and characterizes those automorphisms of $\mathfrak g$ for which equality occurs. This is amounts to an inequality/equality for Thomae's function on the group of automorphisms of $\mathfrak g$. The result has applications to characters of zero weight spaces, graded Lie algebras, and inequalities for adjoint Swan conductors.