An Application of Twisted Character on Signature of a finite-dimensional Representation of a Real reductive Lie group

TL;DR

This paper presents an alternative proof for a signature formula of finite-dimensional representations of complex reductive Lie groups, utilizing twisted character theory to deepen understanding of invariant hermitian forms.

Contribution

It offers a new proof of the signature formula by applying twisted character theory, expanding the theoretical framework for analyzing representations of real reductive Lie groups.

Findings

Provides an alternative proof of the signature formula.

Connects twisted character theory with invariant hermitian forms.

Enhances the theoretical understanding of finite-dimensional representations.

Abstract

Let $G_\cpl$ be a connected complex reductive Lie group, and $G$ be a real form. Let $(\pi,V)$ be a finite-dimensional irreducible representation of $G$. Assume $(\pi,V)$ admits a $G$ invariant hermitian form. {In \cite{Signature-of-a-rep-of-reductive}, an analog of the Weyl dimension formula that instead computes the signature of the invariant form for finite dimensional representations of complex Lie groups is given. The goal of this paper is to give a short alternate proof as an application of a more general twisted character theory of \cite{Dirac-Index-and-Twisted-Characters}.