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

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}.