Signatures for finite-dimensional representations of real reductive Lie groups

TL;DR

This paper derives a closed formula for the signature of invariant Hermitian forms on finite-dimensional irreducible representations of real reductive Lie groups, revealing their indefinite nature and providing bounds on signatures.

Contribution

It introduces a new explicit formula for signatures of invariant Hermitian forms, extending the understanding of representation structures in real reductive Lie groups.

Findings

Signature formulas analogous to Weyl's dimension formula

Signatures are very indefinite with specific bounds

Application of Kostant's Dirac operator kernel computation

Abstract

We present a closed formula, analogous to the Weyl dimension formula, for the signature of an invariant Hermitian form on any finite-dimensional irreducible representation of a real reductive Lie group, assuming that such a form exists. The formula shows in a precise sense that the form must be very indefinite. For example, if an irreducible representation of $GL(n,R)$ admits an invariant form of signature $(p,q)$, then we show that $(p-q)^2 \le p+q$. The proof is an application of Kostant's computation of the kernel of the Dirac operator.