The Dirac operator on compact symmetric spaces

TL;DR

This paper studies the Dirac operator on compact symmetric spaces, characterizing the representations of the group G in its kernel and showing how all irreducible G-representations can be constructed via this operator.

Contribution

It introduces a method to determine G-representations in the kernel of a Dirac operator on symmetric spaces, extending Parthasarathy's approach.

Findings

Identifies G-representations in the Dirac kernel

Shows all irreducible G-representations can be constructed this way

Provides explicit construction method for representations

Abstract

Let G be a compact connected semisimple Lie group and let H\subset G be a closed connected subgroup such that rank(G)=rank(H) and G/H is a symmetric space. Given an irreducible representation of H, we define a Dirac operator D and determine the representations of G in the kernel of D. Moreover, we show that any irreducible representation of G can be constructed in this way. Our approach is similar to that of Parthasarathy.