On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces

TL;DR

This paper computes eigenfunctions of the Dirac operator on spheres and hyperbolic spaces, derives the heat kernel, and explores their group-theoretic properties, advancing understanding of spinor analysis on symmetric spaces.

Contribution

It provides explicit eigenfunctions and heat kernels for the Dirac operator on spheres and hyperbolic spaces, using separation of variables and group-theoretic methods.

Findings

Eigenfunctions explicitly computed for arbitrary dimensions.

Heat kernel derived for the iterated Dirac operator.

Group-theoretic derivation of spinor spherical functions provided.

Abstract

The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates. These eigenfunctions are used to derive the heat kernel of the iterated Dirac operator on these spaces. They are then studied as cross sections of homogeneous vector bundles, and a group-theoretic derivation of the spinor spherical functions and heat kernel is given based on Harish-Chandra's formula for the radial part of the Casimir operator.