Laplacians on spheres

TL;DR

This paper explores the spectral properties of Laplacians on spheres viewed as homogeneous spaces, analyzing their representation-theoretic decompositions to gain insights into related noncompact cases.

Contribution

It reviews known decompositions of $L^2$ spaces on spheres as homogeneous spaces and investigates their implications for noncompact real forms.

Findings

Decomposition of $L^2(G/H)$ reveals structure of Laplacians on spheres.

Insights into noncompact real forms of $G$ and $H$.

Connections between representation theory and spectral analysis.

Abstract

Spheres can be written as homogeneous spaces $G/H$ for compact Lie groups in a small number of ways. In each case, the decomposition of $L^2(G/H)$ into irreducible representations of $G$ contains interesting information. We recall these decompositions, and see what they can reveal about the analogous problem for noncompact real forms of $G$ and $H$.