Higher Laplacians on pseudo-Hermitian symmetric spaces

TL;DR

This paper proves that certain higher Laplacians generate the algebra of invariant differential operators on pseudo-Hermitian symmetric spaces, confirming a conjecture for Hermitian symmetric spaces.

Contribution

It establishes that odd-indexed higher Laplacians form a set of algebraically independent generators for the invariant differential operators, confirming a conjecture by Engliš and Peetre.

Findings

Higher Laplacians $L_1, L_3, ..., L_{2r-1}$ generate the algebra $ ext{D}_G(X)$.

Confirms the conjecture for Hermitian symmetric spaces.

Provides a structural understanding of invariant differential operators on pseudo-Hermitian symmetric spaces.

Abstract

Let $X=G/H$ be a symmetric space for a real simple Lie group $G$, equipped with a $G$-invariant complex structure. Then, $X$ is a pseudo-Hermitian manifold, and in this geometric setting, higher Laplacians $L_m$ are defined for each positive integer $m$, which generalize the ordinary Laplace-Beltrami operator. We show that $L_1,L_3,\ldots, L_{2r-1}$ form a set of algebraically independent generators for the algebra $\mathbb{D}_G(X)$ of $G$-invariant differential operators on $X$, where $r$ denotes the rank of $X$. This confirms a conjecture of Engli\v{s} and Peetre, originally stated for the class of Hermitian symmetric spaces.