Higher Laplacians on pseudo-Hermitian symmetric spaces

Abstract

One of the most fundamental operators studied in geometric analysis is the classical Laplace–Beltrami operator. On pseudo-Hermitian manifolds, higher Laplacians Lm are defined for each positive integer m, where L1 coincides with the Laplace–Beltrami operator. Despite their natural definition, these higher Laplacians have not yet been studied in detail. In this paper, we consider the setting of simple pseudo-Hermitian symmetric spaces, i.e., let X=G/H be a symmetric space for a real simple Lie group G, equipped with a G-invariant complex structure. We show that the higher Laplacians L1,L3,…,L2r−1 form a set of algebraically independent generators for the algebra DG(X) of G-invariant differential operators on X, where r denotes the rank of X. For higher rank, this is the first instance of a set of generators for DG(X) defined explicitly in purely geometric terms, and confirms a conjecture of Engliš and Peetre, originally stated in 1996 for the class of Hermitian symmetric spaces.