Higher Laplacians on pseudo-Hermitian symmetric spaces |
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Authors: | Benjamin Schwarz |
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Affiliation: | Universität Paderborn, Fakultät für Elektrotechnik, Informatik und Mathematik, Institut für Mathematik, Warburger Str. 100, 33098 Paderborn, Germany |
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Abstract: | One of the most fundamental operators studied in geometric analysis is the classical Laplace–Beltrami operator. On pseudo-Hermitian manifolds, higher Laplacians are defined for each positive integer m, where 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 be a symmetric space for a real simple Lie group G, equipped with a G-invariant complex structure. We show that the higher Laplacians form a set of algebraically independent generators for the algebra 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 defined explicitly in purely geometric terms, and confirms a conjecture of Engli? and Peetre, originally stated in 1996 for the class of Hermitian symmetric spaces. |
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Keywords: | primary 32A50 secondary 53C35 32M15 22E46 Pseudo-Hermitian symmetric space Higher Laplacian Invariant differential operator |
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