Symmetries of CR sub-Laplacian

TL;DR

This paper studies the symmetries of the CR sub-Laplacian on a specific CR manifold, constructing all symmetries via ambient methods and analyzing the algebraic structure of the symmetry algebra.

Contribution

It introduces a CR structure on a hyperplane in complex space, constructs all symmetries of the CR sub-Laplacian using ambient techniques, and analyzes the symmetry algebra's structure.

Findings

Constructed all symmetries of the CR sub-Laplacian.

Derived the algebraic decomposition of symmetry representations.

Analyzed the structure of the symmetry algebra.

Abstract

We define a CR structure on a distinguished hyperplane in $\mathbb{C}^{n+1}$ and the CR sub-Laplacian on this CR manifold. We also define symmetries of the CR sub-Laplacian in general and for this special case construct all of them using the ambient construction. Then we investigate the algebra structure of the symmetry algebra of the sub-Laplacian. For this purpose we derive the decomposition of $S^{k}_{0}\mathfrak{sl}(V)$ under the action of $SL(V)$.