# The higher spin Laplace operator in several vector variables

## TL;DR

This paper derives an explicit form of the conformally invariant higher spin Laplace operator acting on vector-valued functions and identifies special polynomial solutions, advancing understanding of higher spin conformal operators.

## Contribution

It provides the first explicit expression for the higher spin Laplace operator in several variables and characterizes specific polynomial solutions.

## Key findings

- Explicit formula for the higher spin Laplace operator.
- Identification of special polynomial solutions.
- Enhanced understanding of conformally invariant operators.

## Abstract

In this paper, an explicit expression is obtained for the conformally invariant higher spin Laplace operator $\mathcal{D}_{\lambda}$, which acts on functions taking values in an arbitrary (finite-dimensional) irreducible representation for the orthogonal group with integer valued highest weight. Once an explicit expression is obtained, a special kind of (polynomial) solutions of this operator is determined.