# Plane wave formulas for spherical, complex and symplectic harmonics

## TL;DR

This paper derives explicit formulas for reproducing kernels of spherical, complex, and symplectic harmonics, and expresses them as integrals over Stiefel manifolds using plane wave formulas and Pizzetti formulas.

## Contribution

It introduces the reproducing kernel for symplectic harmonics and provides plane wave formulas for all three harmonic types.

## Key findings

- Reproducing kernel for symplectic harmonics expressed as a Jacobi polynomial.
- Plane wave formulas for kernels as integrals over Stiefel manifolds.
- Use of Pizzetti formulas to relate integrals to differential operators.

## Abstract

This paper is concerned with spherical harmonics, and two refinements thereof: complex harmonics and symplectic harmonics. The reproducing kernels of the spherical and complex harmonics are explicitly given in terms of Gegenbauer or Jacobi polynomials. In the first part of the paper we determine the reproducing kernel for the space of symplectic harmonics, which is again expressible as a Jacobi polynomial of a suitable argument. In the second part we find plane wave formulas for the reproducing kernels of the three types of harmonics, expressing them as suitable integrals over Stiefel manifolds. This is achieved using Pizzetti formulas that express the integrals in terms of differential operators.