Generalized Bernstein--Reznikov integrals

TL;DR

This paper derives a closed-form expression for a class of triple integrals involving symplectic forms on spheres, providing new proofs and extending results to linear and conformal structures.

Contribution

It introduces a novel closed-form formula for symplectic kernel integrals on spheres, generalizing the Bernstein--Reznikov integral and applying to broader geometric structures.

Findings

Derived a closed-form formula for symplectic form-based integrals

Provided a new proof of the Bernstein--Reznikov integral in the case n=1

Extended the method to linear and conformal structures

Abstract

We find a closed formula for the triple integral on spheres in $\mathbb{R}^{2n}\times\mathbb{R}^{2n}\times\mathbb{R}^{2n}$ whose kernel is given by powers of the standard symplectic form. This gives a new proof to the Bernstein--Reznikov integral formula in the $n=1$ case. Our method also applies for linear and conformal structures.