Symmetric Powers of Symmetric Bilinear Forms, Homogeneous Orthogonal Polynomials on the Sphere and an Application to Compact Hyperk\"ahler Manifolds

The Beauville-Fujiki relation for a compact Hyperk"ahler manifold $X$ of dimension $2k$ allows to equip the symmetric power ${\text{Sym}}^k{H}^2(X)$ with a symmetric bilinear form induced by the Beauville-Bogomolov form. We study some of its properties and compare it to the form given by the Poincar'e pairing. The construction generalizes to a definition for an induced symmetric bilinear form on the symmetric power of any free module equipped with a symmetric bilinear form. We point out how the situation is related to the theory of orthogonal polynomials in several variables. Finally, we construct a basis of homogeneous polynomials that are orthogonal when integrated over the unit sphere ${\mathbb{S}}^d$, or equivalently, over ${\mathbb{R}}^{d+1}$ with a Gaussian kernel.