Symmetric powers of symmetric bilinear forms, homogeneous orthogonal polynomials on the sphere and an application to compact Hyperkähler manifolds

Abstract

The Beauville–Fujiki relation for a compact Hyperkähler manifold [Formula: see text] of dimension [Formula: see text] allows to equip the symmetric power [Formula: see text] 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 Poincaré 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 [Formula: see text], or equivalently, over [Formula: see text] with a Gaussian kernel.