Ultradifferentiable Chevalley theorems and isotropic functions

Abstract

We prove ultradifferentiable Chevelley restriction theorems for a wide range of ultradifferentiable classes. As a special case we find that isotropic functions, i.e., functions defined on the vector space of real symmetric matrices invariant under the action of the special orthogonal group by conjugation, possess some ultradifferentiable regularity if and only if their restriction to diagonal matrices has the same regularity.