A novel approach through spherical functions in the characterization of invariant functions

Abstract

Given a compact subgroup K of the orthogonal group acting on the Euclidean space ‫ޒ‬ n , Gerald Schwarz proved that every smooth K -invariant function on ‫ޒ‬ n can be expressed as a smooth function of a generating set of K -invariant polynomials on n variables.Our goal is to provide an alternative and more straightforward proof of this result, based on Gelfand theory, with a particular focus on spherical functions.This result is very strong because it relates the differentiable structures of two spaces that in principle were only homeomorphic.Indeed, let ρ : ‫ޒ‬ n → ‫ޒ‬ l be given by ρ(x) := (ρ 1 (x), . . ., ρ l (x)),for every x ∈ ‫ޒ‬ n .This map induces a homeomorphism between the space of orbits ‫ޒ‬ n /K and ρ(‫ޒ‬ n ):As explained in [20], ‫ޒ‬ n /K can be given a smooth structure by defining a function on the quotient space ‫ޒ‬ n /K to be smooth if, when lifted to a K -invariant function on ‫ޒ‬ n , it is smooth in the classical sense.