Exercise 30, § 4

Let $n\in \mathbf{N}$. Let G denote the subgroup of the affine group $A(\mathbf{R}^n)$ of the space $\mathbf{R}^n$ formed by the elements $g\in A(\mathbf{R}^n)$ such that the linear mapping $g_0$ associated with $g$ belongs to $\mathbf{S}\mathbf{O}(\mathbf{R}^n)$; it is a real Lie group (cf. A, II, p. 131 and LIE, III, p. 102, example). We write $g= (g_0, a)$ for the elements of G, with $g_0\in \mathbf{S}\mathbf{O}(\mathbf{R}^n)$ and $a\in \mathbf{R}^n$.

a) The pair $(G,\mathbf{S}\mathbf{O}(\mathbf{R}^n))$ is a Gelfand pair.

b) Let $\varphi \in \mathscr{H}(G,\mathbf{S}\mathbf{O}(\mathbf{R}^n))$. Let $f:\mathbf{R}^n\rightarrow \mathbf{C}$ denote the function defined by $f(r) =\varphi (1_{\mathbf{R}^N}, g(e_1))$. The mapping $u:\varphi \mapsto f$ is an injective morphism of complex algebras of $\mathscr{H}(G,K)$ into $\mathscr{K}(\mathbf{R}^n)$; its image is the subspace of $\mathscr{K}(\mathbf{R}^n)$ formed by the functions $f$ such that $f(g_0(x)) =f(x)$ for every $g_0\in \mathbf{S}\mathbf{O}(\mathbf{R}^n)$ and every $x\in \mathbf{R}^n$.

c) Let Δ denote the Laplacian on $\mathbf{R}^n$. A function $\varphi \in \mathscr{K}(G)$ is a spherical function if and only if the function $f=u(\varphi )$ is of class $C^{\infty}$, satisfies $f(0) = 1$, and if there exists $\lambda \in \mathbf{C}$ such that $\Delta f=\lambda f$. (Use the fact that $\Delta (f\circ g_0) = (\Delta f)\circ g_0$ for every $g_0\in \mathbf{S}\mathbf{O}(\mathbf{R}^n).$)