Bochner identities for Fourier transforms

Abstract:

Let G be a compact Lie group and R an orthogonal representation of G acting on ${{\mathbf {R}}^n}$. For any irreducible unitary representation $\pi$ of G and vector v in the representation space of $\pi$ define $\mathcal {S}(\pi ,v)$ to be those functions in $\mathcal {S}({{\mathbf {R}}^n})$ which transform (under the action R) according to the vector v. The Fourier transform $\mathcal {F}$ preserves the class $\mathcal {S}(\pi ,v)$. A Bochner identity asserts that for different choices of G, R, $\pi ,v$ the Fourier transform is the same (up to a constant multiple). It is proved here that for G, R, $\pi ,v$ and $G’,R’,\pi ’,v’$ and a map $T:\mathcal {S}(\pi ,v) \to \mathcal {S}(\pi ’,v’)$ which has the form: restriction to a subspace followed by multiplication by a fixed function, a Bochner identity $\mathcal {F}’Tf = cT\mathcal {F}f$ for all $f \in \mathcal {S}(\pi ,v)$ holds if and only if $\Delta ’Tf = {c_1}T\Delta f$ for all $f \in \mathcal {S}(\pi ,v)$. From this result all known Bochner identities follow (due to Harish-Chandra, Herz and Gelbart), as well as some new ones.