Fourier and Helgason Fourier transforms for Vector Bundle-valued Differential Forms on Homogeneous Spaces

Abstract

We employ the perspective of the functional equation satisfied by the classical Fourier transform to derive the Helgason Fourier transform map $Ω^{l}(G/K,W)\longrightarrowΩ^{k}(G/K\times G/P,V[χ]):f\longmapsto \widehat{f}:G/K\times G/P\mapsto V[χ]:(x,b)\longmapsto\widehat{f}(x,b)$ (for $W-$valued differential forms $f\in Ω^{l}(G/K,W)$) as the $G-$ invariant vector bundle-valued differential form $\widehat{f}$ on the product space $G/K\times G/P$ whose image under the vector bundle-valued Poisson transform is the fibre convolution-integral $\varphi^{U^{σ,ν}}_{τ,l,k}* f$ on $G/K,$ where $\varphi^{U^{σ,ν}}_{τ,l,k}$ is the $W-$valued $τ-$spherical $l-$form on $G/K.$ Explicitly, we prove that $$\widehat{f}_{l,k,\varepsilon(λ)}(x,b)=({\bf C_{o}λ)}^{-1}\circβ^{V}(λ))\circ(\int_{G/K}\varphi^{U^{σν},t}_{λ,l,k}\wedgeπ^{*}_{K}f)(x),$$ where $b\in G/P$ is a consequence of the boundary map $β^{V}(λ),$ ${\bf C_{o}(λ)}$ is the vector bundle-valued Harish-Chandra $c-$function and for some $λ-$linear relation, $\varepsilon(λ).$ The Fourier transform is found to be the map $Ω^{l}G/K,W)\longrightarrowΩ^{k}(G/K\times G/P,W)$ $:f\mapsto f^{\triangle}:$ $G/P\times G/K\longrightarrow W$ $:(b,x)\longmapsto f^{\triangle}(b,x)$ and is then established to be explicitly given as $f^{\triangle}_{l,k,\upsilon(λ)}(b,x)=$ $$\int_{G/P}φ_{k,l,λ}\wedgeπ^{*}_{P}(({\bf C_{o}(λ)}^{-1}\circβ^{V}λ))\circ(\int_{G/K}\varphi^{U^{σν},t}_{λ,l,k}\wedgeπ^{*}_{K}f)(x)),$$ where $\upsilon(λ)$ is some $λ-$linear relation.