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

Abstract

The major aim of studying harmonic analysis on any group or symmetric space is to understand enough of its structure to be able to prove the fundamental theorem on it via the Fourier transform map. To this end we employ the perspective of the functional equation satisfied by the classical Fourier transform to derive the Helgason Fourier transform given as the map $\Omega^{l}(G/K,W)\longrightarrow\Omega^{k}(G/K\times G/P,V[\chi]):f\longmapsto \widehat{f}:G/K\times G/P\mapsto V[\chi]:(x,b)\longmapsto\widehat{f}(x,b)$ (for $W-$valued differential forms $f\in \Omega^{l}(G/K,W)$) 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 $\varphi^{U^{\sigma,\nu}}_{\tau,l,k}* f$ on $G/K,$ where $\varphi^{U^{\sigma,\nu}}_{\tau,l,k}$ is the $W-$valued $\tau-$spherical $l-$form on $G/K.$ Explicitly, we prove that $$\widehat{f}_{l,k,\varepsilon(\lambda)}(x,b)=({\bf C_{o}(\lambda)}^{-1}\circ\beta^{V}(\lambda))\circ(\int_{G/K}\varphi^{U^{\sigma\nu},t}_{\lambda,l,k}\wedge\pi^{*}_{K}f)(x),$$ where $b\in G/P$ is a consequence of the boundary map $\beta^{V}(\lambda),$ ${\bf C_{o}(\lambda)}$ is the vector bundle-valued Harish-Chandra $c-$function and for some $\lambda-$linear relation, $\varepsilon(\lambda).$ The Fourier transform is found to be the map $\Omega^{l}(G/K,W)\longrightarrow\Omega^{k}(G/K\times G/P,W)$ $:f\mapsto f^{\triangle}:$ $G/P\times G/K\longrightarrow W$ $:(b,x)\longmapsto f^{\triangle}(b,x)$ and is then established to be explicitly given as $f^{\triangle}_{l,k,\upsilon(\lambda)}(b,x)=$ $$\int_{G/P}\phi_{k,l,\lambda}\wedge\pi^{*}_{P}(({\bf C_{o}(\lambda)}^{-1}\circ\beta^{V}(\lambda))\circ(\int_{G/K}\varphi^{U^{\sigma\nu},t}_{\lambda,l,k}\wedge\pi^{*}_{K}f)(x)),$$ where $\upsilon(\lambda)$ is some $\lambda-$linear relation.