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

We employ the perspective of the functional equation satisfied by the classical Fourier transform to derive the Helgason Fourier transform map ${\mathrm{\Omega }}^l(G\mathrm{/}K,W)⟶{\mathrm{\Omega }}^k(G\mathrm{/}K\times G\mathrm{/}P,V[\chi ]):f⟼\widehat{f}:G\mathrm{/}K\times G\mathrm{/}P\mapsto V[\chi ]:(x,b)⟼\widehat{f}(x,b)$ (for $W-$valued differential forms $f\in {\mathrm{\Omega }}^l(G\mathrm{/}K,W)$) as the $G-$ invariant vector bundle-valued differential form $\widehat{f}$ on the product space $G\mathrm{/}K\times G\mathrm{/}P$ whose image under the vector bundle-valued Poisson transform is the fibre convolution-integral ${\phi }_{\tau ,l,k}^{U^{\sigma ,\nu }}\ast f$ on $G\mathrm{/}K,$ where ${\phi }_{\tau ,l,k}^{U^{\sigma ,\nu }}$ is the $W-$valued $\tau -$spherical $l-$form on $G\mathrm{/}K\mathrm{.}$ Explicitly, we prove that $${\widehat{f}}_{l,k,\epsilon (\lambda )}(x,b)=({{\mathbf{C}}_{\mathbf{o}}\mathbf{\lambda })}^{-1}\circ {\beta }^V(\lambda ))\circ ({\int }_{G\mathrm{/}K}{\phi }_{\lambda ,l,k}^{U^{\sigma \nu },t}\wedge {\pi }_K^{\ast }f)(x),$$ where $b\in G\mathrm{/}P$ is a consequence of the boundary map ${\beta }^V(\lambda ),$ ${\mathbf{C}}_{\mathbf{o}}(\mathbf{\lambda })$ is the vector bundle-valued Harish-Chandra $c-$function and for some $\lambda -$linear relation, $\epsilon (\lambda )\mathrm{.}$ The Fourier transform is found to be the map ${\mathrm{\Omega }}^{l}G\mathrm{/}K,W)⟶{\mathrm{\Omega }}^k(G\mathrm{/}K\times G\mathrm{/}P,W)$ $:f\mapsto f^{\mathrm{△}}:$ $G\mathrm{/}P\times G\mathrm{/}K⟶W$ $:(b,x)⟼f^{\mathrm{△}}(b,x)$ and is then established to be explicitly given as $f_{l,k,\upsilon (\lambda )}^{\mathrm{△}}(b,x)=$ $${\int }_{G\mathrm{/}P}{\phi }_{k,l,\lambda }\wedge {\pi }_P^{\ast }(({{\mathbf{C}}_{\mathbf{o}}(\mathbf{\lambda })}^{-1}\circ {\beta }^V\lambda ))\circ ({\int }_{G\mathrm{/}K}{\phi }_{\lambda ,l,k}^{U^{\sigma \nu },t}\wedge {\pi }_K^{\ast }f)(x)),$$ where $\upsilon (\lambda )$ is some $\lambda -$linear relation.