Fourier and Helgason Fourier transforms for Vector Bundle-valued Differential Forms on Homogeneous Spaces
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| Format: | Preprint |
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2025
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| _version_ | 1866915266600894464 |
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| author | Oyadare, Olufemi O. |
| author_facet | Oyadare, Olufemi O. |
| contents | 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 $φ^{U^{σ,ν}}_{τ,l,k}* f$ on $G/K,$ where $φ^{U^{σ,ν}}_{τ,l,k}$ is the $W-$valued $τ-$spherical $l-$form on $G/K.$ Explicitly, we prove that $$\widehat{f}_{l,k,\varepsilon(λ)}(x,b)=({\bf C_{o}λ)}^{-1}\circβ^{V}(λ))\circ(\int_{G/K}φ^{U^{σν},t}_{λ,l,k}\wedgeπ^{*}_{K}f)(x),$$ 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 $f^{\triangle}_{l,k,\upsilon(λ)}(b,x)=$ $$\int_{G/P}ϕ_{k,l,λ}\wedgeπ^{*}_{P}(({\bf C_{o}(λ)}^{-1}\circβ^{V}λ))\circ(\int_{G/K}φ^{U^{σν},t}_{λ,l,k}\wedgeπ^{*}_{K}f)(x)),$$ where $\upsilon(λ)$ is some $λ-$linear relation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_18543 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier and Helgason Fourier transforms for Vector Bundle-valued Differential Forms on Homogeneous Spaces Oyadare, Olufemi O. Functional Analysis Differential Geometry Representation Theory 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 $φ^{U^{σ,ν}}_{τ,l,k}* f$ on $G/K,$ where $φ^{U^{σ,ν}}_{τ,l,k}$ is the $W-$valued $τ-$spherical $l-$form on $G/K.$ Explicitly, we prove that $$\widehat{f}_{l,k,\varepsilon(λ)}(x,b)=({\bf C_{o}λ)}^{-1}\circβ^{V}(λ))\circ(\int_{G/K}φ^{U^{σν},t}_{λ,l,k}\wedgeπ^{*}_{K}f)(x),$$ 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 $f^{\triangle}_{l,k,\upsilon(λ)}(b,x)=$ $$\int_{G/P}ϕ_{k,l,λ}\wedgeπ^{*}_{P}(({\bf C_{o}(λ)}^{-1}\circβ^{V}λ))\circ(\int_{G/K}φ^{U^{σν},t}_{λ,l,k}\wedgeπ^{*}_{K}f)(x)),$$ where $\upsilon(λ)$ is some $λ-$linear relation. |
| title | Fourier and Helgason Fourier transforms for Vector Bundle-valued Differential Forms on Homogeneous Spaces |
| topic | Functional Analysis Differential Geometry Representation Theory |
| url | https://arxiv.org/abs/2504.18543 |