The spherical Paley-Wiener theorem on the complex Grassmann manifolds SU(𝑝+𝑞)/S(U_{𝑝}×U_{𝑞})

Camporesi, Roberto

doi:10.1090/S0002-9939-06-08408-5

The spherical Paley-Wiener theorem on the complex Grassmann manifolds $\mbox {SU}(p+q)/\mbox {S}(\mbox {U}_p\times \mbox {U}_q)$
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Proc. Amer. Math. Soc. 134 (2006), 2649-2659 Request permission

Abstract:

We prove the Paley-Wiener theorem for the spherical transform on the complex Grassmann manifolds $U/K=\mbox {SU}(p+q)/\mbox {S}(\mbox {U}_p\times \mbox {U}_q)$. This theorem characterizes the $K$-biinvariant smooth functions $f$ on the group $U$ that are supported in the $K$-invariant ball of radius $R$, with $R$ less than the injectivity radius of $U/K$, in terms of holomorphic extendability, exponential growth, and Weyl invariance properties of the spherical Fourier transforms $\hat {f}$, originally defined on the discrete set $\Lambda _{sph}$ of highest restricted spherical weights.

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