Capelli identity and contiguity relations of Radon hypergeometric function on the Grassmannian

Abstract

The Radon hypergeometric function (Radon HGF) F(z;\a) of type \l on the Grassmannian Gr(m,N), N=rn for some integers r,n>0, is defined as a Radon transform of the character of the universal covering of the Lie group H_{\l}\subset GL(N) specified by a partition \l of n, where \a \in C^{n} is the parameter to in the character. For this Radon HGF, we give the contiguity relations of the form L^{(i,j)}F(z;\a)=\b(\a_{0}^{(j)})F(z;\a+\e^{(i)}-\e^{(j)}) with a differential operator L^{(i,j)} of order r and the polynomial \b(s) called b-function. As its application, we derive the contiguity relation for the beta and gamma functions defined by Hermitian matrix integral. In establishing the contiguity relations for Radon HGF, the classical Capelli identity and Cayley's formula play an essential role.