Explicit formulae for spherical functions on symmetric spaces of type AII (Q1200385)

In this note there is proposed an explicit formula for a spherical function \(\varphi_ \lambda\) for the noncompact type symmetric space \(SU^*(2n)/Sp(n)\): \[ \varphi_ \lambda(g)=b(\lambda)\prod_{i<j}sh^{-2}(x_ i-x_ j)\sum_{w\in S_ n}\text{sgn}(w)\cdot\psi(\sqrt{-1} w\lambda,x), \] where \(\lambda\in\mathbb{R}^ n\), \(b(\lambda)\) is an explicitly written factor, \(g=\exp a\), \(a=\text{diag}(x_ 1,\dots,x_ n,x_ 1,\dots,x_ n)\), \(\sum x_ i=0\). The function \(\psi(k,x)\) is obtained by applying a differential operator \(D\) (its explicit expression is given) to the function \(\exp\sum k_ i x_ i\). Moreover the authors give an explicit inversion formula of the Abel transform for the symmetric space in question. Here they are based on some results of R. Beerends.