Weyl group action on Radon hypergeometric function and its symmetry

For positive integers $r,n,N:=rn$, we consider the Radon hypergeometric function (Radon HGF) associated with a partition $\lambda$ of $n$ defined on the Grassmannian $Gr(m,N)$ for $r<m<N$, which is obtained as the Radon transform of a character of the group $H_{\lambda}\subset G:=GL(N)$. We study its symmetry described by the Weyl group analogue $N_{G}(H_{\lambda})/H_{\lambda}$. We consider the Hermitian matrix integral analogue of the Gauss HGF and its confluent family, which are understood as the Radon HGF on $Gr(2r,4r)$ for partitions $\lambda$ of $4$, we apply the result of symmetry to these particular cases and derive a transformation formula for the Gauss analogue which is known as a part of "24 solutions of Kummer" for the classical Gauss HGF. We derive a similar transformation formula for the analogue Kummer's confluent HGF.