Weyl group action on Radon hypergeometric function and its symmetry

TL;DR

This paper investigates the symmetry properties of Radon hypergeometric functions on Grassmannians, revealing new transformation formulas related to Weyl group actions and classical hypergeometric solutions.

Contribution

It introduces the Weyl group symmetry framework for Radon hypergeometric functions and derives new transformation formulas for their confluent and Gauss analogues.

Findings

Identified Weyl group symmetry in Radon hypergeometric functions.

Derived transformation formulas for Gauss and confluent hypergeometric analogues.

Connected symmetry results to classical solutions of hypergeometric functions.

Abstract

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.