Berezin-Karpelevich formula for chi-spherical functions on complex Grassmannians

Abstract

In [5], Berezin and Karpelevich gave, without a proof, an explicit formula for spherical functions on complex Grassmannian manifolds. A first attempt to give a proof of Berezin-Karpelevich formula was taken, in [16], by Takahashi. His proof contained a gap, which was fixed later, in [10], by Hoogenboom. The aim of this paper is to generalize Berezin-Karpelevich formula to the case of $\chi$-spherical functions on complex Grassmannian manifolds $ SU(p+q)/S\left(U(p)\times U(q)\right)$.