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

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### New Zealand Journal of Mathematics

Vol. 50, (2020), Pages 29-48

Mahmoud Al-Hashami

Department of Mathematics

Sultan Qaboos University

Alkhoud

Sultanate of Oman

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 χ-spherical functions on complex Grassmannian manifolds $SU(p+q)/S\left(U(p)\times U(q)\right)$.

Keywords Spherical Functions, Hermitian Symmetric Spaces, Jacobi Function, Laplace-Beltrami Operator.

Classification (MSC2000) 43A85, 28C10, 43A77, 43A90, 53C35.