Spherical functions on the de Sitter group

TL;DR

This paper analyzes matrix elements and spherical functions of the de Sitter group’s irreducible representations, providing explicit formulas and connections to hypergeometric functions on various homogeneous spaces.

Contribution

It introduces quaternion Euler angles for the universal covering of the de Sitter group and derives explicit forms of key operators and functions in terms of hypergeometric functions.

Findings

Explicit quaternion Euler angles for the de Sitter group

Formulas for Casimir and Laplace-Beltrami operators on homogeneous spaces

Expressions of matrix elements in terms of hypergeometric functions

Abstract

Matrix elements and spherical functions of irreducible representations of the de Sitter group are studied on the various homogeneous spaces of this group. It is shown that a universal covering of the de Sitter group gives rise to quaternion Euler angles. An explicit form of Casimir and Laplace-Beltrami operators on the homogeneous spaces is given. Different expressions of the matrix elements and spherical functions are given in terms of multiple hypergeometric functions both for finite-dimensional and unitary representations of the principal series of the de Sitter group.