The theory of representations of groups $SO_{0}(2,1)$ and $ISO(2,1)$. Wigner coefficients of the group $SO_{0}(2,1)$

TL;DR

This paper analyzes the representations of the groups SO(2,1) and ISO(2,1), deriving explicit formulas for their spherical functions and Wigner coefficients, with implications for cosmology and mathematical physics.

Contribution

It provides explicit expressions for spherical functions and Wigner coefficients of SO(2,1), advancing understanding of their unitary representations and applications in physics.

Findings

Derived irreducible unitary representations of various series.

Obtained explicit formulas for spherical functions using hypergeometric functions.

Computed Wigner coefficients with explicit bilateral series expressions.

Abstract

This paper is devoted to the representations of the groups $SO (2,1)$ and $ISO (2,1)$. Those groups have an important role in cosmology, elementary particle theory and mathematical physics. Irreducible unitary representations of the principal continuous and supplementary as well as discrete series were obtained. Explicit expressions for spherical functions of the group $SO_0(2,1)$ are obtained through the Gauss hypergeometric functions. The Wigner coefficients of the group $SO_{0}(2,1)$ were computed and their explicit expressions using the bilateral series were represented. The results could be used to study the non-degenerate representations of the de Sitter group $SO(3,2)$.