Matrix Elements and Characters of the Discrete Series ("Massive") Unitary Irreducible Representations of Sp(4,R)

TL;DR

This work constructs and analyzes the matrix elements and characters of the discrete-series ("massive") unitary irreducible representations of $Sp(4,\mathbb{R})$, the two-fold cover of $SO_0(2,3)$ relevant to anti-de Sitter spacetime. It employs a holomorphic Segal-Bargmann realization on the Cartan domain $\mathcal{D}^{(3)}$ and provides explicit expansions for $U^{(\varsigma,s)}$ in terms of quaternionic determinants, $SU(2)$ Clebsch–Gordan coefficients, and harmonic polynomials, with separate treatment for the scalar case $s=0$. The paper then computes characters by diagonalizing generic group elements in the complexified group, yielding closed-form traces for both scalar and spin representations on the diagonal class $g_d$. These results supply essential algebraic data for covariant integral quantization of AdS massive systems and illuminate the Poincaré contraction limits to massive UIRs.

Abstract

This paper obtains the matrix elements and characters of the discrete series unitary irreducible representations (UIRs) of the Sp$(4,\mathbb{R})$ group. With an isomorphic relationship to the two-fold covering of SO$_0(2,3)$ (Sp$(4,\mathbb{R}) \sim$ SO$_0(2,3)\times \mathbb{Z}_2$), this group holds particular importance as the kinematical/relativity group within the framework of ($1+3$-dimensional) anti-de Sitter spacetime.