Realization of unitary representations of the Lorentz group on de Sitter space

This paper builds on our previous work in which we showed that, for all connected semisimple linear Lie groups $G$ acting on a non-compactly causal symmetric space $M = G/H$, every irreducible unitary representation of $G$ can be realized by boundary value maps of holomorphic extensions in distributional sections of a vector bundle over $M$. In the present paper we discuss this procedure for the connected Lorentz group $G = SO_{1,d}(R)_e$ acting on de Sitter space $M = dS^d$. We show in particular that the previously constructed nets of real subspaces satisfy the locality condition. Following ideas of Bros and Moschella from the 1990's, we show that the matrix-valued spherical function that corresponds to our extension process extends analytically to a large domain $G_C^{cut}$ in the complexified group $G_C = \SO_{1,d}(C)$, which for $d = 1$ specializes to the complex cut plane $C \setminus (-\infinity, 0]$. A number of special situations is discussed specifically: (a) The case $d = 1$, which closely corresponds to standard subspaces in Hilbert spaces, (b) the case of scalar-valued functions, which for $d > 2$ is the case of spherical representations, for which we also describe the jump singularities of the holomorphic extensions on the cut in de Sitter space, (c) the case $d = 3$, where we obtain rather explicit formulas for the matrix-valued spherical functions.