Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

Abstract

For any pseudo-Riemannian hyperbolic space X over $R,C,H$ or $O$, we show that the resolvent $R\left(z\right)={(\square -z\mathrm{Id})}^{-1}$ of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators $C_c^{\infty }\left(X\right)\to D^{\prime }\left(X\right)$. Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in $D^{\prime }\left(X\right)$ forms a representation of the isometry group of X, which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces.

For Riemannian symmetric spaces analogous results were obtained by Miatello–Will and Hilgert–Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which $R\left(z\right)$ extends, and the residue representations can be infinite-dimensional.