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(z)=(□−zId)−1 of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators Cc∞(X)→D′(X). 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′(X) 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(z) extends, and the residue representations can be infinite-dimensional.