Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

Abstract

For any pseudo-Riemannian hyperbolic space $X$ over $\mathbb{R},\mathbb{C},\mathbb{H}$ or $\mathbb{O}$, we show that the resolvent $R(z)=(\Box-z\operatorname{Id})^{-1}$ of the Laplace-Beltrami operator $-\Box$ on $X$ can be extended meromorphically across the spectrum of $\Box$ as a family of operators $C_c^\infty(X)\to \mathcal{D}'(X)$. Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in $\mathcal{D}'(X)$ 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.