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 $R(z)=(\square -z\mathrm{Id}{)}^{-1}$ of the Laplace-Beltrami operator $-\square$ on $X$ can be extended meromorphically across the spectrum of $\square$ as a family of operators $C_c^{\infty }(X)\to {\mathcal{D}}^{\prime }(X)$. Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in ${\mathcal{D}}^{\prime }(X)$ 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.

23 pages, final published version

References in corpus (1)

• Analytic Continuation of Resolvent Kernels on noncompact Symmetric Spaces