Let $\Delta$ be the Laplacian on a Riemannian symmetric space $X=G/K$ of the noncompact type and let $\sigma(\Delta)\subseteq \mathbb{C}$ denote its spectrum. The resolvent $(\Delta-z)^{-1}$ is a holomorphic function on $\mathbb{C} \setminus \sigma(\Delta)$, with values in the space of bounded operators on $L^2(X)$. If we view it as a function with values in Hom$(C_c^\infty(X), C_c^\infty(X)^*)$, then it often admits a meromorphic continuation beyond $\mathbb{C} \setminus \sigma(\Delta)$. We study this meromorphic continuation as a map defined on a Riemann surface above $\mathbb{C} \setminus \sigma(\Delta)$. The poles of the meromorphically extended resolvent are called resonances. The image of the residue operator at a resonance is a $G$-module. The main problems are the existence and the localization of the resonances as well as the study of the (spherical) representations of $G$ so obtained. In this talk, based on joint works with Joachim Hilgert and Tomasz Przebinda, we will describe a variety of different situations occurring in the rank two case.

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Last updated: 17 Oct 2019