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.