Angela Pasquale (Université de Lorraine) will speak in the geometry seminar on Monday 1st of April, at 2pm in the Salle de Profs. Angela’s title is *Resonances of the Laplacian on products of rank-one symmetric spaces* and her abstract is below.

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 $(\Delta-z)^{-1}$ of $\Delta$ is a bounded operator on $L^2(X)$ which depends holomorphically on the complex variable $z\in \mathbb{C} \setminus \sigma(\Delta)$. Let $C^\infty(X)$ and $C_c^\infty(X)$ respectively denote the space of smooth and compactly-supported smooth functions on $X$. If we view the resolvent as a function on $C_c^\infty(X)$, then it has values in $C^\infty(X)$ and it often admits a meromorphic continuation across the spectrum $\sigma(\Delta)$, to a suitable Riemann surface above $\mathbb{C}$. The poles of the meromorphically extended resolvent are called resonances. The image of the residue operator at a resonance is a spherical $G$-module. The main problems are the existence and the localization of the resonances as well as the study of the corresponding $G$-modules.

Using the Helgason-Fourier analysis, one obtains an explicit formula for the resolvent as a singular intergral, and the study of the resonances of $\Delta$ reduces to the

meromorphic continuation of this integral. In this talk we look at low rank situations in which the space $X$ is a product of rank-one symmetric spaces.This talk is based on ongoing works with Joachim Hilgert (Universit\”at Paderborn) and Tomasz Przebinda (University of Oklahoma).