Speaker: Jean-Philippe Anker (Orleans)
Title: Bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces
Abstract:
Let $X=G/K$ be a Riemannian symmetric space of noncompact type, let $\Gamma$ be a discrete torsion free subgroup of $G$, let $Y\!=\Gamma\backslash G/K$ be the associated locally symmetric space and let $\Delta$ be the Laplace-Beltrami operator on $Y$. In rank one, a celebrated result, due to Elstrodt, Patterson, Sullivan and Corlette, expresses the bottom of the $L^2$ spectrum of $-\Delta$ in terms of the critical exponent of the Poincar\'e series of $\Gamma$ on $X$. A less precise result was obtained later on by Leuzinger in higher rank.
We shall discuss in this talk higher rank analogs of the rank one result, which are obtained by considering suitable Poincar\'e series. This is joint work with Hong-Wei Zhang [arXiv:2006.06473].