A spectral gap property for subgroups of finite covolume in Lie groups

Bachir Bekka ¹   ; Yves Cornulier ²

¹ IRMAR, UMR 6625 du CNRS Université de Rennes 1 Campus Beaulieu F-35042 Rennes Cedex, France
² IRMAR, UMR 6625 du CNRS, Université de Rennes 1 Campus Beaulieu F-35042 Rennes Cedex, France

Colloquium Mathematicum, Tome 118 (2010) no. 1, pp. 175-182

Résumé

Let $G$ be a real Lie group and $H$ a lattice or, more generally, a closed subgroup of finite covolume in $G$. We show that the unitary representation $\lambda _{G/H}$ of $G$ on $L^2(G/H)$ has a spectral gap, that is, the restriction of $\lambda _{G/H}$ to the orthogonal complement of the constants in $L^2(G/H)$ does not have almost invariant vectors. This answers a question of G. Margulis. We give an application to the spectral geometry of locally symmetric Riemannian spaces of infinite volume.

Détail

DOI : 10.4064/cm118-1-9

Keywords: real lie group lattice generally closed subgroup finite covolume unitary representation lambda has spectral gap restriction lambda orthogonal complement constants does have almost invariant vectors answers question margulis application spectral geometry locally symmetric riemannian spaces infinite volume