Lie Groups, Representations and Discrete Mathematics

A pair (G,H), where G is a group and H a subgroup, has relative Property T if every isometric action of G on a Hilbert space has a H-fixed point. In a connected Lie group or a lattice G, we characterize subgroups H such that (G,H) has relative Property T. We will discuss examples showing that the existence of an unbounded subgroup with relative Property T is not the only obstruction to have a proper isometric action on a Hilbert space.