Kazhdan and Haagerup properties in algebraic groups over local fields

Given a Lie algebra, 5, we call Lie s-algebra a Lie algebra endowed with a reductive action of s. We characterize the minimal s-Lie algebras with a nontrivial action of s, in terms of irreducible representations of 5 and invariant alternating forms. As a first application, we show that if g is a Lie algebra over a field of characteristic zero whose amenable radical is not a direct factor, then g contains a subalgebra which is isomorphic to the semidirect product of sl(2) by either a nontrivial irreducible representation or a Heisenberg group (this was essentially due to Cowling, Dorofaeff, Seeger, and Wright). As a corollary, if G is an algebraic group over a local field K of characteristic zero, and if its amenable radical is not, up to isogeny, a direct factor, then G(K) has Property (T) relative to a noncompact subgroup. In particular, G(K) does not have Haagerup's property. This extends a similar result of Cherix, Cowling and Valette for connected Lie groups, to which our method also applies. We give some other applications. We provide a characterization of connected Lie groups all of whose countable subgroups have Haagerup's property. We give an example of an arithmetic lattice in a connected Lie group which does not have Haagerup's property, but has no infinite subgroup with relative Property (T). We also give a continuous family of pairwise non-isomorphic connected Lie groups with with Property (T), with pairwise non-isomorphic (resp. isomorphic) Lie algebras.