Kazhdan and Haagerup Properties in Algebraic Groups over Local Fields

We prove some results about solvable Lie algebras endowed with a reductive action of a fixed Lie algebra. The first application consists in proving that if g is a Lie algebra over a local 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 sl2 by either a nontrivial irreducible representation or a Heisenberg group. 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) is not Haagerup. This extends a similar result of Cherix, Cowling and Valette for Lie groups, to which our method also applies. We give some other applications. We provide a characterization of Lie groups all of whose countable subgroups are Haagerup. We give an example of arithmetic lattice in a Lie group which is not Haagerup, but has no infinite subgroup with relative Property (T). We also give a continuous family of pairwise non-isomorphic Lie groups with Property (T), with pairwise non-isomorphic (resp. isomorphic) Lie algebras.