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Journal of Lie Theory 16 (2006), No. 1, 067--082
Copyright Heldermann Verlag 2006

Kazhdan and Haagerup Properties in Algebraic Groups over Local Fields

Yves de Cornulier
Institut de Géométrie, Algèbre et Topologie, École Polytechnique Fédérale, 1015 Lausanne, Switzerland


\def\g{{\frak g}} \def\s{{\frak s}} Given a Lie algebra $\s$, 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 $\s$ and invariant alternating forms. \endgraf 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 $\frak{sl}_2$ 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 $\bf K$ of characteristic zero, and if its amenable radical is not, up to isogeny, a direct factor, then $G(\bf K)$ has Property (T) relative to a noncompact subgroup. In particular, $G(\bf{K})$ 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. \endgraf 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 Property (T), with pairwise non-isomorphic (resp. isomorphic) Lie algebras.

Keywords: Kazhdan's Property (T), Haagerup Property, a-T-menability.

MSC: 22E50; 22D10, 20G25, 17B05

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