Kazhdan and Haagerup Properties in algebraic groups over local fields
| dc.creator | de Cornulier, Yves | |
| dc.date | 2004-04-29 | |
| dc.date | 2006-01-21 | |
| dc.date.accessioned | 2026-07-07T06:36:44Z | |
| dc.date.available | 2026-07-07T06:36:44Z | |
| dc.description | 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. 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 Property (T), with pairwise non-isomorphic (resp. isomorphic) Lie algebras. | |
| dc.description | 11 pages, no figure | |
| dc.identifier | https://arxiv.org/abs/math/0404534 | |
| dc.identifier | http://arxiv.org/abs/math/0404534 | |
| dc.identifier | J. Lie Theory 16 (2006), 67-82 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100183 | |
| dc.subject | Group Theory | |
| dc.subject | 22E50 (Primary) 22D10, 20G25, 17B05 (Secondary) | |
| dc.title | Kazhdan and Haagerup Properties in algebraic groups over local fields | |
| dc.type | text |