
Journal of Lie Theory 12 (2002), No. 1, 031039 Copyright Heldermann Verlag 2002 On Kazhdan's Property (T) for Sp_{2}(k) M. B. Bekka Dép. des Mathématiques, Université de Metz, Ile du Saulcy, 57045 Metz, France Markus Neuhauser Zentrum Mathematik, Technische Universität, Arcisstrasse 21, 80290 München, Germany The aim of this note is to give a new and elementary proof of Kazhdan's Property (T) for Sp_{2}(k), the symplectic group on 4 variables, for any local field k. The crucial step is the proof that the Dirac measure δ_{0} at 0 is the unique mean on the Borel subsets of the second symmetric power S^{2}(k^{2}) of k^{2} which is invariant under the natural action of SL_{2}(k). In the case where k has characteristic 2, we observe that this is no longer true if S^{2}(k^{2}) is replaced by its dual, the space of the symmetric bilinear forms on k^{2}. [ Fulltextpdf (163 KB)] 