EMIS ELibM Electronic Journals Journal of Lie Theory
Vol. 12, No. 1, pp. 31--39 (2002)

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On Kazhdan's Property (T) for $\Sp_2(k)$

M. B. Bekka and M. Neuhauser

M. B. Bekka
Université de Metz
Département de Mathématiques
Ile du Saulcy, F- 57045 Metz, France

M. Neuhauser
Technische Universität München
Zentrum Mathematik
Arcisstr. 21, D-80290 München, Germany

Abstract: The aim of this note is to give a new and elementary proof of Kazhdan's Property (T) for $\func{Sp}_2\left( {\bf k}\right),$ the symplectic group on 4 variables, for any local field {\bf k}. The crucial step is the proof that the Dirac measure $\delta _{0}$ at $0$ is the unique mean on the Borel subsets of the second symmetric power $S^2({\bf k}^{2})$ of ${\bf k}^{2}$ which is invariant under the natural action of $\func{SL}_{2}\left({\bf k}\right).$ In the case where ${\bf k}$ has characteristic 2, we observe that this is no longer true if $S^2({\bf k}^{2})$ is replaced by its dual, the space of the symmetric bilinear forms on ${\bf k}^{2}.$

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Electronic fulltext finalized on: 30 Oct 2001. This page was last modified: 9 Nov 2001.

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