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

M. B. Bekka and M. Neuhauser

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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