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Journal of Lie Theory 12 (2002), No. 1, 031--039
Copyright Heldermann Verlag 2002

On Kazhdan's Property (T) for Sp2(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 Sp2(k), the symplectic group on 4 variables, for any local field k. The crucial step is the proof that the Dirac measure δ0 at 0 is the unique mean on the Borel subsets of the second symmetric power S2(k2) of k2 which is invariant under the natural action of SL2(k). In the case where k has characteristic 2, we observe that this is no longer true if S2(k2) is replaced by its dual, the space of the symmetric bilinear forms on k2.

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