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We introduce property $(T_{Schur},G,K)$ and prove it for some non-cocompactlattice in $Sp_4$ in a local field of finite characteristic. We show thatproperty $(T_{Schur},G,K)$ for a non-cocompact lattice $\Gamma$ in a higherrank almost simple algebraic group in a local field is an obstacle to provingthe Baum Connes conjecture without coefficient for $\Gamma$ with known methods,and this is stronger than the well-known fact that $\Gamma$ does not have theproperty of rapid decay (property (RD)). It is the first example (as announcedin [Laf10a]) for which all known methods for proving the Baum Connes conjecturewithout coefficient fail.

About the difficulty to prove the Baum Connes conjecture without coefficient for a non-cocompact lattice in $Sp_4$ in a local field