  Let $k$ be an algebraically closed field of characteristic $p>0$ and let
$\ell$ be another prime number. O. Gabber and F. Loeser proved that for any
algebraic torus $T$ over $k$ and any perverse $\ell$-adic sheaf $\calF$ on $T$
the Euler characteristic $\chi(\calF)$ is non-negative.
  We conjecture that the same result holds for any perverse sheaf $\calF$ on a
reductive group $G$ over $k$ which is equivariant with respect to the adjoint
action. We prove the conjecture when $\calF$ is obtained by Goresky-MacPherson
extension from the set of regular semi-simple elements in $G$. From this we
deduce that the conjecture holds for $G$ of semi-simple rank 1.
