In this note we construct a finite analogue of classical Siegel's Space. Our
approach is to look at it as a non commutative Poincare's half plane. The
finite Siegel Space is described as the space of Lagrangians of a 2n
dimensional space over a quadratic extension E of a finite base field F.
The orbits of the action of the symplectic group Sp(n,F) on Lagrangians are
described as homogeneous spaces. Also, Siegel's Space is described as the set
of anti-involutions of the symplectic group.2