It is not clear what you really want: the space $Sym^kG^{\Bbb C}$ is not smooth, and there are several different definitions of symplectic structures on singular spaces.

(For a compact Kahler manifold). If the hbc is strictly positive, your manifold is $CP^n$; if it is non-negative, your manifold is (conjecturally) homogeneous. HBC measures positivity of the tangent bundle.

I did not find this theorem in Kneser (my German is bad, but I can read a bit, and I tried to leaf through this book). Could you be so kind to tell me where in the book I can find it?

The problem with Siegel's Main Theorem at page 304-305 is that it starts with "definite lattice", in the end there is two lines explaining what happens for indefinite lattices, but it's not immediately clear how this implies finiteness of the number of orbits. I hope the Kneser's book is more direct.