Let $G$ be a semi-simple group over complex number; for simplicity let us assume that it is simply laced. Let $X$ be the orbit of the highest root line in the adjoint representation of $G$. This is a projective variety, such that the corresponding cone is equal to the minimal nilpotent in the Lie algebra of $G$.

What is the cohomology of $X$? E.g. can you write explicitly its Poincare polynomial? Is it, for example, true that its Euler characteristic is equal to $dim(G)-rank(G)$? It would be great it you could give me a reference.