  Let G be a connected semisimple complex algebraic group and let P be a
parabolic subgroup. In this paper we define a new (commutative and associative)
product on the cohomology of the homogenous spaces G/P and use this to give a
more efficient solution of the eigenvalue problem and also for the problem of
determining the existence of G-invariants in the tensor product of irreducible
representations of G. On the other hand, we show that this new product is
intimately connected with the Lie algebra cohomology of the nil-radical of P
via some works of Kostant and Kumar. We also initiate a uniform study of the
geometric Horn problem for an arbitrary group $G$ by obtaining two (a priori)
different sets of necessary recursive conditions to determine when a cohomology
product of Schubert classes in
  G/P is non-zero. Hitherto, this was studied largely only for the group
\SL(n).
