Let G be a connected, simply connected, simple, complex, linear algebraic group. Let P be an arbitrary parabolic subgroup of G. Let X= G/P be the G-homogeneous projective space attached to this situation. Let d ∈ H2(X) be a degree. Let M0,3 (X,d) be the (coarse) moduli of three pointed genus zero stable maps to X of degree d. We prove under reasonable assumptions on d that M0,3 (X,d) is quasi-homogeneous under the action of G.
The essential assumption on d is that d is a minimal degree, i.e. that d is a degree which is minimal with the property that qd occurs with non-zero coefficient in the quantum product σu* σv of two Schubert cycles σu and σv where * is the product in the (small) quantum cohomology rinq QH*(X) attached to X. We prove our main result on quasi-homogeneity by constructing an explicit morphism which has a dense open G-orbit in M0,3 (X,d) . To carry out the construction of this morphism, we develop a combinatorial theory of generalized cascades of orthogonal roots which is interesting in its own right.