QUASI-HOMOGENEITY OF THE MODULI SPACE OF STABLE MAPS TO HOMOGENEOUS SPACES (II)

Abstract

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 variety attached to this situation. Let d ∈ H2(X) be a degree. Let $$ {\overline{M}}_{0,3} $$ (X, d) be the (coarse) moduli space of three pointed genus zero stable maps to X of degree d. Building on and improving our previous results [4], we prove that $$ {\overline{M}}_{0,3} $$ (X, d) is quasi-homogeneous under the action of Aut(X) for all minimal degrees d in H2 (X). By a minimal degree in H2(X), we mean a degree d ∈ H2(X) such that there exist Weyl group elements u and v such that d is minimal with the property that qd occurs (with non-zero coefficient) in the quantum product σu ✭ σv of the Schubert classes σu and σv, where ✭ denotes the product in the (small) quantum cohomology ring QH*(X) attached to X. Along the way, we prove that M0;3(X; d) is quasi-homogeneous under the action of G for all minimal degrees d in H2 (X) except for one instance of G, P and d which occurs in type G2.