The SL(<Emphasis Type="Italic">V</Emphasis>)-ample cone of product of flag varieties

Let k be an algebraically closed field, and V be a vector space of dimension n over k. For a set ω = (\(\vec d\)(1), ..., \(\vec d\)(m)) of sequences of positive integers, denote by L _ω the ample line bundle corresponding to the polarization on the product X = Π _i=1 ^m Flag(V, \(\vec n\)(i)) of flag varieties of type \(\vec n\)(i) determined by ω. We study the SL(V)-linearization of the diagonal action of SL(V) on X with respect to L _ω. We give a sufficient and necessary condition on ω such that X ^ss (L _ω) ≠ \(\not 0\) (resp., X ^s (L _ω) ≠ \(\not 0\)). As a consequence, we characterize the SL(V)-ample cone (for the diagonal action of SL(V) on X), which turns out to be a polyhedral convex cone.