Root components for tensor product of affine Kac-Moody Lie algebra modules.

Let \mathfrak {g} be an affine Kac-Moody Lie algebra and let \lambda, \mu be two dominant integral weights for \mathfrak {g}. We prove that under some mild restriction, for any positive root \beta, V(\lambda)\otimes V(\mu) contains V(\lambda +\mu -\beta) as a component, where V(\lambda) denotes the integrable highest weight (irreducible) \mathfrak {g}-module with highest weight \lambda. This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V(\lambda)\otimes V(\mu). Then, we prove the corresponding geometric results including the higher cohomology vanishing on the \mathcal {G}-Schubert varieties in the product partial flag variety \mathcal {G}/\mathcal {P}\times \mathcal {G}/\mathcal {P} with coefficients in certain sheaves coming from the ideal sheaves of \mathcal {G}-sub-Schubert varieties. This allows us to prove the surjectivity of the Gaussian map. [ABSTRACT FROM AUTHOR]