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

Abstract

Let g be an affine Kac-Moody Lie algebra and let $λ, μ$ be two dominant integral weights for g. We prove that under some mild restriction, for any positive root $β$, $V(λ)\otimes V(μ)$ contains $V(λ+μ-β)$ as a component, where $V(λ)$ denotes the integrable highest weight (irreducible) g-module with highest weight $λ$. 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 $V(λ)\otimes V(μ)$. Then, we prove the corresponding geometric results including the higher cohomology vanishing on the G-Schubert varieties in the product partial flag variety G/P X G/P with coefficients in certain sheaves coming from the ideal sheaves of G-sub Schubert varieties. This allows us to prove the surjectivity of the Gaussian map.