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

Abstract:

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.