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

TL;DR

This paper extends the understanding of tensor product components for affine Kac-Moody Lie algebra modules, showing new inclusion results and geometric properties using Virasoro algebra actions and cohomology vanishing.

Contribution

It generalizes Kumar's finite-dimensional results to affine Kac-Moody algebras and establishes geometric vanishing theorems related to tensor products.

Findings

Tensor product contains specific shifted modules under mild restrictions.

Virasoro algebra action is crucial in the proof.

Higher cohomology vanishing on G-Schubert varieties is established.

Abstract

Let g be an affine Kac-Moody Lie algebra and let $\lambda, \mu$ be two dominant integral weights for 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) 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 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.