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

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- by Samuel Jeralds and Shrawan Kumar
- Represent. Theory
**26**(2022), 825-858 - DOI: https://doi.org/10.1090/ert/617
- Published electronically: July 26, 2022
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## 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.## References

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## Bibliographic Information

**Samuel Jeralds**- Affiliation: School of Mathematics and Physics, The University of Queensland, Brisbane, Queensland 4072, Australia
- ORCID: 0000-0003-3109-7733
- Email: s.jeralds@uq.edu.au
**Shrawan Kumar**- Affiliation: Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599-3250
- MR Author ID: 219351
- Email: shrawan@email.unc.edu
- Received by editor(s): July 14, 2021
- Received by editor(s) in revised form: April 21, 2022
- Published electronically: July 26, 2022
- Additional Notes: The second author was partially supported by the NSF grant number DMS-1802328.
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory
**26**(2022), 825-858 - MSC (2020): Primary 14M15, 14M17, 20G44, 22E47
- DOI: https://doi.org/10.1090/ert/617
- MathSciNet review: 4457456