SMRI Algebra and Geometry Online: Kumar -- Root components for tensor product of affine Kac-Moody Lie algebra modules’

SMRI Algebra and Geometry Online 
’Root components for tensor product of affine Kac-Moody Lie algebra modules’
Shrawan Kumar (University of North Carolina) 

Friday Jul 23, 2021 
11:00am-12:30pm (AEST) 
Register:
https://uni-sydney.zoom.us/meeting/register/tZ0sc--qrzkpHdd5eE6IgQUYtXWfnEssOCIC 

This is a joint work with Samuel Jeralds.  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) ⊗ 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) ⊗ 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 × 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.  

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