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