On the tensor semigroup of affine kac-moody lie algebras

In this paper, we are interested in the decomposition of the tensor product of two representations of a symmetrizable Kac-Moody Lie algebra $\mathfrak g$. Let $P_+$ be the set of dominant integral weights. For $\lambda\in P_+$ , $L(\lambda)$ denotes the irreducible, integrable, highest weight representation of g with highest weight $\lambda$. Let $P_{+,\mathbb Q}$ be the rational convex cone generated by $P_+$. Consider the tensor cone $\Gamma(\mathfrak g) := \{(\lambda_1 ,\lambda_2, \mu) ∈ P_{+,\mathbb Q}^3\,| \exists N > 1 L(N\mu) \subset L(N \lambda_1)\otimes L(N \lambda_2)\}$. If $\mathfrak g$ is finite dimensional, $\Gamma(\mathfrak g)$ is a polyhedral convex cone described in 2006 by Belkale-Kumar by an explicit finite list of inequalities. In general, $\Gamma(\mathfrak g)$ is nor polyhedral, nor closed. In this article we describe the closure of $\Gamma(\mathfrak g)$ by an explicit countable family of linear inequalities, when $\mathfrak g$ is untwisted affine. This solves a Brown-Kumar's conjecture in this case. We also obtain explicit saturation factors for the semigroup of triples $(\lambda_1, \lambda_2 , \mu) ∈ P_+^3$ such that $L(\mu) ⊂ L(\lambda_1) \otimes L(\lambda_2)$. Note that even the existence of such saturation factors is not obvious since the semigroup is not finitely generated. For example, in type $A , we prove that any integer $d\geq 2$ is a saturation factor, generalizing the case ${\tilde A}_1$ shown by Brown-Kumar