A study of saturated tensor cone for symmetrizable Kac-Moody algebras

Let $\fg$ be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra $\fh$ and the Weyl group $W$. Let $P_+$ be the set of dominant integral weights. For $\lambda \in P_+$, let $L(\lambda)$ be the irreducible, integrable, highest weight representation of $\fg$ with highest weight $\lambda$. For a positive integer $s$, define the {\em saturated tensor semigroup} as \begin{align*} \Gamma_s:= \{(\lambda_1, \dots, \lambda_s,\mu)\in P_+^{s+1}: \exists\, N>1 \,\,\text{with}\,\, L(N\mu)\subset L(N\lambda_1)\otimes \dots \otimes L(N\lambda_s)\}. \end{align*} The aim of this paper is to begin a systematic study of $\Gamma_s$ in the infinite dimensional symmetrizable Kac-Moody case. In this paper, we produce a set of necessary inequalities satisfied by $\Gamma_s$. We further prove that any integer $d>0$ is a saturation factor for $A^{(1)}_1$ and 4 is a saturation factor for $A^{(2)}_2$.