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Kostant asked the following question: Let $\mathfrak{g}$ be a simple Liealgebra over the complex numbers. Let $\lambda$ be a dominant integral weight.Then, $V(\lambda)$ is a component of $V(\rho)\otimes V(\rho)$ if and only if$\lambda \leq 2 \rho$ under the usual Bruhat-Chevalley order on the set ofweights. In an earlier work with R. Chirivi and A. Maffei the second authorgave an affirmative answer to this question up to a saturation factor. The aimof the current work is to extend this result to untwisted affine Kac-Moody Liealgebra $\mathfrak{g}$ associated to any simple Lie algebra$\mathring{\mathfrak{g}}$ (up to a saturation factor). In fact, we prove theresult for affine $sl_n$ without any saturation factor. Our proof requires someadditional techniques including the Goddard-Kent-Olive construction and studyof the characteristic cone of non-compact polyhedra.

Components of $V(\rho) \otimes V(\rho)$ and dominant weight polyhedra for affine Kac-Moody Lie algebras