Components of $V(\ddot{I})\otimes V(\ddot{I})$ and dominant weight polyhedra for affine Kac-Moody Lie algebras

Abstract

Kostant asked the following question: Let $\mathfrak{g}$ be a simple Lie algebra over the complex numbers. Let $\hat{I}»$ be a dominant integral weight. Then, $V(\hat{I}»)$ is a component of $V(\ddot{I})\otimes V(\ddot{I})$ if and only if $\hat{I}»\le 2\ddot{I}$ under the usual Bruhat-Chevalley order on the set of weights. In an earlier work with R. Chirivi and A. Maffei the second author gave an affirmative answer to this question up to a saturation factor. The aim of the current work is to extend this result to untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ associated to any simple Lie algebra $\mathring{\mathfrak{g}}$ (up to a saturation factor). In fact, we prove the result for affine $s{l}_n$ without any saturation factor. Our proof requires some additional techniques including the Goddard-Kent-Olive construction and study of the characteristic cone of non-compact polyhedra.

16 pages; v2: minor edits