Components of V(ρ)\otimes V(ρ)

Abstract

Kostant asked the following question: Let $\mathfrak{g}$ be a simple Lie algebra over the complex numbers. Let $λ$ be a dominant integral weight. Then, $V(λ)$ is a component of $ V(ρ)\otimes V(ρ)$ if and only if $λ\leq 2ρ$ under the usual Bruhat-Chevalley order on the set of weights. We give an affirmative answer to this question up to a saturation factor. In particular, the question is answered affirmatively for the special linear Lie algebras due to the Saturation Theorem of Knutson-Tao.