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The intersection $\\mu(\\mathbb P(V_\\lambda))^+=\\mu(\\mathbb P(V_\\lambda))\\cap{\\mathfrak t}^+$ of the momentum image with a fixed Weyl chamber is a convex polytope called the momentum polytope of $V_\\lambda$. We construct an affine rational polyhedral convex cone $\\Upsilon_\\lambda$ with vertex $\\lambda$, such that $\\mu(\\mathbb P(V_\\lambda))^+\\subset\\Upsilon_\\lambda \\cap {\\mathfrak t}^+$. 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Tsanov","submitted_at":"2015-04-05T10:33:21Z","abstract_excerpt":"Let $K$ be a connected compact semisimple group and $V_\\lambda$ be an irreducible unitary representation with highest weight $\\lambda$. We study the momentum map $\\mu:\\mathbb P(V_\\lambda)\\to\\mathfrak k^*$. The intersection $\\mu(\\mathbb P(V_\\lambda))^+=\\mu(\\mathbb P(V_\\lambda))\\cap{\\mathfrak t}^+$ of the momentum image with a fixed Weyl chamber is a convex polytope called the momentum polytope of $V_\\lambda$. We construct an affine rational polyhedral convex cone $\\Upsilon_\\lambda$ with vertex $\\lambda$, such that $\\mu(\\mathbb P(V_\\lambda))^+\\subset\\Upsilon_\\lambda \\cap {\\mathfrak t}^+$. 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