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To each finite-dimensional representation of a simple Lie algebra isassociated a multiplicative graph in the sense of Kerov and Vershik definedfromthe decomposition of its tensor powers into irreducible components. Theconditioning of naturalrandom Littelmann paths to stay in their correspondingWeyl chamber is thencontrolled by central measures on this type of graphs.Using the K-theory of associated C*-algebras, Handelman established ahomeomorphism between the set of central measures on these multiplicativegraphs and the weight polytope of theunderlying representation. In the presentpaper, we make explicit this homeomorphism independently of Handelman's resultsby using Littelmann's path model. As a by-product we also get an explicitparametrization of theweight polytope in terms of drifts of random Littelmannpaths. This explicit parametrization yields a complete description of harmonicand c-harmonic functions for this Littelmann paths model.

Central measures on multiplicative graphs, representations of Lie algebras and weight polytopes