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