A rational representation $(G,V)$ of a complex reductive linear algebraic group is called **multiplicity-free** if the decomposition of $\mathbb C[V]$ into irreducible $G$-modules contains each irreducible $G$-module with multiplicity at most one. Part of the importance of such representations lies in the fact that they serve to explain several phenomena in invariant theory and harmonic analysis (e.g. Peter-Weyl theorem).

There are several different characterizations of multiplicity free representations, among them: any Borel subgroup has an open (dense) orbit; the principal orbits of a compact real form are coisotropic submanifolds with respect to an invariant (real) symplectic structure.

If $(G,V)$ is multiplicity free then $G$ has finitely many orbits in $V$.
In particular, it is *visible* in the sense that $V$ has finitely many nilpotent orbits, and hence the multiplicity of any nonzero weight is at most one, see Kac, V. G. Some remarks on nilpotent orbits. J. Algebra 64 (1980), no. 1, 190–213.

It is apparent from the classification of (indecomposable) multiplicity free representations that they are *weight multiplicity free*, namely, the multiplicity of any weight is at most one. **Is there a direct proof of this fact**?

Schur lectures? I don't have a copy close at hand, but he discusses the relations berween MF, WMF and skew-multiplicity free properties. $\endgroup$ – Victor Protsak Aug 23 '16 at 19:47