A Remark on Random Vectors and Irreducible Representations

It was observed in [1] that the expectation of a squared scalar product of two random independent unit vectors that are uniformly distributed on the unit sphere in $R^n $ is equal to $1/n$. It is shown in this paper, that this is a characteristic property of random vectors defined on invariant probability subspaces of unit spheres in irreducible real representations of compact Lie groups.