  For each irreducible module of the symmetric group on $N$ objects there is a
set of parametrized nonsymmetric Jack polynomials in $N$ variables taking
values in the module. These polynomials are simultaneous eigenfunctions of a
commutative set of operators, self-adjoint with respect to certain Hermitian
forms. These polynomials were studied by the author and J.-G. Luque using a
Yang-Baxter graph technique. This paper constructs a matrix-valued measure on
the $N$-torus for which the polynomials are mutually orthogonal. The
construction uses Fourier analysis techniques. Recursion relations for the
Fourier-Stieltjes coefficients of the measure are established, and used to
identify parameter values for which the construction fails. It is shown that
the absolutely continuous part of the measure satisfies a first-order system of
differential equations.
