We seek for the Hilbert series of the ring of invariant polynomials in the $2n+n^2$ variables $\{u_i,v_j,x_{i,j}\}_{i,j=1}^n$ under the action of $GL_n[C]$ by right multiplication on the row vector $u=(u_1,u_2,\ldots ,u_n)$, left multiplication on the column vector $v=(v_1,v_2,\ldots ,v_n)$ and by conjugation on the matrix $\|x_{i,j}\|_{i,j=1}^n$. We reduce the computation of this Hilbert series to the evaluation of the constant term of a certain rational function. Remarkably, the final result hinges on the explicit evaluation of certain Kostka-Foulkes polynomials.