The expected values of invariant polynomials with matrix argument of elliptical distributions

Invariant polynomials with matrix arguments have been defined by the theory of group representation, generalizing the zonal polynomials. They have developed as a useful tool to evaluate certain integrals arising in multivariate distribution theory, which were expanded as power series in terms of the invariant polynomials. Some interesting polynomials has been shown by people working in the field of econometric theory. In this paper, we derive the expected values of C_φ^κ,λ(BR,BU), C _κ(BR)C_λ(BU) and C_κ(B ^-1U), where B=^dX′X and X_nxp is distributed according to an elliptical matrix distribution. We also give their applications in multivariate distribution theory including the related development in econometrics.