For positive integers and , we study the cyclic -module generated by the -th power of the -determinant . This cyclic module is isomorphic to the -th tensor space of the symmetric -th tensor space of for all but finitely many exceptional values of . If is exceptional, then the cyclic module is equivalent to a proper submodule of , i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in . The degeneration of each isotypic component of the cyclic module is described by a matrix whose size is given by a Kostka number and whose entries are polynomials in with rational coefficients. In particular, we determine the matrix completely when . In this case, the matrix becomes a scalar and is essentially given by a classical Jacobi polynomial. Moreover, we prove that these polynomials are unitary.

In the Appendix, we consider a variation of the spherical Fourier transformation for as a main tool for analyzing the same problems, and describe the case where by using the zonal spherical functions of the Gelfand pair .