Alpha-determinant cyclic modules and Jacobi polynomials

For positive integers n and l, we study the cyclic U(gl n )-module generated by the l-th power of the α-determinant det(X). This cyclic module is isomorphic to the n-th tensor space (Sym(C)) of the symmetric l-th tensor space of C for all but finite exceptional values of α. If α is exceptional, then the cyclic module is equivalent to a proper submodule of (Sym(C)), i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in (Sym(C)). The degeneration of each isotypic component of the cyclic module is described by a matrix whose size is given by a Kostka number and entries are polynomials in α with rational coefficients. Especially, we determine the matrix completely when n = 2. In that case, the matrix becomes a scalar and is essentially given by the classical Jacobi polynomial. Moreover, we prove that these polynomials are unitary. In the Appendix, we consider a variation of the spherical Fourier transformation for (Snl,S n l ) as a main tool to analyze the same problems, and describe the case where n = 2 by using the zonal spherical functions of the Gelfand pair (S2l,S 2 l ).