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 (α) (X). This cyclic module is isomorphic to the n-th tensor space S l (ℂ n ) ⊗n of the symmetric l-th tensor space of ℂ n for all but finitely many exceptional values of α. If α is exceptional, then the cyclic module is equivalent to a proper submodule of S l (ℂ n ) ⊗n , i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in S l (ℂ n ) ⊗n . 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 α with rational coefficients. In particular, we determine the matrix completely when n = 2. 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 (O nl , O n l ) as a main tool for analyzing the same problems, and describe the case where n = 2 by using the zonal spherical functions of the Gelfand pair (O 2l , O 2 l ).