  We study the cyclic $U(\mathfrak{gl}_n)$-module generated by the $l$-th power
of the $\alpha$-determinant. When $l$ is a non-negative integer, for all but
finite exceptional values of $alpha$, one shows that this cyclic module is
isomorphic to the $n$-th tensor space $(S^l(\mathbb{C}^n))^{\otimes n}$ of the
symmetric $l$-th tensor space of $\mathbb{C}^n$. If $alpha$ is exceptional,
then the structure of the module changes drastically, i.e. some irreducible
representations which are the irreducible components of the decomposition of
$(S^l(\mathbb{C}^n))^{\otimes n}$ disappear in the decomposition of the cyclic
module. 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 $alpha$ with rational coefficients. As a special case, we
determine the matrix in a full of the detail for the case where $n=2$; the
matrix becomes a scalar and is essentially given by the classical Jacobi
polynomial. Moreover, we prove that these polynomials are unitary.
