Alpha-determinant cyclic modules and Jacobi polynomials

For positive integers $n$ and $l$, we study the cyclic $\mathcal{U}(\mathfrak{gl}_n)$-module generated by the $l$-th power of the $\alpha$-determinant $\det^{(\alpha)}(X)$. This cyclic module is isomorphic to the $n$-th tensor space $\mathcal{S}^l(\mathbb{C}^n)^{\otimes n}$ of the symmetric $l$-th tensor space of $\mathbb{C}^n$ for all but finitely many exceptional values of $\alpha$. If $\alpha$ is exceptional, then the cyclic module is equivalent to a proper submodule of $\mathcal{S}^l(\mathbb{C}^n)^{\otimes n}$, i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in $\mathcal{S}^l(\mathbb{C}^n)^{\otimes n}$. 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 $\alpha$ 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 $(\mathfrak{S}_{nl},\mathfrak{S}_l^n)$ 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 $(\mathfrak{S}_{2l},\mathfrak{S}_l^2)$.