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 emph {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. par 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 $(mathfrak {S}_{2l},mathfrak {S}_l^2)$. [ABSTRACT FROM AUTHOR]