## Abstract

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}(ℂn)^{×} 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})^{×}, i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in S^{l}(ℂ^{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 a 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 (C_{nl},C^{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 (C _{2l},C^{2}_{l} ). copy; 2009 American Mathematical Society.

Original language | English |
---|---|

Pages (from-to) | 6447-6473 |

Number of pages | 27 |

Journal | Transactions of the American Mathematical Society |

Volume | 361 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 2009 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics