For positive integers n and l, we study the cyclic U(gln)-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 Sl(ℂn)× of the symmetric l-th tensor space of Cn for all but finitely many exceptional values of α. If α is exceptional, then the cyclic module is equivalent to a proper submodule of Sl(ℂn)×, i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in Sl(ℂn)×. 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 (Cnl,Cnl ) 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,C2l ). copy; 2009 American Mathematical Society.
All Science Journal Classification (ASJC) codes
- General Mathematics
- Applied Mathematics