Quadratic transformations of Macdonald and Koornwinder polynomials

Details Quadratic transformations of Macdonald and Koornwinder polynomials Quadratic transformations of Macdonald and Koornwinder polynomials

2006-06-09

arxiv.org/abs/math/0606204v1

Mathematics

Representation Theory

32 pages LaTeX, 10 xfig figures

Scientific paper

When one expands a Schur function in terms of the irreducible characters of the symplectic (or orthogonal) group, the coefficient of the trivial character is 0 unless the indexing partition has an appropriate form. A number of q-analogues of this fact were conjectured in math.QA/0112035; the present paper proves most of those conjectures, as well as some new identities suggested by the proof technique. The proof involves showing that a nonsymmetric version of the relevant integral is annihilated by a suitable ideal of the affine Hecke algebra, and that any such annihilated functional satisfies the desired vanishing property. This does not, however, give rise to vanishing identities for the standard nonsymmetric Macdonald and Koornwinder polynomials; we discuss the required modification to these polynomials to support such results.

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