Quadratic transformations of Macdonald and Koornwinder polynomials

TL;DR

This paper proves several conjectured q-analogues related to the expansion of Schur functions in terms of symplectic and orthogonal group characters, using affine Hecke algebra techniques.

Contribution

It establishes most of the conjectured q-analogues and introduces methods involving affine Hecke algebra ideals to prove vanishing properties.

Findings

Most conjectured q-analogues are proven.

Nonsymmetric integral versions are shown to be annihilated by affine Hecke algebra ideals.

Modifications to Macdonald and Koornwinder polynomials are discussed for vanishing identities.

Abstract

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.