Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces

TL;DR

This paper introduces quantum analogues of certain homogeneous spaces and shows that their zonal spherical functions are represented by Macdonald's symmetric polynomials with specific parameter choices.

Contribution

It constructs quantum versions of $ ext{GL}(n)/ ext{SO}(n)$ and $ ext{GL}(2n)/ ext{Sp}(2n)$ and identifies Macdonald polynomials as their zonal spherical functions.

Findings

Quantum analogues of classical homogeneous spaces are defined.

Zonal spherical functions on these spaces are expressed via Macdonald polynomials.

Specific parameter values relate Macdonald polynomials to the quantum spaces.

Abstract

Quantum analogues of the homogeneous spaces $\GL(n)/\SO(n)$ and $\GL(2n)/\Sp(2n)$ are introduced. The zonal spherical functions on these quantum homogeneous spaces are represented by Macdonald's symmetric polynomials $P_{\ld}=P_{\ld}(x_1,\cdots,x_n;q,t)$ with $t=q^{1 \over 2}$ or $t=q^2$.