Quantum Matrix Spherical Functions
Stein Meereboer, Philip Schlösser
TL;DR
This work develops a comprehensive framework for quantum matrix-spherical functions tied to dual Hopf algebras and quantum symmetric pair coideal subalgebras. It proves existence, orthogonality, and a Cartan-decomposition-driven reduction that localizes MSF to Cartan data, yielding matrix weights and enabling radial-operator diagonalization. A key achievement is the identification of vector-valued MSF with Intermediate Macdonald Polynomials, including explicit symmetry and q↔q^{-1} invariance, plus a suite of finite-type examples that realize symmetric and non-symmetric parabolic Macdonald families. The results generalize Letzter’s zonal-spherical connection to a broad vector-valued setting, offering new perspectives on Macdonald theory in the quantum group context and providing concrete, integrable-case examples that align with known parabolic Macdonald polynomials. Collectively, the paper links quantum symmetric-pair representation theory with rich families of orthogonal polynomials, enabling both structural insights and practical computations in finite-type cases.
Abstract
A general theory of matrix-spherical functions for dual Hopf algebras and right coideal subalgebras is developed. We establish their existence and define their orthogonality relations. When specialized to Kolb and Letzter's quantum symmetric pair coideal subalgebras, we associate, to each classical commutative triple, a unique corresponding quantum commutative triple. This leads to families of vector-valued orthogonal polynomials, which diagonalize a commutative algebra of difference-reflection operators and are invariant under sending $q \mapsto q^{-1}$. Various examples of these vector-valued orthogonal polynomials are given and identified with Intermediate Macdonald polynomials