# On generalized Macdonald polynomials

## TL;DR

This paper introduces generalized Macdonald polynomials (GMP), exploring their orthogonality, factorized denominators, and a simplified deformed version called generalized Schur functions, highlighting their mathematical structure and potential applications.

## Contribution

The paper presents the construction and properties of GMP, including their orthogonality, factorized denominators, and introduces generalized Schur functions as a simplified deformed variant.

## Key findings

- GMP are eigenfunctions of deformed Ruijsenaars Hamiltonians.
- Denominators in GMP expansions are fully factorized due to hidden symmetries.
- Introduction of generalized Schur functions as a simplified GMP variant.

## Abstract

Generalized Macdonald polynomials (GMP) are eigenfunctions of specifically-deformed Ruijsenaars Hamiltonians and are built as triangular polylinear combinations of Macdonald polynomials. They are orthogonal with respect to a modified scalar product, which could be constructed with the help of an increasingly important triangular perturbation theory, showing up in a variety of applications. A peculiar feature of GMP is that denominators in this expansion are fully factorized, which is a consequence of a hidden symmetry resulting from the special choice of the Hamiltonian deformation. We introduce also a simplified but deformed version of GMP, which we call generalized Schur functions. Our basic examples are bilinear in Macdonald polynomials.