Orthogonality of spin $q$-Whittaker polynomials

TL;DR

This paper proves the orthogonality of inhomogeneous spin q-Whittaker polynomials, generalizing Macdonald polynomials, and establishes their basis property in symmetric polynomial space, with implications for related special functions.

Contribution

It demonstrates the orthogonality of inhomogeneous spin q-Whittaker polynomials and their basis property, extending Macdonald polynomial theory with new eigenrelations and special cases.

Findings

Proved orthogonality with respect to a Sklyanin measure.

Established basis property in symmetric polynomial space.

Connected to variants of Grothendieck polynomials and spin Whittaker functions.

Abstract

The inhomogeneous spin $q$-Whittaker polynomials are a family of symmetric polynomials which generalize the Macdonald polynomials at $t=0$. In this paper we prove that they are orthogonal with respect to a variant of the Sklyanin measure on the $n$ dimensional torus and as a result they form a basis of the space of symmetric polynomials in $n$ variables. Instrumental to the proof are inhomogeneous eigenrelations, which partially generalize those of Macdonald polynomials. We also consider several special cases of the inhomogeneous spin $q$-Whittaker polynomials, which include variants of symmetric Grothendieck polynomials or spin Whittaker functions.