Weyl modules for $\mathfrak{osp}(1,2)$ and nonsymmetric Macdonald polynomials

TL;DR

This paper establishes a connection between Weyl modules for the Lie superalgebra rak{osp}(1,2) and nonsymmetric Macdonald polynomials, providing explicit formulas and bases, thus linking representation theory with special functions.

Contribution

It introduces a novel relationship between Weyl modules of rak{osp}(1,2) and nonsymmetric Macdonald polynomials of types A_2^{(2)} and A_2^{(2)agger}, including explicit formulas and bases.

Findings

Computed dimensions and constructed bases of Weyl modules.

Derived explicit formulas for t=0 and t=5 specializations.

Connected specializations to Lie superalgebra actions.

Abstract

The main goal of our paper is to establish a connection between the Weyl modules of the current Lie superalgebras (twisted and untwisted) attached to $\mathfrak{osp}(1,2)$ and the nonsymmetric Macdonald polynomials of types $A_2^{(2)}$ and $A_2^{(2)\dagger}$. We compute the dimensions and construct bases of the Weyl modules. We also derive explicit formulas for the t=0 and t=\infty specializations of the nonsymmetric Macdonald polynomials. We show that the specializations can be described in terms of the Lie superalgebras action on the Weyl modules.