Mathematical Research Letters

Volume 24 (2017)

Number 3

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

Pages: 741 – 766

DOI: http://dx.doi.org/10.4310/MRL.2017.v24.n3.a6

Authors

Evgeny Feigin (Department of Mathematics, Higher School of Economics, National Research University, Moscow, Russia; and Skolkovo Institute of Science and Technology, Moscow, Russia)

Ievgen Makedonskyi (Max Planck Institute for Mathematics, Bonn, Germany; and Department of Mathematics, Higher School of Economics, National Research University, Moscow, Russia)

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 $A^{(2)\dagger}_2$. 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.

Full Text (PDF format)

Paper received on 9 July 2015.