Deformations of unitary Howe dual pairs

TL;DR

This paper investigates how certain classical dual pairs involving unitary groups and Lie superalgebras deform within the framework of rational Cherednik algebras, preserving algebraic structures and enabling explicit decompositions.

Contribution

It demonstrates the preservation of Lie (super)algebra structures under deformation and provides explicit decompositions for specific cases involving dihedral groups.

Findings

Lie (super)algebra structures are preserved under deformation.

Decompositions are multiplicity-free.

Complete descriptions are given for dihedral group cases.

Abstract

We study deformations of the Howe dual pairs $(\mathrm{U}(n),\mathfrak{u}(1,1))$ and $(\mathrm{U}(n),\mathfrak{u}(2|1))$ to the context of a rational Cherednik algebra $H_{1,c}(G,E)$ associated with a real reflection group $G$ acting on a real vector space $E$ of even dimension. For each pair, we show that the Lie (super)algebra structure of one partner is preserved under the deformation, which leads to a multiplicity-free decomposition of the standard module or its tensor product with a spinor space. For the case where $E$ is two-dimensional and $G$ is a dihedral group, we provide complete descriptions for the deformed pair and the relevant joint-decomposition.