Supercentralizers for deformations of the Pin osp dual pair

TL;DR

This paper investigates the algebraic structure of supercentralizers in deformations of the Pin osp dual pair, focusing on their generators and relation to Howe duality, especially within rational Cherednik algebra frameworks.

Contribution

It establishes that the supercentralizer is fully characterized, provides a minimal generating set, and clarifies its connection to the Pin osp dual pair and Howe duality.

Findings

Identifies the supercentralizer as the full algebra of elements supercommuting with the osp(1|2) realization.

Provides a minimal set of generators for this supercentralizer.

Describes the relationship between the supercentralizer and the Howe dual pair $( ext{Pin}(d), ext{osp}(2m+1|2n))$.

Abstract

In recent work, we examined the algebraic structure underlying a class of elements supercommuting with realization of the Lie superalgebra $\mathfrak{osp}(1|2)$ inside a generalization of the Weyl Clifford algebra. This generalization contained in particular the deformation by means of Dunkl operators, yielding a rational Cherednik algebra instead of the Weyl algebra. The aim of this work is to show that this is the full supercentralizer, give a (minimal) set of generators, and to describe the relation with the $(\mathrm{Pin}(d),\mathfrak{osp}(2m+1|2n))$ Howe dual pair.