The centre of the Dunkl total angular momentum algebra

TL;DR

This paper studies the structure of the Dunkl total angular momentum algebra associated with rational Cherednik algebras and reflection groups, revealing its center and establishing analogues of Vogan's conjecture for related operators.

Contribution

It characterizes the center of the Dunkl total angular momentum algebra for all parameters and relates it to the structure of the reflection group and Clifford algebra elements.

Findings

Center of the algebra is a univariate polynomial ring.

The generator of the center depends on the presence of $(-1)_V$ in $W$.

Results analogous to Vogan's conjecture are established for certain operators.

Abstract

For a finite dimensional representation $V$ of a finite reflection group $W$, we consider the rational Cherednik algebra $\mathsf{H}_{t,c}(V,W)$ associated with $(V,W)$ at the parameters $t\neq 0$ and $c$. The Dunkl total angular momentum algebra $O_{t,c}(V,W)$ arises as the centraliser algebra of the Lie superalgebra $\mathfrak{osp}(1|2)$ containing a Dunkl deformation of the Dirac operator, inside the tensor product of $\mathsf{H}_{t,c}(V,W)$ and the Clifford algebra generated by $V$. We show that, for every value of the parameter $c$, the centre of $O_{t,c}(V,W)$ is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not $(-1)_V$ is an element of the group $W$. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into $O_{t,c}(V,W)$, we establish results analogous to ``Vogan's conjecture'' for a family of operators depending on suitable elements of the double cover $\tilde{W}$.