The centre of the Dunkl total angular momentum algebra

Abstract

For a finite dimensional representation V of a finite reflection group W , we consider the rational Cherednik algebra H t,c (V, W ) associated with (V, W ) at the parameters t = 0 and c. The Dunkl total angular momentum algebra O t,c (V, W ) arises as the centraliser algebra of the Lie superalgebra osp(1|2) containing a Dunkl deformation of the Dirac operator, inside the tensor product of 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 W .