The Dunkl-Cherednik deformation of a Howe duality

We consider the deformed versions of the classical Howe dual pairs (O(r, C), sl(2, C)) and (O(r, C), spo(2|2, C)) in the context of a rational Cherednik algebra Hc = Hc(W, h) associated to a finite Coxeter group W at the parameters c and t = 1. For the first pair, we compute the centraliser of the well-known copy of s ∼= sl(2, C) inside Hc. For the second pair, we show that the classical copy of g ∼= spo(2|2, C) inside the Weyl-Clifford algebra W ⊗ C deforms to a Lie superalgebra inside Hc ⊗ C and compute its centraliser algebra. For a generic parameter c such that the standard Hc-module is unitary, we compute the joint ((Hc) s , s)- and ((Hc ⊗ C) g , g)-decompositions of the relevant modules