The Dunkl-Cherednik deformation of a Howe duality

Abstract

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.