The Dunkl-Cherednik deformation of a Howe duality

Abstract We consider the deformed versions of the classical Howe dual pairs ( O ( r , C ) , s l ( 2 , C ) ) and ( O ( r , C ) , s p o ( 2 | 2 , C ) ) in the context of a rational Cherednik algebra H c = H c ( 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 ≅ s l ( 2 , C ) inside H c . For the second pair, we show that the classical copy of g ≅ s p o ( 2 | 2 , C ) inside the Weyl-Clifford algebra W ⊗ C deforms to a Lie superalgebra inside H c ⊗ C and compute its centraliser algebra. For a generic parameter c such that the standard H c -module is unitary, we compute the joint ( ( H c ) s , s ) - and ( ( H c ⊗ C ) g , g ) -decompositions of the relevant modules.