The Dunkl-Cherednik Deformation of a Howe duality

Abstract

We consider the deformed versions of the classical Howe dual pairs $(O(r),\mathfrak{sl}(2))$ and $(O(r),\mathfrak{spo}(2|2))$ in the context of a rational Cherednik algebra $H_c=H_c(W,\mathfrak{h})$ 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 $\mathfrak{s}\cong \mathfrak{sl}(2)$ inside $H_c$. For the second pair, we show that the classical copy of $\mathfrak{g}\cong \mathfrak{spo}(2|2)$ inside the Weyl-Clifford algebra $\mathcal{W}\otimes \mathcal{C}$ deforms to a Lie superalgebra inside $H_c\otimes \mathcal{C}$ and compute its centraliser algebra. For a generic parameter $c$ such that the standard $H_c$-module is unitary, we compute the joint $((H_c{)}^{\mathfrak{s}},\mathfrak{s})$- and $((H_c\otimes \mathcal{C}{)}^{\mathfrak{g}},\mathfrak{g})$-decompositions of the relevant modules.

29 pages; version that was accepted for publication. In this revised version we shortened the discussion about Drinfeld orbifold algebras, added a List of symbols and made minor corrections throughout