Degenerate affine Hecke–Clifford algebras and type Q Lie superalgebras

Abstract

We construct the finite dimensional simple integral modules for the (degenerate) affine Hecke–Clifford algebra (AHCA), $${\mathcal{H}_{\mathcal{C}\ell}^{{\rm aff}}(d)}$$ . Our construction includes an analogue of Zelevinsky’s segment representations, a complete combinatorial description of the simple calibrated $${\mathcal{H}_{\mathcal{C}\ell}^{{\rm aff}}(d)}$$ -modules, and a classification of the simple integral $${\mathcal{H}_{\mathcal{C}\ell}^{{\rm aff}}(d)}$$ -modules. Our main tool is an analogue of the Arakawa–Suzuki functor for the Lie superalgebra $${\mathfrak{q}(n)}$$ .