Representations of Hecke-Clifford superalgebras at roots of unity

Abstract

In this article, we give a classification of irreducible completely splittable representations of affine Hecke-Clifford superalgebras $H_n^{\mathrm{aff}}(q)$ when $q^2$ is a primitive $h$-th root of unity. As an application, we derive a necessary and sufficient condition for the finite Hecke-Clifford superalgebra $H_n(q)$ to be semisimple. Specially we show that $H_n(q)$ is semisimple if and only $h >n$ in the case $h$ is odd and $h >2n$ in the case $h$ is even.