Hecke–Clifford Superalgebras and Crystals of Type $D_l^{(2)}$

Abstract

In [BK], Brundan and Kleshchev showed that some parts of the representation theory of the affine Hecke–Clifford superalgebras and its finite-dimensional “cyclotomic” quotients are controlled by the Lie theory of type $A_{2l}^{(2)}$ when the quantum parameter $q$ is a primitive $(2l+1)$-th root of unity. We show that similar theorems hold when $q$ is a primitive $4l$-th root of unity by replacing the Lie theory of type $A_{2l}^{(2)}$ with that of $D_l^{(2)}$.