Crystal Basis Theory for a Quantum Symmetric Pair $(\mathbf{U},\mathbf{U}^{\jmath })$

Abstract

We study the representation theory of a quantum symmetric pair |$(\mathbf{U},\mathbf{U}^{\jmath })$| with two parameters |$p,q$| of type AIII, by using highest weight theory and a variant of Kashiwara’s crystal basis theory. Namely, we classify the irreducible |$\mathbf{U}^{\jmath }$|-modules in a suitable category and associate with each of them a basis at |$p=q=0$|⁠, the |$\jmath $|-crystal basis. The |$\jmath $|-crystal bases have nice combinatorial properties as the ordinary crystal bases do.