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

Crystal basis theory for a quantum symmetric pair $(\mathbf{U},\mathbf{U}^{\jmath})$