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

We study the representation theory of a quantum symmetric pair $(\mathbf{U},{\mathbf{U}}^{\mathrm{ȷ}})$ 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 ${\mathbf{U}}^{\mathrm{ȷ}}$-modules in a suitable category and associate with each of them a basis at $p=q=0$, the $\mathrm{ȷ}$-crystal basis. The $\mathrm{ȷ}$-crystal bases have nice combinatorial properties as the ordinary crystal bases do.