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

Abstract

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

arXiv comment40 pages. The proof of the existence theorem for $ȷ$-crystal bases was modified