Crystal Bases of Modified \(\imath \)quantum Groups of Certain Quasi-Split Types

Abstract

In order to see the behavior of \(\imath \)canonical bases at \(q = \infty \), we introduce the notion of \(\imath \)crystals associated to an \(\imath \)quantum group of certain quasi-split type. The theory of \(\imath \)crystals clarifies why \(\imath \)canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of \(\imath \)crystals whose projective limit can be thought of as the \(\imath \)canonical basis of the modified \(\imath \)quantum group at \(q = \infty \).