Differential operator approach to $ı$quantum groups and their oscillator representations

Abstract

For a quasi-split Satake diagram, we define a modified $q$-Weyl algebra, and show that there is an algebra homomorphism between it and the corresponding $ı$quantum group. In other words, we provide a differential operator approach to $ı$quantum groups. Meanwhile, the oscillator representations of $ı$quantum groups are obtained. The crystal basis of the irreducible subrepresentations of these oscillator representations are constructed.