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For a quasi-split Satake diagram, we define a modified $q$-Weyl algebra, andshow that there is an algebra homomorphism between it and the corresponding$\imath$quantum group. In other words, we provide a differential operatorapproach to $\imath$quantum groups. Meanwhile, the oscillator representationsof $\imath$quantum groups are obtained. The crystal basis of the irreduciblesubrepresentations of these oscillator representations are constructed.

Differential operator approach to $\imath$quantum groups and their oscillator representations