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

TL;DR

This paper introduces a differential operator framework for $ extit{i}$-quantum groups associated with quasi-split Satake diagrams, constructing oscillator representations and their crystal bases.

Contribution

It establishes a novel differential operator approach to $ extit{i}$-quantum groups and develops oscillator representations with crystal bases.

Findings

Defined a modified $q$-Weyl algebra for quasi-split Satake diagrams

Constructed algebra homomorphism linking the $q$-Weyl algebra and $ extit{i}$-quantum groups

Built oscillator representations and their crystal bases

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 $\imath$quantum group. In other words, we provide a differential operator approach to $\imath$quantum groups. Meanwhile, the oscillator representations of $\imath$quantum groups are obtained. The crystal basis of the irreducible subrepresentations of these oscillator representations are constructed.