K-theoretic stable envelopes, quantum loop groups and wall-crossings

The stable envelope is an important tool in both geometric representation theory and the enumerative geometry. One of the most important application is that it generates the geometric quantum loop group via the FRT formalism. In this talk, we will show that the geometric quantum loop group is isomorphic to the Drinfeld double given by the preprojective K-theoretic Hall algebra and the nilpotent K-theoretic Hall algebra. Moreover we will show a more refined result that the wall-crossing for the K-theoretic stable envelope is controlled by the universal R-matrix for the slope subalgebra of the Drinfeld double, which leads to the isomorphism between the wall subalgebra in geometric quantum loop groups and the slope subalgebras in the Drinfeld double. If time permits, I will talk about the recent progress of such isomorphism in the case of the critcial stable envelopes in both critical K-theory and critical cohomology. This is based on the work 2511.02161

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