Equivariant K-theory approach to $\imath$-quantum groups

TL;DR

This paper develops an equivariant K-theory framework to study $ extit{i}$-quantum groups, providing a geometric realization and proving a conjecture for specific cases related to Satake diagrams.

Contribution

It introduces an equivariant K-theory approach to $ extit{i}$-quantum groups associated with Satake diagrams, complementing existing geometric realizations.

Findings

Established an equivariant K-theory model for $ extit{i}$-quantum groups.

Proved Li's conjecture for certain Satake diagram cases.

Connected geometric realization with algebraic structures of $ extit{i}$-quantum groups.

Abstract

Various constructions for quantum groups have been generalized to $\imath$-quantum groups. Such generalization is called $\imath$-program. In this paper, we fill one of parts in the $\imath$-program. Namely, we provide an equivariant K-theory approach to $\imath$-quantum groups associated to the Satake diagram in \eqref{eq1}, which is the Langlands dual picture of that constructed in \cite{BKLW14}, where a geometric realization of the $\imath$-quantum group is provided by using perverse sheaves. As an application of the main results, we prove Li's conjecture \cite{L18} for the special cases with the satake diagram in \eqref{eq1}.