Affine $\mathrm{i}$quantum groups and Steinberg varieties of type C, II

TL;DR

This paper extends geometric realizations of affine iquantum groups of type C using equivariant K-groups of Steinberg varieties, including new cases and alternative realizations avoiding localization.

Contribution

It constructs the quasi-split affine iquantum group of type AIII_{2n}^{(τ)} and offers a new realization for type AIII_{2n-1}^{(τ)} using Steinberg varieties of type D.

Findings

Constructed the affine iquantum group of type AIII_{2n}^{(τ)}.

Provided a new realization of type AIII_{2n-1}^{(τ)} avoiding localization.

Extended geometric realization techniques to new types and cases.

Abstract

A geometric realization of the quasi-split affine iquantum group of type $\mathrm{AIII}_{2n-1}^{(\tau)}$ was given by Wang and the second author, in terms of equivariant K-groups of Steinberg varieties of type C. As a completion of that work, this paper focuses on the previously untreated case. We provide a similar construction of the quasi-split affine iquantum group of type $\mathrm{AIII}_{2n}^{(\tau)}$, using the same equivariant K-groups of Steinberg varieties of type C. In the appendix, we employ Steinberg varieties of type D to give a new realization of the quasi-split affine iquantum group of type $\mathrm{AIII}_{2n-1}^{(\tau)}$, thereby avoiding the localization method adopted in the previous work.