A Geometric Realization of Symmetric Pairs of Type AIII

TL;DR

This paper constructs a geometric realization of symmetric pairs of type AIII using $\sigma$-quiver varieties, extending Nakajima's approach with new geometric techniques involving Hecke correspondences.

Contribution

It introduces a novel geometric realization of type AIII symmetric pairs via $\sigma$-quiver varieties, expanding the framework of Nakajima quiver varieties.

Findings

Established a geometric model for type AIII symmetric pairs

Defined generator actions via pull back and push forward through Hecke correspondences

Verified the relations to ensure the correctness of the realization

Abstract

In this paper, we construct a geometric realization of symmetric pair of type AIII over $\sigma$-quiver varieties as an analogy of the construction over Nakajima quiver varieties. To achieve this, we define the action of generators by pull back and push forward through $\iota$Hecke correspondences and check the relations.