Quasi-split symmetric pairs of $U(\mathfrak{gl}_N)$ and their Schur algebras

TL;DR

This paper establishes explicit isomorphisms between certain algebras and their Schur algebras related to symmetric pairs of type B, providing geometric presentations and clarifying dualities in representation theory.

Contribution

It introduces explicit isomorphisms for algebras associated with quasi-split symmetric pairs of $U(rak{gl}_N)$ and describes their Schur algebras, connecting algebraic and geometric perspectives.

Findings

Explicit algebra isomorphisms established

Geometric presentations of Schur algebras provided

Connections to type B Weyl group actions clarified

Abstract

We establish explicit isomorphisms of two seemingly-different algebras, and their Schur algebras, arising from the centralizers of two different type B Weyl group actions in Schur-like dualities. We provide a presentation of the geometric counterpart of the above Schur algebras in Bao-Kujawa-Li-Wang (2018) specialized at $q=1$.