Geometric construction of Schur algebras

TL;DR

This paper develops a geometric framework for constructing generalized Schur algebras of any type, connecting them to Langlands correspondence, quantum groups, and duality theories.

Contribution

It introduces a new geometric construction of Schur algebras using Borel-Moore homologies and equivariant K-groups, extending their applications to quantum groups and dualities.

Findings

Constructed generalized Schur algebras via geometric methods.

Established a geometric analogue of the local Langlands correspondence.

Realized quasi-split affine $ extit{ extbf{A}}$-type $ extit{ extbf{I}}$ quantum groups in equivariant K-theory.

Abstract

We provide the geometric construction of a series of generalized Schur algebras of any type via Borel-Moore homologies and equivariant K-groups of generalized Steinberg varieties. As applications, we obtain a Schur algebra analogue of the local geometric Langlands correspondence of any type, provide an equivariant K-theoretic realization of quasi-split $\imath$quantum groups of affine type AIII, and establish a geometric Howe duality for affine ($\imath$-)quantum groups.