Application of Schur-Weyl duality to Springer theory

TL;DR

This paper explores the representations of a certain quantum algebra using Schur-Weyl duality and Springer theory, building on prior work connecting homology of Springer fibers to algebraic structures.

Contribution

It introduces a novel approach combining Schur-Weyl duality with Springer theory to analyze representations of fixed point subalgebras of sl_n.

Findings

Representation of $U(sl_n^{ heta})$ via Springer fibers

Connection between Borel-Moore homology and algebraic structures

New insights into the structure of fixed point subalgebras

Abstract

In \cite{FMX19}, it is proved that the convolution algebra of top Borel-Moore homology on Steinberg variety of type $B/C$ realizes $U(sl_n^{\theta})$, where $sl_{n}^{\theta}$ is the fixed point subalgebra of involution on $sl_n$. So top Borel-Moore homology of the partial Springer's fibers gives the representations of $U(sl_n^{\theta})$. In this paper, we study these representations using the Schur-Weyl duality and Springer theory.