On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution

TL;DR

This paper introduces filtered quiver representations, generalizes conjugacy classes to block upper triangular matrices, and explores invariant rings and their relation to the Grothendieck-Springer resolution, proposing future research directions.

Contribution

It systematically generalizes conjugacy classes to filtered quiver representations and analyzes their invariant rings, connecting them to the Grothendieck-Springer resolution.

Findings

Described invariant polynomial rings for finite ADE and affine type A quivers.

Established connections between filtered representations and the Grothendieck-Springer resolution.

Proposed conjectures for future research in the area.

Abstract

We introduce the notion of filtered representations of quivers, which is related to usual quiver representations, but is a systematic generalization of conjugacy classes of $n\times n$ matrices to (block) upper triangular matrices up to conjugation by invertible (block) upper triangular matrices. With this notion in mind, we describe the ring of invariant polynomials for interesting families of quivers, namely, finite $ADE$-Dynkin quivers and affine type $\widetilde{A}$-Dynkin quivers. We then study their relation to an important and fundamental object in representation theory called the Grothendieck-Springer resolution, and we conclude by stating several conjectures, suggesting further research.