Unipotent invariants of filtered representations of quivers and the isospectral Hilbert scheme

TL;DR

This paper studies unipotent invariants in filtered quiver representations, proving finiteness of certain algebraic varieties and establishing a birational link to the isospectral Hilbert scheme, with implications for quantum geometry.

Contribution

It introduces a new framework for unipotent invariants in filtered quiver representations and constructs an ADHM analog related to the isospectral Hilbert scheme.

Findings

Finite-dimensionality of the unipotent invariant algebraic variety.

Construction of an ADHM analog birational to the isospectral Hilbert scheme.

Discussion of quantum Hamiltonian reduction and deformation quantization in nonreductive settings.

Abstract

Given any finite quiver, we consider a complete flag of vector spaces over each vertex. Consider the unipotent invariant subalgebra of the coordinate ring of the filtered quiver representation subspace. We prove that the dimension of the algebraic variety of the unipotent invariant subalgebra is finite. We also construct an ADHM analog for the Borel subalgebra setting, showing its birationality to the isospectral Hilbert scheme. Quiver-graded Steinberg varieties, quantum Hamiltonian reduction, and deformation quantization constructions for the nonreductive setting are discussed, ending with open problems.