Laumon parahoric local models via quiver Grassmannians

TL;DR

This paper explores the structure of Laumon parahoric local models in type A using quiver Grassmannians, describing their irreducible components, stratifications, cellular decompositions, and desingularizations.

Contribution

It provides a detailed analysis of the special fibers of Laumon local models via quiver representations, including cellular decompositions and desingularization properties.

Findings

Description of irreducible components and strata of quiver Grassmannians.

Construction of cellular decompositions and poset of cells.

Analysis of desingularizations and their compatibility with projections.

Abstract

Local models of Shimura varieties in type A can be realized inside products of Grassmannians via certain linear algebraic conditions. Laumon suggested a generalization which can be identified with a family over a line whose general fibers are quiver Grassmannians for the loop quiver and the special fiber is a certain quiver Grassmannian for the cyclic quiver. The whole family sits inside the Gaitsgory central degeneration of the affine Grassmannians. We study the properties of the special fibers of the (complex) Laumon local models for arbitrary parahoric subgroups in type A using the machinery of quiver representations. We describe the irreducible components and the natural strata with respect to the group action for the quiver Grassmannians in question. We also construct a cellular decomposition and provide an explicit description for the corresponding poset of cells. Finally, we study the properties of the desingularizations of the irreducible components and show that the desingularization construction is compatible with the natural projections between the parahoric subgroups.