A relation between Mirkovi\'c-Vilonen cycles and modules over preprojective algebra of Dynkin quiver of type ADE

TL;DR

This paper establishes a partial proof connecting Mirković-Vilonen cycles and modules over preprojective algebras of Dynkin quivers, advancing understanding of their relationship in representation theory.

Contribution

It provides a proof of part of Baumann and Kamnitzer's conjecture relating MV cycles and module varieties, and also proves the reducibility conjecture.

Findings

Partial proof of the conjecture relating MV cycles and module varieties.

Proof of the reducibility conjecture for the associated cycles.

Enhanced understanding of the geometric and algebraic structures in Lie theory.

Abstract

The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian for a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. Given this conjecture, I give a proof of the reduceness conjecture.