Lusztig sheaves, characteristic cycles and the Borel-Moore homology of Nakajima's quiver varieties

TL;DR

This paper links canonical bases of quantum groups to Borel-Moore homology of Nakajima's quiver varieties using characteristic cycles, providing new insights into their structure and integer realizations.

Contribution

It constructs a morphism connecting canonical bases to homology groups and proves Nakajima's conjecture on tensor product varieties.

Findings

Canonical bases correspond to fundamental classes in homology.

Realization of modules over integers is established.

A new proof of Nakajima's conjecture is provided.

Abstract

By using characteristic cycles, we build a morphism from the canonical bases of integrable highest weight modules of quantum groups to the top Borel-Moore homology groups of Nakajima's quiver and tensor product varieties, and compare the canonical bases and the fundamental classes. As an application, we show that Nakajima's realization of irreducible highest weight modules and their tensor products can be defined over integers. We also give a new proof of Nakajima's conjecture on the canonical isomorphism of tensor product varieties.