The characteristic cycles and semi-canonical bases on type $A$ quiver variety

TL;DR

This paper investigates the relationship between canonical and semi-canonical bases via characteristic cycles in type A quiver varieties, proving a conjecture for A2 and reducing the general case to a vanishing Euler characteristic problem.

Contribution

It formulates an approach to the Geiss-Leclerc-Schr{"o}er conjecture and proves it for type A2 quiver, advancing understanding of basis relations in quiver varieties.

Findings

Proved the conjecture for type A2 quiver.

Reduced the general case to a vanishing Euler characteristic problem.

Established a connection between characteristic cycles and basis relations.

Abstract

In this article we study a conjecture of Geiss-Leclerc-Schr{\"o}er, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type $A_2$ quiver. In general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic.