# Generalized orbital varieties for Mirkovic-Vybornov slices as affinizations of Mirkovic-Vilonen cycles

## TL;DR

This paper establishes a correspondence between generalized orbital varieties for Mirkovic-Vybornov slices and semi-standard Young tableaux, linking combinatorial and geometric Lusztig data through the Mirkovic-Vybornov isomorphism.

## Contribution

It introduces a new indexing of generalized orbital varieties by Young tableaux and demonstrates their relation to Mirkovic-Vilonen cycles via the isomorphism.

## Key findings

- Generalized orbital varieties are indexed by semi-standard Young tableaux.
- The Mirkovic-Vybornov isomorphism maps these varieties to dense subsets of MV cycles.
- The combinatorial Lusztig datum matches the geometric Lusztig datum under this correspondence.

## Abstract

We show that generalized orbital varieties for Mirkovic-Vybornov slices can be indexed by semi-standard Young tableaux. We also check that the Mirkovic-Vybornov isomorphism sends generalized orbital varieties to (dense subsets of) Mirkovic-Vilonen cycles, such that the (combinatorial) Lusztig datum of a generalized orbital variety, which it inherits from its tableau, is equal to the (geometric) Lusztig datum of its MV cycle.