Hamiltonian actions on symplectic varieties with invariant Lagrangian subvarieties

TL;DR

This paper explores the relationship between Hamiltonian actions on symplectic varieties and invariant Lagrangian subvarieties, establishing equalities between their moment map images and generalizing known theorems.

Contribution

It proves that the moment map images of a Hamiltonian variety and an invariant Lagrangian subvariety coincide, extending Panyushev's theorem to broader contexts.

Findings

Images of moment maps coincide for Hamiltonian and Lagrangian subvarieties

Complexity and rank relate to corank and defect of the variety

Generalization to invariant coisotropic subvarieties

Abstract

We prove several results on symplectic varieties with a Hamiltonian action of a reductive group having invariant Lagrangian subvarieties. Our main result states that the images of the moment maps of a Hamiltonian variety and of the cotangent bundle over an invariant Lagrangian subvariety coincide. This implies that the complexity and rank of the Lagrangian subvariety are equal to the half of the corank and to the defect of the Hamiltonian variety, respectively. This result generalizes a theorem of Panyushev on the complexity and rank of a conormal bundle. A simple elementary proof of this theorem is also given in the paper. A generalization of the above results to some special class of invariant coisotropic subvarieties is obtained.