Slices for reductive group actions in algebraic and holomorphic symplectic geometry
Peter Crooks, Rebecca Goldin, Yiannis Loizides
TL;DR
The work develops a unified theory of slices for Hamiltonian actions beyond classical compact groups, introducing Poisson and symplectic slices that accommodate complex reductive groups and holomorphic/complex-algebraic contexts. It connects Slodowy slices and decomposition classes to principal-type phenomena, and extends to Hamiltonian actions of symplectic groupoids via an abstract symplectic-slice theorem. The results include principal and natural slices, residual group actions A(x), and root-system descriptions, together with a comprehensive analysis of Slodowy and decomposition-class structures within Poisson geometry. Collectively, these advances provide a robust framework for local normal forms and reductions in algebraic and holomorphic symplectic geometry, with explicit constructions and comparisons to the classical compact-case theory.
Abstract
Symplectic slice theorems elucidate the local structure of symplectic manifolds carrying Hamiltonian actions of compact Lie groups. We generalize these theorems in two natural settings. The first is based on the idea that complex reductive algebraic groups are the natural complex-geometric counterparts of compact Lie groups. Using new definitions of Poisson and symplectic slices, we prove analogues of the classical symplectic slice theorems for Hamiltonian actions of complex reductive algebraic groups. These analogues have versions in the complex-algebraic and holomorphic categories, and make extensive use of Slodowy slices and decomposition classes in complex reductive Lie algebras. The starting point for our second setting is the fact that Hamiltonian Lie group actions are special cases of Hamiltonian symplectic groupoid actions. We generalize the classical symplectic slice theorems to the latter case.