Symplectic complexity of reductive group actions

Abstract

Let a complex algebraic reductive group G act on a complex algebraic manifold X. For a G-invariant subvariety Ξ of the nilpotent cone N(g∗)⊂g∗ we define a notion of Ξ-symplectic complexity of X. This notion generalizes the notion of complexity defined in Vinberg (1986). We prove several properties of this notion, and relate it to the notion of Ξ-complexity defined in Aizenbud and Gourevitch (2024) motivated by its relation with representation theory.