Symplectic complexity of reductive group actions

Abstract

Let a complex algebraic reductive group act on a complex algebraic manifold . For a -invariant subvariety of the nilpotent cone we define a notion of -symplectic complexity of . 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 -complexity defined in Aizenbud and Gourevitch (2024) motivated by its relation with representation theory.