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 $\Xi$ of the nilpotent cone $N\left(g^{\ast }\right)\subset g^{\ast }$ we define a notion of $\Xi$-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 $\Xi$-complexity defined in Aizenbud and Gourevitch (2024) motivated by its relation with representation theory.