Symplectic complexity of reductive group actions

Let a complex algebraic reductive group $\bf G$ act on a complex algebraic manifold $\bf X$. For a $\bf G$-invariant subvariety $\Xi$ of the nilpotent cone $\mathcal{N}(\mathfrak{g}^*)\subset \mathfrak{g}^*$ we define a notion of $\Xi$-symplectic complexity of $\bf X$. This notion generalizes the notion of complexity defined in [Vin86]. We prove several properties of this notion, and relate it to the notion of $\Xi$-complexity defined in [AG] motivated by its relation with representation theory.