Symplectic complexity of reductive group actions

Abstract

Let a complex algebraic reductive group $\mathbf{G}$ act on a complex algebraic manifold $\mathbf{X}$. For a $\mathbf{G}$-invariant subvariety $\Xi$ of the nilpotent cone $\mathcal{N}({\mathfrak{g}}^{\ast })\subset {\mathfrak{g}}^{\ast }$ we define a notion of $\Xi$-symplectic complexity of $\mathbf{X}$. 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.

10 p