Stratifications Associated to Reductive Group Actions on Affine Spaces

For a complex reductive group G acting linearly on a complex affine space V with respect to a characterρ, we show two stratifications ofV associated to this action (and a choice of invariant inner product on the Lie algebra of the maximal compact subgroup ofG) coincide. The first is Hesselink’s stratification by adapted 1-parameter subgroups and the second is the Morse theoretic stratification associated to the norm square of the moment map. We also give a proof of a version of the Kempf–Ness theorem, which states that the geometric invariant theory quotient is homeomorphic to the symplectic reduction (both taken with respect to ρ). Finally, for the space of representations of a quiver of fixed dimension, we show that the Morse theoretic stratification and Hesselink’s stratification coincide with the stratification by Harder–Narasimhan types.