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For a complex reductive group  G  acting linearly on a complex affine space  V  with respect to a character  ρ , we show two stratifications of  V  associated to this action (and a choice of invariant inner product on the Lie algebra of the maximal compact subgroup of  G ) 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.

STRATIFICATIONS ASSOCIATED TO REDUCTIVE GROUP ACTIONS ON AFFINE SPACES