  For a complex reductive group G acting linearly on a complex affine space V
with respect to a character, 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
GIT quotient is homeomorphic to the symplectic reduction (both taken with
respect to the character). 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.
