Stratifications associated to reductive group actions on affine spaces

TL;DR

This paper proves the equivalence of two stratifications in reductive group actions on affine spaces, relates GIT quotients to symplectic reduction, and connects stratifications to Harder-Narasimhan types in quiver representations.

Contribution

It demonstrates the coincidence of Hesselink's and Morse theoretic stratifications for reductive group actions and extends the Kempf-Ness theorem to this setting.

Findings

Hesselink's and Morse stratifications coincide

GIT quotient is homeomorphic to symplectic reduction

Stratifications match Harder-Narasimhan types in quiver representations

Abstract

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.