2012/10/25 by Victoria Hoskins, Hoskins, Victoria · 3 citations
Mathematics · #Advanced Algebra and Geometry #Affine transformation #Algebra over a field #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics #FOS: Mathematics #Group theory #Mathematics #Pure mathematics #Quotient #Reductive group #Stratification (seeds) #Symplectic Geometry (math.SG) #Symplectic geometry #math.AG #math.SG
paper · pdf · doi:10.48550/arxiv.1210.6811
published in Zurich Open Repository and Archive (University of Zurich) (University of Zurich) · 24 pages
arxiv created 2012/10/25 · openalex publication_date 2012/10/25 · arxiv updated 2012/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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.