Stratifications Associated to Reductive Group Actions on Affine Spaces

Abstract

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 for moduli of sheaves and moduli of quiver representations

“…Proof. We deduce the result for a closed G-invariant subvariety V of an affine space W from the result for W given in [10]. Since taking invariants for a reductive group G is exact and V ⊂ W is closed, it follows that V ρ−ss = V ∩ W ρ−ss and the Hesselink stratification of V − V ρ−ss is the intersection of V − V ρ−ss with the Hesselink stratification of W − W ρ−ss ; that is,…”

“…Let V λ + be the closed subvariety of V consisting of points v such that lim t→0 λ(t) · v exists; then we have a natural retraction p λ : V λ + → V λ onto the λ-fixed locus. The following theorem describing the Hesselink strata appears in [10] for the case when V is an affine space. Theorem 2.5.…”

Equivariant Morse Theory for the Norm-Square of a Moment Map on a Variety

“…(1) For projective varieties Condition (1) follows from Kirwan's explicit construction of the critical values in [24]. For affine varieties this follows from the analogous construction in [19].…”

“…Proof. When f = µ 2 is the norm-square of a moment map on a complex vector space, Hoskins [19] gives an explicit description of the Morse strata of f based on Kirwan's description of the strata for moment maps on projective varieties in [24]. In particular, the strata are submanifolds is an open subset of the analytic set V β + := {v ∈ V : lim t→∞ e −iβt ·v exists}.…”

Kempf–Ness type theorems and Nahm equations

“…fixed by the coadjoint action), so we can consider the symplectic reduction µ −1 (ξ)/K. Then, King [22, §6] (see also [20]) showed that µ −1 (ξ)/K is homeomorphic to the twisted GIT quotient M // χ G, i.e. the GIT quotient of M by G with respect to the trivial line bundle M × C with the G-action g · (p, z) = (g · p, χ(g)z).…”