Stratifications associated to reductive group actions on affine spaces (Q2922462)

Let \(G\) be a complex reductive group, and let \(V\subset\mathbb{P}^{d}\) be a complex projective variety on which \(G\) acts linearly. One can consider different stratifications of \(V\). For example, a character \(\rho\) of \(V\) gives rise to a GIT\ quotient \(V//_{\rho}G:=\)Proj\(\,\bigoplus_{n\geq0} \mathbb{C}\left[ V\right] _{\rho^{n}}^{G},\) where \(\mathbb{C}\left[ V\right] _{\rho^{n}}^{G}=\left\{ f\in\mathbb{C}\left[ V\right] f\left( g\cdot v\right) =\rho^{n}\left( g\right) f\left( v\right) \text{ for all }v\in V,g\in G\right\} ;\) this creates an adaptation of the Hesselink stratification. Alternatively, there is a Morse stratification, which depends on an inner product which is invariant on the Lie algebra of the maximal compact subgroup of \(G.\)NEWLINENEWLINEIn the work under review, the author shows that these two stratifications coincide, a result analogous to results obtained by \textit{F. C. Kirwan} [Cohomology of quotients in symplectic and algebraic geometry. Princeton, New Jersey: Princeton University Press (1984; Zbl 0553.14020)], and \textit{L. Ness} [Am. J. Math. 106, 1281--1329 (1984; Zbl 0604.14006)]. Furthermore, when applied to quiver representations the choice of inner product allows the author to consider a Harder-Narasimhan filtration, and it is shown that resulting stratification agrees with the previous two.