Torus Actions on Quotients of Affine Spaces

We study the locus of fixed points of a torus action on a GIT quotient of a complex vector space by a reductive complex algebraic group which acts linearly. We show that, under the assumption that $G$ acts freely on the stable locus, the components of the fixed point locus are again GIT quotients of linear subspaces by Levi subgroups.

💡 Research Summary

The paper investigates the fixed‑point locus of a torus action on a geometric invariant theory (GIT) quotient of a complex vector space. Let (G) be a connected reductive complex algebraic group acting linearly on a finite‑dimensional representation (V). Fix a character (\theta\in X^{*}(G)) and consider the (\theta)-stable locus (V^{\mathrm{st}}(G,\theta)) together with its geometric quotient (V^{\mathrm{st}}(G,\theta)/G). A torus (T\simeq(\mathbb C^{\times})^{r}) is assumed to act linearly on (V) commuting with the (G)-action; consequently the action descends to the quotient.

The central hypothesis is that (G) acts freely on the stable locus. This guarantees, via Luna’s slice theorem, that the quotient map is an étale principal (G)-bundle and that the quotient is smooth. Under this hypothesis the authors describe the fixed‑point set ((V^{\mathrm{st}}(G,\theta)/G)^{T}) completely.

The main construction proceeds as follows. For any homomorphism (\rho:T\to G) (more precisely (\rho:T\to T) where (T) is a fixed maximal torus of (G)), define \