Definition
(Virtual Localization Revisited Defn. A.9). Let \(X\) be an Artin stack over \(S\) with \(G\)-action (see Romagny’s definition) and denote by \(\mathcal X = [X/G]\). The \(G\)-fixed locus \(X^{sG}\subset X\) is the locus where the canonical homomorphism of group algebraic spaces over \(X\) \[I_{\mathcal X/S}\times_{\mathcal X/S} X \to G \times_{S}X \] is surjective. In other words, let \(A\) be an \(S\)-scheme and \(x\in X(A)\) be an \(A\)-point and consider the homomorphism of group algebraic spaces over \(A\) \[\alpha_A:\Aut_{\mathcal X}(x) \to G_A\] obtained by base changing along \(A\). Then \(X^{sG}(A)\) is the collection of \(x \in X(A)\) where \(\alpha_A\) is surjective.
We denote it with the \(sG\) exponent for “stabilizer” to differentiate it from the Romagny fixed stack.