A plethora of inertial products (Q745115)

For an algebraic group \(G\) acting on an algebraic space \(X\), we define the inertia space \(I_G(X)\) which comes with a natural \(G\)-action. The higher inertia spaces \(\mathbb{I}_G^kX\) are defined as the fiber products of \(I_G(X)\) over \(X\). The quotient space \(\mathbb{I}_G^k\mathcal{X} = [\mathbb{I}_G^kX/G]\) and it is the corresponding higher inertia stack. Let \(\mathcal{X}\) be a smooth Deligne-Mumford stack such that the inertia map is finite and every stack has the resolution property. Then \(\mathcal{X} = [X/G]\), where \(G\) is an algebraic group acting on an algebraic space \(X\). An inertia pair for \(\mathcal{X}\) is a pair \((\mathcal{R}, \mathcal{S})\) where \(\mathcal{R}\) is a strongly associative vector bundle over \(\mathbb{I}_G^2X\) and \(\mathcal{S} \in K_G(I_GX)_{\mathbb{Q}}\) is \(\mathcal{R}\)-strongly Chern compatible. In the paper under review, or each inertia pair, inertia products are defined on \(K(I\mathcal{X})\) and \(A^*(I\mathcal{X})\) and Chern character ring homomorphism between them. The idea is the comparison between the \(G\)-eruivariant objects over \(X\) with those non-equivariant ones over \(\mathcal{X}\). The authors construct many such inertia pairs. In particular, for a \(G\)-equivariant bundle \(V\) on \(X\), two new inertia pairs \((\mathcal{R}^{\pm}V, \mathcal{S}^{\pm}V)\) are defined using the methods in [\textit{D. Edidin} et al., Duke Math. J. 153, No. 3, 427--473 (2010; Zbl 1210.14066)]. As special cases, they recover most of the orbitfold products already known. The authors also introduce an entirely new product which they call the localized orbifold product, defined on \(K(I\mathcal{X})\otimes\mathbb{C}\).