ScalingStacks

Remark 4.8. The above discussion should be a shadow of the following conjectural general theory. There should be a Chow weight structure on the category $\operatorname{DM}_{G}^{gm}(\mathcal{N})$ similarly to the non-equivariant case, see Example 2.2(3). The heart of this Chow weight structure should be equivalent to a category of equivariant relative Chow motives $\operatorname{Chow}_{G}(\mathcal{N},\mathbb{Q})$ in which the composition of morphisms is defined via convolution as in (4.1). Since by assumption $\widetilde{\mathcal{N}}_{i}$ is smooth and $\mu_{i}$ is projective the motive $\mu_{!}(\mathbb{Q}_{M_{i}})$ should be in the heart and correspond to the relative Borel–Moore motive $M^{BM}(\widetilde{\mathcal{N}}_{i}/\mathcal{N})$ in the category $\operatorname{Chow}_{G}(N,\mathbb{Q}).$

In the non-equivariant case this is shown to be true by Fangzhou [Fan16]. To define a weight structure in the equivariant case, one would need appropriate $G$-equivariant resolution of singularities or alterations, see [SVW18, Remark II.4.15].

In this article we get around this problem by defining a weight structure on the subcategory $\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q})$ by brute force using the conditions (PT) and (FO).