Subsection 4.1. (0N04) · ScalingStacks

4.1. The setup

Recall that all varieties are over $k=\overline{\mathbb{F}}_{p}.$ Let $G$ be a linear algebraic group. Let $\mu_{i}:\widetilde{\mathcal{N}}_{i}\to\mathcal{N}$ be a collection of $G$ -equivariant proper maps where each $\widetilde{\mathcal{N}}_{i}$ is smooth and connected. Consider the collections of objects

\displaystyle\mathcal{T}^{Spr}=\{\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}})\in\operatorname{DM}_{G}(\mathcal{N})\}\text{ and }\widehat{\mathcal{T}}^{Spr}=\bigcup_{n\in\mathbb{Z}}\mathcal{T}^{Spr}(n)[2n].

0MYP

Definition 4.1. The triangulated category of Springer motives $\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q})$ is the full subcategory of $\operatorname{DM}_{G}(\mathcal{N})$ generated by $\widehat{\mathcal{T}}^{Spr}$ with respect to isomorphism, direct summands and triangles,

\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q})=\langle\widehat{\mathcal{T}}^{Spr}\rangle_{\cong,\inplus,\Delta}\subset\operatorname{DM}_{G}(\mathcal{N}).

We define two conditions, that are assumed in some of the later results.

(PT)

Pure Tate. For each $x\in\mathcal{N},$ the motive $\operatorname{M}(\mu_{i}^{-1}(\{x\})$ is pure Tate.
(FO)

Finite Number of Orbits. The images $\mu_{i}(\widetilde{\mathcal{N}}_{i})\subset\mathcal{N}$ have finitely many $G$ -orbits.

0MYQ

Remark 4.2. Condition (FO) allows for simple induction arguments. In many settings it can be weakened such that all arguments still work. For instance, there is a quite straightforward adaption to ind-varieties with possibly infinitely many orbits.

Jens Niklas Eberhardt, Catharina Stroppel

Original source: arXiv:2109.00305v2