ScalingStacks

Proof. The statement can be shown by an induction on the number of orbits. Denote by $j:G/H\cong\mathcal{O}\hookrightarrow X$ and $i:Z=X\backslash\mathcal{O}\to X$ the inclusion of an open orbit $\mathcal{O}$ and its closed complement $Z.$ Then the localisation triangle induces an exact sequence

Since $i^{*}M$ and $i^{!}N$ are $*$- and $!$-pointwise pure Tate, respectively, the first term of the sequence vanishes by induction. The last term vanishes since by assumption $j^{*}M$ and $j^{!}N$ correspond to objects in $\operatorname{DTM}_{H}(k,\mathbb{Q})^{w=0}$ via the induction equivalence (3.6) and thus have no non-trivial extension by Proposition 3.12. See [SVW18, Corollary II.4.19] for a similar proof. ∎