ScalingStacks

Proof. First, by Proposition 3.11 and induction it suffices to show the statement for objects of the form $\operatorname{Ind}_{G_{0}}^{G}\mathbb{Q}(n)[m].$ If $G$ is connected the homomorphims of objects of the form $\operatorname{Ind}_{G_{0}}^{G}\mathbb{Q}(n)[m]=\mathbb{Q}(n)[m]$ are described in terms of $S^{W}$ in both $\operatorname{DM}_{G}(k,\mathbb{Q}),$ see Proposition 3.12, and in $\operatorname{D}_{G}(\operatorname{pt},\mathbb{Q}_{\ell}),$ see [BL94, Section 13.10], and the statement is easily seen to be true. The case that $G$ is not connected can be handled as described in [SVW18, Theorem A.2.8]. ∎