Proposition 3.15 (0MYD) · ScalingStacks

Proposition 3.15. The equivalence in (3.9) induces an isomorphism

\bigoplus_{n\in\mathbb{Z}}\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q}_{\ell})}(M,N(n))\to\operatorname{Hom}_{\operatorname{D}_{G}(k,\mathbb{Q}_{\ell})}(\operatorname{Real}_{\ell}(M),\operatorname{Real}_{\ell}(N))

for $M,N\in\operatorname{DTM}_{G}(k).$ If $M,N\in\operatorname{DTM}_{G}(k)^{w=0}$ then all summands for $n\neq 0$ vanish and the functor $\operatorname{Real}_{\ell}$ gives an isomorphism

\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q}_{\ell})}(M,N)\to\operatorname{Hom}_{\operatorname{D}_{G}(k,\mathbb{Q}_{\ell})}(\operatorname{Real}_{\ell}(M),\operatorname{Real}_{\ell}(N)).

0MYE

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]. ∎