ScalingStacks

Proposition 3.20. Assume that the $G$-action on $X$ has finitely many orbits. Let $\ell\neq p$ be a prime and $M,N\in\operatorname{DM}_{G}(X).$ Then the natural isomorphisms $\operatorname{Real}_{\ell}\circ(1)\to\operatorname{Real}_{\ell},$ see (3.9), induces isomorphisms

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

if $M,N$ are $*$- and $!$-pointwise mixed Tate, respectively, and

$$\operatorname{Hom}_{\operatorname{DM}_{G}(X,\mathbb{Q}_{\ell})}(M,N)\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{Hom}_{\operatorname{D}_{G}(X,\mathbb{Q}_{\ell})}(\operatorname{Real}_{\ell}(M),\operatorname{Real}_{\ell}(N))$$

if $M,N$ are $*$- and $!$-pointwise pure Tate, respectively.