Let $m,q$ be positive integers.
Let $\mathcal M$ be a stack and $\mathcal L\in \operatorname{Pic}(\mathcal M)\otimes \Bbb Q$.
Let $B\Bbb G_m^q:=[\operatorname{Spec}\Bbb Z/\Bbb G_m^q]$ over a field $k$ be the quotient stack and $f: B\Bbb G_m^q \to \mathcal M$ a map.
Then $f^{*}\mathcal L$ amounts to a rational character in $X^*(B\Bbb G_m^q)\otimes \Bbb Q$.
How can we prove it?
My attempt:
- We know that $f^{*}\mathcal L \to B\Bbb G_m^q$ is equivalent to a pair $(\psi,\phi)$ where $\psi:Y\to f^{*}\mathcal L$ is a $\Bbb G_m^q$-torsor and $\phi:Y\to \operatorname{Spec}\Bbb Z$ is $\Bbb G_m^q$-equivariant (from Example 5.22 p 59). But I don't see how this amounts to a character.
Thank you.