is not uniquely determined by the above properties. But one obtains a full uniqueness statement by an analogous scheme as in Remark \[r11a\]; see [@L10a §3].

\[r15\] Consider any object $A\in \mathcal{D}G$ and suppose that its inverse image $F^*A$ under the Frobenius map is isomorphic to $A$ in $\mathcal{D}G$. Let $\phi \colon F^*A \stackrel{\sim}{\rightarrow} 
A$ be an isomorphism. Then $\phi$ induces a linear map $\phi_{i,g}\colon 
{{\mathcal{H}}}_g^i(A)\rightarrow {{\mathcal{H}}}_g^i(A)$ for each $i$ and $g\in G^F$. This gives rise to a class function $\chi_{A,\phi}\in \mbox{CF}(G^F)$, called “characteristic function” of $A$, defined by $$\chi_{A,\phi}(g)=\sum_i (-1)^i \mbox{Trace}(\phi_{i,g},{{\mathcal{H}}}_g^i(A))\qquad
\mbox{for $g \in G^F$},$$ see [@L2b 8.4]. Note that $\phi$ is unique up to a non-zero scalar; hence, $\chi_{A,\phi}$ is unique up to a non-zero scalar.

Now assume that $A\in\hat{G}$. Then one can choose an isomorphism $\phi_A
\colon F^*A \stackrel{\sim}{\rightarrow} A$ such that the values of $\chi_{A,\phi_A}$ are cyclotomic integers and $\langle \chi_{A,\phi_A},
\chi_{A,\phi_A}\rangle=1$; see [@L2e 25.6, 25.7] (and also the comments in [@L10 3.10]). The precise conditions which guarantee these properties are formulated in [@L2c 13.8], [@L2e 25.1]; note that these conditions specify $\phi_A$ up to multiplication by a root of unity. In the following, we will tacitly assume that $\phi_A$ has been chosen in this way 