Let $F$ be a $p$-adic field with residue field order $q$ and the normalized absolute value $|\cdot|=|\cdot|_F$. Let $\mathbf{G}$ be a connected reductive group over $F$. Let me denote by $G=\mathbf{G}(F)$ the group of $F$-points of $\mathbf{G}$.
Let $X^*(\mathbf{G})$ be the group of $F$-rational characters of $\mathbf{G}$, and $\mathfrak{a}_{\mathbf{G},\mathbb{C}}^*:=X^*(\mathbf{G})\otimes_{\mathbb{Z}}\mathbb{C}$. Let $$X^{\mathrm{ur}}(G)=\{|\chi|^s:\chi\in X^*(\mathbf{G}),s\in\mathbb{C}\}$$ be the group of unramified characters of $G$, where $|\chi|^s(g):=|\chi(g)|_F^s$.
Then we have a canonical surjection $$\mathfrak{a}_{\mathbf{G},\mathbb{C}}^*\rightarrow X^{\mathrm{ur}}(G),\quad \chi\otimes s\mapsto |\chi|^s.$$ The kernel of this surjection is $\left(\frac{2\pi i}{\log q}\right)\cdot R$, where $R$ is a lattice of $\mathfrak{a}_{\mathbf{G},\mathbb{Q}}^*$.
I think that in general $R$ needn't be $X^*(\mathbf{G})$. This is equivalent to saying that, in general, it may happen that $$\forall \chi\in X^*(\mathbf{G}),~q\notin |\chi(G)|=\{|\chi (g)|:g\in G\}.$$
But so far I couldn't write down an explicit example. My question:
- How to construct an explicit example of this phenomenon?
- This looks closely related to Galois cohomology. Is there a cohomological criterion for this phenomenon? Or maybe just a sufficient condition for $q\in |\chi(G)|$? (I guess that the unramified groups $\mathbf{G}$, i.e. quasi-split and splits over an unramified extension, satisfy this.)
Thanks a lot in advance for any help/suggestions!
Let $\mathbf G=\mathrm{Res}_{E/F}\mathbb G_m$, the restriction of scalars, where $E/F$ is a ramified quadratic extension. Then $X^*(\mathbf G)$ is generated by $N_{E/F}\colon \mathbf G\to\mathbb G_m$. The group of unramified characters $X^{\mathrm{ur}}(G)$ is the set of homomorphisms $E^\times\to\mathbb C^\times$ trivial on $\mathcal O_E^\times$.
Now, the kernel of $\mathfrak a_{G,\mathbb C}^*\to X^{ur}(G)$ is generated by $N_{E/F}\otimes\pi i/\log q$, instead of $N_{E/F}\otimes2\pi i/\log q$. Indeed, $N_{E/F}\otimes s$ is mapped to $E^\times\to\mathbb C^\times:x\mapsto |N_{E/F}(x)|_F^s=|x|_E^{2s}$.