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Let $M/F$ be a quadratic extension of number fields, with Galois group $G=\{1,\tau\}$. Consider the following unitary group $U_1(R)=\{z\in (R\otimes_FM)^\times :zz^\tau=1\},$ where $R$ is an $F$-algebra. If we have a character $\eta:U_1(F)\backslash U_1(\mathbb{A}_F)\to\mathbb{C}^\times,$ then we can get a Hecke character $\chi$ on $\mathbb{A}_M$ by setting $\chi(z):=\eta\left(\frac{z}{z^\tau}\right)$.

Question: What would be a necessary and sufficient condition for a Hecke character $\chi$ on $\mathbb{A}_M$ to descend to a character on $U_1(\mathbb{A}_F)$? At first, I thought that anti-Galois invariance (i.e. $(\chi^\tau)^{-1}=\chi)$ would be enough, but I don't see how to prove this.

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    Hilbert's theorem 90 seems to be the mechanism to prove your surmise.2012-07-23

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