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Let $G$ be the group of $F$-points of a connected, reductive group over a $p$-adic field $F$. The unramified character of $G$ are $\chi\circ\psi$ where $\chi$ is an unramified character of $F^{\times}$ and $\psi$ is a \emph{rational character} of $F$. I understand that the determinant is a rational character, but:

What is the definition of a rational character>

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If $\mathbb{G}$ is a reductive group over $F$, and $G$ the group of its $F$-points, then rational characters of $G$ are algebraic characters $\psi : \mathbb{G} \to \mathbb{G}_m$ that are defined over $F$. If you think of $G$ as a subgroup of $\textrm{GL}_n(F)$, this would amount to a morphism $G \to F^*$ which is given by a rational formula with regards to coefficients (so the determinant is indeed a rational character).

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Thanks for the answer. –  Thomas Sep 23 '12 at 22:55

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