Hecke character

Remarks

Example: the idele norm $\abs{x} = \prod_v \abs{x_v}_v$ is continuous and multiplicative, and $\abs{a} = 1$ for $a \in K^\times$ by the product formula; so it is a Hecke character, and so is $\abs{\cdot}^s$ for every $s \in \C$. Non-example: for $K = \Q$, the map $x \mapsto \abs{x_\infty}$ that keeps only the real component is a continuous homomorphism, but it sends the rational number $2$ to $2 \neq 1$. Some authors call $\chi$ a Hecke character only when $\abs{\chi(x)} = 1$ for all $x$, and call the general case a quasi-character or Größencharakter. Every Hecke character is a unitary one times $\abs{\cdot}^\sigma$ for a real number $\sigma$. The Hecke characters of $\Q$ of finite order correspond to the primitive Dirichlet characters.