Let $G$ be a $p$-adic reductive group and $Z\subseteq G$ the center. Let $\pi$ be an irreducible admissible representation of $G$. By Schur's lemma, it is easy to show that there is a group homomorphism $\omega\_\pi:Z\to \mathbb{C}^\times$, namely the central character. But how does one to prove that this group homomorphism is indeed continuous? My understanding is that a character on $Z$ is a continuous homomorphism, not just a homomorphism.



Continuity of central character
Let $K\subset G$ be a compact open subgroup such that $V^K\ne0$. Now $Z$ acts on $V^K$ by the central character $\omega\_\pi$, and the action is trivial on $Z\cap K$. Thus $\omega\_\pi$ is trivial on the compact open subgroup $Z\cap K$, hence is continuous.


