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.
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2$\begingroup$ I would have thought that "admissible representation of G" means in particular a continuous representation? $\endgroup$– Theo Johnson-FreydCommented Jul 1, 2023 at 17:16
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$\begingroup$ This is a very easy exercise. $\pi$ is assumed admissible, whence smooth : any vector has an open stabilizer in $G$. From this it follows that the central character is smooth as well. Read any basic book on representation theory of p-adic groups (Bushnell-Henniart, Renard, Bernstein-Zelevinski,...) $\endgroup$– Paul BroussousCommented Jul 2, 2023 at 9:16
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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.
answered Jul 4, 2023 at 4:00