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[Community Question] Linear-algebra: Unitary Central Character by Schur's Lemma

One of our user asked:

Consider an irreducible smooth representation $\pi$ of the group $G=GL_n(\mathbb{Q}_p)$ with center $Z$. Does there exist a unitary central character for $\pi$?

More precisely, is there a (quasi-)character $\omega: G \to \mathbb{C}^{\times}$ such that $\pi \otimes \omega$ when restricted to the center $Z$ is a unitary character for $Z$? I find this result casually stated in many references, where they say it follows from Schur's lemma. But I am unable to see it directly from Schur's lemma.


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