Unitary Central Character by Schur's Lemma 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.
Recall the following from Garrett's Automorphic Representations and L-functions:

Definition: A central character is simply a continuous group homomorphism $\omega : Z_A \to $ $\mathbb C^\times$ where $Z$ is the center of a reductive linear group $G$ over a field $k$ and $Z_A$ denotes the adele points of $Z$.
Triviality of $\omega \in Z_k$ is often assumed, resulting in the mapping $\omega : Z_A \to Z_k \diagup Z_A \to $ $\mathbb C^\times$. Furthermore, $\omega$ is said to be unitary when, $\forall z \in Z_A$, $|\omega(z)| = 1$

The key theorems are as follows:

Theorem 4.1 (Schur's Lemma over $\mathbb C$). If $V$ is an irreducible complex $G$-representation, then every linear operator $\phi : V \to V$ that commutes with $G$ is a scalar.
Proof. Let $\lambda$ be an eigenvalue of $\phi$ and assume that the eigenspace $E_\lambda$ is $G$-invariant. Then $v \in E_\lambda \implies \phi(v) = \lambda v,$ whence $\phi(gv) = g\phi(v) = g(\lambda v) = \lambda * gv$, so $gv \in E_\lambda$ (g was arbitrary). However, by irreducibility, this implies that $E_\lambda = V$, thus $\phi = \lambda \text{Id}$.

So, consider an irreducible smooth $G$-representation $\pi$.

Theorem 3.8 (Schur's Lemma). For a td-group $G$, any irreducible smooth $G$-representation $\pi$ satisfies End$_G$($\pi$) = $\mathbb C$.

Since a unitary central character is, by definition, a continuous group homomorphism $\omega : Z_A \to \mathbb C^\times$ such that $|\omega(z)| = 1 \space \forall z \in Z_A$, and, by Theorem 3.8, any  irreducible smooth $G$-representation $\pi$ satisfies End$_G$($\pi$) = $\mathbb C$, we can associate a central character $\omega_\pi(z) : Z \to \mathbb C^\times$ such that, for all matrix coefficients $f$ of $\pi$, $f(zh) = \omega_\pi(z)f(h).$ (commuting with G as a scalar, cf. Thm 4.1)
A matrix coefficient defined in Definition 5.1 of 2.
There is a central character $\omega:Z \rightarrow \mathbb{C}^{\times}$ by Schur, but it need not be unitary. E.g., consider $∣\text{det}∣$.