$\DeclareMathOperator\GL{GL}$Let $(\pi,V)$ be an infinite dimensional irreducible admissible representation of $\GL\_2(\mathbb{Q}\_p)$. Let us fix an element $v\_0\in V$ and define a vector space
$$V\_{v\_0} := \left\{\pi\begin{pmatrix}a& \\ &1\end{pmatrix}v\_0,\;\text{s.t.}\;a\in \GL\_1(\mathbb{Q}\_p)\right\}.$$


We define the representation
\begin{align}\GL\_1(\mathbb{Q}\_p)&{}\to \operatorname{Aut}(V\_{v\_0}),\\ a&{}\mapsto \pi\begin{pmatrix}a& \\ &1\end{pmatrix}.\end{align}
It is irreducible and admissible since
$$\begin{pmatrix}\mathbb{Z}\_p^{\times}& \\ &1\end{pmatrix}\subseteq GL\_2(\mathbb{Z}\_p).$$


On the one hand applying Schur's lemma for admissible representations (Lemma 4.2.4 of "[Automorphic forms and representations](https://doi.org/10.1017/CBO9780511609572)" of Bump) then the representation of $\GL\_1(\mathbb{Q}\_p)$ should satisfy that
$$\pi\begin{pmatrix}a& \\ &1\end{pmatrix}v\_0 = \chi(a)v\_0,$$
where $\chi$ is a character of $\GL\_1(\mathbb{Q}\_p)$.


On the other hand, let us suppose that $\pi\simeq \operatorname{Ind}\_{P\_{\GL\_2}}^{\GL\_2}\xi$, where $\xi$ is a character defined on the diagonal elements of $\GL\_2(\mathbb{Q}\_p)$ such that $\pi$ is irreducible. We denote their Satake parameters by $\alpha\_1,\;\alpha\_2$ and the Whittaker functional by $W(\cdot)$. Theorem 4.6.5 of "[Automorphic forms and representations](https://doi.org/10.1017/CBO9780511609572)" of Bump states that
$$W\left(\pi\begin{pmatrix}p^k& \\ &1\end{pmatrix}v\_0\right) = W(v\_0)\frac{\alpha\_1^{k+1}-\alpha\_2^{k+1}}{\alpha\_1-\alpha\_2}.$$
The function
$$p^k\to \frac{\alpha\_1^{k+1}-\alpha\_2^{k+1}}{\alpha\_1-\alpha\_2},$$
is not a character for $(p)$. This contradicts the previous Schur's lemma argument. Where is the mistake in those computations?



Schur lemma and Whittaker functions
$\DeclareMathOperator\GL{GL}$Let me try to clarify. The formula for the Whittaker functional in Theorem 4.6.5 of "[Automorphic forms and representations](https://doi.org/10.1017/CBO9780511609572)" of Bump states that
$$W\left(\pi\begin{pmatrix}p^k& \\ &1\end{pmatrix}v\_0\right) = W(v\_0)\frac{\alpha\_1^{k+1}-\alpha\_2^{k+1}}{\alpha\_1-\alpha\_2},$$
where $v\_0$ is a $\GL\_2(\mathbb{Z}\_p)$-stable vector in the induced representation.


Your claim is then that the cyclic representation generated by $\GL\_1(\mathbb{Q}\_p)$-translates of $v\_0$ is always irreducible as a $\GL\_1(\mathbb{Q}\_p)$ representation. As Aurel points out in the [comments](https://mathoverflow.net/questions/408555/schur-lemma-and-whittaker-functions#comment1047991_408555), this is certainly false in general. In the case related to the above formula, the fact that $v\_0$ is $\GL\_2(\mathbb{Z}\_p)$-stable does imply that $\mathbb{Z}\_p^\times$ acts by a character since
$$\begin{pmatrix}\mathbb{Z}\_p^{\times}& \\ &1\end{pmatrix}\subseteq GL\_2(\mathbb{Z}\_p),$$
but this says nothing about the action of
$$
\begin{pmatrix}p^{k}& \\ &1\end{pmatrix}
$$
for $k>0$. Those values are determined by the formula.


