Non-trivial extension of representations have same central character

Question

Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations of $G$. Assume that the central character of $\pi_1$ is $\phi$, then I'd like to show that $\phi$ is also the central character of $\pi_2$.

Idea: For all $z$ in the centre $Z$ of $G$, we can define a function $h_v$ on the underlying space of $\pi$ by $h_v: v\mapsto \phi(z)v-zv$. If $h_v=0$ for all $v$, then the centre of $G$ acts via $\phi$ on $\pi$, in particular the centre of $G$ acts via $\phi$ on $\pi_2$, which is the quotient of $\pi$. If $h_v\neq0$ for some $v_0$, i.e. there exists $z_0$ and $v_0\in\pi$ such that $\phi(z_0)v_0-z_0v_0\neq0$. In this case, I'd like to know how to deduce that $\pi\cong\pi_2$ and in particular, $\pi_2$ has $\phi$ as central character.

Answer 1

@KentaSuzuki's answer is probably the best way of thinking about it, but here's another argument in line with what you wanted.

Permit me to write $\phi_1$ rather than $\phi$ for the central character of $\pi_1$; to write $V$, $V_1$, and $V_2$ for the spaces of $\pi$, $\pi_1$, and $\pi_2$; and to change your proposed function from $h_v : Z \to V$ to $h_z : V \to V$, but otherwise defined the same. That is, for each $z \in Z$ and $v \in V$, we define $h_z(v) = \phi_1(z)v - z v$.

For each $z \in Z$, we have that $h_z$ annihilates $V_1$, and so induces a map $V_2 \cong V/V_1 \to V$. Let us continue to denote this map by $h_z$.

If $h_z : V_2 \to V$ is $0$ for all $z \in Z$, then $z v$ equals $\phi_1(z)v$ for all $z \in Z$ and $v \in V$, so we are done.

Thus, without loss of generality, there is some $z \in Z$ such that $h_z : V_2 \to V$ is non-$0$. The composition $V_2 \xrightarrow{h_z} V \to V_2$ intertwines the action of $G$ on $V_2$, so must be a scalar. If it were $0$, then we would have a non-$0$ $G$-intertwining map $h_z : V_2 \to V_1$, which is a contradiction since $\pi_1$ and $\pi_2$ are inequivalent; so the scalar is non-$0$. (This business with intertwiners is probably where I'm using the smoothness assumption that @KentaSuzuki pointed out is needed.) That is, up to adjustment by this scalar, $h_z$ is the desired splitting of $V_1 \to V \to V_2$.

Answer 2

You probably should assume all representations are smooth.

You hope to show if $\pi_1$ and $\pi_2$ have two different central characters $\phi_1$ and $\phi_2$ then any extension $0\to \pi_1\to \pi\to \pi_2\to 0$ splits. Certainly $\pi$ splits as a representation of $F^\times\hookrightarrow\mathrm{GL}_2(F)$ (by using that $\pi$ is smooth). So $\pi=\pi_1\oplus V$ where $F^\times$ acts on $V$ as $\phi_2$ (i.e., $V$ is the $\phi_2$-isotypic component).

Then $V$ is a representation of $\mathrm{GL}_2(F)$, and $V\to \pi_2$ is an isomorphism, as desired.

P.S. Of course there is nothing special about $\mathrm{GL}_2$ in the argument above: it applies to any smooth representations of $p$-adic groups.