Let F be a local field, and $\pi$ an absolutely cuspidal representation of $GL_2(F)$. Then the Kirillov model of the representation is given by the space of locally constant functions with compact support in $F^\times$.

My question is in the opposite direction, if we start with the space of locally constant functions with compact support in $F^\times$, this is an absolutely irreducible representation of the group $\left(\begin{smallmatrix} a & b \\ 0 & 1\end{smallmatrix}\right)$. In particular (if we fixed a quasi-character to be our central character) we can induce such representation from the Borel subgroup to the whole $GL_2(F)$. Clearly any absolutely compact representation (as the one before) appears in this big representation.

Question: do they appear with multiplicity one?

The answer should be yes assuming some Frobenius reciprocity, but I am making no assumption on the central quasi character, so I do not know if such result holds in this more general context (this could be the original question as well). I guess that if the central character is a character, then Frobenius might follow from the fact that the representations are unitary.