Question absolutely cuspidal representation

Question

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.

Answer 1

The answer is yes. If $H$ is a closed subgroup of a locally profinite group $G$ then the functor of restriction to $H$ has a right adjoint : the functor of smooth induction (we work in categories of smooth representations). You do not need to assume that your representations are unitary. The reference is :

Bernstein and Zelevinsky : Representations of the group ${\rm GL}(n,F)$, where $F$ is a non-achimedean local field, Russian Math. Surveys, 1976.

where smooth representations are called algebraic. In this paper you can also find the theory of Whittaker (and Kirillov) models for ${\rm GL}(n)$.

In the more general case of ${\rm GL}(n)$, irreducible supercuspidal representations are generic (they have a Whittaker model) and one has multiplicity $1$. This is not true in general (i.e. for certain non supercuspidal representations, or for certain supercuspidal representations of other reductive groups) : representations are not always generic and when they are generic one not necessarily has multiplicity $1$.