Branching laws for smooth representations

Question

Let $E / F$ be a quadratic extension of nonarchimedean local fields (characteristic 0 if it matters), and $\pi$ an irreducible infinite-dimensional smooth representation of $GL_2(E)$. Let $B$ be the upper-triangular Borel of $GL_2$. I'd like to know: do we always have $$\operatorname{dim} Hom_{B(F)}(\pi, \mathbf{C}) = 1 ?$$ I know how to construct a non-zero element in this space, so my concern is to prove its dimension is $\le 1$, i.e. to show that $(GL_2(E), B(F))$ is a Gelfand pair.

Here's why I think it should be true. Firstly, the analogous result is true if we replace the field extension $E$ with $F \oplus F$; this is the main result of the Harris--Scholl paper cited below. Secondly, it's true if $\pi$ is discrete-series: by Frobenius reciprocity we have $Hom_{B(F)}(\pi, \mathbf{C}) = Hom_{GL_2(F)}(\pi, Ind_{B(F)}^{GL_2(F)}(\mathbf{C}))$. The representation $Ind_{B(F)}^{GL_2(F)}(\mathbf{C})$ has a trivial subrepresentation and the quotient is the Steinberg representation $\mathrm{St}$, so we have a left-exact seq $$0 \to Hom_{GL_2(F)}(\pi, \mathbf{C}) \to Hom_{B(F)}(\pi, \mathbf{C}) \to Hom_{GL_2(F)}(\pi, \mathrm{St}).$$ From the results in Prasad's 1992 paper cited below, exactly one of $Hom_{GL_2(F)}(\pi, \mathbf{C})$ and $Hom_{GL_2(F)}(\pi, \mathrm{St})$ is 1-dimensional and the other is zero, so we are done.

However, if $\pi$ is a principal series, we have the same exact sequence but it can happen that $Hom_{GL_2(F)}(\pi, \mathrm{St})$ and $Hom_{GL_2(F)}(\pi, \mathbf{C})$ are both non-zero (this occurs iff $\pi$ is the normalised induction of a pair of characters of $E^\times$ which are distinct, and both trivial on $F^\times$). I tried to bash out this case via Mackey theory, but I couldn't get it to work.

Harris, M.; Scholl, A. J., A note on trilinear forms for reducible representations and Beilinson’s conjectures., J. Eur. Math. Soc. (JEMS) 3, No. 1, 93-104 (2001). Preprint version Published version

Prasad, Dipendra, Invariant forms for representations of $\text{GL}_2$ over a local field, Am. J. Math. 114, No. 6, 1317-1363 (1992).

Answer 1

You can approach the problem via the mirabolic subgroup $P_2(F)\subset B_2(F)$. First we can restrict to $\pi$ with central character trivial on $F^\times$. Then you want to know if in this situation $Hom_{P_2(F)}(\pi,1)$ is of dimension at most $1$. This is indeed the case for unitary representations for example by [Matringe, Pacific Journal 2014, Proposition 2.5]. In particular this seems to take care of the remaining case you had of representations of the form $\chi_1\times \chi_2$ with $(\chi_i)_{|F^\times}=1$, which are clearly unitary as both $\chi_i$'s have to be unitary in this case.