Let $F$ be either a p-adic field or real number field. $GL_{n-1}(F)$ embeds into $GL_n(F)$ on the left upper corner, let $P_n(F)$ be the mirabolic subgroup of $G_n(F)$ consisting of matrices with the last low like $(0,0,...,0,1)$. So we have inclusions $GL_{n-1}(F)\subset P_n(F)$.

When $F$ is p-adic, let $(\pi,V)$ be a generic irreducible admissible smooth representation of $GL_n(F)$, with its Whittaker model $\mathcal{W}(\pi)$. It is well known that for any $W_v(g)\in \mathcal{W}(\pi)$, the restriction map sending $W_v$, as a function of $GL_n$, to $W_v|_{GL(n-1)}$ ,

is injective, and from here we obtain the Kirillov model of $\pi$ as a space of functions on $GL_{n-1}$ .

My question is that when $F=\mathbb{R}$, and $\pi$ is unitary, in which case we also have Kirillov model for $\pi$, but I'm not sure if it comes in the same way as we described above in p-adic case. I don't know the exact reference containing the proof that $\pi$ has Kirillov model, and thus have no idea how it comes.

Another question is that when $F=\mathbb{R}$, $\pi$ a generic smooth irreducible admissible, is it still true for any Whittaker function $W_v$ of $\pi$, the restriction of $W_v$ to $GL(n-1)$ as above is injective, which implies we again have a Kirillov model for $\pi$.

Any answer, comment and reference are appreciated.