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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.

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1 Answer 1

I happened to glance through Cogdell's "L-functions and Converse Theorems for GL(n)" (in Sarnak/Shahidi "Automorphic Forms and Applications), which cites Jacquet and Shalika's "On Euler Products and the Classification of Automorphic Forms I", available from Jacquet's website, as answering your question in the affirmative (see Section 1.3): the map $\xi\rightarrow W_\xi|_P$ in injective (where $P$ is the mirabolic subgroup), moreover, the Kirillov model (of a generic irreducible unitary representation), defined as the space of restrictions of Whittaker functions, is isomorphic to ${\rm Ind}_N^P(\psi)$ as representations of $P$.

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