A note on Kirillov model for representations of ${GL}_n(\mathbb{C})$

Let $G=G{L}_n(\mathbb{C})$ and $1\mathrm{\ne }\psi :\mathbb{C}\to {\mathbb{C}}^{\times }$ be an additive character. Let $U$ be the subgroup of upper triangular unipotent matrices in $G$. Denote by $\theta$ the character $\theta :U\to \mathbb{C}$ given by $$\theta (u):=\psi (u_{1,2}+u_{2,3}+\mathrm{.}\mathrm{.}\mathrm{.}+u_{n-1,n})\mathrm{.}$$ Let $P$ be the mirabolic subgroup of $G$ consisting of all matrices in $G$ with the last row equal to $(0,0,\mathrm{.}\mathrm{.}\mathrm{.},0,1)$. We prove that if $\pi$ is an irreducible generic representation of $G{L}_n(\mathbb{C})$ and $\mathcal{W}(\pi ,\psi )$ is its Whittaker model, then the space ${f|_{P}:P\to $\mathbb{C}$:, f\in $\mathcal{W}$(\pi,\psi)}$ contains the space of infinitely differentiable functions $f:P\to \mathbb{C}$ which satisfy $f(up)=\psi (u)f(p)$ for all $u\in U$and$p\in P$andwhichhaveacompactsupportmodulo$U$. A similar result was proven for $G{L}_n(F)$, where $F$ is a $p$-adic field by Gelfand and Kazhdan in “Representations of the group $GL(n,K)$ where K is a local field”, Lie groups and their representations, Proc. Summer School, Bolyai J'anos Math. Soc., Budapest:95-118, 1975, and for $G{L}_n(\mathbb{R})$ by Jacquet in “Distinction by the quasi-split unitary group”, Israel Journal of Mathematics, 178(1):269-324, 2010.