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

TL;DR

This paper extends the Kirillov model for irreducible generic representations of ${GL}_n( ext{complex})$, showing the space of functions with certain invariance and support properties is rich enough to include all smooth functions satisfying specific conditions.

Contribution

It proves a new Kirillov-type result for ${GL}_n( ext{complex})$, generalizing previous results for $p$-adic and real fields.

Findings

The space of functions contains all smooth functions with prescribed invariance and support properties.

The result parallels known theorems for $p$-adic and real groups.

It advances understanding of Whittaker models for complex general linear groups.

Abstract

Let $G=GL_{n}(\mathbb{C})$ and $1\ne\psi:\mathbb{C}\to\mathbb{C}^{\times}$ 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 \[ \theta(u):=\psi(u_{1,2}+u_{2,3}+...+u_{n-1,n}). \] Let $P$ be the mirabolic subgroup of $G$ consisting of all matrices in $G$ with the last row equal to $(0,0,...,0,1)$. We prove that if $\pi$ is an irreducible generic representation of $GL_{n}(\mathbb{C})$ 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 $p\in P$ and which have a compact support modulo $U$. A similar result was proven for $GL_{n}(F)$, 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 $GL_{n}(\mathbb{R})$ by Jacquet in "Distinction by the quasi-split unitary group", Israel Journal of Mathematics, 178(1):269-324, 2010.