text:
\section{Introduction}\label{Sect1}

`Weyl-Pedersen calculus' is the name proposed in \cite{BB09c}
for the remarkable correspondence $a\mapsto{\rm Op}^\pi(a)$ constructed
by N.V.~Pedersen in \cite{Pe94} as a generalization of
the pseudo-differential Weyl calculus on~${\mathbb R}^n$.
Here $\pi\colon G\to{\mathscr B}({\mathscr H})$ is any unitary irreducible representation
of a connected, simply connected, nilpotent Lie group $G$,
the symbol $a$ can be any tempered distribution on
the coadjoint orbit~${\mathscr O}$ corresponding to $\pi$ by the orbit method
of~\cite{Ki62}, and ${\rm Op}^\pi(a)$ is a linear operator
in the representation space ${\mathscr H}$, which is in general unbounded.
When $\pi$ is the Schr\"odinger representation of
the $(2n+1)$-dimensional Heisenberg group,
the correspondence $a\mapsto{\rm Op}^\pi(a)$ is precisely
the calculus suggested by H.~Weyl in \cite{We28}
for applications to quantum mechanics.
This calculus was later developed by L.~H\"ormander in \cite{Hor79}
and made into a powerful calculus of pseudo-differential operators on~${\mathbb R}^n$;
see \cite{Hor07}.

In the present paper we discuss the classical notion of smooth vectors
---and the related notion of smooth operators---
with a view toward their crucial importance for the Weyl-Pedersen calculus.
We then approach a related circle of ideas that recently emerged in \cite{BB09c},
namely the modulation spaces
for unitary irreducible representations of nilpotent Lie groups.
We take the opportunity of this discussion
to extend some known facts to a setting
where they hold true in a natural degree of generality
(see for instance Theorem~\ref{smooth_ideal} below).
We also take a close look at some new examples of unitary irreducible representations
and find out their related notions which illustrate the main theme
of the present paper: the preduals of the corresponding coadjoint orbits,
their ambiguity function, or their space of smooth vectors
(see Proposition~\ref{double} and Corollary~\ref{double_cor}).

Let us mention that the importance of the Weyl-Pedersen calculus and the related circle of ideas
goes far beyond the framework of representation theory of nilpotent Lie groups.
Many other interesting developments within
the theory of partial differential equations and the finite-dimensional Lie theory
can be found for instance in the references \cite{An69}, \cite{An72},
\cite{How80}, \cite{Mi82}, \cite{Me83}, \cite{How84}, \cite{HRW84}, \cite{HN85}, \cite{Mi86},
\cite{FG92}, \cite{Ma91}, \cite{Ma95},
and \cite{Ma07}.
Moreover, one can use a similar construction
even for representations of certain \emph{infinite-dimensional Lie groups}
in order to provide a geometric explanation for
the gauge covariance for the magnetic Weyl calculus
of \cite{MP04}, \cite{IMP07}, \cite{IMP09}, \cite{MP09} and the references therein.
The representation theoretic approach to the magnetic Weyl calculus
has been taken up in the papers \cite{BB09a}, \cite{BB09b};
see also the survey \cite{BB09d}.

The structure of the present paper is summarized in the following table of contents:

\tableofcontents

\noindent\textbf{Notation and background.}
Throughout the paper we denote by ${\mathscr S}({\mathscr V})$ the Schwartz space
on a finite-dimensional real vector space~${\mathscr V}$.
That is, ${\mathscr S}({\mathscr V})$ is the set of all smooth functions
that decay faster than any polynomial together with
their partial derivatives of arbitrary order.
Its topological dual ---the space of tempered distributions on ${\mathscr V}$---
is denoted by ${\mathscr S}'({\mathscr V})$.
We shall also have the occasion to
use these notions with ${\mathscr V}$ replaced by a coadjoint orbit
of a nilpotent Lie group.
In this situation we need the notion of polynomial structure
on a manifold; see Sect.~1 in \cite{Pe89} for details.
We use $\scalar{\cdot}{\cdot}$ to denote any duality pairing between
finite-dimensional real vector spaces whose meaning is clear
from the context.

We shall also use the convention that the Lie groups are denoted by
upper case Latin letters and the Lie algebras are denoted
by the corresponding lower case Gothic letters.

As regards the background information for the present paper,
we refer to \cite{Hor07}, \cite{Fo89}, and \cite{Gr01} for
basic notions of pseudo-differential Weyl calculus on ${\mathbb R}^n$.
The necessary notions of representation theory for nilpotent Lie groups
(in particular,
the correspondence between the coadjoint orbits and the unitary irreducible representations)
can be found in \cite{Pu67}, \cite{CG90}, and \cite{Ki04};
see also \cite{Wa72} and \cite{Ki76}.
Our references for topological vector spaces, nuclear spaces, and related topics
are \cite{Sch66}, \cite{Tr67}, and \cite{Co68}.

\section{Smooth vectors for Lie group representations}

The smooth vectors have been a basic tool in representation theory
of Lie groups; see for instance the early paper \cite{Ga47} and
the classical monographs \cite{Wa72} and \cite{Ki76}.
In this section we discuss some of the very basic properties
of the smooth vectors for the purpose of providing the necessary
background for the later developments in the present paper.

\begin{notation}
\normalfont
Throughout this section we shall use the following notation:
\begin{itemize}
\item $G$ is a connected unimodular Lie group with the Lie algebra $\gg$;
\item ${\rm d} x$ denotes a fixed Haar measure on $G$;
\item ${\mathscr V}$ and ${\mathscr Y}$ are some complex Banach spaces;
\item $\pi\colon G\to {\mathscr B}({\mathscr Y})$ is a representation
which is continuous, in the sense that for every $x\in {\mathscr Y}$
the mapping $\pi(\cdot)x\colon G\to {\mathscr Y}$ is continuous.
\end{itemize}
\qed
\end{notation}

\subsection{Distribution theory on Lie groups}
Some references for distribution theory on Lie groups
are \cite{Br56} and \cite{Wa72}.
The present subsection just records a few basic notions and properties
needed later.

\begin{definition}
\normalfont
We define the \emph{spaces of test functions} on the Lie group $G$ as follows:
\begin{enumerate}
\item The space
$${\mathscr E}(G,{\mathscr V}):=\{\phi\colon G\to{\mathscr V}\mid \phi\text{ is smooth}\}$$
with the usual topology of a Fr\'echet space
(given by the uniform convergence
on compact sets of functions and their partial derivatives in local charts).
\item The space
$${\mathscr D}(G,{\mathscr V}):=\{\phi\in{\mathscr E}(G,{\mathscr V})\mid {\rm supp}\,\,\phi\text{ is compact}\}$$
with the usual topology of an inductive limit of Fr\'echet spaces.
\end{enumerate}
If ${\mathscr V}={\mathbb C}$ then we denote simply ${\mathscr E}(G,{\mathbb C})={\mathscr E}(G)$ and ${\mathscr D}(G,{\mathbb C})={\mathscr D}(G)$.
For every integer $m\ge1$ we shall also need the function space
$${\mathscr C}^m_0(G):=\{\phi\colon G\to{\mathbb C}\mid \phi\text{ is of class ${\mathscr C}^m$ and }{\rm supp}\,\,\phi\text{ is compact}\} $$
with its usual topology of an inductive limit of Banach spaces.

We then define the \emph{spaces of vector valued distributions}
$${\mathscr D}'^{{\mathscr Y}}(G,{\mathscr V}):=\{u\colon{\mathscr D}(G,{\mathscr V})\to{\mathscr Y}\mid u\text{ is linear and continuous}\}$$
and
$${\mathscr E}'^{{\mathscr Y}}(G,{\mathscr V}):=\{u\colon{\mathscr E}(G,{\mathscr V})\to{\mathscr Y}\mid u\text{ is linear and continuous}\}$$
and endow them with the topology of pointwise convergence.
We denote the evaluation mapping by
$$\langle\cdot,\cdot\rangle\colon
{\mathscr D}'^{{\mathscr Y}}(G,{\mathscr V})\times{\mathscr D}(G,{\mathscr V})\to{\mathscr Y},\quad
\langle u,\phi\rangle:=u(\phi), $$
and similarly for
$\langle\cdot,\cdot\rangle\colon
{\mathscr E}'^{{\mathscr Y}}(G,{\mathscr V})\times{\mathscr E}(G,{\mathscr V})\to{\mathscr Y}$.

For ${\mathscr V}={\mathbb C}$ we denote simply
${\mathscr E}'^{{\mathscr Y}}(G,{\mathbb C})={\mathscr E}'^{{\mathscr Y}}(G)$ and ${\mathscr D}'^{{\mathscr Y}}(G,{\mathbb C})={\mathscr D}'^{{\mathscr Y}}(G)$.
If also ${\mathscr Y}={\mathbb C}$, then we further denote
${\mathscr E}'^{{\mathscr Y}}(G)={\mathscr E}'(G)$ and ${\mathscr D}'^{{\mathscr Y}}(G)={\mathscr D}'(G)$.
\qed
\end{definition}

\begin{definition}
\normalfont
The \emph{support} of the distribution $u\in{\mathscr D}'^{{\mathscr Y}}(G,{\mathscr V})$
is denoted by ${\rm supp}\, u$ and is defined as the intersection
of all the closed sets $F\subseteq G$ such that for every $\phi\in{\mathscr D}(G,{\mathscr V})$
with $F\cap{\rm supp}\,\phi=\emptyset$ we have $\langle u,\phi\rangle=0$.
\qed
\end{definition}

\begin{remark}\label{funct}
\normalfont
Let $L^1_{_{\opn{loc}}}(G)$ denote the linear space of (equivalence classes of)
measurable functions on $G$ which are absolutely integrable with respect to the Haar measure ${\rm d} x$ on every compact subset of~$G$.
Then there exists a natural linear embedding
$L^1_{_{\opn{loc}}}(G)\hookrightarrow{\mathscr D}'(G)$.
Specifically, every $f\in L^1_{{\rm loc}}(G)$ gives rise to
a distribution also denoted by $f$ and defined by
$$(\forall \phi\in{\mathscr D}(G))\quad \langle f,\phi\rangle=\int\limits_G f\phi{\rm d} x. $$
Note that $L^1_{{\rm loc}}(G)$ contains many function spaces on $G$,
like the space of continuous functions,
or the space of smooth functions ${\mathscr E}(G)$,
or the Lebesgue space $L^p(G)$ if $1\le p\le\infty$ etc.
\qed
\end{remark}

\begin{remark}\label{comp_supp}
\normalfont
We have
$${\mathscr E}'(G)=\{u\in{\mathscr D}'(G)\mid{\rm supp}\, u\text{ is compact}\}. $$
For every compact set $K\subseteq G$ we denote
${\mathscr E}'_K(G)=\{u\in{\mathscr D}'(G)\mid{\rm supp}\, u\subseteq K\}. $
\qed
\end{remark}

\begin{remark}\label{tens}
\normalfont
We recall from \cite{Sch66} and \cite{Tr67} that
the locally convex spaces ${\mathscr E}(G)$ and ${\mathscr D}(G)$
are nuclear.
Moreover, we have the linear topological isomorphisms
$${\mathscr E}(G,{\mathscr Y})\simeq{\mathscr E}(G)\widehat{\otimes}{\mathscr Y}
\text{ and }{\mathscr D}(G,{\mathscr Y})\simeq{\mathscr D}(G)\widehat{\otimes}{\mathscr Y},$$
which are natural in the sense that every pair $(\phi,y)\in{\mathscr E}(G)\times{\mathscr Y}$
corresponds to the function $\phi(\cdot)y\in{\mathscr E}(G,{\mathscr Y})$.
Also recall the the linear topological isomorphisms
$${\mathscr E}(G)\widehat{\otimes}{\mathscr E}(G)\simeq{\mathscr E}(G\times G)
\quad\text{ and }\quad
{\mathscr D}(G)\widehat{\otimes}{\mathscr D}(G)\simeq{\mathscr D}(G\times G) $$
that take a pair $(\phi_1,\phi_2)\in{\mathscr D}(G)\times{\mathscr D}(G)$
to the function $\phi_1\otimes\phi_2$ defined by $(x_1,x_2)\mapsto \phi_1(x_1)\phi_2(x_2)$.
\qed
\end{remark}

\begin{example}
\normalfont
Here are some examples of vector valued distributions
that will be needed in the sequel.
\begin{enumerate}
\item For arbitrary $g\in G$
the ${\mathscr Y}$-valued \emph{Dirac distribution} $\delta_g^{{\mathscr Y}}\in{\mathscr E}'^{{\mathscr Y}}(G,{\mathscr Y})$ is defined by
$$\delta_g^{{\mathscr Y}}\colon{\mathscr E}(G,{\mathscr Y})\to{\mathscr Y}, \quad
\langle\delta_g^{{\mathscr Y}},\phi\rangle=\phi(g).  $$
If ${\mathscr Y}={\mathbb C}$ then we denote simply $\delta_g^{{\mathscr Y}}=\delta_g$.
\item By using Remark~\ref{tens}, one can define a canonical linear mapping
$${\mathscr E}'(G)\to{\mathscr E}'^{{\mathscr Y}}(G,{\mathscr Y}),\quad u\mapsto u\otimes{\rm id}_{{\mathscr Y}}, $$
which takes every distribution $u\colon{\mathscr E}(G)\to{\mathbb C}$
to its tensor product with the identity operator ${\rm id}_{{\mathscr Y}}\colon{\mathscr Y}\to{\mathscr Y}$.
\end{enumerate}
\qed
\end{example}

\begin{definition}\label{conv_def}
\normalfont
Let $u_1,u_2\in{\mathscr E}'(G)$.
Then the \emph{tensor product of distributions}
$u_1\otimes u_2\in{\mathscr E}'(G\times G)$
can be defined by using Remark~\ref{tens}
such that $\langle u_1\otimes u_2,\phi_1\otimes\phi_2\rangle
=\langle u_1,\phi_1\rangle\cdot\langle u_2,\phi_2\rangle$.
On the other hand, there exists a continuous linear co-product
$${\mathscr E}(G)\to{\mathscr E}(G\times G),\quad \phi\mapsto\phi^\Delta, $$
where $\phi^\Delta(x_1,x_2):=\phi(x_1x_2)$ whenever $x_1,x_2\in G$
and $\phi\in{\mathscr E}(G)$.
The \emph{convolution product of distributions} $u_1\ast u_2\in{\mathscr E}'(G)$
is defined by
$$(\forall \phi\in{\mathscr E}(G))\quad
\langle u_1\ast u_2,\phi\rangle:=\langle u_1\otimes u_2,\phi^\Delta\rangle.$$
The convolution product makes the distribution space ${\mathscr E}'(G)$
into an associative algebra
whose unit element is the Dirac distribution $\delta_{\1}\in{\mathscr E}'(G)$.
\qed
\end{definition}

\begin{example}\label{conv_ex}
\normalfont
Let us consider a few simple properties of the convolution product.
\begin{enumerate}
\item For arbitrary $g_1,g_2\in G$ we have $\delta_{g_1}\ast\delta_{g_2}=\delta_{g_1g_2}$.
\item For every $u_1,u_2\in{\mathscr E}'(G)$ we have
$${\rm supp}\,(u_1\ast u_2)\subseteq\{x_1x_2\mid x_j\in{\rm supp}\, u_j\text{ for }j=1,2\}.$$
\end{enumerate}
\qed
\end{example}

\begin{definition}\label{univ_def}
\normalfont
We shall think of $\gg$ as a real subalgebra of its \emph{complexification}
$\gg_{{\mathbb C}}:={\mathbb C}\otimes_{{\mathbb R}}\gg$, hence $\gg_{{\mathbb C}}=\gg\dotplus{\rm i}\gg$.
The \emph{universal enveloping algebra} ${\rm U}(\gg_{{\mathbb C}})$ is
the  complex unital associative algebra satisfying the following conditions:
\begin{enumerate}
\item\label{univ_def_item1}
The complexification $\gg_{{\mathbb C}}$ is a Lie subalgebra of ${\rm U}(\gg_{{\mathbb C}})$.
\item\label{univ_def_item2}
For every complex unital associative algebra ${\mathscr A}$ and every linear mapping
$\theta\colon\gg_{{\mathbb C}}\to{\mathscr A}$ satisfying  $\theta([X,Y])=\theta(X)\theta(Y)-\theta(Y)\theta(X)$
for all $X,Y\in\gg_{{\mathbb C}}$ there exists a unique extension of
$\theta$ to a homomorphism of complex unital associative algebras ${\rm U}(\gg_{{\mathbb C}})\to{\mathscr A}$.
\end{enumerate}
One can prove that there always exists an algebra ${\rm U}(\gg_{{\mathbb C}})$ satisfying
these conditions and it is uniquely determined up to an isomorphism of
complex unital associative algebras.
Moreover, there exists a unique (complex-)linear mapping ${\rm U}(\gg_{{\mathbb C}})\to{\rm U}(\gg_{{\mathbb C}})$, $u\mapsto u^\perp$
such that
$$(vw)^\perp=w^\perp v^\perp,\quad (w^\perp)^\perp=w,
\quad\text{and}\quad X^\perp=-X $$
for every $u,v\in{\rm U}(\gg_{{\mathbb C}})$ and $X\in\gg$.
(See \cite{Dix74} for more details).
\qed
\end{definition}

\begin{example}\label{univ_ex}
\normalfont
If $\gg$ is an abelian Lie algebra of dimension $n$,
then ${\rm U}(\gg_{{\mathbb C}})$ is the algebra of polynomials ${\mathbb C}[x_1,\dots,x_n]$
and for every $p\in{\mathbb C}[x_1,\dots,x_n]$ we have $p^\perp(x_1,\dots,x_n)=p(-x_1,\dots,-x_n)$.
\qed
\end{example}

We are going to describe in Remark~\ref{univ_rem}
some realizations of the universal enveloping algebra ${\rm U}(\gg_{{\mathbb C}})$
which are needed later.
To this end we first introduce the regular representations of $G$ on distribution spaces.

\begin{definition}\label{reg}
\normalfont
We shall need the \emph{translation maps} $\lambda_g\colon G\to G$, $x\mapsto gx$
and $\rho_g\colon G\to G$, $x\mapsto xg$
defined by an arbitrary element $g\in G$.
The corresponding \emph{regular representations} of $G$ on the distribution space ${\mathscr D}'(G)$
are defined by
$$\lambda\colon G\to{\rm End}\,({\mathscr D}'(G)),\quad \langle\lambda(g)u,\phi\rangle=\langle u,\phi\circ\lambda_g\rangle$$
and
$$\rho\colon G\to{\rm End}\,({\mathscr D}'(G)),\quad \langle\rho(g)u,\phi\rangle=\langle u,\phi\circ\rho_{g^{-1}}\rangle$$
whenever $g\in G$, $u\in{\mathscr D}'(G)$, and $\phi\in{\mathscr D}(G)$.
For every $X\in\gg$ and $\phi\in{\mathscr E}(G)$ we also define the functions
$$\dot\lambda(X)\phi\colon G\to{\mathbb C}, \quad
(\dot\lambda(X)\phi)(z)=\frac{{\rm d}}{{\rm d} t}\Big\vert_{t=0}\phi(\exp_G(-tX)z)$$
and
$$\dot\rho(X)\phi\colon G\to{\mathbb C}, \quad
(\dot\rho(X)\phi)(z)=\frac{{\rm d}}{{\rm d} t}\Big\vert_{t=0}\phi(z\exp_G(tX)).$$
Then we can define the \emph{derivatives of the regular representations} by
$$\dot\lambda\colon\gg\to{\rm End}\,({\mathscr D}'(G)),\quad
\langle\dot\lambda(X)u,\phi\rangle:=\langle u,\dot\lambda(-X)\phi\rangle$$
and
$$\dot\rho\colon\gg\to{\rm End}\,({\mathscr D}'(G)),\quad
\langle\dot\rho(X)u,\phi\rangle:=\langle u,\dot\rho(-X)\phi\rangle.$$
These derivatives are homomorphisms of Lie algebras,
hence condition~\eqref{univ_def_item2} in Definition~\ref{univ_def}
shows that they can be uniquely extended to unital homomorphisms
of associative algebras ${\rm U}(\gg_{{\mathbb C}})\to{\rm End}\,({\mathscr D}'(G))$.
These extensions will also be denoted by
$\dot\lambda\colon{\rm U}(\gg_{{\mathbb C}})\to{\rm End}\,({\mathscr D}'(G))$ and
$\dot\rho\colon{\rm U}(\gg_{{\mathbb C}})\to{\rm End}\,({\mathscr D}'(G))$, respectively.

For later use, we also introduce the notation $\phi^\perp(x):=\phi(x^{-1})$
for every $\phi\in{\mathscr E}(G)$ and $x\in G$.
This gives rise to the idempotent linear mapping
$${\mathscr D}'(G)\to{\mathscr D}'(G),\quad u\mapsto u^\perp,$$
where $\langle u^\perp,\phi\rangle:=\langle u,\phi^\perp\rangle$
for $u\in{\mathscr D}'(G)$ and $\phi\in{\mathscr D}(G)$.
\qed
\end{definition}

\begin{remark}\label{univ_rem}
\normalfont
With Definition~\ref{reg} at hand,
we can describe some realizations of the universal enveloping algebra ${\rm U}(\gg_{{\mathbb C}})$ as follows.
For the sake of simplicity, let us denote by ${\mathscr E}'_{\1}(G)$ the space of distributions on $G$
with the support contained in $\{\1\}$, thought of as a complex unital associative algebra
with respect to the convolution product, cf.~Example~\ref{conv_ex}.
(This set should actually be denoted by ${\mathscr E}'_{\{\1\}}(G)$ according to Remark~\ref{comp_supp}.)
Recall that $\delta_{\1}\in{\mathscr E}'(G)$ is the Dirac distribution at $\1\in G$.

Both mappings
$$\begin{aligned}
{\rm U}(\gg_{{\mathbb C}})\to{\mathscr E}'_{\1}(G), \quad & w\mapsto\dot\lambda(w)\delta_{\1}, \\
{\rm U}(\gg_{{\mathbb C}})\to{\mathscr E}'_{\1}(G),\quad & w\mapsto\dot\rho(w)\delta_{\1}
  \end{aligned}$$
are isomorphisms of complex unital associative algebras.
(See for instance Th.~1 in Sect.~10.4 of~\cite{Ki76}.)
These isomorphisms are related by the commutative diagram
$$\xymatrix{{\rm U}(\gg_{{\mathbb C}}) \ar[r]^{w\mapsto w^\perp} \ar[d]_{\dot\lambda(\cdot)\delta_{\1}}
      & {\rm U}(\gg_{{\mathbb C}})\ar[d]^{\dot\rho(\cdot)\delta_{\1}} \\
{\mathscr E}'_{\1}(G) \ar[r]^{u\mapsto u^\perp} & {\mathscr E}'_{\1}(G)
}$$
where the horizontal arrows stand for
the mappings introduced in Definitions \ref{univ_def} and~\ref{reg},
respectively.
From now on, \emph{we perform the identification ${\rm U}(\gg_{{\mathbb C}})\simeq{\mathscr E}'_{\1}(G)$ by means of the mapping
$w\mapsto\dot\rho(w)\delta_{\1}$}, by writing simply $w$ instead of $\dot\rho(w)\delta_{\1}$ whenever $w\in{\rm U}(\gg_{{\mathbb C}})$.
\qed
\end{remark}

\begin{proposition}\label{dirac_dec}
For every integer $m\ge1$ and every compact neighbourhood $K$ of $\1\in G$
there exist finitely many elements $u_1,\dots,u_N\in{\rm U}(\gg_{{\mathbb C}})$
and the functions $\phi_1,\dots,\phi_N\in{\mathscr C}^m_0(G)$ with
$\bigcup\limits_{j=1}^N{\rm supp}\,\phi_j\subseteq K$ such that
$\delta_{\1}=\sum\limits_{j=1}^N \phi_j\ast u_j$.
\end{proposition}

\begin{proof}
Use Lemme~2 in \cite{Ro76} or Lemma~2.3 in \cite{DDJP09};
see also the proof of Lemme~1.1 in~\cite{Ca76}.
\end{proof}

\subsection{Smooth vectors}

\begin{definition}\label{deriv_def}
\normalfont
The  \emph{smooth vectors} for
the representation $\pi\colon G\to{\mathscr B}({\mathscr Y})$
are the elements of the linear subspace of ${\mathscr Y}$ defined by
$${\mathscr Y}_\infty:=\{y\in{\mathscr Y}\mid\pi(\cdot)y\in{\mathscr E}(G,{\mathscr Y})\}. $$
The linear space ${\mathscr Y}_\infty$ will be endowed with the linear topology
which makes the linear injective map
$${\mathscr Y}_\infty\to{\mathscr E}(G,{\mathscr Y}),\quad y\mapsto\pi(\cdot)y $$
into a linear topological isomorphism onto its image.

For every distribution $u\in{\mathscr E}'(G)$ and
every smooth vector $y\in{\mathscr Y}_\infty$ we then define
$$\dot\pi(u)y:=\langle u\widehat{\otimes}{\rm id}_{{\mathscr Y}},\pi(\cdot)y\rangle\in{\mathscr Y}.$$
\qed
\end{definition}

\begin{proposition}\label{deriv_prop}
The following assertions hold:
\begin{enumerate}
\item\label{deriv_prop_item0}
The space of smooth vectors ${\mathscr Y}_\infty$ is a Fr\'echet space
and the inclusion map ${\mathscr Y}_\infty\hookrightarrow{\mathscr Y}$ is continuous.
\item\label{deriv_prop_item1}
The space ${\mathscr Y}_\infty$ is dense in ${\mathscr Y}$.
\item\label{deriv_prop_item2}
For every $u\in{\mathscr E}'(G)$ we have $\dot\pi(u){\mathscr Y}_\infty\subseteq{\mathscr Y}_\infty$.
\item\label{deriv_prop_item3}
The mapping $\dot\pi\colon{\mathscr E}'(G)\to{\rm End}\,({\mathscr Y}_\infty)$
is a homomorphism of unital associative algebras.
\item\label{deriv_prop_item4}
For every $X\in\gg$ and $y\in{\mathscr Y}_\infty$
we have $\dot\pi(X)y:=\frac{{\rm d}}{{\rm d} t}\Big\vert_{t=0}\pi(\exp_G(tX))y$.
\item\label{deriv_prop_item5}
For every $y\in{\mathscr Y}_\infty$
we have a smooth mapping $\pi(\cdot)y\colon G\to{\mathscr Y}_\infty$.
\end{enumerate}
\end{proposition}

\begin{proof}
See for instance \cite{Wa72} and Sect.~10.5 in \cite{Ki76}.
\end{proof}

\begin{notation}
\normalfont
We shall always denote by ${\mathscr Y}_{-\infty}$ the space of continuous antilinear functionals on
the Fr\'echet space ${\mathscr Y}_\infty$.
\qed
\end{notation}

\begin{proposition}\label{dm}
For every integer $m\ge1$ there exist finitely many functions $\phi_1,\dots,\phi_N\in{\mathscr C}^m_0(G)$
such that for every $y\in{\mathscr Y}_\infty$ there exist $y_1,\dots,y_N\in{\mathscr Y}$
satisfying the equality $y=\dot\pi(\phi_1)y_1+\cdots+\dot\pi(\phi_N)y_N$.
\end{proposition}

\begin{proof}
Use Proposition~\ref{dirac_dec} to get  $u_1,\dots,u_N\in{\rm U}(\gg_{{\mathbb C}})$
and $\phi_1,\dots,\phi_N\in{\mathscr C}^m_0(G)$ with
$\delta_{\1}=\sum\limits_{j=1}^N \phi_j\ast u_j$.
Then Proposition~\ref{deriv_prop} shows that
$$y=\dot\pi(\delta_{\1})y=\sum_{j=1}^N \dot\pi(\phi_j)\dot\pi(u_j)y
=\sum_{j=1}^N \dot\pi(\phi_j)y_j,$$
where we have denoted $y_j:=\dot\pi(u_j)y$ for $j=1,\dots,N$.
\end{proof}

\begin{remark}
\normalfont
As we already mentioned, the smooth vectors for representations of Lie groups
were discussed in detail in \cite{Wa72}.
Other important references in this connection are \cite{Ga47}, \cite{Ga60}, \cite{Ki76}, \cite{Ca76}, \cite{CGP77}, \cite{DM78}, \cite{RT87}, and \cite{CG90}.
\qed
\end{remark}

\section{Smooth operators for unitary representations}

We are going to discuss here the space of smooth operators for
a given representation of a Lie group.
The method of investigation was suggested in \cite{Pe94}
and relies on exhibiting
this space of operators as the space of smooth vectors
for a suitable representation (see Definition~\ref{suitable}).
The main result is recorded as Theorem~\ref{smooth_ideal}
and it is particularly significant in the case of unitary irreducible representations of nilpotent Lie groups (Corollary~\ref{smooth_ideal_cor}).

\begin{notation}
\normalfont
In this section we shall use the following notation:
\begin{itemize}
\item $G$ is a connected unimodular Lie group with the Lie algebra $\gg$;
\item ${\rm d} x$ denotes a fixed Haar measure on $G$;
\item ${\mathscr H}$ is a complex Hilbert space;
\item $\pi\colon G\to {\mathscr B}({\mathscr H})$ is a continuous unitary representation,
and ${\mathscr H}_\infty$ is the corresponding space of smooth vectors.
\end{itemize}
\qed
\end{notation}

The following notion of smooth operator
was singled out on page~349 in \cite{How77}
and then further developed in \cite{Pe94}.

\begin{definition}\label{smooth_oper}
\normalfont
The set ${\mathscr B}({\mathscr H})_\infty$ of \emph{smooth operators} for the representation~$\pi$
is defined as the set of all operators $T\in{\mathscr B}({\mathscr H})$ satisfying the following conditions:
\begin{enumerate}
\item\label{smooth_oper_item1}
We have $T({\mathscr H})+T^*({\mathscr H})\subseteq{\mathscr H}_\infty$.
\item\label{smooth_oper_item2}
For every $u\in{\rm U}(\gg_{{\mathbb C}})$ the operators
$\dot\pi(u)T$ and $\dot\pi(u)T^*$ are bounded on~${\mathscr H}$.
\end{enumerate}
\qed
\end{definition}

\begin{example}\label{smooth_oper_ex}
\normalfont
For every $x,y\in{\mathscr H}_\infty$ the rank-one operator $(\cdot\mid x)y$ belongs to
the space of smooth operators~${\mathscr B}({\mathscr H})_\infty$.
We shall see in Corollary~\ref{approximation} that
the linear span of these rank-one operators is dense in ${\mathscr B}({\mathscr H})_\infty$
provided that $G$ is a nilpotent Lie group and $\pi$ is an irreducible representation.
\qed
\end{example}

\begin{remark}\label{smooth_oper_rem}
\normalfont
It follows at once by Definition~\ref{smooth_oper} that ${\mathscr B}({\mathscr H})_\infty$ is an associative $*$-subalgebra
of ${\mathscr B}({\mathscr H})$.
\qed
\end{remark}

\begin{definition}\label{char_def}
\normalfont
We shall say that the representation $\pi$ \emph{has a smooth character} if
for every $\phi\in{\mathscr D}(G)$ we have $\pi(\phi)\in{\mathfrak S}_1({\mathscr H})$
and the linear mapping
$${\mathscr D}(G)\to{\mathfrak S}_1({\mathscr H}),\quad \phi\mapsto\dot\pi(\phi) $$
is continuous.
In this case we define the corresponding \emph{character} as
$$\chi_\pi\colon{\mathscr D}(G)\to{\mathbb C},\quad \chi_\pi(\phi):={\rm Tr}\,\pi(\phi).$$
Note that $\chi_\pi\in{\mathscr D}'(G)$.
\qed
\end{definition}

\begin{example}
\normalfont
Every unitary irreducible representation of a nilpotent Lie group has a smooth character;
see for instance Th.~2 in \S5 of Ch.~II, Part.II in \cite{Pu67}.
\qed
\end{example}

\begin{remark}\label{char_rem}
\normalfont
If the representation $\pi$ has a smooth character, then there exists a continuous seminorm $p(\cdot)$
on ${\mathscr D}(G)$ such that
$$(\forall \phi\in{\mathscr D}(G))\quad \Vert\dot\pi(\phi)\Vert_1\le p(\phi).$$
In view of the definition of the topology on ${\mathscr D}(G)$ and of
the fact that ${\mathscr D}(G)$ is dense in ${\mathscr C}^m_0(G)$ for every $m\ge1$,
it then easily follows that for every compact subset $K\subset G$
there exists an integer $m\ge1$
such that for every $\phi\in{\mathscr C}^m_0(G)\cap{\mathscr E}'_K(G)$ we have $\dot\pi(\phi)\in{\mathfrak S}_1({\mathscr H})$,
and moreover the mapping
$${\mathscr C}^m_0(G)\cap{\mathscr E}'_K(G)\to{\mathfrak S}_1({\mathscr H}),\quad \phi\mapsto\dot\pi(\phi) $$
is linear and continuous.
\qed
\end{remark}

\begin{definition}\label{adm}
\normalfont
An \emph{admissible ideal} is a non-trivial two-sided ideal ${\mathscr J}$ of ${\mathscr B}({\mathscr H})$
with the following properties:
\begin{enumerate}
\item\label{adm_item1}
The ideal ${\mathscr J}$ is endowed with a complete norm $\Vert\cdot\Vert_{{\mathscr J}}$ such that
for every $A,B\in{\mathscr B}({\mathscr H})$ and $T\in{\mathscr J}$ we have
$\Vert ATB\Vert_{{\mathscr J}}\le\Vert A\Vert \cdot\Vert T\Vert_{{\mathscr J}}\cdot\Vert B\Vert$
and $\Vert T^*\Vert_{{\mathscr J}}=\Vert T\Vert_{{\mathscr J}}$.
\item\label{adm_item2}
The ideal ${\mathscr F}({\mathscr H})$ of finite-rank operators is a dense subspace of ${\mathscr J}$.
\item\label{adm_item3}
For every $x,y\in{\mathscr H}$ we have
$\Vert(\cdot\mid x)y\Vert_{{\mathscr J}}=\Vert x\Vert\cdot\Vert y\Vert$.
\end{enumerate}
\qed
\end{definition}

\begin{example}
\normalfont
Every Schatten ideal ${\mathfrak S}_p({\mathscr H})$ with $1\le p\le\infty$ is an admissible ideal.
There exist many other examples of admissible ideals; see for instance \cite{GK69}.
\qed
\end{example}

\begin{remark}\label{so}
\normalfont
Let ${\mathscr J}$ be an admissible ideal.
By using condition~\eqref{adm_item2} in Definition~\ref{adm}
with $A={\rm id}_{{\mathscr H}}$ and $B=(\cdot\mid x)x$ for $x\in{\mathscr H}$,
and then taking into account
condition~\eqref{adm_item3}, it follows that
$\Vert Tx\Vert\le\Vert T\Vert_{{\mathscr J}}\cdot\Vert x\Vert$.
That is, for every $T\in{\mathscr J}$ we have $\Vert T\Vert\le\Vert T\Vert_{{\mathscr J}}$.

On the other hand, it follows at once by condition~\eqref{adm_item3} in Definition~\ref{adm} that for every $T\in{\mathscr F}({\mathscr H})$ we have
$\Vert T\Vert_{{\mathscr J}}\le\Vert T\Vert_1$.
Since ${\mathscr F}({\mathscr H})$ is dense in ${\mathfrak S}_1({\mathscr H})$,
we get
$$(\forall T\in{\mathfrak S}_1({\mathscr H}))\quad
\Vert T\Vert\le\Vert T\Vert_{{\mathscr J}}\le\Vert T\Vert_1. $$
In particular, we have ${\mathfrak S}_1({\mathscr H})\subseteq{\mathscr J}$.
\qed
\end{remark}

\begin{definition}\label{suitable}
\normalfont
For every admissible ideal ${\mathscr J}$ we define
a linear representation $\pi^{\otimes 2}_{{\mathscr J}}\colon G\times G\to{\mathscr B}({\mathscr J})$ by
$$\pi^{\otimes 2}_{{\mathscr J}}(g_1,g_2)T:=\pi(g_1)T\pi(g_2)^{-1} $$
for every $g_1,g_2\in G$ and $T\in{\mathscr J}$.
\qed
\end{definition}

\begin{lemma}
The representation $\pi^{\otimes 2}_{{\mathscr J}}\colon G\times G\to{\mathscr B}({\mathscr J})$
is continuous for every admissible ideal ${\mathscr J}$.
\end{lemma}

\begin{proof}
If $T\in{\mathscr F}({\mathscr H})$, then it is straightforward to check that
$\pi^{\otimes 2}_{{\mathscr J}}(\cdot)T\colon G\times G\to{\mathscr J}$ is a continuous mapping.

Now let $T\in{\mathscr J}$ arbitrary.
Since ${\mathscr J}$ is admissible, there exists a sequence $\{T_k\}_{k\ge1}$ in ${\mathscr F}({\mathscr H})$
such that $\lim\limits_{k\to\infty}\Vert T-T_k\Vert_{{\mathscr J}}=0$.
On the other hand, since $\pi$ is a unitary representation, it follows that
for $k=1,2,\dots$ and every $(g_1,g_2)\in G\times G$ we have
$$\Vert\pi^{\otimes 2}_{{\mathscr J}}(g_1,g_2)T-\pi^{\otimes 2}_{{\mathscr J}}(g_1,g_2)T_k\Vert_{{\mathscr J}}\le\Vert T-T_k\Vert_{{\mathscr J}}.$$
Therefore $\pi^{\otimes 2}_{{\mathscr J}}(\cdot)T\colon G\times G\to{\mathscr J}$ is the uniform limit on $G\times G$
of the sequence of continuous mappings $\pi^{\otimes 2}_{{\mathscr J}}(\cdot)T$,
hence it is in turn continuous.
\end{proof}

\begin{theorem}\label{smooth_ideal}
Let $\pi\colon G\to{\mathscr B}({\mathscr H})$ be a continuous unitary representation,
assume that ${\mathscr J}\subset{\mathscr B}({\mathscr H})$ is an admissible ideal,
and denote by ${\mathscr J}_\infty$ the space of smooth vectors
for the corresponding representation~$\pi^{\otimes 2}_{{\mathscr J}}$.
Then the following assertions hold:
\begin{enumerate}
\item\label{smooth_ideal_item1}
We have ${\mathscr J}_\infty\subseteq{\mathscr B}({\mathscr H})_\infty$.
\item\label{smooth_ideal_item2}
If the Fr\'echet space of smooth vectors ${\mathscr H}_\infty$ is nuclear,
then we have
$${\mathscr J}_\infty={\mathscr B}({\mathscr H})_\infty\subseteq{\mathfrak S}_1({\mathscr H}),$$
and the Fr\'echet space ${\mathscr J}_\infty$ does not depend on
the choice of the admissible ideal~${\mathscr J}$.
\end{enumerate}
\end{theorem}

\begin{proof}
\eqref{smooth_ideal_item1}
To prove the inclusion ${\mathscr J}_\infty\subseteq{\mathscr B}({\mathscr H})_\infty$,
let $T\in{\mathscr J}_\infty$ arbitrary,
hence the mapping
$$G\times G\to{\mathscr J},\quad
(g_1,g_2)\mapsto\pi^{\otimes 2}_{{\mathscr J}}(g_1,g_2)T=\pi(g_1)T\pi(g_2)^{-1}$$
is smooth.
In particular,
the mapping $\pi(\cdot)T\colon G\to{\mathscr J}$ is smooth.
On the other hand, it follows by Remark~\ref{so} that
for arbitrary $x\in{\mathscr H}$ we have a continuous linear mapping
${\mathscr J}\to{\mathscr H}$, $T\mapsto Tx$.
Hence the mapping $\pi(\cdot)Tx\colon G\to{\mathscr H}$ will be smooth as
a composition of two smooth mappings.
Thus for arbitrary $x\in{\mathscr H}$ we have $Tx\in{\mathscr H}_\infty$.
Moreover, since the operation of taking the Hilbert space adjoint
is (${\mathbb R}$-linear and) continuous on ${\mathscr J}$ by condition
\eqref{adm_item1} in Definition~\ref{adm},
it follows at once that $T^*\in{\mathscr J}_\infty$.
Hence by the above reasoning with $T$ replaced by $T^*$
we get $T^*x\in{\mathscr H}_\infty$ for arbitrary $x\in{\mathscr H}$.
Thus the operator $T$ satisfies condition~\eqref{smooth_oper_item1}
in Definition~\ref{smooth_oper}.
To check condition~\eqref{smooth_oper_item2} in the same definition
just note that since the mapping $\pi(\cdot)T\colon G\to{\mathscr J}$ is smooth,
it follows that for every $u\in{\rm U}(\gg_{{\mathbb C}})$ we have
$\dot{\pi}(u)T\in{\mathscr J}$, hence $\dot{\pi}(u)T\in{\mathscr B}({\mathscr H})$.
Since we have seen above that $T^*\in{\mathscr J}_\infty$,
it also follows that $\dot{\pi}(u)T^*\in{\mathscr B}({\mathscr H})$.
This completes the proof of the fact that $T\in{\mathscr B}({\mathscr H})_\infty$.

\eqref{smooth_ideal_item2}
If ${\mathscr H}_\infty$ is a nuclear space,
then the inclusion map ${\mathscr H}_\infty\hookrightarrow{\mathscr H}$ is a nuclear operator
(see Prop.~7.2 in Ch.~III of \cite{Sch66}).
Since condition~\eqref{smooth_oper_item1} in Definition~\ref{smooth_oper}
shows an arbitrary operator $T\in{\mathscr B}({\mathscr H})_\infty$
factorizes as
${\mathscr H}\mathop{\longrightarrow}\limits^T{\mathscr H}_\infty\hookrightarrow{\mathscr H}$,
it follows that $T\in{\mathfrak S}_1({\mathscr H})$
(see Cor.~2 to Prop.~7.2 in Ch.~III of \cite{Sch66}).
Thus, by taking into account the above Assertion~\eqref{smooth_ideal_item1},
we get
\begin{equation}\label{smooth_ideal_eq1}
{\mathscr J}_\infty\subseteq{\mathscr B}({\mathscr H})_\infty\subseteq{\mathfrak S}_1({\mathscr H}).
\end{equation}
To see that ${\mathscr J}_\infty$ does not depend on the choice of
the admissible ideal ${\mathscr J}$, we shall prove the equality of Fr\'echet spaces
\begin{equation}\label{smooth_ideal_eq2}
{\mathscr J}_\infty={\mathfrak S}_1({\mathscr H})_\infty,
\end{equation}
where the right-hand side denotes the space of smooth vectors for
the representation
$\pi^{\otimes 2}_{{\mathfrak S}_1({\mathscr H})}\colon G\times G\to{\mathscr B}({\mathfrak S}_1({\mathscr H}))$.
First recall from Remark~\ref{so} that
we have a continuous inclusion map ${\mathfrak S}_1({\mathscr H})\hookrightarrow{\mathscr J}$,
which clearly intertwines the representations
$\pi^{\otimes 2}_{{\mathfrak S}_1({\mathscr H})}$ and $\pi^{\otimes 2}_{{\mathscr J}}$.
It then easily follows by Definition~\ref{deriv_def} that
we have a continuous inclusion map for the corresponding spaces of smooth vectors ${\mathfrak S}_1({\mathscr H})_\infty\hookrightarrow{\mathscr J}_\infty$.
On the other hand, we have already proved that ${\mathscr J}_\infty\subseteq{\mathfrak S}_1({\mathscr H})$,
hence there exists the following  commutative diagram
$$\xymatrix{
 {\mathscr J}_\infty\ar[r] \ar[d] & {\mathfrak S}_1({\mathscr H})\ar[ld]   \\
  {\mathscr J}              }
$$
whose arrows stand for inclusion maps between Fr\'echet or Banach spaces.
The arrows that point to ${\mathscr J}$ are continuous inclusions
(by Remark~\ref{so} and Proposition~\ref{deriv_prop}\eqref{deriv_prop_item0}),
hence the closed graph theorem implies that
the inclusion map ${\mathscr J}_\infty\hookrightarrow{\mathfrak S}_1({\mathscr H})$ is continuous as well.
Since for arbitrary $T\in{\mathscr J}_\infty$
the mapping $G\times G\to{\mathscr J}_\infty$, $(g_1,g_2)\mapsto\pi(g_1)T\pi(g_2)^{-1}$
is smooth by Proposition~\ref{deriv_prop}\eqref{deriv_prop_item5},
it then follows that the mapping
$G\times G\to{\mathfrak S}_1({\mathscr H})$, $(g_1,g_2)\mapsto\pi(g_1)T\pi(g_2)^{-1}$
is also smooth, hence $T\in{\mathfrak S}_1({\mathscr H})_\infty$.
Thus ${\mathscr J}_\infty={\mathfrak S}_1({\mathscr H})_\infty$ as sets.
Since both sides of this equality are Fr\'echet spaces and we have already seen
that the inclusion map ${\mathfrak S}_1({\mathscr H})_\infty\hookrightarrow{\mathscr J}_\infty$
is continuous, it follows by the open mapping theorem that we have
the equality of Fr\'echet spaces in \eqref{smooth_ideal_eq2}.

Finally, note that for arbitrary $T\in{\mathscr B}({\mathscr H})_\infty$ and
every $u\in{\rm U}(\gg_{{\mathbb C}})$ we have $\dot\pi(u)T,\dot\pi(u)T^*\in{\mathscr B}({\mathscr H})_\infty$
(see Definition~\ref{smooth_oper}).
On the other hand,
we have proved above that ${\mathscr B}({\mathscr H})_\infty\subseteq{\mathfrak S}_1({\mathscr H})$,
hence $\dot\pi(u)T,\dot\pi(u)T^*\in{\mathfrak S}_1({\mathscr H})$ for all $u\in{\rm U}(\gg_{{\mathbb C}})$,
and this implies that $T\in{\mathfrak S}_1({\mathscr H})_\infty$.
Thus ${\mathscr B}({\mathscr H})_\infty\subseteq{\mathfrak S}_1({\mathscr H})_\infty$,
and then by using Assertion~\eqref{smooth_ideal_item1}
with ${\mathscr J}={\mathfrak S}_1({\mathscr H})$ we get ${\mathscr B}({\mathscr H})_\infty={\mathfrak S}_1({\mathscr H})_\infty$.
Now by \eqref{smooth_ideal_eq2} and \eqref{smooth_ideal_eq1}
we get ${\mathscr J}_\infty={\mathscr B}({\mathscr H})_\infty\subseteq{\mathfrak S}_1({\mathscr H})$,
and this completes the proof.
\end{proof}

\begin{corollary}\label{smooth_ideal_cor}
Assume that $G$ is a nilpotent Lie group and
$\pi\colon G\to{\mathscr B}({\mathscr H})$ is a unitary irreducible representation.
If ${\mathscr J}\subset{\mathscr B}({\mathscr H})$ is an admissible ideal,
and we denote by ${\mathscr J}_\infty$ the space of smooth vectors
for the corresponding representation~$\pi^{\otimes 2}_{{\mathscr J}}$,
Then the following assertions hold:
\begin{enumerate}
\item\label{smooth_ideal_cor_item1}
The Fr\'echet space of smooth vectors ${\mathscr H}_\infty$ is nuclear.
\item\label{smooth_ideal_cor_item2}
We have
${\mathscr J}_\infty={\mathscr B}({\mathscr H})_\infty\subseteq{\mathfrak S}_1({\mathscr H})$,
and the Fr\'echet space ${\mathscr J}_\infty$ does not depend on
the choice of the admissible ideal~${\mathscr J}$.
\item\label{smooth_ideal_cor_item3}
The space of smooth operators ${\mathscr B}({\mathscr H})_\infty$ has the natural structure of
a nuclear Fr\'echet space.
\end{enumerate}
\end{corollary}

\begin{proof}
Since the representation $\pi$ is irreducible,
there exists a linear topological isomorphism from
the Fr\'echet space ${\mathscr H}_\infty$ onto
the Schwartz space of rapidly decreasing functions ${\mathscr S}({\mathbb R}^{d/2})$,
where $d$ is equal to the dimension of the coadjoint orbit
of $G$ corresponding to the representation~$\pi$.
(This follows by Th.~1 in \S 5 of Ch.~II, Part.~II in \cite{Pu67};
see also the Cor.~to Th.~3.1 in \cite{CGP77}, or \cite{CG90}.)
On the other hand, it is well known that
the Schwartz space ${\mathscr S}({\mathbb R}^{d/2})$
is nuclear (see for instance Ex.~5 in \S 8 of Ch.~III in \cite{Sch66}).
Therefore the Fr\'echet space ${\mathscr H}_\infty$ is nuclear,
and then Theorem~\ref{smooth_ideal} applies.

Finally, by using Assertion~\eqref{smooth_ideal_cor_item2}
when ${\mathscr J}={\mathfrak S}_2({\mathscr H})$ (the Hilbert-Schmidt ideal), it follows that
${\mathscr B}({\mathscr H})_\infty$ is equal to the space of smooth vectors
for the unitary representation $\pi^{\otimes 2}_{{\mathfrak S}_2({\mathscr H})}$,
hence it is a Fr\'echet space in a natural way.
On the other hand, the representation
$\pi^{\otimes 2}_{{\mathfrak S}_2({\mathscr H})}$
is irreducible since so is~$\pi$.
(See for instance the proof of Lemma~2.18(a) in \cite{BB09c}.)
Now the fact that ${\mathscr B}({\mathscr H})_\infty$ is nuclear follows
by the above Assertion~\eqref{smooth_ideal_cor_item1} applied
for the unitary irreducible
representation~$\pi^{\otimes 2}_{{\mathfrak S}_2({\mathscr H})}\colon G\times G\to{\mathscr B}({\mathfrak S}_2({\mathscr H}))$.
\end{proof}

\begin{corollary}\label{regularization}
If $G$ is a nilpotent Lie group and
$\pi\colon G\to{\mathscr B}({\mathscr H})$ is a unitary irreducible representation,
then the operators in ${\mathscr B}({\mathscr H})_\infty$ are precisely the regularizing operators.
That is, $A\in{\mathscr B}({\mathscr H})_\infty$
if and only if $A$ extends to a continuous linear map $A\colon{\mathscr H}_{-\infty}\to{\mathscr H}_\infty$,
so that the diagram
$$\xymatrix{
{\mathscr H}_{-\infty} \ar[r]^{A} & {\mathscr H}_\infty \ar@{^{(}->}[d] \\
{\mathscr H} \ar[r]^A \ar@{^{(}->}[u] & {\mathscr H}
} $$
is commutative
\end{corollary}

\begin{proof}
By the closed graph theorem, it is enough to prove that
if $A\in{\mathscr B}({\mathscr H})_\infty$  and $f\in{\mathscr H}_{-\infty}$,
then $Af\in{\mathscr H}_\infty$,
in the sense that there exists a smooth vector
denoted $Af$ such that for every $\phi\in{\mathscr H}_\infty$ we have
$(f\mid A^*\phi)=(Af\mid\phi)$.
This is a consequence of the above Proposition~\ref{deriv_prop}\eqref{deriv_prop_item2}, Corollary\ref{smooth_ideal_cor}, and  Th.~1.3(b) in~\cite{Ca76}.

Conversely, it follows by Definition~\ref{smooth_oper}
that the restriction to ${\mathscr H}$ of every continuous linear map
$A\colon{\mathscr H}_{-\infty}\to{\mathscr H}_\infty$
belongs to ${\mathscr B}({\mathscr H})_\infty$.
\end{proof}

\begin{corollary}\label{approximation}
Assume that $G$ is a nilpotent Lie group and
$\pi\colon G\to{\mathscr B}({\mathscr H})$ is a unitary irreducible representation.
The linear space spanned by the operators $(\cdot\mid x)y$ with $x,y\in{\mathscr H}_\infty$
is dense in ${\mathscr B}({\mathscr H})_\infty$.
\end{corollary}

\begin{proof}
Let $T\in{\mathscr B}({\mathscr H})_\infty$ arbitrary.
Then Corollary~\ref{smooth_ideal_cor}\eqref{smooth_ideal_cor_item2}
shows that $T$ is a smooth vector for the representation
$\pi^{\otimes 2}_{{\mathfrak S}_2({\mathscr H})}\colon G\times G\to{\mathscr B}({\mathfrak S}_2({\mathscr H}))$.
It follows by Proposition~\ref{dm} that
there exist finitely many functions
$\phi_1,\dots,\phi_N\in{\mathscr C}^m_0(G\times G)$
and the operators $Y_1,\dots,Y_N\in{\mathfrak S}_2({\mathscr H})$ such that
$$T=\pi^{\otimes 2}_{{\mathfrak S}_2({\mathscr H})}(\phi_1)Y_1+\cdots+\pi^{\otimes 2}_{{\mathfrak S}_2({\mathscr H})}(\phi_N)Y_N.$$
Since ${\mathscr D}(G\times G)$ is dense in ${\mathscr C}^m_0(G\times G)$
and ${\mathscr D}(G)\otimes{\mathscr D}(G)$ is dense in ${\mathscr D}(G\times G)$,
it follows by Proposition~\ref{deriv_prop}\eqref{deriv_prop_item3}
that $T$ can be approximated in ${\mathscr B}({\mathscr H})_\infty$ by finite linear combinations of operators of the form
$$\pi^{\otimes 2}_{{\mathfrak S}_2({\mathscr H})}(\psi_1\otimes\psi_2)Y=\pi(\psi_1)Y\pi(\psi_2^\perp)$$
with $\psi_1,\psi_2\in{\mathscr D}(G)$ and $Y\in{\mathfrak S}_2({\mathscr H})$.
On the other hand, such an $Y$ can be approximated in ${\mathfrak S}_2({\mathscr H})$
by finite linear combinations of operators $(\cdot\mid v_2)v_1$
with $v_1,v_2\in{\mathscr H}$.
The corollary now follows by noticing that
$$\pi(\psi_1)((\cdot\mid v_2)v_1)\pi(\psi_2^\perp)=(\cdot\mid \pi(\bar\psi_2^\perp)v_2)\pi(\psi_1)v_1$$
and recalling that $\pi(\psi)v\in{\mathscr H}_\infty$ when $\psi\in{\mathscr D}(G)$ and $v\in{\mathscr H}$
(\cite{Ga47}).
\end{proof}

\begin{remark}\label{smooth_ideal_rem}
\normalfont
Let ${\mathscr J}\subset{\mathscr B}({\mathscr H})$ be any admissible ideal.
If the representation $\pi$ has a smooth character,
then the corresponding space of smooth vectors ${\mathscr H}_\infty$ is nuclear
according to Th.~2.6 in \cite{Ca76},
hence the above Theorem~\ref{smooth_ideal}\eqref{smooth_ideal_item2} applies.

The inclusion ${\mathscr J}_\infty\subseteq{\mathfrak S}_1({\mathscr H})$
can be alternatively proved in this case as follows.
Let $T\in{\mathscr J}_\infty$ arbitrary.
Since the representation $\pi$ has a smooth character,
we have a continuous linear mapping
$${\mathscr D}(G)\to{\mathfrak S}_1({\mathscr H}),\quad \phi\mapsto\dot\pi(\phi).$$
On the other hand, note that for every $\phi_1,\phi_2\in{\mathscr D}(G)$ we have
$\pi^{\otimes 2}_{{\mathscr J}}(\phi_1\otimes\phi_2)T=\dot\pi(\phi_1)T\dot\pi(\phi_2^\perp)$.
Hence for arbitrary $Y\in{\mathscr J}$ we get a jointly continuous trilinear mapping
$${\mathscr D}(G)\times{\mathscr D}(G)\times{\mathscr J}\to{\mathfrak S}_1({\mathscr H}),\quad
(\phi_1,\phi_2,Y)\mapsto\pi^{\otimes 2}_{{\mathscr J}}(\phi_1\otimes\phi_2)Y, $$
which extends to a jointly continuous bilinear mapping
$${\mathscr D}(G\times G)\times{\mathscr J}\to{\mathfrak S}_1({\mathscr H}), \quad
(\phi,Y)\mapsto\pi^{\otimes 2}_{{\mathscr J}}(\phi)Y.$$
(See also Remark~\ref{tens}.)
By an argument similar to the one of Remark~\ref{char_rem},
we can further extend the above mapping to a continuous bilinear mapping
\begin{equation}\label{smooth_ideal_rem_eq1}
({\mathscr C}^m_0(G\times G)\cap{\mathscr E}'_K(G\times G))\times{\mathscr J}\to{\mathfrak S}_1({\mathscr H}), \quad
(\phi,Y)\mapsto\pi^{\otimes 2}_{{\mathscr J}}(\phi)Y
\end{equation}
for a suitable compact neighborhood $K$ of
$(\1,\1)\in G\times G$ and a suitably large integer $m\ge1$.
On the other hand, since $T\in{\mathscr J}_\infty$,
it follows by Proposition~\ref{dm} that
there exist finitely many functions
$\phi_1,\dots,\phi_N\in{\mathscr C}^m_0(G\times G)\cap{\mathscr E}'_K(G\times G)$
and the operators $Y_1,\dots,Y_N\in{\mathscr J}$ such that
$T=\pi^{\otimes 2}_{{\mathscr J}}(\phi_1)Y_1+\cdots+\pi^{\otimes 2}_{{\mathscr J}}(\phi_N)Y_N$,
hence by \eqref{smooth_ideal_rem_eq1} we get $T\in{\mathfrak S}_1({\mathscr H})$.
\qed
\end{remark}

\section{Weyl-Pedersen calculus}

In the present section we provide a brief discussion
of the remarkable Weyl correspondence constructed in \cite{Pe94}
and we shall also describe some complementary results
which were recently obtained in \cite{BB09c}.

\subsection{Preduals for coadjoint orbits}
This subsection records some properties of the coadjoint orbits
of nilpotent Lie groups which play a crucial role for
the construction of the Weyl-Pedersen calculus.

\begin{setting}\label{predual_sett}
\normalfont
We shall use the following notation:
\begin{enumerate}
 \item Let $G$ be a connected, simply connected, nilpotent Lie group with Lie algebra~$\gg$.
 Then the exponential map $\exp_G\colon\gg\to G$ is a diffeomorphism
 with the inverse denoted by $\log_G\colon G\to\gg$.
 \item We denote by $\gg^*$ the linear dual space to $\gg$ and
  by $\hake{\cdot,\cdot}\colon\gg^*\times\gg\to{{\mathbb R}}$ the natural duality pairing.
 \item Let $\xi_0\in\gg^*$ with the corresponding coadjoint orbit ${\mathscr O}:={\rm Ad}_G^*(G)\xi_0\subseteq\gg^*$.
 \item Let $\pi\colon G\to{\mathscr B}({\mathscr H})$ be any unitary irreducible representations
associated with the coadjoint orbit ${\mathscr O}$ by Kirillov's theorem (\cite{Ki62}).
 \item The \emph{isotropy group} at $\xi_0$ is $G_{\xi_0}:=\{g\in G\mid{\rm Ad}_G^*(g)\xi_0=\xi_0\}$
 with the corresponding \emph{isotropy Lie algebra} $\gg_{\xi_0}=\{X\in\gg\mid\xi_0\circ{\rm ad}_{\gg}X=0\}$.
 If we denote the \emph{center} of $\gg$ by ${\mathfrak z}:=\{X\in\gg\mid[X,\gg]=\{0\}\}$,
 then it is clear that ${\mathfrak z}\subseteq\gg_{\xi_0}$.
 \item Let $n:=\dim\gg$ and fix a sequence of ideals in $\gg$,
$$\{0\}=\gg_0\subset\gg_1\subset\cdots\subset\gg_n=\gg$$
such that $\dim(\gg_j/\gg_{j-1})=1$ and $[\gg,\gg_j]\subseteq\gg_{j-1}$
for $j=1,\dots,n$.
 \item Pick any $X_j\in\gg_j\setminus\gg_{j-1}$ for $j=1,\dots,n$,
so that the set $\{X_1,\dots,X_n\}$ will be a \emph{Jordan-H\"older basis} in~$\gg$.
\end{enumerate}
\qed
\end{setting}

\begin{definition}\label{jump_def}
\normalfont
 Consider the set of \emph{jump indices} of the coadjoint orbit ${\mathscr O}$
with respect to the aforementioned Jordan-H\"older basis $\{X_1,\dots,X_n\}\subset\gg$,
$$e:=\{j\in\{1,\dots,n\}\mid \gg_j\not\subseteq\gg_{j-1}+\gg_{\xi_0}\}
=\{j\in\{1,\dots,n\}\mid X_j\not\in\gg_{j-1}+\gg_{\xi_0}\}$$
and then define the corresponding \emph{predual of the coadjoint orbit}~${\mathscr O}$,
$$\gg_e:={\rm span}\,\{X_j\mid j\in e\}\subseteq\gg.$$
We note the direct sum decomposition $\gg=\gg_{\xi_0}\dotplus\gg_e$.
\qed
\end{definition}

\begin{remark}\label{jump_remark}
\normalfont
Let $\{\xi_1,\dots,\xi_n\}\subset\gg^*$ be the dual basis for $\{X_1,\dots,X_n\}\subset\gg$.
Then the coadjoint orbit ${\mathscr O}$ can be described in terms of the jump indices
mentioned in Definition~\ref{jump_def}.
More specifically, if we denote
$$\gg^*_{{\mathscr O}}:={\rm span}\,\{\xi_j\mid j\in e\}
\quad\text{ and }\quad\gg^\perp_{{\mathscr O}}:={\rm span}\,\{\xi_j\mid j\not\in e\},$$
then the coadjoint orbit
${\mathscr O}\subset\gg^*\simeq\gg_e^*\times\gg_e^\perp$
is the graph of a certain \emph{polynomial mapping} $\gg_e^*\to\gg_e^\perp$.
This leads to the following pieces of information on ${\mathscr O}$:
\begin{enumerate}
 \item $\dim{\mathscr O}=\dim\gg_e={\rm card}\, e=:d$;
 \item if we let $j_1<\cdots<j_d$ such that $e=\{j_1,\dots,j_d\}$,
then the mapping
$${\mathscr O}\to{{\mathbb R}}^d,\quad \xi\to(\hake{\xi,X_{j_1}},\dots,\hake{\xi,X_{j_d}}) $$
is a global chart which takes the Liouville measure of ${\mathscr O}$
to a Lebesgue measure on~${{\mathbb R}}^d$.
\end{enumerate}
We define the Fourier transform ${\mathscr S}({\mathscr O})\to{\mathscr S}(\gg_e)$ by
$$(\forall X\in\gg_e)\quad \widehat{a}(X)=\int\limits_{{\mathscr O}}{\rm e}^{-{\rm i}\hake{\xi,X}}a(\xi){\rm d}\xi  $$
for every $a\in{\mathscr S}({\mathscr O})$, where ${\rm d}\xi$ stands for a Liouville measure on ${\mathscr O}$.
This Fourier transform is invertible.
The Lebesgue measure on $\gg_e$ can be normalized
such that the Fourier transform extends to a unitary operator
$$L^2({\mathscr O})\to L^2(\gg_e),\quad a\mapsto\widehat{a},$$
and its inverse is defined by the usual formula.
We shall always consider the predual $\gg_e$ endowed with
this normalized measure.
(See for instance Lemma~1.6.1 in \cite{Pe89} and Lemma~4.1.1 in \cite{Pe94}
for more details and proofs for the above assertions.)
\qed
\end{remark}

\begin{remark}
\normalfont
Some basic references for the geometry of coadjoint orbits
of nilpotent Lie groups include \cite{Pu67}, \cite{Pe84}, \cite{Pe88}, \cite{Pe89},
and \cite{CG90}; see also \cite{BB09d}.
\qed
\end{remark}

\subsection{Weyl-Pedersen calculus and Moyal identities}
We begin this subsection by the general construction of
a Weyl correspondence due to \cite{Pe94}.

\begin{definition}\label{calc_def}
\normalfont
The \emph{Weyl-Pedersen calculus} ${\rm Op}^\pi(\cdot)$ for the unitary representation~$\pi$ is defined
for every $a\in{\mathscr S}({\mathscr O})$ by
$${\rm Op}^\pi(a)=\int\limits_{\gg_e}\widehat{a}(X)\pi(\exp_GX){\rm d} X\in{\mathscr B}({\mathscr H}). $$
We call ${\rm Op}^\pi(a)$ is the \emph{pseudo-differential operator} with
the \emph{symbol} $a\in{\mathscr S}({\mathscr O})$.
\qed
\end{definition}

\begin{theorem}\label{pedersen}
The Weyl-Pedersen calculus has the following properties:
\begin{enumerate}
\item\label{pedersen_item1} For every symbol $a\in{\mathscr S}({\mathscr O})$ we have ${\rm Op}^\pi(a)\in{\mathscr B}({\mathscr H})_\infty$
and the mapping ${\mathscr S}({\mathscr O})\to{\mathscr B}({\mathscr H})_\infty$, $a\mapsto{\rm Op}^\pi(a)$ is
a linear topological isomorphism.
\item\label{pedersen_item1bis}
For every $T\in{\mathscr B}({\mathscr H})_\infty$ we have $T={\rm Op}^\pi(a)$,
where $a\in{\mathscr S}({\mathscr O})$ satisfies the condition $\widehat{a}(X)={\rm Tr}\,(\pi(\exp_G X)^{-1}T)$ for every $X\in\gg_e$.
\item\label{pedersen_item2} For every $a,b\in{\mathscr S}({\mathscr O})$ we have
\begin{enumerate}
\item\label{pedersen_item2a} ${\rm Op}^\pi(\bar{a})={\rm Op}^\pi(a)^*$;
\item\label{pedersen_item2b} ${\rm Tr}\,({\rm Op}^\pi(a))=\int\limits_{{\mathscr O}}a(\xi){\rm d}\xi$;
\item\label{pedersen_item2c} ${\rm Tr}\,({\rm Op}^\pi(a){\rm Op}^\pi(b))=\int\limits_{{\mathscr O}}a(\xi)b(\xi){\rm d}\xi$;
\item\label{pedersen_item2d} ${\rm Tr}\,({\rm Op}^\pi(a){\rm Op}^\pi(b)^*)=\int\limits_{{\mathscr O}}a(\xi)\overline{b(\xi)}{\rm d}\xi$.
\end{enumerate}
\end{enumerate}
\end{theorem}

\begin{proof}
See Th.~4.1.4 and Th.~2.2.7 in \cite{Pe94}.
\end{proof}

\begin{definition}\label{moyal}
\normalfont
Recall from Remark~\ref{smooth_oper_rem} that ${\mathscr B}({\mathscr H})_\infty$ is an involutive associative subalgebra of ${\mathscr B}({\mathscr H})$.
It then follows by Theorem~\ref{pedersen}\eqref{pedersen_item1}
that there exists an uniquely defined bilinear associative \emph{Moyal product}
$${\mathscr S}({\mathscr O})\times{\mathscr S}({\mathscr O})\to{\mathscr S}({\mathscr O}),\quad (a,b)\mapsto a\#^\pi b $$
such that
$$(\forall a,b\in{\mathscr S}({\mathscr O}))\quad {\rm Op}^\pi(a\#^\pi b)={\rm Op}^\pi(a){\rm Op}^\pi(b). $$
Thus ${\mathscr S}({\mathscr O})$ is made into an involutive associative algebra
such that the mapping ${\mathscr S}({\mathscr O})\to{\mathscr B}({\mathscr H})_\infty$, $a\mapsto{\rm Op}^\pi(a)$
is an algebra isomorphism.
\qed
\end{definition}

\begin{notation}\label{distrib_vect}
\normalfont
Recall that ${\mathscr H}_{-\infty}$ is the space of continuous antilinear functionals on ${\mathscr H}_\infty$
and the corresponding pairing will be denoted by
$(\cdot\mid\cdot)\colon{\mathscr H}_{-\infty}\times{\mathscr H}_\infty\to{\mathbb C}$.
just as the scalar product in ${\mathscr H}$, since they agree on ${\mathscr H}_\infty\times{\mathscr H}_{\infty}$
if we think of the natural inclusions ${\mathscr H}_\infty\hookrightarrow{\mathscr H}\hookrightarrow{\mathscr H}_{-\infty}$.
(See for instance \cite{Ca76} for more details.)
\qed
\end{notation}

\begin{definition}\label{amb}
\normalfont
If $f\in{\mathscr H}_{-\infty}$ and $\phi\in{\mathscr H}_\infty$, or $f,\phi\in{\mathscr H}$,
then we define the corresponding \emph{ambiguity function}
$${\mathscr A}(f,\phi)={\mathscr A}_\phi f\colon\gg_e\to{\mathbb C},\quad ({\mathscr A}_\phi f)(X)=(f\mid\pi(\exp_G X)\phi).$$
For $\phi\in{\mathscr H}_{-\infty}$ and $f\in{\mathscr H}_\infty$ we also define
$({\mathscr A}_\phi f)(X)=\overline{(\phi\mid\pi(\exp_G(-X))f)}$ whenever $X\in\gg_e$.

It follows by Proposition~\ref{orthog}\eqref{orthog_item1} below that if $f,\phi\in{\mathscr H}$,
then ${\mathscr A}_\phi f\in L^2(\gg_e)$,
so we can use the aforementioned Fourier transform to define
the corresponding \emph{cross-Wigner distribution} ${\mathscr W}(f,\phi)\in L^2({\mathscr O})$
such that
$\widehat{{\mathscr W}(f,\phi)}:={\mathscr A}_\phi f$.
\qed
\end{definition}

The second equality in Proposition~\ref{orthog}\eqref{orthog_item1} below
could be referred to as the \emph{Moyal identity} since
that classical identity (see for instance \cite{Gr01})
is recovered
in the special case when $G$ is a simply connected Heisenberg group.

\begin{proposition}\label{orthog}
 The following assertions hold:
\begin{enumerate}
 \item\label{orthog_item1}
If $\phi\in{\mathscr H}$, then ${\mathscr A}_\phi f\in L^2(\gg_e)$.
We have
\begin{equation}\label{orthog_eq1}
\begin{aligned}
({{\mathscr A}}_{\phi_1}f_1\mid {{\mathscr A}}_{\phi_2}f_2)_{L^2(\gg_e)}
&=(f_1\mid f_2)_{{\mathscr H}}\cdot(\phi_2\mid\phi_1)_{{\mathscr H}} \\
&=({{\mathscr W}}(f_1,\phi_1)\mid {{\mathscr W}}(f_2,\phi_2))_{L^2({\mathscr O})}
\end{aligned}
\end{equation}
for arbitrary $\phi_1,\phi_2,f_1,f_2\in{\mathscr H}$.
\item\label{orthog_item2}
If $\phi_0\in{\mathscr H}$ with $\Vert\phi_0\Vert=1$,
then the operator ${{\mathscr A}}_{\phi_0}\colon{\mathscr H}\to L^2(\gg_e)$, $f\mapsto {{\mathscr A}}_{\phi_0} f$,
is an isometry and we have
\begin{equation*}
\int\limits_{\gg_e}({{\mathscr A}}_{\phi_0}f)(X)\cdot\pi(\exp_G X)\phi\,{\rm d} X
=(\phi\mid\phi_0)f
\end{equation*}
for every $\phi\in{\mathscr H}_\infty$ and $f\in{\mathscr H}$.
In particular,
\begin{equation*}
\int\limits_{\gg_e}({{\mathscr A}}_{\phi_0} f)(X)\cdot\pi(\exp_G X)\phi_0\,{\rm d} X=f
\end{equation*}
for arbitrary $f\in{\mathscr H}$.
\end{enumerate}
\end{proposition}

\begin{proof}
See \cite{BB09c}.
\end{proof}

\begin{corollary}\label{pseudo_prop}
The following assertions hold:
\begin{enumerate}
\item\label{pseudo_prop_item2}
For each $a\in{\mathscr S}({\mathscr O})$ we have
$$
({\rm Op}^\pi(a)\phi\mid f)_{{\mathscr H}}=(\widehat{a}\mid{\mathscr A}_\phi f)_{L^2(\gg_e)}
=(a\mid{\mathscr W}(f,\phi))_{L^2({\mathscr O})}$$
whenever $\phi,f\in{\mathscr H}$.
Similar equalities hold if $a\in{\mathscr S}'({\mathscr O})$ and $\phi,f\in{\mathscr H}_\infty$.
\item\label{pseudo_prop_item3}
If $\phi_1,\phi_2\in{\mathscr H}_\infty$ and $a:={\mathscr W}(\phi_1,\phi_2)\in{\mathscr S}({\mathscr O})$,
then ${\rm Op}^\pi(a)$ is a rank-one operator, namely
${\rm Op}^\pi(a)=(\cdot \mid\phi_2) \phi_1$.
\end{enumerate}
\end{corollary}

\begin{proof}
See \cite{BB09c}.
\end{proof}

Assertion~\eqref{orthog_distrib_item3} in the following corollary
in the special case of square-integrable representations
reduces to a theorem of \cite{Co84} and \cite{CM96}.
One thus recovers Th.~2.3 in \cite{GZ01}
in the case of the Schr\"odinger representation
of the Heisenberg group.

\begin{corollary}\label{orthog_distrib}
If $\phi_0\in{\mathscr H}_\infty$ with $\Vert\phi_0\Vert=1$,
then the following assertions hold:
\begin{enumerate}
\item\label{orthog_distrib_item1}
For every $f\in{\mathscr H}_{-\infty}$ we have
\begin{equation}\label{orthog_distrib_eq1}
\int\limits_{\gg_e}({{\mathscr A}}_{\phi_0} f)(X)\cdot\pi(\exp_G X)\phi_0\,{\rm d} X=f
\end{equation}
where the integral is convergent in the weak$^*$-topology of ${\mathscr H}_{-\infty}$.
\item\label{orthog_distrib_item2}
If $f\in{\mathscr H}_\infty$, then the above integral converges in
the Fr\'echet topology of ${\mathscr H}_\infty$.
\item\label{orthog_distrib_item3}
If $f\in{\mathscr H}_{-\infty}$, then we have
$f\in{\mathscr H}_\infty$ if and only if ${\mathscr A}_{\phi_0}f\in{\mathscr S}(\gg_e)$.
\end{enumerate}
\end{corollary}

\begin{proof}
See \cite{BB09c}.
\end{proof}

\begin{remark}\label{calc_dual}
\normalfont
Let ${\mathscr B}({\mathscr H})_\infty^*$ be the topological dual of the Fr\'echet space ${\mathscr B}({\mathscr H})_\infty$
and denote by $\hake{\cdot,\cdot}$ either of the duality pairings
$${\mathscr B}({\mathscr H})_\infty^*\times{\mathscr B}({\mathscr H})_\infty\to{\mathbb C}
\text{ and }{\mathscr S}'({\mathscr O})\times{\mathscr S}({\mathscr O})\to{\mathbb C}.$$
Then for every tempered distribution $a\in{\mathscr S}'({\mathscr O})$ we can use Theorem~\ref{pedersen}\eqref{pedersen_item1}
to define
${\rm Op}^\pi(a)\in{\mathscr B}({\mathscr H})_\infty^*$ such that
$$(\forall b\in{\mathscr S}({\mathscr O}))\quad \hake{{\rm Op}^\pi(a),{\rm Op}^\pi(b)}=\hake{a,b} $$
Just as in Definition~\ref{calc_def} we call ${\rm Op}^\pi(a)$ the \emph{pseudo-differential operator} with
the \emph{symbol} $a\in{\mathscr S}'({\mathscr O})$.
Note that if actually $a\in{\mathscr S}({\mathscr O})$,
then the present notation agrees with Definition~\ref{calc_def}
because of Theorem~\ref{pedersen}\eqref{pedersen_item2c}.

The continuity properties of the above pseudo-differential operators
can be investigated by using modulation spaces of symbols; see \cite{BB09c} for details.
Specifically, one can introduce modulation spaces $M^{r,s}_\phi(\pi)$
for every unitary irreducible representation
$\pi\colon G\to{\mathscr B}({\mathscr H})$.
We always have ${\mathscr H}_\infty\subseteq M^{r,s}_\phi(\pi)$
and $M^{2,2}_\phi(\pi)={\mathscr H}$.
There exists a natural representation
$\pi^{\#}\colon G\ltimes G\to{\mathscr B}(L^2({\mathscr O}))$
such that for suitable $\Phi\in{\mathscr S}({\mathscr O})\setminus\{0\}$,
the Weyl calculus ${\rm Op}^\pi(\cdot)$ defines a continuous linear mapping
from the modulation space $M^{\infty,1}_\Phi(\pi^{\#})$ into the space of \emph{bounded} linear operators
on~${\mathscr H}$.
One of the main theorems of \cite{GH99} is recovered in the special case when $\pi$ is the Schr\"odinger representation
of the $(2n+1)$-dimensional Heisenberg group.
Some new results related to this circle of ideas
will be established in Section~\ref{Sect5} below.
\qed
\end{remark}

\begin{remark}[\cite{BB09c}]\label{wig_ext}
\normalfont
We can define the cross-Wigner distribution ${\mathscr W}(f_1,f_2)\in{\mathscr S}'({\mathscr O})$
for arbitrary $f_1,f_2\in{\mathscr H}_{-\infty}$ as follows.
By using Corollary~\ref{regularization} we can define for $f_1,f_2\in{\mathscr H}_{-\infty}$
the continuous antilinear functional
$$T_{f_1,f_2}\colon{\mathscr B}({\mathscr H})_\infty\to{\mathbb C},\quad
T_{f_1,f_2}(A):=(f_1\mid Af_2).$$
That is, $T_{f_1,f_2}\in{\mathscr B}({\mathscr H})_\infty^*$,
and then Th.~4.1.4(5) in \cite{Pe94} shows that
there exists a unique distribution $a_{f_1,f_2}\in{\mathscr S}'({\mathscr O})$
such that ${\rm Op}^\pi(a_{f_1,f_2})=T_{f_1,f_2}$.
Now define
$${\mathscr W}(f_1,f_2):=a_{f_1,f_2}.$$
We can consider the rank-one operator
$S_{f_1,f_2}:=(\cdot\mid f_2)f_1\colon {\mathscr H}_\infty\to{\mathscr H}_{-\infty}$ and
for arbitrary $A\in{\mathscr B}({\mathscr H})_\infty$ thought of as
a continuous linear map $A\colon{\mathscr H}_{-\infty}\to{\mathscr H}_\infty$ as above
we have
$${\rm Tr}\,(S_{f_1,f_2}A)=(f_1\mid Af_2)=T_{f_1,f_2}(A).$$
Thus the trace duality pairing allows us to identify
the functional $T_{f_1,f_2}\in{\mathscr B}({\mathscr H})_\infty^*$
with the rank-one operator $(\cdot\mid f_2)f_1$,
and then we can write
\begin{equation}\label{wig_ext_eq1}
(\forall f_1,f_2\in{\mathscr H}_{-\infty})\quad
{\rm Op}^\pi({\mathscr W}(f_1,f_2))=(\cdot\mid f_2)f_1.
\end{equation}
In particular, it follows that the above extension of
the cross-Wigner distribution to a mapping
${\mathscr W}(\cdot,\cdot)\colon{\mathscr H}_{-\infty}\times{\mathscr H}_{-\infty}\to{\mathscr S}'({\mathscr O})$
allows us to generalize
the assertion of Corollary~\ref{pseudo_prop}\eqref{pseudo_prop_item3}
to arbitrary $\phi_1,\phi_2\in{\mathscr H}_{-\infty}$.
\qed
\end{remark}

\section{Modulation spaces}\label{Sect5}

The modulation spaces play a central role in the time-frequency analysis
(see \cite{Gr01}) and proved to be a very useful tool in the study
of continuity properties of pseudo-differential operators (\cite{GH99}).
These classical ideas can be formulated within the representation theory
of the Heisenberg groups, and this representation theoretic viewpoint
turned out to be very effective in order to extend the corresponding notions
to unitary irreducible representations of arbitrary nilpotent Lie groups
(see \cite{BB09c}).
In the first two subsections of the present section we shall provide
some preparations and then describe the general notion of
modulation spaces introduced in \cite{BB09c}.
We eventually illustrate this notion by discussing a specific class
of irreducible representations on Hilbert spaces of the form $L^2({\mathscr O})$,
where ${\mathscr O}$ is any coadjoint orbit of a nilpotent Lie group
(see Proposition~\ref{double} and Remark~\ref{double_cor2}).

\subsection{Semidirect products}

\begin{definition}\label{sd_def}
\normalfont
Let $G_1$ and $G_2$ be connected Lie groups and assume that we have
 a continuous group homomorphism
$\alpha\colon G_1\to{\rm Aut}\, G_2$, $g_1\mapsto\alpha_{g_1}$.
The corresponding \emph{semidirect product of Lie groups}
$G_1\ltimes_\alpha G_2$ is the connected Lie group whose
underlying manifold is the Cartesian product $G_1\times G_2$
and whose group operation is given by
\begin{equation}\label{sd_prod}
(g_1,g_2)\cdot(h_1,h_2)=(g_1h_1,\theta_{h_1^{-1}}(g_2)h_2)
\end{equation}
whenever $g_j,h_j\in G_j$ for $j=1,2$.

Let us denote by $\dot\alpha\colon\gg_1\to{\rm Der}\,\gg_2$
the homomorphism of Lie algebras defined as the differential
of the Lie group homomorphism $G_1\to{\rm Aut}\,\gg_2$, $g_1\mapsto\text{\bf L}(\alpha_{g_1})$.
Then the \emph{semidirect product of Lie algebras}
$\gg_1\ltimes_{\dot\alpha}\gg_2$
is the Lie algebra whose underlying linear space is the Cartesian product
$\gg_1\times\gg_2$ with the Lie bracket given by
\begin{equation}\label{sd_bracket}
[(X_1,X_2),(Y_1,Y_2)]=([X_1,Y_1],\dot\alpha(X_1)Y_2-\dot\alpha(Y_1)X_2+[X_2,Y_2])
\end{equation}
if $X_j,Y_j\in\gg_j$ for $j=1,2$.
One can prove that $\gg_1\ltimes_{\dot\alpha}\gg_2$ is the Lie algebra
of the Lie group $G_1\ltimes_\alpha G_2$
(see for instance Ch.~9 in \cite{Ho65}).
\qed
\end{definition}

\begin{remark}\label{sd_rem}
\normalfont
Assume that $G_1$ and $G_2$ are nilpotent Lie groups and
$\alpha\colon G_1\to{\rm Aut}\, G_2$ is a \emph{unipotent automorphism}.
That is, for every $X_1\in\gg_1$ there exists an integer $m\ge1$ such that
$\dot\alpha(X_1)^m=0$.
Then an inspection of \eqref{sd_bracket} shows that $\gg_1\ltimes_{\dot\alpha}\gg_2$
is a nilpotent Lie algebra, hence $G_1\ltimes_\alpha G_2$ is a nilpotent Lie group.
\qed
\end{remark}

\begin{example}\label{sd_ex1}
\normalfont
Let $G$ be a nilpotent Lie group.
If we specialize Definition~\ref{sd_def} for $G_1:=G$, $G_2=(\gg,+)$, and
$\alpha:={\rm Ad}_G\colon G\to{\rm Aut}\,\gg$, then we get
the semidirect product $G\ltimes_{{\rm Ad}_G}\gg$ which is a nilpotent Lie group by
Remark~\ref{sd_rem} and is isomorphic to the tangent group $TG$.
The Lie algebra of $G\ltimes_{{\rm Ad}_G}\gg$ is $\gg\ltimes_{{\rm ad}_{\gg}}\gg_0$
(where $\gg_0$ stands for the abelian Lie algebra that has the same
underlying linear space as~$\gg$)
and the corresponding exponential map is given by
$$\begin{aligned}
\exp_{G\ltimes_{{\rm Ad}_G}\gg}(X,Y)
&=(\exp_G X,\int\limits_0^1{\rm Ad}_G(\exp_G(sX))Y\,{\rm d} s) \\
&=(\exp_G X,\int\limits_0^1{\rm e}^{s\cdot{\rm ad}_{\gg}X}Y\,{\rm d} s)
\end{aligned}$$
for every $(X,Y)\in\gg\ltimes_{{\rm ad}_{\gg}}\gg_0$
(see for instance Prop.~2.7(2) in \cite{BB09a}).
\qed
\end{example}

\subsection{Modulation spaces for unitary representations}
In this short subsection we just recall the definition
of the modulation spaces for the unitary irreducible representations of nilpotent Lie groups.
We refer to \cite{BB09c} for a more detailed discussion of this notion.

\begin{definition}\label{modular_def}
\normalfont
Let $\phi\in{\mathscr H}_\infty\setminus\{0\}$ be fixed and
assume that we have a direct sum decomposition
$\gg_e=\gg_e^1\dotplus\gg_e^2$.

Then let $1\le r,s\le\infty$ and
for arbitrary $f\in{\mathscr H}_{-\infty}$ define
$$\Vert f\Vert_{M^{r,s}_\phi}
=\Bigl(\int\limits_{\gg_e^2}
\Bigl(\int\limits_{\gg_e^1}
\vert({\mathscr A}_\phi f)(X_1,X_2)\vert^r{\rm d} X_1 \Bigr)^{s/r}
{\rm d} X_2\Bigr)^{1/s}\in[0,\infty] $$
with the usual conventions if $r$ or $s$ is infinite.
Then we call the space
$$M^{r,s}_\phi(\pi):=\{f\in{\mathscr H}_{-\infty}\mid\Vert f\Vert_{M^{r,s}_\phi}<\infty\}$$
a \emph{modulation space} for the irreducible unitary representation $\pi\colon G\to{\mathscr B}({\mathscr H})$
with respect to the decomposition $\gg_e\simeq\gg_e^1\times\gg_e^2$
and the \emph{window vector} $\phi\in{\mathscr H}_\infty\setminus\{0\}$.
\qed
\end{definition}

\begin{example}\label{mod_L2}
\normalfont
For any choice of $\phi\in{\mathscr H}_\infty\setminus\{0\}$ in Definition~\ref{modular_def} we have
$$M^{2,2}_\phi(\pi)={\mathscr H}.$$
Indeed, this equality holds since
$\Vert{\mathscr A}_\phi f\Vert_{L^2(\gg_e)}=\Vert\phi\Vert\cdot\Vert f\Vert$
for every $f\in{\mathscr H}$
(see Proposition~\ref{orthog} above).
\qed
\end{example}

\subsection{A specific irreducible representation on $L^2({\mathscr O})$}
We are going to construct here some irreducible representations on the Hilbert spaces of the form $L^2({\mathscr O})$,
where ${\mathscr O}$ can be any coadjoint orbit of a nilpotent Lie group.
A different construction involving the Moyal product (see Definition~\ref{moyal})
was used in Def.~2.19 in the paper \cite{BB09c} in order to get a representation $\pi^\#$
with the same representation space $L^2({\mathscr O})$.
The modulation spaces for $\pi^\#$
turned out to be relevant for establishing
the continuity properties of the pseudo-differential operators
obtained by the Weyl-Pedersen calculus for any unitary representation
associated with the coadjoint orbit~${\mathscr O}$.
(See also Remark~\ref{calc_dual}.)

\begin{proposition}\label{double}
Let $Z$ be the center of the connected, simply connected, nilpotent Lie group~$G$
with the corresponding Lie algebra ${\mathfrak z}\subseteq\gg$.
Endow the coadjoint orbit~${\mathscr O}$ with a Liouville measure and define
$$\widetilde{\pi}\colon G\ltimes_{{\rm Ad}}\gg\to{\mathscr B}(L^2({\mathscr O})),\quad
(\widetilde{\pi}(g,Y)f)(\xi)={\rm e}^{{\rm i} \hake{\xi,Y}}f({\rm Ad}^*_G(g^{-1})\xi).$$
Then the following assertions hold:
\begin{enumerate}
\item\label{double_item1}
The group $\widetilde{G}:=G\ltimes_{{\rm Ad}}\gg$
is nilpotent and its center is $Z\times{\mathfrak z}$.
\item\label{double_item2}
$\widetilde{\pi}$ is a unitary irreducible representation
of $\widetilde{G}$.
\item\label{double_item2and1/2}
Let us denote by $\widetilde{\gg}=\gg\ltimes_{{\rm ad}_{\gg}}\gg_0$
the Lie algebra of $\widetilde{G}$
(where $\gg_0$ stands for the abelian Lie algebra with the same
underlying linear space as~$\gg$)
and define
$$\widetilde{X}_j=
\begin{cases}
\hfill (0,X_j) &\text{ for }j=1,\dots,n,\\
(X_{j-n},0) &\text{ for }j=n+1,\dots,2n.
\end{cases}
$$
Then $\widetilde{X}_1,\dots,\widetilde{X}_{2n}$ is a Jordan-H\"older basis
in $\widetilde{\gg}$ and the corresponding predual for
the coadjoint orbit $\widetilde{{\mathscr O}}\subseteq\widetilde{\gg}^*$ associated with
the representation $\widetilde{\pi}$ is $$\widetilde{\gg}_{\widetilde{e}}=\gg_e\times\gg_e\subseteq\widetilde{\gg},$$
where $\widetilde{e}$ is the set of jump indices for~$\widetilde{{\mathscr O}}$.
\item\label{double_item4}
The space of smooth vectors for the representation $\widetilde{\pi}$
is ${\mathscr S}({\mathscr O})$.
\end{enumerate}
\end{proposition}

\begin{proof}
\eqref{double_item1}
Recall that the multiplication in the semi-direct product group $\widetilde{G}$
is given by
$$(g_1,Y_1)\cdot(g_2,Y_2)=(g_1g_2,Y_1+{\rm Ad}_G(g_1)Y_2) $$
while the bracket in the corresponding Lie algebra $\widetilde{\gg}=\gg\ltimes_{{\rm ad}}\gg$ is
defined by
\begin{equation}\label{double_eq0}
[(X_1,Y_1),(X_2,Y_2)]=([X_1,X_2],[X_1,Y_2]-[X_2,Y_1]).
\end{equation}
An inspection of these equations quickly leads to the conclusion that $\widetilde{\gg}$
is a nilpotent Lie algebra with the center ${\mathfrak z}\times{\mathfrak z}$.

\eqref{double_item2}
If $(g_1,Y_1),(g_2,Y_2)\in\widetilde{G}$ and $f\in L^2({\mathscr O})$,
then for $\xi\in{\mathscr O}$ we have
$$\begin{aligned}
\widetilde{\pi}(g_1,Y_1)(\widetilde{\pi}(g_2,Y_2)f)(\xi)
&={\rm e}^{{\rm i}\hake{\xi,Y_1}}(\widetilde{\pi}(g_2,Y_2)f)({\rm Ad}_G^*(g_1^{-1})\xi) \\
&={\rm e}^{{\rm i}\hake{\xi,Y_1}}{\rm e}^{{\rm i}\hake{{\rm Ad}_G^*(g_1^{-1})\xi,Y_2}}f({\rm Ad}_G^*(g_2^{-1}){\rm Ad}_G^*(g_1^{-1})\xi) \\
&={\rm e}^{{\rm i}\hake{\xi,Y_1+{\rm Ad}_G(g_1)Y_2}}f({\rm Ad}_G^*((g_1g_2)^{-1})\xi)
\end{aligned}$$
hence $\widetilde{\pi}(g_1,Y_1)\widetilde{\pi}(g_2,Y_2)=\widetilde{\pi}((g_1,Y_1)(g_2,Y_2))$.
Next note that the representation $\widetilde{\pi}$ is unitary
since the Liouville measure on ${\mathscr O}$ is invariant under the coadjoint
action of $G$.

To see that $\widetilde{\pi}$ is irreducible, let $T\colon L^2({\mathscr O})\to L^2({\mathscr O})$
be any bounded linear operator satisfying $T\widetilde{\pi}(g,Y)=\widetilde{\pi}(g,Y)T$
for arbitrary $(g,Y)\in\widetilde{G}$.
We have to check that $T$ is a scalar multiple of the identity operator on $L^2({\mathscr O})$.
By applying the assumption for $g=\1\in G$ we see that
$T$ belongs to the commutant of the family of multiplication operators
by the functions in the family $\{{\rm e}^{{\rm i}\hake{Y,\cdot}}\mid Y\in\gg\}\subseteq L^\infty({\mathscr O})$.
On the other hand,
we recall that the mapping
$${\mathscr O}\to{\mathbb R}^d,\quad \xi\to(\hake{\xi,X_{j_1}},\dots,\hake{\xi,X_{j_d}}) $$
is a global chart which takes the Liouville measure of ${\mathscr O}$
to a Lebesgue measure on~${\mathbb R}^d$
(see for Remark~\ref{jump_remark}).
Then we can use the Fourier transform to see that the linear subspace generated by $\{{\rm e}^{{\rm i}\hake{Y,\cdot}}\mid Y\in\gg\}$
is weak$^*$-dense in $L^\infty({\mathscr O})$ ($\simeq L^1({\mathscr O})^*$).
Therefore the operator $T\colon L^2({\mathscr O})\to L^2({\mathscr O})$ commutes with
all the multiplication operators by functions in $L^\infty({\mathscr O})$,
and then it has to be in turn the multiplication operator by some function $\phi\in L^\infty({\mathscr O})$.
Now, by using the assumption that $T$ commutes with $\pi(g,0)$ for arbitrary $g\in G$,
it easily follows that $\phi$ has to be a constant function
since the coadjoint action of $G$ on the orbit ${\mathscr O}$ is transitive.

\eqref{double_item2and1/2}
It is straightforward to check that
$\widetilde{X}_1,\dots,\widetilde{X}_{2n}$ is a Jordan-H\"older basis
in $\widetilde{\gg}$.
Next note that ${\mathscr S}({\mathscr O})$ is contained in the space of smooth vectors
for the representation $\widetilde{\pi}$ and for arbitrary
$f\in{\mathscr S}({\mathscr O})$ and $(X,Y)\in\widetilde{\gg}$
we have
\begin{equation}\label{double_proof_eq1}
(\forall\xi\in{\mathscr O})\quad
({\rm d}\widetilde{\pi}(X,Y)f)(\xi)={\rm i}\hake{\xi,Y}f(\xi)
+\frac{{\rm d}}{{\rm d} t}\Big{\vert}_{t=0}f(\xi\circ{\rm e}^{t{\rm ad}_{\gg}X}).
\end{equation}
It then follows by a straightforward application of Prop.~2.4.1 in \cite{Pe84}
and by Lemmas 1.4.1 and 1.5.1 in \cite{Pe89}
that the set of jump indices for the coadjoint orbit $\widetilde{{\mathscr O}}$
is $\widetilde{e}=\{j_1,\dots,j_d,n+j_1,\dots,n+j_d\}$, and then
$\widetilde{\gg}_{\widetilde{e}}=\gg_e\times\gg_e\subseteq\widetilde{\gg}$.

\eqref{double_item4}
It follows by \eqref{double_proof_eq1} and
by Lemmas 1.4.1 and 1.5.1 in \cite{Pe89}
again that there exists a polynomial chart on ${\mathscr O}$ such that
in the corresponding chart, the associative algebra
generated by the image of ${\rm d}\widetilde{\pi}$
contains all the linear partial differential operators with polynomial coefficients.
This implies that the space of smooth vectors for the representation $\widetilde{\pi}$
is equal to ${\mathscr S}({\mathscr O})$, as asserted.
\end{proof}

\begin{corollary}\label{double_cor}
Assume the setting of Proposition~\ref{double}.
The ambiguity function
$$\widetilde{{\mathscr A}}\colon L^2({\mathscr O})\times L^2({\mathscr O})\to L^2(\widetilde{\gg}_{\widetilde{e}})
=L^2(\gg_e\times\gg_e) $$
for the representation $\widetilde{\pi}\colon\widetilde{G}\to{\mathscr B}(L^2({\mathscr H}))$
is given by the formula
$$(\widetilde{{\mathscr A}}_\phi f)(X,Y)
=\int\limits_{{\mathscr O}}{\rm e}^{-{\rm i}\hake{\xi,\int\limits_0^1{\rm e}^{s\cdot{\rm ad}_{\gg}X}Y\,{\rm d} s}}
f(\xi)\overline{\phi(\xi\circ{\rm e}^{{\rm ad}_{\gg}X})} {\rm d}\xi$$
for arbitrary $X,Y\in\gg_e$ and $f,\phi\in L^2({\mathscr O})$.
\end{corollary}

\begin{proof}
For every $f,h\in L^2({\mathscr O})$ we have
\begin{equation}\label{double_cor_proof_eq1}
(\widetilde{{\mathscr A}}_\phi f)(X,Y)=(f\mid\widetilde{\pi}(\exp_{\widetilde{G}}(X,Y))\phi)_{L^2({\mathscr O})}.
\end{equation}
On the other hand, for the element $(X,Y)\in\widetilde{\gg}$ we have
$$\exp_{\widetilde{G}}(X,Y)=(\exp_G X,\int\limits_0^1{\rm Ad}_G(\exp_G(sX))Y\,{\rm d} s)
=(\exp_G X,\int\limits_0^1{\rm e}^{s\cdot{\rm ad}_{\gg}X}Y\,{\rm d} s)$$
(see Example~\ref{sd_ex1})
hence
$$\begin{aligned}
(\widetilde{\pi}(\exp_{\widetilde{G}}(X,Y))\phi)(\xi)
&={\rm e}^{{\rm i}\hake{\xi,\int\limits_0^1{\rm e}^{s\cdot{\rm ad}_{\gg}X}Y\,{\rm d} s}}
\phi({\rm Ad}_G^*(\exp_G(-X))\xi) \\
&={\rm e}^{{\rm i}\hake{\xi,\int\limits_0^1{\rm e}^{s\cdot{\rm ad}_{\gg}X}Y\,{\rm d} s}}
\phi(\xi\circ{\rm e}^{{\rm ad}_{\gg}X})
\end{aligned}$$
and then the conclusion follows by \eqref{double_cor_proof_eq1}.
\end{proof}

\begin{remark}\label{double_cor2}
\normalfont
Assume the setting of the above Proposition~\ref{double}.
It follows by Corollary~\ref{double_cor} along with
Schur's criterion for integral operators that there exists
a constant $C_\Phi>0$ such that for every
$F\in L^2({\mathscr O})$ and $Y\in\gg_e$ we have
$\Vert\widetilde{{\mathscr A}}_\Phi F(\cdot,Y)\Vert_{L^2(\gg_e)}
\le C_\Phi\Vert F\Vert_{L^2({\mathscr O})}$, hence
$\Vert F\Vert_{M^{2,\infty}_\Phi(\widetilde{\pi})}\le
C_\Phi \Vert F\Vert_{L^2({\mathscr O})}$.
Therefore there exists a continuous inclusion map
$L^2({\mathscr O})\hookrightarrow M^{2,\infty}_\Phi(\widetilde{\pi})$.
See also \cite{BB09b} for similar inclusion maps for the modulation spaces
in the setting of the magnetic Weyl calculus on nilpotent Lie groups.
\qed
\end{remark}

\begin{example}\label{two-step}
\normalfont
Assume that $\gg$ is two-step nilpotent Lie algebra, that is, we have $[\gg,[\gg,\gg]]=\{0\}$.
Let ${\mathscr O}\subseteq\gg^*$ be any nontrivial coadjoint orbit
and pick $\xi_0\in{\mathscr O}$.
If we denote by ${\mathfrak z}$ the center of $\gg$, then
$${\mathscr O}=\{\xi\in\gg^*\mid\quad \xi\vert_{{\mathfrak z}}=\xi_0\vert_{{\mathfrak z}}\},$$
since ${\mathscr O}$ is a flat orbit.
Then by Corollary~\ref{double_cor} along with the fact that $[\gg,\gg]\subseteq{\mathfrak z}$ we get
$$\begin{aligned}
(\widetilde{{\mathscr A}}_\phi f)(X,Y)
&=\int\limits_{{\mathscr O}}{\rm e}^{-{\rm i}\hake{\xi,\int\limits_0^1{\rm e}^{s\cdot{\rm ad}_{\gg}X}Y\,{\rm d} s}}
f(\xi)\overline{\phi(\xi\circ{\rm e}^{{\rm ad}_{\gg}X})} {\rm d}\xi \\
&=\int\limits_{{\mathscr O}}{\rm e}^{-{\rm i}\hake{\xi,Y+\frac{1}{2}[X,Y]}}
f(\xi)\overline{\phi(\xi+\xi\circ{\rm ad}_{\gg}X)} {\rm d}\xi \\
&={\rm e}^{-\frac{{\rm i}}{2}\hake{\xi_0,[X,Y]}}\int\limits_{{\mathscr O}}{\rm e}^{-{\rm i}\hake{\xi,Y}}
f(\xi)\overline{\phi(\xi+\xi_0\circ{\rm ad}_{\gg}X)} {\rm d}\xi.
\end{aligned}$$
The above formula and suitable global coordinates on ${\mathscr O}$,
one shows that the ambiguity function of the representation $\widetilde{\pi}$
agrees with the ambiguity function of the Schr\"odinger representation of a certain Heisenberg group,
as defined in \cite{Gr01}.
\qed
\end{example}

\textbf{Acknowledgment.}
Partial financial support from the grant PNII - Programme ``Idei'' (code 1194)
is acknowledged.

redpajama_set_name: RedPajamaArXiv


text:
Entrepreneur's Delight
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A collection of write-ups from writers/authors willing to write on Entrepreneurship, Business, Corporate Leadership & Management.
Truck Suvidha: Redefining the Indian transport Industry
Necessity, goes the adage, is the mother of invention. When Ishu Bansal, saw his family business face problems in transporting their manufactured goods, he was convinced there were others sailing in the same boat too and facing the same hardships as his family. Find out what he did.
Bansal's family had set up a plywood manufacturing unit at Mundia in Badaun district in 2003. The manufacturing unit being based in a small town meant that the Bansals would have to get trucks from a 100-150 km circle to despatch their finished products. They would often end up delivering their promised goods after the deadlines. Resulting into losses due to not being able to deliver on time as well as loss of goodwill. Buyers would walk away from a manufacturer who, more often than not, would fail to deliver on time.
Unable to get reliable and timely transportation facilities, the Bansal family went as far as buying a truck of their own in 2007. That too did not seem to work out.
Recalls Bansal, "A part of the problem did get solved. We were able to deliver on time but the idea was not commercially viable. We were not able to find return loads for trucks, which resulted in huge costs when we took into account the daily expenses of hiring a driver, the diesel for the vehicle and the return freight. At times, the truck drivers would have to wait at one place for 5 to 7 days at times- waiting to get loads for the return journey. The whole setup, we soon realized, though enabled us to deliver on time, had led to another set of headaches. Obviously, not a very pleasant scenario."
Ishu would see the harried family and wanted to create a solution. Not only for his family but for all many others like them who, he was sure, were suffering too. Almost four years back, Ishu was enjoying a cup of tea with Amit Punaini, also a partner in the plywood business, when he brought up the topic of transportation related woes. Amit who also had a successful scrap and metal business behind him knew that the man in front of him, almost ten years his younger, was talking sense.
The idea was thus conceived during a casual chit chat but once the seed had been sown, there was no looking back. Together, the two decided to use technology to rid this highly unorganized sector of, at least, some of its ills. That was way back in 2011. "We knew we wanted to do it. Rather, we HAD to do it. We started refining our thought process, closely looking at the industry, building up the database and collecting feedback from various entities involved in transport business. The research consisted of endless visits to Azadpur Mandi (Delhi), transport houses, truck drivers, dhabas, truck unions and interacting with the people who had been in this industry for a long time. After those long conversations and hearing to the travails of others like us, we finally knew we were following the right thing," says Amit.
Ishu, only 21 at that time, was working with Infosys. The self-propelled entrepreneur pooled in all his savings and together with Amit, who was his family's business associate, set out to work upon the idea.
The two gleam with pride when they mention that the concept was entirely self-funded! The concept was under-the-works for a long three years, during which they zeroed in on every possible aspect of it. The portal was finally launched in September 2014.
Adds Ishu, "With this project, we are targeting the logistics industry in which truckers are the people who are the worst sufferers. Since we were aiming to provide 'suvidha' (convenience) to trucks and truck drivers/ operators, we decided to call it Truck Suvidha.
Keen to know how the whole thing works, we asked Amit to elaborate on it.
"See! There are different parties and aspects to the whole thing. Let us first talk from the transporter's point of view. Whoever wishes to transport a load will need to register on the portal (which has been kept simple and easy) and purchase a subscription. Once the load is available from his home district, he will get an SMS about the same. Or else he can view the status by visiting the website. Here, we give them two options. They can post their own loads and find truck(s) for them. Or take the other option. They can also post their trucks and find loads for them.
Now, from the truck drivers' perspective. They can register under any transporter and find loads for his truck. They can also get return loads by posting truck on the website."
Quips in he, "That's not all. Packers and movers can also post their loads and find trucks for them. Infact, we are going to launch shifting requirements board for packers and movers soon as well."
"Lastly, we also make things easier for customers/industrialists. They can post their loads and find trucks for them. They can directly contact with truck posting owners. I mean, those who posted trucks on the website. Or, search for transporters in the directory."
Truck Suvidha, which now operates from Yamunanagar (Haryana) with 21 full time employees excluding the co-owners and four part timers, is not done with the teething troubles.
"It has not been easy convincing transporters and truck drivers, most of whom are not too well read, that technology can be a driving factor for their business. Worse still, even the educated ones in the transportation sector do not realize the importance of staying in sync with the times. Not many turn to, or think of turning to the internet for business."
Convincing them, getting them to register with their portal has been difficult, he adds. Difficult, but not impossible.
Truck Suvidha helps improve productivity by reducing the time it takes to search for loads or trucks by using a state of the art filtered email notification system and eliminating non-relevant services (links) and animated ads which make most freight matching load board services annoying for those turning to the internet.
Not wanting to leave any stone unturned and to get out of the red as soon as they possibly can, the two co-founders are now focussed on popularizing the concept and raising awareness about the need for it. For that, they are making use of every possible channel. Digital marketing, telemarketing, Search Engine Optimization (SEO), SMO (Social Media Optimization), in person marketing- everything, in short!
Amit, the more experienced one of the two, adds, "The revenues and margins are good for the logistics industry. Truck Suvidha would be earning through advertisements, packages etc. We know we might take a year to break even. Maybe even two. But that doesn't bother us!"
The initial response has been good, they say. Exceeding expectations infact. 16,403 registered transport contractors, 8,084 fleet owners, 5,340 commission agents and 1,729 transport brokers over a seven month period is no mean achievement, after all! Those astoundingly huge numbers speak volumes about the rapid strides Truck Suvidha has been making into this unchartered territory.
The team is also working hard on positioning themselves as a brand. They have a very active Facebook page with more than 11,000 likes and a Twitter fan base of 4,000 users.
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At present, they have clearly demarcated areas in which they are both, in their own right, specialists. As Amit, the seasoned businessman, takes care of the day to day operations, Ishu is the tech guy who looks after the website. Both together work hard to ensure that a one-time visitor to TS not only converts into a loyal customer but also helps spread the good word about them.
The journey so far, inspite of being a bumpy ride on a road covered, for the most part, with potholes, has given the two young entrepreneurs reasons enough to smile. The most memorable moments till date have been a launch at IIT Kanpur.
"The portal which went beta in September 2014 went fully functional during the IITK launch. The TS was finally opened to the public during the India International Trade Fair (IITF). Which was a BIG moment for us," says Ishu.
Amit wants to add, "Later, we also got the first runners up award for presenting our idea at Thadomal Shahani Centre for Management (TSCFM), Mumbai and the second runners up award at another event where we highlighted the social aspect of our initiative- our aim to streamline things for each of the parties involved with transportation through trucks."
India, as we all know, is one of the largest consumer markets in the world and driven primarily by domestic trade. Even in 2012, domestic trade comprised more than half (58%, to be precise) share of the GDP. Land transport accounts for a lion's share of this movement of goods within the country. It is surprising therefore that no one thought of developing a platform like this earlier.
The men at the helm of Truck Suvidha state in unambiguous terms that their vision entails helping all players who need to use trucks by bringing them together come under one umbrella.
This time, we want to join them in saying, "Difficult, of course. But not impossible!"
Copyrights © 2020 Inspiration Unlimited eMagazine
Any facts, figures or references stated here are made by the author & don't reflect the endorsement of iU at all times unless otherwise drafted by official staff at iU. This article was first published here on 16th August 2016.
Shilpa Mahajan
Shilpa Mahajan is a contributing writer at Inspiration Unlimited eMagazine.
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Home > News & Events > Zen Bow > Interconnection and the Alchemy of Performance
Interconnection and the Alchemy of Performance
By Rebecca Gilbert | Zen Bow | Spring 2020, Vol. XLII, No. 1
An Interview with Rebecca Gilbert
The discovery of interconnection, on the stage and in the zendo.
Zen Bow: Were you raised in a religious or spiritual tradition?
Rebecca Gilbert: My mother had more of an interest than my father in making sure we had a spiritual path, so we were enrolled in a Sunday School at a Methodist church in our small town of Lomira, Wisconsin. My parents only went to church regularly during the months my younger sister and I were enrolled in Sunday School. During that time of Sunday School studies, I also sang in the church choir.
ZB: Was yours a musical family?
RG: Not particularly. One of my grandmothers played a little piano, I only ever heard her playing Christmas carols, and I had an aunt who played some accordion.
ZB: What was the first instrument you played?
RG: My first musical experiences were in the church choir.
ZB: You have a decent singing voice?
RG: I wouldn't say that. In a past life I may have been a singer because I have these latent fantasies of being an opera singer. I dream about it a lot, and when I play the flute I think about it as my voice; that is, I feel the sound as an extension of my voice. I never really was trained [as a singer] although I had a teacher in graduate school who, as part of her pedagogical technique, had us take voice lessons, but I never really, really sang.
But I was only in a choir for a short period of time—maybe a year when we were going to church. I was young, maybe ten years old, and it wasn't a children's choir. It was the adult church choir. I have this memory of connecting with everybody who was singing—connecting with their voices—and I still love the sound of a great chorus. I love chanting in the zendo, just being able to hear how we're all resonating together. And that sense of our bodies and our spirit and our energy all combining.
ZB: When did you learn to read music?
RG: I learned in music class at school and I played recorder in school before I played the flute. The music program in our rural Wisconsin community was incredible. You learned to read music, to play the recorder, and when you chose an instrument you had a private lesson once a week instead of recess. That was helpful in keeping us from learning bad habits, and it accelerated our early progress.
ZB: When did you start playing the flute?
RG: In sixth grade as part of a public school band program
ZB: Were you expected to practice every day? For how long?
RG: My parents gave me no parameters about practicing. I had a goal-oriented, type-A personality which kept me self-motivated with my flute practice. But I almost quit after we moved the summer between my sixth and seventh grade years.
We moved to Broken Arrow, Oklahoma, outside of Tulsa, and the culture shock from the rural Midwest to the suburban Southwest was an enormous challenge. The band program at my new school was focused on pep band/marching band music for football games where the focus is on the brass players, this kind of balls-to-the-wall, blow 'em away approach to playing music, which I wasn't interested in.
Plus, in the new school the band wasn't as good and, though my family are rabid Green Bay Packers fans, I had never been interested in sports—and, in particular, hated football. When we moved to Oklahoma I had been playing flute for only a year and that's when I was ready to quit.
ZB: When did you start thinking that you might be a professional musician?
RG: This was a process which came after a series of events that led me to the choice. At that point, when I was so disheartened, my mother saw an opportunity for me to maybe invest a little bit more and she found me a private teacher, my first private teacher outside of school. My teacher was a professor at Oral Roberts University, so I used to go onto this gilded golden campus for my lessons.
At the same time, I remained in the school band. Ironically, the marching band was very competitive and we were the top in the state. It was actually one of the first experiences that taught me what level of artistic excellence you can achieve with hard work. We would get up in the morning and be at band practice at 6:00 in the morning before school started. There were high expectations. It required a very intense commitment. I didn't love the music, the venues, or having to march or any of that, but I did absorb and appreciate what it was like to do hard work as a team together.
And the private lessons worked! There was no orchestra at the school, no strings at the school, but I auditioned and won a spot in the Tulsa Youth Symphony and then I absolutely fell in love with the sound of the orchestra…in particular the lush blend of the string sections and the way the band instruments were positioned as solo instruments in the texture of the sound.
Later in high school I won a position playing Principal Flute in the All-State Orchestra and was recruited by the Flute Professor at Oklahoma State University, who opened the door to what a career as a professional musician would look like.
ZB: Is it "flutist" or "flautist"?
RG: Every person who plays the flute could have a different answer, but the question comes from the European roots of Western Classical music. The word for flute is pronounced with a specific accent depending on the nationality of the player. So if you lived in Italy, the word flute is spelled flauto, pronounced with an Italian accent and then you might choose "flautist." As I am American, and play the flute, I prefer "flutist."
ZB: How long have you been with the RPO? What was your audition like?
RG: I joined the RPO in 1996. The audition process is quite interesting. It starts with applying for the job and receiving a list of required repertoire to prepare. Then you perform for a panel of musicians who sit behind a screen in rounds and sort through the candidates with a voting system.
ZB: Do you have any anxiety before auditions and performances? How do you deal with it?
RG: Early on, I developed a routine including specific steps that set me up to be in a reliable physiological/psychological state for performing. For instance, I always do a cardio workout, refrain from caffeine, and eat the same things before performing. It's like training for an athletic event: you want to neutralize anything that might get in your way (example: caffeine) and feel confident that you have control over your body and mind. Now, of course, I do as much zazen as I can before performing. It helps me feel more energetic and activates my intuition.
Now, as I am getting older, my physical routines have changed a bit and I am doing more yoga and Pilates as well as cardio. Playing the flute requires an asymmetrical body posture, and because of that I have had to pay more attention to physical maintenance in the past few years. I also watch what I eat all the time because I am performing all the time.
Another of my pre-performance rituals, believe it or not, is watching reruns of Little House on the Prairie. I always loved watching them as a girl and find them soothing, grounding, and capable of opening up my heart. Listening to inspiring music also has the same effect on me.
Auditions are a little different because you are playing, literally, to a wall. You have to re-create a performance setting in your mind because of the screen, and you don't have the interconnection with and responsiveness of the audience. When I coach younger musicians before auditions, I emphasize the importance of imagining the audience as they are playing.
ZB: When did you first start meditating or first come to the Zen Center?
RG: I started Zen practice in 1999, only a few years after moving to Rochester. Before finding the Zen Center, I had a sporadic interest in meditation and a simple meditation practice I started a few years before moving to Rochester. I was inspired to look for a community of practitioners after attending a weekend sampler workshop at Kripalu, where I attended a combination yoga/meditation class. I loved the feeling of connecting with the other practitioners in the room and it reminded me of the feeling of connection I feel when I am performing. Also, I had an aspiration to go deeper with my practice and wanted to find a teacher.
ZB: What attracted you to Zen practice?
RG: I'm an introverted extrovert who needs plenty of quiet and grounding for the emotional demands of my performing career. I am also healing from family and sexual assault trauma.
ZB: You're in a very competitive environment. How do you balance professional ambition with the notion of ego attrition?
RG: I think in some ways I escaped some of the pitfalls of pursuing career ambition because I was so naïve. My early musical experiences were well out of the mainstream of the classical music world, and I always just felt lucky that things kept working out for me. I didn't have a lot of expectations, and that set me up for feeling pretty grateful and humble. My last flute mentor, who was the Principal Flutist of the New York Philharmonic, often tried to encourage me to think bigger than I naturally thought.
I love the alchemy of performance and the feeling of connecting with the audience. This has been the most compelling thing about why I gravitated to a performing career. / / /
Rebecca Gilbert joined the Rochester Philharmonic Orchestra as principal flute (The Charlotte Whitney Allen Chair) in the 1996–97 season. She has performed as acting principal flute of the St. Louis Symphony Orchestra and guest assistant principal flute with the Boston Symphony at Tanglewood. She has been a member of the Zen Center since 1999.
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ASC2014 - 2-5 Feb, Brisbane
Info for Presenters
Cathy Reade
Cathy Reade spent her early career working for a range of Canberra-based lobby groups. In 1989 she started working for the Crawford Fund, a non-profit organisation that works to increase Australia's engagement in international agricultural research, development and education for the benefit of developing countries and Australia. Cathy developed and manages a Crawford Fund Master Class in Communication with Stakeholders for scientists in developing countries, with facilitators Jenni Metcalfe and Toss Gascoigne. Additionally, she has a media consultancy concentrating in the not-for-profit sector around agriculture, development/aid, S&T and environment policy, issues and events. Cathy produced development-focussed sessions at the World Conference of Science Journalists in Melbourne in 2007 and at the ASC Conference in Canberra in 2010.
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