# A generalization of Duflo’s conjecture

Hongfeng Zhang  BICMR, Peking University, No. 5 Yiheyuan Road, Haidian
District, Beijing 100871, China<EMAIL_ADDRESS>
###### Abstract.

In this article, we generalize Duflo’s conjecture to understand the branching
laws of non-discrete series. We give a unified description on the geometric
side about the restriction of an irreducible unitary representation $\pi$ of
$\mathrm{GL}_{n}(k)$, $k=\mathbb{R}$ or $\mathbb{C}$, to the mirabolic
subgroup, where $\pi$ is attached to a certain kind of coadjoint orbit.

Mathematics Subject Classification (2010). 22E46 (17B08, 53D20).

Keywords. Kirillov’s conjecture, Duflo’s conjecture, orbit method, moment map.

###### Contents

  1. 1 Introduction
  2. 2 $P_{n}(k)$-coadjoint orbit
    1. 2.1 Coadjoint action, the dual map and the moment map
    2. 2.2 Classification of $P_{n}(k)$-coadjoint orbit
  3. 3 Geometry of the moment map $\mathrm{p}:\mathcal{O}_{f}\to\mathfrak{p}^{*}$
    1. 3.1 Moment map in the $\mathrm{GL}_{n}(\mathbb{C})$ case
      1. 3.1.1 Regular orbits
      2. 3.1.2 General orbits
    2. 3.2 Moment map in the $\mathrm{GL}_{n}(\mathbb{R})$ case
  4. 4 Results of Kirillov’s conjecture
  5. 5 Orbit method
    1. 5.1 Representations attached to coadjoint orbits of $\mathrm{GL}_{n}(k)$
      1. 5.1.1 Jordan decomposition
      2. 5.1.2 Hyperbolic step
      3. 5.1.3 Elliptic step
      4. 5.1.4 Nilpotent step
      5. 5.1.5 Representations of $\mathrm{GL}_{n}(k)$ attached to some kind of coadjoint orbits
    2. 5.2 Representations attached to coadjoint orbits of $P_{n}(k)$
      1. 5.2.1 Coadjoint orbits of algebraic groups
      2. 5.2.2 Construction of unitary dual of algebraic groups
      3. 5.2.3 Duflo’s construction in the $P_{n}(k)$ case
  6. 6 A generalization of Duflo’s conjecture

##  1\. Introduction

One goal of the orbit method is to establish a correspondence between
coadjoint orbits and irreducible unitary representations of Lie groups. One
can benefit a lot from such a correspondence, for example, finding out
irreducible unitary representations through coadjoint orbits; or predicting
the decomposition of $\pi|_{H}$ by looking at the decomposition of
$\mathrm{p}(\mathcal{O}_{\pi})$ as $H$-orbit, where $\pi$ is a unitary
representation of the Lie group $G$ and is attached to the coadjoint orbit
$\mathcal{O}_{\pi}$, $H$ is a closed subgroup of $G$, and $\mathrm{p}$ is the
moment map from $\mathcal{O}_{\pi}$ to $\mathfrak{h}^{*}$ with
$\mathfrak{h}^{*}$ the dual of the Lie algebra of $H$; etc.(see Kirillov [12],
Introduction).

The orbit method is brought up by Kirillov, who attached an irreducible
unitary representation to a coadjoint orbit for nilpotent groups in a perfect
way. Later, Kostant (see Auslander-Kostant [3]) established the theory of
quantization, and built up the orbit method for solvable groups. Duflo [6]
constructed all irreducible unitary representations of almost algebraic groups
from certain coadjoint orbits, assuming one knew unitary dual of reductive
groups. It suggested that to built up the orbit method for general Lie groups,
one should concentrate on reductive groups. The quantization problem of
coadjoint orbits of reductive groups can be reduced to quantizing nilpotent
coadjoint orbits. It has not been solved completely, but there are some
wonderful results (see Vogan [25], Chapter 10).

In spirit by some work on how the branching laws behave in the orbit method,
including Heckman [11] and Guillemin-Sternberg [10] for compact groups,
Fujiwara [9] for exponential solvable groups, and the work of Kobayashi [14]
on the branching laws of reductive groups, Duflo [8] formulated a conjecture
for the branching problem of discrete series of almost algebraic group (see
also Liu [15]).

Let $G$ be an (almost) algebraic group, let $H$ be a closed algebraic subgroup
of $G$. Let $\mathfrak{g}$ (resp. $\mathfrak{h}$) be the Lie algebra of $G$
(resp. of $H$). Let $\mathfrak{g}^{*}$ (resp. $\mathfrak{h}^{*}$) be the dual
of $\mathfrak{g}$ (resp. of $\mathfrak{h}$). Let $\pi$ be a discrete series
representation of $G$. By Duflo [7], $\pi$ is attached to a strongly regular
$G$-coadjoint orbit $\mathcal{O}_{\pi}$. Consider the restriction of $\pi$ to
$H$, denoted by $\pi|_{H}$, Duflo’s conjecture says that

  * (i)

$\pi|_{H}$ is $H$-admissible if and only if the moment map
$\mathrm{p}:\mathcal{O}_{\pi}\to\mathfrak{h}^{*}$ is weakly proper.

  * (ii)

If $\pi|_{H}$ is $H$-admissible, then each irreducible $H$-representation
$\sigma$ which appears in $\pi|_{H}$ is attached to a strongly regular
$H$-coadjoint orbit $\Omega$ which is contained in
$\mathrm{p}(\mathcal{O}_{\pi})$.

  * (iii)

If $\pi|_{H}$ is $H$-admissible, then the multiplicity of each such $\sigma$
can be expressed geometrically in terms of the reduced space of $\Omega$ with
respect to the moment map $\mathrm{p}$.

Let’s explain in details. The notion “almost algebraic group” is explained in
[7]. An element $f\in\mathfrak{g}^{*}$ is called strongly regular if $f$ is
regular (i.e. the coadjoint orbit containing $f$ is of maximal dimension) and
its “reductive factor” $\mathfrak{s}(f):=\\{X\in\mathfrak{g}(f):\mathrm{ad}X\
\text{is semisimple}\\}$ is of maximal dimension among the reductive factors
of all the regular elements in $\mathfrak{g}^{*}$, where $\mathfrak{g}(f)$
denotes the centralizer of $f$ in $\mathfrak{g}$.

Let $\mathcal{O}$ be a $G$-coadjoint orbit in $\mathfrak{g}^{*}$. Then
$\mathcal{O}$ is equipped with the Kirillov-Kostant-Souriau symplectic form
$\omega$ and becomes an $H$-Hamiltonian space. The corresponding moment map is
the natural projection $\mathrm{p}:\mathcal{O}\to\mathfrak{h}^{*}$.

In (i), the notion “$H$-admissible” is due to Kobayashi, which means that
$\pi|_{H}$ is discretely decomposable (i.e. $\pi|_{H}$ is a direct sum of
irreducible unitary representations of $H$) and all irreducible
representations of $H$ has only finite multiplicities in $\pi|_{H}$. The
notion “weak properness” means that the preimage (for $\mathrm{p}$) of each
compact subset which is contained in
$\mathrm{p}(\mathcal{O}_{\pi})\cap\Upsilon_{sr}$ is compact in
$\mathcal{O}_{\pi}$, where $\Upsilon_{sr}$ denotes the set of strongly regular
elements in $\mathfrak{h}^{*}$.

In (iii), the reduce space of $\Omega$ is equal to
$\mathrm{p}^{-1}(\Omega)/H$.

Duflo’s conjecture has been proved in some cases (see Liu [15]), and is also
generalized by Liu-Yu [17], who verified Duflo’s conjecture for the
restriction of tempered representations of $\mathrm{GL}_{n}(k)$ (for
$k=\mathbb{R}$ or $\mathbb{C}$) to its mirabolic subgroup and gave a geometric
interpretation of the Kirillov’s conjecture. Recently, Liu-Oshima-Yu [16]
verifies Duflo’s conjecture for the restriction of irreducible unitary
representations of $\mathrm{Spin}(N,1)$ to its parabolic subgroups.

This article generalizes Duflo’s conjecture in the framework of Kirillov’s
conjecture in spirit by Liu-Yu [17]. We obtain a generalization for the
irreducible unitary representations of $\mathrm{GL}_{n}(k)$ (for
$k=\mathbb{R}$ or $\mathbb{C}$) which are attached to some coadjoint orbits in
section 5.

Let’s say a few words about the Kirillov’s conjecture. The conjecture says
that the restriction of any irreducible unitary representation of
$\mathrm{GL}_{n}(k)$ (for $k=\mathbb{R},\mathbb{C}$ or $Q_{p}$) to the
mirabolic subgroup is also irreducible. It was proved by Bernstein for
$p$-adic fields [5], Sahi [19] for tempered representations of
$\mathrm{GL}_{n}(\mathbb{R})$ or $\mathrm{GL}_{n}(\mathbb{C})$, and Sahi-Stein
[21] for Speh representations of $\mathrm{GL}_{n}(\mathbb{R})$, and Baruch [4]
for the archimedean fields.

Here are the main results of this article. Let $k=\mathbb{R}$ or $\mathbb{C}$,
let $G=\mathrm{GL}_{n}(k)$, and let $P=P_{n}(k)$ be the mirabolic subgroup of
$G$, consisting of the elements whose last row is $(0,\cdots,0,1)$. Let
$\mathfrak{p}$ denote the Lie algebra of $P$ and $\mathfrak{p}^{*}$ denote the
dual of $\mathfrak{p}$. Assume that $\pi$ is an irreducible unitary
representation of $G$ and $\pi$ is attached to the $G$-coadjoint orbit
$\mathcal{O}_{\pi}$ in section 5. It turns out that the image of the moment
map $\mathrm{p}:\mathcal{O}_{\pi}\to\mathfrak{p}^{*}$ contains finite
$P$-coadjoint orbits, and there is a unique dense $P$-coadjoint orbit in
$\mathrm{p}(\mathcal{O}_{\pi})$, denoted by $\Omega$. The moment map
$\mathrm{p}$ is proper over $\Omega$. Moreover, $\pi|_{P}$ is attached to
$\Omega$ in the sense of Duflo. It explores more about the geometry of the
Kirillov’s conjecture.

The article is organized as follows.

In section 2, we present the classification of the $P_{n}(k)$-coadjoint orbits
and its proof, which comes from Liu-Yu [17].

In section 3, we compute the moment map and obtain similar results as Liu-Yu
[17] on the geometric side.

In section 4, we summarize some results of Kirillov’s conjecture, from which
we obtain the restrictions of all irreducible unitary representations of
$\mathrm{GL}_{n}(k)$ to $P_{n}(k)$.

In section 5, firstly, we summarize the orbit method for reductive groups, and
obtain the correspondence between coadjoint orbits and irreducible unitary
representations of $\mathrm{GL}_{n}(k)$. Secondly, we show how to construct
irreducible unitary representations from coadjoint orbits of $P_{n}(k)$ in the
sense of Duflo.

In section 6, our generalization is presented and proved by comparing the
results of the moment maps and the restrictions of irreducible unitary
representations of $\mathrm{GL}_{n}(k)$ to $P_{n}(k)$.

##  2\. $P_{n}(k)$-coadjoint orbit

This section is mainly adapted from Liu-Yu [17], which will be used in the
rest of the article.

###  2.1. Coadjoint action, the dual map and the moment map

Let $k$ be the field $\mathbb{R}$ or $\mathbb{C}$, and let
$n\in\mathbb{Z}_{+}$. Set $G_{n}(k)=\mathrm{GL}_{n}(k)$ and

$P_{n}(k)=\\{\left(\begin{array}[]{cc}A&\alpha\\\
0&1\end{array}\right):A\in\mathrm{GL}_{n-1}(k),\alpha\in k^{n-1}\\}$

be the mirabolic subgroup of $G_{n}(k)$. $\mathfrak{g}_{n}(k)$ denotes the Lie
algebra of $G_{n}(k)$ and

$\mathfrak{p}_{n}(k)=\\{\left(\begin{array}[]{cc}A&\alpha\\\
0&0\end{array}\right):A\in\mathfrak{gl}_{n-1}(k),\alpha\in k^{n-1}\\}$

denotes the Lie algebra of $P_{n}(k)$. $\mathfrak{g}_{n}(k)^{*}$ and
$\mathfrak{p}_{n}(k)^{*}$ denote the dual spaces of $\mathfrak{g}_{n}(k)$ and
$\mathfrak{p}_{n}(k)$, respectively.

$G_{n}(k)$ acts on $\mathfrak{g}_{n}(k)^{*}$ through

$(g\cdot f)\xi=f(g^{-1}\cdot\xi),\forall g\in G_{n}(k),\forall
f\in\mathfrak{g}_{n}(k)^{*},\forall\xi\in\mathrm{g}_{n}(k).$

This is called the coadjoint action, and a $G_{n}(k)$-orbit in
$\mathfrak{g}_{n}(k)^{*}$ is called a coadjoint orbit. Similarly, one can
define the coadjoint action of $P_{n}(k)$ on $\mathfrak{p}_{n}(k)^{*}$ and the
corresponding coadjoint orbit.

By taking trace, we can define a $G_{n}(k)$-invariant nondegenerate bilinear
form on $\mathfrak{g}_{n}(k)\times\mathfrak{g}_{n}(k)$,

$(X,Y)\mapsto\mathrm{tr}(XY),X,Y\in\mathfrak{g}_{n}(k).$

We have a $G_{n}(k)$-module isomorphism

$\mathrm{pr}:\mathfrak{g}_{n}(k)\to\mathfrak{g}_{n}(k)^{*},X\mapsto(Y\mapsto\mathrm{tr}(XY),\forall
Y\in\mathfrak{g}_{n}(k)),\forall X\in\mathfrak{g}_{n}(k).$

Let $\mathrm{pr}^{\prime}:\mathfrak{g}_{n}(k)\to\mathfrak{p}_{n}(k)^{*}$,
$X\mapsto(Y\mapsto\mathrm{tr}(XY),\forall Y\in\mathfrak{p}_{n}(k)),\forall
X\in\mathfrak{g}_{n}(k)$, then

$\ker\mathrm{pr}^{\prime}=\\{\left(\begin{array}[]{cc}0&\alpha\\\
0&t\end{array}\right):\alpha\in k^{n-1},t\in k\\}.$

Let $\bar{\mathfrak{p}}_{n}(k)=\\{\left(\begin{array}[]{cc}A&0\\\
\alpha&0\end{array}\right)|A\in\mathfrak{g}_{n-1}(k),\alpha^{t}\in
k^{n-1}\\}$, then
$\mathfrak{g}_{n}(k)=\ker\mathrm{pr}^{\prime}\oplus\bar{\mathfrak{p}}_{n}(k)$,
and $\mathrm{pr}^{\prime}|_{\bar{\mathfrak{p}}_{n}(k)}$ is an isomorphism.

We use $\mathrm{pr}_{n}$ (resp. $\mathrm{pr}^{\prime}_{n}$) to denote
$\mathrm{pr}$ (resp. $\mathrm{pr}^{\prime}$) if there is ambiguity.

###  2.2. Classification of $P_{n}(k)$-coadjoint orbit

Let

$L_{n}(k)=\\{\left(\begin{array}[]{cc}A&0\\\ 0&1\end{array}\right)\ |\ A\in
G_{n-1}(k)\\}$

and

$N_{n}(k)=\\{\left(\begin{array}[]{cc}I_{n-1}&\alpha\\\ 0&1\end{array}\right)\
|\ \alpha\in k^{n-1}\\}.$

Let $\mathfrak{l}_{n}(k)$ and $\mathfrak{n}_{n}(k)$ denote the Lie algebra of
$L_{n}(k)$ and $N_{n}(k)$, respectively.

Then $L_{n}(k)$ is the Levi subgroup of $P_{n}(k)$, $N_{n}(k)$ is the
unipotent radical of $P_{n}(k)$, and $P_{n}(k)=L_{n}(k)\ltimes N_{n}(k)$.
Moreover, we have the following exact sequence as $P_{n}(k)$-module

$0\to\mathfrak{n}_{n}(k)\to\mathfrak{p}_{n}(k)\to\mathfrak{l}_{n}(k)\to 0,$

so is

$0\to\mathfrak{l}_{n}(k)^{*}\to\mathfrak{p}_{n}(k)^{*}\to\mathfrak{n}_{n}(k)^{*}\to
0.$

We will regard $\mathfrak{l}_{n}(k)^{*}$ (resp. $\mathfrak{n}_{n}(k)^{*}$) as
a $P_{n}(k)$-submodule (resp. quotient module) of $\mathfrak{p}_{n}(k)^{*}$ in
the following.

Let $h\in\mathfrak{p}_{n}(k)^{*}$ such that $h|_{\mathfrak{l}_{n}(k)}=0$,
$h|_{\mathfrak{n}_{n}(k)}\neq 0$. We have following key proposition.

######  Proposition 2.1.

  * (1)

Every element in $\mathfrak{p}_{n}(k)^{*}$ is $P_{n}(k)$-conjugated to an
element in $\mathfrak{l}_{n}(k)^{*}$ or $h+\mathfrak{l}_{n}(k)^{*}$.

  * (2)

$\mathrm{Stab}_{P}(h|_{\mathfrak{n}_{n}(k)})=L_{n}(k)^{h}\ltimes N_{n}(k)$ and
$(h+\mathfrak{l}_{n}(k)^{*})/N_{n}(k)$ is $h+(\mathfrak{l}_{n}(k)^{h})^{*}$.

  * (3)

The stabilizer $L_{n}(k)^{h}\cong P_{n-1}(k)$.

As a result,

$\mathfrak{p}_{n}(k)^{*}/P_{n}(k)\cong\mathfrak{l}_{n}(k)^{*}/L_{n}(k)\bigsqcup\mathfrak{p}_{n-1}(k)^{*}/P_{n-1}(k).$

###### Proof.

(1) Take any $x\in\mathfrak{p}_{n}(k)^{*}$, if $x|_{\mathfrak{n}_{n}(k)}=0$,
then $x\in\mathfrak{l}_{n}(k)^{*}$. Assume that $x|_{\mathfrak{n}_{n}(k)}\neq
0$. As $L_{n}(k)$ acts on $\mathfrak{n}_{n}(k)^{*}$ transitively, $x$ is
conjugated to an element in $h+\mathfrak{l}_{n}(k)^{*}$.

(2) As $N_{n}(k)$ is abelian, $N_{n}(k)$ acts trivially on
$\mathfrak{n}_{n}(k)^{*}$, so

$N_{n}(k)\subset\mathrm{Stab}_{P}(h|_{\mathfrak{n}_{n}(k)}).$

Using $P_{n}(k)=L_{n}(k)\ltimes N_{n}(k)$, we only need to compute
$\mathrm{Stab}_{L_{n}(k)}(h|_{\mathfrak{n}_{n}(k)})$. As
$h|_{\mathfrak{l}_{n}(k)}=0$,
$\mathrm{Stab}_{L_{n}(k)}(h|_{\mathfrak{n}_{n}(k)})=\mathrm{Stab}_{L_{n}(k)}(h)=L_{n}(k)^{h}$.

Let $X\in\mathfrak{n}_{n}(k)$, $\exp(X)\in N_{n}(k)=\exp\mathfrak{n}_{n}(k)$,
$\exp(X)$ acts on $h$ by

$h\mapsto(Y\mapsto h(\mathrm{Ad}(\exp(-X))(Y)),\forall
Y\in\mathfrak{p}_{n}(k)).$

Since $\mathrm{Ad}(\exp(-X))(Y)=Y-[X,Y],\forall Y\in\mathfrak{p}_{n}(k)$, we
get that

$h(\mathrm{Ad}(\exp(-X))(Y))=h(Y)+h(-[X,Y])=(h+\mathrm{ad}(X)h)(Y),\forall
Y\in\mathfrak{p}_{n}(k).$

As the pairing between $\mathfrak{l}_{n}(k)$ and $\mathfrak{l}_{n}(k)^{*}$ is
nondegenerate, we only need to check that $\mathfrak{l}_{n}(k)^{h}$ is the
zero set of $\\{\mathrm{ad}(X)(h)|X\in\mathfrak{n}\\}$. Take any
$l\in\mathfrak{l}_{n}(k)$ such that $l$ vanishes at
$\\{\mathrm{ad}(X)(h)|X\in\mathfrak{n}\\}$, that is,

$l(\mathrm{ad}(X)(h))=h(-[X,l])=-X(\mathrm{ad}(l)(h))=0,$

so $l\in\mathfrak{l}_{n}(k)^{h}$.

(3) It is easy to check directly. ∎

We go to calculate $\mathrm{Stab}_{P_{n}(k)}(x)$ for any
$x\in\mathfrak{p}_{n}(k)^{*}$.

######  Proposition 2.2.

Let $x$ be an element in $h+\mathfrak{l}_{n}(k)^{*}$, and let $[x]$ be the
image of $x$ in $h+(\mathfrak{l}_{n}(k)^{h})^{*}$ as Proposition 2.1.(2) and
$x^{\prime}=[x]-h$. Then
$\mathrm{Stab}_{P_{n}(k)}(x)=\mathrm{Stab}_{L_{n}(k)^{h}}(x^{\prime})$.

###### Proof.

Take any $g\in\mathrm{Stab}_{P_{n}(k)}(x)$, then $g=ln$, $l\in L_{n}(k)$,
$n\in N_{n}(k)$. As $n$ stabilizes $h|_{\mathfrak{n}_{n}(k)}$ and
$\mathfrak{l}_{n}(k)^{*}$, $l$ stabilizes $h|_{\mathfrak{n}_{n}(k)}$, hence
also stabilizes $h$. As a result, we get a group morphism

$\phi:\mathrm{Stab}_{P_{n}(k)}(x)\to\mathrm{Stab}_{L_{n}(k)^{h}}(x^{\prime}),g=ln\mapsto
l.$

We claim that $\phi$ is an isomorphism. If
$l\in\mathrm{Stab}_{L_{n}(k)^{h}}(x^{\prime})$, Proposition 2.1 shows

$\mathfrak{l}_{n}(k)^{*}/\mathrm{ad}(\mathfrak{n}_{n}(k))h=(L_{n}(k)^{h})^{*},$

and one can choose an $n_{l}\in N_{n}(k)$, such that
$ln_{l}\in\mathrm{Stab}_{P_{n}(k)}(x)$. So we get a group morphism

$\psi:\mathrm{Stab}_{L_{n}(k)^{h}}(x^{\prime})\to\mathrm{Stab}_{P_{n}(k)}(x),l\mapsto
ln_{l}.$

It is obvious that $\phi\circ\psi=1$, and it remains to check that $\phi$ is
injective. Let $ln\in\ker(\phi)$, then $n$ stabilizes $h$. It can be checked
directly that $n=1$. ∎

######  Proposition 2.3.

Let $x$ be an element in $(\mathfrak{l}_{n}(k))^{*}$, then
$\mathrm{Stab}_{P_{n}(k)}(x)=\mathrm{Stab}_{L_{n}(k)}(x)\ltimes N_{n}(k)$.

###### Proof.

It is easy to check directly. ∎

In conclusion, we get the following result.

######  Theorem 2.4.

Every element of $\mathfrak{p}_{n}(k)^{*}$ is $P_{n}(k)$-conjugated to an
element of the form $\alpha=\mathrm{pr}^{\prime}(\xi)$, where
$\xi=\left(\begin{array}[]{cc}A&\\\ &J_{j}\end{array}\right),$ where
$A\in\mathfrak{g}_{n-j}(k)$, and

$J_{j}=\left(\begin{array}[]{cccc}0&&&\\\ 1&\ddots&&\\\ &\ddots&\ddots&\\\
&&1&0\end{array}\right)\in\mathfrak{g}_{j}(k).$

The stabilizer $\mathrm{Stab}_{P_{n}(k)}(\alpha)$ is isomorphic to
$\mathrm{Stab}_{G_{n-j}(k)}(A)\ltimes N_{n-j+1}(k)$.

###### Proof.

Choose
$h=\mathrm{pr}^{\prime}(\left(\begin{array}[]{ccc}0_{(n-1)\times(n-2)}&0_{(n-1)\times
1}&0_{(n-1)\times 1}\\\ 0_{1\times(n-2)}&1&0\end{array}\right))$, applying
Proposition 2.1, we get that every element of $\mathfrak{p}_{n}(k)^{*}$ is
$P_{n}(k)$-conjugated to $\mathrm{pr}^{\prime}(\left(\begin{array}[]{cc}A&\\\
&0\end{array}\right))$ for some $A\in\mathfrak{g}_{n-1}(k)$ or
$\mathrm{pr}^{\prime}(\left(\begin{array}[]{cc}A&\\\ &0\end{array}\right))+h$,
where $A\in\bar{\mathfrak{p}}_{n-1}(k)$. Using the result of Proposition 2.1
and by induction, one can prove the first statement. Applying Proposition 2.2
and 2.3, we can get the stabilizer as the statement. ∎

######  Definition 2.1.

Define the depth of such element in $\mathfrak{p}_{n}(k)^{*}$ as $j$. It is
well defined by the proof of Theorem 2.4.

######  Corollary 2.5.

The elements with depth $n$ form a unique dense open $P_{n}(k)$-orbit in
$\mathfrak{p}_{n}(k)^{*}$ and they are all strongly regular elements in
$\mathfrak{p}_{n}(k)^{*}$.

###### Proof.

Applying Theorem 2.4, we have $\alpha\in\mathfrak{p}_{n}(k)^{*}$ has depth $n$
if and only if $\mathrm{Stab}_{P_{n}(k)}(\alpha)$ has minimal dimension, if
and only if $\mathrm{Stab}_{P_{n}(k)}(\alpha)=1$. So we get the statement
immediately. ∎

######  Example 2.1.

Set

$B=\left(\begin{array}[]{cccc}a_{1}&&&\\\ &\ddots&&\\\ &&a_{n-1}&\\\
b_{1}&\cdots&b_{n-1}&0\end{array}\right)\in\mathfrak{g}_{n}(k),$

$a_{i}\neq a_{j}$ when $1\leq i\neq j\leq n-1$, $b_{i}\neq 0,$ when $1\leq
i\leq n-1$, and $f=\mathrm{pr}^{\prime}(B)\in\mathfrak{p}_{n}(k)^{*}$. Then
$f$ has depth $n$.

###### Proof.

It is sufficient to check that $\mathrm{Stab}_{P_{n}(k)}(f)=\\{1\\}.$ See Liu-
Yu [17] for a proof. An alternative proof is given in Lemma 3.2 and Lemma 3.3.
∎

##  3\. Geometry of the moment map
$\mathrm{p}:\mathcal{O}_{f}\to\mathfrak{p}^{*}$

###  3.1. Moment map in the $\mathrm{GL}_{n}(\mathbb{C})$ case

Let $n\in\mathbb{Z}_{+}$, let $G=G_{n}(\mathbb{C})$ and let
$P=P_{n}(\mathbb{C})$ be the mirabolic subgroup of $G$. $\mathfrak{g}$ (resp.
$\mathfrak{p}$) denotes the Lie algebra of $G$ (resp. of $P$).

Let $\mathfrak{g}^{*}$ (resp. $\mathfrak{p}^{*}$) denote the dual space of
$\mathfrak{g}$ (resp. of $\mathfrak{p}$). By the Lie theory, we have

$n=\min_{x\in\mathfrak{g}^{*}}\dim(\mathrm{Stab}_{G}(x)).$

For any element $f\in\mathfrak{g}^{*}$, we call $f$ a regular element in
$\mathfrak{g}^{*}$, or say that $f$ is regular, if
$\dim(\mathrm{Stab}_{G}(f))=n$. And we call a $G$-coadjoint orbit
$\mathcal{O}_{f}$ a regular orbit if $f$ is regular.

We will calculate the moment map
$\mathrm{p}:\mathcal{O}_{f}\to\mathfrak{p}^{*}$ for any
$f\in\mathfrak{g}^{*}$. We find that the result about any regular element
$f\in\mathfrak{g}^{*}$ is similar to the results about semisimple regular
elements, for example, $\mathrm{p}(\mathcal{O}_{f})$ contains the unique
strongly regular orbit of $\mathfrak{p}^{*}$ (see Liu-Yu [17]). Using the
results about the regular elements, we can settle the cases of singular
elements in $\mathfrak{g}^{*}$.

Furthermore, using the results of the case of $\mathfrak{g}_{n}(\mathbb{C})$,
one can settle the case of $\mathfrak{g}_{n}(\mathbb{R})$ immediately.

####  3.1.1. Regular orbits

Write

$J_{s}(a)=\left(\begin{array}[]{cccc}a&&&\\\ 1&a&&\\\ &\ddots&\ddots&\\\
&&1&a\end{array}\right)\in\mathfrak{g}_{s}(\mathbb{C}),s\in\mathbb{Z}\
\text{and}\ s\geq 2,a\in\mathbb{C},$

and $J_{1}(a)=a_{1\times 1},a\in\mathbb{C}$. Set $J_{s}=J_{s}(0)$, $\forall
s\in\mathbb{Z}_{+}$.

Let $a_{i}\in\mathbb{C}$, $s_{i}\in\mathbb{Z}_{+}$, $\forall 1\leq i\leq m$,
such that $a_{i}\neq a_{j}$ when $1\leq i\neq j\leq m$, and
$\sum_{i=1}^{m}s_{i}=n$. Set

$\xi=\mathrm{diag}(J_{s_{1}}(a_{1}),\cdots,J_{s_{m}}(a_{m}))\in\mathfrak{g},$

and set $f=\mathrm{pr}(\xi)\in\mathfrak{g}^{*}$. It is clear that every
regular $G$-coadjoint orbit in $\mathfrak{g}^{*}$ is of the form
$\mathcal{O}_{f}$.

Firstly, we describe all $P$-orbits in $\mathcal{O}_{f}$.

For any $s\in\mathbb{Z}_{+}$, let

$L_{s}=\mathrm{Stab}_{G_{s}(\mathbb{C})}(J_{s})=\\{\left(\begin{array}[]{cccc}x_{1}&&&\\\
x_{2}&x_{1}&&\\\ \vdots&\ddots&\ddots&\\\ \
x_{s}&\cdots&x_{2}&x_{1}\end{array}\right)|\ x_{i}\in\mathbb{C},1\leq i\leq
s,x_{1}\neq 0\\},$

Set

$L=\mathrm{Stab}_{G}(f)=\\{\mathrm{diag}(X_{s_{1}},\cdots,X_{s_{m}})\ |\
X_{s_{i}}\in L_{s_{i}},1\leq i\leq m\\}.$

We have $\mathcal{O}_{f}\cong G/L$ and $P\setminus\mathcal{O}_{f}\cong
P\setminus G/L$.

Define the right action of $G$ on $\mathbb{C}^{n}$ by

$v\cdot g=(x_{1},\cdots,x_{n})A,$

where $v=(x_{1},\cdots,x_{n})^{t}\in\mathbb{C}^{n}$ and $g=A\in
G_{n}(\mathbb{C})$. Set $v_{0}=(0,\cdots,0,1)^{t}$, we get
$\mathrm{Stab}_{G}(v_{0})=P$, and $\mathbb{C}^{n}-\\{0\\}\cong P\setminus G$.

To find the representatives of $P\setminus\mathcal{O}_{f}$, we need to find
the representatives of $(\mathbb{C}^{n}-\\{0\\})/L$.

######  Proposition 3.1.

Set $I_{0}=\\{1,2,\cdots,n\\},K_{0}=\\{1,\cdots,m\\}$ and $s_{0}=0$. For any
$\emptyset\neq K\subseteq K_{0}$ and $r_{k}\in\\{1,2,\cdots,s_{k}\\}$ for
$k\in K$, define

$I=\\{\sum_{j=0}^{k-1}s_{j}+r_{k}|\ k\in K\\}\subseteq I_{0}$

attached to $K$ and $\\{r_{k},k\in K\\}$. Define $v_{I}=(x_{1},\cdots,x_{n})$
by $x_{i}=1$ when $i\in I$ and $x_{i}=0$ when $i\notin I$. Let $I_{a}$ be the
set of all $I$’s constructed as above.

Then $\\{v_{I}|\ I\in I_{a}\\}$ form all different representatives of
$(\mathbb{C}^{n}-\\{0\\})/L$.

###### Proof.

The $L$-orbit of $v_{I}$ is the set

$\displaystyle\\{(y_{1},\cdots,y_{n})^{t}\ $ $\displaystyle|\
y_{i}\in\mathbb{C},1\leq i\leq n;y_{i}\neq 0\ \text{if}\ i\in I;y_{i}=0\
\text{if}\ i\notin J,$ $\displaystyle J=\bigsqcup_{k\in
K}\\{\sum_{j=0}^{k-1}s_{j}+1,\cdots,\sum_{j=0}^{k-1}s_{j}+r_{k}\\}\\}.$

It follows that $\\{v_{I}|\ I\in I_{a}\\}$ are all different representatives
of $(\mathbb{C}^{n}-\\{0\\})/L$. ∎

Let $g_{I}\in G$ be the element corresponding to the representative $v_{I}$
under the isomorphism $\mathbb{C}^{n}-\\{0\\}\cong P\setminus G$, then

$G=\bigsqcup_{I\in I_{a}}Pg_{I}L.$

So there are $\\#I_{a}=\prod_{1\leq i\leq m}(s_{i}+1)-1$ $P$-orbits in
$\mathcal{O}_{f}$,

$\\{P\cdot(g_{I}\cdot f)\ |\ I\in I_{a}\\}.$

Set $S=\\{s_{1},s_{1}+s_{2},\cdots,n\\}$, then $Pg_{S}L$ is the unique dense
open subset of $G$ among $\\{Pg_{I}L,I\in I_{a}\\}$, since $v_{S}\cdot L$ is
the unique dense $L$-orbit in $\mathbb{C}^{n}-\\{0\\}$. As a consequence,
$P\cdot(g_{S}\cdot f)$ is the unique dense open $P$-orbit in
$\mathcal{O}_{f}$.

By definition, we can choose $\\{g_{I}|\ I\in I_{a}\\}$ as follows.

When $n\in I$, $g_{I}=\left(\begin{array}[]{cc}I_{n-1}&0\\\
\beta&1\end{array}\right)$, $\beta=(x_{1},\cdots,x_{n-1})$ is defined by
$x_{i}=1$ when $i\in I-\\{n\\}$ and $x_{i}=0$ when $i\notin I-\\{n\\}$.

When $n\notin I$, let $k=\max\\{i:i\in I\\}$,
$g_{I}=\left(\begin{array}[]{ccc}I_{k-1}&&\\\ &&I_{n-k}\\\
\beta&1&\end{array}\right)$, $\beta=(x_{1},\cdots,x_{k-1})$ is defined by
$x_{i}=1$ when $i\in I-\\{k\\}$ and $x_{i}=0$ when $i\notin I-\\{k\\}$.

We go to calculate the image of each $P$-orbit of $g_{I}\cdot f$ under the
moment map $\mathrm{p}:\mathcal{O}_{f}\to\mathfrak{p}^{*}$ and find that
$\mathrm{p}(\mathcal{O}_{f})$ contains the unique dense open $P$-orbit in
$\mathfrak{p}^{*}$, more precisely, $\mathrm{p}(g_{S}\cdot f)$ is a strongly
regular element in $\mathfrak{p}^{*}$.

######  Lemma 3.2.

We have $\mathrm{Stab}_{P}(g_{S}\cdot f)=1$, so the dimension of $P$-orbit of
$g_{S}\cdot f$ equals to $\mathrm{dim}P=n^{2}-n$.

###### Proof.

It is obvious that

$\mathrm{Stab}_{P}(g_{S}\cdot f)=P\cap g_{S}\mathrm{Stab}_{G}(f)g_{S}^{-1}.$

And

$P\cap g_{S}\mathrm{Stab}_{G}(f)g_{S}^{-1}=P\cap
g_{S}Lg_{S}^{-1}=g_{S}(g_{S}^{-1}Pg_{S}\cap L)g_{S}^{-1}.$

Since $g_{S}^{-1}Pg_{S}$ is the stabilizer of $v_{0}\cdot g_{S}=v_{S}$, we
have $g_{S}^{-1}Pg_{S}\cap L=\mathrm{Stab}_{L}(v_{S})$.

By direct calculation, $\mathrm{Stab}_{L}(v_{S})=1$, so
$\mathrm{Stab}_{P}(g_{S}\cdot f)=1$. ∎

For convenience, let $[U_{1},\cdots,U_{k}]$ denote an $n\times n$ matrix with
$k$ blocks, where the $i$-th block is an $n\times n_{i}$ ($1\leq n_{i}\leq n$)
submatrix $U_{i}$ for $1\leq i\leq k$, and $\sum_{i=1}^{k}n_{i}=n$.

######  Lemma 3.3.

If $x\in\mathfrak{p}^{*}$ is not strongly regular, then for any element
$z\in\mathrm{p}^{\prime-1}(x)$, the $P$-orbit of $z$ has dimension $<n^{2}-n$.
Here $\mathrm{p}^{\prime}:\mathfrak{g}^{*}\to\mathfrak{p}^{*}$ is the natural
projection.

###### Proof.

Take any element $x=\mathrm{pr}^{\prime}([X,0_{n\times
1}])\in\mathfrak{p}^{*}$ which is not strongly regular, by the definition,
$\mathrm{p}^{\prime-1}(x)=\\{\mathrm{pr}([X,\tilde{Y}]),\tilde{Y}\in\mathbb{C}^{n}$}.
Fix an element $z\in\mathrm{p}^{\prime-1}(x)$, $z=\mathrm{pr}([X,Y])$ for some
$Y=(y_{1},\cdots,y_{n})^{t}\in\mathbb{C}^{n}$.

Since the $P$-orbit of $z$ has dimension

$\mathrm{dim}P-\mathrm{dim}(\mathrm{Stab}_{P}(z))=n^{2}-n-\mathrm{dim}(\mathrm{Stab}_{P}(z)),$

we only need to prove $\mathrm{dim}(\mathrm{Stab}_{P}(z))\geq 1$.

Since $\\{\mathrm{pr}([0,\tilde{Y}]),\tilde{Y}\in\mathbb{C}^{n}\\}$ is a
$P$-stable subspace, $\mathrm{Stab}_{P}(z)\subseteq\mathrm{Stab}_{P}(x)$.

Using the result of the classification of $P$-coadjoint orbits in
$\mathfrak{p}^{*}$, we know that $x$ is $P$-conjugated to an element of the
form $\alpha=\mathrm{pr}^{\prime}(\left(\begin{array}[]{cc}A&\\\
&J_{k}\end{array}\right))$, where $A\in\mathfrak{g}_{n-k}(\mathbb{C})$,

$\mathrm{Stab}_{P}(\alpha)\cong\mathrm{Stab}_{G_{n-k}(\mathbb{C})}(A)\ltimes
N_{n-k+1}(\mathbb{C})$

with dimension $\geq 2(n-k)$ and $k\leq n-1$ as $x$ is not strongly regular.
It doesn’t matter to replace $x$ by the elements in its $P$-orbit and we
assume that $x=\alpha$.

By direct calculation, we get

$\displaystyle\mathrm{Stab}_{P}(x)$
$\displaystyle=\\{\left(\begin{array}[]{cc}C&\\\ &I_{k}\end{array}\right)|\
C\in
G_{n-k}(\mathbb{C}),CAC^{-1}=A\\}\ltimes\\{\left(\begin{array}[]{cc}I_{n-k}&N\\\
&I_{k}\end{array}\right)$ $\displaystyle|\ N=[N_{1},\cdots,N_{k}],\
N_{i}\in\mathbb{C}^{n-k},1\leq i\leq k,N_{i}=A^{i-1}N_{1},2\leq i\leq k\\}.$

Let $t=\left(\begin{array}[]{cc}C&CN\\\
&I_{k}\end{array}\right)\in\mathrm{Stab}_{P}(x)$, then
$t\in\mathrm{Stab}_{P}(z)$ if and only if

$\left(\begin{array}[]{cc}C&CN\\\
&I_{k}\end{array}\right)(\left(\begin{array}[]{cc}A&\\\
&J_{k}\end{array}\right)+[0,Y])\left(\begin{array}[]{cc}C^{-1}&-N\\\
&I_{k}\end{array}\right)=\left(\begin{array}[]{cc}A&\\\
&J_{k}\end{array}\right)+[0,Y],$

So

$t\in\mathrm{Stab}_{P}(z)\Leftrightarrow-
CA^{k}N_{1}+C\left(\begin{array}[]{c}y_{1}\\\ \vdots\\\
y_{n-k}\end{array}\right)+CN\left(\begin{array}[]{c}y_{n-k+1}\\\ \vdots\\\
y_{n}\end{array}\right)=\left(\begin{array}[]{c}y_{1}\\\ \vdots\\\
y_{n-k}\end{array}\right).$

Since $1\in\mathrm{Stab}_{P}(z)$, we obtain a nonempty subvariety
$\mathrm{Stab}_{P}(z)$ of $\mathrm{Stab}_{P}(x)$ given by $n-k$ algebraic
equations. Therefore

$\dim(\mathrm{Stab}_{P}(z))\geq\mathrm{dim}(\mathrm{Stab}_{P}(x))-(n-k).$

Since $\mathrm{dim}(\mathrm{Stab}_{P}(x))\geq 2(n-k)$ and $n-k\geq 1$, we get
$\dim(\mathrm{Stab}_{P}(z))\geq 1$. ∎

######  Proposition 3.4.

$\mathrm{p}(g_{S}\cdot f)$ is a strongly regular element in
$\mathfrak{p}^{*}$.

###### Proof.

By Lemma 3.2, the $P$-orbit of $g_{S}\cdot f$ has dimension $n^{2}-n$.
Applying Lemma 3.3, we see that $\mathrm{p}(g_{S}\cdot f)$ is strongly
regular. ∎

For convenience, let $A(I_{1},I_{2})$ denote the submatrix of the matrix $A$
with rows indexed by $I_{1}$ and columns indexed by $I_{2}$, and let
$a(I_{1})$ denote the subvector of the vector $a$ with index $I_{1}$.

With the result above, we can calculate the $P$-orbit of
$\mathrm{p}(g_{I}\cdot f)$ for any $I\in I_{a}$. We find out that
$\\{\mathrm{p}(g_{I}\cdot f)\ |\ I\in I_{a}\\}$ represent all different
$P$-orbits in $\mathrm{p}(\mathcal{O}_{f})$. More precisely, we have the
following result.

######  Proposition 3.5.

For any $\emptyset\neq K\subseteq K_{0}$ and $r_{k}\in\\{1,2,\cdots,s_{k}\\}$
for $k\in K_{0}$, define $I=\\{\sum_{j=0}^{k-1}s_{j}+r_{k}|\ k\in K\\}$ and
$g_{I}$ as above. Then $\mathrm{p}(g_{I}\cdot f)$ has depth $d=\sum_{k\in
K}r_{k}$. Moreover, if we set

$t_{k}=\left\\{\begin{array}[]{ll}r_{k},&k\in K,\\\ 0,&k\in
K_{0}-K,\end{array}\right.$

then $\mathrm{p}(g_{I}\cdot f)$ is $P$-conjugated to
$\mathrm{pr}^{\prime}(\eta)$, where

$\eta=\mathrm{diag}(J_{s_{1}-t_{1}}(a_{1}),\cdots,J_{s_{m}-t_{m}}(a_{m}),J_{d}).$

###### Proof.

Firstly, we change $\mathrm{p}(g_{I}\cdot
f)=\mathrm{pr}^{\prime}(g_{I}\cdot\xi)$ to a good form by a series of explicit
$P$-conjugations, then we use the result of Proposition 3.4 to get the
statement. Set $A_{I}=g_{I}\cdot\xi$.

In the case of $n\in I$, by direct calculation, we have

$A_{I}=\mathrm{diag}(J_{s_{1}}(a_{1}),\cdots,J_{s_{m}}(a_{m}))+\left(\begin{array}[]{c}0_{(n-1)\times
n}\\\ \alpha\end{array}\right),$

where $\alpha=(\alpha_{1},\cdots,\alpha_{m})$,
$\alpha_{k}=(\alpha_{k,1},\cdots,\alpha_{k,s_{k}})$ is defined by

$\alpha_{k}=\left\\{\begin{array}[]{ll}(\underbrace{0,\cdots,0}_{s_{k}}),&k\notin
K-\\{n\\},\\\ (a_{k}-a_{m},\underbrace{0,\cdots,0}_{s_{k}-1}),&k\in
K-\\{n\\},r_{k}=1\\\
(\underbrace{0,\cdots,0}_{r_{k}-2},1,a_{k}-a_{m},\underbrace{0,\cdots,0}_{s_{k}-r_{k}}),&k\in
K-\\{n\\},r_{k}>1.\end{array}\right.$

We claim that $\mathrm{pr}^{\prime}(A_{I})$ is $P$-conjugated to
$\mathrm{pr}^{\prime}(\zeta)$, where

$\zeta=\left(\begin{array}[]{ccccc}J_{t_{1}}(a_{1})&&&&\\\
&J_{s_{1}-t_{1}}(a_{1})&&&\\\ &&\ddots&&\\\ &&&J_{t_{m}}(a_{m})&\\\
&&&&J_{s_{m}-t_{m}}(a_{m})\end{array}\right)+\left(\begin{array}[]{c}0_{(n-1)\times
n}\\\ \alpha\end{array}\right).$

It is sufficient to prove that $\mathrm{pr}^{\prime}(A_{I})$ is $P$-conjugated
to $\mathrm{pr}^{\prime}(\zeta_{k})$ for $0\leq k\leq m$, where

$\zeta_{k}=\left(\begin{array}[]{cccccccc}J_{t_{1}}(a_{1})&&&&&&&\\\
&J_{s_{1}-t_{1}}(a_{1})&&&&&&\\\ &&\ddots&&&&&\\\ &&&J_{t_{k}}(a_{k})&&&&\\\
&&&&J_{s_{k}-t_{k}}(a_{k})&&&\\\ &&&&&J_{s_{k+1}}(a_{k+1})&&\\\
&&&&&&\ddots&\\\
&&&&&&&J_{s_{m}}(a_{m})\end{array}\right)+\left(\begin{array}[]{c}0_{(n-1)\times
n}\\\ \alpha\end{array}\right),$

since $\zeta_{m}=\zeta$.

Let’s prove it by induction on $k$. When $k=0$, it is trivially true. Assume
that $k\geq 1$ and it is true for $k-1$.

If $k\notin K$, or if $k\in K$ and $r_{k}=s_{k}$, then $\zeta_{k}=\zeta_{k-1}$
and so it is true for $k$. Assume that $k\in K,r_{k}<s_{k}$, we go to prove
that it is true for $k$. Since we assume that $\mathrm{pr}^{\prime}(A_{I})$ is
$P$-conjugated to $\mathrm{pr}^{\prime}(\zeta_{k-1})$, it is sufficient to
prove that $\mathrm{pr}^{\prime}(\zeta_{k-1})$ is $P$-conjugated to
$\mathrm{pr}^{\prime}(\zeta_{k})$.

Step 1. We have

$\zeta_{k-1}=\left(\begin{array}[]{ccccc}B_{1}&0&0&0&0\\\ 0&C_{1}&0&0&0\\\
0&D&C_{2}&0&0\\\ 0&0&0&B_{2}&0\\\
v_{1}&w_{1}&0&v_{2}&a_{m}\end{array}\right),$

where

$B_{1}=\mathrm{diag}(J_{t_{1}}(a_{1}),J_{s_{1}-t_{1}}(a_{1}),\cdots,J_{t_{k-1}}(a_{k-1}),J_{s_{k-1}-t_{k-1}}(a_{k-1})),$
$B_{2}=\mathrm{diag}(J_{s_{k+1}}(a_{k+1}),\cdots,J_{s_{m-1}}(a_{m-1}),J_{s_{m}-1}(a_{m})),$
$C_{1}=J_{r_{k}}(a_{k}),C_{2}=J_{s_{k}-r_{k}}(a_{k}),v_{1}=(\alpha_{1},\cdots,\alpha_{k-1}),w_{1}=(\alpha_{k,1},\cdots,\alpha_{k,r_{k}}),$
$v_{2}=\left\\{\begin{array}[]{ll}(\alpha_{k+1},\cdots,\alpha_{m-1}),&s_{m}=1,\\\
(\alpha_{k+1},\cdots,\alpha_{m-1},\underbrace{0,\cdots,0}_{s_{m}-2},1),&s_{m}>1,\end{array}\right.$
$D\ \text{is the}\ (s_{k}-r_{k})\times r_{k}\ \text{matrix with}\ 1\ \text{in
the position}\ (1,r_{k})\ \text{and zero elsewhere}.$

Let

$P_{1}=\left(\begin{array}[]{ccccc}I_{s_{1}+\cdots+s_{k-1}}&0&0&0&0\\\
0&I_{r_{k}}&0&0&0\\\ 0&N_{1}&I_{s_{k}-r_{k}}&0&N_{2}\\\
0&0&0&I_{s_{k+1}+\cdots+s_{m}-1}&0\\\ 0&0&0&0&1\end{array}\right)$

with $N_{1}=[0_{(s_{k}-r_{k})\times(r_{k}-1)},n_{1}]$, $N_{2}=-n_{1}$ and

$n_{1}=(\frac{1}{a_{k}-a_{m}},-\frac{1}{(a_{k}-a_{m})^{2}},\cdots,\frac{(-1)^{s_{k}-r_{k}+1}}{(a_{k}-a_{m})^{s_{k}-r_{k}}})^{t}\in\mathbb{C}^{s_{k}-r_{k}}.$

By direct calculation, we get

$P_{1}\zeta_{k-1}P_{1}^{-1}=\left(\begin{array}[]{ccccc}B_{1}&0&0&0&0\\\
0&C_{1}&0&0&0\\\ N_{2}v_{1}&0&C_{2}&N_{2}v_{2}&a_{m}N_{2}-A_{2}N_{2}\\\
0&0&0&B_{2}&0\\\ v_{1}&w_{1}&0&v_{2}&a_{m}\end{array}\right).$

Step 2. Let

$P_{2}=\left(\begin{array}[]{ccccc}I_{s_{1}+\cdots+s_{k-1}}&0&0&0&0\\\
0&I_{r_{k}}&0&0&0\\\ M_{1}&0&I_{s_{k}-r_{k}}&M_{2}&0\\\
0&0&0&I_{s_{k+1}+\cdots+s_{m}-1}&0\\\ 0&0&0&0&1\end{array}\right),$

$M_{1}$ is an $(s_{k}-r_{k})\times(s_{1}+\cdots+s_{k-1})$ matrix, and $M_{2}$
is an $(s_{k}-r_{k})\times(s_{k+1}+\cdots+s_{m}-1)$ matrix, then
$P_{2}(P_{1}\zeta_{k-1}P_{1}^{-1})P_{2}^{-1}=$

$\left(\begin{array}[]{ccccc}B_{1}&0&0&0&0\\\ 0&C_{1}&0&0&0\\\
N_{2}v_{1}+(M_{1}B_{1}-C_{2}M_{1})&0&C_{2}&N_{2}v_{2}+(M_{2}B_{2}-C_{2}M_{2})&a_{m}N_{2}-A_{2}N_{2}\\\
0&0&0&B_{2}&0\\\ v_{1}&w_{1}&0&v_{2}&a_{m}\end{array}\right).$

By solving the linear equations, one see that there exists $M_{1}$ and
$M_{2}$, such that

$N_{2}v_{1}+(M_{1}B_{1}-C_{2}M_{1})=0,\text{and}\
N_{2}v_{2}+(M_{2}B_{2}-C_{2}M_{2})=0.$

Actually, we can write $B_{1}=D^{\prime}+N^{\prime}$ and
$C_{2}=D^{\prime\prime}+N^{\prime\prime}$ as Jordan decomposition, where
$D^{\prime},D^{\prime\prime}$ are semisimple matrices and
$N^{\prime},N^{\prime\prime}$ are nilpotent matrices. It is easy to see that
the linear map

$M_{1}\mapsto M_{1}N^{\prime}-N^{\prime\prime}M_{1}$

is nilpotent, the linear map

$M_{1}\mapsto M_{1}D^{\prime}-D^{\prime\prime}M_{1}$

is semisimple, and they commute with each other. Moreover, the linear map
$M_{1}\mapsto M_{1}D^{\prime}-D^{\prime\prime}M_{1}$ is nondegenerate since
$B_{1}$ and $C_{2}$ have no common eigenvalues. Therefore, we can easily see
that the equation $N_{2}v_{1}+(M_{1}B_{1}-C_{2}M_{1})=0$ has a (unique)
solution $M_{1}$. Similarly we can get $M_{2}$.

Using such $P_{2}$ defined as above, we have
$\mathrm{pr}^{\prime}(P_{2}(P_{1}\zeta_{k-1}P_{1}^{-1})P_{2}^{-1})=\mathrm{pr}^{\prime}(\zeta_{k})$.
By induction, one can finish the proof of the claim.

Let $\zeta^{\prime}$ be the submatrix of $\zeta$ with the rows and columns
indexed by

$\bigsqcup_{h\in
K}\\{\sum_{j=0}^{h-1}s_{j}+1,\cdots,\sum_{j=0}^{h-1}s_{j}+r_{h}\\}.$

Applying Proposition 3.4, we see that
$\mathrm{pr}^{\prime}_{d}(\zeta^{\prime})$ is $P_{d}(\mathbb{C})$-conjugated
to $\mathrm{pr}^{\prime}_{d}(J_{d})$, where
$\mathrm{pr}^{\prime}_{d}:\mathfrak{g}_{d}(\mathbb{C})\to\mathfrak{p}_{d}(\mathbb{C})^{*}$.
So $\mathrm{pr}^{\prime}(\zeta)$ is $P$-conjugated to the element
$\mathrm{pr}^{\prime}(\eta)$ in the statement, and so is
$\mathrm{p}(g_{I}\cdot\xi)$ .

In the case of $n\notin I,m\in K$, one can prove the statement similarly. We
have

$A_{I}=\left(\begin{array}[]{ccc}B_{1}&0&0\\\ C&B_{2}&v_{1}\\\
v_{2}&0&a_{m}\end{array}\right),$

where

$B_{1}=\mathrm{diag}(J_{s_{1}}(a_{1}),\cdots,J_{s_{m-1}}(a_{m-1}),J_{r_{m}-1}(a_{m})),B_{2}=J_{s_{m}-r_{m}}(a_{m}),$
$C=\left(\begin{array}[]{c}-v_{I}({1,\cdots,n-(s_{m}-r_{m})-1})\\\
0_{(s_{m}-r_{m})\times(n-(s_{m}-r_{m})-1)}\end{array}\right),$
$v_{1}=(1,\underbrace{0,\cdots,0}_{s_{m}-r_{m}-1})^{t},v_{2}=\left\\{\begin{array}[]{ll}(\alpha({1,\cdots,n-s_{m}}),\underbrace{0,\cdots,0}_{r_{m}-2},1),&r_{m}\geq
2\\\ (\alpha({1,\cdots,n-s_{m}}),&r_{m}=1.\end{array}\right.$

We can use the method in the step 2 to show that $\mathrm{p}(A_{I})$ is
$P$-conjugated to $\mathrm{pr}(\zeta^{\prime})$, where

$\zeta^{\prime}=\left(\begin{array}[]{ccc}B_{1}&0&0\\\ 0&B_{2}&v_{1}\\\
v_{2}&0&a_{m}\end{array}\right).$

Then the submatrix of $\zeta^{\prime}$ with rows and columns indexed by
$\\{1,\cdots,n\\}-\\{n-(s_{k}-r_{k}),\cdots,n-1\\}$ is

$\left(\begin{array}[]{cc}B_{1}&0\\\ v_{2}&a_{m}\end{array}\right),$

which is of the form in the case of $n\in I$. Applying the result of the case
of $n\in I$, we can get the statement in the case of $n\notin I,m\in K$.

In the case of $n\notin I,m\notin K$, assume $m_{0}=\max\\{k\in K\\}$. By
directly calculation, we get that the submatrix of $A_{I}$ with rows and
columns indexed by $\\{1,2,\cdots,s_{1}+\cdots+s_{m_{0}}-1\\}\cup\\{n\\}$ is
of the form in the above two cases. Applying the results of the above two
cases, we can get the statement in the case of $n\notin I,m\notin K$.

This completes the proof of the statement. ∎

####  3.1.2. General orbits

We go to calculate the moment map
$\mathrm{p}:\mathcal{O}_{f}\to\mathfrak{p}^{*}$ for any
$f\in\mathfrak{g}^{*}$. Write

$J^{l}_{k}(a)=\mathrm{diag}(\underbrace{J_{k}(a),\cdots,J_{k}(a)}_{l\
\text{blocks}}),$

for any $a\in\mathbb{C}$, $k,l\in\mathbb{Z}_{+}$.

Let $m\in\mathbb{Z}_{+}$, $m\leq n$. Let $a_{1}$, $\cdots$, $a_{m}$
$\in\mathbb{C},a_{i}\neq a_{j}$ when $1\leq i\neq j\leq m$. Set

$J_{k,a_{i}}^{l}=J_{k}^{l}(a_{i})$

for $1\leq i\leq m$, $k,l\in\mathbb{Z}_{+}$. Let

$\xi=\mathrm{diag}(J_{k_{11},a_{1}}^{l_{11}},J_{k_{12},a_{1}}^{l_{12}},\cdots,J_{k_{1r_{1}},a_{1}}^{l_{1r_{1}}},\cdots,J_{k_{m1},a_{m}}^{l_{m1}},J_{k_{m2},a_{m}}^{l_{m2}},\cdots,J_{k_{mr_{m}},a_{m}}^{l_{mr_{m}}}),$

where $r_{j}\in\mathbb{Z}_{+},1\leq j\leq m$; and
$l_{ji},k_{ji}\in\mathbb{Z}_{+}$, $\forall 1\leq j\leq m,1\leq i\leq r_{j}$;
$k_{ji_{1}}<k_{ji_{2}}$ when $1\leq j\leq m$, $1\leq i_{1}<i_{2}\leq r_{j}$;
and

$\sum_{1\leq i\leq m,1\leq i\leq r_{j}}k_{ji}\cdot l_{ji}=n.$

So $\xi$ has $m$ different eigenvalues, denoted by $a_{j},1\leq j\leq m$, and
has $r_{j}$ different kinds of Jordan blocks for the eigenvalue $a_{j}$, with
size $k_{ji}$ and $l_{ji}$ pieces of such size for $1\leq i\leq r_{j}$.

Set $f=\mathrm{pr}(\xi)$. It is clear that every orbit in $\mathfrak{g}^{*}$
is of the form $\mathcal{O}_{f}$.

Now, we fix a $\xi$, and set
$\xi_{j}=\mathrm{diag}(J_{k_{j1},a_{j}}^{l_{j1}},J_{k_{j2},a_{j}}^{l_{j2}},\cdots,J_{k_{jr_{j}},a_{j}}^{l_{jr_{j}}})$,
and $n_{j}=\sum_{r=1}^{r_{j}}k_{jr}\cdot l_{jr}$ for $1\leq j\leq m$. For
convenience, set $n_{0}=0$, $k_{j0}=0,l_{j0}=0$ for $1\leq j\leq m$.

Let $M_{s\times s}(\mathbb{C})$ be the set of all $s\times s$ matrices over
$\mathbb{C}$. Write

$K(x_{1},\cdots,x_{s})=\left(\begin{array}[]{cccc}x_{1}&&&\\\ x_{2}&x_{1}&&\\\
\vdots&\ddots&\ddots&\\\ \
x_{s}&\cdots&x_{2}&x_{1}\end{array}\right),x_{i}\in\mathbb{C},1\leq i\leq s,$

then

$\\{X\in M_{s\times s}(\mathbb{C})\ |\
XJ_{s}=J_{s}X\\}=\\{K(x_{1},\cdots,x_{s})\ |\ x_{i}\in\mathbb{C},1\leq i\leq
s\\},$

and we have the following Proposition.

######  Proposition 3.6.

For any $a\in\mathbb{C}$, $k,l\in\mathbb{Z}_{+}$,

$\displaystyle L_{k,l}$ $\displaystyle=\mathrm{Stab}_{\mathrm{GL}_{k\cdot
l}(\mathbb{C})}(J^{l}_{k}(a))=\\{\left(\begin{array}[]{ccc}K(x_{11,1},\cdots,x_{11,k})&\cdots&K(x_{1l,1},\cdots,x_{1l,k})\\\
\vdots&\ddots&\vdots\\\
K(x_{l1,1},\cdots,x_{l1,k})&\cdots&K(x_{ll,1},\cdots,x_{ll,k})\end{array}\right)$
$\displaystyle|\ x_{ij,h}\in\mathbb{C},\forall 1\leq i,j\leq l,1\leq h\leq
k,\det(x_{ij,1})_{1\leq i,j\leq l}\neq 0\\}.$

###### Proof.

By direct calculation, we see that the determinate of the element on the right
hand side equals to $(\det(x_{ij,1})_{1\leq i,j\leq l})^{k}$ and get the
statement immediately. ∎

For any $k_{1},k_{2}\in\mathbb{Z}_{+}$ such that $k_{1}\leq k_{2}$, and
$x_{1},\cdots,x_{k_{1}}\in\mathbb{C}$, define $H_{1}(x_{1},\cdots,x_{k_{1}})$
as the $k_{1}\times k_{2}$ matrix

$[K(x_{1},\cdots,x_{k_{1}}),0_{k_{1}\times(k_{2}-k_{1})}],$

and define $H_{2}(x_{1},\cdots,x_{k_{1}})$ is the $k_{2}\times k_{1}$ matrix

$\left(\begin{array}[]{c}0_{(k_{2}-k_{1})\times k_{1}}\\\
K(x_{1},\cdots,x_{k_{1}})\end{array}\right).$

######  Proposition 3.7.

For any $a\in\mathbb{C}$, $k_{1},k_{2},l_{1},l_{2}\in\mathbb{Z}_{+}$,
$k_{1}\leq k_{2}$,

$\displaystyle M_{k_{1},k_{2}}^{l_{1},l_{2}}$ $\displaystyle=\\{X\in
M_{k_{1}l_{1}\times k_{2}l_{2}}(\mathbb{C})\ |\
X(J_{k_{2}}^{l_{2}}(a))=(J_{k_{1}}^{l_{1}}(a))X\\}$
$\displaystyle=\\{\left(\begin{array}[]{ccc}H_{1}(x_{11,1},\cdots,x_{11,k_{1}})&\cdots&H_{1}(x_{1l_{2},1},\cdots,x_{1l_{2},k_{1}})\\\
\vdots&\ddots&\vdots\\\
H_{1}(x_{l_{1}1,1},\cdots,x_{l_{1}1,k_{1}})&\cdots&H_{1}(x_{l_{1}l_{2},1},\cdots,x_{l_{1}l_{2},k_{1}})\end{array}\right)$
$\displaystyle|\ x_{ij,h}\in\mathbb{C},\forall 1\leq i\leq l_{1},1\leq j\leq
l_{2},1\leq h\leq k_{1}\\},$ $\displaystyle N_{k_{1},k_{2}}^{l_{1},l_{2}}$
$\displaystyle=\\{X\in M_{k_{2}l_{2}\times k_{1}l_{1}}(\mathbb{C})\ |\
X(J_{k_{1}}^{l_{1}}(a))=(J_{k_{2}}^{l_{2}}(a))X\\}$
$\displaystyle=\\{\left(\begin{array}[]{ccc}H_{2}(x_{11,1},\cdots,x_{11,k_{1}})&\cdots&H_{2}(x_{1l_{1},1},\cdots,x_{1l_{1},k_{1}})\\\
\vdots&\ddots&\vdots\\\
H_{2}(x_{l_{2}1,1},\cdots,x_{l_{2}1,k_{1}})&\cdots&H_{2}(x_{l_{2}l_{1},1},\cdots,x_{l_{2}l_{1},k_{1}})\end{array}\right)$
$\displaystyle|\ x_{ij,h}\in\mathbb{C},\forall 1\leq i\leq l_{2},1\leq j\leq
l_{1},1\leq h\leq k_{1}\\}.$

###### Proof.

It can be checked easily. ∎

For convenience, we fix $j$ and set $r=r_{j}$, $k_{u}=k_{ju},l_{u}=l_{ju}$,
for $1\leq u\leq r_{j}$, in the Proposition 3.8 and its proof.

######  Proposition 3.8.

The stabilizer

$\displaystyle L_{j}$
$\displaystyle=\mathrm{Stab}_{\mathrm{GL}_{n_{j}}(\mathbb{C})}(\xi_{j})=\\{\left(\begin{array}[]{cccc}X_{11}&X_{12}&\cdots&X_{1r}\\\
X_{21}&X_{22}&\cdots&X_{2r}\\\ \vdots&\vdots&\ddots&\vdots\\\
X_{r1}&X_{r2}&\cdots&X_{rr}\end{array}\right)\ |\ X_{uu}\in
L_{k_{u},l_{u}},\forall 1\leq u\leq r,$ $\displaystyle X_{st}\in
M_{k_{s},k_{t}}^{l_{s},l_{t}},\forall 1\leq s<t\leq r,X_{st}\in
N_{k_{s},k_{t}}^{l_{s},l_{t}},\forall 1\leq t<s\leq r\\}.$

###### Proof.

It is clear that

$\displaystyle\mathrm{Stab}_{\mathrm{GL}_{n_{j}}(\mathbb{C})}(\xi_{j})$
$\displaystyle\subseteq\\{\left(\begin{array}[]{cccc}X_{11}&X_{12}&\cdots&X_{1r}\\\
X_{21}&X_{22}&\cdots&X_{2r}\\\ \vdots&\vdots&\ddots&\vdots\\\
X_{r1}&X_{r2}&\cdots&X_{rr}\end{array}\right)\ |\ X_{st}\in
M_{k_{s},k_{t}}^{l_{s},l_{t}},\forall 1\leq s\leq t\leq r,$ $\displaystyle
X_{st}\in N_{k_{s},k_{t}}^{l_{s},l_{t}},\forall 1\leq t<s\leq r\\},$

so we only need to prove the element on the right hand side is invertible if
and only if $X_{uu}$ is invertible for all $1\leq u\leq r$. This follows from
the fact that the determinant of the element on the right hand side is equal
to $\prod_{1\leq u\leq r}\det X_{uu}$. ∎

It is clear that

$\mathrm{Stab}_{G}(f)=\mathrm{Stab}_{G}(\xi)=\\{\mathrm{diag}(X_{1},\cdots,X_{m})\
|\ X_{i}\in L_{i},1\leq i\leq m\\}.$

And we have $\mathcal{O}_{f}\cong G/L$ and $P\setminus\mathcal{O}_{f}\cong
P\setminus G/L$.

Define the right action of $G$ on $\mathbb{C}^{n}$ by

$v\cdot g=(x_{1},\cdots,x_{n})A,$

where $v=(x_{1},\cdots,x_{n})^{t}\in\mathbb{C}^{n}$ and $g=A\in
G_{n}(\mathbb{C})$. Set $v_{0}=(0,\cdots,0,1)^{t}$, we get
$\mathrm{Stab}_{G}(v_{0})=P$, and $\mathbb{C}^{n}-\\{0\\}\cong P\setminus G$.

To calculate $P\setminus G/L$, we need to calculate
$(\mathbb{C}^{n}-\\{0\\})/L$. Actually, we only need to calculate
$(\mathbb{C}^{n_{j}}-\\{0\\})/L_{j}$. Once we obtain all the different
representatives of $(\mathbb{C}^{n_{j}}-\\{0\\})/L_{j}$, denoted by $V_{j}$,
we can get all the different representatives of $(\mathbb{C}^{n}-\\{0\\})/L$
as follows,

$\\{v=(v_{1},\cdots,v_{m})\neq 0\ |\ v_{i}\in\mathbb{C}^{n_{i}},v_{i}\in
V_{i}\ \text{or}\ v_{i}=0,1\leq i\leq m\\}.$

We fix $j$ and adopt the same notation as Proposition 3.8. We have following
proposition about $(\mathbb{C}^{n_{j}}-\\{0\\})/L_{j}$.

######  Proposition 3.9.

Write $R_{0}=\\{1,\cdots,r\\}$. For any $\emptyset\neq R\subset R_{0}$, and
$1\leq x_{i}\leq k_{i}$ for $i\in R$, such that

  * (1)

$x_{i}<x_{i^{\prime}}$ for any $i,i^{\prime}\in R$ such that $i<i^{\prime}$,

  * (2)

$k_{i}-x_{i}<k_{i^{\prime}}-x_{i^{\prime}}$ for any $i,i^{\prime}\in R$ such
that $i<i^{\prime}$,

we define

$I=\\{\sum_{u=0}^{i-1}k_{u}l_{u}+k_{i}(l_{i}-1)+x_{i},i\in
R\\}\subseteq\\{1,2,\cdots,n_{j}\\},$

and $v_{I}=(y_{1},y_{2},\cdots,y_{n_{j}})$, $y_{i}=1$ when $i\in I$ and
$y_{i}=0$ when $i\notin I$. Let $I_{a}$ be the set of all $I$’s constructed as
above. Then $\\{v_{I}\ |\ I\in I_{a}\\}$ represent all different
$L_{j}$-orbits in $\mathbb{C}^{n_{j}}-\\{0\\}$.

###### Proof.

We only need to prove that any $L_{j}$-orbit in $(\mathbb{C}^{n_{j}}-\\{0\\})$
is of the form $v_{I}\cdot L_{j}$ for some $I$ given as above, and such orbits
are different from each other.

For any $k,l\in\mathbb{Z}_{+}$, set
$e(k,l)=(\underbrace{0,\cdots,0}_{k-1},1,\underbrace{0,\cdots,0}_{l-k})\in\mathbb{C}^{l}$.
For $1\leq i\leq r$, set

$t_{i}=\\{\sum_{u=0}^{i-1}k_{u}l_{u}+1,\sum_{u=0}^{i-1}k_{u}l_{u}+2,\cdots,\sum_{u=0}^{i}k_{u}l_{u}\\}.$

Let $z=(z_{1},\cdots,z_{n_{j}})$ $\in\mathbb{C}^{n_{j}}-\\{0\\}$. We use
$z(t_{i})$ to denote the subvector of $z$ with the positions indexed by
$t_{i}$.

For any $1\leq i\leq r$, we show that one can use the right action of $L_{j}$
to change $z$ to $z^{\prime}$, such that $z^{\prime}(t_{i})$ equals $0$ or
$e(k_{i}(l_{i}-1)+x_{i},k_{i}l_{i})$, for some $1\leq x_{i}\leq k_{i}$,
without changing other elements of $z$.

If $z(t_{i})=0$, we don’t need to change $z$ any more. Otherwise, take $v_{i}$
to be the maximal integer such that $1\leq v_{i}\leq k_{i}$ and

$z(\sum_{u=0}^{i-1}k_{u}l_{u}+k_{i}\cdot w+v_{i})\neq 0$

for some integer $0\leq w\leq l_{i}-1$. Applying Proposition 3.6, we can
realize $z(t_{i})$ as the $k_{i}(l_{i}-1)+v_{i}$-th row of some element in
$L_{k_{i},l_{i}}$, denoted by $\tilde{L}$. By the right action of the element

$\mathrm{diag}(I_{k_{1}l_{1}},\cdots,I_{k_{i-1}l_{i-1}},\tilde{L}^{-1},I_{k_{i+1}l_{i+1}},\cdots,I_{k_{r}l_{r}})\in
L_{j},$

one can change $z$ to $z^{\prime}$, such that
$z^{\prime}(t_{i})=e(k_{i}(l_{i}-1)+v_{i},k_{i}l_{i})$, without changing other
elements of $z$.

Now we can get all representatives of $(\mathbb{C}^{n_{j}}-\\{0\\})/L_{j}$.
For any $\emptyset\neq R\subseteq R_{0}$, and $1\leq x_{i}\leq k_{i},i\in R$,
define

$J=\\{\sum_{u=0}^{i-1}k_{u}l_{u}+k_{i}(l_{i}-1)+x_{i},i\in
R\\}\subseteq\\{1,2,\cdots,n_{j}\\}$

and $v_{J}=(y_{1},y_{2},\cdots,y_{n_{j}})$, $y_{i}=1$ when $i\in J$ and
$y_{i}=0$ when $i\notin J$. Then such $v_{J}$’s include all representatives of
$(\mathbb{C}^{n_{j}}-\\{0\\})/L_{j}$. However, some of them may be the same
representatives.

We go to prove that if $x_{i}\geq x_{i^{\prime}}$ for some $i<i^{\prime}$,
then $v_{I^{\prime}}\cdot L_{j}=v_{I}\cdot L_{j}$ with
$I^{\prime}=I-\\{\sum_{u=0}^{i^{\prime}-1}k_{u}l_{u}+k_{i^{\prime}}(l_{i^{\prime}}-1)+x_{i^{\prime}}\\}$.
For convenience, set $a=\sum_{u=0}^{i-1}k_{u}l_{u}+k_{i}(l_{i}-1)$ and
$b=\sum_{u=0}^{i^{\prime}-1}k_{u}l_{u}+k_{i^{\prime}}(l_{i^{\prime}}-1)$. From
Proposition 3.8, we have

$\tilde{L}^{\prime}=I_{n_{j}}+\sum_{1\leq k\leq
k_{i}-x_{i}+x_{i^{\prime}}}E_{a+x_{i}-x_{i^{\prime}}+k,b+k}\in L_{j}.$

Moreover, $v_{I^{\prime}}\cdot\tilde{L}^{\prime}=v_{I}$, so
$v_{I^{\prime}}\cdot L_{j}=v_{I}\cdot L_{j}$.

If $k_{i}-x_{i}\geq k_{i^{\prime}}-x_{i^{\prime}}$ for some $i<i^{\prime}$,
then $v_{I^{\prime\prime}}\cdot L_{j}=v_{I}\cdot L_{j}$,
$I^{\prime\prime}=I-\\{\sum_{u=0}^{i-1}k_{u}l_{u}+k_{i}(l_{i}-1)+x_{i}\\}$.
For convenience, set $a=\sum_{u=0}^{i-1}k_{u}l_{u}+k_{i}(l_{i}-1)$ and
$b=\sum_{u=0}^{i^{\prime}-1}k_{u}l_{u}+k_{i^{\prime}}(l_{i^{\prime}}-1)$. From
Proposition 3.8, we have

$\tilde{L}^{\prime\prime}=I_{n_{i^{\prime}}}+\sum_{1\leq k\leq
k_{i^{\prime}}-x_{i^{\prime}}+x_{i}}E_{b+x_{i^{\prime}}-x_{i}+k,a+k}\in
L_{j}.$

Moreover, $v_{I^{\prime\prime}}\cdot\tilde{L}^{\prime\prime}=v_{I}$, so
$v_{I^{\prime\prime}}\cdot L_{j}=v_{I}\cdot L_{j}$.

Now, we have proved that the set of $\\{v_{I}\ |\ I\in I_{a}\\}$ contains all
representatives of $(\mathbb{C}^{n_{j}}-\\{0\\})/L_{j}$. It remains to prove
that the orbits of $\\{v_{I}\ |\ I\in I_{a}\\}$ are different from each other.
Assume $v_{I}\cdot L_{j}=v_{\tilde{I}}\cdot L_{j}$, $\tilde{I}$ is
corresponding to $\tilde{R}$ and $\tilde{x}_{i}$. Then by direct calculation
of the orbit $v_{I}\cdot L_{j}$ and $v_{\tilde{I}}\cdot L_{j}$, one see that

$r\in R\Rightarrow r\in\tilde{R},\ \text{and then}\ \tilde{x}_{r}=x_{r}.$

And get that $i\in R$ $\Rightarrow$ $i\in\tilde{R}$, and $\tilde{x}_{i}=x_{i}$
for $i=r-1,\cdots,1$, successively. ∎

######  Remark 3.1.

Let $R=\\{r\\}$ and $x_{r}=k_{r}$, define $I^{o}=\\{n_{j}\\}$, and the
corresponding $v_{I^{o}}$. From the structure of $L_{j}$, we can see that the
$L_{j}$-orbit of $v_{I^{o}}$ is the unique dense open orbit in
$\mathbb{C}^{n_{j}}-\\{0\\}$.

######  Remark 3.2.

If we just assume that “$k_{ji}\neq k_{ji^{\prime}}$ for any $1\leq i\neq
i^{\prime}\leq r_{j}$” rather than “$k_{ji}<k_{ji^{\prime}}$ for any $1\leq
i<i^{\prime}\leq r_{j}$”, the conditions (1) and (2) in the Proposition 3.9
should be replaced by (1)’ $x_{i}<x_{i^{\prime}}$ for any $i,i^{\prime}\in R$
such that $k_{ji}<k_{ji^{\prime}}$, and (2)’
$k_{i}-x_{i}<k_{i^{\prime}}-x_{i^{\prime}}$ for any $i,i^{\prime}\in R$ such
that $k_{ji}<k_{ji^{\prime}}$.

With the result above, we can get the representatives of
$(\mathbb{C}^{n}-\\{0\\})/L$ immediately.

######  Proposition 3.10.

Set $M_{0}=\\{1,2,\cdots,m\\}$ and $R_{j0}=\\{1,2,\cdots,r_{j}\\}$, $1\leq
j\leq m$. For any $\emptyset\neq M\subseteq M_{0}$, and $\emptyset\neq
R_{j}\subseteq R_{j0}$ for each $j\in M$, and $1\leq x_{ji}\leq k_{ji}$ for
$i\in R_{j}$, $j\in M$ satisfying the following two conditions,

  * (1)

$x_{ji_{1}}<x_{ji_{2}}$ for any $i_{1},i_{2}\in R_{j}$ such that
$i_{1}<i_{2}$,

  * (2)

$k_{ji_{1}}-x_{ji_{1}}<k_{ji_{2}}-x_{ji_{2}}$ for any $i_{1},i_{2}\in R_{j}$
such that $i_{1}<i_{2}$,

set

$I=\bigsqcup_{j\in
M}\\{\sum_{v=0}^{j-1}n_{v}+\sum_{u=0}^{i-1}k_{ju}l_{ju}+k_{ji}(l_{ji}-1)+x_{ji},i\in
R_{j}\\}\subseteq\\{1,2,\cdots,n\\},$

and $v_{I}=(y_{1},y_{2},\cdots,y_{n}),y_{i}=1$ when $i\in I$ and $y_{i}=0$
when $i\notin I$. Let $I_{a}$ be the set of all such $I$’s. Then $\\{v_{I}\ |\
I\in I_{a}\\}$ form all different representatives of
$(\mathbb{C}^{n}-\\{0\\})/L$.

###### Proof.

It follows from the Proposition 3.9. ∎

######  Remark 3.3.

Set $M=M_{0}$, $R_{j}=\\{r_{j}\\}$ and $x_{jr_{j}}=k_{jr_{j}}$ for $1\leq
j\leq m$. Let $I^{o}$ be the set corresponding to such $M$, $R_{j}$ and
$x_{jr_{j}}$, i.e. $I^{o}=\\{n_{1},n_{1}+n_{2},\cdots,n\\}$. Then $v_{I^{o}}$
is the unique dense open $L$-orbit in $\mathbb{C}^{n}-\\{0\\}.$

We can define $g_{I}$ in the same way as the contents about regular orbits and
such $g_{I}\cdot f$’s form all different representatives of
$P\setminus\mathcal{O}_{f}$ and the orbit of $g_{I^{o}}\cdot f$ is the unique
dense open orbit among them.

We go to calculate the moment map
$\mathrm{p}:\mathcal{O}_{f}\to\mathfrak{p}^{*}$. Let’s fix some $I$
corresponding to the given $M$, $R_{j}(j\in M)$ and $1\leq x_{ji}\leq
k_{ji}(i\in R_{j},j\in M)$. For any $j$, let $q_{j}=\\#R_{j}$, and
$R_{j}=\\{c_{j1},\cdots,c_{jq_{j}}\\}$ with $c_{j1}<c_{j2}<\cdots<c_{jq_{j}}$.
For $j\in M$, set
$\overline{x}_{jp}=x_{jc_{jp}},\overline{k}_{jp}=k_{jc_{jp}}$ for $1\leq p\leq
q_{j}$, $\overline{x}_{j0}=0$, and $d_{j}=\overline{x}_{jq_{j}}$.

######  Proposition 3.11.

Given $I$ as above, then $\mathrm{p}(g_{I}\cdot f)$ has depth $d=\sum_{j\in
M}d_{j}$, and is $P$-conjugated to the element $\mathrm{pr}^{\prime}(\eta)$,
where

$\eta=\mathrm{diag}(A_{1},\cdots,A_{m},J_{d}),$

where $A_{j}=\xi_{j}$ if $j\notin M$; and
$A_{j}=\mathrm{diag}(A_{j1},A_{j2},\cdots,A_{jr_{j}})$, if $j\in M$, where
$A_{ji}$’s are defined as follows, $A_{ji}=J_{k_{ji},a_{j}}^{l_{ji}}$ if
$i\notin R_{j}$, and

$A_{ji}=\mathrm{diag}(J_{k_{ji},a_{j}}^{l_{ji}-1},J_{t,a_{j}})$

with

$t=\overline{k}_{jh}-\overline{x}_{jh}+\overline{x}_{j(h-1)},$

if $i=c_{jh}\in R_{j}$ for $1\leq h\leq q_{j}$.

###### Proof.

As in the proof of Proposition 3.5, by block calculation, we can assume
$l_{ji}=1$, $\forall 1\leq i\leq r_{j}$ and $R_{j}=\\{1,\cdots,r_{j}\\}$ for
all $1\leq j\leq m$ at the beginning. Hence, $d_{j}=x_{jr_{j}}$ for $1\leq
j\leq m$. For convenience, set $A_{I}=g_{I}\cdot\xi$.

In the case of $n\in I$, we have $r_{m}=1$ since $\\{x_{mi},i\in R_{m}\\}$
satisfies condition (1) and (2).

We claim that $\mathrm{pr}^{\prime}(A_{I})$ is $P$-conjugated
$\mathrm{pr}^{\prime}(\zeta_{j})$ for $0\leq j\leq m-1$, where

$\zeta_{j}=\left(\begin{array}[]{cccccccc}A_{1}&&&&&&&\\\
&J_{d_{1},a_{1}}&&&&&&\\\ &&\ddots&&&&&\\\ &&&A_{j}&&&&\\\
&&&&J_{d_{j},a_{j}}&&&\\\ &&&&&\xi_{j+1}&&\\\ &&&&&&\ddots&\\\
&&&&&&&\xi_{m}\end{array}\right)+\left(\begin{array}[]{c}0_{(n-1)\times n}\\\
\alpha_{j}\end{array}\right),$

and

$\alpha_{j}=(\beta_{1}^{\prime},\cdots,\beta_{j}^{\prime},\beta_{j+1},\cdots,\beta_{m}).$

When $1\leq i\leq m-1$, $\beta^{\prime}_{i}$ and $\beta_{i}$ are defined as
follows.

$\beta^{\prime}_{i}=\left\\{\begin{array}[]{ll}(\underbrace{0,\cdots,0}_{n_{i}-2},1,a_{i}-a_{m}),&d_{i}>1,\\\
(\underbrace{0,\cdots,0}_{n_{i}-1},a_{i}-a_{m}),&d_{i}=1,\end{array}\right.$

and $\beta_{i}=(\beta_{i1},\cdots,\beta_{ir_{i}})$, for $1\leq h\leq r_{i}$,

$\beta_{ih}=\left\\{\begin{array}[]{ll}(\underbrace{0,\cdots,0}_{x_{ih}-2},1,a_{i}-a_{m},\underbrace{0,\cdots,0}_{k_{ih}-x_{ih}}),&x_{ih}>1\\\
(a_{i}-a_{m},\underbrace{0,\cdots,0}_{k_{ih}-1}),&x_{ih}=1.\end{array}\right.$

And $\beta_{m}=(\underbrace{0,\cdots,0}_{n_{m}})$.

It is trivially true that $\mathrm{pr}^{\prime}(A_{I})$ is $P$-conjugated
$\mathrm{pr}^{\prime}(\zeta_{j})$ for $j=0$. Assume that it is true for $j-1$,
let’s prove that $\mathrm{pr}^{\prime}(\zeta_{j-1})$ is $P$-conjugated
$\mathrm{pr}^{\prime}(\zeta_{j})$, so we get that
$\mathrm{pr}^{\prime}(A_{I})$ is $P$-conjugated
$\mathrm{pr}^{\prime}(\zeta_{j})$.

Recall that $n_{j^{\prime}}=\sum_{1\leq i\leq
r_{j^{\prime}}}k_{{j^{\prime}}i}$ for $1\leq j^{\prime}\leq m$, $n_{0}=0$. For
convenience, set $r=r_{j}$, $k_{u}=k_{ju}$ for $1\leq u\leq r_{j}$,
$x_{h}=x_{jh}$ for $1\leq h\leq r_{j}$, $x_{0}=0$. Set

$V=\\{\sum_{j^{\prime}=0}^{j-1}n_{j^{\prime}}+1,\sum_{j^{\prime}=0}^{j-1}n_{j^{\prime}}+2,\cdots,\sum_{j^{\prime}=0}^{j-1}n_{j^{\prime}}+\sum_{u=1}^{r}k_{u}\\}$

and $U=V\cup\\{n\\}$.

We claim that one can use $P$-conjugations, which only change the elements in
the positions $(U,V)$ or in the $n$-th column, to change $\zeta_{j-1}$ to $B$
such that

$B(U,V)=\left(\begin{array}[]{c}\mathrm{diag}(J_{e_{1}}(a_{j}),\cdots,J_{e_{r}}(a_{j}),J_{x_{r}}(a_{j}))\\\
\beta_{j}^{\prime}\end{array}\right),$

where $e_{h}=k_{h}-x_{h}+x_{h-1},1\leq h\leq r$.

Step 1. Let

$P_{1}=\left(\begin{array}[]{ccccccc}I_{n_{1}+\cdots+n_{j-1}}&&&&&&\\\
&I_{k_{1}}&&&&&\\\ &E_{1}&I_{k_{2}}&&&&\\\ &&E_{2}&\ddots&&&\\\
&&&\ddots&\ddots&&\\\ &&&&E_{r-1}&I_{k_{r}}&\\\
&&&&&&I_{n_{j+1}+\cdots+n_{m}}\end{array}\right),$

where $E_{i}$ is the $k_{i+1}\times k_{i}$ matrix, with $-1$ in the positions
$(x_{i+1},x_{i}),(x_{i+1}-1,x_{i}-1),\cdots,(x_{i+1}-x_{i}+1,1)$(here we use
$x_{i+1}>x_{i}$) and $0$ in other positions, for $1\leq i\leq r-1$.

Let $B_{1}=P_{1}^{-1}\zeta_{j-1}P_{1}$, then

$B_{1}(U,V)=\left(\begin{array}[]{ccccc}J_{k_{1},a_{j}}&&&&\\\
E^{\prime}_{1}&J_{k_{2},a_{j}}&&&\\\ 0&E^{\prime}_{2}&\ddots&&\\\
\vdots&\ddots&\ddots&\ddots&\\\ 0&0&\cdots&E^{\prime}_{r-1}&J_{k_{r},a_{j}}\\\
0&0&\cdots&0&\beta_{jr}\end{array}\right),$

where $E^{\prime}_{i}$ is $k_{i+1}\times k_{i}$ matrix with $-1$ in the
position $(x_{i+1}+1,x_{i})$ and $0$ in other positions, for $1\leq i\leq
r-1$.

Step 2. Using the method of in Proposition 3.5, one can use $P$-conjugation,
which just change the elements in the positions $(U,V)$ or in the $n$-th
column, to change $B_{1}$ to $B_{2}$ such that

$B_{2}(U,V)=\left(\begin{array}[]{ccccccc}J_{k_{1},a_{j}}&&&&&\\\
E^{\prime}_{1}&J_{k_{2},a_{j}}&&&&\\\ 0&E^{\prime}_{2}&\ddots&&&\\\
\vdots&\ddots&\ddots&\ddots&&\\\
0&0&\ddots&E^{\prime}_{r-2}&J_{k_{r-1},a_{j}}&\\\
0&0&\cdots&0&0&J_{x_{r},a_{j}}&\\\
0&0&\cdots&0&E^{\prime\prime}_{r}&0&J_{k_{r}-x_{r},a_{j}}\\\
0&0&\cdots&0&0&\beta_{j}^{\prime}&0\end{array}\right),$

$E^{\prime\prime}_{r}$ is the $(k_{r}-x_{r})\times k_{r-1}$ matrix with $1$ in
the position $(1,x_{r-1})$ and $0$ in other positions.

Step 3. Let

$P_{2}=\left(\begin{array}[]{ccccccc}I_{n_{1}+\cdots+n_{j-1}}&&&&&&\\\
&I_{k_{1}}&F_{1}&&&&\\\ &&\ddots&\ddots&&&\\\ &&&\ddots&F_{r-2}&&\\\
&&&&I_{k_{r-1}}&F_{r-1}&\\\ &&&&&I_{k_{r}}&\\\
&&&&&&I_{n_{j+1}+\cdots+n_{m}}\end{array}\right),$

$F_{i}$ is the $k_{i}\times k_{i+1}$ matrix with $-1$ in the positions
$(x_{i}+1,x_{i+1}+1),\cdots,(k_{i},x_{i+1}+k_{i}-x_{i})$(here we use
$k_{i+1}-x_{i+1}>k_{i}-x_{i}$), and $0$ in other positions, for $1\leq i\leq
r$.

Let $B_{3}=P_{2}^{-1}B_{2}P_{2}$, then $B_{3}(U,V)=$

$\left(\begin{array}[]{ccccccccccccc}J_{x_{1},a_{j}}&&&&&&&&\\\
0&J_{k_{1}-x_{1},a_{j}}&&&&&&&\\\ 0&0&J_{x_{2},a_{j}}&&&&&&\\\
E^{\prime\prime}_{1}&0&0&J_{k_{2}-x_{2},a_{j}}&&&&&\\\
0&0&0&0&J_{x_{3},a_{j}}&&&&&&\\\
0&0&E^{\prime\prime}_{2}&0&0&J_{k_{3}-x_{3},a_{j}}&&&&\\\
\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&&&&\\\
\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&J_{x_{r-1},a_{j}}&&&\\\
\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&0&J_{k_{r-1}-x_{r-1},a_{j}}&&\\\
0&0&0&0&0&0&\ddots&0&0&J_{x_{r},a_{j}}&\\\
0&0&0&0&0&0&\cdots&E^{\prime\prime}_{r-1}&0&0&J_{k_{r}-x_{r},a_{j}}\\\
0&0&0&0&0&0&\cdots&0&0&\beta_{j}^{\prime}&0\end{array}\right),$

where $E^{\prime\prime}_{i}$ is the $(k_{i+1}-x_{i+1})\times x_{i}$ matrix
with $1$ in the position $(1,x_{i})$ and $0$ in other positions, for $1\leq
i\leq r-1$. Now we can easily see $B_{3}$ is $P$-conjugated to $\zeta_{j}$.
This finishes the proof of the claim.

Applying Proposition 3.5, by block calculation, we can easily show that
$\mathrm{pr}^{\prime}(\zeta_{m-1})$ is $P$-conjugated to
$\mathrm{pr}^{\prime}(\eta)$. This finish the proof of this case.

In the case of $n\notin I$, using similar method we can also get the
statement.

This completes the proof of the statement. ∎

######  Theorem 3.12.

We have that $\mathrm{p}:\mathcal{O}_{f}\to\mathfrak{p}^{*}$ sends different
$P$-orbits of $\mathcal{O}_{f}$ to different $P$-orbits of $\mathfrak{p}^{*}$,
the image $\mathrm{p}(\mathcal{O}_{f})$ contains a unique dense orbit
$P\cdot(g_{I^{o}}\cdot f)$, and $\mathrm{p}$ is proper over
$P\cdot(g_{I^{o}}\cdot f)$. The reduce space of the unique open dense orbit is
singleton.

###### Proof.

The first statement and the last statement is from the Proposition 3.11. Let’s
check the properness.

In the case of $m=1$, set $r=r_{1}$, $k_{i}=k_{1i}$, $l_{i}=l_{1i}$ for $1\leq
i\leq r_{1}$.

As Lemma 3.2, $\mathrm{Stab}_{P}(g_{I^{o}}\cdot
f)\cong\mathrm{Stab}_{L}(v_{I^{o}})$. For convenience, we denote
$G_{k}(\mathbb{C})$ by $G_{k}$ for any $k\in\mathbb{Z}_{+}$. We know that

$L\cong(G_{l_{1}}\times G_{l_{2}}\times\cdots G_{l_{r}})\ltimes
N(\sum_{i=1}^{r-1}k_{i}l_{i}\cdot(2(l_{i+1}+\cdots+l_{r}))+\sum_{i=1}^{r}l_{i}^{2}(k_{i}-1)),$

where $N(a)$ means the unipotent group which is homeomorphic to
$\mathbb{C}^{a}$ as manifold.

And

$\displaystyle\mathrm{Stab}_{L}(v_{I^{o}})\cong$
$\displaystyle(G_{l_{1}}\times G_{l_{2}}\times\cdots G_{l_{r}-1})\ltimes
N(\sum_{i=1}^{r-1}k_{i}l_{i}\cdot(2(l_{i+1}+\cdots+l_{r}))$
$\displaystyle+\sum_{i=1}^{r}l_{i}^{2}(k_{i}-1)+(l_{r}-1)-(\sum_{i=1}^{r-1}k_{i}l_{i}+(k_{r}-1)l_{r})).$

Similarly, we have

$\displaystyle\mathrm{Stab}_{P}(\mathrm{p}(g_{I^{o}}\cdot f))\cong$
$\displaystyle(G_{l_{1}}\times G_{l_{2}}\times\cdots G_{l_{r}-1})\ltimes
N(\sum_{i=1}^{r-1}k_{i}l_{i}\cdot(2(l_{i+1}+\cdots+l_{r}-1))$ $\displaystyle+$
$\displaystyle\sum_{i=1}^{r-1}l_{i}^{2}(k_{i}-1)+(l_{r}-1)^{2}(k_{r}-1)+(\sum_{i=1}^{r-1}k_{i}l_{i}+(l_{r}-1)k_{r})).$

By direct calculation, we have that the quotient

$\mathrm{Stab}_{P}(\mathrm{p}(g_{I^{o}}\cdot
f))/\mathrm{Stab}_{P}(g_{I^{o}}\cdot f)$

is a single element. So the restriction of $\mathrm{p}$ on
$P\cdot(g_{I^{o}}\cdot f)$ is proper.

In the case of $m>1$, one can get the statement immediately. ∎

###  3.2. Moment map in the $\mathrm{GL}_{n}(\mathbb{R})$ case

Now, we can calculate the moment map in the $G_{n}(\mathbb{R})$ case and get
the similar results as the case of $G_{n}(\mathbb{C})$.

Let $n\in\mathbb{Z}_{+}$, let $G=G_{n}(\mathbb{R})$ and let
$P=P_{n}(\mathbb{R})$ be the mirabolic subgroup of $G$. $\mathfrak{g}$ (resp.
$\mathfrak{p}$) denotes the Lie algebra of $G$ (resp. of $P$).

Write

$J_{s}(a)=\left(\begin{array}[]{cccc}a&&&\\\ 1&a&&\\\ &\ddots&\ddots&\\\
&&1&a\end{array}\right)\in\mathfrak{g}_{s}(\mathbb{R}),s\in\mathbb{Z}\
\text{and}\ s\geq 2,a\in\mathbb{R},$

and $J_{1}(a)=a_{1\times 1},a\in\mathbb{R}$. Set $J_{s}=J_{s}(0)$, $\forall
s\in\mathbb{Z}_{+}$. Write

$J_{k}^{l}(a)=\mathrm{diag}(\underbrace{J_{k}(a),\cdots,J_{k}(a)}_{l\
\text{blocks}}),$

$\forall l\in\mathbb{Z}_{+}$. Write

$J_{k}(a,b)=\left(\begin{array}[]{cc}J_{k}(a)&bI_{k}\\\
-bI_{k}&J_{k}(a)\end{array}\right),\forall
a,b\in\mathbb{R},k\in\mathbb{Z}_{+},$

and

$J_{k}^{l}(a,b)=\mathrm{diag}(\underbrace{J_{k}(a,b),\cdots,J_{k}(a,b)}_{l\
\text{blocks}}),$

Let $u,v\in\mathbb{N}$ such that $2u+v\leq n$. Let
$a_{1},\cdots,a_{u+v},b_{1},\cdots,b_{u}\in\mathbb{R}$ such that
$z_{2k-1}=a_{k}+ib_{k}$, $z_{2k}=a_{k}-ib_{k}$, for $k=1,\cdots,u$, and
$z_{2u+k}=a_{u+k}$ for $1\leq k\leq v$, are $2u+v$ distinct complex number,
$\prod_{1\leq i\leq u}b_{i}\neq 0$.

Let

$\xi=\mathrm{diag}(\xi_{1},\cdots,\xi_{u+v}),$

with

$\xi_{j}=\left\\{\begin{array}[]{c}\mathrm{diag}(J_{k_{j1}}^{l_{j1}}(a_{j},b_{j}),J_{k_{j2}}^{l_{j2}}(a_{j},b_{j}),\cdots,J_{k_{jr_{j}}}^{l_{jr_{j}}}(a_{j},b_{j})),1\leq
j\leq u,\\\
\mathrm{diag}(J_{k_{j1}}^{l_{j1}}(a_{j}),J_{k_{j2}}^{l_{j2}}(a_{j}),\cdots,J_{k_{jr_{j}}}^{l_{jr_{j}}}(a_{j})),u+1\leq
j\leq u+v,\end{array}\right.$

where $r_{j}\in\mathbb{Z}_{+},1\leq j\leq u+v$,
$l_{ji},k_{ji}\in\mathbb{Z}_{+}$, $\forall 1\leq j\leq u+v,1\leq i\leq r_{j}$,
and $k_{ji_{1}}<k_{ji_{2}}$ for $1\leq j\leq{u+v}$, $1\leq i_{1}<i_{2}\leq
r_{i}$, and

$\sum_{1\leq j\leq u,1\leq i\leq r_{j}}2k_{ji}l_{ji}+\sum_{u+1\leq j\leq
u+v,1\leq i\leq r_{j}}k_{ji}l_{ji}=n.$

Set $f=\mathrm{pr}(\xi)$. It is clear that every orbit in $\mathfrak{g}^{*}$
is of the form $\mathcal{O}_{f}$.

Now, we fix a $\xi$. Set

$n_{j}=\left\\{\begin{array}[]{ll}\sum_{i=1}^{r_{j}}2k_{ji}\cdot l_{ji},&1\leq
j\leq u,\\\ \sum_{i=1}^{r_{j}}k_{ji}\cdot l_{ji},&u+1\leq j\leq
u+v,\end{array}\right.$

and $n_{0}=0$. For convenience, set $k_{j0}=0,l_{j0}=0$ for $1\leq j\leq u+v$.

Set $L=\mathrm{Stab}_{G}(f)$, then

$P\setminus\mathcal{O}_{f}\cong P\setminus
G/L\cong(\mathbb{R}^{n}-\\{0\\})/L.$

By calculating the representatives of $(\mathbb{R}^{n}-\\{0\\})/L$, one can
get the representatives of the $P$-orbits in $\mathcal{O}_{f}$.

Since $\xi$ is block diagonal,

$L=\mathrm{diag}(X_{1},\cdots,X_{u+v})\ |\
X_{i}\in\mathrm{Stab}_{\mathrm{GL}_{n_{j}}(\mathbb{R})}(\xi_{j})\\}.$

For $u+1\leq j\leq u+v$, one can obtain
$L_{j}=\mathrm{Stab}_{\mathrm{GL}_{n_{j}}(\mathbb{R})}(\xi_{j})$ from the
result in the case of $G_{n}(\mathbb{C})$ by changing the field $\mathbb{C}$
to $\mathbb{R}$ (see Proposition 3.8).

For $1\leq j\leq u$, let’s calculate $L_{j}$. Let $M_{s\times s}(\mathbb{R})$
be the set of all $s\times s$ matrices over $\mathbb{R}$. Write

$K(x_{1},\cdots,x_{s})=\left(\begin{array}[]{cccc}x_{1}&&&\\\ x_{2}&x_{1}&&\\\
\vdots&\ddots&\ddots&\\\ \
x_{s}&\cdots&x_{2}&x_{1}\end{array}\right),x_{i}\in\mathbb{R},1\leq i\leq s,$

then

$\\{X\in M_{s\times s}(\mathbb{R})\ |\
XJ_{s}=J_{s}X\\}=\\{K(x_{1},\cdots,x_{s})\ |\ x_{i}\in\mathbb{R},1\leq i\leq
s\\}.$

For any
$\vec{x}=(x_{1},\cdots,x_{s}),\vec{y}=(y_{1},\cdots,y_{s})\in\mathbb{R}^{s}$,
define $H(\vec{x},\vec{y})$ as the $2s\times 2s$ matrix

$\left(\begin{array}[]{cc}K(x_{1},\cdots,x_{s})&K(y_{1},\cdots,y_{s})\\\
-K(y_{1},\cdots,y_{s})&K(x_{1},\cdots,x_{s})\end{array}\right).$

We have the following proposition.

######  Proposition 3.13.

For any $a,b\in\mathbb{R}$, $b\neq 0$, $k,l\in\mathbb{Z}_{+}$,

$\displaystyle H_{k,l}$ $\displaystyle=\mathrm{Stab}_{\mathrm{GL}_{2k\cdot
l}(\mathbb{R})}(J_{k}^{l}(a,b))$
$\displaystyle=\\{\left(\begin{array}[]{ccc}H(\vec{x}_{11},\vec{y}_{11})&\cdots&H(\vec{x}_{1l},\vec{y}_{1l})\\\
\vdots&\ddots&\vdots\\\
H(\vec{x}_{l1},\vec{y}_{l1})&\cdots&H(\vec{x}_{ll},\vec{y}_{ll})\end{array}\right)|\
\vec{x}_{ij},\vec{y}_{ij}\in\mathbb{R}^{k},\forall 1\leq i,j\leq l,$
$\displaystyle\det\left(\begin{array}[]{cc}A&B\\\ -B&A\end{array}\right)\neq
0,A=(\vec{x}_{ij}(1))_{1\leq i,j\leq l},B=(\vec{y}_{ij}(1))_{1\leq i,j\leq
l}\\}.$

###### Proof.

It can be proved similarly as Proposition 3.6. ∎

For any $k_{1},k_{2}\in\mathbb{Z}_{+}$ such that $k_{2}\geq k_{1}$, and
$\vec{x}=(x_{1},\cdots,x_{k_{1}})$,
$\vec{y}=(y_{1},\cdots,y_{k_{1}})\in\mathbb{R}^{k_{1}}$, define
$H_{1}(\vec{x},\vec{y})$ as the $2k_{1}\times 2k_{2}$ matrix

$\left(\begin{array}[]{cccc}K(x_{1},\cdots,x_{k_{1}})&0_{k_{1}\times(k_{2}-k_{1})}&K(y_{1},\cdots,y_{k_{1}})&0_{k_{1}\times(k_{2}-k_{1})}\\\
-K(y_{1},\cdots,y_{k_{1}})&0_{k_{1}\times(k_{2}-k_{1})}&K(x_{1},\cdots,x_{k_{1}})&0_{k_{1}\times(k_{2}-k_{1})}\end{array}\right),$

and define $H_{2}(\vec{x},\vec{y})$ as the $2k_{2}\times 2k_{1}$ matrix

$\left(\begin{array}[]{cc}0_{(k_{2}-k_{1})\times k_{1}}&0_{(k_{2}-k_{1})\times
k_{1}}\\\ K(x_{1},\cdots,x_{k_{1}})&K(y_{1},\cdots,y_{k_{1}})\\\
0_{(k_{2}-k_{1})\times k_{1}}&0_{(k_{2}-k_{1})\times k_{1}}\\\
-K(y_{1},\cdots,y_{k_{1}})&K(x_{1},\cdots,x_{k_{1}})\end{array}\right).$

######  Proposition 3.14.

For any $a,b\in\mathbb{R}$, $b\neq 0$,
$k_{1},k_{2},l_{1},l_{2}\in\mathbb{Z}_{+}$, $k_{1}<k_{2}$,

$\displaystyle M_{k_{1},k_{2}}^{l_{1},l_{2}}$ $\displaystyle=\\{X\in
M_{2k_{1}l_{1}\times 2k_{2}l_{2}}(\mathbb{R})\ |\
X(J_{k_{2}}^{l_{2}}(a,b))=(J_{k_{1}}^{l_{1}}(a,b))X\\}$
$\displaystyle=\\{\left(\begin{array}[]{ccc}H_{1}(\vec{x}_{11},\vec{y}_{11})&\cdots&H_{1}(\vec{x}_{1l_{2}},\vec{y}_{1l_{2}})\\\
\vdots&\ddots&\vdots\\\
H_{1}(\vec{x}_{l_{1}1},\vec{y}_{l_{1}1})&\cdots&H_{1}(\vec{x}_{l_{1}l_{2}},\vec{y}_{l_{1}l_{2}})\end{array}\right)$
$\displaystyle|\ \vec{x}_{ij},\vec{y}_{ij}\in\mathbb{R}^{k_{1}},\forall 1\leq
i\leq l_{1},1\leq j\leq l_{2}\\},$ $\displaystyle
N_{k_{1},k_{2}}^{l_{1},l_{2}}$ $\displaystyle=\\{X\in M_{2k_{2}l_{2}\times
2k_{1}l_{1}}(\mathbb{R})\ |\
X(J_{k_{1}}^{l_{1}}(a,b))=(J_{k_{2}}^{l_{2}}(a,b))X\\}$
$\displaystyle=\\{\left(\begin{array}[]{ccc}H_{2}(\vec{x}_{11},\vec{y}_{11})&\cdots&H_{2}(\vec{x}_{1l_{1}},\vec{y}_{1l_{1}})\\\
\vdots&\ddots&\vdots\\\
H_{2}(\vec{x}_{l_{2}1},\vec{y}_{l_{2}1})&\cdots&H_{2}(\vec{x}_{l_{2}l_{1}},\vec{y}_{l_{2}l_{1}})\end{array}\right)$
$\displaystyle|\ \vec{x}_{ij},\vec{y}_{ij}\in\mathbb{R}^{k_{1}},\forall 1\leq
i\leq l_{2},1\leq j\leq l_{1}\\}.$

###### Proof.

It is easy to check. ∎

For convenience, we fix a $1\leq j\leq u$ and set $r=r_{j}$,
$k_{i}=k_{ji},l_{i}=l_{ji}$, for $1\leq i\leq r_{j}$ in the Proposition 3.15.

######  Proposition 3.15.

The stabilizer

$\displaystyle L_{j}$
$\displaystyle=\mathrm{Stab}_{\mathrm{GL}_{n_{j}}(\mathbb{R})}(\xi_{j})=\\{\left(\begin{array}[]{cccc}X_{11}&X_{12}&\cdots&X_{1r}\\\
X_{21}&X_{22}&\cdots&X_{2r}\\\ \vdots&\vdots&\ddots&\vdots\\\
X_{r1}&X_{r2}&\cdots&X_{rr}\end{array}\right)\ |\ X_{ss}\in
H_{k_{s},l_{s}},\forall 1\leq s\leq r,$ $\displaystyle X_{st}\in
M_{k_{s},k_{t}}^{l_{s},l_{t}},\forall 1\leq s<t\leq r,X_{st}\in
N_{k_{s},k_{t}}^{l_{s},l_{t}},\forall 1\leq t<s\leq r\\}.$

###### Proof.

It can be checked directly as Proposition 3.8. ∎

Now we go to calculate $(\mathbb{R}^{n}-\\{0\\})/L$. As $L$ is block diagonal,
we only need to calculate $(\mathbb{R}^{n_{j}}-\\{0\\})/L_{j}$. For $u+1\leq
j\leq u+v$, we can get the result from the case of $G_{n}(\mathbb{C})$. For
$1\leq j\leq u$, we have the following result.

We fix a $j$ and adopt the same notation as Proposition 3.15.

######  Proposition 3.16.

Write $R_{0}=\\{1,\cdots,r\\}$. For $\emptyset\neq R\subset R_{0}$, let $1\leq
x_{i}\leq k_{i}$, for $i\in R$, such that

  * (1)

$x_{i}<x_{i^{\prime}}$ for any $i,i^{\prime}\in R$ such that $i<i^{\prime}$,

  * (2)

$k_{i}-x_{i}<k_{i^{\prime}}-x_{i^{\prime}}$ for any $i,i^{\prime}\in R$ such
that $i<i^{\prime}$,

we define

$I=\\{\sum_{t=0}^{i-1}2k_{t}l_{t}+k_{i}(2l_{i}-1)+x_{i},i\in
R\\}\subseteq\\{1,2,\cdots,n_{j}\\},$

and $v_{I}=(y_{1},y_{2},\cdots,y_{n_{j}})$, $y_{i}=1$ when $i\in I$ and
$y_{i}=0$ when $i\notin I$. Let $I_{a}$ be the set of all $I$’s constructed as
above. Then $\\{v_{I}\ |\ I\in I_{a}\\}$ represent all different
$L_{j}$-orbits in $\mathbb{R}^{n_{j}}-\\{0\\}$.

###### Proof.

It can be proved similarly as Proposition 3.9. ∎

Now we can get the representatives of $(\mathbb{R}^{n}-\\{0\\})/L$.

######  Proposition 3.17.

Set $K_{0}=\\{1,\cdots,u+v\\}$, $R_{j0}=\\{1,\cdots,r_{j}\\},\forall 1\leq
j\leq u+v$. Let $\emptyset\neq K\subseteq K_{0}$, and $\emptyset\neq
R_{j}\subseteq R_{j0}$ for any $j\in K$. Choose an integer $1\leq x_{ji}\leq
k_{ji}$ for $j\in K,i\in R_{j}$, such that

  * (1)

$x_{j{i_{1}}}<x_{j{i_{2}}}$, for any $i_{1},i_{2}\in R_{j}$ such that
$i_{1}<i_{2}$,

  * (2)

$k_{j{i_{1}}}-x_{j{i_{1}}}<k_{j{i_{2}}}-x_{j{i_{2}}}$, for any $i_{1},i_{2}\in
R_{j}$ such that $i_{1}<i_{2}$.

Set

$\displaystyle I=$ $\displaystyle\bigsqcup_{j\in K,1\leq j\leq
u}\\{\sum_{l=0}^{j-1}n_{l}+\sum_{p=0}^{i-1}2k_{jp}l_{jp}+k_{ji}(2l_{ji}-1)+x_{ji},i\in
R_{j}\\}\bigsqcup$ $\displaystyle\bigsqcup_{j\in K,u+1\leq j\leq
u+v}\\{\sum_{l=0}^{j-1}n_{l}+\sum_{p=0}^{i-1}k_{jp}l_{jp}+k_{ji}(l_{ji}-1)+x_{ji},i\in
R_{j}\\},$

and $v_{I}=(v_{1},\cdots,v_{n})$ with $v_{i}=1$ when $i\in I$ and $v_{i}=0$
when $i\notin I$. Let $I_{a}$ be the set of all $I$’s constructed above, then
$\\{v_{I}\ |\ I\in I_{a}\\}$ form all different representatives of
$(\mathbb{R}^{n}-\\{0\\})/L$.

###### Proof.

It can be proved similarly as Proposition 3.10. ∎

We can define $g_{I}$ in the same way as the contents about regular orbits and
such $g_{I}\cdot f$’s form all different representatives of
$P\setminus\mathcal{O}_{f}$ and the orbit of $g_{I^{o}}\cdot f$,
$I^{o}=\\{n_{1},n_{1}+n_{2},\cdots,n\\}$, is the unique dense open orbit among
them.

We go to calculate the moment map
$\mathrm{p}:\mathcal{O}_{f}\to\mathfrak{p}^{*}$. Let’s fix some $I$
corresponding to the given $K$, $R_{j}(j\in K)$ and $1\leq x_{ji}\leq
r_{j}(i\in R_{j},j\in K)$. For any $1\leq j\leq u+v$, let $q_{j}=\\#R_{j}$,
and $R_{j}=\\{c_{j1},\cdots,c_{jq_{j}}\\}$, $c_{j1}<c_{j2}<\cdots<c_{jq_{j}}$.
Set
$\overline{x}_{jh}=x_{jc_{jh}},\overline{k}_{jh}=k_{jc_{jh}},\overline{l}_{jh}=l_{jc_{jh}}$
for $1\leq h\leq q_{j}$, set $d_{j}=\overline{x}_{jq_{j}}$ and
$\overline{x}_{j0}=0$.

######  Proposition 3.18.

We have that $\mathrm{p}(g_{I}\cdot f)$ has depth

$d=\sum_{j\in M,1\leq j\leq u}2d_{j}+\sum_{j\in M,u+1\leq j\leq u+v}d_{j},$

and is $P$-conjugated to the element $\mathrm{pr}^{\prime}(\eta)$, where

$\eta=\mathrm{diag}(A_{1},\cdots,A_{m},J_{d}),$

where $A_{j}=\xi_{j}$ if $j\notin M$, and
$A_{j}=\mathrm{diag}(A_{j1},A_{j2},\cdots,A_{jr_{j}})$, if $j\in M$, where
$A_{ji}$’s are defined as follows, when $i\notin R_{j}$,

$A_{ji}=\left\\{\begin{array}[]{ll}J_{k_{ji}}^{l_{ji}}(a_{j},b_{j}),&1\leq
j\leq u,\\\ J_{k_{ji}}^{l_{ji}}(a_{j}),&u+1\leq j\leq u+v,\end{array}\right.$

when $i=c_{jh}\in R_{j}$ for some $1\leq h\leq q_{j}$,

$A_{ji}=\left\\{\begin{array}[]{ll}\mathrm{diag}(J_{k_{ji}}^{l_{ji}-1}(a_{j},b_{j}),J_{t}(a_{j},b_{j})),&1\leq
j\leq u,\\\ \mathrm{diag}(J_{k_{ji}}^{l_{ji}-1}(a_{j}),J_{t}(a_{j})),&u+1\leq
j\leq u+v,\end{array}\right.$

here $t=k_{ji}-\overline{x}_{jh}+\overline{x}_{j(h-1)}$.

###### Proof.

We regard $\mathfrak{p}_{n}(\mathbb{R})$ as the real form of
$\mathfrak{p}_{n}(\mathbb{C})$. Use the result of $G_{n}(\mathbb{C})$, we get
that $\mathrm{p}(g_{I}\cdot f)$ is $P_{n}(\mathbb{C})$-conjugated to
$\mathrm{pr}^{\prime}(\eta)$. So $\mathrm{p}(g_{I}\cdot f)$ is also
$P_{n}(\mathbb{R})$-conjugated to $\mathrm{pr}^{\prime}(\eta)$. ∎

######  Theorem 3.19.

$\mathrm{p}:\mathcal{O}_{f}\to\mathfrak{p}^{*}$ sends different $P$-orbits of
$\mathcal{O}_{f}$ to different $P$-orbits of $\mathfrak{p}^{*}$, the image
$p(\mathcal{O}_{f})$ contains a unique dense orbit $P\cdot(g_{I^{o}}\cdot f)$,
and the restriction of $\mathrm{p}$ on $P\cdot(g_{I^{o}}\cdot f)$ is proper.
The reduce space of the unique open dense orbit is singleton.

###### Proof.

It follows from Proposition 3.18 and the similar result of the
$G_{n}(\mathbb{C})$ case (see Theorem 3.12). ∎

##  4\. Results of Kirillov’s conjecture

Let $k$ be the field $\mathbb{R}$ or $\mathbb{C}$, let $P_{n}(k)$ be the
mirabolic subgroup of $\mathrm{GL}_{n}(k)$. The Kirillov’s conjecture states
that the irreducible unitary representations of $\mathrm{GL}_{n}(k)$ remains
irreducible upon restriction to $P_{n}(k)$ and was proved by Sahi ([19]) for
tempered representations of $\mathrm{GL}_{n}(\mathbb{R})$ or
$\mathrm{GL}_{n}(\mathbb{C})$, and Sahi-Stein ([21]) for Speh representations
of $\mathrm{GL}_{n}(\mathbb{R})$, and Baruch ([4]) in archimedean fields
general. Later Aizenbud-Gourevitch-Sahi ([21]) calculated the adduced
representations of the Speh complementary series.

Sahi’s and Sahi-Stein’s proofs are based on Mackey’s theory of the unitary
representations of semi-direct products and Vogan’s result about the
classification of the irreducible unitary representations of
$\mathrm{GL}_{n}(k)$, and their key method is constructing the intertwining
operator by Fourier transform. Baruch proved by studying the
$P_{n}(k)$-invariant distribution. Aizenbud-Gourevitch-Sahi’s used annihilator
variety and degenerate Whittaker models to get the adduced representations of
the Speh complementary series. Here we summarize some results of Kirillov’s
conjecture.

Firstly, we show how to get the unitary dual of $P_{n}(k)$, denoted by
$\widehat{P_{n}(k)}$. The following two facts are easy to see.

  * (i)

$P_{n}(k)\cong\mathrm{GL}_{n-1}(k)\ltimes k^{n-1}$.

  * (ii)

$\mathrm{GL}_{n-1}(k)$ has two orbits in $(k^{n-1})^{*}$: $\\{0\\}$ and
$(k^{n-1})^{*}-\\{0\\}$. Let $\xi\in(k^{n-1})^{*}-\\{0\\}$ be defined by
$\xi(x_{1},\cdots,x_{n-1})=x_{n-1}$. Then
$\mathrm{Stab}_{\mathrm{GL}_{n-1}(k)}(\xi)\cong P_{n-1}(k)$.

Based on Mackey’s theory, we have the followings.

Every irreducible unitary representation of $P_{n}(k)$ is obtained in one of
the following two ways:

  * (i)

by trivially extending an irreducible unitary representation of
$\mathrm{GL}_{n-1}(k)$,

  * (ii)

by extending an irreducible unitary representation of $P_{n-1}(k)$ to
$P_{n-1}(k)\ltimes k^{n-1}$ by the character $\xi$ and then unitarily inducing
to $P_{n}(k)$.

We use $E$ and $I$ to denote for functors from the above constructions (i) and
(ii) respectively. Then,

$\widehat{P_{n}(k)}=E(\widehat{\mathrm{GL}_{n-1}(k)})\bigsqcup
I(\widehat{P_{n-1}(k)}).$

Moreover, we have the following fact: each irreducible unitary representation
$\tau$ of $P_{n}(k)$ is of the form $\tau=I^{j-1}E\sigma$, where the integer
$j\geq 1$ and $\sigma\in\widehat{\mathrm{GL}_{n-1}(k)}$ are uniquely
determined by $\tau$.

Now we describe the unitary dual of $\mathrm{GL}_{n}(k)$, which is finally
obtained by Vogan [23]. Let $\pi_{1}$ and $\pi_{2}$ be the representations of
$\mathrm{GL}_{n_{1}}(k)$ and $\mathrm{GL}_{n_{2}}(k)$, $\pi_{1}\times\pi_{2}$
denotes the unitary parabolic induction from the representation
$\pi_{1}\otimes\pi_{2}$ of the Levi subgroup
$\mathrm{GL}_{n_{1}}(k)\times\mathrm{GL}_{n_{2}}(k)$ to
$\mathrm{GL}_{n_{1}+n_{2}}(k)$. Then every irreducible unitary representation
of $\mathrm{GL}_{n}(\mathbb{R})$ is a $\times$-product of unitary characters,
Stein representations, Speh representations, and Speh complementary series
representations. And every irreducible unitary representation of
$\mathrm{GL}_{n}(\mathbb{C})$ is a $\times$-product of unitary characters and
Stein representations.

For convenience, we give the descriptions of Speh representations, Stein
representations and Speh complementary series representations. They can be
described as the (subrepresentations of) degenerated principle series.

  * (i)

Speh representations. Let $Q$ be the subgroup of
$\mathrm{GL}_{2n}(\mathbb{R})$ defined as

$Q=\\{\left(\begin{array}[]{cc}A&B\\\ 0&D\end{array}\right)\ |\
A,D\in\mathrm{GL}_{n}(\mathbb{R}),B\in M_{n\times n}(\mathbb{R})\\}.$

For any $m\in\mathbb{Z}_{+}$, we define a character
$\chi_{m}:Q\to\mathbb{C}^{\times}$ as

$\left(\begin{array}[]{cc}A&B\\\
0&D\end{array}\right)\mapsto|\det(A)|^{m/2}\mathrm{sgn}(\det(A))^{m+1}|\det(D)|^{-m/2}.$

The Speh representation of $\mathrm{GL}_{2n}(\mathbb{R})$, denoted by
$\delta(n,m)$, is the unique nonzero irreducible subrepresentation of
$\mathrm{Ind}_{Q}^{G}(\chi_{m})$, where the smooth vector in
$\mathrm{Ind}_{Q}^{G}(\chi_{m})$ is

$\\{f\in C^{\infty}(G)\ |\ f(qg)=\lambda(q)\chi_{m}(q)f(g),\forall q\in Q,g\in
G\\},$

here $\lambda$ is the modular character defined as

$\left(\begin{array}[]{cc}A&B\\\
0&D\end{array}\right)\mapsto|\det(A)|^{n/2}|\det(D)|^{-n/2}.$

Simply speaking, the Speh representation $\delta(n,m)$ is the unique nonzero
irreducible subrepresentation of

$(|\det|^{m/2}\cdot\mathrm{sgn}(\det)^{m+1})|_{\mathrm{GL}_{n}(\mathbb{R})}\times|\det|^{-m/2}|_{\mathrm{GL}_{n}(\mathbb{R})}.$

  * (ii)

Stein representation. For any $s\in(0,1/2)$, we can define the Stein
representation of $\mathrm{GL}_{2n}(\mathbb{R})$, denoted by $\sigma(n,s)$ (or
$\sigma(n,s)|_{\mathrm{GL}_{2n}(\mathbb{R})}$ when there is ambiguity), as the
representation

$|\det|^{s}|_{\mathrm{GL}_{n}(\mathbb{R})}\times|\det|^{-s}|_{\mathrm{GL}_{n}(\mathbb{R})}.$

The Stein representation of $\mathrm{GL}_{2n}(\mathbb{C})$, denoted by
$\sigma(n,s)$(or $\sigma(n,s)|_{\mathrm{GL}_{2n}(\mathbb{C})}$ when there is
ambiguity), can be defined the same as that of $\mathrm{GL}_{2n}(\mathbb{R})$,

$|\det|^{s}|_{\mathrm{GL}_{n}(\mathbb{C})}\times|\det|^{-s}|_{\mathrm{GL}_{n}(\mathbb{C})}.$

  * (iii)

Speh complementary series representation. For any $m\in\mathbb{Z}_{+}$, and
$s\in(0,1/2)$, we can define the Speh complementary series representation of
$\mathrm{GL}_{4n}(\mathbb{R})$, denoted by $\Delta(n,m,s)$, as the
representation

$|\det|^{s}\delta(n,m)\times|\det|^{-s}\delta(n,m).$

We now state some results on Kirillov’s conjecture. Firstly, S.Sahi [19]
established Kirillov’s conjecture of $\mathrm{GL}_{n}(k)$ for the
$\times$-product of unitary characters and Stein representations.

Sahi [19] defined that

  * (i)

a unitary representation of $P_{n}(k)$, denoted by $\tau$, is homogeneous of
depth $j$ if $\tau=I^{j-1}E\sigma$ for some unitary representation $\sigma$ of
$\mathrm{GL}_{n-1}(k)$,

  * (ii)

a unitary representation of $\mathrm{GL}_{n}(k)$, denoted by $\rho$, is
adducible of depth $j$ if $\rho|_{P_{n}(k)}$ is homogeneous of depth $j$, and
if $\rho|_{P_{n}(k)}=I^{j-1}E\sigma$, we shall write $\sigma=A\rho$ and call
it the adduced representation of $\rho$.

Then Sahi obtained the following key fact based on Mackey’s theory and the
partial Fourier transform, and got the result in that paper.

######  Theorem 4.1 ([19], Theorem 2.1).

If $\rho$ and $\sigma$ are adducible representation of $\mathrm{GL}_{r}(k)$
and $\mathrm{GL}_{s}(k)$ of depths $l$ and $m$, then $\rho\times\sigma$ is
adducible of depth $l+m$. Moreover,
$A(\rho\times\sigma)=(A\rho)\times(A\sigma)$.

We also have the following result.

######  Theorem 4.2 ([19], Lemma 3.1).

Let $\pi$ be a unitary character of $\mathrm{GL}_{n}(k)$, then $\pi$ is
adducible of depth $1$ and $A\pi=\pi|_{\mathrm{GL}_{n-1}(k)}$, where
$\mathrm{GL}_{n-1}(k)$ is imbedded on the top left corner of
$\mathrm{GL}_{n}(k)$.

Latter, Sahi [20], Sahi-Stein [21] obtained the adduced representations of
Stein representations and Speh representations. The key method is constructing
the intertwining operator by partial Fourier transform.

######  Theorem 4.3 ([20], 2.4).

Let $\sigma(n+1,s)$ be the Stein representation of
$\mathrm{GL}_{2n+2}(\mathbb{R})$, $s\in(0,1/2)$, then
$A\sigma(n+1,s)=\sigma(n,s)$.

######  Theorem 4.4 ([21], Theorem 3).

Let $\delta(n+1,m)$ be the $m$-th Speh representation of
$\mathrm{GL}_{2n+2}(\mathbb{R})$, $m\in\mathbb{Z}_{+}$, then
$A\delta(n+1,m)=\delta(n,m)$.

After Baruch [4] proved the Kirillov’s conjecture in archimedean fields,
Aizenbud-Gourevitch-Sahi [21] calculated the adduced representations of the
Speh complementary series.

######  Theorem 4.5 ([2], Theorem 4.2.4).

Let $\Delta(n+1,m,s)$ be the Speh complementary series representation of
$\mathrm{GL}_{4n+4}(\mathbb{R})$, $m\in\mathbb{Z}_{+}$ and $s\in(0,1/2)$, then
$A\Delta(n+1,m,s)=\Delta(n,m,s)$.

######  Example 4.1.

In the case of $\mathrm{GL}_{n}(\mathbb{C})$, let
$a_{i}\in\mathbb{R},s_{i}\in\mathbb{Z}_{+}$, $1\leq i\leq m$,
$\sum_{i=1}^{m}s_{i}=n$, let
$\pi=(\det)^{ia_{1}}|_{\mathrm{GL}_{s_{1}}(\mathbb{C})}\times\cdots\times(\det)^{ia_{m}}|_{\mathrm{GL}_{s_{m}}(\mathbb{C})}$
be the irreducible unitary representation of $\mathrm{GL}_{n}(\mathbb{C})$,
then

$\pi|_{P_{n}}=I^{m-1}E((\det)^{ia_{1}}|_{\mathrm{GL}_{s_{1}-1}(\mathbb{C})}\times\cdots\times(\det)^{ia_{m}}|_{\mathrm{GL}_{s_{m}-1}(\mathbb{C})}).$

##  5\. Orbit method

The orbit method tries to establish a close connection existed between
irreducible unitary representations of a Lie group and its orbits in the
coadjoint representation, and to provide a clear geometric picture of
irreducible unitary representations. In 1950’s , before the orbit method
developed, Mackey and others had obtained some wonderful results on the
irreducible unitary representations of Lie groups, showing how to use induced
representation to obtain irreducible unitary representations, see Mackey [18].
In 1960’s, Kirillov found out a way to think about the set of all irreducible
unitary representations of a simply connected nilpotent Lie group,
establishing a bijection between the coadjoint orbits and the irreducible
unitary representations (see Kirillov [12]), and brought up the orbit method.
Later, Kostant developed a quantization theory to obtain irreducible unitary
representations, and generalized the results of Kirillov to solvable Lie
groups (see Auslander-Kostant [3]). In 1970’s-1980’s, Duflo got all the
irreducible unitary representations of (almost) algebraic groups assuming the
unitary dual of reductive Lie groups, see Duflo [6]. The orbit method (or the
quantization problem) of reductive groups is not completely solved until now.
Roughly speaking, the quantization of coadjoint orbits can be reduced to the
case of nilpotent coadjoint orbits (defined below), and such irreducible
unitary representations attached to nilpotent coadjoint orbits, called
unipotent representations, are not easy to understood in general, see Vogan
[22]. We will summarize some results about the orbit method of reductive Lie
groups and algebraic groups in the following two parts. The method of
constructing representations is using induced representation (includes
parabolic induction and cohomological induction).

###  5.1. Representations attached to coadjoint orbits of $\mathrm{GL}_{n}(k)$

####  5.1.1. Jordan decomposition

To begin with, we use Jordan decomposition to describe the coadjoint orbits of
reductive groups (see Vogan [22], Lecture 2).

Let $G$ be a reductive Lie group and let $\mathfrak{g}$ be its Lie algebra.
Let $\theta$ be a Cartan involution of $\mathfrak{g}$ and let

$G=K\exp\mathfrak{s}$

be the Cartan decomposition, where $K=G^{\theta}$ and $\mathfrak{s}=-1$
eigenspace of $\theta$ on $\mathfrak{g}$.

Let $B$ be a nondegenerate $G$-invariant and $\theta$-invariant bilinear form
on $\mathfrak{g}$ such that the quadratic form

$\mathfrak{g}\ni X\mapsto-B(X,\theta X)$

on $\mathfrak{g}$ is positive definite. Let $\mathfrak{g}^{*}$ be the dual of
$\mathfrak{g}$. Then we have a $G$-isomorphism

$\phi:\mathfrak{g}\to\mathfrak{g}^{*},X\mapsto(Y\mapsto B(X,Y),\forall
Y\in\mathfrak{g}),\forall X\in\mathfrak{g}.$

For any $\lambda\in\mathfrak{g}^{*}$, we call it semisimple (resp. nilpotent,
hyperbolic, elliptic) if $\phi^{-1}(\lambda)$ is a semisimple (resp.
nilpotent, hyperbolic, elliptic) element in $\mathfrak{g}$. By the Jordan
decomposition, we can decompose any $\lambda\in\mathfrak{g}^{*}$ uniquely as

$\lambda=\lambda_{h}+\lambda_{e}+\lambda_{n},$

where $\lambda_{h}$ is hyperbolic, $\lambda_{e}$ is elliptic, $\lambda_{n}$ is
nilpotent and $\phi^{-1}(\lambda_{h})$, $\phi^{-1}(\lambda_{e})$,
$\phi^{-1}(\lambda_{n})$ commute with each other.

Set $X_{h}=\phi^{-1}(\lambda_{h})$, $X_{e}=\phi^{-1}(\lambda_{e})$ and
$X_{n}=\phi^{-1}(\lambda_{n})$. Let $G(X)$ (resp. $G(\lambda)$) be the
stabilizer of $X$ (resp. of $\lambda$), and $\mathfrak{g}(X)$ (resp.
$\mathfrak{g}(\lambda)$) be the Lie algebra of $G(X)$ (resp. of $G(\lambda)$).
Then we have the followings.

######  Proposition 5.1 (Vogan [23], Proposition 2.10, 2.11 and 2.12).

  * (i)

Any hyperbolic element in $\mathfrak{g}$ is conjugated into $\mathfrak{s}$. If
$X\in\mathfrak{s}$ is hyperbolic, then $G(X)$ is reductive group with Cartan
involution $\theta|_{G(X)}$.

  * (ii)

There is a bijection between the $G$-orbits of elements in $\mathfrak{g}$,
which hyperbolic part are $G$-conjugated to $X_{h}$, and the $G(X_{h})$-orbits
of elements in $\mathfrak{g}(X_{h})$, which hyperbolic part is zero.

  * (iii)

Any elliptic element in $\mathfrak{g}$ is conjugated into $\mathfrak{k}$. If
$X\in\mathfrak{k}$ is elliptic, then $G(X)$ is a reductive group with Cartan
involution $\theta|_{G(X)}$.

  * (iv)

There is a bijection between the $G$-orbits of elements in $\mathfrak{g}$,
which semisimple part are $G$-conjugated to $X_{h}+X_{e}$, and the
$G(X_{h}+X_{e})$-orbits of elements in $\mathfrak{g}(X_{h}+X_{e})$, which
semisimple part is zero.

Therefore we can assume $X_{h}\in\mathfrak{s}$ and $X_{e}\in\mathfrak{k}$.

An irreducible unitary representation which is attached to a coadjoint orbit
has a kind of “Jordan decomposition”. To attach an irreducible unitary
representation to the orbit of $\lambda$, we firstly attach an irreducible
representation $\pi_{n}$ of $G(\lambda_{h}+\lambda_{e})$ to the
$G(\lambda_{h}+\lambda_{e})$-orbit of
$\lambda_{n}|_{\mathfrak{g}(\lambda_{h}+\lambda_{e})^{*}}$, which is called as
nilpotent step; then using $\lambda_{e}$ and applying cohomological induction
to $\pi_{n}$, we get an irreducible unitary representation $\pi_{h}$ of
$G(\lambda_{h})$, which is called as elliptic step; finally, using
$\lambda_{h}$ and applying parabolic induction to $\pi_{h}$, we get an
irreducible unitary representation of $G$, which is called as hyperbolic step.

####  5.1.2. Hyperbolic step

The easiest step is hyperbolic step, which uses parabolic induction, see Vogan
[22], Lecture 2.

Set $L=G(\lambda_{h})$ with its Lie algebra $\mathfrak{l}$. Set

$\mathfrak{g}_{r}=\\{Y\in\mathfrak{g}|[X_{h},Y]=rY\\},r\in\mathbb{R}$

and $\mathfrak{u}=\oplus_{r>0}\mathfrak{g}_{r}$. Set $U=\exp\mathfrak{u}$ and
$Q=LU$. Define unitary character $\chi(\lambda_{h})$ of $L$ as follows,

$\chi(\lambda_{h})(k\cdot\exp(Z))=\exp(i\lambda_{h}(Z)),k\in L\cap
K,Z\in\mathfrak{l}\cap\mathfrak{s}.$

Suppose that $\pi_{L}$ is any unitary representation of $G(\lambda_{h})$
attached to the $G(\lambda_{h})$-coadjoint orbit of
$(\lambda_{e}+\lambda_{n})|_{\mathfrak{l}}$, then the unitary representation
of $G$ (this unitary representation may be reducible)

$\pi_{G}=\mathrm{Ind}_{Q}^{G}(\pi_{L}\otimes\chi(\lambda_{h}))$

is attached to the coadjoint orbit of $\lambda$.

####  5.1.3. Elliptic step

The elliptic step is more complicated, which uses cohomological induction, see
Vogan [24]. Set $\lambda_{s}=\lambda_{h}+\lambda_{e}$ be semisimple part of
$\lambda$, and set $X_{s}=X_{h}+X_{e}$.

Let $\mathfrak{g}_{\mathbb{C}}$ be the complexification of $\mathfrak{g}$, and
let $\mathfrak{q}$ be the parabolic subalgebra having
$\mathfrak{g}_{\mathbb{C}}(\lambda_{s})$ as a Levi factor with nilpotent
radical $\mathfrak{u}$ such that the eigenvalues of $X_{s}$ acting on
$\mathfrak{u}$ are in

$\\{z\in\mathbb{C}\ |\ \mathrm{Re}(z)>0,\ \text{or}\
\mathrm{Re}(z)=0,\mathrm{Im}(z)>0\\}.$

Let $e^{2\rho(\mathfrak{u})}$ be the character of the adjoint action of
$G(\lambda_{s})$ on the top exterior power of $\mathfrak{u}$.

Assume that $\tau$ is any irreducible representation of $G(\lambda_{s})$ such
that

$d\tau=i\lambda_{s}+\rho(\mathfrak{u}).$

Then there is attached to $\lambda$ a unitary representation $\pi_{\lambda}$
such that the underlying $(\mathfrak{g}_{\mathbb{C}},K)$-module $\Pi(\lambda)$
is defined as follows,

(1)
$\Pi(\lambda)=(\Gamma^{\mathfrak{g}_{\mathbb{C}},K}_{\mathfrak{g}_{\mathbb{C}},G(\lambda_{s})\cap
K})^{d}(\mathrm{pro}_{\mathfrak{q},G(\lambda_{s})\cap
K}^{\mathfrak{g}_{\mathbb{C}},G(\lambda_{s})\cap K}(\tau\otimes\pi_{n})),$

where $\Gamma$ is the Zuckerman functor, $\mathrm{pro}$ is a kind of
$\mathrm{Hom}$ functor, $d$ is the dimension of
$\mathfrak{u}\cap\mathfrak{k}$, $\pi_{n}$ is the unipotent representation of
$G(\lambda_{s})$ which is attached to
$\lambda_{n}|_{(\mathfrak{g}(\lambda_{s}))^{*}}$.

To define the Zuckerman functor (see Knapp-Vogan [13], Introduction), we need
to define the Hecke algebra. Assume that $\mathfrak{h}$ is a complex reductive
Lie algebra of Lie group $H$, $M$ is a compact subgroup of $H$, then the Hecke
algebra $R(\mathfrak{h},M)$ can be defined as the algebra of left $M$-finite
distributions on $H$ with support in $M$, with convolution as multiplication.

Let’s fix some notations to define the Zuckerman functor and the functor
$\mathrm{pro}$. Let $G$ be a linear connected reductive Lie group and
$G_{\mathbb{C}}$ be the complexification of $G$. Let $K$ be the maximal
compact subgroup of $G$. Let $\mathfrak{g}$ denote the Lie algebra of $G$ and
$\mathfrak{g}_{\mathbb{C}}$ be the complexification of $\mathfrak{g}$.

Assume that $T$ is a torus in $G$, and $L=Z_{G}(T)$ with Lie algebra
$\mathfrak{l}$. Let $L_{\mathbb{C}}$ be the analytic subgroup of
$G_{\mathbb{C}}$ with Lie subalgebra $\mathfrak{l}_{\mathbb{C}}$. Let $Q$ be a
parabolic subgroup in $G_{\mathbb{C}}$ containing $L_{\mathbb{C}}$ as Levi
subgroup and let $\mathfrak{q}$ be the Lie algebra of $Q$.

Assume that $V$ is a $(\mathfrak{g}_{\mathbb{C}},L\cap K)$-module, $W$ is an
$(\mathfrak{l}_{\mathbb{C}},L\cap K)$-module, hence is also a
$(\mathfrak{q},L\cap K)$-module by trivial extension to the radical
$\mathfrak{u}$ of $\mathfrak{q}$.

Define the functor $\mathrm{pro}$ from $(\mathfrak{q},L\cap K)$-module to
$(\mathfrak{g}_{\mathbb{C}},L\cap K)$-module as

$\mathrm{pro}_{\mathfrak{q},L\cap K}^{\mathfrak{g}_{\mathbb{C}},L\cap
K}(W)=\mathrm{Hom}_{\mathfrak{q}}(U(\mathfrak{g}_{\mathbb{C}}),W)_{L\cap K}.$

Here the subscript ${L\cap K}$ means taking all the $L\cap K$ finite vectors.

Define the functor from $(\mathfrak{g}_{\mathbb{C}},L\cap K)$-module to
$(\mathfrak{g}_{\mathbb{C}},K)$-module as

$\Gamma^{\mathfrak{g}_{\mathbb{C}},K}_{\mathfrak{g}_{\mathbb{C}},L\cap
K}(V)=\mathrm{Hom}_{R(\mathfrak{g}_{\mathbb{C}},L\cap
K)}(R(\mathfrak{g}_{\mathbb{C}},K),V)_{K}.$

It is left exact and has $i$-th right derivative functors, denoted by
$(\Gamma^{\mathfrak{g}_{\mathbb{C}},K}_{\mathfrak{g}_{\mathbb{C}},L\cap
K})^{i}$.

Let $(^{\mathrm{u}}\mathcal{R}_{\mathfrak{q},L\cap
K}^{\mathfrak{g}_{\mathbb{C}},K})^{i}(W)$ denote

$(\Gamma^{\mathfrak{g}_{\mathbb{C}},K}_{\mathfrak{g}_{\mathbb{C}},K\cap
L})^{i}(\mathrm{pro}_{\mathfrak{q},L\cap K}^{\mathfrak{g}_{\mathbb{C}},L\cap
K}(W)),$

then the equation (1) can be written as

$(^{\mathrm{u}}\mathcal{R}^{\mathfrak{g}_{\mathbb{C}},K}_{\mathfrak{q},G(\lambda_{s})\cap
K})^{d}(\tau\otimes\pi_{n}).$

####  5.1.4. Nilpotent step

We focus on the special unipotent representations which are defined by Arthur,
Barbasch and Vogan (see Adams-Barbasch-Vogan [1], chapter 27).

Let’s introduce notation about partition firstly. Let $n\in\mathbb{Z}_{+}$,
let $r\in\mathbb{Z}_{+}$, $k_{i},l_{i}\in\mathbb{Z}_{+}$ such that
$\sum_{1\leq i\leq r}k_{i}l_{i}=n$ and $k_{i}>k_{j}$ for any $1\leq i<j\leq
r$. We use $\\{k_{1}^{l_{1}}\cdots k_{r}^{l_{r}}\\}$ to denote the partition
of $n$,

$\\{\underbrace{k_{1},\cdots,k_{1}}_{l_{1}},\cdots,\underbrace{k_{r},\cdots,k_{r}}_{l_{r}}\\}.$

Let $P$ denote this partition, we will use $J_{P}$ to denote
$\mathrm{diag}(J_{k_{1}}^{l_{1}},\cdots,J_{k_{r}}^{l_{r}})$, use $J_{P}(a)$ to
denote $\mathrm{diag}(J_{k_{1}}^{l_{1}}(a),\cdots,J_{k_{r}}^{l_{r}}(a))$, use
$J_{P}(a,b)$ to denote
$\mathrm{diag}(J_{k_{1}}^{l_{1}}(a,b),\cdots,J_{k_{r}}^{l_{r}}(a,b))$.

For the case of $G=G_{n}(\mathbb{R})$, let $f=\mathrm{pr}(\xi)$ with
$\xi=J_{P}$. The special unipotent representation attached to
$\mathcal{O}_{f}$ is

$\pi=\mathrm{sgn}(\det)^{w_{1}}|_{G_{t_{1}}(\mathbb{R})}\times\cdots\times\mathrm{sgn}(\det)^{w_{p}}|_{G_{t_{p}}(\mathbb{R})},$

where $\\{t_{1},\cdots,t_{p}\\}$ is the dual partition of
$\\{k_{1}^{l_{1}}\cdots k_{r}^{l_{r}}\\}$ and $w_{i}=0,1$ for $1\leq i\leq p$.

For the case of $G=G_{n}(\mathbb{C})$, let $f=\mathrm{pr}(\xi)$ with
$\xi=J_{P}$. The special unipotent representation attached to
$\mathcal{O}_{f}$ is

$\pi=1|_{G_{t_{1}}(\mathbb{C})}\times\cdots\times 1|_{G_{t_{p}}(\mathbb{C})}$

where $\\{t_{1},\cdots,t_{p}\\}$ is the dual partition of
$\\{k_{1}^{l_{1}}\cdots k_{r}^{l_{r}}\\}$.

In general, we have the following correspondence between coadjoint orbits and
irreducible unitary representations.

For the case of $G=G_{n}(\mathbb{R})$.

When $n=2m$, assume that $P=\\{k_{1}^{l_{1}}\cdots k_{r}^{l_{r}}\\}$ is a
partition of $m$. Let $f\in\mathfrak{g}^{*}$ and $f=\mathrm{pr}(\xi)$ with
$\xi=J_{P}(0,b/2)$, $b\in\mathbb{Z}_{+}$.

Firstly, we use the nilpotent step and attach the orbit
$f_{n}=\mathrm{pr}(\xi_{n})$, $\xi_{n}=J_{P}(0,0)$, with the irreducible
unitary representation $\pi_{n}$ of $G(f_{s})\cong G_{m}(\mathbb{C})$

$1|_{G_{t_{1}}(\mathbb{C})}\times\cdots\times 1|_{G_{t_{p}}(\mathbb{C})},$

where $\\{t_{1},\cdots,t_{p}\\}$ is the dual partition of
$\\{k_{1}^{l_{1}}\cdots k_{r}^{l_{r}}\\}$.

By direct calculation, we have

$\tau=(\frac{\det}{|\det|})^{a+m}|_{G_{m}(\mathbb{C})}\ \text{and}\
e^{\rho(\mathfrak{u})}=(\frac{\det}{|\det|})^{m}|_{G_{m}(\mathbb{C})}.$

Using Vogan [23], Theorem 17.6, we have

$(^{\mathrm{u}}\mathcal{R}^{\mathfrak{g}_{\mathbb{C}},K}_{\mathfrak{q},G(f_{s})\cap
K})^{d}(\tau\otimes\pi_{n})=\mathop{\times}\limits_{i=1}^{p}\delta(t_{i},b).$

So the irreducible unitary representation
$\mathop{\times}\limits_{i=1}^{p}\delta(t_{i},b)$ is attached to the coadjoint
orbit of $f=\mathrm{pr}(\xi)$ with $\xi=J_{P}(0,b/2)$.

When $f=\mathrm{pr}(\xi)$ with

$\xi=\mathrm{diag}(J_{P_{1}}(0,b_{1}/2),\cdots,J_{P_{r}}(0,b_{r}/2),J_{P}),$

where $r\in\mathbb{Z}_{+}$, $b_{i}\in\mathbb{Z}_{+}$, for $1\leq i\leq r$,
$b_{1}>b_{2}>\cdots>b_{r}$, $P_{i}$ is a partition of $n_{i}$ and $P$ is a
partition of $n_{0}$ with $\sum_{i=1}^{r}2n_{i}+n_{0}=n$.

Let $\\{t_{i1},\cdots,t_{is_{i}}\\}$ be the dual partition of $P_{i}$ for
$1\leq i\leq r$. Let $\\{t_{1},\cdots,t_{s}\\}$ be the dual partition of $P$.

Let

$H=G_{2n_{1}}(\mathbb{R})\times\cdots\times G_{2n_{p}}(\mathbb{R})\times
G_{n_{0}}(\mathbb{R}),$

then $G(f_{s})\subset H$. Let $\mathfrak{h}$ be the Lie algebra of $H$ and
$\mathfrak{h}_{\mathbb{C}}$ be the complexification of $\mathfrak{h}$. Let
$\mathfrak{q}_{1}=\mathfrak{h}_{\mathbb{C}}\cap\mathfrak{q}$, and let
$\mathfrak{u}_{1}$ be the nilpotent radical of $\mathfrak{q}_{1}$. Let $Q$ be
the parabolic subgroup containing $H$ and the invertible upper triangular
matrices. Set $d_{1}=\dim(\mathfrak{u}_{1}\cap\mathfrak{k})$.

We use Vogan [23], Theorem 17.6, and get that

$\displaystyle(^{\mathrm{u}}\mathcal{R}^{\mathfrak{g}_{\mathbb{C}},K}_{\mathfrak{q},G(f_{s})\cap
K})^{d}(\tau\otimes\pi_{n})$
$\displaystyle=\mathrm{Ind}_{Q}^{G}((^{\mathrm{u}}\mathcal{R}^{\mathfrak{h}_{\mathbb{C}},K\cap
H}_{\mathfrak{q_{1}},G(f_{s})\cap K})^{d_{1}}(\tau\otimes\pi_{n}))$
$\displaystyle=(\mathop{\times}\limits_{j=1}^{p}\mathop{\times}\limits_{i=1}^{s_{j}}\delta(t_{ji},b_{j}))\times(\mathop{\times}\limits_{j=1}^{s}\mathrm{sgn}(\det)^{w_{j}}|_{G_{t_{j}}(\mathbb{R})}),$

where $w_{j}=0,1$ for any $1\leq j\leq s$.

####  5.1.5. Representations of $\mathrm{GL}_{n}(k)$ attached to some kind of
coadjoint orbits

In general, for the case of $G_{n}(\mathbb{R})$, let $f=\mathrm{pr}(\xi)$ with
$\xi=\mathrm{diag}(\xi_{1},\cdots,\xi_{m})$, $m\in\mathbb{Z}_{+}$, and

$\xi_{i}=\mathrm{diag}(J_{P_{i1}}(a_{i},b_{i1}/2),\cdots,J_{P_{ip_{i}}}(a_{i},b_{ip_{i}}/2),J_{P_{i}}(a_{i})),1\leq
i\leq m,$

where $p_{i}\in\mathbb{Z}_{+}$, $a_{i}\in\mathbb{R}$ for $1\leq i\leq m$,
$a_{1}>\cdots>a_{m}$, $b_{ij}\in\mathbb{Z}_{+}$, and
$b_{i1}>\cdots>b_{ip_{i}}$, $P_{ij}$ is a partition of $n_{ij}$ for $1\leq
i\leq m$ and $1\leq j\leq p_{i}$, $P_{i}$ is a partition of $n_{i}$ for $1\leq
i\leq m$, $n_{ij},n_{i}$ satisfy

$\sum_{1\leq i\leq m}\sum_{1\leq j\leq p_{i}}2n_{ij}+\sum_{1\leq i\leq
m}n_{i}=n.$

Assume that $\\{t_{ij,1},\cdots,t_{ij,s_{ij}}\\}$ is the dual partition of
$P_{ij}$, and $\\{t_{i,1},\cdots,t_{i,s_{i}}\\}$ is the dual partition of
$P_{i}$.

The representation attached to $\mathcal{O}_{f}$ is
$\pi=\pi_{1}\times\cdots\times\pi_{m}$, where

$\pi_{l}=|\det|^{ia_{l}}\otimes((\mathop{\times}\limits_{j=1}^{p_{l}}\mathop{\times}\limits_{k=1}^{s_{lj}}\delta(t_{lj,k},b_{lj}))\times(\mathop{\times}\limits_{k=1}^{s_{l}}\mathrm{sgn}(\det)^{w_{l,k}}|_{G_{t_{l,k}}(\mathbb{R})})),$

with $w_{l,k}=0,1$ for any $1\leq l\leq m$, $1\leq k\leq s_{l}$.

For the case of $G=G_{n}(\mathbb{C})$, let $f=\mathrm{pr}(\xi)$ with

$\xi=\mathrm{diag}(J_{P_{1}}(a_{1}),\cdots,J_{P_{m}}(a_{m})),$

where $a_{i}\in\mathbb{R}$ for any $1\leq i\leq m$, and $a_{1}>\cdots>a_{m}$,
$P_{i}$ is a partition of $n_{i}$ for $1\leq i\leq m$, and $n_{i}$($1\leq
i\leq m$) satisfy $\sum_{1\leq i\leq m}n_{i}=n$.

The representation attached to $\mathcal{O}_{f}$ is

$\pi=\mathop{\times}\limits_{j=1}^{m}\mathop{\times}\limits_{k=1}^{s_{j}}|\det|^{ia_{j}}|_{G_{p_{jk}}(\mathbb{C})},$

where $\\{p_{j1},\cdots,p_{js_{j}}\\}$ is the dual partition of $P_{j}$ for
$1\leq j\leq m$.

###  5.2. Representations attached to coadjoint orbits of $P_{n}(k)$

Given an (almost) algebraic group $G$, Duflo [6] constructed all the
irreducible unitary representations of $G$ assuming the unitary dual of some
reductive groups, basing on the Mackey’s theory. It suggests that the problem
of attaching irreducible unitary representations with coadjoint orbits can be
reduced to the problem in the case of reductive groups. We summarize some
results of Duflo [6] without proofs in the followings.

####  5.2.1. Coadjoint orbits of algebraic groups

Duflo showed how to get all the coadjoint orbits of $\mathfrak{g}^{*}$
assuming that one knew the coadjoint orbits of reductive groups.

Let $\mathfrak{g}$ be the Lie algebra of $G$, and let $\mathfrak{g}^{*}$ be
the dual of $\mathfrak{g}$. For any $x\in\mathfrak{g}^{*}$, let $G(x)$ be the
stabilizer of $x$ in $G$ and let $\mathfrak{g}(x)$ be the Lie algebra of
$G(x)$, let ${}^{u}G(x)$ be the unipotent radical of $G(x)$ and let
${}^{u}\mathfrak{g}(x)$ be the Lie algebra of ${}^{u}G(x)$.

On $\mathfrak{g}/\mathfrak{g}(x)$, there is a natural symplectic form, denoted
by $\omega_{x}$,

$\omega_{x}(\overline{\alpha},\overline{\beta})=x([\alpha,\beta]),$

where $\alpha,\beta\in\mathfrak{g}$ and $\overline{\alpha},\overline{\beta}$
denote the image of $\alpha$ and $\beta$ in $\mathfrak{g}/\mathfrak{g}(x)$.

Firstly, we need to define the linear forms of unipotent type (see [6], I). If
$G$ is unipotent, all of $\mathfrak{g}^{*}$ are of unipotent type, and if $G$
is reductive, $0$ is the unique element of $\mathfrak{g}^{*}$ of unipotent
type. In general, its definition depends on the following things.

######  Definition 5.1.

Let $x\in\mathfrak{g}^{*}$ and $\mathfrak{b}$ be a subalgebra of
$\mathfrak{g}$. We say that

  * (i)

$\mathfrak{b}$ is coisotropic relative to $x$ (or coisotropic) if the
orthogonal of $\mathfrak{b}$ in $\mathfrak{g}$ with respect to $\omega_{x}$,
denoted by $\mathfrak{b}^{\perp}$, is contained in $\mathfrak{b}$.

  * (ii)

$\mathfrak{b}$ is a polarization of $x$ if
$\mathfrak{b}^{\perp}=\mathfrak{b}$.

  * (iii)

An coisotropic subalgebra $\mathfrak{b}$ is of strongly unipotent type if
$\mathfrak{b}$ is algebraic (i.e. there exists algebraic subgroup with Lie
subalgebra $\mathfrak{b}$) and
$\mathfrak{b}=\mathfrak{g}(x)+$${}^{u}\mathfrak{b}$.

######  Definition 5.2.

An element $x$ in $\mathfrak{g}^{*}$ is said to be of unipotent type if it
satisfies

  * (i)

There exists a reductive factor of $\mathfrak{g}(x)$ contained in $\ker x$.

  * (ii)

There is a subalgebra of strongly unipotent type relative to $x$.

Let $x\in\mathfrak{g}^{*}$ be a form of unipotent type, then we can construct
a subalgebra of strongly unipotent type relative to $x$ by induction on
$\dim\mathfrak{g}$ canonically (see [6], I.20). Let
$u=x|_{{}^{u}\mathfrak{g}}$ and let $\mathfrak{h}=\mathfrak{g}(u)$. If
$\mathfrak{h}=\mathfrak{g}$, then set $\mathfrak{b}=\mathfrak{h}$. If
$\mathfrak{h}\neq\mathfrak{g}$, then $y=x|_{\mathfrak{h}}$ is a form of
unipotent type. Let $\mathfrak{l}\subset\mathfrak{h}$ be the canonical
subalgebra of strongly unipotent type relative to $y$, set
$\mathfrak{b}=$${}^{u}\mathfrak{g}+\mathfrak{h}$. Then $\mathfrak{b}$ is the
canonical subalgebra of strongly unipotent type relative to $x$.

Let $D$ be the set of pairs $(x,\lambda)$, where $x$ is a linear form of
unipotent type on $\mathfrak{g}$, and $\lambda$ is an element in $L(x)$,
$L(x)$ is the set of linear forms over $\mathfrak{g}(x)$ whose restriction to
${}^{u}\mathfrak{g}(x)$ is equal to $x|_{{}^{u}\mathfrak{g}(x)}$.

Let $\mathfrak{r}$ be a reductive factor of $\mathfrak{g}(x)$, then the
restriction map establish a bijection of $L(x)$ and $\mathfrak{r}^{*}$,
sending $y\in L(x)$ to $y|_{\mathfrak{r}}$.

Let $(x,\lambda)\in D$. Let $\mathfrak{b}$ be a subalgebra of $\mathfrak{g}$,
which is of unipotent type relative to $x$ (see [6],I.8 for definition, from
[6], I.9, I.22, we have that when $x$ is of unipotent type, a subalgebra is of
unipotent type relative to $x$ if and only if it is strongly unipotent type
relative to $x$). Let $f$ be an element of $\mathfrak{g}^{*}$ whose
restriction to ${}^{u}\mathfrak{b}$ is equal to $x|_{{}^{u}\mathfrak{b}}$, and
whose restriction restriction to $\mathfrak{g}(x)$ is equal to $\lambda$. Such
an element $f$ exists by the definition of $L(x)$.

Let $G$ acts on $D$ naturally, then we have the followings.

######  Proposition 5.2 ([6], II.5).

  * (i)

The orbit $G\cdot f$ in $\mathfrak{g}^{*}$ does not depend on the choices of
$\mathfrak{b}$ and $f$. We denote it by $O_{x,\lambda}$.

  * (ii)

The map $(x,\lambda)\mapsto O_{x,\lambda}$ induces a bijection from $D/G$ to
$\mathfrak{g}^{*}/G$.

####  5.2.2. Construction of unitary dual of algebraic groups

Duflo showed how to construct the irreducible unitary representations of $G$
using the coadjoint orbits and the irreducible unitary representations of some
reductive groups.

The construction of representations used Metaplectic groups.

Since $G(x)$ acts on $\mathfrak{g}/\mathfrak{g}(x)$ and keeps the form
$\omega_{x}$, we get a morphism

$\phi:G(x)\to\mathrm{Sp}(\mathfrak{g}/\mathfrak{g}(x),\omega_{x}).$

Let $\mathrm{Mp}(\mathfrak{g}/\mathfrak{g}(x),\omega_{x})$ be the metaplectic
group, i.e., there is a nontrivial connected double cover

$\varphi:\mathrm{Mp}(\mathfrak{g}/\mathfrak{g}(x),\omega_{x})\to\mathrm{Sp}(\mathfrak{g}/\mathfrak{g}(x),\omega_{x}).$

We define a double cover of $G(x)$ as

$G(x)^{\mathfrak{g}}=\\{(g,t)\ |\ g\in
G(x),t\in\mathrm{Mp}(\mathfrak{g}/\mathfrak{g}(x),\omega_{x})\ \text{and}\
\phi(g)=\varphi(t)\\}.$

Let $\psi:G(x)^{\mathfrak{g}}\to G(x)$ be the covering morphism, and let
$(1,-1)$ denote the nontrivial element of $\ker\psi$.

Similarly, for any group $H$, which acts on a symplectic vector space
$\mathfrak{m}$ and keep the symplectic form, we have a morphism
$\phi^{\prime}:H\to\mathrm{Sp}(\mathfrak{m})$ and a double covering morphism
$\varphi^{\prime}:\mathrm{Mp}(\mathfrak{m})\to\mathrm{Sp}(\mathfrak{m})$.
Define a double cover

$\widetilde{H}=\\{(h,t)\ |\ h\in H,t\in\mathrm{Mp}(\mathfrak{m})\ \text{and}\
\phi^{\prime}(h)=\varphi^{\prime}(t)\\}.$

By Segal-Shale-Weil representation (see Duflo [6], II.6), any
$\hat{s}\in\mathrm{Mp}(\mathfrak{m})$ which is mapped to
$s\in\mathrm{Sp}(\mathfrak{m})$ can be represented by $(s,\theta_{\hat{s}})$
(or just denoted by $(s,\theta)$), where $\theta_{\hat{s}}$ is a function
depended on $\hat{s}$, with complex value module $1$ over the set of maximal
isotropic subspaces in $\mathfrak{m}$. Therefore, any element
$(h,t)\in\widetilde{H}$ ($h\in H,t\in\mathrm{Mp}(\mathfrak{m})$) can be
represented by $(h,\theta_{t})$. We will also use $H^{\mathfrak{m}}$ to denote
$\widetilde{H}$.

Since ${}^{u}G(x)$ is unipotent, ${}^{u}G(x)$ can be embedded into
$G(x)^{\mathfrak{g}}$. Let

$\displaystyle Y^{irr}(x)$ $\displaystyle=\\{\text{irreducible unitary
representation}\ \pi$ $\displaystyle\text{of}\ G(x)^{\mathfrak{g}}/^{u}G(x)\
|\ \pi((1,-1))=-1\\}.$

Let $C$ denote the set of couples $(x,\tau)$, where $x$ is a linear form of
unipotent type, and $\tau\in Y^{irr}(x)$. Then the group $G$ acts on $C$
naturally.

Now, we can construct an irreducible unitary representation $T_{x,\tau}$ of
$G$ (see [6], III for details).

Let $\mathfrak{b}$ be a subalgebra of strongly unipotent type relative to $x$
which is stable under $G(x)$, and let $\mathfrak{v}$ denote the nilpotent
radical of $\mathfrak{b}$. Let $V$ be the subgroup corresponding to
$\mathfrak{v}$. Set $B=G(x)V$. Let $v$ denote the restriction of $x$ to
$\mathfrak{v}$, and $G(x)^{\mathfrak{v}}$ is defined.

Let $\mathfrak{l}\subset\mathfrak{v}$ be a polarization relative to $v$, and
let $L$ be the unipotent subgroup with Lie algebra $\mathfrak{l}$. Define the
irreducible unitary representation of $V$ attached to $v$ as

$T_{v}=\mathrm{Ind}_{L}^{V}e^{iv|_{\mathfrak{l}}}.$

Define

$T_{x,\tau}=\mathrm{Ind}_{B}^{G}(\tau^{\prime}\otimes S_{v}T_{v}),$

where $\tau^{\prime}$ and $S_{v}$ are defined as follows, and we have that
$T_{x,\tau}$ is independent of the choice of $B$ (see [6], III.16).

Define $\tau^{\prime}$ to be the representation of $G(x)^{\mathfrak{v}}$ as

$\tau^{\prime}(y,\psi)=(\varphi\psi^{-1})\overline{\tau}(y,\varphi),$

here $\overline{\tau}=\tau\otimes\chi_{x}$ of
$(G(x)^{\mathfrak{g}}/^{u}G(x))\rtimes$${}^{u}G(x)$ $\cong
G(x)^{\mathfrak{g}}$ with $\chi_{x}$ defined by
$\mathrm{d}\chi_{x}=ix|_{{}^{u}\mathfrak{g}(x)}$, and

$\varphi\psi^{-1}=\varphi(\mathfrak{l}^{\prime}+\mathfrak{g}^{\perp})\psi(\mathfrak{l}^{\prime})^{-1}$

is a constant which is independent on the choice of the maximal totally
isotropic subspace $\mathfrak{l}^{\prime}$ of $\mathfrak{v}$ (see [6], II.8).

Define $S_{v}$ to be the action of $G(x)^{\mathfrak{v}}$ on $T_{v}$ by

$S_{v}(y,\varphi)=\varphi(\mathfrak{l})\cdot S^{\prime}_{v,\mathfrak{l}}(y),$

and $S^{\prime}_{v,\mathfrak{l}}$ is the action of $G(x)$ on $T_{v}$ defined
by

$S^{\prime}_{v,\mathfrak{l}}(y)=||A(y)||^{-1}F_{\mathfrak{l},y\mathfrak{l}}A(y),$

$F_{\mathfrak{l},y\mathfrak{l}}$ is the intertwining operator from $T_{yv}$ to
$T_{v}$, and $A(y)$ is the operator from $T_{v}$ to $T_{yv}$ with

$A(y)\alpha(z)=\alpha(y^{-1}(z)),\alpha\in T_{v},z\in V.$

It can be checked that $S_{v}$ is independent of the choice of $\mathfrak{l}$
(see [6], II.10).

######  Proposition 5.3 ([6], III.12).

For any element $(x,\tau)\in C$, we get an irreducible unitary representation
$T_{x,\tau}$ of $G$. Moreover, the map $(x,\tau)\mapsto T_{x,\tau}$ induces a
bijection from $C/G$ to $\widehat{G}$.

Now, we can obtain the correspondence between coadjoint orbits and irreducible
unitary representations of an algebraic group $G$ (see [6], III.19, III.20).

Let $f\in\mathfrak{g}^{*}$, by Proposition 5.2, we get
$\mathcal{O}_{f}=\mathcal{O}_{x,\lambda}$ for some $x\in\mathfrak{g}^{*}$ of
unipotent type and $\lambda\in
L(x)\cong(\mathfrak{g}(x)/^{u}\mathfrak{g}(x))^{*}$. Assume that $\tau$ is an
irreducible unitary representation of $G(x)^{\mathfrak{g}}/^{u}G(x)$ attached
to $\lambda$ such that $\tau((1,-1))=-1$, then $T_{x,\tau}$ is an irreducible
unitary representation of $G$ attached to $\mathcal{O}_{f}$.

####  5.2.3. Duflo’s construction in the $P_{n}(k)$ case

Set $G=P_{n}(k)$, we show how to attach representations with coadjoint orbits
by the above method (using the same notation).

For any element $f\in\mathfrak{g}^{*}$, applying the results in section 2, we
have that $f$ is conjugated to an element
$\mathrm{pr}^{\prime}(\mathrm{diag}(A,J_{m}))$, for some $m\in\mathbb{Z}_{+}$
and $A\in\mathfrak{gl}_{n-m}(k)$.

Let $x=\mathrm{pr}^{\prime}(\mathrm{diag}(0,J_{m}))$, then $x$ is of unipotent
type. Actually, $\mathfrak{g}(x)$ has reductive factor

$\mathfrak{g}^{\prime}=\\{\mathrm{diag}(X,0_{m\times m})\ |\
X\in\mathfrak{gl}_{n-m}(k)\\}\subset\ker(x).$

Let

$\mathfrak{b}=\\{\left(\begin{array}[]{cc}X&Y\\\ 0&Z\end{array}\right)\ |\
X\in\mathfrak{gl}_{n-m}(k),Y\in M_{(n-m)\times
m}(k),Z\in\mathfrak{u}_{m}(k)\\},$

where $\mathfrak{u}_{m}(k)$ is the set of strictly upper triangular matrices
over $k$, then $\mathfrak{b}$ is a subalgebra of strongly unipotent type
relative to $x$. Therefore, $x$ is of unipotent type.

Let $\lambda^{\prime}=\mathrm{pr}^{\prime}(\mathrm{diag}(A,0))$, then
$\lambda:=\lambda^{\prime}|_{\mathfrak{g}(x)}\in L(x)$, since
$x|_{{}^{u}\mathfrak{g}(x)}=0$ and $\lambda|_{{}^{u}\mathfrak{g}(x)}=0$.
Moreover,

$L(x)\cong(\mathfrak{g}(x)/^{u}\mathfrak{g}(x))^{*}\cong\mathfrak{gl}_{n-m}(k)^{*}.$

Also it is easy to see that $\mathcal{O}_{f}=\mathcal{O}_{x,\lambda}$.

Furthermore, we have that $G(x)^{\mathfrak{g}}$ is a trivial double cover of
$G(x)$, that is $G(x)^{\mathfrak{g}}\cong G(x)\times\mathbb{Z}_{2}$. Actually,

$G(x)=\\{\left(\begin{array}[]{ccc}Y&Z&0\\\ 0&1&0\\\
0&0&I_{m-1}\end{array}\right)\ |\ Y\in\mathrm{GL}_{n-m}(k),Z\in M_{(n-m)\times
1}(k)\\}.$

It is not hard to see that the image of the reductive factor

$\\{\mathrm{diag}(Y,I_{m})\ |\ Y\in\mathrm{GL}_{n-m}(k)\\}$

in

$\mathrm{Sp}(\mathfrak{g}/\mathfrak{g}(x),\omega_{x})\cong\mathrm{Sp}((n-\frac{m}{2})(m-1),k)$

has trivial double cover in the metaplectic group
$\mathrm{Mp}((n-\frac{m}{2})(m-1),k)$. Therefore, $G(x)^{\mathfrak{g}}$ is a
trivial double cover of $G(x)$.

To attach $\mathcal{O}_{f}$ with representation of $P_{n}(k)$, we assume that
$\tau_{0}$ is the irreducible unitary representation of

$G(x)/^{u}G(x)\cong\mathrm{GL}_{n-m}(k)$

which is attached to the $\mathrm{GL}_{n-m}(k)$-orbit of $\lambda\in
L(x)\cong\mathfrak{gl}_{n-m}(k)^{*}$. Define an irreducible unitary
representation of $G(x)/^{u}G(x)\times\mathbb{Z}_{2}$, denoted by $\tau$, such
that $\tau|_{G(x)/^{u}G(x)}=\tau_{0}$ and $\tau((1,-1))=-1$. Then the
irreducible representation of the irreducible unitary representation attached
to $\mathcal{O}_{f}$ is

$T_{x,\tau}=\mathrm{Ind}_{B}^{G}(\tau^{\prime}\otimes S_{v}T_{v}).$

The subalgebra of unipotent type relative to $x$ constructed above,
$\mathfrak{b}$, is the canonical one. And the corresponding subgroup $B$ is

$B=\\{\left(\begin{array}[]{cc}Y&Z\\\ 0&U\end{array}\right)\ |\
Y\in\mathrm{GL}_{n-m}(k),Z\in M_{(n-m)\times m}(k),U\in U_{m}(k)\\}.$

Here $U_{m}(k)$ is the set of upper triangular unipotent matrices over $k$.
Moreover, $\mathfrak{v}=$${}^{u}\mathfrak{b}$ itself is the polarization of
$v=x|_{\mathfrak{v}}$, and we get a one dimensional unitary representation of
$V$, $T_{v}=e^{ix|_{\mathfrak{v}}}$.

Since $\omega_{x}$ is trivial on $\mathfrak{b}$, we get
$G(x)^{\mathfrak{v}}\cong G(x)\times\mathbb{Z}_{2}$ and
$G(x)^{\mathfrak{b}}\cong G(x)\times\mathbb{Z}_{2}$, and therefore
$S_{v}|_{G(x)}$ is the one dimensional trivial representation, so we have

$\tau^{\prime}\otimes S_{v}T_{v}=\tau_{0}\otimes e^{ix}.$

Therefore, the irreducible unitary representation of $G$ attached to
$\mathcal{O}_{f}$ with $f=\mathrm{pr}^{\prime}(\mathrm{diag}(A,J_{m}))$ is

$T_{x,\tau}=\mathrm{Ind}_{B}^{G}(\tau_{0}\otimes
e^{ix})=\mathrm{I}^{m-1}\mathrm{E}\tau_{0},$

where $\tau_{0}$ is an irreducible unitary representation of
$\mathrm{GL}_{n-m}(k)$ attached to the $\mathrm{GL}_{n-m}(k)$-orbit of
$\mathrm{pr}_{n-m}(A)$.

##  6\. A generalization of Duflo’s conjecture

We establish a generalization of Duflo’s conjecture for the restriction of an
irreducible unitary representation $\pi$ of $G=\mathrm{GL}_{n}(k)$ to the
mirabolic subgroup $P=P_{n}(k)$, $k=\mathbb{R}$ or $\mathbb{C}$, where $\pi$
is attached to a $G$-coadjoint orbit of $f\in\mathfrak{g}^{*}$ in section 5.
Set $\mathcal{O}_{\pi}=\mathcal{O}_{f}$. Let
$\mathrm{p}:\mathcal{O}_{\pi}\to\mathfrak{p}^{*}$ be the moment map. Then we
have the following theorem.

######  Theorem 6.1.

There are only finitely many $P$-orbits in $\mathrm{p}(\mathcal{O}_{\pi})$,
including a unique open $P$-orbit $\Omega$ in $\mathrm{p}(\mathcal{O}_{\pi})$.
Moreover,

  * (1)

the moment map $\mathrm{p}:\mathcal{O}_{\pi}\to\mathfrak{p}^{*}$ is proper
over $\Omega$,

  * (2)

the restriction of $\pi$ to $P$, $\pi|_{P}$ is irreducible, and is attached to
$\Omega$ in the sense of Duflo,

  * (3)

the reduced space of $\Omega$ (with respect to the moment map $\mathrm{p}$) is
a single point.

###### Proof.

By Theorem 3.12, Theorem 3.19, we get (1),(3) immediately. To get (2), we
firstly get $\pi|_{P}$ from section 4 and get
$\Omega=\mathrm{p}(g_{I^{o}}\cdot f)$ from the results of moment maps in
section 3. It remains to verify if $\pi_{P}$ is attached to $\Omega$ by
Duflo’s construction in section 5. We verify one example and one can verify
the other cases easily.

Let $1\leq t_{p}\leq\cdots\leq t_{1}\in\mathbb{Z}_{+}$ such that
$\sum_{i=1}^{p}t_{i}=n$. Let $\pi$ be the unipotent representation

$1|_{G_{t_{1}}(\mathbb{C})}\times\cdots\times 1|_{G_{t_{p}}(\mathbb{C})}$

of $G_{n}(\mathbb{C})$, which is attached to the coadjoint orbit of
$f=\mathrm{pr}(\xi)$ with

$\xi=\mathrm{diag}(J_{k_{1}}^{l_{1}},\cdots,J_{k_{r}}^{l_{r}}),$

where $k_{i},l_{i}\in\mathbb{Z}_{+}$ for $1\leq i\leq r$, and
$\\{\underbrace{k_{1},\cdots,k_{1}}_{l_{1}},\cdots,\underbrace{k_{r},\cdots,k_{r}}_{l_{r}}\\}$
is the dual partition of $\\{t_{1},\cdots,t_{p}\\}$ (so $k_{1}\geq\cdots\geq
k_{r}$).

Then

$\pi|_{P}=I^{p}E(1|_{G_{t_{1}-1}(\mathbb{C})}\times\cdots\times
1|_{G_{t_{p}-1}(\mathbb{C})}).$

To get the unique dense open $P$-orbit in $\mathrm{p}(\mathcal{O}_{f})$, we
set $\tilde{f}=\mathrm{pr}(\tilde{\zeta})$ with
$\tilde{\zeta}=\mathrm{diag}(J_{k_{r}}^{l_{r}},\cdots,J_{k_{1}}^{l_{1}})$.
Then $\tilde{f}$ is $G_{n}(\mathbb{C})$-conjugated to $f$, so
$\mathcal{O}_{f}=\mathcal{O}_{\tilde{f}}$. By Theorem 3.12, we have
$\Omega=P\cdot\mathrm{p}(g_{I^{o}}\cdot\tilde{f})$ is the unique dense open
$P$-orbit in $\mathrm{p}(\mathcal{O}_{f})$. Since $I^{o}=\\{n\\}$ and
$g_{I^{o}}=I_{n}$, we have $g_{I^{o}}\cdot\tilde{f}=\tilde{f}$. It is easy to
see that $\mathrm{p}(\tilde{f})$ is $P$-conjugated to the element
$\mathrm{pr}^{\prime}(\zeta)$ with

$\zeta=\mathrm{diag}(J_{k_{1}}^{l_{1}-1},J_{k_{2}}^{l_{2}},\cdots,J_{k_{r}}^{l_{r}},J_{k_{1}}),$

So $\Omega=P\cdot\mathrm{pr}^{\prime}(\zeta)$.

Based on Duflo’s construction, we get that the irreducible unitary
representation of $P$ attached to $\Omega$ is

$I^{k_{1}}E(1|_{G_{s_{1}}(\mathbb{C})}\times\cdots\times
1|_{G_{s_{q}}(\mathbb{C})}),$

where $\\{s_{1},\cdots,s_{q}\\}$ is the dual partition of
$\\{\underbrace{k_{1},\cdots,k_{1}}_{l_{1}-1},\underbrace{k_{2},\cdots,k_{2}}_{l_{2}},\cdots,\underbrace{k_{r},\cdots,k_{r}}_{l_{r}}\\}$.
By definition of dual partition, we have $p=q$, $s_{i}=t_{i}-1$ for $1\leq
i\leq p$, and $k_{1}=p$. Hence we get that $\pi|_{P}$ is attached to $\Omega$.
∎

## Acknowledgements

The author would like to thank Jun Yu for suggesting this problem and many
discussions, thank Daniel Kayue Wong for discussions on the orbit method of
reductive groups and thank Huajian Xue for many discussions, etc.

## References

  * [1] J.Adams, D.Barbasch, D.Vogan: _The Langlands Classification and Irreducible Characters for Real Reductive Groups_ , Progress in Mathematics, 104. Birkhäuser Boston, Inc., Boston, MA, 1992.
  * [2] A.Aizenbud, D.Gourevitch, S.Sahi: _Derivatives for smooth representations of $\mathrm{GL}(n,\mathbb{R})$ and $\mathrm{GL}(n,\mathbb{C})$_, Israel J. Math. 206 (2015), no. 1, 1-38.
  * [3] L.Auslander, B.Kostant: _Quantization and representations of solvable Lie groups_ , Bull. Amer. Math. Soc. 73 (1967), 692-695.
  * [4] E.M.Baruch: _A proof of Kirillov’s conjecture_ , Ann. of Math. (2) 158 (2003), no. 1, 207-252.
  * [5] J.Bernstein: _$P$ -invariant distributions on $\mathrm{GL}(N)$ and the classification of unitary representations of $\mathrm{GL}(N)$ (non-Archimedean case)_, Lie group representations, II (College Park, Md., 1982/1983), 50-102, Lecture Notes in Math., 1041, Springer, Berlin, 1984.
  * [6] M.Duflo: _Théorie de Mackey pour les groupes de Lie algébriques_ , (French) [Mackey theory for algebraic Lie groups] Acta Math. 149 (1982), no. 3-4, 153-213.
  * [7] M.Duflo: _On the Plancherel formula for almost algebraic real Lie groups_ , Lie group representations, III (College Park, Md., 1982/1983) 101-165, Lecture Notes in Math., 1077, Springer, Berlin, 1984.
  * [8] M.Duflo, M.Vergne: _Kirillov’s formula and Guillemin-Sternberg conjecture_ , C. R. Math. Acad. Sci. Paris 349 (2011), no. 23-24, 1213-1217.
  * [9] H.Fujiwara: _Sur les restrictions des représentations unitaires des groupes de Lie résolubles exponentiels_ , (French) [On restrictions of unitary representations of exponential solvable Lie groups] Invent. Math. 104 (1991), no.3, 647-654.
  * [10] V.Guillemin, S.Sternberg: _Geometric quantization and multiplicities of group representations_ , Invent. Math. 67 (1982), no.3, 515-538.
  * [11] G.Heckman: _Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups_ , Invent. Math. 67 (1982) no.2, 333-356.
  * [12] A.Kirillov: _Lectures on the Orbit Method_ , Graduate Studies in Mathematics, 64. American Mathematical Society, Providence, RI, 2004.
  * [13] A.W.Knapp, D.Vogan: _Cohomological induction and unitary representations_ , Princeton Mathematical Series, 45. Princeton University Press, Princeton, NJ, 1995.
  * [14] T. Kobayashi: _Restrictions of unitary representations of real reductive groups_ , Lie theory, 139-207, Progr. Math., 229, Birkhäuser Boston, Boston, MA, 2005.
  * [15] G.Liu: _Duflo’s conjecture for the branching problem to the Iwasawa $AN$-subgroup of a semisimple Lie group_, Math. Ann. 362 (2015), no. 1-2, 107-120.
  * [16] G.Liu, Y.Oshima, J.Yu: _Restriction of irreducible unitary representations of $\mathrm{Spin}(N,1)$ to parabolic subgroups_, arXiv: 2010.01026.
  * [17] G.Liu, J.Yu: _A geometric interpretation of Kirillov’s conjecture_ , arXiv: 1806.06318.
  * [18] G.W.Mackey: _Induced representations of locally compact groups I_ , Ann. of Math. (2) 55 (1952), 101-139.
  * [19] S.Sahi: _On Kirillov’s conjecture for Archimedean fields_ , Compositio Math. 72 (1989), no.1, 67-86.
  * [20] S.Sahi: _A simple construction of Stein’s complementary series representations_ , Proc. Amer. Math. Soc. 108 (1990), no.1, 257-266.
  * [21] S.Sahi, E.M.Stein: _Analysis in matrix space and Speh’s representations_ , Invent. Math. 101 (1990), no.2, 379-393.
  * [22] D.Vogan: _The method of coadjoint orbits for real reductive groups_ , Representation theory of Lie groups (Park City, UT, 1998), 179-238, IAS/Park City Math. Ser., 8, Amer. Math. Soc., Providence, RI, 2000.
  * [23] D.Vogan: _The unitary dual of $\mathrm{GL}(n)$ over an Archimedean field_, Invent.Math. 83 (1986), no. 3, 449-505.
  * [24] D.Vogan: _Unitary representations of reductive Lie groups and the orbit method_ , Based on notes prepared by Jorge Vargas. Progr. Math.,105, New developments in Lie theory and their applications (Córdoba, 1989), 87-114, Birkhäuser Boston, Boston, MA, 1992.
  * [25] D.Vogan: _Unitary Representations of Reductive Lie Groups_ , Annals of Mathematics Studies, 118. Princeton University Press, Princeton, NJ, 1987.