# A note on unfolding manifolds of meromorphic connections on the Riemann
sphere with unramified singularities
Dedicated to Prof. Yoshitsugu Takei on the occasion of his 60th birthday

Kazuki Hiroe111 The author is supported byJSPS KAKENHI Grant Number 20K03648.
Department of Mathematics and Informatics, Chiba University
1-33, Yayoi-cho, Inage-ku, Chiba-shi, Chiba, 263-8522 JAPAN
email<EMAIL_ADDRESS>
###### Abstract

This note explains a construction of a Poisson manifold whose symplectic
foliation describes a deformation of a moduli space of meromorphic connections
with unramified irregular singularities. In particular, this deformation of
the moduli space corresponds to the unfolding of irregular singularities of
the meromorphic connections. This is an announcement of some results in the
forthcoming paper.

###### Contents

  1. 1 Moduli space of meromorphic connections on a trivial bundle over the Riemann sphere with unramified irregular singularities
    1. 1.1 Hukuhara-Turrittin-Levelt normal forms
    2. 1.2 Truncated orbits and extended orbits
      1. 1.2.1 Truncated orbits as coadjoint orbits
      2. 1.2.2 Extended orbits of HTL normal forms
    3. 1.3 Triangular factorization of $\mathbb{O}_{H_{\mathrm{irr}}}$
      1. 1.3.1 Compositions of a natural number and subgroups of $\mathrm{GL}_{n}(\mathbb{C})$
      2. 1.3.2 Sequences of compositions of positive integers
      3. 1.3.3 Spectral types of HTL-normal forms
      4. 1.3.4 Triangular factrization of $\mathbb{O}_{H_{\mathrm{irr}}}$
    4. 1.4 Moduli space of meromorphic connections on a trivial bundle over the Riemann sphere with unramified irregular singularities
      1. 1.4.1 Algebraic meromorphic connections on a trivial bundle over the Riemann sphere
      2. 1.4.2 The space of meromorphic connections as a complex manifold
      3. 1.4.3 Residue maps of truncated orbits
      4. 1.4.4 Moduli space of meromorphic connections on a trivial bundle with unramified irregular sigularities
  2. 2 Construction of unfolding manifold
    1. 2.1 Deformation of HTL normal forms and unfolding of spectral types
      1. 2.1.1 An open subset of $\mathbb{C}^{k+1}$ associated to $H$
      2. 2.1.2 A stratification of $\mathbb{C}^{k+1}$ associated to partitions of a finite set
      3. 2.1.3 Unfolding of an HTL normal form
    2. 2.2 Deformation of $\mathbb{O}_{H_{\mathrm{irr}}}$
      1. 2.2.1 Deformations of $\mathbb{C}[z]_{l}$ and $\mathbb{C}[z^{-1}]_{l}dz$
      2. 2.2.2 Partial fraction decomposition
      3. 2.2.3 Lie groupoid as a deformation of $N_{\mathbf{m}_{l}^{l+1}}^{-}(\mathbb{C}[z]_{l-1})^{1}$
      4. 2.2.4 Symplectic foliation as deformation of $\mathbb{O}_{H_{\mathrm{irr}}}$
    3. 2.3 Local unfolding manifold
      1. 2.3.1 Unfolding manifold of a truncated orbit
      2. 2.3.2 Symplecitc foliation of $(\mathbb{O}_{H})_{\mathbb{D}(H)}$
    4. 2.4 Unfolding manifold of meromorphic connections
      1. 2.4.1 Stratification of the product of local unfolding manifolds
      2. 2.4.2 Unfolding manifold of meromorphic connections
      3. 2.4.3 Example: unfolding manifold of unramified Painlevé phase spaces

## Introduction

This note is an announcement of some results in the forthcoming paper [4].
Detailed descriptions of the statements without proofs in this note shall be
given in [4].

For linear ordinary equations on complex domains, the confluence of singular
points is a classical tools to investigate irregular singularities. For
example, it is well-known that each differential equation of Airy, Hermite-
Weber, and Kummer confluent hypergeometric functions is obtained from the
differential equation

$z(1-z)\frac{d^{2}}{dz^{2}}w+(c-(a+b+1)z)\frac{d}{dz}w-abw=0$

of the Gauss hypergeometric function by the confluence of singular points
$z=0,1,\infty$, and further, analytic properties of these functions can be
related to that of the Gauss hypergeometric function. Similarly for the Heun
differential equation which has an accessory parameter and is known as a
generalization of Gauss hypergeometric differentia equations, there is a well-
known family of differential equations obtained by the confluence of singular
points, confluent Heun, bi-confluent Heun, tri-confluent Heun, and doubly-
confluent Heun equations. Furthermore, for the study of higher dimensional
Painlevé equations, Kawakami-Nakamura-Sakai constructed many confluent
families of differential equations with 4 accessory parameters in [5].

We may notice that these confluence families of differential equations are
considered as deformations of spaces of their accessory parameters, namely, we
can say that the confluence of singularities gives rise to deformations of
moduli spaces of differential equations. Indeed in [2], Chekhov-Mazzocco-
Rubtsov defined a deformation of Painlevé monodromy manifolds associated to
the confluence of singular points of differential equations. Namely, they
considered monodromy manifolds of linear ordinary differential equations on
the Riemann sphere whose isomonodromic deformations give rise to Painlevé
equations, and obtained explicit deformations of these manifolds which
correspond to the confluence of singular points of differential equations. In
case of moduli spaces of connections, Inaba constructed in [8] a one-parameter
deformation of moduli spaces of meromorphic connections with unramifed
irregular singular points whose Hukuhara-Turrittin-Levelt normal forms have
distinct eigenvalues. A similar deformation also considered by Gaiur-Mazzocco-
Rubtsov in [3], in which they defined a deformation of coadjoint orbits of a
Lie algebra of polynomials which they call the Takiff algebra. Furthermore,
Gaiur-Mazzocco-Rubtsov and Inaba considered in their papers [3] and [8] the
confluence of isomonodromic deformation equations.

Based on these preceding works, we shall construct a deformation space of
moduli space of differential equations describing the confluence of their
singularities. Let us explain our main result. We focus on the opposite
operation to the confluence, i.e., the unfolding of irregular singular points.
Namely, we consider the procedure to unfold an unramified irregular singular
points to regular singular ones. Let us consider meromorphic connections on a
trivial bundle defined over the complex projective line $\mathbb{P}^{1}$ with
unramified irregular singularities on a finite subset
$D\subset\mathbb{P}^{1}$. We fix a collection of unramified Hukuhara-
Turrittin-Levelt normal forms $\mathbf{H}=(H_{a})_{a\in D}$ at each singular
point $a\in D$. Then Boalch introduced in [1] the moduli space of irreducible
meromorphic connections with the fixed normal forms $\mathbf{H}$ which we
denote by $\mathcal{M}_{s}^{*}(\mathbf{H})$. Firstly, we introduce the
deformation $\mathbf{H}(\mathbf{c})$,
$\mathbf{c}\in\mathbb{C}^{N_{\mathbf{H}}}$, of the collection of normal forms
$\mathbf{H}$, where $N_{\mathbf{H}}$ is the positive integer uniquely
determined by $\mathbf{H}$. This is a family of unramified Hukuhara-Turrittin-
Levelt normal forms describing the unfolding procedure of unramified irregular
singular points to regular singular ones. Then our main theorem is the
following.

######  Theorem 0.1 (Theorem 2.15, Theorem 2.16).

Suppose that $\mathcal{M}_{s}^{*}(\mathbf{H})\neq\emptyset$. Then, there are a
Zariski open subset
$\mathbb{D}(\mathbf{H})\subset\mathbb{C}^{N_{\mathbf{H}}}$, holomorphic
Poisson manifold $\mathcal{M}_{s}^{*}(\mathbf{H})_{\mathbb{D}(\mathbf{H})}$,
and holomorphic map
$\theta\colon\mathcal{M}_{s}^{*}(\mathbf{H})_{\mathbb{D}(\mathbf{H})}\rightarrow\mathbb{D}(\mathbf{H})$
such that for each $\mathbf{c}\in\mathbb{D}(\mathbf{H})$ there exits an
embedding

$\theta^{-1}(\mathbf{c})\hookrightarrow\mathcal{M}_{s}^{*}(\mathbf{H}(\mathbf{c}))$

onto a Zariski open subset. Namely, the complex manifold
$\mathcal{M}_{s}^{*}(\mathbf{H})_{\mathbb{D}(\mathbf{H})}$ is a deformation of
the moduli space $\mathcal{M}_{s}^{*}(\mathbf{H})$ and moreover each fiber
$\theta^{-1}(\mathbf{c})$ is isomorphic to a Zariski open subset of the moduli
space of meromorphic connections $\mathcal{M}_{s}^{*}(\mathbf{H}(\mathbf{c}))$
obtained by the unfolding of $\mathbf{H}$.

Furthermore, the family of the fibers
$(\theta^{-1}(\mathbf{c}))_{\mathbf{c}\in\mathbb{D}(\mathbf{H})}$ is the
symplectic foliation of the regular Poisson manifold
$\mathcal{M}_{s}^{*}(\mathbf{H})_{\mathbb{D}(\mathbf{H})}$ and the above
embeddings are symplectic morphisms.

Our families $\mathcal{M}_{s}^{*}(\mathbf{H})_{\mathbb{D}(\mathbf{H})}$ of
moduli space cover many of known deformations of differential equations
induced by the confluence of their singular points. For example, confluent
families explained above, i.e., families of Gauss hypergeometric equation,
Heun equation, and equations with 4 accessory parameters appearing in [5] by
Kawakami-Nakamura-Sakai are all contained in our families when we focus on
unramified irregular singularities. Our construction of the deformation is
strongly influenced by Oshima’s work in [12]. In that paper, Oshima defined a
good confluence family of linear differential equations on the Riemann sphere
without accessory parameter which he called the versal family of rigid
differential equations. Our result can be said to be a generalization of his
versal family to non-rigid cases. For instance, Oshima’s families recovers
from our deformation spaces of moduli spaces of rigid connections.

Acknowledgement

The author express his gratitude to Toshio Oshima for his inspiring works and
many variable discussions. He thanks to Hiroshi Kawakami, Akane Nakamura, and
Daisuke Yamakawa. Discussions with them in the early stage help him very much
to press forward this project. He is also grateful to Takuro Mochizuki, Hiraku
Nakajima, and Vladimir Rubtsov for their variable comments.

This article is submitted to RIMS Kôkyûroku Bessatsu, a Festschrift volume in
Honor of Yoshitsugu Takei on the occasion of his 60th birthday. The author
express his configurations.

##  1 Moduli space of meromorphic connections on a trivial bundle over the
Riemann sphere with unramified irregular singularities

In the paper [9], Jimbo, Miwa, and Ueno developed a general study of
symplectic structure of the spaces of accessory parameters of differential
equations with irregular singular points. After their work, in [1], Boalch
introduced a natural construction of the symplectic structure from the
symplectic geometry of coadjoint orbits and the theory of symplectic
reduction. Following Boalch’s method, we give a review of the symplectic
structure of the moduli spaces of meromorphic connections on the trivial
bundle with unramified irregular singular points. A detailed exposition of the
contents in this section can also be found in [16].

###  1.1 Hukuhara-Turrittin-Levelt normal forms

A classification theory of local and formal differential equations is obtained
by Hukuhara, Turrittin, and Levelt, see [7], [13], [10]. We give a quick
review of this classification theory.

######  Definition 1.1.

Let $R$ be either the ring of formal power series $\mathbb{C}[\\![z]\\!]$ or
the field of Laurent series $\mathbb{C}(\\!(z)\\!)$, and $V$ be a free
$R$-module of finite rank. Let us consider a $\mathbb{C}$-linear map
$\nabla\colon V\rightarrow V\otimes_{R}\mathbb{C}(\\!(z)\\!)dz$ satisfying the
Leibniz rule,

$\nabla(fv)=v\otimes\left(df\right)+f\nabla(v),\quad f\in R,\ v\in V,$

which is called meromorphic connection or connection on $V$. Here we denote
the vector space of formal meromorphic 1-forms by
$\mathbb{C}(\\!(z)\\!)dz:=\left\\{f(z)dz\mid
f(z)\in\mathbb{C}(\\!(z)\\!)\right\\}$, and
$d\colon\mathbb{C}(\\!(z)\\!)\rightarrow\mathbb{C}(\\!(z)\\!)dz$ stands for
the exterior derivative defined by $df(z):=\left(\frac{d}{dz}f(z)\right)dz$
for $f(z)\in R$. Then we also call the pair $(\nabla,V)$ a formal meromorphic
connection over $R$.

Let $(\nabla,V)$ be a connection over $\mathbb{C}(\\!(z)\\!)$ of rank $n$ and
consider an algebraic field extension $L$ of $\mathbb{C}(\\!(z)\\!)$. Then
Newton-Puiseux theory tells us that there exists a positive integer
$q\in\mathbb{Z}_{>0}$ and $L\cong\mathbb{C}(\\!(t)\\!)$ with $t^{q}=z$. Then
the inclusion map $\iota\colon\mathbb{C}(\\!(z)\\!)\ni f(z)\rightarrow
f(t^{q})\in\mathbb{C}(\\!(t)\\!)$ induces the map

$\begin{array}[]{cccc}\iota_{*}\colon&\mathbb{C}(\\!(z)\\!)dz&\longrightarrow&\mathbb{C}(\\!(t)\\!)dt\\\
&f(z)dz&\longmapsto&f(t^{q})qt^{q-1}\,dt\end{array}$

which is compatible with the exterior derivative. Then we can define the
connection $(\nabla_{t},V_{t})$ over $\mathbb{C}(\\!(t)\\!)$ which we call the
extension of $(\nabla,V)$ to $\mathbb{C}(\\!(t)\\!)$ as follows. Let
$V_{t}:=\mathbb{C}(\\!(t)\\!)\otimes_{\mathbb{C}(\\!(z)\\!)}V$ be the
extension of scalars of $V$ with the inclusion map $\iota_{V}\colon V\ni
v\mapsto 1\otimes v\in V_{t}$ as $\mathbb{C}(\\!(z)\\!)$-vector spaces. As
well as the above, we have the induced morphism as
$\mathbb{C}(\\!(z)\\!)$-vector spaces,

$\begin{array}[]{cccc}\iota_{V\,*}\colon&V\otimes_{\mathbb{C}(\\!(z)\\!)}\mathbb{C}(\\!(z)\\!)dz&\longrightarrow&V_{t}\otimes_{\mathbb{C}(\\!(t)\\!)}\mathbb{C}(\\!(t)\\!)dt\\\
&v\otimes f(z)dz&\longmapsto&\iota_{V}(v)\otimes\iota_{*}(f(z)dz)\end{array}$

Then we define the following $\mathbb{C}$-linear map $\nabla_{t}\colon
V_{t}\rightarrow V_{t}\otimes_{\mathbb{C}(\\!(t)\\!)}\mathbb{C}(\\!(t)\\!)dt$
by

$\nabla_{t}(f(t)\otimes v):=\iota_{V}(v)\otimes
df(t)+f(t)\otimes\iota_{V\,*}(\nabla v),\quad f(t)\in\mathbb{C}(\\!(t)\\!),\
v\in V.$

The classification theory of connections over the algebraic closure
$\mathbb{C}(\\!(z)\\!)^{\text{alg}}$ was developed by Hukuhara and Turrittin
independently and improved and simplified as in the following manner by Levelt
afterward. We say that a connection $(\nabla,V)$ over $\mathbb{C}(\\!(t)\\!)$
is diagonalizable when $V$ is a direct sum of 1-dimensional $\nabla$-invariant
subspaces $V_{i}$, i.e., $\nabla(V_{i})\subset
V_{i}\otimes\mathbb{C}(\\!(t)\\!)dt$.

######  Theorem 1.2 (Hukuhara, Turrittin, Levelt, [7], [13], [10]).

Let $(\nabla,V)$ be a connection over $\mathbb{C}(\\!(z)\\!)$. Then there
exists an algebraic filed extension $\mathbb{C}(\\!(t)\\!)$ of
$\mathbb{C}(\\!(z)\\!)$ of degree $q$ such that $\nabla_{t}$ has the
decomposition

$t\nabla_{t}=S+N$

where $S$ is a diagonalizable connection on $V_{t}$, $N$ is a nilpotent
$\mathbb{C}(\\!(t)\\!)$-linear map and $S$ and $N$ commute with each other,
i.e., $[S,N]:=SN-NS=0$. The smallest integer $q$ satisfying the above is
called ramification index of the connection. If the ramification index is
$q=1$, we say that $\nabla$ is unramified.

Furthermore, there exists a basis of $V_{t}$ over $\mathbb{C}(\\!(t)\\!)$
under which $S$ and $N$ can be written as

$S=td_{t}-\sum_{i=0}^{k}S_{i}t^{-i}dt,\quad N=-N_{0}\,dt$

with diagonal matrices $S_{i}\in M_{n}(\mathbb{C})$, $i=0,1,\ldots,k$ and a
nilpotent matrix $N_{0}\in M_{n}(\mathbb{C})$. Here $n$ is the rank of $V$ and
$A\,dt$ for $A\in M_{n}(\mathbb{C}(\\!(t)\\!))$ denotes the
$\mathbb{C}(\\!(t)\\!)$-linear map $V_{t}\ni v\mapsto Av\,dt\in
V_{t}\otimes_{\mathbb{C}(\\!(t)\\!)}\mathbb{C}(\\!(t)\\!)dt$.

This leads to the following definition of normal forms in
$M_{n}(\mathbb{C}(\\!(t)\\!))dt$ which are usual Jordan normal forms if $k=1$.

######  Definition 1.3 (HTL normal form).

A Hukuhara-Turrittin-Levelt normal form or HTL normal form for short, is an
element in $M_{n}(\mathbb{C}(\\!(t)\\!))dt$ of the form

$\left(\frac{S_{k}}{t^{k}}+\cdots+\frac{S_{1}}{t}+S_{0}+N_{0}\right)\frac{dt}{t}$

where $S_{i}\in M_{n}(\mathbb{C})$ are diagonal matrices commute with the
nilpotent matrix $N_{0}\in M_{n}(\mathbb{C})$, i.e., $[S_{i},N_{0}]=0$, for
all $i=0,1,\ldots,k-1$. The above theorem says that for a connection
$(V,\nabla)$ over $\mathbb{C}(\\!(z)\\!)$, we can find an extension field
$\mathbb{C}(\\!(t)\\!)$ with $t^{q}=z$ and a HTL normal form $H\in
M_{n}(\mathbb{C}(\\!(t)\\!))$ such that

$\nabla_{t}=d-H$

under a suitable basis of $V_{t}$ over $\mathbb{C}(\\!(t)\\!)$. This normal
form $H$ is called the HTL normal form of $(V,\nabla)$. In particular, we say
that $(V,\nabla)$ is an unramified connection if we can take an HTL normal
form $H$ from $M_{n}(\mathbb{C}(\\!(z)\\!))dz\subset
M_{n}(\mathbb{C}(\\!(t)\\!))dt$. In this case, we also call this HTL normal
form $H\in M_{n}(\mathbb{C}(\\!(z)\\!))dz$ an unramified HTL normal form.

Fix an unramified HTL normal form

$H=\left(\frac{S_{k-1}}{z^{k}}+\cdots+\frac{S_{1}}{z}+S_{0}+N_{0}\right)\frac{dz}{z}\in
M_{n}(\mathbb{C}(\\!(z)\\!)dz)$

and consider an unramified meromorphic connection
$(\nabla_{H},\mathbb{C}[\\![z]\\!]^{\oplus n})$ over $\mathbb{C}[\\![z]\\!]$
defined by

$\nabla_{H}:=d-H.$

Then, a connection $(\nabla,\mathbb{C}[\\![z]\\!]^{\oplus n})$ is isomorphic
to $(\nabla_{H},\mathbb{C}[\\![z]\\!]^{\oplus n})$ if and only if there exists
$g(z)\in\mathrm{GL}_{n}(\mathbb{C}[\\![z]\\!])$ such that

$\nabla=d-g(z)[H],$

where

$g(z)[H]:=g(z)Hg(z)^{-1}+\left(\frac{d}{dz}g(z)\right)g(z)^{-1}dz.$

Thus the isomorphic class of $(\nabla_{H},\mathbb{C}[\\![z]\\!]^{\oplus n})$
is parametrized by the orbit

$O_{H}:=\left\\{g(z)[H]\in
M_{n}(\mathbb{C}(\\!(z)\\!))dz\,\middle|\,g(z)\in\mathrm{GL}_{n}(\mathbb{C}[\\![z]\\!])\right\\}.$

As an analogue of this isomorphic class $O_{H}$, Boalch introduced the
truncated orbit of $H$ in [1].

######  Definition 1.4 (truncated orbit).

Let $H$ be an HTL normal form and we regard $H$ as an element in
$M_{n}(\mathbb{C}(\\!(z)\\!)/\mathbb{C}[\\![z]\\!])dz$. Then the orbit of $H$,

$\mathbb{O}_{H}:=\left\\{gHg^{-1}\in
M_{n}(\mathbb{C}(\\!(z)\\!)/\mathbb{C}[\\![z]\\!])dz\,\middle|\,g\in\mathrm{GL}_{n}(\mathbb{C}[\\![z]\\!])\right\\}$

is called the truncated orbit of $H$.

Note that the truncated orbit of $\mathbb{O}_{H}$ is characterized by $H$ in
the following sense.

######  Proposition 1.5 (Boach [1], Yamakawa [14]).

If an HTL normal form $H^{\prime}$ contained in $\mathbb{O}_{H}$, there exists
$C\in\mathrm{GL}_{n}(\mathbb{C})$ such that $H^{\prime}=CHC^{-1}$.

###  1.2 Truncated orbits and extended orbits

Let a fraktur $\mathfrak{q}$ be a $\mathbb{C}$-subalgebra of
$\mathfrak{gl}_{n}(\mathbb{C})=M_{n}(\mathbb{C})$ which can be seen as a Lie
subalgebra defining the bracket by $[X,Y]=XY-YX$ for $X,Y\in\mathfrak{q}$.
Then we denote the corresponding analytic subgroup of
$\mathrm{GL}_{n}(\mathbb{C})$ by the roman $Q$.

For a positive integer $l$, we consider a $\mathbb{C}[\\![z]\\!]$-module
$\mathbb{C}[z^{-1}]_{l}:=z^{-(l+1)}\mathbb{C}[\\![z]\\!]/\mathbb{C}[\\![z]\\!]$.
Since the annihilator ideal of this module is
$\mathrm{Ann}_{\mathbb{C}[\\![z]\\!]}(\mathbb{C}[z^{-1}]_{l})=\langle
z^{l+1}\rangle_{\mathbb{C}[\\![z]\\!]}$, $\mathbb{C}[z^{-1}]_{l}$ can also be
seen as a $\mathbb{C}[z]_{l}:=\mathbb{C}[\\![z]\\!]/\langle
z^{l+1}\rangle_{\mathbb{C}[\\![z]\\!]}\cong\mathbb{C}[z]/\langle
z^{l+1}\rangle_{\mathbb{C}[z]}$-module. Usually we fix the following basis and
identifications as $\mathbb{C}$-vector spaces,
$\mathbb{C}[z^{-1}]_{l}\cong\\{\sum_{i=0}^{l}\frac{a_{j}}{z^{i+1}}\mid
a_{i}\in\mathbb{C}\\}\cong\mathbb{C}^{l+1}$,
$\mathbb{C}[z]_{l}\cong\\{\sum_{i=0}^{l}a_{i}z^{i}\mid
a_{i}\in\mathbb{C}\\}\cong\mathbb{C}^{l+1}$.

####  1.2.1 Truncated orbits as coadjoint orbits

We can consider the finite dimensional complex Lie group
$\mathrm{GL}_{n}(\mathbb{C}[z]_{l})$ with the corresponding Lie algebra
$\mathfrak{gl}_{n}(\mathbb{C}[z]_{l}):=M_{n}(\mathbb{C}[z]_{l})$. For a
$\mathbb{C}$-subalgebra $\mathfrak{q}\subset\mathfrak{gl}_{n}(\mathbb{C})$, we
define the $\mathbb{C}$-algebra
$\mathfrak{q}(\mathbb{C}[z]_{l}):=\mathfrak{q}\otimes_{\mathbb{C}}\mathbb{C}[z]_{l}$
which is naturally regarded as a a $\mathbb{C}$-subalgebra of
$\mathfrak{gl}_{n}(\mathbb{C}[z]_{l})$. Then we denote the corresponding
analytic subgroup of e complex Lie group $\mathrm{GL}_{n}(\mathbb{C}[z]_{l})$
by $Q(\mathbb{C}[z]_{l})$.

If we regard $M_{n}(\mathbb{C}[z^{-1}]_{k})dz$ as a subspace of
$M_{n}(\mathbb{C}(\\!(z)\\!)/\mathbb{C}[\\![z]\\!])dz$ by the inclusion map
$\mathbb{C}[z^{-1}]_{k}\hookrightarrow\mathbb{C}(\\!(z)\\!)/\mathbb{C}[\\![z]\\!]$,
then the truncated orbit $\mathbb{O}_{H}$ can be seen as the orbit through $H$
in $M_{n}(\mathbb{C}[z^{-1}]_{k})dz$ under the action of
$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$, i.e.,

$\mathbb{O}_{H}=\\{gHg^{-1}\in M_{n}(\mathbb{C}[z^{-1}]_{k})dz\mid
g\in\mathrm{GL}_{n}(\mathbb{C}[z]_{k})\\}.$

Then the trace form

$M_{n}(\mathbb{C}[z]_{k})\times
M_{n}(\mathbb{C}[z^{-1}]_{k})dz\ni(A,Bdz)\longmapsto\mathrm{res}_{z=0}(\mathrm{tr}(AB))dz\in\mathbb{C},$

is non-degenerate and thus this enable us to identify the dual space
$\mathfrak{gl}_{n}(\mathbb{C}[z]_{k})^{*}$ with
$M_{n}(\mathbb{C}[z^{-1}]_{k})dz$. Thus $\mathbb{O}_{H}$ is regarded as the
coadjoint orbit of $\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$ through $H\in
M_{n}(\mathbb{C}[z^{-1}]_{k})dz\cong\mathfrak{gl}_{n}(\mathbb{C}[z]_{k})^{*}$.

####  1.2.2 Extended orbits of HTL normal forms

Here we recall the extended orbit which is introduced by Boalch.

First we recall the fact that the coadjoint orbit of a complex Lie group $G$
through an element $\xi$ of $\mathfrak{g}^{*}$ is isomorphic to a symplectic
reduction of the cotangent bundle $T^{*}G$ as follows. The cotangent bundle
$\theta\colon T^{*}M\rightarrow M$ of a complex manifold $M$ has the standard
holomorphic symplectic form $\Omega_{T^{*}M}:=d\omega_{T^{*}M}$, the
derivation of the canonical $1$-form $\omega_{T^{*}M}$ on $T^{*}M$222Here we
notice that many literature adopt $-d\omega_{T^{*}M}$ as the standard
symplectic form.. For a holomorphic map $f\colon M\to N$ between complex
manifolds, we denote the induced maps $T_{m}M\rightarrow T_{f(m)}N$ and
$T^{*}_{f(m)}N\rightarrow T^{*}_{m}M$ by $(Tf)_{m}$ and $(T^{*}f)_{m}$
respectively for $m\in M$. Sometimes we write $f_{*,m}=(Tf)_{m}$ and
$f^{*}_{m}=(T^{*}f)_{m}$ for simplicity. Now we suppose $M=G$ and denote the
right translation by $R_{g}\colon G\ni x\mapsto xg\in G$ and the left
translation by $L_{g}\colon G\ni x\mapsto gx\in G$ for $g\in G$. We define the
left action of $G$ on $T^{*}G$ by $\rho(g)\colon T^{*}G\ni\alpha\mapsto
R^{*}_{g}(\alpha)\in T^{*}G$ for $g\in G$. Then the map

$\mu_{G}\colon T^{*}G\ni\alpha\longmapsto
L_{\theta(\alpha)}^{*}(\alpha)\in\mathfrak{g}^{*}$

is a moment map with respect to the action $\rho$, and for the coadjoint orbit
$\mathbb{O}_{\xi}$ through $\xi\in\mathfrak{g}^{*}$, there exists an
ismorphism

$\mathbb{O}_{\xi}\cong G_{\xi}\backslash\mu_{G}^{-1}(\xi)$

as holomorphic symplectic manifolds. Here $G_{\xi}$ is the stabilizer subgroup
of $\xi$ in $G$.

Now we consider the specific coadjoint orbit $\mathbb{O}_{H}$ of
$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$ through the unramified HTL normal form
$H$. Note that $\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$ has the description as the
semi-direct product of $\mathrm{GL}_{n}(\mathbb{C})$ and the normal subgroup

$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}:=\\{g(z)\in\mathrm{GL}_{n}(\mathbb{C}[z]_{k})\mid
g(0)=E_{n}\\},$

i.e.,
$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})=\mathrm{GL}_{n}(\mathbb{C})\ltimes\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$.
The Lie algebra of this normal subgroup is denoted by
$\mathfrak{gl}_{n}(\mathbb{C}[z]_{k})^{1}$. Also we denote the inclusion map
of $\mathrm{GL}_{n}(\mathbb{C})$ and $\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$
into $\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$ by $\iota_{0}$ and $\iota_{1}$, and
the projection maps from $\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$ onto
$\mathrm{GL}_{n}(\mathbb{C})$ and $\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$ by
$\pi_{0}$ and $\pi_{1}$ respectively.

Then we can moreover consider the intermediate reduction by considering the
moment map with respect to the normal subgroup
$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$,

$\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})}^{1}:=\iota_{1}^{*}\circ\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})}\colon
T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})\rightarrow(\mathfrak{gl}_{n}(\mathbb{C}[z]_{k})^{1})^{*},$

where
$\iota_{1}^{*}\colon\mathfrak{gl}_{n}(\mathbb{C}[z]_{k})^{*}\rightarrow(\mathfrak{gl}_{n}(\mathbb{C}[z]_{k})^{1})^{*}$
is the induced projection map by $\iota_{1}$.

######  Proposition 1.6.

Let us set $H_{\mathrm{irr}}:=\iota_{1}^{*}(H)$ and call it the irregular type
of $H$. We denote the coadjoint orbit of
$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$ through $H_{\mathrm{irr}}$ by
$\mathbb{O}_{H_{\mathrm{irr}}}$. Then there exists an isomorphism

$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}_{H_{\mathrm{irr}}}\backslash(\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})}^{1})^{-1}(H_{\mathrm{irr}})\cong
T^{*}\mathrm{GL}_{n}(\mathbb{C})\times\mathbb{O}_{H_{\mathrm{irr}}}.$

as holomorphic symplectic manifolds.

###### Proof.

This is a direct consequence of the Hamiltonian reduction by stages with
respect to the semi-direct product
$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})=\mathrm{GL}_{n}(\mathbb{C})\ltimes\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$,
see Theorem 10.5.1 in [11]. ∎

Here the isomorphism is induced from the map between the cotangent bundles
$T^{*}(\iota_{0}\times\iota_{1})\colon
T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})\rightarrow
T^{*}\mathrm{GL}_{n}(\mathbb{C})\times
T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$ associated to the isomorphism
given by the multiplication map
$\iota_{0}\times\iota_{1}\colon\mathrm{GL}_{n}(\mathbb{C})\times\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}\rightarrow\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$.
This intermediate reduction space is called the extended orbit of
$H_{\mathrm{irr}}$.

Let us explain that the truncated orbit $\mathbb{O}_{H}$ is a symplectic
reduction of this extended orbit. Since
$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$ is a normal subgroup of
$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$, the subgroup
$\mathrm{GL}_{n}(\mathbb{C})\subset\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$ acts on
$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$ by the adjoint action. Thus we can
consider the induced action of $\mathrm{GL}_{n}(\mathbb{C})$ on
$(\mathfrak{gl}_{n}(\mathbb{C}[z]_{k})^{1})^{*}$. We denote the stabilizer
subgroup of
$H_{\mathrm{irr}}\in(\mathfrak{gl}_{n}(\mathbb{C}[z]_{k})^{1})^{*}$ in
$\mathrm{GL}_{n}(\mathbb{C})$ by
$\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}$ and put
$\iota_{0,H_{\mathrm{irr}}}\colon\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}\hookrightarrow\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$,
the inclusion map. Then we can define a map from
$T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$ to
$\mathfrak{gl}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}^{*}$, the dual space of the
Lie algebra $\mathfrak{gl}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}$ of
$\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}$, as the composite map

$\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}}^{0}\colon
T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}\xrightarrow{T^{*}\pi_{1}}T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})\\\
\xrightarrow{\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})}}\mathfrak{gl}_{n}(\mathbb{C}[z]_{k})^{*}\xrightarrow{\iota_{0,H_{\mathrm{irr}}}^{*}}\mathfrak{gl}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}^{*}.$

Here $T^{*}\pi_{1}\colon
T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}\rightarrow
T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$ is defined by
$T^{*}\pi_{1}(\alpha):=(T^{*}\pi_{1})_{\iota_{1}(\theta(\alpha))}(\alpha)$ for
$\alpha\in T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$. Then we can show the
following.

######  Proposition 1.7.

The restriction of $\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}}^{0}$ on
$\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}}^{-1}(H_{\mathrm{irr}})$ factors
through the quotient map

$q\colon\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}}^{-1}(H_{\mathrm{irr}})\rightarrow\mathbb{O}_{\mathrm{irr}}=\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}_{H_{\mathrm{irr}}}\backslash\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}}^{-1}(H_{\mathrm{irr}}).$

Namely, there uniquely exists the map
$\mathrm{res}_{\mathbb{O}_{H_{\mathrm{irr}}}}\colon\mathbb{O}_{H_{\mathrm{irr}}}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}^{*}$
such that the diagram

${\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}}^{-1}(H_{\mathrm{irr}})}$${\mathfrak{gl}_{n}(\mathbb{C})^{*}_{H_{\mathrm{irr}}}}$${\mathbb{O}_{H_{\mathrm{irr}}}}$$\scriptstyle{\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}}^{0}}$$\scriptstyle{q}$$\scriptstyle{\mathrm{res}_{\mathbb{O}_{H_{\mathrm{irr}}}}}$

is commutative.

###### Proof.

Since the restriction map
$\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}}^{0}|_{\mu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}}^{-1}(H_{\mathrm{irr}})}$
is $\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}_{H_{\mathrm{irr}}}$-invariant, this
map factors through the quotient map. ∎

Let
$\iota_{H_{\mathrm{irr}}}\colon\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}\hookrightarrow\mathrm{GL}_{n}(\mathbb{C})$
be the inclusion map. Then Boalch showed that the truncated orbit
$\mathbb{O}_{H}$ is a symplectic reduction of the extended orbit
$T^{*}\mathrm{GL}_{n}(\mathbb{C})\times\mathbb{O}_{H_{\mathrm{irr}}}$ as
follows.

######  Theorem 1.8 (Boalch [1]).

Let us consider the map

$\begin{array}[]{cccc}\mu_{\mathrm{ext}}\colon&T^{*}\mathrm{GL}_{n}(\mathbb{C})\times\mathbb{O}_{H_{\mathrm{irr}}}&\longrightarrow&\mathfrak{gl}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}^{*}\\\
&(\alpha,\xi)&\longmapsto&\iota_{H_{\mathrm{irr}}}^{*}\circ\mu_{\mathrm{GL}_{n}(\mathbb{C})}(\alpha)+\mathrm{res}_{\mathbb{O}_{H_{\mathrm{irr}}}}(\xi)\end{array}.$

Then this map is a moment map with respect to the
$\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}$-action which is induced from
the natural $\mathrm{GL}_{n}(\mathbb{C})$-action on
$T^{*}\mathrm{GL}_{n}(\mathbb{C})\times
T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}\cong
T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$. Moreover setting
$H_{\mathrm{res}}:=\iota_{0}^{*}(H)$, we have a symplectic isomorphism

$\mathbb{O}_{H}\cong(\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}})_{H_{\mathrm{res}}}\backslash\mu_{\mathrm{ext}}^{-1}(H_{\mathrm{res}}).$

###  1.3 Triangular factorization of $\mathbb{O}_{H_{\mathrm{irr}}}$

We explain the coadjoint orbit $\mathbb{O}_{H_{\mathrm{irr}}}$ has a nice
affine coordinate system which is very important to define a deformation of
the truncated orbit in the latter section.

For a subgroup $G\subset\mathrm{GL}_{n}(\mathbb{C}[z]_{l})$, we denote the
intersection $G\cap\mathrm{GL}_{n}(\mathbb{C}[z]_{l})^{1}$ by $G^{1}$. Also we
denote the intersection $\mathfrak{gl}_{n}(\mathbb{C}[z]_{l})^{1}$ with a Lie
subalgebra $\mathfrak{g}\subset\mathfrak{gl}_{n}(\mathbb{C}[z]_{l})$ by
$\mathfrak{g}^{1}$.

####  1.3.1 Compositions of a natural number and subgroups of
$\mathrm{GL}_{n}(\mathbb{C})$

Let us recall compositions of natural numbers. Let $n$ and $k$ be strictly
positive integers. Then we denote the set of weak composition of $n$ with $l$
terms by

$\mathcal{C}(n,l):=\left\\{(a_{1},a_{2},\ldots,a_{l})\in\mathbb{Z}_{\geq
0}^{l}\,\middle|\,a_{1}+a_{2}+\cdots+a_{l}=n\right\\}$

where we allow some components $a_{i}$ to be zero. If all components in
$(a_{1},a_{2},\ldots,a_{l})\in\mathcal{C}(n,l)$ are strictly positive
integers, then it is called strict composition or composition shortly and we
denote the set of strict composition of $n$ with $l$ terms by
$\mathcal{C}^{+}(n,l).$ The sets of all weak compositions and strict
compositions of $n$ are denoted by

$\mathcal{C}(n):=\bigsqcup_{l\in\mathbb{Z}_{>0}}\mathcal{C}(n,l),\quad\quad\mathcal{C}^{+}(n):=\bigsqcup_{l\in\mathbb{Z}_{>0}}\mathcal{C}^{+}(n,l),$

respectively.

Let us take $\mathbf{m}=\\{n_{1},n_{2},\ldots,n_{m}\\}\in\mathcal{C}(n)$. Let
$\mathbf{e}_{i}\in\mathbb{C}^{n}$ be the vector with $1$ as the $i$-th entry
and $0$ as the others for each $i=1,2,\ldots,n.$ Then we define inclusion maps

$\begin{array}[]{cccc}\iota_{i}\colon&\mathbb{C}^{n_{i}}&\longrightarrow&\mathbb{C}^{n}\\\
&(a_{1},a_{2},\ldots,a_{n_{i}})&\longmapsto&\displaystyle\sum_{j=n_{1}+n_{2}+\cdots+n_{i-1}+1}^{n_{1}+n_{2}+\cdots+n_{i}}a_{j}\mathbf{e}_{j}\end{array}$

for $i=1,2,\ldots,m$ and an isomorphism

$\begin{array}[]{cccc}\iota_{\mathbf{m}}\colon&\bigoplus_{i=1}^{m}\mathbb{C}^{n_{i}}&\longrightarrow&\mathbb{C}^{n}\\\
&(\mathbf{a}_{i})_{i=1,2,\ldots,m}&\longmapsto&\sum_{i=1}^{m}\iota_{i}(\mathbf{a}_{i})\end{array},$

where if $n_{i}=0$, then we set $\iota_{i}$ as the zero map. We identify
$\mathbb{C}^{n}$ and $\bigoplus_{i=1}^{m}\mathbb{C}^{n_{i}}$ through this
isomorphism.

Let us regard
$\mathrm{Hom}_{\mathbb{C}}(\mathbb{C}^{n_{i}},\mathbb{C}^{n_{i^{\prime}}})$ as
a subspace of $\mathrm{End}_{\mathbb{C}}(\mathbb{C}^{n})=M_{n}(\mathbb{C})$
through the canonical projection
$\mathrm{pr}_{i}\colon\mathbb{C}^{n}\rightarrow\mathbb{C}^{n_{i}}$ and
injection
$\iota_{i^{\prime}}\colon\mathbb{C}^{n_{i^{\prime}}}\rightarrow\mathbb{C}^{n}$.
Then we define the following $\mathbb{C}$-subalgebras of $M_{n}(\mathbb{C})$,

$\displaystyle\mathfrak{l}_{\mathbf{m}}$
$\displaystyle:=\bigoplus_{i\in\\{1,2,\ldots,m\\}}\mathrm{Hom}_{\mathbb{C}}(\mathbb{C}^{n_{i}},\mathbb{C}^{n_{i}}),$
$\displaystyle\mathfrak{n}^{+}_{\mathbf{m}}$
$\displaystyle:=\bigoplus_{\begin{subarray}{c}i,i^{\prime}\in\\{1,2,\ldots,m\\}\\\
i<i^{\prime}\end{subarray}}\mathrm{Hom}_{\mathbb{C}}(\mathbb{C}^{n_{i}},\mathbb{C}^{n_{i^{\prime}}}),$
$\displaystyle\mathfrak{n}^{-}_{\mathbf{m}}$
$\displaystyle:=\bigoplus_{\begin{subarray}{c}i,i^{\prime}\in\\{1,2,\ldots,m\\}\\\
i>i^{\prime}\end{subarray}}\mathrm{Hom}_{\mathbb{C}}(\mathbb{C}^{n_{i}},\mathbb{C}^{n_{i^{\prime}}}),$

and regard them as the Lie subalgebras of
$\mathfrak{gl}_{n}(\mathbb{C})=M_{n}(\mathbb{C})$. Let $L_{\mathbf{m}}$,
$N^{+}_{\mathbf{m}}$, and $N^{-}_{\mathbf{m}}$ be the corresponding analytic
subgroups of $\mathrm{GL}_{n}(\mathbb{C})$.

####  1.3.2 Sequences of compositions of positive integers

Let us take weak compositions
$\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{l}),\,\beta=(\beta_{1},\beta_{2},\ldots,\beta_{m})\in\mathcal{C}(n)$.
Then we say that $\beta$ is a refinement of $\alpha$ if there exists a
surjection $\phi\colon\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots,l\\}$ from
the index set of $\beta$ to that of $\alpha$ such that equations
$\alpha_{i}=\sum_{j\in\phi^{-1}(i)}\beta_{j}$ hold for all $i=1,2,\ldots,l$.
In this case we write $\beta\leq\alpha$ or

$\beta\leq_{\phi}\alpha,$

emphasizing the surjection $\phi$ which we call the refinement map between
$\alpha$ and $\beta$. Let
$\widetilde{\mathbf{m}}=(\mathbf{m}_{1},\mathbf{m}_{2},\ldots,\mathbf{m}_{k})$
be an ascending sequence of compositions of $n$, i.e., entries of
$\widetilde{\mathbf{m}}$ consist of the following ascending sequence of
refinement,

$\mathbf{m}_{1}\leq_{\phi_{1}}\mathbf{m}_{2}\leq_{\phi_{2}}\cdots\leq_{\phi_{k-1}}\mathbf{m}_{k}.$

We write $\mathbf{m}_{i}=\\{n_{\langle i,1\rangle},n_{\langle
i,2\rangle},\ldots,n_{\langle i,m(i)\rangle}\\}$ and define a total ordering
$\preceq_{i}$ on the index set $\\{\langle i,1\rangle,\langle
i,2\rangle,\ldots,\langle i,m(i)\rangle\\}$ of $\mathbf{m}_{i}$ for each
$i=1,2,\ldots,k$. We say that the collection of orderings
$\\{\preceq_{i}\\}_{i=1,2,\ldots,k}$ is the ordering of the sequence
$\widetilde{\mathbf{m}}$ if

$\langle i,j\rangle\preceq_{i}\langle i,j^{\prime}\rangle\ \text{ imply }\
\phi_{i}(\langle i,j\rangle)\preceq_{i+1}\phi_{i}(\langle
i,j^{\prime}\rangle)$

for any $\langle i,j\rangle,\langle i,j^{\prime}\rangle\in\\{\langle
i,1\rangle,\langle i,2\rangle,\ldots,\langle i,m(i)\rangle\\}$ and
$i=1,2,\ldots,k-1$.

Let us fix a surjection $\phi^{1}\colon\\{1,2,\ldots,n\\}\rightarrow\\{\langle
1,1\rangle,\,\langle 1,2\rangle\,\ldots,\,\langle 1,m(1)\rangle\\}$ and define
the maps $\phi^{i}\colon\\{1,2,\ldots,n\\}\rightarrow\\{\langle
i,1\rangle,\,\langle i,2\rangle\,\ldots,\,\langle i,m(i)\rangle\\}$ by
$\phi^{i}:=\phi_{i-1}\circ\phi_{i-2}\circ\cdots\circ\phi_{1}\circ\phi^{1}$ for
$i=2,3,\ldots,k$. Then we can regard the vector space $\mathbb{C}^{n_{\langle
i,j\rangle}}$ as the subspace of $\mathbb{C}^{n}$ by the equation

$\mathbb{C}^{n_{\langle
i,j\rangle}}=\bigoplus_{k\in(\phi^{i})^{-1}(j)}\mathbb{C}\mathbf{e}_{k},$

for each $i=1,2,\ldots,k$ and $j=1,2,\ldots,m(i)$. Then we obtain the
decompositions

$\mathbb{C}^{n}=\bigoplus_{j=1}^{m(i)}\mathbb{C}^{n_{\langle i,j\rangle}}$

for $i=1,2,\ldots,k$ associated to $\widetilde{\mathbf{m}}$.

As we saw in the previous section, a composition of $n$ defines the block
decomposition of $\mathfrak{gl}_{n}(\mathbb{C})=M_{n}(\mathbb{C})$. Therefore
a pair of a sequence $\widetilde{\mathbf{m}}$ of compositions and an ordering
$\\{\preceq_{i}\\}_{i=1,2,\ldots,k}$ of $\widetilde{\mathbf{m}}$ defines the
following Lie subalgebras of $\mathfrak{gl}_{n}(\mathbb{C})$ with inclusion
relations

$\begin{split}&\mathfrak{l}_{\mathbf{m}_{1}}\subset\mathfrak{l}_{\mathbf{m}_{2}}\subset\cdots\subset\mathfrak{l}_{\mathbf{m}_{k}},\\\
&\mathfrak{n}_{\mathbf{m}_{k}}^{\pm}\subset\cdots\subset\mathfrak{n}_{\mathbf{m}_{2}}^{\pm}\subset\mathfrak{n}_{\mathbf{m}_{1}}^{\pm}.\end{split}$
(1)

Moreover we define

$\mathfrak{n}^{\pm}_{\mathbf{m}_{i}^{i^{\prime}}}:=\mathfrak{l}_{\mathbf{m}_{i^{\prime}}}\cap\mathfrak{n}_{\mathbf{m}_{i}},$

for pairs $(i,i^{\prime})$ satisfying $1\leq i<i^{\prime}\leq k+1$, which give
us further decompositions

$\mathfrak{l}_{\mathbf{m}_{i^{\prime}}}=\mathfrak{n}^{-}_{\mathbf{m}_{i}^{i^{\prime}}}\oplus\mathfrak{l}_{\mathbf{m}_{i}}\oplus\mathfrak{n}^{+}_{\mathbf{m}_{i}^{i^{\prime}}}.$

Here we formally put
$\mathfrak{l}_{\mathbf{m}_{k+1}}=\mathfrak{gl}_{n}(\mathbb{C})$. Throughout
this note, when we consider a sequence $\widetilde{\mathbf{m}}$ of
compositions, we always fix an ordering $\\{\preceq_{i}\\}_{i=1,2,\ldots,k}$
of $\widetilde{\mathbf{m}}$.

####  1.3.3 Spectral types of HTL-normal forms

We shall introduce spectral types of HTL normal forms. As the easiest case,
let us first consider a square matrix $X\in M_{n}(\mathbb{C})$ and define the
spectral type of $X$ as follows. Take the Jordan decomposition of $X$,

$X=S+N$

with the diagonalizable matrix $S$ and the nilpotent matrix $N$ satisfying the
relation $[S,N]=0$. Then we have the decomposition
$\mathbb{C}^{n}=\bigoplus_{i=1}^{m}V_{i}$ as the direct sum of eigenspaces of
$S$ which defines a composition of $n$,

$\mathbf{m}:=(\mathrm{dim}_{\mathbb{C}}V_{1},\mathrm{dim}_{\mathbb{C}}V_{2},\ldots,\mathrm{dim}_{\mathbb{C}}V_{m}).$

Let us look at the nilpotent matrix $N$ and notice that the restriction
$N_{i}:=N|_{V_{i}}$ on $V_{i}$ defines a nilpotent endomorphism of $V_{i}$ for
each $i=1,2,\ldots,m$ since $[S,N]=0$. Then we have the partial flag
$\mathcal{F}_{N;V_{i}}$,

$V_{i}\supsetneqq(V_{i})_{1}:=\mathrm{Im}N_{i}\supsetneqq(V_{i})_{2}:=\mathrm{Im}N_{i}^{2}\supsetneqq\cdots\supsetneqq(V_{i})_{d_{i}}:=\mathrm{Im}N_{i}^{d_{i}}=\\{0\\},$

and descending sequence of positive integers

$\mathrm{dim}_{\mathbb{C}}V_{i}>\mathrm{dim}_{\mathbb{C}}(V_{i})_{1}>\mathrm{dim}_{\mathbb{C}}(V_{i})_{2}>\cdots>\mathrm{dim}_{\mathbb{C}}(V_{i})_{d_{i}}=0.$

We call the sequence of the integers

$\sigma(\mathcal{F}_{N;V_{i}}):=(\mathrm{dim}_{\mathbb{C}}V_{i},\mathrm{dim}_{\mathbb{C}}(V_{i})_{1},\mathrm{dim}_{\mathbb{C}}(V_{i})_{2},\ldots,\mathrm{dim}_{\mathbb{C}}(V_{i})_{d_{i}-1})$

the signature of $\mathcal{F}_{N;V_{i}}$. We denote the set of the signatures
by

$\sigma({\mathbf{m}}):=\\{\sigma(\mathcal{F}_{N;V_{i}})\\}_{i=1,2,\ldots,m}$

and call it the signature of $\mathbf{m}$.

######  Definition 1.9 (spectral type of a square matrix).

Let $X\in M_{n}(\mathbb{C})$ be a square matrix with the Jordan decomposition
$X=S+N$. Then we can define the composition $\mathbf{m}$ of $n$ and the
signature $\sigma(\mathbf{m})$ of $\mathbf{m}$ as above. we call the pair

$(\mathbf{m},\sigma(\mathbf{m}))$

the spectral type of $X$. In particular, if $X=S$, i.e., $X$ is a semisimple
matrix, then the signature $\sigma(\mathbf{m})$ is trivial. In this case, we
simply denote the spectral type by $(\mathbf{m},\mathrm{triv}).$

Next we consider an unramified HTL-normal form
$\displaystyle\left(\frac{S_{k}}{z^{k}}+\cdots+\frac{S_{1}}{z}+S_{0}+N_{0}\right)\frac{dz}{z}.$
Then we obtain the decomposition
$\mathbb{C}^{n}=\bigoplus_{j=1}^{m(i)}V_{\langle i;j\rangle}$ as the direct
sum of simultaneous eigenspaces of $(S_{i},S_{i+1},\ldots,S_{k})$ which
defines the following composition of $n$,

$\mathbf{m}_{i}:=(\mathrm{dim\,}_{\mathbb{C}}V_{\langle
i;1\rangle},\mathrm{dim\,}_{\mathbb{C}}V_{\langle
i;2\rangle},\ldots,\mathrm{dim\,}_{\mathbb{C}}V_{\langle i;m(i)\rangle}),$

for each $i=0,1,\ldots,k$. Also for each $i=0,1,\ldots,k-1$, let us define the
map

$\phi_{i}\colon\\{\langle i,1\rangle,\langle i,2\rangle,\ldots,\langle
i,m(i)\rangle\\}\rightarrow\\{\langle i+1,1\rangle,\langle
i+1,2\rangle,\ldots,\langle i+1,m(i+1)\rangle\\}$

between index sets of $\mathbf{m}_{i}$ and $\mathbf{m}_{i+1}$ which satisfies
that

$V_{\langle i+1;j\rangle}=\bigoplus_{\mu\in\phi^{-1}_{i}(j)}V_{\langle
i;\mu\rangle}$

for all $j\in\\{1,2,\ldots,m(i)\\}$. Then we obtain a refinement sequence of
partitions

$\mathbf{m}_{0}\leq_{\phi_{0}}\leq\mathbf{m}_{1}\leq_{\phi_{1}}\cdots\leq_{\phi_{k-1}}\mathbf{m}_{k}.$

The restrictions of nilpotent matrices $N_{j}:=N_{0}|_{V_{\langle
0;j\rangle}}\in\mathrm{End}_{\mathbb{C}}(V_{\langle 0,j\rangle})$ define the
flags $\mathcal{F}(N_{0},V_{\langle 0;j\rangle})$ of $V_{\langle 0;j\rangle}$,

$V_{\langle 0;j\rangle}\supsetneqq(V_{\langle
0;j\rangle})_{1}:=\mathrm{Im}N_{j}\supsetneqq(V_{\langle
0;j\rangle})_{2}:=\mathrm{Im}N_{j}^{2}\supsetneqq\cdots\supsetneqq(V_{\langle
0;j\rangle})_{d_{j}}:=\mathrm{Im}N_{j}^{d_{j}}=\\{0\\},$

with the signature

$\sigma(\mathcal{F}(N_{0},V_{\langle
0;j\rangle}))=(\mathrm{dim}_{\mathbb{C}}V_{\langle
0;j\rangle},\mathrm{dim}_{\mathbb{C}}(V_{\langle
0;j\rangle})_{1},\mathrm{dim}_{\mathbb{C}}(V_{\langle
0;j\rangle})_{2},\ldots,\mathrm{dim}_{\mathbb{C}}(V_{\langle
0;j\rangle})_{d_{j}-1}).$

Also we set

$\sigma(\mathbf{m}_{0}):=(\sigma(\mathcal{F}(N_{0},V_{\langle
0;j\rangle})))_{j=1,2,\ldots,m(0)}.$

######  Definition 1.10 (spectral types of HTL normal forms).

The pair

$\mathrm{sp}(H):=(\mathbf{m}_{0}\leq_{\phi_{0}}\leq\mathbf{m}_{1}\leq_{\phi_{1}}\cdots\leq_{\phi_{k-1}}\mathbf{m}_{k},\sigma(\mathbf{m}_{0}))$

of the refinement sequence of partitions of $n$ and signature
$\sigma(\mathbf{m}_{0})$ of $\mathbf{m}_{0}$ defined above is called the
spectral type of the HTL-normal form

$H=\left(\frac{S_{k}}{z^{k}}+\cdots+\frac{S_{1}}{z}+S_{0}+N_{0}\right)\frac{dz}{z}.$

When $N_{0}=0$, the signature $\sigma(\mathbf{m}_{0})$ is trivial. Thus we
denote the spectral type by

$(\mathbf{m}_{0}\leq_{\phi_{0}}\leq\mathbf{m}_{1}\leq_{\phi_{1}}\cdots\leq_{\phi_{k-1}}\mathbf{m}_{k},\mathrm{triv})$

in this case.

####  1.3.4 Triangular factrization of $\mathbb{O}_{H_{\mathrm{irr}}}$

Firstly, we recall a general fact about coadjoint orbits. It is well-known
that a coadjoint orbit $\mathbb{O}_{\xi}$ of a complex Lie group $G$ through
$\xi\in\mathfrak{g}^{*}$ is a immersed submanifold of $\mathfrak{g}^{*}$ via
the injective immersion

$\mathbb{O}_{\xi}\cong
G/G_{\xi}\ni[g]\longmapsto\mathrm{Ad}^{*}(g)(\xi)\in\mathfrak{g}^{*},$

where $[g]\in G/G_{\xi}$ is the class of $g\in G$. Under the identification
$\mathbb{O}_{\xi}\cong G_{\xi}\backslash\mu_{G}^{-1}(\xi)$, this injective
immersion is written as follows. Let us consider the map

$\nu_{G}\colon T^{*}G\ni\alpha\longmapsto
R^{*}_{\theta(\alpha)}(\alpha)\in\mathfrak{g}^{*},$

where $\theta\colon T^{*}G\rightarrow G$ is the natural projection. Then we
note that this map is invariant under the action $\rho$ of $G$ on $T^{*}G$,
i.e., we have $\nu_{G}\circ\rho(g)=\nu_{G}$ for all $g\in G$. Therefore the
restriction map $\nu_{G}\colon\mu_{G}^{-1}(\xi)\rightarrow\mathfrak{g}^{*}$
factors through the quotient map
$q\colon\mu_{G}^{-1}(\xi)\rightarrow\mathbb{O}_{\xi}$ and there uniquely
exists the map $\mathbb{O}_{\xi}\rightarrow\mathfrak{g}^{*}$ such that the
diagram

${\mu_{G}^{-1}(\xi)}$${\mathfrak{g}^{*}}$${\mathbb{O}_{\xi}}$$\scriptstyle{\nu_{G}}$$\scriptstyle{q}$

is commutative. Then we can check that this map
$\mathbb{O}_{\xi}\rightarrow\mathfrak{g}^{*}$ coincides with the above
injective immersion.

Let us consider the subgroups
$(N^{-})^{(l)}:=N_{\mathbf{m}_{l}^{l+1}}^{-}(\mathbb{C}[z]_{l-1})^{1}\subset\mathrm{GL}_{n}(\mathbb{C}[z]_{l-1})^{1}$
for $l=2,3,\ldots,k$. Then we recall that
$\mathrm{GL}_{n}(\mathbb{C}[z]_{l-1})^{1}$ admits the LU decompositions.

######  Proposition 1.11 (LU decomposition).

Let $\mathbf{m}$ be a composition of $n$. Then for a positive integer $j$, the
multiplication gives the isomorphism as complex manifolds,

$N_{\mathbf{m}}^{-}(\mathbb{C}[z]_{j})^{1}\times
L_{\mathbf{m}}(\mathbb{C}[z]_{j})\times
N_{\mathbf{m}}^{+}(\mathbb{C}[z]_{j})^{1}\ni(n^{-},h,n^{+})\longmapsto
n^{-}\cdot h\cdot n^{+}\in\mathrm{GL}_{n}(\mathbb{C}[z]_{j})^{1}.$

###### Proof.

Since any principal minors of $g\in\mathrm{GL}_{n}(\mathbb{C}[z]_{j})^{1}$ are
units in $\mathbb{C}[z]_{j}$, thus the usual LU decomposition algorithm is
valid in this setting. ∎

Thus we obtain the projection
$\mathrm{pr}_{(N^{-})^{(l)}}\colon\mathrm{GL}_{n}(\mathbb{C}[z]_{l-1})^{1}\rightarrow(N^{-})^{(l)}$
along the LU decomposition. Regarding the projection
$\mathrm{pr}^{k}_{l-1}\colon\mathbb{C}[z]_{k}\rightarrow\mathbb{C}[z]_{l-1}$
as a map of $\mathbb{C}$-vector spaces, we can define a section
$s^{k}_{l-1}\colon\mathbb{C}[z]_{l-1}\ni\sum_{i=0}^{l-1}c_{i}z^{i}\mapsto\sum_{i=0}^{l-1}c_{i}z^{i}\in\mathbb{C}[z]_{k}$.
Then we define the projection

$\pi_{(N^{-})^{(l)}}:=\mathrm{pr}_{(N^{-})^{(l)}}\circ\mathrm{pr}^{k}_{l-1}\colon\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}\rightarrow(N^{-})^{(l)}.$

Also the inclusion map
$i_{(N^{-})^{(l)}}\colon(N^{-})^{(l)}\hookrightarrow\mathrm{GL}_{n}(\mathbb{C}[z]_{l})^{1}$
defines a section

$\iota_{{}_{(N^{-})^{(l)}}}:=s^{k}_{l-1}\circ
i_{(N^{-})^{(l)}}\colon(N^{-})^{(l)}\hookrightarrow\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$

of the projection $\pi_{(N^{-})^{(l)}}$. By using these projection and
section, we obtain a map between cotangent bundles,

$T^{*}\pi_{(N^{-})^{(l)}}\colon
T^{*}(N^{-})^{(l)}\ni\alpha\mapsto(T^{*}\pi_{1,(N^{-})^{(l)}})_{\iota_{(N^{-})^{(l)}}(\theta(\alpha))}(\alpha)\in
T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}.$

For $H_{\mathrm{irr}}\in(\mathfrak{gl}_{n}(\mathbb{C}[z]_{k})^{1})^{*}$, we
set
$\omega_{H_{\mathrm{irr}}}:=(L^{*}_{g^{-1}}(H_{\mathrm{irr}}))_{g\in\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}}$,
the left invariant $1$-form on $\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$
associated to $H_{\mathrm{irr}}$. Define a map
$F\colon\prod_{l=2}^{k}T^{*}(N^{-})^{(l)}\rightarrow
T^{*}\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}$ by

$F((\alpha_{l})_{l=2,3,\ldots,k}):=\omega_{H_{\mathrm{irr}},\theta(\alpha)^{[2;k]}}+\sum_{l=2}^{k}L^{*}_{(\theta(\alpha)^{[l+1;k]})^{-1}}\circ
R^{*}_{(\theta(\alpha)^{[2;l-1]})^{-1}}\circ
T^{*}\pi_{(N^{-})^{(l)}}(\alpha_{l})$

for $(\alpha_{l})_{l=2,3,\ldots,k}\in\prod_{l=2}^{k}T^{*}(N^{-})^{(l)}$, where
we put

$\theta(\alpha)^{[i;j]}:=\begin{cases}\theta(\alpha_{j})\theta(\alpha_{j-1})\cdots\theta(\alpha_{i})&i\leq
j\\\ e&i>j\end{cases}.$

The next theorem shows that $T^{*}(N^{-})^{(l)}$ defines an affine coordinate
system of $\mathbb{O}_{H_{\mathrm{irr}}}$.

######  Theorem 1.12 (Yamakawa [15], cf. Hiroe-Yamakawa [6]).

The map

$\nu_{\mathrm{GL}_{n}(\mathbb{C}[z]_{k})^{1}}\circ
F\colon\prod_{l=2}^{k}T^{*}(N^{-})^{(l)}\rightarrow(\mathfrak{gl}_{n}(\mathbb{C}[z]_{k})^{1})^{*}$

gives a symplectic isomorphism

$\prod_{l=2}^{k}T^{*}(N^{-})^{(l)}\cong\mathbb{O}_{H_{\mathrm{irr}}}.$

Let us give a description of the map
$\mathrm{res}_{\mathbb{O}_{H_{\mathrm{irr}}}}\colon\mathbb{O}_{H_{\mathrm{irr}}}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}^{*}$
under the above identification.

######  Proposition 1.13.

Under the isomorphism
$\prod_{l=2}^{k}T^{*}(N^{-})^{(l)}\cong\mathbb{O}_{H_{\mathrm{irr}}}$, the
image of each
$(\alpha_{l})_{l=2,3,\ldots,k}\in\prod_{l=2}^{k}T^{*}(N^{-})^{(l)}$ by the map
$\mathrm{res}_{\mathbb{O}_{H_{\mathrm{irr}}}}\colon\mathbb{O}_{H_{\mathrm{irr}}}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}^{*}$
is given by

$\mathrm{res}_{\mathbb{O}_{H_{\mathrm{irr}}}}((\alpha_{l})_{l=2,3,\ldots,k})=\\\
\iota_{0,\,H_{\mathrm{irr}}}^{*}\circ\left(\sum_{l=2}^{k}\mathrm{Ad}^{*}(\theta(\alpha)^{[2;l-1]})(T^{*}(\pi_{(N^{-})^{(l)}}\circ\pi_{1})(L^{*}_{\theta(\alpha_{l})}(\alpha_{l})))\right).$

###### Proof.

Directly follows from the definition of
$\mathrm{res}_{\mathbb{O}_{H_{\mathrm{irr}}}}$ and Theorem 1.12. ∎

###  1.4 Moduli space of meromorphic connections on a trivial bundle over the
Riemann sphere with unramified irregular singularities

Now we are ready to define a moduli space of meromorphic connections on a
trivial bundle over the Riemann sphere as a symplectic reduction of a product
of truncated orbits.

####  1.4.1 Algebraic meromorphic connections on a trivial bundle over the
Riemann sphere

Let us denote the sheaf of regular function on $\mathbb{P}^{1}$ by
$\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}}$. Let $z$ be the standard
coordinate on $\mathbb{C}\subset\mathbb{P}^{1}$. For $c\in\mathbb{P}^{1}$, set
$z_{c}:=z-c$ if $c\in\mathbb{C}$ and $\displaystyle z_{\infty}:=\frac{1}{z}$
if $c=\infty$. Let $\widehat{\mathcal{O}}_{c}:=\mathbb{C}[\\![z_{c}]\\!]$ be
the ring of formal power series and
$\widehat{\mathcal{M}}_{c}:=\mathbb{C}(\\!(z_{c})\\!)$ the field of formal
Laurent series around $c\in\mathbb{P}^{1}$. Also set
$\mathcal{O}_{c,i}:=\mathbb{C}[z-c]_{i}$ and
$\mathcal{M}_{c,i}:=\mathbb{C}[(z-c)^{-1}]_{i}$ for a positive integer $i$.

Let us consider a finite set $\\{a_{0},a_{1},\ldots,a_{d}\\}$ of points in
$\mathbb{P}^{1}$ and fix an effective divisor
$D:=\sum_{a\in\\{a_{0},a_{1},\ldots,a_{d}\\}}(k_{a}+1)\cdot a$ with
$k_{a}\in\mathbb{Z}_{\geq 0}$. We denote the set of points by
$|D|:=\\{a_{0},a_{1},\ldots,a_{d}\\}$. Let
$\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}}(*|D|)$ and
$\varOmega^{\mathrm{reg}}_{\mathbb{P}^{1}}(*|D|)$ denote the sheaves of
rational function and of rational 1-forms with poles on $|D|$ respectively.

######  Definition 1.14.

An algebraic meromorphic connection on the trivial bundle is a differential
operator, namely morphism of sheaves of $\mathbb{C}$-vector spaces

$\nabla\colon(\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}})^{n}\longrightarrow(\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}})^{n}\otimes_{\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}}}\varOmega^{\mathrm{reg}}_{\mathbb{P}^{1}}(*|D|),$

satisfying the Leibniz rule

$\nabla(fs)=df\otimes s+f\nabla(s)$

for all open subsets $U\subset\mathbb{P}^{1}$,
$f\in\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}}(U)$, and
$s\in(\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}})^{n}(U)$. On any open subset
$U\subset\mathbb{P}^{1}$, $\nabla$ can be uniquely written as the matrix form,

$\nabla=d-A_{U}(z)dz$

where $A_{U}(z)dz\in
M_{n}(\varOmega^{\mathrm{reg}}_{\mathbb{P}^{1}}(*|D|)(U))$. Thus there
uniquely exists the matrix 1-form $A(z)dz\in
M_{n}(\varOmega^{\mathrm{reg}}_{\mathbb{P}^{1}}(*|D|)(\mathbb{P}^{1}))$
defined globally on $\mathbb{P}^{1}$ such that

$A(z)dz|_{U}=A_{U}(z)dz$

for any open subset $U\subset\mathbb{P}^{1}$. We call this matrix $A(z)$ the
coefficient matrix of the connection $\nabla$.

The automorphism group of the free
$\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}}$-module
$(\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}})^{n}=\mathbb{C}^{n}\otimes_{\mathbb{C}}\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}}$
is isomorphic to $\mathrm{GL}_{n}(\mathbb{C})$. Thus meromorphic connections
$\nabla_{1}=d-A_{1}(z)dz$ and $\nabla_{2}=d-A_{2}(z)dz$ of rank $n$ are
isomorphic if and only if there exists $g\in\mathrm{GL}_{n}(\mathbb{C})$ such
that

$A_{2}(z)=gA_{1}(z)g^{-1}.$

A connection $\nabla$ is said to be irreducible if there is no nontrivial
subspace $W\subset\mathbb{C}^{n}$ such that the sub
$\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}}$-module
$W\otimes_{\mathbb{C}}\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}}\subset\mathbb{C}^{n}\otimes_{\mathbb{C}}\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}}=(\mathcal{O}^{\mathrm{reg}}_{\mathbb{P}^{1}})^{n}$
is $\nabla$-stable.

####  1.4.2 The space of meromorphic connections as a complex manifold

By the projective transformation on $\mathbb{P}^{1}(\mathbb{C})$, we may
suppose that $|D|\subset\mathbb{C}$. Let us consider a space of connections
with coefficients in $\varOmega^{\mathrm{reg}}_{\mathbb{P}^{1}}(*|D|)$, i.e.,

$\mathcal{M}_{D}:=\\{\nabla=d-A(z)dz\mid A(z)dz\in
M_{n}(\varOmega^{\mathrm{reg}}_{\mathbb{P}^{1}}(*|D|)(\mathbb{P}^{1}))\\}.$

For $d-A(z)dz\in\mathcal{M}_{D}$, we can write

$A(z)=\sum_{a\in|D|}\mathrm{prc}_{a}(A(z))=\sum_{a\in|D|}\sum_{i=0}^{k_{a}}\frac{A^{(a)}_{i}}{(z-a)^{i}}\frac{dz}{z-a}.$

with the relation

$\sum_{a\in|D|}A^{(a)}_{0}=0.$

Thus we can identify $\mathcal{M}_{D}$ as a space of matrices,

$M_{D}:=\left\\{\left(X^{(a)}_{i}\right)_{\begin{subarray}{c}a\in|D|,\\\
i=0,1,\ldots,k_{a}\end{subarray}}\in\prod_{a\in|D|}\prod_{i=0}^{k_{a}}M_{n}(\mathbb{C})\,\middle|\,\sum_{a\in|D|}\sum_{i=0}^{k_{a}}X^{(a)}_{i}=0\right\\},$

which is a union of hyperplanes in the affine space
$\prod_{a\in|D|}\prod_{i=0}^{k_{a}}M_{n}(\mathbb{C})$, i.e., $M_{D}$ is a
complex manifold. The $\mathrm{GL}_{n}(\mathbb{C})$-action on
$\mathcal{M}_{D}$ is translated into the diagonal action on $M_{D}$, i.e.,

$\mathrm{GL}_{n}(\mathbb{C})\times
M_{D}\ni\left(g,\left(X^{(a)}_{i}\right)_{\begin{subarray}{c}a\in|D|,\\\
i=0,1,\ldots,k_{a}\end{subarray}}\right)\longmapsto\left(gX^{(a)}_{i}g^{-1}\right)_{\begin{subarray}{c}a\in|D|,\\\
i=0,1,\ldots,k_{a}\end{subarray}}\in M_{D}.$

Under this identification, we regard $\mathcal{M}_{D}$ as a complex manifold
with holomorphic $\mathrm{GL}_{n}(\mathbb{C})$-action. The irreducibility of
connections on $\mathcal{M}_{D}$ corresponds to the following condition on
$M_{D}$, namely, $\left(X^{(a)}_{i}\right)_{\begin{subarray}{c}a\in|D|,\\\
i=0,1,\ldots,k_{a}\end{subarray}}\in\prod_{a\in|D|}\prod_{i=0}^{k_{a}}M_{n}(\mathbb{C})$
is said to be irreducible if it has no nontrivial simultaneous invariant
subspace of $\mathbb{C}^{n}$. That is to say, if a subspace
$W\subset\mathbb{C}^{n}$ satisfies $X^{(a)}_{i}W\subset W$ for all $a\in|D|$
and $i=0,1,\ldots,k_{a}$, then $W=\\{0\\}$ or $W=\mathbb{C}^{n}$. Let us
denote subspaces of irreducible elements in $\mathcal{M}_{D}$ and $M_{D}$ by
$\mathcal{M}_{D}^{s}$ and $M^{s}_{D}$ respectively.

####  1.4.3 Residue maps of truncated orbits

Let $\mathbb{O}_{H}$ be the truncated
$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$-orbit of an unramified HTL normal form
$H\in M_{n}(\mathbb{C}[z^{-1}]_{k})$ and recall that the map

$\mu_{\mathbb{O}_{H}}\colon\mathbb{O}_{H}\cong\mathrm{GL}_{n}(\mathbb{C}[z]_{k})/\mathrm{GL}_{n}(\mathbb{C}[z]_{k})_{H}\ni[g]\longmapsto-\mathrm{Ad}^{*}(g)(H)\in
M_{n}(\mathbb{C}[z^{-1}]_{k}),$

is a moment map with respect to the action of
$\mathrm{GL}_{n}(\mathbb{C}[z]_{k})$. Then the map
$\mu_{\mathbb{O}_{H_{a}}}^{0}:=\mathrm{res}_{z=a}\circ\mu_{\mathbb{O}_{H_{a}}}\colon\mathbb{O}_{H_{a}}\rightarrow
M_{n}(\mathbb{C})\cong\mathfrak{gl}_{n}(\mathbb{C})^{*}$ becomes a moment map
with respect to the action of $\mathrm{GL}_{n}(\mathbb{C}).$ Let us consider
the map

$\nu_{\mathrm{GL}_{n}(\mathbb{C})}\circ\mathrm{pr}_{1}\colon
T^{*}\mathrm{GL}_{n}(\mathbb{C})\times\mathbb{O}_{H_{\mathrm{irr}}}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})^{*}$

and its restriction
$\nu_{\mathrm{GL}_{n}(\mathbb{C})}\circ\mathrm{pr}_{1}|_{\mu_{\mathrm{ext}}^{-1}(H_{\mathrm{res}})}$
on the subspace $\mu_{\mathrm{ext}}^{-1}(H_{\mathrm{res}})\subset
T^{*}\mathrm{GL}_{n}(\mathbb{C})\times\mathbb{O}_{H_{\mathrm{irr}}}$. Then
since
$\nu_{\mathrm{GL}_{n}(\mathbb{C})}\circ\mathrm{pr}_{1}|_{\mu_{\mathrm{ext}}^{-1}(H_{\mathrm{res}})}$
is
$(\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}})_{H_{\mathrm{res}}}$-invariant,
there uniquely exists the map
$\nu_{\mathbb{O}_{H}}\colon\mathbb{O}_{H}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})^{*}$
such that the diagram

${\mu_{\mathrm{ext}}^{-1}(H_{\mathrm{res}})}$${\mathfrak{gl}_{n}(\mathbb{C})^{*}}$${\mathbb{O}_{H}}$$\scriptstyle{\nu_{\mathrm{GL}_{n}(\mathbb{C})}\circ\mathrm{pr}_{1}}$$\scriptstyle{\nu_{\mathbb{O}_{H}}}$

is commutative. Then by Theorem 1.8, we obtain the following.

######  Proposition 1.15.

The map
$-\nu_{\mathbb{O}_{H}}\colon\mathbb{O}_{H}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})^{*}$
defined above coincides with the moment map
$\mu_{\mathbb{O}_{H}}^{0}\colon\mathbb{O}_{H}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})^{*}$.

####  1.4.4 Moduli space of meromorphic connections on a trivial bundle with
unramified irregular sigularities

For each $a\in|D|$, let us take an unramified HTL normal form

$H_{a}:=\left(\frac{S_{k_{a}}^{(a)}}{z_{a}^{k_{a}-1}}+\cdots+\frac{S^{(a)}_{1}}{z_{a}}+S^{(a)}_{0}+N^{(a)}_{0}\right)\frac{dz_{a}}{z_{a}},$

such that $\sum_{a\in|D|}\mathrm{res}_{z=a}\mathrm{tr}(H_{a})=0.$ We denote
the collection of these HTL normal forms by $\mathbf{H}:=(H_{a})_{a\in|D|}$.

Let us consider the map

$\begin{array}[]{cccc}\mu_{\mathbf{H}}\colon&\prod_{a\in|D|}\mathbb{O}_{H_{a}}&\longrightarrow&\mathfrak{gl}_{n}(\mathbb{C})^{*}\\\
&(X_{a})_{a\in|D|}&\longmapsto&\sum_{a\in|D|}\mu_{\mathbb{O}_{H_{a}}}^{0}(X_{a})\end{array}$

which is a moment map with respect to the diagonal action of
$\mathrm{GL}_{n}(\mathbb{C})$ on the product space
$\prod_{a\in|D|}\mathbb{O}_{H_{a}}$.

By the injective immersion

$\begin{array}[]{cccc}\iota_{\prod_{a\in|D|}{\mathbb{O}_{H_{a}}}}\colon&\prod_{a\in|D|}\mathbb{O}_{H_{a}}&\longrightarrow&\prod_{a\in|D|}M_{n}(\mathbb{C}[z_{a}^{-1}]_{k_{a}})\\\
&([g_{a}])_{a\in|D|}&\longmapsto&(\mathrm{Ad}^{*}(g_{a})(H_{a}))_{a\in|D|}\end{array},$

we regard $\prod_{a\in|D|}\mathbb{O}_{H_{a}}$ as a subspace of
$\prod_{a\in|D|}\prod_{i=0}^{k_{a}}M_{n}(\mathbb{C})$ and we denote the space
of irreducible elements in
$\prod_{a\in|D|}\mathbb{O}_{H_{a}}\subset\prod_{a\in|D|}\prod_{i=0}^{k_{a}}M_{n}(\mathbb{C})$
by $\left(\prod_{a\in|D|}\mathbb{O}_{H_{a}}\right)^{s}$.

######  Definition 1.16.

The symplectic quotient space

$\mathcal{M}^{*}_{s}(\mathbf{H}):=\mathrm{GL}_{n}(\mathbb{C})\backslash(\mu_{\mathbf{H}}^{s})^{-1}(0)$

is the moduli space of meromorphic connections on the rank $n$ trivial bundle
on $\mathbb{P}^{1}$ with respect to the unramified HTL normal forms
$\mathbf{H}$. It is known that if
$(\mu_{\mathbf{H}}^{s})^{-1}(0)\neq\emptyset$, then the moduli space
$\mathcal{M}^{*}_{s}(\mathbf{H})$ is a holomorphic symplectic manifold.

##  2 Construction of unfolding manifold

This section is the main part of the note. Let us consider a moduli space of
meromorphic connection $\mathcal{M}^{*}_{s}(\mathbf{H})$ with respect to a
collection of unramified HTL normal forms $\mathbf{H}$. First we introduce a
holomorphic family of collections of unramified HTL normal forms which
describes unfolding procedure of irregular singularities of HTL normal forms
in $\mathbf{H}$. We call this family the unfolding of $\mathbf{H}$. Then we
shall explain the construction of a Poisson manifold whose symplectic leaves
are isomorphic to Zariski open subsets of the moduli spaces of meromorphic
connections with respect to the collections of HTL normal forms appearing in
this unfolding family.

###  2.1 Deformation of HTL normal forms and unfolding of spectral types

####  2.1.1 An open subset of $\mathbb{C}^{k+1}$ associated to $H$

Let us consider an unramified HTL normal form

$H=\left(\frac{S_{k}}{z^{k}}+\cdots+\frac{S_{1}}{z}+S_{0}+N_{0}\right)\frac{dz}{z}\in
M_{n}(\mathbb{C}[z^{-1}]_{k})dz$

with the spectral type

$\mathrm{sp}(H)=(\mathbf{m}_{0}\leq_{\phi_{0}}\leq\mathbf{m}_{1}\leq_{\phi_{1}}\cdots\leq_{\phi_{k-1}}\mathbf{m}_{k},\sigma(\mathbf{m}_{0})).$

Let $\mathbb{C}^{n}=\bigoplus_{j=1}^{m(i)}V_{\langle i,j\rangle}$ be the
direct sum of simultaneous eigenspaces of $(S_{i},S_{i+1},\ldots,S_{k})$
defined in Section 1.3.3. We denote the eigenvalues of $S_{l}$ on $V_{\langle
i,j\rangle}$ by $s_{\langle i,j\rangle}^{(l)}$ for $i\leq l\leq k$. Let us
introduce a distance function $d_{i}(\cdot,\cdot)$ on $\\{1,2,\ldots,m(i)\\}$
for each $i=0,1,\ldots,k$ as follows,

$d_{i}(j,j^{\prime}):=\mathrm{min}\\{l\in\\{i,i+1,\ldots,k\\}\mid\phi^{l}_{i}(j)=\phi^{l}_{i}(j^{\prime})\\}-i,$

where $\phi_{i}^{l}$ is the composition of refinement maps
$\phi_{l}\circ\cdots\circ\phi_{i+1}\circ\phi_{i}$ and we formally set
$\phi^{i}_{i}:=\phi_{i}$ and
$\phi_{k}\colon\\{1,2,\ldots,m(k)\\}\rightarrow\\{1\\}$. Then we note that

$\begin{cases}V_{\langle l,\phi_{i}^{l}(j)\rangle}=V_{\langle
l,\phi_{i}^{l}(j^{\prime})\rangle}&d_{i}(j,j^{\prime})+i\leq l\leq k,\\\
V_{\langle l,\phi_{i}^{l}(j)\rangle}\neq V_{\langle
l,\phi_{i}^{l}(j^{\prime})\rangle}&l=d_{i}(j,j^{\prime})+i-1.\end{cases}$

Since $V_{\langle i,j\rangle}$ is a direct summand of $V_{\langle
l,\phi_{i}^{l}(j)\rangle}$, this implies that eigenvalues $s_{\langle
i,j^{\prime}\rangle}^{(l)}$ satisfy the following relations,

$\begin{cases}s_{\langle i,j^{\prime}\rangle}^{(l)}=s_{\langle
i,j^{\prime}\rangle}^{(l)}&d_{i}(j,j^{\prime})+i\leq l\leq k,\\\
s^{(l)}_{\langle i,j\rangle}\neq s^{(l)}_{\langle
i,j^{\prime}\rangle}&l=d_{i}(j,j^{\prime})+i-1.\end{cases}$ (2)

Let us define polynomials

$\displaystyle\alpha(x_{0},x_{1},\ldots,x_{k})_{\langle
i,j\rangle}:=\sum_{i\leq l\leq k}s^{(l)}_{\langle i,j\rangle}\prod_{l<\nu\leq
k}(x_{i}-x_{\nu}),$

and hypersurfaces

$D_{i;j,j^{\prime}}:=\left\\{\mathbf{c}\in\mathbb{C}^{k+1}\,\middle|\,\alpha(\mathbf{c})_{\langle
i,j\rangle}=\alpha(\mathbf{c})_{\langle i,j^{\prime}\rangle}\right\\}$

for $i=0,1,\ldots,k-1$ and $j\neq j^{\prime}\in\\{1,2,\ldots,m(i)\\}$. Then
$(\ref{eq:eigen})$ implies that

$\begin{split}&\alpha(x_{0},x_{1},\ldots,x_{k})_{\langle
i,j\rangle}-\alpha(x_{0},x_{1},\ldots,x_{k})_{\langle i,j^{\prime}\rangle}\\\
&\quad=\sum_{i\leq l\leq i+d_{i}(j,j^{\prime})-1}(s^{(l)}_{\langle
i,j\rangle}-s^{(l)}_{\langle i,j^{\prime}\rangle})\prod_{l<\nu\leq
k}(x_{i}-x_{\nu})\\\ &\quad=\prod_{d_{i}(j,j^{\prime})+i\leq\mu\leq
k}(x_{i}-x_{\nu})\left(\sum_{i\leq l\leq
i+d_{i}(j,j^{\prime})-1}(s^{(l)}_{\langle i,j\rangle}-s^{(l)}_{\langle
i,j^{\prime}\rangle})\prod_{l<\nu\leq
i+d_{i}(j,j^{\prime})-1}(x_{i}-x_{\nu})\right).\end{split}$

This leads us to define the following polynomials

$\alpha^{*}(x_{0},x_{1},\ldots,x_{k})_{i;j,j^{\prime}}:=\sum_{i\leq l\leq
i+d_{i}(j,j^{\prime})-1}(s^{(l)}_{\langle i,j\rangle}-s^{(l)}_{\langle
i,j^{\prime}\rangle})\prod_{l<\nu\leq
i+d_{i}(j,j^{\prime})-1}(x_{i}-x_{\nu}).$

Then we define the subset of $D_{i;j,j^{\prime}}$ by

$D_{i;j,j^{\prime}}^{*}:=\left\\{\mathbf{c}\in\mathbb{C}^{k+1}\,\middle|\,\alpha^{*}(\mathbf{c})_{i;j,j^{\prime}}=0\right\\}\subset
D_{i;j,j^{\prime}}.$

Then we define the Zariski open subset of $\mathbb{C}^{k+1}$ by

$\mathbb{D}(H):=\mathbb{C}^{k+1}-\left(\bigcup_{i=0}^{k-1}\bigcup_{\begin{subarray}{c}j,j^{\prime}\in\\{1,2,\ldots,m(i)\\}\\\
j<j^{\prime}\end{subarray}}D_{i;j,j^{\prime}}^{*}\right).$ (3)

We note that $\mathbb{D}(H)$ contains the origin
$\mathbf{0}\in\mathbb{C}^{k+1}$.

####  2.1.2 A stratification of $\mathbb{C}^{k+1}$ associated to partitions of
a finite set

Firstly we introduce a stratification of $\mathbb{C}^{k+1}$ which is induced
from partitions of the finite set $\\{0,1,2,\ldots,k\\}$. Let
$\mathcal{I}\colon I_{1},I_{2},\ldots,I_{r}$ be a partition of
$\\{0,1,2,\ldots,k\\}$, i.e., the direct sum decomposition $I_{0}\sqcup
I_{1}\sqcup\cdots\sqcup I_{r}=\\{0,1,2,\ldots,k\\}.$ We may suppose that
elements in $I_{j}=\\{i_{[j,0]},i_{[j,1]},\ldots,i_{[j,k_{j}]}\\}$ are
arranged in the ascending order, i.e.,

$i_{[j,0]}<i_{[j,1]}<\ldots<i_{[j,k_{j}]}$

as positive integers. Also we assume that $0\in I_{0}$. Along with the
partition $\mathcal{I}$, we define an embedding of $\mathbb{C}^{r+1}$ into
$\mathbb{C}^{k+1}$,

$\iota_{\mathcal{I}}\colon\mathbb{C}^{r+1}\ni\mathbf{a}=(a_{0},a_{1},\ldots,a_{r})\mapsto(\iota_{\mathcal{I}}(\mathbf{a})_{0},\iota_{\mathcal{I}}(\mathbf{a})_{1},\ldots,\iota_{\mathcal{I}}(\mathbf{a})_{k})\in\mathbb{C}^{k+1}$

by setting

$\iota_{\mathcal{I}}(\mathbf{a})_{i}:=a_{l}\quad(i\in I_{l}).$

Let us denote the configuration space of $r+1$ points in $\mathbb{C}$ by

$C_{r+1}(\mathbb{C}):=\\{(a_{0},a_{1},\ldots,a_{r})\in\mathbb{C}^{r+1}\mid
a_{i}\neq a_{j}\text{ for }i\neq j\\}.$

Then we define a subspace of $\mathbb{C}^{k}$ by

$C(\mathcal{I}):=\iota_{\mathcal{I}}(C_{r+1}(\mathbb{C}))=\left\\{(a_{0},a_{1},\ldots,a_{k})\in\mathbb{C}^{k+1}\,\middle|\,\begin{array}[]{ll}a_{i}=a_{j}&\text{if
}i,j\in I_{l}\text{ for some }l,\\\ a_{i}\neq
a_{j}&\text{otherwise}\end{array}\right\\}.$

Then we obtain the direct sum decomposition

$\mathbb{C}^{k+1}=\bigsqcup_{\mathcal{I}\in\mathcal{P}_{[k+1]}}C(\mathcal{I}),$
(4)

where $\mathcal{P}_{[k+1]}$ is the set of all partitions of
$\\{0,1,\ldots,k\\}$. The set $\mathcal{P}_{[k+1]}$ of partitions is naturally
equipped with the partial order defined by the refinement of the partitions
and the each direct summand satisfies the closure relation

$\overline{C(\mathcal{I})}=\bigsqcup_{\begin{subarray}{c}\mathcal{I}^{\prime}\in\mathcal{P}_{[k+1]},\\\
\mathcal{I}^{\prime}\leq\mathcal{I}\end{subarray}}C(\mathcal{I}^{\prime}).$

Thus the decomposition $(\ref{eq:stra})$ gives a stratification of
$\mathbb{C}^{k+1}$ associated to the poset $\mathcal{P}_{[k+1]}$.

####  2.1.3 Unfolding of an HTL normal form

######  Definition 2.1 (Unfolding of an HTL normal form).

The unfolding of the irregular type $H_{\mathrm{irr}}$ of $H$ is the function
from $\mathbb{C}^{k+1}$ to $M_{n}(\mathbb{C}(z))dz$ of the form,

$H_{\mathrm{irr}}(c_{0},c_{1},\ldots,c_{k}):=\\\
\left(\frac{S_{k}}{(z-c_{1})\cdots(z-c_{k})}+\frac{S_{k-1}}{(z-c_{1})\cdots(z-c_{k-1})}+\cdots+\frac{S_{1}}{z-c_{1}}\right)\frac{dz}{z-c_{0}}.$

We also define the unfolding of $H$ by

$H(c_{0},c_{1},\ldots,c_{k}):=H_{\mathrm{irr}}(c_{0},c_{1},\ldots,c_{k})+(S_{0}+N_{0})\frac{dz}{z-c_{0}}$

and call $H_{\mathrm{irr}}(c_{0},c_{1},\ldots,c_{k})$ the irregular part of
the unfolding $H(c_{0},c_{1},\ldots,c_{k})$.

Let
$\mathbb{C}^{k+1}=\bigsqcup_{\mathcal{I}\in\mathcal{P}_{[k+1]}}C(\mathcal{I})$
be the stratification of $\mathbb{C}^{k+1}$ defined in the previous section.
We shall compute the spectral types of $H(c_{1},c_{2},\ldots,c_{k})$ for the
parameters $(c_{0},c_{1},\ldots,c_{k})$ on these strata $C(\mathcal{I})$.

Fix a partition $\mathcal{I}\colon
I_{0},I_{1},\ldots,I_{r}\in\mathcal{P}_{[k+1]}$ and consider the embedding
$\iota_{\mathcal{I}}\colon\mathbb{C}^{r+1}\hookrightarrow\mathbb{C}^{k+1}$
defined previously. Then since
$C(\mathcal{I})=\\{\iota_{\mathcal{I}}(\mathsf{c})\in\mathbb{C}^{k+1}\mid\mathsf{c}\in
C_{r+1}(\mathbb{C})\\}$, we can introduce a parameterization of the unfolding
of $H$ on $C(\mathcal{I})$ by setting

$H_{\mathcal{I}}(\mathsf{c}):=H(\iota_{\mathcal{I}}(\mathsf{c}))\quad(\mathsf{c}\in
C_{r+1}(\mathbb{C}))$

Let us write $(c_{0},c_{1},\ldots,c_{r})\in C_{r}(\mathbb{C})$. Then
$H_{\mathcal{I}}(\mathsf{c})\in M_{n}(\mathbb{C}(z))dz$ has poles at $c_{j}$
of order at most $k_{j}+1=|I_{j}|$ for $j=0,1,\ldots,r$. Thus the partial
fractional decomposition algorithm gives the description
$H_{\mathcal{I}}(\mathsf{c})=\sum_{j=0}^{r}\sum_{\nu=0}^{k_{i}}\frac{A^{[j]}_{\nu}}{z_{c_{j}}^{\nu+1}}\,dz_{c_{j}}$
where $A^{[j]}_{\nu}\in M_{n}(\mathbb{C})$ and $z_{c_{j}}=z-c_{j}$. We denote
each component of the sum by

$H_{\mathcal{I}}(\mathsf{c})_{z_{c_{j}}}:=\sum_{\nu=0}^{k_{i}}\frac{A^{[j]}_{\nu}}{z_{c_{j}}^{\nu+1}}\,dz_{c_{j}}.$

######  Proposition 2.2 ([4]).

Let us suppose that $\iota_{\mathcal{I}}(\mathsf{c})\in\mathbb{D}(H)$. Then
$H_{\mathcal{I}}(\mathsf{c})_{z_{c_{j}}}$ become unramified HTL-normal forms
with the spectral types

$\begin{cases}(\mathbf{m}_{i_{[j,0]}}\leq_{\phi_{i_{[j,0]}}^{i_{[j,1]}}}\mathbf{m}_{i_{[j,1]}}\leq_{\phi_{i_{[j,1]}}^{i_{[j,2]}}}\cdots\leq_{\phi_{i_{[j,k_{j}-1]}}^{i_{[j,k_{j}]}}}\mathbf{m}_{i_{[j,k_{j}]}},\mathrm{triv})&\text{
for }j=1,2,\ldots,r,\\\
(\mathbf{m}_{i_{[0,0]}}\leq_{\phi_{i_{[0,0]}}^{i_{[0,1]}}}\mathbf{m}_{i_{[0,1]}}\leq_{\phi_{i_{[0,1]}}^{i_{[0,2]}}}\cdots\leq_{\phi_{i_{[0,k_{0}-1]}}^{i_{[0,k_{0}]}}}\mathbf{m}_{i_{[0,k_{0}]}},\sigma(\mathbf{m}_{0}))&\text{
for }j=0\end{cases}.$

This proposition says that the each fiber of the unfolding of $H$ is a sum of
unramified HTL normal forms $H_{\mathcal{I}}(\mathsf{c})_{z_{c_{j}}}$ and
spectral types of them do not depend on the parameter $\mathsf{c}$ but only on
the stratum $C(\mathcal{I})\subset\mathbb{C}^{k+1}$.

###  2.2 Deformation of $\mathbb{O}_{H_{\mathrm{irr}}}$

In the previous section, we introduced the holomorphic family
$(H(\mathbf{c}))_{\mathbf{c}\in\mathbb{D}(H)}$ and explained that each fiber
$H(\mathbf{c})$ is a sum of unramified HTL normal forms. Next we shall
construct a deformation of the truncated orbit $\mathbb{O}_{H}$ in accordance
with this family $(H(\mathbf{c}))_{\mathbf{c}\in\mathbb{D}(H)}$. For this
purpose, we first explain the construction of a deformation of the orbit
$\mathbb{O}_{H_{\mathrm{irr}}}$ of the irregular type.

####  2.2.1 Deformations of $\mathbb{C}[z]_{l}$ and $\mathbb{C}[z^{-1}]_{l}dz$

To a point $\mathbf{c}=(c_{0},c_{1},\ldots,c_{k})\in\mathbb{C}^{k+1}$, we
associate an effective divisor

$D(\mathbf{c}):=c_{0}+c_{1}+\cdots+c_{k}$

of $\mathbb{A}^{1}(\mathbb{C})=\mathbb{C}$. Then we consider a subspace of
rational 1-forms

$\varOmega^{\mathrm{rat}}(D(\mathbf{c})):=\left\\{f\in\mathbb{C}(z)dz\,\middle|\,\mathrm{ord}_{P}(f)\geq-
n_{P}\text{ for all }P\in\mathbb{C}\right\\}\\\ $

where $n_{P}$ is the coefficient of $P\in\mathbb{C}$ in the formal sum
$D(\mathbf{c})=\sum_{P\in\mathbb{C}}n_{P}P$, and $\mathrm{ord}_{P}(f)$ is the
order of $f$ at $P$. We can regard $\varOmega^{\mathrm{rat}}(D(\mathbf{c}))$
as $\mathbb{C}[z]$-module containing $\mathbb{C}[z]dz$ as a submodule and
consider a quotient module

$\widehat{\varOmega}(\mathbf{c}):=\varOmega^{\mathrm{rat}}_{\mathbb{C}}(D(\mathbf{c}))/\mathbb{C}[z]dz.$

We also define the linear system

$L^{\mathrm{rat}}(-D(\mathbf{c})):=\left\\{f\in\mathbb{C}(z)\mid\mathrm{ord}_{P}(f)\geq
n_{P}\text{ for all }P\in\mathbb{C}\right\\}.$

Since $D(\mathbf{c})$ is an effective divisor, i.e., $n_{P}\geq 0$ for all
$P\in\mathbb{C}$, we may regard $L^{\mathrm{rat}}(-D(\mathbf{c}))$ as an ideal
of $\mathbb{C}[z]$, namely,

$L^{\mathrm{rat}}(-D(\mathbf{c}))=\left\langle\prod_{i=0}^{k}(z-c_{i})\right\rangle_{\mathbb{C}[z]},$

and take the quotient ring

$\widehat{\mathbb{C}[z]}(\mathbf{c}):=\mathbb{C}[z]/L^{\mathrm{rat}}(-D(\mathbf{c})).$

We denote the class of $f(z)\in\mathbb{C}[z]$ in
$\widehat{\mathbb{C}[z]}(\mathbf{c})$ and that of
$g(z)dz\in\varOmega^{\mathrm{rat}}_{\mathbb{C}}(D(\mathbf{c}))$ in
$\widehat{\varOmega}_{\mathbb{C}}(\mathbf{c})$ by the same notations, if it
may not cause any confusion.

######  Definition 2.3 (Residue and evaluation maps).

For $l\in\\{0,1,\ldots,k\\}$, we define the residue map at $c_{l}$ by

$\mathrm{res}_{z=c_{l}}([f(z)dz]):=\mathrm{res}_{z=c_{l}}f(z)dz$

for $[f(z)dz]\in\widehat{\varOmega}(\mathbf{c})$, the class of
$f(z)\in\varOmega_{\mathbb{C}}^{\mathrm{rat}}(D(\mathbf{c}))$. Further, we
define the residue at $\infty$ by

$\mathrm{res}_{z=\infty}([f(z)dz]):=-\mathrm{res}_{w=0}f(1/w)\frac{dw}{w^{2}}.$

We note the all these residue maps are well-defined in
$\widehat{\varOmega}(\mathbf{c})$.

Also we define the evaluation map
$\mathrm{ev}_{c_{j}}\colon\widehat{\mathbb{C}[z]}(\mathbf{c})\rightarrow\mathbb{C}$
at $c_{j}$ by

$\mathrm{ev}_{c_{j}}\colon\widehat{\mathbb{C}[z]}(\mathbf{c})=\mathbb{C}[z]/\langle\prod_{\mu=0}^{k}(z-c_{\mu})\rangle_{\mathbb{C}[z]}\longrightarrow\mathbb{C}[z]/\langle
z-c_{j}\rangle_{\mathbb{C}[z]}=\mathbb{C},$

the natural projection induced by the inclusion
$\langle\prod_{\mu=0}^{k}(z-c_{\mu})\rangle_{\mathbb{C}[z]}\subset\langle
z-c_{j}\rangle_{\mathbb{C}[z]}$. Here we identify $\mathbb{C}[z]/\langle
z-c_{j}\rangle_{\mathbb{C}[z]}=\mathbb{C}$ by the unique $\mathbb{C}$-algebra
isomorphism. Usually we simply write $f(c_{j}):=\mathrm{ev}_{c_{j}}(f(z))$ for
$f(z)\in\widehat{\mathbb{C}[z]}(\mathbf{c})$.

Note that $\widehat{\varOmega}(\mathbf{c})$ is a
$\widehat{\mathbb{C}[z]}(\mathbf{c})$-module, since elements of
$L^{\mathrm{rat}}(-D(\mathbf{c}))\subset\mathbb{C}[z]$ act trivially on
$\widehat{\varOmega}_{\mathbb{C}}(\mathbf{c})$. Since we have the isomorphisms

$\widehat{\mathbb{C}[z]}(\mathbf{0})\cong\mathbb{C}[z]_{k},\quad\quad\widehat{\varOmega}(\mathbf{0})\cong\mathbb{C}[z^{-1}]_{k}dz,$

we can regard $\widehat{\mathbb{C}[z]}(\mathbf{c})$ and
$\widehat{\Omega}(\mathbf{c})$ as deformations of $\mathbb{C}[z]_{k}$ and
$\mathbb{C}[z^{-1}]_{k}dz$.

Let us define a filtration on $\widehat{\varOmega}(\mathbf{c})$ and
cofiltration on $\widehat{\mathbb{C}[z]}(\mathbf{c})$. Consider projection
maps
$\mathrm{pr}^{(l)}\colon\mathbb{C}^{k+1}\ni(x_{0},x_{1},\ldots,x_{k})\mapsto(x_{0},x_{1},\ldots,x_{l})\in\mathbb{C}^{l+1}$
and define divisors $D(\mathbf{c})_{\leq l}:=D(\mathrm{pr}^{(l)}(\mathbf{c}))$
for $l=0,1,\ldots,k$. Then we obtain the following filtration

$\varOmega^{\mathrm{rat}}(D(\mathbf{c})_{\leq
0})\subset\varOmega^{\mathrm{rat}}(D(\mathbf{c})_{\leq
1})\subset\cdots\subset\varOmega^{\mathrm{rat}}(D(\mathbf{c})_{\leq
k})=\varOmega^{\mathrm{rat}}(D(\mathbf{c})).$

######  Definition 2.4 (Standard filtration and basis of
$\widehat{\varOmega}(\mathbf{c})$).

Let us set

$\widehat{\varOmega}(\mathbf{c})_{l}:=\varOmega^{\mathrm{rat}}(D(\mathbf{c})_{\leq
l})/\mathbb{C}[z]dz\text{ for }l=0,1,\ldots,k.$

The filtration

$\widehat{\varOmega}(\mathbf{c})_{0}\subset\widehat{\varOmega}(\mathbf{c})_{1}\subset\cdots\subset\widehat{\varOmega}(\mathbf{c})_{k}=\widehat{\varOmega}(\mathbf{c})$

induced by the above filtration of
$\varOmega^{\mathrm{rat}}_{\mathbb{C}}(D(\mathbf{c}))$ is called the standard
filtration of $\widehat{\varOmega}(\mathbf{c})$. The basis of
$\widehat{\varOmega}(\mathbf{c})$,

$\frac{1}{z-c_{0}},\,\frac{1}{(z-c_{0})(z-c_{1})},\ldots,\frac{1}{(z-c_{0})(z-c_{1})\cdots(z-c_{k})}$

as $\mathbb{C}$-vector space is called the standard basis of
$\widehat{\varOmega}(\mathbf{c})$.

Let us note that
$\frac{1}{z-c_{0}},\,\frac{1}{(z-c_{0})(z-c_{1})},\ldots,\frac{1}{(z-c_{0})(z-c_{1})\cdots(z-c_{l})}$
becomes a basis of the $l$-th component $\widehat{\varOmega}(\mathbf{c})_{\leq
l}$ of the standard filtration for each $l=0,1,\ldots,k$.

Similarly, we can consider the filtration

$\langle
z-c_{0}\rangle_{\mathbb{C}[z]}\supset\langle(z-c_{0})(z-c_{1})\rangle_{\mathbb{C}[z]}\supset\cdots\supset\left\langle\prod_{i=0}^{k}(z-c_{i})\right\rangle_{\mathbb{C}[z]}$

of ideals on $\mathbb{C}[z]$.

######  Definition 2.5 (Standard cofiltration and basis of
$\widehat{\mathbb{C}[z]}(\mathbf{c})$).

Let us set

$\widehat{\mathbb{C}[z]}(\mathbf{c})_{l}:=\mathbb{C}[z]/\langle\prod_{i=0}^{l}(z-c_{i})\rangle_{\mathbb{C}[z]}\text{
for }l=0,1,\ldots,k.$

Then the sequence of projection maps

$\widehat{\mathbb{C}[z]}(\mathbf{c})_{0}\rightarrow\widehat{\mathbb{C}[z]}(\mathbf{c})_{1}\rightarrow\cdots\rightarrow\widehat{\mathbb{C}[z]}(\mathbf{c})_{k}=\widehat{\mathbb{C}[z]}(\mathbf{c})$

induced by the above filtration of ideals of $\mathbb{C}[z]$ is called the
standard cofiltration of $\widehat{\mathbb{C}[z]}(\mathbf{c})$. The basis of
$\widehat{\mathbb{C}[z]}(\mathbf{c})$,

$1,(z-c_{0}),(z-c_{0})(z-c_{1}),\ldots,(z-c_{0})(z-c_{1})\cdots(z-c_{k-1})$

as $\mathbb{C}$-vector space is called the standard basis of
$\widehat{\mathbb{C}[z]}(\mathbf{c})$.

Next we introduce the pairing of $\widehat{\mathbb{C}[z]}(\mathbf{c})_{l}$ and
$\widehat{\varOmega}(\mathbf{c})_{l}$ as $\mathbb{C}$-vector spaces defined by

$\begin{array}[]{cccc}\langle\ ,\
\rangle_{\mathbf{c},l}\colon&\widehat{\mathbb{C}[z]}(\mathbf{c})_{l}\times\widehat{\varOmega}(\mathbf{c})_{l}&\longrightarrow&\mathbb{C}\\\
&(f(z),g(z)dz)&\longmapsto&-\mathrm{res}_{z=\infty}(f(z)\cdot
g(z)dz)\end{array}$

for each $l=0,1,\ldots,k$. Then we can show that the paring $\langle\ ,\
\rangle_{\mathbf{c},l}$ is non-degenerate and the bases

$1,(z-c_{0}),(z-c_{0})(z-c_{1}),\ldots,(z-c_{0})(z-c_{1})\cdots(z-c_{l-1})$

and

$\frac{1}{z-c_{0}},\,\frac{1}{(z-c_{0})(z-c_{1})},\ldots,\frac{1}{(z-c_{0})(z-c_{1})\cdots(z-c_{l})}$

are dual bases with respect to this paring.

####  2.2.2 Partial fraction decomposition

In Section 2.1.3, we saw that each fiber $H_{\mathcal{I}}(\mathsf{c})$ of the
unfolding of $H$ on a stratum $C(\mathcal{I})\subset\mathbb{C}^{k+1}$
decomposes into the sum of unramified HTL normal form by the partial fraction
decomposition algorithm. Now we explain that on each stratum $C(\mathcal{I})$,
the partial fraction decomposition gives an decomposition of
$\widehat{\mathbb{C}[z]}(\mathbf{c})_{l}$-module
$\widehat{\varOmega}(\mathbf{c})_{l}$.

Let us take a partition $\mathcal{I}\colon I_{1},I_{2},\ldots,I_{r}$ of the
finite set $\\{0,1,\ldots,k\\}$. Then for each $l=0,1,\ldots,k$, we define a
partition $\mathcal{I}^{(l)}\colon I_{1}^{(l)},I_{2}^{(l)},\ldots,I_{r}^{(l)}$
of the subset $\\{0,1,\ldots,l\\}\subset\\{0,1,\ldots,k\\}$ by
$I^{(l)}_{j}:=I_{j}\cap\\{0,1,\ldots,l\\}$, allowing $I^{(l)}_{j}=\emptyset$.
Let us set $k^{(l)}_{j}:=|I^{(l)}_{j}|-1.$ Let us take
$\mathsf{c}=(c_{0},c_{1},\ldots,c_{r})\in C_{r+1}(\mathbb{C})$ so that
$\iota_{\mathcal{I}}(\mathsf{c})\in C(\mathcal{I})$ . Then we have
$D(\iota_{\mathcal{I}}(\mathsf{c}))_{\leq
l}=\sum_{j=0}^{r}(k_{j}^{(l)}+1)\cdot c_{j}$. The algorithm of the partial
fraction decomposition of rational functions gives us the direct sum
decomposition

$\widehat{\varOmega}(\iota_{\mathcal{I}}(\mathsf{c}))_{l}=\bigoplus_{j=0}^{r}\mathbb{C}[(z-c_{j})^{-1}]_{k_{j}^{(i)}}$

as $\widehat{\mathbb{C}[z]}(\iota_{\mathcal{I}}(\mathsf{c}))_{l}$-modules.

We also have a similar decomposition of
$\widehat{\mathbb{C}[z]}(\iota_{\mathcal{I}}(\mathsf{c}))_{l}$. Since we have
$c_{i}\neq c_{j}$ for $i\neq j$, it follows that

$\displaystyle\langle(z-c_{i})^{k_{i}^{(l)}}\rangle_{\mathbb{C}[z]}+\langle(z-c_{i^{\prime}})^{k_{i^{\prime}}^{(l)}}\rangle_{\mathbb{C}[z]}=\mathbb{C}[z],\quad
i\neq i^{\prime},$
$\displaystyle\bigcap_{i=0}^{l}\langle(z-c_{i})^{k_{i}^{(l)}}\rangle_{\mathbb{C}[z]}=L^{\mathrm{rat}}(-D(\iota_{\mathcal{I}}(\mathbf{c}))_{\leq
l}).$

Therefore the Chinese remainder theorem implies that the map

$\begin{array}[]{cccc}\prod_{i=0}^{l}\mathrm{pr}_{c_{i}}\colon&\widehat{\mathbb{C}[z]}(\iota_{\mathcal{I}}(\mathsf{c}))_{l}&\longrightarrow&\prod_{i=0}^{l}\mathbb{C}[z-c_{i}]_{k_{i}^{(l)}}\\\
&f(z)&\longmapsto&(\mathrm{pr}_{c_{i}}(f(z))_{i=0,1,\ldots,l}\end{array}$

is an algebra isomorphism. Here

$\mathrm{pr}_{c_{i}}\colon\widehat{\mathbb{C}[z]}(\iota_{\mathcal{I}}(\mathsf{c}))_{l}\longrightarrow\\\
\mathbb{C}[z]/\langle(z-c_{i})^{k_{i}^{(l)}+1}\rangle_{\mathbb{C}[z]}=\mathbb{C}[z-c_{i}]_{k_{i}^{(l)}}$

are the natural projections for $i=0,1,\ldots,l$.

If we note that the ideal
$\langle(z-c_{i})^{k_{i}^{(l)}+1}\rangle_{\mathbb{C}[z]}$ annihilates
$\mathbb{C}[(z-c_{i})^{-1}]_{k_{i}^{(l)}}$, it follows that the diagrams

${\widehat{\mathbb{C}[z]}(\iota_{\mathcal{I}}(\mathsf{c}))_{l}\times\mathbb{C}[(z-c_{i})^{-1}]_{k_{i}^{(l)}}}$${\mathbb{C}[(z-c_{i})^{-1}]_{k_{i}^{(l)}}}$${\mathbb{C}[z-c_{i}]_{k_{i}^{(l)}}\times\mathbb{C}[(z-c_{i})^{-1}]_{k_{i}^{(l)}}}$${\mathbb{C}[(z-c_{i})^{-1}]_{k_{i}^{(l)}}}$$\scriptstyle{\mathrm{pr}_{c_{i}}\times\mathrm{id}}$$\scriptstyle{\cdot}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\cdot}$

are commutative. Here horizontal maps are scalar multiplications on
$\mathbb{C}[z-c_{i}]_{k_{i}^{(l)}}$ as
$\widehat{\mathbb{C}[z]}(\iota_{\mathcal{I}}(\mathsf{c}))_{l}$ and
$\mathbb{C}[z-c_{i}]_{k_{i}^{(l)}}$ modules respectively. Further, we obtain
the following commutative diagram,

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####  2.2.3 Lie groupoid as a deformation of
$N_{\mathbf{m}_{l}^{l+1}}^{-}(\mathbb{C}[z]_{l-1})^{1}$

For a $\mathbb{C}$-subalgebra $\mathfrak{k}\subset M_{n}(\mathbb{C})$ and
$\mathbf{c}=(c_{0},c_{1},\ldots,c_{k+1})\in\mathbb{D}(H)$, we define
subalgebras of
$M_{n}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1}):=M_{n}(\mathbb{C})\otimes_{\mathbb{C}}\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1}$
by

$\displaystyle\mathfrak{k}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})$
$\displaystyle:=\left\\{X=X_{0}+\sum_{i=1}^{l-1}X_{i}(z-c_{0})(z-c_{1})\cdots(z-c_{i-1})\,\middle|\,X_{i}\in\mathfrak{k}\right\\},$
$\displaystyle\mathfrak{k}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}$
$\displaystyle:=\left\\{X\in\mathfrak{k}(\widehat{\mathbb{C}[z]}(\mathbf{c}))_{l-1}\mid
X_{0}=0\right\\}.$

Then we have the natural direct sum decomposition as $\mathbb{C}$-algebras,

$\mathfrak{k}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})=\mathfrak{k}\oplus\mathfrak{k}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}.$

Moreover we define $\mathbb{C}$-vector subspaces of
$M_{n}(\widehat{\varOmega}(\mathbf{c})_{l-1}):=M_{n}(\mathbb{C})\otimes_{\mathbb{C}}\widehat{\varOmega}(\mathbf{c})_{l-1}$
by

$\displaystyle\mathfrak{k}(\widehat{\varOmega}(\mathbf{c})_{l-1})$
$\displaystyle:=\left\\{X=\sum_{i=0}^{l-1}\frac{X_{i}}{(z-c_{0})(z-c_{1})\cdots(z-c_{i})}dz\,\middle|\,X_{i}\in\mathfrak{k}\right\\},$
$\displaystyle\mathfrak{k}(\widehat{\varOmega}(\mathbf{c})_{l-1})^{1}$
$\displaystyle:=\\{X\in\mathfrak{k}(\widehat{\varOmega}(\mathbf{c})_{l-1})\mid
X_{0}=0\\}.$

Especially for $\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}\subset
M_{n}(\mathbb{C})$, we define

$\displaystyle
N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})$
$\displaystyle:=\left\\{E_{n}+X\mid
X\in\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})\right\\},$
$\displaystyle
N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}$
$\displaystyle:=\left\\{E_{n}+X\mid
X\in\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}\right\\},$

which are equipped with the Lie group structures and their corresponding Lie
algebras are isomorphic to
$\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})$
and
$\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}$
respectively.

The paring

$\begin{array}[]{ccc}\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c}))_{l-1}\times\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{+}(\widehat{\varOmega}(\mathbf{c}))_{l-1}&\longrightarrow&\mathbb{C}\\\
(X,Y)&\longmapsto&-\mathrm{tr}(\mathrm{res}_{z=\infty}(X\cdot Y)).\end{array}$
(5)

is shown to be non-degenerate and thus gives the identifications

$\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{+}(\widehat{\varOmega}(\mathbf{c})_{l-1})\cong\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{*},\
\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{+}(\widehat{\varOmega}(\mathbf{c})_{l-1})^{1}\cong(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1})^{*}.$

Similarly, by setting
$\mathfrak{p}_{\mathbf{m}_{l}^{l+1}}^{+}=\mathfrak{l}_{\mathbf{m}_{l}}\oplus\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{+}$
and
$\mathfrak{p}_{\mathbf{m}_{l}^{l+1}}^{-}=\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}\oplus\mathfrak{l}_{\mathbf{m}_{l}}$,
we have

$\mathfrak{p}^{+}_{\mathbf{m}_{l}^{l+1}}(\widehat{\varOmega}(\mathbf{c})_{l-1})\cong(\mathfrak{p}_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c}))_{l-1})^{*}.$

Now let us consider the family
$(N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1})_{\mathbf{c}\in\mathbb{D}(H)}$
of complex Lie groups and equip this family with a structure of complex Lie
groupoid.

######  Definition 2.6 (Complex Lie groupoid).

Let us consider a complex manifold $\Gamma$ and its submanifold $\Gamma_{0}$
with the inclusion map $i\colon\Gamma_{0}\hookrightarrow\Gamma$. Let us also
consider holomorphic surjective submersions
$s,t\colon\Gamma\rightarrow\Gamma_{0}$ satisfying $t\circ i=s\circ
i=\mathrm{id}_{\Gamma_{0}}$. Further, we consider a holomorphic map
$m\colon\Gamma^{(2)}:=\\{(\gamma_{1},\gamma_{2})\in\Gamma\times\Gamma\mid
s(\gamma_{1})=t(\gamma_{2})\\}\rightarrow\Gamma$ satisfying $s\circ
m(g_{1},g_{2})=s(g_{2})$ and $t\circ m(g_{1},g_{2})=t(g_{1})$. Then if the
following conditions are satisfied, the tuple $(\Gamma,\Gamma_{0},s,t,m)$ is
called a complex Lie groupoid.

  1. 1.

Associativity: $m(m(g_{1},g_{2}),g_{3})=m(g_{1},m(g_{2},g_{3}))$ for all
$(g_{1},g_{2},g_{3})$ satisfying $s(g_{1})=t(g_{2})$ and $s(g_{2})=t(g_{3})$.

  2. 2.

Units: $m(i\circ t(g),g)=g=m(g,i\circ s(g))$ for all $g\in\Gamma$.

  3. 3.

Inverses: For all $g\in\Gamma$, there exists $h\in\Gamma$ such that
$s(h)=t(g)$, $t(h)=s(g)$ and $m(g,h),m(h,g)\in\Gamma_{0}$.

The maps $s$ and $t$ are called the source map and target map respectively,
and $m$ is called the multiplication map. Also elements of the submanifold
$i\colon\Gamma_{0}\hookrightarrow\Gamma$ are called units.

Let us consider the trivial vector bundle
$\theta\colon\left(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}\right)^{\oplus
l}\times\mathbb{D}(H)\rightarrow\mathbb{D}(H)$ and introduce a Lie groupoid
structure on this bundle. As both of the source and target maps, we employ the
projection map $\theta$. We regard $\mathbb{D}(H)$ as a submanifold of
$\left(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}\right)^{\oplus
l}\times\mathbb{D}(H)$ by the zero section
$i\colon\mathbb{D}(H)\ni\mathbf{c}\mapsto(\mathbf{0},\mathbf{c})\in\left(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}\right)^{\oplus(l-1)}\times\mathbb{D}(H)$.
Let us consider a linear isomorphism

$\begin{array}[]{cccc}\phi_{\mathbf{c}}\colon&\left(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}\right)^{\oplus
l}&\longrightarrow&N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})\\\
&(X_{i})_{i=0,1,\ldots,l-1}&\longmapsto&(E_{n}+X_{0})+\sum_{i=1}^{l-1}X_{i}(z-c_{0})(z-c_{1})\cdots(z-c_{i-1})\end{array}.$

Then we define the multiplication $m(X(\mathbf{c}),Y(\mathbf{c}))$ of
$X(\mathbf{c}):=((X_{i}),\mathbf{c})$ and
$Y(\mathbf{c}):=((Y_{i}),\mathbf{c})\in\left(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}\right)^{\oplus
l}\times\mathbb{D}(H)$ by

$m(X(\mathbf{c}),Y(\mathbf{c})):=\phi_{\mathbf{c}}^{-1}(\phi_{\mathbf{c}}((X_{i}))\cdot\phi_{\mathbf{c}}((Y_{i}))).$

Here $\cdot$ in the right hand side of the equation is the multiplication in
$N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})$.
Then the tuple

$(N_{\mathbf{m}_{l}^{l+1}}^{-})_{\mathbb{D}(H)}:=\left(\left(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}\right)^{\oplus
l}\times\mathbb{D}(H),\mathbb{D}(H),\theta,\theta,m\right)$

becomes a complex Lie groupoid. Together with this Lie groupoid, we consider
the subbundle
$\left(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}\right)^{\oplus(l-1)}\times\mathbb{D}(H)\rightarrow\mathbb{D}(H)$
with the inclusion

$\begin{array}[]{ccc}\left(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}\right)^{\oplus(l-1)}\times\mathbb{D}(H)&\longrightarrow&\left(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}\right)^{\oplus
l}\times\mathbb{D}(H)\\\
((X_{1},X_{2},\ldots,X_{l-1}),\mathbf{c})&\longmapsto&((0,X_{1},X_{2},\ldots,X_{l-1}),\mathbf{c})\end{array}.$

Then this subbundle has the natural Lie groupoid structure whose fiber at each
$\mathbf{c}\in\mathbb{D}(H)$ is isomorphic to
$N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}$.
We especially denote this Lie subgroupoid by $(N^{-})^{(l)}_{\mathbb{D}(H)}$.

Next we introduce complex Lie algebroids.

######  Definition 2.7 (Complex Lie algebroid).

A complex Lie algebroid over a complex manifold $M$ is a holomorphic vector
bundle $A\rightarrow M$, together with a Lie bracket $[\cdot,\cdot]$ on its
space of holomorphic sections, such that there exists a vector bundle map
$\mathbf{a}\colon A\rightarrow TM$ called the anchor map satisfying the
Leibniz rule

$[\sigma,f\tau]=f[\sigma,\tau]+(\mathbf{a}(\sigma)f)\tau$

for all global sections $\sigma,\tau$ of $A\rightarrow M$ and
$f\in\mathcal{O}_{M}(M)$.

Let $\mathfrak{k}\subset M_{n}(\mathbb{C})$ be a $\mathbb{C}$-subalgebra. We
consider a linear isomorphism

$\begin{array}[]{cccc}\psi_{\mathbf{c}}\colon&\mathfrak{k}^{\oplus
l}&\longrightarrow&\mathfrak{k}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})\\\
&(X_{i})_{i=1,2\ldots,l-1}&\longmapsto&X_{0}+\sum_{i=1}^{l-1}X_{i}(z-c_{0})(z-c_{1})\cdots(z-c_{i-1})\end{array},$

and define the Lie bracket $[X(\mathbf{c}),Y(\mathbf{c})]$ of
$X(\mathbf{c}):=((X_{i}),\mathbf{c})$ and
$Y(\mathbf{c}):=((Y_{i}),\mathbf{c})\in\mathfrak{k}^{\oplus
l}\times\mathbb{D}(H)$ by

$[(X(\mathbf{c}),Y(\mathbf{c}))]:=\psi_{\mathbf{c}}^{-1}([\psi_{\mathbf{c}}((X_{i}),\psi_{\mathbf{c}}((Y_{i}))]).$

Then the vector bundle $\theta\colon\mathfrak{k}^{\oplus
l}\times\mathbb{D}(H)\rightarrow\mathbb{D}(H)$ with the Lie bracket and the
zero map as the trivial anchor map becomes a Lie algebroid which we denote by
$\mathfrak{k}_{\mathbb{D}(H)}$. The direct sum decomposition
$\mathfrak{k}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})=\mathfrak{k}\oplus\mathfrak{k}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}$
as Lie algebras induces the decomposition
$\mathfrak{k}_{\mathbb{D}(H)}=(\mathfrak{k}_{\mathbb{D}(H)})_{0}\oplus(\mathfrak{k}_{\mathbb{D}(H)})_{1}$
as vector bundles. Here $(\mathfrak{k}_{\mathbb{D}(H)})_{0}$ and
$(\mathfrak{k}_{\mathbb{D}(H)})_{1}$ are trivial bundles
$\mathfrak{k}\times\mathbb{D}(H)\rightarrow\mathbb{D}(H)$ and
$\mathfrak{k}^{\oplus(l-1)}\times\mathbb{D}(H)\rightarrow\mathbb{D}(H)$ with
the Lie algebroid structures induced form $\mathfrak{k}$ and
$\mathfrak{k}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}$.

In particular, we consider the cases
$\mathfrak{k}=\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}$ and
$\mathfrak{p}_{\mathbf{m}_{l}^{l+1}}^{-}$. In these cases, we can define the
adjoint action of the Lie groupoid
$(N_{\mathbf{m}_{l}^{l+1}}^{-})_{\mathbb{D}(H)}$ as follows. Let us take
holomorphic sections $n$ and $X$ of
$(N_{\mathbf{m}_{l}^{l+1}}^{-})_{\mathbb{D}(H)}$ and
$\mathfrak{k}_{\mathbb{D}(H)}$ respectively. Then we define the holomorphic
section of $\mathfrak{k}_{\mathbb{D}(H)}$ by

$\mathrm{Ad}_{\mathbb{D}(H)}(n)(X)(\mathbf{c}):=\psi_{\mathbf{c}}^{-1}\circ\mathrm{Ad}_{N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})}(n(\mathbf{c}))(\psi_{\mathbf{c}}(X(\mathbf{c})))\quad(\mathbf{c}\in\mathbb{D}(H)).$

As a dual of this adjoint action, we can define the coadjoint action of
$(N_{\mathbf{m}_{l}^{l+1}}^{-})_{\mathbb{D}(H)}$ on the dual bundle
$\mathfrak{k}_{\mathbb{D}(H)}^{*}$. Namely, for holomorphic sections $n$, $X$,
and $\xi$ of $(N_{\mathbf{m}_{l}^{l+1}}^{-})_{\mathbb{D}(H)}$,
$\mathfrak{k}_{\mathbb{D}(H)}$, and $\mathfrak{k}_{\mathbb{D}(H)}^{*}$, we
define the section of $\mathfrak{k}_{\mathbb{D}(H)}^{*}$ by

$(\mathrm{Ad}_{\mathbb{D}(H)}^{*}(n)(\xi))(X)(\mathbf{c}):=\xi(\mathbf{c})(\mathrm{Ad}_{N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})}(n(\mathbf{c})^{-1})(X(\mathbf{c})))\quad(\mathbf{c}\in\mathbb{D}(H)).$

We denote the subbundle $(\mathfrak{k}_{\mathbb{D}(H)})_{1}$ of
$\mathfrak{k}_{\mathbb{D}(H)}$ by $(\mathfrak{n}^{-})^{(l)}_{\mathbb{D}(H)}$
particularly when $\mathfrak{k}=\mathfrak{n}^{-}_{\mathbf{m}_{l}^{l+1}}$.

The paring $(\ref{eq:paring})$ allows us to identify the fiber of the dual
bundle $\mathfrak{k}_{\mathbb{D}(H)}^{*}$ at each $\mathbf{c}\in\mathbb{D}(H)$
with
$\mathfrak{n}^{+}_{\mathbf{m}_{l}^{l+1}}(\widehat{\varOmega}(\mathbf{c})_{l-1})$
(resp.
$\mathfrak{p}^{+}_{\mathbf{m}_{l}^{l+1}}(\widehat{\varOmega}(\mathbf{c})_{l-1})$)
when $\mathfrak{k}=\mathfrak{n}^{-}_{\mathbf{m}_{l}^{l+1}}$ (resp.
$\mathfrak{k}=\mathfrak{p}^{-}_{\mathbf{m}_{l}^{l+1}}$). Thus we can regard
$((\mathfrak{n}^{-})^{(l)}_{\mathbb{D}(H)})^{*}$ as the subbundle of
$(\mathfrak{p}^{-}_{\mathbf{m}_{l}^{l+1}})_{\mathbb{D}(H)}^{*}$ by the
inclusion
$\mathfrak{n}^{+}_{\mathbf{m}_{l}^{l+1}}(\widehat{\varOmega}(\mathbf{c})_{l-1})^{1}\subset\mathfrak{p}^{+}_{\mathbf{m}_{l}^{l+1}}(\widehat{\varOmega}(\mathbf{c})_{l-1})$
at each fiber.

####  2.2.4 Symplectic foliation as deformation of
$\mathbb{O}_{H_{\mathrm{irr}}}$

In Theorem 1.12, we saw that $\mathbb{O}_{H_{\mathrm{irr}}}$ is isomorphic to
the product of $T^{*}(N^{-})^{(l)}$. Thus deforming $T^{*}(N^{-})^{(l)}$, we
can obtain a deformation of $\mathbb{O}_{H_{\mathrm{irr}}}$ as follows.

Let us consider a relative cotangent bundle

$\theta_{(N^{-})^{(l)}_{\mathbb{D}(H)}/\mathbb{D}(H)}\colon
T^{*}\left((N^{-})^{(l)}_{\mathbb{D}(H)}\right)/\mathbb{D}(H)\rightarrow(N^{-})^{(l)}_{\mathbb{D}(H)},$

i.e., the vector bundle corresponding to the quotient sheaf
$\varOmega_{(N^{-})^{(l)}_{\mathbb{D}(H)}}/\theta^{*}\varOmega_{\mathbb{D}(H)}$.
Here $\theta\colon(N^{-})^{(l)}_{\mathbb{D}(H)}\rightarrow\mathbb{D}(H)$ is
the Lie groupoid defined above and $\varOmega_{M}$ denotes the sheaf of
holomorphic differential forms on a complex manifold $M$. Set
$\mathfrak{n}^{(l)}:=\left(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}\right)^{\oplus(l-1)}$
for simplicity. Recall that if we forget the structure of Lie groupoid from
$(N^{-})^{(l)}_{\mathbb{D}(H)}$, this is the trivial bundle
$\theta_{l}\colon\mathfrak{n}^{(l)}\times\mathbb{D}(H)\rightarrow\mathbb{D}(H)$.
Thus $T^{*}\left((N^{-})^{(l)}_{\mathbb{D}(H)}\right)/\mathbb{D}(H)$ is
isomorphic to

$\theta_{\mathfrak{n}^{(l)}}\times\mathrm{id}_{\mathbb{D}(H)}\colon(T^{*}\mathfrak{n}^{(l)})\times\mathbb{D}(H)\rightarrow\mathfrak{n}^{(l)}\times\mathbb{D}(H),$

where $\theta_{\mathfrak{n}^{(l)}}\colon
T^{*}\mathfrak{n}^{(l)}\rightarrow\mathfrak{n}^{(l)}$ is the standard
projection of the cotangent bundle. Here $T^{*}\mathfrak{n}^{(l)}$ has the
Poisson structure induced from the standard symplectic form
$\Omega_{T^{*}\mathfrak{n}^{(l)}}$ and also $\mathbb{D}(H)$ can be seen as the
Poisson manifold with the trivial Poisson bracket. Thus
$T^{*}\left((N^{-})^{(l)}_{\mathbb{D}(H)}\right)/\mathbb{D}(H)$ becomes a
Poisson manifold as the product of these Poisson manifolds.

######  Definition 2.8 (Deformation of $T^{*}(N^{-})^{(l)}$).

Let us set

$\left(T^{*}(N^{-})^{(l)}\right)_{\mathbb{D}(H)}:=T^{*}\left((N^{-})^{(l)}_{\mathbb{D}(H)}\right)/\mathbb{D}(H)$

and call this Poisson manifold the deformation of $T^{*}(N^{-})^{(l)}$ with
respect to the HTL normal form $H$.

Let us define the projection map
$\theta_{l}^{*}\colon\left(T^{*}(N^{-})^{(l)}\right)_{\mathbb{D}(H)}\rightarrow\mathbb{D}(H)$
by setting
$\theta_{l}^{*}:=\theta_{l}\circ\theta_{(N^{-})^{(l)}_{\mathbb{D}(H)}/\mathbb{D}(H)}$.
Then we note that
$\theta_{l}^{*}\colon\left(T^{*}(N^{-})^{(l)}\right)_{\mathbb{D}(H)}\rightarrow\mathbb{D}(H)$
is still a holomorphic vector bundle regarding $T^{*}\mathfrak{n}^{(l)}$ as a
$\mathbb{C}$-vector space through the isomorphism
$T^{*}\mathfrak{n}^{(l)}\cong\mathfrak{n}^{(l)}\oplus(\mathfrak{n}^{(l)})^{*}$.
For each fiber at $\mathbf{c}\in\mathbb{D}(H)$, there exists the natural
isomorphism

$(\theta_{l}^{*})^{-1}(\mathbf{c})\cong
T^{*}N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}.$

Since in particular we have $(\theta_{l}^{*})^{-1}(0)\cong
T^{*}N_{\mathbf{m}_{l}^{l+1}}^{-}(\mathbb{C}[z](0)_{l-1})^{1}=T^{*}(N^{-})^{(l)}$,
we can regard $\left(T^{*}(N^{-})^{(l)}\right)_{\mathbb{D}(H)}$ as a
deformation of $T^{*}(N^{-})^{(l)}$. It directly follows from the definition
that the family
$((\theta_{l}^{*})^{-1}(\mathbf{c}))_{\mathbf{c}\in\mathbb{D}(H)}$ is the
symplectic foliation of $\left(T^{*}(N^{-})^{(l)}\right)_{\mathbb{D}(H)}$.

Since
$N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}$
is normalized by
$L_{\mathbf{m}_{1}}=\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}$, the
cotangent bundle
$T^{*}N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}$
has the natural $L_{\mathbf{m}_{1}}$-action. Hence
$T^{*}(N^{-})^{(l)}_{\mathbb{D}(H)}$ is equipped with the
$L_{\mathbf{m}_{1}}$-action.

Then as the direct sum of vector bundles
$\theta_{l}^{*}\colon\left(T^{*}(N^{-})^{(l)}\right)_{\mathbb{D}(H)}\rightarrow\mathbb{D}(H)$
we define a deformation of $\mathbb{O}_{H_{\mathrm{irr}}}$ as follows.

######  Definition 2.9 (Deformation of $\mathbb{O}_{H_{\mathrm{irr}}}$).

We consider the direct sum of the vector bundles

$\theta_{\mathbb{O}_{H_{\mathrm{irr}}}}\colon\bigoplus_{l=2}^{k}\left(T^{*}(N^{-})^{(l)}\right)_{\mathbb{D}(H)}\rightarrow\mathbb{D}(H)$

and we denote the total space of this bundle by
$(\mathbb{O}_{H_{\mathrm{irr}}})_{\mathbb{D}(H)}$, which is called the
deformation of $\mathbb{O}_{H_{\mathrm{irr}}}$. This complex manifold
$(\mathbb{O}_{H_{\mathrm{irr}}})_{\mathbb{D}(H)}$ is naturally equipped with
the Poisson structure and $L_{\mathbf{m}_{1}}$-action as well as
$\left(T^{*}(N^{-})^{(l)}\right)_{\mathbb{D}(H)}$.

######  Proposition 2.10.

The family
$(\theta_{\mathbb{O}_{H_{\mathrm{irr}}}}^{-1}(\mathbf{c}))_{\mathbf{c}\in\mathbb{D}(H)}$
is the symplectic foliation of
$(\mathbb{O}_{H_{\mathrm{irr}}})_{\mathbb{D}(H)}$. In particular, the special
fiber at $\mathbf{c}=0$, we have the symplectic isomorphism

$\theta_{\mathbb{O}_{H_{\mathrm{irr}}}}^{-1}(0)\cong\mathbb{O}_{H_{\mathrm{irr}}}.$

###### Proof.

The first assertion is obvious since the family
$((\theta_{l}^{*})^{-1}(\mathbf{c}))_{\mathbf{c}\in\mathbb{D}(H)}$ is the
symplectic foliation of $\left(T^{*}(N^{-})^{(l)}\right)_{\mathbb{D}(H)}$ for
each $l$. The second assertion follows from Theorem 1.12. ∎

The right trivialization

$\begin{array}[]{ccc}N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}\times(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1})^{*}&\longrightarrow&T^{*}N_{\mathbf{m}_{l}^{l+1}}^{-}(\widehat{\mathbb{C}[z]}(\mathbf{c})_{l-1})^{1}\\\
(n,\xi)&\longmapsto&R^{*}_{n^{-1}}(\xi)\end{array}$

gives the isomorphism

$(N^{-})^{(l)}_{\mathbb{D}(H)}\oplus\left((\mathfrak{n}^{-})^{(l)}_{\mathbb{D}(H)}\right)^{*}\cong\left(T^{*}(N^{-})^{(l)}\right)_{\mathbb{D}(H)}.$
(6)

as vector budless. Thus we have the isomorphism

$(\mathbb{O}_{H_{\mathrm{irr}}})_{\mathbb{D}(H)}\cong\bigoplus_{l=2}^{k}\left((N^{-})^{(l)}_{\mathbb{D}(H)}\oplus\left((\mathfrak{n}^{-})^{(l)}_{\mathbb{D}(H)}\right)^{*}\right).$
(7)

###  2.3 Local unfolding manifold

In the previous section, we defined a deformation of
$\mathbb{O}_{H_{\mathrm{irr}}}$. Now we shall construct a deformation of the
truncated orbit $\mathbb{O}_{H}$.

####  2.3.1 Unfolding manifold of a truncated orbit

As we saw in Theorem 1.8, truncated orbit $\mathbb{O}_{H}$ is obtained as a
symplectic reduction of the extended orbit
$T^{*}\mathrm{GL}_{n}(\mathbb{C})\times\mathbb{O}_{H_{\mathrm{irr}}}$. Based
on this fact, we shall define a deformation of the truncated orbit as a
Poisson reduction of
$T^{*}\mathrm{GL}_{n}(\mathbb{C})\times(\mathbb{O}_{H_{\mathrm{irr}}})_{\mathbb{D}(H)}$.
To this purpose, we first consider a deformation of the map
$\mathrm{res}_{\mathbb{O}_{H_{\mathrm{irr}}}}\colon\mathbb{O}_{H_{\mathrm{irr}}}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}^{*}$
as follows. Let
$\iota_{l}\colon\left((\mathfrak{n}^{-})^{(l)}_{\mathbb{D}(H)}\right)^{*}\hookrightarrow(\mathfrak{p}_{\mathbf{m}_{l}^{l+1}}^{-})^{*}_{\mathbb{D}(H)}$
be the inclusion as subbundle. Under the identification $(\ref{eq:deform})$,
we define the map
$\mathrm{res}_{\mathfrak{l}_{\mathbf{m}_{1}}}\colon(\mathbb{O}_{H_{\mathrm{irr}}})_{\mathbb{D}(H)}\rightarrow\mathfrak{l}_{\mathbf{m}_{1}}^{*}$
by setting

$\mathrm{res}_{\mathfrak{l}_{\mathbf{m}_{1}}}(n_{l}(\mathbf{c}),X_{l}(\mathbf{c}))_{l=2,3,\ldots,k}:=\pi_{(\mathfrak{p}_{\mathbf{m}_{l}^{l+1}}^{-})^{*}\downarrow\mathfrak{l}_{\mathbf{m}_{1}}^{*}}\left(\sum_{l=2}^{k}-\mathrm{res}_{x=\infty}\left((\mathrm{Ad}^{*}(n_{l}(\mathbf{c}))(\iota_{l}(X_{l}(\mathbf{c}))))\right)\right).$

for
$(n_{l}(\mathbf{c}),X_{l}(\mathbf{c}))_{l=2,3,\ldots,k}\in\bigoplus_{l=2}^{k}\left(N_{\mathbf{m}_{l}^{l+1}}^{-}(\mathbb{D}(H))^{1}\oplus\left(\mathfrak{n}_{\mathbf{m}_{l}^{l+1}}^{-}(\mathbb{D}(H))^{1}\right)^{*}\right)$.
In the right hand side,
$\pi_{(\mathfrak{p}_{\mathbf{m}_{l}^{l+1}}^{-})^{*}\downarrow\mathfrak{l}_{\mathbf{m}_{1}}^{*}}\colon(\mathfrak{p}_{\mathbf{m}_{l}^{l+1}}^{-})^{*}\rightarrow\mathfrak{l}_{\mathbf{m}_{1}}^{*}$
denotes the natural projection. Then we obtain the following.

######  Proposition 2.11.

  1. 1.

The map $\mathrm{res}_{\mathfrak{l}_{\mathbf{m}_{1}}}$ is
$L_{\mathbf{m}_{1}}=\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}$-equivariant.

  2. 2.

The map $\mathrm{res}_{\mathfrak{l}_{\mathbf{m}_{1}}}$ is a deformation of
$\mathrm{res}_{\mathbb{O}_{H_{\mathrm{irr}}}}$. Namely, under the
identification
$\theta_{\mathbb{O}_{H_{\mathrm{irr}}}}^{-1}(0)\cong\mathbb{O}_{H_{\mathrm{irr}}}$,
we have

$\mathrm{res}_{\mathfrak{l}_{\mathbf{m}_{1}}}|_{\theta_{\mathbb{O}_{H_{\mathrm{irr}}}}^{-1}(0)}=\mathrm{res}_{\mathbb{O}_{H_{\mathrm{irr}}}}.$

###### Proof.

The first assertion follows from the definition and second one from
Proposition 1.13. ∎

Let us consider the map

$\begin{array}[]{cccc}\mu_{\mathrm{ext},\,\mathbb{D}(H)}\colon&T^{*}\mathrm{GL}_{n}(\mathbb{C})\times(\mathbb{O}_{H_{\mathrm{irr}}})_{\mathbb{D}(H)}&\rightarrow&\mathfrak{gl}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}^{*}\\\
&(\alpha,\xi)&\longmapsto&\iota_{H_{\mathrm{irr}}}^{*}\circ\mu_{\mathrm{GL}_{n}(\mathbb{C})}(\alpha)+\mathrm{res}_{\mathfrak{l}_{\mathbf{m}_{1}}}(\xi)\end{array}.$

Then we can see that this map is a
$\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}$-equivariant moment map.

######  Definition 2.12 (Local unfolding manifold).

The Poisson reduction

$(\mathbb{O}_{H})_{\mathbb{D}(H)}:=(\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}})_{H_{\mathrm{res}}}\backslash\mu_{\mathrm{ext},\,\mathbb{D}(H)}^{-1}(H_{\mathrm{res}})$

is called the local unfolding manifold of the truncated orbit
$\mathbb{O}_{H}$. We have the projection map
$\theta_{\mathbb{D}(H)}\colon(\mathbb{O}_{H})_{\mathbb{D}(H)}\rightarrow\mathbb{D}(H)$
induced from the projection
$\theta_{\mathbb{O}_{H_{\mathrm{irr}}}}\colon(\mathbb{O}_{H_{\mathrm{irr}}})_{\mathbb{D}(H)}\rightarrow\mathbb{D}(H)$.

It is easy to see that the
$\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}}$-action on
$T^{*}\mathrm{GL}_{n}(\mathbb{C})\times(\mathbb{O}_{H_{\mathrm{irr}}})_{\mathbb{D}(H)}$
is proper. Moreover a similar argument as on the symplectic reduction shows
that $H_{\mathrm{res}}$ is a regular value of the map
$\mu_{\mathrm{ext},\,\mathbb{D}(H)}$. Thus the local unfolding manifold
$(\mathbb{O}_{H})_{\mathbb{D}(H)}$ is truly a complex manifold.

####  2.3.2 Symplecitc foliation of $(\mathbb{O}_{H})_{\mathbb{D}(H)}$

Take a partition $\mathcal{I}\colon
I_{0},I_{1},\ldots,I_{r}\in\mathcal{P}_{[k+1]}$ and the corresponding
embedding
$\iota_{\mathcal{I}}\colon\mathbb{C}^{r+1}\hookrightarrow\mathbb{C}^{k+1}$ as
before. Then we can consider the deformation
$H_{\mathcal{I}}(\mathsf{c})=H(\iota_{\mathcal{I}}(\mathsf{c}))$ of $H$ on
$C(\mathcal{I})=\\{\iota_{\mathcal{I}}(\mathsf{c})\in\mathbb{C}^{k+1}\mid\mathsf{c}=(c_{0},c_{1},\ldots,c_{r})\in
C_{r+1}(\mathbb{C})\\}$. Then as we saw in Proposition 2.2, the partial
fraction decomposition algorithm gives a description of
$H_{\mathcal{I}}(\mathsf{c})$ as the sum of HLT normal forms
$H_{\mathcal{I}}(\mathsf{c})_{z_{c_{j}}}$ at $z=c_{j}$ with spectral types

$\begin{cases}(\mathbf{m}_{i_{[j,0]}}\leq_{\phi_{i_{[j,0]}}^{i_{[j,1]}}}\mathbf{m}_{i_{[j,1]}}\leq_{\phi_{i_{[j,1]}}^{i_{[j,2]}}}\cdots\leq_{\phi_{i_{[j,k_{j}-1]}}^{i_{[j,k_{j}]}}}\mathbf{m}_{i_{[j,k_{j}]}},\mathrm{triv})&\text{
for }j=1,2,\ldots,r,\\\
(\mathbf{m}_{i_{[0,0]}}\leq_{\phi_{i_{[0,0]}}^{i_{[0,1]}}}\mathbf{m}_{i_{[0,1]}}\leq_{\phi_{i_{[0,1]}}^{i_{[0,2]}}}\cdots\leq_{\phi_{i_{[0,k_{0}-1]}}^{i_{[0,k_{0}]}}}\mathbf{m}_{i_{[0,k_{0}]}},\sigma(\mathbf{m}_{0}))&\text{
for }j=0\end{cases}.$

We denote the coadjoint orbit of
$\mathrm{GL}_{n}(\mathbb{C}[z_{c_{j}}]_{k_{j}})$ through
$H_{\mathcal{I}}(\mathsf{c})_{z_{c_{j}}}$ by
$\mathbb{O}_{H_{\mathcal{I}}(\mathsf{c})_{z_{c_{j}}}}$.

As we saw in Section 2.2.2, the partial fraction decomposition of
$\widehat{\varOmega}(\mathbf{c})_{l}$ is compatible with the decomposition of
$\widehat{\mathbb{C}[z]}(\mathbf{c})_{l}$. Thus this induces the partial
fraction decomposition of the each fiber of $T^{*}N^{(l)}$ through the
trivialization $(\ref{eq:rigthtrivbundle})$, and we obtain a symplectic
holomorphic map

$\varphi_{\iota_{\mathcal{I}}(\mathsf{c})}\colon\theta_{\mathbb{D}(H)}^{-1}(\iota_{\mathcal{I}}(\mathsf{c}))\hookrightarrow\prod_{j=0}^{r}\mathbb{O}_{H_{\mathcal{I}}(\mathsf{c})_{z_{c_{j}}}}.$

Then we obtain the following.

######  Theorem 2.13 ([4]).

  1. 1.

The holomorphic family
$(\theta_{\mathbb{D}(H)}^{-1}(\mathbf{c}))_{\mathbf{c}\in\mathbb{D}(H)}$ is
the symplectic foliation of $(\mathbb{O}_{H})_{\mathbb{D}(H)}$.

  2. 2.

Let $\mathcal{I}\colon I_{0},I_{1},\ldots,I_{r}\in\mathcal{P}_{[k+1]}$ be a
partition. Let us take $\mathsf{c}=(c_{0},c_{1},\ldots,c_{r})\in
C_{r+1}(\mathbb{C})$ so that
$\iota_{\mathcal{I}}(\mathsf{c})\in\mathbb{D}(H)$. Then the partial fraction
decomposition explained in Section 2.2.2 gives the holomorphic symplectic
embedding

$\varphi_{\iota_{\mathcal{I}}(\mathsf{c})}\colon\theta_{\mathbb{D}(H)}^{-1}(\iota_{\mathcal{I}}(\mathsf{c}))\hookrightarrow\prod_{j=0}^{r}\mathbb{O}_{H_{\mathcal{I}}(\mathsf{c})_{z_{c_{j}}}}$

whose image is a Zariski open dense subset.

This shows that the symplectic foliation of the unfolding manifold
$(\mathbb{O}_{H})_{\mathbb{D}(H)}$ gives a deformation of the truncated orbit
$\mathbb{O}_{H}$ along with the deformation
$(H(\mathbf{c}))_{\mathbf{c}\in\mathbb{D}(H)}$ of the HTL normal form $H$
introduced in Definition 2.1.

Let us introduce a deformation of the residue map
$\mu_{\mathbb{O}_{H}}^{0}\colon\mathbb{O}_{H}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})^{*}$
of the truncated orbit defined in Section 1.4.3 as follows. Consider the map

$\nu_{\mathrm{GL}_{n}(\mathbb{C})}\circ\mathrm{pr}_{1}\colon
T^{*}\mathrm{GL}_{n}(\mathbb{C})\times(\mathbb{O}_{H_{\mathrm{irr}}})_{\mathbb{D}(H)}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})^{*}.$

Then as well as in Section 1.4.3, we can see that its restriction
$\nu_{\mathrm{GL}_{n}(\mathbb{C})}\circ\mathrm{pr}_{1}|_{\mu_{\mathrm{ext},\mathbb{D}(H)}^{-1}(H_{\mathrm{res}})}$
on the subspace $\mu_{\mathrm{ext},\mathbb{D}(H)}^{-1}(H_{\mathrm{res}})$ is
$(\mathrm{GL}_{n}(\mathbb{C})_{H_{\mathrm{irr}}})_{H_{\mathrm{res}}}$-invariant.
Thus there uniquely exists the map
$\nu_{(\mathbb{O}_{H})_{\mathbb{D}(H)}}\colon(\mathbb{O}_{H})_{\mathbb{D}(H)}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})^{*}$
such that the diagram

${\mu_{\mathrm{ext},\mathbb{D}(H)}^{-1}(H_{\mathrm{res}})}$${\mathfrak{gl}_{n}(\mathbb{C})^{*}}$${(\mathbb{O}_{H})_{\mathbb{D}(H)}}$$\scriptstyle{\nu_{\mathrm{GL}_{n}(\mathbb{C})}\circ\mathrm{pr}_{1}}$$\scriptstyle{\nu_{(\mathbb{O}_{H})_{\mathbb{D}(H)}}}$

is commutative. Then Proposition 1.15 assures that the map

$\mu_{(\mathbb{O}_{H})_{\mathbb{D}(H)}}^{0}:=-\nu_{(\mathbb{O}_{H})_{\mathbb{D}(H)}}\colon(\mathbb{O}_{H})_{\mathbb{D}(H)}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})^{*}$

is a deformation of the map
$\mu_{\mathbb{O}_{H}}^{0}\colon\mathbb{O}_{H}\rightarrow\mathfrak{gl}_{n}(\mathbb{C})^{*}$,
i.e., we have the equation

$\mu_{(\mathbb{O}_{H})_{\mathbb{D}(H)}}^{0}|_{\theta_{\mathbb{D}(H)}^{-1}(0)}=\mu_{\mathbb{O}_{H}}^{0}$

under the isomorphism $\theta_{\mathbb{D}(H)}^{-1}(0)\cong\mathbb{O}_{H}$.

###  2.4 Unfolding manifold of meromorphic connections

Now we are ready to define the unfolding manifold which is a Poisson manifold
descending the unfolding of the unramified irregular singularities of
connections in the moduli space $\mathcal{M}_{s}^{*}(\mathbf{H})$.

####  2.4.1 Stratification of the product of local unfolding manifolds

Let $\mathbf{H}=(H_{a})_{a\in|D|}$ be a collection of unramified HTL normal
forms,

$H_{a}=\left(\frac{S_{k_{a}}^{(a)}}{z_{a}^{k_{a}-1}}+\cdots+\frac{S^{(a)}_{1}}{z_{a}}+S^{(a)}_{0}+N^{(a)}_{0}\right)\frac{dz_{a}}{z_{a}}$

with the spectral types
$\mathrm{sp}(H_{a}):=(\mathbf{m}_{0}^{[a]}\leq_{\phi_{0}^{[a]}}\leq\mathbf{m}^{[a]}_{1}\leq_{\phi_{1}^{[a]}}\cdots\leq_{\phi_{k-1}^{[a]}}\mathbf{m}_{k_{a}}^{[a]},\sigma(\mathbf{m}_{0}^{[a]}))$.
We consider the moduli space
$\mathcal{M}_{s}^{*}(\mathbf{H})=\mathrm{GL}_{n}(\mathbb{C})\backslash(\mu_{\mathbf{H}}^{s})^{-1}(0)$
of meromorphic connections with respect to $\mathbf{H}$. Then we define a
deformation of $\mathcal{M}_{s}^{*}(\mathbf{H})$ as follows. Let us consider
the product of the local unfolding manifolds

$\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}$

with the natural projection
$\bar{\theta}_{\prod_{a\in|D|}\mathbb{D}(H_{a})}\colon\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}\rightarrow\prod_{a\in|D|}\mathbb{D}(H_{a})$.

As well as in Section 2.1.2, we can consider a stratification

$\prod_{a\in|D|}\mathbb{C}^{k_{a}+1}=\bigsqcup_{(\mathcal{I}_{a})_{a\in|D|}\in\prod_{a\in|D|}\mathcal{P}_{k_{a}+1}}\left(\prod_{a\in|D|}C(\mathcal{I}_{a})\right).$

of $\prod_{a\in|D|}\mathbb{C}^{k_{a}+1}$ associated to collections of
partitions
$(\mathcal{I}_{a})_{a\in|D|}\in\prod_{a\in|D|}\mathcal{P}_{k_{a}+1}$. This
induces the following stratification of
$\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}$,

$\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}=\bigsqcup_{(\mathcal{I}_{a})_{a\in|D|}\in\prod_{a\in|D|}\mathcal{P}_{k_{a}+1}}\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}^{\mathcal{I}_{a}},$

where

$(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}^{\mathcal{I}_{a}}:=\bigcup_{\mathbf{c}_{a}\in\mathbb{D}(H)\cap
C(\mathcal{I}_{a})}\theta_{\mathbb{D}(H_{a})}^{-1}(\mathbf{c}_{a}).$

Let us look at a stratum
$\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}^{\mathcal{I}_{a}}$.
Then Theorem 2.13 says that each fiber
$\prod_{a\in|D|}\theta_{\mathbb{D}(H_{a})}^{-1}(\iota_{\mathcal{I}_{a}}(\mathsf{c}_{a}))$
at $(\mathsf{c}_{a})\in\prod_{a\in|D|}\mathbb{C}^{r_{\mathcal{I}_{a}}+1}$ is
isomorphic to a Zariski open subset of
$\prod_{a\in|D|}\prod_{j=0}^{r_{\mathcal{I}_{a}}}\mathbb{O}_{H_{\mathcal{I}_{a}}(\mathsf{c}_{a})_{z_{c_{j}}}}$.

By these isomorphisms, we can define an open subspace
$\left(\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}^{\mathcal{I}_{a}}\right)^{s}$
of the stratum as the union of the inverse images of
$\left(\prod_{a\in|D|}\prod_{j=0}^{r_{\mathcal{I}_{a}}}\mathbb{O}_{H_{\mathcal{I}_{a}}(\mathsf{c}_{a})_{z_{c_{j}}}}\right)^{s}$.
Further we define

$\left(\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}\right)^{s}:=\bigsqcup_{(\mathcal{I}_{a})_{a\in|D|}\in\prod_{a\in|D|}\mathcal{P}_{k_{a}+1}}\left(\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}^{\mathcal{I}_{a}}\right)^{s}.$

####  2.4.2 Unfolding manifold of meromorphic connections

Let us consider the $\mathrm{GL}_{n}(\mathbb{C})$-equivariant map

$\begin{array}[]{cccc}\mu_{\mathbf{H},\prod_{a\in|D|}\mathbb{D}(H_{a})}^{s}\colon&\left(\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}\right)^{s}&\longrightarrow&\mathfrak{gl}_{n}(\mathbb{C})^{*}\\\
&(X_{a})_{a\in|D|}&\longmapsto&\sum_{a\in|D|}\mu^{0}_{(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}}(X_{a})\end{array}.$

######  Definition 2.14 (Unfolding manifold).

The Poisson quotient space

$\mathcal{M}_{s}^{*}(\mathbf{H})_{\prod_{a\in|D|}\mathbb{D}(H_{a})}:=\mathrm{GL}_{n}(\mathbb{C})\backslash(\mu_{\mathbf{H},\prod_{a\in|D|}\mathbb{D}(H_{a})}^{s})^{-1}(0)$

is called the unfolding manifold of meromorphic connections on the rank $n$
trivial bundle on $\mathbb{P}^{1}$ with respect to the unramifed HTL normal
forms $\mathbf{H}$.

Since the projection map
$\bar{\theta}_{\prod_{a\in|D|}\mathbb{D}(H_{a})}\colon\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}\rightarrow\prod_{a\in|D|}\mathbb{D}(H_{a})$
is $\mathrm{GL}_{n}(\mathbb{C})$-invariant, this map naturally induces the
projection map

$\theta_{\prod_{a\in|D|}\mathbb{D}(H_{a})}\colon\mathcal{M}_{s}^{*}(\mathbf{H})_{\prod_{a\in|D|}\mathbb{D}(H_{a})}\rightarrow\prod_{a\in|D|}\mathbb{D}(H_{a}).$

Then we can show that the closed submanifold
$(\mu_{\mathbf{H},\prod_{a\in|D|}\mathbb{D}(H_{a})}^{s})^{-1}(0)$ of the
Poisson manifold
$\left(\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}\right)^{s}$ has
a clean intersection with both the symplectic leaves
$\bar{\theta}_{\prod_{a\in|D|}\mathbb{D}(H_{a})}^{-1}(\mathbf{c})$ of
$\left(\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}\right)^{s}$ and
the orbits of $\mathrm{GL}_{n}(\mathbb{C})$. Therefore we obtain the following
our first main theorem.

######  Theorem 2.15 ([4]).

Suppose that the moduli space $\mathcal{M}_{s}^{*}(\mathbf{H})$ is not the
empty set. Then the unfolding manifold
$\mathcal{M}_{s}^{*}(\mathbf{H})_{\prod_{a\in|D|}\mathbb{D}(H_{a})}$ is a
Poisson manifold with the symplectic foliation
$(\theta_{\prod_{a\in|D|}\mathbb{D}(H_{a})}^{-1}(\mathbf{c}))_{\mathbf{c}\in\prod_{a\in|D|}\mathbb{D}(H_{a})}.$

Let us define a subspace

$\displaystyle\mathcal{M}_{s}^{*}(\mathbf{H})_{\prod_{a\in|D|}\mathbb{D}(H_{a})}^{(\mathcal{I}_{a})_{a\in|D|}}$
$\displaystyle:=\prod_{a\in|D|}\bigcup_{\mathbf{c}_{a}\in\mathbb{D}(H)\cap
C(\mathcal{I}_{a})}\theta_{\mathbb{D}(H_{a})}^{-1}(\mathbf{c}_{a})$
$\displaystyle=\mathcal{M}_{s}^{*}(\mathbf{H})_{\prod_{a\in|D|}\mathbb{D}(H_{a})}\cap\prod_{a\in|D|}(\mathbb{O}_{H_{a}})_{\mathbb{D}(H_{a})}^{\mathcal{I}_{a}}$

for each $(\mathcal{I}_{a})_{a\in|D|}\in\prod_{a\in|D|}\mathcal{P}_{k_{a}+1}$.
Then we obtain a stratification

$\mathcal{M}_{s}^{*}(\mathbf{H})_{\prod_{a\in|D|}\mathbb{D}(H_{a})}=\bigsqcup_{(\mathcal{I}_{a})_{a\in|D|}\in\prod_{a\in|D|}\mathcal{P}_{k_{a}+1}}\mathcal{M}_{s}^{*}(\mathbf{H})_{\prod_{a\in|D|}\mathbb{D}(H_{a})}^{(\mathcal{I}_{a})_{a\in|D|}}.$

Let us fix a collection $(\mathcal{I}_{a})_{a\in|D|}$ of partitions and
consider the corresponding stratum
$\mathcal{M}_{s}^{*}(\mathbf{H})_{\prod_{a\in|D|}\mathbb{D}(H_{a})}^{(\mathcal{I}_{a})_{a\in|D|}}$.
Then the fibers $\theta_{\prod_{a\in|D|}\mathbb{D}(H_{a})}^{-1}(\mathbf{c})$
contained in this stratum are of the forms
$\theta_{\prod_{a\in|D|}\mathbb{D}(H_{a})}^{-1}((\iota_{\mathcal{I}_{a}}(\mathsf{c}_{a}))_{a\in|D|}))$
with some
$(\mathsf{c}_{a})_{a\in|D|}\in\prod_{a\in|D|}\mathbb{C}^{r_{\mathcal{I}_{a}}+1}$
We associate the collection of HTL normal forms

$\mathbf{H}((\mathsf{c}_{a})_{a\in|D|})^{(\mathcal{I}_{a})_{a\in|D|}}:=(H_{\mathcal{I}_{a}}(\mathsf{c}_{a})_{z_{c^{[a]}_{j}}})_{a\in|D|,\,0\leq
j\leq r_{\mathcal{I}_{a}}}$

to the fiber
$\theta_{\prod_{a\in|D|}\mathbb{D}(H_{a})}^{-1}((\iota_{\mathcal{I}_{a}}(\mathsf{c}_{a}))_{a\in|D|}))$.
Then as a corollary of Theorem 2.13, we obtain the second main theorem which
says that each symplectic leaf
$\theta_{\prod_{a\in|D|}\mathbb{D}(H_{a})}^{-1}(\mathbf{c})$ of the unfolding
manifold is embedded into a Zariski open subset of a moduli space of
connections, namely, the symplectic leaves describe a deformation of the
moduli space of connections and moreover this deformation corresponds to the
unfolding of the irregular singularities of the associated HTL normal forms to
the moduli space.

######  Theorem 2.16 ([4]).

In a stratum
$\mathcal{M}_{s}^{*}(\mathbf{H})_{\prod_{a\in|D|}\mathbb{D}(H_{a})}^{(\mathcal{I}_{a})_{a\in|D|}}$
of the unfolding manifold associated to $(\mathcal{I}_{a})_{a\in|D|}$, each
fiber
$\theta_{\prod_{a\in|D|}\mathbb{D}(H_{a})}^{-1}((\iota_{\mathcal{I}_{a}}(\mathsf{c}_{a}))_{a\in|D|}))$
is isomorphic to a Zariski open subspace of the moduli space
$\mathcal{M}_{s}^{*}(\mathbf{H}((\mathsf{c}_{a})_{a\in|D|})^{(\mathcal{I}_{a})_{a\in|D|}})$
of meromorphic connections on $\mathbb{P}^{1}$.

####  2.4.3 Example: unfolding manifold of unramified Painlevé phase spaces

Let us consider the HTL normal form

$H_{I\\!I}:=\left(\frac{\begin{pmatrix}a_{3}&\\\
&b_{3}\end{pmatrix}}{z^{3}}+\frac{\begin{pmatrix}a_{2}&\\\
&b_{2}\end{pmatrix}}{z^{2}}+\frac{\begin{pmatrix}a_{1}&\\\
&b_{1}\end{pmatrix}}{z^{1}}+\begin{pmatrix}a_{0}&\\\
&b_{0}\end{pmatrix}\right)\frac{dz}{z}$

Then it is known that $\mathcal{M}_{s}^{*}(H_{I\\!I})$ is the phase space of
the Hamiltonian system of the Painlevé $I\\!I$ equation. The unfolding of
$H_{I\\!I}$ is

$\displaystyle H_{I\\!I}(c_{0},c_{1},c_{2},c_{3})=$
$\displaystyle\left(\frac{\begin{pmatrix}a_{3}&\\\
&b_{3}\end{pmatrix}}{(z-c_{1})(z-c_{2})(z-c_{3})}+\frac{\begin{pmatrix}a_{2}&\\\
&b_{2}\end{pmatrix}}{(z-c_{1})(z-c_{2})}+\frac{\begin{pmatrix}a_{1}&\\\
&b_{1}\end{pmatrix}}{(z-c_{1})}+\begin{pmatrix}a_{0}&\\\
&b_{0}\end{pmatrix}\right)\frac{dz}{z-c_{0}}.$

The unfolding parameter space $\mathbb{C}^{4}$ has the following $5$ types of
strata

$\displaystyle C_{I\\!I}$
$\displaystyle:=\\{(c_{i})_{i=0,\ldots,3}\in\mathbb{C}^{4}\mid
c_{0}=c_{1}=c_{2}=c_{3}\\},$ $\displaystyle C_{I\\!I\\!I}^{(i,j|k,l)}$
$\displaystyle:=\\{(c_{i})_{i=0,\ldots,3}\in\mathbb{C}^{4}\mid
c_{i}=c_{j},\,c_{k}=c_{l},\,c_{i}\neq c_{k}\\},$ $\displaystyle
C_{I\\!V}^{(i|j,k,l)}$
$\displaystyle:=\\{(c_{i})_{i=0,\ldots,3}\in\mathbb{C}^{4}\mid
c_{j}=c_{k}=c_{l},\,c_{i}\neq c_{j}\\},$ $\displaystyle C_{V}^{(i|j|k,l)}$
$\displaystyle:=\\{(c_{i})_{i=0,\ldots,3}\in\mathbb{C}^{4}\mid
c_{k}=c_{l},\,c_{i}\neq c_{j},\,c_{i}\neq c_{k},\,c_{j}\neq c_{k}\\},$
$\displaystyle C_{V\\!I}^{(i|j|k|l)}$
$\displaystyle:=\\{(c_{i})_{i=0,\ldots,3}\in\mathbb{C}^{4}\mid c_{s}\neq
c_{t},\,s\neq t\\},$

where $\\{i,j,k,l\\}=\\{0,1,2,3\\}$. The diagram below is the Hasse diagram of
the closure relation of the strata
$C_{I\\!I},C_{I\\!I\\!I}^{(0,1|2,3)},C_{I\\!V}^{(0|1,2,3)},C_{V}^{(0|1|2,3)},C_{V\\!I}^{(0|1|2|3)}$.

$C_{V\\!I}^{(0|1|2|3)}$$C_{V}^{(0|1|2,3)}$$C_{I\\!V}^{(0|1,2,3)}$$C_{I\\!I\\!I}^{(0,1|2,3)}$$C_{I\\!I}$

If $\mathbf{c}=(c_{0},\ldots,c_{3})\in
C_{I\\!I\\!I}^{(i,j|k,l)}\cap\mathbb{D}(H_{I\\!I})$ for example, there exist
holomorphic functions $a^{(\mu)}_{\nu}(\mathbf{c})b^{(\mu)}_{\nu}(\mathbf{c})$
of $\mathbf{c}\in C_{I\\!I\\!I}^{(i,j|k,l)}\cap\mathbb{D}(H_{I\\!I})$ such
that we can write
$H_{I\\!I}(c_{0},c_{1},c_{2},c_{3})=H^{(i,j|k,l)}_{I\\!I\\!I,\,c_{i}}+H^{(i,j|k,l)}_{I\\!I\\!I,\,c_{k}}$
by

$\displaystyle H^{(i,j|k,l)}_{I\\!I\\!I,\,c_{i}}$
$\displaystyle:=\left(\frac{\begin{pmatrix}a^{(i)}_{1}(\mathbf{c})&\\\
&b^{(i)}_{1}(\mathbf{c})\end{pmatrix}}{(z-c_{i})}+\begin{pmatrix}a^{(i)}_{0}(\mathbf{c})&\\\
&b^{(i)}_{0}(\mathbf{c})\end{pmatrix}\right)\frac{d}{z-c_{i}},$ $\displaystyle
H^{(i,j|k,l)}_{I\\!I\\!I,\,c_{k}}$
$\displaystyle:=\left(\frac{\begin{pmatrix}a^{(k)}_{1}(\mathbf{c})&\\\
&b^{(k)}_{1}(\mathbf{c})\end{pmatrix}}{(z-c_{k})}+\begin{pmatrix}a^{(k)}_{0}(\mathbf{c})&\\\
&b^{(k)}_{0}(\mathbf{c})\end{pmatrix}\right)\frac{d}{z-c_{k}}.$

The condition $\mathbf{c}\in
C_{I\\!I\\!I}^{(i,j|k,l)}\cap\mathbb{D}(H_{I\\!I})$ moreover assures that the
moduli space $\mathcal{M}_{s}^{*}(\mathbf{H}_{I\\!I\\!I})$ associated to the
collection
$\mathbf{H}_{I\\!I\\!I}:=(H^{(i,j|k,l)}_{I\\!I\\!I,\,c_{i}},H^{(i,j|k,l)}_{I\\!I\\!I,\,c_{k}})$
is isomorphic to the phase space of Painlevé $I\\!I\\!I$ equation. In this
case, our main theorem says the following.

######  Proposition 2.17.

For each $\mathbf{c}\in C_{I\\!I\\!I}^{(i,j|k,l)}\cap\mathbb{D}(H_{I\\!I})$,
the fiber $\theta_{\mathbb{D}(H_{I\\!I})}^{-1}(\mathbf{c})$ of the unfolding
manifold
$\theta_{\mathbb{D}(H_{I\\!I})}\colon\mathcal{M}_{s}^{*}(H_{I\\!I})_{\mathbb{D}(H_{I\\!I})}\rightarrow\mathbb{D}(H_{I\\!I})$
is isomorphic to a Zariski open subset of
$\mathcal{M}_{s}^{*}(\mathbf{H}_{I\\!I\\!I})$, the phase space of Painlevé
$I\\!I\\!I$ equation.

The same statement holds for each above 5 strata, namely, we have the
following.

######  Theorem 2.18.

The symplectic foliation of
$\mathcal{M}_{s}^{*}(H_{I\\!I})_{\mathbb{D}(H_{I\\!I})}$ consists of Zariski