# On finite generation of fiber ring of invariant jet differentials

Mohammad Reza-Rahmati, Gerardo Flores<EMAIL_ADDRESS>(M. Reza Rahmati),
<EMAIL_ADDRESS>(Gerardo Flores)

###### Abstract.

We prove that the fiber ring of the space of invariant jet differentials of a
projective manifold is finitely generated on the regular locus. Berczi-Kirwan
has partially worked out the question in [2]; however, our method is different
and complementary. The analytic automorphism group of regular $k$-jets of
holomorphic curves on a projective variety $X$ is a non-reductive subgroup of
the general linear group $GL_{k}\mathbb{C}$. In this case, the Chevalley
theorem on the invariant polynomials in the fiber rings fails in general.
Thus, the analysis of Cartan subalgebras of the Lie algebra and its Weyl group
requires different methods. We employ some techniques of algebraic Lie groups
(not necessarily reductive) together with basic results obtained in [2] to
prove the finite generation of the stalk ring at a regular point.

###### Key words and phrases:

Invariant Jet Differentials, Generic Stabilizer, Demailly-Semple bundle, Non-
reductive Invariant Theory, Adjoint Representation, Chevalley Theorem

###### 1991 Mathematics Subject Classification:

14L30, 17B20, 22E46

##  1\. Introduction

In this text, we initiate a general method to study the (non-reductive)
invariant theory of the transformation group of $k$-jets of entire holomorphic
curves on a projective variety $X$, under the reparametrizations of the domain
$(\mathbb{C},0)$. Our method uses the theory of generic stabilizers of Lie
group actions on manifolds. Specifically, we prove several structure theorems
for the Lie group $G_{k}$. We study the adjoint action of $G_{k}$ on its Lie
algebra $\mathfrak{g}_{k}$, and show that the adjoint action has generic
stabilizers. This implies that the Weyl group of the action is finite. We also
apply the same procedure to the action of $G_{k}$ on the
$J_{k}\mathbb{C}^{n}$. Our first result is a consequence of checking Elashvili
conditions for this action to prove the existence of generic stabilizers. We
use a result of Igusa [see Theorem 2.15] and also a general stratification
theorem for quotient spaces to prove the finite generation of the ring of
invariant $k$-jets.

The moduli of $k$-jets of holomorphic curves (denoted by $J_{k}X$) on a
projective variety $X$ (maybe singular) is one of the interesting geometric
construction that reflects some of the topological properties of the variety.
As a classifying space, it may be embedded in a Grassmann variety and
consequently in a projective space. Its compactification and boundary points
have been one of the subjects of study for mathematicians in the last decades.
For geometric reasons, one may look for a coordinate-free definition of the
$k$-jets on a projective variety. That is, one considers moduli of invariant
$k$-jets of complex curves independent to the parametrizing variable in
$\mathbb{C}$. The last moduli is a quotient space of the ordinary moduli of
$k$-jets by a non-reductive subgroup of the Lie group $GL_{k}\mathbb{C}$. The
ring of regular functions on the space of invariant $k$-jets and the stalk of
this ring at a regular jet is meaningful to consider. We want to prove that
the last stalk is a finitely generated polynomial ring. Because a non-
reductive group gives the quotient, the Chevalley type theorems fail to be
true. Thus, the proof of the finite generation of the local ring of invariant
polynomial functions at a regular point is more complicated. We employ some
facts from the general theory of algebraic groups and their actions on
manifolds to show this property.

###  1.1. Motivation

Since the work of Green and Griffiths [16], developed by Demailly [11],
Diverio, Merker, and Rousseau in [14], the action of the local automorphism
group $G_{k}$ of $(\mathbb{C},0)$ on the bundle $J_{k}TX$ of $k$-jets of
holomorphic curves $f:\mathbb{C}\to X$ in a complex projective manifold $X$
has been a focus of the investigation. The transformation group of $k$-jets
under reparametrization of the curve, is a non-reductive subgroup
$G_{k}=\mathbb{C}^{*}\ltimes U_{k}$ of $GL_{k}(\mathbb{C})$, where $U_{k}$ is
the unipotent radical consisting of upper-triangular $k\times k$ matrices of a
certain type. Unlike the reductive case, here, one can not deduce the
Chevalley kind of theorems. Besides, It is still unknown if the ring of
regular invariant functions under the action of $G_{k}$ or its unipotent part
is finitely generated. An attempt toward this conjecture has been made in [2].

###  1.2. The problem and contribution

Assume $X$ is a projective manifold. The Lie group $G_{k}$ of
reparametrizations of a holomorphic curve $f:\mathbb{C}\to X$ is the group of
matrices of the form,

(1) $G_{k}:=\left[\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&...&a_{k}\\\
0&a_{1}^{2}&2a_{1}a_{2}&...&a_{1}a_{k-1}+...+a_{k-1}a_{1}\\\
0&0&a_{1}^{3}&...&3a_{1}^{2}a_{k-2}+...\\\ .&.&.&...&.\\\
0&0&0&...&a_{1}^{k},\end{array}\right]$

where $0\neq a_{1}\in\mathbb{C}^{*}$ and all others $a_{i}\in\mathbb{C}$. The
formula for the entries of the matrix is

(2) $[G_{k}]_{ij}=\sum_{s_{1}+s_{2}+...+s_{i}=j}a_{s_{1}}...a_{s_{i}}$

where $s_{i}\geq 1$. The Lie group $G_{k}$ is a subgroup of $GL_{k}\mathbb{C}$
and can be written as $G_{k}=\ \mathbb{C}^{*}\ltimes U_{k}$, where $U_{k}$ is
the unipotent radical and sits inside $SL_{k}$. The group appears as the
transformation group of the vector Taylor series,

(3)
$f(t)=tf^{\prime}(0)+\frac{t^{2}}{2!}f^{\prime\prime}(0)+...+\frac{t^{k}}{k!}f^{(k)}(0)+O(t^{k+1})$

under the change of parameterization group of $(\mathbb{C},0)$,

(4) $t\longmapsto\phi(t)=a_{1}t+a_{2}t^{2}+...+a_{k}t^{k},\qquad
a_{1}\in\mathbb{C}^{*}.$

By replacing (4) in (3) we have:

(5)
$f(t)=(a_{1}t+a_{2}t^{2}+...+a_{k}t^{k})f^{\prime}(0)+\frac{(a_{1}t+a_{2}t^{2}+...+a_{k}t^{k})^{2}}{2!}f^{\prime\prime}(0)\\\
+...+\frac{(a_{1}t+a_{2}t^{2}+...+a_{k}t^{k})^{k}}{k!}f^{(k)}(0),\qquad\mod
t^{k+1}.$

Regrouping the terms shows that the following matrix multiplication defines
the action of $G_{k}$ on the $k$-jets,

(6)
$[f^{\prime}(0),f^{\prime\prime}(0)/2!,...,f^{(k)}(0)/k!].\left[\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&...&a_{k}\\\
0&a_{1}^{2}&2a_{1}a_{2}&...&a_{1}a_{k-1}+...a_{k-1}a_{1}\\\
0&0&a_{1}^{3}&...&3a_{1}^{2}a_{k-2}+...\\\ .&.&.&...&.\\\
0&0&0&...&a_{1}^{k}\end{array}\right].$

The unipotent elements correspond to the matrices with $a_{1}=1$. The
exponents along the diagonal elements arise for weights under the
reparametrization $\phi(t):t\mapsto\lambda t$ in the Taylor series of $f$. The
action of $G_{k}$ on the generic vectors
$f^{\prime}(0),f^{\prime\prime}(0)/2!,...,f^{(k)}(0)/k!$ induces an action of
$G_{k}$ on the polynomial ring
$\mathbb{C}[f^{\prime}(0),f^{\prime\prime}(0),...,f^{(k)}(0)]^{G_{k}}$, where
we have considered $f^{(k)}(0)$ as a germ of variable. In this text, we attend
to prove the following open question.

######  Conjecture 1.1.

The ring of invariant polynomials
$\mathcal{O}(J_{k}\mathbb{C}^{n})^{G_{k}}=\mathbb{C}[f^{\prime}(0),f^{\prime\prime}(0),...,f^{(k)}(0)]^{G_{k}}$
is finitely generated.

The problem to determine the finite generation of complex, graded algebras
$\mathcal{O}(J_{k})^{G_{k}}$ has been a long-time open problem in this area.
The above ring appears as the local ring of invariant sections of
$J_{k}(X)=J_{k}T_{X}$ at a generic point $x\in X$, where $J_{k}(X)$ is the
bundle of $k$-jets on the tangent bundle of $X$, defined as the bundle of
germs of holomorphic curves $f:\mathbb{C}\to X$ up to equivalence of their
Taylor series $\text{mod}\ t^{k+1}$. The projectivized bundle
$J_{k}(X)/\mathbb{C}^{*}$ is called the Green-Griffiths (GG)-bundle. An
alternative way to define the (GG)-bundles is through their ring of sections.
One can consider a ring of weighted homogeneous polynomials as,

(7) $P(z,\xi)=\sum_{a}A(z)\xi^{a}$

in the variables $\xi=(\xi_{1},...,\xi_{k})$ with weights $1,2,...,k$ and,
$a=(a_{1},...,a_{k})$ respectively, denoted by

(8) $E_{k}^{GG}=\bigoplus_{m}E_{k,m}^{GG},$

where $m$ stands for the weighted degree. One may also work with a holomorphic
subbundle $V$ of the tangent bundle $TX$. We can consider the group of all
analytic reparametrization automorphisms of $J_{k}V$ and the invariant bundle
$(J_{k}V)^{G_{k}}$. We denote by $E_{k,m}V^{*}$ the Demailly-Semple bundle
whose fiber at $x$ consists of $U_{k}$-invariant polynomials on the fiber
coordinates of $J_{k}V$ at $x$ of weighted degree $m$. Also, set

(9) $E_{k}=\bigoplus_{m}E_{k,m}$

to be the Demailly-Semple bundle of graded algebras of invariant jets. The
invariant jets play an essential role in the strategy introduced by Green-
Griffiths [16], and Demailly [11] to prove the Kobayashi hyperbolicity
conjecture [21].

We initiate a new method to study the non-reductive invariant theory of the
Lie group $G_{k}$. We stress several results on the structure theory of
$G_{k}$. Our method employs the theory of generic stabilizers [Pa.PV] in the
action of Lie groups on algebraic varieties. In particular, we prove that the
adjoint action of $G_{k}$ on its Lie algebra and also the action $G_{k}$ on
$J_{k}\mathbb{C}^{n}$ have generic stabilizers. We employ a fundamental
theorem of Igusa [see [24, 30]] to answer the Conjecture 1.1, resolved in
Theorem 4.8 in the paper.

###  1.3. Content

The remainder of this paper is organized as follows. In Section 2 we present
some fundamental aspects of the invariant theory of algebraic groups (not
necessarily reductive Lie groups) and their Lie algebras. Section 3 contains
basic definitions on the jet bundles on projective manifolds and the Lie group
of transformations of $k$-jets of holomorphic curves under reparametrizations
of their domains, namely $G_{k}$. Section 4 consists of the main results of
the paper on the (non-reductive) invariant theory of the Lie group $G_{k}$;
all the material of this section is the paper’s new results and contributions.
Moreover, a proof of the well-known conjecture on finite generation of the
total fiber ring of invariant $k$-jets of entire curves on a projective
manifold is given in such a section. Finally, some conclusions are presented
in Section 5.

##  2\. Preliminaries on non-reductive Lie groups and algebras

This section explains some general facts on the invariant theory of algebraic
groups. We do not stress that the Lie group $G$ be reductive. Where ever this
assumption is necessary, we have specifically mentioned. The primary purpose
of this section is to provide some preliminary materials that are used in
Section 4.

###  2.1. Invariant theory of non-reductive Lie groups

Let $G$ be an algebraic group that acts on an affine variety $X$. Denote by
$\mathbb{C}[X]^{G}$ the algebra of $G$-invariant regular functions on $X$. If
$\mathbb{C}[X]^{G}$ is finitely generated, then we set

(10) $X//G:=\text{Spec}(\mathbb{C}[X]^{G}),$

and we have the quotient morphism

(11) $\pi:X\longrightarrow X//G$

which corresponds to the inclusion
$\mathbb{C}[X]^{G}\longhookrightarrow\mathbb{C}[X]$. Now, denote by $G_{x}$
the stabilizer of the point $x\in X$. Set $\mathfrak{g}=Lie(G)$ and

(12) $\mathfrak{g}_{x}=\text{Lie}(G_{x})\subset\mathfrak{g}.$

Assume $G$ is an affine algebraic group acting on $X$, then by a theorem of
Rosenlicht (see Theorem 2.1 below [24, 33, 34]), there is a dense $G$-stable
subset $U\subset X$ such that the functions in $\mathbb{C}[X]^{G}$ separate
the $G$-orbits in $U$. In this case, one has

(13) $\text{trdegree}(\mathbb{C}(X)^{G})=\dim X-\text{Max}_{x}\dim(G.x)$

We formalize this in the following theorem.

######  Theorem 2.1.

(M. Rosenlicht) [[34, 33], [24] theorem 1.1] Let $K\subset\mathbb{C}(X)^{G}$
be a subfield. Then, $K=\mathbb{C}(X)^{G}$ if $K$ separates the $G$-orbits in
a dense open subset of $X$.

The Theorem 2.1 motivates the following definition, which plays a crucial role
in this text.

######  Definition 2.2.

If in an open dense subset $U$ of $X$, all the stabilizers $\mathfrak{g}_{x}$
are $G$-conjugate, we say the action of $G$ on $X$ has generic stabilizer and
call the points in $U$ to be generic. Similarly, one defines by $G_{x}$ the
generic isotropy subgroups.

The existence of a generic isotropy group implies the existence of a generic
stabilizer. The converse is also true [31, 32, 30]. Let’s define the set of
regular points by,

(14) $X^{\text{reg}}=\\{\ x\in X\ |\ \dim G.x=\text{Max}_{z}\dim G.z\ \\}.$

The definition (14) depends on the algebraic group $G$. Sometimes we denote
this by $G$-regular points. If $G$ is reductive and $X$ is a smooth variety,
the generic stabilizers always exist. We list some results on the existence of
generic stabilizers, in case the algebraic group $G$ is reductive, [see [43]
section 7],

  * •

If $G$ is reductive and $X$ is an irreducible affine variety on which $G$
acts, then the generic stabilizers exist, [Richardson-Luna [31, 32]].

  * •

If $G$ is reductive, $X$ is affine and the action is stable, then the generic
stabilizers exist.

  * •

If $G$ is reductive and $X$ is irreducible smooth projective variety, and
$X^{G}\neq\emptyset$, then the generic stabilizers exist.

  * •

If $G$ is reductive and $X$ irreducible smooth projective variety with
$\text{Pic}(X)=\mathbb{Z}$, then the generic stabilizers exist.

  * •

If $G$ is connected and $X$ is arbitrary irreducible variety, then there are
generic stabilizers of the action of a Borel and a unipotent subgroup of $G$,
[Brion-Luna-Vust [6, 44]].

We wish to consider the generic stabilizer property in a non-reductive case.
In general, we have the following theorem due to Richardson.

######  Theorem 2.3.

(R. W. Richardson) [31] Let $G$ be an algebraic group acting on an irreducible
algebraic variety $X$. For $x\in X$, let $U_{x},L_{x}$ be the unipotent
radical and Levi subgroup of $G_{x}$. There are non-singular subvarieties
$X_{1},...,X_{n}$ of $X$ such that

  * •

$X=\bigcup_{j}X_{j}$.

  * •

$X_{j}$ is open in $X\setminus\bigcup_{l=1}^{j-1}X_{l}$.

  * •

For any $j$ and any $x,y\in X_{j}$ the groups $L_{x},L_{y}$ are conjugate in
$G$.

  * •

The projection of the set $\\{(x,u)|x\in X_{j},u\in U_{x})\\}\subset
X_{j}\times G$ on $X_{j}$ is smooth for any $j$ (has surjective differential).

The theorem states that for a general action of an algebraic group $G$ on an
algebraic variety $X$, we have a stratification of the variety into disjoint
smooth subvarieties $X_{j}$, where on each $X_{j}$ the restriction of quotient
mapping has generic stabilizers. Actually, with only these assumptions, one
still can not deduce if $\mathbb{C}[X]^{G}$ is finitely generated. However, if
each $X_{j}$ may satisfy a codimension two, condition (C.2.C), one can proceed
more for this result.

Assume $G$ is a complex algebraic group (not necessarily reductive) and,

(15) $\rho:G\longrightarrow GL(V)$

is a finite dimensional representation of $G$, and
$\varrho:\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$ is the corresponding
representation of $\mathfrak{g}$. If $v\in V$ let define

(16) $V^{\mathfrak{g}_{v}}=\\{\ x\in V\ |\ \mathfrak{g}_{v}.x=0\ \\}.$

If $\Omega\subset V$, then set

(17) $\displaystyle N(\Omega)$ $\displaystyle=\\{g\in G\ |\
g.\Omega\subset\Omega\\}$ $\displaystyle Z(\Omega)$ $\displaystyle=\\{g\in G\
|\ g.x=x,\ \forall x\in\Omega\\}.$

######  Theorem 2.4 ([24] Lemma 1.2, Lemma 1.3, prop. 1.4).

Assume $\rho:G\longrightarrow GL(V)$ is a representation and
$\Omega=V^{\mathfrak{g}_{x}}$. We have the following:

  * •

$\text{Lie}\ Z(\Omega)=\mathfrak{g}_{x}$.

  * •

$N(\Omega)=N_{G}(Z(\Omega))=N_{G}(\mathfrak{g}_{x})$.

  * •

$Z(\Omega)\unlhd N(\Omega)$.

  * •

If $x\in\Omega^{\text{reg}}$, then $G.x\bigcap\Omega=N(\Omega).x$

  * •

$\mathfrak{g}.x\bigcap\Omega=\mathfrak{n}_{\mathfrak{g}}(\mathfrak{g}_{v}).x$,
holds for $x\in\Omega^{\text{reg}}$.

  * •

$Y=\overline{G.\Omega}$ is a $G$-stable irreducible subvariety of $V$.

  * •

The restriction map yields an isomorphism

(18)
$\mathbb{C}(Y)^{G}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathbb{C}(\Omega)^{N(\Omega)/Z(\Omega)}.$

The induced map on algebras gives an embedding,

(19)
$\mathbb{C}[Y]^{G}\longhookrightarrow\mathbb{C}[\Omega]^{N(\Omega)/Z(\Omega)}.$

which in general is not onto.

The group $W=N(\Omega)/Z(\Omega)$ is called the Weyl group of the action of
$G$ on $V$. It can be an infinite group. We want to characterize the property
of having a generic stabilizer for the action of $G$ on a vector space $V$ and
propose to state Chevalley-type theorems.

######  Proposition 2.5 ((Elashvili) [15], [24] lemma 1.6, see also [43] sec.
7).

If $v\in V$, then $G.V^{\mathfrak{g}_{v}}$ is dense in $V$ iff

(20) $V=\mathfrak{g}.v+V^{\mathfrak{g}_{v}}.$

In order that the group $G_{v}$ be a generic isotropy, it is necessary and
sufficient that

  * •

the set $\\{u\in V^{G_{v}}|G_{v}=G_{u}\\}$ be dense in $V^{G_{v}}$.

  * •

$V=\mathfrak{g}.v+V^{G_{v}}$.

The last criterion in Proposition 2.5 follows by Definition 2.2 and the map
$G\times V^{G_{v}}\to V$ has constant rank on the set $G\times\\{u\in
V^{G_{v}}|G_{v}=G_{u}\\}$. Differentiating this argument yields that, in order
that the tangent algebra $\mathfrak{g}_{v}$ be a generic stabilizer is
necessary and sufficient that:

  * •

the set $\\{u\in V^{\mathfrak{g}_{v}}|\mathfrak{g}_{v}=\mathfrak{g}_{u}\\}$ be
dense in $V^{\mathfrak{g}_{v}}$.

  * •

$V=\mathfrak{g}.v+V^{\mathfrak{g}_{v}}$.

where the argument follows by analyzing the map $G\times
V^{\mathfrak{g}_{v}}\to V$. The existence of a non-trivial generic stabilizer
yields a Chevalley-type theorem for the field of invariants. If $v\in V$ is a
generic point and $\Omega=V^{\mathfrak{g}_{v}}$ then the last item in Theorem
2.4 gives the desired claim.

######  Theorem 2.6.

[30] Consider the adjoint representation,

(21) $Ad:G\longrightarrow GL(\mathfrak{g}),\qquad g\longmapsto
Ad(g):\left(x\mapsto g.x.g^{-1}\right).$

If $G$ is reductive, then, the adjoint representation has the following
properties:

  * •

It is self dual with $\mathfrak{h}$ as a generic stabilizer.

  * •

The algebra $\mathbb{C}[X]^{G}$ is a polynomial ring.

  * •

The restriction homomorphism induces an isomorphism [Chevalley Theorem, see
[24, 30]].

(22) $\mathbb{C}[\mathfrak{g}]^{G}\cong\mathbb{C}[\mathfrak{h}]^{W}.$

  * •

The quotient map $\mathfrak{g}\to\mathfrak{g}//G$ is equi-dimensional.

  * •

The fiber $\pi^{-1}\pi(0)$ is the union of finitely many orbits.

These properties may fail if $G$ is replaced with an arbitrary (possibly non-
reductive) Lie group. Specifically, the first item is refused to be true. The
coadjoint representation of a non-reductive group shows different properties
than the adjoint one. Let

(23) $ad:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g})$

be the derivative of Ad at zero. In this case the stabilizer
$\mathfrak{g}_{x}=\mathfrak{z}_{\mathfrak{g}}(x)$, is the centralizer of $x$.
A point $x\in\mathfrak{g}$ is generic iff [cf. [24]],

(24)
$[\mathfrak{g},x]+\mathfrak{g}^{\mathfrak{z}_{\mathfrak{g}}(x)}=\mathfrak{g}.$

Recall that a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ is a
nilpotent subalgebra such that
$\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h})=\mathfrak{h}$. Cartan subalgebras
always exist and all are conjugate under $G$, [37].

######  Proposition 2.7 ([24] props. 2.1 and 2.5).

The adjoint action of the Lie group $G$ on its Lie algebra $\mathfrak{g}$ has
a generic stabilizer if and only if either of the following equivalent
conditions holds:

  * •

There exists $x\in\mathfrak{g}$ such that the centralizer
$\mathfrak{z}_{\mathfrak{g}}(x)$ is commutative and

(25) $[\mathfrak{g},x]\oplus\mathfrak{z}_{\mathfrak{g}}(x)=\mathfrak{g}$

holds.

  * •

The Cartan subalgebras of $\mathfrak{g}$ are commutative.

If $x\in\mathfrak{g}$ is generic, then
$\mathfrak{n}_{\mathfrak{g}}(\mathfrak{z}_{\mathfrak{g}}(x))=\mathfrak{z}_{\mathfrak{g}}(x)$,
and the commutativity of $\mathfrak{h}$ implies
$\mathfrak{h}=\mathfrak{z}_{\mathfrak{g}}(x)$. We shall assume that a generic
point has a generic isotropy group (always).

######  Theorem 2.8 ([24] theorem 2.6).

Assume $\mathfrak{g}$ has a generic stabilizer (centralizer) under the adjoint
action, and $x\in\mathfrak{g}$ be a generic point where $Z_{G}(x)$ is generic.
Then,

  * •

$Z(\mathfrak{z}_{\mathfrak{g}}(x))=Z_{G}(x)$.

  * •

$\mathbb{C}(\mathfrak{g})^{G}=\mathbb{C}(\mathfrak{z}_{\mathfrak{g}}(x))^{W}$.

  * •

The Weyl group $W=N_{G}(\mathfrak{z}_{\mathfrak{g}}(x))/Z_{G}(x)$ is finite.

Theorem 2.8 provides a way to establish the finite generation of
$\mathbb{C}[\mathfrak{g}]^{G}$ when the Weyl group $W$ is finite.

######  Remark 2.9.

The last property may fail for the coadjoint representation, That is, the
${Ad}^{*}:G\rightarrow\mathrm{Aut}({\mathfrak{g}^{*})}$ is defined by,

(26)
${\displaystyle\langle\mathrm{Ad}_{g}^{*}\,\mu,Y\rangle=\langle\mu,\mathrm{Ad}_{g^{-1}}Y\rangle}\qquad{\displaystyle
g\in G,Y\in{\mathfrak{g}},\mu\in{\mathfrak{g}}^{*},}$

where ${\displaystyle\langle\mu,Y\rangle}$ denotes the value of the linear
functional ${\displaystyle\mu}$ on the vector ${\displaystyle Y}$. The Weyl
group of the coadjoint representation may not be finite, [see [24] for
details].

Assume that the algebraic group $G$ acts on a variety $X$, and let $V$ be a
$G$-module. Consider the module of covariants $(\mathbb{C}[X]\otimes V)^{G}$
and define,

(27) $\text{ev}_{x}:(\mathbb{C}[X]\otimes V)^{G}\longrightarrow V,\qquad
F\otimes v\longmapsto F(x)v.$

A natural question is if the image of the map $\text{ev}_{x}$ captures the
information on $V^{G_{x}}$. The following theorem answers this question under
some codimension $2$-condition for the complement of the $G$-orbit of $x$ in
its closure. The map $ev_{x}$ plays a crucial role in the study of the basic
invariants of the ring $\mathbb{C}[V]^{G_{x}}$.

######  Theorem 2.10 ([27] theorem 1).

Let $G$ be an algebraic group acting on an affine variety $X$. If for some
$x\in X$, the closure orbit $\overline{G.x}$ is a normal subvariety of $X$,
and $\text{codim}_{\overline{G.x}}(\overline{G.x}\setminus G.x)\geq 2$, then
$\text{image}(ev_{x})=V^{G_{x}}$.

We denote by $\mathfrak{g}^{\text{reg}}$ the regular elements of
$\mathfrak{g}$ under the adjoint action of $G$ (cf. (14)). Let $T$ be a
maximal torus of $G$. Assume $G$ and $V$ be as in Theorem 2.10. Let denote by
$Mor_{G}(X,V)$ the set of all $G$-equivariant maps from $X$ to $V$. With the
above nomenclature, we have the following result due to Kostant [21].

######  Theorem 2.11.

(Kostant) [[21] page 385] Assume $G$ is an algebraic group acting on the
vector space $V$ ($V$ is a $G$-module). Then, $\dim V^{G_{x}}=\dim V^{T}$ for
any $x\in\mathfrak{g}^{\text{reg}}$. The module $Mor_{G}(\mathfrak{g},V)$ is
freely generated over $\mathbb{C}[\mathfrak{g}]^{G}$ and has rank equal to
$\dim V^{T}$.

An action of $G$ on $V$ is said to be stable if the union of the closed orbits
is dense in $V$. In this case, a generic stabilizer is reductive and
$\mathbb{C}(V)^{G}$ is the quotient field of $\mathbb{C}[V]^{G}$, [41, 30]. If
$f\in\mathbb{C}[V]$, then, the differential of $f$ defines a map $v\mapsto
df(v)$ in $Hom(V,V^{*})$. If the function $f$ belong to $\mathbb{C}[V]^{G}$,
then the resulting map lies in $Mor_{G}(V,V^{*})$, i.e, $G$-equivariant
morphisms $V\to V^{*}$, [see [24] sec. 4].

######  Theorem 2.12 ((T. Vust) [44], [24] theorem 4.5).

Assume $G$ acts on $V$ such that $\mathbb{C}[V]^{G}$ is a polynomial algebra
and the quotient map $\pi:V\to V//G$ is equidimensional. Let $H$ be a generic
isotropy group for the action, and suppose $N_{G}(H)/H$ is finite. Let
$f_{1},...,f_{r}$ is a set of basic invariants generating $\mathbb{C}[V]^{G}$.
Then, $Mor_{G}(V,V^{*})$ is freely generated by $df_{1},...,df_{r}$.

######  Remark 2.13.

In [24] the properties of the invariant ring $\mathbb{C}[V]^{G}$ has been
studied by considering the module of covariants $[\mathbb{C}[X]\otimes
V]^{G}$. There have been determined the exact conditions under which it is
finitely generated over $\mathbb{C}[\mathfrak{g}]^{G}$. The basic invariants
of $\mathbb{C}[\mathfrak{g}]$ is determined via Theorem 2.12.

We need the following definition, from [24], to state the Igusa theorem.

######  Definition 2.14.

[24, 18] Assume $X$ is an algebraic variety. We say that an open subset
$\Omega\subset X$ is big if $X\setminus\Omega$ contains no divisor.

######  Theorem 2.15.

[(Igusa) [18], [24] lemma 6.1, [30] Theorem 4.12 page 190] Let $G$ be an
algebraic group that regularly acts on an algebraic affine variety $X$.
Suppose $S$ is an integrally closed finitely generated subalgebra of
$\mathbb{C}[X]^{G}$ such that the morphism $\pi:X\to\text{Spec}(S)$ satisfies:

  * •

The fibers of $\pi$ over a dense open subset of $\text{Spec}(S)$ contains a
dense $G$-orbit.

  * •

$\text{im}(\pi)$ contains a big open subset of $\text{Spec}(S)$.

Then, $S=\mathbb{C}[X]^{G}$. In particular, the algebra $\mathbb{C}[X]^{G}$ is
finitely generated.

The first condition in the Igusa Theorem applies with the Rosenlicht Theorem
and implies $\mathbb{C}[x]^{G}=\mathbb{C}[\text{Spec}(S)]$. The first
condition implies that $\pi$ is dominant morphism; hence the induced map on
the structure sheaves is a monomorphism (denoted by $\pi^{*}$). The second
condition employs the Richardson lemma in birational geometry, [[7] 3.2 Lemme
1]. The condition implies that if $f\in\mathbb{C}[X]^{G}$ is constant on the
fibers of $\pi$ in general position, it has the form $f=\pi^{*}g$ for
$g\in\mathbb{C}(Y)$. In a normal variety, the set that a rational function is
not defined is either empty or has codimension $1$. The second condition can
be replaced by ”For any point $s$ of some non-empty open subset of $S$, the
fiber $\pi^{-1}(s)$ contains exactly one closed orbit”, cf. [24, 30] loc. cit.

###  2.2. Representation ring of algebraic groups

We recall some basic facts on Grothendieck ring of $G$-modules for general
connected complex Lie group $G$. Let $G$ be a nontrivial connected complex Lie
group, and $T\subset G$ be a maximal torus. We denote the representation ring
of $G$ by $R(G)=\bigoplus\mathbb{Z}[\rho]$, the free abelian group generated
by the irreducible representations $\rho$ of $G$. It has a ring structure by
tensor product. The ring $R(G)$ is a subring of the ring of class functions on
$G$ by the formation of characters. The Weyl group $W=W(G,T)=N_{G}(T)/T$ acts
naturally on $R(T)$. The restriction of class functions to $T$ makes $R(G)$ a
subring of $R(T)^{W}$. Let $r=\dim(T)$ and $\Phi=\Phi(G,T)$ be the associated
root system with $\mathcal{B}=\\{\alpha_{1},...,\alpha_{r}\\}$ a basis of
$\Phi$. Assume $\varpi_{1},...,\varpi_{r}$ are the fundamental weights with
respect to $(G,T,\mathcal{B})$, and $V_{\varpi_{1}},...,V_{\varpi_{r}}$ are
the fundamental representations of $G$. For any dominant weight
$\lambda=\sum_{j}n_{j}\varpi_{j}$, the irreducible representation
$V_{\lambda}$ of $G$, with highest weight $\lambda$, occurs exactly once in
$V_{\varpi_{1}}^{\otimes n_{1}}\otimes...\otimes V_{\varpi_{r}}^{\otimes
n_{r}}$ due to the theorem of the highest weight. The following theorem is
well known.

######  Theorem 2.16 (see [9] page 291).

If $G$ is semisimple and simply connected, then $R(G)$ is a polynomial ring on
the fundamental representations. More specifically, the map
$\mathbb{Z}[X_{1},...,X_{r}]\to R(G)$ defined by
$X_{j}\mapsto[V_{\varpi_{j}}]$ is an isomorphism.

The representation ring of a compact connected Lie group satisfies the
following finite-type theorem.

######  Theorem 2.17 ([9] page 293).

If $G$ is a compact connected Lie group, then, $R(G)=R(T)^{W}$.

In the case of semisimple and simply connected $G$, one notes that the
character group of $T$ is $X(T)=P=\bigoplus\mathbb{Z}\varpi_{j}$. Thus, the
representation ring of $T$ satisfies $R(T)=\mathbb{Z}[X(T)]=\mathbb{Z}[P]$.
Then, $R(T)^{W}$ is a free $\mathbb{Z}$-module with a basis given by the sums
along the $W$-orbit in $P$. Because of the simple transitive action of the
Weyl group $W$ on the Weyl chambers in $X(T)_{\mathbb{R}}$, we have
$\overline{C(\mathcal{B})}\cap P=\mathbb{Z}_{\geq 0}.\varpi_{j}$, where
$C(\mathcal{B})$ is the Weyl chamber corresponding to the basis $\mathcal{B}$.
In other words, any $W$-orbit in $P$ meets a dominant weight. Besides, in the
composition,

(28) $\mathbb{Z}[X_{1},...,X_{r}]\to R(G)\hookrightarrow R(T)^{W},$

and by considering the above observation, one deduces that the map
$\mathbb{Z}\to R(T)^{W}$ is surjective, cf. [9].

##  3\. The bundle of $k$-jets of holomorphic curves

This section reviews basic definitions and facts about the bundle of $k$-jets
of holomorphic curves on a projective variety. This section introduces the
basic notations in use in the following sections.

The references for this section are [2, 11, 13, 14, 16]. Let $X$ be a complex
projective manifold of dimension $n$. The $k$-jet bundle,

(29) $J_{k}\longrightarrow X$

is the bundle of germs of curves in $X$ whose fiber at $x\in X$ is the set of
equivalence classes of germs of holomorphic curves,

(30) $f:(\mathbb{C},0)\longrightarrow(X,x)$

with equivalence relation,

(31) $f\equiv_{k}g\qquad\Longleftrightarrow\qquad f^{(j)}(0)=g^{(j)}(0),\ \
0\leq j\leq k.$

By choosing local holomorphic coordinates around $x$, the fiber $J_{k,x}$
elements can be represented by the Taylor expansion:

(32)
$f(t)=tf^{\prime}(0)+\frac{t^{2}}{2!}f^{\prime\prime}(0)+...+\frac{t^{k}}{k!}f^{(k)}(0)+O(t^{k+1}).$

Setting $f=(f_{1},...,f_{n})$ on an open neighborhood of $0\in\mathbb{C}$, the
fiber can be given as

(33) $J_{k,x}=\\{(f^{\prime}(0),...,f^{(k)}(0))\\}=\mathbb{C}^{nk}.$

We can regard $J_{k}X$ obtained from a sequence of vector bundles $X_{k}\to
X_{k-1}\to...\to X$, parametrized by the vectors of the components in (33).
The projectivized bundle $J_{k}(X)/\mathbb{C}^{*}$ is called the Green-
Griffiths bundle; [see [11, 13, 12, 16] for details]. Let’s $G_{k}$ be the
group of local analytic reparametrizations of $(\mathbb{C},0)$ of the form:

(34)
$t\longmapsto\phi(t)=\alpha_{1}t+\alpha_{2}t^{2}+...+\alpha_{k}t^{k},\qquad\alpha_{1}\in\mathbb{C}^{*}.$

By applying the change of coordinates $\phi$ on a fiber element produces an
action on the $k$-jets given as the following matrix multiplication,

(35)
$[f^{\prime}(0),f^{\prime\prime}(0)/2!,...,f^{(k)}(0)/k!].\left[\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&...&a_{k}\\\
0&a_{1}^{2}&2a_{1}a_{2}&...&a_{1}a_{k-1}+...+a_{k-1}a_{1}\\\
0&0&a_{1}^{3}&...&3a_{1}^{2}a_{k-2}+...\\\ .&.&.&...&.\\\
0&0&0&...&a_{1}^{k}\end{array}\right].$

Then, $G_{k}$ is the subgroup of $GL_{k}\mathbb{C}$ of transformation $k\times
k$-matrices of the form appearing in (35). Now, denote the unipotent elements
corresponding to the matrices with $a_{1}=1$ by $U_{k}$. There is a short
exact sequence,

(36) $1\to U_{k}\to G_{k}\to\mathbb{C}^{*}\to 0.$

The subgroup $U_{k}$ is the unipotent radical of $G_{k}$, see [2]. The action
of $\mathbb{C}^{*}$ on $k$-jets is

(37)
$\lambda\times\left(f^{\prime}(0),...,f^{(k)}(0)\right)\longmapsto\left(\lambda.f^{\prime}(0),...,\lambda^{k}.f^{(k)}(0)\right).$

An alternative way to define these bundles is through their ring of sections.
In a more general setup, we may consider the jets tangent to $V$, a
holomorphic subbundle of the tangent bundle $TX$ of $X$. Green and Griffiths
[16] introduce the vector bundle:

(38) $E_{k,m}^{GG}V^{*}\longrightarrow X$

whose fibers are polynomials,

(39)
$Q(z,\xi)=\sum_{\alpha}A_{\alpha}(z)\xi_{1}^{\alpha_{1}}...\xi_{k}^{\alpha_{k}}$

of weighted degree $m$, where $z$ is a local variable on $X$ and $\xi_{i}$ is
the variable on the fibers of

(40) $X_{i}\longrightarrow X_{i-1}.$

One can identify $E_{k}^{GG}V^{*}=\bigoplus_{m}E_{k,m}^{GG}V^{*}$ with the
projectivized bundle of germs of $k$-jets of the analytic maps
$f:(\mathbb{C},0)\to(X,x)$ tangent to $V$. The weights attached to
$\xi_{1},...,\xi_{k},...$ are $1,2,...,k,...$, respectively, and the total
weighted degree of a homogeneous germ is defined as

(41) $\mid\alpha\mid=\alpha_{1}+2\alpha_{2}+...+k\alpha_{k}.$

The $\mathbb{C}^{*}$-action on the space of germs of holomorphic maps
$f:\mathbb{C}\to X$ is defined as in (37). Then, we have

(42) $E_{k}^{GG}V^{*}=\bigoplus_{m}E_{k,m}^{GG}V^{*}=J_{k}V/\mathbb{C}^{*}.$

######  Remark 3.1.

The symbol $V^{*}$ used in $E_{k,m}^{GG}V^{*}$ is only a notation reminding
the duality appearing in the fact that $S^{*}V=Sym^{\otimes}V$ can be
understood as polynomial algebra on a basis of $V^{*}$. The analytic structure
on $E_{k,m}^{GG}V^{*}$ comes from the fact that one can arrange the germs to
depend holomorphically on their initial values at $0$.

One can consider the group of all analytic reparametrization automorphisms of
$J_{k}V$ and the invariant bundle $(J_{k}V)^{G_{k}}$. We denote by
$E_{k,m}V^{*}$ the Demailly-Semple bundle whose fiber at $x$ consists of
$U_{k}$-invariant polynomials on the fiber coordinates of $J_{k}V$ at $x$ of
weighted degree $m$. Also, set $E_{k}=\bigoplus_{m}E_{k,m}$ to be the
Demailly-Semple bundle of graded algebras of invariant jets, [11, 12, 13, 16,
2].

Recall that, a $k$-jet $f:\mathbb{C}\to X$ is regular if $f^{\prime}(0)\neq
0$. We call such a jet non-degenerate when it has maximal rank in local
coordinates.

######  Proposition 3.2.

[2] There is a $G_{k}$-invariant algebraic morphism

(43) $\phi:J_{k}^{\text{reg}}\longrightarrow
Hom(\mathbb{C}^{k},\text{Sym}^{\leq k}\mathbb{C}^{n})$

that induces an injective map,

(44) $\Phi^{\text{Gr}}:J_{k}^{reg}/G_{k}\hookrightarrow Grass(k,Sym^{\leq
k}\mathbb{C}^{n})$

given explicitly by

(45)
$(f^{\prime},f^{\prime\prime},...,f^{(k)})\longmapsto\left(f^{\prime},f^{\prime\prime}+(f^{\prime})^{2},...,\sum_{i_{1}+...+i_{s}=d}\frac{1}{i_{1}!...i_{s}!}f^{(i_{1})}...f^{(i_{s})},...\right),$

where we have identified $J_{k}\mathbb{C}^{n}$ with
$Hom(\mathbb{C}^{k},\mathbb{C}^{n})$ by putting the coordinates of $f^{(j)}$
in the $j$-th column of a $n\times k$ matrix. The map $\phi$ also gives an
injective map

(46)
$\phi^{\text{Flag}}:J_{k}^{reg}/G_{k}\hookrightarrow\text{Flag}_{k}(\text{Sym}^{\leq
k}\mathbb{C}^{n})$

defined explicitly by

(47) $(f^{(1)},f^{(2)},...,f^{(k)})\longmapsto\left(im(\phi(f^{(1)})\subset
im(\phi(f^{(2)})\subset...\right),$

where we have considered $f^{(j)}\subset\mathbb{C}^{n}$ and

(48) $\text{Flag}_{k}(\mathbb{C}^{n})=\\{F_{0}\subset F_{1}\subset...\subset
F_{k}\subset\mathbb{C}^{n}\\}$

of flags of length $k$. We also have

(49) $\phi^{\text{Gr}}=\phi^{\text{Flag}}\circ\pi_{k}$

where

(50) $\pi_{k}:\text{Flag}_{k}(\text{Sym}^{\leq k}\mathbb{C}^{n}))\to
Grass(k,Sym^{\leq k})$

is the projection to the $k$-dimensional subspace. Besides, the composition of
$\phi^{\text{Gr}}$ with the Plucker embedding,

(51) $Grass(k,Sym^{\leq
k}\mathbb{C}^{n})\hookrightarrow\mathbb{P}(\bigwedge^{k}Sym^{\leq
k}\mathbb{C}^{n}),$

gives an embedding

(52)
$\phi^{\text{proj}}:J_{k}^{\text{reg}}\hookrightarrow\mathbb{P}(\bigwedge^{k}Sym^{\leq
k}\mathbb{C}^{n}).$

######  Definition 3.3.

[2] Let $e_{1},..,e_{n}$ be the standard basis of $\mathbb{C}^{n}$. Then, a
basis of $Sym^{\leq k}\mathbb{C}^{n}$ consists of

(53) $\\{e_{i_{1}...i_{s}}=e_{i_{1}}...e_{i_{s}},s\leq k\\},$

and a basis of $\mathbb{P}(\wedge^{k}Sym^{\leq k}\mathbb{C}^{n})$ is given by

(54) $\\{e_{i_{1}}\wedge...\wedge e_{i_{s}},i_{1}\leq i_{2}\leq...\leq i_{s},\
s\leq k\\}.$

Then, the set

(55) $\\{\ e_{\epsilon_{1}}\wedge...\wedge e_{\epsilon_{k}}\ |\
\epsilon_{j}\in\Pi_{\leq k}\\},\qquad\Pi_{\leq k}=\\{i_{1}\leq
i_{2}\leq...\leq i_{s}\\}$

represents a basis of $\mathbb{P}(\bigwedge^{k}\text{Sym}^{\leq
k}\mathbb{C}^{n})$. The corresponding coordinates of $x\in Sym^{\leq
k}\mathbb{C}^{n}$ will be denoted by $x_{\epsilon_{1}...\epsilon_{s}}$. Also,
we denote by $A_{n,k}\subset\mathbb{P}(\bigwedge^{k}\text{Sym}^{\leq
k}\mathbb{C}^{n})$ the points whose projection on
$\bigwedge^{k}\mathbb{C}^{n}$ is non-zero, i.e., at least one
$x_{i_{1}...i_{k}}\neq 0$.

######  Proposition 3.4.

[2] Assume $k\leq n$. Let

(56)
$z_{k}=\phi^{\text{proj}}(e_{1},...,e_{k})=[e_{1}\wedge(e_{2}+e_{1}^{2})\wedge...\wedge(\sum
e_{i_{1}}...e_{i_{k}})].$

The group $GL(n)$ acts on $z$ as follows. If $g\in GL(n)$ has the column
vectors $v_{1},...,v_{n}$, then

(57)
$g.z_{k}=\phi^{\text{proj}}(v_{1},...,v_{n})=[v_{1}\wedge(v_{2}+v_{1}^{2})\wedge...\wedge(\sum
v_{i_{1}}...v_{i_{k}})].$

Besides,

  * •

If $n\geq k$, then the image of non-degenerate $k$-jets is the $GL(n)$ orbit
of $z$, and it is denoted by $X_{n,k}$.

  * •

If $n\geq k$, then the image of regular $k$-jets is a finite union of
$GL(n)$-orbits, and it is denoted by $Y_{n,k}$.

  * •

If $k>n$, then $X_{n,k}$ and $Y_{n,k}$ are $GL(n)$-invariant quasi-projective
varieties, with no dense $GL(n)$-orbits.

Regular jets are an open subset $J_{k}\mathbb{C}^{n}$, as the condition
defining the singularity is an algebraic equation. If $n>1$, then one has

(58) $\text{codim}_{J_{k}\mathbb{C}^{n}}(J_{k}\mathbb{C}^{n}\setminus
J_{k}^{\text{reg}}\mathbb{C}^{n})\geq 2$

cf. [2] loc. cit. We use this inequality to apply the Igusa theorem for the
action of $G_{k}$ on $J_{k}\mathbb{C}^{n}$ in the next section.

######  Remark 3.5 ([2], Theorems 6.1 and 6.2).

One of the main results of [2] is the following. For $N$ large enough the
stabilizer of $p_{k}=z_{k}\otimes e_{1}^{\otimes
N}\in\bigwedge^{k}\text{Sym}^{\leq
k}\mathbb{C}^{k}\otimes(\mathbb{C}^{k})^{\otimes N}$ for the
$SL_{k}\mathbb{C}$-action is $U_{k}$. Furthermore, the complement of the
$SL_{k}$-orbit of $p_{k}$ in its closure has codimension at least two.
Although this result supports the proof of our results better, we do not need
to stand on that for the applications of the Igusa theorem, and the inequality
(58) is enough in this case.

##  4\. Main results: The invariant theory of $G_{k}$

In this section, we present the main contributions of this paper. All the
theorems and their proof appear to be new in the literature. We consider the
two actions of $G_{k}$ on $\mathfrak{g}_{k}$ (adjoint action), and also on
$J_{k}\mathbb{C}^{n}$, separately. We shall work with the notations introduced
in Sections 2 and 3.

###  4.1. The adjoint action of $G_{k}$ on $\mathfrak{g}_{k}$

Let $\mathfrak{g}_{k}=\text{Lie}(G_{k})$, and

(59) $\text{Ad}_{k}:G_{k}\longrightarrow GL(\mathfrak{g}_{k})$

be the adjoint representation. Also, set
$\text{ad}_{k}:\mathfrak{g}_{k}\longrightarrow\mathfrak{gl}(\mathfrak{g}_{k})$
be its derivative, i.e., the corresponding representation of its Lie algebra
(bracket). The decomposition $G_{k}=\mathbb{C}^{*}\ltimes U_{k}$ implies:

(60) $\mathfrak{g}_{k}=\mathbb{C}\oplus\mathfrak{u}_{k},$

where $\mathfrak{u}_{k}=\text{Lie}(U_{k})$. Our first theorem expresses the
generic stabilizer property for the action (59).

######  Theorem 4.1.

(Main result.) The adjoint action of $G_{k}$ on the Lie algebra
$\mathfrak{g}_{k}$ has a generic stabilizer. In particular, the Cartan
subalgebras of $\mathfrak{g}_{k}$ are commutative.

###### Proof.

We first check that the Elashvili criterion holds for the adjoint action
$Ad_{k}:G_{k}\longrightarrow GL(\mathfrak{g}_{k})$, that is,

(61)
$[\mathfrak{g}_{k},x]+\mathfrak{g}_{k}^{\mathfrak{z}_{\mathfrak{g}_{k}}(x)}=\mathfrak{g}_{k},$

for some $x\in\mathfrak{g}_{k}$. One can check this in a way to find an
$x\in\mathfrak{g}_{k}$, such that ${\mathfrak{z}_{\mathfrak{g}_{k}}(x)}$ is
commutative and
$[\mathfrak{g}_{k},x]+{\mathfrak{z}_{\mathfrak{g}_{k}}(x)}=\mathfrak{g}_{k}$.
The equation implies
$\text{image}(\text{ad}(x))=\text{image}(\text{ad}^{2}(x))$; this will hold as
long as we choose $x$ to be a semisimple element. In this case,
$\text{image}(\text{ad}(x))$ generates $\mathfrak{u}_{k}$. This is due to
choose $u\in\mathfrak{u}_{k}$ in a suitable basis, then,
$ad(x)(u)=\alpha_{u}.u$ for a complex number $\alpha_{u}$. On the other hand,
if $x$ is semisimple, ${\mathfrak{z}_{\mathfrak{g}_{k}}(x)}$ consists of
diagonal elements. In the latter case, it follows that
${\mathfrak{z}_{\mathfrak{g}_{k}}(x)}$ is commutative. Theorem 2.5 and
Proposition 2.7 complete the proof of having generic stabilizers. The proof of
the last claim is also a consequence of Proposition 2.7, and is equivalent to
the generic stabilizer property of adjoint action of $G_{k}$. In other words,
$\mathfrak{z}_{\mathfrak{g}_{k}}(x)=H$ for $x\in\mathfrak{g}_{k}^{\text{reg}}$
is a Cartan subalgebra of $\mathfrak{g}_{k}$. ∎

Using Theorem 4.1 we can prove the following Chevalley-type theorem for the
adjoint action of $G_{k}$ on $\mathfrak{g}_{k}$, which appear as our following
main result.

######  Theorem 4.2.

(Main result.) Let $H$ be a generic stabilizer of the adjoint action of
$G_{k}$ on $\mathfrak{g}_{k}$, (called generic centralizer in this case), and
$\mathfrak{h}=Lie(H)$. Then, we have:

  * 1.

$\mathbb{C}(\mathfrak{g}_{k})^{G_{k}}=\mathbb{C}(\mathfrak{h})^{W}$.

  * 2.

The Weyl group $W=N_{G_{k}}(H)/H$ (of the action) is finite.

  * 3.

$\mathbb{C}[\mathfrak{g}_{k}]^{G_{k}}=\mathbb{C}[\mathfrak{h}]^{W}$.

  * 4.

The invariant ring $\mathbb{C}[\mathfrak{g}_{k}]^{G_{k}}$ is finitely
generated.

###### Proof.

The first and the second items are direct consequences of Theorem 2.8 and the
fact that $\mathfrak{z}_{\mathfrak{g}_{k}}(x)=H$ for
$x\in\mathfrak{g}_{k}^{\text{reg}}$ is a Cartan subalgebra. For the third
item, we have the inclusion
$\mathbb{C}[\mathfrak{g}_{k}]^{G_{k}}\subset\mathbb{C}[\mathfrak{h}]^{W}$.
This is because for
$U=\mathfrak{g}^{\mathfrak{g}_{x}}=\mathfrak{g}^{\mathfrak{h}}=\mathfrak{h}$
the restriction map easily gives an inclusion,

(62) $\mathbb{C}[\mathfrak{g}_{k}]^{G_{k}}\subset\mathbb{C}[U]^{N(U)/Z(U)}.$

Now, the equality follows from the finiteness of the Weyl group $W$. It also
follows that $\mathbb{C}[\mathfrak{g}_{k}]^{G_{k}}$ is finitely generated,
which is the last item. ∎

Theorems 4.1 and 4.2 imply several important properties for the quotient space
$\mathfrak{g}_{k}/G_{k}$ that we list in the next theorem.

######  Theorem 4.3.

(Main result.) Under the adjoint action of $G_{k}$ on $\mathfrak{g}_{k}$, the
following holds:

  * 1.

$\mathbb{C}[\mathfrak{g}_{k}]^{G_{k}}$ is a polynomial algebra.

  * 2.

$\text{Max}_{x\in\mathfrak{g}_{k}}\dim
G_{k}.x=\dim\mathfrak{g}_{k}-\dim\mathfrak{g}//G_{k}$.

  * 3.

The quotient map $\pi:\mathfrak{g}_{k}\longrightarrow\mathfrak{g}_{k}//G_{k}$
is equi-dimensional.

  * 4.

Let $\Omega=\\{x\in\mathfrak{g}_{k}\ |\ d_{x}\pi\ \text{is onto}\ \\}$, then,
$\mathfrak{g}_{k}\setminus\Omega$ contains no divisor.

  * 5.

The null cone $\pi^{-1}\pi(0)$ is an irreducible complete intersection of
dimension $\dim\mathfrak{g}_{k}-\dim\mathfrak{g}_{k}//G_{k}$.

###### Proof.

The first item follows from Theorem 4.2. The second follows from Theorem 4.1.
The equi-dimensionality of $\pi$ is a consequence of Theorem (4.1), i.e., the
generic stabilizer property [see [24] sections 6-8, also the introduction].
The last item follows from the fact that the ideal of the null cone is
generated by the basic invariants of $\mathbb{C}[\mathfrak{g}_{k}]^{G_{k}}$.
It remains to prove the item 4. We prove this using an argument of [30] in
Theorem 3.11. That is, for a generic $x\in\mathfrak{g}_{k}$, the stabilizer
satisfies $G_{k,x}=H$, where $H$ is a Cartan subgroup of $G_{k}$ and thus is a
reductive subgroup. This implies that $\mathbb{C}[G_{k}/G_{k,x}]$ is finitely
generated. Then, the above-mentioned theorem of Grosshans implies that
$\text{codim}(\overline{G_{k}.x}\setminus G_{k}.x)\geq 2$. Therefore, the
regular points of $\mathfrak{g}_{k}$ have the property that
$\text{codim}(\mathfrak{g}_{k}\setminus\mathfrak{g}_{k}^{\text{reg}})\geq 2$,
and $\overline{G.x}$ is a normal subvariety for
$x\in\mathfrak{g}^{\text{reg}}$. Then, Theorems 2.10 and 2.11 imply that
$\Omega\supset\mathfrak{g}_{k}^{\text{reg}}$. Thus,
$\text{codim}(\mathfrak{g}_{k}\setminus\Omega)\geq 2$. ∎

###  4.2. The action of $G_{k}$ on $J_{k}\mathbb{C}^{n}$

We want to prove similar properties for the action of $G_{k}$ on the space of
$k$-jets. That is, to put $V=J_{k}\mathbb{C}^{n},\ G=G_{k},\
\mathfrak{g}=\mathfrak{g}_{k}=\text{Lie}(G_{k})$. Let’s consider the
representation

(63) $\rho:G_{k}\longrightarrow GL(J_{k}\mathbb{C}^{n})$

be the natural action. Again, the first result is the generic stabilizer
property for the action of $\rho$.

######  Theorem 4.4.

(Main result.) The action of $G_{k}$ on $J_{k}\mathbb{C}^{n}$ has generic
stabilizer.

###### Proof.

We check that the Elashvili criterion holds for the action
$\rho:G_{k}\longrightarrow GL(J_{k}\mathbb{C}^{n})$, and some element $z$
regarded as an element in $J_{k}$, i.e.,

(64)
$J_{k}\mathbb{C}^{n}=\mathfrak{g}_{k}.z+(J_{k}\mathbb{C}^{n})^{\mathfrak{g}_{k,z}}.$

The action of the Lie group $G_{k}$ on the $k$-jets is given by multiplication
on the right as in (35). Thus, the same holds for the action of elements in
the Lie algebra. That is, the Lie algebra also acts by multiplication. As was
explained before, we may successively put the vectors’ entries in a $k$-jet in
the matrix columns. In this case, a $k$-jet on $\mathbb{C}^{n}$ belongs to
$\text{Hom}(\mathbb{C}^{k},\mathbb{C}^{n})$. In fact, we have the equality:

(65) $J_{k}\mathbb{C}^{n}=\text{Hom}(\mathbb{C}^{k},\mathbb{C}^{n}).$

One can choose a generic jet $z=J_{k}\mathbb{C}^{n}$ whose stabilizer is among
the diagonal elements, i.e., $\mathbb{C}$. In this case, the second factor
generates $(J_{k}\mathbb{C}^{n})^{\mathfrak{g}_{k,z}}$. Now, (64) can be
checked easily. Here, generic means as follows. Let’s choose the new
variables:

(66)
$\xi_{1}=f^{\prime}(0),\xi_{2}=f^{\prime\prime}(0),...,\xi_{k}=f^{(k)}(0)$

as generic jet variables. Then, these variables are formally related by the
following equations in the invariant ring,

(67) $A^{T}.\left[\begin{array}[]{ccccc}\xi_{1}\\\ \xi_{2}\\\ ...\\\
\xi_{k}\end{array}\right]=\left[\begin{array}[]{ccccc}\xi_{1}\\\ \xi_{2}\\\
...\\\ \xi_{k}\end{array}\right],\qquad(A\in G_{k}),$

where we regard the variables
$\xi_{i}=\left[\begin{array}[]{ccccc}\xi_{i1}&\xi_{i2}&...&\xi_{in}\end{array}\right]$,
defined in (66) to be generic, i.e., all $\xi_{ij}$ are formal variables.
Plugging into such variables, we get a single matrix differential equation:

(68) $\left[\begin{array}[]{ccccc}\xi_{11}&...&\xi_{1n}\\\
\xi_{21}&...&\xi_{2n}\\\ ...&...&...\\\
\xi_{k1}&...&\xi_{kn}\end{array}\right]\left[\begin{array}[]{ccccc}a_{1}&a_{2}&...&a_{k}\\\
0&a_{1}^{2}&...&a_{1}a_{k-1}+...+a_{k-1}a_{1}\\\
0&0&...&3a_{1}^{2}a_{k-2}+...\\\ .&.&...&.\\\
0&0&...&a_{1}^{k}\end{array}\right]=\left[\begin{array}[]{ccccc}\xi_{11}&...&\xi_{1n}\\\
\xi_{21}&...&\xi_{2n}\\\ ...&...&...\\\
\xi_{k1}&...&\xi_{kn}\end{array}\right].$

Let’s denote,

(69) $X_{k\times n}=\left[\begin{array}[]{ccccc}\xi_{11}&...&\xi_{1n}^{(k)}\\\
\xi_{21}&...&\xi_{2n}^{(k)}\\\ ...&...&...\\\
\xi_{k1}&...&\xi_{kn}^{(k)}\end{array}\right].$

In Lie algebras language, the Lie algebra $\mathfrak{g}_{k}$ is the algebra of
operators $\mathfrak{a}\in\mathfrak{gl}_{k}$, such that
$\mathfrak{a}^{t}.X_{k\times n}^{T}=0$. We mean the column vectors in the
$k$-jet are independent by generic. Then, if a matrix in $G_{k}$ fixes all the
jet components, it is a scalar matrix. The $k$-jet $z$ in the Elashvili
condition is generic in the above sense and has a generic stabilizer. In this
case, all the Lie algebras of stabilizers are conjugated over the dense open
subset of generic points under the adjoint action. Plugging into the
variables, we get a single matrix differential equation:
$\mathfrak{a}^{t}.X_{k\times n}^{T}=0$. ∎

######  Corollary 4.5.

(Main result.) We have:

  * •

The set $U=G_{k}.[J_{k}\mathbb{C}^{n}]^{\mathfrak{g}_{k,x}}$ is dense in
$J_{k}\mathbb{C}^{n}$.

  * •

$\mathbb{C}(J_{k}\mathbb{C}^{n}_{k}/{G_{k}})=\mathbb{C}((J_{k}\mathbb{C}^{n})^{\mathfrak{g}_{k,x}})^{W}$,
where $W=N(U)/Z(U)$ is the Weyl group of the action (at this point, it is not
necessarily finite).

###### Proof.

The proof is a consequence of Theorems 2.8 and 4.4. ∎

We now come to the most important main result of this paper. We want to use
the aforementioned literature to prove the finite generation of the fiber ring
of the moduli of $k$-jets. Our method uses the Igusa Theorem 2.15.

######  Theorem 4.6.

[Main result. Finite generation of fiber ring of regular invariant jets] The
algebra $\mathbb{C}[\widetilde{(J_{k}\mathbb{C}^{n})^{G_{k}}}]$ is finitely
generated, where $\widetilde{J_{k}\mathbb{C}^{n}/G_{k}}\to
J_{k}\mathbb{C}^{n}/{G_{k}}$ is the normalization. In particular, the stalk or
the fiber ring of $J_{k}\mathbb{C}^{n}/G_{k}$ at a regular point is finitely
generated.

###### Proof.

We employ the Igusa Theorem 2.15 for some restrictions of the fibration
$f:J_{k}\mathbb{C}^{n}\to J_{k}\mathbb{C}^{n}/{G_{k}}$. In order to match with
the set-up in the Igusa theorem, we may fix an affine subscheme $U=Spec(S)$
sitting in $J_{k}\mathbb{C}^{n}/{G_{k}}$ as an open subset and restrict $f$ to
its preimage $f|_{f^{-1}(U)}:f^{-1}(U)\to U$. The restriction is also given by
taking the quotient. If the theorem is proved, one can deduce it on the
original $f$, [see the discussion at the end of the proof].

We first check that the fibers of $f$ over a dense open subset of
$J_{k}\mathbb{C}^{n}/{G_{k}}$ contains dense $G_{k}$ orbits. This condition in
the Igusa theorem is instead to be checked for the local restriction of $f$;
it can be checked globally and not just over $U=Spec(S)$. If it gets proven
globally, it certainly holds for the restriction. Now, the locus of regular
jets is the complement of an algebraic set defined by certain algebraic
equations [[2] proposition 4.3]. The same holds in the quotient space
$J_{k}\mathbb{C}^{n}/{G_{k}}$, i.e., invariant regular jets are also open
downstairs. Because the $G_{k}$-action does not change the regularity, it
follows that the regular locus is also a union of $G_{k}$-orbits. By (58) for
the complement, we have,

(70) $\text{codim}_{J_{k}\mathbb{C}^{n}}(J_{k}\mathbb{C}^{n}\smallsetminus
J_{k}^{\text{reg}}\mathbb{C}^{n})\geq 2.$

The regular $k$-jets $J_{k}^{\text{reg}}\mathbb{C}^{n}$ and their quotient
$J_{k}^{\text{reg}}\mathbb{C}^{n}/{G_{k}}$ give the corresponding dense open
sets and the orbits where the Igusa Theorem applies, which are also a union of
$G_{k}$ orbits. The same assertion will hold if we intersect with $U$ and its
preimage. Because the action of $G_{k}$ on $J_{k}\mathbb{C}^{n}$ has a generic
stabilizer, or by (13) one has:

(71) $\text{Max}_{x\in J_{k}\mathbb{C}^{n}}\dim G_{k}.x=\dim
J_{k}\mathbb{C}^{n}-\dim J_{k}\mathbb{C}^{n}/G_{k}.$

The left side of this identity is constant. On the other hand, because
$J_{k}^{\text{reg}}\mathbb{C}^{n}$ is open in $J_{k}\mathbb{C}^{n}$, in the
identity (71) we can replace $J_{k}^{\text{reg}}\mathbb{C}^{n}$. It follows
that

(72) $\dim J_{k}\mathbb{C}^{n}-\dim J_{k}^{\text{reg}}\mathbb{C}^{n}=\dim
J_{k}\mathbb{C}^{n}/G_{k}-\dim J_{k}^{\text{reg}}\mathbb{C}^{n}/G_{k}.$

Therefore with (70) we have,

(73)
$\text{codim}_{J_{k}\mathbb{C}^{n}/G_{k}}([J_{k}\mathbb{C}^{n}/G_{k}]\smallsetminus[J_{k}^{\text{reg}}\mathbb{C}^{n}/G_{k}])\geq
2.$

Thus, the Igusa theorem works over the normalization of
$\widetilde{{J_{k}\mathbb{C}^{n}}/{G_{k}}}$.

The second assertion of the Theorem on the finite generation of the stalk on
the regular jets will follow the previous argument. All these stalks are
isomorphic, so it suffices to obtain the result at one of them, which can be
any point on the regular locus of $U$. The stalk of
$J_{k}\mathbb{C}^{n}/G_{k}$ at the normal points is the same as the stalks of
the normalization.

To complete the proof of the first statement in the Theorem, we note that the
open affine subset $U$ was arbitrary. However, we can cover
$\widetilde{{J_{k}\mathbb{C}^{n}}/{G_{k}}}$ by a finite number of them, due to
the quasi-compactness property that we have on the preimage, plus the
surjectivity of quotient morphism. We need to check if we have a sheaf of
rings over a topological space covered by a finite number of open affine
subvarieties that are the spectrum of finitely generated rings. If so, one can
conclude that the ring of global sections of the sheaf itself is finitely
generated. The latter follows from the fact that $I(U_{1}\bigcup
U_{2}\dots\bigcup U_{N})=I(U_{1})\bigcap\dots\bigcap I(U_{N})$, where $U_{i}$
are open affine varieties and $I(U_{i})$ indicates the ideal of $U_{i}$. ∎

Finally, we have the following analogue of Theorem 4.3 for the $\rho$-action.

######  Theorem 4.7.

(Main result) Under the action of $\rho$ on $V=J_{k}\mathbb{C}^{n}$, the
following holds:

  1. (1)

$\mathbb{C}[J_{k}\mathbb{C}^{n}]^{G_{k}}$ is a polynomial algebra.

  2. (2)

$\text{Max}_{x\in J_{k}\mathbb{C}^{n}}\dim G_{k}.x=\dim
J_{k}\mathbb{C}^{n}-\dim J_{k}\mathbb{C}^{n}//G_{k}$.

  3. (3)

The quotient map $\pi:J_{k}\mathbb{C}^{n}\to J_{k}\mathbb{C}^{n}//G_{k}$ is
equidimensional over the normal locus.

  4. (4)

Let $\Omega=\\{x\in J_{k}\ |\ d_{x}\pi\ \text{is onto}\ \\}$, then,
$J_{k}\mathbb{C}^{n}\setminus\Omega$ contains no divisors.

  5. (5)

The null cone $\pi^{-1}\pi(0)$ is an irreducible complete intersection of
dimension $\dim J_{k}\mathbb{C}^{n}-\dim J_{k}\mathbb{C}^{n}//G_{k}$.

###### Proof.

The proof is analogous to the proof of Theorem 4.3 with the action replaced by
the representation $\rho$, where Theorem 4.4 applies. Item 1 follows from
Theorem 4.4. Item 2 is also a consequence of the fact that the action of
$G_{k}$ on $V=J_{k}\mathbb{C}^{n}$ has generic stabilizers, i.e. Theorem 4.4.
Item 3 also follows Theorem 4.3. Item 4 follows from similar argument in the
proof of Theorem 4.3 and the same item there. If we let $f_{1},...,f_{l}$ be
the basic invariants for the action of $G_{k}$ on
$\mathbb{C}[J_{k}\mathbb{C}^{n}]$, and set,

(74) $X_{r}=\\{x\in J_{k}\mathbb{C}^{n}|\dim\langle
df_{1}(x),...,df_{l}(x)\rangle=r\\},$

then, by $\text{codim}_{J_{k}\mathbb{C}^{n}}(J_{k}\mathbb{C}^{n}\setminus
J_{k}^{\text{reg}}\mathbb{C}^{n})\geq 2$ in (58) and Theorems 2.10 and 2.11,
we get $X_{l}\supset J_{k}^{\text{reg}}\mathbb{C}^{n}$ and we can proceed as
in Theorem 4.3 to get $\text{codim}(J_{k}\setminus X_{l})\geq 2$. Now, the
item 4 follows. Item 5 follows from the fact that the ideal of the null cone
is generated by the basic invariants of the ring
$\mathbb{C}[J_{k}\mathbb{C}^{n}]$ [cf. [24] loc. cit.]. ∎

######  Theorem 4.8.

[Main result. Finite generation of the coordinate ring of invariant jets] The
algebra $\mathbb{C}[{(J_{k}\mathbb{C}^{n})]^{G_{k}}}$ is finitely generated.

###### Proof.

The proof is a mixture of the argument on the application of Igusa theorem in
Theorem 4.6, and a stratification property of the variety
$J_{k}\mathbb{C}^{n}/G_{k}$ due to the action of $G_{k}$, cf. [1]. By Theorem
1.1 of [1] there exists a stratification,

(75) $J_{k}=\bigsqcup_{\beta\in B}S_{\beta}$

into disjoint union of locally closed subschemes $S_{\beta}$, where $\beta\in
B$ are partially ordered and such that $\overline{S_{\beta}}\subset
S_{\beta}\bigcup_{\delta>\beta}S_{\delta}$. The strategy is to apply the Igusa
theorem on each strata $S_{\beta}$. Because $J_{k}^{reg}\cap S_{\beta}$ is
dense in $S_{\beta}$ for each $\beta$, according to what we said in the proof
of Theorem 4.6, it just remains to check the codimension 2 condition in the
Igusa theorem. In fact, by the local closedness of $S_{\beta}$ and openness of
$J_{k}^{reg}$, we still have finite generation of the coordinate ring of
invariant jets,

(76)
$\text{codim}_{S_{\beta}}(S_{\beta}\smallsetminus\left(J_{k}^{\text{reg}}\mathbb{C}^{n}\bigcap
S_{\beta}\right))\geq 2.$

The $G_{k}$-action on the strata $S_{\beta}$ is given by the 1-parameter orbit
$\lambda_{\beta}(t)$ of $G_{k}$ associated to $\beta$, [see [1] for details].
By the same argument as in the proof of Theorem 4.6 and the equations
(71)-(73), where $G_{k}$ is replaced by $\lambda_{\beta}(\mathbb{G}_{m})$ and
$J_{k}$ is replaced by $S_{\beta}$, we obtain the codimension 2 condition in
the Igusa theorem for the fibration on each strata. One also has,

(77)
$\text{codim}_{S_{\beta}/G_{k}}([S_{\beta}/G_{k}]\smallsetminus[\left(J_{k}^{\text{reg}}\mathbb{C}^{n}\bigcap
S_{\beta}\right)/G_{k}])\geq 2.$

The strata $S_{\beta}$ is smooth for all $\beta$. Thus, by the Igusa theorem,
the coordinate rings $\mathbb{C}[S_{\beta}^{G_{k}}]$ are finitely generated.
Because the stratification is disjoint and finite, this implies that
$\mathbb{C}[J_{k}^{G_{k}}]$ is finitely generated. ∎

Finally, we prove a finiteness result for the Grothendieck ring of finitely
generated $G_{k}$-modules.

######  Theorem 4.9.

(Main result.) Assume $G_{k}$ is the group of reparametrizations of $k$-jets
of holomorphic curves. Let $T_{k}$ be a maximal torus of $G_{k}$, and $W_{k}$
the associated Weyl group. Then, the representation ring of $G_{k}$ satisfies
$R(G_{k})=R(T_{k})^{W_{k}}$.

###### Proof.

We have $G_{k}=\mathbb{C}^{*}\times U_{k}$. Let
$G^{\prime}=[G_{k},G_{k}]=U_{k}$ be the derived group of $G_{k}$, and $H\to
G^{\prime}$ be its universal (finite) cover. Then, we have $G_{k}=Z\times
G^{\prime}/\pi_{1}$, where $Z$ is a maximal central torus and $\pi_{1}\subset
Z\times G^{\prime}$ is a finite central and closed subgroup.

On the other hand, as we also mentioned in the proof of the Elashvili
condition in Theorem 4.1, by letting
$x\in\mathfrak{g}_{k}\subset\mathfrak{gl}_{k}$ to be a generic semisimple
element implies that $\text{ad}(x)(\mathfrak{g}_{k})=\mathfrak{u}_{k}$. This
is because the ad function is the same as the one in $\mathfrak{gl}_{k}$, and
we may choose a complete set of root vectors that are eigenvectors of
$\text{ad}(x)$. This proves that
$[\mathfrak{g}_{k},\mathfrak{g}_{k}]=\mathfrak{g}_{k}$. In other words,
$\mathfrak{g}_{k}$ is semisimple. Therefore, $G_{k}$ is semisimple.

By Theorem 4.2, the Weyl group $W_{k}$ is finite. Summing up the above two
paragraphs, we get that $H$ is semisimple and simply connected. Now, the claim
of the Theorem holds for $Z$ and $H$. Therefore, it holds for $Z\times H$. By
the first paragraph in the proof, it remains to show that the same claim
descends to central finite quotients. This has been done in [9] page 294, but
we brief the argument below.

First, we note that if $G_{1}=G_{2}/\pi_{1}$ is such a quotient, then we have
$T_{1}=T_{2}/\pi_{1}$, which implies
$W(G_{1},T_{1})=N_{G_{1}}(T_{1})/T_{1}=N_{G_{2}}(T_{2})/T_{2}=W(G_{2},T_{2})$.
Therefore, we have the following diagram:

(78) $\begin{CD}R(G_{1})@>{}>{}>R(T_{1})^{W_{1}}\\\ @V{}V{}V@V{}V{}V\\\
R(G_{2})@>{}>{}>R(T_{2})^{W_{2}}\end{CD}$

where all the arrows are inclusions. By thinking of class functions,
$R(T_{i})$ is the $\mathbb{Z}$-span of irreducible characters trivial on $Z$.
The same holds for $R(G_{i})$ by Schur lemma. Passing to $W$-invariants, the
class functions in $R(T_{i})^{W_{i}}$ are invariant under $Z$-translation. But
a class function on $G_{i}$ is determined by its restriction on $T_{i}$. Thus,
the $Z$ invariance of the class functions can be checked on the torus. The
claim follows. ∎

##  5\. Conclusions

This text proves the well-known conjecture on the finite generation of fiber
ring of the moduli of $k$-jets of holomorphic curves at a regular point. We
also show that the total coordinate ring of generic $k$-jets, i.e.,
$\mathbb{C}[J_{k}\mathbb{C}^{n}]^{G_{k}}$, is finitely generated, where
$G_{k}$ is the group of reparametrizations of the holomorphic curves. Finally,
some of the main properties of the invariant theory of the non-reductive group
$G_{k}$ have been presented based on the generic stabilizer property of its
action.

## Author contributions statement

M. Reza-Rahmati and G. Flores reviewed the manuscript and contributed equally
to work.

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