# The Chevalley-Weil formula on nodal curves

Yubo TONG  Xiamen University
School of Mathematical Sciences
Siming South Road 422
Xiamen, Fujian 361005 (China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the eigensubspace of the space of the holomorphic
differentials of nodal curves over the algebracally closed field under the
action of finite automorphism groups. We compute the Chevalley-Weil formula
with some additional contidions of the quotient curve and give some examples.

> ###### Contents

  1. 1 Introduction
  2. 2 Preliminary
    1. 2.1 Notations
    2. 2.2 Ramification modules
  3. 3 Irreducible nodal curves
    1. 3.1 The $G$-invariant differentials
    2. 3.2 Chevalley-Weil formula for irreducible nodal curves
  4. 4 Nodal curves with several irreducible components
    1. 4.1 Reduction
    2. 4.2 Chevalley-Weil formula for connected nodal curves

##  1\. Introduction

Let $X$ be a connected projective smooth curve over an algebraically closed
field $k$ and $G\subseteq\operatorname{Aut}(X)$ be a finite subgroup. Then $G$
acts in a natural way on the space of the holomorphic differentials on $X$,
thus we obtain a linear representation
$G\rightarrow\operatorname{GL}\left(H^{0}(X,\omega_{X})\right)$. A basic
problem is to determine how many times a given irreducible representation of
$G$ occurs in $H^{0}(X,\omega_{X})$.

This problem was first considered by Hurwitz [6] for $G$ cyclic over
$k=\mathbb{C}$. Then in the $30$s of the 20th century, Chevalley and Weil [3]
solved this problem for general $G$ when $\pi:X\rightarrow X/G$ is unramified.
Soon after, Weil [15] solved the case for general $\pi$. This result was named
as the Chevalley-Weil formula and it remains valid for any algebraically
closed field $k$ with $\operatorname{char}(k)=p\nmid\\#G$ [7].

When $\operatorname{char}(k)=p>0$ and $p\mid\\#G$, the structure of
$H^{0}(X,\Omega_{X})$ becomes more complicated. Except the tame ramification
case ([7], [13]), or weakly ramified case([8]), people focus on some special
groups ([14] for the case of cyclic groups, [10] for abelian groups, [4] for
$p$-groups or [2] for groups with a cyclic Sylow subgroup).

In the 1980s, Kani studied the projectivity of the logarithmic differentials
space $H^{0}(X,\Omega_{X}(D))$ as $k[G]-$module in the tamely ramified
case[7]. But most of his work was covered by Nakajima’s work([12],[13]). The
latter improved Mumford’s method[11, $II.5$ Lemma 1] to study the
$H^{i}\left(X,\mathcal{G}\right)$ of the coherent $G$-sheaf $\mathcal{G}$ in
the tamely ramified Galois covering for any dimensional projective varieties.
Nevertheless, Kani’s work gives us sereval valuable tools.

For smooth curves, the Chevalley-Weil formula was well understood by now. In
this paper, we will follow Kani’s methods, and generalize the Chevalley-Weil
formula to the nodal curves for one-dimensional $G$-representations with
$\operatorname{char}(k)=0$ or prime to $\\#G$.

Acknowledgements. I would like to thank my supervisor Wenfei Liu for his
support. I am grateful to Professor Qing Liu for writing advice and helpful
discussions during my visit at University of Bordeaux. This work has been
supported by the NSFC (No. 11971399) and by the Presidential Research Fund of
Xiamen University(No. 20720210006).

##  2\. Preliminary

###  2.1. Notations

In this paper, we consider a finite group $G$ acting faithfully on a nodal
curve $X$ over an algebraically closed field $k$. Let $\\#G=n$ and
$\operatorname{char}(k)=p\nmid n$ or $\operatorname{char}(k)=0$, which implies
that $k[G]$ is semi-simple. A _curve_ means an equidimensional reduced
projective scheme of finite type of dimension $1$ over $k$.

Let $X$ be a nodal curve, $\omega_{X}$ the canonical (dualizing) sheaf of $X$.
Let $\hat{X}\overset{\alpha}{\rightarrow}X$ be the normalizaiton of $X$, then
it induces an immersion $H^{0}(X,\omega_{X})\hookrightarrow
H^{0}(\hat{X},\Omega_{\hat{X}}(\hat{S}_{X}))$, where $\hat{S}_{X}$ is the
preimage of singularities(nodes) of X. For a node $P\in\alpha(\hat{S}_{X})$,
we say $\\{P_{1},P_{2}\\}=\alpha^{-1}(P)$ a _pair_ of $P$.

An element $\varphi_{0}\in H^{0}(X,\Omega_{\hat{X}}(\hat{S}_{X}))$ belongs to
$H^{0}(X,\omega_{X})$ if and only if
$\operatorname{Res}_{P_{1}}\varphi_{0}+\operatorname{Res}_{P_{2}}\varphi_{0}=0$
for any pair $\\{P_{1},P_{2}\\}$. Such an element is called a holomorphic
differential of $X$. It is known that $H^{0}(X,\omega_{X})$ is a $k$-vector
space of dimension $p_{a}$, the arithmetic genus of $X$.

Both the rational function field $K(X)$ and $H^{0}(X,\omega_{X})$ are
naturally (right)$k[G]$-modules, and every $1$-dimensional representation is
its character. Our goal is to compute the multiplicity of any $1$-dimensional
representation $\chi$, that is the dimension of $H^{0}(X,\omega_{X})_{\chi}$
over $k$. Note that all the irreducible representations will be
$1$-dimensional when $G$ is abelian.

Let $X$ be smooth for the rest of this section. Now we recall some properties
for smooth curves.

Consider the branched cover $\pi:X\rightarrow X/G=Y$ and let $e_{P}$ be the
ramification index at $P\in X$, then we have the _ramification divisor_

$R_{\pi}=\sum_{P\in X}(e_{P}-1)P.$

For a divisor $D=\sum a_{i}P_{i}\in\operatorname{Div}(X)$, define
$\pi_{*}D\in\operatorname{Div}(Y)$ by

$\pi_{*}D=\sum a_{i}\pi(P_{i}).$

If $D=\sum a_{i}Q_{i}\in\operatorname{Div}(Y)$ is a divisor and
$r\in\mathbb{R}$, then define $\lfloor rD\rfloor\in\operatorname{Div}(Y)$ by

$\lfloor rD\rfloor=\sum\lfloor ra_{i}\rfloor Q_{i},$

where $\lfloor ra_{i}\rfloor$ denotes the greatest integer $\leq ra_{i}$. And
define $\pi^{*}D\in\operatorname{Div}(Y)$ by

$\pi^{*}D=\sum_{i}a_{i}(\sum_{P\in\pi^{-1}(Q_{i})}e_{P}~{}P).$

######  Proposition 2.1 (Kani [7]).

Let $G$ be a finite group acting on a smooth curve $X$ with $R_{\pi}$ the
ramification divisor of $\pi:X\rightarrow X/G=Y$. Consider a $G$-invariant
divisor $D\in\operatorname{Div}(X)$, then for the trivial character
$\chi=1_{G}$, we have

(1) $\displaystyle H^{0}(X,\mathcal{O}_{X}(D))^{G}$
$\displaystyle=\pi^{*}H^{0}(Y,\mathcal{O}_{Y}\left\lfloor
n^{-1}\pi_{*}D\right\rfloor),$ (2) $\displaystyle H^{0}(X,\Omega_{X}(D))^{G}$
$\displaystyle=\pi^{*}H^{0}(Y,\Omega_{Y}\left\lfloor
n^{-1}\pi_{*}(D+R_{\pi})\right\rfloor).$

For a $1$-dimensional character $\chi$, let $f_{\chi}\in K(X)^{*}$ be such
that $\sigma f_{\chi}=\chi(\sigma)f_{\chi}$ for all $\sigma\in G$ (whose
existence is guaranteed by Hilbert’s theorem 90). Then

(3) $\displaystyle H^{0}(X,\mathcal{O}_{X}(D))_{\chi}$
$\displaystyle=f_{\chi}\cdot\pi^{*}H^{0}(Y,\mathcal{O}_{Y}\left\lfloor
n^{-1}\pi_{*}\left(D+\left(f_{\chi}\right)\right)\right\rfloor),$ (4)
$\displaystyle H^{0}(X,\Omega_{X}(D))_{\chi}$
$\displaystyle=f_{\chi}\cdot\pi^{*}H^{0}(Y,\Omega_{Y}\left\lfloor
n^{-1}\pi_{*}\left(D+\left(f_{\chi}\right)+R_{\pi}\right)\right\rfloor).$

###### Proof.

(More details here than in [7].) Note that $D\geq\pi^{*}\left\lfloor
n^{-1}\pi_{*}D\right\rfloor$ and hence
$H^{0}(X,\mathcal{O}_{X}(D))^{G}\supseteq\pi^{*}H^{0}(Y,\mathcal{O}_{Y}\left\lfloor
n^{-1}\pi_{*}D\right\rfloor)$. Conversely, if $f\in
H^{0}(X,\mathcal{O}_{X}(D))^{G}$, then $f=\pi^{*}e$ with some $e\in K(Y)$.
Hence $\pi_{*}((f)+D)=n(e)+\pi_{*}D\geq 0$, which implies $(e)+\left\lfloor
n^{-1}\pi_{*}D\right\rfloor\geq 0$. This proves (1).

To prove (2), fix a meromorphic differential $0\neq\varphi\in\Omega(Y)$, which
always exists by Riemann-Roch. By
$H^{0}(X,\Omega_{X}(D))=H^{0}(X,\mathcal{O}_{X}\left(D+\left(\pi^{*}\varphi\right)\right))\cdot\pi^{*}\varphi=H^{0}(X,\mathcal{O}_{X}\left(D+\pi^{*}(\varphi)+R_{\pi}\right))\cdot\pi^{*}\varphi,$
we have

$\displaystyle H^{0}(X,\Omega_{X}(D))^{G}$
$\displaystyle=H^{0}(X,\mathcal{O}_{X}\left(D+\pi^{*}(\varphi)+R_{\pi}\right))^{G}\cdot\pi^{*}\varphi$
$\displaystyle=\pi^{*}H^{0}(Y,\mathcal{O}_{Y}\left\lfloor
n^{-1}\pi_{*}(\left(D+\pi^{*}(\varphi)+R_{\pi}\right))\right\rfloor)\cdot\pi^{*}\varphi$
$\displaystyle=\pi^{*}[H^{0}(Y,\mathcal{O}_{Y}(\lfloor
n^{-1}\pi_{*}(D+R_{\pi})\rfloor)+(\varphi))\cdot\varphi]$
$\displaystyle=\pi^{*}H^{0}(Y,\Omega_{Y}\left\lfloor
n^{-1}\pi_{*}(D+R_{\pi})\right\rfloor).$

Finally, (3) and (4) for general $\chi$ is followed by

$H^{0}(X,\mathcal{O}_{X}(D))_{\chi}=f_{\chi}\cdot
H^{0}(X,\mathcal{O}_{X}\left(D+\left(f_{\chi}\right)\right))^{G},$
$H^{0}(X,\Omega_{X}(D))_{\chi}=f_{\chi}\cdot
H^{0}\left(X,\Omega_{X}(D+(f_{\chi}))\right)^{G}.$

∎

###  2.2. Ramification modules

[7, Kani] Let $Bl(Y)$ be the _branch locus_ of $\pi:X\rightarrow Y$.

Fix a point $P\in X$, and let $G_{P}$ be the stablizer subgroup of $G$ at $P$,
which is a cyclic of order $e_{P}$. Then there is a unique character
$\theta_{P}:G_{P}\rightarrow k^{*}$ such that for any $f\in K(X)^{*}$,

$\frac{\sigma
f}{f}\equiv\theta_{P}(\sigma)^{v_{P}(f)}(\operatorname{mod}~{}\mathfrak{m}_{P}),\quad\forall\sigma\in
G_{P},$

where $v_{P}$ denotes the _valuation_ at $P$ and $\mathfrak{m}_{P}$ the
maximal ideal of the local ring $\mathcal{O}_{P}$.

Set

$R_{G,P}:=\operatorname{Ind}_{G_{P}}^{G}\left(\bigoplus_{d=0}^{e_{P}-1}d\cdot\theta_{P}^{d}\right).$

######  Definition 2.2.

For a point $Q\in Y$, define _the ramification module of $Q$_

$R_{G,Q}:=\bigoplus_{P\in\pi^{-1}(Q)}R_{G,P},$

and _the ramification module of $\pi$_

$R_{G}:=\bigoplus_{Q\in Y}R_{G,Q_{i}}.$

Note that this is a finite sum because $R_{G,Q}=0$ for $Q\notin Bl(Y)$.

Consider an $f_{\chi}\in K(X)^{*}$ such that $\sigma
f_{\chi}=\chi(\sigma)f_{\chi}$ for any $\sigma\in G$ in Proposition 2.1. Since
$f_{\chi}^{n}\in\pi^{*}k(Y)$, write $\left(f_{\chi}^{n}\right)=\pi^{*}(nA+B)$
where $A,B\in\operatorname{Div}(Y)$ and $\lfloor n^{-1}B\rfloor=0$.

Note that $\operatorname{Supp}(B)\subseteq Bl(Y)$, so we write $B=\sum_{Q\in
Bl}b_{Q}Q$. By definition, we have

$b_{Q}=n\left\langle\frac{v_{Q}\left(f_{\chi}^{n}\right)}{n}\right\rangle,$

where $\langle r\rangle=r-\lfloor r\rfloor$ denotes the fractional part of
$r$. The following lemma shows that this $B$ is independent of the choices of
$f_{\chi}$.

######  Lemma 2.3.

Let $\chi:G\rightarrow k^{*}$ be a $1$-dimensional character. Then for any
$Q\in Bl(Y)$, we have

(5)
$n\left\langle\frac{v_{Q}\left(f_{\chi}^{n}\right)}{n}\right\rangle=\left\langle\chi,R_{G,Q}\right\rangle_{G}.$

###### Proof.

Let $P\in\pi^{-1}(Q)$. Then by Frobenius reciprocity, we have

(6)
$\displaystyle\left\langle\chi,R_{G,P}\right\rangle_{G}=\left\langle\chi|_{G_{P}},\bigoplus_{d=0}^{e_{P}-1}d\cdot\theta_{P}^{d}\right\rangle_{G_{P}}.$

Note that $\theta_{P}^{d}$ runs through are all the irreducible
representations of $G_{P}$, hence we have

(7)
$\left\langle\chi,R_{G,P}\right\rangle_{G}=a\Leftrightarrow\chi|_{G_{P}}=\theta_{P}^{a}$

with $0\leq a<e_{P}$. Choose a generator $\sigma$ of $G_{P}$, then by
definition of $f_{\chi}$, we have

$\sigma f_{\chi}=\chi(\sigma)f_{\chi}=\theta_{P}(\sigma)^{a}f_{\chi}.$

Furthermore, by the definition of $\theta_{P}$, we have

$\theta_{P}(\sigma)^{a}=\frac{\sigma
f_{\chi}}{f_{\chi}}\equiv\theta_{P}(\sigma)^{v_{P}(f_{\chi})}(\operatorname{mod}~{}\mathfrak{m}_{P}),$

which implies $a\equiv v_{P}(f_{\chi})(\operatorname{mod}~{}e_{P})$ since
$\theta_{P}(\sigma)$ has order $e_{P}$ in $k^{*}$. Finally,

$\frac{\left\langle\chi,R_{G,P}\right\rangle_{G}}{e_{P}}=\left\langle\frac{v_{P}(f_{\chi})}{e_{P}}\right\rangle=\left\langle\frac{v_{Q}(f_{\chi}^{n})}{n}\right\rangle=\frac{b_{Q}}{n},$

namely we have $\left\langle\chi,R_{G,Q}\right\rangle_{G}=b_{Q}$ .

∎

##  3\. Irreducible nodal curves

Let $X$ be an irreducible nodal curve in this section.

###  3.1. The $G$-invariant differentials

Let $G$ be a finite group acting on an irreuducible nodal curve $X$ and
$Y=X/G$ the quotient curve. For the space $H^{0}(X,\omega_{X})^{G}$ of
$G$-invariant differentials, it is a classical fact that

######  Proposition 3.1.

If $X$ is smooth, then

$\mathrm{dim}_{k}H^{0}(X,\Omega_{X})^{G}=g_{Y}.$

###### Proof.

With the notations of §2.1, let $e_{Q}:=e_{P}$ for any $P\in\pi^{-1}(Q)$. Note
that

$\lfloor n^{-1}\pi_{*}R_{\pi}\rfloor=\sum_{Q\in
Y}\left\lfloor\frac{e_{Q}-1}{e_{Q}}\right\rfloor Q=0.$

By Proposition 2.1 (2), we have

$H^{0}(X,\Omega_{X})^{G}=\pi^{*}H^{0}(Y,\Omega_{Y}\left\lfloor
n^{-1}\pi_{*}R_{\pi}\right\rfloor)=\pi^{*}H^{0}(Y,\Omega_{Y}).$

∎

Here comes a natural qustion that for the covering $\pi:X\rightarrow X/G=Y$ of
nodal curves, do we still have the equality

(8) $\displaystyle\mathrm{dim}_{k}H^{0}(X,\omega_{X})^{G}=p_{a}(Y)?$

Consider the normalizations $\hat{X}\rightarrow X$ and $\hat{Y}\rightarrow Y$,
respectively. We have $\hat{X}/G=\hat{Y}$, so there is a commutative diagram

${\hat{X}}$${\hat{Y}}$${X}$${Y.}$$\scriptstyle{\hat{\pi}}$$\scriptstyle{\pi}$

This induces the corresponding morphisms of differentials

(9)
${H^{0}(X,\omega_{X})}$${\pi^{*}H^{0}(Y,\omega_{Y})}$${H^{0}(\hat{X},\Omega_{\hat{X}}(\hat{S}_{X}))}$${\hat{\pi}^{*}H^{0}(\hat{Y},\Omega_{\hat{Y}}(\hat{S}_{Y})),}$$\scriptstyle{?}$

because $X\rightarrow X/G$ takes smooth points to smooth points, so
$\hat{\pi}^{-1}(\hat{S}_{Y})\subseteq\hat{S}_{X}$.

######  Lemma 3.2.

The upper row

$\pi^{*}H^{0}(Y,\omega_{Y})\subseteq H^{0}(X,\omega_{X})$

of (9) exists if and only if the ramification indexes $e_{P_{1}}=e_{P_{2}}$
for all pairs $\\{P_{1},P_{2}\\}\subseteq\hat{S}_{X}$.

###### Proof.

Given some $\varphi_{Y}\in H^{0}(Y,\omega_{Y})$, we have
$\operatorname{Res}_{\hat{\pi}(P_{1})}\varphi_{Y}=\operatorname{Res}_{\hat{\pi}(P_{2})}\varphi_{Y}$.
Note that for any $P\in\hat{X}$, we have
$\operatorname{Res}_{P}(\hat{\pi}^{*}\varphi_{Y})=e_{P}\cdot\operatorname{Res}_{\hat{\pi}(P)}\varphi_{Y}$
. Hence for any pair $\\{P_{1},P_{2}\\}\subseteq\hat{S}_{X}$, we have

(10)
$\operatorname{Res}_{P_{1}}(\hat{\pi}^{*}\varphi_{Y})=\operatorname{Res}_{P_{2}}(\hat{\pi}^{*}\varphi_{Y})$

if and only if $e_{P_{1}}=e_{P_{2}}$. ∎

Note that the points of $\hat{S}_{X}-\hat{\pi}^{-1}(\hat{S}_{Y})$ are mapped
to the smooth part of $Y$.

######  Lemma 3.3.

For the left column of (9), we have

$H^{0}(X,\omega_{X})^{G}\hookrightarrow
H^{0}(\hat{X},\Omega_{\hat{X}}(\pi^{-1}(\hat{S}_{Y})))^{G}.$

###### Proof.

Let $\varphi\in H^{0}(X,\omega_{X})^{G}$, and a pair
$\\{P_{1},P_{2}\\}\subseteq\hat{S}_{X}-\hat{\pi}^{-1}(\hat{S}_{Y})$. Then
there exists some $\sigma\in G$ such that $\sigma(P_{1})=P_{2}$, which implies
that
$\operatorname{Res}_{P_{1}}\varphi=\operatorname{Res}_{P_{1}}\sigma\varphi=\operatorname{Res}_{\sigma(P_{1})}\varphi=\operatorname{Res}_{P_{2}}\varphi.$
As $\operatorname{Res}_{P_{1}}\varphi=-\operatorname{Res}_{P_{2}}\varphi$, we
get $\operatorname{Res}_{P_{1}}\varphi=\operatorname{Res}_{P_{2}}\varphi=0$,
and $\varphi$ is holomorphic at $\\{P_{1},P_{2}\\}$. ∎

If for all pairs $\\{P_{1},P_{2}\\}\subseteq\hat{S}_{X}$, we have
$e_{P_{1}}=e_{P_{2}}$, then we can give a positive answer to (8).

######  Proposition 3.4.

With the notations above, we have

$\hat{\pi}^{*}H^{0}(\hat{Y},\Omega_{\hat{Y}}(\hat{S}_{Y}))=H^{0}(\hat{X},\Omega_{\hat{X}}(\pi^{-1}(\hat{S}_{Y})))^{G}.$

Moreover, if the ramification indexes for any pair
$\\{P_{1},P_{2}\\}\subseteq\hat{S}_{X}$ are equal, namely
$e_{P_{1}}=e_{P_{2}}$, then we have the canonical commutative diagram

${H^{0}(X,\omega_{X})^{G}}$${H^{0}(Y,\omega_{Y})}$${H^{0}(\hat{X},\Omega_{\hat{X}}(\pi^{-1}(\hat{S}_{Y})))^{G}}$${H^{0}(\hat{Y},\Omega_{\hat{Y}}(\hat{S}_{Y})),}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\hat{\pi}^{*}}$

and the rows are both isomorphisms. In particular, we have
$\mathrm{dim}_{k}H^{0}(X,\omega_{X})^{G}=p_{a}(Y)$.

###### Proof.

Now we treat $\hat{S}_{Y}$ as a positive divisor here. By Proposition 2.1 (2),
we have

$H^{0}(\hat{X},\Omega_{\hat{X}}(\pi^{-1}(\hat{S}_{Y})))^{G}=\pi^{*}H^{0}(\hat{Y},\Omega_{\hat{Y}}(\left\lfloor
n^{-1}\pi_{*}(\pi^{-1}(\hat{S}_{Y})+R_{\pi})\right\rfloor)).$

Consider the coefficient of prime divisors in $\left\lfloor
n^{-1}\pi_{*}(\pi^{-1}(\hat{S}_{Y})+R_{\pi})\right\rfloor=\sum a_{Q}Q$.
(1) If $Q\in Bl(Y)-\hat{S}_{Y}$, then
$a_{Q}=\lfloor\frac{e_{Q}-1}{e_{Q}}\rfloor=0$;
(2) If $Q\in\hat{S}_{Y}-Bl(Y)$, then $a_{Q}=1$;
(3) If $Q\in\hat{S}_{Y}\cap Bl(Y)$, then
$a_{Q}=\lfloor\frac{1}{e_{Q}}+\frac{e_{Q}-1}{e_{Q}}\rfloor=1$.

So we have $\left\lfloor
n^{-1}\pi_{*}(\pi^{-1}(\hat{S}_{Y})+R_{\pi})\right\rfloor=\hat{S}_{Y}$, hence
the isomorphism on the lower row. Note that both $H^{0}(X,\omega_{X})^{G}$ and
$H^{0}(Y,\omega_{Y})$ are the subspaces satisfying the residue relations, then
we have the isomorphism of the upper row. ∎

###  3.2. Chevalley-Weil formula for irreducible nodal curves

Let $\chi$ be a $1$-dimensional character of $G$. With the notations in (9),
we consider the embedding

$H^{0}(X,\omega_{X})_{\chi}\rightarrow
H^{0}(\hat{X},\Omega_{\hat{X}}(\hat{S}_{X}))_{\chi}.$

######  Proposition 3.5.

Let $Y$ be smooth, set

(11) $\hat{S}_{X}^{\chi}=\left\\{\hat{P}\in\hat{S}_{X}\mid\exists\tau\in
G_{\alpha(\hat{P})}\text{ s.t. }\tau(\hat{P})\neq\hat{P}\text{ and
}\chi(\tau)=-1\right\\}.$

Then the image of $H^{0}(X,\omega_{X})_{\chi}$ in
$H^{0}(\hat{X},\Omega_{\hat{X}}(\hat{S}_{X}))_{\chi}$ is equal to
$H^{0}(\hat{X},\Omega_{\hat{X}}(\hat{S}_{X}^{\chi}))_{\chi}$. So we have an
isomorphism

(12) $\displaystyle
H^{0}(X,\omega_{X})_{\chi}\overset{\sim}{\rightarrow}H^{0}(\hat{X},\Omega_{\hat{X}}(\hat{S}_{X}^{\chi}))_{\chi}~{}.$

We call $\hat{S}_{X}^{\chi}$ _the singular $\chi$-set of $X$_.

###### Proof.

Assume $\varphi\in
H^{0}(\hat{X},\Omega_{\hat{X}}(\hat{S}_{X}^{\chi}))_{\chi}$, and let
$\alpha^{-1}(P)=\\{P_{1},P_{2}\\}\subseteq\hat{S}_{X}^{\chi}$. So there is a
$T\in G_{P}$ with $T(P_{1})=P_{2}$ and $\chi(T)=-1$ by the definition of
$\hat{S}_{X}^{\chi}$. Hence

$-\operatorname{Res}_{P_{1}}\varphi=\operatorname{Res}_{P_{1}}T\varphi=\operatorname{Res}_{T^{-1}(P_{1})}\varphi=\operatorname{Res}_{P_{2}}\varphi.$

Conversely, given some $\varphi_{0}\in H^{0}(X,\omega_{X})_{\chi}$ with poles
on a pair $\alpha^{-1}(P)=\\{P_{1},P_{2}\\}\subseteq\hat{S}_{X}$ and some
$T\in G_{P}$ with $T(P_{1})=P_{2}$, which exists by the smooth of $Y$, we have

$\chi(T)\operatorname{Res}_{P_{1}}(\varphi_{0})=\operatorname{Res}_{P_{1}}(T\varphi_{0})=\operatorname{Res}_{P_{2}}(\varphi_{0})=-\operatorname{Res}_{P_{1}}(\varphi_{0}).$

Hence $\chi(T)=-1$ and $\\{P_{1},P_{2}\\}\subseteq\hat{S}_{X}^{\chi}$. ∎

######  Remark 3.6.

If $\chi=1$ is the trivial representation, then
$\hat{S}_{X}^{\chi}=\varnothing$, which is consistent with _Lemma 3.4_ in the
case $\hat{S}_{Y}=\varnothing$.

######  Remark 3.7.

Suppose $\varphi^{\prime}\in H^{0}(\hat{X},\Omega_{\hat{X}}(S^{\prime}))$ has
a pole at $P_{1}\in S^{\prime}$, namely
$\operatorname{Res}_{P_{1}}\varphi^{\prime}\neq 0$. If $\varphi^{\prime}\in
H^{0}(X,\omega_{X})$, then for the pair $\\{P_{1},P_{2}\\}$, it requires
a) $v_{P_{2}}(\varphi^{\prime})=v_{P_{1}}(\varphi^{\prime})=-1$; b)
$\operatorname{Res}_{P_{2}}\varphi^{\prime}=-\operatorname{Res}_{P_{1}}\varphi^{\prime}$.

For a), in general, we can’t determine $v_{P_{2}}(\varphi^{\prime})$ from the
value of $v_{P_{1}}(\varphi^{\prime})$. But if $Y$ is smooth, then
$v_{P_{2}}(\varphi^{\prime})=v_{P_{1}}(\varphi^{\prime})=v_{P}(\varphi^{\prime})$
for $\forall P\in\hat{\pi}^{-1}\hat{\pi}(P_{1})$.

For b), under the hypothesis of smoothness of $Y$, we use the criterion from
the _singular $\chi$-set_ to delete these points that can not be poles.

Assume $Y=X/G$ is smooth for the rest of this section.

Now we compute the dimension of
$H^{0}(\hat{X},\Omega_{\hat{X}}(\hat{S}_{X}^{\chi}))_{\chi}$. Let $f_{\chi}$
be a rational fuction on $\hat{X}$ such that $\sigma
f_{\chi}=\chi(\sigma)f_{\chi},~{}\forall\sigma\in G$ and set
$D_{\chi}=\left\lfloor
n^{-1}\hat{\pi}_{*}\left(\hat{S}_{X}^{\chi}+(f_{\chi})+R_{\hat{\pi}}\right)\right\rfloor$.
By Proposition 2.1 (4), we have

$H^{0}(\hat{X},\Omega_{\hat{X}}(\hat{S}_{X}^{\chi}))_{\chi}=f_{\chi}\cdot\hat{\pi}^{*}H^{0}(Y,\Omega_{Y}(D_{\chi})).$

By Riemann-Roch Theorem, we have

(13)
$\displaystyle\mathrm{dim}_{k}H^{0}(Y,\Omega_{Y}(D_{\chi}))=\mathrm{dim}_{k}H^{0}(Y,\mathcal{O}_{Y}(-D_{\chi}))+\operatorname{deg}D_{\chi}+g_{Y}-1.$

######  Lemma 3.8.

The space $H^{0}(Y,\mathcal{O}_{Y}(-D_{\chi}))$ vanishes except when
$\chi=1_{G}$, and in this case, we have
$\mathrm{dim}_{k}H^{0}(Y,-D_{1_{G}})=1$.

###### Proof.

We will show that $\operatorname{deg}D_{\chi}>0$ if
$\hat{S}_{X}^{\chi}\neq\varnothing$ and $D_{\chi}$ is principal if and only if
$\chi=1$. Assume

$\hat{\pi}_{*}(f_{\chi})=\sum_{Q\in Y}\frac{n}{e_{Q}}b_{Q}\cdot Q$

where $b_{Q}=v_{P}(f_{\chi})$, $\forall P\in\hat{\pi}^{-1}(Q)$. Note that

$\left\lfloor
n^{-1}\hat{\pi}_{*}((f_{\chi})+R_{\hat{\pi}})\right\rfloor=\sum_{Q}\left\lfloor\frac{b_{Q}+e_{Q}-1}{e_{Q}}\right\rfloor
Q\geq\sum_{Q}\frac{b_{Q}}{e_{Q}}Q,$

hence we have

(14) $\operatorname{deg}\left\lfloor
n^{-1}\hat{\pi}_{*}\left(\hat{S}_{X}^{\chi}+(f_{\chi})+R_{\hat{\pi}}\right)\right\rfloor\geq\operatorname{deg}\left\lfloor
n^{-1}\hat{\pi}_{*}((f_{\chi})+R_{\pi})\right\rfloor\geq
n^{-1}\operatorname{deg}(f_{\chi})=0.$

Wirte $\hat{\pi}_{*}(\hat{S}_{X}^{\chi})=\sum_{Q\in
Y}\frac{n}{e_{Q}}c_{Q}\cdot Q.$ If $\hat{S}_{X}^{\chi}\neq\varnothing$, then
there is some $c_{Q^{\prime}}\geq 1$, hence

(15) $\displaystyle\operatorname{deg}D_{\chi}$ $\displaystyle=\sum_{Q\neq
Q^{\prime}}\left\lfloor\frac{c_{Q}+b_{Q}+e_{Q}-1}{e_{Q}}\right\rfloor+\left\lfloor\frac{c_{Q^{\prime}}+b_{Q^{\prime}}+e_{Q^{\prime}}-1}{e_{Q^{\prime}}}\right\rfloor$
$\displaystyle\geq\sum_{Q\neq
Q^{\prime}}\left\lfloor\frac{c_{Q}+b_{Q}+e_{Q}-1}{e_{Q}}\right\rfloor+\left\lfloor\frac{b_{Q^{\prime}}+e_{Q^{\prime}}}{e_{Q^{\prime}}}\right\rfloor>\sum_{Q}\frac{b_{Q}}{e_{Q}}=0.$

Hence $\mathrm{dim}_{k}H^{0}(Y,\mathcal{O}_{Y}(-D_{\chi}))=0$ provided
$\hat{S}_{X}^{\chi}\neq\varnothing$.

Now we suppose $\hat{S}_{X}^{\chi}=\varnothing$, and
$\operatorname{deg}\left\lfloor
n^{-1}\hat{\pi}_{*}((f_{\chi})+R_{\hat{\pi}})\right\rfloor=0$, then we have

$\left\lfloor\frac{b_{Q}+e_{Q}-1}{e_{Q}}\right\rfloor=\frac{b_{Q}}{e_{Q}},$

which implies $b_{Q}=\lambda_{Q}e_{Q}$ for some integer $\lambda_{Q}$ and
$D_{\chi}=n^{-1}\hat{\pi}_{*}(f_{\chi})$. If $D_{\chi}$ is principal, namely
$\mathrm{dim}_{k}H^{0}(Y,\mathcal{O}_{Y}(-D_{\chi}))=1$, then we have
$D_{\chi}=(h)$ for some rational function $h\in K(Y)$. Hence
$(f_{\chi})=(\hat{\pi}^{*}h)$, which implies $f_{\chi}\in K(X)^{G}$, namely
$\chi=1$. Conversely, if $\chi=1$, then $f_{\chi}=\hat{\pi}^{*}h$ for some
rational function $h\in K(Y)$ and $D_{\chi}=(h)$, hence
$\mathrm{dim}_{k}H^{0}(Y,\mathcal{O}_{Y}(-D_{\chi}))=1$. ∎

######  Definition 3.9.

Let $S\subseteq\hat{X}$ be a finite subset stable by $G$, and define

(16)
$m_{\chi}(S)=\\#\hat{\pi}(S)+\sum_{Q\notin\hat{\pi}(S)}\left\lfloor\frac{e_{Q}-1}{e_{Q}}+\frac{1}{n}\left\langle\chi,R_{G,Q}\right\rangle_{G}\right\rfloor-\frac{1}{n}\left\langle\chi,R_{G}\right\rangle_{G}.$

######  Lemma 3.10.

We have $\operatorname{deg}D_{\chi}=m_{\chi}(\hat{S}_{X}^{\chi})$, which is
independent of the choice of $f_{\chi}$.

###### Proof.

Denote $s_{\chi}=\\#\hat{\pi}(\hat{S}_{X}^{\chi})$. Write
$n^{-1}\hat{\pi}_{*}(\hat{S}_{X}^{\chi})=U^{\chi}+V^{\chi}$, where
$\operatorname{Supp}(V^{\chi})=Bl(Y)\cap\hat{\pi}(\hat{S}_{X}^{\chi})$.
Suppose $\left(f_{\chi}^{n}\right)=\hat{\pi}^{*}(nA+B)$ and $\lfloor
n^{-1}B\rfloor=0$ as in lemma 2.3. Write $B=\sum_{Q\in Bl(Y)}b_{Q}Q$ . By

(17) $\displaystyle\left\lfloor
n^{-1}\pi_{*}\left(\hat{S}_{X}^{\chi}+(f_{\chi})+R_{\hat{\pi}}\right)\right\rfloor=U^{\chi}+A+\left\lfloor
V^{\chi}+n^{-1}B+n^{-1}\hat{\pi}_{*}R_{\hat{\pi}}\right\rfloor,$

and $\operatorname{deg}B=-\operatorname{deg}nA$, we have

(18) $\displaystyle\operatorname{deg}D_{\chi}$
$\displaystyle=\\#U^{\chi}+\sum_{Q\in\hat{\pi}(\hat{S}_{X}^{\chi})}\left\lfloor
1+\frac{b_{Q}}{n}\right\rfloor+\sum_{Q\notin\hat{\pi}(\hat{S}_{X}^{\chi})}\left\lfloor\frac{e_{Q}-1}{e_{Q}}+\frac{b_{Q}}{n}\right\rfloor-\sum_{Q}\frac{b_{Q}}{n}$
$\displaystyle=s_{\chi}+\sum_{Q\notin\hat{\pi}(\hat{S}_{X}^{\chi})}\left\lfloor\frac{e_{Q}-1}{e_{Q}}+\frac{b_{Q}}{n}\right\rfloor-\sum_{Q}\frac{b_{Q}}{n}.$

By Lemma 2.3 (5), we have $b_{Q}=\left\langle\chi,R_{G,Q}\right\rangle_{G}$,
hence

$\operatorname{deg}D_{\chi}=s_{\chi}+\sum_{Q\notin\hat{\pi}(\hat{S}_{X}^{\chi})}\left\lfloor\frac{e_{Q}-1}{e_{Q}}+\frac{1}{n}\left\langle\chi,R_{G,Q}\right\rangle_{G}\right\rfloor-\frac{1}{n}\left\langle\chi,R_{G}\right\rangle_{G}=m_{\chi}(\hat{S}_{X}^{\chi}).$

∎

Now we summary above discussion.

For a quotient map $\pi:X\rightarrow X/G=Y$ from an irreducible nodal curve to
a smooth curve, we have the induced covering of curves
$\hat{\pi}:\hat{X}\rightarrow Y$ with $\hat{S}_{X}\subseteq\hat{X}$ the
preimage of singular locus. Let $Bl(Y)$ be the branch locus, $R_{G}$ the
ramification module of $\hat{\pi}$, and $R_{G,Q}$ the ramification module of
$Q\in Y$.

######  Theorem 3.11 (The Chevalley-Weil formula on irreducible nodal curves).

Let $f:X\rightarrow X/G$ be the quotient map of irreducible nodal curves by a
finite group $G$ of order $n$. Assume $Y=X/G$ is smooth and
$\operatorname{char}(k)\nmid n$, then the multiplicity of a given irreducible
character $\chi$ is given by

(19)
$\displaystyle\operatorname{dim}_{k}H^{0}(X,\omega_{X})_{\chi}=g_{Y}-1+m_{\chi}(\hat{S}_{X}^{\chi})+\langle\chi,1_{G}\rangle.$

where $\hat{S}_{X}^{\chi}$ is the _singular $\chi$-set_ of $X$ defined in
_Proposition 3.5_, and

$m_{\chi}(\hat{S}_{X}^{\chi})=s_{\chi}+\sum_{Q\notin\hat{\pi}(\hat{S}_{X}^{\chi})}\left\lfloor\frac{e_{Q}-1}{e_{Q}}+\frac{1}{n}\left\langle\chi,R_{G,Q}\right\rangle_{G}\right\rfloor-\frac{1}{n}\left\langle\chi,R_{G}\right\rangle_{G}$

is defined in _Definition 3.9_.

In particular, when $\chi=1_{G}$, we have $\hat{S}_{X}^{\chi}=\varnothing$ and
$\operatorname{dim}_{k}H^{0}(X,\omega_{X})^{G}=g_{Y},$ which is a special case
of _Proposition 3.4_.

######  Example 3.12 (Hyperelliptic stable curves).

A hyperelliptic stable curve $C$ is a stable curve with a hyperelliptic
involution $J:C\rightarrow C$, which is an order $2$ automorphism satisfying
$C/\langle J\rangle=\mathbb{P}^{1}$.

Suppose $C$ is an irreducible hyperelliptic stable curve with $N(\geq 1)$
nodes. Let $\hat{C}$ be the normalization of $C$ with genus $g$, then
$\hat{\pi}:\hat{C}\rightarrow\mathbb{P}^{1}$ has $2g+2$ fixed(ramification)
points, which are all of ramification index $2$, hence
$\\#Bl(\mathbb{P}^{1})=2g+2$.

There are no branch points in $\hat{\pi}(\hat{S}_{C})$, and the Galois group
$G=\langle J\rangle\cong\mathbb{Z}_{2}$ has two irreducible representations
$1_{G}$ and $\chi^{-}$ where $\chi^{-}(J)=-1$.

For $1_{G}$, we have

$\operatorname{dim}_{k}H^{0}(C,\omega_{C})^{G}=g({\mathbb{P}^{1}})=0.$

For $\chi^{-}$, we have $\hat{S}_{C}^{\chi^{-}}=\hat{S}_{C}$, hence
$s_{\chi^{-}}=N$.

And for any fixed point $P$, the induced character $\theta_{P}:G_{P}=\langle
J\rangle\rightarrow k^{*},J\mapsto-1$ is the generator of
$\operatorname{Hom}(\mathbb{Z}_{2},k^{*})$. So we have
$\left\langle\chi^{-},R_{G,Q}\right\rangle_{G}=1$ for all $Q\in
Bl(\mathbb{P}^{1})$ and $\left\langle\chi^{-},R_{G}\right\rangle_{G}=2g+2,$
which gives

(20) $\displaystyle m_{\chi^{-}}(\hat{S}_{C}^{\chi^{-}})$
$\displaystyle=s_{\chi^{-}}+\sum_{Q\in
Bl(\mathbb{P}^{1})}\left\lfloor\frac{e_{Q}-1}{e_{Q}}+\frac{1}{2}\left\langle\chi_{-1},R_{G,Q}\right\rangle_{G}\right\rfloor-\frac{1}{2}\left\langle\chi^{-},R_{G}\right\rangle_{G}$
(21) $\displaystyle=N+2g+2-(g+1)=p_{a}(C)+1.$

Hence

$\operatorname{dim}_{k}H^{0}(C,\omega_{C})_{\chi^{-}}=g({\mathbb{P}^{1}})-1+m_{\chi^{-}}(\hat{S}_{C}^{\chi^{-}})=p_{a}(C).$

##  4\. Nodal curves with several irreducible components

In this section, let $X$ be a connected nodal curve and
$X=\cup^{d}_{i=1}X_{i}$ be the decomposition of irreducible components.
Consider

$\alpha:\coprod X_{i}\rightarrow X,$

the partial normalization at the intersection locus, then we have the
immersion

(22) $\displaystyle
H^{0}(X,\omega_{X})\hookrightarrow\oplus_{i=1}^{d}H^{0}(X_{i},\omega_{X_{i}}(I_{i})),~{}\varphi\mapsto(\varphi|_{X_{i}}),$

where $I_{i}$ is the set of intersection points of each $X_{i}$.

###  4.1. Reduction

Let $G$ be a finite subgroup of $\operatorname{Aut}(X)$ and $\pi:X\rightarrow
X/G=Y$ be the quotient map. Suppose $Y$ is irreducible, then $G$ acts
transitively on $\\{X_{1},\cdots,X_{d}\\}$ and all these components are
isomorphic. Let $G_{i}$ be the stablizer subgroup of $G$ at $X_{i}$ and note
that the canonical map $X_{i}/G_{i}\rightarrow X/G=Y$ is an isomorphism.

######  Proposition 4.1.

Given a $1$-dimensional character $\chi$ of $G$, we have a commutative
diagram:

(23)
${H^{0}(X,\omega_{X})_{\chi}}$${\big{[}\oplus_{i=1}^{d}H^{0}(X_{i},\omega_{X_{i}}(I_{i}))\big{]}_{\chi}~{},}$${\varphi}$${(\varphi|_{X_{i}})_{i=1}^{d}}$${H^{0}(X_{1},\omega_{X_{1}}(I_{1}))_{\chi_{1}}}$${\varphi|_{X_{1}}~{},}$$\scriptstyle{p_{1}}$

where $\chi_{1}$ is the restriction of $\chi$ in $G_{1}$. Moreover, the
projection $p_{1}$ is an isomorphism.

###### Proof.

Fix some
$(\varphi_{1},\cdots,\varphi_{d})\in\big{[}\oplus_{i=1}^{d}H^{0}(X_{i},\omega_{X_{i}}(I_{i}))\big{]}_{\chi}$.
Since $G$ acts on $\\{X_{1},\cdots,X_{d}\\}$ transitively, then we always have
some $T_{i}:X_{i}\rightarrow X_{1}$ and
$T_{i}\varphi_{1}=\chi(T_{i})\varphi_{i}$, hence

$(\varphi_{1},\cdots,\varphi_{d})=(~{}\varphi_{1},\chi(T_{2})^{-1}T_{2}\varphi_{1},\cdots,\chi(T_{d})^{-1}T_{d}\varphi_{1}~{}),$

namely $(\varphi_{1},\cdots,\varphi_{d})$ is uniquely determined by
$\varphi_{1}$. Note that $T_{1}\varphi_{1}=\chi(T_{1})\varphi_{1}$ for any
$T_{1}\in G_{1}$, which implies $\varphi_{1}\in
H^{0}(X_{1},\omega_{X_{1}}(I_{1}))_{\chi_{1}}$, so the projection to the first
component $p_{1}$ is injective.

Conversly, we want to show
$(~{}\varphi_{1},\chi(T_{2})^{-1}T_{2}\varphi_{1},\cdots,\chi(T_{d})^{-1}T_{d}\varphi_{1}~{})$
is the preimgae of $\varphi_{1}\in
H^{0}(X_{1},\omega_{X_{1}}(I_{1}))_{\chi_{1}}$. Note that for any two
$\sigma,\tau:X_{i}\rightarrow X_{1}$, we have
$\chi(\sigma)^{-1}\sigma\varphi_{1}=\chi(\tau)^{-1}\tau\varphi_{1}.$ Hence for
any $T\in G$, we have

(24) $\displaystyle
T(~{}\varphi_{1},\chi(T_{2})^{-1}T_{2}\varphi_{1},\cdots,\chi(T_{d})^{-1}T_{d}\varphi_{1}~{})=\chi(T)(~{}\varphi_{1},\chi(T_{2})^{-1}T_{2}\varphi_{1},\cdots,\chi(T_{d})^{-1}T_{d}\varphi_{1}~{}).$

∎

######  Remark 4.2.

By the same arguments as in _Proposition 3.5_ and _Proposition 4.1_, we set

(25) $I_{i}^{\chi}=\left\\{P\in I_{i}\mid\exists\tau\in G_{P}-G_{i}\text{ s.t.
}\chi(\tau)=-1\right\\}$

to be those intersection points that could be the poles of $\varphi|_{X_{i}}$
for $\varphi\in H^{0}(X,\omega_{X})_{\chi}$, and then we have the isomorphisms

(26)
$H^{0}(X,\omega_{X})_{\chi}\overset{\sim}{\rightarrow}[\oplus_{i=1}^{d}H^{0}(X_{i},\omega_{X_{i}}(I_{i}^{\chi}))]_{\chi}\overset{\sim}{\rightarrow}H^{0}(X_{1},\omega_{X_{1}}(I_{1}^{\chi}))_{\chi_{1}}.$

So our research obeject has been reduced to the irreducible nodal curve acting
by the subgroup $G_{1}$ on $X_{1}$.

###  4.2. Chevalley-Weil formula for connected nodal curves

With the notations above, note that $I_{i}^{\chi}=\varnothing$ when
$\chi=1_{G}$. By _Proposition 3.4_, if the ramification indexes of any pair
$\\{P_{1},P_{2}\\}\subseteq\hat{S}_{X_{1}}$ are equal for
$\pi_{1}:X_{1}\rightarrow X_{1}/G_{1}=Y$, then we have

(27)
$H^{0}(X,\omega_{X})^{G}\overset{\sim}{\rightarrow}H^{0}(X_{1},\omega_{X_{1}})^{G_{1}}\overset{\sim}{\rightarrow}H^{0}(Y,\omega_{Y}).$

In this case, we have
$\operatorname{dim}_{k}H^{0}(X,\omega_{X})^{G}=p_{a}(Y)$.

Assume that $Y$ is smooth for the rest of this section.

Let $\hat{\pi}_{1}:\hat{X_{1}}\rightarrow Y$ be the normalization of
$\pi_{1}$, $Bl(Y)$ the branch locus, $R_{G_{1}}$ the ramification module of
$\hat{\pi}_{1}$, and $R_{{G_{1}},Q}$ the ramification module of $Q\in Y$.

Suppose $\hat{S}_{X_{1}}^{\chi_{1}}$ is the singular $\chi_{1}$-set of $X_{1}$
in _Proposition 3.5_, then we have the isomorphisms

(28) $\displaystyle
H^{0}(X,\omega_{X})_{\chi}\overset{\sim}{\rightarrow}H^{0}(X_{1},\omega_{X_{1}}(I_{1}^{\chi}))_{\chi_{1}}\overset{\sim}{\rightarrow}H^{0}(\hat{X_{1}},\Omega_{\hat{X}_{1}}(\hat{S}_{X_{1}}^{\chi_{1}}\cup
I_{1}^{\chi}))_{\chi_{1}}.$

Note that $n_{1}:=\\#G_{1}=n/d$ and by _Proposition 2.1 (4)_ again, we have

$H^{0}(\hat{X_{1}},\Omega_{\hat{X_{1}}}(\hat{S}_{X_{1}}^{\chi_{1}}\cup
I_{1}^{\chi}))_{\chi_{1}}=f_{\chi_{1}}\cdot\hat{\pi}_{1}^{*}H^{0}(Y,\Omega_{Y}\left\lfloor\
n_{1}^{-1}\pi_{*}\left(\hat{S}_{X_{1}}^{\chi_{1}}\cup
I_{1}^{\chi}+(f_{\chi_{1}})+R_{\pi_{1}}\right)\right\rfloor),$

where $f_{\chi_{1}}\in K(X_{1})^{*}$ is such that $\sigma
f_{\chi_{1}}=\chi_{1}(\sigma)f_{\chi_{1}},~{}\forall\sigma\in G_{1}$. The same
argument in Lemma 3.10 shows that

(29) $\operatorname{deg}\left\lfloor\
n_{1}^{-1}\pi_{*}\left(\hat{S}_{X_{1}}^{\chi_{1}}\cup
I_{1}^{\chi}+(f_{\chi_{1}})+R_{\pi_{1}}\right)\right\rfloor=m_{\chi_{1}}(\hat{S}_{X_{1}}^{\chi_{1}}\cup
I_{1}^{\chi})$

where by _Definition 3.9_,

$m_{\chi_{1}}(\hat{S}_{X_{1}}^{\chi_{1}}\cup
I_{1}^{\chi})=\\#\hat{\pi}_{1}(\hat{S}_{X_{1}}^{\chi_{1}}\cup
I_{1}^{\chi})+\sum_{Q\notin\hat{\pi}_{1}(\hat{S}_{X_{1}}^{\chi_{1}}\cup
I_{1}^{\chi})}\left\lfloor\frac{e_{Q}-1}{e_{Q}}+\frac{d}{n}\left\langle\chi_{1},R_{G_{1},Q}\right\rangle_{G_{1}}\right\rfloor-\frac{d}{n}\langle\chi_{1},R_{G_{1}}\rangle_{G_{1}}.$

By the same argument in _Lemma 3.8_, and Riemann-Roch Theorem, we have

######  Theorem 4.3 (The Chevalley-Weil formula on connected nodal curves).

Let $X$ be a connected nodal curve of $d$ irreducible components and $G$ a
finite group of order $n$ acting on $X$.

Assume the quotient curve $Y=X/G$ is smooth(hence irreducible), then we have a
canonical map $X_{1}/rightarrow=X_{1}/G_{1}=Y$, where $X_{1}$ is an
irreducible component and $G_{1}$ is the stablizer subgroup of $G$ at $X_{1}$.

With the notations above, the multiplicity of a $1$-dimensional character
$\chi$ is given by

(30)
$\operatorname{dim}_{k}H^{0}(X,\omega_{X})_{\chi}=g_{Y}-1+m_{\chi_{1}}(\hat{S}_{X_{1}}^{\chi_{1}}\cup
I_{1}^{\chi})+\delta_{\chi},$

where $\delta_{\chi}=0$ or $1$. And $\delta_{\chi}=1$ if and only if
$I_{1}^{\chi}=\varnothing$ and $\chi_{1}=1_{G_{1}}$. In particular, when
$\chi=1_{G}$, we have

$\operatorname{dim}_{k}H^{0}(X,\omega_{X})^{G}=g_{Y}.$

Note that this theorem is exactly the direct generalization of Theorem 3.11,
since if $d=1$, then $I_{1}^{\chi}=\varnothing$.

######  Example 4.4.

Let $C=C_{1}\cup C_{2}$ be a hyperelliptic stable curve, where $C_{1}\approx
C_{2}\approx\mathbb{P}^{1}$, and they intersect in $\\#(C_{1}\cap
C_{2})=m(>2)$ points, then $p_{a}(C)=m-1$. Consider the hyperelliptic
involution $J$ permuting $C_{1}$ and $C_{2}$, we have $\pi:C\rightarrow
C/\langle J\rangle=\mathbb{P}^{1}$.

The covering map $\pi_{1}:C_{1}\rightarrow\mathbb{P}^{1}$ is an identity. For
the representation $\chi^{-}$, we have $\chi^{-}_{1}=id$. Hence
$\hat{S}_{C_{1}}^{id}=\varnothing$ and $m_{id}(\hat{S}_{C_{1}}^{id}\cup
I_{1}^{\chi^{-}})=\\#\pi(I_{1}^{\chi^{-}})=m$.

So by _Theorem 4.3_, we have

(31)
$\operatorname{dim}_{k}H^{0}(C,\omega_{C})_{\chi^{-}}=g_{\mathbb{P}^{1}}-1+m=m-1=p_{a}(C),$

and $\operatorname{dim}_{k}H^{0}(C,\omega_{C})^{G}=p_{a}({\mathbb{P}^{1}})=0$.

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