# Conformal blocks attached to twisted groups

Chiara Damiolini

###### Abstract.

The aim of this paper is to generalize the notion of conformal blocks to the
situation in which the Lie algebra they are attached to is not defined over a
field, but depends on covering data of curves. The result will be a sheaf of
conformal blocks on the stack
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,n}}$ parametrizing
$\Gamma$-coverings of curves. Many features of the classical sheaves of
conformal blocks are proved to hold in this more general setting, in
particular the fusion rules, the propagation of vacua and the WZW connection.

August 27, 2024

##  1\. Introduction

In conformal field theory [TUY89] there is a way to associate to a simple and
simply connected group $G$ over an algebraically closed field $k$ of
characteristic zero, a vector bundle $\mathbb{H}_{\ell}(0)_{X_{univ}}$, called
the _sheaf of conformal blocks_ , on ${\mathcal{M}_{g}}$, the stack
parametrizing smooth curves of genus $g$. The goal of this paper is to
generalize this construction to the case in which the group $G$ is replaced by
a certain type of parahoric Bruhat-Tits group $\mathcal{H}$ arising from
coverings.

### Classical conformal blocks

Before going into the details of the content of this paper, we briefly recall
the properties of the sheaf of conformal blocks. Denote by ${\mathfrak{g}}$
the Lie algebra of $G$ and by $P_{\ell}$ the set of integral dominant weights
of ${\mathfrak{g}}$ of level at most $\ell$. Let $X$ be a (nodal) curve over
$\textrm{Spec}(k)$ of genus $g$, which is stably marked by the points
${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$. Then, to the $2n$-tuple
$({\mathfrak{p}}_{i},\lambda_{i})_{i=1}^{n}$, with $\lambda_{i}\in P_{\ell}$,
it is possible to associate a vector space $\mathbb{H}_{\ell}(\lambda_{i})$.
This construction extends to families of $n$-pointed stable curves of genus
$g$, giving rise to the vector bundle
$\mathbb{H}_{\ell}(\lambda_{i})_{X_{univ}}$ on
${\overline{\mathcal{M}}_{g,n}}$. This is what is called the _sheaf of
conformal blocks_ attached to the weights $\lambda_{i}$’s.

In the case in which all the $\lambda_{i}$’s are zero, the so called
_propagation of vacua_ ensures that the associated sheaf of conformal blocks
is actually independent of the marked points, hence it descends to
${\mathcal{M}_{g}}$. We denote this vector bundle, which is called the _sheaf
of covacua_ , by $\mathbb{H}_{\ell}(0)_{X_{univ}}$.

The rank of $\mathbb{H}_{\ell}(\lambda_{i})_{X_{univ}}$ has been computed with
the _Verlinde formula_ [TUY89] [Fal94] [Sor96]. The main ingredient for this
computation consists in the _fusion rules_ which control the behaviour of the
rank under degeneration of curves. Thanks to this property the computation of
the rank is reduced to the case of the projective line $\mathbb{P}^{1}$ with
three marked points.

These sheaves have played an important role not only as a tool to study
${\overline{\mathcal{M}}_{g,n}}$, but also in the study of
${\textrm{Bun}_{G}}(X)$, the stack parametrizing $G$-bundles on a smooth curve
$X$. In fact, for every $\ell\in\mathbb{N}$ there is a canonical isomorphism

$H^{0}({\textrm{Bun}_{G}}(X),\mathscr{L}^{\otimes\ell})^{*}\cong\mathbb{H}_{\ell}(0)_{X}$

where $\mathscr{L}$ is the determinant line bundle on ${\textrm{Bun}_{G}}(X)$
[BL94] [KNR94]. The key point to prove this isomorphism is the _uniformization
theorem_ which describes ${\textrm{Bun}_{G}}(X)$ as a quotient of the the
affine Grassmannian $\textsf{Gr}(G)$, whose line bundles and the space of
their global sections have been described in terms of representations of
${\mathfrak{g}}$ by Kumar [Kum87] and Mathieu [Mat88]. This theorem, which was
proved initially by Beauville and Laszlo in [BL94] for $G=\text{SL}_{n}$, has
been generalized for parabolic groups by Pauly in [Pau96] and by Laszlo and
Sorger in [LS97]. Finally Heinloth proved the uniformization theorem for
$\textrm{Bun}_{\mathcal{H}}(X)$, for connected parahoric Bruhat-Tits groups
$\mathcal{H}$ in [Hei10], where he also gave a description of the Picard group
of $\textrm{Bun}_{\mathcal{H}}(X)$. Having in hand the notion of the sheaf of
conformal blocks for parahoric Bruhat-Tits groups $\mathcal{H}$ satisfying
factorization rules and propagation of vacua, is then the first step to
describe the space of global sections
$H^{0}(\textrm{Bun}_{\mathcal{H}}(X),\mathscr{L})$ of certain line bundles
$\mathscr{L}$ on $\textrm{Bun}_{\mathcal{H}}(X)$ and achieve, in a second
time, a Verlinde type formula for
$H^{0}(\textrm{Bun}_{\mathcal{H}}(X),\mathscr{L})$, as asked by Pappas and
Rapoport in [PR10].

### Parahoric Bruhat-Tits groups arising from coverings.

As already mentioned, in our generalization we replace the group $G$ with a
parahoric Bruhat-Tits group $\mathcal{H}$ defined over a curve $X$. Since the
group $\mathcal{H}$ depends on the geometry of the curve, our version of the
sheaf of conformal blocks will be in general not defined over
${\overline{\mathcal{M}}_{g,n}}$ but on a moduli space which encodes also the
information on $\mathcal{H}$. Inspired by [BS15], we restrict ourselves to
consider only those groups _arising from coverings_ in the following sense. We
fix the cyclic group $\Gamma:=\mathbb{Z}/p\mathbb{Z}$ of prime order $p$ and a
group homomorphism $\rho\colon\Gamma\to\text{Aut}(G)$. Let
$q\colon{\widetilde{X}}\to X$ be a (ramified) Galois covering of nodal curves
with Galois group $\Gamma$ and denote its moduli stack by
$\overline{{\mathcal{H}\textrm{ur}}}(\Gamma,\xi)_{g}$. We remark that in
contrast to [BR11], we assume that the nodes of $X$ are disjoint from the
branch locus $\mathcal{R}$ of $q$. Then we say that a group $\mathcal{H}$ on
$X$ arises from $q$ and $\rho$ if it is isomorphic to the group of
$\Gamma$-invariants of the Weil restriction of ${\widetilde{X}}\times_{k}G$
along $q$, i.e. $\mathcal{H}=q_{*}({\widetilde{X}}\times_{k}G)^{\Gamma}$.

We observe that the groups $\mathcal{H}$ that we consider are parahoric
Bruhat-Tits groups which in general are not generically split, while in [BS15]
the authors only work in the split situation. This reflects the condition that
in their paper they only allow $\Gamma$ to act on $G$ by inner automorphisms,
i.e. $\rho$ is a group homomorphism $\Gamma\to G$. The following statement is
a particular instance of Theorem A.0.7 which generalizes [BS15, Theorem
4.1.6].

######  Theorem.

Let $q\colon{\widetilde{X}}\to X$ be a $\Gamma$ covering of curves and
$\rho\colon\Gamma\to\text{Aut}(G)$ be a homomorphism of groups. Set
$\mathcal{H}=(q_{*}(X\times G))^{\Gamma}$. Then the functor
$q_{*}(-)^{\Gamma}$ induces an equivalence between
$\textrm{Bun}_{\mathcal{H}}(X)$ and the stack
$\textrm{Bun}_{(G,\Gamma)}^{G}({\widetilde{X}})$ parametrizing $G$-bundles on
${\widetilde{X}}$ equipped with an action of $\Gamma$ _compatible_ with the
one on $G$.

The notion of compatibility stressed in the above Theorem will be clarified in
Appendix A in terms of _local type_ of $(\Gamma,G)$-bundles.

### Main results

In order to define the generalized sheaf of conformal blocks, we first of all
need to introduce the pointed version of
$\overline{{\mathcal{H}\textrm{ur}}}(\Gamma,\xi)_{g}$ and in second place
replace $P_{\ell}$ with an appropriate set of representations of
$\mathcal{H}$. We denote by
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ the stack
parametrizing $\Gamma$-coverings of nodal curves ${\widetilde{X}}\to X$, where
$X$ is marked by a point ${\mathfrak{p}}$ which is disjoint from the branch
locus $\mathcal{R}$. In similar fashion we define
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,n}}$ for $n\geq 1$. Let
$\mathcal{H}$ be the group on ${X_{univ}}$ arising from the universal covering
$({\widetilde{X}_{univ}}\to{X_{univ}},{\mathfrak{p}})$ on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ and the homomorphism
$\rho\colon\Gamma\to\text{Aut}(G)$. Set
${\mathfrak{h}}:=\text{Lie}(\mathcal{H})$ and denote by
${\text{IrRep}_{\ell}}({\mathfrak{h}}|_{\mathfrak{p}})$ the set of irreducible
representations $\mathcal{V}$ of ${\mathfrak{h}}|_{{\mathfrak{p}}}$ of level
at most $\ell$.

In Section 3 we explain how to associate to each representation
$\mathcal{V}\in{\text{IrRep}_{\ell}}({\mathfrak{h}}|_{\mathfrak{p}})$, a
vector bundle ${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ (Proposition 3.2.9 and
Definition 3.2.10). This definition is generalized in Section 4.6 to the case
in which ${X_{univ}}$ is marked by $n$ points
${\mathfrak{p}}_{1},\dots,{\mathfrak{p}}_{n}$, obtaining a vector bundle
${\mathcal{H}_{\ell}({\mathcal{V}_{1},\dots,\mathcal{V}_{n}})}_{X_{univ}}$ on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,n}}$ attached to the
representations
$\mathcal{V}_{i}\in{\text{IrRep}_{\ell}}({\mathfrak{h}}|_{{\mathfrak{p}}_{i}})$.
The vector bundle
${\mathcal{H}_{\ell}({\mathcal{V}_{1},\dots\mathcal{V}_{n}})}_{X_{univ}}$ is
called the _sheaf of conformal blocks_ attached to the representations
$\mathcal{V}_{i}$’s.

As in the classical case, in Proposition 5.2.2 we formulate fusion rules
controlling the rank of the vector bundle under degeneration of the covering:

######  Proposition.

Let $(q\colon{\widetilde{X}}\to
X,{\mathfrak{p}})\in{\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}(k)$
such that $X$ is irreducible and has only one nodal point $x$. Let $X_{N}$ be
its normalization so that $q_{N}\colon{\widetilde{X}}_{N}\to X_{N}$ is a
$\Gamma$-covering with three marked points. Then for any
$\mathcal{W}\in{\text{IrRep}_{\ell}}({\mathfrak{h}}|_{\mathfrak{p}})$ we have
a canonical isomorphism

${\mathcal{H}_{\ell}({\mathcal{W}})}_{X}\cong\bigoplus_{\mathcal{V}\in{\text{IrRep}_{\ell}}({\mathfrak{h}}|_{x})}{\mathcal{H}_{\ell}({\mathcal{W},\mathcal{V},\mathcal{V}^{*}})}_{X_{N}}.$

Moreover, also the propagation of vacua holds (Proposition 5.1.1), leading to
the following statement.

######  Proposition.

Let $\mathcal{V}(0)$ be the trivial representation of
${\mathfrak{h}}|_{{\mathfrak{p}}_{i}}$. Then the bundle
${\mathcal{H}_{\ell}({\mathcal{V}(0),\dots,\mathcal{V}(0)})}_{X_{univ}}$ is
independent of the choice of the marked points, hence it descends to a vector
bundle ${\mathcal{H}_{\ell}(0)}_{X_{univ}}$ on
${\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g}}$.

### Insights into the construction and properties of
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$

We now give an overview of how the twisted conformal blocks are defined,
generalizing the methods used in [Kac90], [TUY89] and [Loo13]. For simplicity
we consider $({\widetilde{X}}\to
X,{\mathfrak{p}})\in{\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,1}}(\textrm{Spec}(k))$
and denote ${\mathfrak{h}}=\text{Lie}(\mathcal{H})$. Observe that since
${\mathfrak{p}}$ is not a branch point, ${\mathfrak{h}}|_{\mathfrak{p}}$ is
isomorphic, although non canonically, to ${\mathfrak{g}}$. It follows that
once we choose such an isomorphism, we can use the classical construction
[Kac90, Chapters 7 and 9] to associate to each representation
$\mathcal{V}\in{\text{IrRep}_{\ell}}({\mathfrak{h}}|_{\mathfrak{p}})\cong
P_{\ell}$ the integrable highest weight representation
${\mathcal{H}_{\ell}({\mathcal{V}})}$ of
${\widehat{\mathfrak{h}}_{\mathcal{L}}}$, a central extension of
${\mathfrak{h}_{\mathcal{L}}}={\mathfrak{h}}\otimes_{k}k((t))$ defined in
terms of Killing form and residue pairing. The key point is to see that this
construction is actually independent of the isomorphism chosen between
${\mathfrak{h}}|_{\mathfrak{p}}$ and ${\mathfrak{g}}$. The Lie algebra
${\mathfrak{h}_{\mathcal{A}}}:={\mathfrak{h}}|_{X\setminus{\mathfrak{p}}}$ is
a sub Lie algebra of ${\widehat{\mathfrak{h}}_{\mathcal{L}}}$ and we set
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X}$ to be the quotient
${\mathfrak{h}_{\mathcal{A}}}{\,\big{\backslash}}{\mathcal{H}_{\ell}({\mathcal{V}})}$.
The construction of the sheaf of conformal blocks runs similarly for any
family $({\widetilde{X}}\to
X,\sigma)\in{\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}(S)$, being
careful that the isomorphism between ${\mathfrak{h}}|_{\sigma(S)}$ and
${\mathfrak{g}}\otimes_{k}S$ exists only locally.

Although it is easy to show that
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ is coherent (Proposition
3.2.12), it is not immediate from its construction that it is also locally
free. Following Looijenga in [Loo13], the first step to achieve this result is
to show the following statement:

######  Proposition (Corollary 4.5.2).

The sheaf ${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ is equipped with a
projective connection with logarithmic singularities along the boundary
$\Delta_{univ}$.

This shows in particular that
${\mathcal{H}_{\ell}({\mathcal{V}})}_{{X_{univ}}}$ is a locally free module
over ${\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,1}}$. The idea is to realise
${\mathcal{H}_{\ell}({\mathcal{V}})}$ as a Fock-type representation of a Lie
algebra of derivations which is a central extension of the sheaf of
logarithmic vector fields of
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ along $\Delta_{univ}$.
It is important to remark that the flat connection we obtain on
$\mathbb{P}({\mathcal{H}_{\ell}(0)}_{X_{univ}})$ descends to
${\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g}}$ and generalizes the _WZW
connection_ defined in terms of conformal field theory. Combining this with
the fusion rules, we are able to prove the local freeness on the whole stack
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ (Corollary 6.2.8).

It is then clear that the fusion rules play a double role in the theory of
conformal blocks. On one side they contribute to show that
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ is locally free on the whole
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$, and on the other side
they are a useful tool to reduce the computation to lower genera curves.

######  Setting and notation.

Unless otherwise stated, throughout the paper we fix the following objects.

  * •

An algebraically closed field $k$ of characteristic zero.

  * •

A prime $p$ and for simplicity of notation we denote the group
$\mathbb{Z}/p\mathbb{Z}$ by $\Gamma$.

  * •

A simple and simply connected algebraic group $G$ over $\textrm{Spec}(k)$.

  * •

A group homomorphism $\rho\colon\Gamma\to\text{Aut}(G)$.

##  2\. Preliminaries on groups arising from coverings and Hurwitz stacks

In this section we introduce the group schemes associated to coverings as
indicated in the introduction. Since we need to work with these groups in
families, we will formulate the definition for families of coverings of
curves. We obtain in this way the family $\mathcal{H}_{univ}$ over the
universal curve ${X_{univ}}$ over the Hurwitz stack parametrizing coverings of
curves.

######  Definition 2.0.1.

Let $\pi\colon X\to S$ be a possibly nodal curve over $S\in\textbf{Sch}_{k}$.
A _Galois covering_ of $X$ with group $\Gamma$, called also _$\Gamma$
-covering_, is the data of

  1. a)

a finite, faithfully flat and generically étale map $q\colon\widetilde{X}\to
X$ between curves;

  2. b)

an isomorphism $\phi\colon\Gamma\cong\text{Aut}_{X}({\widetilde{X}})$;

satisfying the following conditions:

  1. (1)

each fibre of ${\widetilde{X}}$ is a generically étale $\Gamma$-torsor over
$X$;

  2. (2)

the singular locus of $\pi q$, i.e. the set of nodes of ${\widetilde{X}}$, is
contained in the étale locus of $q$.

We want to attach to any $\Gamma$-covering
$({\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S)$ and to the
homomorphism $\rho\colon\Gamma\to\text{Aut}(G)$ a group scheme $\mathcal{H}$
over $X$ in the same fashion as in [BS15, Section 4]. We remark that Balaji
and Seshadri consider $\rho$ to map to the inner automorphisms of $G$ only,
i.e. arising from a morphism $\Gamma\to G$. Without imposing that restriction
we allow also groups $\mathcal{H}$ which are non-split over the generic point
of $X$.

First of all we consider the scheme
$\widetilde{G}:={\widetilde{X}}\times_{k}G$ and let $q_{*}(\widetilde{G})$ be
its Weil restriction along $q$ which is defined as

$q_{*}\widetilde{G}(T):=\textrm{Hom}_{\widetilde{X}}(T\times_{X}{\widetilde{X}},\widetilde{G})$

for every $T\in\textbf{Sch}_{X}$. It follows from [BLR90, Theorem 4 and
Proposition 5, Section 7.6] that $q_{*}\widetilde{G}$ is representable by a
smooth group scheme over $X$. The actions of $\Gamma$ on $G$ and on
${\widetilde{X}}$ induce the action of $\Gamma$ on $q_{*}\widetilde{G}$ given
by

$(\gamma\cdot
f)(t,{\widetilde{x}}):=\rho(\gamma)^{-1}f(\gamma(t,{\widetilde{x}}))=\rho(\gamma)^{-1}f(t,\gamma^{*}({\widetilde{x}}))$

for all $t\in T$ and ${\widetilde{x}}\in{\widetilde{X}}$.

We define $\mathcal{H}$ to be the subgroup of $\Gamma$-invariants of
$q_{*}(\widetilde{G})$, i.e.

$\mathcal{H}:=(q_{*}\widetilde{G})^{\Gamma}.$

We denote by ${\mathfrak{h}}$ the sheaf of Lie algebras of $\mathcal{H}$.
Since $\mathcal{H}$ is smooth, as shown in [Edi92, Proposition 3.4],
${\mathfrak{h}}$ is a vector bundle on $X$ which is moreover equipped with a
structure of Lie algebra.

######  Remark 2.0.2.

The action of $\Gamma$ on $G$ via $\rho$ induces an action on
${\mathfrak{g}}:=\text{Lie}(G)$. We equivalently could have defined
${\mathfrak{h}}$ as the Lie algebra of $\Gamma$-invariants of
$q_{*}({\mathfrak{g}}\otimes_{k}\mathscr{O}_{\widetilde{X}})$.

######  Example 2.0.3.

Let $\rho\colon\Gamma=\mathbb{Z}/2\mathbb{Z}\to\text{Aut}(\text{SL}_{n})$ be
given by $\rho(\gamma)M=(M^{t})^{-1}$ and $q\colon{\widetilde{X}}\to X$ a
$\Gamma$-covering of smooth curves. The group
$\mathcal{H}=(q_{*}(\text{SL}_{n}\times{\widetilde{X}}))^{\Gamma}$ is the
quasi split special unitary group associated to the extension $k(X)\subseteq
k({\widetilde{X}})$. Observe that only in the case $n=2$ this action comes
from inner automorphisms.

The stack $\textrm{Bun}_{\mathcal{H}}$ which parametrizes
$\mathcal{H}$-bundles on $X$ can be described in terms of $G$-bundles over
${\widetilde{X}}$ which admit an action of $\Gamma$ compatible with $\rho$.
This is a corollary of Theorem A.0.7 which holds in a more general setup and
for which we refer to Appendix A.

###  2.1. Properties of $\Gamma$-coverings

In this section we recall the definitions and properties of coverings of
curves. Although the main reference is [BR11], we make the stronger assumption
that the ramification locus of the covering map $q$ consists only of smooth
points.

####  2.1.1. Ramification and branch divisors.

Consider a $\Gamma$-covering
$(f\colon{\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S)$. We define the
_ramification divisor_ ${\widetilde{\mathcal{R}}}$ to be the effective Cartier
divisor $(p-1){\widetilde{X}}^{\Gamma}$, where ${\widetilde{X}}^{\Gamma}$ is
the subscheme of ${\widetilde{X}}$ fixed by $\Gamma$. Equivalently, since
$\Gamma$ does not have proper subgroups, ${\widetilde{X}}^{\Gamma}$ is the
complement of the étale locus of $q$, which is either empty or an effective
Cartier divisor of ${\widetilde{X}}$. The _reduced branch divisor_
$\mathcal{R}$ is the effective divisor given by the image of
${\widetilde{X}}^{\Gamma}$ in $X$. It is the reduced divisor of the proper
pushforward $q_{*}{\widetilde{\mathcal{R}}}$.

######  Remark 2.1.1.

If the map $q$ is not étale both divisors ${\widetilde{\mathcal{R}}}$ and
$\mathcal{R}$ are finite and étale over $S$. This is a proved in [BR11,
Proposition 3.1.1] for the smooth case only and in [BR11, Proposition 4.1.8]
for the general situation.

The ramification divisors are naturally related to tangent bundles of $X$ and
${\widetilde{X}}$. Let $\mathcal{T}_{{\widetilde{X}}/S}$ be the tangent bundle
of ${\widetilde{X}}$ relative to $S$, so that its sections are
$f^{-1}\mathscr{O}_{S}$-linear derivations of $\mathscr{O}_{{\widetilde{X}}}$.
Consider its pushforward to $X$ along $q$ and notice that the action of
$\Gamma$ on $q_{*}\mathscr{O}_{{\widetilde{X}}}$ induces an action on
$q_{*}\mathcal{T}_{{\widetilde{X}}/S}$ by sending a derivation $D$ to $\gamma
D\gamma^{-1}$. The following lemma, which describes the $\Gamma$-invariants of
$q_{*}\mathcal{T}_{{\widetilde{X}}/S}$, follows from [BR11, Proposition
4.1.11] and we report the proof for completeness.

######  Lemma 2.1.2.

The sheaf $(q_{*}\mathcal{T}_{{\widetilde{X}}/S})^{\Gamma}$ over $X$ is
isomorphic to $\mathcal{T}_{X/S}(-\mathcal{R})$.

###### Proof.

Let first observe that the natural map
$d(q)\colon\mathcal{T}_{{\widetilde{X}}/S}\to q^{*}\mathcal{T}_{X/S}$
identifies $\mathcal{T}_{{\widetilde{X}}/S}$ with
$q^{*}\mathcal{T}_{X/S}(-{\widetilde{\mathcal{R}}})$. This is clear outside
${\widetilde{\mathcal{R}}}$. On the formal neighbourhood $R[[t]]$ of a point
${\widetilde{x}}\in{\widetilde{\mathcal{R}}}$ the map $q$ is given by sending
$t$ to $t\zeta$ for a primitive $p$-th root of unity $\zeta$. It follows that
$d(q)\colon R[[t]]d/dt\to R[[t]]d/d(t^{p})$ sends the generator $d/dt$ to
$pt^{p-1}d/d(t^{p})$, concluding the argument.

We now pushforward the isomorphism
$d(q)\colon\mathcal{T}_{{\widetilde{X}}/S}\to
q^{*}\mathcal{T}_{X/S}(-{\widetilde{\mathcal{R}}})$ along $q$ and take
$\Gamma$-invariants obtaining the isomorphism

$(q_{*}d(q))^{\Gamma}\colon(q_{*}\mathcal{T}_{{\widetilde{X}}/S})^{\Gamma}\to(q_{*}(q^{*}\mathcal{T}_{X/S}(-{\widetilde{\mathcal{R}}})))^{\Gamma}.$

Since étale morphisms induce isomorphism on the tangent bundles, this map is
an isomorphism outside the branch divisor $\mathcal{R}$. Since by assumption
the branch points are smooth, we are left to check that the target of the map
equals $\mathcal{T}_{X/S}(-\mathcal{R})$ under the condition that $X\to S$ is
smooth. From the smoothness we deduce that $\mathcal{T}_{X/S}$ is locally
free, so using the projection formula we obtain that
$(q_{*}(q^{*}\mathcal{T}_{X/S}(-{\widetilde{\mathcal{R}}})))^{\Gamma}\cong\mathcal{T}_{X/S}\otimes
q_{*}(\mathscr{O}(-{\widetilde{\mathcal{R}}}))^{\Gamma}$. We are left to prove
that $q_{*}(\mathscr{O}(-{\widetilde{\mathcal{R}}}))^{\Gamma}$ is isomorphic
to $\mathscr{O}(-\mathcal{R})$. Observe, for this purpose, that the sheaf
$q_{*}\mathscr{O}_{{\widetilde{\mathcal{R}}}}$ is supported only at
$\mathcal{R}$, so we only need to compute that its submodule of
$\Gamma$-invariants is one dimensional. Let $x\in\mathcal{R}$ and note that
the formal neighbourhood of $q_{*}\mathscr{O}_{{\widetilde{\mathcal{R}}}}$ at
$x$ is isomorphic to

$R[[t]]/t^{p-1}R[[t]]\cong R\oplus tR\oplus\cdots\oplus t^{p-2}R$

on which any element of $\Gamma$ acts multiplying $t$ by a $p$-th root of
unity. It follows that the only invariant submodule is $R$, hence
$(q_{*}\mathscr{O}_{{\widetilde{\mathcal{R}}}})^{\Gamma}\cong\mathscr{O}_{\mathcal{R}}$.
∎

#### Hurwitz data

The Hurwitz data provide a description of the action of $\Gamma$ at the
ramification points. Before considering families of curves we take
${\widetilde{X}}\to X$, a $\Gamma$-covering of curves over $k$. Let
${\widetilde{x}}\in{\widetilde{X}}(k)^{\Gamma}$ be a ramification point and up
to the choice of a local parameter $t$ the formal disc around
${\widetilde{x}}$ is isomorphic to $\textrm{Spec}(k[[t]])$. Since $\Gamma$
fixes ${\widetilde{x}}$, one of its generators acts on $k[[t]]$ by sending $t$
to $\zeta t$ for a primitive $p$-th root unity $\zeta$. It follows that the
action of $\Gamma$ on $\textrm{Spec}(k[[t]])$ is uniquely determined by non
trivial characters $\chi_{{\widetilde{x}}}\colon\Gamma\to k^{*}$. Let
$\textrm{Char}(\Gamma)^{*}$ be the set of all non trivial characters of
$\Gamma$ and set
$R_{+}(\Gamma):=\oplus_{\chi\in\textrm{Char}(\Gamma)^{*}}\mathbb{Z}\chi$. The
_ramification data_ or _Hurwitz data_ of a $\Gamma$-covering
${\widetilde{X}}\to X$ is the element

$\xi:=\sum_{{\widetilde{x}}\in{\widetilde{X}}^{\Gamma}}\chi_{{\widetilde{x}}}\in
R_{+}(\Gamma).$

The _degree_ of $\xi=\sum b_{i}\chi_{i}$ is $\deg(\xi):=\sum b_{i}$. Note that
$\deg(\xi)=\deg({\widetilde{X}}^{\Gamma})=\deg(\mathcal{R})$.

######  Remark 2.1.3.

In the case $\Gamma=\mathbb{Z}/2\mathbb{Z}$, the Hurwitz data encode only the
number of points which are fixed.

######  Definition 2.1.4.

Let ${\widetilde{X}}\to X\to S$ be a $\Gamma$-covering with $S$ connected. We
say that it has Hurwitz data $\xi\in R_{+}(\Gamma)$ if $\xi$ is the Hurwitz
data of one, hence all ([BR11, Lemme 3.1.3]), of its fibres.

We fix for the next two lemmas, a generator $\gamma$ of $\Gamma$ and $\zeta\in
k$ a primitive $p$-th root of $1$. This identifies the set of characters of
$\Gamma$ with $\\{0,\dots,p-1\\}$.

######  Lemma 2.1.5.

Denote by $\mathscr{E}_{i}$ the $\mathscr{O}_{X}$-submodule of
$q_{*}\mathscr{O}_{{\widetilde{X}}}$ where $\gamma$ acts by multiplication by
$\zeta^{i}$. Then

$q_{*}\mathscr{O}_{{\widetilde{X}}}=\bigoplus_{i=0}^{p-1}\mathscr{E}_{i}\mathscr{E}_{i}\otimes\mathscr{E}_{p-i}\cong\mathscr{O}(-\mathcal{R}).$

###### Proof.

The action of $\Gamma$ on $q_{*}\mathscr{O}_{\widetilde{X}}$ provides the
decomposition with $\mathscr{E}_{0}\cong\mathscr{O}_{X}$. For the second
statement, the tensor product $\mathscr{E}_{i}\otimes\mathscr{E}_{p-i}$ is a
submodule of $\mathscr{E}_{0}\cong\mathscr{O}_{X}$ as $\gamma$ acts there as
the identity. Outside the branch divisor $\mathcal{R}$ this is an isomorphism
so we only need to check what is the image along $\mathcal{R}$. Let
$x\in\mathcal{R}$ and call ${\widetilde{x}}\in{\widetilde{\mathcal{R}}}$ the
point above $x$ so that
$\widehat{\mathscr{O}}_{{\widetilde{X}},{\widetilde{x}}}\cong R[[t]]$, with
$\gamma(t)=\zeta^{n}t$ with $n\in\\{1,\dots,p-1\\}$. If follows that
$(\widehat{\mathscr{E}}_{i})_{x}\cong t^{i/n}R[[t^{p}]]$ and
$(\widehat{\mathscr{E}}_{p-i})_{x}\cong t^{(p-i)/n}R[[t^{p}]]$, where
$i/n\in\\{1,\dots,p-1\\}$. It follows that
$(\widehat{\mathscr{E}}_{i})_{x}\otimes(\widehat{\mathscr{E}}_{p-i})_{x}\cong
t^{p}R[[t^{p}]]$ which is isomorphic to the completion of
$\mathscr{O}(-\mathcal{R})$ at the point $x$. ∎

######  Lemma 2.1.6.

Denote by ${\mathfrak{g}}^{\zeta^{i}}$ the submodule of ${\mathfrak{g}}$ where
$\gamma$ acts by multiplication by $\zeta^{i}$. The sheaf ${\mathfrak{h}}$
decomposes as

${\mathfrak{h}}=\bigoplus_{i=0}^{p-1}{\mathfrak{g}}^{\zeta^{-i}}\otimes_{k}\mathscr{E}_{i}.$

###### Proof.

As the action of $\gamma$ on ${\mathfrak{g}}$ is diagonalizable with
eigenvalues belonging to $\\{1,\zeta,\cdots,\zeta^{p-1}\\}$, we can decompose
${\mathfrak{g}}$ as $\oplus{\mathfrak{g}}^{\zeta^{-i}}$. As ${\mathfrak{h}}$
is the Lie algebra of $\Gamma$-invariants of
$q_{*}(\mathscr{O}_{{\widetilde{X}}}\otimes_{k}{\mathfrak{g}})=q_{*}\mathscr{O}_{{\widetilde{X}}}\otimes_{k}{\mathfrak{g}}$,
we can combine this with the description of
$q_{*}\mathscr{O}_{{\widetilde{X}}}$ provided by Lemma 2.1.5 to obtain the
wanted decomposition of ${\mathfrak{h}}$. ∎

###  2.2. Hurwitz stacks

We define in this section the stack parametrizing $\Gamma$-coverings with
fixed Hurwitz data $\xi\in R_{+}(\Gamma)$. Let $g$ be a non negative integer.

Let $f\colon{\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S$ be a
$\Gamma$-covering of curves and let $\sigma\colon S\to X$ be a section of
$\pi$ with $\sigma(S)$ disjoint from the nodes of $X$ and from the branch
locus $\mathcal{R}$ of $q$. The we say that the covering is _stably marked_ by
$\sigma$ if $(X,\sigma\cup\mathcal{R})$ is a stably marked curve [BR11,
Définition 4.3.4. and Proposition 5.1.3].

######  Definition 2.2.1.

We define the _Hurwitz stack_
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,n}}$ as

${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,n}}(S)=\left\langle
f\colon{\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S,\>\\{\sigma_{j}\colon
S\to X\\}_{j=1}^{n}\text{ such that i and ii hold}\right\rangle$

  1. i.

the map $q\colon{\widetilde{X}}\to X$ is a $\Gamma$-covering of curves with
ramification data $\xi$;

  2. ii.

$(X,\\{\sigma_{j}\\})$ is an $n$-marked curve of genus $g$ with
$\sigma_{j}(S)$ disjoint from the branch divisor $\mathcal{R}$ for all $j$ and
such that the covering is stably marked by $\\{\sigma_{j}\\}$.

When $n=0$ we omit the subscript and use the notation
$\overline{{\mathcal{H}\textrm{ur}}}(\Gamma,\xi)_{g}$. We denote by
${\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,n}}$ the open substack of
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,n}}$ parametrizing
$\Gamma$-coverings of smooth curves.

######  Remark 2.2.2.

We want to remark that the role of the ramification data is to guarantee the
connectedness of ${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,n}}$ and
${\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,n}}$ [BR11, Proposition 2.3.9].

In the previous section we explained how to associate to each
$\Gamma$-covering
$({\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S)\in\overline{{\mathcal{H}\textrm{ur}}}(\Gamma,\xi)_{g}(S)$,
a group $\mathcal{H}$ (resp. a sheaf of Lie algebras ${\mathfrak{h}}$) over
$X$. This defines a group $\mathcal{H}_{univ}$ (resp. a sheaf of Lie algebras
${\mathfrak{h}}_{univ}$) on ${X_{univ}}$, where we denote by
${\widetilde{X}_{univ}}\to{X_{univ}}$ the universal covering on
$\overline{{\mathcal{H}\textrm{ur}}}(\Gamma,\xi)_{g}$. The same construction
works on ${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,n}}$, defining
then $\mathcal{H}_{univ}$ and ${\mathfrak{h}}_{univ}$ on the universal curve
${X_{univ}}$ of ${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,n}}$.

######  Remark 2.2.3.

The complement
$\Delta_{univ}:={\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,n}}\setminus{\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,n}}$
is a normal crossing divisor. First of all observe that
$\Delta_{{\overline{\mathcal{M}}_{g,d}}}:={\overline{\mathcal{M}}_{g,d}}\setminus{\mathcal{M}_{g,d}}$
is a normal crossing divisor: in fact given a nodal curve
$X\to\textrm{Spec}(k)$ with a reduced divisor $D$ of degree $d$, there exists
a versal deformation $\mathcal{X}\to S$ where the locus $\Delta\subset S$
consisting of singular curves is a normal crossing divisor of $S$ [ACG11]. We
now want to compare the deformation theory of a $\Gamma$-covering
$({\widetilde{X}}\to X,\\{\sigma_{i}\\})$ to the one of
$(X,\\{\sigma_{i}\\})$. Following [BR11, Théorème 5.1.5] we see that the
natural map $\delta\colon\textrm{Def}({\widetilde{X}}\to
X,\\{\sigma_{i}\\})\to\textrm{Def}(X,\\{\sigma_{i}\\}\cup\mathcal{R})$ fails
to be an isomorphism only when the intersection between $\mathcal{R}$ and
$X^{sing}$ is not empty, but since by assumption we impose that
$\mathcal{R}\cap X^{sing}=\emptyset$, in our contest this map is always an
isomorphism. This then allow to obtain, from the versal deformation
$\mathcal{X}\to S$ of $(X,\\{\sigma_{i}\\}\cup\mathcal{R})$, the versal
deformation $({\widetilde{\mathcal{X}}}\to\mathcal{X},\\{\varsigma_{i}\\})$ of
${\widetilde{X}}\to X$, and hence deduce from the theory of
${\overline{\mathcal{M}}_{g,n+\deg(\mathcal{R})}}$ that $\Delta_{univ}$ is a
normal crossing divisor.

The following statement, which is given by [BR11, Proposition 2.3.9. and
Théorème 6.3.1], describes the properties of the above stacks.

######  Proposition 2.2.4.

The stacks ${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,n}}$ and
${\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,n}}$ are smooth Deligne-Mumford stacks
which are connected and of finite type over $\textrm{Spec}(k)$.

Instead of marking the curve $X$, we can mark the curve ${\widetilde{X}}$, so
that we define.

######  Definition 2.2.5.

For each $S\in\textbf{Sch}_{k}$ we set

${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)^{n}_{g}}(S)=\left\langle
f\colon{\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S,\>\\{\tau_{j}\colon
S\to{\widetilde{X}}\\}_{j=1}^{n}\text{ such that i and ii hold}\right\rangle$

  1. i.

the map $q\colon{\widetilde{X}}\to X$ is a $\Gamma$-covering of curves with
ramification data $\xi$;

  2. ii.

$({\widetilde{X}},\\{\tau_{j}\\})$ is an $n$-marked curve with $q\tau_{j}(S)$
pairwise disjoint, $\tau_{j}(S)$ disjoint from ${\widetilde{X}}^{\Gamma}$ for
all $j$ and such that the covering $q$ is stably marked by $\\{q\tau_{j}\\}$.

It follows, from the fact that the image of $\tau$ lies in the étale locus of
$q$, that the map

$\textbf{Forg}^{n}_{n}\colon{\mathcal{H}\textrm{ur}(\Gamma,\xi)^{n}_{g}}\to{\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,n}},\qquad({\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S,\>\\{\tau_{j}\\})\mapsto({\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S,\>\\{p\tau_{j}\\})$

is an étale and surjective morphism of stacks. For any $n\in\mathbb{N}_{0}$
and for $g\geq 0$ we also have the forgetful map
$\textbf{Forg}_{n}\colon{\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,n}}\to{\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g}}$
and in more generality
$\textbf{Forg}_{n+m,n}\colon{\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,n+m}}\to{\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,n}}$
for all $n,m\in\mathbb{N}_{0}$.

Let
$({\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S,\tau)\in{\mathcal{H}\textrm{ur}(\Gamma,\xi)^{1}_{g}}(S)$
and write $\sigma:=q\tau$. Fixing $\tau$ allows us to canonically identify
$\mathcal{H}|_{\sigma(S)}$ with $G\times_{k}S$ as explained in the proof of
the following statement.

######  Proposition 2.2.6.

The section $\tau$ induces an isomorphism between ${\sigma}^{*}\mathcal{H}$
and $G\times_{k}S$.

###### Proof.

Construct the cartesian diagram

$\textstyle{\widetilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\sigma}}$$\scriptstyle{q_{S}}$$\textstyle{{\widetilde{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{X}$

and since by assumption the image of $\sigma$ lies in the étale locus of $q$
the left vertical arrow $q_{S}$ is étale and it has a section given by $\tau$.
This implies that $\widetilde{S}$ is isomorphic to
$\coprod_{\gamma_{i}\in\Gamma}S$. Observe that
${q_{S}}_{*}\widetilde{\sigma}^{*}(\widetilde{G})\cong\sigma^{*}q_{*}(\widetilde{G})$
and that taking $\Gamma$-invariants commutes with restriction along $\sigma$.
It follows that

$\sigma^{*}\mathcal{H}=\left(\sigma^{*}q_{*}(\widetilde{G})\right)^{\Gamma}=\left({q_{S}}_{*}\widetilde{\sigma}^{*}(\widetilde{G})\right)^{\Gamma}=\left({q_{S}}_{*}\left(\coprod_{\gamma_{i}\in\Gamma}S\times
G\right)\right)^{\Gamma}=\left(\prod_{\gamma_{i}\in\Gamma}S\times
G\right)^{\Gamma}$

where $\gamma_{j}\in\Gamma$ acts on $\prod_{\gamma_{i}\in\Gamma}S\times G$ by
sending $(s_{i},g_{i})_{\gamma_{i}}$ to
$(s_{i},\gamma_{j}(g_{i}))_{\gamma_{j}\gamma_{i}}$. It follows that the
invariant elements are of the form $(s,\gamma_{i}(g))_{\gamma_{i}}$ for any
$s\in S$ and $g\in G$, so that the projection on any component of $S\times G$
realized an isomorphism between $\sigma^{*}\mathcal{H}$ and $G\times S$. The
map $\tau$ selects a preferred component, giving in this way a canonical
isomorphism. ∎

##  3\. The sheaf of conformal blocks

In this section we define the sheaf of conformal blocks
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ attached to a
representation $\mathcal{V}$ of $\sigma^{*}{\mathfrak{h}}_{univ}$. To define
the sheaf ${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$, we will define it for
any family
$(f\colon{\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S,\sigma)$ over an
affine base $S=\textrm{Spec}(R)$. We assume moreover that
$X\setminus\sigma(S)\to S\to S$ is affine. We will see in Remark 5.1.5 how to
drop this assumption.

For the classical definition of the sheaf of conformal blocks attached to a
represendetation of ${\mathfrak{g}}$ one can refer to [TUY89] or to [Loo13].
We will use the latter as main reference.

######  Warning.

The word _conformal block_ has been used in literature to denote either a
certain vector bundle or its dual. We use here the word sheaf of conformal
blocks to denote what in [TUY89] is called the dual of the sheaf of conformal
blocks. In [Loo13], the author calls this sheaf the _sheaf of covacua_.

Let $X^{*}:=X\setminus\sigma(S)$ and denote by ${\mathcal{A}}$ the pushforward
to $S$ of $\mathscr{O}_{X^{*}}$, i.e.

${\mathcal{A}}:=\pi_{*}{j_{A}}_{*}\mathscr{O}_{X^{*}}$

where $j_{A}$ denotes the open immersion $X^{*}\to X$. Since the map $\pi$
restricted to $X^{*}$ is affine we have that
$X^{*}=\textbf{Spec}({\mathcal{A}})$ and that
${\mathcal{A}}=\pi_{*}\varinjlim_{n}\mathcal{I}_{\sigma}^{-n}=\varinjlim_{n}\pi_{*}\mathcal{I}_{\sigma}^{-n}$
where $\mathcal{I}_{\sigma}=\mathscr{O}_{X}(-\sigma(S))$ is the ideal defining
$\sigma(S)$.

We denote by $\widehat{\mathcal{O}}$ the formal completion of
$\mathscr{O}_{X}$ along $\sigma(S)$: by definition $\sigma$ gives a short
exact sequence

$0\to\mathcal{I}_{\sigma}\to\mathscr{O}_{X}\to\mathscr{O}_{\sigma(S)}\to 0$

of $\mathscr{O}_{X}$-modules. We define

$\widehat{\mathcal{O}}:=\pi_{*}\varprojlim_{n}\mathscr{O}_{X}{\,\big{/}}(\mathcal{I}_{\sigma})^{n}=\varprojlim_{n}\pi_{*}\mathscr{O}_{X}{\,\big{/}}(\mathcal{I}_{\sigma})^{n}$

which is naturally a sheaf of $\mathscr{O}_{S}$-modules. We denote by
${\mathcal{L}}$ the $\mathscr{O}_{S}$-module

${\mathcal{L}}:=\varinjlim_{N\in\mathbb{N}_{0}}\pi_{*}\varprojlim_{n\in\mathbb{N}}\mathcal{I}_{\sigma}^{-N}/\mathcal{I}_{\sigma}^{n}$

which is equipped with a natural filtration
$F^{N}\\!{\mathcal{L}}=\pi_{*}\varprojlim_{n\in\mathbb{N}}\mathcal{I}_{\sigma}^{N}/\mathcal{I}_{\sigma}^{N+n}$
for $N\geq 0$ and
$F^{N}\\!{\mathcal{L}}=\pi_{*}\varprojlim_{n\in\mathbb{N}}\mathcal{I}_{\sigma}^{N}/\mathcal{I}_{\sigma}^{n}$
for $N\leq-1$ taking into account the order of the poles or zeros along
$\sigma(S)$.

######  Remark 3.0.1.

Recall that when $R=k$, the choice of a local parameter $t$, i.e. of a
generator of $\mathcal{I}_{\sigma}$, gives an isomorphism
$\widehat{\mathcal{O}}\cong k[[t]]$ and hence ${\mathcal{L}}\cong k((t))$ and
so $F^{n}\\!{\mathcal{L}}\cong t^{n}k[[t]]$. Since $\mathcal{I}_{\sigma}$ is
locally principal, for every $s\in\sigma(S)$ we can find an open covering $U$
of $X$ containing $s$ and such that $\mathcal{I}_{\sigma}|_{U}$ is principal.
Let denote by $S^{\prime}$ the open of $S$ given by $\sigma^{-1}(U)$ and by
$U^{\prime}$ the open $U\cap\pi^{-1}{S^{\prime}}$. Then
$\mathcal{I}_{\sigma}|_{U^{\prime}}$ is principal and
$\varprojlim_{n}\mathscr{O}_{U^{\prime}}/(\mathcal{I}_{\sigma}|_{U^{\prime}})^{n}$
is isomorphic to $\mathscr{O}_{S^{\prime}}[[t]]$, where $t$ is a generator of
$\mathcal{I}_{\sigma}|_{U^{\prime}}$. This moreover implies that the
completion of $\widehat{\mathcal{O}}$ at a point $s\in S$ is isomorphic to
$\widehat{\mathscr{O}}_{S,s}[[t]]$, where $\widehat{\mathscr{O}}_{S,s}$
denotes the completion of $\mathscr{O}_{S}$ at $s$.

Denote by ${\mathfrak{h}_{\mathcal{A}}}$ the restriction of ${\mathfrak{h}}$
to the open curve $X^{*}$ and by ${\mathfrak{h}_{\mathcal{L}}}$ the
"restriction of ${\mathfrak{h}}$ to the punctured formal neighbourhood around
$\sigma(S)$", and consider both sheaves as $\mathscr{O}_{S}$-modules naturally
equipped with a Lie bracket. In other words we set

$\displaystyle{\mathfrak{h}_{\mathcal{A}}}$
$\displaystyle:=\pi_{*}{j_{A}}_{*}{j_{A}}^{*}({\mathfrak{h}})=\pi_{*}(\varinjlim_{N\in\mathbb{N}_{0}}\mathcal{I}_{\sigma}^{-N}\otimes_{\mathscr{O}_{X}}{\mathfrak{h}})=\varinjlim_{N\in\mathbb{N}_{0}}\pi_{*}(\mathcal{I}_{\sigma}^{-N}\otimes_{\mathscr{O}_{X}}{\mathfrak{h}})$
$\displaystyle{\mathfrak{h}_{\mathcal{L}}}$
$\displaystyle:=\varinjlim_{N\in\mathbb{N}_{0}}\pi_{*}\varprojlim_{n\in\mathbb{N}}\mathcal{I}_{\sigma}^{-N}/\mathcal{I}_{\sigma}^{n}\otimes_{\mathscr{O}_{X}}{\mathfrak{h}}.$

The following observations follow from the definitions.

  1. (1)

The injective morphism
$\mathcal{I}_{\sigma}^{-N}\to\varprojlim_{n}\mathcal{I}_{\sigma}^{-N}/\mathcal{I}_{\sigma}^{n}$
induces the inclusion
${\mathfrak{h}_{\mathcal{A}}}\to{\mathfrak{h}_{\mathcal{L}}}$.

  2. (2)

The filtration on ${\mathcal{L}}$ defines the filtration
$F^{*}{\mathfrak{h}_{\mathcal{L}}}$ as

$F^{N}({\mathfrak{h}_{\mathcal{L}}})=\pi_{*}\varprojlim_{n\in\mathbb{N}}\mathcal{I}_{\sigma}^{N}/\mathcal{I}_{\sigma}^{N+n}\otimes_{\mathscr{O}_{X}}{\mathfrak{h}}F^{-N}({\mathfrak{h}_{\mathcal{L}}})=\pi_{*}\varprojlim_{n\in\mathbb{N}}\mathcal{I}_{\sigma}^{-N}/\mathcal{I}_{\sigma}^{n}\otimes_{\mathscr{O}_{X}}$

for all $N\in\mathbb{N}_{0}$ and we denote
$F^{0}({\mathfrak{h}_{\mathcal{L}}})$ by
${\mathfrak{h}}_{\widehat{\mathcal{O}}}$.

  3. (3)

We could have equivalently defined ${\mathfrak{h}_{\mathcal{A}}}$ as the Lie
subalgebra of $\Gamma$-invariants of
$f_{*}({\mathfrak{g}}\otimes_{k}{j_{\widetilde{A}}}_{*}\mathscr{O}_{{\widetilde{X}}*})$
where $j_{\widetilde{A}}$ denotes the open immersion of
${\widetilde{X}}^{*}:={\widetilde{X}}\times_{X}X^{*}\to{\widetilde{X}}$. This
follows from the equalities

${j_{A}}^{*}{\mathfrak{h}}={j_{A}}^{*}(q_{*}({\mathfrak{g}}\otimes_{k}\mathscr{O}_{{\widetilde{X}}}))^{\Gamma}=({j_{A}}^{*}q_{*}({\mathfrak{g}}\otimes_{k}\mathscr{O}_{{\widetilde{X}}}))^{\Gamma}=q_{*}({j_{\widetilde{A}}}^{*}({\mathfrak{g}}\otimes_{k}\mathscr{O}_{X}^{*}))^{\Gamma}.$

Similarly ${\mathfrak{h}_{\mathcal{L}}}$ is the Lie subalgebra of
$\Gamma$-invariants of ${\mathfrak{g}}\otimes_{k}\widehat{{\mathcal{L}}}$,
where

$\widehat{{\mathcal{L}}}:=\varinjlim_{N}f_{*}\varprojlim_{n}({\mathfrak{g}}\otimes_{k}q^{*}(\mathcal{I}_{\sigma}^{-N})/q^{*}(\mathcal{I}_{\sigma}^{n})).$

######  Remark 3.0.2.

Since $\sigma(S)$ has trivial intersection with $\mathcal{R}$, we can find an
étale cover of $S$ such that $p^{-1}(\sigma(S))=\coprod_{\Gamma}S$ or in other
terms such that the pull back of $\mathcal{I}_{\sigma}$ to the cover totally
splits, i.e.
$q^{*}\mathcal{I}_{\sigma}=\prod_{\gamma_{i}\in\Gamma}\mathcal{I}_{\sigma,i}$.
This implies that

${\mathfrak{h}_{\mathcal{L}}}\cong\left({\mathfrak{g}}\otimes_{k}\bigoplus_{\gamma_{i}\in\Gamma}(\varinjlim_{N}f_{*}\varprojlim_{n}\mathcal{I}_{\sigma,i}^{-N}/\mathcal{I}_{\sigma,i}^{n})\right)^{\Gamma}$

which leads to
${\mathfrak{h}_{\mathcal{L}}}\cong({\mathfrak{g}}\otimes_{k}(\oplus_{\gamma_{i}\in\Gamma}{\mathcal{L}}))^{\Gamma}$
where the action is given by

$\gamma_{j}*\left((X_{i}f_{i})_{\gamma_{i}}\right)=(\gamma_{j}(X_{i})f_{i})_{\gamma_{j}\gamma_{i}}\quad\text{for
all }X_{i}\in{\mathfrak{g}}\text{ and }f_{i}\in{\mathcal{L}}.$

It follows that the invariant elements are combination of elements of the type
$(\gamma_{i}(X)f)_{\gamma_{i}}$. For every $i\in\\{0,\dots,p-1\\}$, the
projection on the $i$-th component

$\text{pr}_{i}\colon{\mathfrak{h}_{\mathcal{L}}}\to{\mathfrak{g}{\mathcal{L}}}:={\mathfrak{g}}\otimes_{k}{\mathcal{L}},\quad(\gamma_{j}(X)f)_{\gamma_{j}}\mapsto\gamma_{i}(X)f$

defines a non canonical isomorphism of sheaves of Lie algebras of
${\mathfrak{h}_{\mathcal{L}}}$ with ${\mathfrak{g}{\mathcal{L}}}$. The inverse
is the map that sends the element $Xf$ of ${\mathfrak{g}{\mathcal{L}}}$ to the
$p$-tuple $(\gamma_{j}(\gamma_{i}^{-1}(X))f)_{\gamma_{j}}$.

###  3.1. The central extension of ${\mathfrak{h}_{\mathcal{L}}}$

Once we have defined ${\mathfrak{h}_{\mathcal{L}}}$ and
${\mathfrak{h}_{\mathcal{A}}}$, in order to define
${\mathcal{H}_{\ell}(0)}_{X_{univ}}$, we need to extend
${\mathfrak{h}_{\mathcal{L}}}$ centrally. Following [Kac90, Chapter 7],
[TUY89] and [Loo13] we construct this central extension using a normalized
Killing form and the residue pairing.

#### Normalized Killing form

We fix once and for all a maximal torus $T$ of $G$ and a Borel subgroup $B$ of
$G$ containing $T$, or equivalently we fix the root system
$R(G,T)=R({\mathfrak{g}},\mathfrak{t})\subseteq\mathfrak{t}^{\vee}:=\textrm{Hom}(\mathfrak{t},k)$
of $G$ and a basis $\Delta$ of positive simple roots, where
$\mathfrak{t}=\text{Lie}(T)$. Given a root $\alpha$ we denote by
$H_{\alpha}\in\mathfrak{t}$ the associated coroot.

Denote by $(\,|\,)\colon{\mathfrak{g}}\otimes{\mathfrak{g}}\to k$ the unique
multiple of the Killing form such that $(H_{\theta}|H_{\theta})=2$ where
$\theta$ is the highest root of ${\mathfrak{g}}$. As ${\mathfrak{g}}$ is
simple, this form gives an isomorphism $(\,|\,)$ between ${\mathfrak{g}}$ and
${\mathfrak{g}}^{\vee}:=\textrm{Hom}({\mathfrak{g}},k)$. Pulling back this
form to ${\widetilde{X}}$ we obtain
$\tilde{(\,|\,)}\colon\widetilde{{\mathfrak{g}}}\otimes\widetilde{{\mathfrak{g}}}\to\mathscr{O}_{\widetilde{X}}$,
where
$\widetilde{{\mathfrak{g}}}:=\mathscr{O}_{{\widetilde{X}}}\otimes_{k}{\mathfrak{g}}$.
We push forward $\tilde{(\,|\,)}$ along $q$ obtaining

$q_{*}\tilde{(\,|\,)}\colon q_{*}(\widetilde{{\mathfrak{g}}})\otimes
q_{*}(\widetilde{{\mathfrak{g}}})\to q_{*}(\mathscr{O}_{\widetilde{X}})$

which is $\Gamma$-equivariant as the Killing form is invariant under
automorphisms of ${\mathfrak{g}}$. Taking $\Gamma$-invariants we obtain the
pairing

$(\,|\,)_{\mathfrak{h}}\colon{\mathfrak{h}}\otimes_{\mathscr{O}_{X}}{\mathfrak{h}}\to\mathscr{O}_{X}$

which however is not perfect because of ramification. Combining this with the
multiplication morphism
$\mathcal{I}_{\sigma}^{-N}/\mathcal{I}_{\sigma}^{N+n}\times\mathcal{I}_{\sigma}^{-N}/\mathcal{I}_{\sigma}^{N+n}\to\mathcal{I}_{\sigma}^{-2N}/\mathcal{I}_{\sigma}^{n}$
and taking the limit on $n$ and $N$ we obtain the perfect pairing
$(\,|\,)_{\mathfrak{h}_{\mathcal{L}}}\colon{\mathfrak{h}_{\mathcal{L}}}\otimes_{{\mathcal{L}}}{\mathfrak{h}_{\mathcal{L}}}\to{\mathcal{L}}$.

#### Residue pairing

We introduce the sheaf $\theta_{{\mathcal{L}}/S}$ of continuous derivations of
${\mathcal{L}}$ which are $\mathscr{O}_{S}$ linear. Denote its
${\mathcal{L}}$-dual by $\omega_{{\mathcal{L}}/S}$: this is the sheaf of
continuous differentials of ${\mathcal{L}}$ relative to $\mathscr{O}_{S}$.

######  Remark 3.1.1.

When $\widehat{\mathcal{O}}\cong R[[t]]$ we have that
$\theta_{{\mathcal{L}}/S}$ is isomorphic to $R((t))d/dt$ and
$\omega_{{\mathcal{L}}/S}$ to $R((t))dt$.

The residue map $\textrm{Res}\colon\omega_{{\mathcal{L}}/S}\to\mathscr{O}_{S}$
is computed locally as $\textrm{Res}(\sum_{i\geq
N}\alpha_{i}t^{i}dt)=\alpha_{-1}$. Composing this with the canonical morphism
${\mathfrak{h}_{\mathcal{L}}}^{\vee}\times{\mathfrak{h}_{\mathcal{L}}}\to{\mathcal{L}}$
we obtain the perfect pairing

$\textrm{Res}_{\mathfrak{h}}\colon\omega_{{\mathcal{L}}/S}\otimes_{\mathcal{L}}{\mathfrak{h}_{\mathcal{L}}}^{\vee}\times{\mathfrak{h}_{\mathcal{L}}}\to\mathscr{O}_{S}.$

#### The differential of a section

Let $d\colon\mathscr{O}_{\widetilde{X}}\to\Omega_{{\widetilde{X}}/S}$ be the
universal derivation, which induces the morphism
$d\colon{\mathfrak{g}}\otimes_{k}\mathscr{O}_{{\widetilde{X}}}\to{\mathfrak{g}}\otimes_{k}\Omega_{{\widetilde{X}}/S}$
by tensoring it with ${\mathfrak{g}}$. Let $U=X\setminus\\{\mathcal{R}\cup
X^{\text{sing}}\\}$ be the open subscheme of $X$ which is smooth over $S$ and
which does not intersect the branch divisor $\mathcal{R}$ of $q$ and call
$\widetilde{U}=U\times_{X}{\widetilde{X}}$. Once we restrict $d$ to
$\widetilde{U}$ and we push it forward along $q$ we obtain the map

$d\colon q_{*}({\mathfrak{g}}\otimes_{k}\mathscr{O}_{\widetilde{U}})\to
q_{*}({\mathfrak{g}}\otimes_{k}\mathscr{O}_{\widetilde{U}})\otimes_{\mathscr{O}_{U}}\Omega_{U/S}$

by using the projection formula. Taking $\Gamma$-invariants one obtains
$d\colon{\mathfrak{h}}|_{U}\to{\mathfrak{h}}|_{U}\otimes_{U}\Omega_{U/S}$ and
since $\sigma(S)\subset U$, this induces the map
$d\colon{\mathfrak{h}_{\mathcal{L}}}\to\omega_{{\mathcal{L}}/S}\otimes_{\mathcal{L}}{\mathfrak{h}_{\mathcal{L}}}$.
We can furthermore compose this map with the morphism
${\mathfrak{h}_{\mathcal{L}}}\to{\mathfrak{h}_{\mathcal{L}}}^{\vee}$ given by
the normalized Killing form $(\,|\,)_{\mathfrak{h}_{\mathcal{L}}}$, obtaining

$d_{\mathfrak{h}_{\mathcal{L}}}\colon{\mathfrak{h}_{\mathcal{L}}}\to\omega_{{\mathcal{L}}/S}\otimes_{\mathcal{L}}{\mathfrak{h}_{\mathcal{L}}}^{\vee}.$

######  Remark 3.1.2.

Given $X,Y\in{\mathfrak{h}_{\mathcal{L}}}$, we simply write $(dX|Y)$ for
$d_{\mathfrak{h}_{\mathcal{L}}}(X)(Y)\in\omega_{L/S}$. Note that the following
equality holds
$d_{{\mathcal{L}}}(X|Y)_{\mathfrak{h}_{\mathcal{L}}}=(dX|Y)+(X|dY)$, where
$d_{{\mathcal{L}}}\colon{\mathcal{L}}\to\omega_{{\mathcal{L}}/S}$ is the
universal derivation.

#### The central extension of ${\mathfrak{h}_{\mathcal{L}}}$

We have introduced all the ingredients we needed to be able to define the
central extension $0\to
c\mathscr{O}_{S}\to{\widehat{\mathfrak{h}}_{\mathcal{L}}}\to{\mathfrak{h}_{\mathcal{L}}}\to
0$ of ${\mathfrak{h}_{\mathcal{L}}}$ where $c$ is a formal variable.

######  Definition 3.1.3.

We define the sheaf of Lie algebras ${\widehat{\mathfrak{h}}_{\mathcal{L}}}$
to be ${\mathfrak{h}_{\mathcal{L}}}\oplus c\mathscr{O}_{S}$ as
$\mathscr{O}_{S}$-module, with $c\mathscr{O}_{S}$ being in the centre of
${\widehat{\mathfrak{h}}_{\mathcal{L}}}$ and with Lie bracket defined as

$[X,Y]:=[X,Y]_{{\mathfrak{h}_{\mathcal{L}}}}+c\textrm{Res}_{\mathfrak{h}}\left(d_{{\mathfrak{h}_{\mathcal{L}}}}(X)\otimes
Y\right)=[X,Y]_{{\mathfrak{h}_{\mathcal{L}}}}+c\textrm{Res}\left(dX|Y\right)$

for all $X,Y\in{\mathfrak{h}_{\mathcal{L}}}$.

The Lie algebra ${\widehat{\mathfrak{h}}_{\mathcal{L}}}$ comes equipped with
the filtration
$F^{i}{\widehat{\mathfrak{h}}_{\mathcal{L}}}=F^{i}{\mathfrak{h}_{\mathcal{L}}}$
for all $i\geq 1$ and
$F^{i}{\widehat{\mathfrak{h}}_{\mathcal{L}}}=F^{i}{\mathfrak{h}_{\mathcal{L}}}\oplus
c\mathscr{O}_{S}$ for $i\leq 0$. As
${\mathfrak{h}_{\mathcal{A}}}\subset{\mathfrak{h}_{\mathcal{L}}}$, one might
wonder which is the Lie algebra structure induced on
$\widehat{{\mathfrak{h}}}_{\mathcal{A}}$. The following two lemmas tell us
that ${\mathfrak{h}_{\mathcal{A}}}$ is a Lie subalgebra of
${\widehat{\mathfrak{h}}_{\mathcal{L}}}$.

######  Lemma 3.1.4.

The image of ${\mathfrak{h}_{\mathcal{A}}}$ via
$d_{{\mathfrak{h}_{\mathcal{L}}}}$ is
$\omega_{\mathcal{A}}\otimes{\mathfrak{h}_{\mathcal{A}}}^{\vee}$.

###### Proof.

We can restrict to the case of family of smooth curves, as on the singular
points the result follows from [Loo13, Lemma 5.1] by identifying
${\mathfrak{h}}$ with ${\mathfrak{g}}$. Recall from Lemma 2.1.6 that
${\mathfrak{h}}=\bigoplus_{i=0}^{p-1}{\mathfrak{g}}^{\zeta^{-i}}\otimes_{k}\mathscr{E}_{i}$,
and note that the image of $\mathscr{E}_{i}$ under $d$ is
$\mathscr{E}_{i}(\mathcal{R})\otimes\Omega_{X}$. Moreover observe that
$(\,|\,)$ gives an isomorphism between ${\mathfrak{g}}^{\zeta^{i}}$ and the
dual of ${\mathfrak{g}}^{\zeta^{-i}}$. Since
$\mathscr{E}_{i}\otimes\mathscr{E}_{p-i}\cong\mathscr{O}(-\mathcal{R})$ for
$i\neq 0$, the normalized killing form $(\,|\,)_{\mathfrak{h}}$ gives an
isomorphism between ${\mathfrak{g}}^{\zeta^{-i}}\otimes_{k}\mathscr{E}_{i}$
and
$({\mathfrak{g}}^{\zeta^{i}}\otimes_{k}\mathscr{E}_{p-i}(\mathcal{R}))^{\vee}$.
It follows that

$d_{\mathfrak{h}_{\mathcal{L}}}({\mathfrak{g}}^{\zeta^{-i}}\otimes_{k}\mathscr{E}_{i})={\mathfrak{g}}^{\zeta^{i}}\otimes_{k}(\mathscr{E}_{i}(\mathcal{R}))^{\vee}\otimes_{\mathscr{O}_{X}}\Omega_{X}={\mathfrak{g}}^{\zeta^{i}}\otimes_{k}\mathscr{E}_{-i}\otimes_{\mathscr{O}_{X}}\Omega_{X}$

which yields
$d_{\mathfrak{h}_{\mathcal{L}}}({\mathfrak{h}_{\mathcal{A}}})=\omega_{\mathcal{A}}\otimes{\mathfrak{h}_{\mathcal{A}}}^{\vee}$.
∎

######  Lemma 3.1.5.

The annihilator of ${\mathfrak{h}_{\mathcal{A}}}$ with respect to the pairing
$\textrm{Res}_{\mathfrak{h}}$, denoted
$\text{Ann}_{\textrm{Res}_{\mathfrak{h}}}({\mathfrak{h}_{\mathcal{A}}})$, is
$\omega_{\mathcal{A}}\otimes{\mathfrak{h}_{\mathcal{A}}}^{\vee}$.

###### Proof.

Before starting with the proof, we remark that this lemma holds if we replace
${\mathfrak{h}_{\mathcal{A}}}$ with any vector bundle $\mathscr{E}$ on $X$ as
it is essentially a consequence of Serre duality. We start by giving a
description of the quotient
${\mathfrak{h}_{\mathcal{A}}}{\,\big{\backslash}}{\mathfrak{h}_{\mathcal{L}}}$,
as the annihilator of ${\mathfrak{h}_{\mathcal{A}}}$ will be the dual of that
quotient with respect to the residue pairing. The double quotient
${\mathfrak{h}_{\mathcal{A}}}{\,\big{\backslash}}{\mathfrak{h}_{\mathcal{L}}}{\,\big{/}}F^{n}{\mathfrak{h}_{\mathcal{L}}}$
computes
$R^{1}\pi_{*}({\mathfrak{h}}\otimes_{\mathscr{O}_{X}}\mathcal{I}_{\sigma}^{n})$.
It follows that the projective limit $\varprojlim_{n\geq
1}R^{1}\pi_{*}({\mathfrak{h}}\otimes_{\mathscr{O}_{X}}\mathcal{I}_{\sigma}^{n}))$
equals $\varprojlim_{n\geq
1}{\mathfrak{h}_{\mathcal{A}}}{\,\big{\backslash}}{\mathfrak{h}_{\mathcal{L}}}{\,\big{/}}F^{n}{\mathfrak{h}_{\mathcal{L}}}$
which is
${\mathfrak{h}_{\mathcal{A}}}{\,\big{\backslash}}{\mathfrak{h}_{\mathcal{L}}}$.
As the residue pairing gives rise to Serre duality, we know that
$R^{1}\pi_{*}({\mathfrak{h}}\otimes_{\mathscr{O}_{X}}\mathcal{I}_{\sigma}^{n})$
is isomorphic to the dual of
$\pi_{*}(\Omega_{X/S}\otimes({\mathfrak{h}}\otimes\mathcal{I}_{\sigma}^{n})^{\vee})$.
It follows that

$\text{Ann}_{\textrm{Res}_{\mathfrak{h}}}({\mathfrak{h}_{\mathcal{A}}})=\varinjlim_{n\geq
1}\pi_{*}\left(\Omega_{X/S}\otimes_{\mathscr{O}_{X}}({\mathfrak{h}}\otimes_{\mathscr{O}_{X}}\mathcal{I}_{\sigma}^{n})^{\vee}\right).$

which equals
$\omega_{{\mathcal{A}}/S}\otimes_{{\mathcal{A}}}{\mathfrak{h}_{\mathcal{A}}}^{\vee}$.
∎

###  3.2. Conformal blocks attached to integrable representations

We have all the ingredients to define the sheaf of conformal blocks. Let
$U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ denote the universal enveloping
algebra of ${\widehat{\mathfrak{h}}_{\mathcal{L}}}$ and recall that
$F^{0}{\mathfrak{h}_{\mathcal{L}}}=\pi_{*}\varprojlim_{n}\mathscr{O}_{X}/\mathcal{I}_{\sigma}^{n}$,
i.e. it’s the subalgebra of ${\mathfrak{h}_{\mathcal{L}}}$ which has no poles
along $\sigma(S)$. Observe that this implies that it is also a Lie sub algebra
of ${\widehat{\mathfrak{h}}_{\mathcal{L}}}$.

######  Definition 3.2.1.

For any $\ell\in\mathbb{N}$ we define the _Verma module of level $\ell$_ to be
the left $U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$-module given by

${\widetilde{\mathcal{H}_{\ell}}({0})}:=U{\widehat{\mathfrak{h}}_{\mathcal{L}}}{\,\big{/}}\left(U{\widehat{\mathfrak{h}}_{\mathcal{L}}}\circ
F^{0}{\mathfrak{h}_{\mathcal{L}}},\,c=\ell\right).$

For what follows we will need a generalization of this module attached to
certain representations of $\sigma^{*}{\mathfrak{h}}$.

######  Definition 3.2.2.

An irreducible finite dimensional representation $\mathcal{V}$ of
$\sigma^{*}{\mathfrak{h}}$ is a locally free $\mathscr{O}_{S}$-module which is
equipped with an action of $\sigma^{*}{\mathfrak{h}}$ which locally étale on
$S$, and up to an isomorphism of $\sigma^{*}{\mathfrak{h}}$ with
${\mathfrak{g}}\times\mathscr{O}_{S}$, is isomorphic to
$V\otimes_{k}\mathscr{O}_{S}$ for an irreducible finite dimensional
representation $V$ of ${\mathfrak{g}}$.

Let $\mathcal{V}$ be an irreducible finite dimensional representation of the
Lie algebra $\sigma^{*}{\mathfrak{h}}$: we will see how this induces a
representation of ${\widehat{\mathfrak{h}}_{\mathcal{L}}}$ with the central
element acting as multiplication by $\ell\in\mathbb{N}$. As first step, note
that the exact sequence

$0\to\mathcal{I}_{\sigma}\to\mathscr{O}_{X}\to\mathscr{O}_{S}\to 0$

defining $\sigma(S)$ gives rise to the map of Lie algebras
$[\star]_{\mathcal{I}_{\sigma}}\colon
F^{0}{\mathfrak{h}_{\mathcal{L}}}\to\sigma^{*}{\mathfrak{h}}$ induced by the
truncation map
$\varprojlim_{n}\mathscr{O}_{X}/\mathcal{I}_{\sigma}^{n}\to\mathscr{O}_{X}/\mathcal{I}_{\sigma}$.
The action of $\sigma^{*}{\mathfrak{h}}$ on $\mathcal{V}$ is then extended to
the action of
$F^{0}{\widehat{\mathfrak{h}}_{\mathcal{L}}}=F^{0}{\mathfrak{h}_{\mathcal{L}}}\oplus
c\mathscr{O}_{S}$ by imposing, for every $v\in\mathcal{V}$ and for every $X\in
F^{0}{\mathfrak{h}_{\mathcal{L}}}$, the relations

$c*v:=\ell vX*v:=[X]_{\mathcal{I}_{\sigma}}v.$

In view of this, once we fix $\ell\in\mathbb{N}$ we always view a
representation $\mathcal{V}$ of $\sigma^{*}{\mathfrak{h}}$ as a
$UF^{0}{\widehat{\mathfrak{h}}_{\mathcal{L}}}$-module with the central part
acting by multiplication by $\ell$.

######  Definition 3.2.3.

For every $\ell\in\mathbb{N}$ we define the _Verma module of level $\ell$
attached to $\mathcal{V}$_ to be left
$U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$-module of level $\ell$ attached to
$\mathcal{V}$, meaning

${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}})}:=U{\widehat{\mathfrak{h}}_{\mathcal{L}}}\underset{UF^{0}{\widehat{\mathfrak{h}}_{\mathcal{L}}}}{\otimes}\mathcal{V}$

where $F^{0}{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ acts on
$U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ by multiplication on the right and
$U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ acts on
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}})}$ by left multiplication.

######  Remark 3.2.4.

Note that when $\mathcal{V}$ is the trivial representation of
$\sigma^{*}{\mathfrak{h}}$, we obtain that
${\widetilde{\mathcal{H}_{\ell}}({\mathscr{O}_{S}})}$ coincides with
${\widetilde{\mathcal{H}_{\ell}}({0})}$ given in Definition 3.2.1.

In the constant case $\sigma^{*}{\mathfrak{h}}\cong{\mathfrak{g}}$, the
properties af ${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}})}$ have been
studied in [Kac90, Chapter 7] when $R=k$ and $\mathcal{V}$ an irreducible
representation of ${\mathfrak{g}}$ of level at most $\ell$, where it is shown
that it has a maximal irreducible quotient
${\mathcal{H}_{\ell}({\mathcal{V}})}$. From this, one generalizes the
construction to families of curves, but still in the constant case
$\sigma^{*}{\mathfrak{h}}\cong{\mathfrak{g}}$, meaning working on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)^{1}_{g}}$. The new step is to
descend ${\mathcal{H}_{\ell}({\mathcal{V}})}$ from
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)^{1}_{g}}$ to
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$.

We then first of all recall the construction in the constant case in Section
3.2.1 and then show how it descends to
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ in Section 3.2.2.

####  3.2.1. Integrable representations of level $\ell$ on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)^{1}_{g}}$

The morphism
$\textbf{Forg}^{1}_{1}\colon{\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)^{1}_{g}}\to{\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,1}}$
is a finite étale covering, so if we want to define a-module on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$, we could first define
it on ${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)^{1}_{g}}$ and later show
that the construction is $\Gamma$-equivariant, hence it descends to a module
on ${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$. As already
written, the advantage of working on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)^{1}_{g}}$, is the
identification of ${\mathfrak{h}_{\mathcal{L}}}$ with
${\mathfrak{g}{\mathcal{L}}}$, which allows us to use representation theory of
${\mathfrak{g}}$ and of the affine Lie algebra
${\widehat{\mathfrak{g}}{\mathcal{L}}}$ [Kac90, Chapter 7].

We recall here some facts about representation theory of ${\mathfrak{g}}$ and
${\widehat{\mathfrak{g}}{\mathcal{L}}}$. Let
$R(G,T)=R({\mathfrak{g}},\mathfrak{t})$ be the root system of ${\mathfrak{g}}$
with basis of positive roots $\Delta$. The _dominant weights_ of
${\mathfrak{g}}$ are those element $\lambda\in\mathfrak{t}^{*}$ such that
$\lambda(H_{\alpha})\in\mathbb{N}_{0}$ for all positive roots $\alpha$. By
[Bou75, Théorème 1, Chapitre VIII, §7] the set of dominant weights $P_{+}$ of
${\mathfrak{g}}$ is in bijection with the isomorphism classes of irreducible
and finite dimensional representations of ${\mathfrak{g}}$. The representation
$V_{\lambda}$ associated with $\lambda$ is characterized by the property of
being generated by a highest weight vector $v_{\lambda}$ which is annihilated
by the elements of ${\mathfrak{g}}^{\alpha}$ for every positive root $\alpha$
and such that $H(v_{\lambda})=\lambda(H)v_{\lambda}$ for every
$H\in\mathfrak{t}$. Let $\theta$ be the highest root and denote by
$H_{\theta}$ the highest coroot of ${\mathfrak{g}}$. Then for every
$\ell\in\mathbb{N}$ we set

$P_{\ell}:=\\{\lambda\in P_{+}|\lambda(H_{\theta})\leq\ell\\}.$

In view of the correspondence $\lambda\leftrightarrow V_{\lambda}$, the set
$P_{\ell}$ represents the equivalence classes of representations of level at
most $\ell$, meaning those representations $V_{\lambda}$ of ${\mathfrak{g}}$
where $X^{\ell+1}$ acts trivially on $V_{\lambda}$ for every nilpotent element
$X\in{\mathfrak{g}}$. In what follows we will use $P_{\ell}$ to denote either
the weights or the representations of level at most $\ell$. Note that the
trivial representation corresponds to the trivial weight $\lambda=0$, so that
it belongs to $P_{\ell}$ for every $\ell$.

######  Remark 3.2.5.

We note that the action of $\Gamma$ on ${\mathfrak{g}}$ induces an action of
$\Gamma$ on $P_{\ell}$ in the following way. Let
$\rho_{\lambda}\colon{\mathfrak{g}}\times V\to V$ be the representation
associated to $\lambda$, then we define the representation
$\rho_{\gamma\lambda}\colon{\mathfrak{g}}\times V\to V$ as
$\rho_{\gamma\lambda}(X,v):=\rho_{\lambda}(\gamma^{-1}X)v$ for all
$X\in{\mathfrak{g}}$ and $v\in V$. The weight $\gamma\lambda$ belongs to
$P_{\ell}$ since $\Gamma$ sends nilpotent elements to nilpotent elements.

Let $V_{\lambda}$ be an irreducible and finite dimensional representation of
${\mathfrak{g}}$ and consider the $U\widehat{{\mathfrak{g}}}k((t))$-module
${\widetilde{\mathcal{H}_{\ell}}({V_{\lambda}})}$ constructed as in Definition
3.2.3. The properties of ${\widetilde{\mathcal{H}_{\ell}}({V_{\lambda}})}$ are
well known and described for example in [Kac90], [KR87], [TUY89] and in
[Bea96]. The main results are collected in the following proposition.

######  Proposition 3.2.6.

Let $V_{\lambda}$ be an irreducible and finite dimensional representation of
${\mathfrak{g}}$ of level at most $\ell$.

  1. (1)

The module ${\widetilde{\mathcal{H}_{\ell}}({V_{\lambda}})}$ contains a
maximal proper $U{\widehat{\mathfrak{g}}{\mathcal{L}}}$ submodule
$\mathcal{Z}_{\lambda}$, so that it has a unique maximal irreducible quotient
${\mathcal{H}_{\ell}({V_{\lambda}})}:={\widetilde{\mathcal{H}_{\ell}}({V_{\lambda}})}/\mathcal{Z}_{\lambda}$.

  2. (2)

The natural map $V_{\lambda}\to{\mathcal{H}_{\ell}({V_{\lambda}})}$ sending
$v$ to $1\otimes v$ identifies $V_{\lambda}$ with the submodule of
${\mathcal{H}_{\ell}({V_{\lambda}})}$ annihilated by
$UF^{1}{\mathfrak{g}{\mathcal{L}}}=U{\mathfrak{g}}tk[[t]]$.

  3. (3)

The module ${\mathcal{H}_{\ell}({V_{\lambda}})}$ is integrable, i.e. for any
$X\in{\mathfrak{g}}$ nilpotent and every $f(t)\in k((t))$, the element $Xf(t)$
acts locally nilpotently on ${\mathcal{H}_{\ell}({V_{\lambda}})}$. This means
that there exists $n\in\mathbb{N}$ such that $Xf(t)^{\circ n}$ acts trivially
on ${\mathcal{H}_{\ell}({V_{\lambda}})}$.

It follows that to every $({\widetilde{X}}\to
X\to\textrm{Spec}(k),\tau)\in{\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)^{1}_{g}}(\textrm{Spec}(k))$
and $\lambda\in P_{\ell}$, we can associate the irreducible
$U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ module
${\mathcal{H}_{\ell}({V_{\lambda}})}$ realized as quotient of
${\widetilde{\mathcal{H}_{\ell}}({V_{\lambda}})}$.

Let $({\widetilde{X}}\to X\to
S,\tau)\in{\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)^{1}_{g}}(S)$ and call
$\sigma$ the composition $p\tau$. An isomorphism of
${\mathfrak{h}_{\mathcal{L}}}$ with ${\mathfrak{g}{\mathcal{L}}}$ is fixed by
$\tau$, as well as an isomorphism of ${\sigma}^{*}{\mathfrak{h}}$ with
${\mathfrak{g}}\otimes_{k}\mathscr{O}_{S}$. Denote by
$\mathcal{V}_{\lambda}:=V_{\lambda}\otimes_{k}\mathscr{O}_{S}$ the extension
of scalars of $V_{\lambda}$ from $k$ to $\mathscr{O}_{S}$, so that
$\mathcal{V}_{\lambda}$ is naturally a representation of
${\mathfrak{g}}\otimes_{k}\mathscr{O}_{S}={\sigma}^{*}{\mathfrak{h}}$. We show
how to construct ${\mathcal{H}_{\ell}({\mathcal{V}_{\lambda}})}$ as quotient
of ${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}$.

Let us assume first that $\widehat{\mathcal{O}}\cong\mathscr{O}_{S}[[t]]$.
This implies that ${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}$
is isomorphic to

$U{\mathfrak{g}}t^{-1}\mathscr{O}_{S}[t^{-1}]\otimes_{\mathscr{O}_{S}}\mathcal{V}_{\lambda}=U{\mathfrak{g}}t^{-1}k[t^{-1}]\otimes_{k}V_{\lambda}\otimes_{k}\mathscr{O}_{S}={\widetilde{\mathcal{H}_{\ell}}({V_{\lambda}})}\otimes_{k}\mathscr{O}_{S}.$

######  Remark 3.2.7.

Observe that the isomorphism that we have obtained between
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}$ and
${\widetilde{\mathcal{H}_{\ell}}({V_{\lambda}})}\otimes_{k}\mathscr{O}_{S}$
does not depend on the choice of the parameter $t$.

It follows that ${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}$
has a unique maximal $U\widehat{{\mathfrak{g}}}\otimes_{k}k((t))$ proper
submodule $\mathcal{Z}_{S}:=\mathcal{Z}_{\lambda}\otimes\mathscr{O}_{S}$,
where $\mathcal{Z}_{\lambda}$ is the maximal proper submodule of
${\mathcal{H}_{\ell}({V_{\lambda}})}$. We define
${\mathcal{H}_{\ell}({\mathcal{V}_{\lambda}})}$ as the quotient
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}{\,\big{/}}\mathcal{Z}_{S}$
or equivalently as
${\mathcal{H}_{\ell}({V_{\lambda}})}\otimes\mathscr{O}_{S}$. This construction
uses a choice of the isomorphism
${\mathfrak{h}_{\mathcal{L}}}\cong{\mathfrak{g}{\mathcal{L}}}$, but since
$\mathcal{Z}_{\lambda}$ and hence $\mathcal{Z}_{S}$ satisfy a maximality
condition, they do not depend on the isomorphism
${\mathfrak{h}_{\mathcal{L}}}\cong{\mathfrak{g}{\mathcal{L}}}$, concluding
that that ${\mathcal{H}_{\ell}({\mathcal{V}_{\lambda}})}$ is the maximal
irreducible quotient representation of
$U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$.

In the general case, we want to show that Zariski locally on $S$ we can reduce
to $\widehat{\mathcal{O}}\cong\mathscr{O}_{S}[[t]]$ so that we can locally
define ${\mathcal{H}_{\ell}({\mathcal{V}_{\lambda}})}$ and then show that this
gives rise to a global object. Since $\mathcal{I}_{\sigma}$ is locally
principal, we can find an open covering $\\{U_{i}\\}$ of $X$ such that
$\mathcal{I}_{\sigma}|_{U_{i}}$ is principal. This implies that
$\varprojlim_{n}\mathscr{O}_{U_{i}}/\left(\mathcal{I}_{\sigma}|_{U_{i}}\right)^{n}$
is isomorphic to $\mathscr{O}_{S_{i}}[[t]]$ where $S_{i}:=\sigma^{-1}(U_{i})$.
Observe that this does not imply that
$\widehat{\mathcal{O}}\otimes_{S}\mathscr{O}_{S_{i}}\cong\mathscr{O}_{S_{i}}[[t]]$,
but only that
$\widehat{\mathcal{O}}\,\widehat{\otimes}_{S}\,\mathscr{O}_{S_{i}}\cong\mathscr{O}_{S_{i}}[[t]]$.
Consider then the sheaf of Lie algebras
${\mathfrak{g}{\mathcal{L}}}_{i}:={\mathfrak{g}}\otimes\mathscr{O}_{S_{i}}((t))\cong{\mathfrak{g}{\mathcal{L}}}\,\widehat{\otimes}\,\mathscr{O}_{S_{i}}$,
and construct the $U{\widehat{\mathfrak{g}}{\mathcal{L}}}_{i}$-module
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}_{i}$.

######  Claim.

The inclusion
${\mathfrak{g}{\mathcal{L}}}\otimes\mathscr{O}_{S_{i}}\to{\mathfrak{g}{\mathcal{L}}}_{i}$
induces an isomorphism of $\mathscr{O}_{S_{i}}$-modules
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}\otimes_{S}\mathscr{O}_{S_{i}}\cong{\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}_{i}$.

###### Proof.

We need to prove that
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}\to\mathscr{O}_{S_{i}}\cong{\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}_{i}$
is surjective. We use induction on the length of the elements of
$U{\mathfrak{g}{\mathcal{L}}}_{i}$, where the length of an element $u\in
U{\mathfrak{g}{\mathcal{L}}}_{i}$ is the minimum $n$ such that
$u\in\oplus_{j=0}^{n}{{\mathfrak{g}{\mathcal{L}}}_{i}}^{\otimes j}$. Let
$aX\in{\mathfrak{g}{\mathcal{L}}}_{i}$ with $X\in{\mathfrak{g}}$ and
$a=\sum_{i\geq-N}a_{i}t^{i}\in\mathscr{O}_{S_{i}}((t))$, and take
$v\in\mathcal{V}_{\lambda}$. The class of $aX\otimes v$ in
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}_{i}$ is the same as
the one of $[aX]\otimes v:=[a]X\otimes v$, where
$[a]=\sum_{i\geq-N}^{0}a_{i}t^{i}$, which then belongs to
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}\otimes_{S}\mathscr{O}_{S_{i}}$.
Let now $Y=Y_{1}\circ\dots\circ Y_{n}$ be an element of
$U{\mathfrak{g}{\mathcal{L}}}_{i}$, and note that in
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}_{i}$ the element
$Y\otimes v$ is equivalent to the class of
$([Y_{n}]\circ\dots\circ[Y_{1}]+u)\otimes v$ where $u$ has length lower than
$n$. Using the induction hypothesis we conclude the proof. ∎

We define the $\mathscr{O}_{S_{i}}$-module
${\mathcal{H}_{\ell}({\mathcal{V}_{\lambda}})}|_{S_{i}}$ to be
${\mathcal{H}_{\ell}({\mathcal{V}_{\lambda}})}_{i}={\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}_{i}{\,\big{/}}\mathcal{Z}_{i}$.
This gives rise to the $\mathscr{O}_{S}$-module
${\mathcal{H}_{\ell}({\mathcal{V}_{\lambda}})}$ because on the intersection
$S_{ij}$ the modules ${\mathcal{H}_{\ell}({\mathcal{V}_{\lambda}})}_{i}$ and
${\mathcal{H}_{\ell}({\mathcal{V}_{\lambda}})}_{j}$ are isomorphic via to the
transition morphisms defining $\mathcal{I}_{\sigma}$. Equivalently we could
have defined $\mathcal{Z}|_{S_{i}}$ to be the image of $\mathcal{Z}_{i}$ in
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}|_{S_{i}}$ and so
${\mathcal{H}_{\ell}({\mathcal{V}_{\lambda}})}|_{S_{i}}$ would be the quotient
of ${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}|_{S_{i}}$ by
$\mathcal{Z}|_{S_{i}}$. The modules $\mathcal{Z}|_{S_{i}}$ glue and give rise
to a ${\widehat{\mathfrak{g}}{\mathcal{L}}}$-module $\mathcal{Z}$ on $S$, so
that ${\mathcal{H}_{\ell}({\mathcal{V}_{\lambda}})}$ is given by
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}_{\lambda}})}{\,\big{/}}\mathcal{Z}$.
This construction is invariant under the action of $\Gamma$, hence it defines
${\widehat{\mathfrak{h}}_{\mathcal{L}}}$ as a
$U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$-module.

####  3.2.2. Integrable representations of level $\ell$ on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$

We show here how to descend ${\mathcal{H}_{\ell}({\mathcal{V}})}$ from
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ to
${\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,1}}$, so let consider
$({\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S,\sigma)\in{\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,1}}(S)$.
The first issue is that, unless we choose an isomorphism between
$\sigma^{*}{\mathfrak{h}}$ and ${\mathfrak{g}}$, we are not able to provide a
representation of $\sigma^{*}{\mathfrak{h}}$ associated to a $\lambda\in
P_{\ell}$. In fact, one obstruction to this, as we noticed in Remark 3.2.5, is
that $\Gamma$ does not in general act trivially on $P_{\ell}$, so it is
impossible to identify $\mathcal{V}$ as $\mathcal{V}_{\lambda}$, with
$\lambda$ independent of the isomorphism between
${\mathfrak{h}_{\mathcal{L}}}$ and ${\mathfrak{g}{\mathcal{L}}}$.

The conclusion is that it seems unreasonable to associate to $\lambda\in
P_{\ell}$ a module ${\mathcal{H}_{\ell}({\mathcal{V}_{\lambda}})}$ on
${\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,1}}$ because $P_{\ell}$ contains only
local information. The following set is what replaces $P_{\ell}$.

######  Definition 3.2.8.

A representation $\mathcal{V}$ of $\sigma^{*}{\mathfrak{h}}$ is said to be of
level at most $\ell$ if for every nilpotent element $X$ of
$\sigma^{*}{\mathfrak{h}}$, then $X^{\ell+1}$ acts trivially on $\mathcal{V}$.
Equivalently this means that locally étale we can identify $\mathcal{V}$ with
$V\otimes\mathscr{O}_{S}$ for a representation $V\in P_{\ell}$. Define
${\text{IrRep}_{\ell}}(\sigma^{\mathfrak{h}})$ or by abuse of notation only
${\text{IrRep}_{\ell}}(\sigma)$ or ${\text{IrRep}_{\ell}}$ to be the set of
isomorphism classes of irreducible and finite dimensional representations
$\mathcal{V}$ of $\sigma^{*}{\mathfrak{h}}$ of level at most $\ell$.

The main step towards the definition of the sheaf of conformal blocks attached
to $\mathcal{V}\in{\text{IrRep}_{\ell}}$ is the following result.

######  Proposition 3.2.9.

Let $\mathcal{V}\in{\text{IrRep}_{\ell}}$. Then there exists a unique maximal
proper $U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ submodule $\mathcal{Z}$ of
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}})}$.

###### Proof.

We show that the maximal proper submodule of
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}})}$ on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)^{1}_{g}}$ descends along
$\textbf{Forg}^{1}_{1}$ to the maximal proper submodule of
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}})}$ on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$. Recall that since
$\sigma(S)$ does not intersect the branch locus of $q$, we can find an étale
covering $S^{\prime}\to S$ such that the pullback of $({\widetilde{X}}\to X\to
S,\sigma)$ lies in the image of $\textbf{Forg}^{1}_{1}$. This implies that to
give $\mathcal{V}\in{\text{IrRep}_{\ell}}$ is equivalent to give an
irreducible and finite dimensional representation $\mathcal{V}^{\prime}$ of
${\sigma^{\prime}}^{*}{\mathfrak{h}}$ and an isomorphism $\phi\colon
p_{1}^{*}\mathcal{V}^{\prime}\to p_{2}^{*}\mathcal{V}^{\prime}$ satisfying the
cocycle conditions on $S^{\prime\prime\prime}$, where $p_{i}\colon
S^{\prime\prime}=S^{\prime}\times_{S}S^{\prime}\to S^{\prime}$ is the $i$-th
projection.

This tells us moreover that ${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}})}$
is obtained by descending
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}^{\prime}})}$ from $S^{\prime}$
to $S$. Observe that up to the choice of an isomorphism
${\sigma^{\prime}}^{*}{\mathfrak{h}}\cong{\mathfrak{g}}\otimes_{k}\mathscr{O}_{S^{\prime}}$,
the representation $\mathcal{V}^{\prime}$ is of the form
$\mathcal{V}_{\lambda}$, so that $\mathcal{Z}^{\prime}$ and
${\mathcal{H}_{\ell}({\mathcal{V}^{\prime}})}$ are well defined. We construct
${\mathcal{H}_{\ell}({\mathcal{V}})}$ by descending $\mathcal{Z}^{\prime}$ to
a module $\mathcal{Z}$ on $S$, so that
${\mathcal{H}_{\ell}({\mathcal{V}})}:={\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}})}{\,\big{/}}\mathcal{Z}$.

Since ${\mathfrak{h}_{\mathcal{L}}}$ is a module on $\mathscr{O}_{S}$, we have
a canonical isomorphism
$\phi_{12}\colon{p_{1}}^{*}{\mathfrak{h}_{\mathcal{L}}}|_{S^{\prime}}\to{p_{2}}^{*}{\mathfrak{h}_{\mathcal{L}}}|_{S^{\prime}}$
satisfying the cocycle conditions on
$S^{\prime\prime\prime}:=S^{\prime\prime}\times_{S}S^{\prime}$. Recall
moreover that $\mathcal{Z}^{\prime}$ is the maximal proper
$U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ submodule of
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}^{\prime}})}$, which is then
$\Gamma$-invariant. This induces an isomorphism between
$p_{1}^{*}\mathcal{Z}^{\prime}$ and $p_{2}^{*}\mathcal{Z}^{\prime}$ which
satisfied the cocycle condition on $S^{\prime\prime\prime}$ and it is
independent of the isomorphism
${\mathfrak{h}_{\mathcal{L}}}\cong{\mathfrak{g}{\mathcal{L}}}$. ∎

######  Definition 3.2.10.

Let $\mathcal{V}\in{\text{IrRep}_{\ell}}$. The maximal irreducible quotient of
${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}})}$ is denoted
${\mathcal{H}_{\ell}({\mathcal{V}})}$ and defines a sheaf

${\mathcal{H}_{\ell}({\mathcal{V}})}_{X}={\mathfrak{h}_{\mathcal{A}}}\circ{\mathcal{H}_{\ell}({\mathcal{V}})}{\,\big{\backslash}}{\mathcal{H}_{\ell}({\mathcal{V}})}$

on $S$ which is called the _sheaf of conformal blocks attached to
$\mathcal{V}$_. When $\mathcal{V}$ is the trivial representation of
$\sigma^{*}{\mathfrak{h}}$, we denote ${\mathcal{H}_{\ell}({\mathcal{V}})}$ by
${\mathcal{H}_{\ell}(0)}$ and its quotient
${\mathfrak{h}_{\mathcal{A}}}{\,\big{\backslash}}{\mathcal{H}_{\ell}(0)}$ is
called the _sheaf of covacua._

The collection $\\{{\mathcal{H}_{\ell}(0)}_{X}\\}_{{\widetilde{X}}\to X\to S}$
determines the sheaf of modules ${\mathcal{H}_{\ell}(0)}_{X_{univ}}$ on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$. In similar way, given
compatible families $\\{\mathcal{V}(\sigma)\\}_{\\{{\widetilde{X}}\to X\to
S,\sigma\\}}$ defining an element $\mathcal{V}$ of
${\text{IrRep}_{\ell}}(\sigma_{univ})$, the collection
${\mathcal{H}_{\ell}({\mathcal{V}(\sigma)})}_{X}$ defines
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$.

Observe that Proposition 3.2.6 generalizes as follows.

######  Corollary 3.2.11.

Let $\mathcal{V}\in{\text{IrRep}_{\ell}}(\sigma)$, then:

  1. (1)

The natural map $\mathcal{V}\to{\mathcal{H}_{\ell}({\mathcal{V}})}$ sending
$v$ to $1\otimes v$ identifies $\mathcal{V}$ with the submodule of
${\mathcal{H}_{\ell}({\mathcal{V}})}$ annihilated by
$UF^{1}{\widehat{\mathfrak{h}}_{\mathcal{L}}}$.

  2. (2)

The module ${\mathcal{H}_{\ell}({\mathcal{V}})}$ is integrable.

Inspired by [Sor96, Section 2.5] we prove the following statement.

######  Proposition 3.2.12.

The $\mathscr{O}_{S}$-module ${\mathcal{H}_{\ell}({\mathcal{V}})}_{X}$ is
coherent. It follows that ${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ is
a coherent module on ${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$.

###### Proof.

This is essentially a consequence of [Sor96, Lemma 2.5.2]. As this is a local
statement, we can assume that ${\mathcal{L}}\cong R((t))$ and we can fix an
isomorphism ${\mathfrak{h}_{\mathcal{L}}}\cong{\mathfrak{g}{\mathcal{L}}}$.
Observe that the quotient
${\mathfrak{h}_{\mathcal{A}}}{\,\big{\backslash}}{\mathfrak{h}_{\mathcal{L}}}{\,\big{/}}F^{0}{\mathfrak{h}_{\mathcal{L}}}$
is a finitely generated $R$-module as it computes $H^{1}(X,{\mathfrak{h}})$
and ${\mathfrak{h}}$ is locally free over $X$. This implies that
${\mathfrak{h}_{\mathcal{A}}}{\,\big{\backslash}}{\widehat{\mathfrak{h}}_{\mathcal{L}}}{\,\big{/}}F^{1}{\mathfrak{h}_{\mathcal{L}}}$
is finitely generated too over $R$ and so we can choose finitely many
generators $e_{1},\dots,e_{n}$ so that we can write

${\widehat{\mathfrak{h}}_{\mathcal{L}}}=F^{1}{\mathfrak{h}_{\mathcal{L}}}+{\mathfrak{h}_{\mathcal{A}}}+\sum_{i=1}^{n}Re_{i}.$

which in terms of enveloping algebras becomes

$U{\widehat{\mathfrak{h}}_{\mathcal{L}}}=\sum_{(N_{1},\dots,N_{n})\in{\mathbb{N}_{0}}^{n}}U({\mathfrak{h}_{\mathcal{A}}})\circ{e_{1}}^{\circ
N_{1}}\circ\cdots\circ{e_{n}}^{\circ N_{n}}\circ
U(F^{1}{\mathfrak{h}_{\mathcal{L}}})$

thing that can be proven using induction on the length of elements of
$U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$.

We can furthermore assume that the elements $e_{i}$ acts locally nilpotently
on ${\mathcal{H}_{\ell}({\mathcal{V}})}$, meaning that there exists
$M\in\mathbb{N}$ such that $e_{i}^{\circ M}$ acts trivially on
${\mathcal{H}_{\ell}({\mathcal{V}})}$. In fact we might use the isomorphism
${\mathfrak{h}_{\mathcal{L}}}$ with ${\mathfrak{g}{\mathcal{L}}}$ and the
Cartan decomposition of ${\mathfrak{g}}=\mathfrak{t}\oplus_{\alpha\in
R({\mathfrak{g}},\mathfrak{t})}{\mathfrak{g}}_{\alpha}$. The algebras
${\mathfrak{g}}_{\alpha}$’s are nilpotent and generate ${\mathfrak{g}}$, so
that ${\mathfrak{g}{\mathcal{L}}}$ is generated by $\oplus_{\alpha\in
R({\mathfrak{g}},\mathfrak{t})}{\mathfrak{g}}_{\alpha}{\mathcal{L}}$. This
means that also the elements $e_{i}$ are generated by elements of
$\oplus_{\alpha\in
R({\mathfrak{g}},\mathfrak{t})}{\mathfrak{g}}^{\alpha}{\mathcal{L}}$ so that,
up to replace $e_{i}$ with a choice of nilpotent generators, we can ensure
that all the $e_{i}$’s live in $\oplus_{\alpha\in
R({\mathfrak{g}},\mathfrak{t})}{\mathfrak{g}}^{\alpha}{\mathcal{L}}$ and so
using Corollary 3.2.11 (2) the $e_{i}$’s will act locally nilpotently on
${\mathcal{H}_{\ell}({\mathcal{V}})}$.

It follows that

${\widetilde{\mathcal{H}_{\ell}}({\mathcal{V}})}=\sum_{(N_{1},\dots,N_{n})\in{\mathbb{N}_{0}}^{n}}U({\mathfrak{h}_{\mathcal{A}}})\circ{e_{1}}^{\circ
N_{1}}\circ\cdots\circ{e_{n}}^{\circ
N_{n}}\underset{c=\ell}{\otimes}\mathcal{V}$

and that

${\mathcal{H}_{\ell}({\mathcal{V}})}=\sum_{(N_{1},\dots,N_{n})\in{\mathbb{N}_{0}}^{n}}U({\mathfrak{h}_{\mathcal{A}}})\circ{e_{1}}^{\circ
N_{1}}\circ\cdots\circ{e_{n}}^{\circ
N_{n}}\underset{c=\ell}{\otimes}\mathcal{V}{\,\big{/}}\mathcal{Z}$

Using induction on $n$ and the fact that the $e_{i}$’s act locally
nilpotently, we can conclude that the sum can be taken over finitely many
$(N_{1},\dots,N_{n})\in{\mathbb{N}_{0}}^{n}$, hence that the quotient
${\mathfrak{h}_{\mathcal{A}}}{\,\big{\backslash}}{\mathcal{H}_{\ell}({\mathcal{V}})}={\mathcal{H}_{\ell}({\mathcal{V}})}_{X}$
is finitely generated.∎

##  4\. The projective connection on
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$

We want to prove that the sheaf of conformal blocks
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ is a vector bundle on the
Hurwitz stack ${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$, so that
its rank will be constant. Since we already know that
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ is coherent, one method to
exhibit local freeness is to provide a projectively flat connection on it. In
this section we provide a projective action of
$\mathcal{T}_{{\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}/k}(-\log(\Delta))$
on ${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$, showing its freeness when
restricted to ${\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,1}}$.

###  4.1. The tangent to
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$

Let $({\widetilde{X}}\to
X,\sigma)\in{\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}(\textrm{Spec}(k))$
and recall that in Remark 2.2.3 we saw that the tangent space of
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ at
$({\widetilde{X}}\to X,\sigma)$ is isomorphic to the tangent space of
${\mathcal{M}_{g,(1+\deg(\xi))}}$ at $(X,\sigma\cup\mathcal{R})$. The latter,
which is the space of infinitesimal deformations of
$(X,\sigma\cup\mathcal{R})$, can be explicitely described as the space
$\textrm{Ext}^{1}(\Omega_{X/k},\mathscr{O}(-\mathcal{R}-\sigma(S))$ [ACG11,
Chapter XI] which sits in the short exact sequence

$0\to
H^{1}(X,\mathcal{T}_{X/k}(-\mathcal{R}-\sigma(S))\to\textrm{Ext}^{1}(\Omega_{X/k},\mathscr{O}(-\mathcal{R}-\sigma(S))\to
H^{0}(X,\mathscr{E}\\!\textit{xt}^{1}(\Omega_{X/k},\mathscr{O}(-\mathcal{R}-\sigma(S)))\to
0$

where the last term is supported on the singular points of $X$.

We now use the assumption that the curve $(X,\sigma\cup\mathcal{R})$ is stably
marked to assume that there exists a versal family $\mathcal{X}\to S$ with a
reduced divisor $\sigma_{\mathcal{X}}+\mathcal{R}_{\mathcal{X}}$ deforming it
and such that the subscheme of $S$ whose fibres are singular is a normal
crossing divisor $\Delta$. Call $s_{0}$ the point of $S$ such that
$\mathcal{X}|_{s_{0}}$ is $X$. The versality condition means that the Kodaira-
Spencer map

$\texttt{KS}\colon\mathcal{T}_{S/k}\to\mathscr{E}\\!\textit{xt}^{1}(\Omega_{\mathcal{X}/S},\mathscr{O}(-\mathcal{R}_{\mathcal{X}}-\sigma_{\mathcal{X}}))$

is an isomorphism, so that we identify the tangent space of
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ at
$({\widetilde{X}}\to X,\sigma)$ with the tangent space of $S$ at $s_{0}$.

The conclusion is that to provide a projective connection on
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ is equivalent to provide an
action of $\mathcal{T}_{S/k}$ on ${\mathcal{H}_{\ell}({\mathcal{V}})}_{X}$ for
every versal family ${\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S$. As
aforementioned, we will however not be able to provide a projective action of
the whole $\mathcal{T}_{S/k}$, but only of the submodule
$\mathcal{T}_{S/k}(-\log(\Delta))$, which via the Kodaira-Spencer map is
identified with $R^{1}\pi_{*}(\mathcal{T}_{X/S}(-\mathcal{R}-\sigma(S)))$.

###  4.2. Tangent bundles and the action of $\Gamma$

In view of the previous observations, we assume that
$({\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S,\sigma)\in{\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}(S)$
is a versal family, so that the locus of points $s$ of $S$ such that the
fibres $X_{s}$ (or equivalently ${\widetilde{X}}_{s}$) are non smooth is a
normal crossing divisor $\Delta$ of $S$. We give in this section a description
of $\mathcal{T}_{S/k}(-\log(\Delta))$ and $\mathcal{T}_{S/k}$, by realizing it
as a quotient of certain sheaves of derivations. Consider, to begin with, the
following situation described by Looienga [Loo13, Section 2]. Let $R\in
k\textbf{-Alg}$ and $L=R((t))$, then we can consider the following two
modules: $\theta_{L/R}$ consisting of continuous $R$-linear derivations of $L$
and $\theta_{L,R}$ consisting of continuous $k$-linear derivations of $L$
which restrict to derivations of $R$ into itself. The quotient
$\theta_{L,R}/\theta_{L/R}$ is canonically identified with the module of
$k$-linear derivations of $R$.

Take
$(f\colon{\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S,\sigma)\in{\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}(S)$.
We already introduced in Subsection 3.1 the $\mathscr{O}_{S}$-module
$\theta_{{\mathcal{L}}/S}$ of continuous $\mathscr{O}_{S}$-linear derivations
of ${\mathcal{L}}$ and we define now $\theta_{{\mathcal{L}},S}$ as the
$\mathscr{O}_{S}$-module of continuous $k$-linear derivations of
${\mathcal{L}}$ which restrict to derivations of $\mathscr{O}_{S}$. Observe
that $\theta_{{\mathcal{L}}/S}$ and $\theta_{{\mathcal{L}},S}$ depend only on
the marked curve $(X\to S,\sigma)$, so the following well known result belongs
to the classical setting.

######  Proposition 4.2.1.

The sequence of $\mathscr{O}_{S}$-modules

$0\to\theta_{{\mathcal{L}}/S}\to\theta_{{\mathcal{L}},S}\to\mathcal{T}_{S/k}\to
0$

is exact.

###### Proof.

As exactness can be checked on formal neighbourhoods, we can assume that
${\mathcal{L}}\cong R((t))$ so that the result follows from the example
presented above.∎

In similar fashion we now describe the subsheaf
$\mathcal{T}_{S/k}(-\log(\Delta))$ as quotient of appropriate sheaves of
derivations.

#### The sheaves $\theta_{{\mathcal{A}}/S}(-\mathcal{R})$ and
$\theta_{{\mathcal{A}},S}(-\mathcal{R})$

Following Looijenga’s notation, we denote by $\theta_{{\mathcal{A}}/S}$ the
sheaf of derivations $f_{*}\mathcal{T}_{X/S}$. Recall that in Lemma 2.1.2 we
have showed that there is an isomorphism between
$(q_{*}\mathcal{T}_{{\widetilde{X}}/S})^{\Gamma}$ and
$\mathcal{T}_{X/S}(-\mathcal{R})$. This implies that
$f_{*}(\mathcal{T}_{{\widetilde{X}}^{*}/S}))^{\Gamma}\cong\theta_{{\mathcal{A}}/S}\otimes_{S}\pi_{*}\mathscr{O}(-\mathcal{R})$
and by abuse of notation we will denote this sheaf by
$\theta_{{\mathcal{A}}/S}(-\mathcal{R})$. In a similar way we consider the
action of $\Gamma$ on the pushforward to $S$ of
$\mathcal{T}_{{\widetilde{X}}^{*},S}$, the sheaf of $k$-linear derivations of
$\mathscr{O}_{{\widetilde{X}}^{*}}$ which restrict to derivations of
$f^{-1}\mathscr{O}_{S}$ and we call $\theta_{{\mathcal{A}},S}(-\mathcal{R})$
the sub module of $\Gamma$-invariants.

######  Remark 4.2.2.

Recall that we defined $\widetilde{{\mathcal{L}}}$ as
$\varinjlim_{N}f_{*}\varprojlim_{n}(q^{*}\mathcal{I}_{\sigma})^{-N}/(q^{*}\mathcal{I}_{\sigma})^{n}$
and define now

$\theta_{\widetilde{{\mathcal{L}}}/S}:=\varinjlim_{N}\pi_{*}q_{*}\varprojlim_{n}\mathcal{T}_{{\widetilde{X}}/S}\otimes_{\mathscr{O}_{\widetilde{X}}}q^{*}\mathcal{I}_{\sigma}^{-N}/q^{*}\mathcal{I}_{\sigma}^{n}$

or equivalently $\theta_{\widetilde{{\mathcal{L}}}/S}$ is the module of
continuous derivations of $\widetilde{{\mathcal{L}}}$ which are
$\mathscr{O}_{S}$ linear. Thanks to Lemma 2.1.2, the
$\mathscr{O}_{S}$-submodule of $\Gamma$-invariants is identified with
$\varinjlim_{N}\pi_{*}\varprojlim_{n}\mathcal{T}_{X/S}(-\mathcal{R})\otimes_{\mathscr{O}_{X}}\mathcal{I}_{\sigma}^{-N}/\mathcal{I}_{\sigma}^{n}$
which equals
$\varinjlim_{N}\pi_{*}\varprojlim_{n}\mathcal{T}_{X/S}\otimes_{\mathscr{O}_{X}}\mathcal{I}_{\sigma}^{-N}/\mathcal{I}_{\sigma}^{n}$
as $\mathcal{R}$ and $\sigma(S)$ are disjoint. The latter is the
$\mathscr{O}_{S}$-module of continuous and $\mathscr{O}_{S}$-linear
derivations of ${\mathcal{L}}$, which is $\theta_{{\mathcal{L}}/S}$.

The previous remark implies moreover that $\theta_{{\mathcal{A}}/S}$ is a
submodule of $\theta_{{\mathcal{L}}/S}$.

######  Remark 4.2.3.

Observe that the action of $\mathcal{T}_{{\widetilde{X}}/S}$ (resp. of
$\mathcal{T}_{{\widetilde{X}},S}$) on
${\mathfrak{g}}\otimes_{k}\mathscr{O}_{\widetilde{X}}$ by coefficientwise
derivation is $\Gamma$-equivariant. This implies that
$(q_{*}\mathcal{T}_{{\widetilde{X}}/S})^{\Gamma}$ (resp.
$(q_{*}\mathcal{T}_{{\widetilde{X}},S})^{\Gamma}$) acts on ${\mathfrak{h}}$
and we will say that the action is by coefficientwise derivation. By
restricting ourselves to ${\widetilde{X}}^{*}$ this implies that
$\theta_{{\mathcal{A}}/S}(-\mathcal{R})$ (resp.
$\theta_{{\mathcal{A}},S}(-\mathcal{R})$) acts on
${\mathfrak{h}_{\mathcal{A}}}$ by coefficientwise derivation. The same holds
for $\theta_{{\mathcal{L}}/S}$ and $\theta_{{\mathcal{L}},S}$ acting on
${\mathfrak{h}_{\mathcal{L}}}$.

######  Example 4.2.4.

We recall in this example the local description of the quotient
$\theta_{{\mathcal{A}},S}{\,\big{/}}\theta_{{\mathcal{A}}/S}$ at nodal points
[Loo13, Section 5]. Let $R$ be a local complete $k$-algebra with maximal ideal
$\mathfrak{m}$ and let $A:=R[[x,y]]/xy-t$ for some $t\in\mathfrak{m}$. We are
interested in the following $R$-modules: $\theta_{A/R}$, the module of
$R$-linear derivations of $A$, which lives inside $\theta_{A,R}$, the module
of $k$-linear derivations of $A$ which restrict to derivations of $R$. We are
interested in understanding the quotient $\theta_{A,R}/\theta_{A/R}$ and we
claim that this gets identified with $\theta_{R/k}(-\log(t))$, the module of
$k$-linear derivations of $R$ which send the element $t$ to an element of the
ideal $tR$. We in fact note that from the relation $xy=t$, each derivation
$D\in\theta_{A,R}$ should send $t$ to a multiple of itself inside $R$, hence
the natural map $\theta_{A,R}\to\theta_{R/k}$, whose kernel is $\theta_{A/R}$,
has image landing inside $\theta_{R/k}(-\log(t))$ and what we need to prove is
that this is exactly the image. Given in fact a derivation
$D\in\theta_{R/k}(-\log(t))$ , we claim that it is possible to extend it to a
derivation of $A$, i.e. to define $D(x)$ and $D(y)$ satisfying
$xD(y)+yD(x)=D(t)$. Since $D(t)=tr=2xyr$ for some $r\in R$, it will be enough
to set $D(y)=yr$ and $D(x)=xr$.

######  Proposition 4.2.5.

The sequence

$0\to\theta_{{\mathcal{A}}/S}(-\mathcal{R})\to\theta_{{\mathcal{A}},S}(-\mathcal{R})\to\mathcal{T}_{S}(-\log(\Delta))\to
0$

is exact.

###### Proof.

As taking $\Gamma$-invariants is an exact functor ($\text{char}(k)=0$) and
$\Gamma$ acts trivially on $\mathcal{T}_{S/k}(-\log(\Delta))$, it suffices to
prove that the sequence

$0\to f_{*}\mathcal{T}_{{\widetilde{X}}^{*}/S}\to
f_{*}\mathcal{T}_{{\widetilde{X}}^{*},S}\to\mathcal{T}_{S}(-\log(\Delta))\to
0$

is exact. This statement does not depend on the covering, and appears in
[Sor96] and [Loo13]. We give the proof of it, by starting observing that in
the case ${\widetilde{X}}^{*}=\mathbb{A}^{1}_{S}$, the result follows by using
the same argument of the proof of Proposition 4.2.1.

Let $U=S\setminus\Delta$ and denote by
${\widetilde{X}}_{U}:={\widetilde{X}}\times_{S}U$. As ${\widetilde{X}}_{U}$ is
smooth over $S$, there exists an affine covering
$\\{\textrm{Spec}(A_{i})={\widetilde{X}}_{i}\\}$ of ${\widetilde{X}}_{U}$ and
$\\{\textrm{Spec}(R_{i})=U_{i}\\}$ of $U$ such that the map
$f_{i}=f|_{{\widetilde{X}}_{i}}\colon{\widetilde{X}}_{i}\to S_{i}$ factors
through $\mathbb{A}^{1}_{R_{i}}$ via an étale map $\phi_{i}$ as in the
diagram:

---
$\textstyle{{\widetilde{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\widetilde{X}}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{i}}$$\scriptstyle{\phi_{i}}$$\textstyle{\mathbb{A}^{1}_{R_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{S}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
U_{i}}$

It follows that the sequence $0\to
q_{*}\mathcal{T}_{\mathbb{A}^{1}_{R_{i}}/U_{i}}\to
q_{*}\mathcal{T}_{\mathbb{A}^{1}_{R_{i}},U_{i}}\to\mathcal{T}_{U_{i}|k}\to 0$
is exact. The étaleness of $\phi_{i}$ provides an isomorphism between
${f_{i}}_{*}\mathcal{T}_{{\widetilde{X}}_{i}/S_{i}}$ and
$q_{*}\mathcal{T}_{\mathbb{A}^{1}_{R_{i}}/U_{i}}$ and similarly for
${f_{i}}_{*}\mathcal{T}_{{\widetilde{X}}_{i},U_{i}}$ and
$q_{*}\mathcal{T}_{\mathbb{A}^{1}_{R_{i}},U_{i}}$, concluding the argument for
the smooth points.

Let $s$ be a geometric point of $\Delta$ and
${\widetilde{x}}\in{\widetilde{X}}_{s}$ a nodal point. This means that there
exists an isomorphism

$\widehat{\mathscr{O}_{{\widetilde{X}},{\widetilde{x}}}}\cong\widehat{\mathscr{O}_{S,s}}[[x,y]]/xy-t$

for some $t\in\widehat{\mathfrak{m}}_{s}$. It follows that the Example 4.2.4
represents the local picture of the situation, where the result holds,
concluding the proof. ∎

###  4.3. The Virasoro algebra of ${\mathcal{L}}$

Now that we can express $\mathcal{T}_{S/k}(-\log(\Delta))$ as
$\theta_{{\mathcal{A}},S}(-\mathcal{R}){\,\big{/}}\theta_{{\mathcal{A}}/S}(-\mathcal{R})$,
we will define a projective action of $\theta_{{\mathcal{A}},S}(-\mathcal{R})$
on ${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ which factors through that
quotient.

In order to achieve this result, we will follow the methods of [Loo13] and
define as first step the Virasoro algebra $\widehat{\theta}_{{\mathcal{L}}/S}$
of ${\mathcal{L}}$ as a central extension of $\theta_{{\mathcal{L}}/S}$. We
report in this section the construction of
$\widehat{\theta}_{{\mathcal{L}}/S}$ as explained in [Loo13, Section 2], for
which we will use the same notation. As mentioned before, the
$\mathscr{O}_{S}$-module $\theta_{{\mathcal{L}}/S}$ does not depend on the
covering, and the same holds for its central extension
$\widehat{\theta}_{{\mathcal{L}}/S}$. We will see in Section 4.4, how
$\widehat{\theta}_{{\mathcal{L}}/S}$ acts on
${\mathcal{H}_{\ell}({\mathcal{V}})}$ and how it induces a central extension
of $\theta_{{\mathcal{L}},S}$.

#### The Lie algebras ${\mathfrak{l}}$ and its central extension
$\widehat{{\mathfrak{l}}}$.

We denote by ${\mathfrak{l}}$ the sheaf of abelian Lie algebras (over $S$)
whose underlying module is ${\mathcal{L}}$. The filtration
$F^{*}\\!{\mathcal{L}}$ gives the filtration $F^{*}{\mathfrak{l}}$. Denote by
$U{\mathfrak{l}}$ the universal enveloping algebra, which is isomorphic to
$\textrm{Sym}({\mathfrak{l}})$ since ${\mathfrak{l}}$ is abelian. This algebra
is not complete with respect to the filtration $F^{*}{\mathfrak{l}}$, so we
complete it on the right obtaining

$\overline{U}{\mathfrak{l}}:=\varprojlim_{n}U{\mathfrak{l}}/U{\mathfrak{l}}\circ
F^{n}{\mathfrak{l}}.$

######  Remark 4.3.1.

Note that in this case the completions on the right
$\varprojlim_{n}U{\mathfrak{l}}/U{\mathfrak{l}}\circ F^{n}{\mathfrak{l}}$ and
on the left $\varprojlim_{n}F^{n}{\mathfrak{l}}\circ
U{\mathfrak{l}}{\,\big{\backslash}}U{\mathfrak{l}}$ coincide because
${\mathfrak{l}}$ is abelian. The element $\sum_{i\in\mathbb{N}}t^{-i}\circ
t^{i}$ belongs to $\overline{U}{\mathfrak{l}}$, as well as
$\sum_{i\in\mathbb{N}}t^{-(i)^{m}}\circ t^{i}$ for every $m\in\mathbb{N}$.
However $\sum_{i\in\mathbb{N}}t^{-i}\circ t$ is not an element of
$\overline{U}{\mathfrak{l}}$.

We extend centrally ${\mathfrak{l}}$ via the residue pairing described in
Section 3.1 defining the Lie bracket on
$\widehat{{\mathfrak{l}}}={\mathfrak{l}}\oplus\hslash\mathscr{O}_{S}$ as

$[f+\hslash r,g+\hslash s]=\hslash\textrm{Res}(gdf)$

for every $f,g\in{\mathfrak{l}}$ and $r,s\in\mathscr{O}_{S}$. The filtration
of ${\mathfrak{l}}$ extends to a filtration of $\widehat{{\mathfrak{l}}}$ by
setting $F^{i}\widehat{{\mathfrak{l}}}=F^{i}{\mathfrak{l}}$ for $i\geq 0$ and
$F^{i}\widehat{{\mathfrak{l}}}=F^{i}{\mathfrak{l}}\oplus\hslash\mathscr{O}_{S}$
for $i\leq 0$. The universal enveloping algebra of $\widehat{{\mathfrak{l}}}$
is denoted by $U\widehat{{\mathfrak{l}}}$ and
$\overline{U}\widehat{{\mathfrak{l}}}$ denotes its completion on the right
with respect to the filtration $F^{*}\widehat{{\mathfrak{l}}}$. Note that
since $\hslash$ is a central element, we have that
$\overline{U}\widehat{{\mathfrak{l}}}$ is an $\mathscr{O}_{S}[\hslash]$
algebra so that we will write $\hslash^{2}$ instead of $\hslash\circ\hslash$
and similarly $\hslash^{n}$ for every $n\in\mathbb{N}$.

######  Remark 4.3.2.

Since $\widehat{{\mathfrak{l}}}$ is no longer abelian, completion on the right
and on the left differ. Take for example the element
$\sum_{i\in\mathbb{N}}t^{i}\circ t^{-i}$ which belongs to
$\overline{U}{\mathfrak{l}}$. It does not belong to
$\overline{U\widehat{{\mathfrak{l}}}}$: an element on the completion on the
right morally should have zeros of increasing order on the right side, but in
this case, in order to "bring the element $t^{i}$ on the right side", we
should use the equality $t^{i}\circ t^{-i}=t^{-i}\circ t^{i}+\hslash i$
infinitely many times, which is not allowed.

####  4.3.1. The Virasoro algebra of ${\mathcal{L}}$

We want to use the residue morphism
$\textrm{Res}\colon\omega_{{\mathcal{L}}/S}\to\mathscr{O}_{S}$ to view
$\theta_{{\mathcal{L}}/S}$ as an $\mathscr{O}_{S}$ submodule of
$\overline{U}\widehat{{\mathfrak{l}}}$ and induce from this a central
extension. Let $D\in\theta_{{\mathcal{L}}/S}$, and since
$\omega_{{\mathcal{L}}/S}$ and $\theta_{{\mathcal{L}}/S}$ are ${\mathcal{L}}$
dual we identify $D$ with the map

$\phi_{D}\colon\omega_{{\mathcal{L}}/S}\times\omega_{{\mathcal{L}}/S}\to\mathscr{O}_{S},\quad(\alpha,\beta)\mapsto\textrm{Res}(D(\alpha)\beta).$

Notice that since $\textrm{Res}(D(\alpha)\beta)=\textrm{Res}(D(\beta)\alpha)$,
we have that $\phi_{D}$ is an element of
$(\textrm{Sym}^{2}(\omega_{{\mathcal{L}}/S}))^{\vee}$. Moreover, since the
latter is canonically isomorphic to the closure of
$\textrm{Sym}^{2}({\mathfrak{l}})$ in $\overline{U}{\mathfrak{l}}$ (see Remark
4.3.3), we will consider $\phi_{D}$ as an element of
$\overline{U}{\mathfrak{l}}$. We define
$C\colon\theta_{{\mathcal{L}}/S}\to\overline{U}{\mathfrak{l}}$ by setting
$2C(D)=\phi_{D}$.

######  Remark 4.3.3.

The fact that $(\textrm{Sym}^{2}(\omega_{{\mathcal{L}}/S}))^{\vee}$ is
canonically isomorphic to the closure of $\textrm{Sym}^{2}({\mathfrak{l}})$ in
$\overline{U}{\mathfrak{l}}$ is essentially a consequence of the fact that
${\mathcal{L}}$ is defined as a limit of finitely generated
$\mathscr{O}_{S}$-modules
$\mathcal{I}_{\sigma}^{-n}/\mathcal{I}_{\sigma}^{m+1}$. We assume, for
simplicity, that ${\mathcal{L}}=\mathscr{O}_{S}((t))$. Write
$\omega_{{\mathcal{L}}/S}$ as $\varinjlim_{n}\varprojlim_{m}\omega^{n}_{m}$
where $\omega^{n}_{m}$ is the free $\mathscr{O}_{S}$-module generated by
$\\{t^{-n}dt,\cdots,t^{m}dt\\}$. The maps defining the projective system are
truncation maps, while the ones for the inductive limit are inclusions. As the
direct limit is given by unions, we deduce that
$(\textrm{Sym}^{2}(\omega_{{\mathcal{L}}/S}))^{\vee}$ is isomorphic to
$\varprojlim_{n}\varinjlim_{m}\textrm{Hom}_{\mathscr{O}_{S}}(\omega^{n}_{m}\circ\omega^{n}_{m},\mathscr{O}_{S})$.
The residue pairing gives the isomorphism between
$\textrm{Hom}_{\mathscr{O}_{S}}(\omega^{n}_{m},\mathscr{O}_{S})$ and the sub
$\mathscr{O}_{S}$-module of ${\mathfrak{l}}$ generated freely by
$\\{t^{-m-1},\cdots,t^{n-1}\\}$ which we denote by
${\mathfrak{l}}^{m+1}_{n-1}$. As these are free modules of finite dimension we
get canonical identification with

$\varprojlim_{n}\varinjlim_{m}\left({\mathfrak{l}}^{m+1}_{n-1}\circ{\mathfrak{l}}^{m+1}_{n-1}\right)=\varprojlim_{n}\left(t^{n-1}\mathscr{O}_{S}[t^{-1}]\circ
t^{n-1}\mathscr{O}_{S}[t^{-1}]\right).$

As the product is symmetric, this is identified with
$\varprojlim_{n}\left(\mathscr{O}_{S}((t))\circ
t^{n-1}\mathscr{O}_{S}[t^{-1}]\right)$. By decomposing $\mathscr{O}_{S}((t))$
as $t^{n-1}\mathscr{O}_{S}[t^{-1}]\oplus t^{n}\mathscr{O}_{S}[[t]]$ this
module equals
$\varprojlim_{n}\dfrac{\mathscr{O}_{S}((t))\circ\mathscr{O}_{S}((t))}{\mathscr{O}_{S}((t))\circ
t^{n}\mathscr{O}_{S}[[t]]}$, which is the closure of
$\textrm{Sym}^{2}({\mathfrak{l}})$ in $\overline{U{\mathfrak{l}}}$.

######  Remark 4.3.4.

Assume for simplicity that $R=k$ and identify ${\mathcal{L}}$ with $k((t))$.
For every $i\in\mathbb{Z}$ we set $\alpha_{i}=t^{-i-1}dt$ and $a_{i}=t^{i}$ so
that $\textrm{Res}(a_{i},\alpha_{j})=\delta_{ij}$ and $\\{\alpha_{i}\\}$ and
$\\{a_{i}\\}$ are linearly independent generators of
$\omega_{{\mathcal{L}}/S}$ and ${\mathcal{L}}$. Then we can write explicitly

$C(D)=\dfrac{1}{2}\sum_{i\in\mathbb{Z}}D(t^{-i-1}dt)\circ t^{i}.$

In general, let $\\{\alpha_{i}\\}$ and $\\{a_{i}\\}$ be linearly independent
generators of $\omega_{{\mathcal{L}}/S}$ and ${\mathcal{L}}$ with the property
that $\textrm{Res}(a_{i},\alpha_{j})=\delta_{ij}$. Then we can write

$C(D)=\dfrac{1}{2}\sum_{i\in\mathbb{Z}}D(\alpha_{i})\circ a_{i}$

which is a well defined object of $\overline{U}{\mathfrak{l}}$ thanks to the
previous remark.

As explained in [Loo13, Section 2], the central extension
$\widehat{{\mathfrak{l}}}$ and the inclusion
$C\colon\theta_{{\mathcal{L}}/S}\to\overline{U}\widehat{{\mathfrak{l}}}$
induce a central extension $\widehat{\theta}_{{\mathcal{L}}/S}$ of
$\theta_{{\mathcal{L}}/S}$. We recall here how this is achieved. Consider now
${\mathfrak{l}}\otimes{\mathfrak{l}}$, and call ${\mathfrak{l}}_{2}$ its image
in $U\widehat{{\mathfrak{l}}}$. This means that
${\mathfrak{l}}_{2}={\mathfrak{l}}\otimes{\mathfrak{l}}\oplus\hslash\mathscr{O}_{S}$
modulo the relation $f\otimes g=g\otimes f+\hslash\textrm{Res}(gdf)$. Denote
by $\overline{{\mathfrak{l}}_{2}}$ its closure in
$\overline{U}\widehat{{\mathfrak{l}}}$ and observe the following diagram

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hslash\mathscr{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\mathfrak{l}}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[-]_{\hslash}}$$\textstyle{\overline{\textrm{Sym}^{2}({\mathfrak{l}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\theta_{{\mathcal{L}}/S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C}$

where $[-]_{\hslash}$ is the reduction modulo the central element
$\hslash\mathscr{O}_{S}$ so that the short sequence is exact.

######  Definition 4.3.5.

We define $\widehat{\theta}_{{\mathcal{L}}/S}$ to be the pullback of
$\theta_{{\mathcal{L}}/S}$ along $[-]_{\hslash}$. Equivalently its elements
are pairs
$(D,u)\in\theta_{{\mathcal{L}}/S}\times\overline{{\mathfrak{l}}_{2}}$ such
that $C(D)=u\mod\hslash\mathscr{O}_{S}$.

Denote by
$\widehat{C}\colon\widehat{\theta}_{{\mathcal{L}}/S}\to\overline{U}\widehat{{\mathfrak{l}}}$
the injection $\widehat{C}(D,u)=u$ and we write $[-]^{\theta}_{\hslash}$ for
the pullback of $[-]_{\hslash}$ along $C$ so that we have the commutative
diagram with exact rows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hslash\mathscr{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\mathfrak{l}}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[-]_{\hslash}}$$\textstyle{\overline{\textrm{Sym}^{2}({\mathfrak{l}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hslash\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathscr{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\theta}_{{\mathcal{L}}/S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{C}}$$\scriptstyle{[-]^{\theta}_{\hslash}}$$\textstyle{\theta_{{\mathcal{L}}/S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C}$$\textstyle{0.}$

Observe, for example using Remark 4.3.4, that the map $C$ is not a Lie algebra
morphism, and so $\widehat{\theta}_{{\mathcal{L}}/S}$ does not arise naturally
as a Lie algebra which centrally extends $\theta_{{\mathcal{L}}/S}$.

We however want to induce a Lie bracket on
$\widehat{\theta}_{{\mathcal{L}}/S}$ from the one of
$\overline{U}\widehat{{\mathfrak{l}}}$ by conveniently modifying
$\widehat{C}$. To understand how to do this, local computations are carried
out.

######  Definition 4.3.6.

Choose a local parameter $t$ so that locally
${\mathcal{L}}\cong\mathscr{O}_{S}((t))$. Define the _normal ordering_
$:\star:\colon\overline{\textrm{Sym}^{2}({\mathfrak{l}})}\to\overline{{\mathfrak{l}}^{2}}$
by setting

$:t^{n}\otimes t^{m}:=\begin{cases}t^{n}\otimes t^{m}&n\leq m\\\ t^{m}\otimes
t^{n}&n\geq m\end{cases}$

and extend it by linearity to every element of
$\overline{\textrm{Sym}^{2}({\mathfrak{l}})}$.

The map $:\star:$ defines a section of $[-]_{\hslash}$, so that
$(\text{Id},:\star:C)$ is a section of $[-]_{\hslash}^{\theta}$. Once we make
the choice of a local parameter defining the ordering $:\star:$, we will
denote by $\widehat{D}$ the element
$(D,:C(D):)\in\widehat{\theta}_{{\mathcal{L}}/S}$. Consider the following
relations which hold in $\overline{U}\widehat{{\mathfrak{l}}}$ and which are
proved in [Loo13, Lemma 2.1].

######  Lemma 4.3.7.

Let $D\in\theta_{{\mathcal{L}}/S}$ and
$D_{i}=t^{i+1}d/dt\in\theta_{{\mathcal{L}}/S}$, Then we have

  1. (1)

$[\widehat{C}(\widehat{D}),f]=-\hslash D(f)$ for every
$f\in{\mathfrak{l}}\subset\widehat{{\mathfrak{l}}}$;

  2. (2)

$[\widehat{C}(\widehat{D}_{k}),\widehat{C}(\widehat{D}_{l})]=-\hslash(l-k)\widehat{C}(\widehat{D}_{k+l})+\dfrac{k^{3}-k}{12}\hslash^{2}\delta_{k,-l}$.
∎

This suggests to rescale the morphism $\widehat{C}$ and to define

$T:=-\dfrac{\widehat{C}}{\hslash}\colon\widehat{\theta}_{{\mathcal{L}}/S}\to\overline{U}\widehat{{\mathfrak{l}}}\left[\dfrac{1}{\hslash}\right]$

which is injective and its image is a Lie subalgebra of the target. Denote by
$c_{0}$ the element $(0,-\hslash)$ which is sent to $1$ by $T$. By
construction we obtain the following result.

######  Proposition 4.3.8.

[Loo13, Corollary-Definition 2.2] The Lie algebra structure induced on
$\widehat{\theta}_{{\mathcal{L}}/S}$ by $T$ is a central extension of the
canonical Lie algebra structure on $\theta_{{\mathcal{L}}/S}$ by
$c_{0}\mathscr{O}_{S}$. This is called the _Virasoro algebra_ of
${\mathcal{L}}$.

###  4.4. Sugawara construction

In this section we generalize to our case, i.e. using
${\widehat{\mathfrak{h}}_{\mathcal{L}}}$ in place of
${\widehat{\mathfrak{g}}{\mathcal{L}}}$, the construction of

$T_{\mathfrak{g}}\colon\widehat{\theta}_{{\mathcal{L}}/S}\to\left(\overline{U}{\widehat{\mathfrak{g}}{\mathcal{L}}}[(c+\check{h})^{-1}]\right)^{\text{Aut}({\mathfrak{g}})}$

described by Looijenga in [Loo13, Corollary 3.2], which essentially represents
the local picture of our situation. In the classical case the idea is to use
the Casimir element of ${\mathfrak{g}}$ to induce, from $\widehat{C}$ the map
$\widehat{C}_{\mathfrak{g}}\colon\widehat{\theta}_{{\mathcal{L}}/S}\to\overline{U}{\widehat{\mathfrak{g}}{\mathcal{L}}}$
which, in turn, will give the map of Lie algebras $T_{\mathfrak{g}}$. When in
place of ${\mathfrak{g}{\mathcal{L}}}$ we have ${\mathfrak{h}_{\mathcal{L}}}$,
we can run the same argument using the element Casimir $\mathfrak{c}$ of
${\mathfrak{h}_{\mathcal{L}}}$. This is the content of this section.

As in Section 3.1, we consider the normalized Killing form defined on
${\mathfrak{g}}$. Recall that it provides an isomorphism between
${\mathfrak{g}}$ and ${\mathfrak{g}}^{\vee}$, hence it gives an identification
of ${\mathfrak{g}}\otimes{\mathfrak{g}}$ with
$\text{End}_{k}({\mathfrak{g}})={\mathfrak{g}}\otimes{\mathfrak{g}}^{\vee}$.
Moreover, as $\sigma(S)$ is disjoint from the ramification locus, we also have
that $(\,|\,)_{\mathfrak{h}_{\mathcal{L}}}$ provides an isomorphism of
${\mathfrak{h}_{\mathcal{L}}}$ with ${\mathfrak{h}_{\mathcal{L}}}^{\vee}$,
giving in this way an identification of
${\mathfrak{h}_{\mathcal{L}}}\otimes_{\mathcal{L}}{\mathfrak{h}_{\mathcal{L}}}$
with $\text{End}_{\mathcal{L}}({\mathfrak{h}_{\mathcal{L}}})$. The _Casimir
element_ of ${\mathfrak{h}_{\mathcal{L}}}$ with respect to the form
$(\,|\,)_{\mathfrak{h}_{\mathcal{L}}}$ is the element in
${\mathfrak{h}_{\mathcal{L}}}\otimes_{\mathcal{L}}{\mathfrak{h}_{\mathcal{L}}}$
corresponding to the identity $\text{Id}_{\mathfrak{h}_{\mathcal{L}}}$ via the
identification provided by $(\,|\,)$. We denote it by $\mathfrak{c}$.

######  Remark 4.4.1.

We could have defined the Casimir element of ${\mathfrak{h}_{\mathcal{L}}}$
via the local isomorphism of ${\mathfrak{h}_{\mathcal{L}}}$ with
${\mathfrak{g}{\mathcal{L}}}$. Let $\mathfrak{c}({\mathfrak{g}})$ be the
Casimir element of ${\mathfrak{g}}$, and observe that via the inclusion
${\mathfrak{g}}\to{\mathfrak{g}{\mathcal{L}}}$, we can see it as an element of
${\mathfrak{g}{\mathcal{L}}}\otimes_{\mathcal{L}}{\mathfrak{g}{\mathcal{L}}}$.
Since $\mathfrak{c}({\mathfrak{g}})$ is invariant under automorphisms, it is
invariant under $\Gamma$, hence it gives an element
${\mathfrak{h}_{\mathcal{L}}}\otimes_{\mathcal{L}}{\mathfrak{h}_{\mathcal{L}}}$
which equals $\mathfrak{c}$.

Since $(\,|\,)$ is a symmetric form, we have that also $\mathfrak{c}$ is a
symmetric element of
${\mathfrak{h}_{\mathcal{L}}}\otimes_{\mathcal{L}}{\mathfrak{h}_{\mathcal{L}}}$
and moreover $\mathfrak{c}$ lies in the centre of
$U_{\mathcal{L}}({\mathfrak{h}_{\mathcal{L}}})$. As
${\mathfrak{h}_{\mathcal{L}}}$ is simple over ${\mathcal{L}}$, this implies
that there exists $\check{h}\in k$ such that
$\text{ad}(\mathfrak{c})X=2\check{{\mathfrak{h}}}X$ for all
$X\in{\mathfrak{h}_{\mathcal{L}}}$, where $\text{ad}(-)$ denotes the adjoint
representation of ${\mathfrak{h}_{\mathcal{L}}}$.

######  Remark 4.4.2.

Locally, for every bases $\\{X_{i}\\}_{i=1}^{\dim({\mathfrak{g}})}$ and
$\\{Y_{i}\\}_{i=1}^{\dim({\mathfrak{g}})}$ of ${\mathfrak{h}_{\mathcal{L}}}$
such that $(X_{i}|Y_{j})_{\mathfrak{h}_{\mathcal{L}}}=\delta_{ij}$ we have the
explicit description of $\mathfrak{c}$ as
$\sum_{i=1}^{\dim({\mathfrak{g}})}X_{i}\circ Y_{i}$. It follows that
$\check{h}$ is given by the equality
$\sum_{i=1}^{\dim({\mathfrak{g}})}[X_{i},[Y_{i},Z]]=2\check{h}Z$ for every
$Z\in{\mathfrak{h}_{\mathcal{L}}}$.

Let denote by $\overline{U}{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ the
completion on the right of $U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ with
respect to the filtration $F^{*}{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ given
by $F^{n}{\mathfrak{h}_{\mathcal{L}}}$ for $n\geq 1$. We now construct
$\widehat{\gamma}_{\mathfrak{c}}\colon\overline{{\mathfrak{l}}}\to\overline{U}{\widehat{\mathfrak{h}}_{\mathcal{L}}}$
which composed with $\widehat{C}$ will give
$\widehat{C}_{{\mathfrak{h}}}\colon\widehat{\theta}_{{\mathcal{L}}/S}\to\overline{U}{\widehat{\mathfrak{h}}_{\mathcal{L}}}$.

Let consider the map
$\gamma_{\mathfrak{c}}\colon{\mathfrak{l}}\otimes_{\mathscr{O}_{S}}{\mathfrak{l}}\to{\mathfrak{h}_{\mathcal{L}}}\otimes_{\mathscr{O}_{S}}{\mathfrak{h}_{\mathcal{L}}}\subset
U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ given by tensoring with
$\mathfrak{c}$. This map uniquely extends to a map of Lie algebras
${\mathfrak{l}}_{2}\to U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ as follows.
Using local bases as in Remark 4.4.2 and the symmetry of ${\mathfrak{c}}$ we
deduce the following equality

$\gamma_{\mathfrak{c}}\left(\hslash\textrm{Res}(gdf)\right)=\gamma_{\mathfrak{c}}(f\circ
g-g\circ
f)=c\dim({\mathfrak{g}})\textrm{Res}(gdf)+c\sum_{i=1}^{\dim({\mathfrak{g}})}\textrm{Res}(fg(dY_{i}|X_{i}))$

and recalling that Remark 3.1.2 implies that $(dX_{i}|Y_{i})=-(dY_{i}|X_{i})$,
we conclude that

$\gamma_{\mathfrak{c}}\left(\hslash\textrm{Res}(gdf)\right)=c\dim({\mathfrak{g}})\textrm{Res}(gdf).$

We then define $\widehat{\gamma}_{\mathfrak{c}}\colon{\mathfrak{l}}_{2}\to
U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ by sending $\hslash$ to
$c\dim({\mathfrak{g}})$ and acting as $\gamma_{\mathfrak{c}}$ on
${\mathfrak{l}}\otimes{\mathfrak{l}}$. Such a map can be extended to the
closure of ${\mathfrak{l}}_{2}$ in $\overline{U}\widehat{{\mathfrak{l}}}$ once
we extend the target to $\overline{U}{\widehat{\mathfrak{h}}_{\mathcal{L}}}$,
obtaining
$\widehat{\gamma}_{\mathfrak{c}}\colon\overline{{\mathfrak{l}}_{2}}\to
U{\widehat{\mathfrak{h}}_{\mathcal{L}}}$. We define
$\widehat{C}_{\mathfrak{h}}$ as the composition

$\widehat{\gamma}_{\mathfrak{c}}\widehat{C}\,\colon\widehat{\theta}_{{\mathcal{L}}/S}\to\overline{U}{\widehat{\mathfrak{h}}_{\mathcal{L}}}.$

As for $\widehat{C}$, also in this case the morphism
$\widehat{C}_{\mathfrak{h}}$ does not preserves the Lie bracket, and thanks to
local computation we understand how to solve this issue. Following [TUY89] we
first of all extend the normal ordering defined in 4.3.6 as as follows.

######  Definition 4.4.3.

Let fix an isomorphism between ${\mathcal{L}}$ and $R((t))$ for a local
parameter $t\in\mathcal{I}_{\sigma}$. Let $Xt^{n}$ and $Yt^{m}$ be elements of
${\mathfrak{g}{\mathcal{L}}}={\mathfrak{g}}\otimes R((t))$. Then we set

${{}_{\circ}^{\circ}}\,Xt^{n}\otimes
Yt^{m}\,{{}_{\circ}^{\circ}}=\begin{cases}Xt^{n}\otimes Yt^{m}&n<m\\\
\dfrac{1}{2}(Xt^{n}\otimes Yt^{m}+Yt^{m}\otimes Xt^{n})&n=m\\\ Yt^{m}\otimes
Xt^{n}&n>m.\end{cases}$

This definition is $\Gamma$-equivariant, hence defines a normal ordering on
${\mathfrak{h}_{\mathcal{L}}}\otimes{\mathfrak{h}_{\mathcal{L}}}$.

This defines a section from the image of $\gamma_{c}$ to the image of
$\widehat{\gamma}_{c}$ which makes the diagram to commute:

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c\mathscr{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{Im}(\widehat{\gamma}_{c})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[-]_{c}}$$\textstyle{\text{Im}(\gamma_{c})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{{}_{\circ}^{\circ}}\star{{}_{\circ}^{\circ}}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hslash\mathscr{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\mathfrak{l}}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[-]_{\hslash}}$$\scriptstyle{\widehat{\gamma}_{c}}$$\textstyle{\overline{\textrm{Sym}^{2}({\mathfrak{l}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{c}}$$\scriptstyle{:\star:}$$\textstyle{0}$$\textstyle{\widehat{\theta}_{{\mathcal{L}}/S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{C}}$$\textstyle{\theta_{{\mathcal{L}}/S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C}$

For any $D\in\theta_{{\mathcal{L}}/S}$, we write
$\widehat{C}_{\mathfrak{h}}(\widehat{D})$ to denote the element
${{}_{\circ}^{\circ}}\,\gamma_{\mathfrak{c}}C(D)\,{{}_{\circ}^{\circ}}\,=\widehat{\gamma}_{\mathfrak{c}}:C(D):$.

######  Remark 4.4.4.

As we have done in Remark 4.3.4 we can write locally the element
$\widehat{C}_{\mathfrak{h}}(\widehat{D})$ in a more explicit way. Consider the
morphism
$1\otimes(\,|\,)\colon\omega_{{\mathcal{L}}/S}\otimes{\mathfrak{h}_{\mathcal{L}}}\to\omega_{{\mathcal{L}}/S}\otimes{\mathfrak{h}_{\mathcal{L}}}^{\vee}$
and, after tensoring it with ${\mathfrak{h}_{\mathcal{L}}}$, compose it with
$\textrm{Res}_{\mathfrak{h}}$ to obtain the pairing
$\textrm{Res}_{(\,|\,)}\colon\omega_{{\mathcal{L}}/S}\otimes_{\mathcal{L}}{\mathfrak{h}_{\mathcal{L}}}\times{\mathfrak{h}_{\mathcal{L}}}\to\mathscr{O}_{S}$.
Let $\\{A_{i}\\}$ and $\\{B_{i}\\}$ be orthonormal bases of
$\omega_{{\mathcal{L}}/S}\otimes{\mathfrak{h}_{\mathcal{L}}}$ and
${\mathfrak{h}_{\mathcal{L}}}$ with respect to $\textrm{Res}_{(\,|\,)}$. Then
for every $D\in\theta_{{\mathcal{L}}/S}$ we have

$\widehat{C}_{\mathfrak{h}}(\widehat{D})=\frac{1}{2}\sum{{}_{\circ}^{\circ}}D(A_{i})\circ
B_{i}{{}_{\circ}^{\circ}}$

where we see $D$ as a linear map $\omega_{{\mathcal{L}}/S}\to{\mathcal{L}}$,
so that $D(A_{i})\in{\mathfrak{h}_{\mathcal{L}}}$.

As in [Loo13, Lemma 3.1] we have the following result.

######  Lemma 4.4.5.

The following equalities hold true in
$\overline{U}{\widehat{\mathfrak{h}}_{\mathcal{L}}}$:

  1. (1)

$[\widehat{C}_{\mathfrak{h}}(\widehat{D}),X]=-(c+\check{h})D(X)$ for all
$X\in{\mathfrak{h}_{\mathcal{L}}}$ and $D\in\theta_{{\mathcal{L}}/S}$;

  2. (2)

$[\widehat{C}_{\mathfrak{h}}(\widehat{D}_{k}),\widehat{C}_{\mathfrak{h}}(\widehat{D}_{l})]=(c+\check{h})(k-l)\widehat{C}_{\mathfrak{h}}(\widehat{D}_{k+l})+c\dim({\mathfrak{g}})(c+\check{h})\dfrac{k^{3}-k}{12}\delta_{k,-l}$
where $D_{i}=t^{i+1}d/dt$.

As Lemma 4.3.7 also Lemma 4.4.5 suggests to rescale
$\widehat{C}_{\mathfrak{h}}$ and consider instead the map

$T_{\mathfrak{h}}:=-\dfrac{\widehat{C}_{\mathfrak{h}}}{c+\check{h}}\colon\widehat{\theta}_{{\mathcal{L}}/S}\to\overline{U}{\widehat{\mathfrak{h}}_{\mathcal{L}}}\left[\dfrac{1}{c+\check{h}}\right]$

which is compatible with the Lie brackets of
$\widehat{\theta}_{{\mathcal{L}}/S}$ and
$\overline{U}{\widehat{\mathfrak{h}}_{\mathcal{L}}}[(c+\check{h})^{-1}]$,
proving the following statement.

######  Proposition 4.4.6.

The map $T_{\mathfrak{h}}$ is a homomorphism of Lie algebras which sends the
central element $c_{0}=(0,-\hslash)$ to
$(c\dim({\mathfrak{g}}))/(c+\check{h})$. We call $T_{\mathfrak{h}}$ the
_Sugawara representation_ of $\widehat{\theta}_{{\mathcal{L}}/S}$.

####  4.4.1. Fock representation

We induce the representation $T_{\mathfrak{h}}$ to the quotient
${\mathcal{F}}^{+}({\mathfrak{h}_{\mathcal{L}}})$ of
$\overline{U}{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ defined as

${\mathcal{F}}^{+}({\mathfrak{h}_{\mathcal{L}}}):=\left(U{\widehat{\mathfrak{h}}_{\mathcal{L}}}{\,\big{/}}U{\widehat{\mathfrak{h}}_{\mathcal{L}}}\circ
F^{1}{\widehat{\mathfrak{h}}_{\mathcal{L}}}\right)\left[\dfrac{1}{c+\check{h}}\right]=\left(\overline{U}{\widehat{\mathfrak{h}}_{\mathcal{L}}}{\,\big{/}}\overline{U}{\widehat{\mathfrak{h}}_{\mathcal{L}}}\circ
F^{1}{\widehat{\mathfrak{h}}_{\mathcal{L}}}\right)\left[\dfrac{1}{c+\check{h}}\right]$

By abuse of notation call $T_{\mathfrak{h}}$ the composition of
$T_{\mathfrak{h}}$ with the projection of
$\overline{U}{\widehat{\mathfrak{h}}_{\mathcal{L}}}$ to
${\mathcal{F}}^{+}({\mathfrak{h}_{\mathcal{L}}})$, so that
${\mathcal{F}}^{+}({\mathfrak{h}_{\mathcal{L}}})$ is a representation of
$\widehat{\theta}_{{\mathcal{L}}/S}$. We can depict the result as follows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{O}_{S}\cdot\text{Id}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textrm{End}({\mathcal{F}}^{+}({\mathfrak{h}_{\mathcal{L}}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}\textrm{End}({\mathcal{F}}^{+}({\mathfrak{h}_{\mathcal{L}}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{O}_{S}c_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\theta}_{{\mathcal{L}}/S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{T}_{\mathfrak{h}}}$$\textstyle{\theta_{{\mathcal{L}}/S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the first vertical arrow maps $c_{0}$ to
$c\dim({\mathfrak{g}})(\check{h}+c)^{-1}\cdot\text{Id}$ and by abuse of
notation we wrote $T_{\mathfrak{h}}$ instead of $T_{\mathfrak{h}}(-)\circ$.

######  Remark 4.4.7.

We give a local description of how the action looks like. Choose for this
purpose a local parameter $t$ of $\mathcal{I}_{\sigma}$ so that we can
associate to $D\in\theta_{{\mathcal{L}}/S}$ the element
$\widehat{D}\in\widehat{\theta}_{{\mathcal{L}}/S}$. Let $X_{r}\circ\cdots\circ
X_{1}$ be representatives of an element of
${\mathcal{F}}^{+}({\mathfrak{h}_{\mathcal{L}}})$ whith
$X_{i}\in{\mathfrak{h}_{\mathcal{L}}}$. Then the action of
$\theta_{{\mathcal{L}}/S}$ is described as follows

$T_{\mathfrak{h}}(\widehat{D})\circ X_{r}\circ\cdots\circ
X_{1}=\sum_{i=1}^{r}X_{r}\circ\cdots\circ D(X_{i})\circ\cdots\circ
X_{1}+X_{r}\circ\cdots\circ X_{1}\circ T_{\mathfrak{h}}(\widehat{D})$

where $D(X_{i})$ denotes the image of $X_{i}$ under coefficientwise derivation
by $D$ (Remark 4.2.3).

####  4.4.2. Projective representation of $\theta_{{\mathcal{L}},S}$

We want to define in this section a map of Lie algebras
$\mathbb{P}T_{{\mathfrak{h}},S}\colon\theta_{L,S}\to\mathbb{P}\textrm{End}({\mathcal{H}_{\ell}({\mathcal{V}})})$
which is induced by $T_{\mathfrak{h}}$ and which will lead, in a second time,
to a projective connection on the sheaves of conformal blocks. The
construction of $\mathbb{P}T_{{\mathfrak{h}},S}$ in the classical case is the
content of [Loo13, Corollary 3.3].

Let $F^{0}\theta_{{\mathcal{L}}/S}$ be the subsheaf of
$\theta_{{\mathcal{L}}/S}$ given by those derivations $D$ such that
$D(F^{1}\\!{\mathcal{L}})\in F^{1}\\!{\mathcal{L}}$, and similarly we set
$F^{0}\theta_{{\mathcal{L}},S}$ to be the subsheaf of
$\theta_{{\mathcal{L}},S}$ whose elements $D$ satisfy
$D(F^{1}\\!{\mathcal{L}})\in F^{1}\\!{\mathcal{L}}$.

######  Remark 4.4.8.

Assume that ${\mathcal{L}}=\mathscr{O}_{S}((t))$ so that every element of
$F^{0}\theta_{{\mathcal{L}}/S}$ is written as $D=\sum_{i\geq
0}a_{i}t^{i}d/dt$. The element $T_{\mathfrak{h}}(\widehat{D})$ acts on
$\mathcal{V}$ as the operator
$\dfrac{a_{0}}{-2(\ell+\check{h})}c_{\mathfrak{h}}$ where $c_{\mathfrak{h}}$
is the Casimir element of ${\mathfrak{h}}$, hence the action is by scalar
multiplication. Combining this with Remark 4.4.7, we obtain that
$F^{0}(\theta_{{\mathcal{L}}/S})$ acts on
${\mathcal{H}_{\ell}({\mathcal{V}})}$ by coefficientwise derivation up to
scalars.

As in the classical case, also in our context this observation is the key
input to define $\mathbb{P}T_{{\mathfrak{h}},S}$. In fact we let
$F^{0}\theta_{{\mathcal{L}},S}$ act on ${\mathcal{H}_{\ell}({\mathcal{V}})}$
by coefficientwise derivation so that we obtain a map

$F^{0}\theta_{{\mathcal{L}},S}\times\theta_{L/S}\to\mathbb{P}\textrm{End}({\mathcal{H}_{\ell}({\mathcal{V}})})$

which uniquely defines the Lie algebra homomorphism

$\mathbb{P}T_{{\mathfrak{h}},S}\colon\theta_{{\mathcal{L}},S}\to\mathbb{P}\textrm{End}({\mathcal{H}_{\ell}({\mathcal{V}})})$

and hence the central extension
$\widehat{\theta}_{{\mathcal{L}},S}\to\theta_{{\mathcal{L}},S}$ and the map
$T_{{\mathfrak{h}},S}\colon\widehat{\theta}_{{\mathcal{L}},S}\to\textrm{End}({\mathcal{H}_{\ell}({\mathcal{V}})})$.

###### Proof.

We only have to prove that the Lie algebra generated by
$F^{0}\theta_{{\mathcal{L}},S}$ and $\theta_{L/S}$ is
$\theta_{{\mathcal{L}},S}$. This can be checked locally, where the choice of a
local parameter $t$ allows us to split the exact sequence

$0\to\theta_{{\mathcal{L}}/S}\to\theta_{{\mathcal{L}},S}\to\mathcal{T}_{S/k}\to
0,$

hence to write $\theta_{{\mathcal{L}},S}$ as
$\theta_{{\mathcal{L}}|S}\oplus\mathcal{T}_{S/k}$. We can in fact decompose
every element $D\in\theta_{{\mathcal{L}},S}$ as $D_{ver}\oplus D_{hor}$ which
are uniquely determined by the conditions

$(\star)\quad
D_{ver}\in\theta_{{\mathcal{L}}|S},D_{hor}(t)=0D_{hor}(s)=D(s)\text{ for all
}s\in\mathscr{O}_{S}.\quad$

This implies that
$F^{0}\theta_{{\mathcal{L}},S}=F^{0}\theta_{{\mathcal{L}}/S}\oplus\mathcal{T}_{S/k}$,
concluding the argument. ∎

######  Remark 4.4.9.

Assume that ${\mathcal{L}}=R((t))$ so that we can write every element
$D\in\theta_{{\mathcal{L}},S}$ as $D=D_{ver}+D_{hor}$ satisfying $(\star)$.
Then Remark 4.4.7 tells us that the action of $D$ on
${\mathcal{H}_{\ell}({\mathcal{V}})}$ is given by componentwise derivation by
$D$ plus right multiplication by $T(\widehat{D}_{ver})$.

######  Remark 4.4.10.

We want to remark that in the case in which $\mathcal{V}$ is the trivial
representation, then the central extension
$\widehat{\theta}_{{\mathcal{L}},S}$ is isomorphic to
$\widehat{\theta}_{{\mathcal{L}}/S}\oplus\mathcal{T}_{S/k}$, viewed as a Lie
subalgebra of ${\mathfrak{gl}}({\mathcal{H}_{\ell}(0)})$, where the action of
$\mathcal{T}_{S/k}$ is by coefficientwise derivation. In fact in the previous
proof we saw that locally on $S$, and up to the choice of a local parameter
this is the case. By looking at Remark 4.4.8 and the previous proof, we note
that the obstruction to deduce this statement globally lies in the action of
the Casimir element $c_{\mathfrak{h}}$ on $\mathcal{V}$. When $\mathcal{V}$ is
the trivial representation $c_{\mathfrak{h}}$ acts as multiplication by zero,
hence there is no obstruction. In particular, the central charge
$c_{0}=(0,-\hslash)\in\widehat{\theta}_{{\mathcal{L}},S}$ acts by
multiplication by $\dim({\mathfrak{g}})\ell/(\ell+\check{h})$.

###  4.5. The projective connection on
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$

The aim of this section is to induce, from $\mathbb{P}T_{{\mathfrak{h}},S}$,
the projectively flat connection
$\nabla\colon\mathcal{T}_{S/k}(-\log(\Delta))\to\mathbb{P}\textrm{End}({\mathcal{H}_{\ell}({\mathcal{V}})}_{X})$.
In Proposition 4.2.5 we realised $\mathcal{T}_{S/k}(-\log(\Delta))$ as the
quotient
$\theta_{{\mathcal{A}},S}(-\mathcal{R})/\theta_{{\mathcal{A}}/S}(-\mathcal{R})$,
so that the content of this section can be summarized in the following
statement.

######  Theorem 4.5.1.

The actions of $\theta_{{\mathcal{A}},S}$ and of
$\theta_{{\mathcal{A}}/S}(-\mathcal{R})$ on
${\mathcal{H}_{\ell}({\mathcal{V}})}$ induce a projective action of
$\mathcal{T}_{S}(-\log(\Delta))$ on ${\mathcal{H}_{\ell}(0)}_{X}$. In
particular ${\mathcal{H}_{\ell}({\mathcal{V}})}_{X}$ is locally free if
restricted to $S\setminus\Delta$.

As a consequence of it, we obtain that
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ is locally free on
${\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,1}}$.

######  Corollary 4.5.2.

The sheaf ${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ on
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ is equipped with a
projective connection with logarithmic singularities along the boundary
$\Delta_{univ}$. In particular
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X_{univ}}$ is locally free on
${\mathcal{H}\textrm{ur}(\Gamma,\xi)_{g,1}}$.

###### Proof.

As pointed out in Subsection 4.1 the tangent space of
${\overline{\mathcal{H}\textrm{ur}}(\Gamma,\xi)_{g,1}}$ at a versal covering
$({\widetilde{X}}\overset{q}{\to}X\overset{\pi}{\to}S,\sigma)$ is identified
via the Kodaira-Spencer map with the tangent bundle
$\mathcal{T}_{S/k}(-\log(\Delta))$. The previous theorem gives the projective
action of the latter on ${\mathcal{H}_{\ell}({\mathcal{V}})}_{X}$, concluding
in this way the argument. ∎

#### Proof of Theorem 4.5.1

We first of all prove that the action of
$\theta_{{\mathcal{A}},S}(-\mathcal{R})$ on
${\mathcal{H}_{\ell}({\mathcal{V}})}$ descends to
${\mathcal{H}_{\ell}({\mathcal{V}})}_{X}$.

######  Proposition 4.5.3.

The projective action of $\theta_{{\mathcal{A}},S}(-\mathcal{R})$ on
${\mathcal{H}_{\ell}({\mathcal{V}})}$ preserves the space
${\mathfrak{h}_{\mathcal{A}}}\circ{\mathcal{H}_{\ell}({\mathcal{V}})}$, hence
induces a projective action on ${\mathcal{H}_{\ell}({\mathcal{V}})}_{X}$.

###### Proof.