2. Twisted affine Kac–Moody algebras

This section is devoted to recalling the definition of twisted affine Kac–Moody Lie algebras and their basic properties (we need).

Let $\sigma$ be an operator of finite order $m$ acting on two vector spaces $V$ and $W$ over $\mathbb {C}$. Consider the diagonal action of $\sigma$ on $V\otimes W$. We have the following decomposition of the $\sigma$-invariant subspace in $V\otimes W$,

\[ (V\otimes W)^\sigma=\bigoplus_{\xi} V_{\xi}\otimes W_{\xi^{-1}} , \]

where the summation is over $m$th roots of unity and $V_{\xi }$ (respectively, $W_{\xi ^{-1}}$) is the $\xi$-eigenspace of $V$ (respectively, $\xi ^{-1}$-eigenspace of $W$). We say $v\otimes w$ is pure or more precisely $\xi$-pure if $v\otimes w\in V_{\xi }\otimes W_{\xi ^{-1}}$. Throughout this paper, if we write $v\otimes w\in (V\otimes W)^\sigma$, we mean $v\otimes w$ is pure.

Let $\mathfrak {l}$ be a Lie algebra over $\mathbb {C}$ and let $A$ be a commutative algebra over $\mathbb {C}$. Let $\sigma$ act on $\mathfrak {l}$ (respectively, $A$) as Lie algebra (respectively, algebra) automorphism of finite orders. For any $x\otimes a\in \mathfrak {l}\otimes A$, we denote it by $x[a]$ for brevity. There is a Lie algebra structure on $\mathfrak {l}\otimes A$ with the Lie bracket given by

\[ [x[a],y[b]]:=[x,y][ab], \quad \text{for any elements } x[a],y[b]\in \mathfrak{l}\otimes A. \]

Then, $(\mathfrak {l}\otimes A)^\sigma$ is a Lie subalgebra.

Let $\mathfrak {g}$ be a simple Lie algebra over $\mathbb {C}$ with a Cartan subalgebra $\mathfrak {h}$ and let $\sigma$ be an automorphism of $\mathfrak {g}$ such that $\sigma ^m=1$ ($\sigma$ is not necessarily of order $m$). Let $\langle \cdot,\cdot \rangle$ be the invariant (symmetric, non-degenerate) bilinear form on $\mathfrak {g}$ normalized so that the induced form on the dual space $\mathfrak {h}^*$ satisfies $\langle \theta,\theta \rangle =2$ for the highest root $\theta$ of $\mathfrak {g}$. The bilinear form $\langle \cdot,\cdot \rangle$ is $\sigma$-invariant since $\sigma$ is a Lie algebra automorphism of $\mathfrak {g}$.

Let $\mathcal {K} =\mathbb {C}((t)):= \mathbb {C} [[t]][t^{-1}]$ be the field of Laurent power series, and let $\mathcal {O}$ be the ring of formal power series $\mathbb {C}[[t]]$ with the maximal ideal $\mathfrak {m}=t\mathcal {O}$. We fix a primitive $m$th root of unity $\epsilon = \epsilon _m= e^{ {2\pi i}/{m}}$ throughout the paper. We define an action of $\sigma$ on $\mathcal {K}$ as field automorphism by setting

\[ \sigma (t)=\epsilon^{-1}t\ \text{and $\sigma$ acting trivially on $\mathbb{C}$}. \]

It gives rise to an action of $\sigma$ on the loop algebra $L(\mathfrak {g}):=\mathfrak {g}\otimes _{\mathbb {C}}\mathcal {K}$. Under this action,

\[ L(\mathfrak{g})^\sigma = \bigoplus_{j=0}^{m-1}\big(\mathfrak{g}_j\otimes \mathcal{K}_j\big), \]

where

(1)\begin{equation} \mathfrak{g}_j:=\{x\in \mathfrak{g}: \sigma (x)=\epsilon^j x\}\quad \text{and}\quad \mathcal{K}_j=\{P\in \mathcal{K}: \sigma (P)=\epsilon^{-j} P\}. \end{equation}

We now define a central extension $\hat {L}(\mathfrak {g},\sigma ):=L(\mathfrak {g})^\sigma \oplus \mathbb {C}C$ of $L(\mathfrak {g})^\sigma$ under the bracket

(2)\begin{equation} [x[P]+z C, x'[P'] +z' C] = [x,x' ][PP'] +m^{-1}\operatorname{\mathrm{Res}}_{t=0} \,\big(({dP}) P'\big) \langle x,x'\rangle C, \end{equation}

for $x[P],x'[P']\in L(\mathfrak {g})^\sigma$, $z, z'\in \mathbb {C}$; where $\operatorname {\mathrm {Res}}_{t=0}$ denotes the coefficient of $t^{-1}\,dt$. Let $\hat {L}(\mathfrak {g},\sigma )^{\geq 0}$ denote the subalgebra

\[ \hat{L}(\mathfrak{g},\sigma)^{\geq 0}:= \bigoplus_{j= 0}^{m-1} \mathfrak{g}_j\otimes \mathcal{O}_j \oplus \mathbb{C} C, \]

where $\mathcal {O}_j=\mathcal {K}_j\cap \mathcal {O}$. We also denote

\[ \hat{L}(\mathfrak{g},\sigma)^+:= \bigoplus_{j = 0}^{m-1} \mathfrak{g}_j\otimes \mathfrak{m}_{j} \quad \text{and}\quad \hat{L}(\mathfrak{g},\sigma)^-:= \bigoplus_{j< 0} \mathfrak{g}_j\otimes t^{j} , \]

where $\mathfrak {m}_j=\mathfrak {m}\cap \mathcal {O}_j$. Then, $\hat {L}(\mathfrak {g},\sigma )^+$ is an ideal of $\hat {L}(\mathfrak {g},\sigma )^{\geq 0}$ and the quotient $\hat {L}(\mathfrak {g},\sigma )^{\geq 0}/ \hat {L}(\mathfrak {g},\sigma )^+$ is isomorphic to $\mathfrak {g}_0\oplus \mathbb {C}C$. Note that $\mathfrak {g}_0$ is the Lie algebra $\mathfrak {g}^\sigma$ of $\sigma$-fixed points in $\mathfrak {g}$. As vector spaces we have

\[ \hat{L}(\mathfrak{g}, \sigma) = \hat{L}(\mathfrak{g},\sigma)^{\geq 0} \oplus \hat{L}(\mathfrak{g},\sigma)^- . \]

By the classification theorem of finite order automorphisms of simple Lie algebras (cf. [Reference KacKac90, Proposition 8.1, Theorems 8.5 and 8.6]), there exists a ‘compatible’ Cartan subalgebra $\mathfrak {h}$ and a ‘compatible’ Borel subalgebra $\mathfrak {b} \supset \mathfrak {h}$ of $\mathfrak {g}$ both stable under the action of $\sigma$ such that

(3)\begin{equation} \sigma=\tau \epsilon^{{\rm ad}\, h}, \end{equation}

where $\tau$ is a diagram automorphism of $\mathfrak {g}$ of order $r$ preserving $\mathfrak {h}$ and $\mathfrak {b}$, and $\epsilon ^{{\rm ad}\, h}$ is the inner automorphism of $\mathfrak {g}$ such that for any root $\alpha$ of $\mathfrak {g}$, $\epsilon ^{{\rm ad}\, h}$ acts on the root space $\mathfrak {g}_\alpha$ by the multiplication $\epsilon ^{\alpha (h)}$, and $\epsilon ^{{\rm ad}\, h}$ acts on $\mathfrak {h}$ by the identity. We consider $\tau = \operatorname {\mathrm {Id}}$ also as a diagram automorphism. Here $h$ is an element in $\mathfrak {h}^\tau$. In particular, $\tau$ and $\epsilon ^{{\rm ad}\,h}$ commute. Moreover, $\alpha (h) \in \mathbb {Z}^{\geq 0}$ for any simple root $\alpha$ of $\mathfrak {g}^\tau$, $\beta (h) \in \mathbb {Z}$ for any simple root $\beta$ of $\mathfrak {g}$ and $\theta _0(h)\leq {m}/{r}$ where $\theta _0\in (\mathfrak {h}^\tau )^*$ denotes the following weight of $\mathfrak {g}^\tau$:

\[ \theta_0=\begin{cases} \text{highest root of } \mathfrak{g}, & \text{if } r=1\\ \text{highest short root of } \mathfrak{g}^\tau, & \text{if } r>1 \text{ and }(\mathfrak{g}, r)\neq (A_{2n},2)\\ 2\cdot \text{highest short root} \text{ of } \mathfrak{g}^\tau, & \text{if } (\mathfrak{g},r)=(A_{2n},2). \end{cases} \]

Observe that $r$ divides $m$, and $r$ can only be $1,2, 3$. Note that $\mathfrak {g}^\sigma$ and $\mathfrak {g}^{\tau }$ share the common Cartan subalgebra $\mathfrak {h}^{\sigma }=\mathfrak {h}^{\tau }$.

Let $I(\mathfrak {g}^\tau )$ denote the set of vertices of the Dynkin diagram of $\mathfrak {g}^\tau$. Let $\alpha _i$ denote the simple root associated to $i\in I(\mathfrak {g}^\tau )$. Let $\hat {I}(\mathfrak {g},\sigma )$ denote the set $I(\mathfrak {g}^\tau )\sqcup \{o\}$, where $o$ is just a symbol. (Observe that $\tau$ is determined from $\sigma$.) Set

\[ s_i=\begin{cases} \alpha_i(h) & \text{if } i\in I(\mathfrak{g}^\tau) \\ \dfrac{m}{r}-\theta_0(h) & \text{if } i=o . \end{cases} \]

Then, $s=\{s_i \,|\, i\in \hat {I}(\mathfrak {g},\sigma ) \}$ is a tuple of non-negative integers. Let $\hat {L}(\mathfrak {g}, \tau )$ denote the Lie algebra with the construction similar to $\hat {L}(\mathfrak {g}, \sigma )$ where $\sigma$ is replaced by $\tau$, $m$ is replaced by $r$ and $\epsilon$ is replaced by $\epsilon ^{ {m}/{r}}$. There exists an isomorphism of Lie algebras (cf. [Reference KacKac90, Theorem 8.5]):

(4)\begin{equation} \phi_\sigma: \hat{ L}(\mathfrak{g}, \tau)\simeq \hat{L}(\mathfrak{g}, \sigma) \end{equation}

given by $C\mapsto C$ and $x[t^j]\mapsto x[t^{( {m}/{r})j+k } ]$, for any $x$ an $\epsilon ^{ ({m}/{r})j}$-eigenvector of $\tau$, and $x$ also a $k$-eigenvector of ${\rm ad}\,h$. We remark that in the case $(\mathfrak {g},r)=(A_{2n},2)$, our labelling for $i=o$ is the same as $i=n$ in [Reference KacKac90, Chapter 8]. It is well-known that $\hat {L}(\mathfrak {g},\tau )$ is an affine Lie algebra, more precisely $\hat {L}(\mathfrak {g}, \tau )$ is untwisted if $r=1$ and twisted if $r>1$.

By Theorem 8.7 in [Reference KacKac90], there exists a $sl_2$-triple $x_i,y_i,h_i \in \mathfrak {g}$ for each $i\in \hat {I}(\mathfrak {g},\sigma )$ where:

• • $x_i\in (\mathfrak {g}^\tau )_{\alpha _i}$, $y_i\in (\mathfrak {g}^\tau )_{-\alpha _i}$ when $i\in I(\mathfrak {g}^\tau )$;

• • $x_{o}$ (respectively, $y_{o}$) is a $(-\theta _0)$(respectively, $\theta _0$)-weight vector with respect to the adjoint action of $\mathfrak {h}^\tau$ on $\mathfrak {g}$, and is also an $\epsilon ^{ {m}/{r}}$ (respectively, $\epsilon ^{- {m}/{r}}$)-eigenvector of $\tau$;

• • $x_i\in \mathfrak {n}$ for $i\in I(\mathfrak {g}^\tau )$ and $x_o \in \mathfrak {n}^-$, where $\mathfrak {n}$ (respectively, $\mathfrak {n}^-$) is the nilradical of $\mathfrak {b}$ (respectively, the opposite Borel subalgebra $\mathfrak {b}^-$); similarly, $y_i\in \mathfrak {n^-}$ for $i\in I(\mathfrak {g}^\tau )$ and $y_o \in \mathfrak {n}$

(see explicit construction of $x_i, y_i, i\in \hat {I}(\mathfrak {g},\sigma )$ in [Reference KacKac90, §§ 7.4, 8.3]), such that

\[ x_i[t^{s_i}],\quad y_i[t^{-s_i}],\quad [x_i[t^{s_i} ],\quad y_i[t^{-s_i}]], \quad i\in \hat{I}(\mathfrak{g},\sigma), \]

are Chevalley generators of $\hat {L}(\mathfrak {g},\sigma )$ and $\{x_i, y_i, [x_i, y_i]\}_{i\in \hat {I}(\mathfrak {g},\sigma ): s_i=0}$ are Chevalley generators of $[\mathfrak {g}^\sigma, \mathfrak {g}^\sigma ]$. We set

\[ \tilde{x}_i:=x_i[t^{s_i}], \quad\tilde{y}_i:=y_i[t^{-s_i}]\quad \text{and}\quad \tilde{h}_i:=[\tilde{x}_i, \tilde{y}_i],\quad \text{for any } i\in \hat{I}(\mathfrak{g},\sigma) . \]

Via the isomorphism $\phi _\sigma$, we have

\[ \phi_\sigma(x_i)= \tilde{x_i},\quad \phi_\sigma(y_i)=\tilde{y}_i,\quad \text{for any } i\in I(\mathfrak{g}^\tau), \]

and

\[ \phi_\sigma(x_o[t])=\tilde{x}_o, \quad \phi_\sigma(y_o[t^{-1}])=\tilde{y}_o . \]

Thus, $\deg \tilde {x}_i=s_i$ and $\deg \tilde {y}_i=-s_i$. The Lie algebra $\hat {L}(\mathfrak {g},\sigma )$ is called an $(s,r)$-realization of the associated affine Lie algebra $\hat {L}(\mathfrak {g},\tau )$.

From the above discussion, for any $i\in \hat {I}(\mathfrak {g}, \sigma )$, we have

(5)\begin{equation} \sigma(x_i)=\epsilon^{s_i} x_i \quad\text{and}\quad \sigma(y_i)=\epsilon^{-s_i}y_i . \end{equation}

We fix a positive integer $c$ called the level or central charge. Let $\operatorname {\mathrm {Rep}}_c$ be the set of isomorphism classes of integrable highest weight (in particular, irreducible) $\hat {L}(\mathfrak {g},\sigma )$-modules with central charge $c$, where in our realization $C$ acts by $c$, the standard Borel subalgebra of $\hat {L}(\mathfrak {g},\sigma )$ is generated by $\{\tilde {x}_i, \tilde {h}_i\}_{i\in \hat {I}(\mathfrak {g}, \sigma )}$ and $\hat {L}(\mathfrak {g},\sigma )^-$ is generated by

\[ \{\tilde{y}_i\}_{\{i\in \hat{I}(\mathfrak{g}, \sigma): s_i> 0\}}\quad (\mbox{cf. Kac90, Theorem 8.7}). \]

Thus, $\hat {L}(\mathfrak {g},\sigma )^{\geq 0}$ is a standard parabolic subalgebra of $\hat {L}(\mathfrak {g},\sigma )$. For any $\mathscr {H}\in \operatorname {\mathrm {Rep}}_c$, let $\mathscr {H}^0$ be the subspace of $\mathscr {H}$ annihilated by $\hat {L}(\mathfrak {g},\sigma )^+$. Then, $\mathscr {H}^0$ is an irreducible finite-dimensional $\mathfrak {g}^\sigma$-submodule of $\mathscr {H}$ with highest weight (say) $\lambda (\mathscr {H})\in (\mathfrak {h}^\sigma )^*=(\mathfrak {h}^\tau )^*$ for the choice of the Borel subalgebra of $\mathfrak {g}^\sigma$ generated by $\mathfrak {h}^\sigma$ and $\{x_i: s_i=0\}$. The correspondence $\mathscr {H}\mapsto \lambda (\mathscr {H})$ sets up an injective map $\operatorname {\mathrm {Rep}}_c \to (\mathfrak {h}^\tau )^*$. Let $D_c$ be its image. For $\lambda \in D_c$, let $\mathscr {H}(\lambda )$ be the corresponding integrable highest weight $\hat {L}(\mathfrak {g},\sigma )$-module with central charge $c$.

For any $\lambda \in D_c$ and $i\in \hat {I}(\mathfrak {g},\sigma )=I(\mathfrak {g}^\tau )\sqcup \{o\}$, we associate an integer $n_{\lambda, i}$ as follows. Set

(6)\begin{equation} n_{\lambda, i} = \lambda ([x_i, y_i]) + \langle x_i,y_i \rangle \frac{s_i c }{ m}. \end{equation}

For $\sigma = \tau$ a diagram automorphism of $\mathfrak {g}$ (including $\tau =\operatorname {\mathrm {Id}}$), by definition $s_i= 0$ for $i\in I(\mathfrak {g}^\tau )$ and $s_o = 1$.

For any diagram automorphism $\tau$ of order $r$ (including $r = 1$), we follow the concrete realization of $x_i, y_i,\ i\in {I}(\mathfrak {g}^\tau )\sqcup \{o\}$ in [Reference KacKac90, § 8.3]. We emphasize that in the case $(\mathfrak {g},r)=(A_{2n, 2})$, our labelling ‘$o$’ corresponds to $i=n$ in [Reference KacKac90, § 8.3]. When $(\mathfrak {g}, r)\not = (A_{2n}, 2)$, we have

(7)\begin{equation} \langle x_i, y_i\rangle =\begin{cases} 1 & \text{if $\alpha_i$ is a long root for $i\in I(\mathfrak{g}^\tau)$,} \\ r & \text{if $i=o$, or $\alpha_i$ is a short root for $i\in I(\mathfrak{g}^\tau)$,} \end{cases} \end{equation}

and when $(\mathfrak {g}, r)= (A_{2n}, 2)$,

(8)\begin{equation} \langle x_i, y_i\rangle =\begin{cases} 1 & \text{if $i=o$,} \\ 2 & \text{if $\alpha_i$ is a long root for $i\in I(\mathfrak{g}^\tau)$,}\\ 4 & \text{if $\alpha_i$ is a short root for $i\in I(\mathfrak{g}^\tau)$.} \end{cases} \end{equation}

Lemma 2.1 The set $D_c$ can be described as follows:

\[ D_c=\{ \lambda \in (\mathfrak{h}^\tau)^* \,| \, n_{\lambda,i}\in \mathbb{Z}_{\geq 0}\quad \text{for any } i\in \hat{I}(\mathfrak{g},\sigma) \} . \]

Define

\[ \bar{s}_i= \langle x_i, y_i \rangle s_i, \quad \text{for any $i\in \hat{I}(\mathfrak{g}, \sigma)$ } \]

and let

\[ \bar{s} := \text{gcd}\{ \bar{s}_i: i\in \hat{I}(\mathfrak{g},\sigma)\}. \]

As an immediate consequence of Lemma 2.1 and the identity (6), we get the following.

We recall the following well-known result.

Let $V(\lambda )$ be the irreducible $\mathfrak {g}^\sigma$-module with highest weight $\lambda$ and highest weight vector $v_+$. Let $\hat {M}(V(\lambda ),c)$ be the generalized Verma module $U( \hat {L}(\mathfrak {g},\sigma )) \otimes _{U( \hat {L}(\mathfrak {g},\sigma )^{\geq 0} ) } V(\lambda )$ with highest weight vector $v_+ =1\otimes v_+$, where the action of $\hat {L}(\mathfrak {g},\sigma )^{\geq 0}$ on $V(\lambda )$ factors through the projection map $\hat {L}(\mathfrak {g},\sigma )^{\geq 0}\to \mathfrak {g}^\sigma \oplus \mathbb {C} C$ and the center $C$ acts by $c$. If $\lambda \in D_c$, then the unique irreducible quotient of $\hat {M}(V(\lambda ),c)$ is the integrable representation $\mathscr {H}(\lambda )$. Let $K_\lambda$ be the kernel of $\hat {M}(V(\lambda ),c)\to \mathscr {H}(\lambda )$. Set

\[ \hat{I}(\mathfrak{g},\sigma)^+:=\{i\in \hat{I}(\mathfrak{g},\sigma)\,|\, s_i>0 \} . \]

Then, as $U( \hat {L}(\mathfrak {g},\sigma ) )$-module, $K_\lambda$ is generated by

(9)\begin{equation} \{ \tilde{y}_i^{n_{\lambda,i} +1} \cdot v_+ \,| \, i\in \hat{I}(\mathfrak{g},\sigma)^+\}\quad \mbox{(cf. Kum02, } \$ \mbox{2.1)}. \end{equation}

Moreover, these elements are highest weight vectors.

Lemma 2.4 Fix $i\in \hat {I}(\mathfrak {g},\sigma )$. Let $f\in \mathcal {K}$ be such that $\sigma (f)=\epsilon ^{- s_i}f$ and $f\equiv t^{s_i}\mod {t^{{s_i}+1}}$. Put $X=x_i[f]$ and $Y=\tilde {y}_i=y_i[t^{-s_i}]$. For any $p> n_{\lambda,i}$ and $q> 0$, there exists $\alpha \neq 0$ such that

\[ Y^{p}\cdot v_+=\alpha X^{q} Y^{p+q}\cdot v_+ \]

in the generalized Verma module $\hat {M}(V(\lambda ),c)$.

Proof. Let $H:=[X,Y]=h_i[t^{-s_i}f]+ ({\bar {s_i }}/{m}) C$. Then, $[H, Y]=-2y_i[t^{-2s_i}f]$ commutes with $Y$. Note that $H\cdot v_+= n_{\lambda, i}v_+$. Then, one can check that, for $d\geq 0$,

\[ HY ^d\cdot v_+= (n_{\lambda,i}-2d)Y^d\cdot v_+ , \]

and, for $p\geq 0$,

\[ XY^{p+1}\cdot v_+= (p+1)(n_{\lambda, i} -p ) Y^p\cdot v_+. \]

By induction on $q$, the lemma follows.

3. Twisted analogue of conformal blocks

By a scheme we mean a quasi-compact and separated scheme over $\mathbb {C}$. By an algebraic curve, we mean a projective, reduced but not necessarily connected curve.

In this section we define the space of twisted covacua attached to a Galois cover of an algebraic curve. We prove that this space is finite dimensional.

For a smooth point $p$ in an algebraic curve $\bar {\Sigma }$ over $\mathbb {C}$, let $\mathcal {K}_p$ denote the quotient field of the completed local ring $\hat {\mathscr {O}}_p$ of $\bar {\Sigma }$ at $p$. We denote by $\mathbb {D}_p$ (respectively, $\mathbb {D}^\times _p$) the formal disc ${\rm Spec} \, \hat {\mathscr {O}}_p$ (respectively, the punctured formal disc ${\rm Spec}\, \mathcal {K}_p$).

For any smooth point $q\in {\Sigma }$, the stabilizer group $\Gamma _q$ of $\Gamma$ at $q$ is always cyclic. The order $e_q:=|\Gamma _q|$ is called the ramification index of $q$. Thus, $q$ is unramified if and only if $e_q=1$. Denote $p=\pi (q)$. We can also write $e_p=e_q$, since $e_{q}=e_{q'}$ for any $q,q'\in \pi ^{-1}(p)$. We also say that $e_p$ is the ramification index of $p$. Denote by $d_p$ the cardinality of the fiber $\pi ^{-1}(p)$. Then $|\Gamma |=e_p\cdot d_p$.

The action of $\Gamma _q$ on the tangent space $T_q{\Sigma }$ induces a primitive character $\chi _q: \Gamma _q\to \mathbb {C}^\times$, i.e. $\chi _q(\sigma _q)$ is a primitive $e_p$th root of unity for any generator $\sigma _q$ in $\Gamma _q$. From now on we shall fix $\sigma _q\in \Gamma _q$ so that

\[ \chi_q(\sigma_q) =e^{2\pi i/e_p}. \]

For any two smooth points $q, q'\in {\Sigma }$, if $\pi (q)=\pi (q')$, then

\[ \Gamma_{q'}=\gamma\Gamma_q\gamma^{-1},\quad \text{for any element } \gamma\in \Gamma \text{ such that } q'=\gamma\cdot q. \]

Moreover,

\[ \chi_{q'}(\gamma\sigma \gamma^{-1}) =\chi_q(\sigma), \quad \text{for any } \sigma\in \Gamma_q. \]

Given a smooth point $p\in \bar {\Sigma }$ such that $\pi ^{-1}(p)$ consists of smooth points in ${\Sigma }$, let $\pi ^{-1}(\mathbb {D}_p)$ (respectively, $\pi ^{-1}(\mathbb {D}^\times _p)$) denote the fiber product of ${\Sigma }$ and $\mathbb {D}_p$ (respectively, $\mathbb {D}_p^\times$) over $\bar {\Sigma }$. Then,

\[ \pi^{-1}(\mathbb{D}_p)\simeq\sqcup_{q\in \pi^{-1}(p) } {\mathbb{D}}_q \quad \text{and} \quad \pi^{-1}(\mathbb{D}^\times_p)\simeq \sqcup_{q\in \pi^{-1}(p) } {\mathbb{D}}^\times_q , \]

where ${\mathbb {D}}_q$ (respectively, ${\mathbb {D}}_q^\times$) denotes the formal disc (respectively, formal punctured disc) in ${\Sigma }$ around $q$.

Let the finite group $\Gamma$ also act on $\mathfrak {g}$ as Lie algebra automorphisms.

Let $\mathfrak {g}[\pi ^{-1}(\mathbb {D}^\times _p) ]^\Gamma$ be the Lie algebra consisting of $\Gamma$-equivariant regular maps from $\pi ^{-1}(\mathbb {D}^\times _p )$ to $\mathfrak {g}$. There is a natural isomorphism $\mathfrak {g}[\pi ^{-1}(\mathbb {D}^\times _p) ]^\Gamma \simeq ( \mathfrak {g}\otimes \mathbb {C}[\pi ^{-1}(\mathbb {D}^\times _p)])^\Gamma$. Let

(10)\begin{equation} \hat{ \mathfrak{g}}_p :=\mathfrak{g}[\pi^{-1}(\mathbb{D}^\times_p) ]^\Gamma\oplus \mathbb{C}C \end{equation}

be the central extension of $\mathfrak {g}[\pi ^{-1}(\mathbb {D}^\times _p) ]^\Gamma$ defined as follows:

(11)\begin{equation} [X, Y] =[X,Y]_0+ \frac{1}{|\Gamma|} \sum_{q\in \pi^{-1}(p) } {\rm Res}_{q} \langle d X, Y \rangle C, \end{equation}

for any $X,Y \in \mathfrak {g}[\pi ^{-1}(\mathbb {D}^\times _p) ]^\Gamma$, where $[,]_0$ denotes the point-wise Lie bracket induced from the bracket on $\mathfrak {g}$. We set the subalgebra

(12)\begin{equation} \hat{\mathfrak{p}}_p :=\mathfrak{g}[\pi^{-1}(\mathbb{D}_p) ]^\Gamma\oplus \mathbb{C}C \end{equation}

and

(13)\begin{equation} \hat{\mathfrak{g}}_p^+ :=\operatorname{\mathrm{Ker}} ( \mathfrak{g}[\pi^{-1}(\mathbb{D}_p) ]^\Gamma \to \mathfrak{g}[\pi^{-1}(p)]^\Gamma ) \end{equation}

obtained by the restriction map $\mathbb {C}[\pi ^{-1}(\mathbb {D}_p)] \to \mathbb {C}[\pi ^{-1}(p)]$, where $\mathfrak {g}[\pi ^{-1}(p) ]^\Gamma$ denotes the Lie algebra consisting of $\Gamma$-equivariant maps $x:\pi ^{-1}(p)\to \mathfrak {g}$. Let $\mathfrak {g}_p$ denote $\mathfrak {g}[ \pi ^{-1}(p) ]^\Gamma$. The following lemma is obvious.

Lemma 3.2 The evaluation map ${\rm ev}_q: \mathfrak {g}_p\to \mathfrak {g}^{\Gamma _q}$ given by

\[ x\mapsto x(q) \]

for any $x\in \mathfrak {g}_p$ and $q\in \pi ^{-1}(p)$ is an isomorphism of Lie algebras.

Let $\sigma _q$ be the generator of $\Gamma _q$ such that $\chi _q(\sigma _q) = e^{ {2\pi i}/{e_p}}$. Let $\hat {L}(\mathfrak {g},\sigma _q)$ denote the affine Lie algebra associated to $\mathfrak {g}$ and $\sigma _q$ as defined in § 2. We denote this algebra by $\hat {L}(\mathfrak {g}, \Gamma _q,\chi _q)$ or $\hat {L}(\mathfrak {g}, \Gamma _q )$ in short.

Lemma 3.3 The restriction map ${\rm res}_q: \hat {\mathfrak {g}}_p\to \hat {L}(\mathfrak {g}, \Gamma _q)$ given by

\[ X\mapsto X_q \quad\text{and}\quad C\mapsto C, \]

is an isomorphism of Lie algebras, where $X\in \mathfrak {g}[\pi ^{-1}(\mathbb {D}^\times _p ) ]^\Gamma$ and $X_q$ is the restriction of $X$ to $\mathbb {D}_q^\times$. Moreover,

\[ {\rm res}_q(\hat{\mathfrak{p}}_p )= \hat{L}(\mathfrak{g},\Gamma_q)^{\geq 0} \quad\text{and}\quad {\rm res}_q(\hat{\mathfrak{g}}^+_p)= \hat{L}(\mathfrak{g},\Gamma_q)^+. \]

Proof. For any $X, Y\in \mathfrak {g}[\pi ^{-1}( \mathbb {D}^\times _p ) ]^\Gamma$, the restriction of $[X,Y]_0$ to $\mathbb {D}_q^\times$ is equal to $[X_q,Y_q]_0$. Note that for any $\gamma \in \Gamma$ and $x,y\in \mathfrak {g}$, we have $\langle \gamma (x), \gamma (y) \rangle = \langle x,y \rangle$, which follows from the Killing form realization of $\langle,\rangle$ on $\mathfrak {g}$. Since $X, Y$ are $\Gamma$-equivariant, for any $q, q'\in \pi ^{-1}(p)$ we have

\[ {\rm Res}_q\langle dX, Y\rangle= {\rm Res}_{q'}\langle dX, Y\rangle. \]

It is now easy to see that ${\rm res}_q: \hat {\mathfrak {g}}_p\to \hat {L}(\mathfrak {g}, \Gamma _q)$ is an isomorphism of Lie algebras, and

\[ {\rm res}_q(\hat{\mathfrak{p}}_p )= \hat{L}(\mathfrak{g},\Gamma_q)^{\geq 0} \quad\text{and}\quad {\rm res}_q(\hat{\mathfrak{g}}^+_p)= \hat{L}(\mathfrak{g},\Gamma_q)^+. \]

By the above lemma, we have a faithful functor ${\rm Rep}_c(\hat {\mathfrak {g}}_p)\to {\rm Rep}(\mathfrak {g}_p)$ from the category of integrable highest weight representations of $\hat {\mathfrak {g}}_p$ of level $c$ to the category of finite-dimensional representations of $\mathfrak {g}_p$. We denote by $D_{c,p}$ the parameter set of (irreducible) integrable highest weight representations of $\hat {\mathfrak {g}}_p$ of level $c$ obtained as the subset of the set of dominant integral weights of $\mathfrak {g}_p$ under the above faithful functor. Let $D_{c,q}$ denote the parameter set of (irreducible) integrable highest weight representations of $\hat {L}(\mathfrak {g}, \Gamma _q)$ as in § 2. Then, we can identify $D_{c,p}$ and $D_{c,q}$ via the restriction isomorphism ${\rm res}_q: \hat {\mathfrak {g}}_p\to \hat {L}(\mathfrak {g}, \Gamma _q)$ as in Lemma 3.3.

From now on we fix an $s$-pointed curve $(\bar {\Sigma }, \vec {p})$ (for any $s\geq 1$), where $\vec {p}=(p_1, \ldots, p_s)$, and a Galois cover $\pi : {\Sigma } \to \bar {\Sigma }$ with the finite Galois group $\Gamma$ such that the fiber $\pi ^{-1}(p_i)$ consists of smooth points for any $i=1,2,\ldots, s$. We also fix a simple Lie algebra $\mathfrak {g}$ and a group homomorphism $\phi : \Gamma \to {\rm Aut}(\mathfrak {g})$, where ${\rm Aut}(\mathfrak {g})$ is the group of automorphisms of $\mathfrak {g}$.

We now fix an $s$-tuple $\vec {\lambda }=(\lambda _{1},\ldots,\lambda _{s})$ of weights with $\lambda _{i}\in D_{c, p_i}$ ‘attached’ to the point $p_i$. To this data, there is associated the space of (twisted) vacua $\mathscr {V}_{{\Sigma },\Gamma, \phi }(\vec {p},\vec {\lambda })^{{{\dagger}} }$ (or the space of (twisted) conformal blocks) and its dual space $\mathscr {V}_{{\Sigma },\Gamma,\phi }(\vec {p},\vec {\lambda })$ called the space of (twisted) covacua (or the space of (twisted) dual conformal blocks) defined as follows.

Definition 3.5 Let $\mathfrak {g}[{\Sigma }\backslash \pi ^{-1}( \vec {p}) ]^\Gamma$ denote the space of $\Gamma$-equivariant regular maps $f:{\Sigma }\backslash \pi ^{-1}(\vec {p}) \to \mathfrak {g}$. Then, $\mathfrak {g}[{\Sigma }\backslash \pi ^{-1}(\vec {p}) ]^\Gamma$ is a Lie algebra under the pointwise bracket.

Set

(14)\begin{equation} \mathscr{H}(\vec{\lambda}):=\mathscr{H}(\lambda_{1})\otimes\cdots \otimes\mathscr{H}(\lambda_{s}), \end{equation}

where $\mathscr {H}(\lambda _i)$ is the integrable highest weight representation of $\hat {\mathfrak {g}}_{p_i}$ of level $c$ with highest weight $\lambda _i\in D_{c, p_i}$.

Define an action of the Lie algebra $\mathfrak {g}[{\Sigma }\backslash \pi ^{-1} ( \vec {p} ) ]^\Gamma$ on $\mathscr {H}(\vec {\lambda })$ as follows:

(15)\begin{equation} X\cdot (v_{1}\otimes\cdots\otimes v_{s}) = \sum^{s}_{i=1}v_{1}\otimes\cdots\otimes X_{p_i}\cdot v_{i}\otimes \cdots\otimes v_{s}, \quad \text{for} \ X\in \mathfrak{g} [{\Sigma}\backslash \pi^{-1}(\vec{p}) ]^\Gamma, \ \text{and} \ v_{i}\in \mathscr{H}(\lambda_{i}), \end{equation}

where $X_{p_{i}}$ denotes the restriction of $X$ to $\pi ^{-1}(\mathbb {D}^\times _{p_i})$, hence $X_{p_i}$ is an element in $\hat {\mathfrak {g}}_{p_i}$.

By the residue theorem [Reference HartshorneHar77, Theorem 7.14.2, Chap. III],

(16)\begin{equation} \sum_{q\in \pi^{-1}( \vec{p}) } \text{~Res}_{q} \langle dX,Y \rangle =0,\quad \text{for any}\ X,Y \in \mathfrak{g}[ {\Sigma} \backslash \pi^{-1} ( \vec{p}) ]^{\Gamma}. \end{equation}

Thus, the action (15) indeed is an action of the Lie algebra $\mathfrak {g}[{\Sigma }\backslash \pi ^{-1}( \vec {p}) ]^\Gamma$ on $\mathscr {H}(\vec {\lambda })$.

Finally, we are ready to define the space of (twisted) vacua

(17)\begin{equation} {\mathscr{V}_{{\Sigma},\Gamma,\phi }(\vec{p},\vec{\lambda})^{{{\dagger}}}} :=\operatorname{\mathrm{Hom}}_{\mathfrak{g}[{\Sigma}\backslash \pi^{-1}(\vec{p})]^\Gamma}(\mathscr{H} (\vec{\lambda}),\mathbb{C}),\end{equation}

and the space of (twisted) covacua

(18)\begin{equation} \mathscr{V}_{{\Sigma},\Gamma,\phi }(\vec{p},\vec{\lambda}):=[\mathscr{H}(\vec{\lambda})]_{\mathfrak{g}[{\Sigma}\backslash \pi^{-1} (\vec{p}) ]^\Gamma},\end{equation}

where $\mathbb {C}$ is considered as the trivial module under the action of $\mathfrak {g}[{\Sigma }\backslash \pi ^{-1} (\vec {p})]^\Gamma$, and $[\mathscr {H} (\vec {\lambda })]_{\mathfrak {g}[{\Sigma }\backslash \pi ^{-1} ( \vec {p} ) ]^\Gamma }$ denotes the space of covariants $\mathscr {H} (\vec {\lambda })/\bigl (\mathfrak {g}[{\Sigma }\backslash \pi ^{-1}( \vec {p} ) ]^\Gamma \cdot \mathscr {H} (\vec {\lambda })\bigr )$. Clearly,

(19)\begin{equation} {\mathscr{V}_{{\Sigma},\Gamma,\phi}(\vec{p},\vec{\lambda})^{{{\dagger}}}} \simeq \mathscr{V}_{{\Sigma},\Gamma,\phi}(\vec{p},\vec{\lambda})^{*}.\end{equation}

Proof. Let $\mathfrak {g}[\pi ^{-1}(\mathbb {D}^\times _{\vec {p}} ) ]^\Gamma$ be the space of $\Gamma$-equivariant maps from the disjoint union of formal punctured discs $\sqcup _{q\in \pi ^{-1}(\vec {p}) } \mathbb {D}^\times _{q}$ to $\mathfrak {g}$. Define a Lie algebra bracket on

(20)\begin{equation} \hat{\mathfrak{g}}_{\vec{p}} := \mathfrak{g}[\pi^{-1}(\mathbb{D}^\times_{\vec{p}} ) ]^\Gamma \oplus \mathbb{C}C , \end{equation}

by declaring $C$ to be the central element and the Lie bracket is defined in the similar way as in (11).

Now, define an embedding of Lie algebras:

\[ \beta:\mathfrak{g}[{\Sigma}\backslash \pi^{-1} ( \vec{p} ) ]^\Gamma \to \hat{\mathfrak{g}}_{\vec{p}}, \quad X\mapsto X_{\vec{p}} \]

where $X_{\vec {p}}$ is the restriction of $X$ to $\pi ^{-1}(\mathbb {D}^\times _{\vec {p}} )$ .

By the residue theorem, $\beta$ is indeed a Lie algebra homomorphism. Moreover, by Riemann–Roch theorem, $\operatorname {\mathrm {Im}} \beta + \mathfrak {g} [\pi ^{-1} (\mathbb {D}_{\vec {p}} )]^\Gamma$ has finite codimension in ${\hat {\mathfrak {g}}}_{\vec {p}}$, where $\pi ^{-1}(\mathbb {D}_{\vec {p}} )$ is the disjoint union $\sqcup _{q\in \pi ^{-1}(\vec {p})}\mathbb {D}_q$. Further, define the following surjective Lie algebra homomorphism from the direct sum Lie algebra

\[ \bigoplus^{s}_{i=1} \hat{\mathfrak{g}}_{p_i} \to {\hat{\mathfrak{g}}}_{\vec{p}} ,\quad \sum^{s}_{i=1} X_i \mapsto \sum^{s}_{i=1} \tilde{X}_i, \quad C_{i}\to C, \]

here $C_{i}$ is the center $C$ of $\hat {\mathfrak {g}}_{p_{i}}$, and the map $X_i\in \mathfrak {g} [ \pi ^{-1} ( \mathbb {D}^\times _{p_i}) ]^\Gamma$ naturally extends to $\tilde {X}_i \in \mathfrak {g}[ \pi ^{-1}(\mathbb {D}^\times _{\vec {p}} )]^\Gamma$ by taking $\pi ^{-1}(\mathbb {D}^\times _{p_j} )$ to $0$ if $j\neq i$.

Now, the lemma follows from [Reference KumarKum02, Lemma 10.2.2].

4. Propagation of twisted vacua

We prove the propagation theorem in this section, which asserts that adding marked points and attaching weight 0 to those points does not alter the space of twisted vacua.

Let ${\Sigma }\to \bar {\Sigma }$ be a $\Gamma$-cover (cf. Definition 3.1). Moreover, $\phi :\Gamma \to {\rm Aut}(\mathfrak {g})$ is a group homomorphism.

Definition 4.1 Let $\vec {o}=(o_{1},\ldots,o_{s})$ and $\vec {p}=(p_{1},\ldots,p_{a})$ be two disjoint non-empty sets of smooth and distinct points in $\bar {\Sigma }$ such that $(\bar {\Sigma }, \vec {o})$ is a $s$-pointed curve and let $\vec {\lambda }=(\lambda _{1},\ldots,\lambda _{s})$, $\vec {\mu }=(\mu _{1},\ldots,\mu _{a})$ be tuples of dominant weights such that $\lambda _i\in D_{c, o_i}$ and $\mu _j\in D_{c,p_j}$ for each $1\leq i\leq s, 1\leq j\leq a$.

We assume that $\pi ^{-1}(o_i)$ and $\pi ^{-1}(p_j)$ consist of smooth points.

Denote the tensor product

(21)\begin{equation} V(\vec{\mu}):=V(\mu_{1})\otimes\cdots\otimes V(\mu_{a}), \end{equation}

where $V(\mu _{k})$ is the irreducible $\mathfrak {g}_{p_k}$-module with highest weight $\mu _k$.

Define a $\mathfrak {g}[{\Sigma }\backslash \pi ^{-1}( \vec {o}) ]^\Gamma$-module structure on $V(\vec {\mu })$ as follows:

(22)\begin{equation} X \cdot (v_{1}\otimes\cdots\otimes v_{a})=\sum^{a}_{k=1}v_{1}\otimes\cdots\otimes X|_{p_k}\cdot v_{k}\otimes\cdots\otimes v_{a}, \end{equation}

for $v_{k}\in V(\mu _{k}), X \in \mathfrak {g}[{\Sigma }\backslash \pi ^{-1}( \vec {o}) ]^\Gamma$, and $X|_{p_k}$ denotes the restriction $X|_{\pi ^{-1}(p_k)} \in \mathfrak {g}_{p_k}$. This gives rise to the tensor product $\mathfrak {g}[\Sigma \backslash \pi ^{-1} ( \vec {o} ) ]^\Gamma$-module structure on $\mathscr {H} (\vec {\lambda })\otimes V(\vec {\mu })$.

The proof of the following lemma was communicated to us by Bernstein.

Theorem 4.3 With the notation and assumptions as in Definition 4.1, assume further that $\Gamma$ stabilizes a Borel subalgebra of $\mathfrak {g}$. Then, the canonical map

\[ \theta:\big[\mathscr{H} (\vec{\lambda})\otimes V(\vec{\mu})\big]_{\mathfrak{g}[\Sigma\backslash \pi^{-1} ( \vec{o}) ]^\Gamma}\to \mathscr{V}_{{\Sigma},\Gamma,\phi }\big((\vec{o},\vec{p}),(\vec{\lambda},\vec{\mu})\big) \]

is an isomorphism, where $\mathscr {V}_{\Sigma,\Gamma,\phi }$ is the space of covacua and the map $\theta$ is induced from the $\mathfrak {g}[{\Sigma }\backslash \pi ^{-1}( \vec {o})]^\Gamma$-module embedding

\[ \mathscr{H} (\vec{\lambda})\otimes V(\vec{\mu})\hookrightarrow \mathscr{H} (\vec{\lambda},\vec{\mu}), \]

with $V(\mu _{j})$ identified as a $\mathfrak {g}_{p_j}$-submodule of $\mathscr {H} (\mu _{j})$ annihilated by $\hat {\mathfrak {g}}_{p_j}^+$. (Observe that since the subspace $V(\mu _{j})\subset \mathscr {H} (\mu _{j})$ is annihilated by $\hat {\mathfrak {g}}_{p_j}^+$, the embedding $V(\mu _{j})\subset \mathscr {H} (\mu _{j})$ is indeed a $\mathfrak {g}[{\Sigma }\backslash \pi ^{-1}( \vec {o} ) ]^\Gamma$-module embedding.)

Proof. By Lemma 4.2, we may assume that $\Gamma$ stabilizes a Borel subalgebra $\mathfrak {b}$ and a Cartan subalgebra $\mathfrak {h}$ contained in $\mathfrak {b}$. From now on we fix such a $\mathfrak {b}$ and $\mathfrak {h}$.

Let $\mathscr {H} :=\mathscr {H} (\vec {\lambda })\otimes V(\mu _{1})\otimes \cdots \otimes V(\mu _{a-1})$. By induction on $a$, it suffices to show that the inclusion $V(\mu _{a})\hookrightarrow \mathscr {H} (\mu _{a})$ induces an isomorphism (abbreviating $\mu _{a}$ by $\mu$ and $p_{a}$ by $p$)

(23)\begin{equation} [\mathscr{H}\otimes V(\mu)]_{\mathfrak{g}[{\Sigma}^o]^\Gamma}\xrightarrow{\sim}[\mathscr{H}\otimes \mathscr{H} (\mu)]_{\mathfrak{g}[{\Sigma}^o\backslash \pi^{-1}(p)]^\Gamma}, \end{equation}

where ${\Sigma }^o := {\Sigma }\backslash \pi ^{-1} (\vec {o})$.

We first prove (23) replacing $\mathscr {H} (\mu )$ by the generalized Verma module $\hat {M}(V(\mu ),c)$ for $\hat {\mathfrak {g}}_p$ and the parabolic subalgebra $\hat {\mathfrak {p}}_p$, i.e.

(24)\begin{equation} [\mathscr{H}\otimes V(\mu)]_{\mathfrak{g}[{\Sigma}^o]^\Gamma}\xrightarrow{\sim}[\mathscr{H}\otimes \hat{M}(V(\mu),c)]_{\mathfrak{g}[{\Sigma}^o\backslash \pi^{-1}(p)]^\Gamma}. \end{equation}

Consider the Lie algebra

(25)\begin{equation} \mathfrak{s}_p := \mathfrak{g}[{\Sigma}^o\backslash \pi^{-1}(p) ]^\Gamma \oplus \mathbb{C}C, \end{equation}

where $C$ is central in $\mathfrak {s}_p$ and

(26)\begin{equation} [X, Y]=[X,Y]_0+ \frac{1}{|\Gamma|} \sum_{q\in \pi^{-1}(p) } \operatorname{\mathrm{Res}}_{q} \langle dX,Y \rangle C,\quad\text{for}\ X, Y\in \mathfrak{g}[{\Sigma}^o \backslash \pi^{-1}(p) ]^\Gamma, \end{equation}

where $[X,Y]_0$ is the point-wise Lie bracket.

Let $\mathfrak {s}_p^{\geq 0}$ be the subalgebra of $\mathfrak {s}_p$:

\[ \mathfrak{s}_p^{\geq 0}:=\mathfrak{g}[{\Sigma}^o ]^\Gamma\oplus \mathbb{C}C . \]

Fix a point $q\in \pi ^{-1}(p)$ and a generator $\sigma _q$ of $\Gamma _q$ such that $\sigma _q$ acts on $T_q{\Sigma }$ by $\epsilon _q :=e^{ {2\pi i}/{e_p}}$ (which is a primitive $e_p$th root of unity). By the Riemann–Roch theorem there exists a formal parameter $z_q$ around $q$ such that $z_q^{-1}$ is a regular function on ${\Sigma }^o\backslash \{q\}$. Moreover, we require $z_q^{-1}$ to vanish at any other point $q'$ in $\pi ^{-1}(p)$. Replacing $z_q^{-1}$ by

\[ \sum_{j=1}^{e_p} \epsilon_q^{-j}\sigma_q^j(z_q^{-1}), \]

we can (and will) assume that

(27)\begin{equation} \sigma_q\cdot z_q^{-1} = \epsilon_qz_q^{-1}. \end{equation}

Recall the Lie algebras $\hat {L}(\mathfrak {g},\Gamma _q)$ and $\hat {L}(\mathfrak {g},\Gamma _q)^-=(z_q^{-1}\mathfrak {g}[z_q^{-1}])^{\Gamma _q}$ from § 2. Since $z_q$ is a formal parameter at $q$ with $\sigma _q\cdot z_q= \epsilon _q^{-1}z_q$, we have

(28)\begin{equation} \hat{L}(\mathfrak{g}, \Gamma_q)=\hat{L}(\mathfrak{g},\Gamma_q)^{\geq 0}\oplus (z_q^{-1}\mathfrak{g}[z_q^{-1}])^{\Gamma_q}. \end{equation}

Define, for any $x\in \mathfrak {g}$ and $k\geq 1$,

\[ A(x[z_q^{-k}]) := \frac{1}{|\Gamma_q|} \sum_{\gamma\in \Gamma}\gamma\cdot (x[z_q^{-k}]) \in \mathfrak{s}_p, \]

and let $V\subset \mathfrak {s}_p$ be the span of $\{A(x[z_q^{-k}])\}_{x\in \mathfrak {g}, k \geq 1}$. It is easy to check that

(29)\begin{equation} \mathfrak{s}_p=\mathfrak{s}_p^{\geq 0}\oplus V. \end{equation}

By Lemmas 3.2 and 3.3, we can view $\hat {M}(V(\mu ),c)$ as a generalized Verma module over $\hat {L}(\mathfrak {g},\Gamma _q)$ induced from $V(\mu )$ as $\hat {L}(\mathfrak {g}, \Gamma _q)^{\geq 0}$-module.

Consider the embedding of the Lie algebra

\[ \mathfrak{s}_p\hookrightarrow \hat{L}(\mathfrak{g},\Gamma_q) \]

by taking $C\mapsto C$ and any $X\mapsto X_q$. We assert that the above embedding $\mathfrak {s}_p \hookrightarrow \hat {L}(\mathfrak {g}, \Gamma _q)$ induces a vector space isomorphism

(30)\begin{equation} \gamma: \mathfrak{s}_p/ \mathfrak{s}_p^{\geq 0} \simeq \hat{L}(\mathfrak{g}, \Gamma_q)/\hat{L}(\mathfrak{g}, \Gamma_q)^{\geq 0}. \end{equation}

To prove the above isomorphism, observe first that $\gamma$ is injective: for $\alpha \in \mathfrak {g}[{\Sigma }^o\setminus \pi ^{-1}(p)]^\Gamma$, if $\gamma (\alpha )\in \hat {L}(\mathfrak {g}, \Gamma _q)^{\geq 0}$, then $\alpha \in \mathfrak {g}[({\Sigma }^o\setminus \pi ^{-1}(p))\cup \{q\}]$. The $\Gamma$-invariance of $\alpha$ forces $\alpha \in \mathfrak {g}[{\Sigma }^o]$, proving the injectivity of $\gamma$. To prove the surjectivity of $\gamma$, take a $\Gamma _q$-invariant $\alpha =x[z_q^{-k}]$ for $k \geq 1$. Thus, $\sigma _q(x)= \epsilon _q^{-k}x$. By the definition, since $z_q^{-1}$ vanishes at any point $q'\in \pi ^{-1}(p)$ different from $q$,

\[ \gamma(A(\alpha))=\alpha + \hat{L}(\mathfrak{g}, \Gamma_q)^{\geq 0}. \]

This proves the surjectivity of $\gamma$. Thus, by the Poincaré–Birkhoff–Witt theorem, as $\mathfrak {s}_p$-modules

(31)\begin{equation} \hat{M}(V(\mu),c)\simeq U(\mathfrak{s}_p)\otimes_{U(\mathfrak{s}_p^{\geq 0} )}V(\mu). \end{equation}

Let $\mathfrak {g}[{\Sigma }^o\backslash \pi ^{-1}(p)]^\Gamma$ act on $\mathscr {H}$ as follows:

\begin{align*} &X \cdot (v_1\otimes \cdots \otimes v_s\otimes w_1\otimes \cdots \otimes w_{a-1}) \\ &\quad= \sum_{i=1}^s v_1\otimes \cdots \otimes X_{o_i}\cdot v_i\otimes \cdots \otimes v_s\otimes w_1\otimes \cdots \otimes w_{a-1} \\ &\qquad + \sum_{j=1}^{a-1} v_1\otimes \cdots \otimes v_s\otimes w_1\otimes \cdots \otimes X|_{p_j}\cdot w_j\otimes \cdots \otimes w_{a-1} \end{align*}

for $X\in \mathfrak {g}[{\Sigma }^o\backslash \pi ^{-1}(p)]^\Gamma, v_i\in \mathscr {H}(\lambda _i)$ and $w_j \in V(\mu _j)$, and let $C$ act on $\mathscr {H}$ by the scalar $-c$. By the residue theorem, these actions combine to make $\mathscr {H}$ into an $\mathfrak {s}_p$-module. Thus, the action of $C$ on the tensor product $\mathscr {H}\otimes \hat {M}(V(\mu ),c)$ is trivial.

Now, by the isomorphism (31) (in the following, $\mathfrak {g}[{\Sigma }^o]^\Gamma$ acts on $V(\mu )$ via its restriction on $\pi ^{-1}(p)$ and $C$ acts via the scalar $c$)

\begin{align*} \big[ \mathscr{H}\otimes \hat{M}(V(\mu),c)\big]_{\mathfrak{g}[{\Sigma}^o\backslash \pi^{-1}(p)]^\Gamma} &= \big[\mathscr{H}\otimes \hat{M}(V(\mu),c)\big]_{\mathfrak{s}_p },\quad \text{since $C$ acts trivially}\\ &\simeq \mathscr{H}\otimes_{U(\mathfrak{s}_p )}\hat{M}(V(\mu),c)\\ &\simeq \mathscr{H}\otimes_{U(\mathfrak{s}_p )}\big(U(\mathfrak{s}_p )\otimes_{U(\mathfrak{g}[{\Sigma}^o]^\Gamma \oplus \mathbb{C}C)}V(\mu)\big)\\ &\simeq \mathscr{H}\otimes_{U(\mathfrak{g}[{\Sigma}^o]^\Gamma \oplus \mathbb{C}C)} V(\mu)\\ &= \mathscr{H}\otimes_{U(\mathfrak{g}[{\Sigma}^o]^\Gamma )}V(\mu)\\ &= [\mathscr{H}\otimes V(\mu)]_{\mathfrak{g}[{\Sigma}^o]^\Gamma}. \end{align*}

This proves (24).

Now, we come to the proof of (23).

Let $K(\mu )$ be the kernel of the canonical projection $\hat {M}(V(\mu ),c)\twoheadrightarrow \mathscr {H} (\mu )$. In view of (24), to prove (23), it suffices to show that the image of

\[ \iota: \big[\mathscr{H}\otimes K(\mu)\big]_{\mathfrak{g}[{\Sigma}^o\backslash \pi^{-1}(p)]^\Gamma}\to \big[\mathscr{H}\otimes \hat{M}(V(\mu),c)\big]_{\mathfrak{g}[{\Sigma}^o\backslash \pi^{-1}(p)]^\Gamma} \]

is zero: from the isomorphism (30), we get

\[ \hat{L}(\mathfrak{g}, \Gamma_q) =\mathfrak{s}_p+\hat{L}(\mathfrak{g},\Gamma_q)^{\geq 0}. \]

Moreover, write

\[ \hat{L}(\mathfrak{g}, \Gamma_q)^{\geq 0}=\hat{L}(\mathfrak{g}, \Gamma_q)^+ + \mathfrak{g}^{\Gamma_q}+ \mathbb{C} C, \]

and observe that any element of $\mathfrak {g}^{\Gamma _q}$ can be (uniquely) extended to an element of $\mathfrak {g}_p:=\mathfrak {g}[\pi ^{-1}(p)]^\Gamma$ (cf. Lemma 3.2). Further, ${\Sigma }^o$ being affine, the restriction map $\mathfrak {g}[{\Sigma }^o]^\Gamma \to \mathfrak {g}_p$ is surjective, and, of course, $\mathfrak {g}[{\Sigma }^o]^\Gamma \subset \mathfrak {s}_p^{\geq 0}$. Thus, we get the decomposition:

\[ \hat{L}(\mathfrak{g}, \Gamma_q) =\mathfrak{s}_p+\hat{L}(\mathfrak{g},\Gamma_q)^+, \]

and, hence, by the Poincaré–Birkhoff–Witt theorem, $U(\hat {L}(\mathfrak {g},\Gamma _q))$ is the span of elements of the form

\[ Y_{1}\ldots Y_{m}\cdot X_{1}\ldots X_{n},\quad \text{for }Y_{i}\in \mathfrak{s}_p, X_{j}\in\hat{L}(\mathfrak{g},\Gamma_q)^+ \text{ and }m, n\geq 0. \]

Consider the decomposition (3) for $\sigma _q$: $\sigma _q=\tau _q \epsilon _q^{{\rm ad}\, h}$, under a choice of $\sigma _q$-stable Borel subalgebra $\mathfrak {b}_q$ containing the same Cartan subalgebra $\mathfrak {h}$ in the sense of § 2. (Since $\Gamma$, in particular $\sigma _q$, stabilizes the pair $(\mathfrak {b}, \mathfrak {h})$, as in the proof of Lemma 2.5, $\mathfrak {h}^{\sigma _q}=\mathfrak {h}^{\tau '_q}$ for some diagram automorphism $\tau '_q$ of $\mathfrak {g}$ associated to the pair $(\mathfrak {b}, \mathfrak {h})$. In particular, the centralizer $Z_{\mathfrak {g}}(\mathfrak {h}^\sigma )$ of $\mathfrak {h}^\sigma$ in $\mathfrak {g}$ equals $\mathfrak {h}$ and, hence, we can take $\mathfrak {h}_q=\mathfrak {h}$.) Under such a choice, there exist $sl_2$-triples $x_i,y_i,h_i \in \mathfrak {g}$ for each $i\in \hat {I}(\mathfrak {g}, \sigma _q)$ such that $\tilde {x}_i:=x_i[z_q^{s_i}], \tilde {y}_i:=y_{i}[z_q^{-s_i}], i\in \hat {I}(\mathfrak {g},\sigma _q)$ are Chevalley generators of $\hat {L}(\mathfrak {g}, \sigma _q)$. Moreover, $x_i,y_i$ satisfy

(32)\begin{equation} \sigma_q(x_i)= \epsilon_q^{s_i} x_i \quad\text{and}\quad \sigma_q(y_i)=\epsilon_q^{-s_i} y_i . \end{equation}

Let $v_+$ be the highest weight vector of $\hat {M}(V(\mu ),c)$. Recall (cf. (9)) that $K(\mu )$ is generated by $\tilde {y}^{n_{\mu,i}+1}_i\cdot v_+$, for $i\in \hat {I}(\mathfrak {g},\sigma _q)^+$ consisting of $i\in \hat {I}(\mathfrak {g}, \sigma _q)$ such that $s_i>0$.

Thus, to prove the vanishing of the map $\iota$, it suffices to show that for any $i\in \hat {I}(\mathfrak {g},\sigma _q)^+$

(33)\begin{equation} \iota\big(h\otimes (X_{1}\ldots X_{n}\cdot \tilde{y}_i^{n_{\mu,i}+ 1}\cdot v_+)\big)=0, \end{equation}

for $h\in \mathscr {H}$, any $n\geq 0$ and $X_{j}\in \hat {L}(\mathfrak {g}, \Gamma _q)^+$. However, $\tilde {y}_i^{n_{\mu,i}+ 1}\cdot v_+$ being a highest weight vector,

\[ \hat{L}(\mathfrak{g}, \Gamma_q)^+\cdot (\tilde{y}_i^{n_{\mu,i}+ 1}\cdot v_+)=0. \]

Thus, to prove (33), it suffices to show that for any $i\in \hat {I}(\mathfrak {g},\sigma _q)^+$

(34)\begin{equation} \iota (h\otimes ( \tilde{y}_i^{n_{\mu,i}+ 1}\cdot v_+))=0,\quad\text{for any}\ h\in \mathscr{H}. \end{equation}

Fix $i\in \hat {I}(\mathfrak {g}, \sigma _q)^+$. Take $f\in \mathbb {C}[{\Sigma }^o]$ such that

\[ f_{q}\equiv z_q^{s_i}(\text{mod~}z_q^{s_i+1}), \]

and the order of vanishing of $f$ at any $q'\neq q\in \pi ^{-1}(p)$ is at least $(n_{\mu,i}+3)s_i$. Moreover, replacing $f$ by $ ({1}/{|\Gamma _q|})\sum _{j=1}^{|\Gamma _q|} \epsilon _q^{s_ij}\sigma _q^j\cdot f$, we can (and will) assume that

\[ \sigma_q\cdot f= \epsilon_q^{-s_i}f . \]

Now, take

\[ Z=\sum_{\gamma \in \Gamma/\Gamma_q} \gamma \cdot (x_i[f]). \]

Then, writing $Y=\tilde {y}_i$,

(35)\begin{align} Z^N Y^{n_{\mu,i}+ N+1}\cdot v_+&=\bigg(\sum_{\gamma \in \Gamma/\Gamma_q} \big( \gamma \cdot (x_i[f])\big)_q\bigg)^NY^{n_{\mu,i}+N+ 1}\cdot v_+ \nonumber\\ &=(x_i[f_q])^NY^{n_{\mu,i}+ N+1}\cdot v_+. \end{align}

To prove the last equality, observe that $\big (\gamma _1 \cdot (x_i[f])\big )_q \dots \big (\gamma _N \cdot (x_i[f])\big )_q$ has zero of order at least $(n_{\mu,i}+3)s_i+(N-1)s_i$ unless each $\gamma _j\cdot \Gamma _q=\Gamma _q$. However, $Y^{n_{\mu,i}+ N+1}$ has order of pole equal to $(n_{\mu,i}+N+1)s_i$. Since $(n_{\mu,i}+3)s_i+(N-1)s_i >(n_{\mu,i}+N+1)s_i$, we get the last equality. Thus, by Lemma 2.4 for $X=x_i[f_q]$ and $Y=\tilde {y}_i$, for any $N\geq 1$, there exists $\alpha \neq 0$ such that

\begin{align*} \iota\big(h\otimes (Y^{n_{\mu,i}+ 1}\cdot v_+)\big) &= \alpha \iota \big(h\otimes X^{N}Y^{n_{\mu,i}+ N+1}\cdot v_+\big)\\ &=\alpha \iota \big(h\otimes Z^{N}Y^{n_{\mu,i}+ N+1}\cdot v_+\big), \quad \text{by (35)}\\ &= (-1)^{N}\alpha \iota\big(Z^{N}\cdot h\otimes Y^{n_{\mu,i} + N+1}\cdot v_+\big)\\ &= 0,\quad\text{by Lemma 2.5 for large $N$ (see the argument below).} \end{align*}

This proves (34) and, hence, completes the proof of the theorem.

We now explain more precisely how Lemma 2.5 implies $Z^N\cdot h=0$. With respect to the pair $(\mathfrak {b},\mathfrak {h})$ stable under $\Gamma$ (note that $\mathfrak {b}$ might not be the same as $\mathfrak {b}_q$ given above (3) though $\mathfrak {h}_q$ is taken to be $\mathfrak {h}$), since $\Gamma$ preserves the pair $(\mathfrak {b},\mathfrak {h})$, the group $\Gamma$ acts on the root system $\Phi (\mathfrak {g},\mathfrak {h})$ of $\mathfrak {g}$ by factoring through the group of outer automorphisms with respect to the pair $(\mathfrak {b}, \mathfrak {h})$. In particular, $\Gamma$ preserves the set of positive (respectively, negative) roots. From the construction of $x_i$ in § 2, $x_i$ is either a linear combination of positive root vectors or a linear combination of negative root vectors with respect to $\mathfrak {b}_q$. Thus, either $\gamma \cdot x_i\in \mathfrak {n}$ for all $\gamma \in \Gamma$, or $\gamma \cdot x_i\in \mathfrak {n}^-$ for all $\gamma \in \Gamma$, where $\mathfrak {b}^-$ is the negative Borel of $\mathfrak {b}$ and $\mathfrak {n}$ (respectively, $\mathfrak {n}^-$) is the nilradical of $\mathfrak {b}$ (respectively, $\mathfrak {b}^-$). Therefore, we may apply Lemma 2.5 to show $Z^N\cdot h=0$.

The following result is the twisted analogue of ‘Propagation of Vacua’ due to Tsuchiya, Ueno, and Yamada [Reference Tsuchiya, Ueno and YamadaTUY89].

5. Factorization theorem

The aim of this section is to prove the factorization theorem which identifies the space of covacua for a genus $g$ nodal curve $\bar {\Sigma }$ with a direct sum of the spaces of covacua for its normalization $\bar {\Sigma }'$ (which is a genus $g-1$ curve).

Let $\pi : {\Sigma }\to \bar {\Sigma }$ be a $\Gamma$-cover of a $s$-pointed curve $(\bar {\Sigma }, \vec {o})$. We do not assume that $\bar {\Sigma }$ is irreducible. Moreover, $\phi :\Gamma \to$ Aut$(\mathfrak {g})$ is a group homomorphism.

Definition 5.1 [Reference Bertin and RomagnyBR11, Définition 4.1.4]

Let ${\Sigma }$ be a reduced (but not necessarily connected) projective curve with at worst only simple nodal singularity. (Recall that a point $P\in {\Sigma }$ is called a simple node if analytically a neighborhood of $P$ in ${\Sigma }$ is isomorphic with an analytic neighborhood of $(0,0)$ in the curve $xy=0$ in $\mathbb {A}^2$.) Then, the action of $\Gamma$ on $\Sigma$ at any simple node $q\in \Sigma$ is called stable if the derivative $\dot {\sigma }$ of any element $\sigma \in \Gamma _q$ acting on the Zariski tangent space $T_q(\Sigma )$ satisfies the following:

(36)\begin{align} \det (\dot{\sigma})&=1\quad \text{ if $\sigma$ fixes the two branches at $q$,}\nonumber\\ &=-1\quad \text{ if $\sigma$ exchanges the two branches.} \end{align}

We say that $\Gamma$ acts stably on $\Sigma$ if it acts stably on each of its nodes.

From now on, by a node we will always mean a simple node.

Assume that $p\in \bar {\Sigma }$ is a node (possibly among other nodes) and also assume that the fiber $\pi ^{-1}(p)$ consists of nodal points. Assume further that the action of $\Gamma$ at the points $q\in \pi ^{-1}(p)$ is stable. Observe that, in this case, since $p$ is assumed to be a node, any $\sigma \in \Gamma _q$ can not exchange the two branches at $q$ for otherwise the point $p$ would be smooth.

We fix a level $c\geq 1$.

Let $\bar {\Sigma }'$ be the curve obtained from $\bar {\Sigma }$ by the normalization $\bar {\nu }:\bar {\Sigma }'\to \bar {\Sigma }$ at only the point $p$. Thus, $\bar {\nu }^{-1}(p)$ consists of two smooth points $p',p''$ in $\bar {\Sigma }'$ and

\[ \bar{\nu}_{|\bar{\Sigma}'\backslash \{ p',p'' \} }:\bar{\Sigma}'\backslash \{ p',p'' \} \to \bar{\Sigma}\backslash \{p\} \]

is a biregular isomorphism. We denote the preimage of any point of $\bar {\Sigma }\backslash \{p\}$ in $\bar {\Sigma }'\backslash \{ p',p'' \}$ by the same symbol. Let $\pi ' : {\Sigma }' \to \bar {\Sigma }'$ be the pull-back of the Galois cover $\pi$ via $\bar {\nu }$. In particular, $\pi '$ is a Galois cover with Galois group $\Gamma$. Thus, we have the following fiber diagram.

Proof. Let $q$ be any point in $\pi ^{-1}(p)$ of ramification index $e_q$. Since $\pi ^{-1}(p)$ consists of nodal points by assumption, there are two branches in the formal neighborhood of $q$. If any $\sigma \in \Gamma _q$ exchanges two branches, then the point $p=\pi (q)$ is smooth in $\bar {\Sigma }$, which contradicts the assumption that $p$ is a nodal point. Thus, $\Gamma _q$ must preserve branches. In particular, since no non-trivial element of $\Gamma$ fixes pointwise any irreducible component of $\Sigma$, $\Gamma _q$ is cyclic. Therefore, by the condition (36), we can choose a formal coordinate system $z',z''$ around the nodal point $q$ such that $\hat {\mathscr {O}}_{{\Sigma }, q }\simeq \mathbb {C}[[z',z'' ]]/(z'z'')$, and a generator $\sigma _q$ of $\Gamma _q$ such that

\[ \sigma_q(z')=\epsilon^{-1}z' \quad\text{and}\quad \sigma_q(z'')=\epsilon z'', \]

where $\epsilon := e^{ {2\pi i}/{e_q}}$ is the standard primitive $e_q$th root of unity. (Observe that $\epsilon$ must be a primitive $e_q$th root of unity, since $\Gamma _q$ acts faithfully on each of the two formal branches through $q$.)

We can choose a formal coordinate system $x'$, $x''$ around $p$ in $\bar {\Sigma }$ such that $\hat {\mathscr {O}}_{{\bar {\Sigma }}, p }\simeq \mathbb {C}[[ x',x'' ]]/(x'x'')$ and the embedding $\hat {\mathscr {O}}_{{\bar {\Sigma }}, p } \hookrightarrow \hat {\mathscr {O}}_{{\Sigma }, q }$ is given by $x'\mapsto (z')^{e_q}, x''\mapsto (z'')^{e_q}$.

The node $p$ splits into two smooth points $p',p''$ via $\bar {\nu }$. Without loss of generality, we can assume $x'$ (respectively, $x''$) is a formal coordinate around $p'$ (respectively, $p''$) in $\bar {\Sigma }'$. Then, $q$ will also split into two smooth points $q',q''$ via the map $\nu$, where $z'$ (respectively, $z''$) is a formal coordinate around $q'$ (respectively, $q''$). It shows that the map $\nu$ is a normalization at every point $q\in \pi ^{-1}(p)$.

The pullback gives a decomposition

\[ (\pi\circ \nu)^{-1} (p)= \nu^{-1}(\pi^{-1}(p) )=\pi'^{-1}(p')\sqcup \pi'^{-1}(p'') . \]

From the definition of the fiber product, there exist $\Gamma$-equivariant canonical bijections:

(37)\begin{equation} \pi'^{-1}(p')\simeq \pi^{-1}(p) \quad \text{and}\quad \pi'^{-1}(p'')\simeq \pi^{-1}(p). \end{equation}

Hence, we get a $\Gamma$-equivariant canonical bijection $\kappa : \pi '^{-1}(p')\simeq \pi '^{-1}(p'')$. For any $q\in \pi ^{-1}(p)$, $\nu ^{-1}(q)=\{q',q''\}$. By the choice of $q',q''$ as above, $\pi '(q')=p'$ and $\pi '(q'')=p''$. Therefore, $\kappa$ maps $q'$ to $q''$. Moreover, from (37), the stabilizer groups $\Gamma _q$, $\Gamma _{q'}$, and $\Gamma _{q'}$ are all the same (and of order $e_q$). Since $q'$ (respectively, $q''$) is a smooth point of $\Sigma '$, $\Gamma _{q'}$ (respectively, $\Gamma _{q''}$) is cyclic.

Let $\mathfrak {g}_p$ denote the Lie algebra $\mathfrak {g}[\pi ^{-1}(p)]^\Gamma$ (observe that we can attach a Lie algebra $\mathfrak {g}_p$ regardless of the smoothness of $p$). Then, the $\Gamma$-equivariant bijections $\nu : \pi '^{-1}(p')\simeq \pi ^{-1}(p)$ and $\nu : \pi '^{-1}(p'')\simeq \pi ^{-1}(p)$ (cf. (37)) induce isomorphisms of Lie algebras $\varkappa ': \mathfrak {g}_{p'}\simeq \mathfrak {g}_p$ and $\varkappa '': \mathfrak {g}_{p''}\simeq \mathfrak {g}_{p}$, respectively. Recall that $p', p''$ are smooth points of $\bar {\Sigma }'$. Let $D_{c,p'}$ (respectively, $D_{c,p''}$) denote the finite set of highest weights of irreducible representations of $\mathfrak {g}_p$ induced via the isomorphism $\varkappa '$ (respectively, $\varkappa ''$) which give rise to integrable highest weight $\hat {\mathfrak {g}}_{p'}$-modules (respectively, $\hat {\mathfrak {g}}_{p''}$-modules) with central charge $c$.

Set

\[ {\Sigma}^o={\Sigma}\backslash \pi^{-1}(\vec{o})\quad \text{and}\quad {\Sigma}^{\prime o}={\Sigma}'\backslash \pi'^{-1}(\vec{o}). \]

The map ${\nu }$ on restriction gives rise to an isomorphism

\[ \nu: {\Sigma}^{\prime o} \backslash \pi'^{-1}\{ p',p''\}\simeq \Sigma^o\backslash \pi^{-1} (p)\hookrightarrow \Sigma^o, \]

which, in turn, gives rise to a Lie algebra homomorphism

\[ \nu^*: \mathfrak{g}[ {\Sigma}^o]^\Gamma \to \mathfrak{g}[{\Sigma}^{\prime o} \backslash \pi'^{-1}\{ p',p''\}]^\Gamma. \]

Let $\vec {\lambda }=(\lambda _1, \ldots, \lambda _s)$ be an $s$-tuple of weights with $\lambda _i\in D_{c, o_i}$ ‘attached’ to $o_i$. We denote the highest weight of the dual representation $V(\mu )^*$ of $\mathfrak {g}_p$ by $\mu ^*$, thus $V(\mu )^* \simeq V(\mu ^*)$.

By Lemma 5.2, there exists a canonical bijection $\kappa : \pi '^{-1}(p')\simeq \pi '^{-1}(p'')$ compatible with the action of $\Gamma$. Thus, it induces an isomorphism of Lie algebras $\mathfrak {g}_{p'}\simeq \mathfrak {g}_{p''}$.