# The centre of the modular affine vertex algebra
at the critical level

Tomoyuki Arakawa, Lewis Topley and Juan J. Villarreal  Department of
Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom
Research Institute for Mathematical Sciences, Kyoto University,
Yoshidahonmachi, Sakyo Ward, Kyoto, 606-8501, Japan
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

A remarkable theorem of Feigin and Frenkel states that, over the complex
numbers, the centre of the universal affine vertex algebra at the critical
level is an infinite rank polynomial algebra, canonically identified with the
algebra of regular functions on the space of opers on the disk. The first
author and W. Wang observed that in positive characteristics, the universal
affine vertex algebra contains a large central subalgebra known as the
$p$-centre. They conjectured that the centre should be generated by the
Feigin–Frenkel centre and the $p$-centre, analogous to Veldkamp’s theorem. In
this paper we prove the conjecture for classical simple Lie algebras for $p$
larger than the Coxeter number, and for exceptional Lie algebras in large
characteristics.

Dedicated to Victor Kac on the occasion of his 80th birthday.

##  1\. Introduction

Let $\mathbbm{k}$ be an algebraically closed field of positive characteristic
$p$. Let $G$ be a connected, simply connected, simple algebraic group scheme
such that $p$ larger than the Coxeter number. Under these assumption,
$\mathfrak{g}=\operatorname{Lie}G$ is a simple Lie algebra, and the normalised
Killing form $\kappa$ is non-vanishing on $\mathfrak{g}$.

Understanding the geometry of the centre $Z(\mathfrak{g})$ of the enveloping
algebra $U(\mathfrak{g})$ is fundamental to describing the representation
theory of $\mathfrak{g}$. For example the maximal dimension of simple
$\mathfrak{g}$-modules is equal to the generic rank of $U(\mathfrak{g})$ over
$Z(\mathfrak{g})$ [Za, Theorem 6], and the smooth locus of
$\operatorname{Spec}Z(\mathfrak{g})$ coincides with the Azumaya locus [BG].
These discoveries underpinned the derived localisation theorem [BMR] which
relates the derived category of $\mathfrak{g}$-modules (with fixed/generalised
central characters) to the derived category of coherent sheaves on a Springer
fibre. The starting point of such a study is describing $Z(\mathfrak{g})$
algebraically.

On one hand, there is the subalgebra $U(\mathfrak{g})^{G}\subseteq
Z(\mathfrak{g})$ which is isomorphic to the algebra of Weyl group invariants
$U(\mathfrak{h})^{W}$ under the twisted Harish–Chandra homomorphism. We call
this the Harish–Chandra centre. On the other hand, there is a large central
subalgebra $Z_{p}(\mathfrak{g})$ which is specific to the modular setting,
known as the $p$-centre. This is best understood as a filtered lifting of the
central subalgebra $\mathbbm{k}[\mathfrak{g}^{*}]^{p}$ of the Poisson algebra
$\mathbbm{k}[\mathfrak{g}^{*}]$. The spectrum of $Z_{p}(\mathfrak{g})$
naturally identifies with the Frobenius twist $(\mathfrak{g}^{*})^{(1)}$.
Veldkamp’s theorem [Ve] states that these two subalgebras generate the centre
$Z(\mathfrak{g})$ and so

(1.1)
$\displaystyle\operatorname{Spec}Z(\mathfrak{g})\overset{\sim}{\longrightarrow}\mathfrak{h}^{*}/W_{\bullet}\times_{(\mathfrak{h}^{*}/W)^{(1)}}(\mathfrak{g}^{*})^{(1)}$

where $W_{\bullet}$ denotes the $\rho$-shifted action (see [BMR, 3.1.6] for
more details). Our goal is to give a similar description of the centre of the
universal affine vertex algebra at the critical level.

The Kac–Moody affinisation $\widehat{\mathfrak{g}}$ of
$\mathfrak{g}=\operatorname{Lie}(G)$ is the non-split central extension of the
loop Lie algebra $\mathfrak{g}\otimes\mathbbm{k}(\\!(t)\\!)$ afforded by
$\kappa$. The vacuum module $V^{k}(\mathfrak{g})$ at level $k$ is the
generalised Verma module induced from a maximal parabolic subalgebra of
$\widehat{\mathfrak{g}}$, and it carries the structure of a vertex algebra
[Bo1]. The level $k=-h^{\vee}$ is called the critical level and the critical
level representation theory is significantly more complicated and more
interesting. As usual, $h^{\vee}$ denotes the dual Coxeter number.

For the moment we briefly leave the modular setting to discuss some features
of the representation theory of $V^{k}(\mathfrak{g})$ over the complex
numbers. Here there is a well-known dichotomy between the critical and non-
critical levels. The differences between these cases are quite stark. The
multiplicity of simple hight weight modules in a Verma module is described by
the Kazhdan-Lusztig polynomials/the inverse Kazhdan-Lusztig polynomials
associated with the affine Weyl groups at the non-critical levels [KT], while
at the critical level the multiplicity of simple hight weight modules in a
restricted Verma module [AF1] is described by the periodic Kazhdan-Lusztig
polynomials [Lus, AF2, FG]. The primary feature which causes these differences
is the nature of the centre $\mathfrak{z}V^{k}(\mathfrak{g})$ [FF]. The centre
$\mathfrak{z}V^{k}(\mathfrak{g})$ is trivial away from the critical level (a
direct consequence of the conformal grading) whilst at the critical level
$\mathfrak{z}V^{-h^{\vee}}(\mathfrak{g})$ is an infinite rank polynomial ring,
canonically identified with the algebra of regular functions on the moduli
space of certain principle bundles on the formal disk, known as opers [BD]. We
remark that the non-negative part of the loop group acts on
$V^{-h^{\vee}}(\mathfrak{g})$ and the centre in the complex setting is equal
to $V^{-h^{\vee}}(\mathfrak{g})^{G[[t]]}$.

Returning to our field $\mathbbm{k}$ of positive characteristic, the study of
vertex algebra in positive characteristic was initiated in [BR, Bo2] and has
been developed in e.g. [DR1, DR2, Mu, AW, LM, JLM, W, HJ]. However, the
representation theory of $V^{k}(\mathfrak{g})$ is much less established
compared to that over $\mathbb{C}$.

In [AW] the first author and Weiqiang Wang introduced the theory of baby
Wakimoto modules. One of the basic observations is that the restricted
structure on $\widehat{\mathfrak{g}}$ leads to a large central subalgebra
$\mathfrak{z}_{p}V^{k}(\mathfrak{g})\subseteq V^{k}(\mathfrak{g})$, also
dubbed the $p$-centre. This subalgebra is naturally isomorphic to the
coordinate ring $\mathbbm{k}[(J_{\infty}\mathfrak{g}^{*})^{(1)}]$ on the
Frobenius twist of the arc space of $\mathfrak{g}^{*}$.

The natural way to understand a non-commutative algebra is to first understand
the semi-classical limit. The Poincaré–Birkoff–Wiit (PBW) filtration of the
enveloping algebra $U(\widehat{\mathfrak{g}})$ gives rise to a filtration of
the vertex algebra $V^{k}(\mathfrak{g})$ and the associated graded algebra is
the classical affine vertex algebra [FB, §16.2]. In Section 3.3 we explain the
minor differences involved in the definition of a Poisson vertex algebra
whilst working in positive characteristic.

The semiclassical limit $\operatorname{gr}V^{k}(\mathfrak{g})$ is
$G[[t]]$-equivariantly identified with the coordinate ring
$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]$ of the arc space on
$\mathfrak{g}^{*}$. Our first main theorem describes the centre of this
Poisson vertex algebra.

######  Theorem 1.1.

Let $G$ be a simply connected, connected simple group scheme over a field
$\mathbbm{k}$ of characteristic $p$ larger than the Coxeter number of $G$.
Then the following holds

  1. (1)

$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{G[[t]]}$ polynomial ring on
infinitely many generators which we describe explicitly (4.7). The subalgebra
$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{p}$ of $p$th powers is also central.

  2. (2)

The centre of $\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]$ is generated by
$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{G[[t]]}$ and
$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{p}$.

  3. (3)

$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{\mathfrak{g}[[t]]}$ is a free
$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{p}$-module and the basis can be
describe explicitly (4.8).

  4. (4)

$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{\mathfrak{g}[[t]]}$ is isomorphic to
the tensor product of $\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{p}$ and
$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{G[[t]]}$ over the intersection.

######  Remark 1.2.

Theorem 1.1 can be proven using our methods under the assumption that $p$ is a
very good prime for $G$. This means that $p$ is good for $G$ (see [Ja1, §6.4]
and that $p\nmid n+1$ in type An. We have worked under the assumption $p>h$
for simplicity, in order to maintain uniform hypotheses throughout the paper.

In the final section of this paper we deduce a description of the centre of
the vertex algebra $V^{-h^{\vee}}(\mathfrak{g})$ at the critical level. Our
result is an analogue of Veldkamp’s theorem.

######  Theorem 1.3.

Suppose that we are in one of the following two situations:

  1. (i)

$\mathfrak{g}$ is a classical simple Lie algebra, and $p$ larger than the
Coxeter number,

  2. (ii)

$\mathfrak{g}$ is an exceptional simple Lie algebra and $p\gg 0$.

Then we have:

  1. (1)

$V^{-h^{\vee}}(\mathfrak{g})^{G[[t]]}$ is a free commutative differential
algebra generated by $r$ elements.

  2. (2)

$\mathfrak{z}V^{-h^{\vee}}(\mathfrak{g})$ is generated by
$V^{-h^{\vee}}(\mathfrak{g})^{G[[t]]}$ and
$\mathfrak{z}_{p}V^{-h^{\vee}}(\mathfrak{g})$ as a commutative algebra.

  3. (3)

$\mathfrak{z}V^{-h^{\vee}}(\mathfrak{g})$ is a free
$\mathfrak{z}_{p}V^{-h^{\vee}}(\mathfrak{g})$-module of finite rank over
$\mathfrak{z}_{p}V^{-h^{\vee}}(\mathfrak{g})$.

  4. (4)

$\mathfrak{z}V^{-h^{\vee}}(\mathfrak{g})$ is isomorphic to the tensor product
of $\mathfrak{z}_{p}V^{-h^{\vee}}(\mathfrak{g})$ and
$V^{-h^{\vee}}(\mathfrak{g})^{G[[t]]}$ over their intersection.

Let
$\mathcal{X}^{-h^{\vee}}:=\operatorname{Spec}V^{-h^{\vee}}(\mathfrak{g})^{G[[t]]}$.
From Theorem 1.3 we deduce a geometric description of the centre

(1.2)
$\displaystyle\operatorname{Spec}\mathfrak{z}V^{-h^{\vee}}(\mathfrak{g})\overset{\sim}{\longrightarrow}\mathcal{X}^{-h^{\vee}}\times_{(J_{\infty}\mathfrak{h}^{*}/W)^{(1)}}(J_{\infty}\mathfrak{g}^{*})^{(1)}$

The map
$(J_{\infty}\mathfrak{g}^{*})^{(1)}\to(J_{\infty}\mathfrak{h}^{*}/W)^{(1)}$ is
the Frobenius twist of the jet morphism of the adjoint quotient
$\mathfrak{g}^{*}\to\mathfrak{h}^{*}/\\!/W$, whilst
$\mathcal{X}^{-h^{\vee}}\to(J_{\infty}\mathfrak{h}^{*}/W)^{(1)}$ arises from
the natural maps
$\mathbbm{k}[(J_{\infty}\mathfrak{g}^{*})^{(1)}]^{G[[t]]}\cong\mathfrak{z}_{p}V^{-h^{\vee}}(\mathfrak{g})^{G[[t]]}\to
V^{-h^{\vee}}(\mathfrak{g})$. This should be compared with (1.1).

Theorem 1.3 is deduced from Theorem 1.1 using a process of reduction modulo
$p$. We make use of explicit formulas for the generators of
$V^{-h^{\vee}}(\mathfrak{g})$ discovered over $\mathbb{C}$ by Molev [Mo1,
Mo2], in classical types. For exceptional types we use a rather primitive form
of modular reduction, depending on the Feigin-Frenkel theorem [FF].

We briefly describe the structure of the paper. Section 2 surveys the required
theory of group schemes, arcs and jets, and sets up our conventions for
defining the Kac-Moody affinisation. We also recall the precise statement of
Veldkamp’s theorem and its semiclassical limit.

In Section 3 we recall the definition of a Poisson vertex algebra and vertex
algebra over $\mathbbm{k}$, paying special attention to the minor differences
to the complex setting, which is better understood. We explain that the vertex
algebra $V^{k}(\mathfrak{g})$ naturall degenerates to
$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]$ and we construct the basic
$G[[t]]$-invariants which freely generate
$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{G[[t]]}$. Next we outline the
procedure of modular reduction for invariants in vertex algebras. Finally we
prove Theorem 1.1 following an approach analogous to [Fr], adapting to the
modular setting using a powerful theorem of Skryabin [Sk].

In the final Section 5 we prove Theorem 1.3 using the tools of modular
reduction and Molev’s formulas [Mo1, Mo2].

We expect that Theorem 1.3 will hold under the assumption that $p$ is very
good for $G$. We note that the methods this paper are already sufficient to
prove the theorem in type G2 for $p>3$: Theorem 1.1 holds under this
hypothesis (Remark 1.2) whilst the formulas for Segal–Sugawara vectors
discovered by Yakimova in [Ya, §6] can be defined over a
$\mathbb{Z}[6^{-1}]$-lattice in $V^{-h^{\vee}}(\mathfrak{g})$, and so the
Theorem follows in this case, following the observations of Section 5.3
verbatim. Similar remarks hold for $\mathfrak{sl}_{N}$ when $p\nmid N$, using
the formulas of [MC], and the remarks of Section 5.2.

### Acknowledgements

We would like to thank Dmitriy Rumynin and Alexander Molev for useful
correspondence on the subject of this paper. The first author is partially
supported by JSPS KAKENHI Grant Number J21H04993. The second and third author
are supported by a UKRI Future Leaders Fellowship, grant numbers grant numbers
MR/S032657/1, MR/S032657/2, MR/S032657/3.

##  2\. Group schemes, jets and arcs

###  2.1. Arcs and jets

Throughout this paper we fix an algebraically closed field $\mathbbm{k}$ of
characteristic $p>0$. All vector spaces, algebras and schemes will be defined
over $\mathbbm{k}$, with the exception of Section 3.7 where we discuss modular
reduction.

Here we discuss jet schemes and arc spaces. We refer the reader to [Is] for
slightly more detail.

A higher derivation of a commutative algebra $A$ is a tuple
$\partial=(\partial^{(i)}\mid i\geq 0)$ of linear endomorphisms of $A$ with
$\partial^{(0)}$ equal to the identity, satisfying the following relations for
$i\geq 0$:

(2.3) $\displaystyle\begin{array}[]{l}\partial^{(i)}(1)=0;\vspace{4pt}\\\
\partial^{(i)}(ab)=\sum_{i_{1}+i_{2}=i}\partial^{(i_{1})}(a)\partial^{(i_{2})}(b).\end{array}$

Similarly a higher derivation of order $m$ is a tuple
$\partial=(\partial^{(i)}\mid i=0.,,,.m)$ satisfying (2.3) for $i,j\leq m$. A
differential algebra (of order $m$) $(A,\partial)$ is an algebra equipped with
a higher derivation (of order $m$).

There is a universal differential algebra $(J_{\infty}A,\partial)$ with a
homomorphism $\alpha:A\to J_{\infty}A$ such that, for every other differential
algebra $(B,\sigma)$ admitting a homomorphism $\beta:A\to B$, there is a
unique homomorphism $\varphi$ of differential algebras such that
$\varphi\circ\alpha=\beta$.

We construct $J_{\infty}A$ as follows. First assume $A=\mathbbm{k}[x_{k}\mid
k\in K]$ is a polynomial ring, where $K$ is an index set. Define
$J_{\infty}A:=\mathbbm{k}[x_{k}^{(j)}\mid k\in K,\ 0\leq j]$. The homomorphism
$\alpha:A\to J_{\infty}A$ is given by $x_{k}\mapsto x_{i}^{(0)}$ and the
differential $\partial$ is determined uniquely by (2.3) along with
$\partial^{(i)}x_{k}^{(0)}=x_{k}^{(i)}$ and

$\partial^{(i)}\partial^{(j)}=\binom{i+j}{i}\partial^{(i+j)}$

for all $i,j,k$ in the appropriate ranges. If $I\subseteq A$ is an ideal then
we can define $J_{\infty}I:=(\partial^{(i)}a\mid 0\leq i,\ a\in I)$, which is
a differential ideal of $J_{\infty}A$.

Now if $A$ has a presentation $A=\mathbbm{k}[x_{k}\mid k\in K]/I$ then we
define $J_{\infty}A:=J_{\infty}\mathbbm{k}[x_{k}\mid k\in K]/J_{\infty}I$. The
map $A\to J_{\infty}A$ and higher derivation $\partial$ are defined as above,
and $A\to J_{\infty}A$ is readily seen to inject. It is not hard to see that
the isomorphism type of $J_{\infty}A$ is independent the presentation, and
that $J_{\infty}A$ satisfies the universal property mentioned above. It
follows immediately that $J_{\infty}$ is an endofunctor on the category of
commutative algebras, and $J_{\infty}A$ is known as the arc algebra of $A$.

If $A$ is a Hopf algebra with comultiplication $\delta:A\to A\otimes A$,
counit $\eta:A\to\mathbbm{k}$ and antipode $S:A\to A$, then there is a natural
Hopf structure on $J_{\infty}A$ extending the structure on $A$. The existence
of such a structure follows from functoriality of $J_{\infty}$, however we can
describe it explicitly: for $a\in A\subseteq J_{\infty}A$ we have

(2.7)
$\displaystyle\begin{array}[]{l}(\Delta(\partial^{(i)}a))=\partial^{(i)}\Delta(a);\\\
S(\partial^{(i)}(a))=\partial^{(i)}S(a);\\\
\eta(a)=a+J_{m}\operatorname{Ker}\eta.\end{array}$

where we equip $A\otimes A$ with the structure of a differential algebra in
the obvious manner, and note that
$J_{\infty}A/J_{\infty}\operatorname{Ker}\eta\cong
J_{\infty}\mathbbm{k}=\mathbbm{k}$.

For $m\in\mathbb{N}$ there is an analogous construction of a universal
differential algebra $J_{m}A$ of order $m$. Briefly,
$J_{m}\mathbbm{k}[x_{k}\mid k\in K]$ is the subalgebra of
$J_{\infty}\mathbbm{k}[x_{k}\mid k\in K]$ generated by $\\{x_{k}^{(j)}\mid
k\in K,0\leq j\leq m\\}$ with differential defined as before, along with the
condition $\partial^{(i)}x_{k}^{(j)}=0$ for $i+j>m$. We have
$J_{\infty}A=\varinjlim J_{m}A$ as algebras.

If $X$ is an affine scheme then the $m$-jet scheme $J_{m}X$ is defined to be
$J_{m}X=\operatorname{Spec}J_{m}A$ and the arc space is
$\operatorname{Spec}J_{\infty}X$. These functors behave well under gluing and
thus $J_{m}$ is a functor from schemes to schemes. If $X$ has finite type then
so do $J_{m}X$ for all $m\in\mathbb{N}$, whilst $J_{m}X$ is of profinite type.

Jet schemes and arc spaces are uniquely determined by the adjunctions

$\displaystyle\operatorname{Hom}_{\operatorname{Sch}/\mathbbm{k}}(Y,J_{m}X):=\operatorname{Hom}_{\operatorname{Sch}/\mathbbm{k}}(Y\times_{\operatorname{Spec}(\mathbbm{k})}\operatorname{Spec}(\mathbbm{k}[t]/(t^{m+1})),X),$
$\displaystyle\operatorname{Hom}_{\operatorname{Sch}/\mathbbm{k}}(Y,J_{\infty}X):=\operatorname{Hom}_{\operatorname{Sch}/\mathbbm{k}}(Y\times_{\operatorname{Spec}(\mathbbm{k})}\operatorname{Spec}(\mathbbm{k}[[t]],X).$

and so the $\mathbbm{k}$-points of $J_{m}X$ are naturally identified with the
$\mathbbm{k}[t]/(t^{m+1})$-points of $X$ for $m\in\mathbb{N}$, and with the
$\mathbbm{k}[[t]]$-points of $X$ for $m=\infty$.

If $G$ is a group scheme then, by our above remarks, both $J_{m}G$ and
$J_{\infty}G$ are group schemes.

######  Lemma 2.1.

Suppose that $G$ is a group scheme which is generated by a collection
$(H_{k})_{k\in K}$ of subgroup schemes. Then $J_{\infty}G$ is generated by
$(J_{\infty}H_{k})_{k\in K}$.

###### Proof.

Let $A$ be a commutative Hopf algebra. Observe that the collection of Hopf
ideals of $A$ is stable under sums, but not intersections. It follows that for
any algebra ideal $I\subseteq A$ there is a unique maximal Hopf ideal subject
to $H(I)\subseteq I$. More precisely it is sum of all Hopf ideals contained in
$I$.

We claim that for any ideal $I$ we have $J_{\infty}H(I)=H(J_{\infty}I)$. Using
properties (2.7) one can show that $J_{\infty}H(I)$ is a Hopf ideal of
$J_{\infty}A$ contained in $J_{\infty}I$, and so $J_{\infty}H(I)\subseteq
H(J_{\infty}I)$. Conversely we have $H(I)\subseteq I\subseteq J_{\infty}I$ and
so $H(I)\subseteq HJ_{\infty}(I)$. In order to obtain $J_{\infty}H(I)\subseteq
HJ_{\infty}(I)$ it suffices to show that $HJ_{\infty}(I)$ is closed under
$\partial$. Using (2.7) once again one can easily show that $H$ preserves the
class of differential-closed ideals, and this completes the proof of the
claim.

For $k\in K$ we let $I_{k}\subseteq\mathbbm{k}[G]$ be the Hopf ideal defining
the subgroup $H_{k}$. The hypothesis of the lemma is equivalent to the
assertion $H(\cap_{k}I_{k})=0$, and the goal is to show that
$H(\cap_{k}J_{\infty}I_{k})=0$. Now we have
$H(\cap_{k}J_{\infty}I_{k})=H(J_{\infty}\cap_{k}I_{k})=J_{\infty}H(\cap_{k}I_{k})=0$,
and the proof is complete. ∎

###  2.2. Simple group schemes

Let $G$ be a connected, simple group scheme of rank $r$, and write
$\mathfrak{g}=\operatorname{Lie}(G)$. The Coxeter number (resp. dual Coxeter
number) is denoted $h$ (resp. $h^{\vee}$).

For the rest of the paper we will assume that $p>h$ which has the following
consequences:

  1. (i)

$2h^{\vee}$ is invertible;

  2. (ii)

The Lie algebra $\mathfrak{g}$ is simple (see [Ka1, Proposition 1.2] for
example);

  3. (iii)

the space of $\mathfrak{g}$-invariant forms on $\mathfrak{g}$ is one
dimensional and every nonzero $\mathfrak{g}$-invariant form is non-degenerate,
by (ii);

  4. (vi)

The Killing form is non-vanishing, hence non-degenerate (see [SS, Theorem
4.8]).

Throughout this article we fix the normalised Killing form on $\mathfrak{g}$
which is defined by

(2.8)
$\displaystyle\kappa(x,y)=\frac{1}{2h^{\vee}}\operatorname{Tr}(\operatorname{ad}(x)\operatorname{ad}(y)).$

When we consider the induced bilinear form on $\mathfrak{h}^{*}$, the
normalisation ensures that the square length of every long root is equal to 2,
which is the standard convention in the literature [Fr, Ka2]. In classical
types this form is just the trace form $x,y\mapsto\operatorname{Tr}(xy)$
associated to the natural representation.

We fix a torus $T\subseteq G$, let $X^{*}(T)$ denote the character lattice,
and write $\Phi\subseteq X^{*}(T)$ for the roots. For $\alpha\in\Phi$ we
consider the morphism of group schemes corresponding to the $\alpha$-root
subgroup $u_{\alpha}:\mathbb{G}_{a}\to G$ (see [Mi, §21.c]).

######  Lemma 2.2.

  1. (1)

$G$ is generated by $\\{u_{\alpha}\mid\alpha\in\Phi\\}$.

  2. (2)

$J_{\infty}G$ is generated by $\\{J_{\infty}u_{\alpha}\mid\alpha\in\Phi\\}$.

###### Proof.

Part (1) is [Mi, Proposition 21.62], whilst part (2) follows from Lemma 2.1. ∎

###  2.3. Chevalley restriction and Veldkamp’s theorem

Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$,
equipped with the Poincaré–Birhoff–Witt (PBW) filtration
$U(\mathfrak{g})=\bigcup_{i\geq 0}U(\mathfrak{g})_{i}$. The associated graded
algebra is
$\operatorname{gr}U(\mathfrak{g})=S(\mathfrak{g})=\mathbbm{k}[\mathfrak{g}^{*}]$
as $G$-algebras.

Write $Z(\mathfrak{g})$ for the centre of $U(\mathfrak{g})$. Both $G$ and
$\mathfrak{g}$ act naturally on $U(\mathfrak{g})$ and
$\mathbbm{k}[\mathfrak{g}^{*}]$ via automorphisms extending the adjoint
representation, and we have the following obvious inclusions
$U(\mathfrak{g})^{G}\subseteq U(\mathfrak{g})^{\mathfrak{g}}=Z(\mathfrak{g})$
and
$\mathbbm{k}[\mathfrak{g}^{*}]^{G}\subseteq\mathbbm{k}[\mathfrak{g}^{*}]^{\mathfrak{g}}$,
as well as
$\operatorname{gr}U(\mathfrak{g})^{G}\subseteq\mathbbm{k}[\mathfrak{g}^{*}]^{G}$
and
$\operatorname{gr}Z(\mathfrak{g})\subseteq\mathbbm{k}[\mathfrak{g}^{*}]^{\mathfrak{g}}$.

We write $\mathbbm{k}[\mathfrak{g}^{*}]^{p}=\\{f^{p}\mid
f\in\mathbbm{k}[\mathfrak{g}^{*}]\\}$ for the subalgebra of $p$th powers.
There is an obvious inclusion
$\mathbbm{k}[\mathfrak{g}^{*}]^{p}\subseteq\mathbbm{k}[\mathfrak{g}^{*}]^{\mathfrak{g}}$
since $\mathfrak{g}$ acts by derivations.

Recall that $\mathfrak{g}$ admits a natural $G$-equivariant restricted
structure $x\mapsto x^{[p]}$, which gives rise to a large central subalgebra
of $U(\mathfrak{g})$ known as the $p$-centre. It is generated by elements
$x^{p}-x^{[p]}$. Since the map $x\mapsto x^{p}-x^{[p]}$ is $p$-semilinear
$Z_{p}(\mathfrak{g})$ is a polynomial ring in $\dim(\mathfrak{g})$ generators,
and its spectrum is naturally identified as a $G$-set with the first Frobenius
twist of the coadjoint representation
$\operatorname{Spec}Z_{p}(\mathfrak{g})=(\mathfrak{g}^{*})^{(1)}$, and
$\operatorname{gr}Z_{p}(\mathfrak{g})=\mathbbm{k}[\mathfrak{g}^{*}]^{p}$ under
the identification
$\operatorname{gr}U(\mathfrak{g})=\mathbbm{k}[\mathfrak{g}^{*}]$; see [Ja1]
for more detail.

The next result, known as Veldkamp’s theorem, describes the centre of
$U(\mathfrak{g})$ for $p>h$; [Ve]. Numerous authors improved the bound on $p$
(see [Ja1, §9]), and in fact the theorem holds under the standard hypotheses
[BG, Theorem 3.5].

The statement presented here can easily be deduced from [BG, Section 3] and
[Sk, Theorem 5.4(1)].

######  Theorem 2.3.

We have the following:

  1. (1)

$\mathbbm{k}[\mathfrak{g}^{*}]^{G}$ is generated by $r$ algebraically
independent elements $P_{1},...,P_{r}$.

  2. (2)

$\mathbbm{k}[\mathfrak{g}^{*}]^{\mathfrak{g}}$ is generated by
$\mathbbm{k}[\mathfrak{g}^{*}]^{G}$ and $\mathbbm{k}[\mathfrak{g}^{*}]^{p}$

  3. (3)

$\mathbbm{k}[\mathfrak{g}^{*}]^{\mathfrak{g}}$ is a free
$\mathbbm{k}[\mathfrak{g}^{*}]^{p}$-module with basis given by restricted
monomials

$\\{P_{1}^{k_{1}}\cdots P^{k_{r}}_{r}\mid 0\leq k_{j}<p\text{ for all }j\\}.$

  4. (4)

$\mathbbm{k}[\mathfrak{g}^{*}]^{\mathfrak{g}}\cong\mathbbm{k}[\mathfrak{g}^{*}]^{G}\otimes_{(\mathbbm{k}[\mathfrak{g}^{*}]^{p})^{G}}\mathbbm{k}[\mathfrak{g}^{*}]^{p}$.

  5. (5)

The $P_{i}$ admit filtered lifts $Z_{1},...,Z_{r}$ and $U(\mathfrak{g})^{G}$
is a polynomial algebra on $r$ generators.

  6. (6)

$Z(\mathfrak{g})$ is generated by $U(\mathfrak{g})^{G}$ and
$Z_{p}(\mathfrak{g})$.

  7. (7)

$Z(\mathfrak{g})$ is a free $Z_{p}(\mathfrak{g})$-module of rank
$p^{\dim\mathfrak{g}}$, with basis given by restricted monomials

$\\{Z_{1}^{i_{1}}\cdots Z_{r}^{i_{r}}\mid 0\leq i_{j}<p\text{ for all }j\\}.$

  8. (8)

$Z(\mathfrak{g})\cong
U(\mathfrak{g})^{G}\otimes_{Z_{p}(\mathfrak{g})^{G}}Z_{p}(\mathfrak{g})$.
$\hfill\qed$

##  3\. Modular vertex algebras

###  3.1. Spaces of formal series

Let $V$ a $\mathbbm{k}$-vector space, and $V[z]$ (resp. $V[[z]]$, resp.
$V(\\!(z)\\!)$) the space of polynomials (resp. formal power series, resp.
formal Laurent series) in $z$ with coefficients in $V$. We warn the reader
that $V\otimes_{\mathbbm{k}}\mathbbm{k}[[z]]\subsetneq V[[z]]$ when $V$ is
infinite dimensional. These spaces are equipped with a higher derivation
$\partial_{z}=(\partial_{z}^{(i)})_{\geq 0}$ where $\partial_{z}^{(0)}$ is the
identity map and $\partial_{z}^{(k)}$ is defined by

(3.1) $\displaystyle\partial^{(i)}z^{m}=\binom{m}{i}z^{m-i}\text{ for
}i\in\mathbb{N},\,\,m\in\mathbb{Z}.$

The reader can easily check that (2.3) holds. Note that if
$i=i_{0}+i_{1}p+\cdots+i_{k}p^{k}$ is the $p$-adic expansion of
$i\in\mathbb{Z}_{>0}$ we have a factorisation

(3.2)
$\displaystyle\partial^{(i)}=(\partial)^{i_{0}}(\partial^{(p)})^{i_{1}}\cdots(\partial^{(p^{k})})^{i_{k}}.$

###  3.2. Modular vertex algebras

Vertex algebras have been widely studied over the complex numbers, and so to
emphasise the fact that our underlying field has positive characteristic we
call them modular vertex algebras. The definition was introduced by Borcherds
in [Bo].

######  Definition 3.1.

A modular vertex algebra is a $\mathbbm{k}$-vector space $V$ equipped with a
family of bilinear products

$\begin{split}&V\otimes V\rightarrow V\\\ &a\otimes b\rightarrow
a_{(n)}b\,,\quad n\in\mathbb{Z}\,,\end{split}$

and a vector ${\boldsymbol{1}}\in V$ such that

  1. (1)

$a_{(n)}b=0$ for $n\gg 0$,

  2. (2)

${\boldsymbol{1}}_{(n)}a=\delta_{n,-1}a$,

  3. (3)

$a_{(-1)}{\boldsymbol{1}}=a$,

  4. (4)

The following identity is satisfied

(3.3)
$\sum_{j\in\mathbb{Z}_{+}}(-1)^{j}\binom{n}{j}\left(a_{(m+n-j)}b_{(k+j)}-(-1)^{n}b_{(n+k-j)}a_{(m+j)})\right)=\sum_{j\in\mathbb{Z}_{+}}\binom{m}{j}(a_{(n+j)}b)_{(m+k-j)}\,.$

We also have the following equivalent definition.

######  Definition 3.2.

A modular vertex algebra is a $\mathbbm{k}$-vector space $V$, equipped with a
vector ${\boldsymbol{1}}\in V$, a family of endomorphism
$T^{(k)}\in\operatorname{End}(V)$, $k\in\mathbb{Z}_{>0}$, an a linear map

$Y\colon V\to\operatorname{Hom}_{\mathbbm{k}}(V,V(\\!(z)\\!))\,,\qquad
a\mapsto Y(a,z)\,,$

subject to the following axioms:

  1. (1′)

$Y({\boldsymbol{1}},z)=I$, $Y(a,z){\boldsymbol{1}}\in V[[z]]$,
$Y(a,z){\boldsymbol{1}}\big{|}_{z=0}=a$.

  2. (2′)

$Y(T^{(k)}a,z)=\partial^{(k)}_{z}Y(a,z)$ and $T^{(k)}{\boldsymbol{1}}=0$.

  3. (3′)

For every $a,b\in V$, there exists $N\in\mathbb{N}$ such that

$\begin{split}(z_{1}-z_{2})^{N}Y(a,z_{1})Y(b,z_{2})=(z_{1}-z_{2})^{N}Y(b,z_{2})Y(a,z_{1})\,.\\\
\end{split}$

The equivalence between the Definition 3.1 and Definition 3.2 is well-known
over $\mathbb{C}$ and is explained in detail in [Mas, Theorem 4.3] in case the
base is an arbitrary ring. We sketch the correspondence here. From Definition
3.1 we define $T^{(k)}\in\operatorname{End}(V)$ for $k\geq 1$ and
$Y:V\rightarrow\operatorname{End}(V)((z))$ as follows

$\begin{split}&Y(a,z)=\sum_{n\in\mathbb{Z}}a_{(n)}z^{-n-1}\,,\qquad
T^{(k)}(a):=a_{(-k-1)}{\boldsymbol{1}}\,,\qquad\text{ for }a\in V.\end{split}$

The conditions (1), (2) and (3) follow easily from (3.3). On the other hand,
using Definition 3.2 we can write
$Y(a,z)=\sum_{n\in\mathbb{Z}}a_{(n)}z^{-n-1}$ and the operators $a_{(i)}$
satisfy the required axioms.

######  Definition 3.3.

For a vertex algebra $V$ the centre is the subspace

$\mathfrak{z}(V):=\\{b\in V\mid a_{(n)}b=0\text{ for all }a\in V\text{ and
}n\geq 0\\}\,.$

Applying (3.3) with $n=0$ we see that the centre coincides with the set of
elements $v\in V$ such that

$[v_{(m)},a_{(k)}]=0\qquad\text{for all }a\in V$

for all $m,k\in\mathbb{Z}$.

###  3.3. Modular Poisson vertex algebras

A commutative vertex algebra $V$ is one such that $u_{(n)}v=0$ for all $u,v\in
V$ and $n\geq 0$. Equivalently it satisfies $Y(\bullet,z):V\to V[[z]]$. In
this case, the product $a,b\mapsto a_{(-1)}b$ is commutative, associative and
unital with unit ${\boldsymbol{1}}$. Moreover, the endomorphisms $T^{(k)}$
define a higher derivation of this commutative algebra (2.3). Moreover the
category of commutative differential algebras is equivalent to the category of
commutative vertex algebras. The key difference between characteristic zero
and the modular setting is that differential algebras over $\mathbb{C}$ are
equipped with a higher differential of order 1, as opposed to a higher
differential over $\mathbbm{k}$. All this was noted in Borcherds’ seminal work
[Bo1].

######  Definition 3.4.

Let $(V,{\boldsymbol{1}},T)$ be a commutative vertex algebra, with
$T=(T^{(k)})_{k\geq 0}$.

For a formal power series $a(z)=\sum_{m\in\mathbb{Z}}a_{m}z^{m}$ the polar
part of $a(z)$ is defined to be $a_{-}(z)=\sum_{m<0}a_{(m)}z^{m}$.

We say that $V$ is a Poisson vertex algebra if there is a linear map

(3.6)
$\displaystyle\begin{array}[]{rcl}Y_{-}&:&V\to\operatorname{Hom}(V,z^{-1}V[z^{-1}]),\\\
&&a\mapsto Y_{-}(a,z)=\sum_{n\geq 0}a_{(n)}z^{-n-1}\end{array}$

subject to the following axioms:

  1. (i)

$Y_{-}(T^{(k)}a,z)=\partial^{(k)}_{z}Y(a,z)$.

  2. (ii)

$Y_{-}(a,z)b=\left(\sum_{k\geq 0}z^{k}T^{(k)}Y(b,-z)a\right)_{-}$.

  3. (iii)

$[a_{(m)},Y_{-}(b,w)]=\sum_{k\geq
0}\binom{m}{k}\left(w^{m-k}Y_{-}(a_{(k)}b,w)\right)_{-}\,.$

  4. (iv)

$Y_{-}(a,w)(b\cdot c)=(Y_{-}(a,w)b)\cdot c+(Y_{-}(a,w)c)\cdot b$ .

We refer the reader to [FB, §16.2] for a good introduction.

Now suppose that $V$ is a graded vertex algebra $V=\bigoplus_{i\geq 0}V_{i}$.
Then we can define a filtration $V=\bigcup_{i\geq 0}F_{i}V$ by letting
$F_{d}V$ be the subspace spanned by

(3.7) $\displaystyle a_{-n_{1}-1}^{1}a_{-n_{2}-1}^{2}\cdots
a_{-n_{m}-1}^{m}{\boldsymbol{1}}$

where $a^{1},...,a^{m}\in V$ with $a^{i}\in V_{d_{i}}$ and
$\sum_{i=1}^{m}d_{i}\leq d.$

######  Theorem 3.5.

[Ar, §3.7] This filtration is compatible with the vertex algebra structure,
and the associated graded algebra $\operatorname{gr}V$ inherits the structure
of a Poisson vertex algebra.

###  3.4. Modular affine vertex algebras

Let $G$ be a connected, simple group scheme over $\mathbbm{k}$ such that the
characteristic of $\mathbbm{k}$ is very good for the root system. Recall that
$\kappa$ denotes the normalised Killing form (2.8) on
$\mathfrak{g}=\operatorname{Lie}(G)$.

The loop Lie algebra is
$\mathfrak{g}(\\!(t)\\!):=\mathfrak{g}\otimes\mathbbm{k}(\\!(t)\\!)$ and the
Kac–Moody affinization
$\widehat{\mathfrak{g}}:=\mathfrak{g}(\\!(t)\\!)\oplus\mathbbm{k}K$ of
$\mathfrak{g}$ is the central extension of $\mathfrak{g}(\\!(t)\\!)$ afforded
by $\kappa$. Note that $K\in\widehat{\mathfrak{g}}$ is central and the Lie
brackets in $\widehat{\mathfrak{g}}$ are determined by
$[xt^{i},yt^{j}]=[x,y]t^{i+j}+\delta_{i,-j}\kappa(x,y)iK$. We use the standard
notation $xt^{i}:=x_{i}$ which may denote either an element or an endomorphism
of $V^{k}(\mathfrak{g})$.

Let $\mathbbm{k}_{k}$ be the one dimensional representation of
$\widehat{\mathfrak{g}}_{+}:=\mathfrak{g}[[t]]\oplus\mathbbm{k}K$ upon which
$\mathfrak{g}[[t]]$ acts trivially and $K$ acts by $k$. The vacuum module for
$\widehat{\mathfrak{g}}$ is the induced module

(3.8)
$V^{k}(\mathfrak{g}):=U(\widehat{{\mathfrak{g}}})\otimes_{\widehat{\mathfrak{g}}_{+}}\mathbbm{k}_{k},$

and $k$ is known as the level of $V^{k}(\mathfrak{g})$.

Equivalently we can define $V^{k}(\mathfrak{g})$ to be the quotient of
$U(\widehat{\mathfrak{g}})$ by the left ideal
$U(\widehat{\mathfrak{g}})\mathfrak{g}_{+}^{k}$ where
$\mathfrak{g}_{+}^{k}:=\mathfrak{g}[[t]]\oplus\mathbbm{k}(K-k)$. We note that
we have a natural adjoint action of $\mathfrak{g}[[t]]$ on
$\widehat{\mathfrak{g}}$, since
$\mathfrak{g}[[t]]\subseteq\widehat{\mathfrak{g}}$ is a subalgebra. This
extends to an action on $U(\widehat{\mathfrak{g}})$ by derivations,
stabilising the left ideal
$U(\widehat{\mathfrak{g}})\widehat{\mathfrak{g}}_{+}^{k}$. The induced adjoint
action on $V^{k}(\mathfrak{g})$ coincides with the left module structure.

The vacuum module carries a canonical structure of modular vertex algebra
[Ka2]. If we pick and ordered basis $\\{x^{i}\mid i=1,...,\dim\mathfrak{g}\\}$
for $\mathfrak{g}$, then the Poincare–Birkhoff–Witt (PBW) theorem implies that
we have a basis given by ordered monomials

(3.9) $x^{i_{1}}_{-n_{1}-1}\cdots x^{i_{m}}_{-n_{m}-1}{\boldsymbol{1}}$

such that $0\leq n_{1}\leq n_{2}\leq\cdots\leq n_{m}$, and $i_{j}\leq i_{j+1}$
whenever $n_{j}=n_{j+1}$. The vacuum vector in $V^{k}(\mathfrak{g})$ is
${\boldsymbol{1}}=1\otimes 1$. For $k\in\mathbb{N}$ the translation operator
$T^{(k)}$ is determined by $T^{(k)}{\boldsymbol{1}}=0$ and

(3.10) $\displaystyle
T^{(k)}x^{i}_{-n-1}{\boldsymbol{1}}=\binom{n+k}{k}x^{i}_{-n-k-1}{\boldsymbol{1}}$

whilst in general, $T^{(k)}$ acts on $V^{k}(\mathfrak{g})$ by

(3.11) $\displaystyle T^{(k)}(x^{i_{1}}_{n_{1}}\cdots
x^{i_{m}}_{n_{m}}{\boldsymbol{1}})=\sum_{j_{1}+\cdots+j_{m}=m}T^{(j_{1})}(x^{i_{1}}_{n_{1}})\cdots
T^{(j_{m})}(x^{a_{m}}_{n_{m}}){\boldsymbol{1}}\,.$

The fields are defined as follows. We have
$Y({\boldsymbol{1}},z)=\operatorname{Id}_{V^{k}(\mathfrak{g})}$ and

$Y(x^{i}_{-1}{\boldsymbol{1}},z)=x^{i}(z):=\sum_{n\in\mathbb{Z}}x^{i}_{n}z^{-n-1},$

whilst in general we have

(3.12) $Y(x^{i_{1}}_{-n_{1}-1}\cdots
x^{i_{m}}_{-n_{m}-1}{\boldsymbol{1}},z)=:\partial_{z}^{(n_{1})}x^{i_{1}}(z)\cdots\partial_{z}^{(n_{m})}x^{i_{m}}(z):$

where $:\ \ :$ denotes the normally ordered product (see [Ka2]). The locality
of the fields is a consequence of the explicit description of Lie brackets in
$\widehat{\mathfrak{g}}$.

In this article we are interested in describing the centre
$\mathfrak{z}(V^{k}({\mathfrak{g}}))$. Consider the space of
$\mathfrak{g}[[t]]$ invariants

$V^{k}({\mathfrak{g}})^{\mathfrak{g}[[t]]}=\\{b\in
V^{\kappa}({\mathfrak{g}})\mid(x_{n})\/b=0\text{ for }n\geq 0\\}\,.$

The following useful property follows from (3.12); see [Fr, Lemma 3.1.1] for
more detail.

######  Lemma 3.6.

$\mathfrak{z}V^{k}(\mathfrak{g})=V^{k}({\mathfrak{g}})^{{\mathfrak{g}}[[t]]}.$
$\hfill\qed$

###  3.5. The $p$-centre of modular affine vertex algebras

We introduce a central subalgebra of $V^{k}({\mathfrak{g}})$, following [AW].
The restricted structure on $\mathfrak{g}$ gives rise to a natural restricted
structure on $\widehat{\mathfrak{g}}$ determined by
$(xt^{i})^{[p]}=x^{[p]}t^{pi}$ and $K^{[p]}=K$. We denote the $p$-centre of
$U(\widehat{\mathfrak{g}})$ by $Z_{p}(\widehat{\mathfrak{g}})$, which is
generated by $\\{x^{p}-x^{[p]}\mid x\in\widehat{\mathfrak{g}}\\}$. It is
isomorphic to a polynomial ring in countably many generators.

We define the $p$-centre $\mathfrak{z}_{p}V^{k}(\mathfrak{g})\subseteq
V^{k}(\mathfrak{g})$ to be the image of the natural map
$Z_{p}(\widehat{\mathfrak{g}})\to U(\widehat{\mathfrak{g}})\to
V^{k}(\mathfrak{g})$. Note that
$\mathfrak{z}_{p}V^{k}({\mathfrak{g}})\subseteq
V^{k}({\mathfrak{g}})^{{\mathfrak{g}}[[t]]}$, and so Lemma 3.6 implies the
following result.

######  Lemma 3.7.

$\mathfrak{z}_{p}V^{\kappa}({\mathfrak{g}})\subseteq\mathfrak{z}V^{\kappa}({\mathfrak{g}}).$
$\hfill\qed$

###  3.6. The current group and the centre

The current group $G[[t]]$ of $\mathbbm{k}[[t]]$-points of $G$ is equal to the
group of $\mathbbm{k}$-points of $J_{\infty}G$, by definition. Let $\epsilon$
be a formal variable satisfying $\epsilon^{2}=0$. The Lie algebra
$\operatorname{Lie}G[[t]]$ of $G[[t]]$ is defined to be the kernel of the
homomorphism $G(\mathbbm{k}[[t]][\epsilon])\to G(\mathbbm{k}[[t]])$ coming
from the map $\mathbbm{k}[[t]][\epsilon]\to\mathbbm{k}[[t]]$ by functoriality.
By [Mi, p. 192] we have $\operatorname{Lie}G[[t]]=\mathfrak{g}[[t]]$.

######  Lemma 3.8.

For any $k\in\mathbbm{k}$ there is a natural rational $G[[t]]$-action on
$V^{k}(\mathfrak{g})$ such that the differential at the identity element is
the left action of $\mathfrak{g}[[t]]$.

###### Proof.

Recall that $\operatorname{ad}\mathfrak{g}[[t]]$ preserves
$\widehat{\mathfrak{g}}_{+}^{k}$ and the adjoint action of $\mathfrak{g}[[t]]$
on $U(\widehat{\mathfrak{g}})$ descends to the left action on the quotient
$V^{k}(\mathfrak{g})$. Let
$\widetilde{\mathfrak{g}}=\mathfrak{g}\otimes\mathbbm{k}[t,t^{-1}]\oplus\mathbbm{k}K\subseteq\widehat{\mathfrak{g}}$
a Lie subalgebra. Note that the composition $U(\widetilde{\mathfrak{g}})\to
U(\widehat{\mathfrak{g}})\to V^{k}(\mathfrak{g})$ is surjective, and if we set
$\widetilde{V}^{k}(\mathfrak{g})=U(\widetilde{\mathfrak{g}})\otimes_{\widetilde{\mathfrak{g}}_{+}}\mathbbm{k}_{k}$
then the composition factors to an isomorphism
$\widetilde{V}^{k}(\mathfrak{g})\cong V^{k}(\mathfrak{g})$.

Now [Mar, Theorem 7.90] implies that there exists a $G[t]$-action on
$\widetilde{\mathfrak{g}}$ and the induced action on
$\widetilde{V}^{k}(\mathfrak{g})$ is locally smooth for the $t$-adic topology,
hence $G[[t]]$ acts on $V^{k}(\mathfrak{g})$ as required. ∎

The next result follows from Lemma 3.6 and Lemma 3.8.

######  Corollary 3.9.

$V^{k}(\mathfrak{g})^{G[[t]]}\subseteq\mathfrak{z}V^{k}(\mathfrak{g})$.
$\hfill\qed$

###  3.7. Integral forms on vertex algebras

Let $G_{\mathbb{Z}}$ be a connected, split simple group such that the base
change of $G_{\mathbb{Z}}$ to $\mathbbm{k}$ is equal to $G$. Also let
$T_{\mathbb{Z}}$ be a choice of split maximal torus whose base change to
$\mathbbm{k}$ is $T$.

Let $R$ be the subring of $\mathbb{Q}$ generated by the reciprocals of
integers between 1 and $h$. Since $p>h$ there is a natural homomorphism
$R\to\mathbbm{k}$. Write $G_{\mathbb{C}}$ for the base change to $\mathbb{C}$,
write $\mathfrak{g}_{\mathbb{C}}=\operatorname{Lie}G_{\mathbb{C}}$. Similarly
we write $G_{R}$ and $\mathfrak{g}_{R}$ for the $R$-defined objects. Since
$\mathfrak{g}$ is simple for $p>h$ we can identify
$\mathfrak{g}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbbm{k}=\mathfrak{g}$.

Since $S(\mathfrak{g}_{R}[t^{-1}]t^{-1})$ is a free $R$-module, it acts as an
intermediary between the symmetric algebras
$S(\mathfrak{g}_{\mathbb{C}}[t^{-1}]t^{-1})$ and
$S(\mathfrak{g}[t^{-1}]t^{-1})$. to be precise we have natural identifications

(3.13) $\displaystyle S(\mathfrak{g}_{R}[t^{-1}]t^{-1})\otimes_{R}\mathbb{C}$
$\displaystyle=$ $\displaystyle S(\mathfrak{g}_{\mathbb{C}}[t^{-1}]t^{-1})$
(3.14) $\displaystyle S(\mathfrak{g}_{R}[t^{-1}]t^{-1})\otimes_{R}\mathbbm{k}$
$\displaystyle=$ $\displaystyle S(\mathfrak{g}[t^{-1}]t^{-1}).$

Now we address the modular reduction of the affine vertex algebra
$V^{-h^{\vee}}(\mathfrak{g}_{\mathbb{C}})$. We let
$\widehat{\mathfrak{g}}_{R}=\mathfrak{g}_{R}\otimes_{R}R[t^{\pm 1}]\oplus RK$,
which is a Lie $R$-subalgebra of $\widehat{\mathfrak{g}}_{\mathbb{C}}$, and a
free $R$-module. The enveloping algebra $U(\widehat{\mathfrak{g}}_{R})$
satisfies the conditions of the PBW theorem, and we can construct the vacuum
module
$V^{-h^{\vee}}(\mathfrak{g}_{R}):=U(\widehat{\mathfrak{g}}_{R})\otimes_{\widehat{\mathfrak{g}}_{+},R}R_{-h^{\vee}}$
(Cf. (3.8)), where $\widehat{\mathfrak{g}}_{+,R}$ is the induced $R$-form on
$\widehat{\mathfrak{g}}_{+}$ and $R_{-h^{\vee}}$ is the rank one free
$R$-module with $\widehat{\mathfrak{g}}_{+,R}$-action given by letting $K$ act
via $-h^{\vee}$ and $\mathfrak{g}_{R}[[t]]$ acts trivially. This is a free
$R$-module and so we have

(3.15) $\displaystyle V^{-h^{\vee}}(\mathfrak{g}_{R})\otimes_{R}\mathbb{C}$
$\displaystyle=$ $\displaystyle V^{-h^{\vee}}(\mathfrak{g}_{\mathbb{C}})$
(3.16) $\displaystyle V^{-h^{\vee}}(\mathfrak{g}_{R})\otimes_{R}\mathbbm{k}$
$\displaystyle=$ $\displaystyle V^{-h^{\vee}}(\mathfrak{g}).$

##  4\. Invariants and Arc spaces

###  4.1. The semiclassical limit of the affine vertex algebra

We equip $V^{k}(\mathfrak{g})$ with the filtration described in (3.7). This
filtration is ascending, exhaustive $V^{k}(\mathfrak{g})=\bigcup_{d\geq
0}F^{d}V^{k}(\mathfrak{g})$, and connected
$F^{0}V^{k}(\mathfrak{g})=\mathbbm{k}{\boldsymbol{1}}$. Furthermore, $G[[t]]$,
$\mathfrak{g}[[t]]$ and $\\{T^{(i)}\mid i\geq 0\\}$ all preserve the filtered
pieces.

This gives rise to an associated graded space

(4.4)
$\displaystyle\begin{array}[]{rcl}\operatorname{gr}V^{k}(\mathfrak{g})&=&\bigoplus_{d\geq
0}F_{d}V^{k}(\mathfrak{g})/F_{d-1}V^{k}(\mathfrak{g})\vspace{5pt}\\\
&=&S(\widehat{\mathfrak{g}})/(\mathfrak{g}_{+}^{0})=S(\mathfrak{g}[t^{-1}]t^{-1})\vspace{5pt}\\\
&=&\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]\end{array}$

where the final identification follows from the remarks of Section 2.1 (see
[Ar, §3.8, (34)] for more detail). There is an induced action of $G[[t]]$ and
$\mathfrak{g}[[t]]$ on $\operatorname{gr}V^{k}(\mathfrak{g})$. Via (4.4) these
actions coincide with the natural ones on
$S(\widehat{\mathfrak{g}})/(\mathfrak{g}_{+}^{0})$ which arise from the
adjoint action of $G[[t]]$ on $\mathfrak{g}(\\!(t)\\!)/\mathfrak{g}[[t]]$, in
the same manner as Lemma 3.8. This is an isomorphism of
$\mathfrak{g}[[t]]$-modules, and we will identify
$\operatorname{gr}V^{k}(\mathfrak{g})=\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]$
in the sequel.

Similarly the translation operator $T$ gives rise to a higher derivation
$\partial=(\partial^{(0)},\partial^{(1)},\partial^{(2)},...)$ on
$S(\mathfrak{g}[t^{-1}]t^{-1})$ by the rule
$\partial^{(i)}(v+F^{d-1}V^{k}(\mathfrak{g})):=T^{(i)}(v)+F^{d-1}V^{k}(\mathfrak{g})$
for $i,d\geq 0$. More explicitly, $\partial^{(i)}$ coincides with
$(-1)^{i}\partial_{t}^{(i)}$ where $\partial_{t}$ is described in Section 3.1.

Combining our remarks with Lemma 3.6 we have the inclusion

(4.5)
$\displaystyle\operatorname{gr}\mathfrak{z}V^{k}(\mathfrak{g})\subseteq\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{\mathfrak{g}[[t]]}.$

We also have

(4.6)
$\displaystyle\operatorname{gr}\mathfrak{z}_{p}V^{k}(\mathfrak{g})=S(\mathfrak{g}[t^{-1}]t^{-1})^{p}.$

By Theorem 2.3 we have a homogeneous invariant generating set
$P_{1},...,P_{r}\in\mathbbm{k}[\mathfrak{g}^{*}]^{G}$. If we let $G[[t]]$ act
on $\mathbbm{k}[\mathfrak{g}^{*}]$ via the homomorphism
$G[[t]]\twoheadrightarrow G$ then the inclusion
$\mathbbm{k}[\mathfrak{g}^{*}]\subseteq\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]$
given by $x\mapsto x_{-1}$ is $G[[t]]$-equivariant. Therefore the elements
$\\{P_{i,-1}\mid i=1,...,r\\}$ we obtain are $G[[t]]$-invariant.

For $j>0$ we define regular functions
$P_{i,-j}\in\mathbbm{k}[J_{\infty}\mathfrak{g}]$ by

(4.7) $P_{i,-j}:=\partial^{(j-1)}P_{i,-1}.$

We can define these regular functions on $J_{\infty}\mathfrak{g}^{*}$
equivalently using fields. For $i=1,...,\dim\mathfrak{g}$ we let
$x^{i}(z)=\sum_{n<0}x^{i}_{n}z^{-n-1}$. The change of variables $x^{i}\mapsto
x^{i}(z)$ produces a map
$\mathbbm{k}[\mathfrak{g}^{*}]\mapsto\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}][z]$
and the image of $P_{i}$ under this map is the generating function for the
series $(P_{i,-n-1})_{n\geq 0}$.

In the following lemma we write $x^{1},...,x^{n}$ for a basis of
$\mathfrak{g}$.

######  Lemma 4.1.

For $P\in\mathbbm{k}[\mathfrak{g}^{*}]$ consider
$P_{-1}\in\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]$ as above. Then

$\frac{\partial}{\partial
x^{i}_{-1-s}}\partial^{(m)}P_{-1}=\begin{cases}\partial^{(m-s)}\frac{\partial
P_{-1}}{\partial x^{i}_{-1}},\qquad m\geq s\geq 0\,,\\\
0\,,\qquad\qquad\qquad\quad m<s\,.\end{cases}$

###### Proof.

It suffices to check the claim for monomials in $x^{1},...,x^{n}$. The case
$m<s$ is clear because $\partial^{(m)}P_{-1}$ only depends on variables
$x^{j}_{-1},...,x^{j}_{-1-m}$ with $j=1,...,\dim\mathfrak{g}$.

We prove the statement for $m\geq s$ by induction on the total degree of $P$.
The degree 1 case is easily verified. Now suppose it has been verified for all
$P$ of degree less than $d$. Any monomial of degree greater than 1 can be
written as a product $PQ$ of monomials of degrees strictly less than $d$.

Using (2.3) have

$\displaystyle\partial^{(m-s)}\frac{\partial}{\partial
x^{i}_{-1}}(P_{-1}Q_{-1})$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{m-s}\big{(}(\partial^{(j)}\frac{\partial
P_{-1}}{\partial
x^{i}_{-1}})(\partial^{(m-s-j)}Q_{-1})+(\partial^{(j)}P_{-1})(\partial^{(m-s-j)}\frac{\partial
Q_{-1}}{\partial x^{i}_{-1}})\big{)}$ $\displaystyle=$
$\displaystyle\sum_{j=s}^{m}(\partial^{(j-s)}\frac{\partial P_{-1}}{\partial
x^{i}_{-1}})(\partial^{(m-j)}Q_{-1)})+\sum_{j=0}^{m-s}(\partial^{(j)}P_{-1})(\partial^{(m-s-j)}\frac{\partial
Q_{-1}}{\partial x^{i}_{-1}}).$

In the second line we shift the indexes for half of the summands.

On the other hand we have

$\displaystyle\frac{\partial}{\partial
x^{i}_{-1-s}}\partial^{(m)}(P_{-1}Q_{-1})$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{m}\big{(}(\frac{\partial}{\partial
x^{i}_{-1-s}}\partial^{(j)}P_{-1})(\partial^{(m-j)}Q_{-1})+(\partial^{(j)}P_{-1})(\frac{\partial}{\partial
x^{i}_{-1-s}}\partial^{(m-j)}Q_{-1})$ $\displaystyle=$
$\displaystyle\sum_{j=s}^{m}(\frac{\partial}{\partial
x^{i}_{-1-s}}\partial^{(j)}P_{-1})(\partial^{(m-j)}Q_{-1})+\sum_{j=0}^{m-s}(\partial^{(j)}P_{-1})(\frac{\partial}{\partial
x^{i}_{-1-s}}\partial^{(m-j)}Q_{-1})$

where the second line uses the vanishing statement of the lemma for $s<m$.
Since the degree of both $P$ and $Q$ is less than the degree of $PQ$ the proof
concludes using the inductive hypothesis. ∎

Over the complex numbers the $G[[t]]$-invariance of the polynomials $P_{i,-j}$
is equivalent to the $\mathfrak{g}[[t]]$-invariance. In our setting this is no
longer the case, and so we check invariance by a process of modular reduction.

######  Lemma 4.2.

$P_{i,-j}\in\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{G[[t]]}$.

###### Proof.

Resume the notation
$R,G_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}},\mathfrak{g}_{R}$ of Section 3.7.
Then $S(\mathfrak{g}_{\mathbb{C}})$ admits an $R$-form $S(\mathfrak{g}_{R})$.
Applying the remarks of [Ja1, 9.6] verbatim, with the symmetric algebra in the
place of the enveloping algebra, we see that
$S(\mathfrak{g}_{R})^{G_{R}}\otimes_{\mathbb{Z}}\mathbbm{k}=S(\mathfrak{g})^{G}$
and
$S(\mathfrak{g}_{R})^{G_{R}}\otimes_{R}\mathbb{C}=S(\mathfrak{g}_{\mathbb{C}})^{G_{\mathbb{C}}}$
(in fact, this holds under weaker hypotheses than $p>h$). Therefore we can
choose a homogeneous generating set
$P_{1}^{\mathbb{C}},...,P_{r}^{\mathbb{C}}$ for
$S(\mathfrak{g}_{\mathbb{C}})^{G_{\mathbb{C}}}$ lying $S(\mathfrak{g}_{R})$
and choose $P_{i}:=P_{i}^{\mathbb{C}}\otimes 1\in S(\mathfrak{g})$. The
elements $P_{1},...,P_{r}$ satisfy Theorem 2.3.

By (3.1) we know that the divided power operators $\partial^{(i)}$ on
$S(\mathfrak{g}_{\mathbb{C}}[t^{-1}]t^{-1})$ preserve
$S(\mathfrak{g}_{R}[t^{-1}]t^{-1})$, and they intertwine the projection map
$S(\mathfrak{g}_{R}[t^{-1}]t^{-1})\to S(\mathfrak{g}[t^{-1}]t^{-1})$. By [Fr,
Theorem 3.4.2] the element
$P^{\mathbb{C}}_{i,-j}=\partial^{(j-1)}P_{i,-1}^{\mathbb{C}}\in
S(\mathfrak{g}_{R}[t^{-1}]t^{-1})$ is $G_{\mathbb{C}}[[t]]$-invariants and
$P_{i,-j}^{\mathbb{C}}\otimes 1=P_{i,-j}$

Now let $S$ be either $\mathbb{C},R$ or $\mathbbm{k}$. By Lemma 2.2 we know
that $J_{\infty}G_{S}$ is generated by
$\\{J_{\infty}u_{\alpha}\mid\alpha\in\Phi\\}$. The group of $S$-points of
$J_{\infty}u_{\alpha}$ is isomorphic to the additive group $S[[t]]$. For each
$m\geq 0$ and $\alpha\in\Phi$ we consider the morphism of schemes
$u_{\alpha,m}^{S}:\mathbb{G}_{a}\to J_{\infty}G_{S}$ determined by setting
$u_{\alpha,m}^{S}(a)=J_{\infty}u_{\alpha}(at^{i})$, for any $S$-algebra $A$
and element $a\in A$. Thus we have a group homomorphism
$u_{\alpha,m}^{S}(S):S\to G_{S}[[t]]$ of $S$-points. If $f\in
S(\mathfrak{g}_{S}[t^{-1}]t^{-1})$ then $u_{\alpha,m}(S)(z)\cdot f$ is a
polynomial expression in $z$ for $z\in S$.

We now fix $i,j$ and $\alpha,m$. The $G_{\mathbb{C}}[[t]]$-invariance of
$P_{i,-j}$ implies that $u_{\alpha,m}^{\mathbb{C}}(\mathbb{C})(z)\cdot
P_{i,-j}^{\mathbb{C}}-P_{i,-j}^{\mathbb{C}}$ vanishes identically for all
values of $z\in\mathbb{C}$. We can regard this as a polynomial identity in
$S(\mathfrak{g}_{R}[t^{-1}]t^{-1})[z]$ where $z$ is a formal variable.
Reducing modulo $p$ we deduce that
$u_{\alpha,m}^{\mathbbm{k}}(\mathbbm{k})(z)\cdot P_{i,-j}-P_{i,-j}=0$ in
$S(\mathfrak{g}[t^{-1}]t^{-1})$. Since this holds for every $\alpha,m$, and
since $G[[t]]$ is generated by the images of the morphisms
$u_{\alpha,m}^{\mathbbm{k}}(\mathbbm{k})$ we deduce that $P_{i,-j}$ is
$G[[t]]$-invariant. ∎

We provide one example to show that the invariance of $P_{i,-j}$ can be
derived directly in special cases.

######  Example 4.3.

Let $G=\operatorname{SL}_{n+1}$ so that

$G[[t]]=\\{g\in\operatorname{Mat}_{n+1}(\mathbbm{k}[[t]]))\mid\det(g)=1\\}$

Let $\\{x^{i,j}\mid 1\leq i,j\leq n+1\\}$ be the standard basis for
$\operatorname{Mat}_{n+1}(\mathbbm{k})$. Define a matrix
$X\in\operatorname{Mat}_{n+1}(\mathbbm{k}[t^{-1}]t^{-1}[[z]])$ whose $i,j$
entry is the series $x^{i,j}(z)=\sum_{m<0}x^{i,j}_{m}z^{-1-m}$. Write

$\det(\lambda-X)=\lambda^{n}+\sum_{i=0}^{n}P_{i}(z)\lambda^{i}$

We identify $\operatorname{Mat}_{n+1}(\mathbbm{k}[t^{-1}]t^{-1})$ with
$\operatorname{Hom}_{\mathbbm{k}}(\operatorname{Mat}_{n+1}(\mathbbm{k}[[t]]),\mathbbm{k})$
via the residue pairing. Then the $z^{j}$ coefficient of the series $P_{i}(z)$
restricts to $P_{i,-1-j}$ on
$J_{\infty}\mathfrak{sl}_{n+1}\subseteq\operatorname{Mat}_{n+1}(\mathbbm{k}[[t]])$.

The group $G[[t]]$ acts on
$\operatorname{Mat}_{n+1}(\mathbbm{k}[t^{-1}]t^{-1})=J_{\infty}\mathfrak{gl}_{n+1}^{*}$
and the invariance of the $P_{i,-j}$ is now a simple consequence of the fact
that the characteristic polynomial is $G[[t]]$-invariant.

Finally, we prove the main theorem of this section, which is a vertex algebra
analogue of Theorem 2.3(1)-(4).

######  Theorem 4.4.

  1. (1)

$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{G[t]]}=\mathbbm{k}[P_{i,-j}\mid
i=1,..,r,\ j<0]$ polynomial ring on infinitely many generators.

  2. (2)

$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{\mathfrak{g}[[t]]}$ is generated by
$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{G[t]]}$ and
$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{p}$.

  3. (3)

$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{\mathfrak{g}[[t]]}$ is a free
$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{p}$-module with basis

(4.8) $\displaystyle\Big{\\{}\prod_{i,j}P_{i,-j}^{k_{i,j}}\mid 0\leq
k_{i,j}<p,\text{ finitely many nonzero}\Big{\\}}.$

  4. (4)

$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{\mathfrak{g}[[t]]}$ is isomorphic to
the tensor product of $\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{p}$ and
$\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]^{G[t]]}$ over the intersection.

###### Proof.

In the proof we make use of the group of $m$-jets $J_{m}G$ which has Lie
algebra $\mathfrak{g}_{m}:=\mathfrak{g}\otimes\mathbbm{k}[t]/(t^{m}+1)$ and
$\mathbbm{k}$-points $G_{m}$, known as the truncated current group.

The map $\mathfrak{g}^{*}[[t]]\twoheadrightarrow\mathfrak{g}_{m}^{*}$ gives
rise to an inclusion
$\mathbbm{k}[\mathfrak{g}_{m}^{*}]\to\mathbbm{k}[J_{\infty}\mathfrak{g}^{*}]$.
This subalgebra is a $G[[t]]$-submodule and the action factors through the
homomorphism $G[[t]]\twoheadrightarrow G_{m}$ arising from the inclusion of
Hopf algebras $J_{m}\mathbbm{k}[G]\hookrightarrow J_{\infty}\mathbbm{k}[G]$
(see (2.7)). Now in order to prove the theorem it suffices to prove the
corresponding claims about $\mathbbm{k}[J_{m}\mathfrak{g}^{*}]^{G_{m}}$ and
$\mathbbm{k}[J_{m}\mathfrak{g}^{*}]^{\mathfrak{g}_{m}}$, mutatis mutandis.

In order to prove (2) and (3), we apply a result of Skryabin [Sk, Theorem
5.4]. To satisfy the conditions of his theorem we must demonstrate that:

  1. i)

$\operatorname{ind}({\mathfrak{g}}_{m})=(m+1)\operatorname{rank}(\mathfrak{g})$

  2. ii)

$\operatorname{codim}_{{\mathfrak{g}}_{m}^{*}}({\mathfrak{g}}_{m}^{*}\setminus
J(P_{i,-j}))\geq 2$

where
$\operatorname{ind}(\mathfrak{g}_{m}):=\min_{\chi\in\mathfrak{g}^{*}}\dim\mathfrak{g}_{m}^{\chi}$
is the index of the Lie algebra, and $J(P_{i,-j})$ denotes the Jacobian locus
of the $P_{i,-j}$, i.e. the open set of $\mathfrak{g}_{m}^{*}$ consisting of
points $\chi$ such that the functions $d_{\chi}P_{i,-j}$ are linearly
independent.

Part (i) is a special case of a theorem of Raïs–Tauvel (where they work over
the complex numbers) and we include a short proof valid in our setting. Pick a
triangular decomposition
$\mathfrak{g}={\mathfrak{n}}^{-}\oplus\mathfrak{h}\oplus{\mathfrak{n}}^{+}$,
which gives
$\mathfrak{g}_{m}={\mathfrak{n}}^{-}_{m}\oplus\mathfrak{h}_{m}\oplus{\mathfrak{n}}_{m}^{+}$.
Let $\mathfrak{h}_{m}^{\circ}=\\{\sum_{i=0}^{m}h_{i}t^{i}\mid
h_{0}\in\mathfrak{h}^{\operatorname{reg}}\\}$.

The differential of the adjoint action
$G_{m}\times\mathfrak{h}_{m}\to\mathfrak{g}_{m}$ at a point $(1,h)$ is the
linear map $\mathfrak{g}_{m}\times\mathfrak{h}_{m}\to\mathfrak{g}_{m}$ given
by $x,z\mapsto[x,h]+z$. When $h\in\mathfrak{h}_{m}^{\circ}$ a simple inductive
argument shows that

(4.9)
$\displaystyle[h,\mathfrak{g}_{m}]={\mathfrak{n}}_{m}^{-}\oplus{\mathfrak{n}}_{m}^{+}$
and $\displaystyle\mathfrak{g}_{m}^{h}=\mathfrak{h}_{m}$

and so the differential is surjective for such a point. Consequently the
conjugates of $\mathfrak{h}_{m}^{\circ}$ are dense in $\mathfrak{g}_{m}$. On
the other hand
$\dim\mathfrak{g}_{m}^{h}=(m+1)\operatorname{rank}(\mathfrak{g})$ by (4.9),
and so by upper-semicontuinuity of dimensions of fibres of morphisms, we have
that
$\min_{x\in\mathfrak{g}_{m}}\dim\mathfrak{g}_{m}^{x}=(m+1)\operatorname{rank}(\mathfrak{g})$.

There is a $\mathfrak{g}_{m}$-invariant bilinear form on $\mathfrak{g}_{m}$
given by $xt^{i},yt^{j}\mapsto\delta_{i,m+1-j}\kappa(x,y)$, and this induces
an isomorphism $\theta:\mathfrak{g}_{m}\to\mathfrak{g}_{m}^{*}$ of
$\mathfrak{g}_{m}$-modules. We deduce that
$\operatorname{ind}(\mathfrak{g}_{m})=(m+1)\operatorname{rank}(\mathfrak{g})$,
which proves (i).

Let $\mathfrak{g}_{m}^{\circ}=\\{\sum_{i=0}^{m}x_{i}t^{i}\mid
x_{0}\in\mathfrak{g}^{\operatorname{reg}}\\}$. Thanks to [BG, Proposition 3.2]
we have
$\operatorname{codim}_{\mathfrak{g}_{m}}(\mathfrak{g}_{m}\setminus\mathfrak{g}_{m}^{\circ})=3$.
We shall show that $\theta(\mathfrak{g}_{m}^{\circ})\subseteq J(P_{i,-j})$,
which shall prove (ii).

Once again we write $x^{1},...,x^{n}$ for a basis of linear functions on
$\mathfrak{g}^{*}$, which gives a basis $\\{x^{i}_{-j}\mid i=1,...,r,\
j=1,...,m+1\\}$ of linear functions on $J_{m}\mathfrak{g}^{*}$. Consider the
Jacobian of the functions $\\{P_{i,-j}\mid 1\leq i\leq r,\ 1\leq j\leq
m+1\\}$, a matrix of size
$(m+1)\dim\mathfrak{g}\times(m+1)\operatorname{rank}\mathfrak{g}$. It has
block form

(4.10) $\left(\begin{matrix}\frac{\partial P_{i,-1}}{\partial
x^{j}_{-1}}&&&0\\\ \frac{\partial P_{i,-2}}{\partial
x^{j}_{-1}}&\frac{\partial P_{i,-2}}{\partial x^{j}_{-2}}&\\\
\vdots&&\ddots&\\\ \frac{\partial P_{i,-1-m}}{\partial
x^{j}_{-1}}&\frac{\partial P_{i,-1-m}}{\partial
x^{j}_{-2}}&\dots&\frac{\partial P_{i,-m-1}}{\partial
x^{j}_{-m-1}}\end{matrix}\right)$

where each block has size
$(\dim\mathfrak{g})\times(\operatorname{rank}\mathfrak{g})$. Using Lemma 4.1
we see that the diagonal blocks are all equal to $(\frac{\partial
P_{i,-1}}{{\partial x^{j}_{-1}}})$. Evaluating at any point of
$\mathfrak{g}_{m}^{\circ}$ we see that this matrix has full rank, thanks to
the differential criterion for regularity (see Lemma 3.4 and Proposition 3.4
of [BG]). We have now shown that $\theta(\mathfrak{g}_{m}^{\circ})\subseteq
J(P_{i,-j})$, which completes the proof of (ii), and part (2) and (3) of the
current theorem.

For part (1) it suffices to show that
$\mathbbm{k}[J_{m}\mathfrak{g}^{*}]^{G_{m}}=\mathbbm{k}[P_{i,-j}\mid
i=1,...,r,\ -m-1\leq-j<0]$ is a polynomial ring. Suppose that
$f\in\mathbbm{k}[J_{m}\mathfrak{g}^{*}]^{G_{m}}$. Then by part (1) we can
write $f$ uniquely as $f=\sum_{b\in B}g_{b}b$ where $B$ denotes the collection
of monomials described in (4.8) (with all $j\leq m+1$) and
$g_{b}\in\mathbbm{k}[J_{m}\mathfrak{g}^{*}]^{p}$. By uniqueness, and using
Lemma 4.2, we see that each $g_{b}$ is $G_{m}$-invariant. Since
$\mathbbm{k}[J_{m}\mathfrak{g}^{*}]$ has no nilpotent elements it follows that
$g_{b}=(\bar{g}_{b})^{p}$ for some
$\bar{g}_{b}\in\mathbbm{k}[J_{m}\mathfrak{g}^{*}]^{G_{m}}$. Now part (1) may
be proven by a downward induction on total degree.

Part (4) follows from [GT, Lemma 2.1], whose proof does not require the free
basis to be finite. ∎

##  5\. Centre of $V^{k}(\mathfrak{g})$ at the critical level

###  5.1. A criterion for the existence of Segal–Sugawara vectors

Recall that $R$ is the subring of $\mathbb{Q}$ generated by $\\{1/n\mid 1\leq
n\leq h\\}$, and now let $Q\subseteq\mathbb{Q}$ be a finitely generated
$R$-algebra. Retain the notation
$G_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}},G_{R},\mathfrak{g}_{R}$ from Section
3.7. In this section we discuss the centre of $V^{-h^{\vee}}(\mathfrak{g})$ at
the critical level, and so we ease notation further by writing
$V_{A}=V^{-h^{\vee}}(\mathfrak{g}_{A})$ whenever
$A\in\\{R,Q,\mathbb{C},\mathbbm{k}\\}$. Similarly we write
$S_{A}=S(\mathfrak{g}_{A}[t^{-1}]t^{-1})$.

Recall that $\mathfrak{z}V_{\mathbbm{k}}$ is a commutative vertex algebra and,
in particular, it is a commutative (differential) algebra. To prove Theorem
1.3 we use the following criterion, which follows directly from Theorem 4.4 by
modular reduction and a standard filtration argument.

######  Proposition 5.1.

Suppose that there is a non-zero homomorphism $Q\to\mathbbm{k}$ and that there
exist elements $S_{i,-1}\in V_{Q}\cap\mathfrak{z}V_{\mathbb{C}}$ such that

(5.1) $\displaystyle\operatorname{gr}S_{i,-1}=P_{i,-1}$

for $i=1,...,r$ . If $S_{i,-j}:=T^{(j-1)}S_{i,-1}$ and denote the image of
$S_{i,-j}$ under $V_{Q}\to V_{\mathbbm{k}}$ by the same symbol. The following
hold:

  1. (1)

$V_{\mathbbm{k}}^{G[[t]]}=\mathbbm{k}[S_{i,-j}\mid i=1,..,r,\ j>0]$ is a
polynomial algebra on infinitely many generators.

  2. (2)

$\mathfrak{z}V_{\mathbbm{k}}$ is generated by $V_{\mathbbm{k}}^{G[[t]]}$ and
$\mathfrak{z}_{p}V_{\mathbbm{k}}$ as a commutative algebra.

  3. (3)

$\mathfrak{z}V_{\mathbbm{k}}$ is a free
$\mathfrak{z}_{p}V_{\mathbbm{k}}$-module of over
$\mathfrak{z}_{p}V_{\mathbbm{k}}$ with basis

(5.2) $\displaystyle\Big{\\{}\prod_{i,j}S_{i,-j}^{k_{i,j}}\mid 0\leq
k_{i,j}<p,\text{ finitely many nonzero}\Big{\\}}.$

  4. (4)

$\mathfrak{z}V_{\mathbbm{k}}$ is isomorphic to the tensor product of
$\mathfrak{z}_{p}V_{\mathbbm{k}}$ and $V_{\mathbbm{k}}^{G[t]]}$ over their
intersection.

###### Proof.

Recall Lemma 3.6, Lemma 3.7 and Corollary 3.9 which state that
$\mathfrak{z}_{p}V_{\mathbbm{k}},V_{\mathbbm{k}}^{G[[t]]}\subseteq\mathfrak{z}V_{\mathbbm{k}}=V_{\mathbbm{k}}^{\mathfrak{g}[[t]]}$.

It is clear that $V_{Q}$ is a vertex $Q$-subalgebra of $V_{\mathbb{C}}$ and
that the translation operators on $V_{Q}$ and $V_{\mathbbm{k}}$ intertwine the
natural map $V_{Q}\to V_{\mathbbm{k}}$. Therefore
$\operatorname{gr}S_{i,-j}=P_{i,-j}$ for all $i,j$. Since $\mathbb{C}$ has
characteristic zero, $\mathfrak{g}_{\mathbb{C}}[[t]]$-invariants are
$G_{\mathbb{C}}[[t]]$-invariant, and so we have $S_{i,-j}\in
V_{\mathbb{C}}^{G_{\mathbb{C}}[[t]]}$.

Using an argument identical to the proof of Lemma 4.2 we see that the images
$S_{i,-j}\in V_{\mathbbm{k}}$ are $G[[t]]$-invariant. Property (5.1) continues
to hold in $V_{\mathbbm{k}}$ and so we have $\mathbbm{k}[S_{i,-j}\mid
i,j]\subseteq V_{\mathbbm{k}}^{G[[t]]}$. By Theorem 4.4 we have
$\operatorname{gr}\mathbbm{k}[S_{i,-j}\mid
i,j]\subseteq\operatorname{gr}V_{\mathbbm{k}}^{G[[t]]}\subseteq
S_{\mathbbm{k}}^{G[[t]]}=\mathbbm{k}[P_{i,-j}\mid
i,j]=\operatorname{gr}\mathbbm{k}[S_{i,-j}\mid i,j]$ and so we must have
equality throughout. This proves (1).

Part (2) follows by a very similar line of reasoning, in view (4.5) and (4.6).
Arguing in the same fashion once again, we see that the elements (5.2) span
$\mathfrak{z}V_{\mathbbm{k}}$ as a $\mathfrak{z}_{p}V_{\mathbbm{k}}$-module.
If there were a $\mathfrak{z}_{p}V_{\mathbbm{k}}$-linear relation amongst them
the the projection into $\operatorname{gr}V_{\mathbbm{k}}$ would give an
$S_{\mathbbm{k}}^{p}$-linear relation amongst $\\{P_{i,-j}\mid i,j\\}$, which
contradicts Theorem 4.4(3). This proves (3).

Part (4) follows from [GT, Lemma 2.1], whose proof does not require the free
basis to be finite. ∎

In order to complete the proof of Theorem 1.3 we must show that the hypotheses
of Proposition 5.1 can be satisfied for each simple finite root datum.

###  5.2. Completing the proof: type A

Let $n\in\mathbb{N}$. The known formulas for generators of the Feigin–Frenkel
centre in type A are expressed inside the vertex algebra
$V^{-h^{\vee}}(\mathfrak{gl}_{N}(\mathbb{C}))$. The Coxeter number $h$ for
$\operatorname{SL}_{N}$ is $N$. Since we have assumed that $p>h$ it follows
that $\mathfrak{gl}_{N}(R)$ is the direct sum (as Lie $R$-algebras) of
$\mathfrak{sl}_{N}(R)$ and the rank 1 centre $RI$.

The normalised Killing form on $\mathfrak{gl}_{N}(R)$ is given by
$\kappa(x,y)=\operatorname{Tr}(xy)-\frac{1}{N}\operatorname{Tr}(\operatorname{ad}(x)\operatorname{ad}(y))$.
The kernel is the centre of $\mathfrak{gl}_{N}(R)$ and the restriction to
$\mathfrak{sl}_{N}(R)$ is the normalised Killing form (2.8).

Let $\mathfrak{c}_{R}=RI\otimes R((t))\oplus
RK\subseteq\widehat{\mathfrak{gl}}_{N}(R)$ a commutative Lie subalgebra. Then
the image $H_{R}$ of $U(\mathfrak{c}_{R})$ in
$V^{-h^{\vee}}(\mathfrak{gl}_{N}(R))$ is a commutative vertex subalgebra,
isomorphic to the semiclassical limit of the Heisenberg vertex algebra.
Furthermore we have an $R$-linear isomorphism of vertex rings
$V^{-h^{\vee}}(\mathfrak{gl}_{N}(R))\cong
V^{-h^{\vee}}(\mathfrak{sl}_{N}(R))\otimes_{R}H_{R}$. Similar remarks hold
over $\mathbbm{k}$ or $\mathbb{C}$. The next lemma follows immediately from
these remarks.

######  Lemma 5.2.

Theorem 1.3 holds for $V^{-h^{\vee}}(\mathfrak{sl}_{N})$ if and only if it
$V^{-h^{\vee}}(\mathfrak{gl}_{N})$. $\hfill\qed$

Generators of $V^{-h^{\vee}}(\mathfrak{gl}_{N}(\mathbb{C}))$ satisfying the
hypotheses of Proposition 5.1 are described [CM, Theorem 3.1] or [Mo2, Theorem
7.1.3]. They are manifestly defined over $R=\mathbb{Z}[1/n\mid 1\leq n\leq N]$
and this completes the proof in type A.

###  5.3. Completing the proof: classical types

Let $G$ be a connected, simply connected, simple classical group scheme of
type B, C, or D. Under the hypothesis $p>h$ there are no inseparable
isogenies, and so it does no harm to replace $G$ by one of the matrix groups
$\operatorname{SO}_{N}$ or $\operatorname{Sp}_{N}$.

In this case the hypotheses of Proposition 5.1 are satisfied with $Q=R$,
thanks to Theorems 8.1.6, 8.1.9 and 8.3.8 of [Mo2]. See also [Mo1] and [Ya].

###  5.4. Completing the proof: exceptional types

Finally we let $G$ be a simply connected, connected exceptional simple group
scheme, and $\mathfrak{g}=\operatorname{Lie}(G)$. Recall the notation from
Section 5.1, and $x^{1},...,x^{n}$ denotes a basis for $\mathfrak{g}$.

Fix $i=1,...,r$. For $m>0$ let $V_{\mathbb{C}}^{[m]}$ be the span of PBW
monomials (3.9) of degree less than $\deg P_{i,-1}$ involving $x^{j}_{-k}$
where $m\geq k\geq 1$ and $j=1,...,n$.

Since $\mathfrak{g}_{\mathbb{C}}[[t]]$ is topologically generated by
$\mathfrak{g}_{\mathbb{C}}$ and $\mathfrak{g}_{\mathbb{C}}\otimes t$, the
existence of an element
$S_{i,-1}\in(V_{\mathbb{C}}^{[m]})^{\mathfrak{g}_{\mathbb{C}}[[t]]}$ such that
$\operatorname{gr}S_{i,-1}=P_{i,-1}$ can be expressed as a solution to a
finite system of affine linear equations with integer coefficients. The
Feigin–Fenkel theorem (see [Fr, Theorem 8.1.3] implies that there exists $m>0$
such that the system has a solution and, by integrality, the existence of a
solution in $(V_{\mathbb{C}}^{[m]})^{\mathfrak{g}_{\mathbb{C}}[[t]]}$ implies
the existence of a solution
$S_{i,-1}\in(V_{\mathbb{Q}}^{[m]})^{\mathfrak{g}_{\mathbb{Q}}[[t]]}$. Writing
$S_{i,-1}$ as a $\mathbb{Q}$-linear span of elements of the PBW basis, we can
choose an integer $q_{i}\in\mathbb{N}$ (say, the maximal denominator appearing
in the coefficients) so that $S_{i,-1}\in V_{R_{i}}$ where
$R_{i}=\mathbb{Z}[q_{i}^{-1}]$. Taking $q=\max\\{q_{i}\\!\mid i=1,...,r\\}$
and setting $Q:=R[q^{-1}]$ we have $S_{i,-1}\in V_{Q}\cap
V_{\mathbb{C}}^{\mathfrak{g}_{\mathbb{C}}[[t]]}$ for all $i=1,...,r$.

Now provided the characteristic of the field $\mathbbm{k}$ is greater than $q$
there is a nonzero homomorphism $Q\to\mathbbm{k}$, and so the hypotheses of
Proposition 5.1 are satisfied. This completes the proof of Theorem 1.3.

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