# A McKay Correspondence in positive characteristic

Christian Liedtke  TU München, Zentrum Mathematik - M11, Boltzmannstr. 3,
85748 Garching bei München, Germany<EMAIL_ADDRESS>
###### Abstract.

We establish a McKay correspondence for finite and linearly reductive subgroup
schemes of ${\mathbf{SL}}_{2}$ in positive characteristic. As an application,
we obtain a McKay correspondence for _all_ rational double point singularities
in characteristic $p\geq 7$.

We discuss linearly reductive quotient singularities and canonical lifts over
the ring of Witt vectors. In dimension 2, we establish simultaneous
resolutions of singularities of these canonical lifts via $G$-Hilbert schemes.

In the appendix, we discuss several approaches towards the notion of
_conjugacy classes_ for finite group schemes: This is an ingredient in McKay
correspondences, but also of independent interest.

###### Key words and phrases:

linearly reductive group scheme, Klein singularity, rational double point,
McKay correspondence, conjugacy class, Hopf algebra.

###### 2020 Mathematics Subject Classification:

14E16, 14L15, 16T05, 14J17, 13A35, 20C20

###### Contents

  1. 1 Introduction
  2. 2 Linearly reductive group schemes
  3. 3 McKay graph and McKay correspondence
  4. 4 Linearly reductive quotient singularities
  5. 5 Hecke correspondences
  6. 6 Conjugacy classes and Ito-Reid correspondence
  7. 7 Derived categories
  8. A Hopf algebras
  9. B Conjugacy classes for finite group schemes

##  1\. Introduction

###  1.1. Klein’s classification and McKay’s correspondence

Felix Klein [Kl84] classified finite subgroups $G$ of
${\mathbf{SL}}_{2}({{\mathbb{C}}})$: up to conjugation, there are two infinite
series and three isolated cases.

  1. (1)

The associated quotient singularity ${{\mathbb{C}}}^{2}/G$ is called a _Klein
singularity_ and the singularities arising this way are precisely the
_rational double point singularities_. Its minimal resolution of singularities
is a union of ${{\mathbb{P}}}^{1}$’s, whose dual intersection graph $\Gamma$
is a simply-laced Dynkin diagram of finite type, that is, of type $A_{n}$,
$D_{n}$, $E_{6}$, $E_{7}$ or $E_{8}$.

  2. (2)

John McKay [Mc80] associated a finite graph $\widehat{\Gamma}$ to
$G\subset{\mathbf{SL}}_{2}({{\mathbb{C}}})$, whose vertices correspond to the
isomorphism classes of the simple representations of $G$. This graph is a
Dynkin diagram of affine type $\widehat{A}_{n}$, $\widehat{D}_{n}$,
$\widehat{E}_{6}$, $\widehat{E}_{7}$ or $\widehat{E}_{8}$.

  3. (3)

After these preparations, the _classical McKay correspondence_ consists of the
following observations:

    1. (a)

The graph $\Gamma$ is obtained from $\widehat{\Gamma}$ by removing the vertex
corresponding to the trivial representation.

    2. (b)

There exists a bijection between conjugacy classes of $G$, vertices of
$\widehat{\Gamma}$, and isomorphism classes of simple representations of $G$.

    3. (c)

There exists a bijection between finite subgroups of
${\mathbf{SL}}_{2}({{\mathbb{C}}})$ up to conjugacy, the above Dynkin diagrams
of affine type, Klein singularities, and the above Dynkin diagrams of finite
type.

By now, there are various approaches to and a vast literature on this subject,
such as [Kn85, Ko85, Mc80, Mc81, St85] and many more. Also, there are now
generalisations into very different directions: higher dimensional algebraic
geometry [Re02], K-theory [GV83], derived categories of coherent sheaves
[BKR01, KV00], representations of quivers [Ki16], non-commutative geometry and
Hopf algebras [CKWZ16], and string theory [Aetal09] \- just to mention a few.

###  1.2. Positive characteristic

Now, let $k$ be an algebraically closed field of characteristic $p>0$.

####  1.2.1. Wild McKay correspondence

The classical McKay correspondence as sketched above is still partially
available over $k$ if $G$ is assumed to be a finite subgroup of
${\mathbf{SL}}_{2}(k)$ of order prime to $p$ \- this is the _tame_ case. If
$p$ divides the order of $G$ \- this is the _modular_ or _wild_ case - then
this correspondence breaks down. We refer to Yasuda’s survey [Ya21] about
conjectures and partial results concerning such wild McKay correspondences.

####  1.2.2. Linearly reductive McKay correspondence

In this article, we show that if $G$ is a finite and _linearly reductive_
subgroup scheme of ${\mathbf{SL}}_{2,k}$, then there is a reasonable version
of the classical McKay correspondence. For example, we obtain a McKay
correspondence for _all_ rational double point singularities if $p\geq 7$.
Instead of considering groups, we allow non-reduced group schemes over $k$,
but we require their categories of $k$-linear and finite-dimensional
representations to be semi-simple. We refer to [TY20] about conjectures and
partial results concerning a McKay correspondence for the group scheme
$\boldsymbol{\alpha}_{p}$, which is not linearly reductive.

Thus, let $G$ be a finite and linearly reductive subgroup scheme of
${\mathbf{SL}}_{2,k}$ with $p\geq 7$. (In this introduction, we will exclude
small characteristics whenever this is makes our discussion easier.) Let $x\in
X:=U/G$ with $U={{\mathbb{A}}}_{k}^{2}$ or
$U=\widehat{{{\mathbb{A}}}}_{k}^{2}$ be the associated Klein singularity,
which is a rational double point. One goal of this article is to define a
notion of conjugacy class for $G$, to construct graphs $\Gamma$ and
$\widehat{\Gamma}$, and to establish bijections as above.

Let us make three comments:

  1. (1)

It is interesting in its own that a McKay correspondence can be extended from
finite group schemes of length prime to $p$, that is, the tame case, to
linearly reductive group schemes.

  2. (2)

What makes this linearly reductive McKay correspondence really interesting is
that the bijection in Theorem 1.1 is _not_ true when considering finite groups
of order prime to $p$ only, see Example 1.2.

  3. (3)

Probably, many more aspects of the classical McKay correspondence can be
carried over to the linearly reductive setting, but rather than writing a
whole monograph, we decided to establish only some basic bijections.

###  1.3. Linearly reductive group schemes

Let $G$ be a _finite and linearly reductive group scheme_ over an
algebraically closed field $k$ of characteristic $p\geq 0$. By definition,
this means that the category $\mathrm{Rep}_{k}(G)$ of $k$-linear and finite-
dimensional representations of $G$ is semi-simple.

If $p=0$, then every finite group scheme over $k$ is étale and linearly
reductive. and in fact, it is the constant group scheme associated to a finite
group. In fact, the functor $G\mapsto G_{\mathrm{abs}}:=G(k)$ induces an
equivalence of categories between finite group schemes over $k$ and finite
groups.

If $p>0$, then every linearly reductive group scheme admits a canonical semi-
direct product decomposition

$G\,\cong\,G^{\circ}\rtimes G^{\mathrm{\acute{e}t}},$

where $G^{\mathrm{\acute{e}t}}$ is a group scheme of length prime to $p$ (and
thus, the constant group scheme associated to a finite group of order prime to
$p$) and where $G^{\circ}$ is infinitesimal and diagonalisable. The latter
implies that $G^{\circ}$ is a product of group schemes of the form
$\boldsymbol{\mu}_{p^{n}}$. Conversely, every such semi-direct product of a
diagonalisable group scheme with the constant group scheme associated to a
finite group of order prime to $p$ is linearly reductive. This structure
result is usually attributed to Nagata [Na61], but see also [AOV08], [Ch92],
and [Ha15]. In particular, the class of finite and linearly reductive group
schemes over $k$ strictly contains the class of constant group schemes
associated to finite groups of order prime to $p$.

####  1.3.1. Abstract groups

Associated to $G$, there is an abstract finite group $G_{\rm abs}$ and we
refer for the slightly technical definition to Section 2.2. The order of
$G_{\rm abs}$ is equal to the length of $G$. For example, if $G$ is étale over
$k$, then we have $G_{\rm abs}\cong G(k)$. The assignment $G\mapsto
G_{\mathrm{abs}}$ establishes an equivalence of categories

(1) $\left\\{\begin{array}[]{l}\mbox{finite linearly reductive}\\\ \mbox{group
schemes over
$k$}\end{array}\right\\}\quad\leftrightarrow\quad\left\\{\begin{array}[]{l}\mbox{finite
groups with a}\\\ \mbox{unique $p$-Sylow subgroup}\end{array}\right\\},$

see [LMM21] and Lemma 2.2.

####  1.3.2. Canonical lifts

In [LMM21], we showed that if $G$ is a finite and linearly reductive group
scheme, then there exists a lift of $G$ over the ring of Witt vectors $W(k)$.
We note that $G^{\circ}$ and $G^{\mathrm{\acute{e}t}}$ even lift uniquely to
$W(k)$ and we define the _canonical lift_ $G_{\rm can}\to\mathrm{Spec}\>K$ of
$G$ to be the unique lift that is a semi-direct product of the lifts of
$G^{\circ}$ and $G^{\mathrm{\acute{e}t}}$. Any lift of $G$ to some extension
field of $K$ becomes isomorphic to $G_{\mathrm{can}}$ after possibly passing
to some further field extension. Moreover, the finite group
$G_{\mathrm{can}}(\overline{K})$ is isomorphic to $G_{\rm abs}$.

####  1.3.3. Representation theory

By [LMM21] and Proposition 2.6, there exist canonical equivalences of
representation categories

(2)
$\mathrm{Rep}_{k}(G)\,\to\,\mathrm{Rep}_{\overline{K}}(G_{\mathrm{can},\overline{K}})\,\to\,\mathrm{Rep}_{{\mathbb{C}}}(G_{\mathrm{abs}})$

that are compatible with degrees, direct sums, tensor products, duals, and
simplicity. These equivalences induce isomorphisms of rings

$K_{k}(G)\,\to\,K_{\overline{K}}(G_{\mathrm{can},\overline{K}})\,\to\,K_{{{\mathbb{C}}}}(G_{\mathrm{abs}}),$

see Corollary 2.8. Here, $K_{F}(G)$ denotes the K-group associated to
$F$-linear and finite-dimensional $G$-representations.

####  1.3.4. Hopf algebras

If $G$ is a finite group scheme over $k$, then the multiplication map turns
$H^{0}(G,{\cal O}_{G})$ into a finite-dimensional _Hopf algebra_ over $k$. We
discuss finite group schemes and among them the linearly reductive ones from
the point of Hopf algebras in Appendix A.

###  1.4. Linearly reductive subgroup schemes of ${\mathbf{SL}}_{2,k}$

Let $k$ be an algebraically closed field of characteristic $p\geq 0$.
Hashimoto [Ha15] extended Klein’s classification [Kl84] of finite subgroups of
${\mathbf{SL}}_{2}({{\mathbb{C}}})$ up to conjugation to the setting of finite
and linearly reductive subgroup schemes of ${\mathbf{SL}}_{2,k}$. If $p\geq
7$, then one obtains a list analogous to Klein’s classical list. If
$p\in\\{2,3,5\\}$, then some classical cases are missing, but there are no new
cases.

###  1.5. McKay graph and McKay correspondence

Let $G$ be a finite and linearly reductive subgroup scheme of
${\mathbf{SL}}_{2,k}$. As in McKay’s original construction [Mc80], we
associate an affine Dynkin diagram $\widehat{\Gamma}$ to $G$, its embedding
into ${\mathbf{SL}}_{2,k}$, and the set of isomorphism classes of simple
representations of $G$. This is the _McKay graph_ associated to this data. In
fact, we will see that it is compatible with the equivalences induced by (1)
and (2) and we refer to Section 3.1 for details. We establish the following
version of McKay’s theorem [Mc80] in positive characteristic.

######  Theorem 1.1 (Theorem 3.6).

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. There
exists a bijection between non-trivial, finite, and linearly reductive
subgroup schemes of ${\mathbf{SL}}_{2,k}$ up to conjugation and affine Dynkin
graphs of type

$\begin{array}[]{lcl}\widehat{A}_{n},\widehat{D}_{n},\widehat{E}_{6},\widehat{E}_{7},\widehat{E}_{8}&\mbox{\quad
if \quad}&p=0\mbox{\quad or \quad}p\geq 7,\\\
\widehat{A}_{n},\widehat{D}_{n},\widehat{E}_{6},\widehat{E}_{7}&\mbox{\quad if
\quad}&p=5,\\\ \widehat{A}_{n},\widehat{D}_{n}&\mbox{\quad if \quad}&p=3,\\\
\widehat{A}_{n}&\mbox{\quad if \quad}&p=2.\end{array}$

By construction, this bijection is compatible with the classical McKay
correspondence via the lifting results and the equivalences (1), (2).

######  Example 1.2.

The linearly reductive group scheme corresponding to $\widehat{A}_{n}$ is
$\boldsymbol{\mu}_{n+1}$. This group scheme is reduced, that is, étale, if and
only if $p$ does not divide $n+1$. In particular, it is crucial to allow non-
reduced group schemes in order to obtain a bijection as in characteristic
zero.

###  1.6. Linearly reductive quotient singularities

Consider ${\mathbf{GL}}_{2,k}$ with its usual linear action on
$U={{\mathbb{A}}}^{2}_{k}$ or $U=\widehat{{{\mathbb{A}}}}_{k}^{2}$. If $G$ is
a finite, linearly reductive, and _very small_ (see Definition 4.1) subgroup
scheme of ${\mathbf{GL}}_{2,k}$, then the associated quotient singularity
$x\in X:=U/G$ is a two-dimensional _linearly reductive quotient singularity_
in the sense of [LMM21]. By loc.cit., such a singularity determines $G$
together with its embedding $G\to{\mathbf{GL}}_{2,k}$ up to isomorphism and
conjugation, respectively.

In Section 4.3, we will see that a minimal resolution of singularities of a
two-dimensional linearly reductive quotient singularity $x\in X=U/G$ is
provided by the $G$-Hilbert scheme

(3) $\pi\,:\,G\mathrm{-Hilb}(U)\,\to\,U/G,$

which generalises work of Ishii, Ito, and Nakamura [IN19, IN99].

######  Remark 1.3.

In dimension two, a linearly reductive quotient singularity is the same as an
F-regular singularity [LMM21]. Thus, every two-dimensional F-regular
singularity can be resolved by a suitable $G$-Hilbert scheme, see also Remark
4.6.

If moreover $G$ is a subgroup scheme of ${\mathbf{SL}}_{2,k}$, then $x\in
X=U/G$ is called a _Klein singularity_. Klein singularities are rational
double point singularities. If $p=0$ or $p\geq 7$, then conversely every
rational double point is a Klein singularity by Hashimoto [Ha15] and,
independently, by [LS14]. If $p\in\\{2,3,5\\}$, then not every rational double
point is a Klein singularity.

###  1.7. Canonical lifts and simultaneous resolutions

Let $G$ be a very small, finite, and linearly reductive subgroup scheme of
${\mathbf{GL}}_{2,k}$ and let $x\in X=U/G$ be the associated linearly
reductive quotient singularity. Assume $p>0$ and let $W(k)$ be the ring of
Witt vectors of $k$. In Section 4.4, we will establish the existence of a
_canonical lift_

$\mathcal{X}_{\mathrm{can}}\,\to\,\mathrm{Spec}\>W(k)$

of $x\in X=U/G$. Using $G$-Hilbert schemes in families, we will see in Section
4.5 that it admits a simultaneous and minimal resolution of singularities

$\widetilde{\pi}\,:\,\mathcal{Y}\,\to\,\mathcal{X}_{\mathrm{can}}\,\to\,\mathrm{Spec}\>W(k).$

We will prove this resolution to be unique, see Theorem 4.10.

###  1.8. McKay correspondence for Klein singularities

Let $G$ be a finite and linearly reductive subgroup scheme of
${\mathbf{SL}}_{2,k}$ and let $x\in X=U/G$ be the associated Klein
singularity.

As discussed in Section 1.5, we have the McKay graph $\widehat{\Gamma}$
associated to $G$, its embedding into ${\mathbf{SL}}_{2,k}$, and the set of
isomorphism classes of simple representations of $G$.

Let $\pi:Y\to X$ be a minimal resolution of singularities. Since $x\in X$ is a
rational double point, the exceptional divisor $\mathrm{Exc}(\pi)$ of $\pi$ is
a configuration of ${{\mathbb{P}}}^{1}$’s, whose dual intersection graph
$\Gamma$ is a simply-laced Dynkin diagram.

######  Theorem 1.4 (Theorem 5.3).

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. Let $G$
be a finite and linearly reductive subgroup scheme of ${\mathbf{SL}}_{2,k}$
and let $x\in X=U/G$ be the associated Klein singularity. Then, there exists a
natural bijection of the graph $\Gamma$ with the graph obtained from
$\widehat{\Gamma}$ by removing the vertex corresponding to the trivial
representation.

Since every rational double point in characteristic $p\geq 7$ is a Klein
singularity, we obtain the following.

######  Corollary 1.5.

There exists a linearly reductive McKay correspondence for rational double
point singularities in every characteristic $p\geq 7$.

To establish this theorem, we use the Ishii-Ito-Nakamura resolution of
singularities (3), as well as Hecke correspondences as in the work of Ito and
Nakamura [IN99] and Nakajima [Na96, Na01].

###  1.9. Generalisations and variants

Let $G$ be a very small, finite, and linearly reductive subgroup scheme of
${\mathbf{GL}}_{2,k}$ and let $x\in X=U/G$ be the associated two-dimensional
linearly reductive quotient singularity. Let $\pi:Y\to X$ be the minimal
resolution of singularities and let $\mathrm{Exc}(\pi)$ be the exceptional
divisor of $\pi$.

  1. (1)

In Theorem 5.4, we associate a representation of $G$ to each point of
$\mathrm{Exc}(\pi)$. This generalises results of Ishii and Nakamura [Is02,
IN19].

  2. (2)

In Theorem 5.6, we establish a bijection between the components of
$\textrm{Exc}(\pi)$ and reflexive ${\cal O}_{X}$-modules. Probably, this
result should be viewed as a theorem on two-dimensional F-regular
singularities, see Remark 5.7. It generalises work of Artin and Verdier
[AV85], Wunram [Wu88], and Ishii and Nakamura [IN19].

  3. (3)

In Theorem 7.1, we establish an equivalence of derived categories of coherent
sheaves ${\mathcal{D}}(Y)$ and ${\mathcal{D}}^{G}(U)$ ($G$-equivariant sheaves
on $U$). This generalises work of Kapranov and Vasserot [KV00], Bridgeland,
King, and Reid [BKR01], and Ishii, Nakamura, and Ueda [Is02, IN19, IU15]. Of
course, these articles themselves generalise work of Gonzalez-Sprinberg and
Verdier [GV83] from K-theory to derived categories of coherent sheaves.

###  1.10. Ito-Reid correspondence

If $G$ is a finite subgroup of ${\mathbf{SL}}_{2}({{\mathbb{C}}})$, then Ito
and Reid [IR96] found a natural bijection, the _Ito-Reid correspondence_ ,
between the conjugacy classes of $G$ and the vertices of the McKay graph
$\widehat{\Gamma}$.

######  Theorem 1.6.

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. There
exists an Ito-Reid correspondence for finite and linearly reductive subgroup
schemes of ${\mathbf{SL}}_{2,k}$.

This can be proven using lifting results and the Ito-Reid correspondence over
${{\mathbb{C}}}$, see Section 6.3 for details.

The main difficulty is to define a notion of conjugacy class for finite and
linearly reductive group schemes that makes such a correspondence work: Let us
recall the ring $K_{k}(G)$ from Section 1.3.3. If $G$ is a finite and linearly
reductive group scheme over $k$, then we define the set of _conjugacy classes_
of $G$ to be

$\mathrm{Spec}\>\left({{\mathbb{C}}}\otimes K_{k}(G)\right).$

At first sight, this might look rather artificial. One should think of it as
defining conjugacy classes to be “dual” to simple representations. Moreover,
our definition is compatible with lifting over $W(k)$ and it induces a
bijection of the conjugacy classes of $G$ with the conjugacy classes of
$G_{\rm abs}$. We refer to Section 6.1 and Appendix B.2 for details.

###  1.11. Conjugacy classes

In Theorem 1.6, we had to find a definition of conjugacy classes that makes
this theorem true. This begs for the question whether there are other
definitions or approaches, which is a question that is interesting in its own.

Let $G$ be a finite group scheme (not necessarily linearly reductive) over an
algebraically closed field $k$. In Appendix B, we study the following
approaches to the notion of the set of conjugacy classes of $G$:

  1. (1)

The set of conjugacy classes $G(k)/\sim$ of the group of $k$-rational points
of $G$.

  2. (2)

The spectrum of $F\otimes K_{k}(G)$, where $F$ is a field of characteristic
zero that contains “sufficiently many” roots of unity.

  3. (3)

The scheme that represents the functor of conjugacy classes from schemes over
$k$ to sets defined by $S\mapsto G(S)/\sim$,

  4. (4)

the isotypical component of the trivial representation of the adjoint
representation of $G$.

  5. (5)

The simple subrepresentations of the extended adjoint representations
${}^{\mathrm{Ad}}A$ and ${}^{\mathrm{Ad}}(A^{*})$, which are defined using the
quantum doubles of the Hopf algebra $A:=H^{0}(G,{\cal O}_{G})$ and its dual
$A^{*}$.

All approaches lead essentially to the “same answer” in characteristic zero.
However, they usually lead to very different notions in positive
characteristic. On the other hand, all approaches have their merits and
drawbacks. For linearly reductive group schemes, the approach (2) leads to a
definition that is compatible with lifting and that leads to an Ito-Reid
correspondence.

###  1.12. Organisation of this article

  1. -

In Section 2, we recall basic facts about finite and linearly reductive group
schemes over algebraically closed fields.

  2. -

In Section 3, we construct the McKay graph $\widehat{\Gamma}$. The main result
is Theorem 3.6, a McKay correspondence.

  3. -

In Section 4, we recall basic facts about linearly reductive quotient
singularities. We establish an Ishii-Ito-Nakamura-type resolution of
singularities, the canonical lift of such a singularity, and a unique
simultaneous resolution of singularities of the canonical lift.

  4. -

In Section 5, we revisit the Ishii-Ito-Nakamura-type resolution $\pi$ of
singularities of $x\in X=U/G$ and we introduce Hecke correspondences, which
leads to Theorem 5.3, a bijection between simple and non-trivial
representations of $G$ and components of $\pi$. We also study generalisations
of this result to two-dimensional linearly reductive quotient singularities.

  5. -

In Section 6, we establish an Ito-Reid correspondence between conjugacy
classes of $G$ and exceptional divisors in the minimal resolution of
singularities of $x\in X=U/G$.

  6. -

In Section 7, we study derived categories of $G$-equivariant sheaves on $U$
and on the minimal resolution $\pi:Y\to X=U/G$.

  7. -

In Appendix A, we recall results on finite group schemes from the point of
view of Hopf algebras. We recall the adjoint representation, quantum doubles,
and the extended adjoint representation.

  8. -

In Appendix B, we study several approaches toward the notion of a conjugacy
class for finite group schemes.

###### Acknowledgements.

I thank Frank Himstedt, Martin Lorenz, Frans Oort, Matt Satriano, and Takehiko
Yasuda for discussions and comments.

##  2\. Linearly reductive group schemes

In this section, we recall a couple of general facts about finite and linearly
reductive group schemes over algebraically closed fields. We discuss the close
relationship between such a group scheme $G$ and a certain abstracted group
$G_{\mathrm{abs}}$ associated to it. For the relationship to Hopf algebras, we
refer to Appendix A.

###  2.1. Group schemes

Let $G$ be a finite group scheme over an algebraically closed field $k$ of
characteristic $p\geq 0$. Since $k$ is perfect, there is a short exact
sequence of finite group schemes over $k$

(4) $1\,\to\,G^{\circ}\,\to\,G\,\to\,G^{\mathrm{\acute{e}t}}\,\to\,1,$

where $G^{\circ}$ is the connected component of the identity and where
$G^{\mathrm{\acute{e}t}}$ is an étale group scheme over $k$. The reduction
$G_{\mathrm{red}}\to G$ provides a canonical splitting of (4) and we obtain a
canonical semi-direct product decomposition $G\cong G^{\circ}\rtimes
G^{\mathrm{\acute{e}t}}$. Since $k$ is algebraically closed,
$G^{\mathrm{\acute{e}t}}$ is the constant group scheme associated to the
finite group $G(k)=G^{\mathrm{\acute{e}t}}(k)$ of $k$-rational points.
Moreover, $G^{\circ}$ is an infinitesimal group scheme of length equal to some
power of $p$. In particular, if $p=0$ or if the length of $G$ is prime to $p$,
then $G^{\circ}$ is trivial and then, $G$ is étale.

If $M$ is a finitely generated abelian group, then the group algebra $k[M]$
carries a Hopf algebra structure and the associated commutative group scheme
is denoted $D(M):=\mathrm{Spec}\>k[M]$. By definition, such group schemes are
called _diagonalisable_. For example, we have
$D({\mathbf{C}}_{n})\cong\boldsymbol{\mu}_{n}$, where ${\mathbf{C}}_{n}$
denotes the cyclic group of order $n$. We have that $\boldsymbol{\mu}_{n}$ is
étale over $k$ if and only if $p\nmid n$.

A finite group scheme $G$ over $k$ is said to be _linearly reductive_ if every
$k$-linear and finite-dimensional representation of $G$ is semi-simple. If
$p=0$, then all finite group schemes over $k$ are étale and linearly
reductive. If $p>0$, then, by a theorem that is often attributed to Nagata
[Na61, Theorem 2] (but see also [AOV08, Proposition 2.10], [Ch92], and [Ha15,
Section 2]), a finite group scheme over $k$ is linearly reductive if and only
if it is an extension of a finite and étale group scheme, whose length is
prime to $p$, by a diagonalisable group scheme.

###  2.2. Abstract groups and canonical lifts

Let $G$ be a finite and linearly reductive group scheme over an algebraically
closed field $k$ of characteristic $p>0$. Following [LMM21, Section 2], we
study the finite group

$G_{\mathrm{abs}}\,:=\,\left(\underline{((G^{\circ})^{D}(k))}_{{{\mathbb{C}}}}\right)^{D}({{\mathbb{C}}})\,\rtimes\,G^{\mathrm{\acute{e}t}}(k).$

Here, $-^{D}$ denotes $\underline{\mbox{Hom}}(-,{{\mathbb{G}}}_{m})$, the
_Cartier dual_ , of a commutative and finite group scheme. If $G$ is étale,
then $G_{\mathrm{abs}}=G(k)$ and if $G=\boldsymbol{\mu}_{p^{n}}$, then
$G_{\mathrm{abs}}={\mathbf{C}}_{p^{n}}$. In any case, the order of
$G_{\mathrm{abs}}$ is equal to the length of $G$.

######  Definition 2.1.

The finite group $G_{\mathrm{abs}}$ is called the _abstract finite group_
associated to $G$.

######  Lemma 2.2.

The functor

$G\,\mapsto\,G_{\mathrm{abs}}$

establishes an equivalence of categories between the category of finite and
linearly reductive group schemes over $k$ and the category of finite groups
with a normal and abelian $p$-Sylow subgroup.

###### Proof.

[LMM21, Lemma 2.1]. ∎

######  Remark 2.3.

A finite group $G$ with a normal and abelian $p$-Sylow subgroup $P$ is the
same as a finite group with a unique and abelian $p$-Sylow subgroup. In this
case, the Schur-Zassenhaus theorem implies that $G$ is isomorphic to a semi-
direct product $P\rtimes G/P$.

Next, we study lifts to characteristic zero: let $W(k)$ be the ring of Witt
vectors of $k$, let $K$ be its field of fractions, and let $\overline{K}$ be
an algebraic closure of $K$. By [LMM21, Proposition 2.4], there exist lifts of
$G$ as finite and flat group scheme over $W(k)$. More precisely, $G^{\circ}$
and $G^{\mathrm{\acute{e}t}}$ even lift uniquely to $W(k)$, but their
extension class usually does not, see also [LMM21, Example 2.6]. However,
there is a unique lift ${\cal G}_{\mathrm{can}}$ of $G$ to $W(k)$ that is
characterised by being a semi-direct product of the unique lift of $G^{\circ}$
with the unique lift of $G^{\mathrm{\acute{e}t}}$.

######  Definition 2.4.

The lift $\mathcal{G}_{\mathrm{can}}\to\mathrm{Spec}\>W(k)$ is called the
_canonical lift_ of $G$. We set
$G_{\mathrm{can}}:=\mathcal{G}_{K}\to\mathrm{Spec}\>K$ and also call it
canonical lift.

Every other lift of $G$ to some extension of $R\supseteq W(k)$ differs from
${\cal G}_{\mathrm{can},R}$ by a twist and thus, there is only one _geometric
lift_ of $G$ to $\overline{K}$ up to isomorphism, namely
$G_{\mathrm{can},\overline{K}}$, see [LMM21, Section 2.2].

Since $\overline{K}$ is algebraically closed and of characteristic zero,
$G_{\mathrm{can},\overline{K}}$ is the constant group scheme associated to the
finite group $G_{\mathrm{can}}(\overline{K})$. In fact, we have
$G^{\circ}\cong\prod_{i}\boldsymbol{\mu}_{p^{n_{i}}}$ for some $n_{i}$’s and
if we set $N:=\max\\{n_{i}\\}_{i}$, fix a primitive $p^{N}$.th root of unity
$\zeta_{p^{N}}$ and set $K_{N}:=K(\zeta_{p^{N}})$, then
$G_{\mathrm{can},K_{N}}$ is the constant group scheme associated to the finite
group $G_{\mathrm{can}}(K_{N})$ and we have
$G_{\mathrm{can}}(K_{N})=G_{\mathrm{can}}(\overline{K})$.

######  Lemma 2.5.

There exist isomorphism of finite groups

$G_{\mathrm{abs}}\,\cong\,G_{\mathrm{can}}(K_{N})\,=\,G_{\mathrm{can}}(\overline{K}).$

In particular, there exist isomorphisms

$G_{\mathrm{can},K_{N}}\,\cong\,\left(\underline{G_{\mathrm{abs}}}\right)_{K_{N}}\mbox{
\quad and
\quad}G_{\mathrm{can},\overline{K}}\,\cong\,\left(\underline{G_{\mathrm{abs}}}\right)_{\overline{K}}$

of finite group schemes over $K_{N}$ and $\overline{K}$, respectively.∎

###  2.3. Representation theory

Let $k$ be an algebraically closed field of characteristic $p>0$, and let
$W(k)$, $K$, and $\overline{K}$ be as in the previous section. The equivalence
of Lemma 2.2 can be extended to representations and K-theory. If $G$ is a
finite group or a finite group scheme over $k$, we denote by ${\rm
Rep}_{k}(G)$ the category of its $k$-linear and finite-dimensional
representations. Let us first recall [LMM21, Corollary 2.11].

######  Proposition 2.6.

Let $G$ be a finite and linearly reductive group scheme over $k$. Then, there
exist canonical equivalences of categories

$\mathrm{Rep}_{k}(G)\,\to\,\mathrm{Rep}_{K_{N}}(G_{\mathrm{can},K_{N}})\,\to\,\mathrm{Rep}_{\overline{K}}(G_{\mathrm{can},\overline{K}})\,\to\,\mathrm{Rep}_{{\mathbb{C}}}(G_{\mathrm{abs}}),$

which are compatible with degrees, direct sums, tensor products, duals, and
simplicity.∎

######  Remark 2.7.

In particular, this allows us to define _characters_ or even a _character
table_ of $G$ via Proposition 2.6 and $G_{\mathrm{abs}}$.

Let $K_{k}(G)$ be $K$-group associated to ${\rm Rep}_{k}(G)$. In fact,
$K_{k}(G)$ has a natural structure of a commutative ring with one, where the
sum (resp. product) structure comes from direct sums (resp. tensor products)
of representations. A straight forward application of Proposition 2.6 is the
following.

######  Corollary 2.8.

There exist isomorphisms of rings

$K_{k}(G)\,\to\,K_{K_{N}}(G_{\mathrm{can},K_{N}})\,\to\,K_{\overline{K}}(G_{\mathrm{can},\overline{K}})\,\to\,K_{{{\mathbb{C}}}}(G_{\mathrm{abs}}).\qed$

##  3\. McKay graph and McKay correspondence

In this section, we introduce the _McKay graph_ associated to a finite and
linearly reductive subgroup scheme $G$ over an algebraically closed field $k$
of characteristic $p\geq 0$ and a representation
$\rho:G\to{\mathbf{GL}}_{n,k}$. This induces a bijection between certain
affine Dynkin diagrams and finite and linearly reductive subgroup schemes of
${\mathbf{SL}}_{2,k}$. As an application, we establish a _linearly reductive
McKay correspondence_.

###  3.1. McKay graph

Let $G$ be a finite and linearly reductive group scheme over an algebraically
closed field $k$ of characteristic $p\geq 0$. Let $\\{\rho_{i}\\}_{i}$ be the
finite set of isomorphism classes of $k$-linear and simple representations of
$G$. Following tradition, we assume that $\rho_{0}$ is the trivial
representation. We fix a representation $\rho:G\to{\mathbf{GL}}(V)$. If $G$ is
a subgroup scheme of ${\mathbf{SL}}_{n,k}$ or ${\mathbf{GL}}_{n,k}$, then
$\rho$ is usually the linear representation corresponding to the embedding of
$G$ into this linear algebraic group. By assumption, $\mathrm{Rep}_{k}(G)$ is
semi-simple. Therefore, there exist unique integers
$a_{ij}\in{{\mathbb{Z}}}_{\geq 0}$ for each $i$, such that we have
isomorphisms of $k$-linear representations

$\rho\otimes\rho_{i}\,\cong\,\bigoplus_{j}\,\rho_{j}^{\oplus a_{ij}}.$

Associated to this data, we define the _McKay graph_ , denoted
$\Gamma(G,\\{\rho_{i}\\},\rho)$:

  1. -

The vertices are the $\\{\rho_{i}\\}_{i}$. (Some sources exclude the the
trivial representation $\rho_{0}$.)

  2. -

There are $a_{ij}$ edges from the vertex corresponding to $\rho_{i}$ to the
vertex corresponding to $\rho_{j}$.

We now establish a couple of elementary properties of this graph, which are
well-known in the classical case and which immediately carry over to the
linearly reductive situation. We leave the proof of the first lemma to the
reader.

######  Lemma 3.1.

We have

$a_{ij}\,=\,\dim_{k}\,\mathrm{Hom}(\rho_{i},\rho_{j}\otimes\rho).$

In particular, if $\rho$ is self-dual, that is, $\rho\cong\rho^{\vee}$, then
$a_{ij}=a_{ji}$ for all $i,j$. In this case, we can consider
$\Gamma(G,\\{\rho_{i}\\},\rho)$ as an undirected graph.∎

######  Lemma 3.2.

Let $\rho:G\to{\mathbf{SL}}_{2,k}$ be a homomorphism of group schemes over
$k$, considered as a 2-dimensional representation. Then, $\rho$ is self-dual.

###### Proof.

Being a 2-dimensional representation, $\rho^{\vee}$ is isomorphic to
$\rho\otimes\det(\rho)$ and the lemma follows. ∎

######  Lemma 3.3.

Let $\rho:G\to{\mathbf{GL}}_{n,k}$ be a faithful representation. Then, every
irreducible representation of $G$ occurs as subrepresentation of
$\rho^{\otimes m}$ for some suitable $m$.

###### Proof.

This is well-known for finite groups. Using Proposition 2.6, it carries over
to finite and linearly reductive group schemes. ∎

######  Corollary 3.4.

If $\rho$ is a faithful representation, then the graph
$\Gamma(G,\\{\rho_{i}\\},\rho)$ is connected.

###### Proof.

It suffices to note that the number of paths in $\Gamma$ of length $m$ that
connect the vertices corresponding to $\rho_{i}$ and $\rho_{j}$ is equal to
the multiplicity of $\rho_{i}$ in $\rho_{j}\otimes\rho^{\otimes m}$, see, for
example, the proof of [Ki16, Theorem 8.13]. ∎

Let $k$ be an algebraically closed field of characteristic $p>0$, and let
$W(k)$, $K$, and $\overline{K}$ be as in Section 2.2. There, we also discussed
the canonical lift $G_{\mathrm{can}}$ of $G$ over $K$ and we saw that there
exists an isomorphism of finite groups $G_{\mathrm{abs}}\cong
G_{\mathrm{can}}(\overline{K})$. Proposition 2.6 implies the following result.

######  Proposition 3.5.

Let $G$ be a finite and linearly reductive group scheme over $k$. Let
$\\{\rho_{i}\\}_{i}$ be the set of isomorphism classes of simple
representations of $G$. Let $\rho$ be a finite-dimensional representation of
$G$.

  1. (1)

Proposition 2.6 yields sets of representations
$\\{\rho_{\mathrm{can},i}\\}_{i}$ and $\\{\rho_{\mathrm{abs},i}\\}_{i}$ of
$G_{\mathrm{can}}$ and $G_{\mathrm{abs}}$, respectively, which are the sets of
isomorphism classes of simple representations of $G_{\mathrm{can}}$ and
$G_{\mathrm{abs}}$, respectively.

  2. (2)

Proposition 2.6 yields representations $\rho_{\mathrm{can}}$ and
$\rho_{\mathrm{abs}}$ of $G_{\mathrm{can}}$ and $G_{\mathrm{abs}}$,
respectively.

This data leads to a bijection of the three McKay graphs

$\Gamma(G,\\{\rho_{i}\\}_{i},\rho),\quad\Gamma(G_{\mathrm{can}},\\{\rho_{\mathrm{can},i}\\}_{i},\rho_{\mathrm{can}}),\quad\Gamma(G_{\mathrm{abs}},\\{\rho_{\mathrm{abs},i}\\}_{i},\rho_{\mathrm{abs}}).\qed$

###  3.2. McKay correspondence

We now run through the classical McKay correspondence [Mc80] in our setting,
where we follow [Ki16, Section 8.3]. Let $k$ be an algebraically closed field
of characteristic $p\geq 0$ and let $G$ be a finite and linearly reductive
subgroup scheme of ${\mathbf{SL}}_{2,k}$.

  1. (1)

The K-group $K_{k}(G)$ carries a symmetric bilinear form

$([V],[W])_{0}\,:=\,\dim_{k}\,\mathrm{Hom}_{G}(V,W).$

  2. (2)

We consider the closed embedding $G\to{\mathbf{SL}}_{2,k}$ as a 2-dimensional
representation $\rho:G\to{\mathbf{SL}}_{2,k}\to{\mathbf{GL}}_{2,k}$ and define
an operator

$A\,:\,K_{k}(G)\,\to\,K_{k}(G),\mbox{\qquad}[V]\mapsto[V]\otimes\rho\,.$

Since $\rho$ is self-dual by Lemma 3.2, it follows that $A$ is symmetric with
respect to $(-,-)_{0}$.

  3. (3)

Using $\rho$, we define a symmetric bilinear form on
$K_{k}(G)\otimes_{{\mathbb{Z}}}{{\mathbb{R}}}$ by

$([V],[W])\,:=\,\left([V],\,(2-A)[W]\right)_{0},$

which is positive semi-definite. The class $\delta\in K_{k}(G)$ of the regular
representation of $G$ generates the radical of $(-,-)$.

  4. (4)

Let $\\{\rho_{i}\\}$ be the set of isomorphism classes of simple
representations of $G$.

    1. (a)

The McKay graph $\widehat{\Gamma}=\Gamma(G,\\{\rho_{i}\\}_{i},\rho)$ is an
affine Dynkin diagram of type $\widehat{A}_{n}$, $\widehat{D}_{n}$ with $n\geq
4$, $\widehat{E}_{6}$, $\widehat{E}_{7}$ or $\widehat{E}_{8}$.

    2. (b)

After removing the vertex corresponding to the trivial representation
$\rho_{0}$ from $\widehat{\Gamma}$, we obtain a finite Dynkin diagram of type
$A_{n}$, $D_{n}$ with $n\geq 4$, $E_{6}$, $E_{7}$ or $E_{8}$, respectively.

If $G$ is a finite subgroup of ${\mathbf{SL}}_{2}({{\mathbb{C}}})$, then these
statements are part of the classical McKay correspondence, see [Mc80] or
[Ki16, Section 8.3]. In our setting of finite and linearly reductive subgroup
schemes of ${\mathbf{SL}}_{2,k}$, the above claims immediately follow from the
classical McKay correspondence together with the lifting results Proposition
2.6 and Proposition 3.5. We then obtain the following analog of McKay’s
theorem [Mc80].

######  Theorem 3.6.

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. There
exists a bijection between finite, non-trivial, and linearly reductive
subgroup schemes of ${\mathbf{SL}}_{2,k}$ up to conjugation and affine Dynkin
diagrams of type

$\begin{array}[]{lcl}\widehat{A}_{n},\widehat{D}_{n},\widehat{E}_{6},\widehat{E}_{7},\widehat{E}_{8}&\mbox{\quad
if \quad}&p=0\mbox{\quad or \quad}p\geq 7,\\\
\widehat{A}_{n},\widehat{D}_{n},\widehat{E}_{6},\widehat{E}_{7}&\mbox{\quad if
\quad}&p=5,\\\ \widehat{A}_{n},\widehat{D}_{n}&\mbox{\quad if
\quad}&p=3,\mbox{ and}\\\ \widehat{A}_{n}&\mbox{\quad if
\quad}&p=2.\end{array}$

###### Proof.

If $p=0$, then this is part of the classical McKay correspondence. If $p>0$,
then this follows from the linearly reductive McKay correspondence just
discussed together with Hashimoto’s classification [Ha15, Theorem 3.8] of
finite and linearly reductive subgroup schemes of ${\mathbf{SL}}_{2,k}$. ∎

######  Remark 3.7.

The linearly reductive group schemes corresponding to $\widehat{E}_{6}$,
$\widehat{E}_{7}$, and $\widehat{E}_{8}$ are étale and correspond to finite
groups of order prime to $p$. The linearly reductive group scheme
corresponding to $\widehat{A}_{n}$ (resp. $\widehat{D}_{n}$) is étale if and
only if $p\nmid(n+1)$ (resp. $p\nmid(n-2)$). We refer to [LS14, Proposition
4.2] for details. Thus, even if $p$ is sufficiently large, then one does not
obtain a bijection in Theorem 3.6 with finite groups of order to prime to $p$
only.

######  Remark 3.8.

Steinberg [St85] established many properties of the McKay graph and the McKay
correspondence for finite subgroups of ${\mathbf{SU}}_{2}({{\mathbb{C}}})$
_without_ using classification lists. It seems plausible to obtain a proof of
Theorem 3.6 along these lines without using lifting results or classification
lists.

##  4\. Linearly reductive quotient singularities

In this section, we recall _linearly reductive quotient singularities_ in the
sense of [LMM21] and some general results, including the _canonical lift_ of
such a singularity over the ring of Witt vectors. In dimension two, we
establish a minimal resolution of singularities using $G$-Hilbert schemes as
in the work of Ishii, Ito, and Nakamura [Is02, IN19, IN99]. We also show that
the canonical lift admits a unique minimal and simultaneous resolution of
singularities. Finally, we discuss _Klein singularities_ and their relation to
rational double point singularities.

###  4.1. Quotient singularities

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. If $G$
is a finite and linearly reductive group scheme over $k$, if $V$ is a finite-
dimensional $k$-vector space, and if $\rho:G\to{\mathbf{GL}}(V)$ is a linear
representation, then we define the _$\lambda$ -invariant_ of $\rho$ as in
[LMM21, Definition 2.7] to be

$\lambda(\rho)\,:=\,\max_{\\{{\mathrm{id}}\\}\neq\boldsymbol{\mu}_{n}\subseteq
G}\dim V^{\boldsymbol{\mu}_{n}},$

where $V^{\boldsymbol{\mu}_{n}}$ denotes the subspace of
$\boldsymbol{\mu}_{n}$-invariants. As explained in [LMM21, Remark 2.8], the
representation $\rho$ is faithful if and only if $\lambda(\rho)\neq\dim V$.
Moreover, $\rho$ contains no pseudo-reflections if and only if
$\lambda(\rho)\leq\dim V-2$ and in this case, the representation $\rho$ is
said to be _small_.

######  Definition 4.1.

The representation $\rho$ is called _very small_ if $\lambda(\rho)=0$.

Note that in dimension two the notions of small and very small coincide. We
refer to [LMM21, Section 3] for the classification of finite and linearly
reductive group schemes that admit a very small representation. By [LMM21,
Proposition 6.5], $\lambda(\rho)$ is equal to the dimension of the non-free
locus of the induced $G$-action on the spectrum of the formal power series
ring $k[[V^{\vee}]]=(k[V^{\vee}])^{\wedge}$. Following [LMM21, Definition 6.4]
and using the linearisation result [LMM21, Proposition 6.3], we define:

######  Definition 4.2.

A _linearly reductive quotient singularity_ over $k$ is an isolated
singularity that is analytically isomorphic to
$\mathrm{Spec}\>k[[V^{\vee}]]^{G}$, where $G$ is a finite and linearly
reductive group scheme over $k$ and where $\rho:G\to{\mathbf{GL}}(V)$ is a
very small representation.

######  Remark 4.3.

We set $U:=\mathrm{Spec}\>k[V^{\vee}]$ or $U:=\mathrm{Spec}\>k[[V^{\vee}]]$
and simply write $x\in X=U/G$ with the assumptions of Definition 4.2
implicitly understood.

Properties of these singularities have been studied in [LMM21]: for their
local étale fundamental groups, class groups, and F-signatures, we refer to
[LMM21, Section 7].

By [LMM21, Theorem 8.1], a linearly reductive quotient singularity determines
the finite and linearly reductive group scheme $G$ together with the very
small representation $\rho:G\to{\mathbf{GL}}(V)$ uniquely up to isomorphism
and conjugacy, respectively. It thus makes sense to refer to
$\rho(G)\subset{\mathbf{GL}}(V)$ as the _finite and linearly reductive
subgroup scheme of ${\mathbf{GL}}(V)$ associated to the linearly reductive
quotient singularity_ $x\in X=U/G$. In particular, the classification of
linearly reductive quotient singularities in dimension $d$ is “the same” as
the classification of very small, finite, and linearly reductive subgroup
schemes of ${\mathbf{GL}}_{d,k}$ up to conjugacy. We refer to [LMM21, Section
3] for details and this classification.

###  4.2. Minimal resolution of singularities

If $x\in X$ is a normal surface singularity, then it admits a unique minimal
resolution of singularities

$\pi\,:\,Y\,\to\,X.$

If $x\in X$ is moreover a rational singularity, then the exceptional locus of
$\pi$ is a union of ${{\mathbb{P}}}^{1}$’s, whose dual intersection graph
$\Gamma$ contains no cycles. The graph $\Gamma$ is called the _type_ of $x\in
X$. If the type determines the singularity up to analytic isomorphism, then
the singularity is said to be _taut_.

Over ${{\mathbb{C}}}$, taut singularities have been classified by Laufer
[La73]. For example, two-dimensional finite quotient singularities over
${{\mathbb{C}}}$ are taut. Since two-dimensional linearly reductive quotient
singularities over algebraically closed fields of positive characteristic are
F-regular (see [LMM21, Proposition 7.1]), they are taut by Tanaka’s theorem
[Ta15].

######  Remark 4.4.

Very small, finite, and linearly reductive subgroup schemes of
${\mathbf{GL}}_{2,k}$ have been classified in [LMM21, Theorem 3.4], extending
Brieskorn’s classification [Br67, Satz 2.9] of small subgroups of
${\mathbf{GL}}_{2}({{\mathbb{C}}})$. The types of the associated quotient
singularities in terms of this classification are given by [Br67, Satz 2.11].

Finite and linearly reductive subgroup schemes of ${\mathbf{SL}}_{2,k}$ are
automatically very small and they have been classified by Hashimoto [Ha15,
Theorem 3.8], extending Klein’s classification [Kl84] of finite subgroups of
${\mathbf{SL}}_{2}({{\mathbb{C}}})$ and we refer to [Ha15, LS14] for the types
of the associated quotient singularities (see also Theorem 4.12 below).

###  4.3. The Ishii-Ito-Nakamura resolution

In [IN99], Ito and Nakamura showed that if $G$ is a finite subgroup of
${\mathbf{SL}}_{2}({{\mathbb{C}}})$, then a minimal resolution of
singularities of ${{\mathbb{C}}}^{2}/G$ is provided by the $G$-Hilbert scheme
$G\mathrm{-Hilb}({{\mathbb{C}}}^{2})$. Ishii [Is02] extended this to quotient
singularities ${{\mathbb{C}}}^{2}/G$ for $G$ a finite and very small subgroup
of ${\mathbf{GL}}_{2}({{\mathbb{C}}})$ and Ishii and Nakamura [IN19] extended
this to quotient singularities $U/G$ for $G$ a finite and very small subgroup
of ${\mathbf{GL}}_{2}(k)$ of order prime to $p$.

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. Let $G$
be a very small, finite, and linearly reductive subgroup scheme of
${\mathbf{GL}}_{2,k}$. Set $U:={{\mathbb{A}}}^{2}_{k}$ or
$\widehat{{{\mathbb{A}}}}_{k}^{2}$ and let $x\in X:=U/G$ be the associated
linearly reductive quotient singularity. By [Bl11], there exists a $G$-Hilbert
scheme $G\mbox{-Hilb}(U)$ over $k$ that parametrises zero-dimensional
$G$-invariant subschemes $Z\subset U$ (so-called _clusters_), such that the
$G$-representation on $H^{0}(Z,{\cal O}_{Z})$ is the regular representation.
Taking a cluster $Z$ to its $G$-orbit (see, for example, [Bl11, Remark 3.3])
induces a morphism

$\pi\,:\,Y\,:=\,G\mbox{-Hilb}(U)\,\to\,U/G\,=\,X$

over $k$.

######  Theorem 4.5.

The morphism $\pi$ is a minimal resolution of singularities.

###### Proof.

If $p=0$, then this is shown in [IN19], extending [IN99]. If $p>0$ and $G$ is
of length prime to $p$, then this is shown in [IN19]. However, this proof also
works if $G$ is linearly reductive and $p$ divides the length of $G$. ∎

######  Remark 4.6.

By [LMM21, Theorem 11.5], every two-dimensional F-regular singularity $x\in X$
is a linearly reductive quotient singularity, say $x\in X=U/G$ for some finite
and linearly reductive subgroup scheme $G$ of ${\mathbf{GL}}_{2,k}$. By
[LMM21, Theorem 8.1], $G$ and its embedding into ${\mathbf{GL}}_{2,k}$ are
unique up to isomorphism and conjugacy, respectively. By Theorem 4.5,
$G\mbox{-Hilb}(U)\to U/G=X\ni x$ is the minimal resolution of singularities.
In particular, every two-dimensional F-regular singularity can be resolved by
a suitable $G$-Hilbert scheme.

###  4.4. Canonical lifts

Let $k$ be an algebraically closed field of characteristic $p>0$, let $W(k)$
be the ring of Witt vectors, let $K$ be its field of fractions, and let
$\overline{K}$ be an algebraic closure. Let $x\in X=U/G$ be a $d$-dimensional
linearly reductive singularity, where $U={{\mathbb{A}}}^{d}_{k}$ or
$U=\widehat{{{\mathbb{A}}}}_{k}^{d}$. Next, we set ${\cal
U}:={{\mathbb{A}}}^{d}_{W(k)}$ or $\widehat{{{\mathbb{A}}}}^{d}_{W(k)}$,
respectively. In Section 2.2, we recalled the canonical lift ${\cal
G}_{\mathrm{can}}$ of $G$ over $W(k)$. By [LMM21, Proposition 2.9], there
exists a unique lift of the $G$-action from $U$ to ${\cal U}$. From this, we
obtain a flat family

(5) $\mathcal{X}_{\mathrm{can}}\,:=\,{\cal U}/{\cal
G}_{\mathrm{can}}\,\to\,\mathrm{Spec}\>W(k)\,.$

of linearly reductive quotient singularities over $W(k)$, whose special fibre
over $k$ is $x\in X=U/G$.

######  Definition 4.7.

The family (5) is called the _canonical lift_ of the linearly reductive
quotient singularity $x\in X=U/G$.

By the Lefschetz principle, the geometric generic fibre of
$\mathcal{X}_{\mathrm{can}}$ can be identified with a finite quotient
singularity of the form ${{\mathbb{C}}}^{d}/G_{\mathrm{abs}}$, where
$G_{\mathrm{abs}}$ is the abstract group associated to $G$ and the embedding
of $G_{\mathrm{abs}}\to{\mathbf{GL}}_{d}({{\mathbb{C}}})$ corresponds to the
embedding $G\to{\mathbf{GL}}_{d,k}$ provided by Proposition 2.6. The canonical
lift is unique in the following sense:

######  Proposition 4.8.

We keep the notations and assumptions. Let $W(k)\subseteq R$ be a finite
extension of complete DVRs and let $\mathcal{X}\to\mathrm{Spec}\>R$ be a lift
of $x\in X$ that is of the form ${\cal V}/{\cal G}\to\mathrm{Spec}\>R$ for
some flat lift ${\cal G}$ of $G$ to $R$ and ${\cal V}\cong{\cal
U}\times_{R}S$. Then, there exists a finite extension $R\subseteq S$ of
complete DVRs, such that

  1. (1)

There exists an isomorphism

${\cal G}_{\mathrm{can}}\times_{W(k)}S\,\cong\,{\cal G}\times_{R}S$

of group schemes over $S$.

  2. (2)

The very small representation $\rho:G\to{\mathbf{GL}}_{d,k}$ corresponding to
the $G$-action on $U$ lifts uniquely to ${\cal G}_{\mathrm{can}}$ and ${\cal
G}$, respectively, and they become conjugate over $S$.

  3. (3)

There is an isomorphism

$\mathcal{X}_{\mathrm{can}}\times_{W(k)}S\,\cong\,\mathcal{X}\times_{R}S$

of deformations of $x\in X$ over $S$.

###### Proof.

Since ${\cal G}$ and ${\cal G}_{\mathrm{can},R}$ are lifts of $G$ to $R$, they
become isomorphic after passing to a finite extension $R\subseteq S$, see also
the discussion in Section 2.2. There, we also saw that the linear
representation $\rho:G\to{\mathbf{GL}}_{d,k}$ lifts uniquely to ${\cal G}$ and
${\cal G}_{\mathrm{can}}$, respectively, and that they become conjugate over
$S$. From this, we deduce an isomorphism
$\mathcal{X}_{\mathrm{can}}\times_{W(k)}S\cong\mathcal{X}\times_{R}S$. ∎

######  Remark 4.9.

If $d\geq 3$, then [LMM21, Corollary 10.10] shows that a $d$-dimensional
linearly reductive quotient singularity $x\in X$ admits precisely one lift
over $W(k)$, namely the canonical lift. If $d=2$, then linearly reductive
quotient singularities usually have positive dimensional deformation spaces
and admit many non-isomorphic lifts to $W(k)$, see [LMM21, Section 12].

###  4.5. Simultaneous resolution of singularities

If $\mathcal{X}\to S$ is a deformation of a rational double point singularity
$x\in X$, then it admits a simultaneous resolution of singularities, but
usually only after some finite base-change $S^{\prime}\to S$, see [Ar74]. In
the case where $\mathcal{X}\to S$ is a family of rational surface
singularities, then such a finite $S^{\prime}\to S$ exists if $S$ maps to the
so-called _Artin component_ inside the versal deformation space of $x\in X$.
Moreover, due to the existence of flops, these simultaneous resolutions (if
they exist) are not unique in general.

In the special case where $x\in X$ is a two-dimensional linearly reductive
quotient singularity over some algebraically closed field $k$ of
characteristic $p>0$ and where
$\mathcal{X}_{\mathrm{can}}\to\mathrm{Spec}\>W(k)$ is the canonical lift of
$x\in X$, we will now show that there exists a simultaneous and minimal
resolution of singularities over $W(k)$ and that it is unique. This
simultaneous resolution can be most elegantly constructed using the Ishii-Ito-
Nakamura resolution from Section 4.3 in families.

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a
finite and linearly reductive group scheme over $k$ and let
$\rho:G\to{\mathbf{GL}}_{2,k}$ be a very small representation. Let ${\cal
G}_{\mathrm{can}}\to\mathrm{Spec}\>W(k)$ be the canonical lift of $G$ and let
$\widetilde{\rho}:{\cal G}_{\mathrm{can}}\to{\mathbf{GL}}_{2,W(k)}$ be the
lift of $\rho$ to $W(k)$. Let ${\cal U}:={{\mathbb{A}}}^{2}_{W(k)}$ and let

$\mathcal{X}_{\mathrm{can}}\,:=\,{\cal U}/{\cal
G}_{\mathrm{can}}\,\to\,\mathrm{Spec}\>W(k)$

be the canonical lift of $x\in X$. By [Bl11], there exists a ${\cal
G}_{\mathrm{can}}$-Hilbert scheme

${\cal G}_{\mathrm{can}}\mbox{-Hilb}({\cal U})\,\to\,\mathrm{Spec}\>W(k)$

that parametrises ${\cal G}_{\mathrm{can}}$-invariant subschemes
$Z\subset{\cal U}$ that are finite and flat over $W(k)$ (so-called ${\cal
G}_{\mathrm{can}}-$_clusters_) and such that the ${\cal
G}_{\mathrm{can}}$-representation on $H^{0}(Z,{\cal O}_{Z})$ is the regular
representation. Taking such a cluster $Z$ to its ${\cal
G}_{\mathrm{can}}$-orbit (see, for example, [Bl11, Remark 3.3]) induces a
morphism

(6) $\widetilde{\pi}\,:\,\mathcal{Y}\,:=\,{\cal
G}_{\mathrm{can}}\mbox{-Hilb}({\cal U})\,\to\,{\cal U}/{\cal
G}_{\mathrm{can}}\,=\,\mathcal{X}_{\mathrm{can}}$

over $W(k)$.

######  Theorem 4.10.

Keeping the assumptions and notations

$\widetilde{\pi}\,:\,\mathcal{Y}\,\to\,\mathcal{X}\,\to\,\mathrm{Spec}\>W(k)$

is a simultaneous minimal resolution of singularities of the canonical lift
$\mathcal{X}\to\mathrm{Spec}\>W(k)$ of the linearly reductive quotient
singularity $x\in X=U/G$.

  1. (1)

The simultaneous resolution $\widetilde{\pi}$ is unique up to isomorphism.

  2. (2)

The exceptional locus of $\widetilde{\pi}$ is a union of
${{\mathbb{P}}}^{1}_{W(k)}$’s meeting transversally and we denote by $\Gamma$
its dual intersection graph.

  3. (3)

The special fibre and the generic fibre are linearly reductive quotient
singularities of type $\Gamma$ and $\widetilde{\pi}$ identifies the components
of the exceptional loci of $\widetilde{\pi}_{k}$ and $\widetilde{\pi}_{K}$.

###### Proof.

We recall that we defined $G_{\mathrm{can}}:={\cal G}_{\mathrm{can},K}$, which
will simplify the notation in the following. The generic and special fibre of
(6) over $K$ and $k$ are isomorphic to

$\begin{array}[]{cccccc}&G_{\mathrm{can}}\mbox{-Hilb}({\cal U}_{K})&\to&{\cal
U}_{K}/G_{\mathrm{can}}&\to&\mathrm{Spec}\>K\\\ \mbox{and
\quad}&G\mbox{-Hilb}(U)&\to&U/G&\to&\mathrm{Spec}\>k,\end{array}$

respectively, by [Bl11, Remark 3.1]. Now,
$G_{\mathrm{can},\overline{K}}\mbox{-Hilb}({\cal U}_{\overline{K}})\to{\cal
U}_{\overline{K}}/G_{\mathrm{can},\overline{K}}$ and $G\mbox{-Hilb}(U)\to
U/G$, are minimal resolutions of singularities by Theorem 4.5. Thus,
$G_{\mathrm{can}}\mbox{-Hilb}({\cal U}_{K})\to{\cal U}_{K}/G_{\mathrm{can}}$
is a resolution of singularities, which is minimal since it is minimal over
$\overline{K}$. Thus, $\mathcal{Y}\to\mathcal{X}\to\mathrm{Spec}\>W(k)$ is a
simultaneous minimal resolution of singularities.

The exceptional fibres $\widetilde{\pi}_{K}$ and $\widetilde{\pi}_{k}$ of
$\widetilde{\pi}$ over $K$ and $k$ are unions of ${{\mathbb{P}}}^{1}$’s that
intersect transversally. The types, that is dual resolution graphs, associated
to $\widetilde{\pi}_{K}$ and $\widetilde{\pi}_{k}$ are the same (see also
Remark 4.4) and we denote this graph by $\Gamma$. In particular, the
exceptional fibres of $\widetilde{\pi}_{K}$ and $\widetilde{\pi}_{k}$ have the
same numbers of irreducible components. In particular, the specialisation maps
of Néron-Severi lattices

$\mathrm{NS}(\mathcal{Y}_{K})\,\leftarrow\,\mathrm{NS}(\mathcal{Y})\,\to\,\mathrm{NS}(\mathcal{Y}_{k})$

are isometries of lattices. These identify the components of the exceptional
loci of $\widetilde{\pi}_{K}$ and $\widetilde{\pi}_{k}$. Moreover, given such
a component of the exceptional locus of $\widetilde{\pi}_{k}$, it is
isomorphic to ${{\mathbb{P}}}^{1}_{k}$ and it extends uniquely to a
${{\mathbb{P}}}^{1}_{W(k)}$ in the exceptional locus of $\widetilde{\pi}$.
This establishes claims (2) and (3).

It remains to prove claim (1): Let
$\mathcal{Y}^{\prime}\to\mathcal{X}\to\mathrm{Spec}\>W(k)$ be a simultaneous
resolution of singularities that coincides with the minimal resolution on
special and generic fibres, respectively. Let
$\alpha:\mathcal{Y}^{\prime}_{K}\,\to\,\mathcal{Y}_{K}$ be an isomorphism over
$\mathcal{X}_{K}$ and choose a relatively (to $\mathcal{X}$) ample invertible
sheaf $\mathcal{L}$ on $\mathcal{Y}_{K}$. Then, $\alpha^{*}\mathcal{L}$ is
relatively ample on $\mathcal{Y}_{K}$. Since the types of the singularities of
$\mathcal{X}_{K}$ and $\mathcal{X}_{k}$ are the same, the fibres of
$\mathcal{Y}^{\prime}_{K}\to\mathcal{X}_{K}$ and
$\mathcal{Y}^{\prime}_{k}\to\mathcal{Y}_{k}$ contain the same number of
exceptional divisors. Thus, the specialisation map of Néron-Severi lattices
$\mathrm{NS}(\mathcal{Y}^{\prime}_{K})\to\mathrm{NS}(\mathcal{Y}^{\prime}_{k})$
is an injective map between two lattices of the same rank and the same
discriminant, whence an isometry. Similarly, the specialisation map of Néron-
Severi lattices $\mathrm{NS}(\mathcal{Y}_{K})\to\mathrm{NS}(\mathcal{Y}_{k})$
is an isometry. In particular, we can identify the components of the
exceptional loci of $\mathcal{Y}_{K}$ and $\mathcal{Y}_{k}$. (If the types of
$\mathcal{X}_{k}$ and $\mathcal{X}_{K}$ differ, then this specialisation map
is usually only injective, there are usually more $(-2)$-curves in the special
fibre and this is also the place where the difficulties with flops begin.) In
particular, $\mathcal{L}$ and $\alpha^{*}\mathcal{L}$ extend to relative ample
invertible sheaves on $\mathcal{Y}$ and $\mathcal{Y}^{\prime}$, respectively.
By [Ko09, Theorem 5.14], the isomorphism $\alpha$ extends to an isomorphism
$\mathcal{Y}^{\prime}\to\mathcal{Y}$ over $\mathcal{X}$ and the claimed
uniqueness follows. ∎

###  4.6. Rational double point singularities and Klein singularities

We specialise Definition 4.2 to the following case.

######  Definition 4.11.

A _Klein singularity_ is a linearly reductive quotient singularity as in
Definition 4.2 with $\dim V=2$ and $\det\rho=1$, that is, $\rho$ is a
homomorphism of $G$ to ${\mathbf{SL}}_{2,k}$.

In particular, a Klein singularity is a two-dimensional and linearly reductive
quotient singularity, it is a rational surface singularity, and since
$\det(\rho)=1$, it is Gorenstein. Quite generally, rational and Gorenstein
surface singularities are precisely the _rational double point singularities_
[Ar66]. We refer to [Du79] or [SSS] for more background on surface
singularities and rational double point singularities.

If $x\in X$ is a rational double point, then its type $\Gamma$ is a simply-
laced finite Dynkin graph. In characteristic zero, every rational double point
singularity is taut. In positive characteristic, a Klein singularity is
F-regular and thus, taut. In the case of rational double points, this also
follows from Artin’s explicit classification [Ar77]. On the other hand,
rational double point singularities that are not F-regular need not be taut.
We have the following relation between rational double point singularities and
Klein singularities in positive characteristic, see [LMM21, Theorem 11.2].

######  Theorem 4.12.

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. Let
$x\in X$ be a normal surface singularity over $k$.

  1. (1)

If $p=0$ or $p\geq 7$, then $x\in X$ is a Klein singularity if and only if
$x\in X$ is a rational double point singularity.

  2. (2)

If $p>0$, then $x\in X$ is a Klein singularity if and only if $x\in X$ is a
rational double point singularity and F-regular. The finite Dynkin graphs of
these singularities are of type

$\begin{array}[]{lcl}A_{n},D_{n},E_{6},E_{7},E_{8}&\mbox{\quad if \quad}&p\geq
7,\\\ A_{n},D_{n},E_{6},E_{7}&\mbox{\quad if \quad}&p=5,\\\
A_{n},D_{n}&\mbox{\quad if \quad}&p=3,\\\ A_{n}&\mbox{\quad if
\quad}&p=2.\end{array}$

A Klein singularity $x\in X=U/G$ is a linearly reductive quotient singularity
and thus, determines $G$ and $\rho:G\to{\mathbf{SL}}_{2,k}$ up to isomorphism
and conjugacy, respectively. Thus, the classification of Klein singularities
boils down to the classification of finite and linearly reductive subgroup
schemes of ${\mathbf{SL}}_{2,k}$ and we refer to Remark 4.4 for this
classification.

######  Remark 4.13.

Theorem 3.6 and Theorem 4.12 rely on the classification of finite and linearly
reductive subgroup schemes of ${\mathbf{SL}}_{2,k}$. It is therefore no
surprise that the classification lists coincide.

##  5\. Hecke correspondences

Let $x\in X=U/G$ be a Klein singularity over an algebraically closed field $k$
of characteristic $p>0$. In Section 2.2, we discussed the canonical lift
$\mathcal{X}\to\mathrm{Spec}\>W(k)$ of $x\in X$ and in Section 4.5 we
established a simultaneous minimal resolution of singularities of the
canonical lift. More precisely, this simultaneous resolution was constructed
using $G$-Hilbert schemes extending previous work of Ishii, Ito, and Nakamura
[Is02, IN19, IN99].

In this section, we refine this resolution of singularities as in the work of
Ito and Nakamura [IN99] and Nakajima [Na96, Na01]: we eventually obtain a
bijection between the components of the minimal resolution of singularities of
the Klein singularity $x\in X=U/G$ and the simple and non-trivial
representations of $G$ using special _Hecke correspondences_.

###  5.1. The Ito-Nakamura resolution revisited

We first slightly extend the setup of Section 4.3. Let $k$ be an algebraically
closed field of characteristic $p>0$ and let $x\in X=U/G$ be a Klein
singularity as in Definition 4.11, that is, we have $U={{\mathbb{A}}}^{2}_{k}$
and $G$ a very small, finite, and linearly reductive subgroup scheme of
${\mathbf{SL}}_{2,k}$.

Let $\\{\rho_{i}\\}_{i\in I}$ be the set of isomorphism classes of simple
representations of $G$. Given a finite-dimensional representation $\rho$ of
$G$, there exist non-negative integers $\nu_{i}\in{{\mathbb{Z}}}_{+}$, such
that $\rho$ is isomorphic to $\bigoplus_{i}\rho_{i}^{\oplus\nu_{i}}$ and we
combine these into a multi-index $\nu=(\\{\nu_{i}\\})\in{{\mathbb{Z}}}_{\geq
0}^{I}$. We set $\dim(\nu):=\sum_{i}\nu_{i}\dim\rho_{i}$, which is the
dimension of the representation associated to $\nu$.

For any integer $n\geq 1$, the $G$-action on $U$ induces a $G$-action on the
Hilbert scheme $\mathrm{Hilb}^{n}(U)$, which parametrises zero-dimensional
subschemes of length $n$ of $U$. We consider the fixed point scheme

$\mathrm{Hilb}^{n,G}(U)\,:=\,\left(\mathrm{Hilb}^{n}(U)\right)^{G},$

that is, the largest subscheme of $\mathrm{Hilb}^{n}(U)$ on which $G$ acts
trivially. It parametrises $G$-invariant and zero-dimensional subschemes of
length $n$ of $U$. Given $\nu=(\\{\nu_{i}\\})_{i}\in{{\mathbb{Z}}}_{\geq
0}^{I}$ with $\dim(\nu)=n$, we define

$H^{\nu}\,=\,\left\\{Z\,\in\,\mathrm{Hilb}^{n,G}(U)\,|\,H^{0}(Z,{\cal
O}_{Z})\,\cong\,\bigoplus\rho_{i}^{\oplus\nu_{i}}\mbox{ as
$G$-representation}\right\\},$

which defines a subscheme of $\mathrm{Hilb}^{n,G}(U)$. Adapting [Ki16, Lemma
12.4], which follows [IN99, Section 9], to our situation, we have the
following.

######  Lemma 5.1.

We keep the assumptions and notations and let $n\geq 1$ be an integer.

  1. (1)

The scheme $\mathrm{Hilb}^{n,G}(U)$ is smooth over $k$.

  2. (2)

We have a decomposition

$\mathrm{Hilb}^{n,G}(U)\,=\,\bigsqcup_{\nu}\,H^{\nu},$

where the disjoint union runs over all multi-indices
$\nu\in{{\mathbb{Z}}}_{\geq 0}^{I}$ of dimension $n$. Each $H^{\nu}$ is a
smooth subscheme of $\mathrm{Hilb}^{n,G}(U)$.

###### Proof.

Since $U$ is two-dimensional, $\textrm{Hilb}^{n}(U)$ is smooth over $k$ by
[Fo68, Theorem 2.4]. To show Claim (1), it remains to show that the fixed
point scheme for the $G$-action is also smooth, which follows from Lemma 5.2
below. Then, Claim (2) is obvious. ∎

######  Lemma 5.2.

Let $k$ be an algebraically closed field, let $X$ be a scheme that is smooth
over $k$, let $G$ be a finite and linearly reductive group scheme over $k$,
and assume that $G$ acts on $X$. Then, the fixed point scheme $X^{G}\subseteq
X$ is smooth over $k$.

###### Proof.

Let $x\in X^{G}$. Passing to the completion of the local ring ${\cal
O}_{X,x}$, using that $X$ is smooth over $k$, and passing to coordinates such
that the $G$-action is linear (this is always possible by [Sa12, proof of
Corollary 1.8]), we may assume that

$\widehat{{\cal O}}_{X,x}\,\cong\,k[[u_{1},...,u_{d}]]$

and that the $G$-action is linear. In this description it is easy to see that
the $G$-invariant subscheme of $\mathrm{Spec}\>\widehat{{\cal O}}_{X,x}$ is
smooth (see also the proof of [IN99, Lemma 9.1]), which implies that $X^{G}$
is smooth near $x$. ∎

An important special case is the _regular representation_ of $G$, where we
have $\nu_{i}=\dim\rho_{i}$ for all $i$ and in this case, we will write
$\delta$ for the corresponding multi-index. The dimension of $\delta$ is equal
to the length of $G$. In this case, we have

$\pi\,:\,H^{\delta}\,=\,G\mbox{-Hilb}(U)\,\to\,U/G\,=\,X,$

which we already studied in Section 4.3. There, we saw that $\pi$ is a minimal
resolution of singularities of the Klein singularity $x\in X=U/G$.

###  5.2. Hecke correspondences

We keep the assumptions and notations of the previous section. We let $n$ be
the length of $G$ and let $\delta$ be the multi-index corresponding to the
regular representation. For $i\in I$, we set $\alpha_{i}:=(0,...,0,1,0...)$
with the non-zero entry in the $i$.th position. We define

$B_{i}\,:=\,\left\\{J_{1}\,\in\,H^{\delta-\alpha_{i}},\quad
J_{2}\,\in\,H^{\delta}\quad|\quad J_{2}\subseteq
J_{1}\right\\}\,\subseteq\,H^{\delta-\alpha_{i}}\times H^{\delta},$

which is a _Hecke correspondence_ , see [Na96], [Na01, Section 6.1] or [Ki16,
Section 12.4]. We let

$E_{i}\,:=\,\mathrm{pr}_{2}(B_{i})\,\subseteq\,H^{\delta}\,=\,G\mathrm{-Hilb}(U)$

be the image under the projection $\mathrm{pr}_{2}$ onto the second factor.

In characteristic zero, the following result is due to Nakajima [Na96] and
independently to Ito and Nakamura [IN99], see also [Ki16, Section 12.4].

######  Theorem 5.3.

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. Let $G$
be a finite and linearly reductive subgroup scheme of ${\mathbf{SL}}_{2,k}$,
let $x\in X:=U/G$ be the associated Klein singularity, and let
$\pi:H^{\delta}\to X$ be the Ito-Nakamura resolution.

  1. (1)

The assignment $i\mapsto E_{i}$ defines a bijection between the set
$\\{\rho_{i}\\}_{i}$ of isomorphism classes of non-trivial simple
representations of $G$ and the set of irreducible components of the
exceptional divisor $\mathrm{Exc}(\pi)$ of $\pi$.

  2. (2)

If $i\neq j$, then

$B_{i}\cdot B_{j}\,=\,A_{ij}$

where the numbering of the $\\{\rho_{i}\\}_{i}$ is as in [IN99, Section 10].

We thus obtain a bijection between the McKay graph of $G$ and the dual
resolution graph of $\pi$.

###### Proof.

The case-by-case analysis of [IN99, Section 12] carries directly over from
finite subgroups of ${\mathbf{SL}}_{2}({{\mathbb{C}}})$ to finite and linearly
reductive subgroup schemes of ${\mathbf{SL}}_{2,k}$. ∎

Here is a second proof of this theorem that uses our lifting results:

###### Proof.

If $p=0$, then the Lefschetz principle allows to reduce to $k={{\mathbb{C}}}$,
where the theorem is due to Ito, Nakajima, and Nakamura [IN99, Na96].

We now assume $p>0$. Let $\mathcal{X}_{\mathrm{can}}={\cal U}/{\cal
G}_{\mathrm{can}}\to\mathrm{Spec}\>W(k)$ be the canonical lift of $x\in X=U/G$
(see Section 4.4) and let
$\widetilde{\pi}:\mathcal{Y}\to\mathcal{X}_{\mathrm{can}}\to\mathrm{Spec}\>W(k)$
be the simultaneous resolution of singularities (see Section 4.5). Moreover,
we have the abstract group $G_{\mathrm{abs}}$ and the canonical lift
$G_{\mathrm{can}}$ associated to $G$ (see Section 2.2) and we have a bijection
of simple representations of $G_{\mathrm{abs}}$, $G_{\mathrm{can}}$, and $G$
by Proposition 2.6.

The generic fibre $\mathcal{X}_{K}$ is isomorphic to ${\cal
U}_{K}/G_{\mathrm{can}}$ and using the Lefschetz principle, the geometric
generic fibre can be identified with ${{\mathbb{C}}}^{2}/G_{\mathrm{abs}}$.
Over ${{\mathbb{C}}}$, we have Theorem 5.3 for $G_{\mathrm{abs}}$,
${{\mathbb{C}}}^{2}/G_{\mathrm{abs}}$, and its minimal resolution of
singularities by [IN99, Na96].

By the Lefschetz principle, we have it for $G_{\mathrm{can}}$,
$\mathcal{X}_{K}={\cal U}_{K}/G_{\mathrm{can}}$ and $\widetilde{\pi}_{K}$.
Using the comparison results Proposition 2.6 and Theorem 4.10, we obtain it
for $\mathcal{X}_{k}={\cal U}_{k}/{\cal G}_{\mathrm{can},k}$ and
$\widetilde{\pi}_{k}$, that is, for $X=U/G$, $G$ and $\pi$. ∎

###  5.3. Local McKay correspondence

We now extend work of Ishii and Nakamura [Is02, IN19] to our linearly
reductive setting. Let $G$ be a very small, finite, and linearly reductive
subgroup scheme of ${\mathbf{GL}}_{2,k}$ and let $x\in X=U/G$ be the
associated two-dimensional linearly reductive quotient singularity. Until the
end of this section, we do not require it to be a Klein singularity. Note that
in dimension two, linearly reductive quotient singularities are the same as
F-regular singularities, see Remark 4.6.

Let $\\{\rho_{i}\\}_{i}$ be the set of simple representations of $G$. Let
$\rho_{0}$ be the trivial representation and we choose our numbering of the
$\rho_{i}$’s to be the one of [IN19, Theorem 3.6]. Let $\pi:Y\to X$ be its
minimal resolution of singularities and let $\\{E_{i}\\}$ be the irreducible
components of the exceptional divisor $\textrm{Exc}(\pi)$ of $\pi$. Then,
there is exists a connection between the $\\{\rho_{i}\\}$ and the exceptional
divisors of $\pi$ as follows - this is yet another version of the McKay
correspondence.

######  Theorem 5.4.

Keeping assumptions and notations, let ${\mathfrak{m}}\subset{\cal O}_{U}$ be
the maximal ideal corresponding to the origin, let $y\in Y$ be a closed point,
let $Z_{y}$ be the $G$-invariant cluster of $U$ corresponding to $y$, and let
${\cal I}_{Z_{y}}\subset{\cal O}_{U}$ be its ideal sheaf. Then, the
$G$-representation on ${\cal I}_{Z_{y}}/{\mathfrak{m}}{\cal I}_{Z_{y}}$ is
given by

$\left\\{\begin{array}[]{ll}\rho_{i}\,\oplus\,\rho_{0}&\mbox{ \quad if }y\in
E_{i}\,\backslash\,\bigcup_{j\neq i}E_{j}\\\
\rho_{i}\,\oplus\,\rho_{j}\,\oplus\,\rho_{0}&\mbox{ \quad if }y\in E_{i}\cap
E_{j}\mbox{ \quad with \quad}i\neq j.\end{array}\right.$

###### Proof.

For $k={{\mathbb{C}}}$, this is [Is02, Theorem 7.1]. For $k$ algebraically
closed of arbitrary characteristic and $G$ is a very small and finite subgroup
of ${\mathbf{GL}}_{2}(k)$, this is [IN19, Theorem 3.6]. However, these proofs
also work if $G$ is a very small and finite and linearly reductive subgroup
scheme of ${\mathbf{GL}}_{2,k}$. ∎

######  Remark 5.5.

In [IN19], the $G$-quiver structure of $G\mathrm{-Hilb}(U)$ was studied in the
case where $G$ is a very small and finite subgroup of ${\mathbf{GL}}_{2}(k)$.
We leave the extension of these results to the linearly reductive case to the
reader.

###  5.4. Reflexive sheaves on the minimal resolution

We end this section by shortly digressing on work of Wunram [Wu88], Ishii and
Nakamura [IN19], which generalises work of Artin and Verdier [AV85]. We keep
the assumptions and notations from Section 5.3. Let $F\subset Y$ be the
_fundamental divisor_ of $\pi$, see [Ar66].

######  Theorem 5.6.

Keeping assumptions and notations, there exists a bijection between

  1. (1)

the set of irreducible components $\\{E_{i}\\}$ of $\mathrm{Exc}(\pi)$ and

  2. (2)

the set of non-trivial indecomposable full ${\cal O}_{Y}$-modules $\\{{\cal
M}_{i}\\}$, special in the sense that $H^{1}(Y,{\cal M}_{i}^{\vee})=0$

This correspondence ${\cal M}_{i}\mapsto E_{i}$ is defined by

$c_{1}({\cal M}_{i})\cdot E_{j}\,=\,\delta_{i,j}\,.$

The rank of ${\cal M}_{i}$ is equal to $c_{1}({\cal M}_{i})\cdot F$, the
multiplicity of $E_{i}$ in $F$.∎

###### Proof.

For $k={{\mathbb{C}}}$, this is [Wu88]. For $k$ algebraically closed of
arbitrary characteristic and $G$ a very small and finite subgroup of
${\mathbf{GL}}_{2,k}$, this is [IN19, Theorem 2.8]. However, these proofs also
work if $G$ is a very small, finite, and linearly reductive subgroup scheme of
${\mathbf{GL}}_{2,k}$. ∎

Here, a _full_ ${\cal O}_{Y}$-module is as defined in [IN19, Definition 2.4].
By [IN19, Corollary 2.5], the assignment ${\cal M}\mapsto\pi_{*}{\cal M}$ sets
up a bijection between the set of (indecomposable) full ${\cal O}_{Y}$-modules
with the set of (indecomposable) reflexive ${\cal O}_{X}$-modules. In
particular, the above theorem yields a bijection between the set of
irreducible components of $\mathrm{Exc}(\pi)$ and the set of reflexive ${\cal
O}_{X}$-modules.

######  Remark 5.7.

The group scheme $G$ plays no rôle in this theorem and the discussion
thereafter. Probably, these results should be viewed as results on two-
dimensional F-regular singularities (see Remark 4.6).

##  6\. Conjugacy classes and Ito-Reid correspondence

For a finite group, the number of isomorphism classes of complex semi-simple
representations is equal to the number of conjugacy classes. Thus, one can
_choose_ a bijection between these two sets. In particular, for a finite
subgroup of ${\mathbf{SL}}_{2}({{\mathbb{C}}})$ one can _choose_ a bijection
between the vertices of its McKay graph and the conjugacy classes of $G$. In
[IR96], Ito and Reid gave a _canonical_ bijection between these two sets. In
this section, we extend this _Ito-Reid correspondence_ to finite and linearly
reductive subgroup schemes of ${\mathbf{SL}}_{2,k}$. The main difficulty is to
define a suitable notion of _conjugacy classes_ for finite and linearly
reductive group schemes.

###  6.1. Conjugacy classes

Given a finite group $G_{\mathrm{abs}}$, a representation
$\rho:G_{\mathrm{abs}}\to{\mathbf{GL}}_{n}({{\mathbb{C}}})$, and an element
$g\in G_{\mathrm{abs}}$, we have the trace
$\mathrm{tr}(\rho(g))\in{{\mathbb{C}}}$. This pairing
$(\rho,g)\mapsto\mathrm{tr}(\rho(g))$ induces a non-degenerate pairing between
isomorphism classes of semi-simple representations of $G_{\mathrm{abs}}$ and
conjugacy classes. Thus, conjugacy classes can be thought of as being “dual”
to semi-simple representations, which can be used to give an unusual
definition of conjugacy classes. This definition generalises to finite and
linearly reductive group schemes. More precisely, we make the following
definition, which is inspired by a result of Serre [Se77, Section 11.4] and
which we discuss in detail in Appendix B.2.

######  Definition 6.1.

Let $G$ be a finite and linearly reductive group scheme over an algebraically
closed field $k$ of characteristic $p\geq 0$. Then, the _set of conjugacy
classes_ is defined to be $\mathrm{Spec}\>{{\mathbb{C}}}\otimes K_{k}(G)$.

We discuss several approaches to conjugacy classes for finite group schemes in
Appendix B \- there, we hope to convince the reader that Definition 6.1 is the
best for the purposes of this article. For example, by Proposition B.2 it is
compatible with canonical lifts and there is a natural bijection with the set
of conjugacy classes of the abstract group $G_{\mathrm{abs}}$ associated to
$G$. Concerning the choice of field ${{\mathbb{C}}}$ in Definition 6.1, we
refer to Remarks B.3.

###  6.2. An explicit bijection

For a finite and linearly reductive subgroup scheme $G$ of
${\mathbf{SL}}_{2,k}$, we have an associated embedding of the finite group
$G_{\rm abs}$ into ${\mathbf{SL}}_{2}({{\mathbb{C}}})$. Let
$\\{\rho_{i}\\}_{i}$ be the set of isomorphism classes of semi-simple
representations of $G$ and let $\\{\rho_{\mathrm{abs},i}\\}_{i}$ be the
corresponding set for $G_{\mathrm{abs}}$ obtained by Proposition 2.6. The
associated McKay graphs $\Gamma(G,\rho,\\{\rho_{i}\\}_{i})$ and
$\Gamma(G_{\mathrm{abs}},\rho_{\mathrm{abs}},\\{\rho_{\mathrm{abs},i}\\}_{i})$
coincide by Proposition 3.5 and we denote both by $\widehat{\Gamma}$. Next,
the group $G_{\mathrm{abs}}$ admits a presentation of the form

$G_{\rm abs}\,=\,\left\langle A,B,C\,|\,A^{r}=B^{s}=C^{t}=ABC\right\rangle$

for suitable non-negative integers $r,s,t$. The non-trivial conjugacy classes
of $G_{\rm abs}$ can be uniquely represented by the following elements

$ABC,\quad A^{i}\mbox{ with }1\leq i\leq r-1,\quad B^{i}\mbox{ with }1\leq
i\leq s-1,\quad C^{i}\mbox{ with }1\leq i\leq t-1.$

This allows to give an explicit bijection between the conjugacy classes of
$G_{\mathrm{abs}}$ and the vertices of $\widehat{\Gamma}$. We refer to [Ki16,
Section 2] for details. Using Proposition B.2, we obtain an explicit bijection
between the conjugacy classes of $G$ and the vertices of $\widehat{\Gamma}$.

###  6.3. The Ito-Reid correspondence

If $G$ is a finite subgroup of ${\mathbf{SL}}_{2}({{\mathbb{C}}})$, then Ito
and Reid [IR96] (see also [Re02]) constructed a _canonical_ bijection. Let us
run through [Re02, Section 2] and show that this can be carried over to our
linearly reductive setting: let $k$ be an algebraically closed field of
characteristic $p>0$ and let $G$ be a finite and linearly reductive subgroup
scheme of ${\mathbf{SL}}_{2,k}$.

  1. (1)

Associated to $G$, we have the abstract group $G_{\mathrm{abs}}$. By
Proposition 2.6, an embedding $G\to{\mathbf{SL}}_{n,k}$ yields an embedding
$G_{\mathrm{abs}}\to{\mathbf{SL}}_{n}({{\mathbb{C}}})$. By Proposition B.2, we
can identify conjugacy classes of $G$ (in the sense of Definition 6.1) and
conjugacy classes of $G_{\mathrm{abs}}$, which allows us to define the _age_
of a conjugacy class of $G$ via the corresponding notion for
$G_{\mathrm{abs}}$ as, for example, in [Re02, Section 2]. A conjugacy class of
age 1 is called _junior_ and if $n=2$, then all conjugacy classes are junior.

######  Remark 6.2.

Using the “toric mechanism” mentioned in [Re02, Section 2], one can define the
age directly and without referring to lifts, but we will not pursue this here.

  1. (2)

Let $x\in X:=U/G$ be the associated Klein singularity. We have the canonical
lift $\mathcal{X}_{\mathrm{can}}\to\mathrm{Spec}\>W(k)$ and the simultaneous
resolution of singularities
$\widetilde{\pi}:\mathcal{Y}\to\mathcal{X}_{\mathrm{can}}\to\mathrm{Spec}\>W(k)$
by Theorem 4.10. Passing to geometric generic fibres and using the Lefschetz
principle, we obtain the minimal resolution of singularities of
${{\mathbb{C}}}^{2}/G_{\mathrm{abs}}$. The special fibre of $\widetilde{\pi}$,
the geometric generic fibre of $\widetilde{\pi}$ and the minimal resolution of
${{\mathbb{C}}}^{2}/G_{\mathrm{abs}}$ are _crepant_ in the sense of [Re87] and
we can identify the exceptional divisors of these three resolutions with each
other, see Theorem 4.10. This way, we obtain an Ito-Reid correspondence
between junior conjugacy classes of $G$ and crepant divisors of the resolution
[Re02, Theorem 2.1].

######  Remark 6.3.

It seems reasonable that one can extend this correspondence to finite and
linearly reductive subgroup schemes of ${\mathbf{SL}}_{n,k}$ with $n\geq 3$,
but we will not pursue this here.

##  7\. Derived categories

Let $G$ be a very small, finite, and linearly reductive subgroup scheme of
${\mathbf{GL}}_{2,k}$, let $x\in X:=U/G$ be the associated linearly reductive
quotient singularity, and let $\pi:Y\to X$ be its minimal resolution of
singularities. Gonzalez-Sprinberg and Verdier [GV83] gave an interpretation of
the McKay correspondence as an isomorphism between the K-groups $K^{G}(U)$ and
$K(Y)$. Kapranov and Vasserot [KV00] and Bridgeland, King, and Reid [BKR01]
generalised this to an equivalence of derived categories
${\mathcal{D}}^{G}(U)$ and ${\mathcal{D}}(Y)$. In this section, we extend this
to our setting, following Ishii, Ito, Nakamura, and Ueda [Is02, IN19, IU15].

We have a commutative diagram

$\textstyle{U\times_{k}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{U}}$$\scriptstyle{\pi_{Y}}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varpi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X.}$

By Theorem 4.5, the minimal resolution $\pi$ can be constructed by the Ishii-
Ito-Nakamura resolution

$G\mathrm{-Hilb}(U)\,\to\,U/G$

We let ${\mathcal{Z}}$ be the universal cluster over $G\mathrm{-Hilb}(U)$, we
identify $G\mathrm{-Hilb}(U)$ with $Y$ and then, we have a commutative diagram

$\textstyle{{\mathcal{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{p}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varpi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X.}$

Let ${\mathcal{D}}(Y)$ be the derived category of coherent sheaves on $Y$. Let
${\mathcal{D}}^{G}(U)$ be the derived category of $G$-equivariant coherent
sheaves on $U$. Following [Is02] and [IN19], we define two functors

$\begin{array}[]{ccccc}\Psi&:&{\mathcal{D}}^{G}(U)&\to&{\mathcal{D}}(Y)\\\
\Phi&:&{\mathcal{D}}(Y)&\to&{\mathcal{D}}^{G}(U)\end{array}$

by

$\Psi(-)\,:=\,[p_{*}\,\mathbf{L}q^{*}\,(-)]^{G}$

and

$\Phi(-)\,:=\,\mathbf{R}\pi_{U,*}\left({\cal
O}_{{\mathcal{Z}}}^{\vee}\,\otimes^{\mathbf{L}}\,\pi_{Y}^{*}(-\otimes\rho_{0})\,\otimes^{\mathbf{L}}\,\pi_{U}^{*}K_{U}\right)[2],$

where ${\cal O}_{{\mathcal{Z}}}^{\vee}:=\mathbf{R}Hom({\cal
O}_{{\mathcal{Z}}},{\cal O}_{Y\times U})$ denotes the dual of ${\cal
O}_{{\mathcal{Z}}}$, where
$-\otimes\rho_{0}:{\mathcal{D}}(Y)\to{\mathcal{D}}^{G}(Y)$ denotes the functor
that attaches the trivial $G$-action, and where $K_{U}$ denotes the canonical
sheaf of $U$. We refer to [IN19, Section 3.1] for details, conventions, and
notations.

######  Theorem 7.1.

Keeping assumptions and notations, $\Phi$ is fully faithful and $\Psi$ is a
left adjoint of $\Phi$.

###### Proof.

For $G$ is a very small subgroup of ${\mathbf{GL}}_{2}({{\mathbb{C}}})$, this
is [Is02, Section 6] and [IU15, Proposition 1.1 and Lemma 2.9]. For $k$
algebraically closed of arbitrary characteristic and $G$ is a very small and
finite subgroup of ${\mathbf{GL}}_{2}(k)$, this is [IN19, Theorem 3.2].
However, these proofs also work if $G$ is a very small, finite, and linearly
reductive subgroup scheme of ${\mathbf{GL}}_{2,k}$. ∎

##  Appendix A Hopf algebras

In the first section of the appendix, we study finite group schemes from the
point of view of finite-dimensional Hopf algebras. We also recall the quantum
double of a Hopf algebra, as well as the adjoint and the extended adjoint
representation. Many results of this section should be well-known to the
experts, but are somewhat scattered over the literature - especially, since
many sources (sometimes implicitly) assume characteristic zero or work even
over the complex numbers.

###  A.1. Generalities

If $G$ is a finite group scheme over a field $k$, then the $k$-algebra
$A:=H^{0}(G,{\cal O}_{G})$ is commutative and finite-dimensional as $k$-vector
space. The multiplication $m:G\times G\to G$, the inverse $i:G\to G$, and the
neutral element $e:\mathrm{Spec}\>k\to G$ induce $k$-algebra homomorphisms
$m^{*}:A\to A\otimes_{k}A$, $i^{*}:A\to A$, and $e^{*}:A\to k$, which turn $A$
into a co-algebra over $k$ with co-multiplication $m^{*}$ and antipode
$S:=i^{*}$, and thus, into a _Hopf algebra_ over $k$. Since $i$ is the inverse
of $G$, the antipode $S$ satisfies $S^{2}={\rm id}_{A}$, that is, $A$ is an
_involutive_ Hopf algebra.

Conversely, if $A$ is a finite-dimensional and commutative Hopf algebra over
$k$, then it is involutive and $\mathrm{Spec}\>A$ is a finite group scheme
over $k$. Moreover, $A$ is a co-commutative Hopf algebra if and only if $G$ is
a commutative group scheme.

######  Example A.1.

Let $G_{\rm abs}$ be a finite group. The group algebra $k[G_{\rm abs}]$
becomes a Hopf algebra by defining the co-multiplication to be
$\Delta(g)=g\otimes g$ and the antipode to be $S(g)=g^{-1}$.

  1. (1)

Clearly, $k[G_{\rm abs}]$ is an involutive and co-commutative Hopf algebra.
Moreover, $k[G_{\rm abs}]$ is commutative if and only if $G_{\rm abs}$ is
commutative.

  2. (2)

There exist examples of non-isomorphic finite groups $H_{\rm abs}$ and $G_{\rm
abs}$, whose group rings $k[H_{\rm abs}]$ and $k[G_{\rm abs}]$ are isomorphic
as $k$-algebras.

However, they are not isomorphic as Hopf algebras: Recall that an element
$x\in B$ in a Hopf algebra $B$ is called _group-like_ if $\Delta(x)=x\otimes
x$. The set of group-like elements of $B$ form a group. If $B=k[G_{\rm abs}]$,
then the group of group-like elements of $B$ is isomorphic to $G_{\rm abs}$
and thus, recovers the group.

  3. (3)

There exists an isomorphism of finite group schemes over $k$

$G\,\cong\,\mathrm{Spec}\>k[G_{\rm abs}]^{*},$

where $k[G_{\rm abs}]^{*}$ denotes the dual Hopf algebra and where $G$ denotes
the constant group scheme over $k$ associated to $G_{\rm abs}$.

###  A.2. Simplicity

A Hopf algebra $A$ is _co-semi-simple_ if its dual Hopf algebra $A^{*}$ is
semi-simple and $A$ is _bi-semi-simple_ if both $A$ and $A^{*}$ are semi-
simple.

Let $G$ be a finite group scheme over $k$ and let $A:=H^{0}(G,{\cal O}_{G})$
be the associated Hopf algebra. To give a finite-dimensional and $k$-linear
representation $\rho:G\to{\rm GL}(V)$ is the same as to give an $A$-co-module
$V\to V\otimes_{k}A$, see [Wa79, Section 3.2] for details. Using this
equivalence, we see that $G$ is linearly reductive if and only if $A$ is co-
semi-simple. For the classification of linearly reductive group schemes (see
Section 2) in the language of Hopf algebras we refer to [Ch92].

The following equivalences are probably well-known to the experts.

######  Proposition A.2.

Let $G$ be a finite group scheme over an algebraically closed field $k$ of
characteristic $p\geq 0$. Let $A:=H^{0}(G,{\cal O}_{G})$ be the Hopf algebra
associated to $G$.

  1. (1)

$A$ is semi-simple if and only if $G$ is étale.

  2. (2)

$A$ is co-semi-simple if and only if $G$ is linearly reductive.

  3. (3)

$A$ is bi-semi-simple if and only if $G$ is of length prime to $p$.

###### Proof.

We already established Claim (2) above.

If $G$ is étale, then $A\cong k[G_{\rm abs}]^{*}$, where $G_{\rm abs}:=G(k)$
is the abstract group associated to $G$. Since group rings are co-semi-simple,
$A$ is semi-simple. Conversely, if $A$ is semi-simple, then
$\langle\varepsilon,\int_{H}^{r}\rangle\neq 0$ by Maschke’s theorem, which
implies that $A$ is a separable $k$-algebra, which implies that $G$ is étale
over $k$. This establishes Claim (1).

Of course, (3) follows immediately from (1) and (2), but we can also give an
independent proof: Being an involutive Hopf algebra, $A$ is bi-semi-simple if
and only if $p$ does not divide $\dim_{k}A$, see [EG98, Corollary 3.2] or
[LR88, Corollary 2.6]. The latter is equivalent to $G$ being of length prime
to $p$. This establishes Claim (3). ∎

###  A.3. Adjoint representation

Let $G_{\rm abs}$ be a finite group and let $k$ be an algebraically closed
field of characteristic $p\geq 0$ ($p$ may or may not divide the order of
$G_{\rm abs}$). Then, the action of $G_{\rm abs}$ on itself by conjugation is
a permutation representation and we denote by

$\rho_{\rm ad}\,:\,G_{\rm abs}\,\to\,{\mathbf{GL}}(V_{\rm ad})$

the associated $k$-linear representation.

######  Remark A.3.

To be more precise: Depending on whether one considers $x\mapsto gxg^{-1}$ or
$x\mapsto g^{-1}xg$, one should speak about _left_ or _right_ adjoint actions
and representations. For our discussion, this is not important, as long as one
chooses one of them and stays with it.

######  Remark A.4.

Let us recall a couple of general results about $\rho_{\rm ad}$, which are
well-known (I thank Frank Himstedt for explaining them to me):

  1. (1)

Let $C$ be the set of conjugacy classes of $G_{\rm abs}$ and let
$\\{g_{c}\\}_{c\in C}$ be a system of representatives. Then, we have an
isomorphism of $k$-linear representations

$\rho_{\mathrm{ad}}\,\cong\,\bigoplus_{c\in
C}\,\mathrm{Ind}_{C(g_{c})}^{G_{\rm abs}}\,\mathbb{1},$

where $C(g_{c})$ denotes the centraliser of $g_{c}\in G_{\rm abs}$, where
$\mathbb{1}$ denotes the one-dimensional trivial representation, and where
$\mathrm{Ind}$ denotes induction from a subgroup. See, for example, [NT89,
page 171 and Exercise 1.5].

  2. (2)

Let $g\in G_{\rm abs}$ and let

$V\,:=\,\mathrm{Ind}_{C(g)}^{G_{\mathrm{abs}}}\,\mathbb{1}\,\cong\,\bigoplus_{i}V_{i}$

be a decomposition into indecomposable representations, which is unique up to
isomorphism and numbering.

There is precisely one summand, say $V_{i_{1}}$, that has a subrepresentation
that is isomorphic to $\mathbb{1}$. Moreover, this subrepresentation is
unique. Also, there is precisely one summand, say $V_{i_{2}}$, that has a
quotient representation that is isomorphic to $\mathbb{1}$. Moreover, this
quotient representation is unique. Then, we we have $i_{1}=i_{2}$ and set

$S(V)\,:=\,V_{i_{1}}\,=\,V_{i_{2}},$

which is called the _Scott representation_ of $V$. See, for example, [NT89,
page 296 and Theorem 8.4].

In particular, the dimension of the largest trivial subrepresentation (resp.
quotient representation) of $\rho_{\rm ad}$ is equal to the number of
conjugacy classes of $G_{\rm abs}$.

If $A$ is a finite dimensional Hopf algebra over $k$, then there is an
_adjoint action_ of $A$ on itself (in fact, a _left_ adjoint action and a
_right_ adjoint action), see, for example, [Mo93, Definition 3.4.1]. We will
write

${\rm ad}\,:\,A\,\to\,{\rm End}(A)$

or simply ${}^{\rm ad}A$ for this representation.

######  Remarks A.5.

  1. (1)

If $G_{\mathrm{abs}}$ is a finite group, then the adjoint representation of
$H:=k[G_{\mathrm{abs}}]$ can be identified with the $k$-linear extension from
$G_{\mathrm{abs}}$ to $H$ of the conjugation action of $G_{\rm abs}$ on
itself.

  2. (2)

Let $G$ be a finite group scheme over $k$ and let $A:=H^{0}(G,{\cal O}_{G})$
be the $A:=H^{0}(G,{\cal O}_{G})$. There is an adjoint representation

$\rho_{\rm ad}\,:\,G\,\to\,{\mathbf{GL}}(V_{\rm ad}),$

which corresponds to the adjoint representation of the _dual_ Hopf algebra
$A^{*}$.

    1. (a)

If $G_{\rm abs}$ is a finite group and $G$ is the constant group scheme
associated to it, then $A$ is isomorphic to the dual of $k[G_{\rm abs}]$
equipped with its usual Hopf algebra structure. This shows that $\rho_{\rm
ad}$ should be defined via the adjoint representation of $A^{*}$ rather than
$A$.

    2. (b)

The representation $\rho_{\rm ad}$ should not be confused with the adjoint
representation of $G$ on its Lie algebra, see for example, [Wa79, page 100].
The latter can be identified with a subquotient of $\rho_{\rm ad}$.

###  A.4. Quantum doubles

If $A$ is a finite-dimensional Hopf algebra over a field $k$, then Drinfeld
[Dr87] defined a Hopf algebra $D(A):=(A^{\rm op})^{*}\bowtie A$, called the
_quantum double_ or _Drinfeld double_ , where the bicrossed product structure
is defined using the co-adjoint representation of $A$ on $A^{*}$ and the co-
adjoint representation of $A^{*}$ on $A$, see also [Mo93, Definition 10.3.1].

######  Remarks A.6.

Let $A$ be a finite-dimensional Hopf algebra over $k$ and let
$D(A):=(A^{\mathrm{op}})^{*}\bowtie A$ be its quantum double.

  1. (1)

As $k$-vector space, $D(A)$ is of dimension $(\dim_{k}A)^{2}$.

  2. (2)

$A$ is a Hopf subalgebra of $D(A)$ via $\varepsilon\bowtie A$ and
$(A^{\mathrm{op}})^{*}$ is a Hopf subalgebra of $D(A)$ via
$(A^{\mathrm{op}})^{*}\bowtie 1$.

  3. (3)

If $A$ is commutative and co-commutative, then we have $A^{\mathrm{op}}=A$ and
using the definition of the quantum double, we obtain isomorphisms

$D(A)\,\cong\,A^{*}\otimes_{k}A\,\cong\,D(A^{*})$

of Hopf algebras over $k$, where the tensor product is the trivial tensor
product of Hopf algebras. If we set $G:=\mathrm{Spec}\>A$, then we have
$G^{D}\cong\mathrm{Spec}\>A^{*}$, where $-^{D}$ denotes the Cartier dual group
scheme, and we have an isomorphism

$\mathrm{Spec}\>D(A)\,\cong\,G\times_{k}G^{D}$

of finite and commutative group schemes over $k$.

  4. (4)

In particular, $D(A)$ is commutative if and only if $A$ is commutative and co-
commutative.

The assertions on simplicity in the next proposition extend results of
Witherspoon [Wi96, Proposition 1.2] from groups to group schemes. They are
also related to general semi-simplicity results of quasi-triangular Hopf
algebras in positive characteristic due to Etingof and Gelaki [EG98] and in
this form, they might be known to the experts.

######  Proposition A.7.

Let $G$ be a finite group scheme over an algebraically closed field $k$ of
characteristic $p\geq 0$ and let $A:=H^{0}(G,{\cal O}_{G})$ be the associated
Hopf algebra. Then, the following are equivalent

_(1)_ |  $D(A)$ is semi-simple | _(1’)_ |  $D(A^{*})$ is semi-simple
---|---|---|---
_(2)_ |  $D(A)$ is co-semi-simple | _(2’)_ |  $D(A^{*})$ is co-semi-simple
_(3)_ |  $A$ is bi-semi-simple | _(3’)_ |  $A^{*}$ is bi-semi-simple
_(4)_ |  $G$ is of length prime to $p$ |  |

###### Proof.

The equivalences (1) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3) and (1’)
$\Leftrightarrow$ (2’) $\Leftrightarrow$ (3’) are shown in [Mo93, Corollary
10.3.13]. The equivalence (3) $\Leftrightarrow$ (3’) is trivial and finally,
the equivalence (3) $\Leftrightarrow$ (4) was shown in Proposition A.2. ∎

###  A.5. Extended adjoint representation

Given a Hopf algebra $A$, we defined the adjoint representation
${}^{\mathrm{ad}}A$ in Section A.3 above. There is a way to extend this to a
representation

$\mathrm{Ad}\,:\,D(A)\,\to\,{\rm End}(A)$

the _extended adjoint representation_ of $A$, which is denoted
${}^{\mathrm{Ad}}A$. For its definition, we refer to Zhu’s article [Zhu97], as
well as [CW11] for subsequent work on this representation. We note that it can
be described as an induced representation

(7) ${}^{\mathrm{Ad}}\,A\,\cong\,{\mathrm{Ind}}_{A}^{D(A)}\,\mathbb{1},$

where $\mathbb{1}$ denotes the trivial one-dimensional representation and
where we consider $A$ as a subalgebra of $D(A)$ via $\varepsilon\bowtie A$,
see [Bu06] or [Ja17, Section 4.3.4].

######  Example A.8.

Let $G_{\rm abs}$ be a finite group and let $H:=k[G_{\rm abs}]$ be the group
algebra equipped with its usual Hopf algebra structure. The elements $g\in
G_{\rm abs}$ form a basis of $H$ as $k$-vector space and we denote by
$\rho_{g}\in H^{*}$ the dual basis elements.

  1. (1)

The multiplication of $D(H)$ is given by

$(\rho_{h}\bowtie
g)\cdot(\rho_{k}\bowtie\ell)\,=\,\delta_{h,gkg^{-1}}\cdot\rho_{h}\bowtie
g\ell,$

see, for example, [Ja17, Example 2.4.2].

    1. (a)

The extended adjoint representation ${}^{\rm Ad}H$ is given by

$(\rho_{h}\bowtie
k)\,\mapsto\,(g\,\mapsto\,\delta_{h^{-1},kgk^{-1}}\,\cdot\,kgk^{-1})$

see, for example, [Ja17, Example 2.5.3].

    2. (b)

The unit of $H^{*}$ is $\varepsilon:=\sum_{g\in G_{\rm abs}}\rho_{g}$ and the
restriction of ${}^{\mathrm{Ad}}H$ to $\varepsilon\bowtie H$ is ${}^{\rm
ad}H$. Thus, the extended adjoint representation extends the adjoint
representation from an $A$-representation to a $D(A)$-representation, whence
the name.

    3. (c)

The extended adjoint representation ${}^{\rm Ad}H$ is semi-simple. More
precisely, the $k$-subvector space of $H$ generated by the elements of a
conjugacy class of $G_{\mathrm{abs}}$ is a simple $D(H)$-subrepresentation of
${}^{\mathrm{Ad}}H$. This gives a bijection between conjugacy classes of
$G_{\mathrm{abs}}$ and simple subrepresentations of ${}^{\mathrm{Ad}}H$. By
[Ja17, Lemma 4.3.2], this is also true if $p$ divides the order of
$G_{\mathrm{abs}}$.

We refer to [Bu06, DPR91, Go93, Wi96] for more results about the
representation theory of $D(H)$ and ${}^{\mathrm{Ad}}H$.

  2. (2)

If we identify $H^{**}$ with $H$, then the multiplication of $D(H^{*})$ is
given by

$(g\bowtie\rho_{h})\cdot(\ell\bowtie\rho_{k})\,=\,\delta_{h,k}\,\cdot\,(g\cdot\ell)\bowtie\rho_{h}$

and the extended adjoint representation ${}^{\rm Ad}(H^{*})$ is given by

$(h\bowtie\rho_{k})\,\mapsto\,(\rho_{g}\,\mapsto\,\rho_{gh^{-1}}),$

which can be identified with the dual of the regular representation of $G_{\rm
abs}$. The restriction of ${}^{\rm Ad}(H^{*})$ to $1\bowtie H^{*}$ is trivial,
which is equal to the adjoint representation ${}^{\rm ad}(H^{*})$, which is
also trivial.

######  Example A.9.

Let $A$ be a finite-dimensional Hopf algebra over $k$ that is commutative and
co-commutative.

  1. (1)

The adjoint representations ${}^{\rm ad}A$ and ${}^{\rm ad}(A^{*})$ are
trivial.

By Remark A.6, we have isomorphisms

$D(A)\,\cong\,A^{*}\otimes_{k}A\,\cong\,D(A^{*})$

of Hopf algebras over $k$, where the tensor product is the trivial tensor
product of Hopf algebras. Thus, to give a representation
$\rho:D(A)\to\mathrm{End}(V)$ is equivalent to giving two representations
$\rho_{1}:A\to\mathrm{End}(V)$ and $\rho_{2}:A^{*}\to\mathrm{End}(V)$, whose
actions on $V$ commute.

  1. (2)

Using the description (7) of ${}^{\rm Ad}A$ as induced representation, we
obtain ${}^{\rm Ad}A$ from the the trivial $A$-action on $A$ (this is
$\rho_{1}$) and the dual of the regular representation of $A^{*}$ (this is
$\rho_{2}$).

  2. (3)

Similarly, we obtain ${}^{\rm Ad}(A^{*})$ from the trivial $A^{*}$-action on
$A^{*}$ and the dual of the regular representation of $A$.

##  Appendix B Conjugacy classes for finite group schemes

Let $G$ be a finite group scheme over an algebraically closed field $k$ of
characteristic $p\geq 0$. In the second section of the appendix, we discuss
several approaches towards the notion of a _conjugacy class_ for $G$. If
$p=0$, then all of them lead to the same notion, namely, the familiar one. All
approaches look reasonable at first sight if $p>0$ and they lead to
essentially “the same” answer if $G$ is of length prime to $p$. In general
however, they lead to different notions, all of which have their merits and
drawbacks. This appendix serves as a motivation for Definition 6.1, but the
discussion and results may be interesting in themselves.

###  B.1. First approach: via rational points

Let $G(k)$ be the group of $k$-rational points. We obtain an equivalence
relation $\sim$ on this set by defining $g_{1}\sim g_{2}$ if and only if there
exists $h\in G(k)$ such that $g_{1}=hg_{2}h^{-1}$. The quotient $G(k)/\sim$ is
the first candidate for the set of conjugacy classes.

######  Example B.1.

If $G$ is étale over $k$, then it is the constant group scheme associated to
the finite group $G_{\rm abs}:=G(k)$. In this case, $G(k)/\sim$ coincides with
the set of conjugacy classes of $G_{\rm abs}$.

If $G$ is étale, which is automatic if $p=0$, then this approach is
satisfactory. However, if $p>0$, then the connected-étale sequence (4) induces
a bijection

$G(k)\,\cong\,G^{\mathrm{\acute{e}t}}(k),$

which is an isomorphism of finite groups. In particular, $G(k)/\sim$ depends
on the maximal étale quotient $G^{\mathrm{\acute{e}t}}$ of $G$ only. For
example, in the extremal case where $G$ is connected, we have $G(k)=\\{1\\}$
and then, $G(k)/\sim$ consists of one element, and we do not gain much
information about $G$.

Concerning functoriality: if $\varphi:G\to H$ is a homomorphism of finite
group schemes over $k$, then we have induced morphisms $G(k)\to H(k)$,
$G^{\mathrm{\acute{e}t}}\to H^{\mathrm{\acute{e}t}}$, and $G(k)/\sim\to
H(k)/\sim$.

###  B.2. Second approach: via representations and K-theory

If $G_{\rm abs}$ is a finite group of order prime to $p$, then the category
$\operatorname{Rep}_{k}(G)$ of $k$-linear and finite-dimensional $G_{\rm
abs}$-representations is semi-simple. In this case, the number of isomorphism
classes of simple representations is equal to the number of conjugacy classes.
More precisely, if $\rho$ is a $k$-linear and finite-dimensional
representation of $G_{\rm abs}$ and $g\in G_{\rm abs}$ is an element, then

$(\rho,g)\,\mapsto\,{\rm Tr}\left(\rho(g)\right)\,\in\,k$

induces a pairing between representations and conjugacy classes. Since the
character table of a finite group is a quadratic and invertible matrix (see,
for example, [Se77, Proposition I.7]), this pairing is non-degenerate and one
can think of conjugacy classes as being “dual” to simple representations.

This idea can be made precise as follows: Let $F$ be a field, let
$\mathrm{Cl}(G_{\mathrm{abs}})$ be the set of conjugacy classes of
$G_{\mathrm{abs}}$, and let $F^{\mathrm{Cl}(G_{\mathrm{abs}})}$ be the ring of
class functions on $G_{\mathrm{abs}}$ with values in $F$. Let $g$ be the order
of $G_{\mathrm{abs}}$, let $\zeta_{g}\in{{\mathbb{C}}}$ be a primitive $g$.th
primitive root of unity, and assume that $F$ contains
${{\mathbb{Q}}}(\zeta_{g})$. Since the characters of $G_{\mathrm{abs}}$ take
values in $F$ (here, we use ${{\mathbb{Q}}}(\zeta_{g})\subseteq F$), we have
injective ring homomorphisms

$F\,\to\,F\otimes_{{\mathbb{Z}}}K_{k}(G_{\mathrm{abs}})\,\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}\,F^{\mathrm{Cl}(G_{\mathrm{abs}})}$

and $\gamma$ is an isomorphism, see [Se77, Section 11.4] and the first
paragraph of [Se77, Section 9.1]. In particular,

$\mathrm{Spec}\>\,F\otimes_{{\mathbb{Z}}}K_{k}(G_{\mathrm{abs}})$

is a finite set consisting of maximal ideals only and the topology is
discrete. Since $\gamma$ is an isomorphism, the cardinality of this set is
equal to that of $\mathrm{Cl}(G_{\mathrm{abs}})$. This carries over to finite
and linearly reductive schemes as follows.

######  Proposition B.2.

Let $G$ be a finite and linearly reductive group scheme over $k$ and let
$G_{\mathrm{abs}}$ be the associated abstract group. Let $g$ be the length of
$G$, fix a $g$.th root of unity $\zeta_{g}\in{{\mathbb{C}}}$, and let $F$ be a
field that contains ${{\mathbb{Q}}}(\zeta_{g})$. Then, there exista canonical
bijection of sets

$\left\\{\begin{array}[]{l}\mbox{conjugacy classes of $G_{\rm
abs}$}\end{array}\right\\}\,\to\,\mathrm{Spec}\>\,F\otimes_{{\mathbb{Z}}}K_{k}(G_{\rm
abs})$

and homeomorphisms

$\mathrm{Spec}\>\,F\otimes_{{\mathbb{Z}}}K_{k}(G_{\rm
abs})\,\leftarrow\,\mathrm{Spec}\>\,F\otimes_{{\mathbb{Z}}}K_{\overline{K}}(G_{\rm
abs})\,\leftarrow\,\mathrm{Spec}\>\,F\otimes_{{\mathbb{Z}}}K_{k}(G).$

###### Proof.

If $c$ is a conjugacy class of $G_{\rm abs}$, then $P_{0,c}$ (notation as in
[Se77, Section 11.4, Proposition 30]) is an element of
$\mathrm{Spec}\>F\otimes K_{k}(G_{\rm abs})$ and by loc. cit. this defines a
bijection. The isomorphisms of Corollary 2.8 induce homeomorphisms and thus,
bijections of spectra as stated. ∎

######  Remarks B.3.

  1. (1)

The maximal abelian extension ${{\mathbb{Q}}}\subset{{\mathbb{Q}}}^{\rm ab}$
is generated by all roots of unity by the Kronecker-Weber theorem. Thus, if we
have ${{\mathbb{Q}}}^{\mathrm{ab}}\subseteq F$, then we have a field that
works independent of the length of $G$. In Definition 6.1, we have chosen
$F={{\mathbb{C}}}$ as this field may be more familiar than
${{\mathbb{Q}}}^{\mathrm{ab}}$.

  2. (2)

In [Se77, Section 11.4], Serre described
$\mathrm{Spec}\>A\otimes_{{\mathbb{Z}}}K_{k}(G_{\rm abs})$, where
$A={{\mathbb{Z}}}[\zeta_{g}]$. Using Corollary 2.8, we obtain a homeomorphism

$\mathrm{Spec}\>A\otimes_{{\mathbb{Z}}}K_{k}(G)\,\to\,\mathrm{Spec}\>A\otimes_{{\mathbb{Z}}}K_{k}(G_{\rm
abs}).$

In particular, Serre’s results from loc.cit. carry over to
$A\otimes_{{\mathbb{Z}}}K_{k}(G)$. For the purposes of this article, we are
only interested in the fibre over $0\in\mathrm{Spec}\>F$ with
$F=\mathrm{Frac}(A)={{\mathbb{Q}}}(\zeta_{g})$, that is, the Zariski open
subset $\mathrm{Spec}\>F\otimes K_{k}(G)\subset\mathrm{Spec}\>A\otimes
K_{k}(G)$.

######  Example B.4.

Let $G$ be the group scheme $\boldsymbol{\alpha}_{p}$ or ${\mathbf{C}}_{p}$
over the algebraically closed field $k$ of characteristic $p>0$. Then,
$\operatorname{Rep}_{k}(G)$ is not semi-simple: the only simple $k$-linear
representation of $G$ is the trivial one-dimensional representation
$\mathbb{1}$. Thus, $\mathbb{1}\mapsto 1$ induces an isomorphism of rings
$K_{k}(G)\cong{{\mathbb{Z}}}$ and
$\mathrm{Spec}\>F\otimes_{{\mathbb{Z}}}K_{k}(G)$ consists of one point only.
The approach to conjugacy classes in this subsection may therefore lead to
somewhat unexpected results if $G$ is not linearly reductive.

######  Proposition B.5.

Let $\varphi:G\to H$ be a morphism of finite and linearly reductive group
schemes over $k$. Let $\varphi_{\rm abs}:G_{\rm abs}\to H_{\rm abs}$ be the
induced homomorphism of their associated abstract groups. Let $F$ be a field
that contains ${{\mathbb{Q}}}(\zeta_{g},\zeta_{h})$, where $g$ (resp. $h$)
denotes the length of $G$ (resp. $H$).

  1. (1)

There maps $\varphi$ and $\varphi_{\mathrm{abs}}$ induce ring homomorphisms
$K_{k}(H)\to K_{k}(G)$ and $K_{\overline{K}}(H_{\mathrm{abs}})\to
K_{\overline{K}}(G_{\mathrm{abs}})$, respectively. We obtain a commutative
diagram of continuous maps

$\textstyle{\mathrm{Spec}\>F\otimes_{{\mathbb{Z}}}K_{k}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Spec}\>F\otimes_{{\mathbb{Z}}}K_{k}(H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Spec}\>F\otimes_{{\mathbb{Z}}}K_{\overline{K}}(G_{\mathrm{abs}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Spec}\>F\otimes_{{\mathbb{Z}}}K_{\overline{K}}(H_{\mathrm{abs}}),}$

whose vertical arrows are the homeomorphisms from Proposition B.2.

  2. (2)

Let $G_{\mathrm{abs}}/\sim\to H_{\mathrm{abs}}/\sim$ be the map on conjugacy
classes induced by $\varphi_{\mathrm{abs}}$. We obtain a commutative diagram
of maps of sets

$\textstyle{G_{\rm
abs}/\sim\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{\rm
abs}/\sim\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Spec}\>F\otimes_{{\mathbb{Z}}}K_{\overline{K}}(G_{\mathrm{abs}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Spec}\>F\otimes_{{\mathbb{Z}}}K_{\overline{K}}(H_{\mathrm{abs}}),}$

whose vertical maps are the bijections from Proposition B.2.

###### Proof.

Clearly, $\varphi$ induces a morphism $K_{k}(H)\to K_{k}(G)$ of rings since
every $H$-representation becomes a $G$-representation via $\varphi$.
Similarly, $\varphi_{\mathrm{abs}}$ induces a ring homomorphism
$\varphi_{K,\mathrm{abs}}\,:\,K_{\overline{K}}(H_{\mathrm{abs}})\to
K_{\overline{K}}(G_{\mathrm{abs}})$. We leave it to the reader to check the
compatibility of these maps with the homeomorphisms of Proposition B.2.

To check commutativity of the second diagram, let $g\in G_{\rm abs}$. With the
notations and definitions of [Se77, Section 11.4], it is easy to see that we
have

$P_{0,\varphi_{\mathrm{abs}}(g)}\,=\,\varphi_{K,\mathrm{abs}}^{-1}(P_{0,g})\,=\,\varphi_{K,\mathrm{abs}}^{\sharp}(P_{0,g}),$

where

$\varphi_{K,\mathrm{abs}}^{\sharp}\,:\,\mathrm{Spec}\>F\otimes
K_{\overline{K}}(G_{\mathrm{abs}})\,\to\,\mathrm{Spec}\>F\otimes
K_{\overline{K}}(H_{\mathrm{abs}})$

is the induced map on spectra. Since the image of the conjugacy class $[g]$ of
$G_{\mathrm{abs}}$ is the conjugacy class $[\varphi_{\mathrm{abs}}(g)]$ of
$H_{\mathrm{abs}}$, the assertion follows. ∎

###  B.3. Third approach: the scheme of conjugacy classes

Just as group schemes generalise the notion of a group, one could try to
replace the _set_ of conjugacy classes by a suitable notion of _scheme_ of
conjugacy classes. More precisely, let $G$ be a finite group scheme over $k$
and let $\underline{\mathrm{Aut}}(G)$ be the automorphism group scheme of $G$.
For every scheme $T\to\mathrm{Spec}\>k$, we have the set $G(T)$ of $T$-valued
points of $G$ and a conjugation action of $G(T)$ on $G(T)$. This induces a
morphism of schemes $G\to\underline{\mathrm{Aut}}(G)$. We will say that two
elements of $G(T)$ are _equivalent_ if they differ by such an automorphism and
we denote the resulting equivalence relation by $\sim$.

We obtain a functor from the category of schemes over $k$ to sets

$\begin{array}[]{cccccc}\underline{{\rm Conj}}_{G}&:&({\rm
Schemes}/k)&\to&({\rm Sets})\\\ &&T&\mapsto&G(T)/\sim&.\end{array}$

This functor should somehow represent the conjugacy classes of $G$ in the
sense of schemes.

######  Proposition B.6.

Let $G$ be a finite group scheme over $k$. Then, the functor $\underline{{\rm
Conj}}_{G}$ is representable by a scheme ${\rm Conj}_{G}$, which is finite
over $\mathrm{Spec}\>k$.

###### Proof.

Set $V:=H^{0}(G,{\cal O}_{G})$ and let $\rho_{{\rm ad}}:G\to{\mathbf{GL}}(V)$
be the adjoint representation of $G$ (see Appendix A.3). Then, the functor
$\underline{\mathrm{Conj}}_{G}$ can be rephrased as the functor that
associates to each $T\to\mathrm{Spec}\>k$ the quotient of $V\times{\cal
O}_{T}$ modulo $G(T)$. This amounts to representing the quotient $V/G$ by a
scheme. Since $V$ is a vector space and thus, can be identified with an affine
scheme, and since $G$ is a finite group scheme, this quotient is representable
by a scheme. In fact, it is representable by the spectrum of the invariant
ring $\mathrm{Spec}\>V^{G}$. ∎

######  Definition B.7.

$\mathrm{Conj}_{G}$ is called the _scheme of conjugacy classes_ of $G$.

######  Remark B.8.

If $\rho_{\mathrm{ad}}:G\to{\mathbf{GL}}(H^{0}(G,{\cal O}_{G}))$ is the
adjoint representation, then the previous proof shows that the length of
$\mathrm{Conj}_{G}$ over $\mathrm{Spec}\>k$ is equal to the dimension of the
maximal trivial subrepresentation of $\rho_{\mathrm{ad}}$. If $G_{\rm abs}$ is
a finite group or if $G$ is a finite and étale group scheme over
$\mathrm{Spec}\>k$, then we gave an explicit description of this maximal
trivial subrepresentation of $\rho_{\mathrm{ad}}$ in Remark A.4.

######  Examples B.9.

  1. (1)

If $G$ is étale over $k$, then $G$ is isomorphic to the constant group scheme
associated to $G_{\mathrm{abs}}:=G(k)$. In this case, $\mathrm{Conj}_{G}$ is a
disjoint union of copies of $\mathrm{Spec}\>k$, one copy for each conjugacy
class of the abstract group $G_{\mathrm{abs}}$.

  2. (2)

If $G$ is commutative, then the adjoint representation is trivial and thus,
$\mathrm{Conj}_{G}$ is isomorphic to the scheme underlying $G$. For example,
if $G=\boldsymbol{\alpha}_{p}$ or $G=\boldsymbol{\mu}_{p}$, then
$\mathrm{Conj}_{G}$ is a non-reduced scheme of length $p$ with reduction
$(\mathrm{Conj}_{G})_{\mathrm{red}}\cong\mathrm{Spec}\>k$.

In characteristic zero, all finite group schemes are étale and Case (1)
applies. However, in characteristic $p>0$, the second example shows that for
non-reduced group schemes $\mathrm{Conj}_{G}$ may also be non-reduced. The
reduction $(\mathrm{Conj}_{G})_{\mathrm{red}}$ is related to our first
approach to conjugacy classes:

######  Proposition B.10.

  1. (1)

A morphism $G\to H$ of finite group schemes over $k$ induces a morphism
$\mathrm{Conj}_{G}\to\mathrm{Conj}_{H}$ of schemes over $k$.

  2. (2)

Assume $p>0$ and let $G\to G^{\mathrm{\acute{e}t}}$ be the maximal étale
quotient of the finite group scheme $G$ over $k$. Then, reduction identifies
$G_{\mathrm{red}}$ with $G^{\mathrm{\acute{e}t}}$ and the induced natural
inclusion $G^{\mathrm{\acute{e}t}}\to G$ induces an isomorphism

$\mathrm{Conj}_{G^{\mathrm{\acute{e}t}}}\,\to\,\left(\mathrm{Conj}_{G}\right)_{\mathrm{red}}$

of schemes over $k$. In particular, $(\mathrm{Conj}_{G})_{\mathrm{red}}$ is a
disjoint union of copies of $\mathrm{Spec}\>k$ with one copy for each
conjugacy class of the abstract group
$G_{\mathrm{abs}}:=G(k)=G^{\mathrm{\acute{e}t}}(k)$.

###### Proof.

We first prove Claim (1). If $G_{\rm abs}\to H_{\rm abs}$ is a homomorphism of
groups, then we get a well-defined induced map of conjugacy classes. Thus, if
$G\to H$ is as in (1) and if $T$ is a scheme over $k$, then we get a well-
defined map $(G(T)/\sim)\to(H(T)/\sim)$. This induces a morphism of functors
$\underline{\mathrm{Conj}}_{G}\to\underline{\mathrm{Conj}}_{H}$ and thus, a
morphism of schemes $\mathrm{Conj}_{G}\to\mathrm{Conj}_{H}$.

Let $G$ be as in Claim (2). Let $G_{\mathrm{red}}\to G$ be the reduction,
which is a morphism of group schemes. We thus obtain canonical homomorphisms
$G^{\mathrm{\acute{e}t}}\to G\to G^{\mathrm{\acute{e}t}}$ of group schemes
over $k$, which (by Claim (1)) induce morphisms of their associated schemes of
conjugacy classes

$\mathrm{Conj}_{G^{\mathrm{\acute{e}t}}}\,\to\,\mathrm{Conj}_{G}\,\to\,\mathrm{Conj}_{G^{\mathrm{\acute{e}t}}}.$

The composition is the identity. Since
$\mathrm{Conj}_{G^{\mathrm{\acute{e}t}}}$ is reduced, we obtain a
factorisation

(8)
$\mathrm{Conj}_{G^{\mathrm{\acute{e}t}}}\,\to\,\left(\mathrm{Conj}_{G}\right)_{\mathrm{red}}\,\to\,\mathrm{Conj}_{G^{\mathrm{\acute{e}t}}}.$

All these schemes are reduced and finite over the algebraically closed field
$k$. Thus, all of them are finite disoint unions of copies of
$\mathrm{Spec}\>k$. To prove that the morphisms in (8) are isomorphisms, it
suffices to check that the induced maps on $k$-rational points are bijections.
This follows easily from the fact that the maps $G_{\mathrm{red}}(k)\to
G(k)\to G^{\mathrm{\acute{e}t}}(k)$ are bijections. ∎

If $G$ is moreover linearly reductive, then the length of $\mathrm{Conj}_{G}$
is related to the approach to conjugacy classes from Section B.2.

######  Proposition B.11.

Let $G$ be a finite and linearly reductive group scheme over $k$, let
$G_{\mathrm{abs}}$ be the abstract group associated to $G$, and let $F$ be a
field as in Proposition B.2. Then,

$\mathrm{length}_{k}\,\mathrm{Conj}_{G}\,=\,|\mathrm{Spec}\>F\otimes_{{\mathbb{Z}}}K_{k}(G)|$

and this length agrees with the number of conjugacy classes of
$G_{\mathrm{abs}}$.

###### Proof.

By Remark B.8, the length of $\mathrm{Conj}_{k}(G)$ is equal to the dimension
of the largest trivial subrepresentation of $\rho_{\mathrm{ad}}$. Since ${\rm
Rep}_{k}(G)$ is semi-simple, this is the same as the multiplicity of
$\mathbb{1}$ in $\rho_{\mathrm{ad}}$. Using the isomorphism from Prop 2.6,
this multiplicity is the same as the multiplicity of $\mathbb{1}$ in the
adjoint representation of $G_{\mathrm{abs}}$. This latter multiplicity is
equal to the number of conjugacy classes of $G_{\mathrm{abs}}$, see, for
example, Remark A.4. By Proposition B.2, this number is equal to the
cardinality of $\mathrm{Spec}\>F\otimes_{{\mathbb{Z}}}K_{k}(G)$. ∎

###  B.4. Fourth approach: via lifting to characteristic zero

Another idea might be to use lifting to characteristic zero: Let $k$ be an
algebraically closed field of characteristic $p>0$, let $W(k)$ be the ring of
Witt vectors of $k$, and let $K$ be field of fractions of $W(k)$.

######  Example B.12 (Mumford–Oort).

Let
$\varphi:\boldsymbol{\mu}_{p}\to\underline{\mathrm{Aut}}(\boldsymbol{\alpha}_{p})\cong{{\mathbb{G}}}_{m}$
be a non-trivial homomorphism and let
$G:=\boldsymbol{\alpha}_{p}\rtimes_{\varphi}\boldsymbol{\mu}_{p}$ be the
corresponding semi-direct product group scheme. Then, $G$ is a non-commutative
group scheme of length $p^{2}$ over $k$. Oort and Mumford [MO68, Introduction,
Example (-B)] (but see also [Oo71, page 266]) showed that there does _not_
exist a lift of $G$ to any extension of $W(k)$.

In particular, one _cannot_ define conjugacy classes by first lifting $G$ over
some possibly ramified extension of $W(k)$, then passing to the geometric
generic fibre of the lift, which would be the constant group associated to an
abstract finite group $G_{\mathrm{abs}}$, and finally use the conjugacy
classes of $G_{\mathrm{abs}}$ as a replacement for the conjugacy classes of
$G$. The reason is simply that lifts may not exist to start with.

In the following two cases, lifts do exist and we leave the straightforward
proofs of our assertions to the reader:

######  Remark B.13.

Assume that $G$ is étale over $k$. Then, there exists a unique flat lift of
$G$ over $W(k)$, which is the constant group scheme associated to
$G_{\mathrm{abs}}:=G(k)$ over $W(k)$. The constant scheme associated to the
set is the unique a flat lift of the scheme $\mathrm{Conj}_{G}$ over $W(k)$,
which is étale over $W(k)$. More precisely, it is the constant scheme
associated to the set of conjugacy classes of $G_{\mathrm{abs}}$ over $W(k)$.

######  Remark B.14.

Assume that $G$ is linearly reductive. Let $G_{\mathrm{abs}}$ abstract group
associated to $G$ and recall that their representation categories are
equivalent by Proposition 2.6. Proposition B.2 shows that Definition 6.1 is
compatible with $G_{\rm abs}$.

If ${\cal G}\to\mathrm{Spec}\>W(k)$ is a flat lift of $G$ over $W(k)$, for
example, the canonical lift ${\cal G}_{\mathrm{can}}$, then

$\mathrm{Spec}\>\,H^{0}({\cal G},{\cal O}_{{\cal G}})^{{\cal
G}}\,\to\,\mathrm{Spec}\>W(k),$

where the invariants are taken with respect to the adjoint representation, is
a flat lift of $\mathrm{Conj}_{G}$ over $W(k)$, whose geometric generic fibre
is the constant scheme associated to the set of conjugacy classes of
$G_{\mathrm{abs}}$. (Flatness follows from the fact that the special and the
geometric generic fibre have the same length by Remark B.8.)

The length of $\mathrm{Conj}_{G}$ is equal to the number of conjugacy classes
of $G_{\rm abs}$. The length of $\mathrm{Conj}_{G}$ is at least the number of
$k$-rational points $\mathrm{Conj}_{G}(k)$ and in general not equal, since
$\mathrm{Conj}_{G}$ may not be étale over $k$.

If $G$ is of length prime to $p$, then it is étale and linearly reductive and
in this case, all previous approaches yield essentially the same notion of
conjugacy class. On the other hand, the examples of the above subsections show
that if $G$ is étale of length divisible by $p$ or if $G$ is linearly
reductive but not étale, then the various approaches of the above subsections
usually lead to different notions of conjugacy classes.

###  B.5. Fifth approach: via adjoint representation

Let $G_{\rm abs}$ be a finite group and let
$\rho_{\mathrm{ad}}:G_{\mathrm{abs}}\to{\mathbf{GL}}(V_{\mathrm{ad}})$ be its
adjoint representation over $k$. By Remark A.4, the dimension of the largest
trivial subrepresentation of $\rho_{\mathrm{ad}}$ is equal to the dimension of
largest trivial quotient representation of $\rho_{\mathrm{ad}}$ (even if $p$
divides the order of $G_{\mathrm{abs}}$) and these dimensions are equal to the
number of conjugacy classes of $G_{\mathrm{abs}}$. Unfortunately, these
trivial sub- or quotient representations do not admit _canonical_
decompositions into one-dimensional subspaces. Thus, there is no canonical
bijection between the conjugacy classes of $G_{\mathrm{abs}}$ with trivial
sub- or quotient representations of $\rho_{\mathrm{ad}}$.

######  Remark B.15.

Let $H:=k[G_{\mathrm{abs}}]$ be the group algebra with its usual Hopf algebra
structure. The set of group-like elements of $H$ recovers the group
$G_{\mathrm{abs}}$, see Example A.1. This suggests to use the adjoint
representation ${}^{\mathrm{ad}}H$ together with the Hopf algebra structure of
$H$ \- in particular, its co-algebra structure - to define a useful notion of
conjugacy class.

As we have seen (somewhat implicitly) in Section B.3 above, this works: Let
$G:=\mathrm{Spec}\>H^{*}$ be the constant group scheme associated to
$G_{\mathrm{abs}}$. Then, the topological space underlying $\mathrm{Conj}_{G}$
is the set of group-like elements of the Hopf algebra $H$ modulo the adjoint
representation. This set can be identified with the set of conjugacy classes
of $G_{\mathrm{abs}}$.

Unfortunately, the previous remark does not carry over to finite group schemes
that are not étale, as the following example shows.

######  Example B.16.

Let $G=\boldsymbol{\mu}_{p}$ or $G=\boldsymbol{\alpha}_{p}$ over the
algebraically closed field $k$ of characteristic $p>0$.

  1. (1)

The adjoint representation $\rho_{\mathrm{ad}}$ of $G$ is trivial of dimension
$p$, but there is no canonical way to decompose it into one-dimensional
subspaces. This would suggest to have $p$ conjugacy classes, but without being
able to distinguish them.

  2. (2)

Moreover, there is only one group-like element in the Hopf algebra
$H^{0}(G,{\cal O}_{G})^{*}$ and the adjoint representation on this element is
trivial as well. This would suggest to have one conjugacy class only, see also
Example B.9.(2).

###  B.6. Sixth approach: via extended adjoint representation

Given a finite-dimensional Hopf algebra $A$ over a field $k$, we have the
adjoint representation ${}^{\rm ad}A$, see Appendix A.3. In Appendix A.4, we
recalled the quantum double $D(A)=(A^{\mathrm{op}})^{*}\bowtie A$. In Appendix
A.5, we recalled that ${}^{\mathrm{ad}}A$ can be extended to a representation
of $D(A)$ on $A$, the extended adjoint representation ${}^{\mathrm{Ad}}A$.

Generalising work of Witherspoon [Wi99], Cohen and Westreich [CW11] defined
the set of conjugacy classes of $A$ for a finite-dimensional semi-simple Hopf
algebra over ${{\mathbb{C}}}$ to be the set of simple subrepresentations of
${}^{\mathrm{Ad}}A$. In this context, we also refer to the work of Jacoby
[Ja17] and Zhu [Zhu97].

If $G$ is a finite group scheme over $k$ with Hopf algebra $A:=H^{0}(G,{\cal
O}_{G})$, then $D(A)$ is semi-simple if and only if $D(A^{*})$ is semi-simple
if and only if $G$ is of length prime to $p$, see Proposition A.7. In
particular, the (extended) adjoint representations of $A$ or $A^{*}$ may not
be semi-simple. We will now study these representations and their relation to
conjugacy classes.

######  Example B.17.

Let $G$ be étale over $k$. Then it is the constant group scheme associated to
$G_{\mathrm{abs}}:=G(k)$. If $A:=H^{0}(G,{\cal O}_{G})$ is the associated Hopf
algebra, then we have $A^{*}\cong k[G_{\mathrm{abs}}]$. As seen in Example
A.8, the extended adjoint representation ${}^{\mathrm{Ad}}(A^{*})$ is semi-
simple and there is a natural bijection between conjugacy classes of
$G_{\mathrm{abs}}$ and simple subrepresentations of ${}^{\mathrm{Ad}}(A^{*})$.
We stress that this is also true if $p$ divides the order of
$G_{\mathrm{abs}}$.

######  Example B.18.

Let $G$ be a finite and linearly reductive group scheme over $k$, let
$G_{\mathrm{abs}}$ be the associated abstract finite group, and let
$A:=H^{0}(G,{\cal O}_{G})$ be the associated Hopf algebra. Since $A$ is
commutative, ${}^{\mathrm{ad}}A$ is trivial. Using the description (7) of
${}^{\mathrm{Ad}}A$ as an induced representation, we can identify it with the
dual of the regular representation

$\rho_{\mathrm{reg}}^{\vee}\,:\,A^{*}\,\to\,\mathrm{End}(A)$

together with the trivial representation of $A$. By Proposition 2.6,
representations of $A^{*}$ can be identified with representations of
${{\mathbb{C}}}[G_{\mathrm{abs}}]$ and thus, representations of
$G_{\mathrm{abs}}$. Thus, we can decompose ${}^{\rm Ad}A$ like the dual of the
regular representation of $G_{\mathrm{abs}}$: the number of isotypical
components of this representation is equal to the number of conjugacy classes
of $G_{\mathrm{abs}}$. Thus, one can think of the conjugacy classes of
$G_{\mathrm{abs}}$ as being “dual” to these isotypical components similarly to
Appendix B.2.

The upshot of this discussion is the following: Let $G$ be a finite group
scheme over $k$ and let $A:=H^{0}(G,{\cal O}_{G})$ be the associated Hopf
algebra.

  1. (1)

If $G$ is étale over $k$, then simple subrepresentations of
${}^{\mathrm{Ad}}(A^{*})$ are in bijection with conjugacy classes of
$G_{\mathrm{abs}}:=G(k)$.

  2. (2)

If $G$ is linearly reductive, then the isotypical components of
${}^{\mathrm{Ad}}A$ give a reasonably good definition for the “dual” of a
conjugacy class.

The following example shows that in general, neither ${}^{\mathrm{Ad}}A$ nor
${}^{\mathrm{Ad}}(A^{*})$ leads to a good approach toward the notion of a
conjugacy class - at least none that is better than the ones already
discussed.

######  Example B.19.

Let $G$ be a finite and commutative group scheme over $k$ and let
$A:=H^{0}(G,{\cal O}_{G})$ be the associated Hopf algebra. As a consequence of
Example A.9, we have the following.

  1. (1)

If $G=\boldsymbol{\mu}_{p}$, then we have $A=k[{\mathbf{C}}_{p}]$. Thus,
${}^{\mathrm{Ad}}A$ splits into the direct sum of $p$ pairwise non-isomorphic
one-dimensional representations, which correspond to the characters of $G$. On
the other hand, ${}^{\mathrm{Ad}}(A^{*})$ is a non-trivial successive $p$-fold
and non-split extension of the trivial representation $\mathbb{1}$, whose
semi-simplification is trivial of dimension $p$.

  2. (2)

If $G={\mathbf{C}}_{p}$, then we obtain the same as before with the rôles of
$A$ and $A^{*}$ interchanged.

  3. (3)

If $G=\boldsymbol{\alpha}_{p}$, then $A\cong A^{*}$ and
${}^{\mathrm{Ad}}A\cong{}^{\mathrm{Ad}}(A^{*})$ is a successive $p$-fold and
non-split extension of the trivial representation $\mathbb{1}$.

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