# Comparison of different definitions of pseudocharacters

Kathleen Emerson and Sophie Morel  CNRS et Unité De Mathématiques Pures Et
Appliquées, ENS de Lyon, 69342 Lyon Cedex 07, France. http://perso.ens-
lyon.fr/sophie.morel/<EMAIL_ADDRESS>
(Date: .)

###### Abstract.

We prove that the definitions of a $d$-dimensional pseudocharacter (or
pseudorepresentation) given by Chenevier and V. Lafforgue agree over any ring.
We also compare the scheme of Lafforgue’s $G$-pseudocharacters of a group with
its $G$-character variety.

###### Key words and phrases:

pseudorepresentations, pseudocharacters, character variety, invariant theory

###### 2020 Mathematics Subject Classification:

Primary 14M35, 13A50; Secondary 14L24, 14R20

Pseudocharacters (also known as pseudorepresentations) were first introduced
by Wiles ([19]) and Taylor ([17]) to study congruences modulo power of prime
numbers between representations of Galois groups. They defined congruence
between representations as congruence between their characters, and this led
them to the introduction of pseudocharacters, which are functions on a group
whose properties mimick those of the character of a finite-dimensional
representation. More precisely, let $\Gamma$ be a group, $A$ be a commutative
unital ring and $d$ be a positive integer. A map $T:\Gamma\rightarrow A$ is a
_$d$ -dimensional pseudocharacter_ if $d!\in A^{\times}$, $T(1)=d$,
$T(\gamma_{1}\gamma_{2})=T(\gamma_{2}\gamma_{1})$ for all
$\gamma_{1},\gamma_{2}\in\Gamma$ and $T$ satisfies the $d$-dimensional
_pseudo-character identity_ : for all
$\gamma_{1},\ldots,\gamma_{d+1}\in\Gamma$,

$\sum_{\sigma\in\mathfrak{S}_{d+1}}\operatorname{sgn}(\sigma)T^{\sigma}(\gamma_{1},\ldots,\gamma_{d+1})=0,$

where, if the decomposition into cycles of $\sigma\in\mathfrak{S}_{d+1}$
(including trivial cycles) is

$\sigma=(i_{1,1}i_{1,2}\ldots i_{1,n_{1}})\ldots(i_{k,1}i_{k,2}\ldots
i_{k,n_{k}}),$

then

$T^{\sigma}(\gamma_{1},\ldots,\gamma_{d+1})=T(\gamma_{i_{1,1}}\gamma_{i_{1,2}}\ldots\gamma_{i_{1,n_{1}}})\ldots
T(\gamma_{i_{k,1}}\gamma_{i_{k,2}}\ldots\gamma_{i_{k,n_{n}}}).$

We denote by $\operatorname{PChar}_{\Gamma,d}(A)$ the set of $d$-dimensional
pseudo-characters on $\Gamma$ with values in $A$. This is a contravariant
functor from the category of commutative $\mathbb{Z}[1/d!]$-algebras to that
of sets, and it is not hard to see that it is representable, unlike the
functor sending $A$ to the set of equivalence classes of $d$-dimensional
representations $\Gamma\rightarrow{\bf{GL}}_{d}(A)$. Moreover, if $A$ is an
algebraically closed field (in which $d!$ is invertible), then Taylor has
shown in [17, Theorem 1] that the trace induces a bijection from the set of
equivalence classes of $d$-dimensional _semi-simple_ representations of
$\Gamma$ over $A$ and $\operatorname{PChar}_{\Gamma,d}(A)$ and, if $k$ is a
field of characteristic $0$, then Chenevier has shown in [5, Proposition 2.3]
that $\operatorname{PChar}_{\Gamma,d,\operatorname{Spec}k}$ is naturally
isomorphic to the ${\bf{GL}}_{d}$-character variety of $\Gamma$ over
$\operatorname{Spec}k$.

The original definition of a $d$-dimensional pseudocharacter does not work
well in characteristic $p$ dividing $d!$, because a semi-simple
$d$-dimensional representation is no longer determined by its trace. However,
it is determined if we know all the coefficients of its characteristic
polynomial, and Chenevier ([4]) came up with a compact way of packaging all
that data in his theory of determinants. Determinants make sense over an
arbitrary ring $A$ without the requirement that $d!\in A^{\times}$, and the
functor $\operatorname{Det}_{\Gamma,d}$ that they define is representable.
Moreover, if $\rho:\Gamma\rightarrow{\bf{GL}}_{d}(A)$ is a morphism, then it
defines a determinant $D_{\rho}\in\operatorname{Det}_{\Gamma,d}(A)$. Chenevier
shows that, if $k$ is an algebraically closed field, then $\rho\mapsto
D_{\rho}$ is a bijection from the set of $d$-dimensional semi-simple
representations of $\Gamma$ over $k$ to $\operatorname{Det}_{\Gamma,k}(k)$.
However, it is not known whether the determinant functor
$\operatorname{Det}_{\Gamma,d}$ is isomorphic to the ${\bf{GL}}_{d}$-character
variety of $\Gamma$.

Vincent Lafforgue also gave a very general definition of pseudocharacters,
that this time makes sense for representations with values in any affine group
scheme $G$ over a base ring $C$. The definition of his pseudocharacters is
less compact, as it requires data for every invariant function on any power
$G$. Again, we obtain a contravariant functor $\operatorname{LPC}_{\Gamma,G}$
on the category of commutative $C$-algebras, which turns out to be
representable by an affine scheme. Also, if $\rho:\Gamma\rightarrow G(A)$ is a
morphism, then it defines a Lafforgue pseudocharacter
$\Theta_{\rho}\in\operatorname{LPC}_{\Gamma,G}(A)$. Lafforgues proves in [8,
Proposition 11.7] (see also [9, Proposition 8.2]) that, if $A=k$ is an
algebraically closed field, then $\rho\mapsto\Theta_{\rho}$ is a bijection
from the set of $G(k)$-conjugacy classes of $G$-completely reducible morphisms
$\Gamma\rightarrow G(k)$ to $\operatorname{LPC}_{\Gamma,G}(k)$; see also
Theorem 4.5 of the paper [2] of Böckle, Harris, Khare and Thorne for the
details of the proof in positive characteristic, where they also give the more
general definition of a Lafforgue pseudocharacter (Lafforgue himself was
working over a field).

In Section 2, we construct a is morphism of schemes from
$\operatorname{LPC}_{\Gamma,G}$ to the $G$-character variety of $\Gamma$
compatible with the bijections of the previous paragraph. We prove that, for
$G$ a geometrically reductive group scheme with connected geometric fibers
over a Dedekind domain or a field, this morphism is an _adequate
homeomorphism_ in the sense of Alper (see [1, Definition 3.3.1]), that is, an
integral universal homeomorphism which is a local isomorphism at all points
with residue characteristic $0$; see Proposition 2.11.

We come back to the case $G={\bf{GL}}_{d}$. The main result of this note is
that, over any ring, Lafforgue pseudocharacters of $\Gamma$ are in bijection
with $d$-dimensional determinants. More precisely, we prove the following
theorem:

######  Theorem .

(See Theorem 4.1.) There exists an isomorphism
$\alpha:\operatorname{LPC}_{\Gamma,{\bf{GL}}_{d}}\rightarrow\operatorname{Det}_{\Gamma,d}$
such that, for any commutative ring $A$ and any morphism
$\rho:\Gamma\rightarrow{\bf{GL}}_{d}(A)$, $\alpha$ sends $\Theta_{\rho}$ to
$D_{\rho}$.

The plan of the paper is as follows. In Section 1, we recall Chenevier’s
definition of determinants and the properties of determinants that we will
need; in particular, we give Vaccarino’s formula for the universal determinant
ring of a free monoid (see [18] and [5, Theorem 1.15]). In Section 2, we
define Lafforgue pseudocharacters and give some of their basic properties, in
particular the (easy) fact that they define a representable functor and the
relationship with the character variety (Proposition 2.11). We construct the
morphism $\alpha$ in Section 3 and, in Section 4, we prove that this morphism
is invertible. The main ingredients are Donkin’s ([6]) description of
generators of
$\mathcal{O}({\bf{GL}}_{d,\mathbb{Z}}^{n})^{{\bf{GL}}_{d,\mathbb{Z}}}$ and the
result of Vaccarino recalled in Section 1.

We thank Julian Quast and Olivier Taïbi for useful discussions.

##  1\. Determinants of algebras

In this section, we recall the definition of determinants by Chenevier ([5])
and some basic properties of the functor of determinants. Throughout this
section, $A$ denotes a commutative unital ring, and $A$-algebras are always
supposed to be associative and unital. Let $\mathcal{C}_{A}$ be the category
of commutative $A$-algebras and $\mathbf{Set}$ be the category of sets. We
first recall Roby’s definition of a polynomial law ([14], I.2), a homogeneous
polynomial law ([14], I.8) and a multiplicative polynomial law ([15], III.4).

######  Definition 1.1.

Let $M$ and $N$ be $A$-modules. A _polynomial law_ $f:M\rightarrow N$ is a
morphism of functors from $\mathcal{C}_{A}$ to $\mathbf{Set}$ between the
functors $B\mapsto M\otimes_{A}B$ and $B\mapsto N\otimes_{A}B$. For every
object $B$ of $\mathcal{C}_{A}$, we denote the resulting map
$M\otimes_{A}B\rightarrow N\otimes_{A}B$ by $f_{B}$.

######  Definition 1.2.

Let $M$ and $N$ be $A$-modules, and let $f:M\rightarrow N$ be a polynomial
law. Let $d\in\mathbb{N}$.

  * (1)

We say that $f$ is _homogeneous of degree $d$_ if, for all
$B\in\mathcal{C}_{A}$, $x\in M\otimes_{A}B$ and $b\in B$, we have

$f_{B}(bx)=b^{d}f_{B}(x).$

  * (2)

Suppose that $M$ and $N$ are $A$-algebras. We say that $f$ is _multiplicative_
if, for every $B\in\mathcal{C}_{A}$, we have $f_{B}(1)=1$ and
$f_{B}(xy)=f_{B}(x)f_{B}(y)$ if $x,y\in M\otimes_{A}B$.

If $M$ and $N$ are $A$-algebras and $d\in\mathbb{N}$, we write
$\mathcal{M}_{A}^{d}(M,N)$ for the set of multiplicative polynomial laws from
$M$ to $N$ that are homogeneous of degree $d$.

The following definition is due to Chenevier (see [5], 1.5).

######  Definition 1.3.

Let $R$ be an $A$-algebra. An _$A$ -valued $d$-dimensional determinant on $R$_
is an element of $\mathcal{M}_{A}(R,A)$. When $R=A[\Gamma]$ for some group
$\Gamma$, we also talk about _$A$ -valued $d$-dimensional determinants on
$\Gamma$_.

We write $\operatorname{Det}_{R,d}(A)$ (resp.
$\operatorname{Det}_{\Gamma,d}(A)$) for the set of $A$-valued $d$-dimensional
determinants on $R$ (resp. $\Gamma$); so
$\operatorname{Det}_{\Gamma,d}(A)=\operatorname{Det}_{A[\Gamma],d}(A)$.

######  Example 1.4.

  * (1)

Let $X$ be a set, and write $A\\{X\\}$ for the free unital associative
$A$-algebra on $X$. Any map $\rho:X\rightarrow M_{d}(A)$ gives rise to an
$A$-valued $d$-dimensional determinant $D_{\rho}$ on $A\\{X\\}$ as follows.
For any object $B$ of $\mathcal{C}_{A}$, the map $\rho$ induces a morphism of
$B$-algebras $\rho_{B}:B\\{X\\}\rightarrow M_{d}(B)$, where $M_{d}$ is the
scheme of $d\times d$ matrices; moreover, if $B\rightarrow B^{\prime}$ is a
morphism of commutative $A$-algebras, then we get a commutative diagram:

$\textstyle{B\\{X\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{B}}$$\textstyle{M_{d}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}\\{X\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{B^{\prime}}}$$\textstyle{M_{d}(B^{\prime})}$

We define $D_{\rho,B}:B\\{X\\}\rightarrow B$ by
$D_{\rho,B}(\omega)=\det(\rho_{B}(\omega))$, where $\det$ is the usual
determinant on $M_{d}(B)$.

  * (2)

Let $\Gamma$ be a group and $\rho:\Gamma\rightarrow{\bf{GL}}_{d}(A)$ be a
morphism of groups. Then we get an $A$-valued $d$-dimensional determinant
$D_{\rho}$ on $A[\Gamma]$ in a similar way: for every commutative $A$-algebra
$B$ and every $x\in B[\Gamma]$, we set $D_{\rho,B}=\det(\rho_{B}(x))$, where
$\rho_{B}:B[\Gamma]\rightarrow M_{d}(B)$ sends an element
$\sum_{i=1}^{n}b_{i}\gamma_{i}$ of $B[\Gamma]$, with $b_{i}\in B$ and
$\gamma_{i}\in\Gamma$, to $\sum_{i=1}^{n}b_{i}\rho(\gamma_{i})$.

######  Example 1.5.

Let $R$ be an $A$-algebra and $D\in\mathcal{M}_{A}^{d}(R,A)$. Following
Chenevier ([5], 1.10), we can define polynomial laws $\Lambda_{i}:R\rightarrow
A$, for $0\leq i\leq d$, as follows. For any commutative $A$-algebra $B$ and
any $x\in R\otimes_{A}B$, set

$D_{B[T]}(T-x)=\sum_{i=0}^{d}\Lambda_{i,B}(x)T^{d-i}.$

Note that $\Lambda_{i}$ is homogeneous of degree $i$. If $D=D_{\rho}$ for
$\rho$ as in Example 1.4(1) or (2), then $\Lambda_{i,B}(x)$ is the coefficient
of the degree $d-i$ term in the characteristic polynomial of the matrix
$\rho_{B}(x)\in M_{d}(B)$.

######  Definition 1.6.

Let $R$ be an $A$-algebra and $d\in\mathbb{N}$. The _determinant functor_
$\operatorname{Det}_{R,d}:\mathcal{C}_{A}\rightarrow\mathbf{Set}$ is defined
by

$\operatorname{Det}_{R,d}(B)=\mathcal{M}_{B}^{d}(R\otimes_{A}B,B)=\mathcal{M}_{A}^{d}(R,B)$

(the equality $\mathcal{M}_{B}^{d}(R\otimes_{A}B)=\mathcal{M}_{A}^{d}(R,B)$ is
proved in [5, 3.4]).

If $R=A[\Gamma]$ with $\Gamma$ a group, we also write
$\operatorname{Det}_{\Gamma,d}$ instead of $\operatorname{Det}_{R,d}$.

######  Remark 1.7.

If $D_{1}\in\operatorname{Det}_{R,d_{1}}$ and
$D_{2}\in\operatorname{Det}_{R,d_{2}}$, we can define $D_{1}\times
D_{2}\in\operatorname{Det}_{R,d_{1}+d_{2}}$ as follows. For every commutative
$A$-algebra $B$ and every $x\in R\otimes_{A}B$, take

$(D_{1}\times D_{2})_{B}(x)=D_{1,B}(x)D_{2,B}(x).$

If $\Gamma$ is a group, $\rho_{1}:\Gamma\rightarrow{\bf{GL}}_{d_{1}}(A)$,
$\rho_{2}:\Gamma\rightarrow{\bf{GL}}_{d_{2}}(A)$ are representations and
$D_{1}=D_{\rho_{1}}$, $D_{2}=D_{\rho_{2}}$, then $D_{1}\times
D_{2}=D_{\rho_{1}\oplus\rho_{2}}$.

######  Example 1.8.

Let $\Gamma$ be a group, and let $X$ be the underlying set of $\Gamma$. Then
the canonical surjective $A$-algebra morphism $A\\{X\\}\rightarrow A[\Gamma]$
gives rise to an injective morphism of functors
$\operatorname{Det}_{\Gamma,d}\rightarrow\operatorname{Det}_{A\\{X\\},d}$.

######  Theorem 1.9 (Roby, see [14] III.1 and [5] 1.6).

Let $R$ be an $A$-algebra and $d\in\mathbb{N}$. Then the functor
$\operatorname{Det}_{R,d}$ is representable by the $A$-algebra
$(\Gamma^{d}_{A}(R))^{\mathrm{ab}}$, where $\Gamma^{d}_{A}(R)$ is the
$A$-algebra of divided powers of order $d$ relative to $A$ and
$(-)^{\mathrm{ab}}$ denotes the abelianization.

See Roby’s papers ([14] III.1 and [15] II) for the definition of
$\Gamma^{d}_{A}(M)$ if $M$ is an $A$-module, and the $A$-algebra structure on
$\Gamma^{d}_{A}(R)$ if $R$ is an $A$-algebra.

We will also write $\operatorname{Det}_{R,d}$ for the affine scheme
representing the functor $\operatorname{Det}_{R,d}$.

######  Remark 1.10.

The scheme $\operatorname{Det}_{R,d}$ is a contravariant functor of the
$A$-algebra $R$. Indeed, if $u:R\rightarrow S$ is a morphism of $R$-algebras,
then, for every commutative $A$-algebra $B$ and every
$D\in\operatorname{Det}_{S,d}(B)=\mathcal{M}_{A}^{d}(S,B)$, the maps
$u^{*}(D)_{C}:R\otimes_{A}C\rightarrow B\otimes_{A}C$, $x\mapsto
D((u\otimes\mathrm{id}_{C})(x))$, for $C$ an object of $\mathcal{C}_{A}$,
define an element of $\mathcal{M}_{A}^{d}(R,B)=\operatorname{Det}_{R,d}(B)$.

In particular, the scheme $\operatorname{Det}_{A\\{X\\},d}$ (resp.
$\operatorname{Det}_{\Gamma,d}$) is a contravariant functor of the set $X$
(resp. the group $\Gamma$).

######  Remark 1.11.

For every commutative $A$-algebra $B$, we have a natural morphism of schemes
$\operatorname{Det}_{R\otimes_{A}B,d}\rightarrow\operatorname{Det}_{R,d}\times_{\operatorname{Spec}A}\operatorname{Spec}B$.
This is an isomorphism by Theorem III.3 on page 262 of [14].

Let $X$ be a set. We recall Vaccarino’s construction of the universal ring of
$\operatorname{Det}(\mathbb{Z}\\{X\\},d)$, following Chenevier’s presentation
in [5, 1.15]. Let $F_{X}(d)=\mathbb{Z}[x_{i,j},\ x\in X,1\leq i,j\leq d]$ be
the ring of polynomials on the variables $x_{i,j}$ for $x\in X$ and
$i,j\in\\{1,2,\ldots,d\\}$. The _generic matrices representation_ is the ring
morphism

$\rho^{\mathrm{univ}}:\mathbb{Z}\\{X\\}\rightarrow M_{d}(F_{X}(d))$

sending each $x\in X$ to the matrix $(x_{ij})_{1\leq i,j\leq d}$. This defines
a degree $d$ homogeneous multiplicative polynomial law
$D_{\rho^{\mathrm{univ}}}:\mathbb{Z}\\{X\\}\rightarrow E_{X}(d)$, where
$E_{X}(d)$ is the subring of $F_{X}(d)$ generated by the coefficients of the
characteristic polynomials of the $\rho^{\mathrm{univ}}(p)$,
$p\in\mathbb{Z}\\{X\\}$.

######  Theorem 1.12 (Vaccarino, see Theorem 1.15 of [5]).

The ring $E_{X}(d)$ and
$D_{\rho^{\mathrm{univ}}}\in\operatorname{Det}_{\mathbb{Z}\\{X\\},d}(E_{X}(d))$
represent the functor $\operatorname{Det}_{\mathbb{Z}\\{X\\},d}$.

Using results of Donkin, we deduce the following corollary.

######  Corollary 1.13.

The morphism $M_{d}^{X}\rightarrow\operatorname{Det}_{\mathbb{Z}\\{X\\},d}$
sending $\rho$ to $D_{\rho}$ (see Example 1.4(1)) induces an isomorphism
$\operatorname{Spec}(\mathcal{O}(M_{d}^{X})^{{\bf{GL}}_{d}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{Det}_{\mathbb{Z}\\{X\\},d}$.

###### Proof.

Note that $M_{d}^{X}=\operatorname{Spec}(F_{X}(d))$, and that the morphism
$M_{d}^{X}\rightarrow\operatorname{Det}_{\mathbb{Z}\\{X\\},d}$ of the
statement corresponds to the multiplicative polynomial law
$\mathbb{Z}\\{X\\}\stackrel{{\scriptstyle
D_{\rho^{\mathrm{univ}}}}}{{\rightarrow}}E_{X}(d)\subset F_{X}(d)$. On the
other hand, by the results of Donkin on the generators of
$\mathcal{O}(M_{d}^{X})^{{\bf{GL}}_{d}}$ (see [6, 3.1]), we have
$E_{X}(d)=\mathcal{O}(M_{d}^{X})^{{\bf{GL}}_{d}}$. The corollary follows.

∎

##  2\. Lafforgue’s definition of pseudocharacters

Let $A$ be a commutative unital ring and $G$ be an affine group scheme over
$A$.

As in Section 1, we denote by $\mathcal{C}_{A}$ the category of commutative
unital $A$-algebras. If $B$ is a commutative $A$-algebra and $X$ is a set, we
write $\mathcal{C}(X,B)$ for the $B$-algebra of functions $X\rightarrow B$.

For every set $X$ (resp. integer $n\in\mathbb{N}$), we make $G$ act on the
scheme $G^{X}$ (resp. $G^{n}$) by diagonal conjugation, and we denote by
$\mathcal{O}(G^{X})^{G}$ (resp. $\mathcal{O}(G^{n})^{G}$) the $A$-algebra of
$G$-invariant regular functions on $G^{X}$ (resp. $G^{n}$).

######  Definition 2.1 (See [8] Proposition 11.7 and [2] Definition 4.1).

Let $X$ be a set and $B$ be an object of $\mathcal{C}_{A}$. A _$B$ -valued
$G$-pseudocharacter for the set $X$_ is a family of $A$-algebra morphisms
$\mathcal{O}(G^{n})^{G}\rightarrow\mathcal{C}(X,B)$, for $n\geq 1$, satisfying
the following condition:

  * (LPC1)

For all $n,m\geq 1$, every map
$\zeta:\\{1,2,\ldots,m\\}\rightarrow\\{1,2,\ldots,n\\}$ and every
$f\in\mathcal{O}(G^{m})^{G}$, if we define
$f^{\zeta}\in\mathcal{O}(G^{n})^{G}$ by
$f^{\zeta}(g_{1},\ldots,g_{n})=f(g_{\zeta(1)},\ldots,g_{\zeta(m)})$, then we
have

$\Theta_{m}(f)(x_{\zeta(1)},\ldots,x_{\zeta(m)})=\Theta_{n}(f^{\zeta})(x_{1},\ldots,x_{n})$

for all $x_{1},\ldots,x_{n}\in X$.

Suppose that $X$ is the underlying set of a group $\Gamma$. Then a _$B$
-valued $G$-pseudocharacter for the group $\Gamma$_ is a $B$-valued
$G$-pseudocharacter for the set $X$ satisfying the following additional
condition:

  * (LPC2)

For all $n\geq 1$ and $f\in\mathcal{O}(G^{n})^{G}$, if we define
$\widehat{f}\in\mathcal{O}(G^{n+1})^{G}$ by
$\widehat{f}(g_{1},\ldots,g_{n+1})=f(g_{1},\ldots,g_{n-1},g_{n}g_{n+1})$, then
we have

$\Theta_{n+1}(\widehat{f})(\gamma_{1},\ldots,\gamma_{n+1})=\Theta_{n}(f)(\gamma_{1},\ldots,\gamma_{n-1},\gamma_{n}\gamma_{n+1})$

for all $\gamma_{1},\ldots,\gamma_{n+1}\in\Gamma$.

We denote by $\operatorname{LPC}^{1}_{X,G}$ (resp.
$\operatorname{LPC}_{\Gamma,G}$) the functor
$\mathcal{C}_{A}\rightarrow\mathbf{Set}$ sending $B$ to the set of $B$-valued
$G$-pseudocharacters for the set $X$ (resp. for the group $\Gamma$). If $X$ is
the underlying set of a group $\Gamma$, then $\operatorname{LPC}_{\Gamma,G}$
is a subfunctor of $\operatorname{LPC}^{1}_{X,G}$.

######  Example 2.2.

  * (1)

If $X$ is a set and $\rho:X\rightarrow G(A)$ is a map, then we define an
$A$-valued $G$-pseudocharacter $\Theta^{1}_{\rho}$ for the set $X$ by

$\Theta^{1}_{\rho,n}(f)(x_{1},\ldots,x_{n})=f(\rho(x_{1}),\ldots,\rho(x_{n})),$

for every $n\geq 1$, every $f\in\mathcal{O}(G^{n})^{G}$ and all
$x_{1},\ldots,x_{n}\in X$. Note that $\Theta^{1}_{\rho}$ only depends on the
conjugacy class of $\rho$.

  * (2)

Similarly, if $\Gamma$ is a group and $\rho:\Gamma\rightarrow G(A)$ is a
morphism of groups, then we define an $A$-valued $G$-pseudocharacter
$\Theta_{\rho}$ for the group $\Gamma$ by

$\Theta_{\rho,n}(f)(\gamma_{1},\ldots,\gamma_{n})=f(\rho(\gamma_{1}),\ldots,\rho(\gamma_{n})),$

for every $n\geq 1$, every $f\in\mathcal{O}(G^{n})^{G}$ and all
$\gamma_{1},\ldots,\gamma_{n}\in\Gamma$. Again, $\Theta_{\rho}$ only depends
on the conjugacy class of $\rho$.

######  Remark 2.3.

The functor $\operatorname{LPC}^{1}_{X,G}$ depends contravariantly on the set
$X$. Indeed, if $u:X\rightarrow Y$ is a map, then, for every commutative
$A$-algebra $B$ and every $\Theta=(\Theta_{n})_{n\geq
1}\in\operatorname{LPC}^{1}_{Y,G}(B)$, the family
$u^{*}(\Theta)=(u^{*}(\Theta)_{n})_{n\geq 1}$ defined by

$u^{*}(\Theta)_{n}(f)(x_{1},\ldots,x_{n})=\Theta_{n}(f)(u(x_{1}),\ldots,u(x_{n}))$

for $n\geq 1$, $f\in\mathcal{O}(G^{n})^{G}$ and $x_{1},\ldots,x_{n}\in X$ is a
$B$-valued $G$-pseudocharacter for the set $X$.

Similary, the functor $\operatorname{LPC}_{\Gamma,G}$ depends contravariantly
on the group $\Gamma$.

If $X$ is a set, $x_{1},\ldots,x_{n}\in X$ and $f\in\mathcal{O}(G^{n})$, we
define a regular function $f_{x_{1},\ldots,x_{n}}\in\mathcal{O}(G^{X})$ by

$f_{x_{1},\ldots,x_{n}}((g_{x})_{x\in X})=f(g_{x_{1}},\ldots,g_{x_{n}}).$

Note that $f\in\mathcal{O}(G^{n})^{G}$ if and only if
$f_{x_{1},\ldots,x_{n}}\in\mathcal{O}(G^{X})^{G}$.

######  Proposition 2.4.

Let $X$ be a set and let $R^{1}_{X,G}=\mathcal{O}(G^{X})^{G}$. Consider the
element $\Theta^{\mathrm{univ}}$ of
$\operatorname{LPC}^{1}_{X,G}(R^{1}_{X,G})$ defined by

$\Theta^{\mathrm{univ}}_{n}(f)(x_{1},\ldots,x_{n})=f_{x_{1},\ldots,x_{n}}\in
R^{1}_{X,G},$

for every $n\geq 1$, every $f\in\mathcal{O}(G^{n})^{G}$ and all
$x_{1},\ldots,x_{n}\in X$. Then $R^{1}_{X,G}$ represents the functor
$\operatorname{LPC}^{1}_{X,G}$, and
$\Theta^{\mathrm{univ}}\in\operatorname{LPC}^{1}_{X,G}(R^{1}_{X,G})$ is the
universal element.

###### Proof.

The morphism of functors
$\operatorname{Spec}(R^{1}_{X,G})\rightarrow\operatorname{LPC}^{1}_{X,G}$
corresponding to $\Theta^{\mathrm{univ}}$ sends a morphism of rings
$u:R^{1}_{X,G}\rightarrow B$ to the $G$-pseudocharacter $\Theta_{u}$ defined
by

$\Theta_{u,n}(f)(x_{1},\ldots,x_{n})=u(\Theta^{\mathrm{univ}}_{n}(f)(x_{1},\ldots,x_{n})),$

for $n\geq 1$, $f\in\mathcal{O}(G^{n})^{G}$ and $x_{1},\ldots,x_{n}\in X$. We
check that it is an isomorphism. The central point is that, by definition of
the product $G^{X}$, we have $\mathcal{O}(G^{X})=\varinjlim_{Y\subset X\
\mathrm{finite}}\mathcal{O}(G^{Y})$; in particular, every element of
$\mathcal{O}(G^{X})^{G}$ is of the form $f_{x_{1},\ldots,x_{n}}$, for some
$n\geq 1$, $x_{1},\ldots,x_{n}\in X$ and $f\in\mathcal{O}(G^{n})^{G}$.

Let $B$ be a commutative $A$-algebra. Let $u,v:R^{1}_{X,G}\rightarrow B$ be
two morphisms of $A$-algebras such that $\Theta_{u}=\Theta_{v}$. Let $h\in
R^{1}_{X,G}$, and choose an integer $n\geq 1$, $x_{1},\ldots,x_{n}\in X$ and
$f\in\mathcal{O}(G^{n})^{G}$ such that $h=f_{x_{1},\ldots,x_{n}}$. Then

$u(h)=\Theta_{u,n}(f)(x_{1},\ldots,x_{n})=\Theta_{v,n}(f)(x_{1},\ldots,x_{n})=v(h).$

This proves that $u=v$. Now let $\Theta\in\operatorname{LPC}^{1}_{X,G}(B)$; we
want to find a morphism of $A$-algebras $w:R^{1}_{X,G}\rightarrow B$ such that
$\Theta=\Theta_{w}$. Let $Y$ be a finite subset of $X$. If
$\zeta:Y\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\\{1,2,\ldots,n\\}$ is a
bijection, then we get an isomorphism of $A$-algebras
$\mathcal{O}(G^{Y})^{G}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}(G^{n})^{G}$
sending $h\in\mathcal{O}(G^{Y})^{G}$ to the regular function
$h^{\zeta}:(g_{1},\ldots,g_{n})\mapsto h((g_{\zeta(y)})_{y\in Y})$, and we
define $w_{Y}:\mathcal{O}(G^{Y})^{G}\rightarrow B$ by

$w_{Y}(h)=\Theta_{n}(h^{\zeta})(\zeta^{-1}(1),\ldots,\zeta^{-1}(n)).$

By condition (LPC1), this does not depend on the choice of $\zeta$ and, if
$Y\subset Z$ are finite subsets of $X$, then
$w_{Z\mid\mathcal{O}(G^{Y})^{G}}=w_{Y}$. So the family $(w_{Y})_{Y\subset X\
\mathrm{finite}}$ defines a morphism of $A$-algebras $w:R^{1}_{X,G}\rightarrow
B$, and it follows immediately from the definition of $\Theta^{\mathrm{univ}}$
that we have $\Theta=\Theta_{w}$.

∎

Let $\Gamma$ be a group. For all $\gamma,\delta\in\Gamma$, we define a
$G$-equivariant morphism of $A$-modules
$\varphi_{\gamma,\delta}:\mathcal{O}(G\times
G^{\Gamma})\rightarrow\mathcal{O}(G^{\Gamma})$ by

$\varphi_{\gamma,\delta}(f)((g_{\alpha})_{\alpha\in\Gamma})=f(g_{\gamma\delta},(g_{\alpha})_{\alpha\in\Gamma})-f(g_{\gamma}g_{\delta},(g_{\alpha})_{\alpha\in\Gamma}).$

We have a morphism of $A$-algebras
$\mathcal{O}(G^{\Gamma})\rightarrow\mathcal{O}(G\times G^{\Gamma})$ induced by
the projection of $G\times G^{\Gamma}$ on its second factor, and this morphism
is equivariant for the action of $G$ by diagonal conjugation. The map
$\varphi_{\gamma,\delta}$ becomes $\mathcal{O}(G^{\Gamma})$-linear for this
action of $\mathcal{O}(G^{\Gamma})$ on $\mathcal{O}(G\times G^{\Gamma})$. In
particular, $\varphi_{\gamma,\delta}(\mathcal{O}(G\times G^{\Gamma}))$ (resp.
$\varphi_{\gamma,\delta}(\mathcal{O}(G\times G^{\Gamma})^{G})$) is an ideal of
$\mathcal{O}(G^{\Gamma})$ (resp. $\mathcal{O}(G^{\Gamma})^{G}$).

######  Proposition 2.5.

  1. (i)

Let $J_{\Gamma,G}\subset\mathcal{O}(G^{\Gamma})$ be the sum of the images of
the $\varphi_{\gamma,\delta}$, for all $\gamma,\delta\in\Gamma$. Then
$J_{\Gamma,G}$ is an ideal of $\mathcal{O}(G^{\Gamma})$, and
$\operatorname{Spec}(\mathcal{O}(G^{\Gamma})/J_{\Gamma,G})\subset\operatorname{Spec}(\mathcal{O}(G^{\Gamma}))=G^{\Gamma}$
sends any commutative $B$-algebra $A$ to the set of maps
$\rho:\Gamma\rightarrow G(B)$ that are morphisms of groups.

  2. (ii)

Let $I_{\Gamma,G}\subset J_{\Gamma,G}^{G}$ be the sum of the
$\varphi_{\gamma,\delta}(\mathcal{O}(G\times G^{\Gamma})^{G})$, for all
$\gamma,\delta\in\Gamma$. Then $I_{\Gamma,G}$ is an ideal of
$R^{1}_{\Gamma,G}=\mathcal{O}(G^{\Gamma})^{G}$, and, if we set
$R_{\Gamma,G}=R^{1}_{\Gamma,G}/I_{\Gamma,G}$, then the isomorphism
$\operatorname{Spec}(R^{1}_{\Gamma,G})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{LPC}^{1}_{\Gamma,G}$
of Proposition 2.4 induces an isomorphism between the closed subscheme
$\operatorname{Spec}(R_{\Gamma,G})$ of $\operatorname{Spec}(R^{1}_{\Gamma,G})$
and $\operatorname{LPC}_{\Gamma,G}\subset\operatorname{LPC}^{1}_{\Gamma,G}$.

###### Proof.

We already know that $J_{\Gamma,G}$ and $I_{\Gamma,G}$ are ideals, because
they are sums of ideals.

Let $B$ be a commutative $A$-algebra and $\rho:\Gamma\rightarrow G(B)$ be a
map; this corresponds to a morphism of $A$-algebras
$u:\mathcal{O}(G^{\Gamma})\rightarrow B$, and we have

$u(f)=f((\rho(\gamma))_{\gamma\in\Gamma})$

for every $f\in\mathcal{O}(G^{\Gamma})$. We want to prove that $\rho$ is a
morphism of groups if and only if $u(J_{\Gamma,G})=0$. Suppose that $\rho$ is
a morphism of groups. Let $\gamma,\delta\in\Gamma$ and
$f\in\mathcal{O}(G\times G^{\Gamma})$. Then

$\displaystyle
u(\varphi_{\gamma,\delta}(f))=f(\rho(\gamma\delta),(\rho(\alpha)_{\alpha\in\Gamma}))-f(\rho(\gamma)\rho(\delta),(\rho(\alpha)_{\alpha\in\Gamma}))=0$

as $\rho$ is a morphism. This shows that
$J_{\Gamma,G}\subset\operatorname{Ker}u$. Conversely, suppose that
$J_{\Gamma,G}\subset\operatorname{Ker}u$. Let $\gamma,\delta\in\Gamma$. Then,
for every $f\in\mathcal{O}(G)$, we have

$0=u(\varphi_{1,(\gamma,\delta)}(f))=f(\rho(\gamma\delta))-f(\rho(\gamma)\rho(\delta)),$

hence $\rho(\gamma\delta)=\rho(\gamma)\rho(\delta)$. This finishes the proof
of (i).

We now prove the second statement of (ii). Let $B$ be a commutative
$A$-algebra, let $\Theta\in\operatorname{LPC}^{1}_{\Gamma,G}(B)$, and let
$u:R^{1}_{\Gamma,G}\rightarrow B$ be the morphism of $A$-algebras
corresponding to $\Theta$. Suppose that $\Theta$ satisfies condition (LPC2).
Let $\gamma,\delta\in\Gamma$ and $f\in\mathcal{O}(G\times G^{\Gamma})^{G}$.
Choose a finite subset $\\{\gamma_{1},\ldots,\gamma_{n}\\}$ of $\Gamma$ and
$h\in\mathcal{O}(G^{n+1})^{G}$ such that

$f(g,(g_{\alpha})_{\alpha\in\Gamma})=h(g_{\gamma_{1}},\ldots,g_{\gamma_{n}},g).$

Then we have

$f(g_{\gamma}g_{\delta},(g_{\alpha})_{\alpha\in\Gamma})=\widehat{h}((g_{\alpha})_{\alpha\in\Gamma},g_{\gamma},g_{\delta}),$

so

$\displaystyle
u(\varphi_{\gamma,\delta}(f))=\Theta_{n+1}(h)(\gamma_{1},\ldots,\gamma_{n},\gamma\delta)-\Theta_{n+2}(\widehat{h})(\gamma_{1},\ldots,\gamma_{n},\gamma,\delta)=0.$

This proves that $I_{\Gamma,G}\subset\operatorname{Ker}u$.

Conversely, suppose that $I_{\Gamma,G}\subset\operatorname{Ker}u$. Let $n\geq
1$, $h\in\mathcal{O}(G^{n})^{G}$ and
$\gamma_{1},\ldots,\gamma_{n+1}\in\Gamma$. Define $f\in\mathcal{O}(G\times
G^{\Gamma})^{G}$ by

$f(g,(g_{\alpha})_{\alpha\in\Gamma})=h(g_{\gamma_{1}},\ldots,g_{\gamma_{n-1}},g).$

Then

$0=u(\varphi_{\gamma_{n},\gamma_{n+1}}(f))=\Theta_{n}(h)(\gamma_{1},\ldots,\gamma_{n-1},\gamma_{n}\gamma_{n+1})-\Theta_{n}(\widehat{h})(\gamma_{1},\ldots,\gamma_{n-1},\gamma_{n},\gamma_{n+1}).$

This implies that $\Theta$ satisfies condition (LPC2).

∎

Next we discuss the behavior of the functors $\operatorname{LPC}^{1}_{X,G}$
and $\operatorname{LPC}_{\Gamma,G}$ under change of the base ring $A$.

We will use the notion of _adequate homeomorphism_ defined by Alper (see [1,
Definition 3.3.1]): a morphism of schemes is an adequate homeomorphism if it
is integral, a universal homeomorphism and a local isomorphism at all points
whose residue field is of characteristic $0$. We will also use Alper’s notion
of _geometrically reductive group schemes_ , see Definition 9.1.1 of [1]),
which generalizes that of reductive group schemes. In particular, if $G$ is an
affine smooth algebraic group over a field $k$, then it is geometrically
reductive if and only if it is reductive (Lemma 9.2.8 of [1]), and, if
$G\rightarrow S$ is a smooth group scheme with connected fibers, then it is a
geometrically reductive group scheme if and only if it is reductive (Theorem
9.7.5 of [1]).

Fix a commutative ring $A$ and a flat affine group scheme $G$ over $A$. Let
$B$ be a commutative $A$-algebra. For every $A$-module $V$ with an action of
$G$, we have $V^{G}\otimes_{A}B\subset(V\otimes_{A}B)^{G_{B}}$. As
$\mathcal{O}(G^{X})\otimes_{A}B=\mathcal{O}(G_{B}^{X})$ for every set $X$, we
deduce that $J_{\Gamma,G_{B}}=J_{\Gamma,G}\otimes_{A}B$ and
$I_{\Gamma,G_{B}}\supset I_{\Gamma,G}\otimes_{A}B$. So we get morphisms of
$B$-algebras, for $X$ a set and $\Gamma$ a group,

$R^{1}_{X,G}\otimes_{A}B=\mathcal{O}(G^{X})^{G}\otimes_{A}B\rightarrow\mathcal{O}(G_{B}^{X})^{G_{B}}=R^{1}_{X,G_{B}}$

and

$R_{\Gamma,G}\otimes_{A}B=(\mathcal{O}(G^{\Gamma})^{G}\otimes_{A}B)/(I_{\Gamma,G}\otimes_{A}B)\rightarrow
R_{\Gamma,G_{B}}$

and corresponding morphisms of schemes

$\beta_{X}:\operatorname{LPC}^{1}_{X,G_{B}}\rightarrow\operatorname{LPC}^{1}_{X,G}\otimes_{A}B$

and

$\beta_{\Gamma}:\operatorname{LPC}_{\Gamma,G_{B}}\rightarrow\operatorname{LPC}_{\Gamma,G}\otimes_{A}B.$

######  Proposition 2.6.

  1. (i)

The morphisms $\beta_{X}$ and $\beta_{\Gamma}$ are isomorphisms in the
following cases:

     * (a)

$B$ is a flat $A$-algebra;

     * (b)

$A$ is a Dedekind domain and $G$ is geometrically reductive over $A$ and has
connected geometric fibers.

  2. (ii)

If $G$ is geometrically reductive over $A$ and has connected geometric fibers,
then $\beta_{X}$ and $\beta_{\Gamma}$ are always adequate homeomorphisms.

###### Proof.

Point (ii) follows from Proposition 5.2.9(3) of [1]. We prove (i). Suppose
first that $B$ is flat over $A$. Then, by Lemma 2 of Seshadri’s paper [16],
for every $A$-module $V$ with an action of $G$, the canonical morphism
$V^{G}\otimes_{A}B\rightarrow(V\otimes_{A}B)^{G_{B}}$ is an isomorphism.
Applying this lemma to $\mathcal{O}(G^{X})$, we see that $\beta_{X}$ is an
isomorphism. As the functor $(-)\otimes_{A}B$ is right exact, Seshadri’s lemma
also implies that $I_{\Gamma,G}\otimes_{A}B=I_{\Gamma,G_{B}}$, hence that
$\beta_{\Gamma}$ is an isomorphism. We finally assume that $A$ is a principal
ideal domain and that $G$ is geometrically reductive over $A$. By Lemma 2.7,
the morphism
$\mathcal{O}(G^{Y})^{G}\otimes_{A}B\rightarrow\mathcal{O}(G_{B}^{Y})^{G_{B}}$
is an isomorphism for every set $Y$; as in the proof of (i)(a), we conclude
that $\beta_{X}$ and $\beta_{\Gamma}$ are isomorphisms.

∎

######  Lemma 2.7.

Let $A$ be a Dedekind domain and $G$ be a geometrically reductive group scheme
with connected geometric fibers over $A$. Then, for every set $Y$, the
injective morphism
$\mathcal{O}(G^{Y})^{G}\otimes_{A}B\rightarrow\mathcal{O}(G_{B}^{Y})^{G_{B}}$
is an isomorphism.

######  Remark 2.8.

Lemma 2.7 and its application to the functor $\operatorname{LPC}$ are probably
well-known to many people in some form. We found version of it in a note by
Chen (see [3]), as well as in a preprint by Quast (see Section 1.3 of [12]).

###### Proof of Lemma 2.7.

We will use a number of results about the cohomology of algebraic groups that
are gathered in Jantzen’s book [7]; the particular results that we need are
due to Donkin and Mathieu, see [7] for the original references.

First, as the tensor product and the functor of invariants commute with direct
limits and as $\mathcal{O}(G^{Y})$ is the direct limit of the
$\mathcal{O}(G^{Z})$ for $Z\subset Y$ finite, we may assume that $Y$ is
finite. By the universal coefficients theorem (Proposition 4.18 in Chapter I
of [7]), we have an exact sequence

$0\rightarrow\mathcal{O}(G^{Y})^{G}\otimes_{A}B\rightarrow\mathcal{O}(G_{B}^{Y})^{G_{B}}\rightarrow\operatorname{Tor}_{1}^{A}(\mathrm{H}^{1}(G,\mathcal{O}(G^{Y})),B)\rightarrow
0.$

So it suffices to show that $\mathrm{H}^{1}(G,\mathcal{O}(G^{Y}))=0$. For
this, it suffices to proves that the localization of
$\mathrm{H}^{1}(G,\mathcal{O}(G^{Y}))$ at every ideal of $A$ is zero; as
localizations are flat, by the universal coefficient theorem again, we may
replace $A$ by one of its localizations, hence assume that $A$ is a field or a
discrete valuation ring. Now, by Lemma B.9 of [7], it suffices to prove that,
for every maximal ideal $\mathfrak{m}$ of $A$, if $k=A/\mathfrak{m}$, then
$\mathcal{O}(G_{k}^{Y})$ has a good filtration (in the sense of II.4.16 of
[7]) as a $G_{k}$-module. This follows immediately from Propositions 4.20 and
4.21 of Chapter II of [7].

∎

We finally investigate the relationship between pseudo-characters and the
character variety.

######  Definition 2.9.

Let $A$ be a commutative ring, $G$ be a flat affine group scheme over $A$ and
$\Gamma$ be a group. The _$G$ -character variety of $\Gamma$_ is the affine
scheme
$\operatorname{Char}_{\Gamma,G}=\operatorname{Spec}((\mathcal{O}(G^{\Gamma})/J_{\Gamma,G})^{G})$.

If $B$ is an $A$-algebra, then we have $I_{\Gamma,G_{B}}\supset
I_{\Gamma,G}\otimes_{A}B$, so we get a morphism of $B$-algebras

$(\mathcal{O}(G^{\Gamma})/J_{\Gamma,G})^{G}\otimes_{A}B\rightarrow((\mathcal{O}(G^{\Gamma})/J_{\Gamma,G})\otimes_{A}B)^{G_{B}}=(\mathcal{O}(G_{B}^{\Gamma})/I_{\Gamma,G_{B}})^{G_{B}},$

and a corresponding morphism of schemes

$\beta_{\Gamma}^{\prime}:\operatorname{Char}_{\Gamma,G_{B}}\rightarrow\operatorname{Char}_{\Gamma,G}\otimes_{A}B.$

######  Lemma 2.10.

Suppose that $G$ is geometrically reductive over $A$ and has connected
geometric fibers.

  1. (i)

If $B$ is a flat $A$-algebra, then $\beta^{\prime}_{\Gamma}$ is an
isomorphism.

  2. (ii)

In general, $\beta^{\prime}_{\Gamma}$ is an adequate homeomorphism.

###### Proof.

Point (i) follows from Proposition 5.2.9(1) of [1], and point (ii) from
Proposition 5.2.9(3) of the same paper.

∎

######  Proposition 2.11.

Suppose that $G$ is a geometrically reductive group scheme with connected
geometric fibers over $A$ and that $A$ is a Dedekind domain or a field. Let
$\iota$ be the morphism
$\operatorname{Char}_{\Gamma,G}\rightarrow\operatorname{LPC}_{\Gamma,G}$
induced by
$\mathcal{O}(G^{\Gamma})^{G}/I_{\Gamma,G}\twoheadrightarrow\mathcal{O}(G^{\Gamma})^{G}/J_{\Gamma,G}^{G}\hookrightarrow(\mathcal{O}(G^{\Gamma})/J_{\Gamma,G})^{G}$.

  1. (i)

If $A$ is a field of characteristic $0$, then $\iota$ is an isomorphism.

  2. (ii)

In general, $\iota$ is an adequate homeomorphism.

The statement of the proposition is very close to that of Proposition 11.7 of
[8] and Theorem 4.5 of [2], and its proof uses the same kind of ideas. We
still include it here because it is not very hard.

###### Proof.

We prove (i). As $A$ is a field, the group $G$ is reductive over $A$. So, for
every algebraic representation $V$ of $G$ over $A$ (not necessarily finite-
dimensional), we have the Reynolds operator $E=E_{V}:V\rightarrow V$ (see [10]
Definition 1.5), which is a $G$-equivariant projection with image $V^{G}$,
compatible with any morphism of representations $V\rightarrow W$; also, if $V$
is a $A$-algebra and the action of $G$ preserves its multiplication, then
$E_{V}$ is $V^{G}$-linear. We claim that $I_{\Gamma,G}=J_{\Gamma,G}^{G}$. We
already know that $I_{\Gamma,G}\subset J_{\Gamma,G}^{G}$. Conversely, let
$h\in J_{\Gamma,G}^{G}$. Write
$h=\sum_{i=1}^{n}\varphi_{\gamma_{i},\delta_{i}}(f_{i})$, with
$\gamma_{i},\delta_{i}\in\Gamma$ and $f_{i}\in\mathcal{O}(G\times
G^{\Gamma})$. As the $\varphi_{\gamma_{i},\delta_{i}}$ are $G$-equivariant
morphisms, they are compatible with the Reynolds operators, so

$h=E(h)=\sum_{i=1}^{n}\varphi_{\gamma_{i},\delta_{i}}(E(f_{i}))\in
I_{\Gamma,G}.$

It remains to prove that the injective morphism
$\mathcal{O}(G^{\Gamma})^{G}/J_{\Gamma,G}^{G}\rightarrow(\mathcal{O}(G^{\Gamma})/J_{\Gamma,G})^{G}$
is also surjective. Let $f\in\mathcal{O}(G^{\Gamma})$, and suppose that the
class of $f$ modulo $J_{\Gamma,G}$ is $G$-invariant. As the canonical
surjection
$\mathcal{O}(G^{\Gamma})\rightarrow\mathcal{O}(G^{\Gamma})/J_{\Gamma,G}$ is
$G$-equivariant, it is compatible with the Reynolds operators, so we deduce
that $f-E(f)\in J_{\Gamma,G}$, which means that $f+J_{\Gamma,G}$ is in the
image of $\mathcal{O}(G^{\Gamma})^{G}/J_{\Gamma,G}^{G}$.

We prove (ii). We know that
$\operatorname{Char}_{\Gamma,G}\rightarrow\operatorname{Spec}(\mathcal{O}(G^{\Gamma})^{G}/J_{\Gamma,G}^{G})$
is an adequate homeomorphism by Lemma 5.2.12 of [1] (see Remark 5.2.14 of
_loc. cit._). So it remains to prove that
$\iota^{\prime}:\operatorname{Spec}(\mathcal{O}(G^{\Gamma})^{G}/J_{\Gamma,G}^{G})\rightarrow\operatorname{Spec}(\mathcal{O}(G^{\Gamma})^{G}/I_{\Gamma,G})$
is an adequate homeomorphism. This morphism is a closed embedding, hence it is
integral, universally injective and universally closed. If
$\operatorname{Frac}(A)$ is of characteristic $0$, then $\iota^{\prime}$
becomes an isomorphism after we tensor it by $\operatorname{Frac}(A)$ by
Proposition 2.6(ii); otherwise, its source and target have no point with
residue field of characteristic $0$. So it remains to show that
$\iota^{\prime}$ is surjective, which is equivalent to the fact that $\iota$
is surjective.

Let $x$ be a point of
$\operatorname{Spec}(\mathcal{O}(G^{\Gamma})^{G}/I_{\Gamma,G})$, which
corresponds to a morphism of $A$-algebras
$u:\mathcal{O}(G^{\Gamma})^{G}/I(\Gamma,G)\rightarrow K$, with $K$ a field. We
want to find an extension $L$ of $K$ and a morphism of $A$-algebras
$v:(\mathcal{O}(G^{\Gamma})/J_{\Gamma,G})^{G}\rightarrow L$ such that
$v\circ{\iota}^{*}=u$. We may always enlarge $K$ and $L$, so we may assume
that they are algebraically closed. Then, by Proposition 2.6(ii) and Lemma
2.10(ii), $\operatorname{Char}_{\Gamma,G}$ and
$\operatorname{Char}_{\Gamma,G_{K}}$ (resp. $\operatorname{LPC}_{\Gamma,G}$
and $\operatorname{LPC}_{\Gamma,G_{K}}$) have the same points over any
algebraically closed extension of $K$, so we may assume that $A=K$ (and so
that $G$ is reductive over $K$). By Theorem 5.13 of [11] (or Lemma 5.2.1 and
Remark 5.2.2 of [1]), there exists an extension $L$ of $K$ and a morphism of
$K$-algebras $w:\mathcal{O}(G^{\Gamma})\rightarrow L$ such that $u$ is induced
by $w_{\mid\mathcal{O}(G^{\Gamma})^{G}}$. The morphism $w$ corresponds to an
element $(g_{\gamma})\in G^{\Gamma}(L)$, and we denote by $H$ the closed
subgroup of $G_{L}$ generated by the set $\\{g_{\gamma},\ \gamma\in\Gamma\\}$.
We choose $w$ such that the dimension of the maximal tori in $Z_{G}(H)$ is
maximal. We claim that $H$ is then strongly reductive in $G$ in the sense of
Definition 16.1 of Richardson’s paper [13], that is, $H$ is not contained in
any proper parabolic subgroup of $Z_{G}(S)$, where $S$ is a maximal torus of
$Z_{G}(H)$. Indeed, let $\lambda$ be a cocharacter of $Z_{G}(S)$, and suppose
that $H$ is contained in $P(\lambda):=\\{g\in Z_{G}(S)\mid\lim_{t\to
0}\lambda(t)g\lambda(t)^{-1}\ \mathrm{exists}\\}$. For every
$\gamma\in\Gamma$, let $g^{\prime}_{\gamma}=\lim_{t\to
0}\lambda(t)g_{\gamma}\lambda(t)^{-1}\in G(L)$. Then
$(g_{\gamma})_{\gamma\in\Gamma}$ and $(g^{\prime}_{\gamma})_{\gamma\in\Gamma}$
have the same image in $\operatorname{Spec}(\mathcal{O}(G^{\Gamma})^{G})(L)$,
so the morphism $w^{\prime}:\mathcal{O}(G^{\Gamma})\rightarrow L$
corresponding to $(g^{\prime}_{\gamma})_{\gamma\in\Gamma}\in G^{\Gamma}(L)$
satisfies
$w^{\prime}_{\mid\mathcal{O}(G^{\Gamma})^{G}}=w_{\mid\mathcal{O}(G^{\Gamma})^{G}}$.
On the other hand, the closed subgroup $H^{\prime}$ of $G$ generated by the
$g^{\prime}_{\gamma}$ is contained in the centralizer of $\lambda$ in
$Z_{G}(S)$, so $\lambda$ has to be central in $Z_{G}(S)$, otherwise
$Z_{G}(H^{\prime})$, which contains the group generated by $S$ and the image
of $\lambda$, would have a maximal torus of dimension greater than $\dim(S)$.
So $H$ is not contained in any proper parabolic subgroup of $Z_{G}(S)$. Also,
as $H$ is a Noetherian scheme, for every big enough finite subset $X$ of
$\Gamma$, the closed subgroup of $G$ generated by $\\{g_{\gamma},\ \gamma\in
X\\}$ is equal to $H$.

We now prove that, for this choice of $w$, the map $\gamma\mapsto g_{\gamma}$
is a morphism of groups; this implies that $w$ extends to
$\mathcal{O}(G^{\Gamma})/J_{\Gamma,G}$, hence defines a point $y$ of
$\operatorname{Char}_{\Gamma,G}(L)$ such that $\iota(y)=x$. Let
$\gamma,\delta\in\Gamma$. Choose a finite subset $X$ of $\Gamma$ such that
$\gamma,\delta,\gamma\delta\in X$ and the closed subgroup of $G$ generated by
$\\{g_{\alpha},\ \alpha\in X\\}$ is equal to $H$. As $w$ vanishes on
$I_{\Gamma,G}$, the images of
$(g_{\gamma\delta},(g_{\alpha})_{\alpha\in\Gamma})$ and
$(g_{\gamma}g_{\delta},(g_{\alpha})_{\alpha\in\Gamma})$ by the map $(G\times
G^{\Gamma})(L)\rightarrow\operatorname{Spec}(\mathcal{O}(G\times
G^{\Gamma})^{G})(L)$ are equal, so the images of
$(g_{\gamma\delta},(g_{\alpha})_{\alpha\in X})$ and
$(g_{\gamma}g_{\delta},(g_{\alpha})_{\alpha\in X})$ by the map $(G\times
G^{X})(L)\rightarrow\operatorname{Spec}(\mathcal{O}(G\times G^{X})^{G})(L)$
are also equal. The closed subgroups of $G$ generated by the families
$(g_{\gamma\delta},(g_{\alpha})_{\alpha\in X})$ and
$(g_{\gamma}g_{\delta},(g_{\alpha})_{\alpha\in X})$ are both equal to $H$,
hence strongly reductive in $G$; so, by Theorem 16.4 of Richardson’s paper
[13], the $G$-orbits of these families in $(G\times G^{X})(L)$ are closed. As
they have the same image in $\operatorname{Spec}(\mathcal{O}(G\times
G^{X})^{G})(L)$, and as $G\times G^{X}$ is of finite type over $K$, this
implies that they are in the same $G$-conjugacy class. Let $h\in G(L)$ such
that $h(g_{\gamma\delta},(g_{\alpha})_{\alpha\in
X})h^{-1}=(g_{\gamma}g_{\delta},(g_{\alpha})_{\alpha\in X})$. Then $h$
centralizes all the $g_{\alpha}$ for $\alpha\in X$, so
$g_{\gamma}g_{\delta}=hg_{\gamma\delta}h^{-1}=g_{\gamma\delta}$. This finishes
the proof.

∎

######  Remark 2.12.

The morphism
$\operatorname{Char}_{\Gamma,G}\rightarrow\operatorname{LPC}_{\Gamma,G}$ of
Proposition 2.11 is an isomorphism if and only if both morphisms
$\mathcal{O}(G^{\Gamma})^{G}/I_{\Gamma,G}\twoheadrightarrow\mathcal{O}(G^{\Gamma})^{G}/J_{\Gamma,G}^{G}$
and
$\mathcal{O}(G^{\Gamma})^{G}/J_{\Gamma,G}^{G}\hookrightarrow(\mathcal{O}(G^{\Gamma})/J_{\Gamma,G})^{G}$
are isomorphisms. This seems unlikely, but we cannot offer a counterexample.

##  3\. A Lafforgue pseudocharacter gives rise to a determinant

Let $A$ be a commutative unital ring and $d$ be a positive integer. If $X$ is
a set and $\Gamma$ is a group, we write $\operatorname{LPC}^{1}_{X,d}$ and
$\operatorname{LPC}_{\Gamma,d}$ instead of
$\operatorname{LPC}^{1}_{X,{\bf{GL}}_{d,A}}$ and
$\operatorname{LPC}_{\Gamma,{\bf{GL}}_{d,A}}$. We will also use the notation
of Example 2.2.

Let $X$ be a set, $d$ be a positive integer, $B$ be a commutative $A$-algebra
and $\Theta=(\Theta_{n})$ be an element of $\operatorname{LPC}^{1}_{X,d}(B)$.
We want to construct an element $\alpha^{1}_{X}(\Theta)$ of
$\operatorname{Det}_{A\\{X\\},d}(B)$, that is, a degree $d$ homogeneous
multiplicative polynomial law from $A\\{X\\}$ to $B$ (seen as $A$-algebras).

Let $Y,Z$ be sets and $\sigma:Y\rightarrow Z$ be a map. If $C$ is a
commutative $A$-algebra, then we define a map
$\det_{\sigma,C}:C\\{Y\\}=A\\{Y\\}\otimes_{A}C\rightarrow\mathcal{O}({{\bf{GL}}}_{d,C}^{Z})^{{\bf{GL}}_{d,C}}$
in the following way. Let $p\in C\\{Y\\}$. Then $\det_{\sigma,C}(p)$ is the
regular function on ${{\bf{GL}}}_{d,C}^{Z}$ sending $(g_{z})_{z\in Z}$ to
$\det(p((g_{\sigma(y)})_{y\in Y}))$ (where $\det$ is the usual determinant on
${\bf{GL}}_{d}$), which is clearly ${\bf{GL}}_{d,C}$-invariant. Note that this
construction is functorial in $C$, and that we have
$\mathcal{O}({\bf{GL}}_{d,C}^{X})^{{\bf{GL}}_{d,C}}=\mathcal{O}({{\bf{GL}}}_{d,A}^{Z})^{{\bf{GL}}_{d,A}}\otimes_{A}C$
by Lemma 2.7. Hence the family
$(\det_{\sigma,C})_{C\in\operatorname{\mathrm{Ob}}(\mathcal{C}_{A})}$ defines
a degree $d$ homogeneous multiplicative polynomial law from $A\\{Y\\}$ to
$\mathcal{O}({\bf{GL}}_{d,A}^{Z})^{{\bf{GL}}_{d,A}}$, which we denote by
$\det_{\sigma}$.

We come back to our $\Theta\in\operatorname{LPC}^{1}_{X,d}(B)$. By Proposition
2.4, it corresponds to a morphism of $A$-algebras
$u_{\Theta}:\mathcal{O}({\bf{GL}}_{d,A}^{X})^{G}\rightarrow B$, and we send it
to the polynomial law

$\alpha^{1}_{X}(\Theta)=u_{\Theta}\circ\det\nolimits_{\mathrm{id}_{X}}:A\\{X\\}\rightarrow
B.$

In other words, for every commutative $A$-algebra $C$ and every $p\in
C\\{X\\}$, the element $\alpha^{1}_{X}(\Theta)_{C}(p)$ of $B\otimes_{A}C$ is
the image by $u_{\Theta}\otimes\mathrm{id}_{C}$ of the element $(g_{x})_{x\in
X}\rightarrow\det(p(g_{x})_{x\in X})$ of
$\mathcal{O}({\bf{GL}}_{d,C}^{X})^{{\bf{GL}}_{d,C}}$.

The functoriality of $\alpha^{1}_{X}(\Theta)_{C}$ in $C$ follows immediately
from its definition, and the fact that it defines a degree $d$ homogeneous
multiplicative polynomial law follows from the properties of the determinant
on ${\bf{GL}}_{d}$.

######  Proposition 3.1.

  1. (i)

Let $X$ be a set and $d\geq 1$. Then the maps
$\operatorname{LPC}^{1}_{X,d}(B)\rightarrow\operatorname{Det}_{A\\{X\\},d}(B)=\mathcal{M}_{A}(A\\{X\\},B)$,
$\Theta\mapsto\alpha^{1}_{X}(\Theta)$, form a morphism of functors
$\alpha^{1}_{X}:\operatorname{LPC}^{1}_{X,d}\rightarrow\operatorname{Det}_{A\\{X\\},d}$
such that, for every commutative $A$-algebra $B$ and every map
$\rho:X\rightarrow{\bf{GL}}_{d}(B)$, we have
$\alpha^{1}_{X}(\Theta^{1}_{\rho})=D_{\rho}$. Moreover, the morphisms
$\alpha^{1}_{X}$ are natural in $X$.

  2. (ii)

Let $\Gamma$ be a group and $d\geq 1$. We denote by $X$ the underlying set of
$\Gamma$. Then the morphism
$\alpha_{X}:\operatorname{LPC}^{1}_{X,d}\rightarrow\operatorname{Det}_{A\\{X\\},d}$
restricts to a morphism
$\alpha_{\Gamma}:\operatorname{LPC}_{\Gamma,d}\rightarrow\operatorname{Det}_{A[\Gamma],d}$
such that, for every commutative $A$-algebra $B$ and every morphism of groups
$\rho:\Gamma\rightarrow{\bf{GL}}_{d}(B)$, we have
$\alpha_{\Gamma}(\Theta_{\rho})=D_{\rho}$. Moreover, the morphisms
$\alpha_{\Gamma}$ are natural in $\Gamma$.

###### Proof.

  1. (i)

The fact that $\alpha^{1}_{X}$ is a morphism of functors and the naturality in
$X$ follow easily from the definition of $\alpha^{1}_{X}(\Theta)$. Let $B$ be
a commutative $A$-algebra and $\rho:X\rightarrow{\bf{GL}}_{d}(B)$ be a map;
then the morphism of $A$-algebra
$u:\mathcal{O}({\bf{GL}}_{d,A}^{X})^{{\bf{GL}}_{d,A}}\rightarrow B$
corresponding to $\Theta_{\rho}$ is given by $u(f)=f((\rho(x))_{x\in X})$. So,
if $C$ is a commutative $A$-algebra and $p\in C\\{X\\}$. we have

$\alpha^{1}_{X}(\Theta)_{C}(p)=\det(p((\rho(x)))_{x\in
X})=\det(\rho(p))=D_{\rho}(p).$

  2. (ii)

It suffices to prove that $\alpha_{X}$ sends $\operatorname{LPC}_{\Gamma,d}$
to $\operatorname{Det}_{A[\Gamma],d}$, the other statements then follow
immediately from (i).

Let $B$ be a commutative $A$-algebra and
$\Theta\in\operatorname{LPC}_{\Gamma,d}(B)$. As $\Theta$ is also in
$\operatorname{LPC}^{1}_{X,d}(B)$, we have a degree $d$ homogeneous polynomial
law $D=\alpha^{1}_{X}(\Theta):A\\{X\\}\rightarrow B$. Saying that $D$ is in
the image of $\mathcal{M}_{A}(A[\Gamma],B)$ means that, for every commutative
$A$-algebra $C$, the map $D_{C}:C\\{X\\}\rightarrow B\otimes_{A}C$ factors
through the obvious surjection $\pi_{C}:C\\{X\\}\rightarrow C[\Gamma]$. Fix a
commutative $A$-algebra $C$. We want to prove that, for all $p,q\in C\\{X\\}$
such that $\pi_{C}(p)=\pi_{C}(q)$, we have $D_{C}(p)=D_{C}(q)$. For $p\in
C\\{X\\}$, let $n(p)$ be the sum over all the nonconstant monomials $m$
appearing in $p$ of $\deg(m)-1$; so $n(p)=0$ if and only if $p$ is of degree
$\leq 1$. We claim that, for every $p\in C\\{X\\}$ such that $n(p)\geq 1$,
there exists $p_{1}\in C\\{X\\}$ such that $n(p_{1})=n(p)-1$,
$\pi_{C}(p_{1})=\pi_{C}(p)$ and $D_{C}(p_{1})=D_{C}(p)$. This claim implies
the desired result; indeed, if $p,q\in C\\{X\\}$ are such that
$\pi_{C}(p)=\pi_{C}(q)$, then, by the claim, we can find $p_{1},q_{1}\in
C\\{X\\}$ such that $n(p_{1})=n(q_{1})=0$,
$\pi_{C}(p_{1})=\pi_{C}(p)=\pi_{C}(q)=\pi_{C}(q_{1})$, $D_{C}(p_{1})=D_{C}(p)$
and $D_{C}(q_{1})=D_{C}(q)$; as $n(p_{1})=n(q_{1})=0$, the polynomials $p_{1}$
and $q_{1}$ are of degree $\leq 1$, so $\pi_{C}(p_{1})=\pi_{C}(q_{1})$ implies
that $p_{1}=q_{1}$, and then we have
$D_{C}(p)=D_{C}(p_{1})=D_{C}(q_{1})=D_{C}(q)$.

So it suffices to prove the claim. Let $p\in C\\{X\\}$ such that $n(p)\geq 1$,
and let $m$ be a monomial of degree $\geq 2$ appearing in $p$. Write
$m=cx_{\gamma_{1}}\ldots x_{\gamma_{k}}$ with $c\in C\setminus\\{0\\}$, $k\geq
2$ and $\gamma_{1},\ldots,\gamma_{k}\in\Gamma$, where, for every
$\gamma\in\Gamma$, we denote by $x_{\gamma}$ the corresponding element of $X$.
Let $r=p-m$ and set $p_{1}=r+cx_{\gamma_{1}\gamma_{2}}x_{\gamma_{3}}\ldots
x_{\gamma_{k}}$. Then $n(p_{1})=n(p)-1$ and $\pi_{C}(p_{1})=\pi_{C}(p)$. It
remains to prove that $D_{C}(p_{1})=D_{C}(p)$. Let
$u_{\Theta}:\mathcal{O}({\bf{GL}}_{d,A}^{X})^{{\bf{GL}}_{d,A}}\rightarrow B$
be the morphism of $A$-algebras corresponding to the image of $\Theta$ in
$\operatorname{LPC}^{1}_{\Gamma,d}(B)$. We have

$D_{C}(p)=\alpha_{X}^{1}(\Theta)_{C}(p)=u_{\Theta}(\det\nolimits_{\mathrm{id}_{X}}(p))\quad\mbox{and}\quad
D_{C}(p_{1})=u_{\Theta}(\det\nolimits_{\mathrm{id}_{X}}(p_{1}).$

As $\Theta$ is in $\operatorname{LPC}_{\Gamma,d}(B)$, the morphism
$u_{\Theta}$ vanishes on $I_{\Gamma,G}$. So, to show that
$D_{C}(p)=D_{C}(p_{1})$, it suffices to note that
$p_{1}-p=\varphi_{\gamma_{1},\gamma_{2}}(F)$, with
$F\in\mathcal{O}({\bf{GL}}_{d,A}\times{\bf{GL}}_{d,A}^{X})^{{\bf{GL}}_{d,A}}$
the function
$(g,(g_{\gamma})_{\gamma\in\Gamma})\mapsto\det\left(r((g_{\gamma})_{\gamma\in\Gamma})+cgg_{\gamma_{3}}\ldots
g_{\gamma_{k}}\right)$.

∎

##  4\. From determinants to pseudocharacters

The main result of this note is the following theorem.

######  Theorem 4.1.

Let $A$ be a commutative ring and $d$ be a positive integer.

  1. (i)

Let $X$ be a set. The morphism
$\alpha^{1}_{X}:\operatorname{LPC}^{1}_{X,d}\rightarrow\operatorname{Det}(A\\{X\\},d)$
of Proposition 3.1(i) corresponds by the isomorphisms
$\operatorname{LPC}^{1}_{X,d}\simeq\operatorname{Spec}(\mathcal{O}({\bf{GL}}_{d}^{X})^{{\bf{GL}}_{d}})$
and
$\operatorname{Det}_{A\\{X\\},d}\simeq\operatorname{Spec}(\mathcal{O}(M_{d}^{X})^{{\bf{GL}}_{d}})$
of Proposition 2.4 and Corollary 1.13 to the restriction morphism
$\mathcal{O}(M_{d}^{X})^{{\bf{GL}}_{d}}\rightarrow\mathcal{O}({\bf{GL}}_{d}^{X})^{{\bf{GL}}_{d}}$.
Moreover, $\alpha^{1}_{X}$ is injective, and, for every commutative
$A$-algebra $B$, an element $D$ of $\operatorname{Det}_{A\\{X\\},d}(B)$ is in
the image of $\alpha^{1}_{X}$ if and only if $D_{B}(x)\in B^{\times}$ for
every $x\in X$.

  2. (ii)

For every set $\Gamma$, the morphism
$\alpha_{\Gamma}:\operatorname{LPC}_{\Gamma,d}\rightarrow\operatorname{Det}(A[\Gamma],d)$
of Proposition 3.1(ii) is an isomorphism.

###### Proof.

By Remark 1.11 and Proposition 2.6(i)(b), we may assume that $A=\mathbb{Z}$.

Let $X$ be a set. As at the end of Section 1, we denote the universal matrices
representation by $\rho^{\mathrm{univ}}:X\rightarrow M_{d}(F_{X}(d))$, where
$F_{X}(d)=\mathbb{Z}[x_{i,j},\ x\in X,\ 1\leq i,j\leq d]$. The universal
element of $\operatorname{Det}_{\mathbb{Z}\\{X\\},d}$ is then
$D^{\mathrm{univ}}=D_{\rho^{\mathrm{univ}}}:\mathbb{Z}\\{X\\}\rightarrow
E_{X}(d)$, where $E_{X}(d)$ is generated by the coefficients of the
characteristic polynomials of the $\rho^{\mathrm{univ}}(p)$,
$p\in\mathbb{Z}\\{X\\}$.

We prove (i). The first statement follows from the fact that
$\alpha^{1}_{X}(D_{\rho})=\Theta_{\rho}$ for every commutative ring $A$ and
every map $\rho:\Gamma\rightarrow{\bf{GL}}_{d}(A)$. For the injectivity of
$\alpha^{1}_{X}$, it suffices to prove that
$\mathcal{O}({\bf{GL}}_{d}^{X})^{{\bf{GL}}_{d}}$ is a localization of
$\mathcal{O}(M_{d}^{X})^{{\bf{GL}}_{d}}$, but this follows from the fact that
$\mathcal{O}({\bf{GL}}_{d}^{X})$ is the localization of
$\mathcal{O}(M_{d}^{X})$ by the multiplicative set generated by the functions
$\det_{x}:(g_{y})_{y\in X}\mapsto\det(g_{x})$, $x\in X$, which are in
$\mathcal{O}(M_{d}^{X})^{{\bf{GL}}_{d}}$. Finally, let $A$ be a commutative
ring, let $D\in\operatorname{Det}_{\mathbb{Z}\\{X\\},d}(A)$, and let
$u:\mathcal{O}(M_{d}^{X})^{{\bf{GL}}_{d}}\rightarrow A$ be the morphism of
rings corresponding to $D$. Then $D$ is the image of $\alpha^{1}_{X}$ if and
only if $u$ extends to a morphism of rings
$\mathcal{O}({\bf{GL}}_{d}^{X})^{{\bf{GL}}_{d}}\rightarrow A$, which is
equivalent to the condition that $u(\det_{x})\in A^{\times}$ for every $x\in
A$. As $u(\det_{x})=u(\det(\rho^{\mathrm{univ}}(x)))=D_{A}(x)$ for every $x\in
X$, we get the conclusion.

We prove (ii). Let $X$ be the underlying set of $\Gamma$. We have a
commutative diagram

$\textstyle{\operatorname{LPC}_{\Gamma,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{\Gamma}}$$\textstyle{\operatorname{Det}_{\mathbb{Z}[\Gamma],d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{LPC}^{1}_{\Gamma,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{1}_{X}}$$\textstyle{\operatorname{Det}_{\mathbb{Z}\\{X\\},d}}$

and $\alpha^{1}_{X}$ is injective by (i), so it suffices to check that the
image of $\alpha_{\Gamma}$ is $\operatorname{Det}_{\mathbb{Z}[\Gamma],d}$. For
this, we check that the universal element of
$\operatorname{Det}_{\mathbb{Z}[\Gamma],d}$ is in the image of
$\alpha_{\Gamma}$. Let
$\varphi:E_{X}(d)\simeq(\Gamma_{\mathbb{Z}}^{d}(\mathbb{Z}\\{X\\}))^{\mathrm{ab}}\rightarrow
A:=(\Gamma_{\mathbb{Z}}(\mathbb{Z}[\Gamma]))^{\mathrm{ab}}$ be induced by the
quotient map $\pi:\mathbb{Z}\\{X\\}\rightarrow\mathbb{Z}[\Gamma]$. The
universal element $D$ of $\operatorname{Det}_{\mathbb{Z}[\Gamma],d}$ is
defined by $D\circ\pi=\varphi\circ D^{\mathrm{univ}}$, and its preimage by
$\alpha^{1}_{X}$ is the element $\Theta$ of
$\operatorname{LPC}^{1}_{\Gamma,d}$ defined by

$\Theta_{n}(f)(\gamma_{1},\ldots,\gamma_{n})=\varphi(\Theta^{\mathrm{univ}}_{n}(f)(\gamma_{1},\ldots,\gamma_{n})),$

where $\Theta^{\mathrm{univ}}=(\alpha^{1}_{X})^{-1}(D^{\mathrm{univ}})$. It
suffices to show that $\Theta$ satisfies condition (LPC2); we will then have
$\Theta\in\operatorname{LPC}_{\Gamma,d}(A)$ and $\alpha_{\Gamma}(\Theta)=D$.
Let $n$ be a positive integer and $\gamma_{1},\ldots,\gamma_{n+1}\in\Gamma$.
We want to check that
$\Theta_{n}(f)(\gamma_{1},\ldots,\gamma_{n-1},\gamma_{n}\gamma_{n+1})=\Theta_{n+1}(\widehat{f})(\gamma_{1},\ldots,\gamma_{n+1})$
for every $f\in\mathcal{O}({\bf{GL}}_{d}^{n})^{{\bf{GL}}_{d}}$. As both sides
are $\mathbb{Z}$-algebra morphisms in $f$, it suffices to check it on
generators of $\mathcal{O}({\bf{GL}}_{d}^{n})^{{\bf{GL}}_{d}}$. So, by results
of Donkin (cf. [6, 3.1]), we may assume that

$f(g_{1},\ldots,g_{n})=\Lambda_{k}(g_{i_{1}}\ldots g_{i_{r}}),$

where $k\in\\{1,\ldots,n\\}$, $\Lambda_{k}$ is the $k$th coefficient of the
characteristic polynomial (i.e.
$\Lambda_{k}(g)=(-1)^{k}\operatorname{Tr}(\Lambda^{k}g)$), $r\in\mathbb{N}$
and $i_{1},\ldots,i_{r}\in\\{1,2,\ldots,n\\}$. For every $\gamma\in\Gamma$, we
denote by $x_{\gamma}$ the element of $X$ corresponding to $\gamma$. For
$i\in\\{1,2,\ldots,n-1\\}$, we set $y_{i}=z_{i}=x_{\gamma_{i}}$, and we also
set $y_{n}=x_{\gamma_{n}\gamma_{n+1}}$,
$z_{n}=x_{\gamma_{n}}x_{\gamma_{n+1}}$; these are all elements of
$\mathbb{Z}\\{X\\}$. We then have

$\displaystyle\Theta_{n}(f)(\gamma_{1},\ldots,\gamma_{n-1},\gamma_{n}\gamma_{n+1})$
$\displaystyle=\varphi(\Theta^{\mathrm{univ}}_{n}(f)(\gamma_{1},\ldots,\gamma_{n-1},\gamma_{n}\gamma_{n+1}))$
$\displaystyle=\varphi(f(\rho^{\mathrm{univ}}(x_{\gamma_{1}}),\ldots,\rho^{\mathrm{univ}}(x_{\gamma_{n-1}}),\rho^{\mathrm{univ}}(x_{\gamma_{n}\gamma_{n+1}})))$
$\displaystyle=\varphi(\Lambda_{k}(\rho^{\mathrm{univ}}(y_{i_{1}}\ldots
y_{i_{r}})))$

and

$\displaystyle\Theta_{n+1}(\widehat{f})(\gamma_{1},\ldots,\gamma_{n+1})$
$\displaystyle=\varphi(\Theta^{\mathrm{univ}}_{n+1}(\widehat{f})(\gamma_{1},\ldots,\gamma_{n+1}))$
$\displaystyle=\varphi(\widehat{f}(\rho^{\mathrm{univ}}(x_{\gamma_{1}}),\ldots,\rho^{\mathrm{univ}}(x_{\gamma_{n+1}})))$
$\displaystyle=\varphi(\Lambda_{k}(\rho^{\mathrm{univ}}(z_{i_{1}}\ldots
z_{i_{r}}))).$

As $\pi\circ\det\circ\rho^{\mathrm{univ}}=\varphi\circ
D^{\mathrm{univ}}=D\circ\pi$ as polynomial laws, we get a commutative diagram

$\textstyle{\mathbb{Z}\\{X\\}[t]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\det\circ\rho^{\mathrm{univ}}}$$\scriptstyle{\pi\otimes\mathrm{id}_{\mathbb{Z}[t]}}$$\textstyle{E_{X}(d)[t]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi\otimes\mathrm{id}_{\mathbb{Z}[t]}}$$\textstyle{\mathbb{Z}[\Gamma][t]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D}$$\textstyle{(\Gamma_{\mathbb{Z}}^{d}(\mathbb{Z}[\Gamma]))^{\mathrm{ab}}[t]}$

As $(\pi\otimes\mathrm{id}_{\mathbb{Z}[t]})(t-y_{i_{1}}\ldots
y_{i_{r}})=(\pi\otimes\mathrm{id}_{\mathbb{Z}[t]})(t-z_{i_{1}}\ldots
z_{i_{r}})$, we get that

(**)
$(\varphi\otimes\mathrm{id}_{\mathbb{Z}[t]})(\det\circ\rho^{\mathrm{univ}}(t-y_{i_{1}}\ldots
y_{i_{r}}))=(\varphi\otimes\mathrm{id}_{\mathbb{Z}[t]})(\det\circ\rho^{\mathrm{univ}}(t-z_{i_{1}}\ldots
z_{i_{r}})).$

But, for every $p\in\mathbb{Z}\\{X\\}$, we have

$\det\circ\rho^{\mathrm{univ}}(t-p)=t^{d}+\sum_{k=1}^{d}\Lambda_{k}(\rho^{\mathrm{univ}}(p))t^{d-k}\in
E_{X}(d)[t],$

so identity (** ‣ 4) implies that

$\varphi(\Lambda_{k}(\rho^{\mathrm{univ}}(y_{i_{1}}\ldots
y_{i_{r}})))=\varphi(\Lambda_{k}(\rho^{\mathrm{univ}}(z_{i_{1}}\ldots
z_{i_{r}}))),$

as desired.

∎

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