$M_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R}$ is a
condensed $R$-module.222Fargues–Scholze in [28, Section VIII] call such
modules _relatively discrete_ $R$-modules.

######  Lemma 4.6.

Let $N$ be an abstract $R$-module. Then
$N_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R}$ is an
inductive limit of profinite submodules along quasi-compact injections. In
particular,
$N_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R}$ is
quasi-separated.

###### Proof.

We write $N=\varinjlim_{i}M_{i}$ where $M_{i}$ are finitely generated
$R$-submodules of $N$. As the tensor product of condensed
$R_{\operatorname{disc}}$-modules is left adjoint to the internal Hom functor
of condensed $R_{\operatorname{disc}}$-modules, it commutes with arbitrary
colimits. Thus

$\displaystyle
N_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R}=(\varinjlim\nolimits_{i}(M_{i})_{\operatorname{disc}})\otimes_{R_{\operatorname{disc}}}\underline{R}=\varinjlim\nolimits_{i}((M_{i})_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R}).$

Since $M_{i}$ are finitely generated they are profinite and we have a natural
isomorphism
$(M_{i})_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R}\cong\underline{M_{i}}$.
Moreover, the transition maps are quasi-compact injections, so
$N_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R}$ is
quasi-separated by A.5. ∎

Let $X$ be an affine scheme of finite type over $\mathcal{O}$ as before and
let $S$ be a profinite set. If $C$ is a commutative condensed
$\mathcal{O}$-algebra then by evaluating it at a profinite set $T$ we obtain a
commutative $\mathcal{O}$-algebra $C(T)$ and then we may evaluate $X$ at
$C(T)$ to obtain a set $X(C(T))$. One may show that $X(C):=X\circ C$ is a
condensed set. If $C$ is quasi-separated, then $X(C)$ is quasi-separated (see
A.6).

######  Definition 4.7.

We will say that a map of sets $f:S\rightarrow X(A)$ is _$R$ -condensed_ if
there is morphism of condensed sets $\tilde{f}:\underline{S}\to
X(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})$,
whose map on underlying sets is $f$.

######  Lemma 4.8.

Let $f:S\rightarrow X(A)$ be a map of sets. Then $f$ is $R$-condensed if and
only if the equivalent conditions of Lemma 4.2 hold.

###### Proof.

Suppose $f$ is $R$-condensed. To verify (1) of 4.2, let $b\in B$ where
$X=\operatorname{Spec}(B)$. This determines a map of schemes
$\varphi_{b}:X\to\mathbb{A}^{1}$. We obtain a map of condensed sets

$\underline{S}\xrightarrow{\tilde{f}}X(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})\xrightarrow{\varphi_{b}}\mathbb{A}^{1}(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})=A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R}.$

By 4.6 $A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R}$
is an inductive limit of condensed $R$-submodules of the form $\underline{M}$
for some finitely generated $R$-submodule $M\subseteq A$. By A.1
$\underline{S}$ is a compact object in the category of condensed sets. In
particular there exists a finitely generated $R$-submodule $M_{b}\subseteq A$,
such that $\varphi_{b}\circ\tilde{f}$ factors over a map
$\underline{S}\to\underline{M_{b}}$. It follows, that $\varphi_{b}(f(S))$ is
contained in $M_{b}$. We deduce condition (1) by applying the universal
property of product spaces.

Assume (2) of 4.2, i.e. there is a closed immersion
$\tau:X\hookrightarrow\mathbb{A}^{n}$ of $\mathcal{O}$-schemes, a finitely
generated $R$-submodule $M\subseteq\mathbb{A}^{n}(A)$, such that
$\tau(f(S))\subseteq M$ and $\tau\circ f:S\to M$ is continuous. We obtain a
map of condensed sets

$g:\underline{S}\to\underline{M}=M_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R}\to\mathbb{A}^{n}(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R}).$

By A.8 the map
$X(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})\to\mathbb{A}^{n}(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})$
is a quasi-compact injection. It follows from [47, Proposition 4.13], that

$X(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})(S)\cong\mathbb{A}^{n}(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})(S)\times_{\mathrm{Map}(S,\mathbb{A}^{n}(A))}\mathrm{Map}(S,X(A)).$

The pair $(g,f)$ is an element in the right hand side. So by Yoneda there
exists a map of condensed sets $\tilde{f}:\underline{S}\to
X(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})$ with
underlying map $f$. ∎

######  Lemma 4.9.

Let $A$ be a topological $R$-algebra and let $f:S\to X(A)$ be an $R$-condensed
map. Then $f$ is continuous.

###### Proof.

Let $\tau:X\hookrightarrow\mathbb{A}^{n}$ be a closed immersion. Since $X(A)$
has the subspace topology of $A^{n}$, it is enough to show that
$\tau\circ\rho:\Gamma\to A^{n}$ is continuous for the product topology on
$A^{n}$. Since $\rho$ is $R$-condensed, there is a finitely generated
$R$-submodule $N\subseteq A^{n}$, such that $\tau(\rho(\Gamma))\subseteq N$
and such that $\tau\circ\rho:\Gamma\to N$ is continuous for the unique
Hausdorff topology on $N$. Let us write $N^{s}$ for $N$ equipped with the
subspace topology of $A^{n}$. We claim, that the identity
$\operatorname{id}_{N}:N\to N^{s}$ is continuous for the Hausdorff topology on
the source. It is enough to show that the projection $N\to
N^{s}/\overline{\\{0\\}}$ is continuous, as every closed subset of $N^{s}$ is
a union of translates of $\overline{\\{0\\}}$, where the closure is taken in
$N^{s}$. But the quotient topology on $N^{s}/\overline{\\{0\\}}$ is Hausdorff,
so $N\to N^{s}/\overline{\\{0\\}}$ is continuous. ∎

###  4.3. $R$-condensed representations

Let $\Gamma$ be a profinite group, and let $G$ be an affine group scheme of
finite type over $\mathcal{O}$.

######  Definition 4.10.

A representation $\rho:\Gamma\rightarrow G(A)$ is _$R$ -condensed_ if $\rho$
is $R$-condensed as a map of sets.

######  Remark 4.11.

Lemma 4.8 implies that $\rho$ is $R$-condensed if and only if there is a
closed immersion $\tau:G\hookrightarrow\mathbb{A}^{n}$ of
$\mathcal{O}$-schemes such that $\tau(\rho(\Gamma))$ is contained in a
finitely generated $R$-submodule of $A^{n}=\mathbb{A}^{n}(A)$. For the
purposes of the paper we could take this last statement as the definition of
$R$-condensed representations, and proceed without actually making use of
condensed mathematics. The main point of Lemma 4.2 is that this definition is
independent of the chosen embedding $\tau$.

######  Lemma 4.12.

Let $\rho:\Gamma\to G(A)$ be an $R$-condensed representation. Then there is a
unique homomorphism of condensed groups $\tilde{\rho}:\underline{\Gamma}\to
G(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})$
whose map on underlying sets is $\rho$.

###### Proof.

Since $\rho$ is $R$-condensed, there is a map of condensed sets

$\tilde{\rho}:\underline{\Gamma}\to
G(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})$

whose map on underlying sets is $\rho$. Moreover, this map is unique by Lemma
A.7. We need to show, that the diagram of condensed sets

(27)
${\underline{\Gamma}\times\underline{\Gamma}}$${G(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})\times
G(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})}$${\underline{\Gamma}}$${G(A_{\operatorname{disc}}\otimes_{R_{\operatorname{disc}}}\underline{R})}$$\scriptstyle{\tilde{\rho}\times\tilde{\rho}}$$\scriptstyle{\tilde{\rho}}$

commutes, where the vertical maps are the multiplications. The diagram
commutes on underlying sets, so the claim follows from uniqueness A.7. ∎

######  Lemma 4.13.

Let $\rho:\Gamma\rightarrow G(A)$ be a representation and let
$\Gamma^{\prime}$ be an open subgroup of $\Gamma$. Then the following are
equivalent:

  1. (1)

$\rho$ is $R$-condensed;

  2. (2)

the restriction of $\rho$ to $\Gamma^{\prime}$ is $R$-condensed.

###### Proof.

Since $\rho(\Gamma^{\prime})\subseteq\rho(\Gamma)$ part (1) trivially implies
part (2).

Since $\Gamma^{\prime}$ is an open subgroup of $\Gamma$ we may write
$\Gamma=\bigcup_{i}c_{i}\Gamma^{\prime}$ for $c_{1},\dots,c_{r}\in\Gamma$ a
finite set of coset representatives. Let
$\tau:G\hookrightarrow\operatorname{GL}_{d}$ be a homomorphism of
$\mathcal{O}$-group schemes, which is a closed immersion and let
$\delta:\operatorname{GL}_{d}\rightarrow M_{d}\times M_{d}$ be the closed
immersion $g\mapsto(g,g^{-1})$. Then $\delta\circ\tau:G\hookrightarrow
M_{d}\times M_{d}$ is a closed immersion. If (2) holds then
$\delta(\tau(\rho(\Gamma^{\prime}))$ is contained in a finitely generated
$R$-submodule of $M_{d}(A)\times M_{d}(A)$. Since the map

$M_{d}(A)\times M_{d}(A)\rightarrow M_{d}(A)\times
M_{d}(A),\quad(X,Y)\mapsto(\tau(\rho(c_{i}))X,Y\tau(\rho(c_{i}))^{-1})$

is $R$-linear, we conclude that $\rho(\Gamma)$ maps into a finitely generated
$R$-submodule of $M_{d}(A)\times M_{d}(A)$, which implies that $\rho$ is
$R$-condensed. ∎

######  Lemma 4.14.

Let $A$ be a Hausdorff topological $R$-algebra, such that $A$ is a filtered
union of closed $R$-subalgebras $B\subseteq A$, where each $B$ contains an
open $R$-subalgebra $B^{0}\subseteq B$, which is a finitely generated
$R$-module. Assume, that $\Gamma$ is a topologically finitely generated
profinite group and let $\rho:\Gamma\to G(A)$ be a homomorphism. Then the
following are equivalent:

  1. (1)

$\rho$ is continuous;

  2. (2)

$\rho$ is $R$-condensed.

###### Proof.

(1) implies (2). Let $\gamma_{1},\dots,\gamma_{N}\in\Gamma$ be generators of a
dense subgroup of $\Gamma$. By assumption there is an $R$-subalgebra
$B\subseteq A$, which contains an $R$-finite open $R$-subalgebra $B^{0}$, such
that $\rho(\gamma_{1}),\dots,\rho(\gamma_{N})\in G(B)$. Since $G(B)$ is closed
in $G(A)$, we have $\rho(\Gamma)\subseteq G(B)$. Since $G$ is of finite type
over $\mathcal{O}$, $G(B^{0})$ is an open subgroup of $G(B)$. Thus the
preimage $\Gamma^{\prime}:=\rho^{-1}(G(B_{0}))$ is an open subgroup of
$\Gamma$. If $\tau:G\rightarrow\mathbb{A}^{n}$ is a closed immersion then
$\tau(\rho(\Gamma^{\prime}))$ is contained in $\mathbb{A}^{n}(B^{0})$, which a
finitely generated $R$-module. Thus the restriction of $\rho$ to
$\Gamma^{\prime}$ is $R$-condensed and Lemma 4.13 implies that $\rho$ is
$R$-condensed.

(2) implies (1) by 4.9. ∎

######  Lemma 4.15.

Let $H$ be a closed subgroup scheme of $G$, let $\\{\gamma_{i}\\}_{i\in I}$ a
set of topological generators of $\Gamma$ and let $\rho:\Gamma\rightarrow
G(A)$ be an $R$-condensed representation. Then the following are equivalent:

  1. (1)

$\rho$ factors through $H(A)\subseteq G(A)$;

  2. (2)

$\rho(\gamma_{i})\in H(A)$ for all $i\in I$.

###### Proof.

Clearly (1) implies (2). For the converse, let $\Delta$ be an abstract
subgroup of $\Gamma$ generated by $\\{\gamma_{i}\\}_{i\in I}$. Let
$\tau:G\hookrightarrow\mathbb{A}^{n}$ be a closed immersion. Since $\rho$ is
$R$-condensed $\rho(\Gamma)$ is contained in a finitely generated
$R$-submodule $M\subseteq\mathbb{A}^{n}(A)$ and the map
$\rho:\Gamma\rightarrow M$ is continuous. The assumption in (2) implies that
$\rho(\Delta)\subseteq H(A)$. Since $H$ is closed in $G$ the restriction of
$\tau$ to $H$ is a closed immersion. It follows from Lemma 4.1 that $M\cap
H(A)$ is a closed subset of $M$. Its preimage in $\Gamma$ will be a closed
subset of $\Gamma$ containing $\Delta$. Since $\Delta$ is dense in $\Gamma$ we
conclude that $\rho(\Gamma)\subseteq M\cap H(A)$, which implies part (1). ∎

######  Lemma 4.16.

Let $\varphi:G\rightarrow H$ be a morphism of $\mathcal{O}$-group schemes of
finite type, and let $\rho:\Gamma\rightarrow G(A)$ be a representation. Then
the following hold:

  1. (1)

if $\rho$ is $R$-condensed then $\varphi\circ\rho:\Gamma\rightarrow H(A)$ is
$R$-condensed;

  2. (2)

if $\varphi\circ\rho$ is $R$-condensed and $\varphi$ is a closed immersion
then $\rho$ is $R$-condensed.

###### Proof.

Lemma 4.3 implies part (1); part (2) holds trivially. ∎

##  5\. The space of generic matrices

Let $G$ be a smooth affine $\mathcal{O}$-group scheme and let
$\overline{\rho}:\Gamma\rightarrow G(k)$ be a continuous representation, where
$\Gamma$ is a profinite group satisfying Mazur’s $p$-finiteness condition. We
will introduce a space of generic matrices for $G$, which will depend on
$\overline{\rho}$ and a faithful representation
$\tau:G\to\operatorname{GL}_{d}$.

###  5.1. Definition of $X^{\mathrm{gen}}_{\operatorname{GL}_{d}}$ and
$X^{\mathrm{gen},\tau}_{G}$

We fix333Its existence follows from [30, Exp. $\mathrm{VII}_{B}$, Rem.
11.11.1]. a closed immersion $\tau:G\to\operatorname{GL}_{d}$ of
$\mathcal{O}$-group schemes. By composing it with $\overline{\rho}$ we obtain
a continuous representation
$\tau\circ\overline{\rho}:\Gamma\rightarrow\operatorname{GL}_{d}(k)$. The
linearisation $(\tau\circ\overline{\rho})^{\mathrm{lin}}:k[\Gamma]\to k$
extends to the completed group ring
$(\tau\circ\overline{\rho})^{\mathrm{lin}}:k\llbracket\Gamma\rrbracket\to k$.
Let $\overline{D}:k\llbracket\Gamma\rrbracket\rightarrow k$ be the continuous
determinant law associated to $(\tau\circ\overline{\rho})^{\mathrm{lin}}$.
(See [16] and [57, Section 3].)

Let $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ be the universal deformation
ring of $\overline{D}$ introduced in [16, Section 3.1] and let
$D^{u}:R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket\rightarrow
R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ be the extension of the universal
$d$-dimensional determinant law lifting $\overline{D}$. In particular,
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ is a complete local noetherian
$\mathcal{O}$-algebra with residue field $k$, and we equip it with its
$\mathfrak{m}$-adic topology. We will write $D^{u}_{|A}$ for the
specialisation of $D^{u}$ along
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\rightarrow A$. We let
$X^{\mathrm{ps}}_{\operatorname{GL}_{d}}:=\operatorname{Spec}(R^{\mathrm{ps}}_{\operatorname{GL}_{d}})$.
Note, that $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ corresponds to
$R^{\mathrm{ps}}$ of [8] when we take the residual representation to be
$\tau\circ\overline{\rho}$.

Let $\mathrm{CH}(D^{u})$ be the two-sided ideal of
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket$ defined in
[16, Section 1.17], so that
$E^{u}:=R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket/\mathrm{CH}(D^{u})$
is the largest quotient of
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket$ on which
$D^{u}$ descends to a Cayley–Hamilton determinant law. Since $\Gamma$
satisfies Mazur’s $p$-finiteness condition, $E^{u}$ is a finitely generated
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-module and $\mathrm{CH}(D^{u})$ is
closed by [57, Proposition 3.6]. We write $D^{u}_{E^{u}}:E^{u}\to
R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ for the determinant law induced on
$E^{u}$ by $D^{u}$ and
$\pi_{E^{u}}:R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket\to
E^{u}$ for the projection.

If $f:E^{u}\rightarrow M_{d}(A)$ is a homomorphism of
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebras for a commutative
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra $A$ then we say $f$ is a
_homomorphism of Cayley–Hamilton algebras_ if $\det\circ(f\otimes
A):E^{u}\otimes_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}A\rightarrow A$ is
equal to $D^{u}_{E^{u}}\otimes A$.

######  Lemma 5.1.

The composition $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}[\Gamma]\rightarrow
R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket\twoheadrightarrow
E^{u}$ is surjective.

###### Proof.

Let $S$ be the image of $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}[\Gamma]$ in
$E^{u}$. Since $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}[\Gamma]$ is a dense
subring of
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket$, $S$ is
also dense in $E^{u}$. It is enough to show that $S$ is closed in $E^{u}$.

It follows from [57, Proposition 3.6] that $E^{u}$ is a finitely generated
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-module. This implies that the
quotient topology on $E^{u}$ coincides with the topology inherited from
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ as a finitely generated
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-module, see the proof of [8, Lemma
3.2]. Every $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-submodule of $E^{u}$ is
closed with respect to this topology. Hence, $S$ is closed in $E^{u}$, and
thus $S=E^{u}$. ∎

######  Lemma 5.2.

Let $R\rightarrow A$ be a map of
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebras such that $R$ is a
noetherian profinite topological
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra. Let
$\rho:\Gamma\to\operatorname{GL}_{d}(A)$ be a representation satisfying
$\det\circ\rho^{\mathrm{lin}}=D^{u}_{|A}$. Then the following are equivalent:

  1. (1)

$\rho^{\mathrm{lin}}:R^{\mathrm{ps}}_{\operatorname{GL}_{d}}[\Gamma]\rightarrow
M_{d}(A)$ extends to a homomorphism of
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebras
$\widehat{\rho}:R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket\rightarrow
M_{d}(A)$;

  2. (2)

$\rho^{\mathrm{lin}}:R^{\mathrm{ps}}_{\operatorname{GL}_{d}}[\Gamma]\rightarrow
M_{d}(A)$ factors through a homomorphism of
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebras $E^{u}\rightarrow
M_{d}(A)$;

  3. (3)

$\rho$ is $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-condensed;

  4. (4)

$\rho$ is $R$-condensed.

Moreover, if the equivalent conditions hold then $E^{u}\rightarrow M_{d}(A)$
in $(2)$ is a homomorphism of Cayley–Hamilton algebras.

###### Proof.

(1) implies (2). The homomorphism $\widehat{\rho}\otimes
A:R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket\otimes_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}A\to
M_{d}(A)$ induces a $d$-dimensional $A$-valued Cayley–Hamilton determinant law

$\det\circ(\widehat{\rho}\otimes
A):R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket\otimes_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}A\to
A$

which satisfies $\det\circ(\widehat{\rho}\otimes A)=D^{u}_{|A}$ by assumption.

It follows, that $D^{u}_{|A}$ is Cayley–Hamilton, hence $\widehat{\rho}\otimes
A$ factors through a homomorphism
$E^{u}\otimes_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}A\to M_{d}(A)$ and so
$\rho^{\mathrm{lin}}|_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}[\Gamma]}$
factors through a homomorphism of Cayley–Hamilton algebras $E^{u}\rightarrow
M_{d}(A)$.

(2) implies (3). Since $E^{u}$ is finitely generated as
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-module by considering the closed
embedding $\delta:\operatorname{GL}_{d}\hookrightarrow M_{d}\times M_{d}$,
$g\mapsto(g,g^{-1})$ we obtain that $\rho$ is
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-condensed.

(3) implies (4) by Lemma 4.5.

(4) implies (1). Since $\rho$ is $R$-condensed, $\delta(\rho(\Gamma))$ is
contained in a finitely generated $R$-submodule of $M_{d}(A)\times M_{d}(A)$.
By projecting onto the first factor we deduce that the the image of
$\rho^{\mathrm{lin}}:R^{\mathrm{ps}}_{\operatorname{GL}_{d}}[\Gamma]\rightarrow
M_{d}(A)$ is contained in a finitely generated $R$-module $M$. Since $M$ is
profinite the map extends to a continuous map of profinite
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-modules
$\widehat{\rho}:R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket\rightarrow
M$. Since $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}[\Gamma]$ is a dense
subalgebra of
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket$ we deduce
that $\rho^{\mathrm{lin}}$ extends to a map of
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebras
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket\rightarrow
M_{d}(A)$. ∎

######  Definition 5.3.

Let
$X^{\mathrm{gen}}_{\operatorname{GL}_{d}}:R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\text{-}\mathrm{alg}\to\operatorname{Set}$
be the functor which sends $A$ to the set of representations
$\rho:\Gamma\rightarrow\operatorname{GL}_{d}(A)$ such that
$\det\circ\rho^{\mathrm{lin}}=D^{u}\otimes_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}A$
and $\rho$ satisfies the equivalent conditions of Lemma 5.2.

We give an alternative description of the points of
$X^{\mathrm{gen}}_{\operatorname{GL}_{d}}$. By the following Lemma, we see
that $X^{\mathrm{gen}}_{\operatorname{GL}_{d}}$ identifies canonically with
the scheme $X^{\mathrm{gen}}$ of [8].

######  Lemma 5.4.

Let $A$ be a commutative $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra.

  1. (1)

$X^{\mathrm{gen}}_{\operatorname{GL}_{d}}(A)$ is in natural bijection with the
set of homomorphisms of Cayley–Hamilton
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebras $f:E^{u}\to M_{d}(A)$. In
particular, $X^{\mathrm{gen}}_{\operatorname{GL}_{d}}$ is representable by a
finitely generated commutative
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra
$A^{\mathrm{gen}}_{\operatorname{GL}_{d}}$.

  2. (2)

If $A$ is a topological $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra then
every representation $\rho\in X^{\mathrm{gen}}_{\operatorname{GL}_{d}}(A)$ is
continuous.

###### Proof.

Part (1). Recall, that in [8, Lemma 3.1] the
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra $A^{\mathrm{gen}}$ is proved
to represent the functor, that maps an
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra $A$ to the set of
homomorphisms of Cayley–Hamilton algebras $f:E^{u}\to M_{d}(A)$. Once we prove
this functor to be naturally isomorphic to
$X^{\mathrm{gen}}_{\operatorname{GL}_{d}}$ as in 5.3, the claim about
representability follows and $X^{\mathrm{gen}}_{\operatorname{GL}_{d}}$ is
naturally isomorphic to the $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-scheme
$X^{\mathrm{gen}}=\operatorname{Spec}A^{\mathrm{gen}}$ from [8].

If $\rho\in X^{\mathrm{gen}}_{\operatorname{GL}_{d}}(A)$ then by Lemma 5.2 (2)
we obtain a natural homomorphism of Cayley–Hamilton algebras
$f_{\rho}:E^{u}\to M_{d}(A)$. Conversely, a map of Cayley–Hamilton algebras
$f:E^{u}\to M_{d}(A)$ gives rise to a representation by means of the
composition
$\rho_{f}:\Gamma\to(E^{u})^{\times}\overset{f}{\to}\operatorname{GL}_{d}(A).$
By restriction along
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma\rrbracket\otimes_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}A\to
E^{u}\otimes_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}A$, we see that

$\det\circ\rho_{f}^{\mathrm{lin}}=\det\circ((f\circ\pi_{E^{u}})\otimes
A)=D^{u}_{|A}.$

Lemma 5.2 (1) implies that $\rho_{f}\in
X^{\mathrm{gen}}_{\operatorname{GL}_{d}}(A)$. The maps $\rho\mapsto f_{\rho}$
and $f\mapsto\rho_{f}$ are mutually inverse.

Part (2) follows from [8, Lemma 3.2] or Lemma 4.9. ∎

######  Lemma 5.5.

There is a topologically finitely generated quotient $\Gamma\twoheadrightarrow
Q$ such that
$R^{\mathrm{ps},\Gamma}_{\operatorname{GL}_{d}}=R^{\mathrm{ps},Q}_{\operatorname{GL}_{d}}$
and every $\rho\in X^{\mathrm{gen}}_{\operatorname{GL}_{d}}(A)$, where $A\in
R^{\mathrm{ps}}_{G}\text{-}\mathrm{alg}$, factors as $\rho:\Gamma\rightarrow
Q\rightarrow\operatorname{GL}_{d}(A)$.

###### Proof.

Let $Q=\Gamma/K$, where $K\subseteq\ker(\overline{\rho})$ is the smallest
closed normal subgroup of $\Gamma$ such that $\ker(\overline{\rho})/K$ is
pro-$p$. Since $\ker(\overline{\rho})$ is open in $\Gamma$ and $\Gamma$
satisfies Mazur’s $p$-finiteness condition, $Q$ is topologically finitely
generated. Lemma 3.8 in [16] implies that
$R^{\mathrm{ps},\Gamma}_{\operatorname{GL}_{d}}=R^{\mathrm{ps},Q}_{\operatorname{GL}_{d}}$
and the Cayley–Hamilton algebra $E^{u}$ does not change if we replace $\Gamma$
by $Q$. The assertion follows from Lemma 5.4 (1). ∎

As already explained above, [57, Proposition 3.6] implies that $E^{u}$ is a
finitely generated $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-module, and hence
$k\otimes_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}E^{u}$ is a finite
dimensional $k$-vector space.

######  Lemma 5.6.

Let $\\{\gamma_{1},\ldots,\gamma_{N}\\}$ be a subset of $\Gamma$, such that
its image via the map in Lemma 5.1 spans
$k\otimes_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}E^{u}$ as a $k$-vector
space. Then the map

$X^{\mathrm{gen}}_{\operatorname{GL}_{d}}\rightarrow\operatorname{GL}_{d}^{N}\times
X^{\mathrm{ps}}_{\operatorname{GL}_{d}},\quad\rho\mapsto(\rho(\gamma_{1}),\ldots,\rho(\gamma_{N}))$

is a closed immersion.

###### Proof.

It follows from Part (1) of Lemma 5.4 that
$X^{\mathrm{gen}}_{\operatorname{GL}_{d}}$ coincides with the scheme
$X^{\mathrm{gen}}$, that was contructed in [8, Lemma 3.1] as a closed
subscheme of $M_{d}^{n}\times X^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ by
choosing generators $\delta_{1},\ldots,\delta_{n}$ of $E^{u}$ as an
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-module and mapping $\rho$ to the
$n$-tuple $(\rho(\delta_{1}),\ldots,\rho(\delta_{n}))$. Since
$\rho(\gamma_{i})$ are invertible matrices, it is enough to show that the
images of $\gamma_{1},\ldots,\gamma_{N}$ generate $E^{u}$ as an
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-module. This follows from Nakayama’s
lemma. ∎

Let $Q$ be a quotient of $\Gamma$ as in Lemma 5.5. We fix an $N$-tuple
$(\gamma_{1},\ldots,\gamma_{N})\in\Gamma^{N}$, such that the entries generate
a dense subgroup of $Q$ and its image spans
$E^{u}\otimes_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}k$ as a $k$-vector
space.

######  Definition 5.7.

We define the $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-scheme
$X^{\mathrm{gen},\tau}_{G}$ as the fibre product

(28)
$X^{\mathrm{gen},\tau}_{G}:=X^{\mathrm{gen}}_{\operatorname{GL}_{d}}\times_{\operatorname{GL}_{d}^{N}}G^{N},$

where $G^{N}\rightarrow\operatorname{GL}_{d}^{N}$ is the map
$(g_{1},\ldots,g_{N})\mapsto(\tau(g_{1}),\ldots,\tau(g_{N}))$.

Thus $X^{\mathrm{gen},\tau}_{G}$ is a closed
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-subscheme of
$X^{\mathrm{gen}}_{\operatorname{GL}_{d}}$, and hence a closed subfunctor. We
note that $X^{\mathrm{gen},\tau}_{G}$ is non-empty, as
$X^{\mathrm{gen},\tau}_{G}(k)$ contains the representation $\overline{\rho}$
that we have started with.

######  Remark 5.8.

By construction $X^{\mathrm{gen}}_{\operatorname{GL}_{d}}$ is a closed
subscheme of $\operatorname{GL}_{d}^{N}\times
X^{\mathrm{ps}}_{\operatorname{GL}_{d}}$. Thus $X^{\mathrm{gen},\tau}_{G}$ is
a closed subscheme of $G^{N}\times X^{\mathrm{ps}}_{\operatorname{GL}_{d}}$,
which can interpreted as follows.

Let $\mathcal{F}_{N}$ be a free group generated by elements
$x_{1},\ldots,x_{N}$. Let $\operatorname{Rep}_{G}^{\mathcal{F}_{N},\square}$
be the affine $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-scheme, which
represents the functor, that maps a commutative
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra $A$ to the set of (abstract)
group homomorphisms $\rho:\mathcal{F}_{N}\to G(A)$. Sending $\rho$ to the
$N$-tuple $(\rho(x_{1}),\ldots,\rho(x_{N}))$ induces an isomorphism of
$X^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-schemes

$\operatorname{Rep}_{G}^{\mathcal{F}_{N},\square}\cong G^{N}\times
X^{\mathrm{ps}}_{\operatorname{GL}_{d}}.$

Mapping $x_{i}$ to $\gamma_{i}$ induces a group homomorphism
$\varphi:\mathcal{F}_{N}\rightarrow Q$ and it follows from above, that mapping
$\rho$ to $\rho\circ\varphi$ induces a closed immersion
$X^{\mathrm{gen},\tau}_{G}\rightarrow\operatorname{Rep}_{G}^{\mathcal{F}_{N},\square}$.

######  Remark 5.9.

If $L^{\prime}$ is a finite extension of $L$ with then ring of integers
$\mathcal{O}^{\prime}$ then it follows from the definition that
$(X^{\mathrm{gen},\tau}_{G})_{\mathcal{O}^{\prime}}\cong
X^{\mathrm{gen},\tau}_{G_{\mathcal{O}^{\prime}}}$. The main technical result
of the paper is the computation of the dimension of
$X^{\mathrm{gen},\tau}_{G}$. Since $\mathcal{O}^{\prime}$ is finite and flat
over $\mathcal{O}$ the dimensions of
$(X^{\mathrm{gen},\tau}_{G})_{\mathcal{O}^{\prime}}$ and
$X^{\mathrm{gen},\tau}_{G}$ are the same and using Proposition 2.8 we may
assume that $G$ is split, $G/G^{0}$ is constant and
$G(\mathcal{O})\rightarrow(G/G^{0})(\mathcal{O})$ is surjective.

We are now ready to give a description of the points of
$X^{\mathrm{gen},\tau}_{G}$.

######  Lemma 5.10.

Let $A$ be a commutative $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra.

  1. (1)

$X^{\mathrm{gen},\tau}_{G}(A)$ is the set of representations
$\rho:\Gamma\rightarrow G(A)$, such that $\tau\circ\rho\in
X^{\mathrm{gen}}_{\operatorname{GL}_{d}}(A)$.

  2. (2)

If $A$ is a topological $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra then
every representation $\rho\in X^{\mathrm{gen},\tau}_{G}(A)$ is continuous.

###### Proof.

Part (1) follows from Lemmas 4.15 and 5.5.

For part (2), we observe that part (1) and Lemma 5.4 (2) imply that that the
composition $\tau\circ\rho:\Gamma\to G(A)\to\operatorname{GL}_{d}(A)$ is
continuous. Since $\tau$ is a closed immersion the topology on $G(A)$ is the
subspace topology induced by $\operatorname{GL}_{d}(A)$ and thus $\rho$ is
continuous. ∎

######  Remark 5.11.

It follows from Lemma 5.10 that the definition of $X^{\mathrm{gen},\tau}_{G}$
does not depend on the choice of the quotient $Q$ and
$\gamma_{1},\ldots,\gamma_{N}$. However, it will depend a priori on the choice
of the faithful representation $\tau$. It will follow from Corollaries 6.12
and 8.4 that the connected component of $X^{\mathrm{gen},\tau}_{G}$ containing
the point corresponding to the representation
$\overline{\rho}:\Gamma\rightarrow G(k)$ is independent of $\tau$.

If $\mathfrak{p}$ is the maximal ideal of
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ then $\kappa(\mathfrak{p})=k$ and we
equip the algebraic closure $\overline{\kappa(\mathfrak{p})}$ with the
discrete topology. If $\mathfrak{p}\in
P_{1}R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ then $\kappa(\mathfrak{p})$ is a
local field and the valuation topology on $\kappa(\mathfrak{p})$ makes it into
a topological $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra by [8, Lemma
3.17]. We equip the algebraic closure $\overline{\kappa(\mathfrak{p})}$ with
the topology induced by a valuation, which extends the valuation on
$\kappa(\mathfrak{p})$.

######  Lemma 5.12.

Let $\mathfrak{p}$ be a prime of $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$
such that $\dim R^{\mathrm{ps}}_{\operatorname{GL}_{d}}/\mathfrak{p}\leq 1$.
The set $X^{\mathrm{gen},\tau}_{G}(\overline{\kappa(\mathfrak{p})})$ is
naturally in bijection with the set of continuous representations
$\rho:\Gamma\rightarrow G(\overline{\kappa(\mathfrak{p})})$, with the topology
on $\overline{\kappa(\mathfrak{p})}$ as above, such that the $d$-dimensional
determinant law $\det\circ(\tau\circ\rho)^{\mathrm{lin}}$ is equal to the
specialisation of $D^{u}$ along
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\rightarrow\overline{\kappa(\mathfrak{p})}$.

###### Proof.

4.14 implies that a representation $\rho:\Gamma\rightarrow
G(\overline{\kappa(\mathfrak{p})})$ is continuous if and only if it is
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-condensed. The assertion then
follows from 5.10. ∎

Let $A^{\mathrm{gen}}_{\operatorname{GL}_{d}}\twoheadrightarrow
A^{\mathrm{gen},\tau}_{G}$ be the quotient representing
$X^{\mathrm{gen},\tau}_{G}$. We will denote by
$\overline{X}^{\mathrm{gen},\tau}_{G}$ the special fibre of
$X^{\mathrm{gen},\tau}_{G}$.

######  Lemma 5.13.

Let $y:R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\rightarrow\Omega$ be a ring
homomorphism into an algebraically closed field $\Omega$ and and let $x\in
X^{\mathrm{gen},\tau}_{G}(\Omega)$ be a point above $y$. Let $H$ be the
smallest Zariski closed subgroup of $G_{\Omega}$ containing
$\rho_{x}(\gamma_{1}),\ldots,\rho_{x}(\gamma_{N})$, where $\rho_{x}$ is the
specialisation at $x$ of the universal representation
$\rho:\Gamma\rightarrow\operatorname{GL}_{d}(A^{\mathrm{gen}}_{\operatorname{GL}_{d}})$.
Then $H$ is equal to the Zariski closure of $\rho_{x}(\Gamma)$ in
$G_{\Omega}$.

###### Proof.

Lemmas 4.15 and 5.5 imply that $\rho_{x}(\Gamma)=\rho_{x}(Q)\subseteq
H(\Omega)$. Hence, its Zariski closure is contained in $H$. The other
inclusion follows from the definition of $H$. ∎

###  5.2. Completion at maximal ideals and deformation problems

Let $Y\subset X^{\mathrm{gen},\tau}_{G}$ be the preimage of the closed point
of $X^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ and let $x$ be either a closed
point of $Y$ or a closed point of $X^{\mathrm{gen},\tau}_{G}\setminus Y$ and
let $y$ be its image in
$\operatorname{Spec}R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$. It follows from
[8, Lemmas 3.17, 3.18] that $\kappa(x)$ is a finite extension of $\kappa(y)$
and there are the following possibilities:

  1. (1)

if $x\in Y$ then $\kappa(x)$ is a finite extension of $k$;

  2. (2)

if $x\in X^{\mathrm{gen},\tau}_{G}[1/p]$ then $\kappa(x)$ is a finite
extension of $L$;

  3. (3)

if $x\in\overline{X}^{\mathrm{gen},\tau}_{G}\setminus Y$ then $\kappa(x)$ is a
local field of characteristic $p$.

The universal property of $A^{\mathrm{gen},\tau}_{G}$ gives us a continuous
Galois representation

$\rho_{x}:\Gamma\rightarrow G(\kappa(x))$

for the natural topology on the target. In this section we want to relate the
completion of the local ring $\mathcal{O}_{X^{\mathrm{gen},\tau}_{G},x}$ to a
deformation problem for $\rho_{x}$.

######  Lemma 5.14.

Let $\kappa$ be either a finite or a local $\mathcal{O}$-field and let
$\rho:\Gamma\rightarrow G(\kappa)$ be a continuous representation. Let
$\Lambda$ be a coefficient ring of $\kappa$ and let
$(A,\mathfrak{m}_{A})\in\mathfrak{A}_{\Lambda}$. If
$\gamma_{1},\ldots,\gamma_{N}$ topologically generate $\Gamma$ then the map

$\rho_{A}\mapsto(\tau\circ\rho_{A},(\rho(\gamma_{1}),\ldots,\rho(\gamma_{N})))$

induces a bijection of sets $D^{\square}_{\rho,G}(A)\longrightarrow
D^{\square}_{\tau\circ\rho,\operatorname{GL}_{d}}(A)\times_{\operatorname{GL}_{d}^{N}(A)}G^{N}(A).$

###### Proof.

Since $\tau:G(A)\rightarrow\operatorname{GL}_{d}(A)$ is injective, the map is
an injection. A point in
$D^{\square}_{\tau\circ\rho,\operatorname{GL}_{d}}(A)\times_{\operatorname{GL}_{d}^{N}(A)}G^{N}(A)$
corresponds to continuous representation
$\rho_{A}:\Gamma\rightarrow\operatorname{GL}_{d}(A)$ such that
$\rho_{A}\equiv\tau\circ\rho\pmod{\mathfrak{m}_{A}}$ and
$(g_{1},\ldots,g_{N})\in G^{N}(A)$ such that $\rho(\gamma_{i})=\tau(g_{i})$
for $1\leq i\leq N$. Since $\tau(G(A))$ is a closed subgroup of
$\operatorname{GL}_{d}(A)$, its preimage $\rho_{A}^{-1}(\tau(G(A)))$ is a
closed subgroup of $\Gamma$. Since it contains the topological generators
$\gamma_{1},\ldots,\gamma_{N}$ we deduce that the image of $\rho_{A}$ is
contained in $\tau(G(A))$. This implies that the map is surjective. ∎

######  Lemma 5.15.

Let $(A,\mathfrak{m})$ be a local (possibly non-noetherian) ring and let $I$
be a finitely generated ideal of $A$. If the $\mathfrak{m}$-adic completion
$\hat{A}$ of $A$ is noetherian then $\widehat{(A/I)}=\hat{A}/I\hat{A}$.

###### Proof.

It follows from [54, Tag 0BNG] that $A/\mathfrak{m}^{n}$ is a quotient of
$\hat{A}$ for all $n\geq 1$. Since $\hat{A}$ is noetherian,
$A/\mathfrak{m}^{n}$ is noetherian, and hence artinian for all $n\geq 1$. By
choosing generators of $I=(a_{1},\ldots,a_{s})$ we obtain a surjection
$A^{s}\twoheadrightarrow I$. Let
$K_{n}=\operatorname{Ker}((A/\mathfrak{m}^{n})^{s}\twoheadrightarrow(I+\mathfrak{m}^{n})/\mathfrak{m}^{n})$.
The modules $K_{n}$ are artinian, since they are finitely generated over
artinian ring $A/\mathfrak{m}^{n}$, and thus form a Mittag-Leffler system,
which implies that the map
$\hat{A}^{s}=\varprojlim_{n}(A/\mathfrak{m}^{n})^{s}\rightarrow\varprojlim_{n}(I+\mathfrak{m}^{n})/\mathfrak{m}^{n}$
is surjective.

The map $\hat{A}\rightarrow\widehat{(A/I)}$ is surjective by [54, Tag 0315]
and the kernel is equal to
$\varprojlim_{n}(I+\mathfrak{m}^{n})/\mathfrak{m}^{n}$, [54, Tag 02N1]. It
follows from above that this projective limit is equal to $I\hat{A}$, which
implies the assertion. ∎

######  Proposition 5.16.

Let $x\in X^{\mathrm{gen},\tau}_{G}(\kappa)$, where $\kappa$ is either a
finite or a local $\mathcal{O}$-field. Let $\mathfrak{q}$ be the kernel of the
map

$\Lambda\otimes_{\mathcal{O}}A^{\mathrm{gen},\tau}_{G}\rightarrow\kappa,\quad\lambda\otimes
a\mapsto\bar{\lambda}\bar{a},$

where $\bar{\lambda}$ and $\bar{a}$ denote the images of $\lambda$ and $a$ in
$\kappa$ and $\Lambda$ is a coefficient ring for $\kappa$. The completion of
$\Lambda\otimes_{\mathcal{O}}A^{\mathrm{gen},\tau}_{G}$ with respect to
$\mathfrak{q}$ is naturally isomorphic to $R^{\square}_{\rho_{x},G}$.

###### Proof.

If $\rho_{A}\in D^{\square}_{\rho,G}(A)$ then [8, Proposition 3.33] implies
that $A$ is naturally an $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra and
$\tau\circ\rho\in X^{\mathrm{gen}}_{\operatorname{GL}_{d}}(A)$. Lemma 5.5
implies that $\rho_{A}$ factors through the topologically finitely generated
quotient $Q$ of $\Gamma$. Hence, we may assume without loss of generality that
$\gamma_{1},\ldots,\gamma_{N}$ are topological generators of $\Gamma$.

Let $I$ be the kernel of the surjection
$\mathcal{O}(\operatorname{GL}_{d}^{N})\twoheadrightarrow\mathcal{O}(G^{N})$.
We note that $I$ is a finitely generated ideal. Let $\mathfrak{m}_{G}$ and
$\mathfrak{m}_{\operatorname{GL}_{d}}$ be the maximal ideals of
$R^{\square}_{\rho_{x},G}$ and
$R^{\square}_{\tau\circ\rho_{x},\operatorname{GL}_{d}}$, respectively. The
rings $R^{\square}_{\rho_{x},G}/\mathfrak{m}_{G}^{n}$ and
$R^{\square}_{\tau\circ\rho_{x},\operatorname{GL}_{d}}/\mathfrak{m}_{\operatorname{GL}_{d}}^{n}$
represent the restrictions of functors $D^{\square}_{\rho_{x},G}$ and
$D^{\square}_{\tau\circ\rho_{x},\operatorname{GL}_{d}}$ to
$\mathfrak{A}_{\Lambda,n}$, respectively. It follows from Lemma 5.14 that

$R^{\square}_{\rho_{x},G}/\mathfrak{m}_{G}^{n}\cong(R^{\square}_{\tau\circ\rho_{x},\operatorname{GL}_{d}}/\mathfrak{m}_{\operatorname{GL}_{d}}^{n})\otimes_{\mathcal{O}(\operatorname{GL}_{d}^{N})}\mathcal{O}(G^{N}).$

By passing to the limit and using Lemma 5.15 (in a noetherian setting) we
obtain

$R^{\square}_{\rho_{x},G}\cong
R^{\square}_{\tau\circ\rho_{x},\operatorname{GL}_{d}}\otimes_{\mathcal{O}(\operatorname{GL}_{d}^{N})}\mathcal{O}(G^{N}).$

If $G=\operatorname{GL}_{d}$ and $\tau=\operatorname{id}$ then the proposition
is [8, Proposition 3.33], so that
$R^{\square}_{\tau\circ\rho_{x},\operatorname{GL}_{d}}$ is naturally
isomorphic to the completion of the local ring of
$\Lambda\otimes_{\mathcal{O}}A^{\mathrm{gen}}_{\operatorname{GL}_{d}}$ at
$\mathfrak{q}$. Since
$A^{\mathrm{gen},\tau}_{G}=A^{\mathrm{gen}}_{\operatorname{GL}_{d}}\otimes_{\mathcal{O}(\operatorname{GL}_{d}^{N})}\mathcal{O}(G^{N})=A^{\mathrm{gen}}_{\operatorname{GL}_{d}}/IA^{\mathrm{gen}}_{\operatorname{GL}_{d}}$
by definition, and $R^{\square}_{\tau\circ\rho_{x},\operatorname{GL}_{d}}$ is
noetherian by Proposition 3.8, the assertion follows from Lemma 5.15 applied
to
$A=(\Lambda\otimes_{\mathcal{O}}A^{\mathrm{gen}}_{\operatorname{GL}_{d}})_{\mathfrak{q}}$.
∎

######  Lemma 5.17.

Let $X:=X^{\mathrm{gen},\tau}_{G}$ and let $x$ be either a closed point of $Y$
or a closed point of $X\setminus Y$. Let $\mathcal{O}_{X,x}$ be the local ring
at $x$ and let $\hat{\mathcal{O}}_{X,x}$ be its completion with respect to the
maximal ideal. Then we have the following isomorphisms of local rings:

  1. (1)

if $x\in X[1/p]$ or $x\in Y$ then $\hat{\mathcal{O}}_{X,x}\cong
R^{\square}_{G,\rho_{x}}$;

  2. (2)

if $x\in\overline{X}\setminus Y$ then $\hat{\mathcal{O}}_{X,x}\llbracket
T\rrbracket\cong R^{\square}_{G,\rho_{x}}$.

In particular, $\mathcal{O}_{X,x}$ is complete intersection if and only if
$R^{\square}_{G,\rho_{x}}$ is.

###### Proof.

The assertion follows from Proposition 5.16 and [8, Lemmas 3.36, 3.37]. ∎

##  6\. GIT quotients

Let $G$ be a generalised reductive $\mathcal{O}$-group scheme. If $A$ is an
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra then the group $G(A)$ acts
on the set $X^{\mathrm{gen},\tau}_{G}(A)$ by conjugation. This defines an
action of $G$ on $X^{\mathrm{gen},\tau}_{G}$. We let

(29) $X^{\mathrm{git},\tau}_{G}:=X^{\mathrm{gen},\tau}_{G}\sslash
G^{0},\quad\overline{X}^{\mathrm{git},\tau}_{G}:=\overline{X}^{\mathrm{gen},\tau}_{G}\sslash
G^{0}.$

######  Remark 6.1.

The closed immersion $\overline{X}^{\mathrm{gen},\tau}_{G}\hookrightarrow
X^{\mathrm{gen},\tau}_{G}$ induces a morphism
$\overline{X}^{\mathrm{git},\tau}_{G}\rightarrow(X^{\mathrm{git},\tau}_{G})_{k}$,
which is finite by [52, Theorem 2] and induces a bijection on geometric points
by [52, Theorem 3]. We deduce that this map is a homeomorphism.

We will show that $X^{\mathrm{git},\tau}_{G}$ is the spectrum of a complete
semi-local $\mathcal{O}$-algebra. We use its connected components to decompose
$X^{\mathrm{gen},\tau}_{G}$ and show in Lemma 6.12 that the resulting
decomposition is the decomposition of $X^{\mathrm{gen},\tau}_{G}$ into
connected components. The component containing the point corresponding to a
representation $\overline{\rho}:\Gamma_{F}\rightarrow G(k)$ that we have
started with in Section 5.1 will be denoted
$X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$. We then evaluate
both functors at finite and local fields and study the corresponding Galois
representations.

In Corollary 6.5 we prove an important finiteness result, which is used in an
essential way in the induction argument in Section 13. This in turn rests on
the following fundamental finiteness theorem proved by Vinberg [56], when $S$
is a spectrum of a field of characteristic zero, by Martin [40], when $S$ is a
spectrum of a field of characteristic $p$, and by Cotner [22] in general.

######  Theorem 6.2 ([22]).

Let $S$ be a locally noetherian scheme and let $f:H\rightarrow G$ be a finite
morphism of generalised reductive smooth affine $S$-group schemes. Then the
induced map

$H^{N}\sslash H\rightarrow G^{N}\sslash G$

is finite.

######  Corollary 6.3.

Let us assume the setting of Theorem 6.2. Then the induced map

$H^{N}\sslash H^{0}\rightarrow G^{N}\sslash G^{0}$

is also finite.

###### Proof.

Since $G^{N}\sslash G^{0}\rightarrow S$ and $H^{N}\sslash H^{0}\rightarrow S$
are affine and $S$ is locally noetherian, we may assume that $S$ is affine and
noetherian. Then $G$, $H$, $G^{N}$, $H^{N}$ as well as their respective GIT
quotients are all affine schemes. Since $H$ is generalised reductive $H/H^{0}$
is a finite étale $S$-group scheme. Since

$H^{N}\sslash H=(H^{N}\sslash H^{0})\sslash(H/H^{0}),$

[30, Exp. V, Thm. 4.1 (ii)] applied with $X_{0}=H^{N}\sslash H^{0}$ and
$X_{1}=(H/H^{0})\times_{S}X_{0}$, implies that the morphism $H^{N}\sslash
H^{0}\rightarrow H^{N}\sslash H$ is integral. It follows from [3, Theorem
6.3.3] applied to the quotient stack $[H^{N}/H^{0}]$ that $H^{N}\sslash H^{0}$
is of finite type over $S$. (If $S=\operatorname{Spec}R$, where $R$ is of
finite type over a universally Japanese ring then the assertion also follows
from [52, Theorem 2 (i)]. This setting is sufficient for our paper as we only
need it over $\mathcal{O}$.) Hence, $H^{N}\sslash H^{0}\rightarrow
H^{N}\sslash H$ is of finite type, and [54, Tag 01WJ] implies that
$H^{N}\sslash H^{0}\rightarrow H^{N}\sslash H$ is finite.

Since the composition of finite morphisms is finite, using Theorem 6.2 we
obtain that $H^{N}\sslash H^{0}\rightarrow G^{N}\sslash G$ is finite. Since
this map factors as $H^{N}\sslash H^{0}\rightarrow G^{N}\sslash
G^{0}\rightarrow G^{N}\sslash G$ and all the schemes are affine we obtain the
claim. ∎

######  Proposition 6.4.

The maps $X^{\mathrm{git},\tau}_{G}\to
X^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ and
$\overline{X}^{\mathrm{git},\tau}_{G}\rightarrow\overline{X}^{\mathrm{ps}}_{\operatorname{GL}_{d}}$
are finite.

###### Proof.

We consider the following diagram:

${X^{\mathrm{git},\tau}_{G}}$${(X^{\mathrm{ps}}_{\operatorname{GL}_{d}}\times
G^{N})\sslash
G^{0}}$${X^{\mathrm{ps}}_{\operatorname{GL}_{d}}}$${(X^{\mathrm{ps}}_{\operatorname{GL}_{d}}\times\operatorname{GL}_{d}^{N})\sslash\operatorname{GL}_{d}}$

By 6.3 the right map is finite. Since the map $X^{\mathrm{gen},\tau}_{G}\to
X^{\mathrm{ps}}_{\operatorname{GL}_{d}}\times G^{N}$ defined in 5.7 is a
closed immersion, it follows from [52, Theorem 2 (ii)], that the induced map
of GIT quotients, which is the top map in the diagram, is finite. It follows,
that the left map in the diagram is finite. The same argument over $k$ proves
the last assertion. ∎

######  Corollary 6.5.

Let $H$ be a closed generalised reductive subgroup scheme of $G$ (resp.
$G_{k}$). Then the map $X^{\mathrm{git},\tau}_{H}\rightarrow
X^{\mathrm{git},\tau}_{G}$ (resp.
$\overline{X}^{\mathrm{git},\tau}_{H}\rightarrow\overline{X}^{\mathrm{git},\tau}_{G}$)
is finite.

###### Proof.

It follows from Proposition 6.4 that the map
$X^{\mathrm{git},\tau}_{H}\rightarrow X^{\mathrm{ps}}_{\operatorname{GL}_{d}}$
is finite. Moreover, it factors as $X^{\mathrm{git},\tau}_{H}\rightarrow
X^{\mathrm{git},\tau}_{G}\rightarrow X^{\mathrm{ps}}_{\operatorname{GL}_{d}}$.
Since these are affine schemes we deduce that the map
$X^{\mathrm{git},\tau}_{H}\rightarrow X^{\mathrm{git},\tau}_{G}$ is finite.
The same argument over $k$ gives finiteness of
$\overline{X}^{\mathrm{git},\tau}_{H}\rightarrow\overline{X}^{\mathrm{git},\tau}_{G}$.
∎

######  Proposition 6.6 ([22, Lemma 4.3]).

Let $\kappa$ be a local $\mathcal{O}$-field and let
$\rho:\Gamma_{F}\rightarrow G(\kappa)$ be a continuous representation for the
topology on the target induced by the valuation on $\kappa$. Then there exists
a finite extension $\kappa^{\prime}$ of $\kappa$ and $g\in
G^{0}(\kappa^{\prime})$ such that $g\rho(\gamma)g^{-1}\in
G(\mathcal{O}_{\kappa^{\prime}})$ for all $\gamma\in\Gamma_{F}$.

###  6.1. Points in $X^{\mathrm{gen},\tau}_{G}$

We recall some terminology following [40]. Let $\kappa$ be an algebraically
closed field, which is an $\mathcal{O}$-algebra. Let $H$ be a closed reduced
subgroup of $G_{\kappa}$. We say that $H$ is _$G$ -irreducible_ in
$G_{\kappa}$ if it is not contained in any proper R-parabolic subgroup of
$G_{\kappa}$. Let $S$ be a maximal torus in the centraliser $Z_{G}(H)$. We say
that $H$ is _strongly reductive_ in $G$ if $H$ is $Z_{G}(S)$-irreducible in
$Z_{G}(S)$.

######  Proposition 6.7.

Let $x$ be a geometric point of $X^{\mathrm{gen},\tau}_{G}$, and let $H$ be
the Zariski closure of $\rho_{x}(\Gamma_{F})$ in $G(\kappa(x))$. Then the
following are equivalent

  1. (1)

the orbit $G^{0}\cdot x$ is closed;

  2. (2)

the orbit $G\cdot x$ is closed;

  3. (3)

$H$ is strongly reductive in $G_{\kappa(x)}$;

  4. (4)

$\rho_{x}:\Gamma_{F}\rightarrow G(\kappa(x))$ is $G$-completely reducible.

###### Proof.

Since $G^{0}$ is of finite index in $G$ we may pick $g_{1},\ldots,g_{n}\in
G(\kappa(x))$ such that

(30) $G\cdot
x=\bigcup_{i=1}^{n}G^{0}\cdot(g_{i}x)=\bigcup_{i=1}^{n}g_{i}\cdot(G^{0}\cdot
x).$

It follows from (30) that (1) implies (2). If $G\cdot x$ is closed, then as it
is $G^{0}$-invariant it will contain a closed $G^{0}$-orbit. We deduce from
(30) that $G^{0}\cdot x$ is closed, so that (2) implies (1). We have
established in Lemma 5.13 that $H$ is equal to the Zariski closure in
$G(\kappa(x))$ of the subgroup generated by the $N$-tuple
$(\rho_{x}(\gamma_{1}),\ldots,\rho_{x}(\gamma_{N}))$. The equivalence of parts
(2), (3) has been established in [46, Theorem 16.4]. The equivalence of (3)
and (4) follows from [5, Theorem 3.1], see the begining of [5, Section 6.3]
for the non-connected case. ∎

######  Lemma 6.8.

Let $y$ be a geometric point in $X^{\mathrm{git},\tau}_{G}$. Then the fibre
$(X^{\mathrm{gen},\tau}_{G})_{y}$ contains a unique closed $G^{0}$-orbit.

###### Proof.

This follows from [52, Theorem 3]. ∎

######  Proposition 6.9.

Let $x$ be a geometric point of $X^{\mathrm{gen},\tau}_{G}$. Then there is
$z\in\overline{G^{0}\cdot x}$ such that the $G^{0}$-orbit of $\rho_{z}$ is
equal to the $G$-semisimplification of $\rho_{x}$. Moreover, $G^{0}\cdot z$ is
the unique closed $G^{0}$-orbit contained in $\overline{G^{0}\cdot x}$.

###### Proof.

Let $P_{\lambda}$ be a minimal R-parabolic containing $\rho_{x}(\Gamma_{F})$.
Then

$\lim_{t\rightarrow
0}\lambda(t)\cdot(\rho_{x}(\gamma_{1}),\ldots,\rho_{x}(\gamma_{N}))$

exists and defines a point $z\in G^{N}(\kappa(x))$. Thus $z$ is contained in
the closure of $G^{0}\cdot x$ in $G^{N}(\kappa(x))$. Since
$X^{\mathrm{gen},\tau}_{G}$ is closed in
$X^{\mathrm{ps}}_{\operatorname{GL}_{d}}\times G^{N}$ by 5.8, we deduce that
$z\in X^{\mathrm{gen},\tau}_{G}(\kappa(x))$. Lemma 5.13 implies that the
$G^{0}$-orbit of $\rho_{z}$ is the $G$-semisimplification of $\rho_{x}$, thus
$\rho_{z}$ is $G$-completely reducible by Proposition 2.23. Thus the orbit
$G\cdot z$ is closed by Proposition 6.7. Let $y$ be the image of $x$ in
$X^{\mathrm{git},\tau}_{G}$. The fibre $(X^{\mathrm{gen},\tau}_{G})_{y}$ is a
closed $G^{0}$-invariant subscheme $X^{\mathrm{gen},\tau}_{G}$ and thus it
will contain $\overline{G^{0}\cdot x}$. The last assertion follows from Lemma
6.8. ∎

######  Corollary 6.10.

Let $y$ be a geometric point of $X^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ and
let $x,x^{\prime}\in X^{\mathrm{gen},\tau}_{G}(\kappa(y))$. Then $x$ and
$x^{\prime}$ map to the same point in $X^{\mathrm{git},\tau}_{G}$ if and only
if $\rho_{x}$ and $\rho_{x^{\prime}}$ have the same $G$-semisimplification.

###### Proof.

By [52, Theorem 3 (ii)] $x$ and $x^{\prime}$ map to the same point in
$X^{\mathrm{git},\tau}_{G}$ if and only if the closures of their
$G^{0}$-orbits intersect non-trivially. If $\overline{G^{0}\cdot
x}\cap\overline{G^{0}\cdot x^{\prime}}\neq\emptyset$ then it contains a point
$z$ such that $G^{0}\cdot z$ is closed. It follows from Proposition 6.9 that
the $G$-semisimplification of $\rho_{x}$ and of $\rho_{x^{\prime}}$ equal to
the $G^{0}$-orbit of $\rho_{z}$. The other implication follows from directly
from Proposition 6.9. ∎

######  Corollary 6.11.

Let $y$ be a geometric point of $X^{\mathrm{git},\tau}_{G}$. Assume that $G$
is split and let $T$ be a maximal split torus of $G$ defined over
$\mathcal{O}$. Then there is an R-Levi $L$ of $G$ defined over $\mathcal{O}$,
which contains $T$, and $x\in X^{\mathrm{gen},\tau}_{G}(\kappa(y))$ such that
the following hold:

  1. (1)

$x$ maps to $y$;

  2. (2)

$\rho_{x}(\Gamma_{F})\subseteq L(\kappa(y))$;

  3. (3)

$\rho_{x}$ is $L_{\kappa(y)}$-irreducible.

Moreover, if $x^{\prime}\in X^{\mathrm{gen},\tau}_{G}(\kappa(y))$ maps to $y$
and $Q$ is a minimal R-parabolic of $G_{\kappa(y)}$ containing
$\rho_{x^{\prime}}(\Gamma_{F})$ with an R-Levi $M$ then there is $g\in
G^{0}(\kappa(y))$ such that $M$ and $L_{\kappa(y)}=gMg^{-1}$ and
$\rho_{x}(\gamma)=gc_{Q,M}(\rho_{x^{\prime}}(\gamma))g^{-1}$ for all
$\gamma\in\Gamma_{F}$.

###### Proof.

Lemma 6.8 implies that there exists $x\in
X^{\mathrm{gen},\tau}_{G}(\kappa(y))$ such that $G^{0}\cdot x$ is a closed.
The representation $\rho_{x}$ is $G$-completely reducible by Proposition 6.7.
Let $P$ be a minimal R-parabolic of $G_{\kappa(x)}$ containing
$\rho_{x}(\Gamma_{F})$. Since $\rho_{x}$ is $G$-completely reducible the image
of $\rho_{x}(\Gamma_{F})$ is contained in an R-Levi $L$ of $P$. Lemma 2.19
implies that after replacing $L$ and $P$ by a conjugate by $g\in
G^{0}(\kappa(y))$ we may assume that $L$ and $P$ are defined over
$\mathcal{O}$ and $L$ contains $T$.

Let $Q$ be a minimal R-parabolic containing $\rho_{x}$ and let $M$ be an
R-Levi of $Q$. Since $x$ and $x^{\prime}$ both map to $y$, Corollary 6.10
implies that $\rho_{x}$ and $\rho_{x^{\prime}}$ have the same
$G$-semisimplification. It follows from Proposition 2.23 that $\rho_{x}$ is
conjugate to $\rho^{\prime}:=c_{Q,M}\circ\rho_{x^{\prime}}$ by $g\in
G^{0}(\kappa(y))$. After replacing $(Q,M)$ by a conjugate, we may assume that
$\rho_{x}=\rho^{\prime}$, so that $\rho_{x}(\Gamma_{F})\subseteq M$, and $Q$
is minimal with respect to the property containing $\rho_{x}(\Gamma_{F})$. Let
$H$ be the Zariski closure of $\rho_{x}(\Gamma_{F})$ in $G(\kappa(y))$.
Minimality of $P$ and $Q$ implies that $H$ is both $L$-irreducible and
$M$-irreducible. Proposition 2.27 implies that $L$ and $M$ are conjugate by an
element $h\in Z_{G}(H)^{0}(\kappa(y))\subseteq G^{0}(\kappa(y))$. Since the
image of $\rho^{\prime}$ is contained in $H$, we have
$h\rho^{\prime}(\gamma)h^{-1}=\rho^{\prime}(\gamma)=\rho(\gamma)$ for all
$\gamma\in\Gamma_{F}$. ∎

###  6.2. Definition of
$X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$

6.4 implies that the map
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\rightarrow(A^{\mathrm{gen},\tau}_{G})^{G^{0}}$
is finite. Since $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ is a complete local
ring with residue field $k$, the Chinese remainder theorem implies that

(31) $R^{\mathrm{git},\tau}_{G}:=(A^{\mathrm{gen},\tau}_{G})^{G^{0}}\cong
R_{1}\times\ldots\times R_{n},$

where each $R_{j}$ is a complete local noetherian
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra with residue field a finite
extension of $k$. The product can be indexed by the set of
$\operatorname{Gal}(\overline{k}/k)$-orbits in
$X^{\mathrm{git},\tau}_{G}(\overline{k})$. From (31) we obtain a
$G^{0}$-equivariant decomposition

(32)
$X^{\mathrm{gen},\tau}_{G}\cong\coprod_{j}X^{\mathrm{gen},\tau}_{G}\times_{X^{\mathrm{git},\tau}_{G}}\operatorname{Spec}R_{j}.$

Moreover, we have

(33)
$(X^{\mathrm{gen},\tau}_{G}\times_{X^{\mathrm{git},\tau}_{G}}\operatorname{Spec}R_{j})\sslash
G^{0}\cong\operatorname{Spec}R_{j}.$

######  Lemma 6.12.

Each
$X^{\mathrm{gen},\tau}_{G}\times_{X^{\mathrm{git},\tau}_{G}}\operatorname{Spec}R_{j}$
is connected. In particular, (32) is the decomposition of
$X^{\mathrm{gen},\tau}_{G}$ into connected components.

###### Proof.

Since a connected component is a union of irreducible components, [8, Lemma
2.1] implies that the connected components of
$X^{\mathrm{gen},\tau}_{G}\times_{X^{\mathrm{git},\tau}_{G}}\operatorname{Spec}R_{j}$
are $G^{0}$-invariant. It follows from [52, Theorem 3 (iii)] that if there
were more than one then their images in $\operatorname{Spec}R_{j}$ under (33)
would disconnect it. This is not possible as $R_{j}$ is a local ring. ∎

The representation $\overline{\rho}:\Gamma_{F}\rightarrow G(k)$ that we have
fixed in Section 5 gives us a point $x_{0}\in X^{\mathrm{gen},\tau}_{G}(k)$.
We denote the unique connected component of $X^{\mathrm{gen},\tau}_{G}$ on
which $x_{0}$ lies on by
$X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$, where
$\overline{\rho}^{\mathrm{ss}}$ is the $G$-semisimplification of the
representation
$\Gamma_{F}\overset{\overline{\rho}}{\longrightarrow}G(k)\hookrightarrow
G(\overline{k})$. The justification for this notation is given in the lemma
below. Let

$R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}:=R_{j_{0}},\quad
X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}:=\operatorname{Spec}R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}},$

where $j_{0}$ is the index corresponding to the image of $x_{0}$ in
$X^{\mathrm{git},\tau}_{G}(k)\subseteq
X^{\mathrm{git},\tau}_{G}(\overline{k})$. Since $x_{0}$ is $k$-rational, the
residue field of $R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ is
equal to $k$.

######  Lemma 6.13.

Let $x$ be a geometric point of $X^{\mathrm{gen},\tau}_{G}$ above the closed
point of $X^{\mathrm{ps}}_{\operatorname{GL}_{d}}$. Then $x\in
X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ if and only if the
$G$-semisimplification of $\rho_{x}$ is equal to
$\overline{\rho}^{\mathrm{ss}}$.

###### Proof.

Let $z\in X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}(\kappa(x))$
be a point above $x_{0}$. It follows from Lemma 6.12 that $x$ and $z$ lie on
the same connected component of $X^{\mathrm{gen},\tau}_{G}$ if and only if
their images in $X^{\mathrm{git},\tau}_{G}$ lie on the same connected
component of $X^{\mathrm{git},\tau}_{G}$. Since $z\in
X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ and the residue field
of $R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ is $k$,
$X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}(\kappa(x))$ is a
point, and thus this equivalent to $z$ and $x$ mapping to the same point in
$X^{\mathrm{git},\tau}_{G}$. Corollary 6.10 implies that this is equivalent to
$\rho_{x}$ and $\rho_{z}$ having the same $G$-semisimplification. ∎

######  Remark 6.14.

After replacing $L$ by an unramified extension we may assume that all the
factors in (31) have residue field $k$. It follows from Lemma 6.13 that in
this case the connected components of $X^{\mathrm{gen},\tau}_{G}$ are indexed
by $G^{0}(\overline{k})$-conjugacy classes of continuous $G$-completely
reducible representations $\overline{\rho}^{\mathrm{ss}}:\Gamma_{F}\rightarrow
G(\overline{k})$ such that
$\det\circ(\tau\circ\overline{\rho}^{\mathrm{ss}})^{\mathrm{lin}}=\overline{D}$.
Lemma 6.12 implies that after replacing $L$ by a finite unramified extension
we may assume that every connected component of $X^{\mathrm{gen},\tau}_{G}$
has a $k$-rational point.

##  7\. Lafforgue’s $G$-pseudocharacters

In his work on the local Langlands correspondence for function fields [36]
Vincent Lafforgue introduced a notion of $G$-_pseudocharacter_ for a reductive
group $G$. If $G=\operatorname{GL}_{d}$ then Emerson and Morel [25] have shown
that $\operatorname{GL}_{d}$-pseudocharacter after Lafforgue is equivalent to
a determinant law after Chenevier [16]. If $d!$ is invertible in the base ring
then the trace of representation determines its
$\operatorname{GL}_{d}$-pseudocharacter. In the arithmetic setting this was
first used by Wiles [58] and Taylor [55].

###  7.1. Definition of $G$-pseudocharacter

The definition of $G$-pseudocharacter we use is a slight modification of
Lafforgue’s original definition [36, Section 11], in that we work over the
base ring $\mathcal{O}$ and allow arbitrary disconnected generalised reductive
group schemes for $G$.

######  Definition 7.1.

Let $\Gamma$ be an abstract group and let $A$ be a commutative
$\mathcal{O}$-algebra. A _$G$ -pseudocharacter_ $\Theta$ of $\Gamma$ over $A$
is a sequence $(\Theta_{n})_{n\geq 1}$ of $\mathcal{O}$-algebra maps

$\Theta_{n}:\mathcal{O}[G^{n}]^{G^{0}}\to\mathrm{Map}(\Gamma^{n},A)$

for $n\geq 1$, satisfying the following conditions444Here $G$ acts on $G^{n}$
by $g\cdot(g_{1},\dots,g_{n})=(gg_{1}g^{-1},\dots,gg_{n}g^{-1})$. This induces
a rational action of $G$ on the affine coordinate ring $\mathcal{O}[G^{n}]$ of
$G^{n}$. The submodule $\mathcal{O}[G^{n}]^{G^{0}}\subseteq\mathcal{O}[G^{n}]$
is defined as the rational invariant module of the $G^{0}$-representation
$\mathcal{O}[G^{n}]$. It is an $\mathcal{O}$-subalgebra, since $G$ acts by
$\mathcal{O}$-linear automorphisms.:

  1. (1)

For each $n,m\geq 1$, each map $\zeta:\\{1,\dots,m\\}\to\\{1,\dots,n\\}$,
$f\in\mathcal{O}[G^{m}]^{G^{0}}$ and $\gamma_{1},\dots,\gamma_{n}\in\Gamma$,
we have

$\Theta_{n}(f^{\zeta})(\gamma_{1},\dots,\gamma_{n})=\Theta_{m}(f)(\gamma_{\zeta(1)},\dots,\gamma_{\zeta(m)})$

where $f^{\zeta}(g_{1},\dots,g_{n})=f(g_{\zeta(1)},\dots,g_{\zeta(m)})$.

  2. (2)

For each $n\geq 1$, for each $\gamma_{1},\dots,\gamma_{n+1}\in\Gamma$ and each
$f\in\mathcal{O}[G^{n}]^{G^{0}}$, we have

$\Theta_{n+1}(\hat{f})(\gamma_{1},\dots,\gamma_{n+1})=\Theta_{n}(f)(\gamma_{1},\dots,\gamma_{n}\gamma_{n+1})$

where $\hat{f}(g_{1},\dots,g_{n+1})=f(g_{1},\dots,g_{n}g_{n+1})$.

We denote the set of $G$-pseudocharacters of $\Gamma$ over $A$ by
$\mathrm{PC}_{G}^{\Gamma}(A)$. If $f:A\to B$ is a homomorphism of
$\mathcal{O}$-algebras, then there is an induced map
$f_{*}:\mathrm{PC}_{G}^{\Gamma}(A)\to\mathrm{PC}_{G}^{\Gamma}(B)$. For
$\Theta\in\mathrm{PC}_{G}^{\Gamma}(A)$, the image $f_{*}(\Theta)$ is called
the _specialisation_ of $\Theta$ along $f$ and is denoted by
$\Theta\otimes_{A}B$. When $\varphi:G\to H$ is a homomorphism of generalised
reductive $\mathcal{O}$-group schemes, the induced maps
$\varphi^{*}_{n}:\mathcal{O}[H^{n}]^{H^{0}}\to\mathcal{O}[G^{n}]^{G^{0}}$ give
rise to an $H$-pseudocharacter $(\Theta_{n}\circ\varphi^{*}_{n})_{n\geq 1}$.
By analogy with the notation for representations we denote this
$H$-pseudocharacter by $\varphi\circ\Theta$. So we also have an induced map
$\mathrm{PC}_{G}^{\Gamma}(A)\to\mathrm{PC}_{H}^{\Gamma}(A)$. It is easy to
verify that specialisation along $f:A\to B$ commutes with composition with
$\varphi$, i.e.
$(\varphi\circ\Theta)\otimes_{A}B=\varphi\circ(\Theta\otimes_{A}B)$.

When $\Gamma$ is a topological group, $A$ is a topological ring and for all
$n\geq 1$, the map $\Theta_{n}$ has image in the set
$\mathcal{C}(\Gamma^{n},A)$ of continuous maps $\Gamma^{n}\to A$, we say that
$\Theta$ is _continuous_. We denote the set of continuous $G$-pseudocharacters
of $\Gamma$ with values in $A$ by $\mathrm{cPC}_{G}^{\Gamma}(A)$.

When $\mathcal{O}^{\prime}$ is a commutative $\mathcal{O}$-algebra the natural
maps $\mathcal{O}[G^{n}]^{G^{0}}\to\mathcal{O}^{\prime}[G^{n}]^{G^{0}}$ induce
a map
$\mathrm{PC}_{G_{\mathcal{O}^{\prime}}}^{\Gamma}(A)\to\mathrm{PC}_{G}^{\Gamma}(A)$.
This map is always a bijection. When $\mathcal{O}^{\prime}$ is flat over
$\mathcal{O}$ the map
$\mathcal{O}[G^{n}]^{G^{0}}\otimes_{\mathcal{O}}\mathcal{O}^{\prime}\to\mathcal{O}^{\prime}[G^{n}]^{G^{0}}$
is an isomorphism by the universal coefficient theorem in rational cohomology.
The case of a general $\mathcal{O}$-algebra is treated in [44, Proposition
3.22]. We will use this fact whenever we need to extend the base ring
$\mathcal{O}$.

###  7.2. Deformations of $G$-pseudocharacters

We adopt the deformation theoretic setup of subsection 5.2. Let $\kappa$ be
either a finite or a local field, which is an $\mathcal{O}$-algebra, equipped
with its natural topology. Let $\Lambda$ be the coefficient ring for $\kappa$
and $\mathfrak{A}_{\Lambda}$ is the category of local artinian
$\Lambda$-algebras with residue field $\kappa$, equipped with the natural
topology.

In this section $G$ is a generalised reductive group over $\mathcal{O}$. We
consider the following deformation problem for $G$-pseudocharacters.

######  Definition 7.2.

Let $\overline{\Theta}\in\mathrm{cPC}^{\Gamma}_{G}(\kappa)$ be a continuous
$G$-pseudocharacter of $\Gamma$ with values in $\kappa$. We define the
_deformation functor_

$D_{\overline{\Theta}}:\mathfrak{A}_{\Lambda}\to\operatorname{Set},\quad
A\mapsto\\{\Theta\in\mathrm{cPC}_{G}^{\Gamma}(A)\mid\Theta\otimes_{A}\kappa=\overline{\Theta}\\}$

that sends an object $A\in\mathfrak{A}_{\Lambda}$ to the set of continuous
$G$-pseudocharacters $\Theta$ of $\Gamma$ over $A$ with
$\Theta\otimes_{A}\kappa=\overline{\Theta}$.

The deformation problem for $G$-pseudocharacters admits a universal
deformation ring, which we will denote by $R^{\mathrm{ps}}_{G}$ when
$\overline{\Theta}=\Theta_{\overline{\rho}}$. We will denote the universal
$G$-pseudocharacter by $\Theta^{u}_{G}$ and will drop the index $G$, when it
is clear which group scheme we are working with.

###  7.3. Comparison of
$X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ with
$X^{\mathrm{ps}}_{G}$

In this section we assume that $\Gamma$ is a profinite topologically finitely
generated group. It has been shown by the second author in [44, Theorem A]
that $R^{\mathrm{ps}}_{G}$ is a complete noetherian local $\Lambda$-algebra.

Let $A$ be a commutative $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra. We
have a natural transformation

$\vartheta_{A}:X^{\mathrm{gen},\tau}_{G}(A)\to\mathrm{PC}_{G}^{\Gamma}(A),\quad\rho\mapsto\Theta_{\rho},$

that attaches to a representation $\rho$ its $G$-pseudocharacter
$\Theta_{\rho}$. By [44, Theorem 3.20] $\mathrm{PC}_{G}^{\Gamma}$ is
representable by an affine scheme, so
$\vartheta:X^{\mathrm{gen},\tau}_{G}\to\mathrm{PC}_{G}^{\Gamma}$ factors over
$\theta:X^{\mathrm{git},\tau}_{G}\to\mathrm{PC}_{G}^{\Gamma}$.

######  Lemma 7.3.

Let $\rho:\Gamma\rightarrow
G(A^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}})$ be the universal
representation. Then
$\Theta_{\rho}\in\mathrm{cPC}_{G}^{\Gamma}(R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}})$
for the $\mathfrak{m}$-adic topology on
$R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$.

###### Proof.

5.10 implies that $\rho$ is continuous for the
$\mathfrak{m}_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}$-adic topology on
$A^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$. The
$G$-pseudocharacter $\Theta_{\rho}$ takes values in
$R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ and is thus
continuous for the subspace topology on
$R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$. Since

$\mathfrak{m}_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}^{n}R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}\subseteq\mathfrak{m}_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}^{n}A^{\mathrm{gen},\tau}_{G}\cap
R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}},\quad\forall n\geq 1,$

$\Theta_{\rho}$ is continuous for the
$\mathfrak{m}_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}$-adic topology on
$R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$. Since by 6.4,
$R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ is a finite
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra, this topology coincides
with the topology defined by the maximal ideal of
$R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$. ∎

Let $\Theta_{\rho}$ be the $G$-pseudocharacter of the universal representation
over $A^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ as in Lemma
7.3. Recall that $R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ has
residue field $k$ and
$\Theta_{\rho}\otimes_{R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}}k=\Theta_{\overline{\rho}}$.
It follows from Lemma 7.3, that $\Theta_{\rho}$ is indeed a continuous
deformation of $\Theta_{\overline{\rho}^{\mathrm{ss}}}$, so we have a local
homomorphism $R^{\mathrm{ps}}_{G}\to
R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ which induces a map

(34) $\nu:X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}\to
X^{\mathrm{ps}}_{G}$

of $\mathcal{O}$-schemes.

######  Proposition 7.4.

$\nu:X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}\to
X^{\mathrm{ps}}_{G}$ is a finite universal homeomorphism.

###### Proof.

By [54, Tag 04DF], it suffices to show, that $\nu$ is finite, universally
injective and surjective. By 6.4
$R_{G,\overline{\rho}^{\mathrm{ss}}}^{\mathrm{git},\tau}$ is a finite
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-module. From the embedding
$\tau:G\to\operatorname{GL}_{d}$ we obtain a factorisation
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\to R^{\mathrm{ps}}_{G}\to
R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$. In particular,
$R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ is a finite
$R^{\mathrm{ps}}_{G}$-module.

The closed point of $X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$
is mapped to the closed point of $X^{\mathrm{ps}}_{G}$, so we only need to
show surjectivity of
$\dot{\nu}:U^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}:=X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}\setminus\\{*\\}\to
U^{\mathrm{ps}}_{G}:=X^{\mathrm{ps}}_{G}\setminus\\{*\\}$. By [54, Tag 02J6]
$U^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ and
$U^{\mathrm{ps}}_{G}$ are Jacobson. By Chevalley’s theorem [54, Tag 054K] the
image of $\dot{\nu}$ is locally constructible and thus also its complement.
Therefore for surjectivity of $\nu$ it is sufficient to show, that the closed
points of $U^{\mathrm{ps}}_{G}$ are all contained in the image of $\dot{\nu}$.

Let $y\in U^{\mathrm{ps}}_{G}$ be a closed point. By [8, Lemma 3.17]
$\kappa(y)$ is a local field. By [44, Theorem 5.7], $R^{\mathrm{ps}}_{G}$ is
noetherian, and thus $\kappa(y)$ with its natural topology is a topological
$R^{\mathrm{ps}}_{G}$-algebra and the image of
$y:R^{\mathrm{ps}}_{G}\rightarrow\kappa(y)$ is contained in the ring of
integers $\mathcal{O}_{\kappa(y)}$ by [8, Lemma 3.17]. We obtain a continuous
$G$-pseudocharacter
$\Theta_{y}\in\mathrm{cPC}_{G}^{\Gamma}(\mathcal{O}_{\kappa(y)})$, which is a
deformation of $\Theta_{\overline{\rho}}$ to $\mathcal{O}_{\kappa(y)}$. By the
continuous reconstruction theorem [44, Theorem 3.8], there is a continuous
representation $\rho^{\prime}:\Gamma\to G(\overline{\kappa})$ with
$\Theta_{\rho^{\prime}}=\Theta_{y}$, where $\overline{\kappa}$ is an algebraic
closure of $\kappa(y)$. Lemma 5.12 implies that $\rho^{\prime}\in
X^{\mathrm{gen},\tau}_{G}(\overline{\kappa})$. The
$G(\overline{\kappa})$-conjugacy class of $\rho^{\prime}$ defines a point
$y^{\prime}\in X^{\mathrm{git},\tau}_{G}(\overline{\kappa})$ which maps to
$y$. We have to show that $y^{\prime}$ lies on the same connected component of
$X^{\mathrm{git},\tau}_{G}$ as $\overline{\rho}^{\mathrm{ss}}$. Since
$X^{\mathrm{gen},\tau}_{G}$ is of finite type over
$X^{\mathrm{ps}}_{\operatorname{GL}_{d}}$, $\rho^{\prime}$ takes values in
$G(\kappa^{\prime})$ for some finite extension of $\kappa(y)$ in
$\overline{\kappa}$. Proposition 6.6 implies that after conjugating
$\rho^{\prime}$ with an element of $G^{0}(\overline{\kappa})$ we may assume
that $\rho^{\prime}$ takes values in $G(\mathcal{O}_{\overline{\kappa}})$. Let
$\overline{\rho}^{\prime}$ be the composition of $\rho^{\prime}$ with the
reduction map $G(\mathcal{O}_{\overline{\kappa}})\rightarrow G(\overline{k})$.
Then
$\Theta_{\overline{\rho}^{\prime}}=\Theta_{y}\otimes_{\mathcal{O}_{\kappa(y)}}\overline{k}=\Theta_{\overline{\rho}^{\mathrm{ss}}}$.
Uniqueness part of the reconstruction theorem implies that
$G$-semisimplification of $\overline{\rho}^{\prime}$ is equal to
$\overline{\rho}^{\mathrm{ss}}$, and Lemma 6.12 implies that $y^{\prime}\in
X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$.

For universal injectivity it suffices by [54, Tag 01S4] to show that the
diagonal morphism $X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}\to
X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}\times_{X^{\mathrm{ps}}_{G}}X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$
is surjective. Let $x\in
X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}\times_{X^{\mathrm{ps}}_{G}}X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$.
We obtain by 6.7 and 6.8 two $G$-semisimple representations
$\rho_{x},\rho_{x}^{\prime}\in
X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}(\overline{\kappa(x)})$,
such that their associated $G$-pseudocharacters coincide. This means by the
uniqueness part of the reconstruction theorem [44, Theorem 3.7] and 6.10,
their image in $X^{\mathrm{git}}_{G}(\overline{\kappa(x)})$ coincides and
therefore $x$ is in the image of the diagonal. ∎

######  Lemma 7.5.

$\nu\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}:X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}[1/p]\to
X^{\mathrm{ps}}_{G}[1/p]$ is a closed immersion.

###### Proof.

Let $\mathcal{F}_{N}$ be a free group generated by elements
$x_{1},\ldots,x_{N}$. Let $\operatorname{Rep}_{G}^{\mathcal{F}_{N},\square}$
be the affine $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-scheme, which
represents the functor, that maps a commutative
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra $A$ to the set of (abstract)
homomorphisms $\rho:\mathcal{F}_{N}\to G(A)$. Mapping $x_{i}$ to $\gamma_{i}$
induces a group homomorphism $\mathcal{F}_{N}\rightarrow\Gamma$ and it follows
from Remark 5.8, that we have a closed immersion
$X^{\mathrm{gen},\tau}_{G}\rightarrow\operatorname{Rep}_{G}^{\mathcal{F}_{N},\square}$
and hence a closed immersion
$X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}\rightarrow\operatorname{Rep}_{G}^{\mathcal{F}_{N},\square}$.

By [25, Proposition 2.11 (i)] the vertical map on the left hand side of the
following diagram is an isomorphism.

(35)
${R^{\mathrm{ps}}_{\operatorname{GL}_{d}}[1/p][\mathrm{PC}_{G}^{\mathcal{F}_{N}}]}$${R^{\mathrm{ps}}_{G}[1/p]}$${R^{\mathrm{ps}}_{\operatorname{GL}_{d}}[1/p][\operatorname{Rep}_{G}^{\mathcal{F}_{N},\square}]^{G^{0}}}$${R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}[1/p]}$$\scriptstyle{\cong}$$\scriptstyle{\nu^{*}}$

The bottom map is surjective, since
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}[1/p][\operatorname{Rep}_{G}^{\mathcal{F}_{N},\square}]\to
A^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}[1/p]$ is surjective as
explained above and taking $G^{0}$-invariants is exact in characteristic $0$.
Since the diagram commutes the right vertical arrow is surjective, and we
obtain the claim. ∎

######  Corollary 7.6.

The map $\nu\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$ induces an isomorphism of
underlying reduced schemes. In particular, if $X^{\mathrm{ps}}_{G}[1/p]$ is
reduced, then $\nu\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$ is an isomorphism.

###### Proof.

Proposition 7.4 implies that $\nu\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$ is a
universal homeomorphism and Lemma 7.5 that it is a closed immersion, which
implies the first claim. The second claim is then immediate. ∎

##  8\. Functoriality

The main result of this section is Proposition 8.3, which gives an intrinsic
description of $X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ as a
functor. As a consequence in Section 8.1 we show that
$X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ is independent of
the choice of an embedding $\tau:G\hookrightarrow\operatorname{GL}_{d}$ and is
functorial in $G$. In Section 8.2 we investigate functoriality in $G$ under
finite maps. In Section 8.3 we study the case when $G$ is a product of two
generalised reductive $\mathcal{O}$-groups. In Section 8.4 we study the
morphisms induced by restriction of pseudocharacters and representations to an
open subgroup of $\Gamma$.

We assume in this section that $\Gamma$ is topologically finitely generated.
This assumption ensures that $R^{\mathrm{ps}}_{G}$ is noetherian. If $\Gamma$
is only assumed to satisfy Mazur’s $p$-finiteness condition then results of
this section can be reformulated by replacing $\Gamma$ by a topologically
finitely generated quotient as in Lemma 5.5. We note that $\Gamma_{F}$ is
topologically finitely generated by [34, Satz 3.6].

###  8.1. Independence of $\tau$ and functoriality in $G$

######  Lemma 8.1.

Let $A$ be an $R^{\mathrm{ps}}_{G}$-algebra and let $\rho\in
X^{\mathrm{gen},\tau}_{G}(A)$. Then $\rho\in
X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}(A)$ if and only if the
$G$-pseudocharacter $\Theta_{\rho}$ is equal to the specialisation of
$\Theta^{u}$ at $A$.

###### Proof.

Let $\rho\in X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}(A)$. Then
$\rho=\rho^{u}\otimes_{A^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}}A$
where $\rho^{u}\in
X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}(A^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}})$
is the universal representation. Taking $G$-pseudocharacters, we obtain
$\Theta_{\rho}=\Theta_{\rho^{u}}\otimes_{A^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}}A$.
Using the map $X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}\to
X^{\mathrm{ps}}_{G}$ given by composing the map in (34) with the GIT quotient
map, we see that
$\Theta_{\rho^{u}}=\Theta^{u}\otimes_{R^{\mathrm{ps}}_{G}}A^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$,
so $\Theta_{\rho}=\Theta^{u}\otimes_{R^{\mathrm{ps}}_{G}}A$.

Conversely, let $\rho\in X^{\mathrm{gen},\tau}_{G}(A)$ satisfy
$\Theta_{\rho}=\Theta^{u}\otimes_{R^{\mathrm{ps}}_{G}}A$. By factoring $\rho$
as $\Gamma\to G(A^{\mathrm{gen},\tau}_{G})\to G(A)$ over the universal
representation on $A^{\mathrm{gen},\tau}_{G}$ and using that
$A^{\mathrm{gen},\tau}_{G}$ is of finite type over
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$, we see that $\rho$ actually takes
values in a finitely generated $R^{\mathrm{ps}}_{G}$-subalgebra
$A^{\prime}\subseteq A$. We want to show, that the image of
$\operatorname{Spec}(A^{\prime})\to X^{\mathrm{gen},\tau}_{G}$ is entirely
contained in the component
$X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$, which amounts to
showing that an idempotent $e\in A^{\mathrm{gen},\tau}_{G}$ associated to a
component $X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}_{2}}$
associated with a residual representation $\overline{\rho}^{\mathrm{ss}}_{2}$
which is non-conjugate to $\overline{\rho}^{\mathrm{ss}}$ acts as zero on
$A^{\prime}$. By contradiction, we assume that $eA^{\prime}$ is nonzero. As
$A^{\prime}$ is of finite type over $R^{\mathrm{ps}}_{G}$, there is a
dimension $\leq 1$ prime ideal
$\mathfrak{p}\in\operatorname{Spec}(eA^{\prime})$ with residue field
$\kappa(\mathfrak{p})$ finite or local by [8, Lemma 3.17]. By 5.12, we know
that the representation $\rho_{\mathfrak{p}}:\Gamma\to
G(\overline{\kappa(\mathfrak{p})})$ is continuous and that
$\Theta_{\rho_{\mathfrak{p}}}=\Theta^{u}\otimes_{R^{\mathrm{ps}}_{G}}\overline{\kappa(\mathfrak{p})}$.
If $\kappa(\mathfrak{p})$ is a finite field then we deduce that the
$G$-semisimplification of $\rho_{\mathfrak{p}}$ is
$\overline{\rho}^{\mathrm{ss}}$, which yields a contradiction to the choice of
$e$ via Lemma 6.13. If $\kappa(\mathfrak{p})$ is a local field, by 6.6 the
image of $\rho_{\mathfrak{p}}$ lies up to conjugation in the ring of integers
of a finite extension of $\kappa(\mathfrak{p})$. The $G$-semisimplification of
the reduction modulo the maximal ideal is $\overline{\rho}^{\mathrm{ss}}$,
since $\Theta_{\rho_{\mathfrak{p}}}$ specialises $\Theta^{u}$, which yields a
contradiction to the choice of $e$ via Lemma 6.13. We conclude, that $\rho\in
X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}(A^{\prime})\subseteq
X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}(A)$. ∎

######  Lemma 8.2.

Let $R\rightarrow A$ be a map of $R^{\mathrm{ps}}_{G}$-algebras such that $R$
is a noetherian profinite topological $R^{\mathrm{ps}}_{G}$-algebra. Let
$\rho:\Gamma\to G(A)$ be a representation satisfying
$\Theta_{\rho}=\Theta^{u}_{|A}$. Then the following are equivalent:

  1. (1)

$(\tau\circ\rho)^{\mathrm{lin}}|_{R^{\mathrm{ps}}_{G}[\Gamma]}$ factors
through
$R^{\mathrm{ps}}_{G}\otimes_{R^{\mathrm{ps}}_{\operatorname{GL}_{d}}}E^{u}$;

  2. (2)

$\rho$ is $R^{\mathrm{ps}}_{G}$-condensed;

  3. (3)

$\rho$ is $R$-condensed.

###### Proof.

It follows from Lemma 4.16 that $\rho$ is $R$-condensed if and only if
$\tau\circ\rho$ is $R$-condensed. As explained in section 7.3 the assumption
$\Theta_{\rho}=\Theta^{u}_{|A}$ implies that
$\det\circ(\tau\circ\rho)^{\mathrm{lin}}=D^{u}_{|A}$. The equivalence now
follows from Lemma 5.2. ∎

######  Proposition 8.3.

Let $A$ be an $R^{\mathrm{ps}}_{G}$-algebra. There is canonical bijection
between the following sets:

  1. (1)

$X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}(A)$;

  2. (2)

the set of representations $\rho:\Gamma\to G(A)$, such that the
$G$-pseudocharacter $\Theta_{\rho}$ is equal to the specialisation of
$\Theta^{u}$ at $A$ and the equivalent conditions of Lemma 8.2 hold.

###### Proof.

If $\rho\in X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}(A)$ then
$\Theta_{\rho}=\Theta^{u}_{|A}$ by Lemma 8.1. Moreover, $\tau\circ\rho$ is
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-condensed by Lemma 5.10 (1). Lemmas
4.16 and 4.5 implies that $\rho$ is $R^{\mathrm{ps}}_{G}$-condensed.

If $\rho:\Gamma\rightarrow G(A)$ satisfies the conditions of part (2) then
$\det\circ(\tau\circ\rho)^{\mathrm{lin}}=D^{u}_{|A}$ and $\tau\circ\rho$ is
$R^{\mathrm{ps}}_{G}$-condensed. Thus $\rho$ satisfies part (4) of Lemma 5.2
with $R=R^{\mathrm{ps}}_{G}$ and thus $\rho\in X^{\mathrm{gen},\tau}_{G}(A)$
by Lemma 5.10 (1). Since $\Theta_{\rho}=\Theta^{u}_{|A}$ Lemma 8.1 implies
that $\rho\in X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}(A)$. ∎

######  Corollary 8.4.

The $R^{\mathrm{ps}}_{G}$-algebra
$A^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ does not depend on
the chosen embedding $\tau:G\hookrightarrow\operatorname{GL}_{d}$.

###### Proof.

The assertion follows from Proposition 8.3, as the condition in part (2) of
Lemma 8.2 is independent of $\tau$. ∎

As a consequence of the Corollary we will drop $\tau$ from the notation and
write $A^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$,
$R^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}$,
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$,
$X^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}$ for
$A^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$,
$R^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$,
$X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ and
$X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$, respectively.

We now turn to functoriality.

######  Proposition 8.5.

Let $\varphi:G\to H$ be a homomorphism of generalised reductive
$\mathcal{O}$-group schemes. Then there is a natural map of
$R^{\mathrm{ps}}_{H}$-schemes
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}\to
X^{\mathrm{gen}}_{H,\varphi\circ\overline{\rho}^{\mathrm{ss}}}$, such that for
every $R^{\mathrm{ps}}_{G}$-algebra $A$, the map
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}(A)\to
X^{\mathrm{gen}}_{H,\varphi\circ\overline{\rho}^{\mathrm{ss}}}(A)$ is given by
$\rho\mapsto\varphi\circ\rho$.

###### Proof.

Let $\Theta^{u}_{G}$ be the universal $G$-pseudocharacter deforming
$\Theta_{\overline{\rho}^{\mathrm{ss}}}$ and let $\Theta^{u}_{H}$ be the
universal $H$-pseudocharacter deforming
$\Theta_{\varphi\circ\overline{\rho}^{\mathrm{ss}}}$. Since
$\varphi(\Theta_{G}^{u})$ is a deformation of
$\Theta_{\varphi\circ\overline{\rho}^{\mathrm{ss}}}$ to $R^{\mathrm{ps}}_{G}$,
we obtain a natural homomorphism of local $\mathcal{O}$-algebras
$R^{\mathrm{ps}}_{H}\rightarrow R^{\mathrm{ps}}_{G}$ such that
$\varphi(\Theta^{u}_{G})=\Theta^{u}_{H}\otimes_{R^{\mathrm{ps}}_{H}}R^{\mathrm{ps}}_{G}$.

If $\rho\in X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}(A)$ then
$\Theta_{\rho}=\Theta^{u}_{G}\otimes_{R^{\mathrm{ps}}_{G}}A$ and by applying
$\varphi$, we get

(36)
$\Theta_{\varphi\circ\rho}=\varphi(\Theta_{\rho})=\varphi(\Theta^{u}_{G})\otimes_{R^{\mathrm{ps}}_{G}}A=\Theta^{u}_{H}\otimes_{R^{\mathrm{ps}}_{H}}A.$

Lemma 4.16 implies that $\varphi\circ\rho$ is $R^{\mathrm{ps}}_{G}$-condensed.
Since $\varphi\circ\rho$ satisfies (36), Lemma 8.2 (3) applied with
$R=R^{\mathrm{ps}}_{G}$ and Proposition 8.3 imply that $\varphi\circ\rho\in
X^{\mathrm{gen}}_{H,\varphi\circ\overline{\rho}^{\mathrm{ss}}}(A)$. ∎

###  8.2. Finite maps

Let $\varphi:G\rightarrow H$ be a morphism of generalised reductive
$\mathcal{O}$-group schemes. By functoriality of
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ proved in Propositions
8.5 we obtain a morphism
$f:X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}\rightarrow
X^{\mathrm{gen}}_{H,\varphi\circ\overline{\rho}^{\mathrm{ss}}}$.

######  Proposition 8.6.

If $\varphi$ is finite then the map $X^{\mathrm{ps}}_{G}\rightarrow
X^{\mathrm{ps}}_{H}$ is also finite.

###### Proof.

By topological Nakayama’s lemma it is enough to show, that the fibre ring
$C:=k\otimes_{R^{\mathrm{ps}}_{H}}R^{\mathrm{ps}}_{G}$ is a finite-dimensional
$k$-vector space. Since $\Gamma$ is topologically finitely generated, we know
by [44, Theorem 5.7], that $R^{\mathrm{ps}}_{G}$ is noetherian, hence $C$ is a
complete local noetherian ring and we are left to show, that $C$ has Krull
dimension $0$.

If $C$ is not $0$-dimensional then there is a point
$x\in\operatorname{Spec}(C)$ of dimension $1$. The residue field $\kappa(x)$
is a local field of characteristic $p$ by [8, Lemma 3.17] and the
corresponding homomorphism $C\to\kappa(x)$ gives a $G$-pseudocharacter
$\Theta\in\mathrm{PC}_{G}^{\Gamma}(\kappa(x))$, such that
$\varphi\circ\Theta\in\mathrm{PC}_{H}^{\Gamma}(\kappa(x))$ arises by scalar
extension from the residual pseudocharacter attached to
$\varphi\circ\overline{\rho}$.

For all $m\geq 0$, the map
$(\varphi\circ\Theta)_{n}=\Theta_{n}\circ\varphi^{*}$ is the composition

$\mathcal{O}[H^{m}]^{H^{0}}\xrightarrow{\varphi^{*}}\mathcal{O}[G^{m}]^{G^{0}}\xrightarrow{\Theta_{n}}\mathrm{Map}(\Gamma^{m},\kappa(x))$

and has image in $\mathrm{Map}(\Gamma^{m},k)$. By 6.3 $\varphi^{*}$ is finite,
so $\Theta_{n}$ has image in $\mathrm{Map}(\Gamma^{m},\overline{k})$. This
implies that $\kappa(x)$ is algebraic over $k$ and contradicts the assumption
that $x$ is a dimension $1$ point. ∎

######  Proposition 8.7.

If $\varphi:G\rightarrow H$ is finite then
$f:X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}\rightarrow
X^{\mathrm{gen}}_{H,\varphi\circ\overline{\rho}^{\mathrm{ss}}}$ is also
finite.

###### Proof.

The induced map $X^{\mathrm{ps}}_{G}\rightarrow X^{\mathrm{ps}}_{H}$ is finite
by 8.6. Since $G\rightarrow H$ is finite by assumption the map
$G^{N}_{X^{\mathrm{ps}}_{G}}\rightarrow H^{N}_{X^{\mathrm{ps}}_{H}}$ is also
finite. We have a commutative diagram

(37)
${X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}}$${G^{N}_{X^{\mathrm{ps}}_{G}}}$${X^{\mathrm{gen}}_{H,\varphi\circ\overline{\rho}^{\mathrm{ss}}}}$${H^{N}_{X^{\mathrm{ps}}_{H}}.}$$\scriptstyle{f}$

with horizontal arrows closed immersions. Since closed immersions are finite
maps and the composition of finite maps are again finite, the diagram implies
the assertion. ∎

###  8.3. Products

In this subsection suppose that $G=G_{1}\times G_{2}$, where the fibre product
is taken over $\operatorname{Spec}\mathcal{O}$ and $G_{1}$ and $G_{2}$ are
generalised reductive group schemes over $\mathcal{O}$. Let
$p_{i}:G\rightarrow G_{i}$ denote the projection map onto the $i$-th
component. Then $\overline{\rho}^{\mathrm{ss}}$ is of the form
$\overline{\rho}^{\mathrm{ss}}_{1}\times\overline{\rho}^{\mathrm{ss}}_{2}$,
where $\overline{\rho}^{\mathrm{ss}}_{i}:\Gamma\rightarrow G_{i}(k)$ is
$G_{i}$-semisimple.

######  Lemma 8.8.

The map $\Theta\mapsto(p_{1}\circ\Theta,p_{2}\circ\Theta)$ induces a bijection
of functors
$\mathrm{PC}_{G}^{\Gamma}\overset{\sim}{\rightarrow}\mathrm{PC}_{G_{1}}^{\Gamma}\times\mathrm{PC}_{G_{2}}^{\Gamma}$
and
$\mathrm{cPC}_{G}^{\Gamma}\overset{\sim}{\rightarrow}\mathrm{cPC}_{G_{1}}^{\Gamma}\times\mathrm{cPC}_{G_{2}}^{\Gamma}$.

###### Proof.

For an integer $m\geq 0$, we have
$\mathcal{O}[G^{m}]^{G^{0}}=\mathcal{O}[G_{1}^{m}]^{G_{1}^{0}}\otimes_{\mathcal{O}}\mathcal{O}[G_{2}^{m}]^{G_{2}^{0}}$,
since the $\mathcal{O}[G_{i}^{m}]$ are flat $\mathcal{O}$-modules.

Recall that a $G$-valued pseudocharacter is a family of $\mathcal{O}$-algebra
homomorphisms
$\Theta_{m}:\mathcal{O}[G^{m}]^{G^{0}}\to\mathrm{Map}(\Gamma^{m},A)$
satisfying certain compatibility conditions. Restriction along the map
$\mathcal{O}[G_{i}^{m}]^{G_{i}^{0}}\to\mathcal{O}[G^{m}]^{G^{0}}$ defines
$G_{i}$-pseudocharacters $\Theta^{(i)}$ satisfying the same compatibility
conditions; this can be directly seen from the description of pseudocharacters
in terms of functors on the category of finitely generated free groups [44,
Proposition 3.19]. Conversely given
$\Theta^{(i)}\in\mathrm{PC}_{G_{i}}^{\Gamma}(A)$, we can define a
$G$-pseudocharacter $\Theta^{(1)}\otimes\Theta^{(2)}$ by defining
$(\Theta^{(1)}\otimes\Theta^{(2)})_{m}:\mathcal{O}[G^{m}]^{G^{0}}\to\mathrm{Map}(\Gamma^{m},A)$
as the tensor product
$\Theta^{(1)}_{m}\otimes\Theta^{(2)}_{m}:\mathcal{O}[G_{1}^{m}]^{G_{1}^{0}}\otimes_{\mathcal{O}}\mathcal{O}[G_{2}^{m}]^{G_{2}^{0}}\to\mathrm{Map}(\Gamma^{m},A)$.
This implies that the first map is bijective.

If $A$ is a topological $\mathcal{O}$-algebra then a map
$f:\Gamma^{m}\rightarrow G(A)$ is continuous if and only if $p_{1}\circ f$ and
$p_{2}\circ f$ are continuous. This implies the second assertion. ∎

######  Lemma 8.9.

$R^{\mathrm{ps}}_{G}\cong
R^{\mathrm{ps}}_{G_{1}}\operatorname{\widehat{\otimes}}_{\mathcal{O}}R^{\mathrm{ps}}_{G_{2}}$.

###### Proof.

Let $\overline{\Theta}_{i}$ be the $G_{i}$-pseudocharacter associated to
$\overline{\rho}^{\mathrm{ss}}_{i}$; $i=1,2$. The ring
$R^{\mathrm{ps}}_{G_{1}}\operatorname{\widehat{\otimes}}_{\mathcal{O}}R^{\mathrm{ps}}_{G_{2}}$
pro-represents the product functor $D_{\overline{\Theta}_{1}}\times
D_{\overline{\Theta}_{2}}:\mathfrak{A}_{\mathcal{O}}\to\operatorname{Set}$. If
$A\in\mathfrak{A}_{\mathcal{O}}$ then the bijection between
$\mathrm{cPC}_{G_{1}}^{\Gamma}(A)\times\mathrm{cPC}_{G_{2}}^{\Gamma}(A)$ and
$\mathrm{cPC}_{G}^{\Gamma}(A)$ established in 8.8 induces a bijection between
$D_{\overline{\Theta}_{1}}(A)\times D_{\overline{\Theta}_{2}}(A)$ and
$D_{\overline{\Theta}}(A)$, which implies the assertion. ∎

The map from the usual tensor product into the completed tensor product
induces a morphism of schemes $X^{\mathrm{ps}}_{G}\rightarrow
X^{\mathrm{ps}}_{G_{1}}\times X^{\mathrm{ps}}_{G_{2}}$ over
$\operatorname{Spec}\mathcal{O}$.

######  Proposition 8.10.

The natural maps $X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}\to
X^{\mathrm{gen}}_{G_{i},\overline{\rho}^{\mathrm{ss}}_{i}}$ constructed in 8.5
induce an isomorphism of schemes over $X^{\mathrm{ps}}_{G}$:

(38) $\displaystyle
X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}\xrightarrow{\sim}(X^{\mathrm{gen}}_{G_{1},\overline{\rho}^{\mathrm{ss}}_{1}}\times
X^{\mathrm{gen}}_{G_{2},\overline{\rho}^{\mathrm{ss}}_{2}})\times_{X^{\mathrm{ps}}_{G_{1}}\times
X^{\mathrm{ps}}_{G_{2}}}X^{\mathrm{ps}}_{G},$

where the fibre products without subscript are taken over
$\operatorname{Spec}\mathcal{O}$.

###### Proof.

Let $A$ be an $R^{\mathrm{ps}}_{G}$-algebra. By Proposition 8.3 and Lemma 8.2
(3) applied with $R=R^{\mathrm{ps}}_{G}$, the $A$-valued points of the right
hand side of (38) are pairs of $R^{\mathrm{ps}}_{G}$-condensed representations
$\rho_{1}:\Gamma\to G_{1}(A)$ and $\rho_{2}:\Gamma\to G_{2}(A)$, such that
$\Theta^{u}_{G_{i}}\otimes_{R^{\mathrm{ps}}_{G_{i}}}A=\Theta_{\rho_{i}}$ for
$i=1,2$. 4.4 implies, that the representation
$\rho:=\rho_{1}\times\rho_{2}:\Gamma\to G(A)$ is
$R^{\mathrm{ps}}_{G}$-condensed. It follows from Lemma 8.9 that the bijection

$\mathrm{PC}^{\Gamma}_{G}(R^{\mathrm{ps}}_{G})\xrightarrow{\sim}\mathrm{PC}^{\Gamma}_{G}(R^{\mathrm{ps}}_{G})\times\mathrm{PC}^{\Gamma}_{G}(R^{\mathrm{ps}}_{G})$

maps $\Theta^{u}_{G}$ to
$(\Theta^{u}_{G_{1}}\otimes_{R^{\mathrm{ps}}_{G_{1}}}R^{\mathrm{ps}}_{G},\Theta^{u}_{G_{2}}\otimes_{R^{\mathrm{ps}}_{G_{2}}}R^{\mathrm{ps}}_{G})$.
Hence, $(\Theta^{u}_{G})_{|A}\in\mathrm{PC}^{\Gamma}_{G}(A)$ is mapped to
$(\Theta_{\rho_{1}},\Theta_{\rho_{2}})\in\mathrm{PC}^{\Gamma}_{G}(A)\times\mathrm{PC}^{\Gamma}_{G}(A)$
and hence $(\Theta^{u}_{G})_{|A}=\Theta_{\rho}$ by Lemma 8.9. Proposition 8.3
implies that $\rho\in X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}(A)$
and hence (38) is surjective. Since $p_{1}\circ\rho$ and $p_{2}\circ\rho$
uniquely determine $\rho\in
X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}(A)$, (38) is also
injective. ∎

###  8.4. Functoriality in $\Gamma$

Let $\Gamma^{\prime}$ be an open subgroup of $\Gamma$. The restriction of
$G$-pseudocharacters of $\Gamma$ to $\Gamma^{\prime}$ defines a morphism
$X^{\mathrm{ps},\Gamma}_{G}\rightarrow X^{\mathrm{ps},\Gamma^{\prime}}_{G}$.
Similarly, 4.13 implies that the restriction of representations of $\Gamma$ to
$\Gamma^{\prime}$ induces a morphism
$X^{\mathrm{gen},\Gamma}_{G,\overline{\rho}^{\mathrm{ss}}}\rightarrow
X^{\mathrm{gen},\Gamma^{\prime}}_{G,\overline{\rho}^{\mathrm{ss}}}$.

######  Lemma 8.11.

The morphism $X^{\mathrm{ps},\Gamma}_{G}\rightarrow
X^{\mathrm{ps},\Gamma^{\prime}}_{G}$ is finite.

###### Proof.

If $G=\operatorname{GL}_{d}$ then this is proved in [8, Lemma A.3]. In the
general case, one could either use that
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\rightarrow R^{\mathrm{ps}}_{G}$ is
finite, which follows from Propositions 6.4 and 7.4, or observe that the proof
of [8, Lemma A.3] carries over to the more general setting. ∎

######  Lemma 8.12.

The morphism
$X^{\mathrm{gen},\Gamma}_{G,\overline{\rho}^{\mathrm{ss}}}\rightarrow
X^{\mathrm{gen},\Gamma^{\prime}}_{G,\overline{\rho}^{\mathrm{ss}}}$ is of
finite type.

###### Proof.

The assertion follows from Lemma 8.11 and the fact that
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ is of finite type over
$X^{\mathrm{ps}}_{G}$. ∎

##  9\. Absolutely irreducible locus

Let $G$ be a generalised reductive $\mathcal{O}$-group scheme and let
$\pi_{G}:G\rightarrow G/G^{0}$ be the quotient map. Naively, the absolutely
irreducible locus should be an open subscheme $V$ of
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ such that a geometric
point $x$ of $X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ lies in $V$
if and only if the image of $\rho_{x}$ is not contained in any proper
$R$-parabolic subgroup of $G_{\kappa(x)}$. It turns out that if
$\pi_{G}\circ\overline{\rho}^{\mathrm{ss}}:\Gamma\rightarrow(G/G^{0})(\overline{k})$
is not surjective then the naively defined absolutely irreducible locus might
be empty, see Example 9.14. To overcome this problem we first show that there
is a closed generalised reductive subgroup scheme $H$ of $G$, such that
$H^{0}=G^{0}$, $\overline{\rho}^{\mathrm{ss}}$ factors through $H$,
$\pi_{H}\circ\overline{\rho}^{\mathrm{ss}}:\Gamma\rightarrow(H/H^{0})(\overline{k})$
is surjective and
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}=X^{\mathrm{gen}}_{H,\overline{\rho}^{\mathrm{ss}}}$.
In particular, we may always assume that $G/G^{0}$ is a finite constant group
scheme.

We then define the absolutely irreducible locus in
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ under the assumption that
$G$ is split and
$\pi_{G}\circ\overline{\rho}^{\mathrm{ss}}:\Gamma\rightarrow(G/G^{0})(\overline{k})$
is surjective, see Proposition 9.7 and Definition 9.12. It is convenient to
define it as a subscheme of
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}\setminus
Y_{\overline{\rho}^{\mathrm{ss}}}$, so that points in
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}(\overline{k})$ are never
in the absolutely irreducible locus, even if the image of the corresponding
representation $\rho_{x}:\Gamma\rightarrow G(\overline{k})$ is not contained
in any proper R-parabolic subgroup of $G$.

The main result proved in Corollary 9.10 says that if the absolutely
irreducible locus is non-empty then the difference between its dimension and
the dimension of its GIT quotient by $G^{0}$ is equal to $\dim G_{k}-\dim
Z(G_{k})$. We also define schemes $V_{LG}$ and $U_{LG}$ for an R-Levi $L$,
which along with Corollary 9.10 play an important role in the inductive
argument in section 13.

The results of this section apply to any profinite group $\Gamma$ satisfying
Mazur’s $p$-finiteness condition. Lemma 5.5 implies that we may replace
$\Gamma$ with a topologically finitely generated quotient without changing
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$. Hence, without loss of
generality we assume that $\Gamma$ is topologically finitely generated.

###  9.1. Trimming the component group

Let $\rho_{z}:\Gamma\rightarrow G(\overline{k})$ be a representation
corresponding to a point $z\in
X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}(\overline{k})$. Then the
composition $\pi_{G}\circ\rho_{z}:\Gamma\rightarrow(G/G^{0})(\overline{k})$
depends only on the $G$-semisimplification of $\rho_{z}$. Since
$\rho_{z}^{\mathrm{ss}}=\overline{\rho}^{\mathrm{ss}}$ by Lemma 6.13 the map
$\pi_{G}\circ\overline{\rho}^{\mathrm{ss}}$ is well defined. Let $\Delta$ be
the image of

$\pi_{G}\circ\overline{\rho}^{\mathrm{ss}}:\Gamma\rightarrow(G/G^{0})(\overline{k}).$

Since $G$ is generalised reductive $G/G^{0}$ is finite étale over
$\mathcal{O}$ by Remark 2.6. We may think of finite étale $\mathcal{O}$-groups
schemes as finite groups together with
$\operatorname{Gal}(\overline{k}/k)$-action by group automorphisms. Since
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}(k)$ is non-empty, $\Delta$
is contained in $(G/G^{0})(k)$ and hence the constant group scheme
$\underline{\Delta}$ defined by $\Delta$ is a finite étale subgroup scheme of
$G/G^{0}$.

By construction $X^{\mathrm{gen},\tau}_{G}$ is a closed subscheme of
$G_{X^{\mathrm{ps}}_{\operatorname{GL}_{d}}}^{N}$. The map $\pi_{G}$ induces a
$G^{0}$-equivariant morphism of schemes
$\nu:X^{\mathrm{gen},\tau}_{G}\rightarrow(G/G^{0})^{N}_{X^{\mathrm{ps}}_{\operatorname{GL}_{d}}}$
for the trivial action on the target. If $A$ is an
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra then the map $\nu$ on
$A$-points is given by

(39)
$\nu:X^{\mathrm{gen},\tau}_{G}(A)\rightarrow(G/G^{0})_{X^{\mathrm{ps}}_{\operatorname{GL}_{d}}}^{N}(A),\quad\rho\mapsto(\pi_{G}(\rho(\gamma_{1})),\ldots,\pi_{G}(\rho(\gamma_{N}))).$

Since $\nu$ is $G^{0}$-equivariant it factors through
$X^{\mathrm{git},\tau}_{G}\rightarrow(G/G^{0})^{N}_{X^{\mathrm{ps}}_{\operatorname{GL}_{d}}}$.
We view an $N$-tuple
$\underline{\delta}=(\delta_{1},\ldots,\delta_{N})\in\Delta^{N}$ as a closed
constant subscheme of
$\underline{\Delta}^{N}_{X^{\mathrm{ps}}_{\operatorname{GL}_{d}}}$ over
$X^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ and define

$X^{\mathrm{git},\tau}_{G,\underline{\delta}}:=X^{\mathrm{git},\tau}_{G}\times_{(G/G^{0})^{N}_{X^{\mathrm{ps}}_{\operatorname{GL}_{d}}}}\underline{\delta}.$

If $X$ is a scheme over $X^{\mathrm{git},\tau}_{G}$ we let

(40)
$X_{\underline{\delta}}:=X\times_{X^{\mathrm{git},\tau}_{G}}X^{\mathrm{git},\tau}_{G,\underline{\delta}},\quad
X_{\overline{\rho}^{\mathrm{ss}}}:=X\times_{X^{\mathrm{git},\tau}_{G}}X^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}.$

######  Lemma 9.1.

If
$\underline{\delta}=(\pi_{G}(\overline{\rho}^{\mathrm{ss}}(\gamma_{1})),\ldots,\pi_{G}(\overline{\rho}^{\mathrm{ss}}(\gamma_{N})))$
then

$X^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}=((X^{\mathrm{git},\tau}_{G})_{\overline{\rho}^{\mathrm{ss}}})_{\underline{\delta}}=((X^{\mathrm{git},\tau}_{G})_{\underline{\delta}})_{\overline{\rho}^{\mathrm{ss}}}.$

In particular, for any scheme $X$ over $X^{\mathrm{git},\tau}_{G}$ we have
$(X_{\underline{\delta}})_{\overline{\rho}^{\mathrm{ss}}}=X_{\overline{\rho}^{\mathrm{ss}}}$.

###### Proof.

If $R$ is a complete local noetherian $\mathcal{O}$-algebra with residue field
$k$ then the map $R\rightarrow k$ induces a bijection
$(G/G^{0})(R)\overset{\sim}{\longrightarrow}(G/G^{0})(k)$ by [2, Exp. I, Cor.
6.2]. We apply this observation with
$R=R^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}$ and let
$\underline{\delta}\in(G/G^{0})^{N}(k)$ be the $N$-tuple corresponding to
$\alpha:X^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}\rightarrow
X^{\mathrm{git},\tau}_{G}\overset{\nu}{\rightarrow}(G/G^{0})^{N}_{X^{\mathrm{ps}}_{\operatorname{GL}_{d}}}$.
Since the closed point in $X^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}$
maps to
$(\pi_{G}(\overline{\rho}^{\mathrm{ss}}(\gamma_{1})),\ldots,\pi_{G}(\overline{\rho}^{\mathrm{ss}}(\gamma_{N})))\in(G/G^{0})^{N}_{X^{\mathrm{git},\tau}_{G}}(k)$
under $\alpha$ we conclude that
$\underline{\delta}=(\pi_{G}(\overline{\rho}^{\mathrm{ss}}(\gamma_{1})),\ldots,\pi_{G}(\overline{\rho}^{\mathrm{ss}}(\gamma_{N})))$.

The image of $\alpha$ is contained in $\underline{\delta}$, which implies the
first equality. The second equality follows from the symmetry of a tensor
product. The last part follows from a manipulation with fibre products. ∎

######  Lemma 9.2.

Let $H$ be a closed subgroup scheme of $G$, such that $H^{0}=G^{0}$. If
$(\pi_{G}\circ\overline{\rho}^{\mathrm{ss}})(\Gamma)\subseteq(H/H^{0})(k)$
then
$X^{\mathrm{git}}_{H,\overline{\rho}^{\mathrm{ss}}}=X^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}$
and
$X^{\mathrm{gen}}_{H,\overline{\rho}^{\mathrm{ss}}}=X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}.$
Moreover, there exists $H$ as above, such that
$\pi_{H}\circ\overline{\rho}^{\mathrm{ss}}:\Gamma\rightarrow(H/H^{0})(\overline{k})$
is surjective.

###### Proof.

We may write $X^{\mathrm{gen},\tau}_{H}$ as the fibre product of
$X^{\mathrm{gen},\tau}_{G}$ and $(H/H^{0})^{N}$ over $(G/G^{0})^{N}$. The
assumption implies that $X^{\mathrm{gen},\tau}_{H}$ contains
$X^{\mathrm{gen},\tau}_{G,\underline{\delta}}$ as a closed subscheme with
$\underline{\delta}$ as in Lemma 9.1. The assertion follows from Lemma 9.1.
Since $H^{0}=G^{0}$ by passing to GIT quotients we obtain
$X^{\mathrm{git}}_{H,\overline{\rho}^{\mathrm{ss}}}=X^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}$.
For the last part we may take $H$ to be the preimage of $\underline{\Delta}$
in $G$. ∎

If $H$ is a closed generalised reductive subgroup scheme of $G$ (resp.
$G_{k}$) we let $X^{\mathrm{git},\tau}_{HG}$ (resp.
$\overline{X}^{\mathrm{git},\tau}_{HG}$) be the scheme theoretic image of
$X^{\mathrm{git},\tau}_{H}\rightarrow X^{\mathrm{git},\tau}_{G}$ (resp.
$\overline{X}^{\mathrm{git},\tau}_{H}\rightarrow\overline{X}^{\mathrm{git},\tau}_{G}$).
Since these maps are finite by Corollary 6.5 we have

(41) $\dim X^{\mathrm{git},\tau}_{HG}\leq\dim
X^{\mathrm{git},\tau}_{H},\quad\dim\overline{X}^{\mathrm{git},\tau}_{HG}\leq\dim\overline{X}^{\mathrm{git},\tau}_{H}.$

######  Remark 9.3.

We note that if $H$ is a closed generalised reductive subgroup of $G$ then the
special fibre of $X^{\mathrm{git},\tau}_{HG}$ is homeomorphic to
$\overline{X}^{\mathrm{git},\tau}_{H_{k}G}$, see Remark 6.1. We will denote
both by $\overline{X}^{\mathrm{git},\tau}_{HG}$. This is harmless, since in
our arguments we only care about dimensions.

We denote by $X^{\mathrm{gen}}_{H,\overline{\rho}^{\mathrm{ss}}}$ (resp.
$X^{\mathrm{git}}_{H,\overline{\rho}^{\mathrm{ss}}}$,
$X^{\mathrm{git}}_{HG,\overline{\rho}^{\mathrm{ss}}}$) the schemes
$(X^{\mathrm{gen},\tau}_{H})_{\overline{\rho}^{\mathrm{ss}}}$ (resp.
$(X^{\mathrm{git},\tau}_{H})_{\overline{\rho}^{\mathrm{ss}}}$,
$(X^{\mathrm{git},\tau}_{HG})_{\overline{\rho}^{\mathrm{ss}}}$) with the
notation introduced in (40). 9.1 implies that these schemes are independent of
$\tau$ and only depend on the map $H\to G$.

######  Lemma 9.4.

Let $H$ be a generalised reductive subgroup scheme of $G$ (resp. $G_{k}$). If
$X^{\mathrm{git}}_{HG,\overline{\rho}^{\mathrm{ss}}}$ (resp.
$\overline{X}^{\mathrm{git}}_{HG,\overline{\rho}^{\mathrm{ss}}}$) is non-empty
then there exist

  * (a)

a finite unramified extension $L^{\prime}$ of $L$ with ring of integers
$\mathcal{O}^{\prime}$ and residue field $k^{\prime}$;

  * (b)

finitely many generalised reductive subgroup schemes $H_{i}$ of
$H_{\mathcal{O}^{\prime}}$;

  * (c)

continuous representations $\overline{\rho}_{i}:\Gamma\rightarrow
H_{i}(k^{\prime})$;

such that the following hold:

  1. (1)

$H_{i}^{0}=H^{0}_{\mathcal{O}^{\prime}}$ and
$\pi_{H_{i}}\circ\overline{\rho}_{i}:\Gamma\rightarrow(H_{i}/H_{i}^{0})(\overline{k})$
is surjective;

  2. (2)

$(X^{\mathrm{git}}_{H,\overline{\rho}^{\mathrm{ss}}})_{\mathcal{O}^{\prime}}$
(resp.
$(\overline{X}^{\mathrm{git}}_{H,\overline{\rho}^{\mathrm{ss}}})_{k^{\prime}}$)
is a disjoint union of
$X^{\mathrm{git}}_{H_{i},\overline{\rho}^{\mathrm{ss}}_{i}}$ (resp.
$\overline{X}^{\mathrm{git}}_{H_{i},\overline{\rho}^{\mathrm{ss}}_{i}}$);

  3. (3)

$(X^{\mathrm{gen}}_{H,\overline{\rho}^{\mathrm{ss}}})_{\mathcal{O}^{\prime}}$
(resp.
$(\overline{X}^{\mathrm{gen}}_{H,\overline{\rho}^{\mathrm{ss}}})_{k^{\prime}}$)
is a disjoint union of
$X^{\mathrm{gen}}_{H_{i},\overline{\rho}^{\mathrm{ss}}_{i}}$ (resp.
$\overline{X}^{\mathrm{gen}}_{H_{i},\overline{\rho}^{\mathrm{ss}}_{i}}$);

  4. (4)

$(X^{\mathrm{git}}_{HG,\overline{\rho}^{\mathrm{ss}}})_{\mathcal{O}^{\prime}}$
(resp.
$(\overline{X}^{\mathrm{git}}_{HG,\overline{\rho}^{\mathrm{ss}}})_{k^{\prime}}$)
is a union of scheme theoretic images

$X^{\mathrm{git}}_{H_{i},\overline{\rho}^{\mathrm{ss}}_{i}}\rightarrow(X^{\mathrm{git},\tau}_{G})_{\mathcal{O}^{\prime}},\quad\text{(resp.
}\overline{X}^{\mathrm{git}}_{H_{i},\overline{\rho}^{\mathrm{ss}}_{i}}\rightarrow(X^{\mathrm{git},\tau}_{G})_{k^{\prime}}).$

###### Proof.

The map $X^{\mathrm{git},\tau}_{H}\rightarrow X^{\mathrm{git},\tau}_{G}$
corresponds to the map between finite products of local rings $\prod_{i\in
I}R_{i}\rightarrow\prod_{j\in J}R_{j}^{\prime}$ by (31). The representation
$\overline{\rho}^{\mathrm{ss}}$ defines an idempotent $e\in\prod_{i}R_{i}$,
such that
$e(\prod_{i}R_{i})=R^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}$. Then
$X^{\mathrm{git}}_{H,\overline{\rho}^{\mathrm{ss}}}$ is the spectrum of
$e(\prod_{j\in J}R_{j}^{\prime})$. Since $R_{j}^{\prime}$ are local rings we
have either $e(R_{j}^{\prime})=R^{\prime}_{j}$ or $e(R^{\prime}_{j})=0$. Thus
$e(\prod_{j\in J}R_{j}^{\prime})=\prod_{j\in J^{\prime}}R^{\prime}_{j}$ for
some subset $J^{\prime}\subseteq J$. We may identify $J^{\prime}$ with the set
of $\operatorname{Gal}(\overline{k}/k)$-orbits in
$X^{\mathrm{git}}_{H,\overline{\rho}^{\mathrm{ss}}}(\overline{k})$.

After replacing $L$ by a finite unramified extension we may ensure that every
connected component of $X^{\mathrm{gen}}_{H,\overline{\rho}^{\mathrm{ss}}}$
has a $k$-rational point, see Remark 6.14. Then (31) yields decompositions:

$X^{\mathrm{git}}_{H,\overline{\rho}^{\mathrm{ss}}}\cong\coprod_{i}X^{\mathrm{git}}_{H,\overline{\rho}^{\mathrm{ss}}_{i}},\quad
X^{\mathrm{gen}}_{H,\overline{\rho}^{\mathrm{ss}}}\cong\coprod_{i}X^{\mathrm{gen}}_{H,\overline{\rho}^{\mathrm{ss}}_{i}}$

for finitely many continuous representations $\rho_{i}:\Gamma\rightarrow
H(k)$. The existence of $H_{i}$ follows from Lemma 9.2. Parts (2), (3) and (4)
then follows immediately. If $H$ is a closed subgroup of $G_{k}$ the argument
is the same. ∎

###  9.2. Absolutely irreducible locus

In this subsection we omit the fixed embedding
$\tau:G\hookrightarrow\operatorname{GL}_{d}$ from the notation and write
$X^{\mathrm{gen}}_{G}$, $X^{\mathrm{git}}_{G}$, $X^{\mathrm{gen}}_{HG}$ and
$X^{\mathrm{git}}_{HG}$ for $X^{\mathrm{gen},\tau}_{G}$,
$X^{\mathrm{git},\tau}_{G}$, $X^{\mathrm{gen},\tau}_{HG}$ and
$X^{\mathrm{git},\tau}_{HG}$. We note that as a consequence of Corollary 8.4
$X^{\mathrm{gen}}_{HG,\overline{\rho}^{\mathrm{ss}}}$ and
$X^{\mathrm{git}}_{HG,\overline{\rho}^{\mathrm{ss}}}$ are independent of
$\tau$. In particular, the schemes $U_{LG,\overline{\rho}^{\mathrm{ss}}}$ and
$V_{LG,\overline{\rho}^{\mathrm{ss}}}$ defined below are independent of
$\tau$.

It follows from Proposition 6.4 that $X^{\mathrm{git}}_{G}$ has only finitely
many closed points, we denote this subscheme by $\mathrm{pts}$. We let $Y$ be
the preimage of these closed points in $X^{\mathrm{gen}}_{G}$. Then $Y$ is a
closed subscheme of $X^{\mathrm{gen}}_{G}$ contained in
$\overline{X}^{\mathrm{gen}}_{G}$.

We assume for the rest of this subsection that $G^{0}$ is split and fix a
maximal split torus $T$ defined over $\mathcal{O}$. Let $L$ be an R-Levi
subgroup of $G$ containing $T$. Then we define

$U_{LG}:=X^{\mathrm{git}}_{LG}\setminus(\mathrm{pts}\cup\bigcup_{M}X^{\mathrm{git}}_{MG}),\quad\overline{U}_{LG}:=\overline{X}^{\mathrm{git}}_{LG}\setminus(\mathrm{pts}\cup\bigcup_{M}\overline{X}^{\mathrm{git}}_{MG})$

where the union is taken over all R-Levi subgroups $M$ of $G$, such that
$T\subseteq M\subsetneq L$. Lemma 2.18 implies that there are only finitely
many such $M$. Thus $U_{LG}$ is an open subscheme of
$X^{\mathrm{git}}_{LG}\setminus\mathrm{pts}$ and $\overline{U}_{LG}$ is an
open subscheme of $\overline{X}^{\mathrm{git}}_{LG}\setminus\mathrm{pts}$.
Hence, $U_{LG}$ and $\overline{U}_{LG}$ are locally closed subschemes of
$X^{\mathrm{git}}_{G}\setminus\mathrm{pts}$. We let $V_{LG}$ be the preimage
of $U_{LG}$ in $X^{\mathrm{gen}}_{G}$ and $\overline{{V}}_{LG}$ be the
preimage of $\overline{U}_{LG}$ in $X^{\mathrm{gen}}_{G}$. We then have

(42)
$X^{\mathrm{gen}}_{G}=Y\cup\bigcup_{L}V_{LG},\quad\overline{X}^{\mathrm{gen}}_{G}=Y\cup\bigcup_{L}\overline{V}_{LG},$

where the union is taken over all $R$-Levi subgroups $L$ of $G$ containing
$T$.

######  Definition 9.5.

We say that a geometric point $x\in X^{\mathrm{gen}}_{G}$ is _$G$ -stable_, if
the orbit $G\cdot x$ is closed and $Z_{G}(x)^{0}=Z(G)^{0}_{\kappa(x)}$, where
$Z_{G}(x)$ is the $G_{\kappa(x)}$-centraliser of $x$.

######  Remark 9.6.

The action of $G$ on $X^{\mathrm{gen}}_{G}$ factors through the action of
$G/Z(G)$. The stability condition in the definition is equivalent to $G\cdot
x$ closed and $\dim G\cdot x=\dim(G/Z(G))_{\kappa(x)}$. Hence, it is
equivalent to $G/Z(G)$-stability in the sense of [52, Definition 1].

######  Proposition 9.7.

Let $x$ be a geometric point of $X^{\mathrm{gen}}_{G}\setminus Y$, and let $H$
be the Zariski closure of $\rho_{x}(\Gamma)$ in $G_{\kappa(x)}$. Then the
following statements are equivalent:

  1. (1)

$x\in V_{GG}$;

  2. (2)

$\rho_{x}$ is $G$-irreducible;

  3. (3)

$H$ is $G$-irreducible;

  4. (4)

$x$ is $G$-stable.

Moreover, $H$ is reductive and $Z_{G}(H)^{0}=Z(G_{\kappa(x)})^{0}$.

###### Proof.

The equivalence of (1) and (2) follows from Corollary 6.11. The equivalence of
(2) and (3) follows from Definition 2.26. The equivalence of (3) and (4)
follows from [46, Proposition 16.7]. The last part follows from [40, Lemma
6.2]. ∎

######  Corollary 9.8.

Let $L$ be an R-Levi of $G$ and let $H$ be a closed generalised reductive
subgroup of $L_{k^{\prime}}$, where $k^{\prime}$ is a finite extension of $k$.
If $\overline{X}^{\mathrm{git}}_{HG}\cap U_{LG}$ is non-empty then $\dim
Z(H)\leq\dim Z(L_{k})$.

###### Proof.

After extending scalars we may assume that $k=k^{\prime}$. Since $H$ is
contained in $L$ the map
$\overline{X}^{\mathrm{git}}_{H}\rightarrow\overline{X}^{\mathrm{git}}_{G}$
factors through
$\overline{X}^{\mathrm{git}}_{H}\rightarrow\overline{X}^{\mathrm{git}}_{L}\rightarrow\overline{X}^{\mathrm{git}}_{G}$.
The assumption that $\overline{X}^{\mathrm{git}}_{HG}\cap U_{LG}$ is non-empty
implies that $\overline{X}^{\mathrm{git}}_{HL}\cap U_{LL}$ is non-empty. It
follows from Proposition 9.7 that $H_{\kappa(x)}$ is not contained in any
R-parabolic subgroup of $L_{\kappa(x)}$, where $x$ is any geometric point of
$V_{LL}$, such that its image in $\overline{X}^{\mathrm{git}}_{L}$ is
contained in $\overline{X}^{\mathrm{git}}_{HL}$. Moreover, we have

$\dim Z(H)=\dim Z(H_{\kappa(x)})\leq\dim Z(L_{\kappa(x)})=\dim Z(L_{k}),$

where the inequality in the middle follows from [40, Lemma 6.2], as the centre
of $H_{\kappa(x)}$ is contained in the $L_{\kappa(x)}$-centraliser of
$H_{\kappa(x)}$, and the other two follow from Lemma 2.2. ∎

######  Corollary 9.9.

Let $X^{\mathrm{gen}}_{G,y}$ be the fibre at a point $y\in U_{GG}$. Then

$\dim X^{\mathrm{gen}}_{G,y}=\dim G-\dim Z(G)=\dim G_{k}-\dim Z(G_{k}).$

###### Proof.

Since extending scalars from $\kappa(y)$ to its algebraic closure does not
change the dimension, we may assume that $y$ is a geometric point of $U_{GG}$
and denote its residue field by $\kappa$.

Let $x$ be any closed point in the fibre $X^{\mathrm{gen}}_{G,y}$. Then the
$G$-orbit $G\cdot x$ is closed, as $x$ is $G$-stable by Proposition 9.7.
Proposition 6.7 implies that the $G^{0}$-orbit $G^{0}\cdot x$ is also closed.
Since $X^{\mathrm{gen}}_{G,y}$ contains a unique closed $G^{0}$-orbit by
Proposition 6.9 we conclude that $G^{0}(\kappa)$ acts transitively on
$X^{\mathrm{gen}}_{G,y}(\kappa)$. Thus the orbit map

$G^{0}_{\kappa}/Z_{G^{0}}(x)\rightarrow X^{\mathrm{gen}}_{G,y}$

induces a bijection on $\kappa$-points. Since both schemes are of finite type
over $\kappa$, we deduce that $\dim X^{\mathrm{gen}}_{G,y}=\dim
G^{0}_{\kappa}/Z_{G^{0}}(x)=\dim G^{0}_{\kappa}-\dim Z_{G^{0}}(x)$. By passing
to neutral components we obtain
$Z_{G^{0}}(x)^{0}=Z_{G}(x)^{0}=Z(G)_{\kappa}^{0}$, where the last equality
holds as $x$ is $G$-stable. Thus $\dim Z_{G^{0}}(x)=\dim Z(G)_{\kappa}$ and we
conclude that

$\dim X^{\mathrm{gen}}_{G,y}=\dim G_{\kappa}-\dim Z(G)_{\kappa}.$

The other assertions follow from 2.9 and 2.1. ∎

######  Corollary 9.10.

Let $U$ be a non-empty locally closed subspace of $U_{GG}$ (resp.
$\overline{U}_{GG}$) and let $V$ be the preimage of $U$ in
$X^{\mathrm{gen}}_{G}$ (resp. $\overline{X}^{\mathrm{gen}}_{G}$). Then

$\dim V=\dim U+\dim G_{k}-\dim Z(G_{k}).$

###### Proof.

Corollary 9.9 says that $\dim V_{y}=d$, where $d:=\dim G_{k}-\dim Z(G_{k})$,
for all $y\in U$. The inequality $\dim V\leq\dim U+d$ follows directly from
[8, Lemma 3.18 (6)] (or alternatively [54, Tag 0BAG]).

To prove the reverse inequality we may assume that $U$ is irreducible and,
since $V\rightarrow U$ is surjective by [52, Theorem 3 (ii)], we may replace
$V$ by an irreducible component, whose image contains the generic point of
$U$. Lemma 2.1 in [8] implies that $V$ is $G^{0}$-invariant and [52, Theorem 3
(iii)] implies that the image of $V$ in $U$ is closed, and hence $V\rightarrow
U$ is surjective. It follows from [54, Tag 02JU] applied with $s\in U$ a
closed point such that $\dim U=\dim\mathcal{O}_{U,s}$ and $x\in V$ a closed
point, which maps to $s$, that $\dim V\geq\dim U+d$. ∎

######  Lemma 9.11.

Assume that
$\pi_{G}\circ\overline{\rho}^{\mathrm{ss}}:\Gamma\rightarrow(G/G^{0})(\overline{k})$
is surjective. Let $L$ be an R-Levi of $G$ containing $T$ and let $P$ be an
R-parabolic subgroup of $G$ with R-Levi $L$ and unipotent radical $N$. Assume
that one of the following is non-empty:
$X^{\mathrm{gen}}_{L,\overline{\rho}^{\mathrm{ss}}}$,
$X^{\mathrm{git}}_{L,\overline{\rho}^{\mathrm{ss}}}$,
$V_{LG,\overline{\rho}^{\mathrm{ss}}}$,
$U_{LG,\overline{\rho}^{\mathrm{ss}}}$. Then the following hold:

  1. (1)

$L/L^{0}\rightarrow G/G^{0}$ is an isomorphism;

  2. (2)

if $\dim L=\dim G$ then $L=G$;

  3. (3)

If $L\neq G$ then $\dim N_{k}\geq 1$.

###### Proof.

It follows from Lemma 2.13 that $L^{0}=L\cap G^{0}$, so that the map in (1) is
injective. The assumption in all the cases implies that
$X^{\mathrm{git}}_{L,\overline{\rho}^{\mathrm{ss}}}$ is non-empty. Lemma 9.1
implies that $X^{\mathrm{git}}_{L,\underline{\delta}}$ is non-empty, where
$\underline{\delta}$ is as in Lemma 9.1. Thus
$\underline{\delta}\in(L/L^{0})(k)^{N}$ and thus
$\pi_{G}(\overline{\rho}^{\mathrm{ss}}(\gamma_{i}))\in(L/L^{0})(k)$ for all
$1\leq i\leq N$. Since $\pi_{G}\circ\overline{\rho}^{\mathrm{ss}}$ is
continuous and surjects onto $(G/G^{0})(\overline{k})$ and
$\gamma_{1},\ldots\gamma_{N}$ generate $\Gamma$ topologically, the entries of
$\underline{\delta}$ generate $(G/G^{0})(\overline{k})$ as a group. Thus
$(L/L^{0})(k)=(G/G^{0})(k)=(G/G^{0})(\overline{k})$. Thus $G/G^{0}$ is a
constant group scheme and we deduce part (1).

As explained in Lemma 2.13, $P^{0}$ is a parabolic of $G^{0}$ with Levi
$L^{0}$. Thus $\dim L=\dim G$ implies that $\dim L^{0}=\dim G^{0}$ and hence
$L^{0}=G^{0}$. Combining this with part (1) we deduce part (2).

Part (3) follows from $\dim G_{k}-\dim L_{k}=\dim G_{k}^{0}-\dim
L_{k}^{0}=2\dim N_{k}$. ∎

######  Definition 9.12.

If
$\pi_{G}\circ\overline{\rho}^{\mathrm{ss}}:\Gamma\rightarrow(G/G^{0})(\overline{k})$
is surjective then we define the _absolutely $G$-irreducible_ locus in
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ as
$V_{GG,\overline{\rho}^{\mathrm{ss}}}$.

######  Remark 9.13.

The justification for calling $V_{GG,\overline{\rho}^{\mathrm{ss}}}$
absolutely $G$-irreducible locus comes from Proposition 9.7. We are not sure
if this is a good definition if $\pi_{G}\circ\overline{\rho}^{\mathrm{ss}}$ is
not surjective, and we discuss an example below. However, one can get around
the problem by replacing $G$ by a closed subgroup scheme $H$ such that
$G^{0}=H^{0}$ and $\pi_{G}\circ\overline{\rho}^{\mathrm{ss}}$ surjects onto
$(H/H^{0})(\overline{k})$ and using Lemma 9.2.

######  Example 9.14.

Assume that $p>2$ and let $G$ be the normaliser of the subgroup of diagonal
matrices in $\operatorname{GL}_{2}$. Then
$G\cong\mathbb{G}_{m}^{2}\rtimes\mathbb{Z}/2\mathbb{Z}$ as in the Example
2.14. Let $E$ be a quadratic Galois extension of $F$ and let
$c\in\operatorname{Gal}(E/F)$ be non-trivial. Let
$\overline{\rho}_{1},\overline{\rho}_{2}:\Gamma_{F}\rightarrow G(k)$ be
representations, such that $\Gamma_{E}$ is mapped to identity and

$\overline{\rho}_{1}(c)=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\quad\overline{\rho}_{2}(c)=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}.$

These representations are $G$-semisimple. Since $\overline{\rho}_{1}(c)$ and
$\overline{\rho}_{2}(c)$ are conjugate in $\operatorname{GL}_{2}(k)$ the
representations $\tau\circ\overline{\rho}_{1}$ and
$\tau\circ\overline{\rho}_{2}$ will have the same
$\operatorname{GL}_{2}$-pseudocharacter, where
$\tau:G\hookrightarrow\operatorname{GL}_{2}$ is the natural inclusion. The
image of $\overline{\rho}_{1}$ is contained in $G^{0}(k)$ and $G^{0}$ is a
proper $R$-parabolic subgroup of $G$, as explained in Example 2.14, hence
$V_{GG,\overline{\rho}_{1}}$ is empty, and hence the $G$-irreducible locus in
$X^{\mathrm{gen}}_{G,\overline{\rho}_{1}}$ is also empty. On the other hand,
$V_{GG,\overline{\rho}_{2}}$ is non-empty. We also would like to point out
that in this case $Z(G^{0})$ and $Z(G)$ have different dimensions, and this
results in a difference between the dimensions of
$V_{G^{0}G,\overline{\rho}_{1}}\sslash G^{0}$ and
$V_{GG,\overline{\rho}_{2}}\sslash G^{0}$.

##  10\. Bounds for the dimensions of reductive subgroups

Let $G$ be a generalised reductive group over an an algebraically closed field
$\kappa$. If $H$ is a closed reduced subgroup scheme of $G$, such that the
unipotent radical of $H^{0}$ is trivial then $H$ is also generalised reductive
by 2.7. Let $G^{\prime}$ denote the derived subgroup of $G^{0}$. Then
$G^{\prime}$ is semisimple and $G^{0}=Z(G^{0})G^{\prime}$, [11, 14.2].
Moreover, $G^{\prime}$ is equal to its own derived subgroup, [17, Example
1.1.16].

######  Proposition 10.1.

Let $H$ be a closed reductive subgroup of $G$. Then one of the following
holds:

  * (i)

$H$ contains $G^{\prime}$;

  * (ii)

$\dim G^{\prime}-\dim(H\cap G^{\prime})\geq 2$.

###### Proof.

After replacing $G$ by $G^{\prime}$ we may assume that $G$ is connected,
reductive and is equal to its own derived subgroup. One may check that the
unipotent radical of $(G^{\prime}\cap H)^{0}$ is normal in $H^{0}$. Since
$H^{0}$ is reductive, $(G^{\prime}\cap H)^{0}$ is also reductive. Thus after
replacing $H$ by $H\cap G^{\prime}$ we may assume that $H$ is closed reductive
subgroup of $G$. Let us assume that $H\neq G$. Then we have to show that $\dim
G/H>1$.

There exists a finite dimensional representation $V$ of $G$ and a
$1$-dimensional subspace $\ell$ of $V$ such that $H$ is equal to the
$G$-stabiliser of $\ell$ in $\mathbb{P}(V)$, [33, 11.2]. Moreover, the
$G$-action on $\mathbb{P}(V)$ induces an isomorphism of homogeneous spaces
between $G/H$ and the $G$-orbit of $\ell$ in $\mathbb{P}(V)$, [33, 12.1].

Since $H$ is reductive $G/H$ is affine, [45], and since $\dim G/H\geq 1$,
$G\cdot\ell$ cannot be closed in $\mathbb{P}(V)$. Its closure contains a
closed $G$-orbit $G\cdot\ell^{\prime}$ of strictly smaller dimension, [33,
8.3].

If $\dim G\cdot\ell^{\prime}\geq 1$ then $\dim G\cdot\ell\geq 2$ and we are
done. If $\dim G\cdot\ell^{\prime}=0$ then since $G$ is connected we deduce
that $G$ fixes $\ell^{\prime}$. Let $v$ be a basis vector for $\ell$ and
$v^{\prime}$ a basis vector of $\ell^{\prime}$. Since $G$ is equal to its
derived subgroup we deduce that $G$ fixes $v^{\prime}$. Let $\bar{\ell}$ be
the image of $\ell$ in $V/\ell^{\prime}$. Since $G\neq H$,
$\ell\neq\ell^{\prime}$ and hence $\bar{\ell}$ is $1$-dimensional. Let
$\bar{H}$ be the $G$-stabiliser of $\bar{\ell}$ in
$\mathbb{P}(V/\ell^{\prime})$. Then $H\subseteq\bar{H}\subseteq G$.

If $\dim G=\dim\bar{H}$ then $\dim G/\bar{H}=0$ and since $G$ is connected
$G=\bar{H}$. Since $G$ is equal to its derived subgroup it fixes
$v+\ell^{\prime}$, and thus $gv=v+a(g)v^{\prime}$ for all $g\in G$. Since $G$
acts trivially on $v^{\prime}$, $a$ defines a group homomorphism
$a:G\rightarrow\mathbb{G}_{a}$. Since $G$ is equal to its own derived subgroup
we get that $a=0$ and hence $G$ fixes $v$, which yields a contradiction as the
$G$-stabiliser of $v$ is $H$. Thus $\dim\bar{H}<\dim G$.

If $\dim H<\dim\bar{H}$ then we obtain $\dim G-\dim H\geq 2$ and we are done.

If $\dim H=\dim\bar{H}$ then $H^{0}=\bar{H}^{0}$, hence $\bar{H}$ is
reductive. We may now repeat the argument with $V/\ell^{\prime}$ instead of
$V$. Since $V$ is finite dimensional this process has to terminate eventually
yielding $\dim G-\dim H\geq 2$. ∎

######  Corollary 10.2.

Assume that we are in the situation of part (ii) of Proposition 10.1. Then
$\dim G-\dim H\geq 2$.

###### Proof.

The $G^{\prime}$-orbit of $H$ in $G/H$ is isomorphic to
$G^{\prime}/(G^{\prime}\cap H)$ and hence

$\dim G/H\geq\dim G^{\prime}/(G^{\prime}\cap H)\geq 2.$

∎

######  Corollary 10.3.

Let $V$ be a finite dimensional representation of $G$ and let $H$ be a closed
reductive subgroup of $G$. If $V^{G^{\prime}}=0$ and $V^{H}\neq 0$ then $\dim
G-\dim H\geq 2$.

###### Proof.

If $H$ contains $G^{\prime}$ then $V^{H}$ is contained in $V^{G^{\prime}}$ and
so $V^{H}\neq 0$ implies that $V^{G^{\prime}}\neq 0$. The assertion follows
from Corollary 10.2. ∎

Since $G^{\prime}$ is a normal subgroup of $G$, $G$ acts on
$\operatorname{Lie}(G^{\prime})$ by conjugation.

######  Lemma 10.4.

If $\pi_{1}(G^{\prime})$ is étale then
$((\operatorname{Lie}G^{\prime})^{*})^{G^{\prime}}=0$.

###### Proof.

We may assume that $\operatorname{Lie}G^{\prime}$ is non-zero. Let
$G^{\prime}_{\operatorname{sc}}\rightarrow G^{\prime}$ be the universal simply
connected cover of $G^{\prime}$. Since $\pi_{1}(G^{\prime})$ is étale the
exact sequence:

$0\rightarrow\pi_{1}(G^{\prime})\rightarrow
G^{\prime}_{\operatorname{sc}}\rightarrow G^{\prime}\rightarrow 0$

induces a $G^{\prime}_{\operatorname{sc}}$-equivariant isomorphism of Lie
algebras
$\operatorname{Lie}G^{\prime}_{\operatorname{sc}}\cong\operatorname{Lie}G^{\prime}$.
We thus may assume that $G^{\prime}$ is simply connected. Let $T$ be a maximal
torus in $G^{\prime}$.

If $\dim T=1$ then $G^{\prime}=\operatorname{SL}_{2}$ by [53, 7.2.4], as the
assumption that $G^{\prime}$ is simply connected rules out
$\operatorname{PGL}_{2}$. If $\mathrm{char}(\kappa)\neq 2$ then the action of
$G^{\prime}$ on $\operatorname{Lie}G^{\prime}$ is an irreducible
representation, isomorphic to $\operatorname{Sym}^{2}(\mathrm{Std})$, where
$\mathrm{Std}$ is the standard $2$-dimensional representation of
$\operatorname{SL}_{2}$. If $\mathrm{char}(\kappa)=2$ then
$\operatorname{Lie}G^{\prime}$ is a non-split extension
$0\rightarrow\mathbf{1}\rightarrow\operatorname{Lie}G^{\prime}\rightarrow
V\rightarrow 0$, with $\mathbf{1}$ equal to the subspace of scalar matrices
and $V$ an irreducible $2$-dimensional representation. In both cases we obtain
the claim.

Let $\dim T$ be arbitrary and let
$\chi:\operatorname{Lie}G^{\prime}\rightarrow\mathbf{1}$ be
$G^{\prime}$-invariant. Let us abbreviate
$\mathfrak{g}=\operatorname{Lie}G^{\prime}$. We have
$\mathfrak{g}=\mathfrak{g}^{T}\oplus\bigoplus\nolimits_{\alpha\in\Phi}\mathfrak{g}_{\alpha}$,
where $\Phi$ is the set of roots, and
$\mathfrak{g}^{T}=\operatorname{Lie}(T)$, [11, 13.18(1)]. Let $\alpha\in\Phi$
and let $G_{\alpha}$ be the centraliser of $(\operatorname{Ker}\alpha)^{0}$ in
$G^{\prime}$. Then $G_{\alpha}$ is reductive of semisimple rank $1$, [11,
13.18(4)]. Let $G^{\prime}_{\alpha}$ be the derived subgroup of $G_{\alpha}$.
The assumption that $G^{\prime}$ is simply connected implies that
$G^{\prime}_{\alpha}\cong\operatorname{SL}_{2}$ by [17, 5.3.9]. It follows
from [17, 1.2.7] that
$\operatorname{Lie}G_{\alpha}^{\prime}=\mathfrak{g}_{-\alpha}\oplus\operatorname{Lie}(\alpha^{\vee}(\mathbb{G}_{m}))\oplus\mathfrak{g}_{\alpha}$.
The restriction of $\chi$ to $\operatorname{Lie}G^{\prime}_{\alpha}$ is
$G^{\prime}_{\alpha}$-invariant and hence zero by the previous part. Hence,
$\chi$ is zero on $\operatorname{Lie}(\alpha^{\vee}(\mathbb{G}_{m}))$ and
$\mathfrak{g}_{\alpha}$ for every coroot $\alpha^{\vee}$.

Let $T_{1}=\prod_{\alpha\in\Delta}\mathbb{G}_{m}$, where $\Delta$ is the set
of positive simple roots with respect to a Borel subgroup containing $T$.
Since $G^{\prime}$ is simply connected the map
$\prod_{\alpha\in\Delta}\alpha^{\vee}:T_{1}\rightarrow T$ induces an
isomorphism on the character groups, [20, Example 1.3.10], and hence an
isomorphism between tori. Thus the subspaces
$\operatorname{Lie}(\alpha^{\vee}(\mathbb{G}_{m}))$ for $\alpha\in\Delta$ span
$\operatorname{Lie}(T)$. Thus $\chi$ vanishes on $\mathfrak{g}^{T}$ and
$\mathfrak{g}_{\alpha}$ for every $\alpha\in\Phi$ and hence $\chi=0$. ∎

######  Lemma 10.5.

Let $G^{\prime}_{\operatorname{sc}}\rightarrow G^{\prime}$ be the simply
connected central cover of $G^{\prime}$. Then there is a natural action of $G$
on $\operatorname{Lie}(G^{\prime}_{\operatorname{sc}})$ such that the map
$\operatorname{Lie}(G^{\prime}_{\operatorname{sc}})\rightarrow\operatorname{Lie}G^{\prime}$
is $G$-equivariant. Moreover, $G^{\prime}$-invariants in
$(\operatorname{Lie}(G^{\prime}_{\operatorname{sc}}))^{\ast}$ are zero.

###### Proof.

The action has been defined over $\mathcal{O}$ in 2.11 and the same holds over
$\kappa$. The $G^{\prime}_{\operatorname{sc}}$-invariants in
$(\operatorname{Lie}(G^{\prime}_{\operatorname{sc}}))^{*}$ are zero by 10.4.
The assertion follows as $G^{\prime}_{\operatorname{sc}}\to G^{\prime}$ is a
central extension and therefore this action factors through the action of
$G^{\prime}$. ∎

Let $P$ be an $R$-parabolic subgroup of $G$ and let $L$ be its Levi subgroup
and $U$ its unipotent radical. Then $P^{0}=P\cap G^{0}$ is a parabolic
subgroup of $G^{0}$ and $L^{0}=L\cap G^{0}$ is a Levi subgroup of $P^{0}$,
[40, Proposition 5.2]. In particular, $L^{0}$ is reductive. We let
$L^{\prime}$ be the derived subgroup of $L^{0}`$.

######  Lemma 10.6.

Let $H$ be a closed reductive subgroup of $L$. If
$((\operatorname{Lie}U)^{*})^{H}\neq 0$ then one of the following holds:

  * (i)

$\dim L-\dim H\geq 2$;

  * (ii)

$\dim L-\dim H=1$ and $\dim_{\kappa}((\operatorname{Lie}U)^{*})^{H}=1$.

###### Proof.

We may assume that $H$ contains $L^{\prime}$ since otherwise (i) holds by
Corollary 10.2. By replacing $H$ by $H\cap L^{0}$ and $L$ by $L^{0}$ we may
assume that $L$ is connected. Then $L=Z(L)H$. Let $T$ be a maximal torus in
$L$. Then $T$ contains $Z(L)$ and hence $L/H\cong T/(T\cap H)$. The action of
$T$ on $\operatorname{Lie}U$ is semisimple, multiplicity free and
$(\operatorname{Lie}U)^{T}=0$, [11, 13.18]. Thus $\dim L>\dim H$. If
$\dim_{\kappa}((\operatorname{Lie}U)^{*})^{H}\geq 2$, then
$\dim_{\kappa}((\operatorname{Lie}U)^{*})^{T\cap H}\geq 2$. Let
$\alpha,\beta\in\Phi$ be distinct roots such that the dual characters
$\alpha^{*}$ and $\beta^{*}$ appear in $((\operatorname{Lie}U)^{*})^{T\cap
H}$. We note that both $\alpha$ and $\beta$ are positive with respect to a
Borel subgroup of $G^{\prime}$ containing $T$ and $U$. Thus
$\alpha\neq-\beta$. Now $\\{\alpha,-\alpha\\}$ are the roots of the
centraliser of $(\operatorname{Ker}\alpha)^{0}$ in $G^{\prime}$, [11,
13.18(4d)]. Thus $\operatorname{Ker}\alpha\neq\operatorname{Ker}\beta$ and
hence $\dim T-\dim(\operatorname{Ker}\alpha\cap\operatorname{Ker}\beta)=2$.
Thus $\dim T-\dim T\cap H\geq 2$, as $T\cap H$ is contained in
$\operatorname{Ker}\alpha\cap\operatorname{Ker}\beta$. Thus part (i) holds. ∎

######  Corollary 10.7.

$((\operatorname{Lie}U)^{*})^{L^{0}}=0$.

###### Proof.

This follows from Lemma 10.6 as $\dim L=\dim L^{0}$. ∎

######  Example 10.8.

Let $G=G^{\prime}=G_{1}\times\operatorname{SL}_{2}$, where $G_{1}$ is any
semisimple group, $L=G_{1}\times\mathbb{G}_{m}$ and $P=G_{1}\times B$, where
$B$ is the subgroup of upper triangular matrices in $\operatorname{SL}_{2}$.
If $H=G_{1}\times\\{1\\}$ then part (ii) of Lemma 10.6 holds.

##  11\. R-Levi subgroups of codimension $2$

Let $G$ be a generalised reductive group over an algebraically closed field
$\kappa$. Let $P$ be an R-parabolic subgroup of $G$ with R-Levi $L$ and
unipotent radical $U$. We have $\dim G-\dim L=2\dim U$. In this section we
assume that $\dim U=1$ and the natural map $L/L^{0}\rightarrow G/G^{0}$ is an
isomorphism. The main result of this section is Proposition 11.5, which is a
criterion for a representation $\rho:\Gamma_{F}\rightarrow L(\kappa)$ to have
$h^{0}(\Gamma_{F},(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{*}(1))=0$,
where $G^{\prime}_{\operatorname{sc}}$ is the simply connected central cover