# Irreducible components of the moduli space of Langlands parameters

Jack Shotton

###### Abstract.

Let $F/\mathbb{Q}_{p}$ be finite and let $\mathfrak{X}_{G}$ be the moduli
space of Langlands parameters valued in $G$, in characteristic distinct from
$p$. First, we determine the irreducible components of $\mathfrak{X}_{G}$.
Then, we determine the local structure around tamely ramified points for which
the image of ‘tame inertia’ is regular. This local structure is related to the
endomorphism rings of Gelfand–Graev representations, by work of Li. Lastly, we
determine an open dense set in $\mathfrak{X}_{M}$, when $M$ is a Levi subgroup
of $G$, such that the natural map of moduli stacks
$[\mathfrak{X}_{M}/M]\to[\mathfrak{X}_{G}/G]$ is smooth on this set.

##  1\. Introduction

Let $p$ be a prime and let $F$ be a finite extension of $\mathbb{Q}_{p}$ with
Weil group $W_{F}$. Let

$G=G^{\circ}\rtimes\Gamma$

with $G^{\circ}$ a split reductive group scheme111Recall that these are
defined to have connected fibres. over $\mathbb{Z}[1/p]$ and $\Gamma$ a finite
quotient of $W_{F}$ equipped with an action on $G^{\circ}$ by automorphisms
preserving a split Borel pair $(B,T)$. Dat–Helm–Kurinczuk–Moss [DHKM20] have
shown that the functor on $\mathbb{Z}[1/p]$-algebras

$R\mapsto\\{\text{$L$-homomorphisms $W_{F}^{0}\to G(R)$}\\}$

is representable by a scheme $\mathfrak{X}_{G}$, locally of finite type and
flat over $\mathbb{Z}[1/p]$ — here $W_{F}^{0}$ is a certain choice of
discretization of $W_{F}$.222In [DHKM20], they write ${}^{L}G$ for our $G$ and
$\hat{G}$ for our $G^{\circ}$. This is more in keeping with the usual notation
in the Langlands program, but leads to a proliferation of ‘$L$’s and hats that
we prefer to avoid.

For $L$ a field of characteristic zero, $\mathfrak{X}_{G,L}$ is known to be
generically reduced by work of Bellovin–Gee [BG19] and so it has a smooth open
dense subset (which may be made explicit). For $L$ of characteristic $l$, this
is far from being true, even for $G=GL_{1}$. However, one can still hope for a
nice description at an open dense set of points. For $G=GL_{n}$, in the tame
case, we did this in [Sho22], finding a local description of
$\mathfrak{X}_{G}$ around a dense subset of its fibre at each prime $l$. This
description turns out to be related to the endomorphism algebra of the
(integral) Gelfand-Graev representation, and we applied this to the $l\neq p$
“Breuil–Mézard” conjecture.

Our aim here is to extend the geometric part of that work to general groups
$G$. We do not quite succeed. Roughly, we can only deal with components on
which the $L$-parameter factors through a Levi subgroup in which ‘inertia is
regular’. For these components, we obtain a description similar to that for
$GL_{n}$, again related to the endomorphism algebras of Gelfand-Graev
representations by work of Li [Li21]. For $G=GL_{n}$ this is everything, but
for general $G$ there are components which fall outside our description. A
limitation is that we restrict to $G$ unramified; however, as in section 5.2
of [DHKM20], in many situations one may reduce to this case. (We also require
$G^{\circ}$ to have smooth centre and simply connected derived subgroup).

The first task, of independent interest, is to determine the geometrically
irreducible components of $\mathfrak{X}_{G}$ and its reductions modulo primes
$l\neq p$. The (geometric) _connected_ components of $\mathfrak{X}_{G}$, as
well as of its base change to $\mathbb{Z}_{l}$ (equivalently,
$\mathbb{F}_{l}$) for primes $l\neq p$, are determined in [DHKM20]; our result
takes a similar form.

We now outline the contents of this paper in more detail. First, the
determination of irreducible components. For $G$ an algebraic group acting on
a scheme $X$, let $G_{x}$ denote the stabiliser of a point $x\in X(R)$ (it is
a closed subscheme of $G_{R}$).

######  Theorem (Corollary 2.7).

Let $L$ be an algebraically closed field of characteristic distinct from $p$.
There is a bijection between the set of irreducible components of
$\mathfrak{X}_{G,L}$ and the set of $G^{\circ}(L)$-conjugacy classes of pairs
$(\eta,\alpha)$ where:

  * •

$\eta$ is a continuous333For the discrete topology on $L$. $L$-homomorphism
$I^{0}_{F}\to G(L)$ over $L$ that extends to $W_{F}$; and

  * •

$\alpha$ is a component of the $(G^{\circ})_{\eta}$-torsor of extensions of
$\eta$ to $W^{0}_{F}$.

Note that two pairs $(\eta,\alpha)$ and $(\eta,\alpha^{\prime})$ are
$G^{\circ}(L)$-conjugate if and only if $\alpha$ and $\alpha^{\prime}$ are
conjugate under $\pi_{0}((G^{\circ})_{\eta})$.

The critical case (by the ‘tame reduction’ of [DHKM20] sections 3 and 4.5 and
the discussion following Lemma 5.7) is when the action of $W_{F}$ on
$G^{\circ}$ is unramified, and $\eta$ is tamely ramified and _unipotent_ , so
determined by a unipotent matrix $\Sigma\in G^{\circ}(L)$. In this case the
irreducible components of type $\Sigma$ are in bijection with twisted
conjugacy classes of the component group of the centralizer of $\Sigma$ in
$G^{\circ}$.

Assume, therefore, that $\Gamma$ corresponds to an unramified extension of
$F$, and define $\mathfrak{X}^{\mathrm{tame}}_{G}$ to be the moduli space of
_tamely ramified_ $L$-parameters. We have a presentation

$W_{F}^{0}/P_{F}\cong\left\langle\operatorname{Fr},\sigma:\operatorname{Fr}\sigma\operatorname{Fr}^{-1}=\sigma^{q}\right\rangle$

Ḋefine $\mathfrak{X}_{G}^{\operatorname{\Sigma-reg}}$ to be the open subset of
parameters $\rho$ where $\Sigma=\rho(\sigma)\in G^{\circ}$ is regular. Let $T$
be a maximal split torus in $G^{\circ}$ preserved by the action of $W_{F}$.
Let $T/\\!\\!/W$ be the GIT quotient of $T$ by the action of the Weyl group
$W$. There is a morphism, the Chevalley morphism,

$\mathrm{Ch}:G^{\circ}\to
G^{\circ}/\\!\\!/G^{\circ}\xrightarrow{\sim}T/\\!\\!/W.$

Let

$(T/\\!\\!/W)^{\operatorname{Fr}^{-1}[q]}$

be the closed subscheme of $T/\\!\\!/W$ fixed by $\operatorname{Fr}^{-1}[q]$,
where $[q]$ is the $q$th power map. Our first result is:

######  Theorem (Theorem 3.2).

Suppose that $G^{\circ}_{\operatorname{der}}$ is simply connected and that the
centre $Z_{G^{\circ}}$ is smooth. Then the morphism

$\mathfrak{X}_{G}^{\operatorname{\Sigma-
reg}}\to(T/\\!\\!/W)^{\operatorname{Fr}^{-1}[q]}$

sending $\rho$ to $\mathrm{Ch}(\rho(\sigma))$ is smooth.

######  Remark 1.1.

Suppose that $G$ is the $L$-group of an unramified group with smooth integral
model $\mathbb{G}/\mathcal{O}_{F}$. In this case, the coordinate ring of
$(T/\\!\\!/W)^{\operatorname{Fr}^{-1}[q]}$ has been shown by Li [Li21] in
characteristic not dividing $|W|$, and Li and the author [LS22] in the case of
good characteristic dividing $|W|$, to be isomorphic to the endomorphism ring
of the Gelfand–Graev representation of $\mathbb{G}(k_{F})$. This is an
inertial shadow of the (conjectural) local Langlands correspondence in
families.

######  Remark 1.2.

Under suitable assumptions on $G$ (as in [XZ19] Remark 4.3.4) we might expect
the a similar result when $G$ is tamely ramified. See Remark 3.8 below.

######  Remark 1.3.

In [DHKM20] Proposition 5.12 one finds conditions under which points of the
special fibre with $\Sigma$ regular unipotent are smooth (at least for $l>h$,
where $h$ is the Coxeter number of $G$). The condition takes the form
$l\nmid\chi_{G}(q)$ where $\chi_{G}$ is a certain product of cyclotomic
polynomials depending on $G$. Our result provides a clean description of
$\mathfrak{X}_{G}$ at such points when $l\mid\chi_{G}(q)$, as long as
$G^{\circ}_{\operatorname{der}}$ is simply connected.

Finally, we can go beyond the $\Sigma$-regular locus with the following
result, for which we can drop the assumption that $G$ is unramified. By a
standard Levi subgroup of ${}^{L}G$ we mean a subgroup of the form
$M=M^{\circ}\rtimes\Gamma$ where $M^{\circ}$ is a $\Gamma$-stable standard
Levi subgroup of $G^{\circ}$ (relative to our fixed split maximal torus).

######  Theorem (Theorem 4.2).

Suppose that $M$ is a Levi subgroup of $G$. Consider the morphism

$c:G^{\circ}\times\mathfrak{X}_{M}\to\mathfrak{X}_{G}$

sending $(g,\rho)$ to $g\rho g^{-1}$. Then there is an open subset
$U\subset\mathfrak{X}_{M}$ such that $c|_{U}$ is smooth, and $U$ intersects
each fibre of $\mathfrak{X}_{M}\to\operatorname{Spec}\mathbb{Z}[1/p]$ in a
dense open subset.

######  Remark 1.4.

The theorem is in the spirit of Lemma 5.14 of [DHKM20], which achieves a
similar reduction for their study of unobstructedness.

For $G=GL_{n}$ the image of

(1)
$\bigcup_{M}\operatorname{im}(G^{\circ}\times\mathfrak{X}^{\operatorname{\Sigma-
reg}}_{M}\to\mathfrak{X}^{\mathrm{tame}}_{G}),$

where $M$ runs over conjugacy classes of standard Levi subgroups, is fibrewise
dense in $\mathfrak{X}_{G}$ (see [Sho22]). In general this is not true; there
are two issues coming from the disconnectedness of component groups and the
existence of non-regular distinguished unipotent conjugacy classes. We give
examples in section 5.

In [Sho22] we were able to use this and the desciption of the coordinate ring
of $(T/\\!\\!/W)^{[q]}$ as the endomorphism ring of a Gelfand–Graev
representation444In fact, we just need the weaker result after
$\otimes\overline{\mathbb{Q}}$. to give a local proof of the author’s “$l\neq
p$ Breuil–Mézard conjecture” in the tame case. It should be possible to do
something similar for unramified groups, but only for modular $L$-parameters
which lie only on components in the image of (1). Given this serious
restriction, we will not discuss this point further here.

###  1.1. Acknowledgments

We thank David Helm and Tzu-Jan Li for helpful conversations, particularly
concerning (respectively) the content of section 2 and the significance of
$(T/\\!\\!/W)^{\operatorname{Fr}^{-1}[q]}$.

##  2\. The moduli space of parameters

###  2.1. Notation

As in the introduction, let $F/\mathbb{Q}_{p}$ be a finite extension with ring
of integers $\mathcal{O}_{F}$ and residue field $k_{F}$ of order $q$. Let
$W_{F}$ be the Weil group of $F$ with inertia group $I_{F}$ and wild inertia
group $P_{F}$. We choose a lift $\operatorname{Fr}\in W_{F}$ of arithmetic
Frobenius and a lift $\sigma\in I_{F}$ of a topological generator of
$I_{F}/P_{F}$ in such a way that
$\operatorname{Fr}\sigma\operatorname{Fr}^{-1}=\sigma^{q}$ (this can be done
using the Schur–Zassenhaus theorem and an argument as in [CHT08] Lemma
2.4.10). Let $W_{F}^{0}$ be the ubgroup of $W_{F}$ generated by
$\sigma,\operatorname{Fr}$, and $P_{F}$; its intersection with $I_{F}$ is
denoted $I_{F}^{0}$. We have that $I_{F}^{0}/P_{F}\cong\mathbb{Z}[1/p]$ via an
isomorphism sending $\sigma$ to $1$. We topologise $W_{F}^{0}$ such that the
open subsets are cosets of open subsets of $P_{F}$ with its profinite
topology.

Let $G$ be an algebraic group over $\mathbb{Z}[1/p]$ of the form
$G^{\circ}\rtimes\Gamma$ where: $G^{\circ}$ is a split reductive group scheme
over $\mathbb{Z}[1/p]$, $\Gamma$ is a finite quotient of $W_{F}$, and $\Gamma$
acts on $G^{\circ}$ via automorphisms preserving a split Borel pair $(B,T)$.
We write the action of $W_{F}$ on $G^{\circ}$ using left superscripts:
$(w,g)\mapsto{}^{w}g$. Finally, we let
$(X^{*}(T),\Delta,X_{*}(T),\Delta^{\vee})$ be the root datum associated to
$G^{\circ}$, together with its action of $W_{F}$.

If $H$ is a subgroup of $W_{F}$ and $R$ is a $\mathbb{Z}[1/p]$-algebra, then
an $L$-homomorphism $H\to G(R)$ is a continuous (for the discrete topology on
$G(R)$) homomorphism

$\rho:H\to G(R)$

such that the composite of $\rho$ with $G(R)\to\Gamma$ agrees with $H\to
W_{F}\to\Gamma$ coming from the given surjection. For $h\in H$, we write
$\rho(h)=(\rho^{\circ}(h),h)$ (slightly loosely using the same letter $h$ for
an element of $H$ and its image in $\Gamma$). If $H\subset H^{\prime}$ then by
an extension of $\rho$ to $H^{\prime}$ we will mean an $L$-homomorphism
$\tilde{\rho}:H^{\prime}\to G(R)$ whose restriction to $H^{\prime}$ is $\rho$.

When $G^{\circ}$ acts on a scheme $X$ and $\eta\in X(S)$, we write $G_{\eta}$
for the stabiliser group scheme (defined over $S$) and $G^{\circ}_{\eta}$ for
the connected component of the identity in $G_{\eta}$ (we only use this over a
field). Note that $G^{\circ}_{\eta}=(G_{\eta})^{\circ}$ and not
$(G^{\circ})_{\eta}$!

###  2.2. $L$-parameters

Let $\Lambda$ be a $\mathbb{Z}[1/p]$ algebra. One of the main results of
[DHKM20] is the following.

######  Theorem 2.1 (Dat–Helm–Kurinczuk–Moss).

The functor sending a $\Lambda$-algebra $R$ to the set of $L$-homomorphisms
$W_{F}^{0}\to G(R)$ is representable by a scheme $\mathfrak{X}_{G}$ over
$\Lambda$. Moreover, $\mathfrak{X}_{G}$ has the following properties:

  1. (1)

The morphism $\mathfrak{X}_{G}\to\operatorname{Spec}\Lambda$ is syntomic
(flat, locally of finite presentation, with fibres that are local complete
intersections) of relative dimension $\dim G+1$.

  2. (2)

If $\Lambda$ is a field of characteristic zero then $\mathfrak{X}_{G}$ is
reduced.555This part relies crucially on prior work of Bellovin and Gee
[BG19].

  3. (3)

The connected components of $\mathfrak{X}_{G}$ are finitely presented over
$\Lambda$.

In the following $\Lambda$ will be vary according to the context. We will
sometimes write $\mathfrak{X}_{G,\Lambda}$ for clarity, noting that
$\mathfrak{X}_{G,\Lambda}=\mathfrak{X}_{G,\mathbb{Z}[1/p]}\times_{\operatorname{Spec}\mathbb{Z}[1/p]}\operatorname{Spec}\Lambda$.

###  2.3. Connected components

Fix a prime $l\neq p$, and let $I^{l}_{F}\subset I_{F}$ be the maximal closed
subgroup of pro-order coprime to $l$.

######  Theorem 2.2 (Dat–Helm–Kurinczuk–Moss).

Let $\Lambda=W(\overline{\mathbb{F}}_{l})$. There are bijections between the
sets of:

  1. (1)

Connected components of $\mathfrak{X}_{G}$;

  2. (2)

Connected components of $\mathfrak{X}_{G,\overline{\mathbb{F}}_{l}}$; and

  3. (3)

$G^{\circ}(\overline{\mathbb{F}}_{l})$-conjugacy classes of pairs
$(\eta,\alpha)$ where $\eta:I^{l}_{F}\to G(\overline{\mathbb{F}}_{l})$ is an
$L$-homomorphism and $\alpha$ is a connected component of the variety of
extensions of $\eta$ to $W_{F}^{0}I^{l}_{F}$.

If instead $L$ is an algebraically closed field of characteristic distinct
from $p$, then we have bijections between the sets of:

  1. (4)

Connected components of $\mathfrak{X}_{G,L}$; and

  2. (5)

$G^{\circ}(L)$-conjugacy classes of pairs $(\eta,\alpha)$ where $\eta:I_{F}\to
G(L)$ is $G$-semisimple (see [DHKM20] Definition 4.12) and $\alpha$ is a
connected component of the variety of extensions of $\eta$ to $W_{F}$.

###### Proof.

The first three parts correspond to sets (1), (2) and (4) in [DHKM20] Theorem
4.21, upon noting that:

  * •

The subset of admissible elements of the set $\Sigma(\eta)$ defined there is
the same as the set of connected components of the variety of extensions
considered above.

  * •

We may work over $\Lambda$ rather than the base $\Lambda_{e}$ by Theorem 4.8
of [DHKM20].

The bijection between (4) and (5) appears in section 1.4 of [DHKM20]. ∎

######  Remark 2.3.

Since $\eta$ in (5) has finite image (since it is assumed to be continuous for
the discrete topology on $L$), if $L$ has characteristic zero then the
$G$-semisimplicity condition is automatic.

###  2.4. Irreducible components

Let $\Lambda=L$ be an algebraically closed field of charracteristic distinct
from $p$. We aim to determine the irreducible components of
$\mathfrak{X}_{G}=\mathfrak{X}_{G,L}$. _For this section, we base change
everything to $L$ — so $G=G_{L}$ etc._

######  Definition 2.4.

An $L$-homomorphism $\eta:I_{F}^{0}\to G(L)$ is _admissible_ if it extends to
an $L$-homomorphism $W_{F}^{0}\to G(L)$.

Given an admissible $\eta$, we consider the affine $L$-scheme $Y_{\eta}$
defined on $L$-algebras $R$ by

$Y_{\eta}(R)=\\{\rho:W_{F}^{0}\to G(R)\text{ extending $\eta$}\\}.$

We write $\Sigma(\eta)$ for its set of connected components (so we use the
same notation as that for a similarly-defined set in [DHKM20] section 3.2).

If we write $\Phi=\rho(\operatorname{Fr})$ then

$Y_{\eta}\cong\\{\Phi\in
G:\Phi\eta(\gamma)\Phi^{-1}=\eta(\operatorname{Fr}\gamma\operatorname{Fr}^{-1})\text{
for all $\gamma\in I_{F}^{0}$}\\}$

and we see that $Y_{\eta}$ is a left $G^{\circ}_{\eta}$-torsor via left
multiplication on $\Phi$. If we fix $\rho_{0}$ extending $\eta$ with
$\rho_{0}(\operatorname{Fr})=\Phi_{0}$, then we can identify $Y_{\eta}\cong
G^{\circ}_{\eta}$, with a general point having
$\rho(\operatorname{Fr})=\phi\Phi_{0}$ for $\phi\in G^{\circ}_{\eta}$.

There is a second action of $G^{\circ}_{\eta}$ on $Y_{\eta}$ given by
conjugation. On choosing $\rho_{0}$ as in the previous paragraph, this may be
identified with the action of $G^{\circ}_{\eta}$ on itself by
$\operatorname{Ad}_{\Phi_{0}}$-twisted conjugation.

If $\mathfrak{I}$ is the moduli space of $L$-homomorphisms $I_{F}^{0}\to G(R)$
(for $L$-algebras $R$), then $\mathfrak{I}$ is a union of connected components
that are each affine of finite type over $L$. The $G^{\circ}$-orbit of
$\eta\in\mathfrak{I}(L)$, $G^{\circ}\cdot\eta$, then naturally has the
structure of a quasiaffine variety and, by [Mil17] Corollary 7.13, there is an
isomorphism

$G^{\circ}/G^{\circ}_{\eta}\xrightarrow{\sim}G^{\circ}\cdot\eta.$

There is a natural morphism $p_{I}:\mathfrak{X}_{G}\to\mathfrak{I}_{L}$ and we
write

$\mathfrak{X}_{\eta}=p_{I}^{-1}(G^{\circ}\cdot\eta),$

a locally closed subscheme of $\mathfrak{X}_{\eta}$.

######  Theorem 2.5.

For each admissible $L$-homomorphism $\eta:I_{F}^{0}\to G(L)$, every connected
component of $\mathfrak{X}_{\eta}$ is irreducible, and its Zariski closure in
$\mathfrak{X}_{G}$ is an irreducible component of $\mathfrak{X}_{G}$.

If $\eta$ is chosen as above, then there is a bijection between:

  1. (1)

The set of connected (or irreducible) components of $\mathfrak{X}_{\eta}$;

  2. (2)

The set of $\pi_{0}(G_{\eta}^{\circ})$-conjugacy classes in $\Sigma(\eta)$.

Choosing a $\rho_{0}:W^{0}_{F}\to G(L)$ extending $\eta$, then these are in
bijection with the $\operatorname{Ad}_{\rho_{0}(\operatorname{Fr})}$-twisted
conjugacy classes in $\pi_{0}(G^{\circ}_{\eta})$.

######  Remark 2.6.

From the proof of the theorem we will see that $|\mathfrak{X}_{\eta}|$ is an
open subset of an irreducible component of $|\mathfrak{X}_{G}|$. However, if
$\operatorname{char}L\neq 0$, then neither $\mathfrak{X}_{\eta}$ nor
$\mathfrak{X}_{G}$ need be reduced, and the non-reduced structures will
typically not agree.

###### Proof.

Note that we have an isomorphism

$\displaystyle Y_{\eta}\times G^{\circ}$
$\displaystyle\cong\mathfrak{X}_{\eta}\times_{G^{\circ}\cdot\eta}G^{\circ}$
$\displaystyle(\rho,g)$ $\displaystyle\mapsto(g\rho g^{-1},g)$

by definition. Taking the quotient by $G^{\circ}_{\eta}$ and using that
$G^{\circ}/G^{\circ}_{\eta}\cong G^{\circ}\cdot\eta$, we obtain an isomorphism

$\mathfrak{X}_{\eta}\cong G^{\circ}\times^{G^{\circ}_{\eta}}Y_{\eta}.$

It follows that $\mathfrak{X}_{\eta}$ is a disjoint union of irreducible
connected components corresponding to the $G^{\circ}_{\eta}$-orbits in
$\pi_{0}(Y_{\eta})=\Sigma(\eta)$, which are the same as the
$\pi_{0}(G_{\eta}^{\circ})$-orbits.

Each component of $Y_{\eta}$ has dimension $\dim G^{\circ}_{\eta}$ and so the
dimension of every component of $\mathfrak{X}_{\eta}$ is

$\dim G^{\circ}+\dim G^{\circ}_{\eta}-\dim G^{\circ}_{\eta}=\dim
G^{\circ}=\dim\mathfrak{X}_{G}.$

Since $G^{\circ}\cdot\eta$ is locally closed, the same is true for
$\mathfrak{X}_{\eta}$. It follows that the underlying topological space of
each connected component of $\mathfrak{X}_{\eta}$ is an open subset of an
irreducible component of $\mathfrak{X}_{\eta}$.

∎

######  Corollary 2.7.

There are bijections between:

  1. (1)

The set of irreducible components of $\mathfrak{X}_{G}$;

  2. (2)

The set of $G^{\circ}$-conjugacy classes of pairs $(\eta,C)$ where
$\eta:I_{F}^{0}\to G(L)$ is an admissible $L$-homomorphism and $C$ is a
connected component of $\Sigma(\eta)$.

Let $\mathcal{T}$ be a set of representatives of the $G^{\circ}$-conjugacy
classes $\eta$ as in (2). For each $\eta\in\mathcal{T}$, choose an extension
$\rho_{\eta}$ of $\eta$ to $W_{F}^{0}$.

Then these sets are in bijection with the set

$\\{\text{$(\eta,C)\;:\;\eta\in\mathcal{T}$, $C$ an
$\operatorname{Ad}_{\rho_{\eta}(\operatorname{Fr})}$-twisted conjugacy class
of $\pi_{0}(G^{\circ}_{\eta})$}\\}.$

###### Proof.

By Theorem 2.5, it is enough to prove that every irreducible component of
$\mathfrak{X}_{G}$ arises as the closure of an irreducible component of some
$\mathfrak{X}_{\eta}$, for $\eta$ unique up to $G^{\circ}$-conjugacy.

To this end, for each irreducible component $\mathcal{C}$ of
$\mathfrak{X}_{G}$, let $\rho$ be an $L$-point that lies only on that
component and take $\eta=\rho|_{I_{F}^{0}}$. Then one of the components of
$\mathfrak{X}_{\eta}$ contains $\rho$, and its closure is an irreducible
component of $\mathfrak{X}_{G}$ containing $\rho$, which must therefore be
equal to $\mathcal{C}$.

Finally, to see that $\eta$ is unique, note that if $\mathcal{C}$ is the
closure of a connected component of $\mathfrak{X}_{\eta}$ then it has an open
dense subset of points for which the restriction to $I_{F}^{0}$ is
$G^{\circ}$-conjugate to $\eta$. This can clearly happen for only one
$G^{\circ}$-conjugacy class of $\eta$. ∎

##  3\. The $\Sigma$-regular locus

###  3.1. Regular elements and the Chevalley isomorphism

_For this subsection only,_ let $G$ be a split connected reductive group
defined over a ring $\Lambda$. Let $T$ be a split maximal torus and $B\supset
T$ a Borel subgroup. Let $W$ be the Weyl group of $G$ relative to $T$. Let $r$
be the rank of $T$.

Consider the conjugation action of $G$ on itself, and denote the GIT quotient
$\operatorname{Spec}\Lambda[G]^{G}$ by $G/\\!\\!/G$. We also define

$T/\\!\\!/W=\operatorname{Spec}\Lambda[T]^{W}=\operatorname{Spec}\Lambda[X^{*}(T)]^{W}.$

######  Theorem 3.1.

The restriction morphism

$G/\\!\\!/G\to T/\\!\\!/W$

is an isomorphism.

###### Proof.

Over a field this is a theorem of Steinberg, [Ste65]. Over a general ring see
[Lee15]. ∎

If $L$ is a field (and a $\Lambda$-algebra) then, for any $g\in G(L)$, its
centraliser $G_{g}$ has dimension $\geq r$, and we say that $g$ is regular if
equality holds. Then

$G^{\operatorname{reg}}=\\{g\in G:\text{$g$ is regular in $G(\kappa(g))$}\\}$

is an open subset of $G$ (by semicontinuity, [Ber65] Proposition 4.1).

######  Theorem 3.2.

Suppose that $G_{\operatorname{der}}$ is simply connected. Then

  1. (1)

The natural morphism

$\mathrm{Ch}:G\to G/\\!\\!/G$

is faithfully flat, and its restriction to $G^{\operatorname{reg}}$ is smooth.

  2. (2)

There is an isomorphism

$G/\\!\\!/G\cong\mathbb{A}^{r}_{\Lambda}\times G/G_{\operatorname{der}}.$

###### Proof.

  1. (1)

By the fibrewise criterion for flatness, we may check this after base-change
to a field. It then follows from [BR22] Propositions 2.4 and 2.5.

  2. (2)

This is [BR22] Lemma 2.3 over a field; however, the proof works over $\Lambda$
if one takes $\chi_{\alpha}$ in that proof to be the character of
$\operatorname{Ind}_{B}^{G}\omega_{\alpha}$ rather than its socle. (In the
semisimple simply connected case, see also [Lee15] Corollary 5.2).

∎

Finally, we recall the statement of [Cot22] Theorem 7.13 (noting that, if
$G_{\operatorname{der}}$ is simply connected, then ‘regular’ and ‘strongly
regular’ in [Cot22] are equivalent).

######  Proposition 3.3.

(Cotner) Suppose that $G_{\operatorname{der}}$ is simply connected, that
$\Lambda=L$ is a field, and that $Z_{G}$ is smooth. Then, for every regular
element $g\in G(L)$, the centralizer $G_{g}$ is smooth.

###  3.2. Application to $\mathfrak{X}_{G}$

Return to the notation of section 2.1. For this subsection we assume that $G$
is _unramified_ , in the sense that $I_{F}$ acts trivially on $G^{\circ}$. We
then may assume that $I_{F}\in\ker(W_{F}\to\Gamma)$, and consider only _tame_
parameters: those that factor through $W_{F}^{0}/P_{F}$. Define
$\mathfrak{X}^{\mathrm{tame}}_{G}$ to represent the functor on
$\Lambda$-algebras

$\mathfrak{X}^{\mathrm{tame}}_{G}(R)=\\{\text{$L$-parameters
$\rho:W_{F}^{\circ}/P_{F}\to G(R)$}\\}.$

######  Remark 3.4.

By [DHKM20] (4.4) and (4.5), the critical case to consider is that of tame
parameters for tamely ramified $G$ — and when considering
$\mathfrak{X}^{\mathrm{tame}}_{G,\mathbb{Z}_{l}}$ it is possible to assume
that the order of the automorphism $\sigma$ is a power of $l$. In particular,
if $G$ is semisimple and simply connected, then it is no great loss to assume
that $G$ is unramified so long as $l>3$.

We have that

$W_{F}^{0}/P_{F}=\left\langle\sigma,\operatorname{Fr}:\operatorname{Fr}\sigma\operatorname{Fr}^{-1}=\sigma^{q}\right\rangle$

and, for a parameter $\rho:W_{F}^{0}\to G(R)$, we write

$\rho(\sigma)=\Sigma_{\rho}\in G^{\circ}$

(recalling that $G$ is unramified) and

$\rho(\operatorname{Fr})=(\Phi_{\rho},\operatorname{Fr}).$

Let $p_{\Sigma}$ and $p_{\Phi}$ be the morphisms
$\mathfrak{X}^{\mathrm{tame}}_{G}\to G^{\circ}$ taking $\rho$ to
$\Sigma_{\rho}$ and $\Phi_{\rho}$ respectively.

Let $A_{G}=G^{\circ}/\\!\\!/G^{\circ}\cong T/\\!\\!/W$. By Theorem 3.1 we have
a morphism

$\mathrm{Ch}:G^{\circ}\to A_{G}.$

Note that $\operatorname{Fr}$ acts on $W$, $T$, and $A_{G}$. Let
$[q]:G^{\circ}\to G^{\circ}$ be the $q$th power map.

The morphisms $\operatorname{Fr}$ and $[q]$ each descend to morphisms, also
denoted $\operatorname{Fr}$ and $[q]$, from $G^{\circ}/\\!\\!/G^{\circ}$ to
$G^{\circ}/\\!\\!/G^{\circ}$. We let

$B_{G}=(G^{\circ}/\\!\\!/G^{\circ})^{\operatorname{Fr}^{-1}[q]}=A_{G}^{\operatorname{Fr}^{-1}[q]}$

be the fixed-point subscheme of $A_{G}$ under the map
$\operatorname{Fr}^{-1}[q]$.

######  Lemma 3.5.

The morphism $\mathfrak{X}^{\mathrm{tame}}_{G}\to A_{G}$ sending $\rho$ to
$\mathrm{Ch}(\Sigma_{\rho})$ factors through a map

$\mathrm{Ch}_{\Sigma}:\mathfrak{X}^{\mathrm{tame}}_{G}\to B_{G}.$

###### Proof.

If $\rho$ is an $R$-point of $\mathfrak{X}^{\mathrm{tame}}_{G}$ for a
$\Lambda$-algebra $R$, then
$\rho(\operatorname{Fr})\Sigma_{\rho}\rho(\operatorname{Fr})^{-1}=\Sigma_{\rho}^{q}$.
This is equivalent to

$\Phi_{\rho}{}^{\operatorname{Fr}}\Sigma_{\rho}\Phi_{\rho}^{-1}=\Sigma_{\rho}^{q}$

and so

$\mathrm{Ch}({}^{\operatorname{Fr}}\Sigma_{\rho})=\mathrm{Ch}(\Sigma_{\rho}^{q})=[q]\mathrm{Ch}(\Sigma_{\rho}).$

Thus
$\mathrm{Ch}(\Sigma_{\rho})\in(G^{\circ}/\\!\\!/G^{\circ})^{\operatorname{Fr}=[q]}=B_{G}$
as required. ∎

Let $\mathfrak{X}_{G}^{\operatorname{\Sigma-
reg}}=p_{\Sigma}^{-1}(G^{\circ,\operatorname{reg}})$. It is an open subscheme
of $\mathfrak{X}^{\mathrm{tame}}_{G}$.

######  Theorem 3.6.

Suppose that $G^{\circ}_{\operatorname{der}}$ is simply connected and that
$Z_{G^{\circ}}$ is smooth over $\Lambda$. Then the restriction of
$\mathrm{Ch}_{\Sigma}$ to $\mathfrak{X}_{G}^{\operatorname{\Sigma-reg}}$ is
smooth.

###### Proof.

Let

$\displaystyle\mathfrak{Z}^{reg}$
$\displaystyle=G^{\circ,\operatorname{reg}}\times_{A_{G}}B_{G}$
$\displaystyle=\\{\Sigma\in
G^{\circ,\operatorname{reg}}:\mathrm{Ch}({}^{\operatorname{Fr}}\Sigma)=\mathrm{Ch}(\Sigma^{q}).\\}$

Then the map $\mathrm{Ch}:\mathfrak{Z}^{\operatorname{reg}}\to B_{G}$ is
smooth by Theorem 3.2 (2), and
$p_{\Sigma}|_{\mathfrak{X}^{\operatorname{\Sigma-reg}}}$ factors through a map

$p_{\Sigma}:\mathfrak{X}^{\operatorname{\Sigma-
reg}}\to\mathfrak{Z}^{\operatorname{reg}}$

that we wish to show is smooth.

Note that there is a closed immersion

$\displaystyle\iota:\mathfrak{Z}^{reg}$ $\displaystyle\to
G^{\circ,\operatorname{reg}}\times_{A_{G}}G^{\circ,\operatorname{reg}}$
$\displaystyle\Sigma$
$\displaystyle\mapsto({}^{\operatorname{Fr}}\Sigma,\Sigma^{q}).$

If we let

$c:G^{\circ}\times G^{\operatorname{reg}}\to
G^{\circ,\operatorname{reg}}\times_{A_{G}}G^{\circ,\operatorname{reg}}$

be the conjugation morphism $c(\gamma,g)=(g,\gamma g\gamma^{-1})$ then the
pullback of $\mathfrak{Z}^{reg}$ along $c$ is

$\displaystyle\\{(\gamma,g,\Sigma)\in G^{\circ}\times
G^{\circ,\operatorname{reg}}\times
G^{\circ,\operatorname{reg}}:g={}^{\operatorname{Fr}}\Sigma,\gamma
g\gamma^{-1}=\Sigma^{q}\\}$
$\displaystyle=\\{(\gamma,\Sigma):\gamma{}^{\operatorname{Fr}}\Sigma\gamma^{-1}=\Sigma^{q}\\}$
$\displaystyle=\mathfrak{X}_{G}^{\operatorname{\Sigma-reg}}.$

It therefore suffices to show that $c$ is smooth. We think of $c$ as a
morphism of $A_{G}$-schemes where $G^{\circ}\times G^{\operatorname{reg}}$ is
regarded as an $A_{G}$-scheme via the second projection. Since $G\times
G^{\operatorname{reg}}$ is flat over $A_{G}$ by Theorem 3.2 (1), we may apply
the fibrewise criterion. It is therefore enough to show that, for every
geometric point $s:\kappa\to A_{G}$,

$c_{s}:G^{\circ}_{\kappa}\times G^{\circ,\operatorname{reg}}_{s}\to
G^{\circ,\operatorname{reg}}_{s}\times G^{\circ,\operatorname{reg}}_{s}$

is smooth.

By Theorem 3.2 (2), $G^{\circ,\operatorname{reg}}_{s}$ is smooth and by [BR22]
Proposition 2.5 it is a single $G^{\circ}$-orbit. The fibre of $c_{s}$ above a
point $(g,g^{\prime})\in G^{\circ,\operatorname{reg}}_{s}(\kappa)\times
G^{\circ,\operatorname{reg}}_{s}(\kappa)$ is then a
$G^{\circ}_{\kappa,g}$-torsor. It follows from the miracle flatness theorem
that $c_{s}$ is flat, and from Proposition 3.3 that it is smooth. ∎

As an application, we generalise [Hel20] Proposition 5.3.

######  Corollary 3.7.

Suppose that $\Lambda=W(\mathbb{F}_{l})$ and that the hypotheses of Theorem
3.6 hold. Let $S=\mathcal{O}(\mathfrak{X}^{\mathrm{tame}}_{G})^{G^{\circ}}$
and $R=\mathcal{O}(B_{G})$. Then the natural map
$\mathrm{Ch}_{\Sigma}^{*}:R\to S$ is injective with saturated image.

###### Proof.

Injectivity follows from the faithful flatness of $\mathrm{Ch}_{\Sigma}$
restricted to the regular locus. In fact, the same argument shows that
$\mathrm{Ch}_{\Sigma}^{*}\otimes\mathbb{F}_{l}$ is injective. Since $R$ is a
finite rank $\Lambda$-algebra, this implies that the image of
$\mathrm{Ch}^{*}_{\Sigma}$ is saturated. ∎

######  Remark 3.8.

Remark 3.4 notwithstanding, it would be desirable to generalise these results
to the case that $G$ is tamely ramified. In this case
$G^{\circ}/\\!\\!/G^{\circ}$ should be replaced by

$A_{G,\sigma}=(G^{\circ}\rtimes\sigma)/\\!\\!/G^{\circ}=G^{\circ}/\\!\\!/_{\sigma}G^{\circ}$

(where $/\\!\\!/_{\sigma}$ denotes the quotient for $G^{\circ}$ acting by
$\sigma$-twisted conjugation). The $q$-power map and Frobenius map define
$G^{\circ}$-equivariant morphisms

$G^{\circ}\rtimes\sigma\to G^{\circ}\rtimes\sigma^{q}$

and we let $B_{G,\sigma}=A_{G,\sigma}^{\operatorname{Fr}^{-1}[q]}$ be their
equaliser. There is a notion of $\sigma$-regularity, for which see [XZ19]
section 5. We then expect that, under appropriate hypotheses on $G^{\circ}$
(perhaps those of [XZ19] Remark 4.3.4 together with a hypothesis on the
centre) there will be a smooth morphism
$\mathfrak{X}_{G}^{\operatorname{\Sigma-reg}}\to B_{G,\sigma}$. For this we
would require generalizations of the results of section 3.1: that is, the
results of [XZ19] sections 4 and 5 over a general base and for a reductive
(rather than semisimple) group, and Proposition 3.3 in the twisted context.

######  Remark 3.9.

If $G$ is the $L$-group of an unramified group then we have a representation-
theoretic interpretation of $B_{G}$, see Remark 1.1. It would be interesting
to have a similar representation-theoretic interpretation of $B_{G,\sigma}$,
as defined in Remark 3.8, in the tamely ramified case.

##  4\. Levi subgroups

For this section, $\Lambda$ is any $\mathbb{Z}[1/p]$-algebra, and we no longer
assume that $G$ is unramified. Let $M\subset G$ be a standard Levi subgroup:
that is, a subgroup of the form $M^{\circ}\rtimes\Gamma$ where $M^{\circ}$ is
a $\Gamma$-stable standard (with respect to our chosen $T$) Levi subgroup of
$G^{\circ}$ (see [Bor79] section 3). Let $P^{\circ}=M^{\circ}U^{\circ}$ be the
associated standard parabolic subgroup and unipotent radical, and
$U^{\circ,{-}}$ the opposite of $U^{\circ}$. We have a decomposition
$\operatorname{Lie}(G)=\operatorname{Lie}(M)\oplus\operatorname{Lie}(U)\oplus\operatorname{Lie}(U^{-})$
that is stable under the action of $W_{F}^{0}$ on $\operatorname{Lie}(G)$. We
have a natural closed immersion

$\mathfrak{X}_{M}\hookrightarrow\mathfrak{X}_{G}$

and a conjugation map

$\displaystyle c:G^{\circ}\times\mathfrak{X}_{M}$
$\displaystyle\to\mathfrak{X}_{G}$ $\displaystyle(g,\rho)$
$\displaystyle\mapsto g\rho g^{-1}$

We will show that there is an open subset $U\subset\mathfrak{X}_{M}$ such that
$c:G^{\circ}\times U\to\mathfrak{X}_{G}$ is smooth. If we fix an exhaustive
filtration $(P^{d}_{F})_{d\geq 0}$ of $P_{F}$ by compact open subgroups then

$\mathfrak{X}_{M}^{d}=\\{\rho\in\mathfrak{X}_{M}:\rho|_{P_{F}^{d}}\text{ is
trivial}\\}$

is a union of connected components of $\mathfrak{X}_{M}$.

Let $d\geq 1$ and let $r$ be an integer, depending on $d$, such that:

  * •

$\operatorname{Fr}^{r}=e$ in $\Gamma$;

  * •

The conjugation action of $\operatorname{Fr}^{r}$ on $P_{F}/P_{F}^{d}$ is
trivial; and

  * •

For any semisimple $s\in G(\overline{\mathbb{Q}})\rtimes\sigma$ such that $s$
is conjugate to $s^{q}$, we have $s=s^{q^{r}}$.

Such an integer must exist; for instance, choose a faithful representation
$G\hookrightarrow GL_{N}$ and take
$r=n!\operatorname{ord}_{\Gamma}(\operatorname{Fr})|\operatorname{Aut}(P_{F}/P_{F}^{d})|$.666See
also [DHKM20] Lemma 2.2. Say that $m\in M^{\circ}(R)\rtimes\operatorname{Fr}$
is _avoidant at depth $d$_ if $\operatorname{ad}_{m}-1$ and
$\operatorname{ad}_{m}-q$ are invertible on $\operatorname{Lie}(U)(R)$ and
$\operatorname{Lie}(U^{-})(R)$, and the following condition holds: if $P$ is
the characteristic polynomial of $\operatorname{ad}_{m^{r}}$ on
$\operatorname{Lie}(M)(R)\oplus\operatorname{Lie}(U)(R)$, and
$\gamma\in\Gamma$, then $P(\operatorname{ad}_{{}^{\gamma}m^{r}})$ is
invertible on $\operatorname{Lie}(U^{-})$; similarly with the roles of $U$ and
$U^{-}$ reversed. The relevance of the first two conditions will become clear,
while the third is used via the following lemma.

######  Lemma 4.1.

Suppose that $a,b\in M^{\circ}(\overline{\mathbb{Q}})$ satisfy the condition
that, if $P$ is the characteristic polynomial of $\operatorname{ad}_{a}$ on
$\operatorname{Lie}M\oplus\operatorname{Lie}P$, then
$P(\operatorname{ad}_{b})$ is invertible on $\operatorname{Lie}(U^{-})$, and
similarly with the roles of $U$ and $U^{-}$ reversed.

If $a=gbg^{-1}$ for some $g$ in $G^{\circ}(\overline{\mathbb{Q}})$, then $g\in
M^{\circ}(\overline{\mathbb{Q}})$.

###### Proof.

Let $g\in G^{\circ}$ such that $gag^{-1}=b$. Then $\operatorname{ad}_{g}$
takes each (generalised) eigenspace of $\operatorname{ad}_{a}$ to a
generalised eigenspace of $\operatorname{ad}_{b}$ with the same eigenvalue.
The assumption on $a$ and $b$ then implies that
$\operatorname{Lie}(M)\oplus\operatorname{Lie}(U)$ is stable under
$\operatorname{ad}_{g}$, from which it follows that the parabolic subgroup
$M^{\circ}U$ is normalised by $g$. Since a parabolic subgroup is its own
normaliser, we see that $g\in M^{\circ}U$. Arguing similarly with $U$ and
$U^{-}$ reversed, we see that $g\in M^{\circ}U\cap M^{\circ}U^{-}=M^{\circ}$
as required. ∎

Write $M^{d,a}$ for the open subscheme of $M^{\circ}\rtimes\operatorname{Fr}$
consisting of those $\Phi$ avoidant at depth $d$. Let
$\mathfrak{X}_{M}^{d,a}=\\{\rho\in\mathfrak{X}^{d}_{M}:\rho(\operatorname{Fr})\in
M^{a}\\}$, and let $\mathfrak{X}_{M}^{a}=\cup_{d\geq
1}\mathfrak{X}_{M}^{d,a}$. We have a morphism

$c:G^{\circ}\times\mathfrak{X}_{M}\to\mathfrak{X}_{G}$

given by $c(g,\rho)=g\rho g^{-1}$. This descends to a morphism
$\overline{c}:G^{\circ}\times^{M^{\circ}}\mathfrak{X}_{M}\to\mathfrak{X}_{G}$.

######  Theorem 4.2.

  1. (1)

The open subscheme $\mathfrak{X}_{M}^{a}\subset\mathfrak{X}_{M}$ is fibrewise
dense.

  2. (2)

The restriction of $c$ to $G^{\circ}\times\mathfrak{X}_{M}^{a}$ is smooth.

  3. (3)

The restriction of $\overline{c}$ to
$G^{\circ}\times^{M^{\circ}}\mathfrak{X}_{M}^{a}$ is étale.

###### Proof.

If $M=G$ then both statements are trivial. So assume that $M$ is a _proper_
standard Levi subgroup. It is enough to prove the theorem for
$\mathfrak{X}_{M}^{d,a}$ for each fixed $d\geq 1$, so fix such a $d$.

  1. (1)

Let $J\subsetneq\Delta$ be the set of simple roots associated to $M$. As in
the proof of [Bor79] Lemma 3.5, if we let

$S=\\{s\in Z(M^{\circ})^{\circ}\cap
G^{\circ}_{\operatorname{der}}:\alpha(s)=\beta(s)\text{for all
$\alpha,\beta\in\Delta\setminus J$}\\},$

then $S$ is a rank one torus and $M=Z_{G}(S)$. In particular, $W_{F}^{0}$ acts
trivially on $S$. There is a character $\lambda:S\to\mathbb{G}_{m}$ such that
the weights of $S$ acting adjointly on $U$ (respectively, $U^{-}$) are
positive (respectively, negative) powers of $\lambda$. For $s\in S$, let
$\chi_{s}$ be the unramified character of $W_{F}^{0}$ sending
$\operatorname{Fr}$ to $s$.

Suppose that $x\in\operatorname{Spec}\Lambda$ and that $\rho$ is a closed
point of the fibre $\mathfrak{X}_{M,x}$ with residue field $\kappa$. Consider
the map

$\displaystyle S_{\kappa}$ $\displaystyle\to\mathfrak{X}_{M,\kappa}$
$\displaystyle s$ $\displaystyle\mapsto\chi_{s}\rho.$

The eigenvalues of $\operatorname{ad}_{s\rho(\operatorname{Fr})}$ acting on
$\operatorname{Lie}(U)$ are of the form $\lambda(s)^{n}\mu$ for $\mu$ an
eigenvalue of $\operatorname{ad}_{\rho(\operatorname{Fr})}$ and $n$ positive.
Similarly, for each $\gamma\in\Gamma$, the eigenvalues of
$\operatorname{ad}_{s{}^{\gamma}\rho(\operatorname{Fr})}$ on
$\operatorname{Lie}(U^{-})$ are of the form $\lambda(s)^{n}\mu^{\prime}$ for
$\mu^{\prime}$ an eigenvalue of
$\operatorname{ad}_{{}^{\gamma}\rho(\operatorname{Fr})}$ and $n$ _negative_.
From this and the same considerations with the roles of $U$ and $U^{-}$
reversed, it follows that $s\rho(\operatorname{Fr})$ is avoidant for all but
finitely many $s\in S_{\kappa}$. In particular, $\rho$ is in the closure of
$\mathfrak{X}_{M}^{d,a}$ as required.

  2. (2)

Let $Z$ be the open subscheme of $\mathfrak{X}^{d}_{G}$ given by

$Z=\\{\rho\in\mathfrak{X}^{d}_{G}:\rho(\operatorname{Fr})\in M^{d,a}\\}.$

    1. (a)

We first show that the morphism $G^{\circ}\times Z\to\mathfrak{X}_{G}$ sending
$(\gamma,\rho)$ to $\gamma\rho\gamma^{-1}$ is smooth. This map is the pullback
of the conjugation map $c:G^{\circ}\times M^{a}\to G$ along the projection
$\mathfrak{X}_{G}\to G$ sending $\rho$ to $\rho(\operatorname{Fr})$. It is
enough to show that $c$ is smooth. Since $G^{\circ}\times M^{d,a}$ and $G$ are
smooth over $\Lambda$, it is enough by Lemma 4.4 to show that the derivative
$Dc$ is surjective on tangent spaces. Since the map $c$ is equivariant for the
action of $G^{\circ}$ we may check this at a point $(e,m)\in G^{\circ}\times
M^{d,a}$ with residue field $\kappa$. Identify the tangent spaces of
$G^{\circ}_{\kappa}\times M^{a}_{\kappa}$ (resp. $G_{\kappa}$) at $(e,m)$
(resp. $m$) with
$\operatorname{Lie}G_{\kappa}\oplus\operatorname{Lie}M_{\kappa}$ (resp.
$\operatorname{Lie}G_{\kappa}$) via left multiplication by $(e,m)$ (resp.
$m$). A computation shows that for
$(X,Y)\in\operatorname{Lie}G_{\kappa}\oplus\operatorname{Lie}M_{\kappa}$

$Dc(X,Y)=(\operatorname{ad}_{m}^{-1}-1)X+Y,$

so that $Dc$ is surjective by the assumption that $m$ is avoidant.

    2. (b)

We must show that the closed immersion $\iota:\mathfrak{X}_{M}^{d,a}\to Z$ is
an isomorphism. Since $\mathfrak{X}_{G}$ is $\Lambda$-flat and reduced, $Z$
also has these properties. Since $\iota$ is a closed immersion of
$\Lambda$-flat schemes, it is enough to show that
$\iota\otimes_{\Lambda}\mathbb{Q}$ is an isomorphism.

    3. (c)

Since $Z\otimes_{\Lambda}\mathbb{Q}$ is reduced and of finite type over
$\mathbb{Q}$, it is enough to show that $\iota$ is a bijection on closed
points. Thus we must show that, if $\rho\in Z(\overline{\mathbb{Q}})$, then
$\rho\in\mathfrak{X}_{M}(\overline{\mathbb{Q}})$. In other words, if $\rho$ is
an $L$-parameter with $\rho(\operatorname{Fr})\in
M^{a}(\overline{\mathbb{Q}})$, then we also have $\rho(\sigma)\in
M(\overline{\mathbb{Q}})$ and $\rho(g)\in M(\overline{\mathbb{Q}})$ for all
$g\in P_{F}/P_{F}^{d}$.

Let $\rho$ be such a parameter, and let
$\rho(\sigma)=(\Sigma_{s},\sigma)\Sigma_{u}$ be the Jordan decomposition of
$\rho(\Sigma)$ (we must have that $\rho(\sigma)_{u}\in
G^{\circ}(\overline{\mathbb{Q}})$ since $\overline{\mathbb{Q}}$ has
characteristic zero).

    4. (d)

We show that $\Sigma_{u}\in M^{\circ}(\overline{\mathbb{Q}})$. Write
$\Sigma_{u}=\exp(N)$ for $N\in\operatorname{Lie}(G)$. Then (by uniqueness of
Jordan decomposition) we have
$\operatorname{ad}_{\rho(\operatorname{Fr})}(N)=qN$. By our assumption that
$\rho(\operatorname{Fr})$ is avoidant, we have
$N\in\operatorname{Lie}(M)(\overline{\mathbb{Q}})$ and so $\Sigma_{u}\in
M^{\circ}(\overline{\mathbb{Q}})$.

    5. (e)

We next show that $\Sigma_{s}\in M(\overline{\mathbb{Q}})$. Let $r$ be as
above. Then

$\rho(\operatorname{Fr})^{r}(\Sigma_{s},\sigma)\rho(\operatorname{Fr})^{-r}=(\Sigma_{s},\sigma)^{q^{r}}=(\Sigma_{s},\sigma),$

which may be rewritten as

$(\Sigma_{s})^{-1}\rho(\operatorname{Fr})^{r}\Sigma_{s}={}^{\sigma}\rho(\operatorname{Fr})^{r}.$

Applying Lemma 4.1, we see that $\Sigma_{s}^{\circ}\in
M^{\circ}(\overline{\mathbb{Q}})$ as required.

    6. (f)

Finally, we show that $\rho(g)\in M(\overline{\mathbb{Q}})$ for $g\in
P_{F}/P_{F}^{d}$. This is identical to the previous step: if
$\pi=\rho(g)^{\circ}$ we obtain

$\pi^{-1}\rho(\operatorname{Fr})^{r}\pi={}^{g}\rho(\operatorname{Fr})^{r}$

and therefore that $\pi\in M^{\circ}(\overline{\mathbb{Q}})$.

  3. (3)

It follows from part 2 that $\overline{c}$ is smooth of relative dimension
zero, and hence étale.

∎

######  Remark 4.3.

In fact, $\overline{c}$ descends further to a map
$G^{\circ}\times^{N_{G^{\circ}}(M^{\circ})}\mathfrak{X}_{M}^{a}\to\mathfrak{X}_{G}$
which one can show is an open immersion. This implies that $\overline{c}$ is a
torsor under

$N_{G^{\circ}}(M^{\circ})/M^{\circ}\cong N_{W}(W_{M^{\circ}})/W_{M^{\circ}}$

(see [MT11] Corollary 12.11 for this isomorphism).

In the previous proof, we used the following standard lemma for which we do
not know a reference.

######  Lemma 4.4.

Let $S$ be a scheme, and let $f:X\to Y$ be a morphism of smooth $S$-schemes.
Suppose that, for each closed point $x\in X$, the derivative $Df:T_{x}X\to
T_{f(x)}Y$ is surjective. Then $f$ is smooth.

###### Proof.

If $S$ is a field, this is [Har77] Proposition III.10.4. In particular,
$f_{s}$ is smooth, and hence flat, for each $s\in S$. By the fibrewise
criterion for flatness [SPA17, Lemma 039E], $f$ is flat. Since $f$ is flat,
locally of finite presentation, and has smooth fibres, it is smooth [SPA17,
Lemma 01V8]. ∎

##  5\. (Non-)Examples

In all the following examples we consider unramified $G$ and tamely ramified
$L$-homomorphisms only. If $\kappa$ is an algebraically closed field of
characteristic $\neq p$, then the irreducible components of
$\mathfrak{X}_{G,\kappa}$ are in bijection with $G^{\circ}(\kappa)$-conjugacy
classes of pairs $(\Sigma,C)$ where: $\Sigma\in G^{\circ}(\kappa)$, and $C$ is
an element of

$\pi_{0}(\\{\Phi\in
G^{\circ}_{\kappa}:\Phi{}^{\operatorname{Fr}}\Sigma\Phi^{-1}=\Sigma^{q}\\}).$

If we fix $\Sigma$ and fix $\Phi_{0}$ with
$\Phi_{0}\Sigma\Phi_{0}^{-1}=\Sigma^{q}$, then the equivalence classes of
pairs $(\Sigma,C)$ are in bijection with
$\operatorname{Ad}_{\Phi_{0}}$-twisted conjugacy classes in
$\pi_{0}(C_{G^{\circ}}(\Sigma))$. We call the corresponding components the
components of type $\Sigma$.

Let $\Lambda=W(\overline{\mathbb{F}}_{l})$. Our main interest is in describing
the local geometry of $\mathfrak{X}_{G,\Lambda}$ around generic points of
components of the special fibre. By [DHKM20] section 4.2, it is possible to
reduce the general case to that of tame parameters such that the semisimple
part of $\Sigma$ has $l$-power order — and so, modulo $l$, $\Sigma$ is
unipotent. We therefore focus on such components in the discussion below. We
call the irreducible components of
$\mathfrak{X}^{\mathrm{tame}}_{\overline{\mathbb{F}}_{l}}$ on which $\Sigma$
is unipotent the _unipotent components_. We have the morphism
$\mathrm{Ch}:\mathfrak{X}_{G}^{\mathrm{tame}}\to(T/\\!\\!/W)^{\operatorname{Fr}^{-1}[q]}$
and the unipotent components are those in the preimage of the identity
$\overline{e}\in(T/\\!\\!/W)^{\operatorname{Fr}^{-1}[q]}(\overline{\mathbb{F}}_{l})$.

###  5.1. $GL_{n}$

Since centralizers are connected in $G=GL_{n}$, the irreducible unipotent
components of $\mathfrak{X}_{G,\overline{\mathbb{F}}_{l}}$ correspond to
unipotent conjugacy classes in $GL_{n}$, which are parametrized by partitions
of $n$ giving the Jordan canonical form of $\Sigma$. For every conjugacy class
$[\Sigma]$, we can find a representative such that $\Sigma$ is regular in a
standard Levi subgroup $M$ (the one corresponding to the same partition of
$n$). By Theorems 4.2 and 3.2, we have open subsets $U\subset\mathfrak{X}_{M}$
and $V\subset\mathfrak{X}_{G}$, each containing the generic point of the
component corresponding to $\Sigma$, and smooth morphisms

$G\times U\to V$

and

$U\to(T/\\!\\!/W_{M})^{[q]}.$

In particular, the completed local ring of $\mathfrak{X}_{M}$ at a point in
$V(\overline{\mathbb{F}}_{l})$ will be formally smooth over
$(T/W_{M})_{\overline{e}}^{[q]}$. We therefore recover the geometric results
of [Sho22] in this case (and there is no real need to assume that $\Sigma$ is
unipotent here).

###  5.2. $SL_{2}$

Suppose that $G=SL_{2}$ (with $\Gamma$ trivial). There are two conjugacy
classes of unipotent $\Sigma$.

If $\Sigma=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$, then we may take
$\Phi_{0}=\begin{pmatrix}q&0\\\ 0&1\end{pmatrix}$. We have
$\pi_{0}(C_{G^{\circ}}(\Sigma))=Z(SL_{2})=\mu_{2}$, and so there are two
components of $\mathfrak{X}_{\overline{\mathbb{F}}_{l}}$ of type $\Sigma$
unless $l=2$, in which case there is only one. If $l\neq 2$, then there is an
open subset $U$ of $\mathfrak{X}_{G}$ intersecting each of these components
such that the map

$U\to(T/\\!\\!/W)^{[q]}=\operatorname{Spec}\mathbb{Z}_{l}[x+x^{-1}]/((x^{q}+x^{-q}-(x+x^{-1}))]$

is smooth.

If $\Sigma=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$, then there is only one
component $C_{nr}$ of type $\Sigma$. There is an open subset $V$ of
$\mathfrak{X}^{\mathrm{tame}}_{T}\cong\mu_{q-1}\times\mathbb{G}_{m}$ such that
the morphism $G\times V\to\mathfrak{X}_{G}$ is smooth and has image
intersecting $C_{nr}$ (necessarily in an open subset).

###  5.3. Unitary groups

We take $G$ to be the $L$-group of a unramified quasisplit unitary group over
$F$. Thus,

$G=G^{\circ}\rtimes\operatorname{Gal}(E/F)$

where $G^{\circ}=GL_{n}$, $E/F$ is the unramified quadratic extension, and the
nontrivial element $c\in\operatorname{Gal}(E/F)$ acts on $G^{\circ}$ as

${}^{c}g=Jg^{-T}J^{-1}$

for $J$ an antidiagonal matrix of alternating $1$s and $-1$s. Then $c$
preserves the standard maximal torus $T$ and Borel $B$ in $G^{\circ}$.

Since centralizers in $G^{\circ}$ are connected, the irreducible components of
$\mathfrak{X}_{G,\overline{F}_{l}}$ are in bijection with
$G^{\circ}(\overline{\mathbb{F}}_{l})$-conjugacy classes of $\Sigma\in
G^{\circ}(\overline{\mathbb{F}}_{l})$ such that ${}^{c}\Sigma\sim\Sigma^{q}$.
In particular, the unipotent irreducible components are in bijection with the
unipotent conjugacy classes in $G^{\circ}(\overline{\mathbb{F}}_{l})\cong
GL_{n}(\overline{\mathbb{F}}_{l})$, since ${}^{c}\Sigma\sim\Sigma$ for
$\Sigma$ unipotent.

Note that $\operatorname{Fr}$ acts on $T/\\!\\!/W$ in the same way as the map
induced by $t\mapsto t^{-1}$. We therefore have a formally smooth morphism
$\mathfrak{X}_{G}^{\operatorname{reg}}\to(T/\\!\\!/W)^{[-q]}$, by Theorem 3.2.

However, it is no longer true that every unipotent conjugacy class of
$G^{\circ}$ contains an element regular in a standard Levi subgroup of $G$.
Indeed, such Levi subgroups must be preserved by $c$, and so we only get
partitions of $n$ for which all except at most one part occurs an even number
of times.

For example, when $n=3$, the unipotent components of $\mathfrak{X}_{G}$ to
which Theorem 4.2 applies are those corresponding to the partitions $(1,1,1)$
and $(3)$; we do not obtain a description of the local geometry of
$\mathfrak{X}_{G}$ about points for which $\Sigma$ is unipotent of type
$(2,1)$.

###  5.4. $GSp_{4}$

Let $G=\mathrm{GSp}_{4}$, with $l>2$. We take the symplectic form to be that
given by the matrix

$J=\begin{pmatrix}&&&1\\\ &&1&\\\ &-1&&\\\ -1&&&\end{pmatrix}.$

We describe the unipotent components of
$\mathfrak{X}_{\overline{\mathbb{F}}_{l}}$. The conjugacy class of a unipotent
matrix $u\in\mathrm{GSp}_{4}$ is determined by the rank of $u-1$. The
centralizer of $u$ is connected unless the rank of $u-1$ is two, in which case
the centralizer has two connected components (see the table on p400 of
[Car85]). We label the unipotent components by the rank of $\Sigma-I$ at a
generic point, so that they are $C_{0}$, $C_{1}$, $C_{2A}$, $C_{2B}$, and
$C_{3}$.

Firstly, on the unramified component $C_{0}$ we have $\Sigma=I$. We may apply
Theorem 4.2 to this component, with the Levi subgroup being the standard
maximal torus, to see that there is an étale neighbourhood $U$ of the generic
point of $C_{0}$ such that $U$ is smooth over $T^{[q]}\cong\mu_{q-1}^{3}$.

Secondly, we have the component $C_{1}$ on which $\Sigma-I$ has rank $1$
(generically). There is a unique such component since the centralizer of any
$\Sigma$ such that $\Sigma-I$ has rank 1 is connected. Letting $M\cong
GL_{2}\times\mathbb{G}_{m}$ be the Levi subgroup of block diagonal matrices
with block sizes 1, 2, 1, there is an étale neighbourhood $U$ of the generic
point of $C_{1}$ such that $U$ is smooth over $(T/\\!\\!/W_{M})^{[q]}$.

Thirdly, we have the component $C_{3}$ on which $\Sigma-I$ has rank $3$
(generically). There is an open neighbourhood $U$ of the generic point of
$C_{3}$ that is smooth over $(T/\\!\\!/W)^{[q]}$.

Lastly, consider

$u=\begin{pmatrix}1&1&0&0\\\ 0&1&0&0\\\ 0&0&1&-1\\\ 0&0&0&1\end{pmatrix}.$

Then $C_{G}(u)$ has two components. The Frobenius-twisted conjugacy classes of
$\pi_{0}(C_{G}(u))\cong\mathbb{Z}/2$ are in this case just the conjugacy
classes and there are therefore two components of $\mathfrak{X}_{u}$. One of
them, which we call $C_{2A}$, contains the points with $\Sigma=u$ and

$\rho(\operatorname{Fr})=\begin{pmatrix}\lambda q&0&0&0\\\ 0&\lambda&0&0\\\
0&0&q&0\\\ 0&0&0&1\end{pmatrix},$

while the other, which we call $C_{2B}$, contains the points with $\Sigma=u$
and

$\rho(\operatorname{Fr})=\begin{pmatrix}0&0&-\lambda q&0\\\ 0&0&0&\lambda\\\
-q&0&0&0\\\ 0&1&0&0\end{pmatrix}.$

The component $C_{2A}$ may be understood by applying the results of 4 with the
Levi subgroup $M\cong GL_{2}\times_{\mathbb{G}_{m}}GL_{2}$ consisting of block
diagonal matrices with block sizes (2, 2), and we see that an étale
neighourhood $U$ of the generic point of $C_{2A}$ will be formally smooth over
$(T/\\!\\!/W_{M})^{[q]}_{1}$. The component $C_{2B}$, however, falls outside
the scope of our results.

######  Remark 5.1.

We can show that, if $q=-1\bmod l$, then any open neighbourhood of the generic
point of $C_{2B}$ intersects the components of the generic fibre of type
$C_{2B}$ _and_ $C_{3}$, so that no statement like 4.2 can be true.

If $q=1\bmod l$ and $l\geq 5$, then one can actually prove a statement like
Theorem 4.2 for the component $C_{2B}$ with $M$ taken to be the pseudo-Levi
subgroup (in the sense of [MS03])

$\displaystyle M$ $\displaystyle=Z_{\mathrm{GSp}_{4}}(\Phi_{0})$ where
$\displaystyle\Phi_{0}$ $\displaystyle=\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\
0&0&1&0\\\ 0&0&0&-1\end{pmatrix}.$

For this, we choose $\lambda=1$ and make a change of basis such that
$\rho(\operatorname{Fr})=\Phi_{0}$ and

$\Sigma=\begin{pmatrix}1&0&1&0\\\ 0&1&0&1\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix}\in M.$

One finds that an étale neighbourhood of $C_{2B}$ is smooth over
$(T/\\!\\!/W_{M})^{[q]}$.

###  5.5. Distinguished unipotent elements

The Bala–Carter theorem (see [Pre03]) says that, in good characteristic, every
unipotent element of a reductive group is a _distinguished_ unipotent element
of a Levi subgroup. For $G=GL_{n}$, the notions of distinguished and regular
coincide, but in general they do not.

Concretely, suppose that $G=\mathrm{GSp}_{6}$, $l>2$, and that

$\Sigma=\begin{pmatrix}1&1&&&&\\\ &1&&&1&\\\ &&1&1&&\\\ &&&1&&\\\ &&&&1&-1\\\
&&&&&1\end{pmatrix}.$

Then there is a single component of type $\Sigma$. This $\Sigma$ is
distinguished unipotent, so is contained in no proper Levi subgroup of $G$,
but is not regular.777Indeed, it has Jordan blocks of shape $(4,2)$ but no
proper Levi subgroup of $G$ contains elements of this shape. This component
therefore falls outside the scope of our results, and we do not understand the
deformation rings at generic points of these components (except for _minimal_
deformations as in [Boo19]).

######  Remark 5.2.

If $q\equiv 1\mod l$ then the trivial representation lies on the component of
type $\Sigma$. In particular, to apply the Ihara avoidance method of [Tay08]
to $\mathrm{GSp}_{2n}$, $n>2$, one would need to understand something about
deformations of representations of type $\Sigma$. For $\mathrm{GSp}_{4}$,
where Ihara avoidance was worked out in section 7.4.5 of [BCGP21], the issue
does not arise, as $C_{2B}$ does not contain the trivial representation.

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