# The trianguline variety, tangent spaces and the Grothendieck-Springer
resolution

Seginus Mowlavi  Université Paris-Saclay, CNRS, Laboratoire de mathématiques
d’Orsay, 91405, Orsay, France

###### Abstract.

By the work of Breuil-Hellmann-Schraen, we know that the trianguline variety
contains crystalline companion points which are parametrised by pairs
$w\preceq w_{\mathrm{sat}}$ of permutations where $\preceq$ is the Bruhat
order. We first define and study a certain combinatorial property of a pair
$w^{\prime}\preceq w$ of permutations in the context of Weyl groups of root
systems. We call good pairs the pairs satisfying this property (which is the
vast majority of pairs) and bad pairs the other ones. We then give an exact
formula for the dimension of the tangent space to the trianguline variety at
(generic) crystalline companion points such that $w\preceq w_{\mathrm{sat}}$
is a good pair. The method (due to Breuil-Hellmann-Schraen) is to first
compute an analogous dimension for a local model of the trianguline variety
built out of Grothendieck’s simultaneous resolution. To achieve this, we prove
a conjecture of Breuil-Hellmann-Schraen, describing the intersection of the
closure of a Schubert cell with another Schubert cell on this local model (in
the context of an arbitrary split reductive group), when this pair of cells is
parametrised by a good pair of permutations. We give counter-examples to this
conjecture for an infinite family of pairs of cells (associated to bad pairs).

###### Contents

  1. 1 Introduction
  2. 2 Some combinatorics in Weyl groups
    1. 2.1 Minimal generating root subsystems
    2. 2.2 Interpretation for a split reductive group
    3. 2.3 Pairs of elements in the Weyl group
    4. 2.4 Systems of type $A$ and pattern avoidance
  3. 3 A conjecture on a Schubert-like decomposition
    1. 3.1 A scheme related to the Springer resolution
    2. 3.2 Statement of the conjecture
    3. 3.3 Application to tangent spaces on the trianguline variety
  4. 4 The case of good pairs
    1. 4.1 Preliminaries: Levi subgroups and the adjoint action
    2. 4.2 Proof of Theorem 4.1
  5. 5 Counter-examples for bad pairs when $G=\mathrm{GL}_{n}$
    1. 5.1 Grassmannians, the flag variety and Schubert cells
    2. 5.2 Explicit equations for $\widetilde{X}$ and $\widetilde{V}_{w}$
    3. 5.3 Taking the Zariski closure
  6. A Jordan decomposition and orbits in $\mathfrak{g}$

##  1\. Introduction

Let $k$ be a field of characteristic $0$. In their study [8] of the
trianguline variety $X_{\mathrm{tri}}^{\square}(\bar{r})$ of Hellmann [12],
Breuil-Hellmann-Schraen introduced a variety $X$ over $k$ associated to a
split reductive group $G$ over $k$. This variety admits a stratification
$X=\coprod_{w\in W}V_{w}$ indexed by the Weyl group $W$ of $G$, similar to the
Schubert decomposition of the flag variety. Denoting by $X_{w}$ the Zariski-
closure in $X$ of the locally closed subvarieties $V_{w}$, the authors of [8]
have linked the dimension of the tangent spaces of
$X_{\mathrm{tri}}^{\square}(\bar{r})$ at certain points of interest to the
dimension of $T_{X_{w},x}$ at particular points $x\in X_{w}\cap
V_{w^{\prime}}$ for $w,w^{\prime}\in W$ such that $w^{\prime}\preceq w$ where
$\preceq$ is the Bruhat order. They also formulate a conjecture describing
$X_{w}\cap V_{w^{\prime}}$, which if true allows to compute the dimension of
$T_{X_{w},x}$. This conjecture is the object of the present paper.

A special case of a variant of this conjecture was already known by the work
of Ginsburg [11] using the Beilinson-Bernstein correspondence. By applying
this special case to certain Levi subgroups of $G$, we are able to expand this
result into a proof of many cases of the conjecture.

These cases are characterised by a combinatorial condition on the pair
$(w^{\prime},w)$ that we call being a “good pair”. We develop the theory of
good pairs in the context of Weyl groups of root systems and establish along
the way different criteria for good pairs. We also show that the set of good
pairs is related to pattern avoidance.

Finally, by spelling out explicit equations defining $V_{w}$ as a
quasiprojective variety in the case $G=\operatorname{GL}_{n/k}$, we give a
family of counter-examples to the conjecture.

We now explain our results in more detail.

#### Good pairs in a Weyl group

Let $\Phi$ be an abstract root system; we write $W\coloneqq W(\Phi)$ for its
Weyl group. For $w\in W$, let
$\Gamma_{w}\coloneqq\sum_{\alpha\in\Phi}\mathbb{Z}(w(\alpha)-\alpha)$, let
$E_{w}\coloneqq\Gamma_{w}\otimes_{\mathbb{Z}}\mathbb{Q}$ and finally let
$\Phi_{w}\coloneqq\Phi\cap E_{w}$. We call $\Phi_{w}$ the _minimal generating
root subsystem of $w$_. It has a clear interpretation in terms of reductive
groups:

######  Proposition 1 (Proposition 2.2.2).

Let $(G,T)$ be a connected split reductive algebraic group over a field $k$ of
characteristic 0 whose root system is isomorphic to $\Phi$. Then the
centraliser $\left(C_{G}((T^{w})^{\circ}),T\right)$ is a connected split
reductive group whose root system is isomorphic to $\Phi_{w}$.

Fixing a basis $I$ of $\Phi$ gives a length function $\lg$ and a Bruhat order
$\prec$ on $W$. The central notion we study in §2 and use in §4 is the
following:

######  Definition 2 (Definition 2.3.1).

Let $(w_{1},w_{2})\in W$ such that $w_{1}\preceq w_{2}$. If there exist roots
$\alpha_{1},\ldots,\alpha_{r}\in\Phi_{w_{1}w_{2}^{-1}}$ which give rise to an
ascending sequence $w_{1}\prec w_{1}s_{\alpha_{1}}\prec\ldots\prec
w_{1}s_{\alpha_{1}}\ldots s_{\alpha_{r}}=w_{2}$, then $(w_{1},w_{2})$ is
called a _good pair_ ; otherwise, it is called a _bad pair_.

In §2.1, we establish a few combinatorial results about minimal generating
subsystems $\Phi_{w}$. In §2.3 we prove a criterion for good pairs which is
motivated by the fact that, for any $w\in W$, the subsystem $\Phi_{w}$ is
conjugated to a standard subsystem, _i.e._ a root subsystem
$\Phi_{J}\subseteq\Phi$ generated by some subset $J\subseteq I$ of simple
roots. More precisely, for $J\subseteq I$, let $W_{J}$ be the subgroup of $W$
generated by $\left\\{s_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in
J\right\\}$, and let $W^{J}$ be the set of representatives of minimal length
for $W_{J}\backslash W$. Then one can write $\Phi_{w}=u^{J}(\Phi_{J})$ for
some $J\subseteq I$ and $u^{J}\in W^{J}$. The criterion is:

######  Proposition 3 (Proposition 2.3.7).

Let $(w_{1},w_{2})\in W^{2}$. Write $\Phi_{w_{2}w_{1}^{-1}}=u^{J}(\Phi_{J})$
for some $J\subseteq I$ and $u^{J}\in W^{J}$. Then there exist $v^{J}\in
W^{J}$ and $w_{J,1},w_{J,2}\in W_{J}$ such that

$w_{1}=u^{J}w_{J,1}(v^{J})^{-1}\,,\quad w_{2}=u^{J}w_{J,2}(v^{J})^{-1}\,,$

and the pair $(w_{1},w_{2})$ is good if and only if $w_{J,1}\preceq w_{J,2}$.

We conclude in §2.4 with an application of this last criterion in the special
case where $\Phi$ is of type $A_{n-1}$ for some $n\in\mathbb{Z}_{>1}$, _i.e._
$W=\mathcal{S}_{n}$ is the symmetric group. It is stated in terms of what is
called _pattern avoidance_. Here we say that a permutation
$w\in\mathcal{S}_{n}$ has a pattern $f\in\mathcal{S}_{m}$ when there exist
$1\leq i_{1}<\ldots<i_{m}\leq n$ such that the $m$-uples
$(w(i_{1}),\ldots,w(i_{m}))$ and $(f(1),\ldots,f(m))$ are in the same order;
see Definition 2.4.8. This can in fact be stated in the general setting of an
abstract root system, and many properties such as smoothness of Schubert
varieties have been found to be instances of pattern avoidance; see for
example [3, §2, §4]. We prove that, at least for root systems of type $A$,
being a good pair is also an instance of pattern avoidance:

######  Theorem 4 (Theorem 2.4.10).

Let $w\in\mathcal{S}_{n}$.

  1. _(1)_

The pair $(w^{\prime},w)$ is good for any $w^{\prime}\in\mathcal{S}_{n}$ if
and only if $w$ avoids the four patterns $[4231]$, $[42513]$, $[35142]$ and
$[351624]$.

  2. _(2)_

The pair $(w,w^{\prime\prime})$ is good for any
$w^{\prime\prime}\in\mathcal{S}_{n}$ if and only if $w$ avoids the four
patterns $[1324]$, $[24153]$, $[31524]$ and $[426153]$.

Linking these patterns with those deciding smoothness of Schubert varieties,
we obtain that, if $(w,w^{\prime})\in(\mathcal{S}_{n})^{2}$ is a bad pair,
then the Schubert variety $\overline{BwB/B}$ in the flag variety $G/B$ is
singular (but not necessarily at the point $w^{\prime}B\in\overline{BwB/B}$);
see Corollary 2.4.11.

#### The conjecture of [8]

Let $G$ be a connected split reductive group over $k$, $T$ a split maximal
torus inside $G$, $B$ a Borel subgroup of $G$ containing $T$ and $U$ the
unipotent subgroup of $B$. We write $\mathfrak{g}$, $\mathfrak{t}$,
$\mathfrak{b}$ and $\mathfrak{u}$ for their respective Lie algebras, and
$W\coloneqq N_{G}(T)/T$ for the Weyl group of $(G,T)$. For $w\in W$, we choose
a lifting of $w$ in $N_{G}(T)$ which we still call $w$. We have the Schubert
decomposition $G/B=\coprod_{w\in W}BwB/B$ of $G/B$ into locally closed strata
$BwB/B$, where a cell $Bw^{\prime}B/B$ intersects the Zariski-closure
$\overline{BwB/B}$ of $BwB/B$ if and only if $w^{\prime}\preceq w$ for the
Bruhat order on $W$, in which case $Bw^{\prime}B/B\subseteq\overline{BwB/B}$.

Let $G\times^{B}\mathfrak{g}$ be the quotient of $G\times\mathfrak{g}$ by the
right-action of $B$ given by $(g,\psi)b=(gb,\operatorname{Ad}(b^{-1})\psi)$.
We define the morphisms $q\colon
G\times^{B}\mathfrak{g}\longrightarrow\mathfrak{g}$ and $\widetilde{\pi}\colon
G\times^{B}\mathfrak{g}\longrightarrow G/B$ by
$q(g,\psi)=\operatorname{Ad}(g)\psi$ and $\widetilde{\pi}(g,\psi)=gB$ ($q$ is
called Grothendieck’s simultaneous resolution of singularities). We then
define the algebraic variety $\widetilde{X}\coloneqq q^{-1}(\mathfrak{b})$.
For $w\in W$, we consider the locally closed subvariety
$\widetilde{V}_{w}\coloneqq\widetilde{X}\cap\widetilde{\pi}^{-1}(BwB/B)$ of
$\widetilde{X}$ as well as its Zariski-closure $\widetilde{X}_{w}$ in
$\widetilde{X}$; they give a cell decomposition of $\widetilde{X}$.

When $L$ is another split reductive group over $k$, we add the subscript $L$
to $q,\widetilde{\pi},\widetilde{X},\widetilde{V}_{w},\widetilde{X}_{w}$ (for
$w\in W(L)$) to mean the same definition with the reductive group $L$ in place
of the reductive group $G$. When nothing is specified, we mean $G$.

Similarly to Schubert cells in $G/B$, the intersection
$\widetilde{V}_{w^{\prime}}\cap\widetilde{X}_{w}$ is non-empty if and only if
$w^{\prime}\preceq w$. The difference is that in this case
$\widetilde{V}_{w^{\prime}}$ is not contained in $\widetilde{X}_{w}$. More
precisely, it is not difficult to show that
$\widetilde{V}_{w^{\prime}}\cap\widetilde{X}_{w}\subseteq\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\mathfrak{t}^{ww^{\prime-1}}+\mathfrak{u})$ ([8, Lemma 2.3.4]) where
$\mathfrak{t}^{ww^{\prime-1}}\subseteq\mathfrak{t}$ is the closed subvariety
of fixed points by $ww^{\prime-1}$ and Breuil-Hellmann-Schraen conjecture that
this is an equality:

######  Conjecture 5 (Conj. 2.3.7 of [8]).

Let $w,w^{\prime}\in W$. If $w^{\prime}\preceq w$, then

$\widetilde{V}_{w^{\prime}}\cap\widetilde{X}_{w}=\widetilde{V}_{w^{\prime}}\cap
q^{-1}\left(\mathfrak{t}^{ww^{\prime-1}}+\mathfrak{u}\right)\,.$

Conjecture 5 first appeared in the arithmetic context of the study of the
rigid analytic _trianguline variety_ $X_{\mathrm{tri}}^{\square}(\bar{r})$
introduced by Hellmann [12] (here $\bar{r}$ is a mod $p$ representation of
$\operatorname{Gal}(\overline{K}/K)$ where $p$ is a prime number and $K$ a
finite extension of $\mathbb{Q}_{p}$). More precisely, Breuil-Hellmann-Schraen
proved in [8] that the varieties $\widetilde{X}_{w}$ are local models of
$X_{\mathrm{tri}}^{\square}(\bar{r})$ at _crystalline regular generic points_
, which allow them to derive a number of consequences about the local geometry
of $X_{\mathrm{tri}}^{\square}(\bar{r})$ at these points.

We show in §5 that Conj. 5 is actually _false_ in general! In §4, we
nonetheless prove it in the case of good pairs:

######  Theorem 6 (Theorem 4.1).

Let $w,w^{\prime}\in W$ such that $w^{\prime}\preceq w$ and $(w^{\prime},w)$
is a good pair. Then, for all $t\in\mathfrak{t}$ fixed by $ww^{\prime-1}$, we
have

(1.1) $\widetilde{V}_{w^{\prime}}\cap
q^{-1}(t+\mathfrak{u})\subseteq\widetilde{X}_{w}\,.$

Hence Conj. 5 for $(w^{\prime},w)$ is true.

Going back to the motivation of [8], Theorem 6 has the following application:

######  Corollary 7 (Theorem 3.3.2).

Let $x\in X_{\mathrm{tri}}^{\square}(\bar{r})$ be a crystalline regular
generic point and $T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}$ the tangent
space to $X_{\mathrm{tri}}^{\square}(\bar{r})$ at $x$. Let $w_{x},w$ be as in
[8, §4.1] ($w_{x}$ and $w$ are elements of a fixed product of Weyl groups of
type $A$) and assume that $(w_{x},w)$ is a good pair. Then

(1.2) $\dim_{k(x)}T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}=\dim
X_{\mathrm{tri}}^{\square}(\bar{r})-d_{ww_{x}^{-1}}+\dim_{\mathbb{Q}_{p}}T_{\overline{(BwB/B)},w_{x}B}-\operatorname{lg}(w_{x})$

where $k(x)$ is the residue field of $X_{\mathrm{tri}}^{\square}(\bar{r})$ at
$x$, $\overline{BwB/B}$ is the Schubert variety associated to $w$ and where we
refer to Definition 2.1.1 for the integer $d_{ww_{x}^{-1}}$.

In [21], we proved Corollary 7 under the assumption of modularity lifting
conjectures; here we do not need this assumption. Interestingly, while they
use fundamentally different methods, both works prove the exact same cases:
the notions of good pair are stated differently (here as in Proposition 2.3.7,
and in [21] as in Definition 2.3.1) but in equivalent ways. Note that it was
already known that the right hand side in (1.2) is an upper bound for
$\dim_{k(x)}T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}$ (even if $(w_{x},w)$ is
a bad pair), see [8, Proposition 4.1.5(ii)]. We do not know if this upper
bound becomes strict for bad pairs (see however Proposition 3.3.3).

Before outlining the proof of Theorem 6, we remark that it suffices to prove
(1.1) for all “sufficiently generic” (closed) points
$t\in\mathfrak{t}^{ww^{\prime-1}}$: see Proposition 4.1.1 and Lemma 4.1.2.
Without loss of generality, we can also assume $k$ algebraically closed; see
§4.2. The proof of Theorem 6 then proceeds in three steps.

Firstly, as $(w^{\prime},w)$ is a good pair, Proposition 3 gives a subset
$J\subseteq I$ of simple roots of $\Phi(G,T)$ and elements $u^{J},v^{J}\in
W^{J}$, $w_{J},w^{\prime}_{J}\in W_{J}$, such that $w=u^{J}w_{J}(v^{J})^{-1}$,
$w^{\prime}=u^{J}w^{\prime}_{J}(v^{J})^{-1}$ and $w^{\prime}_{J}\preceq w_{J}$
in $W_{J}$. Let $L$ be the Levi subgroup of the standard parabolic subgroup
$P_{J}$ of $G$, then
$L=C_{G}\big{(}\big{(}(u^{J})^{-1}T^{ww^{\prime-1}}u^{J}\big{)}^{\circ}\big{)}$
using Proposition 1.

It is proved in [8, §2.4] using the Beilinson-Bernstein localisation that
(1.1) holds for $t=0$ for all $w^{\prime}\preceq w$ (even in the case of bad
pairs). Since $w^{\prime}_{J}\preceq w_{J}$, we can apply this to the
connected split reductive group $L$ with split maximal torus $T$, Borel
subgroup $B_{L}\coloneqq B\cap L$ and unipotent subgroup $U\coloneqq U\cap L$.
For any
$t\in\mathfrak{t}_{J}=\operatorname{Ad}(u^{J})^{-1}\mathfrak{t}^{ww^{\prime-1}}$,
we deduce by translating by $t$ that $\widetilde{V}_{L,w^{\prime}_{J}}\cap
q^{-1}(t+\mathfrak{u}_{L})\subset\widetilde{X}_{L,w_{J}}$.

Secondly, by carefully decomposing the adjoint action of $G$ on some elements
of $\mathfrak{l}\cap\mathfrak{b}$ (where $\mathfrak{l}$ is the Lie algebra of
$L$), we show:

######  Proposition 8 (Proposition 4.2.2).

Let $u^{J},v^{J}\in W^{J}$. One can define a morphism
$\iota_{t,u^{J},v^{J}}\colon L\times^{L\cap
B}\mathfrak{l}\cap\mathfrak{b}\longrightarrow G\times^{B}\mathfrak{b}$ by
$\iota_{t,u^{J},v^{J}}(l,\psi_{L}))=\left(u^{J}l(v^{J})^{-1},\operatorname{Ad}(v^{J})\psi_{L}\right)$.
For any $t\in\operatorname{Ad}(u^{J})^{-1}\mathfrak{t}^{ww^{\prime-1}}$
sufficiently generic and any $w_{J}\in W_{J}$, this morphism
$\iota_{t,u^{J},v^{J}}$ induces an isomorphism of algebraic varieties

$\iota_{t,u^{J},w_{J},v^{J}}\colon\widetilde{V}_{L,w_{J}}\cap
q_{L}^{-1}(t+\mathfrak{u}_{L})\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\widetilde{V}_{u^{J}w_{J}(v^{J})^{-1}}\cap
q^{-1}\left(\operatorname{Ad}(u^{J})(t+\mathfrak{u}_{L})\right)\,.$

Thirdly, using standard results about Jordan decompositions in $\mathfrak{g}$
(that we show in Appendix A as we couldn’t totally find them in the
literature), we prove:

######  Lemma 9 (Lemma 4.2.3).

Let $t\in\operatorname{Ad}(u^{J})^{-1}\mathfrak{t}^{ww^{\prime-1}}$ be
sufficiently generic, let $u^{J}\in W^{J}$. Then there is a surjective
morphism given by

$f\colon\begin{array}[]{ccc}U\times\left(\operatorname{Ad}(u^{J})(t+\mathfrak{u}_{L})\right)&\relbar\joinrel\twoheadrightarrow&\operatorname{Ad}(u^{J})t+\mathfrak{u}\\\
(u,x)&\longmapsto&\operatorname{Ad}(u)x\,.\end{array}$

The surjective morphism $f$ of Lemma 9 combined with the isomorphism of
Proposition 8 induces two morphisms of varieties
$U\times\left(\widetilde{V}_{L,w_{J}}\cap
q_{L}^{-1}(t+\mathfrak{u}_{L})\right)\relbar\joinrel\twoheadrightarrow\widetilde{V}_{w}\cap
q^{-1}\left(\operatorname{Ad}(u^{J})t+\mathfrak{u}\right)$ and
$U\times\widetilde{X}_{L,w_{J}}\longrightarrow\widetilde{X}_{w}$, which fit
into a commutative diagram

${U\times\left(\widetilde{V}_{L,w^{\prime}_{J}}\cap
q_{L}^{-1}(t+\mathfrak{u}_{L})\right)}$${U\times\widetilde{X}_{L,w_{J}}}$${\widetilde{X}_{w}}$${\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\operatorname{Ad}(u^{J})t+\mathfrak{u})}$${\widetilde{X}}$$\scriptstyle{\subseteq}$$\scriptstyle{\subseteq}$$\scriptstyle{\subseteq}$

where the top horizontal arrow is given by the first step. The surjectivity of
the left vertical arrow then proves that the image of the bottom horizontal
arrow is contained in $\widetilde{X}_{w}$, which is the statement (1.1).

#### Counter-examples

In the last §5, we give explicit equations and inequations for the projective
varieties $\widetilde{X}$ and $\widetilde{V}_{w}$ in the case
$G=\operatorname{GL}_{n/k}$. Even though we do not have such an explicit
description for the varieties $\widetilde{X}_{w}$, this is enough to find a
family of counter-examples to Conj. 5.

We use that $\widetilde{X}$ can be seen as the subvariety
$\left\\{(gB,\psi)\in
G/B\\!\times\\!\mathfrak{b}\mathrel{}\middle|\mathrel{}\operatorname{Ad}(g^{-1})\psi\in\mathfrak{b}\right\\}$
of $G/B\times\mathfrak{b}$, see (5.5), which is easier to handle than a
quotient variety. Indeed $\mathfrak{b}$ is an affine space, and the flag
variety $G/B$ is very well-understood. Specifically, $G/B$ is a subvariety of
a product of Grassmannians defined by incidence relations; and there are well-
known equations defining the Grassmannian as a subvariety of a projective
space _via_ the Plücker embedding. Therefore $G/B\times\mathfrak{b}$ is well-
understood as a subvariety of
$\prod_{d=1}^{n-1}\mathbb{P}(\bigwedge^{d}k^{n})\times\mathbb{A}^{n(n+1)/2}$.
Details are spelled out in §5.1.

We then work out a set of equations defining $\widetilde{X}$ inside
$G/B\times\mathfrak{b}$; this is Lemma 5.2.3. The next step is to use the
(in)equations defining each Schubert cell $BwB/B$ inside $G/B$ in order to
find a set $\\{F_{i}=0,G_{j}\neq 0\\}_{i\in I,j\in J}$ of (in)equations
defining $\widetilde{V}_{w}$ inside
$\prod_{d=1}^{n-1}\mathbb{P}(\bigwedge^{d}k^{n})\times\mathbb{A}^{n(n+1)/2}$.
Doing so by brute force is not too difficult; however the challenge is to have
enough equations so that, hopefully, the set of equations $\\{F_{i}=0\\}_{i\in
I}$ describes the Zariski-closure $\widetilde{X}_{w}$ of $\widetilde{V}_{w}$
inside
$\prod_{d=1}^{n-1}\mathbb{P}(\bigwedge^{d}k^{n})\times\mathbb{A}^{n(n+1)/2}$.
The result we achieve in Proposition 5.2.4 is quite satisfying, as Conj. 5 is
equivalent to $\widetilde{X}_{w}$ being the closed subset $\\{F_{i}=0\\}_{i\in
I}$; see Proposition 5.3.1.

On the other hand, through technical simplifications in §5.3, we improve on
Proposition 5.2.4 by finding unexpected additional equations satisfied on
$\widetilde{V}_{w}$. As a consequence, we get for a certain family of couples
$(w,w^{\prime})\in(\mathcal{S}_{n})^{2}$, a beautifully simple equation
satisfied on $\widetilde{X}_{w}\cap\widetilde{V}_{w^{\prime}}$ but not on
$\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\mathfrak{t}^{ww^{\prime-1}}+\mathfrak{u})$, hence the following
result:

######  Theorem 10 (Theorem 5.3.7, Remark 5.3.8, Remark 5.3.9).

For any $n\in\mathbb{Z}_{\geq 4}$, there exists a bad pair $(w,w^{\prime})$ in
$\mathcal{S}_{n}$ such that

$\dim\left(\widetilde{X}_{w}\cap\widetilde{V}_{w^{\prime}}\right)<\dim\left(\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\mathfrak{t}^{ww^{\prime-1}}+\mathfrak{u})\right)\,.$

In particular, Conj. 5 is false.

To our knowledge, the question of deciding whether or not Conj. 5 is false for
_all_ bad pairs remains open (we know that this holds at least for
$\operatorname{GL}_{4}$ and $\operatorname{GL}_{5}$). Another area of further
investigation is to find accurate estimates for the difference
$\dim\big{(}\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\mathfrak{t}^{ww^{\prime-1}}+\mathfrak{u})\big{)}-\dim\big{(}\widetilde{X}_{w}\cap\widetilde{V}_{w^{\prime}}\big{)}$.

#### Notation

We list here some of the main notation of the paper.

Throughout the paper, $k$ is a field of characteristic $0$ with algebraic
closure $k^{\mathrm{a}}$. We denote by $(G,T)$ a connected split reductive
algebraic group over $k$, $B$ a Borel subgroup of $G$ containing $T$ and
$U\subset B$ the subgroup of unipotent elements in $B$.

We write $\mathfrak{g}$ for the Lie algebra of $G$, by which we mean the
affine scheme over $k$ such that $\mathfrak{g}(k)$, as a finite-dimensional
vector space over $k$, is naturally isomorphic to the tangent space $T_{e}G$
of $G$ at the neutral element $e\in G$. Similarly, $\mathfrak{t}$,
$\mathfrak{b}$, $\mathfrak{u}$, $\mathfrak{u}_{\alpha}$ are the Lie algebras
of $T$, $B$, $U$, $U_{\alpha}$ respectively ($\mathfrak{u}_{\alpha}$ is
sometimes also denoted $\mathfrak{g}_{\alpha}$). We write
$\operatorname{Ad}\colon G\longrightarrow\operatorname{GL}(\mathfrak{g})$ and
$\operatorname{ad}\colon\mathfrak{g}\longrightarrow\operatorname{\mathfrak{gl}}(\mathfrak{g})$
for the adjoint representations.

We denote by $X(T)\coloneqq\operatorname{Hom}(T,\mathbb{G}_{\mathrm{m}})$ the
group of algebraic characters of $T$ viewed as a $\mathbb{Z}$-module, and by
$\Phi\subset X(T)$ the set of roots of $(G,T)$, _i.e._ the set of $\alpha\in
X(T)\setminus\\{0\\}$ such that the eigenspace
$\mathfrak{g}_{\alpha}\coloneqq\left\\{x\in\mathfrak{g}\mathrel{}\middle|\mathrel{}\operatorname{Ad}(t)x=\alpha(t)x\;\forall
t\in T\right\\}$ is non-zero. For any root $\alpha\in\Phi$, we have
$\dim\mathfrak{g}_{\alpha}=1$ (see [20, Thm. 21.11(b)]). We write
$U_{\alpha}\subset G$ for the root subgroup of $\alpha$ (see [20, 21.10,
21.11]); its Lie algebra $\mathfrak{u}_{\alpha}$ satisfies
$\mathfrak{u}_{\alpha}(k)=\mathfrak{g}_{\alpha}$.

Next, we write $\Gamma\subseteq X(T)$ for the $\mathbb{Z}$-submodule generated
by $\Phi$, and $E\coloneqq\Gamma\otimes_{\mathbb{Z}}\mathbb{Q}$. We then write
$I$ for the basis of the root system $\Phi$ defined by the Borel $B$ and
$\Phi_{+}$ for the set of positive roots defined by $I$.

We write $W\coloneqq W(\Phi)$ for the Weyl group of $\Phi$. Recall that
$W(G,T)\coloneqq N_{G}(T)/T$ is a finite group with a natural faithful action
on $T$ and hence on $X(T)$, which makes $W(G,T)$ equal to $W$ as groups of
automorphisms of $X(T)$ (see [20, Cor. 21.38]). We therefore make the
identification $W=N_{G}(T)/T$. For $w\in W$, we write $\operatorname{Ad}(w)$
for the conjugation morphism on $T$ by any $\dot{w}\in N_{G}(T)$ which lifts
$w$, _i.e._ $\operatorname{Ad}(w)t=\dot{w}t\dot{w}^{-1}$ for $t\in T$. Then
the action of $W$ on $\Phi$ can be written as
$(w\alpha)(t)=\alpha(\operatorname{Ad}(w^{-1}){t})$ for $\alpha\in\Phi$, $w\in
W$ and $t\in T$. Finally, $T^{w}\coloneqq\left\\{t\in
T\mathrel{}\middle|\mathrel{}\operatorname{Ad}(w)t=t\right\\}$ denotes the
subgroup of $T$ of invariants under $\operatorname{Ad}(w)$ and
$(T^{w})^{\circ}$ denotes its identity component.

In §2 (except §2.2), $E$ is simply defined as a vector space over $\mathbb{Q}$
and $\Phi$ as an abstract root system in $E$, independently of any split
reductive group $G$. In §5, we restrict to the case
$G=\operatorname{GL}_{n/k}$ for some $n\in\mathbb{Z}_{>0}$; then $T$ and $B$
are respectively taken to be the subgroups of diagonal and upper triangular
matrices.

From §3 onwards, we deal with algebraic varieties over $k$, _i.e._ reduced
separated schemes of finite type over $k$. Morphisms of varieties are then
uniquely determined by their value on ${k^{\mathrm{a}}}$-points, and their
“image” (more precisely, the surjectivity onto a subvariety of the target) can
be checked on closed points (see [27, Ex. 7.4.C]). Given two varieties $X_{1}$
and $X_{2}$ equipped with immersions into a third variety $X$, we write

$X_{1}\cap X_{2}\coloneqq\left(X_{1}\times_{X}X_{2}\right)^{\mathrm{red}}$

when there is no ambiguity. By base change, the natural map $X_{1}\cap
X_{2}\longrightarrow X_{1}$ is a closed (resp. locally closed) immersion when
$X_{2}\lhook\joinrel\relbar\joinrel\rightarrow X$ is closed (resp. locally
closed), and similarly for $X_{1}\cap X_{2}\longrightarrow X_{2}$. In the same
way, given a morphism of varieties $f\colon X\longrightarrow Y$ and a closed
(resp. locally closed) immersion
$Y^{\prime}\lhook\joinrel\relbar\joinrel\rightarrow Y$, we write

$f^{-1}(Y^{\prime})\coloneqq\left(X\times_{Y}Y^{\prime}\right)^{\mathrm{red}}\,,$

and the natural map $f^{-1}(Y^{\prime})\longrightarrow X$ is a closed (resp.
locally closed) immersion.

#### Acknowledgements

The results of this paper were obtained during the author’s PhD at the
Laboratoire de Mathématiques d’Orsay. Much credit is due to his PhD advisor
Christophe Breuil for his supervision and for providing key ideas for the
proof of Theorem 6. The author would like to thank Ariane Mézard for numerous
exchanges which have enlightened the search of counter-examples to Conjecture
5. The author would also like to thank Benjamin Schraen and Anne Moreau for
their help, as well as Gabriel Dospinescu and Ruochuan Liu for their careful
review of the author’s PhD dissertation.

##  2\. Some combinatorics in Weyl groups

In this section, we study particular subsystems of a root system $\Phi$
associated with an element of the Weyl group $W(\Phi)$. We then define the
notions of _good pairs_ and _bad pairs_ in $W(\Phi)$. Finally, we explicitly
describe the case of systems of type $A$.

###  2.1. Minimal generating root subsystems

Let $E$ be a finite dimensional a vector space over $\mathbb{Q}$ and let
$\Phi$ be a root system inside $E$ (see [5, Chapter VI, §1]). Write the
(finite) Weyl group of $\Phi$ as $W\subset\operatorname{End}_{\mathbb{Q}}(V)$.
For a root $\alpha\in\Phi$, we denote by $s_{\alpha}\in W$ its associated
reflection.

######  Definition 2.1.1.

Let $w\in W$, $\Gamma_{w}$ the $\mathbb{Z}$-submodule of $E$ generated by
$w(\alpha)-\alpha$ for $\alpha\in\Phi$,
$E_{w}\coloneqq\Gamma_{w}\otimes_{\mathbb{Z}}\mathbb{Q}=\operatorname{im}(w-\operatorname{id}_{E})$
and $\Phi_{w}\coloneqq\Phi\cap E_{w}$. We call $\Phi_{w}$ the _minimal
generating (root) subsystem_ of $w$. We also define
$d_{w}\coloneqq\operatorname{rk}_{\mathbb{Z}}\Gamma_{w}=\dim_{\mathbb{Q}}E_{w}$.

Being the intersection of $\Phi$ with a subspace $E_{w}$ of $E$, the set
$\Phi_{w}$ is a root subsystem in $\Phi$, _i.e._ it is a root system in the
subspace of $E$ that it generates (see the Corollary to Proposition 4 of [5,
Chapter VI, §1.1]). Note that this latter subspace may be strictly smaller
than $E_{w}$; however we will see later in Corollary 2.1.5 that this never
occurs. It therefore makes sense to consider the Weyl group $W(\Phi_{w})$ of
$\Phi_{w}$ as a subgroup of $W$, with the convention $W(\emptyset)=\\{1\\}$.

######  Proposition 2.1.2.

Let $w\in W$. Then $w$ is in the Weyl group $W(\Phi_{w})$ of its minimal
generating subsystem.

###### Proof.

Any root system comes from some connected split reductive group $G$ over
$\mathbb{Q}$ (see for instance the discussion in [14, §33.6]). Hence the
statement follows from Corollary 2.2.3 below. ∎

We equip $E$ with a non-degenerate symmetric bilinear form
$(\,\cdot\mid\cdot\,)$ invariant by $W$ (see Proposition 3 of [5, Chapter VI,
§1.1]). Then every root $\alpha\in\Phi$ satisfies $(\alpha\mid\alpha)\neq 0$,
and $s_{\alpha}(x)=x-2\frac{(x\mid\alpha)}{(\alpha\mid\alpha)}\alpha$ for all
$x\in E$. For any $x\in E$, we also note $x^{\perp}\coloneqq\ker(\,\cdot\mid
x)$ the orthogonal subspace of $x$.

######  Lemma 2.1.3.

Let $w\in W$ and $x\in E$. Then $x\in E_{w}$ if and only if
$\ker(w-\operatorname{id}_{E})\subseteq x^{\perp}$.

###### Proof.

Suppose $x\in E_{w}$; take $y\in E$ such that $x=w(y)-y$. Then, for all
$z\in\ker(w-\operatorname{id})=\ker(w^{-1}-\operatorname{id})$, we have

$(z\mid x)=(z\mid w(y)-y)=(z\mid w(y))-(z\mid y)=(w^{-1}(z)-z\mid y)=0$

hence $z\in x^{\perp}$.

Conversely, suppose that $\ker(w-\operatorname{id})\subseteq x^{\perp}$. Then
the linear form $(\,\cdot\mid x)$ factors through a linear form $l$ on
$E/\ker(w-\operatorname{id})$. The map $w-\operatorname{id}$ also factors
through an isomorphism $f\colon
E/\ker(w-\operatorname{id})\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\operatorname{im}(w-\operatorname{id})$.
The linear form $l\circ f^{-1}$ on $\operatorname{im}(w-\operatorname{id})$
can be extended arbitrarily to a linear form $\widetilde{l}$ on $E$. Since
$(\,\cdot\mid\cdot\,)$ is non-degenerate, there exists $y\in E$ such that
$\widetilde{l}=(\,\cdot\mid y)$. Then, for all $z\in E$,

$(z\mid x)=(w(z)-z\mid y)=(w(z)\mid y)-(z\mid y)=(z\mid w^{-1}(y)-y)$

thus, by non-degeneracy, $x=w^{-1}(y)-y=(w-\operatorname{id})(-w^{-1}(y))\in
E_{w}$. ∎

######  Proposition 2.1.4.

Let $w\in W$ and $\alpha\in\Phi$. Let $w^{\prime}$ be either $s_{\alpha}w$ or
$ws_{\alpha}$.

  1. _(1)_

If $\alpha\notin\Phi_{w}$, then $E_{w^{\prime}}=\mathbb{Q}\alpha\oplus E_{w}$.
In particular, $d_{w^{\prime}}=d_{w}+1$.

  2. _(2)_

If $\alpha\in\Phi_{w}$, then $E_{w}=\mathbb{Q}\alpha\oplus E_{w^{\prime}}$. In
particular, $d_{w^{\prime}}=d_{w}-1$.

###### Proof.

Assume that $w^{\prime}=ws_{\alpha}$. Using the decomposition
$E=\mathbb{Q}\alpha\oplus\alpha^{\perp}$, one can write

(2.1) $E_{w}=\mathbb{Q}(w(\alpha)-\alpha)+(w-\operatorname{id})\alpha^{\perp}$

and $E_{w}=(w-\operatorname{id})\alpha^{\perp}$ if and only if
$\ker(w-\operatorname{id})\not\subseteq\alpha^{\perp}$. From Lemma 2.1.3, this
happens if and only if $\alpha\notin E_{w}$. Similarly, because
$w^{\prime}(\alpha)=-w(\alpha)$ and
$w^{\prime}|_{\alpha^{\perp}}=w|_{\alpha^{\perp}}$,

(2.2)
$E_{w^{\prime}}=\mathbb{Q}(w(\alpha)+\alpha)+(w-\operatorname{id})\alpha^{\perp}$

and $E_{w^{\prime}}=(w-\operatorname{id})\alpha^{\perp}$ if and only if
$\alpha\notin E_{w^{\prime}}$.

Suppose first that $\alpha\notin\Phi_{w}$, or equivalently $\alpha\notin
E_{w}$. Then

$w(\alpha)-\alpha\in E_{w}=(w-\operatorname{id})\alpha^{\perp}\subseteq
E_{w^{\prime}}\,.$

Since $w(\alpha)+\alpha\in E_{w^{\prime}}$, we get
$\alpha=\frac{1}{2}(w(\alpha)+\alpha)-\frac{1}{2}(w(\alpha)-\alpha)\in
E_{w^{\prime}}$. We also know from (2.2) that
$E_{w}=(w-\operatorname{id})\alpha^{\perp}\subseteq E_{w^{\prime}}$ and that
$\dim{E_{w^{\prime}}}\leq\dim{E_{w}}+1$. Altogether, this yields
$E_{w^{\prime}}=\mathbb{Q}\alpha\oplus E_{w}$, so (1) is proved.

Now suppose that $\alpha\in\Phi_{w}$, or equivalently $\alpha\in E_{w}$. Take
$x\in E$ such that $\alpha=w(x)-x$. Then

$(w(x)+x\mid\alpha)=(w(x)+x\mid w(x)-x)=(w(x)\mid w(x))-(x\mid x)=0$

so $w(x)+x\in\alpha^{\perp}$, hence
$x=\frac{1}{2}(-\alpha+w(x)+x)\in-\frac{1}{2}\alpha+\alpha^{\perp}$. Therefore

$2\alpha=2(w(x)-x)\in\alpha-w(\alpha)+(w-\operatorname{id})\alpha^{\perp}$

which can be rewritten as
$w(\alpha)+\alpha\in(w-\operatorname{id})\alpha^{\perp}$. From (2.2), this
means $E_{w^{\prime}}=(w-\operatorname{id})\alpha^{\perp}$, so $\alpha\notin
E_{w^{\prime}}$. As before, this also implies, using (2.1), that
$E_{w^{\prime}}\subseteq E_{w}$ and $\dim{E_{w}}\leq\dim{E_{w^{\prime}}}+1$.
We deduce that $E_{w}=\mathbb{Q}\alpha\oplus E_{w^{\prime}}$, so (2) is
proved.

If $w^{\prime}=s_{\alpha}w$, then one can also write
$w^{\prime}=ww^{-1}s_{\alpha}w=ws_{w^{-1}(\alpha)}$. Since $\alpha\in\Phi_{w}$
if and only if $w^{-1}(\alpha)\in\Phi_{w}$, and similarly for
$\Phi_{w}^{\prime}$, the problem reduces to the previous case. ∎

Let
$R\coloneqq\left\\{s_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in\Phi\right\\}$
be the set of reflections of $W$, which generates $W$ as a group. We can then
view any element $w\in W$ as a word on the alphabet $R$ (in a non-unique way);
we call _length of $w$ on $R$_ the minimal length of such a word, and _reduced
decomposition of $w$ on $R$_ any such word of minimal length. The following
corollary to Proposition 2.1.4 interprets minimal generating subsystems using
reduced decompositions on $R$.

######  Corollary 2.1.5.

Let $w\in W$.

  1. _(1)_

The number $d_{w}$ is the length of $w$ on the alphabet $R$.

  2. _(2)_

The subsystem $\Phi_{w}$ is the set of roots $\alpha\in\Phi$ such that
$s_{\alpha}$ appears in a reduced decomposition of $w$ on the alphabet $R$.

  3. _(3)_

The subsystem $\Phi_{w}$ spans $E_{w}$ as a vector space.

###### Proof.

If $w=1$, then all claims are true because
$E_{w}=\operatorname{im}(\operatorname{id}_{E}-\operatorname{id}_{E})=\\{0\\}$.
We now suppose that $w\neq 1$.

Let $w=s_{\alpha_{r}}\ldots s_{\alpha_{1}}$ be a decomposition of $w$ on $R$,
with $\alpha_{i}\in\Phi$ for all $1\leq i\leq r$. By Proposition 2.1.4, we
know that $d_{s_{\alpha_{i}}\ldots s_{\alpha_{1}}}\leq
d_{s_{\alpha_{i-1}}\ldots s_{\alpha_{1}}}+1$ for all $1\leq i\leq d$, with
equality if and only if $E_{s_{\alpha_{i}}\ldots
s_{\alpha_{1}}}=\mathbb{Q}\alpha_{i}\oplus E_{s_{\alpha_{i-1}}\ldots
s_{\alpha_{1}}}$. An induction on $i$ gives $d_{w}=d_{s_{\alpha_{r}}\ldots
s_{\alpha_{1}}}\leq r$, with equality if and only if

(2.3) $E_{w}=\mathbb{Q}\alpha_{r}\oplus\ldots\oplus\mathbb{Q}\alpha_{1}\,.$

Therefore the three claims will be proved if we can show that $\Phi_{w}$ is
non-empty and that for any $\alpha\in\Phi_{w}$, there is a decomposition of
$w$ on $R$ of length $d_{w}$ containing $s_{\alpha}$.

The non-emptiness of $\Phi_{w}$ is a consequence of Proposition 2.1.2. Now
take $\alpha\in\Phi_{w}$. The wanted decomposition can be obtained by
induction on $d_{w}$: Proposition 2.1.4 implies that
$d_{s_{\alpha}w}=d_{w}-1$, and if $s_{\alpha}w=s_{\alpha_{r}}\ldots
s_{\alpha_{1}}$ is a decomposition of length $r\coloneqq d_{s_{\alpha}w}$,
then $w=s_{\alpha}s_{\alpha_{r}}\ldots s_{\alpha_{1}}$ is a decomposition of
length $d_{w}$. ∎

The following corollary explains the terminology of minimal generating
subsystem introduced in Definition 2.1.1.

######  Corollary 2.1.6.

Let $w\in W$ and let $E^{\prime}$ be a subspace of $E$. Let
$\Phi^{\prime}\coloneqq\Phi\cap E^{\prime}$, which is a root system in
$E^{\prime}$ with Weyl group $W(\Phi^{\prime})\subseteq W$. Then $w\in
W(\Phi^{\prime})$ if and only if $\Phi_{w}\subseteq\Phi^{\prime}$, if and only
if $E_{w}\subseteq E^{\prime}$.

###### Proof.

The implication $E_{w}\subseteq
E^{\prime}\implies\Phi_{w}\subseteq\Phi^{\prime}$ is a mere consequence of the
definitions, and the implication $\Phi_{w}\subseteq\Phi^{\prime}\implies w\in
W(\Phi^{\prime})$ is Proposition 2.1.2.

Now suppose that $w\in W(\Phi^{\prime})$. Let $\Phi^{\prime}_{w}$ be the
minimal generating subsystem of $w$ in $\Phi^{\prime}$ and let
$E^{\prime}_{w}\coloneqq\operatorname{im}(w|_{E^{\prime}}-\operatorname{id}_{E^{\prime}})\subseteq
E^{\prime}\cap E_{w}$. From Corollary 2.1.5(1) applied to $\Phi^{\prime}$, $w$
has a reduced decomposition on
$R^{\prime}\coloneqq\left\\{s_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in\Phi^{\prime}\right\\}$
of length $\dim{E^{\prime}_{w}}$. Since $R^{\prime}\subseteq R$, the same
result applied to $\Phi$ shows that $\dim{E^{\prime}_{w}}\geq\dim{E_{w}}$.
Therefore, the chain of inclusions $E^{\prime}_{w}\subseteq E^{\prime}\cap
E_{w}\subseteq E_{w}$ is a chain of equalities. In particular, $E^{\prime}\cap
E_{w}=E_{w}$; or equivalently $E_{w}\subseteq E^{\prime}$. ∎

Fix $I\subset\Phi$ a basis of simple roots of $\Phi$, and write
$S\coloneqq\left\\{s_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in
I\right\\}\subseteq W$ for the set of simple reflections corresponding to the
simple roots. These define a length function $\lg$ on $W$. The following
result is a generalisation of [6, Lemma 2.7] where it was proved for root
systems $\Phi$ of type $A$.

######  Corollary 2.1.7.

Let $w\in W$. Then

$d_{w}\leq\lg(w)$

with equality if and only if $w$ is a product of distinct simple reflections.

###### Proof.

Let $w=s_{\alpha_{\lg(w)}}\ldots s_{\alpha_{1}}$ be a reduced decomposition of
$w$ on $S$, with $\alpha_{i}\in I$ for all $1\leq i\leq\lg(w)$. Corollary
2.1.5(1) and its proof give $d_{w}\leq\lg(w)$, with equality if and only if
$E_{w}=\mathbb{Q}\alpha_{\lg(w)}\oplus\ldots\oplus\mathbb{Q}\alpha_{1}$ (see
(2.3)). Therefore, in the equality case, the $\alpha_{i}$ must be distinct.

Conversely, suppose that $w$ is a product of distinct simple reflections.
Write $w=s_{\alpha_{r}}\ldots s_{\alpha_{1}}$ with $\alpha_{i}$ for $1\leq
i\leq r$ being distinct elements of $I$; then $r\geq\lg(w)$ by definition.
Also, since the elements of $I$ are linearly independent, the sum
$\mathbb{Q}\alpha_{r}+\ldots+\mathbb{Q}\alpha_{1}$ is direct. The proof of
Corollary 2.1.5 implies that $r=d_{w}$; hence $d_{w}\geq\lg(w)$, so we have
equality. ∎

###  2.2. Interpretation for a split reductive group

For $\Phi$ a root system and $(G,T)$ a connected split reductive group giving
rise to $\Phi$, we give an interpretation of the minimal generating subsystem
$\Phi_{w}\subseteq\Phi$ for $w\in W(\Phi)$ in terms of Levi groups. We use the
notation of the introduction.

We start by recalling a classical lemma.

######  Lemma 2.2.1.

Let $N\geq 1$ be an integer and $M$ be a $\mathbb{Z}[\frac{1}{N}]$-module
equipped with a linear action of the group $\mathbb{Z}/N\mathbb{Z}$. Then
$H^{1}(\mathbb{Z}/N\mathbb{Z},M)=0$.

###### Proof.

The $\mathbb{Z}[\frac{1}{N}]$-module $H^{1}(\mathbb{Z}/N\mathbb{Z},M)$ is
killed by $N$ (see [22, §VII.7, Proposition 6]), hence it must be $0$. ∎

######  Proposition 2.2.2.

For $w\in W$, the centraliser $\left(C_{G}((T^{w})^{\circ}),T\right)$ is a
connected split reductive group whose root system is the minimal generating
subsystem $\Phi_{w}$ of $w$.

###### Proof.

$(T^{w})^{\circ}$ is a subtorus of $T$ (see the discussion in [20, Notation
12.29]; note that $T^{w}$ is always reduced in characteristic $0$), hence
$C_{G}((T^{w})^{\circ})$ is connected and split reductive by [20, Corollary
17.59]. Note that it always contains $T$, so
$\left(C_{G}((T^{w})^{\circ}),T\right)$ is split reductive.

Let $\alpha\in\Phi$ be a root of $(G,T)$. The eigenspace
$\mathfrak{g}_{\alpha}$ has dimension $1$, so $\alpha$ is a root of
$\left(C_{G}((T^{w})^{\circ}),T\right)$ if and only if $\mathfrak{g}_{\alpha}$
is in the Lie group of $C_{G}((T^{w})^{\circ})$. By [20, Thm. 21.11(c)], we
thus have

$\Phi^{\prime}\coloneqq\Phi\left(C_{G}((T^{w})^{\circ}),T\right)=\left\\{\alpha\in\Phi\mathrel{}\middle|\mathrel{}\alpha|_{(T^{w})^{\circ}}=1\right\\}\,.$

Let $N$ be the order of $w$ in $W$. The morphism of algebraic varieties
$T\longrightarrow T,\,t\longmapsto\prod_{k=0}^{N-1}\operatorname{Ad}(w^{-k})t$
has a connected image (because $T$ is connected) included in $T^{w}$. Hence,
$\prod_{k=0}^{N-1}\operatorname{Ad}(w^{-k})t\in(T^{w})^{\circ}$ for all $t\in
T$. Then, for $\alpha\in\Phi^{\prime}$ and $t\in T$, we have

$\left(\sum_{k=0}^{N-1}w^{k}\alpha\right)(t)=\alpha\left(\prod_{k=0}^{N-1}\operatorname{Ad}(w^{-k})t\right)=1$

hence $(\sum_{k=0}^{N-1}w^{k})\alpha=0$. Conversely, if
$\alpha\in\ker(\sum_{k=0}^{N-1}w^{k})$, then for all $t\in(T^{w})^{\circ}$,

$\alpha(t^{N})=\alpha\left(\prod_{k=0}^{N-1}\operatorname{Ad}(w^{-k})t\right)=1$

so, since the group of characters on the torus $(T^{w})^{\circ}$ has no
torsion, we necessarily have $\alpha(t)=1$. Hence
$\alpha|_{(T^{w})^{\circ}}=1$. Therefore
$\Phi^{\prime}=\Phi\cap\ker(\sum_{k=0}^{N-1}w^{k})$. Since $H^{1}(\langle
w\rangle,E)=0$ by Lemma 2.2, we have
$\Phi^{\prime}=\Phi\cap\operatorname{im}(w-\operatorname{id}_{E})=\Phi_{w}$. ∎

######  Corollary 2.2.3.

Let $w\in W$. Then $w\in W(\Phi_{w})$.

###### Proof.

Seeing $w$ as an element of $N_{G}(T)/T$, we have $w\in
C_{G}((T^{w})^{\circ})/T$, so that actually $w\in
N_{C_{G}((T^{w})^{\circ})}(T)/T=W(C_{G}((T^{w})^{\circ}),T)=W(\Phi_{w})$ where
the last equality comes from Proposition 2.2.2. ∎

###  2.3. Pairs of elements in the Weyl group

Fix a basis of simple roots $I\subset\Phi$ and write $\Phi_{+}$ for the subset
of positive roots that it generates. Write
$S\coloneqq\left\\{s_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in
I\right\\}\subseteq W$ for the set of simple reflections corresponding to the
simple roots. They define a length function $\lg$ and a Bruhat order $\preceq$
on $W$.

######  Definition 2.3.1.

Let $(w_{1},w_{2})\in W^{2}$ such that $w_{1}\preceq w_{2}$. Let
$\Phi_{w_{1}w_{2}^{-1}}\subseteq\Phi$ be the minimal generating subsystem of
$w_{1}w_{2}^{-1}$. We say that $(w_{1},w_{2})$ is a _good pair_ if there
exists a sequence $(\alpha_{1},\ldots,\alpha_{r})$ of roots in
$\Phi_{w_{1}w_{2}^{-1}}$ such that $w_{2}=s_{\alpha_{r}}s_{\alpha_{r-1}}\ldots
s_{\alpha_{1}}w_{1}$ and that for each $1\leq i\leq r$, the relation
$s_{\alpha_{i}}\ldots s_{\alpha_{1}}w_{1}\succ s_{\alpha_{i-1}}\ldots
s_{\alpha_{1}}w_{1}$ holds. If such a sequence does not exist, we say that
$(w_{1},w_{2})$ is a _bad pair_.

######  Remark 2.3.2.

The set of reflections
$R=\left\\{s_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in\Phi\right\\}$ of
$W$ can be rewritten as $R=\bigcup_{w\in W}wSw^{-1}$. On the other hand, using
(2) of Corollary 2.1.5, one sees that the subset
$\left\\{s_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in\Phi_{w_{1}w_{2}^{-1}}\right\\}$
of $R$ is the set of reflections that appear in some reduced decomposition of
$w$ on the alphabet $R$. Hence the notion of good pair is intrinsic to the
Coxeter system $(W,S)$.

Let $w_{0}\in W$ be the maximal element for the Bruhat order. Using that
$w\mapsto ww_{0}$ and $w\mapsto w_{0}w$ are antiautomorphisms of $W$ for the
Bruhat order, one easily proves the following result.

######  Proposition 2.3.3.

Let $(w_{1},w_{2})\in W^{2}$. Then $(w_{1},w_{2})$ is a good pair (resp. bad
pair) if and only if $(w_{2}w_{0},w_{1}w_{0})$ is a good pair (resp. bad
pair). Similarly, $(w_{1},w_{2})$ is a good pair (resp. bad pair) if and only
if $(w_{0}w_{2},w_{0}w_{1})$ is a good pair (resp. bad pair).

######  Proposition 2.3.4.

Let $w_{1},w_{2}\in W$ with $w_{1}\preceq w_{2}$. Then

$d_{w_{2}w_{1}^{-1}}\leq\lg(w_{2})-\lg(w_{1})$

and if there is equality, then the pair $(w_{1},w_{2})$ is good.

###### Proof.

By the chain property in $W$ (see [4, Thm. 2.2.6]), there is a sequence of
size $r=\lg(w_{2})-\lg(w_{1})$ of roots $(\alpha_{1},\ldots,\alpha_{r})$ in
$\Phi$ with $w_{2}=s_{\alpha_{r}}s_{\alpha_{r-1}}\ldots s_{\alpha_{1}}w_{1}$
and such that for each $1\leq i\leq r$, we have $\lg(s_{\alpha_{i}}\ldots
s_{\alpha_{1}}w_{1})=\lg(s_{\alpha_{i-1}}\ldots s_{\alpha_{1}}w_{1})+1$ (hence
automatically $s_{\alpha_{i}}\ldots s_{\alpha_{1}}w_{1}\succ
s_{\alpha_{i-1}}\ldots s_{\alpha_{1}}w_{1}$). Then
$w_{2}w_{1}^{-1}=s_{\alpha_{r}}s_{\alpha_{r-1}}\ldots s_{\alpha_{1}}$ is a
decomposition of $w_{2}w_{1}^{-1}$ on
$R\coloneqq\left\\{s_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in\Phi\right\\}$.
Corollary 2.1.5(1) then yields $d_{w_{2}w_{1}^{-1}}\leq
r=\lg(w_{2})-\lg(w_{1})$. If there is equality, Corollary 2.1.5(2) gives
$\alpha_{i}\in\Phi_{w_{2}w_{1}^{-1}}$ for all $1\leq i\leq r$, which implies
that $(w_{1},w_{2})$ is a good pair. ∎

Let $J\subseteq I$ be a set of simple roots. We write
$S_{J}\coloneqq\left\\{s_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in
J\right\\}$ for the set of simple reflections associated to the elements of
$J$, and $W_{J}\coloneqq\left\langle s\mathrel{}\middle|\mathrel{}s\in
S_{J}\right\rangle$ for the subgroup of $W$ it spans. Note that
$W_{J}=W(\Phi_{J})$, where $\Phi_{J}$ is the root subsystem of $\Phi$
generated by $J$, and the Bruhat order on $W_{J}$ is the same as the
restriction of the Bruhat order on $W$. We also write
$W^{J}\coloneqq\left\\{w\in W\mathrel{}\middle|\mathrel{}ws\succ
w\mathrel{}\forall s\in S_{J}\right\\}$. It is known that $W^{J}$ is the set
of representatives of minimal length for the left cosets representatives
$W/W_{J}$, and that for all $w\in W$, the decomposition $w=w^{J}w_{J}$ with
$w^{J}\in W^{J}$ and $w_{J}\in W_{J}$ satisfies $\lg(w)=\lg(w^{J})+\lg(w_{J})$
(see [4, §2.4]).

######  Lemma 2.3.5.

Let $J\subseteq I$ be a subset of the simple roots, let $u^{J}$ and $v^{J}$ be
elements of $W^{J}$, let $w_{J}$ be an element of $W_{J}$ and let $s\in W_{J}$
be a reflection, _i.e._ an element of $\bigcup_{w\in W_{J}}wS_{J}w^{-1}$. Then
$sw_{J}\succ w_{J}$ if and only if $u^{J}sw_{J}(v^{J})^{-1}\succ
u^{J}w_{J}(v^{J})^{-1}$.

###### Proof.

Let $R\coloneqq\left\\{wSw^{-1}\mathrel{}\middle|\mathrel{}w\in W\right\\}$ be
the set of reflections of $W$ and consider the map

$\eta\colon\;\begin{aligned} W\times R\;&\longrightarrow\;\\{-1,1\\}\\\
(w,t)\;&\longmapsto\;\begin{cases}1&\mathrm{if}\;tw\succ w\\\
-1&\mathrm{if}\;tw\prec w\end{cases}\end{aligned}$

(see Theorem 1.4.3 of [4] and its proof). Our goal is then to prove that

$\eta(u^{J}w_{J}(v^{J})^{-1},u^{J}s(u^{J})^{-1})=\eta(w_{J},s).$

For any $w\in W$, $t\in R$ and $\varepsilon\in\\{-1,1\\}$, define

$\pi_{w}(t,\varepsilon)\coloneqq(wtw^{-1},\varepsilon\eta(w^{-1},t))\in
R\times\\{-1,1\\}\,.$

This defines a map $\pi$ from $W$ to the finite group of bijections from
$R\times\\{-1,1\\}$ to itself, and $\pi$ is a group morphism (_loc. cit._ ,
Theorem 1.3.2(i)). Therefore, for any $v,w\in W$ and $t\in R$, we can write
$\pi_{w^{-1}v^{-1}}(t,1)=\pi_{w^{-1}}(\pi_{v^{-1}}(t,1))$ and get the formula

$\eta(vw,t)=\eta(v,t)\eta(w,v^{-1}tv)\,.$

First apply this formula to $v=u^{J}$, $w=w_{J}(v^{J})^{-1}$ and
$t=u^{J}s(u^{J})^{-1}$:

(2.4)
$\eta(u^{J}w_{J}(v^{J})^{-1},u^{J}s(u^{J})^{-1})=\eta(u^{J},u^{J}s(u^{J})^{-1})\eta(w_{J}(v^{J})^{-1},s)\,.$

Now, since $u^{J}\in W^{J}$ and $w_{J}^{-1},s\in W_{J}$ and using the fact
that $w\mapsto w^{-1}$ preserves the length on $W$, we have

$\displaystyle\lg(sw_{J}(v^{J})^{-1})$ $\displaystyle=$
$\displaystyle\lg(v^{J}w_{J}^{-1}s)=\lg(v^{J})+\lg(w_{J}^{-1}s)=\lg(v^{J})+\lg(sw_{J})$
$\displaystyle\lg(w_{J}(v^{J})^{-1})$ $\displaystyle=$
$\displaystyle\lg(v^{J}w_{J}^{-1})=\lg(v^{J})+\lg(w_{J}^{-1})=\lg(v^{J})+\lg(w_{J})$

from which we deduce $\eta(w_{J}(v^{J})^{-1},s)=\eta(w_{J},s)$. Similarly,
$u^{J}\in W^{J}$ and $s\in W_{J}$, therefore

$\lg(u^{J}s(u^{J})^{-1}u^{J})=\lg(u^{J}s)=\lg(u^{J})+\lg(s)>\lg(u^{J})\,,$

which means that $\eta(u^{J},u^{J}s(u^{J})^{-1})=1$. By (2.4), this completes
the proof. ∎

######  Lemma 2.3.6.

Let $w\in W$. There is $u\in W$ and $J\subseteq I$ such that
$\Phi_{w}=u(\Phi_{J})$.

###### Proof.

Given any fixed basis $J^{\prime}$ of $\Phi_{w}$, there is a basis
$I^{\prime}$ of $\Phi$ containing $J^{\prime}$ (see Proposition 24 in §1.8 of
[5, Chapter VI]). The sets $I$ and $I^{\prime}$ are in the same $W$-orbit (see
Remark 4 in §1.5 of _loc. cit._), so we get $u\in W$ such that
$I^{\prime}=u(I)$ and $J\coloneqq u^{-1}(J^{\prime})$ is the subset of $I$ we
are looking for. ∎

The following proposition provides a direct way of seeing if a given pair is
good.

######  Proposition 2.3.7.

Let $(w_{1},w_{2})\in W^{2}$. Let $\Phi_{w_{1}w_{2}^{-1}}\subseteq\Phi$ be the
minimal generating subsystem of $w_{1}w_{2}^{-1}$ and $J\subseteq I$ be such
that $\Phi_{w_{1}w_{2}^{-1}}$ is in the $W$-orbit of the root subsystem
$\Phi_{J}\subseteq\Phi$ (see Lemma 2.3.6). Then we can write

$w_{1}=u^{J}w_{J,1}(v^{J})^{-1}\,,\quad w_{2}=u^{J}w_{J,2}(v^{J})^{-1}$

for some $u^{J},v^{J}\in W^{J}$ and $w_{J,1},w_{J,2}\in W_{J}$. Such a
decomposition of $w_{1}$ and $w_{2}$, as well as the choice of $J$, may not be
unique. Furthermore, for one such (or equivalently any such) decomposition of
$w_{1}$ and $w_{2}$, we have $w_{J,1}\preceq w_{J,2}$ if and only if the pair
$(w_{1},w_{2})$ is good.

###### Proof.

Take $u\in W$ such that $\Phi_{w_{1}w_{2}^{-1}}=u(\Phi_{J})$. Let $u^{J}\in
W^{J}$ be the representative of $uW_{J}$, then
$\Phi_{w_{1}w_{2}^{-1}}=u^{J}(\Phi_{J})$. We know from Proposition 2.1.2 that
$w_{1}w_{2}^{-1}\in W(\Phi_{w_{1}w_{2}^{-1}})=u^{J}W_{J}(u^{J})^{-1}$.
Therefore the cosets $w_{1}^{-1}u^{J}W_{J}$ and $w_{2}^{-1}u^{J}W_{J}$ are the
same. Let $v^{J}\in W^{J}$ be the representative for that coset; we then have
elements $w_{J,1}$ and $w_{J,2}$ of $W_{J}$ such that
$w_{1}^{-1}u^{J}=v^{J}w_{J,1}^{-1}$ and $w_{2}^{-1}u^{J}=v^{J}w_{J,2}^{-1}$,
_i.e._ $w_{1}=u^{J}w_{J,1}(v^{J})^{-1}$, $w_{2}=u^{J}w_{J,2}(v^{J})^{-1}$.

Now assume that $w_{J,1}\preceq w_{J,2}$. Then, by the chain property in
$W_{J}$ (see [4, Thm. 2.2.6]) there is a sequence
$(\alpha_{1},\ldots,\alpha_{r})$ of roots in $\Phi_{J}$ with
$w_{J,2}=s_{\alpha_{r}}s_{\alpha_{r-1}}\ldots s_{\alpha_{1}}w_{J,1}$ such that
$s_{\alpha_{i}}\ldots s_{\alpha_{1}}w_{1}\succ s_{\alpha_{i-1}}\ldots
s_{\alpha_{1}}w_{J,1}$ for each $1\leq i\leq r$. Let
$\beta_{i}=u^{J}(\alpha_{i})\in u^{J}(\Phi_{J})=\Phi_{w_{1}w_{2}^{-1}}$ for
each $i$, then $s_{\beta_{i}}=u^{J}s_{\alpha_{i}}(u^{J})^{-1}$ and we have

$w_{2}=u^{J}w_{J,2}(v^{J})^{-1}=s_{\beta_{r}}\ldots
s_{\beta_{1}}u^{J}w_{J,1}(v^{J})^{-1}=s_{\beta_{r}}\ldots
s_{\beta_{1}}w_{1}\,,$

and for each $i$ the relation by Lemma 2.3.5:

$s_{\beta_{i}}\ldots s_{\beta_{1}}w_{1}=u^{J}s_{\alpha_{i}}\ldots
s_{\alpha_{1}}w_{J,1}(v^{J})^{-1}\succ u^{J}s_{\alpha_{i-1}}\ldots
s_{\alpha_{1}}w_{J,1}(v^{J})^{-1}=s_{\beta_{i-1}}\ldots s_{\beta_{1}}w_{1}.$

Since this chain implies in particular that $w_{1}\preceq w_{2}$, we have
shown all the conditions for the pair $(w_{1},w_{2})$ to be good. Conversely,
going through all the steps backwards and using Lemma 2.3.5, we see that if
$(w_{1},w_{2})$ is a good pair then $w_{J,1}\preceq w_{J,2}$. ∎

###  2.4. Systems of type $A$ and pattern avoidance

We provide a concrete description of minimal generating subsystems
(Proposition 2.4.1) and of good pairs (Proposition 2.4.5 and Corollary 2.4.7)
for root systems of type $A$.

Let $n>1$ an integer, $(e_{i})_{1\leq i\leq n}$ the standard basis of
$\mathbb{Q}^{n}$ and
$\Phi=\left\\{e_{i}-e_{j}\mathrel{}\middle|\mathrel{}1\leq i,j\leq n\;,\;i\neq
j\right\\}\subset\mathbb{Q}^{n}$ a root system of type $A_{n-1}$. The Weyl
group of $\Phi$ is the symmetric group $\mathcal{S}_{n}$ which acts on
$\mathbb{Q}^{n}$, and on $\Phi$, by $w(e_{i})=e_{w(i)}$ for
$w\in\mathcal{S}_{n}$ and $1\leq i\leq n$. We write any $w\in\mathcal{S}_{n}$
as $[w(1),\ldots,w(n)]$. Given a root $\alpha=e_{i}-e_{j}$ with $i\neq j$, the
associated reflection $s_{\alpha}=(i,j)$ is given by $s_{\alpha}(i)=j$,
$s_{\alpha}(j)=i$, $s_{\alpha}(k)=k$ $\forall\ k\neq i,j$. We set the standard
basis $I\coloneqq\left\\{e_{i}-e_{i+1}\mathrel{}\middle|\mathrel{}1\leq i\leq
n-1\right\\}$. It generates the set
$\Phi_{+}\coloneqq\left\\{e_{i}-e_{j}\mathrel{}\middle|\mathrel{}1\leq i<j\leq
n\right\\}$ of positive roots, and the set of simple reflections is
$S=\left\\{[1\ldots(i-1)(i+1)i(i+2)\ldots n]\mathrel{}\middle|\mathrel{}1\leq
i\leq n-1\right\\}$.

######  Proposition 2.4.1.

Let $w\in\mathcal{S}_{n}$. Write $\coprod_{1\leq k\leq r}\Omega_{k}$ for the
partition of $\\{1,\ldots,n\\}$ into orbits under the action of $w$. For
$1\leq k\leq r$, let

$f_{k}\colon\;\begin{aligned} \mathbb{Q}^{n}\;&\longrightarrow\;\mathbb{Q}\\\
(x_{i})_{1\leq i\leq
n}\;&\longmapsto\;\sum_{i\in\Omega_{k}}x_{i}\end{aligned}\,.$

Then $E_{w}=\bigcap_{1\leq k\leq r}\ker(f_{k})$ and $\Phi_{w}=\bigcup_{1\leq
k\leq
r}\left\\{e_{i}-e_{j}\mathrel{}\middle|\mathrel{}i,j\in\Omega_{k}\;,\;i\neq
j\right\\}$.

###### Proof.

The inclusion $E_{w}\subseteq\bigcap_{1\leq k\leq r}\ker(f_{k})$ is
straightforward. Conversely, Proposition 2.2.2 gives $\bigcup_{1\leq k\leq
r}\left\\{e_{i}-e_{j}\mathrel{}\middle|\mathrel{}i,j\in\Omega_{k}\;,\;i\neq
j\right\\}=\Phi(C_{\operatorname{GL}_{n}}((T^{w})^{\circ}))=\Phi_{w}$, hence
$\bigcap_{1\leq k\leq r}\ker(f_{k})\subseteq E_{w}$ by taking the generated
$\mathbb{Q}$-vector space. ∎

######  Definition 2.4.2.

Let $d$ be an integer such that $1\leq d\leq n$. We put a partial order on the
set of size $d$ subsets of $\\{1,\ldots,n\\}$ as follows: for any such subsets
$A$ and $A^{\prime}$, we say that $A\preceq A^{\prime}$ if and only if $\lvert
A\cap\\{1,\ldots,m\\}\rvert\geq\lvert A^{\prime}\cap\\{1,\ldots,m\\}\rvert$
for all $1\leq m\leq n$. Equivalently, $A\preceq A^{\prime}$ if and only if
$\lvert A\cap\\{m,\ldots,n\\}\rvert\leq\lvert
A^{\prime}\cap\\{m,\ldots,n\\}\rvert$ for all $1\leq m\leq n$.

The relation $A\preceq A^{\prime}$ can be seen concretely as follows: if
$A=\\{a_{1}<\ldots<a_{d}\\}$ and
$A^{\prime}=\\{a^{\prime}_{1}<\ldots<a^{\prime}_{d}\\}$, then $A\preceq
A^{\prime}$ if and only if $a_{k}\leq a^{\prime}_{k}$ for all
$k\in\\{1,\ldots,d\\}$.

######  Definition 2.4.3.

Let $w\in\mathcal{S}_{n}$. For $1\leq i,j\leq n$, we write

$w[i,j]\coloneqq\left\lvert
w(\\{1,\ldots,i\\})\cap\\{j,\ldots,n\\}\right\rvert\,.$

For a subset $\Sigma$ of $\\{1,\ldots,n\\}$, we write

$w[i,j]_{\Sigma}\coloneqq\left\lvert
w(\\{1,\ldots,i\\})\cap\\{j,\ldots,n\\}\cap\Sigma\right\rvert\,.$

######  Example 2.4.4.

For any $w,w^{\prime}\in\mathcal{S}_{n}$, we have $w^{\prime}\preceq w$ if and
only if $w^{\prime}[i,j]\leq w[i,j]$ for all $1\leq i,j\leq n$ (see for
example [4, Thm. 2.1.5]). Equivalently, $w^{\prime}\preceq w$ if and only if
$\\{w(1),\ldots,w(d)\\}\preceq\\{w^{\prime}(1),\ldots,w^{\prime}(d)\\}$ for
all $1\leq d\leq n$.

The following proposition is an application of Proposition 2.3.7 to the type
$A_{n-1}$.

######  Proposition 2.4.5.

Let $w_{1}$, $w_{2}$ be elements of $\mathcal{S}_{n}$. Then the pair
$(w_{1},w_{2})$ is good if and only if, for any orbit $\Omega$ of
$w_{1}w_{2}^{-1}$ under the action of $\mathcal{S}_{n}$ on $\\{1,\ldots,n\\}$
and any $1\leq d\leq n$, we have $\Omega\cap
w_{1}(\\{1,\ldots,d\\})\succeq\Omega\cap w_{2}(\\{1,\ldots,d\\})$ (as subsets
of $\\{1,\ldots,n\\}$ in the sense of Definition 2.4.2). Equivalently, the
pair $(w_{1},w_{2})$ is good if and only if, for any orbit $\Omega$ of
$w_{1}w_{2}^{-1}$ and any $1\leq i,j\leq n$, we have $w_{1}[i,j]_{\Omega}\leq
w_{2}[i,j]_{\Omega}$.

###### Proof.

This is a consequence of Proposition 2.3.7 and Example 2.4.4. ∎

This result gives a way to visually recognise good and bad pairs through
flattenings, which we now define.

######  Definition 2.4.6.

Let $m,n$ be integers with $0<m\leq n$, and let
$\Sigma=\\{i_{1}<\ldots<i_{m}\\}$ be a subset of $\\{1,\ldots,n\\}$. We define
the _flattening map_
$\mathrm{fl}_{\Sigma}\colon\mathcal{S}_{n}\longrightarrow\mathcal{S}_{m}$ as
follows. For any $w\in\mathcal{S}_{n}$, we take $\mathrm{fl}_{\Sigma}(w)$ as
the unique $f\in\mathcal{S}_{m}$ such that $(w(i_{1}).\ldots,w(i_{m}))$ is in
the same relative order as $(f(1),\ldots,f(m))$. This means that
$w\in\mathcal{S}_{n}$ flattens to $f\in\mathcal{S}_{m}$ through $\Sigma$ if
and only if for any $1\leq k<l\leq m$, we have $w(i_{k})<w(i_{l})$ if and only
if $f(k)<f(l)$.

######  Corollary 2.4.7.

Let $w_{1},w_{2}\in\mathcal{S}_{n}$. Then the pair $(w_{1},w_{2})$ is good if
and only if for any orbit $\Omega$ of $w_{1}^{-1}w_{2}$, we have
$\mathrm{fl}_{\Omega}(w_{1})\preceq\mathrm{fl}_{\Omega}(w_{2})$.

###### Proof.

This is a consequence of Proposition 2.4.5 and Example 2.4.4). ∎

We now give a criterion that relates good pairs to pattern avoidance for root
systems of type $A$.

######  Definition 2.4.8.

Let $m,n\leq 1$ be integers. For $w\in\mathcal{S}_{n}$ and
$f\in\mathcal{S}_{m}$, we say that _w has the pattern f_ if and only if $w$
flattens to $f$ (through some $\Sigma\subseteq\\{1,\ldots,n\\}$ of size $m$),
_i.e._ there exists a flattening map
$\mathrm{fl}_{\Sigma}\colon\mathcal{S}_{n}\longrightarrow\mathcal{S}_{m}$ (see
Definition 2.4.6) such that $f=\mathrm{fl}_{\Sigma}(w)$.

######  Example 2.4.9.

If $f=[3412]\in\mathcal{S}_{4}$ (resp. $f=[4231]$), the permutation
$w\in\mathcal{S}_{n}$ has the pattern $f$ if and only if there exist $1\leq
a<b<c<d\leq n$ such that $w(c)<w(d)<w(a)<w(b)$ (resp. $w(d)<w(b)<w(c)<w(a)$).

######  Theorem 2.4.10.

Let $w\in\mathcal{S}_{n}$.

  1. _(1)_

There exists $w^{\prime}\in\mathcal{S}_{n}$ such that $(w^{\prime},w)$ is a
bad pair if and only if $w$ has at least one of the four patterns $[4231]$,
$[42513]$, $[35142]$ or $[351624]$.

  2. _(2)_

There exists $w^{\prime\prime}\in\mathcal{S}_{n}$ such that
$(w,w^{\prime\prime})$ is a bad pair if and only if $w$ has at least one of
the four patterns $[1324]$, $[24153]$, $[31524]$ or $[426153]$.

###### Proof.

The second statement is equivalent to the first by applying Proposition 2.3.3.
Also, if $w$ has one of the four patterns $[4231]$, $[42513]$, $[35142]$ and
$[351624]$, then we can easily construct a $w^{\prime}$ such that
$(w^{\prime},w)$ is a bad pair, using Corollary 2.4.7 and the fact that
$([1324],[4231]),([13245],[42513]),([12435],[35142]),([124356],[351624])$ are
bad pairs. We thus only have to prove that for any bad pair $(w^{\prime},w)$,
the permutation $w$ has one of the four patterns.

Let $(w^{\prime},w)$ be such a bad pair. From Corollary 2.4.7, there exist an
orbit $\Omega$ of $ww^{\prime-1}$ and $(i,j)\in\\{1,\ldots,n\\}^{2}$ such that
$w[i,j]_{\Omega}<w^{\prime}[i,j]_{\Omega}$; we fix one such $\Omega$ for the
rest of the proof, and set $\Sigma=\\{1,\ldots,n\\}\setminus\Omega$. The
strategy will be to find the patterns on the plot graph of $w_{1}$ using this
fact. The different elements of the proof are represented in Figure 1.

The first step is to divide the graph into a few regions. First consider
$i_{1}$ minimal with the property that there exists some $j$ such that
$w[i_{1},j]_{\Omega}<w^{\prime}[i_{1},j]_{\Omega}$. Then take $j_{1}$ maximal
such that $w[i_{1},j_{1}]_{\Omega}<w^{\prime}[i_{1},j_{1}]_{\Omega}$, and
$j_{2}$ minimal such that
$w[i_{1},j_{2}]_{\Omega}<w^{\prime}[i_{1},j_{2}]_{\Omega}$. Finally take
$i_{2}$ maximal such that
$w[i_{2},j_{2}]_{\Omega}<w^{\prime}[i_{2},j_{2}]_{\Omega}$.

The second step is to identify up to six points of interest on the graph of
$w$. There exists $A=(i_{A},j_{A})$ such that $i_{A}\leq i_{1}$, $j_{A}\geq
j_{1}$ and $w(i_{A})=j_{A}\in\Sigma$. Indeed,
$w[i_{1},j_{1}]=w[i_{1},j_{1}]_{\Omega}+w[i_{1},j_{1}]_{\Sigma}$ and similarly
with $w^{\prime}$, hence
$w[i_{1},j_{1}]_{\Sigma}>w^{\prime}[i_{1},j_{1}]_{\Sigma}$ and in particular
$w[i_{1},j_{1}]_{\Sigma}\geq 1$. Similarly, there exists $F=(i_{F},j_{F})$
such that $i_{F}>i_{2}$, $j_{F}<j_{2}$ and $w(i_{F})=j_{F}\in\Sigma$. Indeed,
$w[i_{2},j_{2}]_{\Sigma}>w^{\prime}[i_{2},j_{2}]_{\Sigma}$, hence there are
strictly more (resp. strictly fewer) points
$(i,j)\in\\{1,\ldots,i_{2}\\}\times\\{1,\ldots,j_{2}-1\\}$ (resp.
$\in\\{i_{2}+1,\ldots,n\\}\times\\{1,\ldots,j_{2}-1\\}$) with $j=w(i)$ than
with $j=w^{\prime}(i)$. Also define $B=(i_{B},j_{B})$, $C=(i_{C},j_{C})$,
$D=(i_{D},j_{D})$, $E=(i_{E},j_{E})$ respectively by $i_{B}=w^{-1}(j_{2}-1)$
and $j_{B}=j_{2}-1$; $i_{C}=i_{1}$ and $j_{C}=w(i_{1})$; $i_{D}=i_{2}+1$ and
$j_{D}=w(i_{2}+1)$; $i_{E}=w^{-1}(j_{1})$ and $j_{E}=j_{1}$. Note that we
indeed have
$A,B,C,D,E,F\in\left\\{(i,j)\mathrel{}\middle|\mathrel{}w(i)=j\right\\}$.

By definition of $i_{1},j_{1},i_{2},j_{2}$ have
$w(i_{1}),j_{1},w(i_{2}+1),j_{2}-1\in\Omega$: for instance,
$w(i_{1})\notin\Omega$ implies
$w[i_{1}-1,j_{1}]_{\Omega}=w[i_{1},j_{1}]_{\Omega}$ and
$w^{\prime}[i_{1}-1,j_{1}]_{\Omega}=w^{\prime}[i_{1},j_{1}]_{\Omega}$, which
contradicts the minimality of $i_{1}$. We then deduce $i_{B}\leq i_{1}$,
$j_{C}<j_{2}$, $j_{D}\geq j_{2}$, $i_{E}>i_{1}$ by definition of $j_{2}$,
$j_{1}$, $i_{2}$ and $j_{1}$ respectively: for instance, $i_{B}>i_{1}$ implies
$w[i_{1},j_{2}-1]_{\Omega}=w[i_{1},j_{2}]_{\Omega}<w^{\prime}[i_{1},j_{2}]_{\Omega}\leq
w^{\prime}[i_{1},j_{2}-1]_{\Omega}$ which contradicts the minimality of
$j_{2}$. Also, $i_{A}<i_{1}$ and $j_{A}>j_{1}$ since we already know
$i_{A}\leq i_{1}$ and $j_{A}\geq j_{1}$ and $A\in w^{-1}(\Sigma)\times\Sigma$
while $(i,j)\in w^{-1}(\Omega)\times\Omega$. Similarly, $i_{F}>i_{2}+1$ and
$j_{F}>j_{2}+1$.

Note that the sets of points $\\{A\\}$, $\\{B,C\\}$, $\\{D,E\\}$, $\\{F\\}$
lie in distinct quadrants delimited by $i_{1}$ and $j_{2}$. Hence $B=C$ and
$D=E$ are the only equalities that may happen between those six points. Also
note that since those points are all on the graph of $w$, unless two of those
are equal, both of their coordinates are distinct.

$i_{1}$$i_{2}$$j_{2}$$j_{1}$$A$$B_{iii}$$B_{ii}$$C$$D_{2}$$D_{1}$$E_{1}$$E_{2}$$F_{i}$$F_{iii}$
Figure 1. The black dots are on the graph $j=w(i)$; different possibilities
for the position of a given point are connected by an arrow. Gray bands are
either of $i$-coordinate in $w^{-1}(\Omega)$ or of $j$-coordinate in $\Omega$.

The third and final step is to distinguish cases according to the relative
positions of the points.

Case 1: $j_{D}>j_{A}$ and $i_{E}>i_{F}$:

Necessarily $D\neq E$ since $j_{D}>j_{A}=j_{E}$.

Case 1.i: $j_{F}<j_{C}$:

$ACDFE$ gives the pattern $[42513]$; that is to say, $w$ flattens to $[42513]$
through $\\{i_{A},i_{C},i_{D},i_{F},i_{E}\\}$.

Case 1.ii: $i_{B}>i_{A}$:

$ABDFE$ gives the pattern $[42513]$.

Case 1.iii: $j_{F}>j_{C}$ and $i_{B}<i_{A}$:

In this case $B\neq C$ because $i_{B}<i_{A}<i_{1}=i_{C}$, and $BACDFE$ gives
the pattern $[351624]$.

Case 2: $j_{D}<j_{A}$ or $i_{E}<i_{F}$:

Then, $D$ in the case $j_{D}<j_{A}$, or $E$ in the case $i_{E}<i_{F}$, have
the same relative position with respect to the other four points: that is to
say, between $C$ and $F$ along the $i$-axis and between $B$ and $A$ along the
$j$-axis. We can therefore just assume that $j_{D}<j_{A}$.

Case 2.i: $j_{F}<j_{C}$:

$ACDF$ gives the pattern $[4231]$.

Case 2.ii: $i_{B}>i_{A}$:

$ABDF$ gives the pattern $[4231]$.

Case 2.iii: $j_{F}>j_{C}$ and $i_{B}<i_{A}$:

As in case 1.iii, we have $B\neq C$; and here $BACDF$ gives the pattern
$[35142]$. ∎

######  Corollary 2.4.11.

Let $w,w^{\prime}\in\mathcal{S}_{n}$ such that $(w^{\prime},w)$ is a bad pair.
Then the Schubert variety $\overline{BwB/B}$ is singular (where $B$ is the
subgroup of upper triangular matrices and each element of $W=\mathcal{S}_{n}$
can be lifted to $G$).

###### Proof.

Since the patterns $[42513]$, $[35142]$ and $[351624]$ all contain the pattern
$[3412]$, this is a direct consequence of Theorem 2.4.10 and of [18, Thm. 1].
∎

######  Remark 2.4.12.

Given a bad pair $(w^{\prime},w)$ in $\mathcal{S}_{n}$, the point
$w^{\prime}B/B$ has no reason to be in the singular locus of
$\overline{BwB/B}$. For instance, when $n=4$, the pair $([1324],[4231])$ is
bad but the point $[1324]B/B$ is smooth in $\overline{B[4231]B/B}$; in fact
the singular locus of $\overline{B[4231]B/B}$ is $\overline{B[2143]B/B}$ (see
[18, §3]).

##  3\. A conjecture on a Schubert-like decomposition

We recall Conjecture 2.3.7 of [8], whose study is the main topic of this
paper.

###  3.1. A scheme related to the Springer resolution

We recall the construction of a reduced scheme $X$ associated to the split
reductive group $G$ and equipped with a stratification similar to the Schubert
decomposition.

First recall the Schubert decomposition on the flag variety $G/B$. Let $w\in
W$ and $\dot{w}\in N_{G}(T)(k)$ be any closed point lifting $w$. The left-
action of $B$ on $G$ by multiplication induces an action on $G/B$, write
$BwB/B$ for the orbit of $\dot{w}B/B$ under this action. Then $BwB/B$ is a
smooth locally closed subvariety of $G/B$ called a _Schubert cell_ , and we
have the decomposition

(3.1) $G/B=\coprod_{w\in W}BwB/B$

(see [20, Thm. 21.73]). The cells $BwB/B$ are stable under the left action of
$B$, thus their Zariski closure $\overline{BwB/B}$ too, and this action is
transitive on $BwB/B$. Thus, a closed cell $\overline{BwB/B}$ intersects a
cell $Bw^{\prime}B/B$ if and only if
$Bw^{\prime}B/B\subseteq\overline{BwB/B}$, which happens if and only if
$w^{\prime}\preceq w$ for the Bruhat order (see [1, Thm. 2.11]). Therefore
(3.1) gives a good stratification of $G/B$, and we have a more general
decomposition into locally closed strata for all $w\in W$

$\overline{BwB/B}=\coprod_{w^{\prime}\preceq w}Bw^{\prime}B/B\,.$

One can get a similar stratification on $G/B\times G/B$: consider the
isomorphism

(3.2)
$f\colon\begin{array}[]{ccc}G\times^{B}G/B&\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}&G/B\times
G/B\\\ (g_{1},g_{2}B)&\longmapsto&(g_{1}B,g_{1}g_{2}B)\end{array}$

where $G\times^{B}G/B$ is the quotient of $G\times G/B$ by the right
$B$-action defined by $(g_{1},g_{2}B)b\coloneqq(g_{1}b,b^{-1}g_{2}B)$ (see
[23, §3.7]). For $w\in W$, the right $B$-action leaves $G\times(BwB/B)$
invariant, so we can write

$U_{w}\coloneqq f(G\times^{B}(BwB/B))\subseteq G/B\times G/B\,.$

Equivalently, $U_{w}$ is the orbit of $(B/B,\dot{w}B/B)$ under the left
$G$-action on $G/B\times G/B$ given by diagonal left multiplication. Then
$G/B\times G/B=\coprod_{w\in W}U_{w}$ and for any $w\in W$, the decomposition
into locally closed strata

$\overline{U_{w}}=\coprod_{w^{\prime}\preceq w}U_{w^{\prime}}$

is a consequence of the analogous one in $G/B$.

As in [8], let $X$ be the following algebraic variety:

$X\coloneqq\left\\{(g_{1}B,g_{2}B,\psi)\in G/B\times
G/B\times\mathfrak{g}\mathrel{}\middle|\mathrel{}\operatorname{Ad}(g_{1}^{-1})\psi\in\mathfrak{b}\,,\,\operatorname{Ad}(g_{2}^{-1})\psi\in\mathfrak{b}\right\\}\,.$

Let us write $\pi$ for the composition
$X\lhook\joinrel\relbar\joinrel\rightarrow G/B\times
G/B\times\mathfrak{b}\relbar\joinrel\twoheadrightarrow G/B\times G/B$. For
$w\in W$, define

$V_{w}\coloneqq\pi^{-1}(U_{w})\subset X\,,\quad
X_{w}\coloneqq\overline{V_{w}}\,.$

Note that $V_{w}$ is open in $X_{w}$ but not in $X$, since it is only locally
closed. We also define two morphisms

$\kappa_{1}\colon\;\begin{aligned} X\;&\longrightarrow\;\mathfrak{t}\\\
(g_{1}B,g_{2}B,\psi)\;&\longmapsto\;\overline{\operatorname{Ad}(g_{1}^{-1})\psi}\end{aligned}\,,\quad\kappa_{2}\colon\;\begin{aligned}
X\;&\longrightarrow\;\mathfrak{t}\\\
(g_{1}B,g_{2}B,\psi)\;&\longmapsto\;\overline{\operatorname{Ad}(g_{2}^{-1})\psi}\end{aligned}$

where $\overline{\operatorname{Ad}(g_{1}^{-1})\psi}$ (resp.
$\overline{\operatorname{Ad}(g_{2}^{-1})\psi}$) denotes the image of
$\operatorname{Ad}(g_{1}^{-1})\psi$ (resp.
$\operatorname{Ad}(g_{2}^{-1})\psi$) under the projection
$\mathfrak{b}\relbar\joinrel\twoheadrightarrow\mathfrak{b}/\mathfrak{u}=\mathfrak{t}$.

We have the left action of $G$ on $G/B\times G/B$ by diagonal left
multiplication, as well as the action of $G$ on $\mathfrak{g}$ by adjunction.
This gives a diagonal action on $G/B\times G/B\times\mathfrak{g}$ by
$g\cdot(g_{1}B,g_{2}B,\psi)=(gg_{1}B,gg_{2}B,\operatorname{Ad}(g)\psi)$. For
this action, $X$ is stable and $\pi\colon X\longrightarrow G/B\times G/B$ is
$G$-equivariant. It therefore induces an action on $V_{w}$ for any given $w\in
W$.

The cell decompositions $\overline{U_{w}}=\coprod_{w^{\prime}\preceq
w}U_{w^{\prime}}$ for $w\in W$ imply that $X_{w}\cap V_{w^{\prime}}=\emptyset$
unless $w^{\prime}\preceq w$, and that (set theoretically)

$X_{w}=\coprod_{w^{\prime}\preceq w}X_{w}\cap V_{w^{\prime}}\,.$

###  3.2. Statement of the conjecture

######  Conjecture 3.2.1 (Conjecture 2.3.7 of [8]).

Let $w,w^{\prime}\in W$ such that $w^{\prime}\preceq w$. Then

$X_{w}\cap
V_{w^{\prime}}=V_{w^{\prime}}\cap{\kappa_{1}}^{-1}(\mathfrak{t}^{ww^{\prime-1}})$

as reduced closed subschemes of $X$.

A consequence of this conjecture is the following proposition, that is
mentioned in [8, Remark 2.5.4] at least when $w=w_{0}$ is the element of
maximal length in $W$. We provide here a proof for the general case.

######  Proposition 3.2.2.

Let $w,w^{\prime}\in W$ and $x=(\pi(x),0)\in G/B\times G/B\times\mathfrak{b}$
be a closed point of $X_{w}\cap V_{w^{\prime}}$. Let $T_{X_{w},x}$ (resp.
$T_{\overline{U_{w}},\pi(x)}$) be the tangent space of $X_{w}$ at $x$ (resp.
of $\overline{U_{w}}$ at $\pi(x)$) which is a vector space over the residue
field $k(x)$ of $X_{w}$ at $x$ (resp. the residue field $k(\pi(x))$ of
$\overline{U_{w}}$ at $\pi(x)$). If Conjecture 3.2.1 is true for
$(w^{\prime},w)$, then

(3.3)
$\dim_{k(x)}T_{X_{w},x}=\dim_{k(\pi(x))}T_{\overline{U_{w}},\pi(x)}+\dim_{k(x)}\mathfrak{t}^{ww^{\prime-1}}(k(x))+\operatorname{lg}(w^{\prime}w_{0})\,.$

###### Proof.

Since the action of $G$ on $U_{w^{\prime}}$ is transitive and $\pi$ is
$G$-equivariant, we can assume $\pi(x)=(B,w^{\prime}B)$ (in particular the
residue field $k(x)$ is $k$). Consider the composition
$X_{w}\lhook\joinrel\relbar\joinrel\rightarrow
X\lhook\joinrel\relbar\joinrel\rightarrow G/B\times G/B\times\mathfrak{g}$; it
factors through a closed immersion $\iota\coloneqq
X_{w}\lhook\joinrel\relbar\joinrel\rightarrow\overline{U_{w}}\times\mathfrak{g}$.
Because $\pi(x)=(B,w^{\prime}B)$, the proof of [8, Proposition 2.5.3] shows
that the differential $\mathop{}\mathopen{}\mathrm{d}_{x}\iota\colon
T_{X_{w},x}\lhook\joinrel\relbar\joinrel\rightarrow
T_{\overline{U_{w}},\pi(x)}\oplus
T_{\mathfrak{g},0}=T_{\overline{U_{w}},\pi(x)}\oplus\mathfrak{g}(k)$ factors
through an injection of $k$-vector spaces

(3.4) $\mathop{}\mathopen{}\mathrm{d}_{x}\iota\colon
T_{X_{w},x}\lhook\joinrel\relbar\joinrel\rightarrow
T_{\overline{U_{w}},\pi(x)}\oplus\mathfrak{t}^{ww^{\prime-1}}(k)\oplus(\mathfrak{u}(k)\cap\operatorname{Ad}(w^{\prime})\mathfrak{u}(k))\,.$

Assuming Conjecture 3.2.1, we wish to prove the surjectivity of this map;
comparing dimensions would then give the desired equality (3.3).

Consider the closed immersion $G/B\times
G/B\lhook\joinrel\relbar\joinrel\rightarrow X$ given by
$(g_{1}B,g_{2}B)\longmapsto(g_{1}B,g_{2}B,0)$. Its restriction to
$\overline{U_{w}}$ factors through a closed immersion
$\iota_{1}\coloneqq\overline{U_{w}}\lhook\joinrel\relbar\joinrel\rightarrow
X_{w}$. Since $\iota\circ\iota_{1}$ is the closed immersion
$\overline{U_{w}}\longrightarrow\overline{U_{w}}\times\mathfrak{g}$ given by
$u\longrightarrow(u,0)$, the composition
$\mathop{}\mathopen{}\mathrm{d}_{x}\iota\circ\mathop{}\mathopen{}\mathrm{d}_{\pi(x)}\iota_{1}$
is the inclusion
$T_{\overline{U_{w}},\pi(x)}\lhook\joinrel\relbar\joinrel\rightarrow
T_{\overline{U_{w}},\pi(x)}\oplus\mathfrak{g}(k)$. Hence, in (3.4), the first
summand of the right-hand side is contained in the image of
$\mathop{}\mathopen{}\mathrm{d}_{x}\iota$.

Now, we know from [8, Proposition 2.2.1] that $\pi\colon
V_{w^{\prime}}\longrightarrow U_{w^{\prime}}$ is a geometric vector bundle,
and the fiber of $\pi(x)$ in $X$ is

(3.5)
$\pi^{-1}(\pi(x))=(B,w^{\prime}B)\times(\mathfrak{t}\oplus(\mathfrak{u}\cap\operatorname{Ad}(w^{\prime})\mathfrak{u}))\lhook\joinrel\relbar\joinrel\rightarrow
U_{w^{\prime}}\times\mathfrak{g}\lhook\joinrel\relbar\joinrel\rightarrow
G/B\times G/B\times\mathfrak{g}\,.$

Take
$Y\coloneqq(B,w^{\prime}B)\times(\mathfrak{t}^{ww^{\prime-1}}\oplus(\mathfrak{u}\cap\operatorname{Ad}(w^{\prime})\mathfrak{u}))$.
The closed immersion of reduced schemes
$Y\lhook\joinrel\relbar\joinrel\rightarrow X$ factors through
$Y\lhook\joinrel\relbar\joinrel\rightarrow V_{w^{\prime}}$ and
$Y\lhook\joinrel\relbar\joinrel\rightarrow\kappa_{1}^{-1}(\mathfrak{t}^{ww^{\prime-1}})$.
This gives a closed immersion

$Y\lhook\joinrel\relbar\joinrel\rightarrow\left(V_{w^{\prime}}\times_{X}{\kappa_{1}}^{-1}(\mathfrak{t}^{ww^{\prime-1}})\right)^{\mathrm{red}}=V_{w^{\prime}}\cap{\kappa_{1}}^{-1}(\mathfrak{t}^{ww^{\prime-1}})\,.$

Conjecture 3.2.1 of [8] gives a closed immersion
$V_{w^{\prime}}\cap{\kappa_{1}}^{-1}(\mathfrak{t}^{ww^{\prime-1}})\lhook\joinrel\relbar\joinrel\rightarrow
X_{w}$. The composition $\iota_{2}\coloneqq
Y\lhook\joinrel\relbar\joinrel\rightarrow X_{w}$ is such that
$\iota\circ\iota_{2}$ is the composition
$Y\lhook\joinrel\relbar\joinrel\rightarrow
U_{w^{\prime}}\times\mathfrak{g}\lhook\joinrel\relbar\joinrel\rightarrow\overline{U_{w}}\times\mathfrak{g}$.
Therefore, by definition of $Y$, the differential
$\mathop{}\mathopen{}\mathrm{d}_{x}\iota\circ\mathop{}\mathopen{}\mathrm{d}_{x}\iota_{2}$
composed with the projection
$T_{\overline{U_{w}}\times\mathfrak{g},x}\relbar\joinrel\twoheadrightarrow
T_{\mathfrak{g},0}$ has image
$\mathfrak{t}^{ww^{\prime-1}}\oplus(\mathfrak{u}\cap\operatorname{Ad}(w^{\prime})\mathfrak{u})$.
This gives the second and third summands of (3.4). ∎

We now formulate a conjecture equivalent to Conjecture 3.2.1 using a slightly
simpler scheme than $X$. Recall from [23, §4.7] or [15, §VI.8] _Grothendieck’s
simultaneous resolution of singularities_

(3.6) $q\colon\;\begin{aligned}
G\times^{B}\mathfrak{b}\;&\longrightarrow\;\mathfrak{g}\\\
(g,\psi)\;&\longmapsto\;\operatorname{Ad}(g)\psi\end{aligned}$

where $G\times^{B}\mathfrak{b}$ is the quotient of $G\times\mathfrak{b}$ by
the right $B$-action given by
$(g,\psi)b\coloneqq(gb,\operatorname{Ad}(b^{-1})\psi)$. There is also a map
$\widetilde{\pi}\colon G\times^{B}\mathfrak{b}\longrightarrow G/B$ given by
$\widetilde{\pi}(g,\psi)=gB$.

Define, for $w\in W$:

$\widetilde{X}\coloneqq
q^{-1}(\mathfrak{b})\,,\quad\widetilde{V}_{w}\coloneqq\widetilde{X}\cap{\widetilde{\pi}}^{-1}(BwB/B)\,,\quad\widetilde{X}_{w}\coloneqq\overline{\widetilde{V}_{w}}\,.$

As for $X$, we have the decomposition
$\widetilde{X}_{w}=\coprod_{w^{\prime}\preceq
w}\widetilde{X}_{w}\cap\widetilde{V}_{w^{\prime}}$ for any $w\in W$, and
$\widetilde{X}_{w}\cap\widetilde{V}_{w^{\prime}}\neq\emptyset$ implies that
$w^{\prime}\preceq w$.

######  Conjecture 3.2.3 (Breuil-Hellmann-Schraen).

Let $w,w^{\prime}\in W$. If $w^{\prime}\preceq w$, then the locally closed
immersion $\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\mathfrak{t}^{ww^{\prime-1}}+\mathfrak{u})\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}$
factors into a locally closed immersion

$\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\mathfrak{t}^{ww^{\prime-1}}+\mathfrak{u})\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}_{w}\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}\,.$

######  Proposition 3.2.4.

Conjecture 3.2.1 and Conjecture 3.2.3 are equivalent.

###### Proof.

We have an isomorphism

(3.7)
$G\times^{B}\widetilde{X}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}X,\quad(g_{1},(g_{2},\psi))\longmapsto(g_{1}B,g_{1}g_{2}B,\operatorname{Ad}(g_{1}g_{2})\psi)$

where $B$ acts on the right on $G\times\widetilde{X}$ by
$(g_{1},(g_{2},\psi))b\coloneqq(g_{1}b,(b^{-1}g_{2},\psi))$. This isomorphism
commutes with $\kappa_{1}$ (on the right) and with the composition of
$(g_{1},(g_{2},\psi))\mapsto q(g_{2},\psi)\in\mathfrak{b}$ with
$\mathfrak{b}\longrightarrow\mathfrak{b}/\mathfrak{u}=\mathfrak{t}$ (on the
left), identifies $G\times^{B}\widetilde{V}_{w^{\prime}}$ with
$V_{w^{\prime}}$ and identifies $G\times^{B}\widetilde{X}_{w}$ with $X_{w}$.
Thus everything translates from $\widetilde{X}$ to $X$. Therefore Conjecture
3.2.3 is equivalent to having a locally closed immersion
$V_{w^{\prime}}\cap\kappa_{1}^{-1}(\mathfrak{t}^{ww^{\prime-1}})\lhook\joinrel\relbar\joinrel\rightarrow
X_{w}\cap V_{w^{\prime}}$, and [8, Lemma 2.3.4] shows that it has to be an
isomorphism. ∎

Using the formulation of Conjecture 3.2.3, we prove in §4 that Conjecture
3.2.1 is true in the case of good pairs (in the sense of Definition 2.3.1),
and we exhibit in §5 counter-examples showing that it is false in general (for
bad pairs).

###  3.3. Application to tangent spaces on the trianguline variety

We recall the setting in which the variety $X$ was originally introduced by
Breuil-Hellmann-Schraen [8] who were motivated by the study of the
_trianguline variety_ from $p$-adic Hodge theory.

Let $p$ be a prime number and $K,L$ be finite extensions of $\mathbb{Q}_{p}$
such that $\Sigma\coloneqq\operatorname{Hom}_{\mathbb{Q}_{p}}(K,L)$ has
$[K:\mathbb{Q}_{p}]$ elements. Let $K_{0}$ be the maximal unramified extension
of $\mathbb{Q}_{p}$ inside $K$. We write $\mathcal{G}_{K}$ for the absolute
Galois group of $K$ and $k_{L}$ for the residual field of $L$. We fix an
integer $n\in\mathbb{Z}_{>0}$ and a continuous representation
$\bar{r}\colon\mathcal{G}_{K}\longrightarrow\operatorname{GL}_{n}(k_{L})$. We
set $G$ to be the split reductive group
$G\coloneqq\left(\operatorname{Res}_{K/\mathbb{Q}_{p}}G^{\prime}_{K}\right)\times_{\mathbb{Q}_{p}}L\simeq\prod_{\tau\in\Sigma}G^{\prime}_{L}$
over $L$, $T\coloneqq\prod_{\tau\in\Sigma}T^{\prime}_{L}$ and
$B\coloneqq\prod_{\tau\in\Sigma}B^{\prime}_{L}$, where $G^{\prime}$ is the
split reductive group $\operatorname{GL}_{n,\mathbb{Q}_{p}}$ over
$\mathbb{Q}_{p}$, $T^{\prime}$ is the maximal torus in $G^{\prime}$ of
diagonal matrices and $B^{\prime}$ is the Borel subgroup of $G^{\prime}$ of
upper-triangular matrices. Note that the Weyl group of $G$ is
$(\mathcal{S}_{n})^{\Sigma}$.

Let $R_{\bar{r}}^{\square}$ be the framed deformation ring of $\bar{r}$
constructed by Mazur [19, §1.2] and Kisin [16]. Taking its formal spectrum
$\operatorname{Spf}(R_{\bar{r}}^{\square})$ and then applying Berthelot’s
rigidification functor [2, §0.2] gives a rigid analytic space
$\mathfrak{X}^{\square}_{\bar{r}}\coloneqq\operatorname{Spf}(R_{\bar{r}}^{\square})^{\mathrm{rig}}$
over $L$. Let
$\mathcal{T}_{L}\coloneqq\widehat{K^{\times}}\times_{\mathbb{Q}_{p}}L$ be the
rigid analytic space over $L$ parametrising the continuous characters of
$K^{\times}$. For any $\mathbf{k}=(k_{\tau})\in\mathbb{Z}^{\Sigma}$, we write
$z^{\mathbf{k}}\in\mathcal{T}_{L}(L)$ for the character
$z\longmapsto\prod_{\tau\in\Sigma}\tau(z)^{k_{\tau}}$, and $\lvert
z\rvert_{K}\in\mathcal{T}_{L}(L)$ for the $p$-adic absolute value normalised
by $\lvert p\rvert_{K}=p^{-[K:\mathbb{Q}_{p}]}$. A character
$\underline{\delta}=(\delta_{i})\in\mathcal{T}_{L}^{n}$ is called regular if,
for all $1\leq i\neq j\leq n$, the character $\delta_{i}\delta_{j}^{-1}$ is
not of the form $z^{-\mathbf{k}}$ or
$N_{K/\mathbb{Q}_{p}}(z)z^{\mathbf{k}}\lvert z\rvert_{K}$ for some
$\mathbf{k}\in\mathbb{Z}_{\geq 0}^{\Sigma}$, where $N_{K/\mathbb{Q}_{p}}$ is
the norm map. We write
$\mathcal{T}^{n}_{\mathrm{reg}}\subset\mathcal{T}_{L}^{n}$ for the subset of
regular characters, which is Zariski-open.

######  Definition 3.3.1.

Let $U_{\mathrm{tri}}^{\square}(\bar{r})$ be the set of points
$x=(r,\underline{\delta})\in\mathfrak{X}^{\square}_{\bar{r}}\times\mathcal{T}^{n}_{\mathrm{reg}}$
such that the representation
$r\colon\mathcal{G}_{K}\longrightarrow\operatorname{GL}_{n}(k(x))$ is
trianguline in the sense of Colmez with parameter
$\underline{\delta}\colon(K^{\times})^{n}\longrightarrow k(x)^{\times}$ (see
_e.g._[8, Definition 3.3.8]). The _trianguline variety_
$X_{\mathrm{tri}}^{\square}(\bar{r})$ is defined as the Zariski-closure of
$U_{\mathrm{tri}}^{\square}(\bar{r})$ in
$\mathfrak{X}^{\square}_{\bar{r}}\times\mathcal{T}_{L}^{n}$.

The geometry of $U_{\mathrm{tri}}^{\square}(\bar{r})$ is well understood:
$U_{\mathrm{tri}}^{\square}(\bar{r})$ is a Zariski-dense and Zariski-open
subset of $X_{\mathrm{tri}}^{\square}(\bar{r})$ which is smooth of dimension
$n^{2}+[K:\mathbb{Q}_{p}]\frac{n(n+1)}{2}$; see [7, Thm. 2.6]. In particular,
$X_{\mathrm{tri}}^{\square}(\bar{r})$ is also equidimensional of dimension
$n^{2}+[K:\mathbb{Q}_{p}]\frac{n(n+1)}{2}$.

Now let $x=(r,\underline{\delta})\in X_{\mathrm{tri}}^{\square}(\bar{r})$ be a
crystalline point, _i.e._ a point such that $r$ is a crystalline
representation in the sense of Fontaine [9]. Recall that there is a module
$\mathbf{D}_{\mathrm{dR}}(r)$ over
$K\otimes_{\mathbb{Q}_{p}}L\simeq\prod_{\tau\in\Sigma}L$ canonically
associated to $r$, equipped with a (canonical) decreasing filtration
$\operatorname{Fil}^{\bullet}\mathbf{D}_{\mathrm{dR}}(r)$; for each
$\tau\in\Sigma$, there are exactly $n$ integers $h$ such that
$\operatorname{Fil}^{-h}\mathbf{D}_{\mathrm{dR}}(r)\otimes_{K,\tau}L\neq\operatorname{Fil}^{-h+1}\mathbf{D}_{\mathrm{dR}}(r)\otimes_{K,\tau}L$
(with multiplicities), called the $\tau$-Hodge-Tate weights of $r$ and denoted
$h_{\tau,1}\leq\ldots\leq h_{\tau,n}$. Also recall that there is a
$K_{0}\otimes L$-module $\mathbf{D}_{\mathrm{crys}}(r)$ canonically associated
to $r$, equipped with a canonical endomorphism $\varphi$ which is semilinear
with respect to the Frobenius of $K_{0}$. We assume that $r$ is generic in the
sense of [8], which means that for each $\tau\in\Sigma$, the $\tau$-Hodge-Tate
weights of $r$ are all distinct, and that
$\varphi/\varphi^{\prime}\notin\\{1,p^{[K_{0}:\mathbb{Q}_{p}]}\\}$ for any two
eigenvalues $\varphi,\varphi^{\prime}$ of the linearised Frobenius
$\Phi\coloneqq\varphi^{[K_{0}:\mathbb{Q}_{p}]}$ on
$\mathbf{D}_{\mathrm{crys}}(r)$ (with multiplicities).

From [6, Lemma 2.1], we know that $\underline{\delta}$ is determined by the
data of (i) an ordering $\underline{\varphi}=(\varphi_{1},\ldots,\varphi_{n})$
of the eigenvalues of $\Phi$ and (ii) for each $\tau\in\Sigma$, a permutation
$w_{\tau}$ in $\mathcal{S}_{n}$. More precisely, one can write
$\underline{\delta}=z^{w(\mathbf{h})}\operatorname{unr}(\underline{\varphi})$,
where the character
$\operatorname{unr}(\underline{\varphi})=(\operatorname{unr}(\varphi_{i}))_{i}$
of $(K^{\times})^{n}$ is defined by
$\operatorname{unr}(\varphi_{i})|_{\mathcal{O}_{K}^{\times}}=1$ and
$\operatorname{unr}(\varphi_{i})(\varpi)=\varphi_{i}$ for a uniformiser
$\varpi$ of $K$, and where $w\coloneqq(w_{\tau})\in(\mathcal{S}_{n})^{\Sigma}$
and
$w(\mathbf{h})\coloneqq\big{(}h_{\tau,w_{\tau}^{-1}(i)}\big{)}_{\tau,i}\in(\mathbb{Z}^{n})^{\Sigma}$.
Moreover, there is a unique $w_{\mathrm{sat}}\in(\mathcal{S}_{n})^{\Sigma}$
such that the point
$x_{\mathrm{sat}}\coloneqq\left(r,z^{w_{\mathrm{sat}}(\mathbf{h})}\operatorname{unr}(\underline{\varphi})\right)\in\mathfrak{X}^{\square}_{\bar{r}}\times\mathcal{T}_{L}^{n}$
lies in $U_{\mathrm{tri}}^{\square}(\bar{r})$.

Breuil-Hellmann-Schraen have proved that $X_{w}$ at a certain point
$x_{\mathrm{pdR}}\in X_{w}\cap V_{w_{\mathrm{sat}}}$ is a _local model_ of
$X_{\mathrm{tri}}^{\square}(\bar{r})$ at the point $x$, see [8, Thm. 1.6] for
more details. As a consequence, they find that the pair
$(w_{\mathrm{sat}},w)\in\left((\mathcal{S}_{n})^{\Sigma}\right)^{2}$ attached
to $x=(r,\underline{\delta})\in X_{\mathrm{tri}}^{\square}(\bar{r})$ as above
satisfies $w_{\mathrm{sat}}\preceq w$ for the product Bruhat order on
$(\mathcal{S}_{n})^{\Sigma}$ (_loc. cit._ , Theorem 1.8). They also find an
upper bound for $\dim T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}$ (_loc. cit._
, Proposition 4.1.5(ii)). Conjecture 3.2.1 has the following consequence:

######  Theorem 3.3.2.

If Conjecture 3.2.1 is true for the pair $(w_{\mathrm{sat}}\preceq w)$, then

(3.8) $\dim_{k(x)}T_{X_{\mathrm{tri}}^{\square}(\bar{r}),x}=\dim
X_{\mathrm{tri}}^{\square}(\bar{r})-d_{ww_{\mathrm{sat}}^{-1}}+\dim_{\mathbb{Q}_{p}}T_{\overline{(BwB/B)},w_{\mathrm{sat}}B}-\operatorname{lg}(w_{\mathrm{sat}})$

where $k(x)$ is the residue field of $X_{\mathrm{tri}}^{\square}(\bar{r})$ at
$x$.

###### Proof.

From (3.2) and the definition of $U_{w}$ in §3.1, one derives
$\dim_{\mathbb{Q}_{p}}T_{\overline{(BwB/B)},w_{\mathrm{sat}}B}=\dim_{\mathbb{Q}_{p}}T_{\overline{U_{w}},(B,w_{\mathrm{sat}}B)}-[K:\mathbb{Q}_{p}]\frac{n(n-1)}{2}=\dim_{\mathbb{Q}_{p}}T_{\overline{U_{w}},\pi(x_{\mathrm{pdR}})}-[K:\mathbb{Q}_{p}]\frac{n(n-1)}{2}$.
Hence (3.8) is a consequence of Proposition 3.2.2 and the proof of [8,
Proposition 4.1.5]. ∎

We prove in §4 below that the hypothesis of Theorem 3.3.2 (and hence (3.8)) is
always satisfied if $(w_{\mathrm{sat}},w)$ is a good pair in
$(\mathcal{S}_{n})^{\Sigma}=\Phi(G,T)$ in the sense of Definition 2.3.1. The
following result gives a very partial converse.

######  Proposition 3.3.3.

If (3.8) is true and $X_{\mathrm{tri}}^{\square}(\bar{r})$ is smooth at $x$,
then $(w_{\mathrm{sat}},w)$ is a good pair in $(\mathcal{S}_{n})^{\Sigma}$ and
$\overline{(BwB/B)}$ is smooth at $w_{\mathrm{sat}}B$.

###### Proof.

Since $\dim{\overline{(BwB/B)}}=\lg(w)$, this is a direct consequence of
Proposition 2.3.4. ∎

##  4\. The case of good pairs

This section is devoted to proving the following theorem.

######  Theorem 4.1.

Let $(w^{\prime},w)$ be a good pair in $W$ in the sense of Definition 2.3.1.
Then Conjecture 3.2.1 and Conjecture 3.2.3 are true for the pair
$(w^{\prime},w)$, _i.e._ the locally closed immersion
$\widetilde{V}_{w^{\prime}}\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}$
induces

(4.1) $\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\mathfrak{t}^{ww^{\prime-1}}+\mathfrak{u})\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}_{w}\,.$

###  4.1. Preliminaries: Levi subgroups and the adjoint action

We establish a few technical lemmas about Levi subgroups and the derived
adjoint action, which are used in the proof of Theorem 4.1.

Let $J\subseteq I$ be a subset of simple roots of $G$, write
$\Phi_{J}\subseteq\Phi$ for the root subsystem generated by $J$ and denote by
$T_{J}\coloneqq\left(\bigcap_{\alpha\in\Phi_{J}}(\ker\alpha)\right)^{\circ}\subseteq
T$ the largest subtorus of $T$ such that $\alpha(T_{J})=1$ for all
$\alpha\in\Phi_{J}$ (or equivalently $\alpha\in J$). It is indeed a torus, as
it is a reduced (the characteristic is $0$) and connected diagonalisable
group; see the discussion in [20, Notation 12.29].

Let $L\coloneqq C_{G}(T_{J})$ be the standard Levi subgroup of $G$ associated
to $J$. According to [20, Proposition 21.90], $L$ is a connected reductive
group with $T$ as a maximal torus and root system $\Phi_{J}$. Therefore
$L=\left\langle
T,U_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in\Phi_{J}\right\rangle$,
_i.e._ $L$ is the subgroup of $G$ generated by $T$ and the $U_{\alpha}$ for
$\alpha\in\Phi_{J}$. Write $U_{L}\coloneqq\left\langle
U_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in\Phi_{J}\cap\Phi_{+}\right\rangle$
for the unipotent radical of $L\cap B$ and $\mathfrak{t}_{J}$, $\mathfrak{l}$,
$\mathfrak{u}_{L}$ for the Lie algebras of $T_{J}$, $L$, $U_{L}$ respectively.

For a root $\alpha\colon T\longrightarrow\mathbb{G}_{\mathrm{m}}$, consider
its differential
$\mathop{}\mathopen{}\mathrm{d}_{e}\alpha\colon\mathfrak{t}\longrightarrow k$;
we can see it as $\mathrm{Lie}(\alpha)$, where $\mathrm{Lie}$ is the Lie
algebra functor from algebraic groups over $k$ to Lie algebras over $k$. Since
$\mathrm{Lie}$ commutes with finite limits, and in particular with kernels and
fibred products (see [20, 10.14]), we have

$\mathfrak{t}_{J}=\bigcap_{\alpha\in\Phi_{J}}(\ker\mathop{}\mathopen{}\mathrm{d}_{e}\alpha)\,.$

######  Proposition 4.1.1.

There is a Zariski-dense open subscheme $\mathfrak{t}_{J}^{\mathrm{gen}}$ of
the affine scheme $\mathfrak{t}_{J}$ such that, for any closed point
$t\in\mathfrak{t}_{J}^{\mathrm{gen}}$:

  1. _(1)_

$L=C_{G}(t)^{\circ}\coloneqq\left\\{g\in
G\mathrel{}\middle|\mathrel{}\operatorname{Ad}(g)t=t\right\\}^{\circ}$;

  2. _(2)_

$\mathfrak{l}=\mathfrak{z}_{\mathfrak{g}}(t)\coloneqq\left\\{x\in\mathfrak{g}\mathrel{}\middle|\mathrel{}[x,t]=0\right\\}$.

###### Proof.

For any closed point $t\in\mathfrak{t}$, the centraliser $C_{G}(t)$ contains
$T$ and is normalised by $T$, therefore the same is true of the identity
component $C_{G}(t)^{\circ}$ since $T$ is connected. By [20, 21.66], we then
have

(4.2) $C_{G}(t)^{\circ}=\left\langle
T,U_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in\Phi,\,U_{\alpha}\subseteq
C_{G}(t)^{\circ}\right\rangle=\left\langle
T,U_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in\Phi,\,U_{\alpha}\subseteq
C_{G}(t)\right\rangle$

since each $U_{\alpha}$ is connected.

Let $\alpha\in\Phi$ be a root and $t\in\mathfrak{t}({k^{\mathrm{a}}})$ be a
${k^{\mathrm{a}}}$-point such that $U_{\alpha}\subseteq C_{G}(t)$. Then the
morphism

$\phi\colon U_{\alpha}\longrightarrow\mathfrak{g}\,,\quad
u\longmapsto\operatorname{Ad}(u)t$

is constant, so its differential

$\mathop{}\mathopen{}\mathrm{d}_{e}\colon\mathfrak{u}_{\alpha}\longrightarrow\mathfrak{g}\,,\quad
u_{\alpha}\longmapsto\operatorname{ad}(u_{\alpha})t$

is zero. But for any $u_{\alpha}\in\mathfrak{u}_{\alpha}({k^{\mathrm{a}}})$,
we have
$\operatorname{ad}(u_{\alpha})t=[u_{\alpha},t]=-\mathop{}\mathopen{}\mathrm{d}_{e}\alpha(t)u_{\alpha}$.
This shows that $\mathop{}\mathopen{}\mathrm{d}_{e}\alpha(t)=0$ and hence
$t\in\ker\mathop{}\mathopen{}\mathrm{d}_{e}\alpha$.

Not let $\alpha\in\Phi\setminus\Phi_{J}$. Since $\Phi_{J}$ is the root system
of $L$, necessarily $U_{\alpha}\not\subseteq L=C_{G}(T_{J})$. Therefore there
exist $u_{\alpha}\in U_{\alpha}({k^{\mathrm{a}}})$ and $t\in
T_{J}({k^{\mathrm{a}}})$ such that $u_{\alpha}t\neq tu_{\alpha}$, which means
$u_{\alpha}\neq tu_{\alpha}t^{-1}$. This implies $\alpha(t)\neq 1$ (see [20,
21.19]). Hence $\ker(\alpha)\cap T_{J}\neq T_{J}$. Applying the $\mathrm{Lie}$
functor (which preserves finite limits), we get
$\ker(\mathop{}\mathopen{}\mathrm{d}_{e}\alpha)\cap\mathfrak{t}_{J}\neq\mathfrak{t}_{J}$,
so
$\mathfrak{t}_{J}\setminus(\ker(\mathop{}\mathopen{}\mathrm{d}_{e}\alpha)\cap\mathfrak{t}_{J})$
is non-empty hence dense open in $\mathfrak{t}_{J}$. We thus define

$\mathfrak{t}_{J}^{\mathrm{gen}}\coloneqq\mathfrak{t}_{J}\setminus\bigg{(}\bigcup_{\alpha\in\Phi\setminus\Phi_{J}}\ker(\mathop{}\mathopen{}\mathrm{d}_{e}\alpha)\cap\mathfrak{t}_{J}\bigg{)}=\bigcap_{\alpha\in\Phi\setminus\Phi_{J}}\left(\mathfrak{t}_{J}\setminus\left(\ker(\mathop{}\mathopen{}\mathrm{d}_{e}\alpha)\cap\mathfrak{t}_{J}\right)\right)$

which is dense open in $\mathfrak{t}_{J}$.

Now, fix a closed point $t\in\mathfrak{t}_{J}^{\mathrm{gen}}$. For any root
$\alpha\in\Phi$, the previous discussion shows that $U_{\alpha}\subseteq
C_{G}(t)$ implies $\alpha\in\Phi_{J}$. By (4.2), we have

$C_{G}(t)\subseteq\left\langle
T,U_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in\Phi_{J}\right\rangle=L\,.$

The converse $L=C_{G}(T_{J})\subseteq C_{G}(t)$ is true for any
$t\in\mathfrak{t}_{J}$, so (1) is proved.

Statement (2) is a consequence of (1) and [20, Proposition 17.76]. ∎

The following lemma explains the relevance of Proposition 4.1.1.

######  Lemma 4.1.2.

Let $\mathfrak{t}^{\prime}\subseteq\mathfrak{t}^{ww^{\prime-1}}$ be a Zariski-
dense subscheme. Then the algebraic variety (over $k$)

$\widetilde{V}(w^{\prime},\mathfrak{t}^{\prime})\coloneqq\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\mathfrak{t}^{\prime}+\mathfrak{u})$

is a dense subvariety of

$\widetilde{V}(w^{\prime},\mathfrak{t}^{ww^{\prime-1}})\coloneqq\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\mathfrak{t}^{ww^{\prime-1}}+\mathfrak{u})\,.$

###### Proof.

Recall the morphism $\widetilde{\pi}\colon
G\times^{B}\mathfrak{b}\longrightarrow G/B$, which restricts to two morphisms

$\widetilde{\pi}_{1}\colon
q^{-1}(\mathfrak{t}^{ww^{\prime-1}}+\mathfrak{u})\longrightarrow G/B\
,\quad\widetilde{\pi}_{1}^{\prime}\colon
q^{-1}(\mathfrak{t}^{\prime}+\mathfrak{u})\longrightarrow G/B\,.$

Then $\widetilde{V}(w^{\prime},\mathfrak{t}^{ww^{\prime-1}})$ (resp.
$\widetilde{V}(w^{\prime},\mathfrak{t}^{\prime})$) is just the fibre of
$\widetilde{\pi}_{1}$ (resp. $\widetilde{\pi}_{1}^{\prime}$) above
$Bw^{\prime}B/B$. It is thus enough to check density on fibres above closed
points of $Bw^{\prime}B/B$: let $gB\in Bw^{\prime}B/B\subset G/B$ with $g\in
G$, we want to show that $\widetilde{\pi}_{1}^{\prime-1}(gB)$ is dense in
$\widetilde{\pi}_{1}^{-1}(gB)$. Now the left-action of $B$ on
$G\times\mathfrak{b}$ given by $b(g,\psi)=(bg,\psi)$ induces a left-action on
$G\times^{B}\mathfrak{b}$. This action stabilises
$q^{-1}(\mathfrak{t}^{ww^{\prime-1}}+\mathfrak{u})$ and
$q^{-1}(\mathfrak{t}^{\prime}+\mathfrak{u})$ because $\operatorname{Ad}(b)$
stabilises $t+\mathfrak{u}$ for any $b\in B({k^{\mathrm{a}}})$ and
$t\in\mathfrak{t}({k^{\mathrm{a}}})$. Moreover, this action induces an action
on the set of fibres of $\widetilde{\pi}_{1}$ which is continuous in the sense
that, for any $b\in B$ and $gB\in G/B$, we have an isomorphism of algebraic
varieties
$\widetilde{\pi}_{1}^{-1}(gB)\longrightarrow\widetilde{\pi}_{1}^{-1}(bgB),\,x\longmapsto
bx$; and similarly for $\widetilde{\pi}_{1}^{\prime}$. Also, this action on
the fibres is obviously transitive on the subset of fibres at points of
$Bw^{\prime}B/B$.

Therefore it suffices to consider the fibre at $w^{\prime}B\in G/B$: we want
to check that $\widetilde{\pi}_{1}^{\prime-1}(ww^{\prime}B)$ is dense in
$\widetilde{\pi}_{1}^{-1}(w^{\prime}B)$. Combining the formula (3.5) from [8,
Proposition 2.2.1] with the isomorphism (3.7) gives

$\widetilde{\pi}_{1}^{\prime-1}(ww^{\prime}B)=\left\\{(w^{\prime},\psi)\in
G\times^{B}\mathfrak{b}\mathrel{}\middle|\mathrel{}\psi\in\mathfrak{t}^{\prime}\oplus(\mathfrak{u}\cap\operatorname{Ad}(w^{\prime})\mathfrak{u})\right\\}$

and

$\widetilde{\pi}_{1}^{-1}(ww^{\prime}B)=\left\\{(w^{\prime},\psi)\in
G\times^{B}\mathfrak{b}\mathrel{}\middle|\mathrel{}\psi\in\mathfrak{t}^{ww^{\prime-1}}\oplus(\mathfrak{u}\cap\operatorname{Ad}(w^{\prime})\mathfrak{u})\right\\}\,,$

from which we easily deduce the desired density statement. ∎

We introduce other objects associated to $J$. Let $P\coloneqq
P_{J}=\left\langle
U_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in\Phi_{J}\cup\Phi_{+}\right\rangle$
be the parabolic subgroup associated to $J$ and $U_{P}\coloneqq\left\langle
U_{\alpha}\mathrel{}\middle|\mathrel{}\alpha\in\Phi_{+}\setminus\Phi_{J}\right\rangle$
be the unipotent radical of $P$. Write $W_{J}\coloneqq W(\Phi_{J})\subseteq W$
for the Weyl group of $\Phi_{J}$; therefore $W_{J}=N_{L}(T)/T$ is also the
Weyl group of $L$. Write $W^{J}\subset W$ for the set of $w\in W$ such that
$ws_{\alpha}\succ w$ for all $\alpha\in\Phi_{J}$, _i.e._
$w(\alpha)\in\Phi_{+}$ for all $\alpha\in\Phi_{J}\cap\Phi_{+}$. The reductive
group $L$ is the Levi factor of $P$, and we have the decomposition $P=LU_{P}$
(see [20, Thm. 21.91]). Also recall from §2.3 the decomposition
$W=W^{J}W_{J}$.

######  Lemma 4.1.3.

Let $l\in L$ and $w^{J}\in W^{J}$ be such that $w^{J}l(w^{J})^{-1}\in B$. Then
$l\in L\cap B$.

###### Proof.

From the Bruhat decomposition in $L$, let $w_{J}\in W_{J}$ be such that
$l\in(L\cap B)w_{J}(L\cap B)$. The hypothesis $w^{J}\in W^{J}$ can be
reformulated as $w^{J}(L\cap B)(w^{J})^{-1}\subseteq B$. Thus we have
$w^{J}l(w^{J})^{-1}\in Bw^{J}w_{J}(w^{J})^{-1}B$. The Bruhat decomposition in
$G$ and our hypothesis then give $w^{J}w_{J}(w^{J})^{-1}=1$, _i.e._ $w_{J}=1$,
hence $l\in B$. ∎

######  Lemma 4.1.4.

Let $t\in\mathfrak{t}_{J}^{\mathrm{gen}}({k^{\mathrm{a}}})$ be a
${k^{\mathrm{a}}}$-point. Let $w^{J}\in W^{J}$, $u_{P}\in
U_{P}({k^{\mathrm{a}}})$, $l\in L({k^{\mathrm{a}}})$ and
$x\in\mathfrak{u}_{L}({k^{\mathrm{a}}})$ be such that
$\operatorname{Ad}(w^{J}u_{P}l)(t+x)\in\mathfrak{b}({k^{\mathrm{a}}})$. Then
$\operatorname{Ad}(l)x\in\mathfrak{u}_{L}({k^{\mathrm{a}}})$.

###### Proof.

Since $U_{P}$ is normal in $P$, the action of $L$ on $P$ by conjugation (resp.
on the Lie algebra of $P$ by adjunction) keeps $U_{P}$ (resp.
$\mathfrak{u}_{P}$) stable. Also, the action of $U_{P}$ by conjugation on $B$
induces a trivial action on $B/U_{P}$, hence on
$\mathfrak{b}/\mathfrak{u}_{P}({k^{\mathrm{a}}})$ as well. Therefore, setting
$u_{P}^{\prime}\coloneqq l^{-1}u_{P}l\in U_{P}({k^{\mathrm{a}}})$, we have
using $t+x\in\mathfrak{b}({k^{\mathrm{a}}})$:

$y\coloneqq\operatorname{Ad}(u_{P}^{\prime})(t+x)-(t+x)\in\mathfrak{u}_{P}({k^{\mathrm{a}}})\,.$

Hence, setting
$z\coloneqq\operatorname{Ad}(l)y\in\mathfrak{u}_{P}({k^{\mathrm{a}}})$, we
have using $\operatorname{Ad}(l)t=t$:

$\operatorname{Ad}(w^{J}u_{P}l)(t+x)=\operatorname{Ad}(w^{J}lu_{P}^{\prime})(t+x)=\operatorname{Ad}(w^{J})(t+\operatorname{Ad}(l)x+z)$

which is in
$\mathfrak{b}({k^{\mathrm{a}}})=\mathfrak{t}({k^{\mathrm{a}}})\oplus\bigoplus_{\alpha\in\Phi_{+}}\mathfrak{u}_{\alpha}({k^{\mathrm{a}}})$
by assumption.

Now we have
$\operatorname{Ad}(l)x\in\bigoplus_{\alpha\in\Phi_{J}}\mathfrak{u}_{\alpha}({k^{\mathrm{a}}})$
and
$z\in\mathfrak{u}_{P}({k^{\mathrm{a}}})\subset\bigoplus_{\alpha\in\Phi\setminus\Phi_{J}}\mathfrak{u}_{\alpha}({k^{\mathrm{a}}})$,
therefore $\operatorname{Ad}(w^{J})\operatorname{Ad}(l)x$ and
$\operatorname{Ad}(w^{J})z$ have disjoint components in the sum
$\bigoplus_{\alpha\in\Phi}\mathfrak{u}_{\alpha}({k^{\mathrm{a}}})$. Thus they
each cannot have a non-zero component in
$\mathfrak{u}_{\alpha}({k^{\mathrm{a}}})$ for $\alpha\notin\Phi_{+}$, so

$\operatorname{Ad}(w^{J})\operatorname{Ad}(l)x\in\mathfrak{u}({k^{\mathrm{a}}})\,.$

However, $w^{J}\in W^{J}$ sends negative roots in $\Phi_{J}$ to negative roots
in $\Phi$. Because
$\operatorname{Ad}(l)x\in\bigoplus_{\alpha\in\Phi_{J}}\mathfrak{u}_{\alpha}({k^{\mathrm{a}}})$,
this necessarily implies
$\operatorname{Ad}(l)x\in\bigoplus_{\alpha\in\Phi_{J}\cap\Phi_{+}}\mathfrak{u}_{\alpha}({k^{\mathrm{a}}})=\mathfrak{u}_{L}({k^{\mathrm{a}}})$.
∎

######  Lemma 4.1.5.

With the setting of Lemma 4.1.4, we have $w^{J}u_{P}(w^{J})^{-1}\in
U({k^{\mathrm{a}}})$.

###### Proof.

We know that $\operatorname{Ad}(l)x\in\mathfrak{u}_{L}({k^{\mathrm{a}}})$ and
that $\operatorname{Ad}(l)t=t$, so replacing $x$ by $\operatorname{Ad}(l)x$ if
necessary we can assume that
$\operatorname{Ad}(w^{J}u_{P})(t+x)\in\mathfrak{b}({k^{\mathrm{a}}})$.

Partition $\Phi_{+}\setminus(\Phi_{J}\cap\Phi_{+})$ into the two sets

$\displaystyle\Phi_{1}$
$\displaystyle\coloneqq\left\\{\alpha\in\Phi_{+}\setminus(\Phi_{J}\cap\Phi_{+})\mathrel{}\middle|\mathrel{}w^{J}(\alpha)>0\right\\}\,,$
$\displaystyle\Phi_{2}$
$\displaystyle\coloneqq\left\\{\alpha\in\Phi_{+}\setminus(\Phi_{J}\cap\Phi_{+})\mathrel{}\middle|\mathrel{}w^{J}(\alpha)<0\right\\}$

and put a total order $\prec$ on $\Phi_{2}$ such that $\alpha\preceq\beta$
(with $\alpha,\beta\in\Phi_{2}$) whenever $\alpha>\beta$ (recall that by
definition $\alpha>\beta$ if and only if $\alpha-\beta$ is in the
$\mathbb{Z}_{\geq 0}$-span of $I$). Then, for any total order on $\Phi_{1}$,
there is an isomorphism of affine schemes

$\begin{array}[]{ccc}\left(\prod_{\alpha\in\Phi_{1}}U_{\alpha}\right)\times\left(\prod_{\alpha\in\Phi_{2}}U_{\alpha}\right)&\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}&U_{P}\\\
\left(u_{\alpha_{1}},\ldots,u_{\alpha_{\lvert\Phi_{1}\rvert+\lvert\Phi_{2}\rvert}}\right)&\longmapsto&u_{\alpha_{1}}\ldots
u_{\alpha_{\lvert\Phi_{1}\rvert+\lvert\Phi_{2}\rvert}}\end{array}\,;$

this can be seen by applying [20, Thm. 21.68] to $U$ and to $U_{L}$, and using
the Levi decomposition $U=U_{L}U_{P}$. Let $u_{P,1}$ be the component of
$u_{P}$ in $\prod_{\alpha\in\Phi_{1}}U_{\alpha}({k^{\mathrm{a}}})$; by
definition of $\Phi_{1}$, we have $w^{J}u_{P,1}(w^{J})^{-1}\in
U({k^{\mathrm{a}}})$. In particular,

$\operatorname{Ad}\left(w^{J}u_{P,1}^{-1}u_{P}\right)(t+x)=\operatorname{Ad}\left(\left(w^{J}u_{P,1}(w^{J})^{-1}\right)^{-1}\right)\operatorname{Ad}\left(w^{J}u_{P}\right)(t+x)\in\mathfrak{b}({k^{\mathrm{a}}})\,.$

This means that replacing $u_{P}$ by $u_{P,1}^{-1}u_{P}$ changes neither our
hypothesis nor our aim.

Therefore, we can assume that $u_{P}=\prod_{\alpha\in\Phi_{2}}u_{\alpha}$ for
some uniquely determined $u_{\alpha}\in U_{\alpha}({k^{\mathrm{a}}})$. Write
also $x=\sum_{\beta\in\Phi_{J}\cap\Phi_{+}}x_{\beta}$, for uniquely determined
$x_{\beta}\in\mathfrak{u}_{\beta}({k^{\mathrm{a}}})$. Fix some
$\alpha\in\Phi_{2}$. For any root $\beta\in\Phi_{J}\cap\Phi_{+}$, we have
$\alpha+\beta\neq 0$ (because both are positive roots), so [24, Proposition
8.2.3] implies that

$\operatorname{Ad}(u_{\alpha})x_{\beta}-x_{\beta}\in\bigoplus_{i,j>0}\mathfrak{u}_{i\alpha+j\beta}({k^{\mathrm{a}}})\,.$

This is indeed the case with $\alpha=\beta$ as well since
$\operatorname{Ad}(u_{\alpha})x_{\alpha}-x_{\alpha}=0$. As
$i\alpha+j\beta>\alpha$ for $i,j>0$, we can sum over the $x_{\beta}$ and get

(4.3)
$\operatorname{Ad}(u_{\alpha})x-x\in\bigoplus_{\gamma>\alpha}\mathfrak{u}_{\gamma}({k^{\mathrm{a}}})\,.$

Now suppose that $u_{P}\neq 1$; write
$u_{P}=u_{\alpha_{r}}u_{\alpha_{r-1}}\ldots u_{\alpha_{0}}$ for uniquely
determined $r\geq 0$ and $u_{\alpha_{i}}\neq 1$ in
$\mathfrak{u}_{\alpha_{i}}$. Remember that the ordering on $\Phi_{2}$ forbids
(in particular) to have $\alpha_{0}>\alpha_{i}$ for $i>0$. Applying (4.3), we
get, for each $1\leq i\leq r$

$\operatorname{Ad}(u_{\alpha_{i}})\operatorname{Ad}(u_{\alpha_{i-1}}\ldots
u_{\alpha_{0}})x-\operatorname{Ad}(u_{\alpha_{i-1}}\ldots
u_{\alpha_{0}})x\in\bigoplus_{\gamma>\alpha_{i}}u_{\gamma}({k^{\mathrm{a}}})$

and summing over $i$ yields

(4.4)
$\operatorname{Ad}(u_{P})x-x\in\bigoplus_{\gamma>\alpha_{r}}\mathfrak{u}_{\gamma}({k^{\mathrm{a}}})\oplus\ldots\oplus\bigoplus_{\gamma>\alpha_{0}}\mathfrak{u}_{\gamma}({k^{\mathrm{a}}})\subseteq\bigoplus_{\gamma\neq\alpha_{0}}\mathfrak{u}_{\gamma}({k^{\mathrm{a}}})\,.$

Since
$(u_{\alpha_{0}}t^{\prime}u_{\alpha_{0}}^{-1})t^{\prime-1}=u_{\alpha_{0}}(t^{\prime}u_{\alpha_{0}}^{-1}t^{\prime-1})\in
U_{\alpha_{0}}({k^{\mathrm{a}}})$ for $t^{\prime}\in T({k^{\mathrm{a}}})$, we
have
$\operatorname{Ad}(u_{\alpha_{0}})t-t\in\mathfrak{u}_{\alpha_{0}}({k^{\mathrm{a}}})$,
which is non-zero because
$u_{\alpha_{0}}\notin\mathfrak{z}_{\mathfrak{g}}(t)({k^{\mathrm{a}}})$ by
Proposition 4.1.1(2). Applying again (4.3) repeatedly and summing as in (4.4)
gives

(4.5)
$\operatorname{Ad}(u_{P})t-t\in\left(\mathfrak{u}_{\alpha_{0}}({k^{\mathrm{a}}})\setminus\\{0\\}\right)\oplus\bigoplus_{\gamma\neq\alpha_{0}}\mathfrak{u}_{\gamma}({k^{\mathrm{a}}})\,.$

As $x\in\mathfrak{u}_{L}({k^{\mathrm{a}}})$, we have in particular obviously
$t+x\in\mathfrak{t}({k^{\mathrm{a}}})\oplus\bigoplus_{\gamma\neq\alpha_{0}}\mathfrak{u}_{\gamma}({k^{\mathrm{a}}})$.
Combining with (4.4) and (4.5), we see that $\operatorname{Ad}(u_{P})(t+x)$
has a non-zero component in $\mathfrak{u}_{\alpha_{0}}({k^{\mathrm{a}}})$.
Therefore, $\operatorname{Ad}(w^{J})(u_{P})(t+x)$ has a non-zero component in
$\mathfrak{u}_{w^{J}(\alpha_{0})}({k^{\mathrm{a}}})$. This contradicts the
hypothesis that
$\operatorname{Ad}(w^{J})(u_{P})(t+x)\in\mathfrak{b}({k^{\mathrm{a}}})$
because $\alpha_{0}\in\Phi_{2}$ means that $w^{J}(\alpha_{0})<0$. As a
conclusion, $u_{P}=1$ so our claim is true. ∎

The two lemmas 4.1.4 and 4.1.5 can be nicely summarised as:

######  Lemma 4.1.6.

Let $t\in\mathfrak{t}_{J}^{\mathrm{gen}}({k^{\mathrm{a}}})$, $w^{J}\in W^{J}$,
$u_{P}\in U_{P}({k^{\mathrm{a}}})$, $l\in L({k^{\mathrm{a}}})$ and
$x\in\mathfrak{u}_{L}({k^{\mathrm{a}}})$. Then
$\operatorname{Ad}(w^{J}u_{P}l)(t+x)\in\mathfrak{b}({k^{\mathrm{a}}})$ if and
only if $w^{J}u_{P}(w^{J})^{-1}\in U({k^{\mathrm{a}}})$ and
$\operatorname{Ad}(l)x\in\mathfrak{u}_{L}({k^{\mathrm{a}}})$.

###### Proof.

The “only” way is the combination of the two previous lemmas. For the “if”
way, assume that $w^{J}u_{P}(w^{J})^{-1}\in U({k^{\mathrm{a}}})$ and
$\operatorname{Ad}(l)x\in\mathfrak{u}_{L}({k^{\mathrm{a}}})$. Then since
$w^{J}\in W^{J}$, we have
$\operatorname{Ad}(w^{J})\operatorname{Ad}(l)x\in\mathfrak{b}({k^{\mathrm{a}}})$,
so
$\operatorname{Ad}(w^{J}l)(t+x)=\operatorname{Ad}(w^{J})(t+\operatorname{Ad}(l)x)\in\mathfrak{b}({k^{\mathrm{a}}})$,
hence
$\operatorname{Ad}(w^{J}u_{P}(w^{J})^{-1})\operatorname{Ad}(w^{J}l)(t+x)\in\mathfrak{b}({k^{\mathrm{a}}})$
which is what we want to prove. ∎

###  4.2. Proof of Theorem 4.1

This part contains the heart of the proof of Theorem 4.1. Before beginning the
proof, we make the following two observations about base change to the
algebraic closure ${k^{\mathrm{a}}}$.

  1. 1.

If $S$ is a scheme over $k$, then
$(S_{k^{\mathrm{a}}})^{\mathrm{red}}=(S^{\mathrm{red}})_{k^{\mathrm{a}}}$;
indeed $(S^{\mathrm{red}})_{k^{\mathrm{a}}}$ is reduced by [25, Lemma 020I],
hence it satisfies the universal property defining
$(S_{k^{\mathrm{a}}})^{\mathrm{red}}$. Therefore our definition of
intersections of varieties and inverse image of morphism of varieties commute
with base change. In particular, starting from the reductive group
$G_{k^{\mathrm{a}}}$ over ${k^{\mathrm{a}}}$ with Borel subgroup
$B_{k^{\mathrm{a}}}$ and respective Lie algebras
$\mathfrak{g}_{k^{\mathrm{a}}}$ and $\mathfrak{b}_{k^{\mathrm{a}}}$, one can
define the morphism $q_{k^{\mathrm{a}}}\colon
G_{k^{\mathrm{a}}}\times^{B_{k^{\mathrm{a}}}}\mathfrak{b}_{k^{\mathrm{a}}}\longrightarrow\mathfrak{g}_{k^{\mathrm{a}}}$
and the varieties $\widetilde{X}_{k^{\mathrm{a}}}$,
$\widetilde{V}_{w,{k^{\mathrm{a}}}}$ and $\widetilde{X}_{w,{k^{\mathrm{a}}}}$
for $w\in W$, similarly as for the split reductive group $G$ over $k$. We then
easily see that these constructions commute with base change, and that there
is a natural isomorphism for any $w^{\prime}\in W$

$\widetilde{V}_{w^{\prime},{k^{\mathrm{a}}}}\cap
q_{k^{\mathrm{a}}}^{-1}(\mathfrak{t}^{ww^{\prime-1}}_{k^{\mathrm{a}}}+\mathfrak{u}_{k^{\mathrm{a}}})\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\left(\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\mathfrak{t}^{ww^{\prime-1}}+\mathfrak{u})\right)\times_{k}{k^{\mathrm{a}}}.$

  2. 2.

If $Z\lhook\joinrel\relbar\joinrel\rightarrow S$ is a closed immersion of
$k$-schemes, then a morphism $S^{\prime}\longrightarrow S$ of $k$-schemes
factors through $Z$ if and only if the base change
$S^{\prime}_{k^{\mathrm{a}}}\longrightarrow S_{k^{\mathrm{a}}}$ factors
through $Z_{k^{\mathrm{a}}}$. Indeed, it suffices to check this for affine
schemes, _i.e._ to check that if $I$ is an ideal of a $k$-algebra $A$, then
any morphism of $k$-algebras $A\longrightarrow B$ factors through $A/I$ if and
only if the extension of scalars $A\otimes_{k}{k^{\mathrm{a}}}\longrightarrow
B\otimes_{k}{k^{\mathrm{a}}}$ factors through
$(A/I)\otimes_{k}{k^{\mathrm{a}}}=(A\otimes_{k}{k^{\mathrm{a}}})/(I\otimes_{k}{k^{\mathrm{a}}})$.

Therefore we can, and do, assume that $k$ is algebraically closed for the rest
of §4. In particular, the set of closed points of any $k$-scheme $S$ is in
natural bijection with the set $S(k)$ of $k$-points of $S$.

We use the following notation. Let $L$ be the Levi subgroup of a standard
parabolic subgroup $P$ of $G$ (_i.e._ such that $B\subseteq P$); there exists
$J\subseteq I$ such that $P=P_{J}$. Since $L$ is a connected reductive closed
subgroup of $G$ and $L\cap B$ is a Borel subgroup of $L$, one can define
$q_{L}\colon L\times^{L\cap
B}\mathfrak{l}\cap\mathfrak{b}\longrightarrow\mathfrak{l}$ and
$\widetilde{X}_{L}$ in the same way as for $G$, as well as
$\widetilde{V}_{L,w}$ and $\widetilde{X}_{L,w}$ for any $w\in W_{J}$. These
objects are always assumed to be defined for $G$ by default when $L$ is not
written in subscript.

#### Preview of the proof

Theorem 4.1 can be stated as follows: for all good pairs $(w^{\prime},w)$ in
$W$ and closed points $t\in\mathfrak{t}^{ww^{\prime-1}}$, we have the
inclusion

(4.6) $\widetilde{V}_{w^{\prime}}\cap
q^{-1}(t+\mathfrak{u})\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}_{w}\,.$

The proof of (4.6) will proceed in four steps.

The first step is a result of [8] stating that (4.6) is true when $t=0$,
regardless of whether $(w^{\prime},w)$ is a good pair or not. We then apply
this to a conjugate $L$ of the Levi subgroup
$C_{G}((T^{ww^{\prime-1}})^{\circ})$ of $G$, to get the inclusion
$\widetilde{V}_{L,w^{\prime}_{L}}\cap
q_{L}^{-1}(\mathfrak{u}_{L})\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}_{L,w_{L}}$
for any $w^{\prime}_{L},w_{L}$ in the Weyl group of $L$. It is of particular
interest because we can then translate by any suitable closed point
$t\in\mathfrak{t}^{ww^{\prime-1}}$ when such a $t$ is centralised by $L$;
therefore we have, in fact, $\widetilde{V}_{L,w^{\prime}_{L}}\cap
q_{L}^{-1}(t+\mathfrak{u}_{L})\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}_{L,w_{L}}$
for such a $t$.

The second step establishes a connection between $X_{L}$ and $X$. More
precisely, we construct an isomorphism between $\widetilde{V}_{L,w_{L}}\cap
q_{L}^{-1}(\mathfrak{u}_{L})$ (or equivalently, by the previous paragraph,
$\widetilde{V}_{L,w_{L}}\cap q_{L}^{-1}(t^{\prime}+\mathfrak{u}_{L})$ for any
$t^{\prime}$ centralised by $L$ and conjugate to some
$t\in\mathfrak{t}^{ww^{\prime-1}}$) and a closed subvariety of
$\widetilde{V}_{w}\cap q^{-1}(t+\mathfrak{u})$ for some $w_{L}$ in the Weyl
group of $L$ and $w\in W$. We momentarily call $\widetilde{V}^{\prime}_{w}$
this subvariety of $\widetilde{V}_{w}\cap q^{-1}(t+\mathfrak{u})$.

The third step bridges the last gap from $\widetilde{V}^{\prime}_{w}$ to
$\widetilde{V}_{w}\cap q^{-1}(t+\mathfrak{u})$: an action of $U$ on
$\widetilde{V}_{w}\cap q^{-1}(t+\mathfrak{u})$ is defined such that all the
orbits intersect $\widetilde{V}^{\prime}_{w}$.

The fourth and last step articulates the previous steps into a proof: the good
pair condition on some given $(w^{\prime}\preceq w)$ allows us to find a
standard Levi subgroup $L$ of $G$ and an ordered pair $(w^{\prime}_{L}\preceq
w_{L})$ of elements in the Weyl group of $L$, such that the second and third
steps can reduce the problem to the situation of the first.

#### Step 1

######  Theorem 4.2.1 (Beilinson-Bernstein, Ginsburg).

Let $w,w^{\prime}\in W$ such that $w^{\prime}\preceq w$. Then the locally
closed immersion
$\widetilde{V}_{w^{\prime}}\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}$
induces $\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\mathfrak{u})\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}_{w}$.

###### Proof.

It follows from the proof of Proposition 3.2.4 that the theorem is equivalent
to
$V_{w^{\prime}}\cap\kappa_{1}^{-1}(0)\lhook\joinrel\relbar\joinrel\rightarrow
X_{w}$ for all $w^{\prime}\preceq w$. This statement is a consequence of the
Beilinson-Bernstein correspondence (see [13, §11]) and a result of Ginsburg
[11]; the whole argument is laid out in [8, §2.4] (see in particular the proof
of [8, Thm. 2.4.7]). ∎

#### Step 2

######  Proposition 4.2.2.

Let $t\in\mathfrak{t}^{\mathrm{gen}}_{J}$ be a closed point, and let
$u^{J},v^{J}\in W^{J}$.

  1. _(1)_

There is a morphism $\iota_{t,u^{J},v^{J}}$ given by

$\iota_{t,u^{J},v^{J}}\colon\;\begin{aligned} L\times^{L\cap
B}\mathfrak{l}\cap\mathfrak{b}\;&\longrightarrow\;G\times^{B}\mathfrak{b}\\\
(l,\psi_{L})\;&\longmapsto\;\left(u^{J}l(v^{J})^{-1},\operatorname{Ad}(v^{J})(t+\psi_{L})\right)\end{aligned}$

and it is a closed immersion.

  2. _(2)_

The closed immersion $\iota_{t,u^{J},v^{J}}$ induces, for any $w_{J}\in
W_{J}$, an isomorphism

$\iota_{t,u^{J},w_{J},v^{J}}\colon\widetilde{V}_{L,w_{J}}\cap
q_{L}^{-1}(\mathfrak{u}_{L})\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\widetilde{V}_{u^{J}w_{J}(v^{J})^{-1}}\cap
q^{-1}\left(\operatorname{Ad}(u^{J})(t+\mathfrak{u}_{L})\right)\,.$

###### Proof.

We first prove statement (1). We begin by checking that
$\iota_{t,u^{J},v^{J}}$ is well defined. Let $b\in L\cap B$. Since $L\cap
B=T\prod_{\alpha_{\in}\Phi_{J}\cap\Phi_{+}}U_{\alpha}$ and $v^{J}\in W^{J}$,
we have

$b^{\prime}\coloneqq v^{J}b(v^{J})^{-1}\in B\,.$

Thus, for any $l\in L$ and $\psi_{L}\in\mathfrak{l}\cap\mathfrak{b}$, we have

$\left(u^{J}lb(v^{J})^{-1},\operatorname{Ad}(v^{J})(t+\operatorname{Ad}(b)\psi_{L})\right)=\left(u^{J}l(v^{J})^{-1}b^{\prime},\operatorname{Ad}(b^{\prime})\operatorname{Ad}(v^{J})(t+\psi_{L})\right)$

(note that $b$ fixes $t$ as $b\in L$), which shows that
$\iota_{t,u^{J},v^{J}}$ is well defined.

Now we check that $\iota_{t,u^{J},v^{J}}$ is a closed immersion. First
consider the morphism

(4.7)
$\begin{array}[]{ccc}L\times\mathfrak{l}\cap\mathfrak{b}&\longrightarrow&(u^{J}L(v^{J})^{-1}B)/B\times\mathfrak{l}\\\
(l,\psi_{L})&\longmapsto&(u^{J}l(v^{J})^{-1}B,\operatorname{Ad}(l)\psi_{L})\end{array}\,.$

From $u^{J}lb(v^{J})^{-1}=u^{J}l(v^{J})^{-1}b^{\prime}$, we know that for any
$b\in L\cap B$ and $(l,\psi_{L})\in L\times\mathfrak{l}\cap\mathfrak{b}$, the
images of $(l,\psi_{L})$ and $(lb,\operatorname{Ad}(b^{-1})\psi_{L})$ under
(4.7) are the same. Conversely, assume that $(l,\psi_{L})$ and
$(l^{\prime},\psi_{L}^{\prime})$ have the same image. This means that

$\operatorname{Ad}(l)\psi_{L}=\operatorname{Ad}(l^{\prime})\psi_{L}^{\prime}$

and that there is $b\in B$ such that

$u^{J}l(v^{J})^{-1}=u^{J}l^{\prime}(v^{J})^{-1}b\,.$

Write $b^{\prime}=l^{\prime-1}l$. Then $v^{J}b^{\prime}(v^{J})^{-1}=b$, so
$b^{\prime}\in L\cap B$ according to Lemma 4.1.3. We also have
$l=l^{\prime}b^{\prime}$, so
$\psi_{L}=\operatorname{Ad}({b^{\prime}}^{-1})\psi_{L}^{\prime}$ and
$(l,\psi_{L})=(l^{\prime},\psi_{L}^{\prime})$ in $L\times^{L\cap
B}\mathfrak{l}\cap\mathfrak{b}$. Therefore, by definition of the quotient
space $L\times^{L\cap B}\mathfrak{l}\cap\mathfrak{b}$, the morphism (4.7)
factorises as

$f_{L}\colon\begin{array}[]{ccc}L\times^{L\cap
B}\mathfrak{l}\cap\mathfrak{b}&\longrightarrow&\left(u^{J}L(v^{J})^{-1}B\right)/B\times\mathfrak{l}\\\
\left(l,\psi_{L}\right)&\longmapsto&\left(u^{J}l(v^{J})^{-1}B,\operatorname{Ad}(l)\psi_{L}\right)\,.\end{array}$

Together with the morphism

$f_{G}\colon\begin{array}[]{ccc}G\times^{B}\mathfrak{b}&\longrightarrow&G/B\times\mathfrak{g}\\\
(g,\psi)&\longmapsto&\left(gB,\operatorname{Ad}(g)\psi\right)\,,\end{array}$

the morphism $\iota_{t,u^{J},v^{J}}$ and the closed immersion

$\iota\colon\begin{array}[]{ccc}(u^{J}L(v^{J})^{-1}B)/B\times\mathfrak{l}&\lhook\joinrel\relbar\joinrel\rightarrow&G/B\times\mathfrak{g}\\\
(gB,\psi_{L})&\longmapsto&\left(gB,\operatorname{Ad}(u^{J})(t+\psi_{L})\right),\end{array}$

we get a commutative diagram

(4.8) ${L\times^{L\cap
B}\mathfrak{l}\cap\mathfrak{b}}$${G\times^{B}\mathfrak{b}}$${(u^{J}L(v^{J})^{-1}B)/B\times\mathfrak{l}}$${G/B\times\mathfrak{g}\,.}$$\scriptstyle{f_{L}}$$\scriptstyle{\iota_{t,u^{J},v^{J}}}$$\scriptstyle{f_{G}}$$\scriptstyle{\iota}$

This diagram is a cartesian square. Indeed, let $(g,\psi)\in
G\times^{B}\mathfrak{b}$ be such that $(gB,\operatorname{Ad}(g)\psi)$ is in
the image of $\iota$. Then, modifying $(g,\psi)\in G\times\mathfrak{b}$ if
necessary (so that its image in $G\times^{B}\mathfrak{b}$ is the same), there
is $l\in L$ such that $g=u^{J}l(v^{J})^{-1}$, and $\psi_{L}\in\mathfrak{l}$
such that
$\operatorname{Ad}(g)\psi=\operatorname{Ad}(u^{J}l(v^{J})^{-1})\psi=\operatorname{Ad}(u^{J})(t+\psi_{L})$.
From this last equality we get

$\psi=\operatorname{Ad}(v^{J}l^{-1})(t+\psi_{L})=\operatorname{Ad}(v^{J})(t+\operatorname{Ad}(l^{-1})\psi_{L})$

with
$\operatorname{Ad}(l^{-1})\psi_{L}\in\mathfrak{l}=\mathfrak{t}\oplus\bigoplus_{\alpha\in\Phi_{J}}\mathfrak{u}_{\alpha}$.
Furthermore, $v^{J}\in W^{J}$ implies that

$\operatorname{Ad}(v^{J})\bigg{(}\bigoplus_{\alpha\in\Phi_{J}\cap\Phi_{-}}\mathfrak{u}_{\alpha}\bigg{)}\subseteq\bigoplus_{\alpha\in\Phi_{-}}\mathfrak{u}_{\alpha}\,,$

hence, as
$\operatorname{Ad}(v^{J})\operatorname{Ad}(l^{-1})\psi_{L}=\psi-\operatorname{Ad}(v^{J}l^{-1})t\in\mathfrak{b}$,
we necessarily have

$\operatorname{Ad}(l^{-1})\psi_{L}\in\mathfrak{t}\oplus\bigoplus_{\alpha\in\Phi_{J}\cap\Phi_{+}}\mathfrak{u}_{\alpha}=\mathfrak{l}\cap\mathfrak{b}\,.$

This gives
$(g,\psi)=\iota_{t,u^{J},v^{J}}\big{(}l,\operatorname{Ad}(l^{-1})\psi_{L}\big{)}$
as we wanted. Since the diagram (4.8) is cartesian and $\iota$ is a closed
immersion, so is $\iota_{t,u^{J},v^{J}}$ by base change.

As for statement (2), we had already established the inclusion $u^{J}(L\cap
B)(u^{J})^{-1}\subseteq B$ as a consequence of $u^{J}\in W^{J}$, and similarly
for $v^{J}$. From this we easily see that $\iota_{t,u^{J},v^{J}}$ restricts to
a closed immersion of algebraic varieties

$\widetilde{V}_{L,w_{J}}\cap
q_{L}^{-1}(\mathfrak{u}_{L})\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{V}_{u^{J}w_{J}(v^{J})^{-1}}\cap
q^{-1}\left(\operatorname{Ad}(u^{J})(t+\mathfrak{u}_{L})\right)$

which we want to be an isomorphism. For this, let $(g,\psi)\in
q^{-1}\left(\operatorname{Ad}(u^{J})(t+\mathfrak{u}_{L})\right)$. We know that

$G=\coprod_{\begin{subarray}{c}w^{J}\in W^{J}\\\ w_{J}\in
W_{J}\end{subarray}}Bw^{J}w_{J}B=\coprod_{w^{J}\in
W^{J}}Bw^{J}P=\coprod_{w^{J}\in W^{J}}Bw^{J}U_{P}L\,,$

where the first equality comes from the Bruhat decomposition of $G$, the
second from the Bruhat decomposition of $P=P_{J}$ and the third from the Levi
decomposition of $P$ (see [20, Thm. 21.91] for the last two). Hence, there are
$b\in B$, $w^{J}\in W^{J}$, $u_{P}\in U_{P}$ and $l\in L$ such that
$g^{-1}u^{J}=bw^{J}u_{P}l$. Changing the lift of $(g,\psi)$ in
$G\times\mathfrak{b}$ if necessary, we can assume $b=1$, and by assumption
there is $x\in\mathfrak{u}_{L}$ such that

$(g,\psi)=\left(u^{J}l^{-1}u_{P}^{-1}(w^{J})^{-1},\operatorname{Ad}(w^{J}u_{P}l)(t+x)\right)\,.$

Since $\psi\in\mathfrak{b}$, Lemma 4.1.6 implies that
$w^{J}u_{P}(w^{J})^{-1}\in U\subset B$ and that

$\psi_{L}\coloneqq\operatorname{Ad}(l)x\in\mathfrak{u}_{L}\,,$

so we actually have in $G\times^{B}\mathfrak{b}$

(4.9)
$(g,\psi)=\left(u^{J}l^{-1}(w^{J})^{-1},\operatorname{Ad}(w^{J}l)(t+x)\right)=\left(u^{J}l^{-1}(w^{J})^{-1},\operatorname{Ad}(w^{J})(t+\psi_{L})\right)\,.$

Finally, $l^{-1}\in(L\cap B)w_{J}^{\prime}(L\cap B)$ for some
$w_{J}^{\prime}\in W_{J}$, which gives as usual $g=u^{J}l^{-1}(w^{J})^{-1}\in
Bu^{J}w_{J}^{\prime}(w^{J})^{-1}B$. If we add the additional condition that
$(g,\psi)\in\widetilde{V}_{u^{J}w_{J}(v^{J})^{-1}}$, or equivalently that
$g\in Bu^{J}w_{J}(v^{J})^{-1}B$, then we have
$w_{J}^{\prime}(w^{J})^{-1}=w_{J}(v^{J})^{-1}$. This forces
$w_{J}^{\prime}=w_{J}$ and $w^{J}=v^{J}$, thus (4.9) translates to

$(g,\psi)=\iota_{t,u^{J},v^{J}}(l^{-1},\psi_{L})$

with $(l^{-1},\psi_{L})\in\widetilde{V}_{L,w_{J}}$. This finishes the proof of
Proposition 4.2.2. ∎

#### Step 3

The proof of the following lemma uses standard results about Jordan
decompositions, which can be found in Appendix A below.

######  Lemma 4.2.3.

Let $t\in\mathfrak{t}_{J}^{\mathrm{gen}}$ be a closed point and let $u^{J}\in
W^{J}$. Then there is a surjective morphism given by

$f\colon\begin{array}[]{ccc}U\times\left(\operatorname{Ad}(u^{J})(t+\mathfrak{u}_{L})\right)&\relbar\joinrel\twoheadrightarrow&\operatorname{Ad}(u^{J})t+\mathfrak{u}\\\
(u,x)&\longmapsto&\operatorname{Ad}(u)x\,.\end{array}$

###### Proof.

The existence of the map $f$ is a direct consequence of the fact that
$\operatorname{Ad}(u)$ sends any $t^{\prime}\in\mathfrak{t}$ to
$t^{\prime}+\mathfrak{u}$ when $u\in U$. Thus we only have to prove the
surjectivity

Let $y\in\operatorname{Ad}(u^{J})t+\mathfrak{u}$, and consider its Jordan
decomposition $y=y_{\mathrm{s}}+y_{\mathrm{n}}$ in $\mathfrak{g}$ (see Theorem
A.2). From Proposition A.9, there is $g\in G$ such that

$y_{\mathrm{s}}=\operatorname{Ad}(g)\operatorname{Ad}(u^{J})t\,.$

We use the decomposition

$G=\coprod_{w^{J}\in W^{J}}Bw^{J}P=\coprod_{w^{J}\in
W^{J}}UTw^{J}P=\coprod_{w^{J}\in W^{J}}Uw^{J}U_{P}L\,.$

Let $u\in U$, $w^{J}\in W^{J}$ and $u_{P}\in U_{P}$ such that $gu^{J}\in
uw^{J}u_{P}L$, from the fact that $t$ is centralised by $L$ we get:

$y_{\mathrm{s}}=\operatorname{Ad}(uw^{J}u_{P})t\,.$

Now, from Proposition A.9(1), we have

(4.10) $y_{\mathrm{n}}\in\mathfrak{u},\quad y_{\mathrm{s}}\in
y+\mathfrak{u}\subseteq\mathfrak{b}\,.$

Hence,
$\operatorname{Ad}(w^{J}u_{P})t=\operatorname{Ad}(u^{-1})y_{\mathrm{s}}\in\mathfrak{b}$,
so Lemma 4.1.5 implies that $w^{J}u_{P}(w^{J})^{-1}\in U$. Set
$u^{\prime}\coloneqq uw^{J}u_{P}(w^{J})^{-1}\in U$. We can now write

(4.11)
$y_{\mathrm{s}}=\operatorname{Ad}(u^{\prime}w^{J})t\in\operatorname{Ad}(w^{J})t+\mathfrak{u}\,.$

Also, (4.10) and the assumption on $y$ give
$y_{\mathrm{s}}\in\operatorname{Ad}(u^{J})t+\mathfrak{u}$. Together with
(4.11), we deduce that

$\operatorname{Ad}(w^{J})t=\operatorname{Ad}(u^{J})t\,,$

hence $(u^{J})^{-1}w^{J}\in C_{G}(t)$. If $(u^{J})^{-1}w^{J}\notin W_{J}$,
then there exists $\alpha\in\Phi_{J}$ such that
$\beta\coloneqq(u^{J})^{-1}w^{J}(\alpha)\notin\Phi_{J}$. Since
$U_{\alpha}\subseteq C_{G}(t)$, we have

$U_{\beta}=(u^{J})^{-1}w^{J}U_{\alpha}(w^{J})^{-1}u^{J}\subseteq C_{G}(t)$

so $U_{\beta}\subseteq C_{G}(t)^{\circ}=L$ because $U_{\beta}$ is connected,
which is a contradiction as $\beta$ is not a root of $L$. Hence
$(u^{J})^{-1}w^{J}\in W_{J}$, from which we deduce $w^{J}=u^{J}$ as $W^{J}$ is
a set of representatives of $W/W_{J}$.

Moreover, the commutation $[y_{\mathrm{s}},y_{\mathrm{n}}]=0$ together with
$y_{\mathrm{s}}=\operatorname{Ad}(u^{\prime}w^{J})t$ (see (4.11)) and
Proposition 4.1.1(2) imply
$y_{\mathrm{n}}\in\operatorname{Ad}(u^{\prime}w^{J})\mathfrak{l}$. Since
$y_{\mathrm{n}}\in\mathfrak{u}$ and $w^{J}\in W^{J}$, we necessarily get
$y_{\mathrm{n}}\in\operatorname{Ad}(u^{\prime}w^{J})\mathfrak{u}_{L}$. Thus,

$y=y_{\mathrm{s}}+y_{\mathrm{n}}\in\operatorname{Ad}(u^{\prime}w^{J})(t+\mathfrak{u}_{L})=\operatorname{Ad}(u^{\prime})\operatorname{Ad}(u^{J})(t+\mathfrak{u}_{L})$

which concludes the proof. ∎

######  Proposition 4.2.4.

Let $t\in\mathfrak{t}_{J}^{\mathrm{gen}}$ be a closed point, let
$u^{J},v^{J}\in W^{J}$ and let $w_{J}\in W_{J}$. Write $w\coloneqq
u^{J}w_{J}(v^{J})^{-1}$. Then there is a surjective morphism of algebraic
varieties

$\mu_{t,u^{J},w_{J},v^{J}}\colon\begin{array}[]{ccc}U\times\left(\widetilde{V}_{L,w_{J}}\cap
q_{L}^{-1}(\mathfrak{u}_{L})\right)&\relbar\joinrel\twoheadrightarrow&\widetilde{V}_{w}\cap
q^{-1}\left(\operatorname{Ad}(u^{J})t+\mathfrak{u}\right)\\\
\left(u,(l,\psi_{L})\right)&\longmapsto&\left(uu^{J}l(v^{J})^{-1}B,\operatorname{Ad}(v^{J})(t+\psi_{L})\right).\end{array}$

###### Proof.

Remember that the surjectivity of a morphism of algebraic varieties can be
checked on closed points ([27, Exercise 7.4.C]). The morphism of algebraic
varieties

(4.12) $\begin{array}[]{ccc}U\times
q^{-1}(\operatorname{Ad}(u^{J})(t+\mathfrak{u}_{L}))&\longrightarrow&q^{-1}(\operatorname{Ad}(u^{J})t+\mathfrak{u})\\\
(u,(g,x))&\longmapsto&(ug,x)\end{array}$

is surjective as a consequence of Lemma 4.2.3. Since $U\subseteq B$, for any
$u\in U$ and $g\in G$, we have $g\in BwB$ if and only if $ug\in BwB$.
Therefore (4.12) induces a surjective morphism of algebraic varieties

$\begin{array}[]{ccc}U\times\left(\widetilde{V}_{w}\cap
q^{-1}\left(\operatorname{Ad}(u^{J})(t+\mathfrak{u}_{L})\right)\right)&\longrightarrow&\widetilde{V}_{w}\cap
q^{-1}\left(\operatorname{Ad}(u^{J})t+\mathfrak{u}\right)\\\
\left(u,(g,x)\right)&\longmapsto&(ug,x)\end{array}.$

Composing with the bijective morphism
$\operatorname{id}\times\iota_{t,u^{J},w_{J},v^{J}}$ of Proposition 4.2.2(2)
yields the desired morphism $\mu_{t,u^{J},w_{J},v^{J}}$. This concludes the
proof. ∎

#### Step 4

We now prove Theorem 4.1.

Let $\Phi$ be the root system of $G$ with basis $I$ and
$\Phi_{ww^{\prime-1}}\subseteq\Phi$ the minimal generating subsystem of
$ww^{\prime-1}$ in the sense of Definition 2.1.1. According to Proposition
2.2.2, it is the root system of the Levi subgroup
$C_{G}((T^{ww^{\prime-1}})^{\circ})$. Proposition 2.3.7 gives a subset $J$ of
the basis $I$ of roots, and elements $u^{J},v^{J}\in W^{J}$,
$w^{\prime}_{J},w_{J}\in W$ satisfying

$w=u^{J}w_{J}(v^{J})^{-1}\,,\quad
w^{\prime}=u^{J}w^{\prime}_{J}(v^{J})^{-1}\,,\quad w^{\prime}_{J}\preceq
w_{J}$

and such that $u^{J}L(u^{J})^{-1}=C_{G}((T^{ww^{\prime-1}})^{\circ})$, where
$L\coloneqq L_{J}$ is the standard Levi subgroup associated to $J$. Then
$L=C_{G}((T^{w_{J}(w^{\prime}_{J})^{-1}})^{\circ})$, with
$(T^{w_{J}(w^{\prime}_{J})^{-1}})^{\circ}=T_{J}$ and
$\mathfrak{t}^{w_{J}(w^{\prime}_{J})^{-1}}=\mathfrak{t}_{J}$ (using the
notation of §4.1).

The variety $\mathfrak{t}^{\mathrm{gen}}_{J}$ is Zariski-dense in
$\mathfrak{t}_{J}=\mathfrak{t}^{w_{J}(w^{\prime}_{J})^{-1}}$, therefore
$\operatorname{Ad}(u^{J})\mathfrak{t}^{\mathrm{gen}}_{J}$ is Zariski-dense in
$\operatorname{Ad}(u^{J})\mathfrak{t}_{J}=\mathfrak{t}^{ww^{\prime-1}}$.
According to Lemma 4.1.2, we thus only need to prove that
$\widetilde{V}_{w^{\prime}}\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}$
induces

$\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\operatorname{Ad}(u^{J})t+\mathfrak{u})\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}_{w}$

for all closed points $t\in\mathfrak{t}^{\mathrm{gen}}_{J}$. In the following,
fix one such $t$.

The closed immersion $\iota_{t,u^{J},v^{J}}$ of Proposition 4.2.2 restricts to
a closed immersion

$\iota_{t,u^{J},v^{J}}\colon\widetilde{X}_{L}=q_{L}^{-1}(\mathfrak{l}\cap\mathfrak{b})\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}=q^{-1}(\mathfrak{b})\,.$

Indeed, one has to check that
$\operatorname{Ad}(u^{J})\operatorname{Ad}(l)\psi_{L}\in\mathfrak{b}$ for all
$(l,\psi_{L})\in L\times(\mathfrak{l}\cap\mathfrak{b})$ such that
$\operatorname{Ad}(l)\psi_{L}\in\mathfrak{l}\cap\mathfrak{b}$, which is true
because $u^{J}\in W^{J}$ means that
$\operatorname{Ad}(u^{J})(\mathfrak{l}\cap\mathfrak{b})\subseteq\mathfrak{b}$.
In the same way, we can further restrict to

$\iota_{t,u^{J},v^{J}}\colon\widetilde{V}_{L,w_{J}}\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{V}_{w}$

because $u^{J}\in W^{J}$ means that $u^{J}(L\cap B)(u^{J})^{-1}\subseteq B$
and similarly for $v^{J}$. Taking the Zariski closures gives a closed
immersion
$\iota_{t,u^{J},v^{J}}\colon\widetilde{X}_{L,w_{J}}\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}_{w}$.
There is also a morphism

$\displaystyle U\times\widetilde{X}_{w}$
$\displaystyle\longrightarrow\;\widetilde{X}_{w}$ $\displaystyle(u,(g,x))$
$\displaystyle\longmapsto\;(ug,x)$

which, when composed with $\operatorname{id}\times\iota_{t,u^{J},v^{J}}\colon
U\times\widetilde{X}_{L,w_{J}}\longrightarrow U\times\widetilde{X}_{w}$, gives
a morphism

$\bar{\mu}_{t,u^{J},w_{J},v^{J}}\colon
U\times\widetilde{X}_{L,w_{J}}\longrightarrow\widetilde{X}_{w}$

which can be defined by a similar formula to the one for the surjective
morphism $\mu_{t,u^{J},w_{J},v^{J}}$ of Proposition 4.2.4.

We also have an obvious locally closed immersion

$\varphi_{G}\colon\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\operatorname{Ad}(u^{J})t+\mathfrak{u})\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}$

as well as a less obvious locally closed immersion

$\varphi_{L}\colon\widetilde{V}_{L,w^{\prime}_{J}}\cap
q_{L}^{-1}(\mathfrak{u}_{L})\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}_{L,w_{J}}$

given by theorem 4.2.1 applied to the reductive group $L$ together with the
good pair condition $w^{\prime}_{J}\preceq w_{J}$.

All these morphisms fit into a commutative diagram of algebraic varieties over
$k$

${U\times\left(\widetilde{V}_{L,w^{\prime}_{J}}\cap
q_{L}^{-1}(\mathfrak{u}_{L})\right)}$${U\times\widetilde{X}_{L,w_{J}}}$${\widetilde{X}_{w}}$${\widetilde{V}_{w^{\prime}}\cap
q^{-1}(\operatorname{Ad}(u^{J})t+\mathfrak{u})}$${\widetilde{X}}$$\scriptstyle{\operatorname{id}\times\varphi_{L}}$$\scriptstyle{\mu_{t,u^{J},w^{\prime}_{J},v^{J}}}$$\scriptstyle{\bar{\mu}_{t,u^{J},w_{J},v^{J}}}$$\scriptstyle{\varphi_{G}}$

where the surjectivity of the left map $\mu_{t,u^{J},w^{\prime}_{J},v^{J}}$
(Proposition 4.2.4) proves that the image of the bottom map $\varphi_{G}$
lands in $\widetilde{X}_{w}$. This finishes the proof of Theorem 4.1.

##  5\. Counter-examples for bad pairs when $G=\mathrm{GL}_{n}$

In §4, we proved Conjecture 3.2.1 in the case of good pairs. The present
section explores the case of bad pairs for the split reductive group
$G=\operatorname{GL}_{n/k}$ where $n\geq 1$ and $k$ is any field of
characteristic $0$. From now on $G=\operatorname{GL}_{n/k}$, $B$ is the
subgroup of upper triangular matrices in $G$, $T$ is the subgroup of diagonal
matrices in $G$, $U$ is the subgroup of upper triangular matrices in $G$ whose