# Products of conjugacy classes in simple algebraic groups in terms of
diagrams

Iulian I. Simion  Department of Mathematics
Babeș-Bolyai University
Str. Ploieşti 23-25, Cluj-Napoca 400157, Romania
and Department of Mathematics
Technical University of Cluj-Napoca
Str. G. Bariţiu 25, Cluj-Napoca 400027, Romania<EMAIL_ADDRESS>
###### Abstract.

For a simple algebraic group $G$ over an algebraically closed field we study
products of normal subsets. For this we mark the nodes of the Dynkin diagram
of $G$. We use two types of labels, a binary marking and a labeling with non-
negative integers. The first is used to recognize large conjugacy classes
which appear in a product of two conjugacy classes while the second is used to
keep track of multiplicities of regular diagrams. In particular, we formulate
sufficient conditions in terms of marked diagrams, for a product of normal
subsets in $G$ to contain regular semisimple elements.

###### Key words and phrases:

conjugacy class, simple algebraic group, Dynkin diagram

###### 2020 Mathematics Subject Classification:

Primary 20G99; Secondary 05E16

I am grateful to Prof. Attila Maróti for many discussions on this topic. This
work was supported by a grant of the Ministry of Research, Innovation and
Digitalization, CNCS/CCCDI–UEFISCDI, project number PN-
III-P1-1.1-TE-2019-0136, within PNCDI III

##  1\. Introduction

A normal subset of a group $G$ is defined to be a non-empty union of conjugacy
classes in $G$. By a result of Liebeck and Shalev [17, Theorem 1.1], there is
a universal constant $c$ such that whenever $N$ is a non-central normal subset
in a non-abelian finite simple group $G$ then $N^{k}=G$ for any integer $k$ at
least $c\cdot(\log_{2}|G|/\log_{2}|N|)$. This was generalized by Maróti and
Pyber in [21, Theorem 1.2], where they prove that there exists a universal
constant $c$ such that if $N_{1},\ldots,N_{k}$ are non-central normal subsets
in a non-abelian finite simple group $G$ satisfying
$\prod_{i=1}^{m}|N_{i}|\geq|G|^{c}$, then $N_{1}\cdots N_{k}=G$.

The _covering number $\operatorname{cn}(G,N)$_ of a group $G$ by a normal
subset $N$ of $G$ is the smallest integer $k$ such that $N^{k}=G$ or $\infty$
if no such $k$ exists. The _covering number $\operatorname{cn}(G)$_ of $G$ is
the smallest integer $k$ such that $N^{k}=G$ for any normal subset $N$ of $G$
which is not contained in any proper normal subgroup of $G$. The _extended
covering number $\operatorname{ecn}(G)$_ of $G$ is the smallest integer $k$
such that $N_{1}\cdots N_{k}=G$ whenever $N_{1},\ldots,N_{k}$ are normal
subsets of $G$ not contained in any proper normal subgroup of $G$ [8].

By results of Gordeev [7], if $G$ is a simple algebraic group defined over an
algebraically closed field of characteristic $0$ then
$\operatorname{cn}(G)\leq 4\cdot\operatorname{rk}(G)$ where
$\operatorname{rk}(G)$ is the Lie rank of $G$. This result was extended by
Ellers, Gordeev and Herzog [5] to the case of quasisimple Chevalley groups.
More precisely, they show that for such a group $G$ we have
$\operatorname{cn}(G)\leq 2^{13}\cdot\operatorname{rk}(G)$. Gordeev and Saxl
[8] show that there is a constant $c$ such that for any Chevalley group $G$
defined over any field, we have $\operatorname{ecn}(G)\leq
c\cdot\operatorname{rk}(G)$. Moreover, if $G$ is a Chevalley group defined
over an algebraically closed field, they show that $\operatorname{ecn}(G)\leq
4\cdot\operatorname{rk}(G)$. Extending [17, Theorem 1.1], upper bounds on
covering numbers of unipotent conjugacy classes are given in terms of their
dimension and in terms of their (co)ranks in [25] for $G$ a simple algebraic
group over an algebraically closed field $k$ of good characteristic. Recall
that the characteristic $p$ of $k$ is good for $G$ if $p\neq 2$ when $G$ is
not of type $A$, $p\neq 3$ if $G$ is an exceptional group and $p\neq 5$ if $G$
is of type $E_{8}$. Theorem C in [25] is extended and improved by [18, Theorem
1] where it is shown that there exists an absolute constant $c\leq 120$ such
that whenever $C$ is a non-central conjugacy class of a simple algebraic group
$G$ then $C^{k}=G$ for any integer $k$ at least $c\cdot(\dim(G)/\dim(C))$.

The methods used in the context of Chevalley groups give explicit constants
for the upper bounds on the (extended) covering numbers which are missing in
[17, Theorem 1.1] and [21, Theorem 1.2]. On the other hand the results in [17,
21] take into account the size of the normal subsets and should have analogous
statements for Chevalley groups. Bridging the two types of results should
entail an analysis of simple algebraic groups over algebraically closed fields
which are easier to deal with than finite simple groups of Lie type or
Chevalley groups over arbitrary fields, yet resembles these type of groups
closely through the BN-pair structure. Central in this context is to measure
the size of the normal subset $C_{1}C_{2}$ for two conjugacy classes $C_{1}$
and $C_{2}$ of $G$.

The product of small conjugacy classes in simple groups grows rapidly in the
following sense. By a result of Liebeck, Schul and Shalev [15, Theorem 1.3],
given any $\epsilon>0$, there exists $\delta>0$ such that if $N_{1}$, $N_{2}$
are normal subsets of a non-abelian finite simple group $G$ satisfying
$|N_{i}|\leq{|G|}^{\delta}$ for $i=1$, $2$, then
$|N_{1}N_{2}|\geq(|N_{1}||N_{2}|)^{1-\epsilon}$. An analogue of this statement
for algebraic groups is [15, Theorem 1.5]. Given any $\epsilon>0$, there
exists $\delta>0$ such that if $C_{1}$ and $C_{2}$ are conjugacy classes in a
simple algebraic group $G$ defined over an algebraically closed field and
satisfying $\dim(C_{i})\leq\delta\dim(G)$ for $i=1$, $2$, then the product
$C_{1}C_{2}$ contains a conjugacy class of dimension at least
$(1-\epsilon)(\dim(C_{1})+\dim(C_{2}))$.

Products of large conjugacy classes cover $G$ rapidly in the following sense.
By a result of Gow [9, Theorem 2], if $G$ is a finite simple group of Lie
type, then for any two regular semisimple conjugacy classes $C_{1}$ and
$C_{2}$ the product $C_{1}C_{2}$ contains any non-identity semisimple element
of $G$. Hence, the product of four such classes equals $G$. Many of the
results on large classes are motivated by Thomposon’s conjecture. The analogue
for a simple algebraic group $G$ states that there exists a conjugacy class
$C$ such that $C^{2}=G$. We refer to the survay in [19] for more background on
this. For a simple algebraic group $G$ over an algebraically closed field the
product of $4$ regular conjugacy classes equals $G$ (see for example [18,
Lemma 2.1]). When considering a product $N_{1}\cdots N_{k}$ of several normal
subsets which equals $G$ one would like to understand which normal subsets can
make up such a product. By the above, if $k=4$ then regular classes are
possible. As $k$ increases, one would like to understand which smaller classes
can be used in such a product.

From a different perspective, the Arad-Herzog conjecture states that the
product of two non-trivial conjugacy classes in a non-abelian finite simple
group $G$ is never a conjugacy class in $G$. Guralnick, Malle and Tiep [11]
prove a strong version of the Arad-Herzog conjecture for simple algebraic
groups and in particular show that almost always the product of two conjugacy
classes in a simple algebraic group consists of infinitely many conjugacy
classes. Guralnick and Malle [10] classify pairs of conjugacy classes in
almost simple algebraic groups whose product consists of finitely many
classes.

In this paper, $G$ denotes a simple algebraic group defined over an
algebraically closed field of characteristic $p>0$. To a normal subset
$N\subseteq G$ we attach a set of marked diagrams by means of the
representatives in a Borel subgroup $B$. More precisely, for $g\in N\cap B$,
the marked diagram of $g$ is the Dynkin diagram of $G$ in which we mark the
node corresponding to a simple root $\alpha$ if the projection of $g$ on the
root group $U_{\alpha}$ is not $1$ (see §3). The set of all these diagrams -
obtained from elements in $N\cap B$ \- is denoted by ${\mathcal{D}}(N)$. If
$p$ is a good prime for $G$, i.e. $p\neq 2$ if $G$ is not of type $A$, $p\neq
3$ if $G$ is an exceptional group and $p\neq 5$ if $G$ is of type $E_{8}$,
then marked diagrams extend the notion of distinguished diagrams used in the
Bala-Carter classification of unipotent conjugacy classes [4, §5.11].

Marked diagrams offer a way of measuring the ‘size of a conjugacy class’ not
only for algebraic groups over algebraically closed fields but for simple
groups of Lie type and Chevalley groups over arbitrary fields as well: for a
small conjugacy class $C$ the set ${\mathcal{D}}(C)$ contains diagrams with
few marked nodes while for a large class $C$ there are diagrams with many
marked nodes in ${\mathcal{D}}(C)$. If ${\mathcal{D}}^{\circ}$ denotes the
diagram with all nodes marked then ${\mathcal{D}}^{\circ}\in{\mathcal{D}}(N)$
whenever $N$ contains a regular conjugay class (see Propositions 14 and 15).
Moreover, for two classes $C_{1}$ and $C_{2}$ of $G$ if
$D_{1}\in{\mathcal{D}}(C_{1})$ and $D_{2}\in{\mathcal{D}}(C_{2})$ then
$D_{1}\boxplus D_{2}\subseteq{\mathcal{D}}(C_{1}C_{2})$ where $D_{1}\boxplus
D_{2}$ is the marked diagram obtained by marking exactly those nodes which are
marked in both $D_{1}$ and $D_{2}$ (see Proposition 19).

Marked diagrams can also be viewed as elements
$\sum_{\alpha\in\Delta}n_{\alpha}\alpha$ of the monoid
${\mathbb{N}}\Delta={\mathbb{N}}^{|\Delta|}$ where $\Delta$ is a set of simple
roots and $n_{\alpha}\in{\mathbb{N}}$ (see §3.1). In this notation
${\mathcal{D}}^{\circ}$ is the marked diagram $\sum_{\alpha\in\Delta}\alpha$.
Let $D=\sum_{\alpha\in\Delta}n_{\alpha}\alpha$ and
$D^{\prime}=\sum_{\alpha\in\Delta}m_{\alpha}\alpha$ be two diagrams with
$n_{\alpha},m_{\alpha}\in{\mathbb{N}}$. Their sum is
$D+D^{\prime}=\sum_{\alpha\in\Delta}(n_{\alpha}+m_{\alpha})\alpha$. There is a
partial order $\geq$ on ${\mathbb{N}}\Delta$ defined by $D\geq D^{\prime}$ if
and only if $n_{\alpha}-m_{\alpha}\geq 0$ for all $\alpha\in\Delta$. This
gives a way of addressing questions on products of classes in terms of
calculations in the monoid ${\mathbb{N}}\Delta$. A first statement in this
direction is the following.

######  Proposition A.

Let $G$ be a simple algebraic group, defined over an algebraically closed
field. Let $N_{1},\dots,N_{k}$ be normal subsets of $G$ and let
$D_{i}\in{\mathcal{D}}(N_{i})$ for all $1\leq i\leq k$. If
$\sum_{i=1}^{k}D_{i}\geq
12\cdot\operatorname{rk}(G)\cdot{\mathcal{D}}^{\circ}$, then
$\prod_{i=1}^{k}N_{i}=G$.

Comparing this result to [8], where $\operatorname{ecn}(G)$ is shown to be
less than $4\cdot\operatorname{rk}(G)$, we don’t obtain anything new in terms
of the extended covering number of $G$. The condition on the diagrams $D_{i}$
implies that there are at least $12\cdot\operatorname{rk}(G)$ conjugacy
classes in the product, which by [8] has to equal $G$.

Since the product of two open subsets of $G$ equals $G$, it is natural to ask
when a product $N_{1}\cdots N_{k}$ of normal subsets contains an open subset
of $G$. For this, notice that any set $A$ of diagrams is partially ordered.
For a diagram $D\in A$, let $A(\geq D):=\\{E\in A:E\geq D\\}$. We say that $A$
is of type $(r,s)$ with respect to the diagrams $D_{1},\ldots,D_{r}\in A$ if
there is a partition of $A$ into $r$ subsets $A_{1},\dots,A_{r}$ such that
$|A_{i}\cap A(\geq D_{i})|\geq s$ for all $1\leq i\leq r$.

######  Proposition B.

Let $G$ be a simple algebraic group, defined over an algebraically closed
field. Let $N_{1},\dots,N_{k}$ be normal subsets of $G$ and let
$D_{i}\in{\mathcal{D}}(N_{i})$ for all $1\leq i\leq k$. If the set
$\\{D_{i}:1\leq i\leq k\\}$ is of type $(r,6)$ with respect to
$D_{i_{1}},\dots,D_{i_{r}}$ and
$\sum_{j=1}^{r}D_{i_{j}}\geq{\mathcal{D}}^{\circ}$ then
$\dim\prod_{i=1}^{m}N_{i}=\dim G$.

Since the product of four subsets of $G$, each of which contains a regular
conjugacy class, equals $G$, it is natural to ask when a product of normal
subsets $N_{1}\cdots N_{n}$ contains a regular element.

######  Theorem C.

Let $G$ be a simple algebraic group, defined over an algebraically closed
field. Let $N_{1},\dots,N_{k}$ be normal subsets of $G$ and let
$D_{i}\in{\mathcal{D}}(N_{i})$ for all $1\leq i\leq k$. If
$\sum_{i=1}^{k}D_{i}\geq 16{\mathcal{D}}^{\circ}$, then $\prod_{i=1}^{k}N_{i}$
contains regular semisimple elements.

Proposition B shows in particular that if $\sum_{i=1}^{k}D_{i}\geq
4{\mathcal{D}}^{\circ}$ and each diagram $D_{i}$ appears at least $6$ times
then the product of the corresponding normal subsets contains an open subset
of $G$. It follows that if we replace $6$ by $12$ the product of the
corresponding subsets is $G$. Theorem C shows that if the diagrams can be
partitioned into $4$ subsets, each of which sum up to a regular diagram (a
diagram which is greater than or equal to ${\mathcal{D}}^{\circ}$) then the
product of the corresponding subsets is $G$. This suggests that it should be
possible to remove $\operatorname{rk}(G)$ in Proposition A.

######  Question D.

Let $G$ be a simple algebraic group, defined over an algebraically closed
field. Let $N_{1},\dots,N_{k}$ be normal subsets of $G$ and let
$D_{i}\in{\mathcal{D}}(N_{i})$ for all $1\leq i\leq k$. If
$\sum_{i=1}^{k}D_{i}\geq c\cdot{\mathcal{D}}^{\circ}$ for some constant which
does not depend on $G$, does it follow that $\prod_{i=1}^{k}N_{i}=G$?

An affirmative answer to this question would give in particular a means of
recognizing which conjugacy classes can appear in a product $N_{1}\cdots
N_{k}=G$ for a fixed $k$. By Proposition 10, conjugacy classes in the same
Jordan class have the same marked diagrams and Proposition 13 exhibits
conjecturally maximal marked diagrams of a conjugacy class. Notice that it
suffices to give an answer to the above question for classical groups of heigh
rank, a treatment of the bounded rank case is needed for a good upper bound on
the constant $c$.

The paper is structured as follows: in §2 we fix notation, we collect results
on conjugacy classes in algebraic groups which are relevant to marked diagrams
and we recall a factorization of $G$ which is needed in the sequel. In §3 we
introduce the notion of a marked diagram and the associated monoids. In §4 we
describe the link between marked diagrams and unipotent elements. The proofs
of Propositions A and B and of Theorem C are given in §5.

##  2\. Preliminaries

In this paper $G$ denotes a simple algebraic group of rank
$r=\operatorname{rk}(G)$ defined over an algebraically closed field $F$ of
characteristic $p>0$. We fix a Borel subgroup $B$ with unipotent radical $U$
and maximal torus $T$. We let $\Phi$ denote the roots of $G$ with respect to
$T$, the set of positive roots $\Phi^{+}$ are with respect to $U$ and $\Delta$
denotes the set of simple roots of $\Phi$ in $\Phi^{+}$. We denote by $U^{-}$
the radical of the Borel subgroup opposite to $B$, i.e. $U=U^{\dot{w}_{0}}$
for some representative $\dot{w}_{0}\in N_{G}(T)$ of the longest element (with
respect to $\Delta$) of the Weyl group $N_{G}(T)/T$.

For each root $\alpha\in\Phi$ let $u_{\alpha}:F\rightarrow U_{\alpha}$ be an
isomorphism from the additive group of the ground field $F$ onto the root
subgroup $U_{\alpha}$. For each $\alpha\in\Phi$ we denote by
$\alpha^{\vee}:F^{\times}\rightarrow T$ the cocharacter corresponding to the
root $\alpha$. Then

(1)
${}^{\alpha^{\vee}(t)}u_{\beta}(x)=\alpha^{\vee}(t)u_{\beta}(x)\alpha^{\vee}(t)^{-1}=u_{\beta}(\beta(\alpha^{\vee}(t))x)=u_{\beta}(t^{\langle\beta,\alpha\rangle}x)$

for all $\alpha,\beta\in\Phi$, $t\in F^{\times}$, $x\in F$ (see [14, II§1.3]
and [3, Ch.7]).

Any element $g\in B$ has a unique factorization of the form
$g=s\prod_{\alpha\in\Phi^{+}}u_{\alpha}(x_{\alpha})$ for some $s\in T$,
$x_{\alpha}\in F$ and where the product is in a fixed (but arbitrary) ordering
of $\Phi^{+}$ (see for example [20, Theorem 11.1]). The projections $g\mapsto
u_{\alpha}(x_{\alpha})$ depend in general on the ordering of $\Phi^{+}$.
However, the projections on simple root groups do not depend on this order as
can be seen from the commutator relations (see for example [20, Theorem
11.8]). We point out that
$[U,U]\subseteq\prod_{\alpha\in\Phi^{+}-\Delta}U_{\alpha}$ (this can be
deduced from [20, Proposition 11.5] and the commutator relations). In what
follows we make use of this fact without further notice. For a unipotent
element $u=\prod_{\alpha\in\Phi^{+}}u_{\alpha}(x_{\alpha})$ we denote by
$\operatorname{supp}(u)$ the set of simple roots $\alpha$ with the property
that $x_{\alpha}\neq 0$. For an element $s\in T$ we denote by
$\operatorname{supp}(s)$ the set of simple roots $\alpha$ with the property
that $\alpha(s)\neq 1$.

For an element $g\in G$ we denote by $g_{s}$ and $g_{u}$ the semisimple and
the unipotent part in the Jordan decomposition of $g$ respectively:
$g=g_{s}g_{u}=g_{u}g_{s}$. In any algebraic group, all Borel subgroups,
respectively all maximal tori are conjugate and any element in $G$ is
conjugate to an element in $B$. Thus, conjugating if necessary, we may assume
that $g$ lies in $B$ and that $g_{s}$ lies in $T$: we may conjugate $B$ and
$T$ by the same element in $G$, or, when $g$ is a representative of a
conjugacy classe, we may replace $g$ by a $G$-conjugate with the above
property.

For a set of roots $I\subseteq\Phi$, let $\Phi_{I}$ be the root subsystem
generated by $I$, i.e. $\Phi_{I}={\mathbb{Z}}I\cap\Phi$. We denote by $L_{I}$
the subgroup $\langle T,U_{\alpha}:\alpha\in\Phi_{I}\rangle$ of $G$. If the
roots in $I$ are simple then $L_{I}$ is a standard Levi subgroup. In this
case, we denote by $P_{I}$ the standard parabolic subgroup with Levi factor
$L_{I}$. When we need to specify the ambient group $G$, we write $L_{I}^{G}$
or $P_{I}^{G}$. Notice that the notation $L_{I}^{G}$ and $P_{I}^{G}$ makes
sense in the more general case of a reductive algebraic group $G$. In the
particular case of $I=\\{\alpha\\}\subseteq\Delta$ we denote by $P_{\alpha}$
the parabolic subgroup $P_{I}$ and by $G_{\alpha}$ the subgroup generated by
$U_{\pm\alpha}$.

###  2.1. Semisimple conjugacy classes

For an element $g\in G$ we have $C_{G}(g)=C_{C_{G}(g_{s})}(g_{u})$. Hence,
describing the conjugacy class of $g$ entails two parts: the description of
$C_{G}(g_{s})$ and the description of unipotent conjugacy classes in
$C_{G}(g_{s})$. The structure of the centralizer of a semisimple element in
$G$ is known. In the following theorem we extract a combinatorial description
which we use in the description of Jordan classes in Section 2.3. Recall that
the connected components of the centralizers of semisimple elements in $G$ are
called pseudo-Levi subgroups [22]. The Levi-envelope of a pseudo-Levi $H$ is
the minimal Levi subgroup $L$ of $G$ containing $H$ such that
$Z(H)^{\circ}=Z(L)^{\circ}$ (see [2, §3]).

######  Theorem 1 (Centralizers of semisimple elements).

Let $G$ be a simple algebraic group and let $s$ be a semisimple element
contained in the maximal torus $T$. There is a subset $I\subseteq\Delta$ such
that exactly one of the following holds:

  1. (1)

$C_{G}(s)^{\circ}$ is the Levi-subgroup $L_{I}$, or

  2. (2)

there is a root $\gamma\notin\Delta$ and a simple root $\beta\in\Delta$ such
that $C_{G}(s)^{\circ}$ is the proper pseudo-Levi subgroup
$L_{I\cup\\{\gamma\\}}$ with Levi-envelope $L_{I\cup\\{\beta\\}}$.

###### Proof.

The subgroup $M_{s}=C_{G}(s)^{\circ}$ is the pseudo-Levi subgroup given by
$\langle T,U_{\alpha}:\alpha(g_{s})=1\rangle$ [26, II §4.1]. Let $Z_{s}$
denote the center of $M_{s}$. Then $L_{s}=C_{G}(Z_{s}^{\circ})$ is a Levi
subgroup, the Levi envelope of $M_{s}$ (see [2, §3]). Conjugating, we may
assume that it is a standard Levi subgroup, i.e. $L_{s}=L_{J}$ for some
$J\subseteq\Delta$.

The pseudo-Levi $M_{s}$ is a subgroup of $L_{s}$ and the torus $Z_{s}^{\circ}$
is a maximal central torus of $M_{s}$ and of $L_{s}$ [2, Lemma 3.7]. Factoring
we obtain the semisimple subgroup $M_{s}/Z_{s}^{\circ}$ of the semisimple
group $L_{s}/Z_{s}^{\circ}$. Under the projection $L_{s}\rightarrow
L_{s}/Z_{s}^{\circ}$, $x\mapsto\bar{x}$, the centralizer of $\bar{s}$ in
$\bar{L}_{s}$ is $\bar{M}_{s}$.

Since $Z_{s}^{\circ}$ is contained in the maximal torus $T$ which lies in
$M_{s}$ and $L_{s}$, the projection $x\mapsto\bar{x}$ induces a bijection
$\alpha\mapsto\bar{\alpha}$ on the roots of $L_{s}$ w.r.t. $T$ and the roots
of $\bar{L}_{s}$ w.r.t. $\bar{T}$. The root system $\bar{\Phi}_{J}$ decomposes
into a direct sum of irreducible root systems $\bar{\Phi}_{i}$ with $0\leq
i\leq m$ for some integer $m$. Since $G$ is a simple algebraic group and
$L_{s}$ is a standard Levi subgroup of $G$, at most one of the
$\bar{\Phi}_{i}$ is not of type $A$. Renumbering, we may assume that
$\bar{\Phi}_{0}$ has this property.

Conjugating if necessary, we may assume that the root system of $\bar{M}_{s}$
with respect to $\bar{T}$ has a basis
$\bar{K}\subseteq\bar{J}\cup\\{\bar{\gamma}\\}$ where $-\bar{\gamma}$ is the
highest root of $\bar{\Phi}_{0}$ [22, Proposition 30].

The rank of $\bar{M}_{s}$ is at most that of $\bar{L}_{s}$, i.e.
$|\bar{K}|\leq|\bar{J}|$. We claim that $|\bar{K}|=|\bar{J}|$. If this is not
the case, then $\bar{M}_{s}$ has rank at most $|\bar{J}|-1$, i.e. a maximal
torus of $\bar{M}_{s}$ has dimension at most $|\bar{J}|-1$. However the
$|\bar{J}|$-dimensional torus $\bar{T}$ lies in $\bar{M}_{s}$, which is a
contradiction.

It follows that $\bar{K}$ is either $\bar{J}$ or
$\\{\bar{\gamma}\\}\cup(\bar{J}-\\{\bar{\beta}\\})$ for some
$\bar{\beta}\in\bar{J}\cap\bar{\Phi}_{0}$. If $\bar{K}=\bar{J}$ then (1) holds
with $I=K$. If $\bar{K}=\\{\bar{\gamma}\\}\cup(\bar{J}-\\{\bar{\beta}\\})$
then (2) holds with $I=J-\\{\beta\\}$. ∎

For subsets $I\subseteq\Delta$ and $I^{\prime}\subseteq\Phi$ the pair
$(I,I^{\prime})$ is called _of proper pseudo-Levi type_ if
$[L_{I^{\prime}},L_{I^{\prime}}]$ is a proper maximal rank subsystem subgroup
of $[L_{I},L_{I}]$. The pair $(I,I^{\prime})$ is called _of proper Levi type_
if $I^{\prime}=I$ is a Levi subgroup. The pair $(I,I^{\prime})$ is called _of
pseudo-Levi type_ if it is of proper pseudo-Levi type or of proper Levi type.
Examples can be constructed with the Borel-de Siebenthal algorithm (see for
example [20, §13.2]). In particular for type $B_{r}$ one may choose $I=\Delta$
and $I^{\prime}=\Delta\setminus\\{\alpha_{r}\\}\cup\\{-\alpha_{0}\\}$ where
$\alpha_{0}$ is the highest root, in order to obtain
$[L_{I^{\prime}},L_{I^{\prime}}]$ of type $D_{r}$. Notice also that for type
$A_{r}$ the semisimple conjugacy classes are of proper Levi type.

###  2.2. Unipotent conjugacy classes

For our purposes, we use the Bala-Carter-Pommerening classification of
unipotent conjugacy classes [1, 23] (see also [4, Theorem 5.9.6 and §5.11]).
For this, we require the characteristic of the ground field to be good for
$G$, i.e. $p\neq 2$ if $G$ is not of type $A$, $p\neq 3$ if $G$ is of
exceptional type and $p\neq 5$ if $G$ is of type $E_{8}$. The classification
of unipotent classes in the case where the characteristic of the ground field
is bad for $G$ was achieved through the contribution of many authors. The
state of the art for unipotent conjugacy classes is available in [16].

For an element $g\in G$, the group $C_{G}(g_{s})^{\circ}$ is connected
reductive, hence a central product of simple algebraic groups and a central
torus with no non-trivial unipotent elements in the center. Therefore, the
conjugacy class of $g_{u}$ is a product of unipotent conjugacy classes in the
simple factors of $C_{G}(g_{s})^{\circ}$. The following theorem is well known
and translates directly to the case where $G$ is a connected reductive
algebraic group. The statement is implicit in [16, Theorem 1].

Recall that a unipotent element is distinguished if $C_{G}(u)^{\circ}$ is
unipotent. For a parabolic subgroup $P=LQ$ with Levi factor $L$ and unipotent
radical $Q$ we have $\dim L\geq\dim(Q/[Q,Q])$. The group $P$ is a
distinguished parabolic subgroup if $\dim L=\dim(Q/[Q,Q])$ [16, §2.5-6]. An
element $g$ of a parabolic subgroup $P$ is called a Richardson element of $P$
if the $P$-conjugacy class of $g$ intersects $Q$ in an open subset of $Q$.

######  Theorem 2 (Conjugacy classes of unipotent elements).

Let $G$ be a simple algebraic group defined over an algebraically closed field
of good characteristic. There is a bijective correspondence between unipotent
conjugacy classes of $G$ and $G$-classes of pairs $(L,P)$, where $L$ is a Levi
subgroup of $G$ and $P$ is a distinguished parabolic subgroup of $[L,L]$. The
$G$-class of $(L,P)$ corresponds to the $G$-conjugacy class containing a
Richardson element of $P$.

For two subsets of simple roots $K\subseteq J\subseteq\Delta$, the pair
$(J,K)$ is called _distinguished_ if $P_{K}^{[L_{J},L_{J}]}$ is a
distinguished parabolic subgroup of $[L_{J},L_{J}]$.

###  2.3. Jordan classes

For algebraic groups, Jordan classes were introduced in [2] inspired by the
similar notion for the adjoint action of a group on its Lie algebra (see [2,
§4]). Conjugacy classes in $G$ can be grouped together as follows. Two
conjugacy classes are equivalent [2, §4] if they have representative $g$ and
$h$ respectively, such that $C_{G}(h_{s})^{\circ}=C_{G}(g_{s})^{\circ}$,
$h_{s}\in g_{s}Z(C_{G}(g_{s})^{\circ})^{\circ}$ and $h_{u}$ is conjugate to
$g_{u}$ in $C_{G}(g_{s})^{\circ}$. The unions of elements in the corresponding
equivalence classes are called Jordan classes. The set of Jordan classes is
denoted by ${\mathcal{J}}$. They partition $G$ into a finite number of
irreducible normal subsets and the conjugacy classes in a Jordan class have
the same dimension.

In view of the previous sections we assume in this section that the
characteristic of $F$ is good for $G$ and consider the set

${\mathcal{I}}=\Big{\\{}(I,I^{\prime},J,K):(I,I^{\prime})\text{ is of pseudo-
Levi type},J\subseteq I^{\prime},(J,K)\text{ is distinguished}\Big{\\}}.$

We define a map

$\phi:{\mathcal{I}}\rightarrow{\mathcal{J}}$

as follows. Fix $(I,I^{\prime},J,K)\in{\mathcal{I}}$. By [22, Proposition 32],
since the characteristic of $F$ is good for $G$, for a given pair
$(I,I^{\prime})$ of pseudo-Levi type, there exists a semisimple element $s$ in
$G$ such that $C_{G}(s)^{\circ}$ is $L_{I^{\prime}}$. By [24, Proposition 1]
there exists a unipotent element $u\in C_{G}(s)^{\circ}$ such that $u$ is a
Richardson element of $P_{K}^{[L_{J},L_{J}]}$. Define $\phi(I,I^{\prime},J,K)$
to be the Jordan class of $su$. For any pair $(s,u)$ as above, we say that
_$su$ realizes the data $(I,I^{\prime},J,K)$_.

######  Corollary 3.

The map $\phi:{\mathcal{I}}\rightarrow{\mathcal{J}}$ is well-defined and
surjective.

###### Proof.

Fix an element $(I,I^{\prime},J,K)\in{\mathcal{I}}$. If $s_{1}$ and $s_{2}$
are semisimple elements such that
$C_{G}(s_{1})^{\circ}=L_{I^{\prime}}=C_{G}(s_{2})^{\circ}$ then, by
definition, for any $u_{1},u_{2}\in L_{I^{\prime}}$, the elements $s_{1}u_{1}$
and $s_{2}u_{2}$ are in the same Jordan class if and only if $u_{1}$ and
$u_{2}$ are in the same $L_{I^{\prime}}$-conjugacy class. This is the case
since both $u_{1}$ and $u_{2}$ are Richardson elements of
$P_{K}^{[L_{J},L_{J}]}$. Hence $\phi$ is well defined.

Let $g$ be a representative of some Jordan class. We may assume that $g_{s}$
lies in $T$. By Theorem 1 there is a pair $(I,I^{\prime})$ of pseudo-Levi type
such that $C_{G}(g_{s})^{\circ}=L_{I^{\prime}}$. By [26, III §1.14], since we
assume that the characteristic of $F$ is good for $G$ we have $g_{u}\in
L_{I^{\prime}}$. By Theorem 2, $g_{u}$ is a Richardson element of some
distinguished parabolic subgroup $P_{K}^{[L_{J},L_{J}]}$. Hence $\phi$ is
surjective. ∎

###  2.4. Unipotent factorization of $G$

The following result is due to Vavilov, Smolensky and Sury [28]. A proof can
also be obtained with the method used in [6] for finite groups of Lie type. We
refer to [6] and the references therein for more background on this result.

######  Theorem 4.

Let $G$ be a simple algebraic group, $B\subseteq G$ a Borel subgroup with
maximal torus $T$ and unipotent radical $U$. We have $U\cdot
U^{\dot{w}_{0}}\cdot U\cdot U^{\dot{w}_{0}}=G$ where $\dot{w}_{0}$ is a
representative in $N_{G}(T)$ of the longest element of the Weyl group
$N_{G}(T)/T$.

######  Corollary 5.

For any simple algebraic group $G$ we have ${\mathcal{U}}^{3}=G$ where
${\mathcal{U}}$ denotes the variety of unipotent elements in $G$.

###### Proof.

We have $G=U\cdot U^{\dot{w}_{0}}\cdot U\cdot U^{\dot{w}_{0}}\subseteq
U^{U^{\dot{w}_{0}}}\cdot U^{U^{\dot{w}_{0}}}\cdot
U^{\dot{w}_{0}}\subseteq{\mathcal{U}}^{3}$ by Theorem 4. ∎

##  3\. Marked diagrams

The _marked diagram_ of an element $g\in B$ is the Dynkin diagram of $G$ where
we mark the nodes corresponding to the simple roots $\alpha$ for which the
projection of $g$ on the root subgroup $U_{\alpha}$ is not $1$. More
precisely, any $g\in B$ has a factorization of the form
$s\prod_{\alpha\in\Phi^{+}}u_{\alpha}(x_{\alpha})$ with $s\in T$ and
$x_{\alpha}\in F$, and we mark the nodes corresponding to those
$\alpha\in\Delta$ for which $x_{\alpha}\neq 0$. We denote this diagram with
${\mathcal{D}}(g)$ and write ${\mathcal{D}}^{\circ}$ for the marked diagram
having all nodes marked. We denote by $\operatorname{supp}(D)$ the set of
simple roots corresponding to marked nodes of a diagram $D$. Notice that for
$s\in T$ and $u\in U$, we have
$\operatorname{supp}({\mathcal{D}}(su))=\operatorname{supp}(u)$. In what
follows we mention a few examples of marked diagrams.

Unipotent elements. With the obvious choice of $T$ and $B$, the unipotent
element of $\operatorname{SL}_{5}(F)$

$\begin{bmatrix}1&1&0&0&0\\\ 0&1&1&0&0\\\ 0&0&1&0&0\\\ 0&0&0&1&1\\\
0&0&0&0&1\\\ \end{bmatrix}\quad\text{has marked diagram}\quad\leavevmode\hbox
to91.36pt{\vbox to6.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.0pt\lower-3.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{85.35828pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.0pt}{0.0pt}\pgfsys@curveto{3.0pt}{1.65688pt}{1.65688pt}{3.0pt}{0.0pt}{3.0pt}\pgfsys@curveto{-1.65688pt}{3.0pt}{-3.0pt}{1.65688pt}{-3.0pt}{0.0pt}\pgfsys@curveto{-3.0pt}{-1.65688pt}{-1.65688pt}{-3.0pt}{0.0pt}{-3.0pt}\pgfsys@curveto{1.65688pt}{-3.0pt}{3.0pt}{-1.65688pt}{3.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@moveto{31.45276pt}{0.0pt}\pgfsys@curveto{31.45276pt}{1.65688pt}{30.10963pt}{3.0pt}{28.45276pt}{3.0pt}\pgfsys@curveto{26.79588pt}{3.0pt}{25.45276pt}{1.65688pt}{25.45276pt}{0.0pt}\pgfsys@curveto{25.45276pt}{-1.65688pt}{26.79588pt}{-3.0pt}{28.45276pt}{-3.0pt}\pgfsys@curveto{30.10963pt}{-3.0pt}{31.45276pt}{-1.65688pt}{31.45276pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@moveto{59.90552pt}{0.0pt}\pgfsys@curveto{59.90552pt}{1.65688pt}{58.5624pt}{3.0pt}{56.90552pt}{3.0pt}\pgfsys@curveto{55.24864pt}{3.0pt}{53.90552pt}{1.65688pt}{53.90552pt}{0.0pt}\pgfsys@curveto{53.90552pt}{-1.65688pt}{55.24864pt}{-3.0pt}{56.90552pt}{-3.0pt}\pgfsys@curveto{58.5624pt}{-3.0pt}{59.90552pt}{-1.65688pt}{59.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@moveto{59.90552pt}{0.0pt}\pgfsys@curveto{59.90552pt}{1.65688pt}{58.5624pt}{3.0pt}{56.90552pt}{3.0pt}\pgfsys@curveto{55.24864pt}{3.0pt}{53.90552pt}{1.65688pt}{53.90552pt}{0.0pt}\pgfsys@curveto{53.90552pt}{-1.65688pt}{55.24864pt}{-3.0pt}{56.90552pt}{-3.0pt}\pgfsys@curveto{58.5624pt}{-3.0pt}{59.90552pt}{-1.65688pt}{59.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{85.35828pt}{0.0pt}\pgfsys@moveto{88.35828pt}{0.0pt}\pgfsys@curveto{88.35828pt}{1.65688pt}{87.01515pt}{3.0pt}{85.35828pt}{3.0pt}\pgfsys@curveto{83.7014pt}{3.0pt}{82.35828pt}{1.65688pt}{82.35828pt}{0.0pt}\pgfsys@curveto{82.35828pt}{-1.65688pt}{83.7014pt}{-3.0pt}{85.35828pt}{-3.0pt}\pgfsys@curveto{87.01515pt}{-3.0pt}{88.35828pt}{-1.65688pt}{88.35828pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{85.35828pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
Let $g$ be a unipotent element $g$ of $G=\operatorname{SL}_{n}(F)$ in Jordan
form, i.e. $g=\bigoplus_{i}J_{i}$ where the $J_{i}$ are Jordan blocks. Again,
with the obvious choice of $T$ and $B$, the number of connected components of
${\mathcal{D}}(g)$, obtained after removing the non-marked nodes, equals the
number of Jordan blocks $J_{i}$ and each connected component $D_{i}$
corresponds to a Jordan block $J_{i}$ such that the number of nodes in $D_{i}$
is one less than the length of $J_{i}$.

If $g\in B$ is a distinguished unipotent element, then ${\mathcal{D}}(g)$ is
obtained from the labeled Dynkin diagram of $G$ by marking the nodes with
label ‘2’. For a list of these diagrams see [4, §5.9].

Semisimple elements. If $g\in T$, then $g$ is conjugate to an element
$g^{\prime}\in B$ such that ${\mathcal{D}}(g^{\prime})$ has those simple roots
$\alpha$ marked for which $\alpha(g^{\prime})\neq 1$, i.e.
$\operatorname{supp}({\mathcal{D}}(g^{\prime}))=\operatorname{supp}(g)$ (see
Proposition 9).

Regular elements. A regular element is conjugate to an element which has fully
marked diagrams (this follows from Proposition 9).

The _marked diagrams_ of a normal subset $N$ of $G$ are the marked diagrams of
elements in $N\cap B$ and we denote this set by ${\mathcal{D}}(N)$. Note that
${\mathcal{D}}(N)$ is the union of ${\mathcal{D}}(C)$ where $C$ runs over the
conjugacy classes in $N$. Notice also that, by Corollary 3, any conjugacy
class in the Jordan class $\phi(I,I^{\prime},J,K)$ has a marked diagram $D$
with $\operatorname{supp}(D)=(J-K)\cap\Delta$. In Proposition 13 we show that
any conjugacy class in the Jordan class $\phi(I,I^{\prime},J,K)$ has a marked
diagram $D$ with $\operatorname{supp}(D)=(I-\Delta)\cup((J-K)\cap\Delta)$.

###  3.1. Monoids corresponding to marked diagrams

For an abelian monoid $M$ we use additive notation and we denote by $M\Delta$
the free monoid $M^{|\Delta|}$ with basis $\Delta$. We may view marked
diagrams as elements of $M\Delta$ by identifying the basis element $\alpha$
with the marked diagram having only the node corresponding to $\alpha$ marked.
We call the elements of $M\Delta$ marked diagrams.

Denote by ${\mathbb{N}}_{\operatorname{OR}}=(\\{0,1\\},\boxplus)$ the monoid
with two elements where the operation is bitwise OR. With
${\mathbb{N}}_{\operatorname{OR}}\Delta$ one can keep track of whether nodes
were marked. Example:

$\leavevmode\hbox to91.56pt{\vbox to6.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.2pt\lower-3.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{85.35828pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.0pt}{0.0pt}\pgfsys@curveto{3.0pt}{1.65688pt}{1.65688pt}{3.0pt}{0.0pt}{3.0pt}\pgfsys@curveto{-1.65688pt}{3.0pt}{-3.0pt}{1.65688pt}{-3.0pt}{0.0pt}\pgfsys@curveto{-3.0pt}{-1.65688pt}{-1.65688pt}{-3.0pt}{0.0pt}{-3.0pt}\pgfsys@curveto{1.65688pt}{-3.0pt}{3.0pt}{-1.65688pt}{3.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.0pt}{0.0pt}\pgfsys@curveto{3.0pt}{1.65688pt}{1.65688pt}{3.0pt}{0.0pt}{3.0pt}\pgfsys@curveto{-1.65688pt}{3.0pt}{-3.0pt}{1.65688pt}{-3.0pt}{0.0pt}\pgfsys@curveto{-3.0pt}{-1.65688pt}{-1.65688pt}{-3.0pt}{0.0pt}{-3.0pt}\pgfsys@curveto{1.65688pt}{-3.0pt}{3.0pt}{-1.65688pt}{3.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@moveto{31.45276pt}{0.0pt}\pgfsys@curveto{31.45276pt}{1.65688pt}{30.10963pt}{3.0pt}{28.45276pt}{3.0pt}\pgfsys@curveto{26.79588pt}{3.0pt}{25.45276pt}{1.65688pt}{25.45276pt}{0.0pt}\pgfsys@curveto{25.45276pt}{-1.65688pt}{26.79588pt}{-3.0pt}{28.45276pt}{-3.0pt}\pgfsys@curveto{30.10963pt}{-3.0pt}{31.45276pt}{-1.65688pt}{31.45276pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@moveto{59.90552pt}{0.0pt}\pgfsys@curveto{59.90552pt}{1.65688pt}{58.5624pt}{3.0pt}{56.90552pt}{3.0pt}\pgfsys@curveto{55.24864pt}{3.0pt}{53.90552pt}{1.65688pt}{53.90552pt}{0.0pt}\pgfsys@curveto{53.90552pt}{-1.65688pt}{55.24864pt}{-3.0pt}{56.90552pt}{-3.0pt}\pgfsys@curveto{58.5624pt}{-3.0pt}{59.90552pt}{-1.65688pt}{59.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@moveto{59.90552pt}{0.0pt}\pgfsys@curveto{59.90552pt}{1.65688pt}{58.5624pt}{3.0pt}{56.90552pt}{3.0pt}\pgfsys@curveto{55.24864pt}{3.0pt}{53.90552pt}{1.65688pt}{53.90552pt}{0.0pt}\pgfsys@curveto{53.90552pt}{-1.65688pt}{55.24864pt}{-3.0pt}{56.90552pt}{-3.0pt}\pgfsys@curveto{58.5624pt}{-3.0pt}{59.90552pt}{-1.65688pt}{59.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{85.35828pt}{0.0pt}\pgfsys@moveto{88.35828pt}{0.0pt}\pgfsys@curveto{88.35828pt}{1.65688pt}{87.01515pt}{3.0pt}{85.35828pt}{3.0pt}\pgfsys@curveto{83.7014pt}{3.0pt}{82.35828pt}{1.65688pt}{82.35828pt}{0.0pt}\pgfsys@curveto{82.35828pt}{-1.65688pt}{83.7014pt}{-3.0pt}{85.35828pt}{-3.0pt}\pgfsys@curveto{87.01515pt}{-3.0pt}{88.35828pt}{-1.65688pt}{88.35828pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{85.35828pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad\boxplus\quad\leavevmode\hbox
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{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{85.35828pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@moveto{31.45276pt}{0.0pt}\pgfsys@curveto{31.45276pt}{1.65688pt}{30.10963pt}{3.0pt}{28.45276pt}{3.0pt}\pgfsys@curveto{26.79588pt}{3.0pt}{25.45276pt}{1.65688pt}{25.45276pt}{0.0pt}\pgfsys@curveto{25.45276pt}{-1.65688pt}{26.79588pt}{-3.0pt}{28.45276pt}{-3.0pt}\pgfsys@curveto{30.10963pt}{-3.0pt}{31.45276pt}{-1.65688pt}{31.45276pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@moveto{59.90552pt}{0.0pt}\pgfsys@curveto{59.90552pt}{1.65688pt}{58.5624pt}{3.0pt}{56.90552pt}{3.0pt}\pgfsys@curveto{55.24864pt}{3.0pt}{53.90552pt}{1.65688pt}{53.90552pt}{0.0pt}\pgfsys@curveto{53.90552pt}{-1.65688pt}{55.24864pt}{-3.0pt}{56.90552pt}{-3.0pt}\pgfsys@curveto{58.5624pt}{-3.0pt}{59.90552pt}{-1.65688pt}{59.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
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With ${\mathbb{N}}\Delta$ we take into account multiplicities of marked nodes.
For two diagrams $D_{1}=\sum_{\alpha\in\Delta}n_{\alpha}\alpha$ and
$D_{2}=\sum_{\alpha\in\Delta}m_{\alpha}\alpha$, the sum is
$D_{1}+D_{2}=\sum_{\alpha\in\Delta}(n_{\alpha}+m_{\alpha})\alpha$. Example:

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Note that we have a partial ordering on ${\mathbb{N}}\Delta$: if $x=\sum
n_{\alpha}\Delta_{\alpha}$ and $y=\sum m_{\alpha}\Delta_{\alpha}$ then $x\geq
y$ if and only if $n_{\alpha}-m_{\alpha}\geq 0$ for all $\alpha\in\Delta$.

An element $\sum_{\alpha\in\Delta}n_{\alpha}\Delta_{\alpha}$ of $M\Delta$ is
called _regular_ if $n_{\alpha}\neq 0$ for all $\alpha$. The fully marked
diagram ${\mathcal{D}}^{\circ}$ is regular.

While the monoid ${\mathbb{N}}_{\operatorname{OR}}\Delta$ is better suited for
the description of conjugacy classes which appear in a product of conjugacy
classes (as in Proposition 19), the monoid ${\mathbb{N}}\Delta$ is used to
approximate the number of occurences of a regular conjugacy class in a product
of conjugacy classes (as in Theorem C). It is clear from the context which
monoid is used. A connection between the two is given by the following lemma.

######  Lemma 6.

There is a morphism of monoids
$\psi:{\mathbb{N}}\Delta\rightarrow{\mathbb{N}}_{\operatorname{OR}}\Delta$
such that $\psi(D)$ is regular if and only if $D$ is regular.

###### Proof.

For $D=\sum_{\alpha\in\Delta}n_{\alpha}\alpha\in{\mathbb{N}}\Delta$ define
$\psi(D)$ to be
$\sum_{\alpha\in\Delta}m_{\alpha}\alpha\in{\mathbb{N}}_{\operatorname{OR}}\Delta$
with $m_{\alpha}=1$ if and only if $n_{\alpha}\neq 0$. The claims follow
directly from the definitions. ∎

The following lemma is needed for the proof of Proposition A.

######  Lemma 7.

Let $r=|\Delta|$ and let $D_{1},\dots,D_{k}$ be marked diagrams such that
$\sum_{i=1}^{k}D_{i}\geq mr{\mathcal{D}}^{\circ}$ for some integer $m\geq 1$.
There is a partition of $\\{1,\dots,k\\}$ into $m$ subsets $I_{1},\dots,I_{m}$
such that $\sum_{i\in I_{j}}D_{i}\geq{\mathcal{D}}^{\circ}$ for all $j$ with
$1\leq j\leq m$.

###### Proof.

We prove the lemma by induction on $m$. The case $m=1$ is trivial. Assume that
$m\geq 2$. Since $\sum_{i=1}^{k}D_{i}\geq mr{\mathcal{D}}^{\circ}$, there is a
subset $I$ of $\\{1,2,\dots,k\\}$ such that $\sum_{i\in
I}D_{i}\geq{\mathcal{D}}^{\circ}$. Renumbering the diagrams, we may assume
that $I=\\{1,2,\dots,k_{0}\\}$ where $k_{0}=|I|$. Let $I$ be minimal with this
property, i.e. for any $j\in I$, $S_{j}:=\sum_{i\in
I-\\{j\\}}D_{i}\ngeq{\mathcal{D}}^{\circ}$. Since
${\mathcal{D}}^{\circ}=\sum_{\alpha\in\Delta}\alpha$, for every $j\in I$ there
is at least one $\alpha_{j}\in\Delta$ such that the coefficient of
$\alpha_{j}$ in $S_{j}$ is zero. In other words, $D_{j}$ is the only diagram
in $\\{D_{i}\\}_{i\in I}$ which has the node corresponding to $\alpha_{j}$
marked. This gives an injective map $I\rightarrow\Delta:j\mapsto\alpha_{j}$,
hence $|I|\leq|\Delta|=r$. It follows that $\sum_{i\in I}D_{i}\leq
r{\mathcal{D}}^{\circ}$ and that
$\sum_{i=k_{0}+1}^{k}D_{i}\geq(m-1)r{\mathcal{D}}^{\circ}$. The claim follows
by induction. ∎

##  4\. Marked diagrams and unipotent elements

###  4.1. A map from marked diagrams to $U$

Marked diagrams attach combinatorial data to conjugacy classes by means of
elements in $B$. For the reverse direction we consider the map from marked
diagrams to unipotent elements given by

(2) $u:M\Delta\rightarrow U,\quad
u(D):=\prod_{\alpha\in\operatorname{supp}(D)}u_{\alpha}(1).$

It is easy to see that ${\mathcal{D}}(u(D))=D$ for any marked diagram $D$.

######  Lemma 8.

Let $\Delta^{\prime}$ be a subset of $\Delta$. If $u\in U$ is such that
$\operatorname{supp}(u)=\Delta^{\prime}$ then there is a $T$-conjugate of $u$
in $\left(\prod_{\alpha\in\Delta^{\prime}}u_{\alpha}(1)\right)[U,U]$.

###### Proof.

Let $u=\prod_{\alpha\in\Phi^{+}}u_{\alpha}(x_{\alpha})$ with $x_{\alpha}\in
F$. Consider the parabolic subgroup $P_{\Delta^{\prime}}$ with Levi factor
$L_{\Delta^{\prime}}$ and unipotent radical $Q$. It follows that
$u=\check{u}\tilde{u}$ with
$\check{u}\in[L_{\Delta^{\prime}},L_{\Delta^{\prime}}]$ and $\tilde{u}\in Q$.
For a simple root $\alpha$ we have $x_{\alpha}\neq 0$ if and only if
$\alpha\in\Delta^{\prime}$. Thus $\tilde{u}\in Q\cap[U,U]$. Since $T$ contains
a maximal torus $T^{\prime}$ of $[L_{\Delta^{\prime}},L_{\Delta^{\prime}}]$
which normalizes $Q\cap[U,U]$, it suffices to show that $\check{u}$ is
$T^{\prime}$-conjugate to an element in
$\left(\prod_{\alpha\in\Delta^{\prime}}u_{\alpha}(1)\right)[U^{\prime},U^{\prime}]$
where $U^{\prime}=U\cap L_{\Delta^{\prime}}$.

The subgroup $[L_{\Delta^{\prime}},L_{\Delta^{\prime}}]$ is a semisimple
algebraic group, i.e. a central product of simple algebraic groups. Its
maximal torus $T^{\prime}$ contains a maximal torus of each simple factor,
hence we may assume that $[L_{\Delta^{\prime}},L_{\Delta^{\prime}}]$ is a
simple algebraic group. In other words we may assume that
$\Delta=\Delta^{\prime}$ and $T=T^{\prime}$.

Let $\bar{u}$ be the projection of $u$ on $\prod_{\alpha\in\Delta}U_{\alpha}$
and denote by $v$ the element $\prod_{\alpha\in\Delta}u_{\alpha}(1)$. By [27,
§3.7 Theorem 1], the elements $\bar{u}$ and $v$ are regular. Since $\dim
C_{T}(\bar{u})=0$ the orbit
$\bar{u}^{T}\subseteq\prod_{\alpha\in\Delta}U_{\alpha}$ has dimension $\dim
T=\dim\prod_{\alpha\in\Delta}U_{\alpha}$, hence it is open in
$\prod_{\alpha\in\Delta}U_{\alpha}$. Similarly for $v$. Since
$\left(\prod_{\alpha\in\Delta}U_{\alpha}\right)\times\left(\prod_{\alpha\in\Phi-\Delta}U_{\alpha}\right)$
is a factorization of $U$ into algebraic sets it follows that
$\bar{u}^{T}[U,U]=(\bar{u}[U,U])^{T}=(u[U,U])^{T}$ and
$v^{T}[U,U]=(v[U,U])^{T}$ are open in $U$. Hence the two sets have a non-
trivial intersection and the claim follows. ∎

######  Proposition 9.

Let $g\in B$ with $g_{s}\in T$ and let
$\Delta_{s}\subseteq\operatorname{supp}(g_{s})$. There is an element
$\tilde{u}\in[U,U]$ such that $g$ is conjugate by an element of $B$ to
$g_{s}u(D)\tilde{u}$ where $D$ is the diagram with
$\operatorname{supp}(D)=\Delta_{s}\cup\operatorname{supp}(g_{u})$.

###### Proof.

Let $D_{s}$ be the marked diagram with marked nodes corresponding to the
simple roots in $\Delta_{s}$ and note that
$\operatorname{supp}(D)=\Delta_{s}\cup\operatorname{supp}(g_{u})$ is a
disjoint union since $[g_{s},g_{u}]=1$.

Conjugating by an element of $T$ we may assume that the projection of $g_{u}$
on $U_{\alpha}$ is $u_{\alpha}(1)$ for all simple roots $\alpha\in D(g_{u})$
(by Lemma 8), i.e. we may assume that $g_{u}=u({\mathcal{D}}(g_{u}))$ modulo
$[U,U]$. Thus $u(D)=g_{u}u(D_{s})$ modulo $[U,U]$. Denote $u(D_{s})$ by
$\check{u}$ and notice that
$g_{s}u(D)=g_{s}g_{u}\check{u}\hat{u}=g\check{u}\hat{u}$ for some element
$\hat{u}\in[U,U]$.

By [12, §2.4], the element $g\check{u}$ is conjugate by an element $v\in U$ to
$g_{s}u^{\prime}$ with $u^{\prime}\in C_{U}(g_{s})$. The closed connected
group $U$ factors as a variety: $U\cong C_{U}(g_{s})\times Q$ where $Q$ is a
closed connected subvariety of $U$ and where the isomorphism is given by the
product in $G$. Hence $v=v_{0}v_{1}$ with $v_{0}\in C_{U}(g_{s})$ and
$v_{1}\in Q$. It follows that there exists $q\in Q$ such that

$g_{s}u^{\prime}=(g\check{u})^{v}=(g_{s}^{v_{0}v_{1}})(g_{u}\check{u})^{v}=(g_{s}^{v_{1}})(g_{u}\check{u})^{v}\in
g_{s}qg_{u}\check{u}[U,U].$

Therefore

$u^{\prime}\in qg_{u}\check{u}[U,U]=g_{u}q\check{u}[U,U],\quad\text{and
thus}\quad g_{u}^{-1}u^{\prime}\in q\check{u}[U,U].$

Since $g_{u}^{-1}u^{\prime}\in C_{U}(g_{s})$ the projection of
$g_{u}^{-1}u^{\prime}$ on $U_{\alpha}$ is $1$ for all
$\alpha\in\operatorname{supp}(g_{s})$. Hence the projection of
$q\check{u}[U,U]$ on $U_{\alpha}$ is $1$ for all
$\alpha\in\operatorname{supp}(g_{s})$. In other words $q\check{u}\in
C_{U}(g_{s})$ modulo $[U,U]$. But $q,\check{u}\in Q$ hence
$q\check{u}\in[U,U]$. It follows that $g_{u}^{-1}u^{\prime}\in[U,U]$ and so
$\tilde{u}=\hat{u}^{-1}((u^{\prime})^{-1}g_{u})^{v^{-1}}\in[U,U]$. Hence

$\displaystyle(g_{s}u(D)\tilde{u})^{v}=(g\check{u}((u^{\prime})^{-1}g_{u})^{v^{-1}})^{v}=(g\check{u})^{v}(u^{\prime})^{-1}g_{u}=g_{s}u^{\prime}(u^{\prime})^{-1}g_{u}=g.$

######  Proposition 10.

Two conjugacy classes in the same Jordan class have the same sets of marked
diagrams.

###### Proof.

For a conjugacy class $C$ fix a diagram $D\in{\mathcal{D}}(C)$ and let $g\in
B\cap C$ be such that $D={\mathcal{D}}(g)$. Let $g=su$ be the factorization of
$g$ with $s\in T$ and $u\in U$. By [12, §2.4] we may conjugate $g$ by an
element of $U$ to obtain $\tilde{g}=s\tilde{u}$ with $\tilde{u}\in C_{G}(s)$.
Since we conjugated by an element in $U$,
$\operatorname{supp}(D)=\Delta_{s}\cup\operatorname{supp}(\tilde{u})$ for some
$\Delta_{s}\subseteq\operatorname{supp}(s)$.

Any other conjugacy class $C^{\prime}$ in the same Jordan class is represented
by $h=h_{s}h_{u}$ with $h_{u}=\tilde{u}$ and $h_{s}\in
sZ(C_{G}(s)^{\circ})^{\circ}$ with $C_{G}(h_{s})^{\circ}=C_{G}(s)^{\circ}$
(see §2.3). In particular $\operatorname{supp}(h_{s})=\operatorname{supp}(s)$
and $\operatorname{supp}(h_{u})=\operatorname{supp}(\tilde{u})$. By
Proposition 9, $h$ is conjugate to an element $\tilde{h}$ with
$\operatorname{supp}({\mathcal{D}}(\tilde{h}))=\Delta_{s}\cup\operatorname{supp}(\tilde{u})$,
i.e. $C^{\prime}$ contains the element $\tilde{h}$ with
${\mathcal{D}}(\tilde{h})=D$. ∎

######  Lemma 11.

Let $\Delta^{\prime}$ be a subset of $\Delta$. Consider the standard parabolic
subgroup $P_{\Delta^{\prime}}=L_{\Delta^{\prime}}Q$ with Levi factor
$L_{\Delta^{\prime}}$ and unipotent radical $Q$. Any open subset $V$ of $Q$
contains an element $u$ with $\operatorname{supp}(u)=\Delta-\Delta^{\prime}$.

###### Proof.

Let $\Phi_{Q}$ denote the set of roots $\Phi^{+}-\Phi_{\Delta^{\prime}}$. We
have $Q=\prod_{\alpha\in\Phi_{Q}}U_{\alpha}$ as subgroups of $G$. By
uniqueness of expression, this is a direct product of algebraic sets. The
subset $\tilde{V}=\prod_{\alpha\in\Phi_{Q}}(U_{\alpha}-\\{1\\})$ is open in
$Q$, hence $\tilde{V}\cap V\neq\emptyset$. ∎

######  Lemma 12.

For a semisimple element $s\in T$ and any element
$u\in\prod_{\alpha\in\operatorname{supp}(s)}U_{\alpha}$ we have
$\operatorname{supp}(u)=\operatorname{supp}([s,u])$.

###### Proof.

An element $u$ in $\prod_{\alpha\in\operatorname{supp}(s)}U_{\alpha}$ is
determined by the scalars $x_{\alpha}\in F$ for which
$u=\prod_{\alpha\in\operatorname{supp}(s)}u_{\alpha}(x_{\alpha})$. We have

$[s,u]=(u^{-1})^{s}u=\left(\prod_{\alpha\in\operatorname{supp}(s)}u_{\alpha}(-\alpha(s)x_{\alpha}+x_{\alpha})\right)\tilde{u}$

for some $\tilde{u}\in[U,U]$. For $\alpha\in\operatorname{supp}(s)$ we have
$\alpha(s)\neq 1$. Thus $-\alpha(s)x_{\alpha}+x_{\alpha}\neq 0$ whenever
$x_{\alpha}\neq 0$ and the claim follows. ∎

######  Proposition 13.

Any conjugacy class in the Jordan class $\phi(I,I^{\prime},J,K)$ has a marked
diagram $D$ with $\operatorname{supp}(D)=(\Delta-I)\cup((J-K)\cap\Delta)$.

###### Proof.

Let $g\in G$ be an element which realizes the data $(I,I^{\prime},J,K)$ of the
Jordan class which it represents (see §2.3). The $[L_{J},L_{J}]$-conjugacy
class of $u$ intersects the unipotent radical of $P_{K}^{[L_{J},L_{J}]}$ in an
open set. Since $L_{J}\subseteq L_{I^{\prime}}=C_{G}(g_{s})^{\circ}$, by Lemma
11, we may assume that the projections of $g_{u}$ on the root groups
$U_{\alpha}$ with $\alpha\in J-K$ are non-trivial. We therefore have
$\operatorname{supp}(g)=J-K$. Since $C_{G}(g_{s})^{\circ}\subseteq L_{I}$, the
simple roots $\Delta-I$ are in $\operatorname{supp}(g_{s})$. Let
$u=\prod_{\alpha\in\Delta-I}u_{\alpha}(1)$. By Lemma 12,
$\operatorname{supp}([g_{s}^{-1},u])=\operatorname{supp}(u)=\Delta-I$.
Moreover
$g^{u^{-1}}=(g_{s}g_{u})^{u^{-1}}=g_{s}[g_{s}^{-1},u](g_{u})^{u^{-1}}\in
g_{s}[g_{s}^{-1},u]g_{u}[U,U]$. Hence, since
$\operatorname{supp}(g_{s})\cap\operatorname{supp}(g_{u})=\emptyset$ and
$\operatorname{supp}(u)\subseteq\operatorname{supp}(g_{s})$, it follows that
${\mathcal{D}}(g^{u^{-1}})$ has those nodes marked which belong to
$(\Delta-I)\cup((J-K)\cap\Delta)$. ∎

###  4.2. $U$-regular elements

An element of $G$ is called _$U$ -regular_ if it is conjugate to an element of
the form $su$ with $s\in T$ and $u\in U$ a regular unipotent element. A normal
subset is called $U$-regular if it contains a $U$-regular element.

######  Proposition 14.

A normal subset $N\subseteq G$ is $U$-regular if and only if
${\mathcal{D}}(N)$ contains ${\mathcal{D}}^{\circ}$.

###### Proof.

By [27, §3.7 Theorem 1] an element $u\in U$ is regular if and only if it has
non-trivial projections on $U_{\alpha}$ for all $\alpha\in\Delta$. This is
equivalent to $su$ having non-trivial projections on $U_{\alpha}$ for all
$\alpha\in\Delta$ whenever $s$ is an element of $T$. By definition, this is
equivalent to ${\mathcal{D}}(su)={\mathcal{D}}^{\circ}$ for any $s\in T$. ∎

######  Proposition 15.

Regular conjugacy classes are $U$-regular.

###### Proof.

Let $g$ be a representative of a regular conjugacy class such that $g_{s}\in
T$ and $g_{u}\in U$. We may assume that $g$ realizes the data
$(I,I^{\prime},J,K)$ of the Jordan class which it represents. The element
$g\in G$ is regular if and only if $g_{u}$ is regular in
$C_{G}(g_{s})^{\circ}$ [27, §3.5 Proposition 5]. We have
$\operatorname{supp}(g_{s})=\Delta-I^{\prime}$. Since $g_{u}$ is regular in
$C_{G}(g_{s})$ we also have that $J=I^{\prime}$ and $K=\emptyset$ [27, §3.7
Theorem 1]. Thus
$\operatorname{supp}(g_{u})=(J-K)\cap\Delta=I^{\prime}\cap\Delta$. It follows
from Proposition 9 that there is an element $\tilde{u}\in[U,U]$ such that $g$
is conjugate to $g_{s}u(D)\tilde{u}$ where $D$ is the diagram with
$\operatorname{supp}(D)=\operatorname{supp}(g_{s})\cup\operatorname{supp}(g_{u})=\Delta$.
∎

######  Proposition 16.

$U$-regular conjugacy classes of proper Levi type are regular.

###### Proof.

Let $h=su$ be a representative of a $U$-regular conjugacy class with $s\in T$
and $u$ a regular element in $U$. The element $h$ is $U$-conjugate to $g$ with
$g_{s}=s\in T$ and $g_{u}\in C_{U}(g_{s})$ [12, §2.4]. Conjugating in
$C_{G}(g_{s})$, we may assume that $g$ realizes the data $(I,I^{\prime},J,K)$
of the Jordan class which it represents. Since $h$ is $U$-conjugate to $g$, we
have $\operatorname{supp}(g_{u})=\operatorname{supp}(u)\cap I$. Since the
class is of proper Levi type, we have $I=I^{\prime}$, hence
$\operatorname{supp}(g_{u})=\operatorname{supp}(u)\cap I^{\prime}$. Since $g$
is $U$-regular, $\operatorname{supp}(g_{u})=\operatorname{supp}(u)\cap I=I$,
hence $g_{u}$ is regular in the Levi subgroup $C_{G}(g_{s})^{\circ}$ [27, §3.7
Theorem 1]. The proposition follows from [27, §3.5 Proposition 5]. ∎

######  Proposition 17.

If $C_{1},C_{2},\dots,C_{6}$ are $U$-regular conjugacy classes then their
product contains all regular semisimple conjugacy classes of $G$.

###### Proof.

We may partition the set of simple roots $\Delta$ into two sets, $\Delta_{1}$
and $\Delta_{2}$, such that $\alpha$ and $\beta$ are orthogonal for any two
distinct roots $\alpha,\beta\in\Delta_{1}$ or $\alpha,\beta\in\Delta_{2}$.
Thus, for any two such roots $[G_{\alpha},G_{\beta}]=1$. Let $L_{1}$ be the
standard Levi subgroup $T\prod_{\alpha\in\Delta_{1}}G_{\alpha}$ and let
$Q_{1}$ be the unipotent radical of the parabolic subgroup $L_{1}U$.

For $i\in\\{1,\dots,6\\}$, since $C_{i}$ is $U$-regular, it is represented by
an element $s_{i}u_{i}$ with $s_{i}\in T$ and $u_{i}\in U$ regular. Moreover,
$u_{i}=q_{i}v_{i}$ with $v_{i}=\prod_{\alpha\in\Delta_{1}}v_{i,\alpha}$ for
some $q_{i}\in Q$ and $v_{i,\alpha}\in U_{\alpha}=G_{\alpha}\cap U$.
Conjugating by $L$, and since the simple factors of $L$ commute, we may choose
the elements $v_{i,\alpha}$ to be any regular unipotent element in
$G_{\alpha}$. The product $C_{1}C_{2}C_{3}$ contains

$(s_{1}q_{1}v_{1})(s_{2}q_{2}v_{2})(s_{3}q_{3}v_{3})=s_{1}s_{2}s_{3}q_{1}^{s_{2}s_{3}}q_{2}^{(v_{1}^{s_{2}})^{-1}s_{3}}q_{3}^{(v_{1}^{s_{2}s_{3}})^{-1}(v_{2}^{s_{3}})^{-1}}v_{1}^{s_{2}s_{3}}v_{2}^{s_{3}}v_{3}.$

Since the set of non-trivial unipotent elements in $G_{\alpha}$ is stable by
$T$-conjugation the product $v_{1}^{s_{2}s_{3}}v_{2}^{s_{3}}v_{3}$ ranges over
all products of tripples of unipotent elements in
$\prod_{\alpha\in\Delta_{1}}G_{\alpha}$. By Corollary 5, and since
$[G_{\alpha},G_{\beta}]=1$ for distinct roots $\alpha,\beta\in\Delta$, there
are $v_{1}$, $v_{2}$ and $v_{3}$ such that
$b=v_{1}^{s_{2}s_{3}}v_{2}^{s_{3}}v_{3}$ for any element
$b\in\prod_{\alpha\in\Delta_{1}}G_{\alpha}$. Since $Q_{1}$ is stable under
conjugation by $L$, the element
$q_{1}^{s_{2}s_{3}}q_{2}^{(v_{1}^{s_{2}})^{-1}s_{3}}q_{3}^{(v_{1}^{s_{2}s_{3}})^{-1}(v_{2}^{s_{3}})^{-1}}$
lies in $Q_{1}$. In particular, for each
$t\in\prod_{\alpha\in\Delta_{1}}(T\cap G_{\alpha})$ there is $q_{t}\in Q_{1}$
such that

$s_{1}s_{2}s_{3}tq_{t}\in C_{1}C_{2}C_{3}.$

Similarly, for each $t\in\prod_{\alpha\in\Delta_{2}}(T\cap G_{\alpha})$ there
is $q_{t}\in U$ such that

$s_{4}s_{5}s_{6}tq_{t}\in C_{4}C_{5}C_{6}.$

Since $T=\prod_{\alpha\in\Delta}(T\cap G_{\alpha})$, it follows that for any
$t\in T$ the normal subset $C_{1}\cdots C_{6}$ contains $tq_{t}$ for some
$q_{t}\in U$. In particular, if $t\in T=s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}T$ is
regular, then $tq_{t}$ is conjugate to $t$, hence $C_{1}\cdots C_{6}$ contains
all regular semisimple elements of $T$. ∎

##  5\. Marked diagrams and products of conjugacy classes

######  Lemma 18.

Let $\Delta^{\prime}$ be a subset of $\Delta$. If $u_{1},u_{2}\in U$ are such
that $\operatorname{supp}(u_{1})=\operatorname{supp}(u_{2})=\Delta^{\prime}$
then

$u_{1}^{T}u_{2}^{T}[U,U]=\left(\prod_{\alpha\in\Delta^{\prime}}U_{\alpha}\right)[U,U].$

###### Proof.

The sets $u_{1}^{T}u_{2}^{T}$ and $\prod_{\alpha\in\Delta^{\prime}}U_{\alpha}$
lie in the subsystem subgroup $[L_{\Delta^{\prime}},L_{\Delta^{\prime}}]$. As
in the proof of Lemma 8, it suffices to prove the claim for
$\Delta^{\prime}=\Delta$. Again, as in the proof of Lemma 8, $u_{i}^{T}[U,U]$
is open in $U$. Since the product of two open sets in a connected algebraic
group equals the whole group,

$\displaystyle U=(u_{1}^{T}[U,U])(u_{2}^{T}[U,U])=u_{1}^{T}u_{2}^{T}[U,U].$

######  Proposition 19.

Let $C_{1}$ and $C_{2}$ be two conjugacy classes of $G$ represented by $g_{1}$
and $g_{2}$ respectively. If $g_{1},g_{2}\in B$, then

${\mathcal{D}}(g_{1})\boxplus{\mathcal{D}}(g_{2})\in{\mathcal{D}}(C_{1}C_{2}).$

###### Proof.

For $i=1,2$ let $g_{i}=s_{i}u_{i}$ with $s_{i}\in T$ and $u_{i}\in U$ be the
factorization of the element $g_{i}$ in $B=TU$. Note that $C_{i}$ contains
$(s_{i}u_{i})^{T}=s_{i}(u_{i})^{T}$. Let $u_{i}^{1}$ be the projection of
$u_{i}$ on $\prod_{\alpha\in\Delta}U_{\alpha}$ and $u_{i}^{0}\in[U,U]$ be such
that $u_{i}=u_{i}^{1}u_{i}^{0}$. Then $C_{i}$ contains
$s_{i}(u_{i}^{1})^{T}(u_{i}^{0})^{T}$ hence, by the commutator relations, the
product $C_{1}C_{2}$ contains
$s_{1}(u_{1}^{1})^{T}(u_{1}^{0})^{T}s_{2}(u_{2}^{1})^{T}(u_{2}^{0})^{T}$ which
lies in $s_{1}s_{2}(u_{1}^{1})^{T}(u_{2}^{1})^{T}[U,U]$.

Let $\Delta_{i}=\operatorname{supp}(u_{i}^{1})$, let $v_{i}$ be the projection
of $u_{i}^{1}$ on
$\prod_{\alpha\in\Delta-(\Delta_{1}\cap\Delta_{2})}U_{\alpha}$ and let $w_{i}$
be the projection of $u_{i}^{1}$ on
$\prod_{\alpha\in\Delta_{1}\cap\Delta_{2}}U_{\alpha}$. We have

$s_{1}s_{2}(u_{1}^{1})^{T}(u_{2}^{1})^{T}[U,U]=s_{1}s_{2}(v_{1})^{T}(w_{1})^{T}(w_{2})^{T}(v_{2})^{T}[U,U]$

and, by Lemma 18, this further equals

$s_{1}s_{2}(u_{1}^{1})^{T}(u_{2}^{1})^{T}[U,U]=s_{1}s_{2}(v_{1})^{T}\left(\prod_{\alpha\in\Delta_{1}\cap\Delta_{2}}U_{\alpha}\right)(v_{2})^{T}[U,U].$

Since $\operatorname{supp}(v_{1})$, $\operatorname{supp}(v_{2})$ and
$\operatorname{supp}(w_{1})=\operatorname{supp}(w_{2})$ are pairwise disjoint,
the above set contains an element with non-trivial projections on $U_{\alpha}$
exactly when $\alpha\in\Delta_{1}\cup\Delta_{2}$, i.e. it contains an element
with diagram ${\mathcal{D}}(g_{1})\boxplus{\mathcal{D}}(g_{2})$. ∎

######  Corollary 20.

Let $N_{1},\dots,N_{k}$ be normal subsets of $G$. If
$\sum_{i=1}^{k}{\mathcal{D}}(N_{i})$ contains a regular diagram of
${\mathbb{N}}\Delta$, then the normal subset $N_{1}\cdots N_{k}$ is
$U$-regular.

###### Proof.

Let $D_{i}\in{\mathcal{D}}(N_{i})$ be diagrams such that
$D=\sum_{i=1}^{k}D_{i}$ is a regular diagram. By Lemma 6, $D$ is regular if
and only if $\psi(D)=\boxplus_{i=1}^{k}\psi(D_{i})$ is regular. Let
$C_{i}\subseteq N_{i}$ be conjugacy classes such that
$\psi(D_{i})\in{\mathcal{D}}(C_{i})$. By Proposition 19,
$\psi(D_{1})\boxplus\psi(D_{2})$ is a diagram in ${\mathcal{D}}(C_{1}C_{2})$.
By induction ${\mathcal{D}}(C_{1}\cdots C_{k})$ contains $\psi(D)$. By
Proposition 14, the normal subset $\prod_{i\in I_{j}}C_{i}$ is $U$-regular. ∎

###### Proof of Proposition A.

As before $r=\operatorname{rk}(G)$. Let $D_{i}\in{\mathcal{D}}(N_{i})$ be
diagrams such that $\sum_{i=1}^{k}D_{i}\geq 12r{\mathcal{D}}^{\circ}$. Let
$C_{i}\subseteq N_{i}$ be a conjugacy class such that
$D_{i}\in{\mathcal{D}}(C_{i})$. It is enough to show that
$\prod_{i=1}^{k}C_{i}=G$. By Lemma 7, there is a partition of
$\\{1,\dots,k\\}$ into subsets $I_{1},\dots,I_{12}$ such that $\sum_{i\in
I_{j}}D_{i}\geq{\mathcal{D}}^{\circ}$ for all $1\leq j\leq 12$. By Corollary
20, the normal subset $\prod_{i\in I_{j}}C_{i}$ is $U$-regular, i.e. it
contains a $U$-regular conjugacy class $\tilde{C}_{j}$. By Proposition 17, the
product $\tilde{C}_{1}\cdots\tilde{C}_{12}$ contains $S^{2}$ where $S$ is the
union of all regular semisimple conjugacy classes. Since $S$ is open in $G$
[27, §3.5 Corollary], we have $S^{2}=G$. ∎

###### Proof of Proposition B.

Let $D_{i}\in{\mathcal{D}}(N_{i})$ be diagrams such that $\\{D_{i}:1\leq 1\leq
k\\}$ is of type $(r,6)$ with respect to $D_{i_{1}},\dots,D_{i_{r}}$ and such
that $\sum_{j=1}^{r}D_{i_{j}}\geq{\mathcal{D}}^{\circ}$. Let $C_{i}\subseteq
N_{i}$ be a conjugacy class such that $D_{i}\in{\mathcal{D}}(C_{i})$.
Eventually after reindexing, we may assume that $N_{1}\cdots N_{k}$ contains

$\overbrace{\underbrace{C_{1}C_{2}\cdots C_{6}}_{D_{1}\leq
D_{2},\dots,D_{6}}}^{\tilde{N}_{1}}\overbrace{\underbrace{C_{7}C_{8}\cdots
C_{12}}_{D_{7}\leq
D_{8},\dots,D_{12}}}^{\tilde{N}_{7}}\cdots\overbrace{\underbrace{C_{k}C_{k+1}\cdots
C_{k+6}}_{D_{k}\leq D_{k+1},\dots,D_{k+6}}}^{\tilde{N}_{k}}\cdots$

where $D_{6j+1}=D_{i_{j}}$. Let $\Delta_{j}=\operatorname{supp}(D_{i_{j}})$.
As in the proof of Proposition 17, $\tilde{N}_{i}$ contains the torus
$T_{j}=T\cap G(\Delta_{j})$. Thus, the product
$\tilde{N}_{1}\cdots\tilde{N}_{k}$ contains $T_{1}\cdots T_{k}$ and since
$\sum_{j=1}^{r}D_{i_{j}}\geq{\mathcal{D}}^{\circ}$, the torus $T_{1}\cdots
T_{k}$ is a maximal torus. Hence $\tilde{N}_{1}\cdots\tilde{N}_{k}$ contains
an open set of semisimple elements.∎

In what follows we use the standard numering of the simple roots in $\Delta$
\- see for instance [16, p.10] or [20, p.67].

######  Lemma 21.

Let $G$ be of type $A_{r}$. If $\Delta=\\{\alpha_{1},\dots,\alpha_{r}\\}$ are
the simple roots ordered as above then the torus $\prod_{i=1}^{\lceil
r/2\rceil}\alpha_{2i-1}^{\vee}(k^{\times})$ is regular.

###### Proof.

Let $\tilde{T}=\prod_{i=1}^{\lceil r/2\rceil}\alpha_{2i-1}^{\vee}(k^{\times})$
and let $\tilde{\Delta}=\\{\alpha_{1},\alpha_{3},\dots\alpha_{2\lceil
r/2\rceil-1}\\}$. Since $\tilde{T}\subseteq T$, by [26, II §4.1] we have
$C_{G}(\tilde{T})^{\circ}=\langle T,U_{\alpha}:\alpha(t)=1\text{ for all
}t\in\tilde{T}\rangle$. For a root $\alpha\in\Phi$ we have
$\alpha(t)=t^{\langle\alpha,\beta\rangle}$ for any $t\in k^{\times}$ as in
(1). Hence, $\alpha(t)=1$ for all $t\in\tilde{T}$ if and only if
$\langle\alpha,\beta\rangle=0$ for all $\beta\in\tilde{\Delta}$, i.e. if and
only if $\alpha$ is orthogonal to $\tilde{\Delta}$. One checks, for example
with [13, §2.10], that no root $\alpha$ is orthogonal to $\tilde{\Delta}$.
Hence $C_{G}(\tilde{T})^{\circ}=T$, i.e. $\tilde{T}$ is a regular torus. ∎

######  Lemma 22.

Let $\tilde{T}\subseteq T$ be a regular torus of $G$. For any $\tilde{z}\in
T$, the set $\tilde{z}\tilde{T}$ contains an open subset of regular elements.

###### Proof.

Fix $\alpha\in\Phi$. Since $\tilde{T}$ and $\tilde{z}\tilde{T}$ lie in $T$,
the root subgroup $U_{\alpha}$ is stable by conjugation under the elements of
these closed subsets. Since $\tilde{T}$ is a regular torus, it has an open
orbit on $U_{\alpha}$. Thus, restricting the conjugation map to
$\tilde{z}\tilde{T}$ one also obtains an open orbit on $U_{\alpha}$. Hence
$\dim(C_{G}(U_{\alpha})\cap\tilde{z}\tilde{T})\lneq\dim(z\tilde{T})$, i.e.
$z\tilde{T}\setminus C_{G}(U_{\alpha})$ is an open subset of
$\tilde{z}\tilde{T}$. Consider
$V=\tilde{z}\tilde{T}\setminus\cup_{\alpha\in\Phi}C_{G}(U_{\alpha})=\cap_{\alpha\in\Phi}\tilde{z}\tilde{T}\setminus
C_{G}(U_{\alpha})$. Since $|\Phi|$ is finite, $V$ is an open subset of
$\tilde{z}\tilde{T}$. Since the ground field is algebraically closed, $V$ is
non-empty. For any $t\in V$ and any root subgroup $U_{\alpha}$ we have
$[t,U_{\alpha}]\neq 1$, thus, by [26, II §4.1], all elements of $V$ are
regular. ∎

######  Lemma 23.

Let $G$ be of type $A_{r}$. Let $\tilde{\Delta}=\\{\alpha_{2i-1}:1\leq
i\leq\lceil r/2\rceil\\}$ and let $\tilde{T}=\prod_{i=1}^{\lceil
r/2\rceil}\alpha_{2i-1}^{\vee}(k^{\times})$. Let $N_{1},\dots,N_{k}$ be normal
subsets of $G$ and let $D_{i}\in{\mathcal{D}}(N_{i})$ for all $1\leq i\leq k$.
If $\sum_{i=1}^{k}D_{i}\geq 2{\mathcal{D}}^{\circ}$, then
$\prod_{i=1}^{k}N_{i}$ contains an open subset of $\tilde{T}\tilde{z}$ for
some $\tilde{z}\in T$. In particular, Theorem C holds for $G$ of type $A_{r}$.

###### Proof.

Let $D_{i}\in{\mathcal{D}}(N_{i})$ be diagrams such that
$\sum_{i=1}^{k}D_{i}\geq 2{\mathcal{D}}^{\circ}$. Let $C_{i}\subseteq N_{i}$
be a conjugacy class such that $D_{i}\in{\mathcal{D}}(C_{i})$. It is enough to
show that $\prod_{i=1}^{k}C_{i}$ contains a regular semisimple element. Let
$g_{i}\in C_{i}$ be such that ${\mathcal{D}}(g_{i})=D_{i}$.

Notice that $\tilde{T}$ is a maximal torus of the derived group
$[L_{\tilde{\Delta}},L_{\tilde{\Delta}}]$ of the Levi subgroup
$L_{\tilde{\Delta}}$ of type $A_{1}^{\lceil r/2\rceil}$. Thus, the
representatives factor as $g_{i}=l_{i}q_{i}$ for some $l_{i}\in L$ and
$q_{i}\in Q$. Moreover $l_{i}=(\prod_{j=1}^{\lceil r/2\rceil}l_{i,j})z_{i}$
for some $l_{i,j}\in G_{\alpha_{j}}\cap B$ and $z_{i}\in Z(L)$. For $x\in L$
we have

$\prod_{i=1}^{k}g_{i}^{x}=\prod_{i=1}^{k}(\prod_{j=1}^{\lceil
r/2\rceil}l_{i,j}^{x})(z_{i}q_{i})^{x}=\left[\prod_{i=1}^{k}\prod_{j=1}^{\lceil
r/2\rceil}l_{i,j}^{x}\right]\tilde{z}\tilde{q}=\left[\prod_{j=1}^{\lceil
r/2\rceil}\prod_{i=1}^{k}l_{i,j}^{x}\right]\tilde{z}\tilde{q}$

for some $\tilde{z}\in Z(L)$ and $\tilde{q}\in Q$. Since
$\sum_{i=1}^{k}D_{i}\geq 2{\mathcal{D}}^{\circ}$, for each $j$ there are at
least $2$ elements in $\\{l_{i,j}:1\leq i\leq\lceil r/2\rceil\\}$ which are
not central in $G_{\alpha_{j}}$. Then $\prod_{i=1}^{k}l_{i,j}^{L}$ contains an
open subset of $G_{\alpha_{2i-1}}\subseteq[L,L]$ by [8, Theorem 2]. Thus
$\\{\prod_{j=1}^{\lceil r/2\rceil}\prod_{i=1}^{k}l_{i,j}^{x}:x\in L\\}$
contains an open subset of $\tilde{T}$. Notice that, since $z_{i}\in Z(L)$,
$\tilde{z}$ depends only on the choice of representatives $g_{i}$ and not on
$x\in L$. By Lemma 22, there is an open subsets of regular (semisimple)
elements $V\subseteq\tilde{T}\tilde{z}$. Thus, for any $\tilde{t}\tilde{z}\in
V$, $\tilde{t}\tilde{z}\tilde{q}$ is conjugate to $\tilde{t}\tilde{z}$. Hence,
the normal subset $\prod_{i=1}^{k}N_{i}$ contains $V$. Since the ground field
is algebraically closed, $V$ is non-empty. ∎

###### Proof of Theorem C.

Let $D_{i}\in{\mathcal{D}}(N_{i})$ be diagrams such that
$\sum_{i=1}^{k}D_{i}\geq 16{\mathcal{D}}^{\circ}$. Let $C_{i}\subseteq N_{i}$
be a conjugacy class such that $D_{i}\in{\mathcal{D}}(C_{i})$. It is enough to
show that $\prod_{i=1}^{k}C_{i}$ contains a regular semisimple element. Let
$g_{i}\in C_{i}$ be such that ${\mathcal{D}}(g_{i})=D_{i}$. Notice that $k\geq
16$ with equality if and only if all $D_{i}$ are regular. If $k=16$ the claim
follows from Proposition 17. Let $k\geq 17$. If $\operatorname{rk}(G)\leq 8$
then $\prod_{i=1}^{k}C_{i}$ contains an open subset of $G$ by [8, Theorem 2]
and the claim follows.

Suppose $r=\operatorname{rk}(G)>8$ and let
$\Delta_{1}=\\{\alpha_{1},\dots,\alpha_{r-4}\\}$ and
$\Delta_{2}=\\{\alpha_{r-2},\alpha_{r-1},\alpha_{r}\\}$. Consider the
parabolic subgroup $P_{\Delta_{1}\cup\Delta_{2}}$ with Levi factor $L$ and
unipotent radical $Q$. Then $g_{i}=l_{i}q_{i}$ for some $l_{i}\in L$ and
$q_{i}\in Q$. Moreover, $L$ is a connected reductive group with a simple
factor $H_{1}$ of type $A_{r-4}$ corresponding to $\Delta_{1}$ and a simple
factor $H_{2}$ corresponding to $\Delta_{2}$. Thus

$g_{i}=g_{i,1}g_{i,2}z_{i}q_{i}$

for some $g_{i,1}\in H_{1}$, $g_{i,2}\in H_{2}$ and $z_{i}\in Z(L)$. For $x\in
L$ we have

$\prod_{i=1}^{k}g_{i}^{x}=\prod_{i=1}^{k}g_{i,1}^{x}g_{i,2}^{x}(z_{i}q_{i})^{x}=\left[\prod_{i=1}^{k}g_{i,1}^{x}g_{i,2}^{x}\right]\tilde{z}\tilde{q}=\left[\prod_{i=1}^{k}g_{i,1}^{x}\right]\left[\prod_{i=1}^{k}g_{i,2}^{x}\right]\tilde{z}\tilde{q}$

for some $\tilde{q}\in Q$ and $\tilde{z}\in Z(L)$. Notice that, since
$z_{i}\in Z(L)$, $\tilde{z}$ depends only on the choice of representatives
$g_{i}$ and not on $x\in L$.

Let $\Delta_{1}^{\prime}=\\{\alpha_{2i-1}:1\leq i\leq\lceil r/2-2\rceil\\}$,
let $\tilde{T}=\prod_{i=1}^{\lceil
r/2-2\rceil}\alpha_{2i-1}^{\vee}(k^{\times})$ and let $T_{\Delta_{2}}=T\cap
H_{2}$. Since $\sum_{i=1}^{k}D_{i}\geq 16{\mathcal{D}}^{\circ}$, the elements
$g_{i,1}$ and $g_{i,2}$ are non-central in $H_{1}$ and $H_{2}$ respectively.
Moreover $k\geq 17$. Since $H_{1}$ and $H_{2}$ commute, by Lemma 21 and [8,
Theorem 2], $\prod_{i=1}^{k}C_{i}$ contains $\tilde{T}T_{\Delta_{2}}$ up to
multiplication on the right by elements in $\tilde{z}Q$.

We claim that the torus $\tilde{T}T_{\Delta_{2}}$ is regular. If this is the
case then, by Lemma 22, there is a regular element
$\tilde{t}\tilde{z}\in\tilde{T}T_{\Delta_{2}}\tilde{z}$, thus, all elements in
$\tilde{t}\tilde{z}Q$ are regular semisimple and the proof is finished.

For the proof of the claim, let
$\Delta^{\perp}=\\{\alpha\in\Phi:\alpha\perp\beta\text{ for all
}\beta\in\Delta_{1}^{\prime}\cup\Delta_{2}\\}$. We have
$C_{G}(\tilde{T}T_{\Delta_{2}})^{\circ}=\langle
T,U_{\alpha}:\alpha\in\Delta^{\perp}\rangle$ and need to show that
$\Delta^{\perp}=\emptyset$. For this we use the explicit construction of root
systems as described in [13, §2.10]. If $G$ is of type $A$, let
$\alpha=\varepsilon_{i}-\varepsilon_{j}$ ($1\leq i\neq j\leq r+1$) then either
$\beta=\varepsilon_{i-1}-\varepsilon_{i}$ or
$\beta=\varepsilon_{i}-\varepsilon_{i+1}$ is an element of
$\Delta_{1}^{\prime}\cup\Delta_{2}$, hence $\Delta^{\perp}=\emptyset$. If $G$
is of type $B$ and $\alpha$ is the short root $\pm\varepsilon_{i}$ ($1\leq
i\leq r$) then either $\beta=\varepsilon_{i-1}-\varepsilon_{i}$ or
$\beta=\varepsilon_{i}-\varepsilon_{i+1}$ or $\beta=\varepsilon_{i}$ is an
element of $\Delta_{1}^{\prime}\cup\Delta_{2}$, hence
$\alpha\not\perp\Delta_{1}^{\prime}\cup\Delta_{2}$. If $\alpha$ is the long
root $\pm\varepsilon_{i}\pm\varepsilon_{j}$ ($1\leq i<j\leq r$), again
$\beta=\varepsilon_{i-1}-\varepsilon_{i}$ or
$\beta=\varepsilon_{i}-\varepsilon_{i+1}$ or $\beta=\varepsilon_{i}$ is an
element of $\Delta_{1}^{\prime}\cup\Delta_{2}$, hence
$\alpha\not\perp\Delta_{1}^{\prime}\cup\Delta_{2}$. If $G$ is of type $C$ or
$D$, the argument is similar and the claim follows. ∎

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