# Strong integrality of inversion subgroups of
Kac–Moody groups

Abid Ali  Department of Mathematics, Rutgers University, Piscataway, NJ
08854-8019, U.S.A<EMAIL_ADDRESS>,  Lisa Carbone  Department of
Mathematics, Rutgers University, Piscataway, NJ 08854-8019, U.S.A.
<EMAIL_ADDRESS>,  Dongwen Liu  School of Mathematical Sciences,
Zhejiang University, Hangzhou 310027, P.R. China<EMAIL_ADDRESS>and  Scott
H. Murray  Department of Mathematics, Rutgers University, Piscataway, NJ
08854-8019, U.S.A<EMAIL_ADDRESS>
###### Abstract.

Let $A$ be a symmetrizable generalized Cartan matrix with corresponding
Kac–Moody algebra $\mathfrak{g}$ over ${\mathbb{Q}}$. Let $V=V^{\lambda}$ be
an integrable highest weight $\mathfrak{g}$-module and let
$V_{\mathbb{Z}}=V^{\lambda}_{\mathbb{Z}}$ be a ${\mathbb{Z}}$-form of $V$. Let
$G$ be an associated minimal representation-theoretic Kac–Moody group and let
$G({\mathbb{Z}})$ be its integral subgroup. Let $\Gamma({\mathbb{Z}})$ be the
Chevalley subgroup of $G$, that is, the subgroup that stabilizes the lattice
$V_{{\mathbb{Z}}}$ in $V$. For a subgroup $M$ of $G$, we say that $M$ is
integral if $M\cap G({\mathbb{Z}})=M\cap\Gamma({\mathbb{Z}})$ and that $M$ is
strongly integral if there exists $v\in V^{\lambda}_{\mathbb{Z}}$ such that,
for all $g\in M$, $g\cdot v\in V_{\mathbb{Z}}$ implies $g\in G({\mathbb{Z}})$.
We prove strong integrality of inversion subgroups $U_{(w)}$ of $G$ where, for
$w\in W$, $U_{(w)}$ is the group generated by positive real root groups that
are flipped to negative roots by $w^{-1}$. We use this to prove strong
integrality of subgroups of the unipotent subgroup $U$ of $G$ generated by
commuting real root groups. When $A$ has rank 2, this gives strong integrality
of subgroups $U_{1}$ and $U_{2}$ where $U=U_{1}{\Large{*}}\ U_{2}$ and each
$U_{i}$ is generated by ‘half’ the positive real roots.

2000 Mathematics subject classification. Primary 20G44, 81R10; Secondary
22F50, 17B67

The second author’s research is partially supported by the Simons Foundation,
Mathematics and Physical Sciences-Collaboration Grants for Mathematicians,
Award Number 422182. The third author’s research is partially supported by NSF
of Zhejiang Province LZ22A010006 and NSFC 12171421.

##  1\. Introduction

This paper concerns the generalization to infinite-dimensional symmetrizable
Kac–Moody algebras of the construction of Chevalley groups associated to
finite-dimensional semisimple Lie algebras.

Let $A$ be a symmetrizable generalized Cartan matrix. Let $\mathfrak{g}$ be a
Kac–Moody algebra over ${\mathbb{Q}}$ associated with $A$. We denote simple
roots by $\alpha_{1},\dots,\alpha_{\ell}$, the set of roots by $\Delta$, the
set of positive (respectively negative) roots by $\Delta_{\pm}$ , and the set
of real roots by $\Delta^{\operatorname{re}}$. If $A$ is not of finite type,
then $\mathfrak{g}$ is infinite dimensional. Let $V=V^{\lambda}$ be an
integrable highest-weight $\mathfrak{g}$-module with highest weight $\lambda$,
highest weight vector $v_{\lambda}$, and defining homomorphism
$\rho:\mathfrak{g}\to\operatorname{End}(V^{\lambda})$. Let
$\chi_{\pm\alpha_{i}}(t)\in\operatorname{Aut}(V)$ denote $\exp(t\rho(e_{i}))$
(resp. $\exp(t\rho(f_{i}))$), for each $i\in\\{1,\dots,\ell\\}$, where
$t\in{\mathbb{Q}}$ and $e_{i}$ (resp. $f_{i}$) is the Chevalley–Serre
generator corresponding to the simple root $\pm\alpha_{i}$. A (split) _minimal
representation-theoretic Kac–Moody group_ is

$G({\mathbb{Q}})=G^{\lambda}({\mathbb{Q}})=\langle\chi_{\alpha_{i}}(t),\
\chi_{-\alpha_{i}}(t)\mid i\in\\{1,\dots,\ell\\},\ t\in{\mathbb{Q}}\rangle.$

We let ${\mathcal{U}}_{{\mathbb{Z}}}\subseteq{\mathcal{U}}$ denote the
${\mathbb{Z}}$-form of the universal enveloping algebra of $\mathfrak{g}$
([Tit87]; see also Section 3 below). As in [CG03], we set
$V_{\mathbb{Z}}=\mathcal{U}_{\mathbb{Z}}\cdot v_{\lambda}.$ The subgroup of
$G({\mathbb{Q}})$ generated over ${\mathbb{Z}}$ is

$\displaystyle G({\mathbb{Z}})$ $\displaystyle=\langle\chi_{\alpha_{i}}(t),\
\chi_{-\alpha_{i}}(t)\mid i\in\\{1,\dots,\ell\\},\ t\in{\mathbb{Z}}\rangle$

and the _Chevalley subgroup_ of $G({\mathbb{Q}})$ is the subgroup the
stabilizes the integral lattice $V_{\mathbb{Z}}$:

$\Gamma({\mathbb{Z}})=\\{g\in G({\mathbb{Q}})\mid g\cdot
V_{\mathbb{Z}}=V_{\mathbb{Z}}\\}.$

This work concerns the question of integrality of $G({\mathbb{Q}})$. By this
we mean:

‘Is $G({\mathbb{Z}})=\Gamma({\mathbb{Z}})$?’

It is easy to show that $G({\mathbb{Z}})\subseteq\Gamma({\mathbb{Z}})$. Thus
it remains to determine if $\Gamma({\mathbb{Z}})\subseteq G({\mathbb{Z}})$.

The answer to this question has not yet been established for any family of
Kac–Moody groups. Here we consider integrality of certain subgroups of
$G({\mathbb{Q}})$. We call a subgroup $M$ of $G({\mathbb{Q}})$ _integral_ if
$M\cap G({\mathbb{Z}})=M\cap\Gamma({\mathbb{Z}})$. This is equivalent to
showing that for all $g\in M$, $g\cdot V_{\mathbb{Z}}=V_{\mathbb{Z}}$ implies
that $g\in G({\mathbb{Z}})$. Note that $M_{1}\subseteq M_{2}$ and $M_{2}$
integral implies that $M_{1}$ is integral. However $M_{1}$ and $M_{2}$
integral _does not_ imply that the subgroup $\langle M_{1},M_{2}\rangle$
generated by $M_{1}$ and $M_{2}$ is integral.

In the finite dimensional case, the property of integrality for semi-simple
algebraic groups $G({\mathbb{Q}})$, with root system $\Delta$ and Lie algebra
$\mathfrak{g}$, was established in [Che95]. Chevalley constructed (what was
his notion of) an affine group scheme associated to $G({\mathbb{Q}})$ and
$V_{\mathbb{Z}}$ and showed that its coordinate algebra is generated over
${\mathbb{Z}}$. This was a starting point for Grothendieck and Demazure who
organized the SGA3 seminar ([DG70]) where they developed the theory of split
reductive group schemes over arbitrary base schemes. Demazure and
Grothendieck’s theory assumed known the classification of reductive groups
over an algebraically closed field due primarily to Chevalley, which also
holds over ${\mathbb{Z}}$.

Let $L$ be a lattice satisfying $Q\subseteq L\subseteq P$ where $Q$ is the
root lattice and $P$ is the weight lattice of $\mathfrak{g}$. Chevalley showed
that ($\Delta$, $L$) defines an affine group scheme $G_{\mathbb{Z}}$ over
${\mathbb{Z}}$ such that

$\Gamma({\mathbb{Z}})=G_{\mathbb{Z}}({\mathbb{Z}})=\operatorname{Hom}(\operatorname{Spec}({\mathbb{Z}}),G_{{\mathbb{Z}}})$

is its set of integral points ([Che95]). The group
$G_{{\mathbb{Z}}}({\mathbb{Q}})\cong G({\mathbb{Q}})$ is now known as a
Chevalley group. If $L=Q$, then $G_{{\mathbb{Z}}}({\mathbb{Q}})$ is called
adjoint and if $L=P$, then $G_{\mathbb{Z}}({\mathbb{Q}})$ is called simply
connected. It follows that the group $\Gamma({\mathbb{Z}})$ is generated by
the automorphisms

$\\{\chi_{\alpha_{i}}(t),\chi_{-\alpha_{i}}(t)\mid\ t\in{\mathbb{Z}}\\},$

so if $G_{\mathbb{Z}}({\mathbb{Q}})$ is simply connected, it is integral.

To our knowledge, there is no proof in the literature that
$G({\mathbb{Z}})=\Gamma({\mathbb{Z}})$ for semi-simple linear algebraic groups
$G({\mathbb{Q}})$ that doesn’t use the language of group schemes. In certain
cases, it is straightforward to verify integrality using properties of the
linear algebraic group and underlying representation. For example, it is easy
to see that $\operatorname{SL}_{n}({\mathbb{Z}})$ is the stabilizer of the
standard lattice ${\mathbb{Z}}^{n}$ in ${\mathbb{Q}}^{n}$. See also [Sou07],
where Soulé showed that the group $\mathbb{E}_{7}({\mathbb{Z}})$ coincides
with $\mathbb{E}_{7}({\mathbb{R}})\cap\operatorname{Sp}_{56}({\mathbb{Z}})$
where $\mathbb{E}_{7}({\mathbb{R}})$ is the non-compact real form and
$\operatorname{Sp}_{56}({\mathbb{Z}})$ is the stabilizer of the standard
lattice (and the canonical symplectic form) in the fundamental representation
of dimension 56 of the Lie algebra $\mathfrak{e}_{7}$.

The methods used to prove integrality in the finite dimensional case do not
extend to Kac–Moody groups $G({\mathbb{Q}})$. Here we address the question of
integrality of subgroups of $G({\mathbb{Q}})$ using only the definition of
$G({\mathbb{Q}})$ acting on the integrable highest weight module $V$. We
introduce the following slight modification of the notion of integrality.

######  Definition 1.1.

Let $M$ be a subgroup of the Kac–Moody group $G({\mathbb{Q}})$ and let
$v_{0}\in V_{\mathbb{Z}}$. We say that $M$ is _strongly integral with respect
to $v_{0}\in V_{\mathbb{Z}}$_ if, for all $g\in M$, $g\cdot v_{0}\in
V_{\mathbb{Z}}$ implies that $g\in G({\mathbb{Z}})$.

This is stronger than integrality since it only requires us to test the action
on a single vector $v_{0}$, rather than on the entire set $V_{\mathbb{Z}}$. It
is easy to see that strong integrality implies integrality (Lemma 4.1).

The (positive) unipotent subgroup of $G({\mathbb{Q}})$ is

$U=U({\mathbb{Q}})=\langle\chi_{\alpha_{i}}(t)\mid i\in\\{1,\dots,\ell\\},\
t\in{\mathbb{Q}}\rangle.$

Unipotent subgroups play an important role in the study of the structure and
representation theory of Kac–Moody groups. They are constituents of group
decompositions such as Iwasawa and Birkhoff decompositions, which provide
important tools for studying Kac–Moody groups and their applications.

In this paper, as a first step towards proving integrality of $U$, we prove
strong integrality of inversion subgroups of $U$, which are defined as
follows. For $w$ in the Weyl group $W$, the inversion subgroup $U_{(w)}$ is
given as

$U_{(w)}=\langle U_{\beta}\mid\beta\in\Phi_{(w)}\rangle,$

where $\Phi_{(w)}=\\{\beta\in\Delta_{+}\mid w^{-1}\beta\in\Delta_{-}\\}$ and
$U_{\beta}=\\{\chi_{\beta}(t)\mid t\in{\mathbb{Q}}\\}$ for $\chi_{\beta}(t)$
as in Subsection 3.2.

We write $\widetilde{w}$ for the usual lift of $w$ to $G({\mathbb{Q}})$ (see
Subsection 3.2). Our main result is the following (Section 6).

######  Theorem 1.2.

(Strong integrality of inversion subgroups) Let $G({\mathbb{Q}})$ be as above.
Let $U$ be the positive unipotent subgroup of $G({\mathbb{Q}})$. For $w\in W$,
$U_{(w)}$ is strongly integral with respect to $\widetilde{w}v_{\lambda}$.

If the matrix $A$ is of finite type (see Subsection 2.1), then the group

$G({\mathbb{Q}})=G^{\lambda}({\mathbb{Q}})=\langle\chi_{\alpha_{i}}(t),\
\chi_{-\alpha_{i}}(t)\mid i\in\\{1,\dots,\ell\\},\ t\in{\mathbb{Q}}\rangle$

is a semi-simple linear algebraic group. Furthermore, if the set of weights of
$V=V^{\lambda}$ contains all the fundamental weights, then $G({\mathbb{Q}})$
coincides with the simply connected Chevalley group
$G_{\mathbb{Z}}({\mathbb{Q}})$ of type $A$.

Our methods give a proof of integrality of unipotent subgroups of simply
connected Chevalley groups $G_{\mathbb{Z}}({\mathbb{Q}})\cong
G({\mathbb{Q}})$, using only the action of $G({\mathbb{Q}})$ on $V$. As a
corollary of Theorem 1.2, we have the following (Subsection 7.1).

######  Corollary 1.3.

Suppose that the matrix $A$ has finite type. Suppose also that the set of
weights of $V$ contains all the fundamental weights. Then the positive
unipotent subgroup $U$ of $G({\mathbb{Q}})$ is integral.

Theorem 1.2 also yields integrality of other subgroups of $U$. In Subsection
7.2 we prove:

######  Theorem 1.4.

If $M$ is a subgroup of $U$ generated by commuting real root subgroups, then
$M$ is integral.

In Section 7.3, we consider the case where $G$ has rank 2. In this case
$U=U_{1}{\Large{*}}\,U_{2}$ where each $U_{i}$ is generated by ‘half’ the
positive real roots (see [CKMS21]).

######  Theorem 1.5.

Let $G({\mathbb{Q}})$ be of rank 2 with positive unipotent subgroup
$U=U_{1}{\Large{*}}\,U_{2}$ where $U_{i}$ are abelian if $A$ is symmetric, and
nilpotent of class 2 if $A$ is not symmetric. Then the groups $U_{i}$ are
integral.

Though our methods do not currently extend to a proof of integrality for $U$,
we conjecture111An earlier paper by a subset of the authors
(arXiv:1803.11204v2 [math.RT] Proposition 6.4) contains an error. that
integrality holds for $U$. It would then be straightforward to show that
$G({\mathbb{Z}})=\Gamma({\mathbb{Z}})$ in the symmetrizable Kac–Moody case.

The authors would like to thank Shrawan Kumar for his interest in this work
and for helpful discussions.

##  2\. Kac–Moody Algebras

In this section, we review some facts about Kac-Moody algebras and their
representations.

###  2.1. Generalized Cartan matrices

Let $I=\\{1,2,\dots,\ell\\}$ and let $A=(a_{ij})_{i,j\in I}$ be a generalized
Cartan matrix. That is,

$\displaystyle a_{ij}$ $\displaystyle\in{{\mathbb{Z}}};$ $\displaystyle
a_{ii}$ $\displaystyle=2;$ $\displaystyle a_{ij}$ $\displaystyle\leq 0,\text{
for }i\neq j;\text{ and}$ $\displaystyle a_{ij}$ $\displaystyle=0\iff
a_{ji}=0,$

for all $i,j\in I$. The matrix $A$ is _symmetrizable_ if there exist positive
rational numbers $q_{1},\dots,q_{\ell}$, such that the matrix
$\operatorname{diag}(q_{1},\dots,q_{\ell})A$ is symmetric. We say that $A$ is
of _finite type_ if $A$ is positive definite, and that $A$ is of affine type
if $A$ is positive semidefinite but not positive definite. If $A$ is not of
finite or affine type, we say that $A$ has _indefinite type_. In particular,
$A$ is of _hyperbolic type_ if $A$ is neither of finite nor affine type, but
every proper, indecomposable submatrix is either of finite or of affine type.

We assume throughout that $A$ is a symmetrizable generalized Cartan matrix.

###  2.2. Generators and relations for $\mathfrak{g}$

Let $\mathfrak{h}$ be a ${\mathbb{Q}}$-vector space of dimension
$2\ell-\operatorname{rank}(A)$, and let
$\langle\circ,\circ\rangle:\mathfrak{h}^{\ast}\times\mathfrak{h}\to{\mathbb{Q}}$
denote the natural nondegenerate bilinear pairing between $\mathfrak{h}$ and
its dual. Fix _simple roots_
$\Pi=\\{\alpha_{1},\dots,\alpha_{\ell}\\}\subseteq\mathfrak{h}^{\ast}$ and
_simple coroots_
$\Pi^{\vee}=\\{\alpha_{1}^{\vee},\dots,\alpha_{\ell}^{\vee}\\}\subseteq\mathfrak{h}$
such that $\Pi$ and $\Pi^{\vee}$ are linearly independent, and
$\langle\alpha_{j},\alpha_{i}^{\vee}\rangle=\alpha_{j}(\alpha_{i}^{\vee})=a_{ij}$.

As in [Moo67], [Moo68], and [Kac90, Theorem 9.11], the associated _Kac–Moody
algebra_ $\mathfrak{g}$ is the Lie algebra over ${\mathbb{Q}}$ with generating
set $\mathfrak{h}\cup\\{e_{i},f_{i}\mid i\in I\\}$ and defining relations:

$\displaystyle[h,h^{\prime}]$ $\displaystyle=0;$ $\displaystyle[h,e_{i}]$
$\displaystyle=\langle\alpha_{i},h\rangle e_{i};$ $\displaystyle[h,f_{i}]$
$\displaystyle=-\langle\alpha_{i},h\rangle f_{i};$
$\displaystyle[e_{i},f_{i}]$ $\displaystyle=\alpha_{i}^{\vee};$
$\displaystyle[e_{i},f_{j}]$ $\displaystyle=0;$
$\displaystyle(\operatorname{ad}e_{i})^{-a_{ij}+1}(e_{j})$ $\displaystyle=0;$
$\displaystyle(\operatorname{ad}f_{i})^{-a_{ij}+1}(f_{j})$ $\displaystyle=0;$

for $h,h^{\prime}\in\mathfrak{h}$, and $i,j\in I$ with $i\neq j$.

###  2.3. Roots and the Weyl group

The _roots_ of $\mathfrak{g}$ are the nonzero $\alpha\in\mathfrak{h}^{\ast}$
for which the corresponding root space

$\mathfrak{g}_{\alpha}:=\\{x\in\mathfrak{g}\mid[h,x]=\alpha(h)x\text{ for all
$h\in\mathfrak{h}$}\\}$

is nontrivial. The simple roots $\alpha_{i}$ have root spaces
$\mathfrak{g}_{\alpha_{i}}={\mathbb{Q}}e_{i}$. Every root $\alpha$ can be
written in the form $\alpha=\sum_{i=1}^{\ell}k_{i}\alpha_{i}$ where the
$k_{i}$ are integers, with either all $k_{i}\geq 0$, in which case $\alpha$ is
called positive, or all $k_{i}\leq 0$, in which case $\alpha$ is called
negative. We denote the set of roots by $\Delta$ and the set of positive
(resp. negative) roots by $\Delta_{+}$ (resp. $\Delta_{-}$). The _root
lattice_ is $Q:=\mathbb{Z}\Pi\subseteq\mathfrak{h}^{*}$ and the _coroot
lattice_ is $Q^{\vee}:=\mathbb{Z}\Pi^{\vee}\subseteq\mathfrak{h}$.

The Lie algebra $\mathfrak{g}$ has a triangular decomposition ([Kac90, Theorem
1.2])

$\mathfrak{g}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+},$

where

$\mathfrak{n}^{+}=\bigoplus_{\alpha\in\Delta_{+}}\mathfrak{g}_{\alpha}\quad\text{and}\quad\mathfrak{n}^{-}=\bigoplus_{\alpha\in\Delta_{-}}\mathfrak{g}_{\alpha}.$

Since $A$ is symmetrizable, $\mathfrak{g}$ admits a nondegenerate symmetric
invariant bilinear form $(\circ,\circ)$ satisfying

$\displaystyle\left(\alpha^{\vee}_{i},\alpha^{\vee}_{j}\right)$
$\displaystyle=a_{ij},$ $\displaystyle\left({e}_{i},{f}_{j}\right)$
$\displaystyle=\delta_{ij},$
$\displaystyle\left(\mathfrak{h},\mathfrak{n}_{+}\right)$ $\displaystyle=0,$
$\displaystyle\left(\mathfrak{h},\mathfrak{n}_{-}\right)$ $\displaystyle=0.$

We define the _simple reflection_
$w_{i}:\mathfrak{h}^{\ast}\to\mathfrak{h}^{\ast}$ by

$w_{i}(v)=v-\langle v,\alpha_{i}^{\vee}\rangle\alpha_{i}.$

The group $W\subseteq\operatorname{GL}(\mathfrak{h}^{\ast})$ generated by the
simple reflections is called the Weyl group. The induced bilinear form on
${\mathfrak{h}^{\ast}}$ is $W$-invariant. The group $W$ comes equipped with a
length function $\ell\colon W\longrightarrow\mathbb{Z}_{\geq 0}$ defined as
$\ell(w)=k$, where $k$ is the smallest number such that $w$ is the product of
$k$ simple root reflections. Such a minimal length word

$w=w_{i_{1}}w_{i_{2}}\cdots w_{i_{k}}$

is called a _reduced word_.

A root $\alpha\in\Delta$ is called _real_ if there exists $w\in W$ such that
$w\alpha$ is a simple root. A root $\alpha$ which is not real is called
_imaginary_. We denote by $\Delta^{\operatorname{re}}$ the set of real roots
and $\Delta^{\operatorname{im}}$ the set of imaginary roots. We have
$\Delta^{\operatorname{re}}=W\Pi$. Similarly define sets
$\Delta^{\operatorname{re}}_{\pm}:=\Delta^{\operatorname{re}}\cap\Delta_{\pm}$
and
$\Delta^{\operatorname{im}}_{\pm}:=\Delta^{\operatorname{im}}\cap\Delta_{\pm}$.
For $\alpha\in\Delta^{\operatorname{re}}$, the reflection in $\alpha$ is
defined by

$w_{\alpha}(v)=v-\langle v,\alpha^{\vee}\rangle\alpha,$

where $\alpha^{\vee}$ is the usual coroot of $\alpha$ defined using the
bilinear form $(\circ,\circ)$ on $\mathfrak{h}$. Since $w\alpha=\alpha_{i}$
for some $w\in W$, $i\in I$, we have $w_{\alpha}=ww_{i}w^{-1}\in W$.

###  2.4. Integrable highest weight modules

The weight lattice $P\subseteq\mathfrak{h}^{\ast}$ is the dual lattice of
$Q^{\vee}$, that is,

$P=\\{\lambda\in\mathfrak{h}^{\ast}\mid\langle\lambda,\alpha_{i}^{\vee}\rangle\in{\mathbb{Z}},\
i\in I\\}.$

An element $\lambda\in P$ is dominant if
$\langle\lambda,\alpha_{i}^{\vee}\rangle\geq 0$ for all $i\in I$, and is
regular if $\langle\lambda,\alpha_{i}^{\vee}\rangle\neq 0$ for all $i\in I$.
We denote the set of dominant elements of $P$ by $P_{+}$ and write
$Q_{+}=\sum_{i=1}^{\ell}{\mathbb{Z}}_{\geq 0}\alpha_{i}$. The _dominance
order_ on $\mathfrak{h}^{\ast}$ is defined by $\lambda<\mu$ if $\mu-\lambda\in
Q_{+}$.

Let $V$ be a $\mathfrak{g}$-module with defining homomorphism
$\rho:\mathfrak{g}\to\operatorname{End}(V)$. Then the _weight space_ of
$\mu\in\mathfrak{h}^{\ast}$ is

$V_{\mu}:=\\{v\in V\mid\rho(h)(v)=\mu(h)v\text{ for all
$h\in\mathfrak{h}$}\\},$

and the set of _weights_ of $V$ is

$\operatorname{wts}(V)=\left\\{\mu\in\mathfrak{h}^{\ast}\mid V_{\mu}\neq
0\right\\}.$

Recall also that $x\in\mathfrak{g}$ acts _locally nilpotently_ on $V$ if, for
each $v\in V$, there is a natural number $n$ such that $\rho(x)^{n}(v)=0$.
Then $V$ is called an _integrable_ module if it is a direct sum of its weight
spaces, that is $V=\bigoplus_{\mu\in\operatorname{wts}(V)}V_{\mu}$, and each
generator $e_{i}$ or $f_{i}$ acts locally nilpotently on $V$.

We call $\lambda\in\operatorname{wts}(V)$ the _highest weight_ of $V$ if it is
largest element of $\operatorname{wts}(V)$ in the dominance order. If $V$ has
a highest weight we call it a _highest weight module_. If $\mu$ is a weight of
a highest weight module $V$, then $\mu=\lambda-\beta$ for some $\beta\in
Q_{+}$. Among all modules with highest weight $\lambda\in\mathfrak{h}^{\ast}$,
there is a unique irreducible $\mathfrak{g}$–module (Prop. 9.3, [Kac90]),
which we denote by $V^{\lambda}$. The module $V^{\lambda}$ is integrable if
and only if $\lambda\in P_{+}$, that is, $\lambda$ is a dominant weight (Lemma
10.1, [Kac90]). In this case, we get $\operatorname{wts}(V^{\lambda})\subseteq
P$.

From now on, we take $V=V^{\lambda}$ to be an integrable highest weight
$\mathfrak{g}$-module over $\mathbb{Q}$ for a fixed dominant, regular weight
$\lambda$ and a fixed highest weight vector $v_{\lambda}$. For
$x\in\mathfrak{g}$ and $v\in V$, we denote the action of $x$ on $v$ by $x\cdot
v:=\rho(x)(v)$.

##  3\. Representation-theoretic Kac–Moody groups

Let $\mathfrak{g}$ be the Kac–Moody algebra over ${\mathbb{Q}}$ with
symmetrizable generalized Cartan matrix $A$. Let $V=V^{\lambda}$ to be an
integrable highest weight $\mathfrak{g}$-module over $\mathbb{Q}$ for a fixed
dominant, regular weight $\lambda$ and let $v_{\lambda}$ be a fixed highest
weight vector .

The Chevalley involution $\omega$ is an automorphism of $\mathfrak{g}$ with
$\omega^{2}=1$, $\omega(h)=-h$ for all $h\in\mathfrak{h}$, and
$\omega(\mathfrak{g}_{\alpha})=\mathfrak{g}_{-\alpha}$ for all
$\alpha\in\Delta$.

###  3.1. ${\mathbb{Z}}$–forms

Let ${\mathcal{U}}={\mathcal{U}}_{{\mathbb{Q}}}(\mathfrak{g})$ be the
universal enveloping algebra of $\mathfrak{g}$. Let

  * •

${\mathcal{U}}^{+}_{{\mathbb{Z}}}$ be the ${\mathbb{Z}}$-subalgebra generated
by $\dfrac{{e_{i}}^{m}}{m!}$ for $i\in I$ and $m\geq 0$;

  * •

${\mathcal{U}}^{-}_{{\mathbb{Z}}}$ be the ${\mathbb{Z}}$-subalgebra generated
by $\dfrac{{f_{i}}^{m}}{m!}$ for $i\in I$ and $m\geq 0$; and

  * •

${\mathcal{U}}^{0}_{{\mathbb{Z}}}$ be the ${\mathbb{Z}}$-subalgebra generated
by

$\left(\begin{matrix}h\\\
m\end{matrix}\right):=\dfrac{h(h-1)\dots(h-m+1)}{m!},$

for $h\in Q^{\vee}$ and $m\geq 0$.

As in [CG03] and [Tit81], [Tit87], the _${\mathbb{Z}}$ -form of $\mathcal{U}$_
is ${\mathcal{U}}_{{\mathbb{Z}}}$, the ${\mathbb{Z}}$-subalgebra generated by
${\mathcal{U}}^{+}_{{\mathbb{Z}}}$, ${\mathcal{U}}^{-}_{{\mathbb{Z}}}$, and
${\mathcal{U}}^{0}_{{\mathbb{Z}}}$. By [Tit87], we have
$\mathcal{U}^{-}_{{\mathbb{Z}}}\otimes\mathcal{U}_{{\mathbb{Z}}}^{0}\otimes\mathcal{U}^{+}_{{\mathbb{Z}}}\cong\mathcal{U}_{{\mathbb{Z}}}$.

We define

$\mathfrak{g}_{{\mathbb{Z}}}\ =\
\mathfrak{g}\cap{\mathcal{U}}_{{\mathbb{Z}}}\quad\text{and}\quad\mathfrak{h}_{{\mathbb{Z}}}\
=\ \mathfrak{h}\cap{\mathcal{U}}_{{\mathbb{Z}}}.$

Note that
$\mathfrak{g}=\mathfrak{g}_{{\mathbb{Z}}}\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$
and $\mathfrak{h}=\mathfrak{h}_{\mathbb{Z}}\otimes_{\mathbb{Z}}{\mathbb{Q}}$.

Fix the highest weight vector $v_{\lambda}$ in $V$. We have
$\mathcal{U}^{+}_{\mathbb{Z}}\cdot v_{\lambda}=\mathbb{Z}v_{\lambda}$ since
every $x_{\alpha}$ with $\alpha\in\Delta^{\operatorname{re}}_{+}$ annihilates
$v_{\lambda}$. Also $\mathcal{U}^{0}_{\mathbb{Z}}\cdot
v_{\lambda}=\mathbb{Z}v_{\lambda}$ since ${\mathcal{U}}^{0}_{{\mathbb{Z}}}$
acts on $v_{\lambda}$ as scalar multiplication by an integer. Therefore
$\mathcal{U}_{\mathbb{Z}}\cdot v_{\lambda}=\mathcal{U}^{-}_{\mathbb{Z}}\cdot
v_{\lambda}.$ We define the _${\mathbb{Z}}$ -form of $V$_ to be

$V_{\mathbb{Z}}:=\mathcal{U}_{\mathbb{Z}}\cdot
v_{\lambda}=\mathcal{U}^{-}_{\mathbb{Z}}\cdot v_{\lambda}$

and note that $V=V_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Q}.$ For
$\mu\in\operatorname{wts}(V)$, set $V_{\mu,\mathbb{Z}}=V_{\mu}\cap
V_{\mathbb{Z}}$.

###  3.2. The groups $G({\mathbb{Q}})$ and $G({\mathbb{Z}})$

For $i\in I$ and $t\in\mathbb{Q}$ set

$\chi_{\alpha_{i}}(t)=\exp(t\rho(e_{i}))=\sum^{\infty}_{m=0}\frac{t^{m}\rho\left(e_{i}\right)^{m}}{m!},\quad\chi_{-\alpha_{i}}(t)=\exp(t\rho(f_{i}))=\sum^{\infty}_{m=0}\frac{t^{m}\rho\left(f_{i}\right)^{m}}{m!},$

where $\rho$ is the defining homomorphism for $V=V^{\lambda}$. Since $V$ is
integrable, the $\chi_{\alpha_{i}}(t)$ are elements of $\operatorname{GL}(V)$.
We let $G(\mathbb{Q})\leq\operatorname{GL}(V)$ be the group generated by
$\chi_{\pm\alpha_{i}}(t)$ for $i\in I$ and $t\in\mathbb{Q}$. We refer to
$G({\mathbb{Q}})=G^{\lambda}(\mathbb{Q})$ as a _minimal representation-
theoretic Kac–Moody group_. Similarly we define $G(\mathbb{Z})$ as the
subgroup of $G({\mathbb{Q}})$ generated by $\chi_{\pm\alpha_{i}}(t)$ for $i\in
I$ and $t\in\mathbb{{\mathbb{Z}}}$. We denote the action of $g\in G$ on $v\in
V$ by $g\cdot v$.

For each $i\in I$ and $t\in\mathbb{Q}^{\times}$, we define

$\widetilde{w}_{i}(t)=\chi_{\alpha_{i}}(t)\chi_{-\alpha_{i}}(-t^{-1})\chi_{\alpha_{i}}(t)$

and set $\widetilde{w}_{i}=\widetilde{w}_{i}(1)$. Next set

$h_{\alpha_{i}}(t)\ =\ \widetilde{w}_{i}(t)\widetilde{w}_{i}(1)^{-1}$

for $t\in Q^{\times}$ and $i\in I$, and let $H\subseteq G$ be the subgroup
generated by the $h_{\alpha_{i}}(t)$. Let $N$ denote the subgroup of
$G({\mathbb{Q}})$ generated by $H$ and the $\widetilde{w}_{i}$. By Lemma 4 in
Section 5.4 of [Tit87], there exists a unique endomorphism $N\longrightarrow
W$ with kernel $H$. For $w\in W$, take a reduced word
$w=w_{i_{1}}w_{i_{2}}\cdots w_{i_{k}}$ and define

$\widetilde{w}:=\widetilde{w}_{i_{1}}\widetilde{w}_{i_{2}}\cdots\widetilde{w}_{i_{k}}.$

Now $\widetilde{w}_{i}\in G(\mathbb{Z})\subseteq\Gamma({\mathbb{Z}})$ and so

$\displaystyle\widetilde{w}\cdot V_{\mathbb{Z}}=V_{\mathbb{Z}}$

for all $w\in W$.

Note that the adjoint representation makes $\mathfrak{g}$ into a $G$-module
and we also get
$\text{Ad}(\widetilde{w})(\mathfrak{g}_{\mathbb{Z}})=\mathfrak{g}_{\mathbb{Z}}$
as in [MP95]. Let $\alpha\in\Delta_{+}^{\operatorname{re}}$, then
$\alpha=w\alpha_{i}$ for some $w\in W$ and $i\in I$. We define the _root
vectors_ ([MP95], Section 4.1) by

$\displaystyle
x_{\alpha}:=\text{Ad}(\widetilde{w})(e_{i})\in\mathfrak{g}_{\alpha}\cap\mathfrak{g}_{\mathbb{Z}}\quad\text{and}\quad
x_{-\alpha}:=\text{Ad}(\widetilde{w})(f_{i})\in\mathfrak{g}_{-\alpha}\cap\mathfrak{g}_{\mathbb{Z}},$

so that $x_{-\alpha}=-\omega(x_{\alpha})$ and
$[x_{\alpha},x_{-\alpha}]=\alpha^{\vee}$. In particular,
$x_{\alpha_{i}}=e_{i}$ and $x_{-\alpha_{i}}=f_{i}$. The set
$\\{x_{\alpha},x_{-\alpha},\ \alpha^{\vee}\\}$ forms a basis for a subalgebra
isomorphic to $\mathfrak{sl}_{2}$.

######  Lemma 3.1.

Let $w\in W$, $\mu\in\operatorname{wts}(V)$,
$\alpha\in\Delta_{+}^{\operatorname{re}}$, $m\in{\mathbb{N}}$.

  1. (a)

$\widetilde{w}\cdot V_{\mu,{\mathbb{Z}}}=V_{w\mu,{\mathbb{Z}}}$.

  2. (b)

$V_{w\lambda,{\mathbb{Z}}}=\widetilde{w}\cdot
V_{\lambda,{\mathbb{Z}}}={\mathbb{Z}}v_{w\lambda}$ where
$v_{w\lambda}=\widetilde{w}\cdot v_{\lambda}$.

  3. (c)

$\dfrac{{x_{\pm\alpha}}^{m}}{m!}\in\mathcal{U}_{\mathbb{Z}}$ and
$\dfrac{{x_{\pm\alpha}}^{m}}{m!}\cdot V_{\mu,{\mathbb{Z}}}\subseteq V_{\mu\pm
m\alpha,{\mathbb{Z}}}$.

###### Proof.

Lemma 3.8(a) in [Kac90] implies $\widetilde{w_{i}}\cdot V_{\mu}=V_{w_{i}\mu}$
for $i\in I$, and so $\widetilde{w}\cdot V_{\mu}=V_{w\mu}$ by induction.
Combining this with $\widetilde{w}\cdot V_{\mathbb{Z}}=V_{\mathbb{Z}}$ and
$V=\bigoplus_{\mu\in\operatorname{wts}(V)}V_{\mu}$ we get (a). Parts (b) and
(c) are now straightforward. ∎

For $t\in{\mathbb{Q}}$, set

$\displaystyle\chi_{\alpha}(t):=\exp(t\rho(x_{\alpha}))=\sum^{\infty}_{m=0}\frac{t^{m}\rho\left(x_{\alpha}\right)^{m}}{m!}\in\operatorname{GL}(V).$

We have
$\chi_{\alpha}(t)=\widetilde{w}\chi_{\alpha_{i}}(t)\widetilde{w}^{-1}\in G$.
We define the (real) _root group_ and its ${\mathbb{Z}}$-subgroup as

$U_{\alpha}=\\{\chi_{\alpha}(t)\mid t\in\mathbb{Q}\\}\quad\text{and}\quad
U_{\alpha}(\mathbb{Z})=\\{\chi_{\alpha}(t)\mid t\in\mathbb{Z}\\}.$

Given a set of positive real roots
$\Omega\subseteq\Delta_{+}^{\operatorname{re}}$, we define

$U_{\Omega}=\langle U_{\alpha}\mid\alpha\in\Omega\rangle\quad\text{and}\quad
U_{\Omega}({\mathbb{Z}})=\langle
U_{\alpha}({\mathbb{Z}})\mid\alpha\in\Omega\rangle.$

In particular, the _unipotent subgroup_ is
$U:=U_{\Delta_{+}^{\operatorname{re}}}$ and its ${\mathbb{Z}}$-subgroup is
$U({\mathbb{Z}}):=U_{\Delta_{+}^{\operatorname{re}}}({\mathbb{Z}})$. It is
clear that $U_{\Omega}({\mathbb{Z}})\subseteq U_{\Omega}\cap G({\mathbb{Z}})$.
The reverse inclusion is conjectured to be true for reasonable choices of
$\Omega$ but will not be needed for the results in this paper.

We end this section with the following:

######  Lemma 3.2.

For all $u\in U$, $u\cdot v_{\lambda}=v_{\lambda}$.

###### Proof.

It suffices to prove this statement for a generator $\chi_{\alpha}(t)$ of
$U_{\alpha}\subseteq U$. Now

$\displaystyle\chi_{\alpha}(t)\cdot v_{\lambda}=v_{\lambda}+tx_{\alpha}\cdot
v_{\lambda}+t^{2}\,\frac{{x_{\alpha}}^{2}}{2!}\cdot
v_{\lambda}+\dots+t^{m}\,\frac{{x_{\alpha}}^{m}}{m!}\cdot v_{\lambda},$

for some $m$ since $x_{\alpha}$ acts nilpotently. But ${x_{\alpha}}^{i}\cdot
v_{\lambda}\in V_{\lambda+i\alpha}=0$, since $\lambda+i\alpha$ is not a weight
of $V$ for $i>0$. Thus, $\chi_{\alpha}(t)\cdot v_{\lambda}=v_{\lambda}.$ ∎

##  4\. Strong integrality of real root groups

Throughout this section, fix $\alpha\in\Delta^{\operatorname{re}}$. The main
result of this section is to prove strong integrality of the real root group

$U_{\alpha}=\\{\chi_{\alpha}(t)\mid t\in\mathbb{Q}\\}.$

This provides the base case for Theorem 1.2. Integrality of $U_{\alpha}$ will
follow from the following lemma.

######  Lemma 4.1.

Strong integrality implies integrality.

###### Proof.

Let $M\subseteq G$ be strongly integral with respect to $v_{0}\in
V^{\lambda}$. Let $g\in M\cap\Gamma({\mathbb{Z}})$. Then $g\cdot v\in
V^{\lambda}_{\mathbb{Z}}$ for all $v\in V^{\lambda}_{\mathbb{Z}}$. In
particular $g\cdot v_{0}\in V^{\lambda}_{\mathbb{Z}}$ and so $g\in
G({\mathbb{Z}})$ by strong integrality. ∎

Define

$x_{\alpha}^{[m]}:=\frac{{x_{\alpha}}^{m}}{m!}.$

The following technical lemma is essential to our integrality results:

######  Lemma 4.2.

Let $\mu$ be a weight such that $\mu+\alpha$ is not a weight of $V$. Further
assume that $v_{\mu}\in V_{\mu}$ and $n:=\langle\mu,\alpha^{\vee}\rangle>0$.
Then

$x_{\alpha}^{[n]}x_{-\alpha}^{[n]}\cdot v_{\mu}=v_{\mu}.$

###### Proof.

First we prove that

$[x_{\alpha},{x_{-\alpha}}^{k}]=k{x_{-\alpha}}^{k-1}\left(\alpha^{\vee}-(k-1)\right)$

by induction on $k\in{\mathbb{N}}$:

$\displaystyle[x_{\alpha},{x_{-\alpha}}^{k}]$
$\displaystyle=x_{\alpha}{x_{-\alpha}}^{k}-{x_{-\alpha}}^{k}x_{\alpha}$
$\displaystyle=x_{\alpha}{x_{-\alpha}}^{k}-{x_{-\alpha}}^{k-1}x_{\alpha}{x_{-\alpha}}+{x_{-\alpha}}^{k-1}x_{\alpha}{x_{-\alpha}}-{x_{-\alpha}}^{k}x_{\alpha}$
$\displaystyle=[x_{\alpha},{x_{-\alpha}}^{k-1}]{x_{-\alpha}}+{x_{-\alpha}}^{k-1}[x_{\alpha},{x_{-\alpha}}]$
$\displaystyle=(k-1){x_{-\alpha}}^{k-2}\left(\alpha^{\vee}-k+2\right){x_{-\alpha}}+{x_{-\alpha}}^{k-1}\alpha^{\vee}$
$\displaystyle=(k-1){x_{-\alpha}}^{k-2}(\alpha^{\vee}x_{-\alpha})-(k-1)(k-2){x_{-\alpha}}^{k-1}+{x_{-\alpha}}^{k-1}m\alpha^{\vee}$
$\displaystyle=(k-1){x_{-\alpha}}^{k-2}(x_{-\alpha}\alpha^{\vee}-2x_{-\alpha})-(k-1)(k-2){x_{-\alpha}}^{k-1}+{x_{-\alpha}}^{k-1}\alpha^{\vee}$
$\displaystyle=k{x_{-\alpha}}^{k-1}\alpha^{\vee}-k(k-1){x_{-\alpha}}^{k-1}.$

Note that $x_{\alpha}\cdot v_{\mu}\in V_{\mu+\alpha}=0$, so $x_{\alpha}\cdot
v_{\mu}=0$. Also $\alpha^{\vee}\cdot v_{\mu}=\langle\mu,\alpha^{\vee}\rangle
v_{\mu}=nv_{\mu}$. Next we prove

$\displaystyle{x_{\alpha}}^{m}{x_{-\alpha}}^{m}\cdot
v_{\mu}=m!\prod_{j=0}^{m-1}(n-j)\cdot v_{\mu}.$

by induction on $m$:

$\displaystyle{x_{\alpha}}^{m}{x_{-\alpha}}^{m}\cdot v_{\mu}$
$\displaystyle={x_{\alpha}}^{m-1}\left([x_{\alpha},{x_{-\alpha}}^{m}]+{x_{-\alpha}}^{m}x_{\alpha}\right)\cdot
v_{\mu}$ $\displaystyle={x_{\alpha}}^{m-1}[x_{\alpha},{x_{-\alpha}}^{m}]\cdot
v_{\mu}$
$\displaystyle={x_{\alpha}}^{m-1}{x_{-\alpha}}^{m-1}m\left(\alpha^{\vee}-(m-1)\right)\cdot
v_{\mu}$
$\displaystyle=m\left(n-(m-1)\right){x_{\alpha}}^{m-1}{x_{-\alpha}}^{m-1}\cdot
v_{\mu}$ $\displaystyle=m!\prod_{j=0}^{m-1}(n-j)\cdot v_{\mu}.$

Now take $m=n$ to get ${x_{\alpha}}^{n}{x_{-\alpha}}^{n}\cdot
v_{\mu}=n!\,n!\,v_{\mu}$, so $x_{\alpha}^{[n]}x_{-\alpha}^{[n]}\cdot
v_{\mu}=v_{\mu}$. ∎

######  Lemma 4.3 (Lemma 7.2 of [Ste67]).

Let $\alpha\in\Delta^{\operatorname{re}}$ and $t\in{\mathbb{Q}}$. If $\mu$ is
a weight of $V$ and $v\in V_{\mu}$, then

$\chi_{\alpha}(t)\cdot v=v+\sum_{m\in\mathbb{N}}t^{m}(x_{\alpha}^{[m]}\cdot
v),$

where $x_{\alpha}^{[m]}\cdot v\in V_{\mu+m\alpha}$, and only finitely many of
the terms $x_{\alpha}^{[m]}\cdot v$ are non-zero. If moreover $v\in
V_{\mu,{\mathbb{Z}}}$, then $x_{\alpha}^{[m]}\cdot v\in
V_{\mu+m\alpha,{\mathbb{Z}}}$.

###### Proof.

If $v\in V_{\mu,{\mathbb{Z}}}$, then $x_{\alpha}^{[m]}\cdot v\in
V_{\mu+m\alpha,{\mathbb{Z}}}$ by Lemma 3.1(c). ∎

######  Proposition 4.4.

Let $\alpha\in\Delta^{\operatorname{re}}$. If
$\chi_{\alpha}(t)\widetilde{w}_{\alpha}\cdot v_{\lambda}\in V_{\mathbb{Z}}$,
then $\chi_{\alpha}(t)\in U_{\alpha}({\mathbb{Z}})$.

###### Proof.

Let $\alpha$ be positive. Suppose $\chi_{\alpha}(t)\cdot v_{0}\in
V_{\mathbb{Z}}$ where $v_{0}:=\widetilde{w}_{\alpha}\cdot v_{\lambda}$. Let
$n=\langle\lambda,\alpha^{\vee}\rangle$ so that
$w_{\alpha}\lambda=\lambda-n\alpha$. Note that $n>0$ since $\lambda$ is
regular. Now $V_{\lambda-n\alpha,{\mathbb{Z}}}={\mathbb{Z}}v_{0}$ by Lemma
3.1(b). But $x_{-\alpha}^{[n]}\cdot v_{\lambda}$ is also in
$V_{\lambda-n\alpha,{\mathbb{Z}}}$ by Lemma 3.1(c), so $x_{-\alpha}^{[n]}\cdot
v_{\lambda}=mv_{0}$ for some integer $m$. We have

$\chi_{\alpha}(t)\cdot v_{0}=v_{0}+tx_{\alpha}\cdot
v_{0}+t^{2}x_{\alpha}^{[2]}\cdot v_{0}+\dots+t^{n}x_{\alpha}^{[n]}\cdot
v_{0}.$

Now consider the $V_{\lambda,{\mathbb{Z}}}$ component:

$t^{n}x_{\alpha}^{[n]}\cdot
v_{0}=\frac{t^{n}}{m}x_{\alpha}^{[n]}x_{-\alpha}^{[n]}\cdot
v_{\lambda}=\frac{t^{n}}{m}v_{\lambda}$

by Lemma 4.2. Hence $\frac{t^{n}}{m}$ is an integer, so $t^{n}$ is an integer,
and so $t$ is an integer. The proof for negative $\alpha$ is similar. ∎

Since $U_{\alpha}({\mathbb{Z}})\subseteq U_{\alpha}\cap G({\mathbb{Z}})$, we
get:

######  Corollary 4.5.

If $\alpha\in\Delta^{\operatorname{re}}$, then $U_{\alpha}$ is strongly
integral with respect to $\widetilde{w}_{\alpha}\cdot v_{\lambda}$. Hence
$U_{\alpha}$ is integral.

It follows immediately that for $\alpha\in\Delta^{\operatorname{re}}$ and
$u\in U({\mathbb{Z}})$, if $\chi_{\alpha}(s)u\in\Gamma({\mathbb{Z}})$ then
$s\in{\mathbb{Z}}$.

##  5\. Inversion sets and subgroups

###  5.1. Inversion sets and orderings

Let $w\in W$. We define the corresponding _inversion set_ of roots by

$\displaystyle\Phi_{(w)}$ $\displaystyle=$
$\displaystyle\\{\beta\in\Delta_{+}\mid
w^{-1}\beta\in\Delta_{-}\\}=\Delta_{+}\cap w(\Delta_{-}).$

We note the following standard properties of $\Phi_{(w)}$, which can be proven
by induction on $\ell(w)$.

######  Lemma 5.1.

Suppose that $w\in W$ has the reduced expression $w_{i_{1}}w_{i_{2}}\cdots
w_{i_{k}}$.

  1. (a)

$\Phi_{(w)}=\\{\alpha_{i_{1}},\;w_{i_{1}}\alpha_{i_{2}},\;w_{i_{1}}w_{i_{2}}\alpha_{i_{3}},\;\dots,\;w_{i_{1}}w_{i_{2}}\cdots
w_{i_{k-2}}w_{i_{k-1}}\alpha_{k}\\}\subseteq\Delta_{+}^{\operatorname{re}}$.

  2. (b)

$\Phi_{(w)}$ has cardinality $k=\ell(w)$.

  3. (c)

If $w^{\prime}=w_{i_{1}}w_{i_{2}}\cdots w_{i_{k-1}}$, then
$\Phi_{(w)}=\Phi_{(w^{\prime})}\sqcup\\{w^{\prime}\alpha_{i_{k}}\\}.$

  4. (d)

If ${w}^{\prime\prime}=w_{i_{2}}\cdots w_{i_{k}}$, then
$\Phi_{(w)}=\\{\alpha_{i_{1}}\\}\sqcup{w_{i_{1}}}\Phi_{({w^{\prime\prime}})}$.

  5. (e)

For $\mu\in\mathfrak{h}^{\ast}$,
$w\mu=\mu-\langle\mu,\alpha_{i_{1}}\rangle\alpha_{i_{1}}-\langle\mu,\alpha_{i_{2}}\rangle
w_{i_{1}}\alpha_{i_{2}}-\dots-\langle\mu,\alpha_{i_{k}}\rangle
w_{i_{1}}w_{i_{2}}\cdots w_{i_{k-1}}\alpha_{i_{k}}.$

We also note the immediate consequences of Lemma 5.1:

######  Lemma 5.2.

  1. (a)

For $\alpha\in\Delta_{+}^{\operatorname{re}}$, there exists $w\in W$ such that
$\alpha\in\Phi_{(w)}$.

  2. (b)

For a finite subset $\Omega\subseteq\Delta^{\operatorname{re}}_{+}$, there
exists a finite set $T\subseteq W$ such that $\Omega\subseteq\bigcup_{w\in
T}\Phi_{w}.$

We equip $\Phi_{(w)}$ with an ordering

$\alpha_{i_{1}}\prec w_{i_{1}}\alpha_{i_{2}}\prec
w_{i_{1}}w_{i_{2}}\alpha_{i_{3}}\prec\cdots\prec w_{i_{1}}w_{i_{2}}\cdots
w_{i_{k-2}}w_{i_{k-1}}\alpha_{k},$

as described by Papi in [Pap94]. Note that this ordering depends of the choice
of reduced word for $w$.

###  5.2. Inversion subgroups

For $w\in W$, the inversion subgroup is defined as

$U_{(w)}=U_{\Phi_{(w)}}=\langle U_{\alpha}\mid\alpha\in\Phi_{(w)}\rangle.$

######  Lemma 5.3.

Each element $u_{(w)}\in U_{(w)}$ can be written uniquely as

$u_{(w)}=\prod_{\gamma\in\Phi_{w}}u_{\beta},$

where $u_{\beta}\in U_{\beta}$ and the product is in the ordering on
$\Phi_{(w)}$.

######  Corollary 5.4.

$U_{(w)}({\mathbb{Z}})=U_{(w)}\cap U({\mathbb{Z}})$.

For $w\in W$, set $U^{(w)}=U\cap w^{-1}Uw$. We end this section with the
following lemma from Subsection 4.2.6 of [PP19].

######  Lemma 5.5.

$U=U^{(w)}U_{(w)}=U_{(w)}U^{(w)}.$

##  6\. Integrality of inversion subgroups

To prove Theorem 1.2, we need a preliminary result, which we discuss next.
Define

$x_{\beta_{1},\beta_{2},\dots,\beta_{k}}^{[i_{1},i_{2},\dots,i_{k}]}:=x_{\beta_{1}}^{[i_{1}]}x_{\beta_{2}}^{[i_{2}]}\cdots
x_{\beta_{k}}^{[i_{k}]}.$

The proof of the following is similar to Lemma 4.3:

######  Lemma 6.1.

Let $\beta_{i}\in\Delta^{\operatorname{re}}_{+}$ and $t_{i}\in\mathbb{Q}$ for
$i=1,\dots,k$. If $\mu$ is a weight of $V$ and $v\in V_{\mu}$, then

${\prod}_{i=1}^{k}\chi_{\beta_{i}}(t_{i})\cdot
v={{\sum}}_{i_{1},i_{2},\dots,i_{k}\in\mathbb{Z}_{\geq
0}}t_{1}^{i_{1}}t_{2}^{i_{2}}\cdots
t_{k}^{i_{k}}\left(x_{\beta_{1},\beta_{2},\dots,\beta_{k}}^{[i_{1},i_{2},\dots,i_{k}]}\cdot
v\right)$

where
$x_{\beta_{1},\beta_{2},\dots,\beta_{k}}^{[i_{1},i_{2},\dots,i_{k}]}\cdot v\in
V_{\mu+i_{1}\beta_{1}+i_{2}\beta_{2}+i_{k}\beta_{k}}$ and only finitely of the
terms
$x_{\beta_{1},\beta_{2},\dots,\beta_{k}}^{[i_{1},i_{2},\dots,i_{k}]}\cdot v$
are non-zero. If moreover $v\in V_{\mu,\mathbb{Z}}$, then
$x_{\beta_{1},\beta_{2},\dots,\beta_{k}}^{[i_{1},i_{2},\dots,i_{k}]}\cdot v\in
V_{\mu+i_{1}\beta_{1}+i_{2}\beta_{2}+i_{k}\beta_{k},\mathbb{Z}}$.

######  Proposition 6.2.

Let $w\in W$ and $u_{(w)}\in U_{(w)}$. If $u_{(w)}\widetilde{w}\cdot
v_{\lambda}\in V_{\mathbb{Z}}$, then $u_{(w)}\in U_{(w)}({\mathbb{Z}})$.

Theorem 1.2 follows immediately from this result since
$U_{(w)}({\mathbb{Z}})\subseteq U_{(w)}\cap G({\mathbb{Z}})$.

###### Proof.

We proceed by induction on $k=\ell(w)$. The base case $\ell(w)=1$ is just
Proposition 4.4 for a simple root $\alpha_{i}$. So assume $\ell(w)\geq 2$ and
the inductive hypothesis. Let $\Phi_{(w)}=\\{\beta_{1},\dots\beta_{k}\\}$ in
the usual order. So $\beta:=\beta_{1}$ is a simple root and
$w=w_{\beta}w^{\prime\prime}$ with $\ell(w^{\prime\prime})=k-1$, and
$\Phi_{(w^{\prime\prime})}=\\{w_{\beta}\beta_{2},\dots,w_{\beta}\beta_{k}\\}$
by Lemma 5.1(d). In particular, $w_{\beta}\beta_{2},\dots,w_{\beta}\beta_{k}$
are positive roots. Write

$u_{(w)}=\prod_{i=1}^{k}\chi_{\beta_{i}}(t_{i})=\chi_{\beta}(t)\prod_{i=2}^{k}\chi_{\beta_{i}}(t_{i}),$

for $t=t_{1},t_{2},\dots,t_{k}\in{\mathbb{Q}}$.

Write $\mu=w^{\prime\prime}\lambda$ and
$v_{\mu}=\widetilde{w}^{\prime\prime}\cdot v_{\lambda}$. Let
$n=\langle\mu,\beta^{\vee}\rangle$, so that
$w\lambda=w_{\beta}\mu=\mu-n\beta$. Let
$v_{\mu-n\beta}=\widetilde{w}_{\beta}\cdot v_{\mu}=\widetilde{w}\cdot
v_{\lambda}$. Then

$\displaystyle u_{(w)}\widetilde{w}\cdot v_{\lambda}$
$\displaystyle=u_{(w)}\cdot
v_{\mu-n\beta}={{\sum}}_{\begin{subarray}{c}{i_{1},i_{2},\dots,i_{k}}\in\mathbb{Z}_{\geq
0}\end{subarray}}t_{1}^{i_{1}}t_{2}^{i_{2}}\cdots
t_{k}^{i_{k}}\left(x^{[i_{2},\dots,i_{k}]}_{\beta_{2},\dots,\beta_{k}}\cdot
v_{\mu-n\beta}\right)$ $\displaystyle=t^{n}x_{\beta}^{[n]}\cdot
v_{\mu-n\beta}+{{\sum}}_{\begin{subarray}{c}{i\in{\mathbb{Z}}_{\leq
n},\;i_{2},\dots,i_{k}\in\mathbb{Z}_{\geq 0}}\\\ \text{at least one non-
zero}\end{subarray}}t^{n-i}t_{2}^{i_{2}}\cdots
t_{k}^{i_{k}}\left(x^{[n-i,i_{2},\dots,i_{k}]}_{\beta,\beta_{2},\dots,\beta_{k}}\cdot
v_{\mu-n\beta}\right),$

where we have substituted $i=n-i_{1}$ and extracted the term with
$i,i_{2},\dots,i_{k}=0$ from the sum. Now $t^{n}x_{\beta}^{[n]}\cdot
v_{\mu-n\beta}\in V_{\mu}$ and
$x^{[n-i,i_{2},\dots,i_{k}]}_{\beta,\beta_{2},\dots,\beta_{k}}\cdot
v_{\mu-n\beta}\in V_{\nu}$ for $\nu=\mu-i\beta+\sum_{j=2}^{k}i_{j}\beta_{j}$.
Now

$\displaystyle(w^{\prime\prime})^{-1}(\nu)$
$\displaystyle=\lambda-(w^{\prime\prime})^{-1}(i\beta)+(w^{\prime\prime})^{-1}\left(\sum_{j=2}^{k}i_{j}\beta_{j}\right)\geq\lambda-i(w^{\prime\prime})^{-1}(\beta),$

but $(w^{\prime\prime})^{-1}(\beta)>0$, so we must have $i\geq 0$ for $\nu$ to
be a weight of $V$. So we can take $i$ to be in the range $0\leq i\leq n$. If
$\mu=\nu$, then $i\beta=\sum_{j=2}^{k}i_{j}\beta_{j}$, but
$w_{\beta}\left(i_{j}\beta_{j}\right)\geq 0$ and $w_{\beta}(i\beta)\leq 0$, so
$i,i_{1},\dots,i_{k}=0$. Hence the $V_{\mu,{\mathbb{Z}}}$ component of
$u_{(w)}\widetilde{w}\cdot v_{\lambda}\in V_{\mathbb{Z}}$ is just
$t^{n}x_{\beta}^{[n]}\cdot v_{\mu-n\beta}$.

Now $V_{\mu,{\mathbb{Z}}}={\mathbb{Z}}v_{\mu}$, so $t^{n}x_{\beta}^{[n]}\cdot
v_{\mu-n\beta}=mv_{\mu}$ for some integer $m$. Also $x_{-\beta}^{[n]}\cdot
v_{\mu}\in V_{\mu-n\beta,{\mathbb{Z}}}={\mathbb{Z}}v_{\mu-n\beta},$ so
$x_{-\beta}^{[n]}\cdot v_{\mu}=m^{\prime}v_{\mu-n\beta}$ for some integer
$m^{\prime}$. Hence

$mm^{\prime}v_{\mu-n\beta}=mx_{-\beta}^{[n]}\cdot
v_{\mu}=t^{n}x_{-\beta}^{[n]}x_{\beta}^{[n]}\cdot
v_{\mu-n\beta}=t^{n}v_{\mu-n\beta}$

by Lemma 4.2 for $\alpha=-\beta$, since
$\mu-n\beta-\beta\notin\operatorname{wts}(V)$ as
$w^{-1}(\mu-n\beta-\beta)=w^{-1}(w\lambda-\beta)=\lambda-w^{-1}\beta>\lambda$.
Hence $t^{n}=mm^{\prime}$ is an integer, so $t$ is an integer.

Now

$u_{(w)}\widetilde{w}=\chi_{\beta}(t)\prod_{i=2}^{k}\chi_{\beta_{i}}(t_{i})\,\widetilde{w}_{\beta}\widetilde{w}^{\prime\prime}=\chi_{\beta}(t)\widetilde{w}_{\beta}u_{(w^{\prime\prime})}\widetilde{w}^{\prime\prime},$

where
$u_{(w^{\prime\prime})}=\prod_{i=2}^{k}\chi_{w_{\beta}\beta_{i}}(t_{i})\in
U_{(w^{\prime\prime})}$. Now $\chi_{\beta}(t)\widetilde{w}_{\beta}\in
G({\mathbb{Z}})$, so

$u_{(w^{\prime\prime})}\widetilde{w}^{\prime\prime}\cdot
v_{\lambda}=\left(\chi_{\beta}(t)\widetilde{w}_{\beta}\right)^{-1}\cdot\left(u_{(w)}\widetilde{w}\cdot
v_{\lambda}\right)\in V_{\mathbb{Z}},$

and so $u_{(w^{\prime\prime})}\in U_{(w^{\prime\prime})}({\mathbb{Z}})$ by
induction. Hence
$u_{(w)}=\chi_{\beta}(t)\widetilde{w}_{\beta}u_{(w^{\prime\prime})}\widetilde{w}_{\beta}^{-1}\in
U_{(w)}({\mathbb{Z}})$. ∎

##  7\. Further integrality results

###  7.1. Integrality of finite dimensional unipotent groups

For this section, we assume that $A$ has finite type. In this case, $A$ is a
Cartan matrix and $\mathfrak{g}$ is a finite dimensional semi-simple Lie
algebra over ${\mathbb{Q}}$. The group construction in Section 3 can be
carried out with the same external data: a highest weight representation $V$
with highest weight $\lambda$, a ${\mathbb{Z}}$-form of the universal
enveloping algebra (constructed by Cartier and Kostant, see [Kos66]) and a
lattice $V_{{\mathbb{Z}}}$ in $V$ (called an admissible lattice in [Che95]).

The group $G({\mathbb{Q}})$ is then a semi-simple algebraic group and
$G({\mathbb{Q}})\cong G_{\mathbb{Z}}({\mathbb{Q}})$, where
$G_{\mathbb{Z}}({\mathbb{Q}})$ is Chevalley’s group scheme, now known as the
Chevalley–Demazure group scheme ([Che05]).

Our construction of the group $G({\mathbb{Q}})$ does not explicitly involve a
lattice $L$ between the root lattice and the weight lattice, as in Chevalley’s
construction. With the assumption that the set of weights of $V$ contains all
the fundamental weights, our group $G({\mathbb{Q}})$ coincides with
Chevalley’s simply connected group $G_{\mathbb{Z}}({\mathbb{Q}})$.

As a corollary of Theorem 1.2, we have the following.

######  Corollary 7.1.

Suppose that the matrix $A$ has finite type. Suppose also that the set of
weights of $V$ contains all the fundamental weights. Then the unipotent
subgroup $U$ of $G({\mathbb{Q}})$ is integral.

###### Proof.

Since $A$ has finite type, the Weyl group $W$ is finite. Moreover, $W$
contains the longest element, denoted $w_{0}$, which flips all positive roots
to negative roots. That is $\Phi_{w_{0}}=\Delta^{+}$, which gives
$U_{w_{0}}=U$. Combining this with Theorem 1.2 and Lemma 4.1, it follows that
$U$ is integral. ∎

###  7.2. Integrality of groups generated by commutating real root subgroups

We can now prove the result from Section 1 that gives integrality of subgroups
of $U$ generated by commuting real root groups.

######  Theorem 1.4.

Let $U$ be the positive unipotent subgroup of $G=G^{\lambda}({\mathbb{Q}})$.
If $M$ is a subgroup of $U$ generated by commuting real root subgroups, then
$M$ is integral.

###### Proof.

Assume, for the sake of contradiction, that

$\displaystyle
u=\chi_{\beta_{1}}(t_{\beta_{1}})\chi_{\beta_{2}}(t_{\beta_{2}})\cdots\chi_{\beta_{N}}(t_{\beta_{N}})\in
M$

is the shortest word such that $u\cdot V_{\mathbb{Z}}=V_{\mathbb{Z}}$ but
$u\notin U({\mathbb{Z}})$. By Corollary 4.5, we must have $N>1$. Now
$\Omega:=\\{\beta_{1},\beta_{2},\dots,\beta_{N}\\}\subseteq\Delta^{\operatorname{re}}_{+}$
is a finite set of real roots whose corresponding root subgroups commute with
each other. By part (b) of Lemma 5.2, there exists $w\in W$ such that
$\Omega\cap\Phi_{(w)}\neq\emptyset$. Using the commutating of the real root
groups $U_{\beta_{j}}$, we can rearrange the product above as

$u=u_{(w)}u^{(w)}=u^{(w)}u_{(w)},$

where $u_{(w)}=\prod_{\beta\in\Omega\cap\Phi_{(w)}}\chi_{\beta}(t_{\beta})$
and
$u^{(w)}=\prod_{\beta\in\Omega\setminus\Phi_{(w)}}\chi_{\beta}(t_{\beta})$.

Now we have

$\displaystyle u\widetilde{w}\cdot
v_{\lambda}=u_{(w)}u^{(w)}\widetilde{w}\cdot
v_{\lambda}=u_{(w)}\widetilde{w}\cdot(\widetilde{w}^{-1}u^{(w)}\widetilde{w}\cdot
v_{\lambda}).$

But $\widetilde{w}^{-1}u^{(w)}\widetilde{w}\in U$, so
$\widetilde{w}^{-1}u^{(w)}\widetilde{w}\cdot v_{\lambda}=v_{\lambda}$ by Lemma
3.2. So $u_{(w)}\widetilde{w}\cdot v_{\lambda}=u\widetilde{w}\cdot
v_{\lambda}\in V_{\mathbb{Z}}$. But $U_{(w)}$ is strongly integral with
respect to $\widetilde{w}\cdot v_{\lambda}$ by Theorem 1.2, so $u_{(w)}\in
U(\mathbb{Z})$.

Hence ${u_{(w)}}^{-1}\in U(\mathbb{Z})$ and so

$u^{(w)}\cdot V_{\mathbb{Z}}=u\cdot({u_{(w)}}^{-1}\cdot V_{\mathbb{Z}})=u\cdot
V_{\mathbb{Z}}=V_{\mathbb{Z}}.$

Since $\Omega\cap\Phi_{(w)}$ is nonempty,
$u^{(w)}=\prod_{\beta\in\Omega\setminus\Phi_{(w)}}\chi_{\beta}(t_{\beta})$ is
a shorter word than $u$ and so our assumption implies $u^{(w)}\in
U(\mathbb{Z})$. But now $u=u_{(w)}u^{(w)}\in U(\mathbb{Z})$, which is a
contradiction. ∎

###  7.3. Integrality of subgroups of $U$ in the rank $2$ Kac–Moody case

We start with a method for constructing infinite-dimensional integral
unipotent subgroups.

######  Proposition 7.2.

Let
$\Omega=\bigcup_{n=1}^{\infty}\Omega_{n}\subseteq\Delta_{+}^{\operatorname{re}}$
where $\Omega_{1}\subseteq\Omega_{2}\subseteq\cdots$ and assume that each
$U_{\Omega_{n}}$ is integral. Then $U_{\Omega}=\langle
U_{\alpha}\mid\alpha\in\Omega\rangle$ is integral.

###### Proof.

Take $u\in U_{\Omega}$ with $u\cdot V_{\mathbb{Z}}=V_{\mathbb{Z}}$. Then
$u=\prod_{i=1}^{N}\chi_{\beta_{i}}(a_{i})$ for some $\beta_{i}\in\Omega$,
$a_{i}\in{\mathbb{Q}}$. But now the finite set of roots
$\\{\beta_{1},\dots,\beta_{N}\\}$ must be contained in $\Omega_{n}$ for some
$n$. Hence $u\in U_{\Omega_{n}}$, which is an integral subgroup. So $u\in
G({\mathbb{Z}})$. ∎

One way to construct such sets is by taking a sequence of simple reflections
$w_{i_{1}},w_{i_{2}},\dots$ such that every initial subsequence gives a
reduced word $w_{i_{1}}\cdots w_{i_{n}}\in W$. Then each group
$U_{(w_{i_{1}}\cdots w_{i_{n}})}$ is integral by Theorem 1.2. Now take

$\displaystyle\Omega$
$\displaystyle:=\bigcup_{n=1}^{\infty}\Phi_{(w_{i_{1}}\cdots
w_{i_{n}})}=\\{\alpha_{i_{1}},\;w_{i_{1}}\alpha_{i_{2}},\;w_{i_{1}}w_{i_{2}}\alpha_{i_{3}},\;\dots,\;w_{i_{1}}w_{i_{2}}\cdots
w_{i_{k-2}}w_{i_{k-1}}\alpha_{k},\;\dots\\}.$

Then the subgroup $U_{\Omega}$ is integral.

Now suppose that $\mathfrak{g}$ has rank 2. A detailed description of the
structure of $\Delta^{\operatorname{re}}$ and $U$ in rank 2 can be found in
[CKMS21]. We define

$\displaystyle\Omega_{1}:=\\{\alpha_{1},\ w_{1}\alpha_{2},\
w_{1}w_{2}\alpha_{1},\;\dots\\}\quad\text{and}\quad\Omega_{2}:=\\{\alpha_{2},\
w_{2}\alpha_{1},\ w_{2}w_{1}\alpha_{2},\;\dots\\}.$

In fact, $\Omega_{1}$ (resp. $\Omega_{2}$) is the set of all positive real
roots on the lower (resp. upper) branches of the hyperbolas defined in
[CKMS21].

In this case $U=U_{1}{\Large{*}}\,U_{2}$, where for $i=1,2$, the groups
$U_{i}:=U_{\Omega_{i}}$ are abelian if $A$ is symmetric, and nilpotent of
class 2 if $A$ is not symmetric (see [CKMS21]).

The following theorem follows immediately from Proposition 7.2.

######  Theorem 1.5.

Suppose that $G({\mathbb{Q}})$ has rank 2 with $U=U_{1}{\Large{*}}\,U_{2}$,
where $U_{i}=U_{\Omega_{i}}$. Then the groups $U_{i}$ are integral.

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