# On a “Wonderful" Bruhat-Tits group scheme

Vikraman Balaji  Chennai Mathematical Institute, Plot number H1, Sipcot IT
Park, Siruseri, Chennai, India<EMAIL_ADDRESS>and  Yashonidhi Pandey
Indian Institute of Science Education and Research, Mohali Knowledge city,
Sector 81, SAS Nagar, Manauli PO 140306, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper we make a universal construction of Bruhat-Tits group scheme on
wonderful embeddings of adjoint groups in the absolute and relative settings.
We make a similar construction for the wonderful embeddings of adjoint Kac-
Moody groups. These have natural classifying properties reflecting the orbit
structure on the wonderful embeddings.

###### Key words and phrases:

Bruhat-Tits group scheme, parahoric group, Loop groups, Wonderful
compactification

###### 2000 Mathematics Subject Classification:

14L15,14M27,14D20

The support of Science and Engineering Research Board under Mathematical
Research Impact Centric Support File number: MTR/2017/000229 is gratefully
acknowledged.

###### Contents

  1. 1 Introduction
    1. 1.1 Group embeddings and buildings
    2. 1.2 Statement of main results
      1. 1.2.1 The case of Tits building
      2. 1.2.2 The relative case of the Bruhat-Tits buiding
  2. 2 Preliminaries
    1. 2.0.1 Lie-data of $G/k$
    2. 2.0.2 Apartment data
    3. 2.0.3 Loop groups and their parahoric subgroups
    4. 2.0.4 Bruhat-Tits group scheme
    5. 2.0.5 Standard Parahoric subgroups
  3. 3 A Bruhat-Tits group scheme on the wonderful compactification
    1. 3.0.1 The structure of ${\bf X}$
    2. 3.0.2 The local toric structure of ${\bf X}$
    3. 3.0.3 Construction of a Lie-algebra bundle $\mathcal{R}$ on ${\bf Y}_{{}_{0}}$
    4. 3.0.4 Construction of a Lie-algebra bundle $\mathcal{R}$ on ${\bf Y}$
    5. 3.0.5 Weil restrictions and Lie algebras
    6. 3.0.6 Restriction of $\mathcal{R}$ to infinitesimal standard curves $U_{I}$
    7. 3.0.7 The Lie algebra bundle $\mathcal{R}$ on $\bf X$
    8. 3.0.8 Bruhat-Tits group scheme ${\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}}$ on $\bf X$
  4. 4 The Weyl alcove and apartment case
    1. 4.0.1 On the torus-embedding ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}$
    2. 4.0.2 Construction of a finite-dimensional Lie algebra bundle $J$ on ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}$ together with parabolic structures
    3. 4.0.3 The parahoric group scheme on the torus embedding ${{\bf Y}^{{}^{\text{aff}}}}$
  5. 5 The Bruhat-Tits group scheme on ${\bf X}^{{}^{{aff}}}$
  6. 6 The Bruhat-Tits group scheme on ${\bf X}^{{}^{{poly}}}$
  7. 7 An analogue of a construction of Mumford
  8. 8 Appendix on parabolic and equivariant bundles
    1. 8.0.1 The group scheme situation
    2. 8.0.2 Kawamata Coverings

##  1\. Introduction

Let $G$ be an almost simple, simply-connected group over an algebraically
closed field $k$ of characteristic zero and let
$G_{{}_{{}_{\text{ad}}}}:=G/Z(G)$. The aim of this note is to construct
certain universal group schemes on

  * •

the De Concini-Procesi wonderful compactification [7] $\bf X$ of
$G_{{}_{{}_{\text{ad}}}}$,

  * •

the loop “wonderful embedding" ${\bf X}^{{}^{\text{aff}}}$ of the adjoint
affine Kac-Moody group $G^{{}^{\text{aff}}}_{{}_{\text{ad}}}$, constructed by
Solis [16],

  * •

certain toroidal embeddings $\bar{G}_{{}_{{}_{\text{ad,A}}}}$ of the relative
group scheme $G_{{}_{{}_{\text{ad,A}}}}$ modeled after [13].

The group schemes we construct are sufficiently universal to be called
“wonderful". A new point of view, playing a central role in this work, is that
parabolic vector bundle on logarithmic schemes can be used as a tool to make
geometric constructions. Hitherto, they have been objects of study occuring as
points in certain moduli spaces.

Let us briefly motivate the theme of this note.

###  1.1. Group embeddings and buildings

Let us recall that Tits’ buildings are basically of two types. The first one
is the “absolute" Tits building or spherical building which is attached to a
semi-simple group over a general field. This simplicial complex is built from
simplices which correspond to parabolic subgroups. The apartments of the
building correspond to parabolic subgroups containing a fixed maximal torus.
This is built up out of Euclidean spaces decomposed by the usual Weyl
chambers. The second one is the Bruhat-Tits building which is the “relative"
building attached to a semi-simple group over a complete-valued field. This is
based on its parahoric subgroups and built up out of Euclidean spaces
decomposed into affine Weyl chambers.

The two types of buildings can also be seen from an algebro-geometric
perspective. In the absolute case, we work with a semisimple group
$G_{{}_{\text{ad}}}$ of adjoint type. In this setting one has the wonderful
embedding $G_{{}_{\text{ad}}}\subset{\bf X}$ where $G_{{}_{\text{ad}}}$ sits
as an open dense subset of $\bf X$. The complement ${\bf X}\setminus
G_{{}_{\text{ad}}}$ is stratified by subsets $Y$ and there is a bijection
$Y\mapsto\\{P_{{}_{Y}}|B\subset P_{{}_{Y}}\\}$ from these strata to parabolic
subgroups $P_{{}_{Y}}\subset G$ containing a fixed Borel subgroup $B$.
Furthermore, this bijection extends to an isomorphism between the Tits
building and the canonical complex associated with the toroidal embedding
$G_{{}_{\text{ad}}}\subset{\bf X}$ (see Mumford [13, Page 178]).

A second perspective is when the ground field is endowed with a complete non-
archimedean discrete valuation. Let $A=k\llbracket z\rrbracket$ be a complete
discrete valuation ring, $K=k(\\!(z)\\!)$ its quotient field. In this setting
our basic model was constructed by Mumford [13]. He constructs a toroidal
embedding $G_{{}_{\text{ad, A}}}\subset\bar{G}_{{}_{\text{ad, A}}}$ of the
split group scheme $G_{{}_{\text{ad, A}}}=G_{{}_{\text{ad, A}}}\times{\rm
Spec}\,A$. The strata of $\bar{G}_{{}_{\text{ad, A}}}\setminus
G_{{}_{\text{ad, A}}}$ correspond bijectively to parahoric subgroups of $G(K)$
in a way that naturally extends to an isomorphism of the graph of the
embedding $G_{{}_{\text{ad, A}}}\subset\bar{G}_{{}_{\text{ad, A}}}$ with the
Bruhat-Tits building of $G\times{\rm Spec}\,A$ over $A$.

###  1.2. Statement of main results

Classical Bruhat-Tits theory associates, to each facet $\Sigma$ of the Bruhat-
Tits building, a smooth group scheme $\mathcal{G}_{{}_{\Sigma}}$ on ${\rm
Spec}\,~{}A$, with connected fibres and whose generic fibre is
$G\times_{{}_{{\rm Spec}\,k}}{\rm Spec}\,~{}K$. We call
$\mathcal{G}_{{}_{\Sigma}}$ a Bruhat-Tits group scheme on ${\rm Spec}\,A$. The
$A$-valued points $\mathcal{G}_{{}_{\Sigma}}(A)\subset G(K)$ are precisely the
parahoric subgroups of $G(K)$. In this paper we construct universal analogues
of the Bruhat-Tits group scheme.

####  1.2.1. The case of Tits building

In the first setting, namely in the case of the Tits building, we construct an
affine group scheme $\mathcal{G}_{{}_{\bf X}}$ over $\bf X$ whose restriction
along each curve transversal to a strata of ${\bf X}\setminus
G_{{}_{\text{ad}}}$ corresponds to the Bruhat-Tits group scheme associated to
the parabolic subgroup under the bijection $Y\mapsto\\{P_{{}_{Y}}|B\subset
P_{{}_{Y}}\\}$ mentioned above.

To state our theorem, we introduce some relevant notations and notions. Let
${\bf X}:=\overline{G_{{}_{{}_{\text{ad}}}}}$ be the wonderful
compactification of $G_{{}_{{}_{\text{ad}}}}$. We construct a locally free
sheaf of Lie algebras on $\bf X$. This construction is essentially toric in
the sense that it is first constructed on the toric varieties based on the
negative Weyl chambers and then the ones on the bigger spaces is deduced from
this construction. We endow the locally free sheaf $\mathcal{R}$ with a
canonical parabolic structure at the generic points of the divisor ${\bf
X}\setminus G_{{}_{{}_{\text{ad}}}}$ together with a compatible loop Lie
algebra structure.

We fix data $(T,B,G)$ of $G$. Let $S$ denote the set of simple roots of $G$
and $\mathbb{S}=S\cup\\{\alpha_{0}\\}$ denote the set of affine simple roots.
For $\emptyset\neq\mathbb{I}\subset\mathbb{S}$ let $\mathcal{G}_{\mathbb{I}}$
denote the associated Bruhat-Tits group scheme on a dvr and for any $I\subset
S$ let $\mathcal{G}^{{}^{st}}_{I}=\mathcal{G}_{\mathbb{I}}$ where
$\mathbb{I}=I\cup\\{\alpha_{0}\\}$. For ${I}\subset S$, let $Z_{I}$ denote the
corresponding strata of $\bf X$ and for any point $z_{{}_{I}}\in Z_{{}_{I}}$,
let $C_{{}_{I}}\subset{\bf X}$ be a smooth curve with generic point in
$G_{{}_{\text{ad}}}$ and closed point $z_{{}_{I}}$. Let $U_{{}_{z}}\subset
C_{{}_{I}}$ be a formal neighbourhood of $z_{{}_{I}}$.

We these notations, Theorem 3.10 is as follows.

######  Theorem 1.1.

There exists an affine “wonderful" Bruhat-Tits group scheme
${\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}}$ on $\bf X$ satisfying the following
classifying properties.

  1. (1)

There is an identification of the Lie-algebra bundles
$\text{Lie}({\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}})\simeq\mathcal{R}$.

  2. (2)

For $\emptyset\neq I\subset S$ the restriction of ${\mathcal{G}}_{{}_{{\bf
X}}}^{{}^{\varpi}}$ to the formal neighbourhood $U_{{}_{z_{{}_{I}}}}$ of
$z_{{}_{I}}$ in $C_{{}_{I}}$ §(3.0.3) is isomorphic to the standard Bruhat-
Tits group scheme $\mathcal{G}^{{}^{st}}_{{}_{I}}$ §(2.0.5).

In (3.11), we have indicated generalizations to fields of positive
characteristics.

####  1.2.2. The relative case of the Bruhat-Tits buiding

In the second scenario we work in the setting of loop groups and construct an
affine group scheme over a “wonderful" embedding ${\bf X}^{{}^{\text{aff}}}$
constructed by Solis [16]. In the relative case the group scheme is obtained
by “integrating" a locally free sheaf of Lie algebras $\bf R$ on ${\bf
X}^{{}^{\text{aff}}}$. Its construction is achieved by constructing a locally
free sheaf of Lie-algebras $J$ on a finite dimensional scheme ${\bf
Y}^{{}^{\text{aff}}}$ which is the closure of a torus-embedding and whose
translates build up the ind-scheme ${\bf X}^{{}^{\text{aff}}}$. The sheaf $J$
comes equipped with a canonical parabolic structure on a normal crossing
divisor. The sheaf $J$ plays the role analogous to $\mathcal{R}$ (on $\bf X$)
once we view ${\bf Y}^{{}^{\text{aff}}}$ as built out of affine Weyl group
translates of the affine space $\mathbb{A}^{\ell+1}$ whose standard coordinate
hyperplanes play the role of strata of largest dimension of $\bf X$. Using
this perspective, we then briefly consider a toroidal embedding
$\bar{G}_{{}_{ad,A}}$ the structure of which is modeled after Mumford’s
construction in [13]. We then define an affine group scheme over
$\bar{G}_{{}_{ad,A}}$ which has properties analogous to those of
$\mathcal{G}_{{}_{\bf X}}$.

We first introduce some notation to state our theorem more precisely. Let $LG$
denote the loop group of $G$. Let $L^{\ltimes}G=\mathbb{G}_{m}\ltimes LG$
where the rotational torus $\mathbb{G}_{m}$ acts on $LG$ by acting on the
uniformizer (cf. §4). Let $G^{{}^{\text{aff}}}$ denote the Kac-Moody group
associated to the affine Dynkin diagram of $G$. Recall that
$G^{{}^{\text{aff}}}$ is given by a central extension of $L^{\ltimes}G$ by
$\mathbb{G}_{m}$.

Let $T_{{}_{\text{ad}}}:=T/Z(G)$ and let us denote by
$T^{\ltimes}_{{}_{\text{ad}}}$ the torus $\mathbb{G}_{m}\times
T_{{}_{\text{ad}}}\subset G^{{}^{\text{aff}}}_{{}_{\text{ad}}}$. In ${\bf
X}^{{}^{\text{aff}}}$, the closure ${\bf
Y}^{{}^{\text{aff}}}:=\overline{T^{\ltimes}_{{}_{\text{ad}}}}$ gives a torus-
embedding. The complement $Z:={\bf Y}^{{}^{\text{aff}}}\setminus
T^{\ltimes}_{{}_{\text{ad}}}$ is a union
$\cup_{{}_{\alpha\in\mathbb{S}}}H_{{}_{\alpha}}$ of $\ell+1$ smooth divisors
meeting at normal crossings. For $\alpha\in\mathbb{S}$, let
$\xi_{{}_{\alpha}}\in H_{{}_{\alpha}}$ denote the generic points of the
divisors $H_{{}_{\alpha}}$’s. Let $A_{{}_{\alpha}}=\mathcal{O}_{{}_{{\bf
Y}^{{}^{\text{aff}}},\xi_{{}_{\alpha}}}}$ be the dvr’s obtained by localizing
at the height $1$-primes given by the $\xi_{{}_{\alpha}}$’s. Let
$Y_{{}_{\alpha}}:={\rm Spec}\,(A_{{}_{\alpha}})$. In Theorem 4.1 we show the
existence of a finite dimensional Lie-algebra bundle $J$ on ${\bf
Y}^{{}^{\text{aff}}}$ which extends the trivial bundle with fiber
$\mathfrak{g}$ on the open dense subset
${T^{\ltimes}_{{}_{\text{ad}}}}\cap{\bf Y}^{{}^{\text{aff}}}\subset{\bf
Y}^{{}^{\text{aff}}}$ and for each $\alpha\in\mathbb{S}$ we have
$L^{+}(J_{{}_{Y_{{}_{\alpha}}}})\simeq
L^{+}({\text{Lie}}(\mathcal{G}_{{}_{\alpha}}))$. Then in Proposition 5.1 we
show the existence of a finite dimensional Lie-algebra bundle $\bf R$ on ${\bf
X}^{{}^{\text{aff}}}$ which extends the trivial Lie algebra bundle
$G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\times\mathfrak{g}$ on the open dense
subset $G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\subset{\bf X}^{{}^{\text{aff}}}$
and whose restriction to ${\bf Y}^{{}^{\text{aff}}}$ is $J$.

The ind-scheme ${\bf X}^{{}^{\text{aff}}}$ has divisors $D_{\alpha}$ for
$\alpha\in\mathbb{S}$ such that the complement of their union is ${\bf
X}^{{}^{\text{aff}}}\setminus G_{ad}^{aff}$. With these notations, let us
state Theorem 5.2.

######  Theorem 1.2.

There exists an affine “wonderful" Bruhat-Tits group scheme
${\mathcal{G}}_{{}_{{\bf X}^{{}^{\text{aff}}}}}^{{}^{\varpi}}$ on ${\bf
X}^{{}^{\text{aff}}}$ together with a canonical isomorphism
$\text{Lie}({\mathcal{G}}_{{}_{\bf X^{aff}}}^{{}^{\varpi}})\simeq{\bf R}$. It
further satisfies the following classifying property:

For $h\in{\bf X}^{{}^{\text{aff}}}\setminus G_{ad}^{aff}$, let
$\mathbb{I}\subset\mathbb{S}$ be the subset such that $h\in\cap_{\alpha\in
I}D_{\alpha}$. Let $C_{{}_{\mathbb{I}}}\subset{\bf X}^{{}^{\text{aff}}}$ be a
smooth curve with generic point in $G^{aff}_{{}_{\text{ad}}}$ and closed point
$h$. Let $U_{{}_{h}}\subset C_{{}_{\mathbb{I}}}$ be a formal neighbourhood of
the closed point $h$. Then, the restriction ${\mathcal{G}}_{{}_{{\bf
X}^{{}^{\text{aff}}}}}^{{}^{\varpi}}|_{{}_{U_{{}_{h}}}}$ is isomorphic to the
Bruhat-Tits group scheme $\mathcal{G}_{{}_{\mathbb{I}}}$ §(2.0.4) on
$U_{{}_{h}}$.

######  Acknowledgments .

We thank J. Heinloth and M.Brion for their questions and comments. They have
helped us very much in expressing our results with greater precision.

##  2\. Preliminaries

####  2.0.1. Lie-data of $G/k$

Let $G$ be an almost simple, simply-connected group over $k$ (see (3.11)) with
the data $(T,B,G)$. Let $X(T)\,=\,\text{Hom}(T,\mathbb{G}_{{}_{m}})$ be the
group of characters of $T$ and $Y(T)\,=\,\text{Hom}(\mathbb{G}_{m},T)$ be the
group of all one–parameter subgroups of $T$. Let
$G_{{}_{{}_{\text{ad}}}}:=G/Z(G)$ and let $\mathfrak{g}$ denote the Lie-
algebra. We denote $\Phi^{+},\Phi^{-}\subset\Phi$ the set of positive and
negative roots with respect to $B$. Let
$S=\\{\alpha_{{}_{1}},\ldots,\alpha_{{}_{\ell}}\\}$ denote the set of simple
roots of $G$, where $\ell$ is the rank of $G$. Let $\alpha^{\vee}$ denote the
coroot corresponding to $\alpha\in S$ .

####  2.0.2. Apartment data

Let $\mathbb{S}=S\cup\\{\alpha_{0}\\}$ denote the set of affine simple roots.
Let $\mathcal{A}_{T}$ denote the affine apartment corresponding to $T$. It can
be identified with the affine space
${\mathbb{E}}=Y(T)\otimes_{\mathbb{Z}}\mathbb{R}$ together with its origin
$0$.

Let $\mathbf{a}_{0}$ be the unique Weyl alcove of $G$ whose closure contains
$0$ and which is contained in the dominant Weyl chamber corresponding to $B$.
Under the natural pairing between $Y(T)\otimes_{\mathbb{Z}}\mathbb{Q}$ and
$X(T)\otimes_{\mathbb{Z}}\mathbb{Q}$, the integral basis elements dual to $S$
are called the fundamental co-weights $\\{\omega^{\vee}_{\alpha}|\alpha\in
S\\}$. Let $c_{\alpha}$ be the coefficient of $\alpha$ in the highest root.
The vertices of the Weyl alcove $\mathbf{a}_{0}$ are indexed by $0$ and

$\displaystyle\theta_{{}_{\alpha}}:={{\omega^{\vee}_{\alpha}}\over{c_{\alpha}}},\alpha\in
S.$ (2.0.1)

Indeed, any rational point $\theta\in\mathbf{a}_{0}$ can be expressed as
$\theta_{{}_{\lambda}}$ so that there is a unique pair
$(d,\lambda)\in\mathbb{N}\times Y(T)$ defined by the condition that $d$ is the
least positive integer such that

$\lambda=d.\theta_{{}_{\lambda}}\in Y(T).$ (2.0.2)

Thus, if $e_{\alpha}$ is the order of $\omega^{\vee}_{\alpha}$ in the quotient
of the co-weight lattice by $Y(T)$, it follows that for
$\theta_{{}_{\alpha}}$, the number $d$ is:

$d_{{}_{\alpha}}:=e_{\alpha}.c_{\alpha}.$ (2.0.3)

An affine simple root $\alpha\in\mathbb{S}$ may be viewed as an affine
functional on $\mathcal{A}_{T}$. Any non-empty subset
$\mathbb{I}\subset\mathbb{S}$ defines the facet
$\Sigma_{\mathbb{I}}\subset\overline{\mathbf{a}_{0}}$ where exactly the
$\alpha$’s not lying in $\mathbb{I}$ vanish. So $\mathbb{S}$ corresponds to
the interior of the alcove and the vertices of the alcove correspond to
$\alpha\in\mathbb{S}$. Conversely any facet
$\Sigma\subset\overline{\mathbf{a}_{0}}$ defines non-empty subset
$\mathbb{I}\subset\mathbb{S}$. For $\emptyset\neq\mathbb{I}\subset\mathbb{S}$,
the barycenter of $\Sigma_{\mathbb{I}}$ is given by

$\theta_{\mathbb{I}}:=\frac{1}{|\mathbb{I}|}\sum_{\alpha\in\mathbb{I}}\theta_{{}_{\alpha}}.$
(2.0.4)

####  2.0.3. Loop groups and their parahoric subgroups

Let $k=\mathbb{C}$, $A=k\llbracket z\rrbracket$ and let $R$ denote an
arbitrary $k$-algebra. We denote the field of Laurent polynomials by
$K=k(\\!(z)\\!)=k\llbracket z\rrbracket[z^{-1}]$. The loop group $LG$ on a
$k$-algebra $R$ is given by $G(R(\\!(t)\\!))$. Similarly the loop Lie algebra
$L\mathfrak{g}$ is given by $L\mathfrak{g}(R)=\mathfrak{g}(R(\\!(t)\\!))$. We
can similarly define the positive loops (also called jet groups in the
literature) $L^{+}(G)$ to be the subfunctor of $LG$ defined by
$L^{+}(G)(R):=G(R\llbracket z\rrbracket)$. The positive loop construction
extends more generally to any group scheme $\mathcal{G}\rightarrow{\rm
Spec}\,(A)$ and any vector bundle $\mathfrak{P}\rightarrow{\rm Spec}\,(A)$
whose sheaf of sections carries a Lie-bracket.

Let $L^{\ltimes}G=\mathbb{G}_{m}\ltimes LG$ where the rotational torus
$\mathbb{G}_{m}$ acts on $LG$ by acting on the uniformizer via the loop
rotation action as follows: $u\in\mathbb{G}_{m}(R)$ acts on $\gamma(z)\in
LG(R)=G(R(\\!(t)\\!))$ by $u\gamma(z)u^{-1}=\gamma(uz)$. A maximal torus of
$L^{\ltimes}G$ is $T^{\ltimes}=\mathbb{G}_{m}\times T$. A $\eta\in
Hom(\mathbb{G}_{m},T^{\ltimes})\otimes_{\mathbb{Z}}\mathbb{Q}$ over $R(s)$ can
be viewed as a rational $1$-PS, i.e., a $1$-PS $\mathbb{G}_{m}\rightarrow
T^{\ltimes}$ over $R(w)$ where $w^{n}=s$ for some $n\geq 1$. In this case for
$\gamma(z)\in L^{\ltimes}G(R)$ we will view $\eta(s)\gamma(z)\eta(s)^{-1}$ as
an element in $L^{\ltimes}G(R(w))$. Hence, by the condition on $\gamma(z)\in
L^{\ltimes}G(R)$

“$\lim_{{}_{s\rightarrow 0}}\eta(s)\gamma(z)\eta(s)^{-1}$ exists in
$L^{\ltimes}G(R)$",

we mean that there exists a $n\geq 1$ such that for $w^{n}=s$, we have

$\eta(s)\gamma(z)\eta(s)^{-1}\in L^{\ltimes}G(R\llbracket w\rrbracket).$
(2.0.5)

If $\eta=(a,\theta)$ for $a\in\mathbb{Q}$ and $\theta$ a rational $1$-PS of
$T$, then we have

$\eta(s)\gamma(z)\eta(s)^{-1}=\theta(s)\gamma(s^{a}z)\theta(s)^{-1}.$ (2.0.6)

We note further that for any $0<d\in\mathbb{N}$, setting
$\eta=(\frac{a}{d},\frac{\theta}{d})$ we have

$\eta(s)\gamma(z)\eta(s)^{-1}=\theta(s^{\frac{1}{d}})\gamma(s^{\frac{a}{d}}z)\theta(s^{\frac{1}{d}})^{-1}.$
(2.0.7)

So for $\eta=\frac{1}{d}(1,\theta)$ observe that the statement
"$\lim_{s\rightarrow 0}\eta(s)\gamma(z)\eta(s)^{-1}$ exists " is a condition
which is equivalent to

$\theta(s)\gamma(s)\theta(s)^{-1}\in L^{\ltimes}G(R\llbracket w\rrbracket).$
(2.0.8)

In other words, the condition of existence of limits is independent of $d$,
and we may further set $z=s$ in $\gamma(z)$. More generally, if $a>0$ the
condition for $(a,\theta)$ and $(1,\frac{\theta}{a})$ are equivalent. We may
write this condition on $\gamma(s)\in L^{\ltimes}G(R)$ or $LG(R)$ as

$"\lim_{s\rightarrow 0}\theta(s)\gamma(s)\theta(s)^{-1}\quad\text{ exists
in}\quad L^{\ltimes}G(R)\quad\text{or}\quad
LG(R)\quad\text{if}\quad\gamma(z)\in LG(R)"$ (2.0.9)

For $r\in\Phi$, let $u_{r}:\mathbb{G}_{a}\rightarrow G$ denote the root
subgroup. If for some $b\in\mathbb{Z}$ and $t(z)\in R\llbracket z\rrbracket$
we take $\gamma(z):=u_{r}(z^{b}t(z))\in Lu_{r}(R)$ , and $\eta:=(1,\theta)$,
then

$\eta(s)u_{r}(z^{b}t(z))\eta(s)^{-1}=\theta(s)u_{r}((sz)^{b}t(sz))\theta(s)^{-1}=u_{r}(s^{r(\theta)}(sz)^{b}t(sz)).$
(2.0.10)

So for $\eta=(1,\theta)$ the condition that the limit exists is equivalent to

$r(\theta)+b\geq 0\iff\lfloor r(\theta)+b\rfloor=\lfloor
r(\theta)\rfloor+b\geq 0\iff b\geq-\lfloor r(\theta)\rfloor.$ (2.0.11)

We note the independence of the above implications on the number $d$ occurring
in the equation $\eta:=\frac{1}{d}(1,\theta)$.

Let $\pi_{{}_{1}}:T^{\ltimes}\rightarrow\mathbb{G}_{m}$ be the first
projection. For any rational $1$-PS $\eta:\mathbb{G}_{m}\rightarrow
T^{\ltimes}$ we say $\eta$ is positive if $\pi_{{}_{1}}\circ\eta>0$, negative
if $\pi_{{}_{1}}\circ\eta<0$ and non-zero if it is either positive or
negative.

Any non-zero $\eta=(a,\theta)$ defines the following positive-loop functors
from the category of $k$-algebras to the category of groups and Lie-algebras:

$\displaystyle{\mathcal{P}}_{{}_{\eta}}^{\ltimes}(R):=\\{\gamma\in
L^{\ltimes}G(R)|\quad\lim_{s\rightarrow
0}\eta(s)\gamma(z)\eta(s)^{-1}\quad\text{exists in}\quad L^{\ltimes}G(R)\\},$
(2.0.12) $\displaystyle{\mathfrak{P}}_{{}_{\eta}}^{\ltimes}(R):=\\{h\in
L^{\ltimes}\mathfrak{g}(R)|\quad\lim_{s\rightarrow
0}{Ad}(\eta(s))(h(z))\quad\text{exists in}\quad
L^{\ltimes}\mathfrak{g}(R)\\},$ (2.0.13)
$\displaystyle{\mathcal{P}}_{{}_{\eta}}(R):=\\{\gamma\in
LG(R)|\quad\lim_{s\rightarrow 0}\eta(s)\gamma(z)\eta(s)^{-1}\quad\text{exists
in}\quad LG(R)\\},$ (2.0.14)
$\displaystyle{\mathfrak{P}}_{{}_{\eta}}(R)=\\{h\in
L\mathfrak{g}(R)|\quad\lim_{s\rightarrow 0}Ad(\eta(s))(h(z))\quad\text{exists
in}\quad L\mathfrak{g}(R)\\},$ (2.0.15)
$\displaystyle\text{Thus},~{}~{}\mathcal{P}_{{}_{\eta}}:=\mathcal{P}_{{}_{\eta}}^{\ltimes}\cap(1\times
LG)\quad\text{and}\quad{\mathfrak{P}}_{{}_{\eta}}:={\mathfrak{P}}_{{}_{\eta}}^{\ltimes}\cap(0\oplus
L\mathfrak{g}).$ (2.0.16)

A parahoric subgroup of $L^{\ltimes}G$ (resp. $LG$) is a subgroup that is
conjugate to $\mathcal{P}^{\ltimes}_{{}_{\eta}}$ (resp.
$\mathcal{P}_{{}_{\eta}}$) for some $\eta$. In this paper, we will mostly be
using only the case when $\eta=\frac{1}{d}(1,\theta)$. In this case, letting
$L\mathfrak{g}(R)=\mathfrak{g}(R(\\!(s)\\!))$ as in (2.0.9), we may
reformulate (2.0.16) as

${\mathfrak{P}}_{{}_{\eta}}(R)=\\{h\in
L\mathfrak{g}(R)|\quad\lim_{s\rightarrow
0}Ad(\theta(s))(h(s))\quad\text{exists in}\quad L\mathfrak{g}(R)\\}.$ (2.0.17)

By the conditions (2.0.12) and (2.0.16), using (2.0.11), we may express
$\mathcal{P}_{{}_{\eta}}$ in terms of generators as follows:

$\mathcal{P}_{{}_{\eta}}(R)=<T(R\llbracket
z\rrbracket),u_{r}(z^{-\lfloor(r,\theta)\rfloor}R\llbracket
z\rrbracket),r\in\Phi>,$ (2.0.18)

Let $\mathfrak{t}$ denote the Cartan sub-algebra of $T$ and $\mathfrak{u}_{r}$
denote the root algebras associated to $u_{r}$. Then the Lie-algebra functor
of $\mathcal{P}_{{}_{\eta}}$ in terms of generators is given by

$Lie(\mathcal{P}_{{}_{\eta}})(R)=<\mathfrak{t}(R\llbracket
z\rrbracket),\mathfrak{u}_{r}(z^{-\lfloor(r,\theta)\rfloor}R\llbracket
z\rrbracket),r\in\Phi>.$ (2.0.19)

Thus, for any $\eta$ of the type $(1,\theta)$, we get the equality

$\text{Lie}(\mathcal{P}_{{}_{\eta}})={\mathfrak{P}}_{{}_{\eta}}.$ (2.0.20)

This can be seen by (2.0.13) and (2.0.16) and then replacing the conjugation
in (2.0.6) by $Ad(\eta(s))$.

For a rational $1$-PS $\theta$ of $T$, with $\eta=(1,\theta)$ we will have the
notation:

${\mathfrak{P}}_{{}_{\theta}}:={\mathfrak{P}}_{{}_{\eta}}.$ (2.0.21)

In particular, for $r\in\Phi$ let $\mathfrak{g}_{{}_{r}}\subset\mathfrak{g}$
be the root subspace. Let $m_{{}_{r}}(\theta):=-\lfloor{(r,\theta)}\rfloor$.
The parahoric Lie algebra
${\mathfrak{P}}_{{}_{\theta}}(k)\subset\mathfrak{g}(K)$ has $T$-weight space
decomposition as:

$\displaystyle{\mathfrak{P}}_{{}_{\theta}}(k)=\mathfrak{t}(A)\bigoplus
z^{{}^{m_{{}_{r}}(\theta)}}\mathfrak{g}_{{}_{r}}(A).$ (2.0.22)

####  2.0.4. Bruhat-Tits group scheme

To each facet $\Sigma_{\mathbb{I}}\subset\overline{\mathbf{a}_{0}}$, Bruhat-
Tits theory associates a smooth group scheme $\mathcal{G}_{{}_{\mathbb{I}}}$
on ${\rm Spec}\,~{}A$, with connected fibres and whose generic fibre is
$G\times_{{}_{{\rm Spec}\,k}}{\rm Spec}\,~{}K$. We call
$\mathcal{G}_{{}_{\mathbb{I}}}$ a Bruhat-Tits group scheme on ${\rm Spec}\,A$.
To $\mathcal{G}_{{}_{\mathbb{I}}}$, we can associate a pro-algebraic group
$L^{+}(\mathcal{G}_{{}_{\mathbb{I}}})$ over $k$ as follows:

$L^{+}(\mathcal{G}_{{}_{\mathbb{I}}})(R):=\mathcal{G}_{{}_{\mathbb{I}}}(R\llbracket
z\rrbracket).$ (2.0.23)

The $L^{+}(\mathcal{G}_{{}_{\mathbb{I}}})$ also characterise
$\mathcal{G}_{{}_{\mathbb{I}}}$. We thus have the identifications:

$L^{+}(\mathcal{G}_{{}_{\mathbb{I}}})=\mathcal{P}_{{}_{\eta}}\quad\text{and}\quad\text{Lie}(L^{+}(\mathcal{G}_{{}_{\mathbb{I}}}))={\mathfrak{P}}_{{}_{\eta}}$
(2.0.24)

for $\eta=(1,\theta)$ where $\theta$ is any rational $1$-PS lying in
$\Sigma_{\mathbb{I}}$.

####  2.0.5. Standard Parahoric subgroups

The standard parahoric subgroups of $G(K)$ are parahoric subgroups of the
distinguished hyperspecial parahoric subgroup $G(A)$. These are realized as
inverse images under the evaluation map $ev:G(A)\to G(k)$ of standard
parabolic subgroups of $G$. In particular, the standard Iwahori subgroup
${\mathfrak{I}}$ is a standard parahoric and indeed,
${\mathfrak{I}}=ev^{-1}(B)$. Denoting by $Q_{{}_{I}}\subset G$ the parabolic
subgroup associated to the subset $I\subset S$ we will denote by
$\mathcal{G}^{{}^{st}}_{{}_{{I}}}$ the standard parahoric subgroups of $G(A)$
determined by

$L^{+}(\mathcal{G}^{{}^{st}}_{{}_{{I}}})(k):=ev^{-1}(Q_{{}_{I}}).$ (2.0.25)

Thus, for ${I}\subset S$, setting $\mathbb{I}:=I\cup\\{\alpha_{0}\\}$ we have

$L^{+}(\mathcal{G}^{{}^{st}}_{{}_{{I}}})(k)=L^{+}(\mathcal{G}_{{}_{\mathbb{I}}})(k)\quad\text{and}\quad
L^{+}(\mathfrak{P}_{{}_{I}}^{{}^{st}})=L^{+}(\mathfrak{P}_{{}_{\mathbb{I}}}).$
(2.0.26)

##  3\. A Bruhat-Tits group scheme on the wonderful compactification

####  3.0.1. The structure of ${\bf X}$

Let ${\bf X}:=\overline{G_{{}_{{}_{\text{ad}}}}}$ be the wonderful
compactification of $G_{{}_{{}_{\text{ad}}}}$. Let $\\{D_{\alpha}|\alpha\in
S\\}$ denote the irreducible smooth divisors of ${\bf X}$. Let
$D:=\cup_{{\alpha\in S}}D_{\alpha}$. Then $X\setminus
G_{{}_{{}_{\text{ad}}}}=D$. The pair $({\bf X},D)$ is the primary example of a
$(G_{{}_{{}_{\text{ad}}}}\times G_{{}_{{}_{\text{ad}}}})$-homogenous pair [5,
Brion]. Let $P_{{}_{I}}$ be the standard parabolic subgroup defined by subsets
$I\subset S$ , the notation being such that the Levi subgroup $L_{{}_{I,ad}}$
containing $T_{{}_{\text{ad}}}$ has root system with basis $S\setminus I$.
Recall that the $(G_{{}_{{}_{\text{ad}}}}\times
G_{{}_{{}_{\text{ad}}}})$-orbits in $\bf X$ are indexed by subsets $I\subset
S$ and have the following description:

$Z_{{}_{I}}=(G_{{}_{{}_{\text{ad}}}}\times
G_{{}_{{}_{\text{ad}}}})\times_{{}_{P_{{}_{I}}\times
P^{-}_{{}_{I}}}}L_{{}_{I,ad}}.$

Then by [4, Proposition A1], each $Z_{{}_{I}}$ contains a unique base point
$z_{{}_{I}}$ such that $(B\times B^{-}).z_{{}_{I}}$ is dense in $Z_{{}_{I}}$
and there is a $1$-PS $\lambda$ of $T$ satisfying $P_{{}_{I}}=P(\lambda)$ and
$\lim_{{}_{t\to 0}}\lambda(t)=z_{{}_{I}}$. The closures of these $\lambda$
define curves $C_{{}_{I}}\subset{\bf X}$ which meet the strata $Z_{{}_{I}}$
transversally at $z_{{}_{I}}$. In particular, if $I=\\{\alpha\\}$ a singleton,
then the divisor $D_{{}_{\alpha}}$ is the orbit closure $\bar{Z}_{{}_{I}}$ and
the $1$-PS can be taken to be the fundamental co-weight
$\omega^{\vee}_{\alpha}$. The closure of the $1$-PS
$\omega^{\vee}_{\alpha}:\mathbb{G}_{{}_{m}}\to G_{{}_{\text{ad}}}$ defines the
curve $C_{{}_{\alpha}}\subset\bf X$ transversal to the divisor
$D_{{}_{\alpha}}$ at the point $z_{{}_{\alpha}}$.

####  3.0.2. The local toric structure of ${\bf X}$

Let ${\bf Y}:=\overline{T_{{}_{\text{ad}}}}$ be the closure of
$T_{{}_{\text{ad}}}$ in ${\bf X}$. Recall that ${\bf Y}$ is a projective toric
variety associated to the fan of Weyl chambers. In what follows, we will
mostly work with the affine toric embedding ${\bf
Y}_{{}_{0}}:=\overline{T_{{}_{ad,0}}}\simeq{\mathbb{A}}^{{}^{\ell}}$ which is
the toric variety associated to the negative Weyl chamber. Let $U$ (resp
$U^{{}^{-}}$) be the unipotent radical (resp. its opposite) of the Borel
subgroup $B\subset G$. We also recall that one may identify $U\times
U^{{}^{-}}\times{\bf Y}_{{}_{0}}$ with an open subset of $\bf X$ and moreover
the $G\times G$-translates of $U\times U^{{}^{-}}\times{\bf Y}_{{}_{0}}$
covers the whole of $\bf X$.

####  3.0.3. Construction of a Lie-algebra bundle $\mathcal{R}$ on ${\bf
Y}_{{}_{0}}$

The aim of this section is to construct a Lie algebra bundle on ${\bf
Y}_{{}_{0}}$ (see (3.3)), which may be termed “parahoric". We close by making
a similar construction on $\bf Y$ and $\bf X$. The construction is motivated
by the structure of the kernel of the Tits fibration in [5].

We begin with a couple of elementary lemmas which should be well known.

######  Lemma 3.1.

Let $E$ be a locally free sheaf on an irreducible smooth scheme $X$. Let
$\xi\in X$ be the generic point and let $W\subset E_{{}_{\xi}}$ be an
$\mathcal{O}_{{}_{\xi}}$-submodule. Then there exists a unique coherent
subsheaf $F\subset E$ such that $F_{{}_{\xi}}=W$ and
$Q:=Coker(F\hookrightarrow E)$ is torsion-free. Moreover $F$ is a reflexive
sheaf.

###### Proof.

Define $\tilde{F}$ on the affine open $U\subset X$ by $\tilde{F}(U):=E(U)\cap
W$. Then it is easily seen that $\tilde{F}$ defines a coherent subsheaf
$F\subset E$ and it is the maximal coherent subsheaf of $E$ whose fiber over
$\xi$ is $W$. To check $Q$ is torsion-free, let $T$ be the torsion submodule
of $Q$. Let $K:=Ker(E\to Q/T)$. Then since $\xi\notin Supp(T)$, so
$K_{{}_{\xi}}=W$ and hence, by the maximality of $F$ we have $K=F$. Since $E$
is locally free, we have $F^{\vee\vee}/F\hookrightarrow E/F$. But since
$F^{\vee\vee}/F$ is only torsion, and $E/F=Q$ is torsion-free, it follows that
$F$ is automatically a reflexive sheaf.∎

######  Lemma 3.2.

Let $X$ be as above and $i:U\hookrightarrow X$ an open subset such that
$X\setminus U$ has codimension $\geq 2$ in $X$. Let $F_{{}_{U}}$ be a
reflexive sheaf on $U$. Then $i_{{}_{*}}(F_{{}_{U}})$ is a reflexive sheaf on
$X$ which extends $F_{{}_{U}}$.

###### Proof.

. By [11, Corollaire 9.4.8], there exists a coherent
$\mathcal{O}_{{}_{X}}$-module $F_{1}$ such that $F_{1}|_{{}_{U}}\simeq
F_{{}_{U}}$. Set $F:=F_{1}^{{}^{**}}$ to be the double dual. Then $F$ is
reflexive and also since $F_{{}_{U}}$ is reflexive, $F|_{{}_{U}}\simeq
F_{{}_{U}}$. Hence, we have $i_{{}_{*}}(F_{{}_{U}})=F$ [12, Proposition 1.6].
∎

Let $\lambda=\sum_{\alpha\in S}k_{{}_{\alpha}}\omega_{{}_{\alpha}}^{{\vee}}$
be a dominant $1$-PS of $T_{{}_{\text{ad}}}$. It defines the curve
$C_{{}_{\lambda}}\subset\bf Y$. When the $k_{{}_{\alpha}}$ are constrained to
be in $\\{0,1\\}$, then these curves are the standard curves
$C_{{}_{I}},I\subset S$ considered in §(3.0.1), which meet the strata
transversally at the points $z_{{}_{I}}$. We call these dominant $\lambda$’s
as standard. In particular, the curve defined by
$\omega_{{}_{\alpha}}^{{\vee}}$ meets the divisor $H_{{}_{\alpha}}$
transversally at $z_{{}_{\alpha}}$. The formal neighbourhood of the closed
point $z_{{}_{\lambda}}$ in $C_{{}_{\lambda}}$ identifies with the spectrum
$U_{{}_{\lambda}}:={\rm Spec}\,(A_{{}_{\lambda}})$ of
$A_{{}_{\lambda}}=k\llbracket s\rrbracket$ with quotient field
$K_{{}_{\lambda}}=k(\\!(s)\\!)$. We set

$\theta_{\lambda}:=\sum_{\alpha\in
S}k_{{}_{\alpha}}\frac{\theta_{{}_{\alpha}}}{\ell+1}.$ (3.0.1)

When $k_{\alpha}\in\\{0,1\\}$, then $\theta_{\lambda}$ does not lie on the far
wall of $\mathbf{a}_{0}$. So
$\mathfrak{P}^{{}^{st}}_{{}_{\theta_{\lambda}}}=\mathfrak{P}_{{}_{\theta_{\lambda}}}$
§(2.0.5).

######  Theorem 3.3.

There exists a canonical Lie-algebra bundle $\mathcal{R}$ on ${\bf
Y}_{{}_{0}}$ which extends the trivial bundle with fiber $\mathfrak{g}$ on
${T_{{}_{\text{ad}}}}\subset{\bf Y}_{{}_{0}}$ and such that for
$K_{{}_{\alpha}}\in\\{0,1\\}$ we have the identification of functors from the
category of $k$-algebras to $k$-Lie-algebras:

$L^{+}(\mathfrak{P}^{{}^{st}}_{{}_{\theta_{\lambda}}})=L^{+}(\mathcal{R}\mid_{{}_{U_{{}_{\lambda}}}}).$
(3.0.2)

###### Proof.

With notations as in (2.0.3), consider the inclusion of lattices:

$\bigoplus_{{}_{\alpha\in
S}}{\mathbb{Z}}.\omega_{{}_{\alpha}}^{{\vee}}\hookrightarrow\bigoplus_{{}_{\alpha\in
S}}{\mathbb{Z}}.\frac{{\theta_{{}_{\alpha}}}}{e_{{}_{\alpha}}.(\ell+1)}.$
(3.0.3)

This induces an inclusion of $k$-algebras $B_{{}_{0}}\subset B$, where

$B:=k[y_{\alpha},y_{\alpha}^{-1}]_{{}_{\alpha\in S}},~{}~{}\\\
B_{{}_{0}}:=k[x_{\alpha},x_{\alpha}^{-1}]_{{}_{\alpha\in S}}$ (3.0.4)

and $B_{{}_{0}}\subset B$ is given by the equations:
$\\{{y_{{}_{\alpha}}^{{}^{(\ell+1)d_{{}_{\alpha}}}}=x_{{}_{\alpha}}}\\}_{{}_{\alpha\in
S}}$. Let ${\bf T}:={\rm Spec}\,(B),\text{and}T_{{}_{\text{ad}}}={\rm
Spec}\,(B_{{}_{0}})$ and let $p:{\bf T}\to T_{{}_{\text{ad}}}$ be the natural
morphism. We define the “roots" map:

$\displaystyle\mathfrak{r}:{\bf T}\to T_{{}_{\text{ad}}},~{}~{}~{}\text{as}$
(3.0.5)
$\displaystyle{\mathfrak{r}}^{\\#}\big{(}\big{(}x_{\alpha}\big{)}\big{)}:=\big{(}y_{\alpha}\big{)}.$
(3.0.6)

Note that as a map between tori, $\mathfrak{r}$ is an isomorphism. We consider
the map

$\displaystyle Ad\circ\mathfrak{r}:{\bf T}\times\mathfrak{g}\rightarrow{\bf
T}\times\mathfrak{g}$ (3.0.7) $\displaystyle({\bf t},x)\mapsto\big{(}{{\bf
t}},Ad\big{(}\mathfrak{r}({\bf t})\big{)}(x)\big{)}.$ (3.0.8)

We define the embedding of modules

$j:\mathfrak{g}(B_{{}_{0}})\hookrightarrow\mathfrak{g}(B)\stackrel{{\scriptstyle
Ad\circ\mathfrak{r}}}{{\rightarrow}}\mathfrak{g}(B)$ (3.0.9)

where the second map is the one induced by (3.0.7) on sections. When
$\mathfrak{g}(B_{0})$ is viewed as sections of the trivial bundle on
$T_{{}_{\text{ad}}}$ with fibers $\mathfrak{g}$, then it also has a
$T_{{}_{\text{ad}}}$-weight space decomposition.

Let $B^{+}=k[y_{{}_{\alpha}}]_{{}_{\alpha\in S}}$ and
$B_{{}_{0}}^{+}:=k[x_{{}_{\alpha}}]_{{}_{\alpha\in S}}$. Taking intersection
as Lie submodules of $\mathfrak{g}(B)$ we define the following
$B_{{}_{0}}^{+}$-module

$\mathcal{R}(B_{{}_{0}}^{+}):=j(\mathfrak{g}(B_{{}_{0}}))\cap\mathfrak{g}(B^{+}).$
(3.0.10)

Observe that $\mathcal{R}$ is a reflexive sheaf (3.1) on the affine embedding
$T_{{}_{\text{ad}}}\hookrightarrow{\bf Y}_{{}_{0}}:={\rm
Spec}\,(B_{{}_{0}}^{+})(\simeq{\mathbb{A}}^{{}^{\ell}})$. Further, this is a
sheaf of Lie algebras with its Lie bracket induced from $\mathfrak{g}(B)$.

We now check that $\mathcal{R}$ is locally-free on ${\bf Y}_{{}_{0}}$. Observe
that the intersection (3.0.10) also respects $T_{{}_{\text{ad}}}$-weight space
decomposition on the sections. More precisely, for a root $r\in\Phi$, since by
definition we have the identification
$\mathcal{R}(B_{{}_{0}}^{+})_{r}=j(\mathfrak{g}_{r}(B_{{}_{0}}))\cap\mathfrak{g}_{r}(B^{+})$,
we get the following equalities:

$j(\mathfrak{g}(B_{{}_{0}}))\cap\mathfrak{g}_{r}(B^{+})=j(\mathfrak{g}_{r}(B_{{}_{0}}))\cap\mathfrak{g}(B^{+})=\mathcal{R}(B_{{}_{0}}^{+})_{r}.$
(3.0.11)

Since $\mathcal{R}$ is reflexive, so there is an open subset $U\subset{\bf
Y}_{{}_{0}}$, whose complement is of codimension at least two such that the
restriction $\mathcal{R}^{\prime}:=\mathcal{R}|_{{}_{U}}$ is locally free.
Clearly $U$ contains $T_{{}_{\text{ad}}}$ and the generic points
$\zeta_{{}_{\alpha}}$ of the divisors $H_{{}_{\alpha}}$. This gives a
decomposition on $\mathcal{R}^{\prime}$ obtained by restriction from
$\mathcal{R}$. The locally free sheaf $\mathcal{R}^{\prime}$ is a direct sum
of the trivial bundle $\text{Lie}(T_{{}_{\text{ad}}})\times U$ (coming from
the $0$-weight space) and the invertible sheaves coming from the root
decomposition. Now since invertible sheaves extend across codimension $\geq
2$, the reflexivity of $\mathcal{R}$ implies that this direct sum
decomposition of $\mathcal{R}^{\prime}$ extends to ${\bf Y}_{{}_{0}}$ (see
[12, Proposition 1.6, page 126]). Whence $\mathcal{R}$ is locally free.

Let $\lambda:\mathbb{G}_{{}_{m}}\to T_{{}_{\text{ad}}}$ be a $1$-PS. This
defines a rational $1$-PS $\theta_{{}_{\lambda}}:\mathbb{G}_{{}_{m}}\to
T_{{}_{\text{ad}}}$. The map $\mathfrak{r}:{\bf T}\to T_{{}_{\text{ad}}}$ is
abstractly an isomorphism of tori and we can therefore consider the rational
$1$-PS $\boldsymbol{\theta_{{}_{\lambda}}}:\mathbb{G}_{{}_{m}}\to{\bf T}$
defined by
$\boldsymbol{\theta_{{}_{\lambda}}}:=\mathfrak{r}^{{}^{-1}}\circ\theta_{{}_{\lambda}}$.
Let $p:{\bf T}\to T_{{}_{\text{ad}}}$ be the canonical map induced by
$B_{{}_{0}}\subset B$. We observe the following:

  1. (1)

$p\circ\boldsymbol{\theta_{{}_{\lambda}}}=\lambda$,

  2. (2)

$\mathfrak{r}\circ\boldsymbol{\theta_{{}_{\lambda}}}=\theta_{{}_{\lambda}}.$

Let $\lambda=\sum k_{{}_{\alpha}}.\omega_{{}_{\alpha}}^{{\vee}}$ be a standard
dominant $1$-PS of $T_{{}_{\text{ad}}}$. For example, the case
$\lambda=\omega_{{}_{\alpha}}^{{\vee}}$, viewed as a $1$-PS, may be expressed
in $\ell$-many coordinates as follows:

$\omega_{{}_{\alpha}}^{{\vee}}(s)=(1,\ldots,s,\ldots 1)$ (3.0.12)

with $s$ at the coordinate corresponding to $\alpha\in S$. Thus, we may
express $\lambda$ as:

$\lambda(s)=\prod_{\alpha\in
S}\omega_{{}_{\alpha}}^{{\vee}}(s^{{}^{k_{{}_{\alpha}}}}).$ (3.0.13)

Set $f_{{}_{\alpha}}:=\frac{k_{{}_{\alpha}}}{d_{{}_{\alpha}}.(\ell+1)}$. In
this case we have the expression:

$\boldsymbol{\theta_{{}_{\lambda}}}(s)=\prod_{\alpha\in
S}\omega_{{}_{\alpha}}^{{\vee}}(s^{{}^{f_{{}_{\alpha}}}}).$ (3.0.14)

We now check the isomorphism
$L^{+}({\mathcal{R}}|_{{}_{U_{{}_{\lambda}}}})\simeq
L^{+}(\mathfrak{P}_{{}_{\theta_{\lambda}}}^{{}^{st}})$ by first evaluating at
$k$-valued points. By (3.0.10), a section s of $\mathcal{R}$ over
$U_{{}_{\lambda}}={\rm Spec}\,A_{{}_{\lambda}}$, is firstly given by a local
section $\text{\cursive s}_{{}_{K}}$ over the generic point of
$A_{{}_{\lambda}}=k\llbracket s\rrbracket$. This is firstly an element in
$j(\mathfrak{g}(K))$. Observe that this element may be written as

$\text{\cursive
s}_{{}_{K}}=Ad\big{(}\mathfrak{r}({\boldsymbol{\theta_{{}_{\lambda}}}(s)}\big{)}(x_{{}_{K}}))=Ad\big{(}\theta_{{}_{\lambda}}(s)\big{)}(x_{{}_{K}})).$

Let $d$ be any positive integer such that $d.\theta_{\lambda}$ becomes a
$1$-PS of $T_{{}_{\text{ad}}}$. Let $(\tilde{B},w,L)$ be defined by taking a
$d$-th root of the uniformizer $s$ of $A_{{}_{\lambda}}$ and let
$\tilde{U}_{{}_{\lambda}}={\rm Spec}\,(\tilde{B})$. By (3.0.14), we may
express $\boldsymbol{\theta_{{}_{\lambda}}}$ in terms of $w$ as

$\boldsymbol{\theta_{{}_{\lambda}}}(s)=\prod_{\alpha\in
S}\omega_{{}_{\alpha}}^{{\vee}}(w^{{}^{d.f_{{}_{\alpha}}}}).$

In other words,
$\theta_{\lambda}(s)=\theta_{\lambda}(w^{d})=(d\theta_{{}_{\lambda}})(w)$ has
become integral in $w$. Therefore, the membership of s in
$\mathfrak{g}(B^{+})$ (3.0.10) gets interpreted as follows:

$Ad(\theta_{\lambda}(s))(x_{K})\in L\mathfrak{g}(\tilde{B}).$ (3.0.15)

But this is exactly the Lie-algebra version of the condition
"$\lim_{s\rightarrow 0}$" condition for $\eta=(1,\theta)$ in the observation
(2.0.8). In our situation we have assumed $R=k$ and so (3.0.15) is equivalent
to

$\text{\cursive s}\in\mathfrak{P}_{{}_{\theta_{{}_{\lambda}}}}(k).$ (3.0.16)

But by (2.0.26),
$\mathfrak{P}_{{}_{\theta_{{}_{\lambda}}}}(k)=\mathfrak{P}_{{}_{\theta_{{}_{\lambda}}}}^{{}^{st}}(k)$
because $\theta_{{}_{\lambda}}$ lies in the alcove $\bf a_{{}_{0}}$ but not on
the far wall. Thus, we get the equality

$\mathcal{R}|_{{}_{U_{{}_{\lambda}}}}(A_{{}_{\lambda}})=\mathfrak{P}_{{}_{\theta_{{}_{\lambda}}}}^{{}^{st}}(k).$
(3.0.17)

This proves the assertion for $k$-valued points. The above proof goes through
for all $k$-algebras because the Lie-bracket is already defined on $k$, and so
is the underlying module structure. ∎

######  Remark 3.4.

Let $\lambda$ be an arbitrary dominant $1$-PS of $T_{{}_{\text{ad}}}$. In
general, the coefficients $k_{{}_{\alpha}}$ in its expression in terms of the
$\omega_{{}_{\alpha}}^{\vee}$ need not be such that $\sum
k_{{}_{\alpha}}/(\ell+1)<1$. Then we get the identification
$L^{+}(\mathfrak{P}_{{}_{\theta_{\lambda}}})=L^{+}(\mathcal{R}\mid_{{}_{U_{{}_{\lambda}}}})$
without the standardness of the parahoric.

######  Remark 3.5.

In the setting of Theorem 3.3, we may view $p:{\rm Spec}\,(B^{{}^{+}})\to{\rm
Spec}\,(B_{{}_{0}}^{{}^{+}})$ as a ramified covering space of affine toric
varieties induced by the inclusion (3.0.3) of lattices. The Galois group for
this covering is the dual of the quotient of lattices. The computation in 3.3
can be seen in the light of [1], in the sense that via an "invariant direct
image" process, one is able to recover the complete data of all standard
parahoric Lie algebras from this explicit Kawamata cover.

####  3.0.4. Construction of a Lie-algebra bundle $\mathcal{R}$ on ${\bf Y}$

In this subsection, we deduce the existence of Lie algebra bundles on the
projective non-singular toric variety $\bf Y$ with the classifying properties.
The variety $\bf Y$ is the toric variety for ${T_{{}_{\text{ad}}}}$ with fan
consisting of the Weyl chambers. The complement ${\bf
Y}\setminus{T_{{}_{\text{ad}}}}$ is a union of translates of the hyperplanes
$H_{{}_{\alpha}}\subset{\bf Y}_{{}_{0}}$ by the Weyl group $W$. Thus, for each
$w\in W$, we can take the locally free sheaf $\mathcal{R}$ of Lie-algebras on
${\bf Y}_{{}_{0}}$ and its $w$-translate $w.\mathcal{R}$ on ${\bf
Y}_{{}_{w}}:=w{\bf Y}_{{}_{0}}$ in $\bf Y$.

Let $Y^{\prime\prime}\subset{\bf Y}_{{}_{0}}$ be the open subset consisting of
the open orbit $T_{{}_{\text{ad}}}$ and of the orbits of codimension 1. In
other words, $Y^{\prime\prime}$ is obtained from ${\bf Y}_{{}_{0}}$ by
removing all the $T_{{}_{\text{ad}}}$-orbits of codimension at least 2. The
data above defines a locally free sheaf of Lie algebras on $\cup_{{}_{w\in
W}}w.Y^{\prime\prime}$ because $w.Y^{\prime\prime}\cap Y^{\prime\prime}$ is
either $Y^{\prime\prime}$ or $T_{{}_{\text{ad}}}$. Using (3.2), we get a
reflexive sheaf which by abuse of notation can be called $\mathcal{R}$ on the
whole of ${\bf Y}$. That $\mathcal{R}$ on $\bf Y$ is locally free is immediate
since its restrictions to the open cover given by ${\bf Y}_{{}_{0}}$ and its
$W$-translates is locally free. The Lie algebra structure also extends since
it is already there on an open subset with complement of codimension $\geq 2$.
Thus, we conclude:

######  Corollary 3.6.

There exists a canonical Lie-algebra bundle $\mathcal{R}$ on ${\bf Y}$ which
extends the trivial bundle with fiber $\mathfrak{g}$ on
${T_{{}_{\text{ad}}}}\subset{\bf Y}$ with properties as in (3.3) at the
translates of the hyperplanes $H_{{}_{\alpha}}$.

####  3.0.5. Weil restrictions and Lie algebras

Let $X$ be an arbitrary $k$-scheme. For an affine (or possibly ind-affine)
group scheme $\mathcal{H}\rightarrow X$, we denote $\text{Lie}(\mathcal{H})$
the sheaf of Lie-algebras on $X$ whose sections on $U\rightarrow X$ are given
by

$\text{Lie}(\mathcal{H})(U)=\text{ker}(\mathcal{H}(U\times
k[\epsilon])\rightarrow\mathcal{H}(U)).$ (3.0.18)

######  Lemma 3.7.

Let $p:\tilde{X}\rightarrow X$ be a finite flat map of noetherian schemes. Let
$Res_{\tilde{X}/X}$ denote the “Weil restriction of scalars" functor. Let
$\mathcal{H}\rightarrow\tilde{X}$ be an affine group scheme. Then we have a
natural isomorphism

${\text{Lie}}(Res_{\tilde{X}/X}\mathcal{H})\simeq
Res_{\tilde{X}/X}{\text{Lie}}(\mathcal{H}).$ (3.0.19)

When $p$ is also Galois with Galois group $\Gamma$, then in characteristic
$0$, we have

${\text{Lie}}((Res_{\tilde{X}/X}\mathcal{H})^{\Gamma})\simeq(Res_{\tilde{X}/X}{\text{Lie}}(\mathcal{H}))^{\Gamma}.$
(3.0.20)

###### Proof.

See [9, Page 533] and [10, 3.1, page 293] respectively. ∎

####  3.0.6. Restriction of $\mathcal{R}$ to infinitesimal standard curves
$U_{I}$

Let $U_{{}_{I}}$ be the formal neighbourhood of the standard curve
$C_{{}_{I}}$ §(3.0.3) at its closed point. We call the corresponding dominant
$1$-PS $\lambda$ of $T_{{}_{\text{ad}}}$ to be standard. Let
$\theta_{{}_{\lambda}}$ be the point in $\mathbf{a}_{0}$ (3.0.1).

Recall that by [1, Proposition 5.1.2] for each $\theta_{{}_{\lambda}}$ there
exists a ramified cover $q_{{}_{\lambda}}:U^{\prime}_{{}_{\lambda}}\to
U_{{}_{\lambda}}$ (of ramification index $d$ as in (2.0.2)) together with a
$\Gamma_{{}_{\lambda}}$-equivariant $G$-torsor $E_{{}_{\lambda}}$ such that
the adjoint group scheme $\mathcal{H}_{{}_{\lambda}}=E_{{}_{\lambda}}(G)$ has
simply-connected fibers isomorphic to $G$ and we have the identification of
$U_{{}_{\lambda}}$-group schemes:

$\displaystyle\big{(}\text{Res}_{{}_{U^{\prime}/U}}(\mathcal{H}_{{}_{\lambda}})\big{)}^{{}^{\Gamma_{{}_{\lambda}}}}\simeq\mathcal{G}^{{}^{st}}_{{}_{\theta_{\lambda}}}.$
(3.0.21)

We therefore get the following useful corollary to (3.3).

######  Corollary 3.8.

For any standard dominant $1$-PS $\lambda$ of $T_{{}_{ad}}$, with notations as
above we have an isomorphism as sheaves of Lie algebras:

$\mathcal{R}|_{{}_{U_{{}_{\lambda}}}}\simeq
q^{{}^{\Gamma}}_{{}_{\lambda,*}}(E_{{}_{\lambda}}(\mathfrak{g})).$ (3.0.22)

###### Proof.

This is an immediate consequence of (3.3) and [1, Proposition 5.1.2]. ∎

All results in this subsection generalize suitably to non-standard curves
corresponding to dominant $1$-PS in $T_{{}_{\text{ad}}}$ by (3.4).

####  3.0.7. The Lie algebra bundle $\mathcal{R}$ on $\bf X$

Let $\\{D_{\alpha}|\alpha\in S\\}$ denote the irreducible smooth boundary
divisors of ${\bf X}$. Set $H_{{}_{\alpha}}:=D_{{}_{\alpha}}\cap{\bf
Y}_{{}_{0}}$ for each $\alpha$. Recall that $Z:={\bf Y}_{{}_{0}}\setminus
T_{{}_{\text{ad}}}$ is a union $\cup_{{}_{\alpha\in S}}H_{{}_{\alpha}}$ which
are $\ell$ smooth hyperplanes meeting at simple normal crossings. For
$\alpha\in S$, let $\zeta_{{}_{\alpha}}$’s denote the generic points of the
divisors $H_{{}_{\alpha}}$’s. Let

$A_{{}_{\alpha}}=\mathcal{O}_{{}_{{{\bf Y}_{{}_{0}}},\zeta_{{}_{\alpha}}}}$
(3.0.23)

be the dvr’s obtained by localizing at the height $1$-primes given by the
$\zeta_{{}_{\alpha}}$’s and let $Y_{{}_{\alpha}}:={\rm
Spec}\,(A_{{}_{\alpha}})$. Base changing by the local morphism
$Y_{{}_{\alpha}}\to\bf Y_{{}_{0}}$, we have a Lie algebra bundle
$\mathcal{R}|_{{}_{Y_{{}_{\alpha}}}}$ for each $\alpha$. Moreover, the Lie
algebra bundle $\mathcal{R}$ on ${\bf Y}_{{}_{0}}$ gives canonical gluing data
to glue $\mathcal{R}|_{{}_{Y_{{}_{\alpha}}}}$ with the trivial bundle
$\mathfrak{g}\times T_{{}_{\text{ad}}}$.

Let $\xi_{{}_{\alpha}}\in D_{{}_{\alpha}},\alpha\in S$ denote the generic
points of the divisors $D_{{}_{\alpha}}$’s and let

$B_{{}_{\alpha}}:=\mathcal{O}_{{}_{{\bf X},\xi_{{}_{\alpha}}}}$ (3.0.24)

be the dvr’s obtained by localizing at the height $1$-primes given by the
$\xi_{{}_{\alpha}}$’s. By a transport of structures, for each $\alpha$ the
gluing datum of $\mathcal{R}|_{{}_{Y_{{}_{\alpha}}\cap T_{{}_{\text{ad}}}}}$,
gives a Lie-algebra bundle gluing datum on ${\rm Spec}\,(B_{{}_{\alpha}})\cap
G_{{}_{\text{ad}}}$. Thus, the gluing data of the bundle $\mathcal{R}$ on
${\bf Y}_{{}_{0}}$ at the the $\zeta_{{}_{\alpha}}$’s can be now used to
extend the trivial bundle $\mathfrak{g}\times G_{{}_{\text{ad}}}$ to the
$\xi_{{}_{\alpha}}$’s as a locally free sheaf of Lie algebras. The rest of the
proof is as for $\bf Y$, together with the observation that the $G\times
G$-translates of the open subset $U\times U^{{}^{-}}\times{\bf Y}_{{}_{0}}$
cover $\bf X$, where the bundle is simply the pull-back from ${\bf
Y}_{{}_{0}}$. Thus we have:

######  Corollary 3.9.

There exists a canonical Lie-algebra bundle $\mathcal{R}_{{}_{\bf X}}$ on
${\bf X}$ which extends the trivial bundle with fiber $\mathfrak{g}$ on
${G_{{}_{\text{ad}}}}\subset{\bf X}$ with properties as in (3.3).

####  3.0.8. Bruhat-Tits group scheme ${\mathcal{G}}_{{}_{\bf
X}}^{{}^{\varpi}}$ on $\bf X$

Let $L_{{}_{\alpha}}$ be the quotient field of $B_{{}_{\alpha}}$. Let
$X_{{}_{\alpha}}:={\rm Spec}\,(B_{{}_{\alpha}})$. By (3.8), for each $\alpha$
there exist a $\Gamma_{{}_{\alpha}}$-equivariant $G$-torsor $E_{\alpha}$ on a
ramified cover $q_{{}_{\alpha}}:X^{\prime}_{{}_{\alpha}}\to X_{{}_{\alpha}}$
such that the adjoint group scheme $\mathcal{H}_{{}_{\alpha}}=E_{\alpha}(G)$
has simply-connected fibers isomorphic to $G$ and we have the identification
of $B_{{}_{\alpha}}$-group schemes:

$\displaystyle\big{(}\text{Res}_{{}_{X^{\prime}_{{}_{\alpha}}/X_{{}_{\alpha}}}}(\mathcal{H}_{{}_{\alpha}})\big{)}^{{}^{\Gamma_{{}_{\alpha}}}}\simeq\mathcal{G}^{{}^{st}}_{{}_{\theta{{}_{{}_{\alpha}}}}}.$
(3.0.25)

Before stating the main result of this section, we make a few remarks which
might help the reader. The basic underlying principle in these constructions
is that the combinatorial data encoded in the triple consisting of the Weyl
chamber, the fan of Weyl chambers, and the Tits building, is geometrically
replicated by the inclusion ${\bf Y_{{}_{0}}}\subset{\bf Y}\subset\bf X$. The
"wonderful" Bruhat-Tits group scheme which arises on $\bf X$ has its local
Weyl-chamber model on the affine toric variety $\bf Y_{{}_{0}}$. Indeed, in
this case, the Kawamata cover is even explicit (3.5). In particular, the
theorem below, can be executed for ${\bf Y}_{{}_{0}}$ but this will give the
group scheme associated to the data coming from the Weyl chamber alone.

######  Theorem 3.10.

There exists an affine “wonderful" Bruhat-Tits group scheme
${\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}}$ on $\bf X$ satisfying the following
classifying properties.

  1. (1)

There is an identification of the Lie-algebra bundles
$\text{Lie}({\mathcal{G}}_{{}_{\bf
X}}^{{}^{\varpi}})\simeq\mathcal{R}_{{}_{\bf X}}$.

  2. (2)

For $\emptyset\neq I\subset S$ the restriction of ${\mathcal{G}}_{{}_{{\bf
X}}}^{{}^{\varpi}}$ to the formal neighbourhood $U_{{}_{z_{{}_{I}}}}$ of
$z_{{}_{I}}$ in $C_{{}_{I}}$ §(3.0.3) is isomorphic to the standard Bruhat-
Tits group scheme $\mathcal{G}^{{}^{st}}_{{}_{I}}$ §(2.0.5).

###### Proof.

Let $X_{{}_{\alpha}}$ be as above. Note that we can identify the open subset
${\rm Spec}\,(L_{{}_{\alpha}})$ with ${G_{{}_{\text{ad}}}}\cap
X_{{}_{\alpha}}$. Set

$X^{\prime}:=G_{{}_{\text{ad}}}\cup_{\alpha\in S}X_{{}_{\alpha}}.$ (3.0.26)

Hence ${\bf X}\setminus X^{\prime}$ is a colimit of closed subschemes of $\bf
X$ of codimension at least $2$. Consider the fpqc morphism:

$G_{{}_{{}_{\text{ad}}}}\bigsqcup_{\alpha\in
S}X_{{}_{\alpha}}\to{X^{\prime}}.$ (3.0.27)

We restrict $\mathcal{R}=\mathcal{R}_{{}_{\bf X}}$ further to $X^{\prime}$. By
(3.9) , over the open subset $G_{{}_{\text{ad}}}\subset X^{\prime}$, we have
$\mathcal{R}\simeq G_{ad}\times\mathfrak{g}$. Further, the transition
functions of $\mathcal{R}$ on the intersections ${\rm
Spec}\,(L_{{}_{\alpha}}):=G_{{}_{{}_{\text{ad}}}}\cap{\rm
Spec}\,(B_{{}_{\alpha}})$ take values in $\text{Aut}(\mathfrak{g})$. Since
$\text{Aut}(\mathfrak{g})=\text{Aut}(G)$, it follows that the effective
descent datum provided by pulling back $\mathcal{R}$ to
$G_{{}_{{}_{\text{ad}}}}\bigsqcup_{\alpha\in S}X_{{}_{\alpha}}$ gives a
descent datum to glue the trivial group scheme $G\times
G_{{}_{{}_{\text{ad}}}}$ on $G_{{}_{{}_{\text{ad}}}}$ with
$\mathcal{G}^{{}^{st}}_{{}_{\alpha}}$ on ${\rm Spec}\,(B_{{}_{\alpha}})$ along
${\rm Spec}\,(L_{{}_{\alpha}})$. Since the group schemes are affine, this
descent datum is also effective. In other words, we get the group scheme

$\mathcal{G}^{\circ}\rightarrow X^{\prime}.$ (3.0.28)

This can also be seen without “descent" theory as follows. Since the group
schemes $\mathcal{G}^{{}^{st}}_{{}_{\alpha}}$ on ${\rm
Spec}\,(B_{{}_{\alpha}})$ are of finite type, they can be extended to an
affine subscheme $\bf X_{{}_{f}}$. By a further shrinking of this
neighbourhood of $\xi_{{}_{\alpha}}$ one can glue it to $G\times
G_{{}_{{}_{\text{ad}}}}$ along the intersection. So we can think of
$X^{\prime}$ as an honest open subset of $X$ with complement of codimension at
least $2$.

By (3.8), the Lie algebra bundle $\mathcal{R}$ gets canonical parabolic
structures at the generic points $\xi_{{}_{\alpha}}$ of the divisors
$\\{D_{{}_{\alpha}}\\}_{{}_{\alpha\in S}}$. For a parabolic vector bundle with
prescribed rational weights such as $\mathcal{R}$, by [2] (see §8) we get the
following data:

  * •

a global Kawamata cover (8.0.2) $p:Z\rightarrow{\bf X}$ ramified over $D$ with
ramification prescribed by the weights $\\{d_{{}_{\alpha}}\\}$ (2.0.3), with
Galois group $\Gamma$ which “realizes the local ramified covers
$q_{{}_{\alpha}}$ at the points $\xi_{{}_{\alpha}}$" i.e. the isotropy
subgroup of $\Gamma$ at $\xi_{{}_{\alpha}}$ is $\Gamma_{{}_{\alpha}}$ and

  * •

an equivariant vector bundle $V$ on $Z$ such that

$p_{{}_{*}}^{\Gamma}(V)\simeq\mathcal{R}.$ (3.0.29)

Let $Z^{\prime}:=p^{-1}(X^{\prime})$ and $V^{\prime}:=V|_{{}_{Z^{\prime}}}$.
Gluing the trivial $G$-torsor with the $\\{E_{\alpha}\\}$ by the transition
functions of $V^{\prime}$, we make a $\Gamma$-equivariant principal
$\text{Aut}(G)$-torsor $\mathcal{E}^{\circ}$. Its associated group scheme
$\mathcal{E}^{\circ}\times^{{}^{\text{Aut}(G)}}G=\mathcal{H}^{\circ}$ is a now
group scheme on $Z^{\prime}$ with simply-connected fibers $G$, albeit with
transition functions in $\text{Aut}(G)$. Moreover, we have

$\big{(}\text{Res}_{{}_{Z^{\prime}/X^{\prime}}}(\mathcal{H}^{\circ})\big{)}^{{}^{\Gamma}}\simeq\mathcal{G}^{\circ}.$
(3.0.30)

Further, as locally-free sheaves, we also have

$\displaystyle\text{Lie}(\mathcal{H}^{\circ})=V^{\prime}.$ (3.0.31)

We transport the structure of Lie-bracket from
$\text{Lie}(\mathcal{H}^{\circ})$ to $V^{\prime}$ on $Z^{\prime}$. This is
non-degenerate everywhere on $Z^{\prime}$ since $\mathcal{H}^{\circ}$ is the
adjoint group scheme of $\mathcal{E}^{\circ}$. Hence $V^{\prime}$ has fiber
type $\mathfrak{g}$. Since $\text{codim}(Z\setminus Z^{\prime})\geq 2$, the
Lie bracket $[.,.]$ on $V^{\prime}$ extends to a Lie bracket on the locally
free sheaf $V$ with the Killing form being non-degenerate on the whole of $Z$.
In other words $V$ is now a locally free sheaf of Lie algebras on the whole of
$Z$ with semisimple fibres; these fibres are isomorphic to the Lie algebra
$\mathfrak{g}$ by the rigidity of semisimple Lie algebras.

We now wish to extend $\mathcal{E}^{\circ}$ as an equivariant
$\text{Aut}(G)$-torsor $\mathcal{E}$ on the whole of $Z$ such that its
associated Lie-algebra bundle
$\mathcal{E}(\mathfrak{g}):=\mathcal{E}\times^{{}^{\text{Aut}(G)}}\mathfrak{g}$
becomes isomorphic $V$. Since $G$ is simply-connected, making the
identification $\text{Aut}(\mathfrak{g})=\text{Aut}(G)$, we see that the
transition functions of $V$ give the gluing for defining $\mathcal{E}$ on $Z$
satisfying our desired requirements.

Let $\mathcal{H}:=\mathcal{E}\times^{{}^{\text{Aut}(G)}}G$ denote the group
scheme associated to the adjoint group scheme of $\mathcal{E}$. This is an
equivariant group scheme on $Z$ and we define

$\displaystyle{\mathcal{G}}_{{}_{\bf
X}}^{{}^{\varpi}}:=\big{(}\text{Res}_{{}_{Z/{\bf
X}}}(\mathcal{H})\big{)}^{{}^{\Gamma}}.$ (3.0.32)

Then ${\mathcal{G}}_{{}_{\bf
X}}^{{}^{\varpi}}|_{{X^{\prime}}}=\mathcal{G}^{\circ}$. Since by Lemma (3.7),
the functor “invariant direct image" commutes with taking Lie algebras, we
moreover get isomorphisms of locally-free sheaves of Lie algebras

$\text{Lie}({\mathcal{G}}_{{}_{\bf X}}^{{}^{\varpi}})\simeq
p_{{}_{*}}^{\Gamma}(\mathcal{E}(\mathfrak{g}))\simeq\mathcal{R}.$ (3.0.33)

This proves the first claim in the theorem.

Finally, let us verify the classifying property of the group scheme
$\mathcal{H}$. For $\alpha\in S$, at the closed points $z_{{}_{\alpha}}$ of
the curves ${\rm Spec}\,(A_{{}_{\alpha}})$ we have $\mathcal{H}|{{}_{{\rm
Spec}\,(A_{{}_{\alpha}})}}\simeq\mathcal{H}_{{}_{\alpha}}$. The classifying
property is tautologically valid here because this was designed expressly in
(3.0.25).

For the closed points $z_{{}_{\lambda}}$ of $C_{{}_{\lambda}}$ corresponding
to strata of lower dimension, we proceed as follows. Consider the base change
of the Kawamata cover $p:Z\rightarrow{\bf X}$ to the curve
$C_{{}_{\lambda}}\subset{\bf X}$ and further to the formal neighbourhood
$U_{{}_{z_{{}_{\lambda}}}}\subset C_{{}_{\lambda}}$ of $z_{{}_{\lambda}}$.

Let $p_{{}_{z}}:W_{{}_{z}}\to U_{{}_{z}}$ be the restriction of $p$ to a
connected component of $Z\times_{{}_{U_{{}_{z}}}}{\bf X}$. Then, $p_{{}_{z}}$
gives a Galois cover with Galois group some cyclic group
$\mu_{{}_{d}}\subset\Gamma$ of order $d$.

By (3.0.32), the restriction ${\mathcal{G}}_{{}_{{\bf
Y}}}^{{}^{\varpi}}|_{{}_{U_{{}_{z}}}}$ is the “invariant direct image" of
$\mathcal{H}|_{{}_{W_{{}_{z}}}}$. Also, we have the isomorphism
$\mathcal{R}|_{{}_{U_{{}_{z}}}}\simeq
p_{{}_{*}}^{\mu_{{}_{d}}}(\mathcal{E}(\mathfrak{g})_{{}_{W_{{}_{z}}}})$.
Further, by (3.8), $L^{+}(\mathcal{R}|_{{}_{U_{{}_{z_{{}_{\lambda}}}}}})\simeq
L^{+}(\mathfrak{P}^{{}^{std}}_{{}_{\theta_{{}_{\lambda}}}})$.

By (3.8), we have another ramified cover
$q_{{}_{\lambda}}:U^{\prime}_{{}_{z}}\to U_{{}_{z}}$, but now with Galois
group $\Gamma_{{}_{\lambda}}$ and an equivariant
$(\Gamma_{{}_{\lambda}},G)$-torsor $E_{{}_{\lambda}}$ on
$U^{\prime}_{{}_{z}}$, such that
$q^{{}^{\Gamma_{{}_{\lambda}}}}_{{}_{*}}(E_{{}_{\lambda}}(G))\simeq\mathcal{G}^{{}^{st}}_{{}_{\theta_{{}_{\lambda}}}}$
and
$q^{{}^{\Gamma_{{}_{\lambda}}}}_{{}_{*}}(E_{{}_{\lambda}}(\mathfrak{g}))\simeq\mathcal{R}|_{{}_{U_{{}_{z_{{}_{\lambda}}}}}}$.

To finish the proof, we need to show the following isomorphism of group
schemes:

${\mathcal{G}}_{{}_{{\bf
X}}}^{{}^{\varpi}}|_{{}_{U_{{}_{z}}}}\simeq\mathcal{G}^{{}^{st}}_{{}_{\theta_{{}_{\lambda}}}}.$
(3.0.34)

Over a common cover of $U^{\prime}_{{}_{z}}$ and $W_{{}_{z}}$ (which will
continue to be a Kawamata cover of $U_{{}_{z}}$), we can identify the pull-
back of Lie sheaves $E_{{}_{\lambda}}(\mathfrak{g})$ and
$\mathcal{E}(\mathfrak{g})|_{{}_{W_{{}_{z}}}}$ as equivariant Lie sheaves
since both give invariant direct images isomorphic to
$\mathcal{R}|_{{}_{U_{{}_{z_{{}_{\lambda}}}}}}$. Therefore, on the same common
cover we have an identification of pull-backs of
$E_{{}_{\lambda}}\times^{G}\mathfrak{g}$ with
$\mathcal{E}(\mathfrak{g})|_{{}_{W_{{}_{z}}}}=\mathcal{E}\times^{Aut(G)}\mathfrak{g}|_{{}_{W_{{}_{z}}}}$.
Hence we have an identification of pull-backs of $E_{{}_{\lambda}}\times^{G}G$
with $\mathcal{E}\times^{Aut(G)}G|_{{}_{W_{{}_{z}}}}$ as equivariant group
schemes with simply-connected fibers. Now the invariant direct image of the
first group scheme is a standard parahoric group scheme because the curve
$C_{{}_{\lambda}}$ is standard. On the other hand, the second group scheme is
$\mathcal{H}|_{{}_{W_{{}_{z}}}}$ whose invariant direct image by construction
is ${\mathcal{G}}_{{}_{{\bf X}}}^{{}^{\varpi}}|_{{}_{U_{{}_{z}}}}$. Thus, we
have proven (3.0.34).

∎

######  Remark 3.11.

The above construction goes through without change over an algebraically
closed field $k$ of characteristic $p$ coprime to the $d_{{}_{\alpha}}$’s
(2.0.3). The existence of $\bf X$ is known from the works of Strickland [17]
and De Concini-Springer [8] and Kawamata covering works under the above
conditions on characteristics.

##  4\. The Weyl alcove and apartment case

We continue to use the notations as in previous sections. Let
$G^{{}^{\text{aff}}}$ denote the Kac-Moody group associated to the affine
Dynkin diagram of $G$. Recall that $G^{{}^{\text{aff}}}$ is given by a central
extension of $L^{\ltimes}G$ by $\mathbb{G}_{m}$. Analogous to the wonderful
compactification of $G_{{}_{{}_{\text{ad}}}}$, P. Solis in [16] has
constructed a wonderful embedding ${\bf X}^{{}^{\text{aff}}}$ for
$G^{{}^{\text{aff}}}_{{}_{\text{ad}}}:=G^{{}^{\text{aff}}}/Z(G^{{}^{\text{aff}}})=\mathbb{G}_{m}\ltimes
LG/Z(G)$. It is an ind-scheme containing
$G^{{}^{\text{aff}}}_{{}_{\text{ad}}}$ as a dense open ind-scheme and carrying
an equivariant action of $L^{\ltimes}G\times L^{\ltimes}G$.

Let $T_{{}_{\text{ad}}}:=T/Z(G)$,
$T^{\ltimes}_{{}_{\text{ad}}}:=\mathbb{G}_{m}\times T_{{}_{\text{ad}}}\subset
G^{{}^{\text{aff}}}_{{}_{\text{ad}}}$, where $\mathbb{G}_{m}$ is the
rotational torus. In ${\bf X}^{{}^{\text{aff}}}$, the closure ${\bf
Y}^{{}^{\text{aff}}}:=\overline{T^{\ltimes}_{{}_{\text{ad}}}}$ gives a torus-
embedding. It is covered by the affine Weyl group
$W^{{}^{\text{aff}}}$-translates of the affine torus embedding ${\bf
Y}_{{}_{0}}^{{}^{\text{aff}}}:=\overline{T^{\ltimes}_{{}_{\text{ad},0}}}\simeq\mathbb{A}^{\ell+1}$
given by the negative Weyl alcove.

####  4.0.1. On the torus-embedding ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}$

Recall that $Z:={\bf Y}_{{}_{0}}^{{}^{\text{aff}}}\setminus
T^{\ltimes}_{{}_{\text{ad}}}$ is a union
$\cup_{{}_{\alpha\in\mathbb{S}}}H_{{}_{\alpha}}$ of $\ell+1$ many standard
coordinate hyperplanes meeting at normal crossings. For $\alpha\in\mathbb{S}$,
let the $\zeta_{{}_{\alpha}}$’s denote the generic points of the divisors
$H_{{}_{\alpha}}$’s. Let

$A_{{}_{\alpha}}=\mathcal{O}_{{}_{{\bf
Y}_{{}_{0}}^{{}^{\text{aff}}},\zeta_{{}_{\alpha}}}}$ (4.0.1)

be the dvr’s obtained by localizing at the height $1$-primes given by the
$\zeta_{{}_{\alpha}}$’s. Let $K_{{}_{\alpha}}$ be the quotient field of
$A_{{}_{\alpha}}$. Let $Y_{{}_{\alpha}}:={\rm Spec}\,(A_{{}_{\alpha}})$. Note
that we can identify the open subset ${\rm Spec}\,(K_{{}_{\alpha}})$ with
${T^{\ltimes}_{{}_{\text{ad}}}}\cap Y_{{}_{\alpha}}$. Let

$Y^{\prime}:={T^{\ltimes}_{{}_{\text{ad}}}}\cup_{\alpha}Y_{{}_{\alpha}}.$
(4.0.2)

The complement ${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}\setminus Y^{\prime}$ can
again be realized as a colimit of open subsets of ${\bf
Y}_{{}_{0}}^{{}^{\text{aff}}}$ whose codimension is at least $2$ in ${\bf
Y}_{{}_{0}}^{{}^{\text{aff}}}$.

####  4.0.2. Construction of a finite-dimensional Lie algebra bundle $J$ on
${\bf Y}_{{}_{0}}^{{}^{\text{aff}}}$ together with parabolic structures

This construction is exactly analogous to the construction of $\mathcal{R}$ on
$\bf Y$ in §3.0.3. We let $T^{\ltimes}_{{}_{\text{ad}}}$ and $\mathbb{S}$ play
the role of $T_{{}_{\text{ad}}}$ and $S$.

More precisely, let
$\lambda=\sum_{\alpha\in\mathbb{S}}k_{{}_{\alpha}}\omega_{{}_{\alpha}}^{{\vee}}$
be a non-zero dominant $1$-PS of $T_{{}_{\text{ad}}}^{\ltimes}$. We set

$\eta_{\lambda}:=\sum_{\alpha\in\mathbb{S}}k_{{}_{\alpha}}\frac{(1,\theta_{{}_{\alpha}})}{\ell+1}\quad\text{and}\quad\theta_{\lambda}:=\sum_{\alpha\in\mathbb{S}}\frac{k_{{}_{\alpha}}}{\sum_{{}_{\alpha\in\mathbb{S}}k_{{}_{\alpha}}}}\theta_{{}_{\alpha}}$
(4.0.3)

and $C_{{}_{\lambda}}$ the curve in ${\bf Y}^{{}^{\text{aff}}}$ defined by
$\eta_{\lambda}$ and $U_{{}_{\lambda}}$ the formal neighbourhood of the closed
point of $z_{{}_{\lambda}}$ of $C_{{}_{\lambda}}$. So if we substitute the
rational number $a$ of §2.0.3 by the number
$\frac{\sum_{\alpha\in\mathbb{S}}k_{{}_{\alpha}}}{\ell+1}$, we have the
parahoric group identification

$\mathfrak{P}_{{}_{\eta_{\lambda}}}=\mathfrak{P}_{{}_{\theta_{\lambda}}}.$
(4.0.4)

We may prove the following theorem exactly like (3.3). There we viewed the
standard alcove as a cone over the far wall. Note here that unlike the
situation in (3.3), we do not expect to get standard parahoric structures down
all the strata. However, for strata contained in the divisor associated to
$\alpha_{{}_{0}}\in\mathbb{S}$, the parahorics which occur will be standard as
before.

######  Theorem 4.1.

There exists a canonical Lie-algebra bundle $J$ on ${\bf
Y}_{{}_{0}}^{{}^{\text{aff}}}$ which extends the trivial bundle with fiber
$\mathfrak{g}$ on ${T^{\ltimes}_{{}_{\text{ad}}}}\subset{\bf
Y}^{{}^{\text{aff}}}$ and such that for $k_{{}_{\alpha}}\in\\{0,1\\}$ we have
the identification of functors from the category of $k$-algebras to $k$-Lie-
algebras:

$L^{+}(\mathfrak{P}_{{}_{\eta_{{}_{\lambda}}}})=L^{+}(\mathcal{R}\mid_{{}_{U_{{}_{\lambda}}}}).$
(4.0.5)

######  Corollary 4.2.

The Lie algebra bundle $J_{{}_{{\bf Y}^{{}^{\text{aff}}}}}$ gets canonical
parabolic structures (§8) at the generic points $\xi_{{}_{\alpha}}$ of the
$W^{{}^{\text{aff}}}$-translates of the divisors $H_{{}_{\alpha}}\subset{\bf
Y}_{{}_{0}}^{{}^{\text{aff}}}$, $\alpha\in\mathbb{S}$.

###### Proof.

We prescribe ramification indices $d_{\alpha}$ (2.0.3) on the divisor
$H_{{}_{\alpha}}$. Then as in the proof of (3.6), the identification (3.0.22)
of the Lie algebra structures of $J_{{}_{{\bf Y}^{{}^{\text{aff}}}}}$ and the
parahoric Lie algebra structures on the localizations of the generic points of
$H_{{}_{\alpha}}$ allows us to endow parabolic structures at the generic
points of the divisors.∎

####  4.0.3. The parahoric group scheme on the torus embedding ${{\bf
Y}^{{}^{\text{aff}}}}$

######  Theorem 4.3.

There exists an affine “wonderful" Bruhat-Tits group scheme
${\mathcal{G}}_{{}_{{\bf Y}^{{}^{\text{aff}}}}}^{{}^{\varpi}}$ on $\bf Y$
together with a canonical isomorphism $\text{Lie}({\mathcal{G}}_{{}_{{\bf
Y}^{{}^{\text{aff}}}}}^{{}^{\varpi}})\simeq J$. It further satisfies the
following classifying property:

For any point $h\in{{\bf Y}^{{}^{\text{aff}}}}\setminus T^{\ltimes}_{ad}$, let
$\mathbb{I}\subset\mathbb{S}$ be a subset such that
$h\in\cap_{\alpha\in\mathbb{I}}H_{{}_{\alpha}}$. Let
$C_{{}_{\mathbb{I}}}\subset{{\bf Y}^{{}^{\text{aff}}}}$ be a smooth curve with
generic point in $T^{\ltimes}_{{}_{\text{ad}}}$ and closed point $h$. Let
$U_{{}_{h}}\subset C_{{}_{\mathbb{I}}}$ be a formal neighbourhood of the
closed point $h$. Then, the restriction ${\mathcal{G}}_{{}_{{\bf
Y}^{{}^{\text{aff}}}}}^{{}^{\varpi}}|_{{}_{U_{{}_{h}}}}$ is isomorphic to the
Bruhat-Tits group scheme $\mathcal{G}_{{}_{\mathbb{I}}}$ on $U_{{}_{h}}$.

###### Proof.

Recall that ${{\bf Y}^{{}^{\text{aff}}}}$ is covered by affine spaces
$Y_{{}_{w}}\simeq\mathbb{A}^{\ell+1}$ parametrized by the affine Weyl group
$W^{{}^{\text{aff}}}$. Each $Y_{{}_{w}}$ is the translate of ${{\bf
Y}_{{}_{0}}^{{}^{\text{aff}}}}$. The translates of the divisors
$H_{{}_{\alpha}}$ meet each $Y_{{}_{w}}$ in the standard hyperplanes on
$\mathbb{A}^{\ell+1}$ and thus we can prescribe the same ramification data at
the hyperplanes on each of the $Y_{{}_{w}}$’s. On the other hand, although we
have simple normal crossing singularities, we do not have an analogue of the
Kawamata covering lemma for schemes such as ${{\bf Y}^{{}^{\text{aff}}}}$. The
lemma is known only in the setting of quasi-projective schemes. So to
construct the group scheme, we employ a different approach.

We observe firstly that, the formalism of Kawamata coverings applies in the
setting of the affine spaces $Y_{{}_{w}}\subset{{\bf Y}^{{}^{\text{aff}}}}$.
Let $p_{w}:Z_{{}_{w}}\rightarrow Y_{{}_{w}}$ be the associated Kawamata cover
(8.0.2) with Galois group $\Gamma_{w}$. In Corollary 4.2 we observed that the
Lie-algebra bundle $J$ has a canonical parabolic structure like $\mathcal{R}$
in the proof of Theorem 3.10. Letting $Y_{{}_{w}}$ play the role of ${\bf
Y_{{}_{0}}}$ and using all arguments in the proof of Theorem 3.10 we obtain
$\mathcal{H}_{{}_{w}}\rightarrow Z_{{}_{w}}$ which is $\Gamma_{w}$-group
scheme with fibers isomorphic to $G$ whose invariant direct image is a group
scheme $\mathcal{G}_{w}$ such that
$\text{Lie}(\mathcal{G}_{w})=J|_{Y_{{}_{w}}}$. The induced parabolic structure
on $J|_{Y_{{}_{w}}}$ is the restriction of the one on $J$. Indeed, by (4.2)
these parabolic structures are essentially given at the local rings at the
generic points $\xi_{\alpha}$ of the divisors $H_{{}_{\alpha}}$ and hence
these parabolic structures on $J$ agree on the intersections
$Y_{{{}_{uv}}}:=Y_{{}_{u}}\cap Y_{{}_{v}}$.

Let $Z_{{}_{uv}}:=p_{u}^{-1}(Y_{{}_{uv}})$. Let $\tilde{Z}_{{}_{uv}}$ be the
normalization of a component of $Z_{{}_{uv}}\times_{Y_{{}_{uv}}}Z_{{}_{vu}}$.
Then $\tilde{Z}_{{}_{uv}}$ serves as Kawamata cover (8.0.2) of $Y_{{}_{uv}}$
(see [18, Corollary 2.6, page 56]). We consider the morphisms
$\tilde{Z}_{{}_{uv}}\to Z_{{}_{u}}$ (resp.$\tilde{Z}_{{}_{uv}}\to Z_{{}_{v}}$)
and let $\mathcal{H}_{{}_{u,\tilde{Z}}}$ (resp.
$\mathcal{H}_{{}_{v,\tilde{Z}}}$) denote the pull-backs of
$\mathcal{H}_{{}_{u}}$ (resp. $\mathcal{H}_{{}_{v}}$) to
$\tilde{Z}_{{}_{uv}}$.

Let $\Gamma$ denote the Galois group for $\tilde{Z}_{{}_{uv}}\rightarrow
Y_{{}_{uv}}$. Then by Lemma 3.7, the invariant direct images of the
equivariant Lie algebra bundles $\text{Lie}(\mathcal{H}_{{}_{u,\tilde{Z}}})$
and $\text{Lie}(\mathcal{H}_{{}_{v,\tilde{Z}}})$ coincide with the Lie algebra
structure on $J$ restricted to the ${Y_{{}_{uv}}}$ and also as isomorphic
parabolic bundles. Therefore we have a natural isomorphism of equivariant Lie-
algebra bundles

$\text{Lie}(\mathcal{H}_{u,\tilde{Z}})\simeq\text{Lie}(\mathcal{H}_{v,\tilde{Z}}).$
(4.0.6)

As in the proof of Theorem 3.10, this gives a canonical identification of the
equivariant group schemes $\mathcal{H}_{{}_{u,\tilde{Z}}}$ and
$\mathcal{H}_{{}_{v,\tilde{Z}}}$ on $\tilde{Z}_{{}_{uv}}$. Since the invariant
direct image of both the group schemes $\mathcal{H}_{{}_{u,\tilde{Z}}}$ and
$\mathcal{H}_{{}_{v,\tilde{Z}}}$ are the restrictions
$\mathcal{G}_{{}_{u,{Y_{{}_{uv}}}}}$ and $\mathcal{G}_{{}_{v,{Y_{{}_{uv}}}}}$,
it follows that on $Y_{{}_{uv}}=Y_{{}_{u}}\cap Y_{{}_{v}}$ we get a canonical
identification of group schemes:

$\displaystyle\mathcal{G}_{{}_{u,{Y_{{}_{uv}}}}}\simeq\mathcal{G}_{{}_{v,{Y_{{}_{uv}}}}}.$
(4.0.7)

These identifications are canonically induced from the gluing data of the Lie
algebra bundle $J$ for the cover $Y_{{}_{w}}$’s. Therefore the cocycle
conditions are clearly satisfied and the identifications (4.0.7) glue to give
the group scheme ${\mathcal{G}}_{{}_{{\bf
Y}^{{}^{\text{aff}}}}}^{{}^{\varpi}}$ on ${\bf Y}^{{}^{\text{aff}}}$. The
verification of the classifying property follows exactly as in the proof of
Theorem 3.10.

∎

##  5\. The Bruhat-Tits group scheme on ${\bf X}^{{}^{{aff}}}$

Let ${\bf X}^{{}^{{aff}}}$ as in §4. We begin with a generality. Let
$\mathbb{X}$ be an ind-scheme. By an open-subscheme
$i:\mathbb{U}\hookrightarrow\mathbb{X}$ we mean an ind-scheme such that for
any $f:{\rm Spec}\,(A)\rightarrow\mathbb{X}$, the natural morphism
$\mathbb{U}\times_{\mathbb{X}}{\rm Spec}\,(A)\rightarrow{\rm Spec}\,(A)$ is an
open immersion. For a sheaf $\mathbb{F}$ on $\mathbb{U}$ by
$i_{{}_{*}}(\mathbb{F})$ we mean the sheaf associated to the pre-sheaf on the
“big site" of $\mathbb{X}$, whose sections on $f:{\rm
Spec}\,(A)\rightarrow\mathbb{X}$ are given by
$\mathbb{F}(\mathbb{U}\times_{\mathbb{X}}{\rm Spec}\,(A))$.

The ind-scheme ${\bf X}^{{}^{\text{aff}}}$ has a certain open subset ${\bf
X}_{{}_{0}}$ whose precise definition is somewhat technical [16, Page 705].
Let us mention the properties relevant for us.

Recall that ${\bf
X}^{{}^{\text{aff}}}=(G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\times
G_{{}_{\text{ad}}}^{{}^{\text{aff}}})~{}{\bf X}_{{}_{0}}$ and in fact ${\bf
X}^{{}^{\text{aff}}}=(G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\times
G_{{}_{\text{ad}}}^{{}^{\text{aff}}})~{}{{\bf Y}^{{}^{\text{aff}}}}$. Further,
the torus-embedding ${\bf Y}^{{}^{\text{aff}}}$ is covered by ${\bf
Y}^{{}^{\text{aff}}}_{{}_{w}}\simeq\mathbb{A}^{{}^{\ell+1}}$ which are
$W^{{}^{\text{aff}}}$-translates of ${\bf Y}^{{}^{\text{aff}}}_{{}_{0}}$,
where ${\bf Y}^{{}^{\text{aff}}}_{{}_{0}}={{\bf Y}^{{}^{\text{aff}}}}\cap{\bf
X}_{{}_{0}}$. We remark that, analogous to the case of ${\bf
Y}_{{}_{0}}\subset{\bf Y}\subset{\bf X}$, just as the toric variety ${\bf
Y}_{{}_{0}}$ was associated to the negative Weyl chamber, the toric variety
${\bf Y}^{{}^{\text{aff}}}_{{}_{0}}$ is associated to the negative Weyl
alcove.

So let us denote by $0$ the neutral element of $W^{{}^{\text{aff}}}$ also. Let
$U^{\pm}\subset B^{\pm}$ be the unipotent subgroups. Let
$\mathcal{U}^{\pm}:=ev^{-1}(U^{\pm})$ where $ev:G(A)\rightarrow G(k)$ is the
evalution map. Further by [16, Prop 5.3]

${\bf X}_{{}_{0}}=\mathcal{U}\times{\bf
Y}^{{}^{\text{aff}}}_{{}_{0}}\times\mathcal{U}^{-}.$ (5.0.1)

Let ${\bf X}_{{}_{w}}:=\mathcal{U}\times{\bf
Y}^{{}^{\text{aff}}}_{{}_{w}}\times\mathcal{U}^{-}$. These cover
$\mathcal{U}\times{{\bf Y}^{{}^{\text{aff}}}}\times\mathcal{U}^{-}$. For $g\in
G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\times
G_{{}_{\text{ad}}}^{{}^{\text{aff}}}$, let

${\bf X}_{g}:=g{\bf X}_{0}\quad{\bf X}_{{}_{g,w}}:=g{\bf X}_{{}_{w}}\quad{\bf
Y}^{{}^{\text{aff}}}_{{}_{g,w}}:=g{\bf Y}^{{}^{\text{aff}}}_{{}_{w}}.$ (5.0.2)

Note that we have the projection ${\bf X}_{{}_{g,w}}\to{\bf
Y}^{{}^{\text{aff}}}_{{}_{g,w}}$ which is a
$\mathcal{U}\times\mathcal{U}^{-}$-bundle.

By [16, Theorem 5.1] the ind-scheme ${\bf X}^{{}^{\text{aff}}}$ has divisors
$D_{\alpha}$ for $\alpha\in\mathbb{S}$ such that the complement of their union
is ${\bf X}^{{}^{\text{aff}}}\setminus G_{{}_{\text{ad}}}^{{}^{\text{aff}}}$.
The next proposition shows the existence of a finite dimensional Lie algebra
bundle on ${\bf X}^{{}^{\text{aff}}}$ which is analogous to the bundle
$\mathcal{R}$ over ${\bf X}$.

######  Proposition 5.1.

There is a finite dimensional Lie algebra bundle $\bf R$ on ${\bf
X}^{{}^{\text{aff}}}$ which extends the trivial Lie algebra bundle
$G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\times\mathfrak{g}$ on the open dense
subset $G_{{}_{\text{ad}}}^{{}^{\text{aff}}}\subset{\bf X}^{{}^{\text{aff}}}$
and whose restriction to ${\bf Y}^{{}^{\text{aff}}}$ is $J$.

###### Proof.

Following the arguments for (3.6) and (3.9), let $J^{\prime}$ be the locally
free sheaf obtained on the open subset ${\bf X^{\prime}}$ obtained by the
union of $G^{{}^{\text{aff}}}$ and the height one prime ideals of ${\bf
X}^{{}^{\text{aff}}}$. Let ${\bf R}$ denote its push-forward. To check that
the push-forward is locally free, without loss of generality we may consider
its restriction to ${\bf X}_{0}$. But on ${\bf X}_{0}$, the pushforward of
$J^{\prime}$ restricts to the pull-back of a Lie-algebra bundle $J$ on ${\bf
Y}^{{}^{\text{aff}}}$ constructed in (4.1), which completes the argument. The
rest of the properties follows immediately. ∎

######  Theorem 5.2.

There exists an affine “wonderful" Bruhat-Tits group scheme
${\mathcal{G}}_{{}_{{\bf X}^{{}^{\text{aff}}}}}^{{}^{\varpi}}$ on ${\bf
X}^{{}^{\text{aff}}}$ together with a canonical isomorphism
$\text{Lie}({\mathcal{G}}_{{}_{\bf X^{aff}}}^{{}^{\varpi}})\simeq{\bf R}$. It
further satisfies the following classifying property:

For any point $h\in{\bf X}^{{}^{\text{aff}}}\setminus
G_{{}_{\text{ad}}}^{{}^{\text{aff}}}$, let $\mathbb{I}\subset\mathbb{S}$ be
defined by the condition $h\in\cap_{\alpha\in\mathbb{I}}D_{\alpha}$. Let
$C_{{}_{\mathbb{I}}}\subset{\bf X}$ be a smooth curve with generic point in
$G_{{}_{\text{ad}}}^{{}^{\text{aff}}}$ with closed point $h$. Let
$U_{{}_{h}}\subset C_{{}_{\mathbb{I}}}$ be a formal neighbourhood of $h$.
Then, the restriction ${\mathcal{G}}_{{}_{{\bf
X}^{{}^{\text{aff}}}}}^{{}^{\varpi}}|_{{}_{U_{{}_{h}}}}$ is isomorphic to the
Bruhat-Tits group scheme $\mathcal{G}_{{}_{\mathbb{I}}}$ on ${\rm Spec}\,(A)$.

###### Proof.

We begin by observing that ${\bf X}_{{}_{g,w}}$ and ${\bf R}\rightarrow{\bf
X}^{{}^{\text{aff}}}$ play the role of ${{\bf
Y}_{{}_{g,w}}^{{}^{\text{aff}}}}$ and $J\rightarrow{{\bf
Y}^{{}^{\text{aff}}}}$ in the proof of Theorem 4.3. Therefore the group scheme
$\mathcal{G}_{{}_{g,w}}$ glue together and we obtain the global group scheme
${\mathcal{G}}_{{}_{{\bf X}^{{}^{\text{aff}}}}}^{{}^{\varpi}}$. The
verification of the classifying property follows exactly as in the proof of
Theorem 3.10.

∎

##  6\. The Bruhat-Tits group scheme on ${\bf X}^{{}^{{poly}}}$

In [16], apart from the loop group case Solis also discusses the polynomial
loop group situation which in fact is relatively simpler. Consequently, the
analysis as well as the results in the loop group situation go through with
some simplifications. Let $L_{{}_{\text{poly}}}G$ be defined by

$L_{{}_{\text{poly}}}G(R^{\prime})=G(R^{\prime}[z^{{}^{\pm}}])$ (6.0.1)

and we define $L_{{}_{\text{poly}}}^{\ltimes}G$ similarly as
$L_{{}_{\text{poly}}}^{\ltimes}G:=\mathbb{G}_{m}\ltimes
L_{{}_{\text{poly}}}G$. Solis constructs a “wonderful" embedding of the
polynomial loop group $L_{{}_{\text{poly}}}^{\ltimes}G/Z(G)$. We denote this
space by ${\bf X}^{{}^{\text{poly}}}$. It has an open subset ${\bf
X}^{{}^{\text{poly}}}_{{}_{0}}$ whose definition is somewhat technical [16,
§5.3], but, which plays the role completely analogus to ${\bf X}_{{}_{0}}$ for
${\bf X}^{{}^{\text{aff}}}$. We will continue to denote by $\bf
Y:=\overline{T^{\ltimes}_{{}_{\text{ad}}}}$ the toral embedding of
$T^{\ltimes}_{{}_{\text{ad}}}$ obtained by taking its closure in ${\bf
X}^{{}^{\text{poly}}}$. As in the loop group case, in the polynomial case too
$\bf Y$ is covered by infinitely many affine spaces $\mathbb{A}^{\ell+1}$
parametrized by the affine Weyl group $W^{{}^{\text{aff}}}$. In fact, if
$t^{-\alpha_{i}}$ are the regular functions on $T^{\ltimes}_{{}_{\text{ad}}}$
given by the character $-\alpha_{i}$, then

$\overline{T^{\ltimes}_{{}_{ad,0}}}:=\overline{T^{\ltimes}_{{}_{\text{ad}}}}\cap{\bf
X}_{{}_{0}}^{{}^{poly}}\simeq{\rm
Spec}\,\mathbb{C}[t^{-\alpha_{0}},\ldots,t^{-\alpha_{\ell}}].$

We index these affine spaces by the affine Weyl group $W^{{}^{\text{aff}}}$
and denote them as before by $Y_{{}_{w}}$’s, $w\in W^{{}^{\text{aff}}}$. As
before, we may construct a finite-dimensional Lie-algebra bundle $J$ on ${{\bf
Y}^{{}^{\text{aff}}}}$, then ${\bf R}$ on ${\bf X}^{{}^{\text{poly}}}$ and
then construct a group scheme ${\mathcal{G}}_{{}_{{\bf
X}^{{}^{\text{poly}}}}}^{{}^{\varpi}}$. Since the inductive structure of ${\bf
X}^{{}^{\text{poly}}}$ is identical to ${\bf X}^{{}^{\text{aff}}}$, the proofs
are also identical. Therefore we omit them.

Now we outline a construction in the polynomial loop situation which will be
generalized in the next section. For any affine space $Y_{{}_{w}}$, we have an
affine embeddings

$\displaystyle
i_{w}:T_{{}_{\text{ad},w}}\times\mathbb{G}_{m}=T^{\ltimes}_{{}_{\text{ad},w}}\hookrightarrow
Y_{{}_{w}}$ (6.0.2)

Furthermore, the projection
$p:T_{{}_{\text{ad},w}}\times\mathbb{G}_{m}\to\mathbb{G}_{m}$ extends to a
morphism $p_{w}:Y_{{}_{w}}\to\mathbb{A}^{1}$ such that $p^{{}^{-1}}(0)$ is the
union of the hyperplanes. At the level of coordinates, this extension is
simply the product of the coordinate functions. Further $i_{w}$ and $p_{w}$
glue to give

(6.0.7)

##  7\. An analogue of a construction of Mumford

In [13] towards the very end Mumford gives a beautiful construction of the
geometric realization of the relative case of buildings via toroidal
embeddings. He deals with the general situation of an arbitrary discrete
valuation ring $R$ with algebraically closed residue field $k=R/\mathfrak{m}$.
Our aim in this section is limited to the case when $k=\mathbb{C}$,
$\mathfrak{m}=(z)\subset\mathbb{C}[z]$ and
$R=\mathbb{C}[z]_{{}_{\mathfrak{m}}}$. Let $K=\mathbb{C}(z)$ and let $S={\rm
Spec}\,R$ and $\eta$ be the generic point and $o\in S$ the closed point. For
schemes $X$ over $S$, $X_{{}_{\eta}}$ will be the fibre over $\eta$ and
$X_{{}_{o}}$ the fibre over $o$.

We will briefly sketch an analogue of Mumford’s construction in our setting
and outline of the relationship between this construction to Solis’s approach
and as a consequence construct a “wonderful" Bruhat-Tits group scheme on the
toroidal embedding of $G_{{}_{ad,}}\times S$.

For the purposes of this section alone so as to remain consistent with the one
in [13], we will have the following set of notations.

$\displaystyle H:=G_{{}_{\text{ad}}}$ (7.0.1) $\displaystyle
H_{{}_{S}}:=G\times S$ (7.0.2) $\displaystyle T_{{}_{S}}\subset H_{{}_{S}}$
(7.0.3) $\displaystyle T_{{}_{S}}:=T_{{}_{\text{ad}}}\times S$ (7.0.4)
$\displaystyle T_{{}_{K}}:=T_{{}_{\text{ad}}}\times{\rm Spec}\,K$ (7.0.5)

i.e. $H_{{}_{S}}$ is a split adjoint semisimple group scheme over $S$,
$T_{{}_{S}}\subset H_{{}_{S}}$ a fixed split maximal torus and $G$ will as
before stand for the simply connected group. We will denote
$U^{\pm}_{S}:=U^{\pm}\times S$.

Base-changing (6.0.7) by ${\rm Spec}\,R\rightarrow\mathbb{A}^{1}$, by the
definition of $\bf Y$ we obtain:

(7.0.10)

We continue to denote by $Y_{{}_{w}}$ the base-change. The orbit space
$H_{{}_{S}}/T_{{}_{S}}$ exists as an affine scheme over $S$ and $H_{{}_{S}}$
via the quotient map $\pi:H_{{}_{S}}\to H_{{}_{S}}/T_{{}_{S}}$ is a locally
free principal $T_{{}_{S}}$-bundle over $H_{{}_{S}}/T_{{}_{S}}$. We now
consider the associated fibre bundle:

(7.0.15)

Note that $T_{\eta}=Y_{w,\eta}$ and hence
$H_{\eta}=(H_{S}\times^{{}^{T_{S}}}Y_{{}_{w}})_{{}_{\eta}}$. Let us denote by

$\displaystyle\overline{Z}_{w}:=H_{S}\times^{{}^{T_{S}}}Y_{{}_{w}}$ (7.0.16)

and observe that $U_{{}_{S}}^{-}\times Y_{{}_{w}}\times U_{{}_{S}}=Z_{{}_{w}}$
is an open subset of $\overline{Z}_{w}$. In fact, over $U_{{}_{S}}^{-}\times
U_{{}_{S}}\subset H_{{}_{S}}/T_{{}_{S}}$, the quotient map
$\pi:H_{S}\rightarrow H_{S}/T_{S}$ is trivial.

Analogous to Mumford’s definition [13, page 206], we define

$\displaystyle\overline{H_{{}_{S}}}:=\bigcup_{{}_{x\in
H(K)}}(H_{S}\times^{{}^{T_{S}}}Y_{{}_{w}}).x$ (7.0.17)

where the generic fibre $H_{\eta}$ is identified in each copy
$(H_{S}\times^{{}^{T_{S}}}Y_{{}_{w}}).x$ by a translate of $H_{\eta}$ by right
multiplication by $x$ in the Iwahori subgroup. It takes a bit to prove that
there is such an action. Note that we have the identification:

$\displaystyle H_{\eta}=\overline{H_{{}_{\eta}}}$ (7.0.18)

Consider the inclusion $G_{{}_{\text{ad}}}\times\mathbb{G}_{m}\hookrightarrow
L_{{}_{\text{poly}}}^{\ltimes}G/Z(G)$. Let ${\bf
M}=\overline{G_{{}_{\text{ad}}}\times\mathbb{G}_{m}}$ in ${\bf X}^{{}^{poly}}$
as a scheme over $\mathbb{C}$. It is locally of finite type.

######  Proposition 7.1.

The $S$-scheme $\overline{H_{{}_{S}}}$ as a $\mathbb{C}$-scheme is isomorphic
to $\bf M$.

In particular, $\overline{H_{{}_{S}}}$ is a regular scheme over $\mathbb{C}$
such that the closed fibre $\overline{H_{o}}$ is a complete scheme over
$\mathbb{C}$ which is a union of smooth components meeting at normal
crossings. The embedding $H_{{}_{S}}\subset\overline{H_{{}_{S}}}$ is a
toroidal embedding without self-intersection. Finally, the strata of
$\overline{H_{{}_{S}}}-H_{{}_{S}}$ are precisely the parahoric subgroups of
$H(K)$. This bijection between the strata and the parahoric subgroups extends
to an isomorphism of the graph of the embedding
$H_{{}_{S}}\subset\overline{H_{{}_{S}}}$ with the Bruhat-Tits building of $H$
over $S$.

By the methods in this note we immediately deduce the existence of a
“wonderful" Bruhat-Tits group scheme $\mathcal{G}_{{}_{\bf M}}^{{}^{\varpi}}$
on $\bf M$ which behaves naturally with respect to the strata.

######  Remark 7.2.

A difference between Mumford’s construction and ours is that in Mumford’s
construction the fibre over the closed point in $S$ is a reducible divisor but
not necessarily with simple normal crossing singularities. In fact, the
components associated to the parahorics which are not hyperspecial come with
multiplicity being the coefficient $c_{{}_{\alpha}}$ of the associated simple
root in the expression of the highest root. So a Kawamata lemma is not
immediately applicable in that situation.

######  Remark 7.3.

Mumford’s construction works over algebraically closed fields of any
characteristic while we assumed the characteristic to be $0$. We did this to
make the construction in the loop group situation following [16]. On the other
hand, it seems that with a bit more work the whole construction of
$\mathcal{G}_{{}_{\bf M}}^{{}^{\varpi}}$ will go through for characteristics
$p$ such that $p$ is coprime to the $d_{{}_{\alpha}}$’s in (2.0.3).

##  8\. Appendix on parabolic and equivariant bundles

In this section we recall and summarize results on parabolic bundles and
equivariant bundles on Kawamata covers. These play a central role in the
constructions of the Bruhat-Tits group schemes made above. Consider a pair
$(X,D)$, where $X$ is a smooth quasi-projective variety and
$D=\sum_{{}_{j=0}}^{{}^{\ell}}D_{{}_{j}}$ is a reduced normal crossing divisor
with non-singular components $D_{{}_{j}}$ intersecting each other
transversely. The basic examples we have in mind in the note are discrete
valuation rings with its closed point, the wonderful compactification $\bf X$
with its boundary divisors or affine toric varieties.

Let $E$ be a locally free sheaf on $X$. Let $n_{{}_{j}},j=0\ldots,\ell$ be
positive integers attached to the components $D_{{}_{j}}$. Let $\xi$ be a
generic point of $D$.

Let
$E_{{}_{\xi}}:=E\otimes_{{}_{\mathcal{O}_{{}_{X}}}}\mathcal{O}_{{}_{X,\xi}}$
and $\bar{E}_{{}_{\xi}}:=E_{{}_{\xi}}/\mathfrak{m}_{{}_{\xi}}E_{{}_{\xi}}$,
$\mathfrak{m}_{{}_{\xi}}$ the maximal ideal of $\mathcal{O}_{{}_{X,\xi}}$.

######  Definition 8.1.

A parabolic structure on $E$ consists of the following data:

  * •

A flag $\bar{E}_{{}_{\xi}}=F^{{}^{1}}\bar{E}_{{}_{\xi}}\supset\ldots\supset
F^{r{{}_{{}_{j}}}}\bar{E}_{{}_{\xi}}$, at the generic point $\xi$ of each of
the components $D_{{}_{j}}$ of $D$,

  * •

Weights $d_{{}_{s}}/n_{{}_{j}}$, with $0\leq d_{{}_{s}}<n_{{}_{j}}$ attached
to $F^{{}^{s}}\bar{E}_{{}_{\xi}}$ such that $d_{1}<\cdots<d_{r_{j}}$

By saturating the flag datum on each of the divisors we get for each component
$D_{{}_{j}}$ a filtration:

$\displaystyle E_{{}_{D_{{}_{j}}}}=F^{{}^{1}}_{{}_{j}}\supset\ldots\supset
F^{r{{}_{{}_{j}}}}_{{}_{j}}$ (8.0.1)

of sub-sheaves on $D_{{}_{j}}$. Define the coherent subsheaf
$\mathcal{F}^{s}_{{}_{j}}$, where $0\leq j\leq\ell$ and $1\leq s\leq
r_{{}_{j}}$, of $E$ by:

$\displaystyle 0\to\mathcal{F}^{s}_{{}_{j}}\to E\to
E_{{}_{D_{{}_{j}}}}/F^{s}_{{}_{j}}\to 0$ (8.0.2)

the last map is by restriction to the divisors. In [2] it is shown that there
is a Kawamata covering of $p:(Y,\tilde{D})\to(X,D)$ with suitable ramification
data and Galois group $\Gamma$. It is also shown that there is a
$\Gamma$-equivariant vector bundle $V$ on $Y$ such that the invariant direct
image $p^{\Gamma}_{{}_{*}}(V)=E$ which also recovers the filtrations
$\mathcal{F}^{s}_{{}_{j}}$.

For the sake of completeness, we give a self-contained ad hoc argument for
this construction which is more in the spirit of the present note. The data
given in (8.1) is as in [14] which deals with points on curves. Note that in
our setting we have rational weights. Under these conditions, the aim is to
set up an equivalence

$\displaystyle
p^{\Gamma}_{{}_{*}}:\\{\Gamma-\text{equivariant~{}bundles~{}on}~{}Y\\}\to\\{\text{parabolic~{}bundles~{}on}~{}X\\}$
(8.0.3)

and as the notation suggests, this is achieved by taking invariant direct
images. Since $p:Y\to X$ is finite and flat, if $V$ is locally free on $Y$
then so is $p^{\Gamma}_{{}_{*}}(V)$. An equivariant bundle $V$ on $Y$ is
defined in an analytic neighbourhood of a generic point $\zeta$ of a component
by a representation of the isotropy group $\Gamma_{{}_{\zeta}}$ and locally
(in the analytic sense) the action of $\Gamma_{{}_{\zeta}}$ is the product
action. This is called the local type in [15] or [14]. By appealing to the
$1$-dimensional case, we get canonical parabolic structures on $E$ at the
generic points $\xi$ of the components $D_{{}_{j}}$ in $X$.

This construction is an equivalence. Given a parabolic bundle $E$ on $(X,D)$,
to get the $\Gamma$-equivariant bundle $V$ on $Y$, such that
$p^{\Gamma}_{{}_{*}}(V)=E$, we proceed as follows. If we did know the
existence of such a $V$ then we can consider the inclusion $p^{*}(E)\subset
V$. Taking its dual (and since $V/p^{*}(E)$ is torsion, taking duals is an
inclusion):

$\displaystyle V^{*}\hookrightarrow(p^{*}(E))^{*}=p^{*}(E^{*})$ (8.0.4)

By the $1$-dimensional case (obtained by restricting to the height $1$-primes
at the generic points) we see that the quotient
$T_{{}_{\zeta}}:=p^{*}(E^{*})_{{}_{\zeta}}/V_{{}_{\zeta}}^{*}$ is a torsion
$\mathcal{O}_{{}_{Y,\zeta}}$-module and $T_{{}_{\zeta}}$ is completely
determined by the parabolic structure on $E$.

The parabolic structure on $E$ therefore determines canonical quotients:

$\displaystyle p^{*}(E^{*})_{{}_{\zeta}}\to T_{{}_{\zeta}}$ (8.0.5)

for each generic point $\zeta$ of components in $Y$ above the components
$D_{{}_{j}}$ in $X$.

The discussion above suggest how one would construct such a $V$; we begin with
these quotients (8.0.5). Then we observe that there is a maximal coherent
subsheaf $V^{\prime}\subset p^{*}(E^{*})$ such that

$\displaystyle
p^{*}(E^{*})_{{}_{\zeta}}/V_{{}_{\zeta}}^{\prime}=T_{{}_{\zeta}}.$ (8.0.6)

Away from $p^{{}^{-1}}(D)$ in $Y$, the inclusion $V^{\prime}\hookrightarrow
p^{*}(E^{*})$ is an isomorphism. Dualizing again, we get an inclusion
$p^{*}(E)\hookrightarrow(V^{\prime})^{{}^{*}}$ which is an isomorphism away
from $p^{{}^{-1}}(D)$. Set $W=(V^{\prime})^{{}^{*}}$. Since $W$ is the dual of
a coherent sheaf, it is reflexive. It is not hard to check that $W$ coincides
with the vector bundle $V$ that Biswas constructs in codimension $2$ and hence
everywhere by [12, Proposition 1.6, page 126]. This gives us the desired
equivalence (8.0.3).

####  8.0.1. The group scheme situation

Let $(X,D)$ be as above. Let $\xi$ be the generic point of a component
$D_{{}_{j}}$ of $D$ and let $A:=\mathcal{O}_{{}_{X,\xi}}$ and $K$ be the
quotient field of $A$. Let $\mathcal{G}_{{}_{\theta}}$ be a Bruhat-Tits group
scheme on ${\rm Spec}\,(A)$ associated to a vertex $\theta$ of the Weyl
alcove. We always assume that these group scheme are generically split.

Let $B=\mathcal{O}_{{}_{Y,\zeta}}$ where $\zeta$ is the generic point of a
component of $Y$ above $D_{{}_{j}}$ and let $L$ be the quotient field of $B$.
We assume that the local ramification data for the Kawamata covering has
numbers $d_{{}_{\alpha}}$ (2.0.3). Let $\Gamma_{{}_{\zeta}}$ be the stabilizer
of $\Gamma$ at $\zeta\in Y$.

The results of [1, Proposition 5.1.2 and Remark 2.3.3] show that there exists
an equivariant group scheme $\mathcal{H}_{{}_{B}}$ on ${\rm Spec}\,(B)$ with
fiber isomorphic to the simple connected group $G$ and such that
$\text{Res}_{{}_{B/A}}(\mathcal{H}_{{}_{B}})^{{}^{\Gamma_{{}_{\zeta}}}}\simeq\mathcal{G}_{{}_{\theta}}$.

Suppose that we have a group scheme $\mathcal{G}_{{}_{X^{\prime}}}$ on an open
$X^{\prime}\subset X$ which includes all the height $1$ primes coming from the
divisors $D_{{}_{j}}$’s with the following properties:

  * •

Away from the divisor $D\subset X$, $\mathcal{G}$ is the constant group scheme
with fiber $G$.

  * •

The restrictions $\mathcal{G}\mid_{{}_{{\rm Spec}\,(A)}}$ at the generic
points $\xi$ are isomorphic to the Bruhat-Tits group scheme
$\mathcal{G}_{{}_{\theta}}$ for varying $\xi$ and $\theta$.

In other words, $\mathcal{G}_{{}_{X^{\prime}}}$ is obtained by a gluing of the
constant group schemes with $\mathcal{G}_{{}_{\theta}}$’s along ${\rm
Spec}\,(K)$ by an automorphism of constant group scheme $G_{{}_{K}}$.

Now consider the inverse image of the constant group scheme
$p^{{}^{*}}(G_{{}_{X-D}})\simeq G\times p^{*}(X-D)$. Then using the gluing on
$X^{\prime}$, we can glue the constant group scheme $p^{{}^{*}}(G_{{}_{X-D}})$
with the local group schemes $\mathcal{H}_{{}_{B}}$ for each generic point
$\zeta$ to obtain a group scheme $\mathcal{H}_{{}_{Y^{\prime}}}$ on
$Y^{\prime}=p^{{}^{-1}}(X^{\prime})$ such that:

$\displaystyle\text{Res}_{{}_{Y^{\prime}/X^{\prime}}}(\mathcal{H}_{{}_{Y^{\prime}}})^{{}^{\Gamma}}\simeq\mathcal{G}_{{}_{X^{\prime}}}.$
(8.0.7)

####  8.0.2. Kawamata Coverings

Let $X$ be a smooth quasi-projective variety and let
$D=\sum_{{}_{i=0}}^{{}^{\ell}}D_{i}$ be the decomposition of the simple or
reduced normal crossing divisor $D$ into its smooth components (intersecting
transversally). The “Covering Lemma” of Y. Kawamata (see [18, Lemma 2.5, page
56]) says that, given positive integers $n_{{}_{0}},\ldots,n_{{}_{\ell}}$,
there is a connected smooth quasi-projective variety $Z$ over $\mathbb{C}$ and
a Galois covering morphism

$\displaystyle\kappa:Z\to X$ (8.0.8)

such that the reduced divisor
$\kappa^{{}^{*}}{D}:=\,({\kappa}^{{}^{*}}D)_{{}_{\text{red}}}$ is a normal
crossing divisor on $Z$ and furthermore,
${\kappa}^{{}^{*}}D_{{}_{i}}=n_{{}_{i}}.({\kappa}^{{}^{*}}D_{i})_{{}_{\text{red}}}$.
Let $\Gamma$ denote the Galois group for the covering map $\kappa$.

The isotropy group of any point $z\in Z$, for the action of $\Gamma$ on $Z$,
will be denoted by ${\Gamma}_{{}_{z}}$. It is easy to see that the stabilizer
at generic points of the irreducible components of
$(\kappa^{{}^{*}}D_{i})_{{}_{\text{red}}}$ are cyclic of order $n_{{}_{i}}$.

## References

  * [1] V.Balaji and C.S. Seshadri, Moduli of parahoric $\mathcal{G}$–torsors on a compact Riemann surface, J. Algebraic Geometry, 24, (2015),1-49.
  * [2] I. Biswas, Chern classes for parabolic bundles. J. Math. Kyoto Univ. 37 (1997), no. 4, 597–613. doi:10.1215/kjm/1250518206.
  * [3] S. Bosch, W.Lutkebohmert and M.Raynaud, Neron Models, Ergebnisse 21, Springer Verlag, (1990).
  * [4] M. Brion, The behaviour at infinity of the Bruhat decomposition, Comm. Math. Helv. 73 (1998)137-174.
  * [5] M. Brion, Log homogeneous varieties, Actas del XVI Coloquio Latinoamericano de Álgebra, 1–39, Biblioteca de la Revista Matematica Iberoamericana, Madrid, 2007.
  * [6] F.Bruhat and J.Tits, Groupes réductifs sur un crops local II: Schémas en groupes. Existence d’une donnée radicielle valuée, Publications Mathématiques de l’IHÉS 60 (1984) 5-184.
  * [7] C. De Concini and C. Procesi, Complete symmetric varieties, in Invariant theory (Montecatini, 1982), pp. 1–44, Lecture Notes in Math. 996, Springer, 1983.
  * [8] C. De Concini and T. Springer, Compactification of Symmetric Varieties, Transform. Groups 4 (1999), no. 2-3, pp. 273-300.
  * [9] B. Conrad, O.Gabber and Gopal Prasad, Pseudo-reductive groups, Cambridge University Press, New Mathematical Monographs, No 17.(2010), 2nd Edition.
  * [10] B. Edixhoven, Neron models and tame ramification, Compositio Mathematica, 81, (1992), 291-306.
  * [11] A. Grothendieck, Élements de Ǵeométrie Algébrique, Math. Publ. IHES 4, (1960), 5-228.
  * [12] R. Hartshorne, Stable Reflexive Sheaves, Mathematische Annalen 254 (1980), 121-176.
  * [13] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal Embeddings I, Springer Lecture Notes 339, 1973.
  * [14] V. Mehta and C.S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann, 248, (1980), 205-239.
  * [15] C.S. Seshadri, Moduli of $\pi$–vector bundles over an algebraic curve, Questions On algebraic Varieties, C.I.M.E, Varenna, (1969), 139-261.
  * [16] P. Solis, A wonderful embedding of the loop group. Advances in Mathematics 313, , (2017), 689-717.
  * [17] E. Strickland, A vanishing Theorem for Group Compactifications, Math. Ann. 277 (1987), pp. 165-171.
  * [18] E. Vieweg, Projective Moduli for Polarized Manifolds, , Ergebnisse, Springer(1995).