###### Lemma 3.14

For every $\mathcal{G}$ torsor $\mathcal{E}$ there exists $c_{\mathcal{E}}>0$
such that $\frac{\operatorname{wt}(\lambda)}{||\lambda||}\leq c$ for all
$\lambda\colon{\mathbb{G}}_{m,\eta}\to\mathcal{G}^{\mathcal{E}}_{\eta}$.

  * Proof.

As in the classical case for every reduction $\mathcal{E}_{\mathcal{P}}$ of
$\mathcal{E}$ to a parabolic subgroup we have that
$H^{0}(\operatorname{ad}(\mathcal{E}_{\mathcal{P}}))\subset
H^{0}(\operatorname{ad}(\mathcal{E}))$. By Riemann-Roch this implies that the
degree of the unipotent radical
$\deg(\mathcal{E}_{\mathcal{P}}\times^{\mathcal{P}}\mathfrak{u}_{\mathcal{P}})$
is uniformly bounded above for all reductions. In turn this gives for every
reduction to a Borel subgroup $\mathcal{B}\subset\mathcal{G}^{\mathcal{E}}$ an
upper bound for the weight $\operatorname{wt}(\omega_{i})$ for all dominant
coweights $\omega_{i}$. As any dominant $\lambda$ is a positive linear
combination of these, this gives the required bound. $\Box$

Notation. For any $\mathcal{G}$-torsor $\mathcal{E}$ over an algebraically
closed field we define

$\mu_{\max}(\mathcal{E}):=\sup_{\lambda}\frac{\operatorname{wt}_{\mathcal{E}}(\lambda)}{||\lambda||}.$

####  3.F.a. Comparing weights of different reductions

To characterize the canonical reduction we need to compare the weights of
different reductions. Suppose
$\mathcal{B},\mathcal{B}^{\prime}\subset\mathcal{G}^{\mathcal{E}}$ are two
Borel subgroups. Any two such subgroups share a maximal split torus
$S\subset\mathcal{T}_{\eta}\subset\mathcal{B}_{\eta}\cap\mathcal{B}^{\prime}_{\eta}$
([13, Proposition 4.4]). Let $\\{a_{1},\dots a_{n}\\}$ be the positive simple
roots of $\mathcal{B}$. We say that $\mathcal{B},\mathcal{B}^{\prime}$ are
neighboring reductions if there exists $i_{0}$ such that $-a_{i_{0}}$ is a
positive simple root of $\mathcal{B}^{\prime}$ and $a_{i}$ are positive roots
of $\mathcal{B}^{\prime}$ for all $i\neq i_{0}$. In this case
$\mathcal{B},\mathcal{B}^{\prime}$ generate a parabolic $\mathcal{P}_{1}$ that
is minimal among the parabolic subgroups that are not Borel subgroups. Denote
by $\mathcal{L}_{1}$ the corresponding Levi quotient.

###### Lemma 3.15

Let $\mathcal{B},\mathcal{B}^{\prime}$ be neighboring parabolic subgroups.
Then we have:

  1. (1)

$\operatorname{wt}_{\mathcal{B}}(\lambda)=\operatorname{wt}_{\mathcal{B}^{\prime}}(\lambda)$
for all $\lambda\in Z(\mathcal{L}_{I})$.

  2. (2)

$\operatorname{wt}_{\mathcal{B}}(\check{\alpha}_{i})+\operatorname{wt}_{\mathcal{B}^{\prime}}(-\check{\alpha}_{i})\leq
0$

  * Proof.

As the weight only depends on the Rees construction for $\mathcal{E}$ we may
replace $\mathcal{E}$ by the Rees construction applied to any $\lambda$ that
is dominant for $\mathcal{B}$ and $\mathcal{B}^{\prime}$. Thus we may assume
that
$\mathcal{E}=\mathcal{E}_{\mathcal{L}_{1}}\times^{\mathcal{L}_{1}}\mathcal{G}^{\mathcal{E}}$
is induced from a $\mathcal{L}_{1}$ torsor. In this case (1) is immediate from
the definition, as the Rees construction for $\lambda\in Z(L_{1})$ does not
change $\mathcal{E}_{\mathcal{L}_{1}}$. Moreover, $\mathcal{L}_{1}$ is of
semisimple rank $1$ and $\mathcal{B}_{1}:=\mathcal{B}\cap\mathcal{L}_{1}$ and
$\mathcal{B}^{\prime}_{1}\cap\mathcal{L}_{1}$ are Borel subgroups that are
opposite over the generic point $\eta\in C$. Let us denote by
$\mathfrak{u},\mathfrak{u}^{\prime}$ the Lie algebras of the corresponding
unipotent radicals in $\mathcal{L}_{1}$ and by
$\mathfrak{u}^{-}:=\operatorname{Lie}(\mathcal{L}_{1})/\operatorname{Lie}(\mathcal{B}_{1})$
and similarly $\mathfrak{u}^{\prime,-}$. Then we get an injective homomorphism

$\mathcal{E}_{\mathcal{B}_{1}}\times^{\mathcal{B}_{1}}\mathfrak{u}\to\operatorname{ad}(\mathcal{E}_{L_{1}})\to\operatorname{ad}(\mathcal{E}_{L_{1}})/\operatorname{ad}(\mathcal{E}_{\mathcal{B}^{\prime}_{1}})\cong\mathcal{E}_{\mathcal{B}^{\prime}_{1}}\times^{\mathcal{B}^{\prime}_{1}}\mathfrak{u}^{-,\prime}.$

If $a$ is a multipliable root then the homomorphism respects the filtration on
$\mathfrak{u},\mathfrak{u}^{\prime,-}$ given by the roots $a,2a$. So we find:

$\displaystyle\deg(\mathfrak{u}_{a}^{\mathcal{B}})$
$\displaystyle\leq\deg(\mathfrak{u}_{a}^{\mathcal{B}^{\prime}})=-\deg(\mathfrak{u}_{-a}^{\mathcal{B}^{\prime}})-\sum_{x\in\operatorname{Ram}(\mathcal{G})}f_{-a,x}^{\mathcal{B}^{\prime}}$
$\displaystyle-\deg(\mathfrak{u}_{a}^{\mathcal{B}})-\sum_{x\in\operatorname{Ram}(\mathcal{G})}f_{a,x}^{\mathcal{B}}=\deg(\mathfrak{u}_{-a}^{\mathcal{B}})$
$\displaystyle\geq\deg(\mathfrak{u}_{-a}^{\mathcal{B}^{\prime}})$

an the same hods for $\mathfrak{u}_{2a}$ if $2a$ is also a root. Thus:

$2(\deg(\mathfrak{u}_{a}^{\mathcal{B}})+\deg(\mathfrak{u}_{-a}^{\mathcal{B}^{\prime}}))\leq\sum_{x\in\operatorname{Ram}(\mathcal{G})}f_{-a,x}^{\mathcal{B}^{\prime}}+f_{a,x}^{\mathcal{B}}.$

Moreover, the map
$\mathfrak{u}_{a}^{\mathcal{B}}\to\mathfrak{u}_{a}^{\mathcal{B}^{\prime}}$ is
an isomorphism at $x$ if and only if the groups
$\mathcal{B}_{x}^{\mathcal{E}}$ and $\mathcal{B}_{x}^{\prime,\mathcal{E}}$ are
opposite in $\mathcal{G}^{\mathcal{E}}_{x}$. And in this case
$\langle\chi_{x}^{\mathcal{B}_{1}},\check{\alpha}_{i_{0}}\rangle=-\langle\chi_{x}^{\mathcal{B}_{1}^{\prime}},-\check{\alpha}_{i_{0}}\rangle$.
If the morphism is not an isomorphism, then the Borel subgroups are parallel
in $x$, so that
$\langle\chi_{x}^{\mathcal{B}_{1}},\check{\alpha}_{i_{0}}\rangle=\langle\chi_{x}^{\mathcal{B}_{1}^{\prime}},-\check{\alpha}_{i_{0}}\rangle$.
Thus if
$2\langle\chi_{x},\check{a}\rangle\leq\operatorname{rk}{\mathfrak{u}}_{a}\langle
a,\check{a}\rangle$ for all roots $a$ we find:

$2(\deg(\mathfrak{u}_{a}^{\mathcal{B}})+\deg(\mathfrak{u}_{-a}^{\mathcal{B}^{\prime}}))\langle
a,\check{a}\rangle\leq\langle\sum_{x\in\operatorname{Ram}(\mathcal{G})}f_{-a,x}^{\mathcal{B}^{\prime}}+f_{a,x}^{\mathcal{B}}+\chi_{x}^{\mathcal{B}_{1}}-\chi_{x}^{\mathcal{B}_{1}^{\prime}},\check{a}\rangle.$

And this means:

$\displaystyle\operatorname{wt}_{\mathcal{B}_{1}}(\check{\alpha}_{i_{0}})+\operatorname{wt}_{\mathcal{B}_{1}^{\prime}}(-\check{\alpha}_{i_{0}})$
$\displaystyle\leq 0$

if $\underline{\chi}$ is admissible. To compute
$\operatorname{wt}_{\mathcal{B}}(\check{\alpha}_{i_{0}})+\operatorname{wt}_{\mathcal{B}^{\prime}}(-\check{\alpha}_{i_{0}})$
we note that
$\mathcal{E}_{\mathcal{P}_{1}}\cong\mathcal{E}_{\mathcal{B}_{1}}\times^{\mathcal{B}_{1}}\mathcal{P}_{1}\cong\mathcal{E}_{\mathcal{B}_{1}^{\prime}}\times^{\mathcal{B}^{\prime}_{1}}\mathcal{P}_{1}$
and the unipotent radical of $\operatorname{Lie}(\mathcal{P}_{1})$ has a
filtration by $\mathcal{L}$-invariant subspaces such that over the generic
point the associated graded pieces are isomorphic to the unipotent groups
$\mathfrak{u}_{b*,\eta}=\oplus_{c=b+na}\mathfrak{u}_{c,\eta}^{\mathcal{B}}=\oplus_{c=b-na}\mathfrak{u}_{c,\eta}^{\mathcal{B}^{\prime}}$.
Moreover since $a$ is a positive root for $\mathcal{B}_{1}$ and a negative
root for $\mathcal{B}^{\prime}_{1}$ over $C$ the isomorphic bundles
$\mathcal{E}_{\mathcal{B}_{1}}\times^{\mathcal{B}_{1}}\mathfrak{u}_{b*}$ and
$\mathcal{E}_{\mathcal{B}_{1}^{\prime}}\times^{\mathcal{B}_{1}^{\prime}}\mathfrak{u}_{b*}$
come with canonical filtrations that are opposite at the generic point.
Therefore we find again that

$\sum_{c=b+na}(\deg(\mathfrak{u}_{c,\eta}^{\mathcal{B}})-\deg(\mathfrak{u}_{c,\eta}^{\mathcal{B}^{\prime}}))\langle
c,\check{\alpha}_{i_{0}}\rangle\leq 0$

Summing over all $b$ we obtain the result. $\Box$

We define $\mathcal{L}-\deg(\mathcal{E}_{\mathcal{B}})$ by the formula

$(\mathcal{L}-\deg,\lambda):=-\operatorname{wt}_{\mathcal{E}_{\mathcal{B}}}(\lambda).$

Then the previous lemma shows that $\mathcal{L}-\deg$ defines a complementary
polyhedron as defined by Behrend [8]. As in [26, Section 4.3] we therefore
obtain:

###### Proposition 3.16

Suppose $\mathcal{L}_{\det,\underline{\chi}}$ is an admissible line bundle and
$\mathcal{E}$ a $\mathcal{G}$-torsor. Then there exists a reduction
$\lambda\colon{\mathbb{G}}_{m}\to\mathcal{G}_{\eta}^{\mathcal{E}}$ such that
$\mathcal{E}_{\mathcal{P}_{\lambda}}$ is a reduction for which
$\frac{\operatorname{wt}(\lambda)}{||\lambda||}$ is maximal and such that for
every other such reduction to a parabolic subgroup
$\mathcal{Q}_{\lambda^{\prime}}$ we have
$\mathcal{Q}_{\lambda^{\prime}}\subset\mathcal{P}_{\lambda}$.

######  Lemma 3.17 (Semicontinuity of instability)

Let $\mathcal{L}_{\det,\underline{\chi}}$ be an admissible line bundle on
$\operatorname{Bun}_{\mathcal{G}}$. Let $R/k$ be a discrete valuation ring
with fraction field $K$ and residue field $\kappa$ and let $\mathcal{E}_{R}$
be a $\mathcal{G}$-torsor on $C_{R}$.

  1. (1)

If $\mathcal{E}_{K}$ is unstable and the canonical reduction is defined over
$K$, then $\mathcal{E}_{\kappa}$ is unstable and

$\mu_{max}(\mathcal{E}_{K})\leq\mu_{max}(\mathcal{E}_{\kappa}).$

The equality is strict, unless the canonical reduction of $\mathcal{E}_{K}$
extends to $R$.

  2. (2)

If $\mathcal{E}_{K}$ is semistable but not stable then $\mathcal{E}_{\kappa}$
is also not stable.

  * Proof.

The first part follows as in [26, Lemma 4.4.2]. This also shows that if
$\mathcal{E}_{K}$ admits a reduction of weight $0$, then
$\mathcal{E}_{\kappa}$ also cannot be stable. Finally suppose
$\dim\operatorname{Aut}(\mathcal{E}_{K})>0$. We know that
$\mathcal{G}^{\mathcal{E}_{R}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}C_{R}$ is an affine group scheme of finite type over
$C_{R}$. The group of global automorphisms of this group scheme is
$\operatorname{Spec}p_{*}(\mathcal{O}_{\mathcal{G}^{\mathcal{E}_{R}}})$, so
the generic fiber of this is not a finite $K$-algebra. But then by semi
continuity also the special fiber will not be finite. $\Box$

###  3.G. Boundedness for stable $\mathcal{G}$-torsors

With the canonical reduction at hand we can now deduce:

###### Proposition 3.18

Let $\mathcal{L}_{\det,\underline{\chi}}$ be an admissible line bundle on
$\operatorname{Bun}_{\mathcal{G}}$. Then the stacks
$\operatorname{Bun}_{\mathcal{G}}^{st}\subset\operatorname{Bun}_{\mathcal{G}}^{sst}\subset\operatorname{Bun}_{\mathcal{G}}$
of $\mathcal{L}_{\det,\underline{\chi}}$-(semi)-stable $\mathcal{G}$-torsors
are open substacks of finite type.

  * Proof.

By Lemma 3.17 instability and strict semistability are stable under
specialization. Therefore we only need to show that
$\operatorname{Bun}_{\mathcal{G}}^{sst}$ is constructible and contained in a
substack of finite type. Again we argue as in [8]. First we show that for any
$c\geq 0$ the stack of $\mathcal{G}$-torsors of fixed degree satisfying
$\mu_{\max}(\mathcal{E})\leq c$ is of finite type. To prove this we may
suppose that $\chi=0$, as the linear function
$\lambda\mapsto\langle\chi,\lambda\rangle$ only changes
$\frac{\operatorname{wt}(\lambda)}{||\lambda||}$ by a finite constant. By [8]
the claim holds if $\mathcal{G}=\mathcal{G}_{0}$ is a reductive group scheme
over $C$. Moreover if $\mathcal{G}^{\prime}\to\mathcal{G}_{0}$ is a parahoric
group scheme mapping to $\mathcal{G}_{0}$ such that $\mathcal{G}^{\prime}$ is
an Iwahori group scheme at all $x\in\operatorname{Ram}(\mathcal{G}^{\prime})$
then the morphism
$\operatorname{Bun}_{\mathcal{G}^{\prime}}\to\operatorname{Bun}_{\mathcal{G}_{0}}$
is a smooth morphism with fibers isomorphic to a product of flag varieties
$\mathcal{G}_{0,x}/\mathcal{B}_{x}$. Moreover the cokernel of
$\operatorname{Lie}(\mathcal{G}^{\prime})\to\operatorname{Lie}(\mathcal{G}_{0})$
is of finite length. Thus there exists a constant $d$ such that for any
$\mathcal{E}$ is a $\mathcal{G}^{\prime}$ torsor we have that if
$\mathcal{E}\times^{\mathcal{G}^{\prime}}\mathcal{G}_{0}$ admits a reduction
of slope $\mu(\mathcal{E}\times^{\mathcal{G}^{\prime}}\mathcal{G}_{0})>c+d$
then this reduction induces a reduction of $\mathcal{E}$ of slope
$\mu(\mathcal{E})>c$. Therefore the result also holds for
$\mathcal{G}^{\prime}$. Now any parahoric group scheme contains an Iwahori
group scheme so that by the same reasoning the result also holds if
$\mathcal{G}$ is any parahoric Bruhat–Tits group scheme such that
$\mathcal{G}|_{C-\operatorname{Ram}(\mathcal{G})}$ admits an unramified
extension. Now choose $\pi\colon\tilde{C}\to C$ a finite Galois covering with
group $\Gamma$ such that
$\pi^{*}\mathcal{G}|_{C-\operatorname{Ram}(\mathcal{G})}$ is a generically
split reductive group scheme on
$\tilde{C}-\pi^{-1}(\operatorname{Ram}(\mathcal{G}))$, equipped with a
$\Gamma$-action. We can extend this group scheme to a Bruhat–Tits group scheme
$\tilde{\mathcal{G}}$ that is $\Gamma$ equivariant and admits an morphism
$\pi^{*}(\mathcal{G})\to\tilde{\mathcal{G}}$ that is an isomorphism over
$\tilde{C}-\pi^{-1}(\operatorname{Ram}(\mathcal{G}))$. We already know the
result for $\operatorname{Bun}_{\tilde{\mathcal{G}}}$. Now if $\mathcal{E}$ is
a $\mathcal{G}$-torsor then
$\tilde{\mathcal{E}}:=\pi^{*}{\mathcal{E}}\times^{\pi^{*}\mathcal{G}}\tilde{\mathcal{G}}$
is a $\Gamma$-equivariant $\tilde{\mathcal{G}}$ torsor. Now if
$\tilde{\mathcal{E}}$ admits a canonical reduction to a parabolic subgroup
$\tilde{\mathcal{P}}$ this will define an equivariant reduction of
$\pi^{*}\mathcal{E}|_{C-\operatorname{Ram}(\mathcal{G})}$ and therefore a
reduction of $\mathcal{E}$. Again we can compare the weights of the reductions
because
$\pi_{*}\operatorname{ad}(\pi^{*}(\mathcal{E}))=\operatorname{ad}(\mathcal{E})\otimes\pi_{*}(\mathcal{O}_{\tilde{C}})$.
Therefore the determinant of the cohomology
$\det(H^{*}(\tilde{C},\operatorname{ad}(\pi^{*}\mathcal{E}))=\det
H^{*}(C,\operatorname{ad}(\mathcal{E})\otimes\pi_{*}\mathcal{O}_{\tilde{C}})$
defines a power of $\mathcal{L}_{\det}$ on $\operatorname{Bun}_{\mathcal{G}}$.
This implies that the weight of the reduction of $\pi^{*}\mathcal{E}$ is a
just $\deg(\pi)$-times the weight of the induced reduction of $\mathcal{E}$.
So again a very destabilizing canonical reduction of $\tilde{\mathcal{E}}$
induces a destabilizing reduction of $\mathcal{E}$. We are left to show that
the stable and semistable loci are constructible. We saw that unstable bundles
admit a canonical reduction to some $\mathcal{P}\subset\mathcal{G}$, i.e.,
they lie in the image of the natural morphism
$\operatorname{Bun}_{\mathcal{P}}\to\operatorname{Bun}_{\mathcal{G}}$. From
Remark 3.10 we know that it suffices to consider the image of this morphism
for finitely many $\mathcal{P}$. To conclude, we need to see that for every
substack of finite type $U\subset\operatorname{Bun}_{\mathcal{G}}$ only
finitely many connected components of $\operatorname{Bun}_{\mathcal{P}}$
contain canonical reductions that map to $U$. We just proved that on $U$ the
slope $\mu_{\max}$ is bounded above. We claim that for the canonical reduction
of a $\mathcal{G}$-bundle with $\mu_{\max}\leq c$ the degree of the
corresponding $\mathcal{P}$-bundle
$\underline{\deg}\in(X^{*}(\mathcal{P}))^{\vee}$ lies in a finite set. By
construction the canonical reduction was obtained from a complementary
polyhedron and therefore the $\mathcal{L}$-degree of the reduction that was
defined from the weight of the reduction that is bounded below when evaluated
at fundamental weights. As we bounded the $\mu_{max}$ this will also be
bounded above. Now we saw that the from equation 3.1 that the weight of the
reduction can be computed from the degree of the reduction and a local term
only depending on the group $\mathcal{P}$. Therefore we find obtain our bound
for the degree of the $\mathcal{P}$-bundle. To conclude we use that the number
of connected components of $\operatorname{Bun}_{\mathcal{P}}$ of a fixed
degree is finite. This is known for $\mathcal{G}$-bundles (see Proposition 5.2
). From this we can deduce our statement by looking at the Levi quotient
$\mathcal{P}\to\mathcal{L}$ and the morphism
$\operatorname{Bun}_{\mathcal{P}}\to\operatorname{Bun}_{\mathcal{L}}$, which
is smooth with connected fibers, because the kernel
$\mathcal{U}\to\mathcal{P}$ has a filtration by additive groups. For the stack
of stable points we can argue in the same way, oserving that strictly
semistable bundles admit a reduction to a parabolic with maximal slope equal
to $0$. $\Box$

###  3.H. Conclusion for $\mathcal{G}$-torsors

###### Theorem 3.19

Let $\mathcal{G}$ be a parahoric Bruhat–Tits group scheme that splits over a
tamely ramified extension $\tilde{k(C)}/k(C)$ and let
$\mathcal{L}=\mathcal{L}_{\det,\underline{\chi}}$ be an admissible line bundle
on $\operatorname{Bun}_{\mathcal{G}}$. Then the stack of $\mathcal{L}$-stable
$\mathcal{G}$ bundles $\operatorname{Bun}_{\mathcal{G}}^{st}$ admits a
separated coarse moduli space of finite type over $k$.

  * Proof.

This now follows from Proposition 2.8, because the stack of stable
$\mathcal{G}$-torsors $\operatorname{Bun}_{\mathcal{G}}^{st}$ is an open
substack by Proposition 3.18 and it satisfies the conditions of Proposition
2.6 by Proposition 3.3. $\Box$

###### Remark 3.20

For admissible line bundles $\mathcal{L}=\mathcal{L}_{\det,\underline{\chi}}$
the results on the existence of canonical reductions for $\mathcal{G}$-bundles
allow to copy the proof of the semistable reduction theorem using Langton’s
algorithm from [28] and [29]. In particular in those cases where
$\mathcal{L}$-semistability is equivalent to $\mathcal{L}$-stability this then
implies that the coarse moduli spaces are proper.

##  4\. Appendix: Fixing notations for Bruhat–Tits group schemes

In this appendix we collect the results from [17] on the structure of
Bruhat–Tits group schemes that we use. As the definition is local let us fix a
discrete valuation ring $R$ with fraction field $K$ and residue field $k$ and
$G_{K}$ a reductive group over $K$. In general the groups are defined by
descent starting from the quasi-split case over an unramified extension of
$R$. In our applications we can always extend the base field and if
$k={\overline{k}}$ the group $G_{K}$ is quasi-split by the theorem of
Steinberg [12, Section 8.6]. We will therefore assume that $G_{K}$ is quasi-
split. We choose a maximal split torus $S_{K}\subset G_{K}$ and denote
$T_{K}:=Z(S_{K})$ the centralizer of $S$ which is a maximal torus of $G_{K}$
because $G_{K}$ was quasi-split.111In [7] Balaji and Seshadri study the case
where $S_{K}=T_{K}$ is a split maximal torus, i.e. the case where $G_{K}$ is a
split reductive group, which already shows many interesting features. To
construct models of $G_{K}$ over $R$ Bruhat–Tits first extend the torus
$T_{K}$ to a scheme over $R$ and then the root subgroups of $G_{K}$ using a
pinning of $G_{K}$ that they upgrade to a Chevalley-Steinberg valuation
$\varphi$. ([17] Section 4.2.1 and 4.1.3). Let us recall these notions:

###  4.A. Chevalley–Steinberg systems, pinnings and valuations

If $G_{K}$ is split, a Chevalley–Steinberg system is simply a pinning of our
group, i.e., an identification
$x_{\alpha}\colon{\mathbb{G}}_{a}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}U_{\alpha}$
which is compatible for $\alpha,-\alpha$ in the sense that it comes from an
embedding of $\zeta_{\alpha}\colon\textsf{SL}_{2,K}\to G_{K}$ identifying
${\mathbb{G}}_{a}$ with the strict upper and lower triangular matrices [17,
Paragraph 3.2.1] and is compatible with reflections [17, Paragraph 3.2.2,
(Ch1),(Ch2)]. If $G_{K}$ is not split, we can split it over a Galois extension
$\tilde{K}$ and choose an equivariant pinning [17, Paragraph 4.1.3]: Let
$\Gamma:=\operatorname{Gal}(\tilde{K}/K)$ and denote by $\tilde{\Phi}$ the
roots with respect to $T_{\tilde{K}}$ of $G_{\tilde{K}}$ and $\Phi$ the roots
for $S$. Then $X^{*}(S)=X^{*}(S_{\tilde{K}})$ and the elements of $\Phi$ are
the restrictions of elements of $\tilde{\Phi}$ to $S$. The root subgroups
$U_{a}$ can then be described as follows. For any root ray $a\in\Phi$ denote
$\tilde{\Delta}_{a}\subset\tilde{\Phi}$ the set of simple roots that restrict
to $a$. The analog of the $\textsf{SL}_{2}$ defined by a root is a morphism
$\zeta_{a}\colon G^{a}_{K}\to G_{K}$. If $a$ is not a multiple root we have

$G^{a}_{K}\cong\operatorname{Res}_{L_{\alpha}/K}\textsf{SL}_{2},$

where $L_{\alpha}\subset\tilde{K}$ is the field obtained by the stabilizer of
any $\alpha\in\tilde{\Delta}_{a}$. In this case a pinning is an isomorphism

$\operatorname{Res}_{L_{\alpha}/K}{\mathbb{G}}_{a}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}U_{a},$

which is again assumed to be compatible for $a$ and $-a$. ( [17, Paragraph
4.1.7,4.1.8]) If $a$ is a multiple root ray $G^{a}$ is the Weil restriction of
a unitary group222This happens if the Galois group interchanges neighboring
roots in the Dynkin diagram.. In this case for any pair
$\alpha,\alpha^{\prime}\in\tilde{\Delta}_{a}$ such that
$\alpha+\alpha^{\prime}$ is a root let
$L_{\alpha}=\tilde{K}^{\operatorname{Stab}{\alpha}}$ which is a quadratic
extension of $L_{\alpha+\alpha^{\prime}}=:L_{2}$. This extension defines the
unitary group $SU_{3}(L_{\alpha}/L_{2}))$ over $L_{2}$ (with respect to the
standard hermition form). Then

$G^{a}=\operatorname{Res}_{L_{2}/K}SU_{3}(L_{\alpha}/L_{2}).$

In this case the root subgroups $U_{a},U_{-a}$ are of the form

$U_{a}(L_{2})=\Big{\\{}x_{a}(u,v)=\left(\begin{array}[]{ccc}1&-u^{\sigma}&-v\\\
0&1&u\\\ 0&0&1\end{array}\right)\Big{\\}},\quad
U_{-a}(L_{2})=\Big{\\{}x_{-a}(u,v)=\left(\begin{array}[]{ccc}1&0&0\\\ u&1&0\\\
-v&-u^{\sigma}&1\end{array}\right)\Big{\\}}$

with $v+v^{\sigma}=uu^{\sigma}.$ This has a filtration $U_{2a}\cong\\{v\in
L_{\alpha}/L_{2}|\operatorname{tr}(v)=0\\}$ and $U_{a}/U_{2a}\cong
L_{\alpha}$. The Chevalley pinning induces valuations on the groups $U_{a}$.
For non multipliable roots one sets $\phi_{\alpha}(x_{\alpha}(u)):=|u|$ and
for multiple roots one defines $\phi_{a}(x_{a}(u,v)):=\frac{1}{2}|v|$ and
$\phi_{2a}(x_{a}(0,v)):=|v|$. (These choices define a valued root system and
identify the standard appartment $A\cong X_{*}(S)_{{\mathbb{R}}}$ in the
Bruhat–Tits building of $G$).

###  4.B. Parahoric group schemes

With this notation, we can recall the construction of an open part of parhoric
group schemes. For the torus $T_{K}$ Bruhat and Tits choose a version of the
Néron model as extension. In order to be consistent with the conventions from
[37] we choose the connected Néron model $\mathcal{T}_{R}$ as an extension
over $\operatorname{Spec}R$. For the unipotent groups $U_{a}$, the valuations
introduced above can be used to define extensions of $U_{a}$ to group schemes
$\mathcal{U}_{a,k}$ over $\operatorname{Spec}R$ for any $k\in{\mathbb{R}}$.
For non-multiple roots the pinning identifies the abstract group
$U_{a,k}:=\\{u\in U_{a}(K)|\phi_{a}(u)\leq k\\}\cong\\{u\in L_{a}||u|\leq
k\\}$ and this can be equipped with the structure of a groups scheme
isomorphic to $\operatorname{Res}_{R_{a}/R}{\mathbb{G}}_{a}$. For multiple
roots this is slightly more complicated to spell out, but again these group
schemes always correspond to free $R$ modules [17, 4.3.9]. The open subset of
a parahoric group scheme will be of the form ([17, Theorem 3.8.1]):

$\prod_{a\in\Phi^{-}}\mathcal{U}_{a,f(a)}\times\mathcal{T}\times\prod_{a\in\Phi^{+}}\mathcal{U}_{a,f(a)}.$

Now a facet $\Omega\subset A$ defines a valuation of the root system (4.6.26)

$f(a):=\inf\\{k\in{\mathbb{R}}|a(x)+k\geq 0\forall a\in\Omega\\},$

here we used our Chevalley-Steinberg valuation, which defines an isomorphism
$-\phi\colon A\cong X_{*}(S)_{\mathbb{R}}$. The product described above
carries a birational group law ([33, Propostion 5.12]) and thus one can use
[14, Theorem 5] to construct a the group scheme $\mathcal{G}_{\Omega}$
(denoted by $\mathcal{G}_{\Omega}^{\circ}$ in [17]), containing the product as
an open neighborhood of the identity. For our computations this is sufficient
as these only use the Lie-algebra of $\mathcal{G}$.

##  5\. Appendix: Basic results on $\operatorname{Bun}_{\mathcal{G}}$

As in Section 3 we fix a smooth projective geometrically connected curve $C$
over an algebraically closed field $k$ and $\mathcal{G}\to C$ a parahoric
Bruhat-Tits group scheme. In this appendix we collect some results on the
stack of $\mathcal{G}$-bundles $\operatorname{Bun}_{\mathcal{G}}$ for which we
could not find a reference. The basic tool will be the Beilinson–Drinfeld
Grassmannian $\operatorname{GR}_{\mathcal{G}}$ i.e., the ind-projective scheme
representing the functor of $\mathcal{G}$-bundles together with a
trivialization outside of a finite divisor on $C$:

$\operatorname{GR}_{\mathcal{G}}(T)=\left\\{(\mathcal{E},D,\phi)\left|{\mathcal{E}\in\operatorname{Bun}_{\mathcal{G}}(T),D\in
C^{(d)}(T)\text{ for some d}\atop\phi\colon\mathcal{E}|_{C\times
T-D}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{G}\times_{C}(C\times
T-D)}\right.\right\\}.$

It comes equipped with a natural forgetful maps

$p\colon\operatorname{GR}_{\mathcal{G}}\to\operatorname{Bun}_{\mathcal{G}},$
$\operatorname{supp}\colon\operatorname{GR}_{\mathcal{G}}\to\coprod_{d}C^{(d)}.$

For Bruhat-Tits groups it will be useful to consider the open subfunctor,
where the divisor $D$ does not intesect some fixed finite subset of $C$, e.g.,
the ramification points of the group $\mathcal{G}$. Given $S\subset C$ finite,
we will denote by
$\operatorname{GR}_{\mathcal{G},C-S}:=\operatorname{supp}^{-1}\big{(}\coprod(C-S)^{(d)}\big{)}$,
i.e. the ind-scheme parametrizing $\mathcal{G}$ bundles together with a
trivialization outside of a divisor $D$ that is disjoint form $S$. Similarly
we denote by $\operatorname{Gr}_{\mathcal{G},x}:=\operatorname{supp}^{-1}(x)$,
the space parametrizing $\mathcal{G}$-bundles equipped with a trivialization
on $C-\\{x\\}$. The following is a probably well-known geometric version of a
weak approximation theorem:

###### Proposition 5.1

For any Bruhat-Tits scheme $\mathcal{G}\to C$ and any finite subscheme
$S\subset C$ the natural forgetful map
$p_{C-S}\colon\operatorname{GR}_{\mathcal{G},C-S}\to\operatorname{Bun}_{\mathcal{G}}$
is surjective, i.e., after possibly passing to a flat extension, any
$\mathcal{G}$-bundle admits a trivialization on an open neighborhood of $S$.

  * Proof.

For $S=\emptyset$ this follows from a theorem of Steinberg and Borel-Springer
[12], stating that for any algebraically closed field $\overline{K}$ any
$\mathcal{G}$-bundle on $C_{\overline{K}}$ is trivial over the generic point
$\overline{K}(C)$. As any such trivialization is defined over an open subset
this allows to deduce that
$p\colon\operatorname{GR}_{\mathcal{G}}\to\operatorname{Bun}_{\mathcal{G}}$ is
surjective in the fppf-topology. To deduce the result for $p_{C-S}$ we can
argue as in [27, Theorem 5]: As
$\operatorname{GR}_{\mathcal{G},C-S}\subset\operatorname{GR}_{\mathcal{G}}$ is
a dense open subfunctor, it follows that the generic point of any connected
component of $\operatorname{Bun}_{\mathcal{G}}$ is in the image of $p_{C-S}$.
Let $\mathcal{E}\in\operatorname{Bun}_{\mathcal{G}}$ be any bundle. For the
inner form
$\mathcal{G}^{\mathcal{E}}:=\operatorname{Aut}_{\mathcal{G}}(\mathcal{E})$ of
$\mathcal{G}$ we can apply the same reasoning and find that the image of the
map
$\operatorname{GR}_{\mathcal{G}^{\mathcal{E}},C-S}\to\operatorname{Bun}_{\mathcal{G}^{\mathcal{E}}}\cong\operatorname{Bun}_{\mathcal{G}}$
also contains an open subset of every connected component. In particular there
exist $\mathcal{G}$-bundles $\mathcal{E}^{\prime}$ in the image that are also
in the image of $p_{C-S}$. For such any such bundle $\mathcal{E}^{\prime}$
there exist divisors $D_{1},D_{2}$ on $C-S$ such that
$\mathcal{E}^{\prime}|_{C-D_{1}}\cong\mathcal{G}|_{C-D_{1}}$ and
$\mathcal{E}^{\prime}|_{C-D_{2}}\cong\mathcal{E}|_{C-D_{2}}$. Composing these
isomorphisms we find that $\mathcal{E}|_{C-(D_{1}\cup
D_{2})}\cong\mathcal{G}|_{C-(D_{1}\cup D_{2})}$, i.e., $\mathcal{E}$ is also
in the image of $p_{C-S}$. Thus $p_{C-S}$ is surjective on geometric points.
As it is the restriction of a flat morphism to an open subfunctor this implies
thst it is again fppf surjective. $\Box$

To formulate properties of the set of connected components of
$\operatorname{Bun}_{\mathcal{G}}$ we will need some more notation. The
generic point of $C$ will be denoted by $\eta=\operatorname{Spec}k(C)$ and
$\overline{\eta}$ will be a geometric generic point. From [37, Theorem 0.1] we
know that for any $x\in C$ there is a natural isomorphism
$\pi_{0}(\operatorname{Gr}_{\mathcal{G},x})\cong\pi_{1}(\mathcal{G}_{\overline{\eta}})_{I}$
where $I=\operatorname{Gal}({\overline{K}_{x}/K_{x}})$ is the inertia group at
$x$ and $\pi_{1}(\mathcal{G}_{\overline{\eta}})$ is the algebraic fundamental
group, i.e., the quotient of the coweight lattice by the coroot lattice of
$\mathcal{G}_{\overline{\eta}}$. We denote by
$X^{*}(\mathcal{G})=\operatorname{Hom}(\mathcal{G},{\mathbb{G}}_{m,C})$ the
group of characters of $\mathcal{G}$. As $\mathcal{G}$ is a smooth group
scheme with connected fibers $X^{*}(\mathcal{G})\cong
X^{*}(\mathcal{G}_{\eta})=X^{*}(\mathcal{G}_{\overline{\eta}})^{\operatorname{Gal}(\overline{\eta}/\eta)}$.
As any character $\chi\colon\mathcal{G}\to{\mathbb{G}}_{m}$ defines a morphism
$\operatorname{Bun}_{\mathcal{G}}\to\operatorname{Pic}_{X}$ given by
$\mathcal{E}\mapsto\mathcal{E}(\chi):=\mathcal{E}\times^{\mathcal{G}}{\mathbb{G}}_{m}$
it defines a degree $\underline{d}\colon\operatorname{Bun}_{\mathcal{G}}\to
X^{*}(\mathcal{G})^{\vee}$ by $\mathcal{E}\mapsto\deg(\mathcal{E}(\chi))$ and
we will denote by
$\operatorname{Bun}_{\mathcal{G}}^{\underline{d}}\subset\operatorname{Bun}_{\mathcal{G}}$
the substack of bundles of degree $\underline{d}$, which is open and closed
because the degree of line bundles is locally constant in families.

###### Proposition 5.2

  1.   2. (1)

The natural map
$\pi_{1}(\mathcal{G}_{\overline{\eta}})_{\operatorname{Gal}(\overline{\eta}/\eta)}\to\pi_{0}(\operatorname{Bun}_{\mathcal{G}})$
is surjective.

  3. (2)

For any $\underline{d}\in\operatorname{Hom}(X^{*}(\mathcal{G}),{\mathbb{Z}})$
the stack $\operatorname{Bun}_{\mathcal{G}}^{\underline{d}}$ has finitely many
connected components.

  * Proof.

The first part follows from the surjectivity of
$\operatorname{GR}_{\mathcal{G}}\to\operatorname{Bun}_{\mathcal{G}}$ and the
description of the connected components
$\pi_{0}(\operatorname{Gr}_{\mathcal{G},x})\cong\pi_{1}(\mathcal{G}_{\overline{\eta}})_{I}$
from [37] as follows. Let $\operatorname{GR}^{d}_{\mathcal{G}}$ denote the
components of $\operatorname{GR}_{\mathcal{G}}$ mapping to $C^{(d)}$. As the
projection $\operatorname{GR}^{d}_{\mathcal{G}}$ is ind-projective every
connected component intersects the fibers over the diagonal $C\subset
C^{(d)}$. The preimage of the diagonal is isomorphic to
$\operatorname{GR}_{\mathcal{G}}^{1}$. For
$\operatorname{GR}_{\mathcal{G}}^{1}$ the fiber wise isomorphism
$\pi_{0}(\operatorname{Gr}_{\mathcal{G},x})\cong\pi_{1}(\mathcal{G}_{\overline{\eta}})_{I}$
is given by the Kottwitz homomorphism which by construction is induced from a
Galois-equivariant map $\pi_{1}(\mathcal{G}_{\overline{\eta}})$ to the sheaf
of connected components of the fibers of $p$. Thus this induces a surjection
$\pi_{1}(\mathcal{G}_{\overline{\eta}})_{\operatorname{Gal}(\overline{\eta}/\eta)}\to\pi_{0}(\operatorname{GR}_{\mathcal{G}}^{1})\to\pi_{0}(\operatorname{Bun}_{\mathcal{G}})$,
which proves (1). This implies (2), because the map
$\operatorname{Hom}(X^{*}(\mathcal{G}),{\mathbb{Z}})^{I}\to\operatorname{Hom}(X^{*}(\mathcal{G}),{\mathbb{Z}})_{I}$
induces an isomorphism up to torsion. $\Box$

## References

  * [1] D. Abramovich, A. Corti, and A. Vistoli, _Twisted bundles and admissible covers_ , Comm. Algebra 31 (2003), no. 8, Special issue in honor of Steven L. Kleiman, 3547–3618. MR-2007376
  * [2] D. Abramovich, T. Graber, and A. Vistoli, _Gromov-Witten theory of Deligne-Mumford stacks_ , Amer. J. Math. 130 (2008), no. 5, 1337–1398. MR-2450211
  * [3] J. Alper, _Good moduli spaces for Artin stacks_ , Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2349–2402. MR-3237451
  * [4] J. Alper, J. Hall, and D. Rydh, _A Luna étale slice theorem for algebraic stacks_ , preprint 2015. arXiv:1504.06467
  * [5] L. Álvarez-Cónsul, O. García-Prada, and A. Schmitt, _On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces_ , Int. Math. Res. Papers (2006), Art. ID 73597, 82 pp. MR-2253535
  * [6] A. Asok, _Equivariant vector bundles on certain affine G-varieties_ , Pure Appl. Math. Q. 2 (2006), no. 4, Special Issue in honor of Robert D. MacPherson, part 2, 1085–1102. MR-2282414
  * [7] V. Balaji and C.S. Seshadri, _Moduli of parahoric $\mathcal{G}$-torsors on a compact Riemann surface_, J. Algebraic Geom. 24 (2015), no. 1, 1–49. MR-3275653
  * [8] K.A. Behrend, _Semi-stability of reductive group schemes over curves_ , Math. Ann. 301 (1995), no. 2, 281–305. MR-1314588
  * [9] A. Beilinson and V. Drinfel’d, _Quantization of Hitchin’s integrable system and Hecke eigensheaves_ , preprint, available under http://www.ma.utexas.edu/users/benzvi/
  * [10] B. Bhatt and D. Halpern-Leistner, _Tannaka duality revisited_ , preprint 2015. arXiv:1507.01925
  * [11] I. Biswas and N. Hoffmann, _The line bundles on moduli stacks of principal bundles on a curve_ , Documenta Math. 15 (2010), 35–72. MR-2628848
  * [12] A. Borel and T.A. Springer, _Rationality properties of linear algebraic groups. II_ , Tohoku Math. J. (2) 20 (1968), 443–497. MR-0244259
  * [13] A. Borel and J. Tits, _Groupes réductifs_ , Publ. Math. Inst. Hautes Études Sci. 27 (1965), 55–150. MR-0207712
  * [14] S. Bosch, W. Lütkebohmert, and M. Raynaud, _Néron models_ , Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 21, Springer–Verlag, Berlin, 1990. MR-1045822
  * [15] S. Bradlow, _Coherent systems: A brief survey. With an appendix by H. Lange._ In: London Math. Soc. Lecture Note Ser., vol. 359, Moduli spaces and vector bundles, pp. 229–264, Cambridge Univ. Press, Cambridge, 2009. MR-2537071
  * [16] F. Bruhat and J. Tits, _Groupes réductifs sur un corps local_ , Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–251. MR-0327923
  * [17] F. Bruhat and J. Tits, _Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée._ Publ. Math. Inst. Hautes Études Sci. 60 (1984), 197–376. MR-0756316
  * [18] P-H. Chaudouard and G. Laumon, _Le lemme fondamental pondéé. I. Constructions géométriques._ Compositio Math. 146 (2010), no. 6, 1416–1506. MR-2735371
  * [19] B. Conrad, _The Keel–Mori theorem via stacks_ , preprint 2005. http://math.stanford.edu/ $\sim$conrad/papers/coarsespace.pdf
  * [20] G. Faltings, _Algebraic loop groups and moduli spaces of bundles_ , J. Eur. Math. Soc. 5 (2003), no. 1, 41–68. MR-1961134
  * [21] D. Gaitsgory and J. Lurie, _Weil’s conjecture over function fields_ , preprint. http://www.math. harvard.edu/$\sim$lurie/papers/tamagawa.pdf
  * [22] D. Halpern-Leistner, _On the structure of instability in moduli theory_ , preprint 2014. arXiv: 1411.0627
  * [23] D. Halpern-Leistner and A. Preygel, _Mapping stacks and categorical notions of properness_ , preprint 2014. arXiv:1402.3204
  * [24] G. Harder, _Halbeinfache Gruppenschemata über vollständigen Kurven_ ,Invent. Math. 6 (1968), 107–149. MR-0263826
  * [25] G. Harder and U. Stuhler, _Canonical parabolic subgroups for Arakelov group schemes_ , unpublished manuscript.
  * [26] J. Heinloth and A. Schmitt, _The cohomology rings of moduli stacks of principal bundles over curves_ , Doc. Math. 15 (2010), 423–488. MR-2657374
  * [27] J. Heinloth, _Uniformization of $\mathcal{G}$-bundles_, Math. Ann. 347 (2010), no. 3, 499–528. MR-2640041
  * [28] J. Heinloth, _Semistable reduction for G-bundles on curves_ , J. Algebraic Geom. 17 (2008), no. 1, 167–183. MR-2357683
  * [29] J. Heinloth, _Addendum to “Semistable reduction of G-bundles on curves”_ , J. Algebraic Geom. 19 (2010), no. 1, 193–197. MR-2551761
  * [30] W. Hesselink, _Concentration under actions of algebraic groups_. In: Paul Dubreil and Marie-Paul Malliavin Algebra Seminar, 33rd year (Paris, 1980), pp. 55–89, Lecture Notes in Mathematics, vol. 867, Springer, Berlin, 1981. MR-0633514
  * [31] I. Iwanari, _Stable points on algebraic stacks_ , Adv. Math. 223 (2010), no. 1, 257–299. MR-2563218
  * [32] S. Keel and S. Mori, _Quotients by groupoids_ , Ann. of Math. (2) 145 (1997), no. 1, 193–213. MR-1432041
  * [33] E. Landvogt, _A compactification of the Bruhat-Tits building_ , Lecture Notes in Mathematics, vol. 1619, Springer-Verlag, Berlin, 1996. MR-1441308
  * [34] G. Laumon and L. Moret-Bailly, _Champs algébriques_ , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 39, Springer-Verlag, Berlin, 2000. MR-1771927
  * [35] J. Martens and M. Thaddeus, _Compactifications of reductive groups as moduli stacks of bundles_ , Compositio Math. 152 (2016), no. 1, 62–98. MR-3453388
  * [36] D. Mumford, J. Fogarty, and F. Kirwan, _Geometric invariant theory. Third edition._ Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34, Springer-Verlag, Berlin, 1994. MR-1304906
  * [37] G. Pappas and M. Rapoport, _Some questions about $\mathcal{G}$-bundles on curves_. In: Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), pp. 159–171, Adv. Stud. Pure Math., vol. 58, Math. Soc. Japan, Tokyo, 2010. MR-2676160
  * [38] D. Rydh, _Existence and properties of geometric quotients_ , J. Algebraic Geom. 22 (2013), no. 4, 629–669. MR-3084720
  * [39] A. Schmitt, _Moduli problems of sheaves associated with oriented trees_ , Algebr. Represent. Theory 6 (2003), no. 1, 1–32. MR-1960511
  * [40] Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Schémas en groupes II. Lecture Notes in Mathematics, vol. 153, Springer-Verlag, Berlin–New York, 1970. MR-0274459
  * [41] T .A. Springer, _Linear algebraic groups_ , Modern Brikhäuser Classics, Birkhäuser, Boston, MA, 2009. MR-2458469
  * [42] The Stacks Project Authors.  The stacks project.  http://stacks.math.columbia.edu
  * [43] R. Steinberg, _Regular elements of semisimple algebraic groups_ , Publ. Math. Inst. Hautes Études Sci. 25 (1965), 49–80. MR-0180554
  * [44] X. Zhu, _On the coherence conjecture of Pappas and Rapoport_ , Ann. of Math. (2) 180 (2014), no. 1, 1–85. MR-3194811