$(\lambda_{1},\mathcal{G}_{\lambda_{1}})$. We can use this to define, on each
component of $H_{1}$, a relative $1^{st}$ associated graded Levi
$\rho_{\lambda_{1}}$-sheaf, in exactly the same way as in the construction of
the GHN filtration in Subsection 8.3. Putting all of these together for all
connected components, we obtain what we can call a relative $1^{st}$
associated graded Levi $\rho_{\lambda_{1}}$-sheaf
$\text{gr}_{1}\vcentcolon=\text{gr}_{1}(\mathcal{F}^{\bullet}|_{X_{H_{1}}},\sigma|_{X_{H_{1}}})$
defined on $H_{1}$, where we allow a different choice of the cocharacter
$\lambda_{1}$ over each connected component of $H_{1}$.

Second Step: We can apply Proposition 4.37 to the restriction of
$\text{gr}_{1}$ on each connected component in order to obtain a refinement
$H_{2}$ of the stratification $H_{1}$. The scheme $H_{2}$ is universal for the
leading term HN filtration of $\text{gr}_{1}$, hence we can choose a
representative $f_{2}$ for the universal filtration. By the last statement of
Proposition 4.37, the restriction of $f_{2}$ to each connected component of
$H_{2}$ comes from a weighted reduction
$(\lambda_{2},\mathcal{G}_{\lambda_{2}})$ of $\text{gr}_{1}$. The procedure to
construct the GHN filtration can be applied verbatim to this relative
situation. In particular, we can construct a relative lexicographic
$\mathbb{Z}^{2}$-filtration of
$(\mathcal{F}^{\bullet}|_{X_{H_{2}}},\sigma|_{X_{H_{2}}})$. This lexicographic
filtration comes from a 2-weighted parabolic reduction
$((\lambda_{1},\lambda_{2}),\mathcal{G}_{(\lambda_{1},\lambda_{2})})$. By
construction, $H_{2}$ is universal for lexicographic
$\mathbb{Z}^{2}$-filtrations obtained in the second step of the construction
of the GHN filtration. We can define a forgetful morphism
$\operatorname{RelGies}(\mathcal{F}^{\bullet},\sigma)\rightarrow H_{2}$ in the
same way as in the case of the first step $H_{1}$. In this case we remember
the first two coordinates of the $\mathbb{Z}^{q}$-filtration and take the
union over the other coordinates.

Recursion: Now we can iterate as in the construction of the (pointwise) GHN
filtration. This process must terminate by the same reasoning as in Subsection
8.3, namely, the dimensions of the vector spaces in the tuple $V^{\bullet}$
are finite. If it terminates at the $q^{th}$ step, then the corresponding
locally closed stratification $H_{q}$ is universal for GHN filtrations of
$(\mathcal{F}^{\bullet},\sigma)$. Furthermore, on each connected component of
$H_{q}$ the restriction of (a representative up to scaling of) the universal
GHN filtration comes from a relative multi-weighted parabolic reduction. ∎

######  Remark 8.19.

It follows from the proof of Theorem 8.18 that this stratification of the
stack is obtained by recursively taking intersections of certain
$\Theta$-stratifications. At each step we are using the leading term HN
$\Theta$-stratification defined on what is called “the center” of the
$\Theta$-stratum in the previous step. In this case each center can be thought
of as
$\operatorname{Bun}_{\rho_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{i})}}(X)$
for some representation $\rho_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{i})}$
of a Levi subgroup. This is what allows the iterative procedure.

##  9\. Examples

In this section we give some examples of the theory developed.

###  9.1. Torsion-free sheaves

Suppose that $G=\operatorname{GL}(V)$ and $\rho$ is the identity. Then
$\operatorname{Bun}_{\rho}(X)$ is equivalent to the stack
$\operatorname{Coh}_{r}^{tf}(X)$ of torsion-free sheaves of rank $r$ on $X$.
In this case Proposition 3.8 gives a description of a filtration of a torsion-
free sheaf $\mathcal{F}$ in $\operatorname{Bun}_{\rho}(X)$ as a decreasing
$\mathbb{Z}$-filtration of $\mathcal{F}$ by saturated subsheaves
$\mathcal{F}_{m}\subset\mathcal{F}$. The proof of [HLHJ21, Prop. 5.16] shows
that a torsion-free sheaf in $\operatorname{Bun}_{\rho}(X)$ is Gieseker
semistable in our sense if and only if it is Gieseker semistable in the sense
of [HL97, Defn. 1.2.4].

The leading term HN filtration for torsion-free sheaves in
$\operatorname{Bun}_{\rho}(X)$ recovers the weighted leading term filtration
described in [HLHJ21, Thm. 5.11], which we reproduce here. Let

$0=\mathcal{F}^{\text{HN}}_{0}\subset\mathcal{F}^{\text{HN}}_{1}\subset\mathcal{F}^{\text{HN}}_{2}\subset\cdots\subset\mathcal{F}^{\text{HN}}_{n}=\mathcal{F}$

denote the Gieseker-Harder-Narasimhan filtration of $\mathcal{F}$ (c.f. [HL97,
Thm. 1.3.4]). Then we have

$\overline{p}_{\mathcal{F}_{1}^{\text{HN}}}>\overline{p}_{\mathcal{F}_{2}^{\text{HN}}/\mathcal{F}_{1}^{\text{HN}}}>\cdots>\overline{p}_{\mathcal{F}_{n}^{\text{HN}}/\mathcal{F}_{n-1}^{\text{HN}}}$

Let $i$ denote the minimal index such that

$\widehat{\mu}_{h}(\mathcal{F}_{1}^{\text{HN}})=\widehat{\mu}_{h}(\mathcal{F}_{2}^{\text{HN}}/\mathcal{F}_{1}^{\text{HN}})=\cdots=\widehat{\mu}_{h}(\mathcal{F}_{n}^{\text{HN}}/\mathcal{F}_{n-1}^{\text{HN}})$

for all $h>i$. In other words, $i$ is the largest integer such that the
$i^{th}$ slopes of the graded pieces of the filtration are not all equal. We
have

$\widehat{\mu}_{i}(\mathcal{F}_{1}^{\text{HN}})\geq\widehat{\mu}_{i}(\mathcal{F}_{2}^{\text{HN}}/\mathcal{F}_{1}^{\text{HN}})\geq\cdots\geq\widehat{\mu}_{i}(\mathcal{F}_{n}^{\text{HN}}/\mathcal{F}_{n-1}^{\text{HN}})$

where at least one of the inequalities above is strict.

We define indexes $h_{0}<h_{1}<h_{2}<\cdots<h_{j}=n$ as follows. We let
$h_{0}=0$, and let

$\\{h_{m}\\}_{m=1}^{j}=\left\\{\;1\leq l\leq
n\;\left|\;\widehat{\mu}_{i}(\mathcal{F}_{l+1}^{\text{HN}}/\mathcal{F}_{l}^{\text{HN}})<\widehat{\mu}_{i}(\mathcal{F}_{l}^{\text{HN}}/\mathcal{F}_{l-1}^{\text{HN}})\right.\right\\}$

This ensures that
$\widehat{\mu}_{i}(\mathcal{F}_{h_{k}}^{\text{HN}}/\mathcal{F}_{h_{k-1}}^{\text{HN}})>\widehat{\mu}_{i}(\mathcal{F}_{h_{k+1}}^{\text{HN}}/\mathcal{F}_{h_{k}}^{\text{HN}})$.
For each $k$, we set
$\mathcal{F}^{\text{l-term}}_{k}=\mathcal{F}^{\text{HN}}_{h_{k}}$.

######  Definition 9.1.

The filtration

$0=\mathcal{F}^{\text{l-term}}_{0}\subset\mathcal{F}^{\text{l-term}}_{1}\subset\mathcal{F}^{\text{l-term}}_{2}\subset\cdots\subset\mathcal{F}^{\text{l-term}}_{j}=\mathcal{F}$

is called the unweighted leading term filtration of $\mathcal{F}$. We call $i$
the leading term index.

We remark that, if $\mathcal{F}$ is slope unstable, then $i=d-1$ (recall that
$\widehat{\mu}_{d-1}(\mathcal{F})$ is the slope of $\mathcal{F}$), and the
leading term filtration is the slope-Harder-Narasimhan filtration.

######  Definition 9.2.

Let
$\mathcal{F}\in\operatorname{Bun}_{\rho}(X)=\operatorname{Coh}_{r}^{tf}(X)$
and assume that $\mathcal{F}$ is unstable. Suppose that the unweighted leading
term filtration for $\mathcal{F}$ is given by

$0=\mathcal{F}^{\text{l-term}}_{0}\subset\mathcal{F}^{\text{l-term}}_{1}\subset\mathcal{F}^{\text{l-term}}_{2}\subset\cdots\subset\mathcal{F}^{\text{l-term}}_{j}=\mathcal{F}$

with leading slope $\widehat{\mu}_{i}$. Set
$\mu=\widehat{\mu}_{i}(\mathcal{F})$,
$\mu_{k}=\widehat{\mu}_{i}\left(\mathcal{F}^{\text{l-term}}_{k}/\mathcal{F}^{\text{l-term}}_{k-1}\right)$.
Choose a positive rational number $L$ such that $L\cdot\mu\in\mathbb{Z}$ and
$L\cdot\mu_{k}\in\mathbb{Z}$ for all $k$.

Then, we define the weighted leading term filtration
$(\mathcal{F}^{\text{Can}})_{m\in\mathbb{Z}}$ as follows (up to scaling of the
weights).

  1. (1)

$\mathcal{F}^{\text{Can}}_{m}=0$ for $m>L(\mu_{1}-\mu)$.

  2. (2)

Let $1\leq k\leq j-1$. For each $L(\mu_{k}-\mu)\geq m>L(\mu_{k+1}-\mu)$, we
set $\mathcal{F}^{\text{Can}}_{m}=\mathcal{F}^{\text{l-term}}_{k}$.

  3. (3)

$\mathcal{F}^{\text{Can}}_{m}=\mathcal{F}$ for $L(\mu_{j}-\mu)\geq m$.

In other words, the canonical filtration is given by the leading term
filtration and the $k^{th}$ jump happens at the integer $L(\mu_{k}-\mu)$.

We end this subsection by explicitly describing the procedure for the
construction of the GHN filtration in this case. Let
$0=\mathcal{F}^{\text{l-term}}_{0}\subset\mathcal{F}^{\text{l-term}}_{1}\subset\mathcal{F}^{\text{l-term}}_{2}\subset\cdots\subset\mathcal{F}^{\text{l-term}}_{j_{1}}=\mathcal{F}$
be the unweighted leading term filtration, with leading term index $i_{1}$. By
assigning weights as above, this yields the leading term HN filtration
$(\,^{(1)}\mathcal{F}_{m})_{m\in\mathbb{Z}}$ of $\mathcal{F}$. The $1^{st}$
associated graded Levi $\rho_{\lambda_{1}}$-sheaf consists of the tuple

$\text{gr}_{1}(\mathcal{F})=\left(\mathcal{F}^{\text{l-term}}_{i}/\mathcal{F}^{\text{l-term}}_{i-1}\right)_{i=1}^{j_{1}}$

The leading term HN filtration of $\text{gr}_{1}(\mathcal{F})$ is obtained by
taking the leading term filtration of each element
$\mathcal{F}^{\text{l-term}}_{i}/\mathcal{F}^{\text{l-term}}_{i-1}$ in the
tuple. Each corresponding leading term index will be strictly smaller than
$i_{1}$. The sheaves appearing in the leading term filtration of
$\mathcal{F}^{\text{l-term}}_{i}/\mathcal{F}^{\text{l-term}}_{i-1}$ are of the
form $\mathcal{F}^{\text{HN}}_{h}/\mathcal{F}^{\text{l-term}}_{i-1}$ for some
index $1\leq h\leq n$. We denote the corresponding lexicographic
$\mathbb{Z}^{2}$-filtration of $\mathcal{F}$ by
$(\,^{(2)}\mathcal{F}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{2}}$. Let
$\vec{m}_{1}>\vec{m}_{2}>\cdots>\vec{m}_{j_{2}}$ denote the jumping points of
$(\,^{(2)}\mathcal{F}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{2}}$. The unweighted
filtration

$0\subset\,^{(2)}\mathcal{F}_{\vec{m}_{1}}\subset\,^{(2)}\mathcal{F}_{\vec{m}_{2}}\subset\cdots\subset\,^{(2)}\mathcal{F}_{\vec{m}_{j_{2}}}=\mathcal{F}$

is coarser than the Gieseker-Harder-Narasimhan filtration, but finer than the
leading term filtration. The reduced Hilbert polynomials of the graded pieces
are strictly increasing, and each graded piece
${}^{(2)}\mathcal{F}_{\vec{m}_{i}}/\,^{(2)}\mathcal{F}_{\vec{m}_{i-1}}$ is
semistable up to a degree $i_{2}$ in the Hilbert polynomial that is strictly
smaller than the degree $i_{1}$ in the previous step. The $2^{nd}$ associated
graded Levi sheaf is
$\text{gr}_{2}(\mathcal{F})=\left({}^{(2)}\mathcal{F}_{\vec{m}_{i}}/\,^{(2)}\mathcal{F}_{\vec{m}_{i-1}}\right)_{i=1}^{j_{2}}$.
We iterate this procedure, at each step we take the leading term filtration of
the graded pieces.

Suppose that this process terminates at the $q^{th}$ step. Let
$(\,^{(q)}\mathcal{F}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ be the GHN
filtration obtained at this $q^{th}$ step. Denote by
$\vec{m}_{1}>\vec{m}_{2}>\cdots>\vec{m}_{j_{q}}$ the jumping points of the GHN
filtration. Consider the unweighted filtration

(14)
$0\subset\,^{(q)}\mathcal{F}_{\vec{m}_{1}}\subset\,^{(q)}\mathcal{F}_{\vec{m}_{2}}\subset\cdots\subset\,^{(q)}\mathcal{F}_{\vec{m}_{j_{q}}}=\mathcal{F}.$

The reduced Hilbert polynomials of the graded pieces of this filtration are
strictly increasing by construction. Moreover, each graded piece must be
Gieseker semistable, because the $q^{th}$ step is the last one of the
procedure. Hence this filtration is the classical Gieseker-Harder-Narasimhan
filtration of $\mathcal{F}$. The associated $q$-weights in the GHN filtration
encode information about how high is the degree of the difference of reduced
Hilbert polynomials between two consecutive graded pieces.

We have therefore shown the following.

######  Proposition 9.3.

Let $\mathcal{F}$ be a torsion-free sheaf. Let
$(\,^{(q)}\mathcal{F}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ be the GHN
filtration. The corresponding unweighted filtration (14) coincides with the
classical Gieseker-Harder-Narasimhan filtration of $\mathcal{F}$. ∎

###  9.2. Example of Gieseker-Harder-Narasimhan filtration

We compare the different filtrations that we have considered in an example.
Let $G=\operatorname{SO}(7)$ and let
$\rho:\operatorname{SO}(7)\to\operatorname{GL}(7)$ be the standard
representation. Set $X=\mathbb{P}^{3}$, and let
$L\cong\mathbb{P}^{1}\subset\mathbb{P}^{3}$ be a fixed line. Choose two
different points $Z=\\{p,p^{\prime}\\}\subset\mathbb{P}^{3}$. We denote by
$\mathcal{I}_{L}$ the ideal sheaf of $L$, $\mathcal{I}_{p}$ the ideal sheaf of
$p$, and $\mathcal{I}_{Z}$ the ideal sheaf of $Z$. Set

$\mathcal{F}=\mathcal{I}_{L}\oplus\mathcal{I}_{Z}\oplus\mathcal{I}_{p}\oplus\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}.$

Let $\varphi:\mathcal{F}\otimes\mathcal{F}\to\mathcal{O}$ be the symmetric
bilinear form defined by the matrix

$\left(\begin{array}[]{ccccccc}0&\\!\cdots\\!&&&&&1\\\
\vbox{\hbox{.}\hbox{.}\hbox{.}}&&&&&1&\\\ &&&&1&&\\\ &&&1&&&\\\ &&1&&&&\\\
&1&&&&&\vbox{\hbox{.}\hbox{.}\hbox{.}}\\\ 1&&&&&\\!\cdots\\!&0\\\
\end{array}\right)$

Set $U=\mathbb{P}^{3}\setminus(L\cup Z)$. The sheaf $\mathcal{F}|_{U}$ is
locally free, and in fact trivial. Let $\varphi^{\prime}$ be the canonical
trivialization of $\det\mathcal{F}$. Clearly
$\det\varphi=\varphi^{\prime}{}^{2}$, hence the pair
$(\varphi,\varphi^{\prime})$ induces a reduction of the trivial
$\operatorname{GL}(7)$-bundle to $\operatorname{SO}(7)$. The rational
$\operatorname{SO}(7)$-bundle $\mathcal{G}\to U$ which it defines is trivial.
Let $\psi$ denote the canonical isomorphism between $\rho_{*}(\mathcal{G})$
and $\mathcal{F}|_{U}$.

The principal $\rho$-sheaf $(\mathcal{G},\mathcal{F},\psi)$ is slope
semistable (Definition 10.7). The rational principal
$\operatorname{SO}(7)$-bundle $\mathcal{G}$ is trivial, hence also slope
semistable (Definition 10.3). Hence the Harder-Narasimhan canonical reduction
defined in [AAB02] is trivial.

The Hilbert polynomials are

$\displaystyle P_{\mathcal{O}}(n)$
$\displaystyle=\frac{1}{6}(n^{3}+6n^{2}+10n+3)$ $\displaystyle
P_{\mathcal{I}_{p}}(n)$ $\displaystyle=\frac{1}{6}(n^{3}+6n^{2}+10n-3)$
$\displaystyle P_{\mathcal{I}_{Z}}(n)$
$\displaystyle=\frac{1}{6}(n^{3}+6n^{2}+10n-9)$ $\displaystyle
P_{\mathcal{I}_{L}}(n)$ $\displaystyle=\frac{1}{6}(n^{3}+6n^{2}-\;6n-3)$

Note that the three first polynomials only differ in the last term, and the
last polynomial differs in the last two terms.

The principal $\rho$-sheaf $(\mathcal{G},\mathcal{F},\psi)$ is Gieseker
unstable (Definition 10.2). We will now calculate the leading term HN
filtration. We work in the standard basis where the bilinear form is given by
the matrix above. We will use cocharacters of $\operatorname{SO}(7)$ of the
form

$\operatorname{diag}\operatorname{(}t^{a},t^{b},t^{c},1,t^{-c},t^{-b},t^{-a})$

with $a\geq b\geq c\geq 0$. The associated filtration $\mathcal{F}_{m}$
satisfies $\mathcal{F}_{m}^{\perp}=\mathcal{F}_{-m+1}$. Recall the definition
of orthogonal of a subsheaf $\mathcal{F}_{i}\subset\mathcal{F}$

$\mathcal{F}_{i}{}^{\perp}:=\ker\big{(}\mathcal{F}\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}\mathcal{F}^{\vee}\longrightarrow\mathcal{F}_{i}{}^{\vee}\big{)}.$

The leading term HN filtration is

$\overbrace{\mathcal{F}}^{\mathcal{F}_{-1}}\quad\supset\quad\overbrace{\mathcal{I}_{Z}\oplus\mathcal{I}_{p}\oplus\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}}^{\mathcal{F}_{0}}\quad\supset\quad\overbrace{\mathcal{O}}^{\mathcal{F}_{1}}$

where $\mathcal{F}_{1}$ is the last summand in $\mathcal{F}$, and
$\mathcal{F}_{1}^{\perp}=\mathcal{F}_{0}$. By definition of leading term HN
filtration, this is the unique filtration, with
$\mathcal{F}_{m}^{\perp}=\mathcal{F}_{-m+1}$, which gives a maximum of the
numerical function (3).

This filtration has been obtained by the cocharacter
$\lambda_{1}=\operatorname{diag}(t,1,1,1,1,1,t^{-1})$. The associated Levi is
$L_{\lambda_{1}}=\mathbb{G}_{m}\times\operatorname{SO}(5)$, the representation
$\rho_{\lambda_{1}}$ is

$\begin{array}[]{rccc}\rho_{\lambda_{1}}:&\mathbb{G}_{m}\times\operatorname{SO}(5)&\longrightarrow&\operatorname{GL}(1)\times\operatorname{GL}(5)\times\operatorname{GL}(1)\\\
&(t,g)&\mapsto&(t,g,t^{-1})\\\ \end{array}$

and the associated graded sheaf is

$\overbrace{\mathcal{I}_{L}}^{\mathcal{F}^{-1}:=\mathcal{F}_{-1}/\mathcal{F}_{0}}\quad\oplus\quad\overbrace{\mathcal{I}_{Z}\oplus\mathcal{I}_{p}\oplus\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}}^{\mathcal{F}^{0}:=\mathcal{F}_{0}/\mathcal{F}_{-1}}\quad\oplus\quad\overbrace{\mathcal{O}}^{\mathcal{F}^{1}:=\mathcal{F}_{1}}.$

The $\operatorname{SO}(5)$-sheaf defined by $\mathcal{F}^{0}$ is not Gieseker
stable. Its leading term HN filtration is given by the cocharacter
$\lambda_{2}=\operatorname{diag}(t^{2},t,1,t^{-1},t^{-2})$ and it is

$\displaystyle\overbrace{\mathcal{F}^{0}}^{\mathcal{F}^{0}_{-2}}\quad\supset\quad\overbrace{\mathcal{I}_{p}\oplus\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}}^{\mathcal{F}^{0}_{-1}}\quad\supset\quad\overbrace{\mathcal{O}\oplus\mathcal{O}\oplus\mathcal{O}}^{\mathcal{F}^{0}_{0}}\quad\supset\quad\overbrace{\mathcal{O}\oplus\mathcal{O}}^{\mathcal{F}^{0}_{1}}\quad\supset\quad\overbrace{\mathcal{O}}^{\mathcal{F}^{0}_{2}}$

Finally, the (multi-weighted) Gieseker-Harder-Narasimhan filtration is

$\displaystyle\overbrace{\mathcal{F}}^{\mathcal{F}_{(-1,0)}}\quad\supset\quad\overbrace{\mathcal{I}_{Z}\oplus\mathcal{I}_{p}\oplus\mathcal{O}^{\oplus
4}}^{\mathcal{F}_{(0,-2)}}\quad\supset\quad\overbrace{\mathcal{I}_{p}\oplus\mathcal{O}^{\oplus
4}}^{\mathcal{F}_{(0,-1)}}\quad\supset\quad\overbrace{\mathcal{O}^{\oplus
4}}^{\mathcal{F}_{(0,0)}}\quad\supset\quad\overbrace{\mathcal{O}^{\oplus
3}}^{\mathcal{F}_{(0,1)}}\quad\supset\quad\overbrace{\mathcal{O}^{\oplus
2}}^{\mathcal{F}_{(0,2)}}\quad\supset\quad\overbrace{\mathcal{O}}^{\mathcal{F}_{(1,0)}}$

The process stops here because the associated Levi principal sheaf is Gieseker
semistable. Indeed, the Levi is
$L_{\lambda_{2}}=\mathbb{G}_{m}\times(\mathbb{G}_{m}\times\mathbb{G}_{m})\subset\mathbb{G}_{m}\times\operatorname{SO}(5)$,
the representation is

$\begin{array}[]{rccc}\rho_{\lambda_{2}}:&\mathbb{G}_{m}\times(\mathbb{G}_{m}\times\mathbb{G}_{m})&\longrightarrow&\operatorname{GL}(1)^{\times
7}\\\ &(a,b,c)&\mapsto&(a,b,c,1,c^{-1},b^{-1},a^{-1})\\\ \end{array}$

and the associated graded is isomorphic to $\mathcal{F}$.

##  10\. Comparison with previous work

In this section we compare our notions of stability and Harder-Narasimhan
filtrations of principal $\rho$-sheaves with other definitions in the
literature.

###  10.1. Comparison with previous notions of semistability

The problem of finding a compactification of the moduli space by introducing
singular degenerations of principal bundles has already been considered in the
literature, and the objects have been called (honest) singular principal
bundles, principal sheaves or principal $\rho$-sheaves ([Sch02, GS03, GS05,
GLSS08, GLSS10]). All notions fit the same framework: we fix a representation
$\rho$ of $G$ (sometimes $\rho$ is not required to be injective) and the
objects are triples consisting of a principal $G$-bundle $P$ on a big open
subset $U\subset X$, a torsion free sheaf $\mathcal{E}$ on $X$, and an
isomorphism $\psi$ on $U$ between the vector bundle associated to $P$ using
the representation $\rho$ and the restriction $\mathcal{E}|_{U}$.

Unfortunately, the names are not always the same (for instance, singular
principal bundles in [GLSS08] where called honest singular principal bundles
in [Sch02]), and the vector bundle that appears in the isomorphism $\psi$ is
sometimes $P(\rho^{\vee})$, using the dual of the representation $\rho$ (in
[GLSS08, Sch02, GLSS10]) and sometimes $P(\rho)$, using directly the
representation (in [GS05, GS03]). In particular, we warn the reader that, in
[GLSS10, §1.2], a _principal $\rho$-sheaf_ was defined using the dual
representation, and $\rho$ was not required to be injective. In this
subsection we will see that our definition of Gieseker semistability for
principal $\rho$-sheaves coincides with the definitions that have already
appeared.

Let us recall some of the main definitions of this article.

######  Definition 10.1.

Let $G$ be a connected reductive group. Let

$\rho=(\rho_{1},\ldots,\rho_{b}):G\longrightarrow\operatorname{GL}(V^{1})\times\cdots\times\operatorname{GL}(V^{b})$

be an injective homomorphism. Let $S$ be a $k$-scheme. A principal
$\rho$-sheaf on $X_{S}$ is a triple
$(\mathcal{G},\mathcal{F}^{\bullet},\psi^{\bullet})$ where $\mathcal{G}$ is a
principal $G$-bundle on a big open set $U\subset X_{S}$,
$\mathcal{F}^{\bullet}=(\mathcal{F}^{i})_{i=1}^{b}$ is a tuple of torsion free
sheaves on $X_{S}$, and $\psi^{\bullet}=(\psi^{i})_{i=1}^{b}$ is a tuple of
isomorphisms $(\rho_{i})_{*}\mathcal{G}\cong\mathcal{F}^{i}|_{U}$.

We generalize Definition 4.28 in the following way, using the summation by
parts argument as in Equation (12).

######  Definition 10.2.

Let $K\supset k$ be an algebraically closed field. A principal $\rho$-sheaf on
$X_{K}$ is Gieseker semistable if for all cocharacters
$\lambda:(\mathbb{G}_{m})_{K}\longrightarrow G_{K}$ and all weighted parabolic
reductions $(\lambda,\mathcal{G}_{\lambda})$ with associated underlying
filtrations $(\mathcal{F}^{i}_{m})_{m\in\mathbb{Z}}$ of $\mathcal{F}^{i}$, (as
in Definition 3.15) we have

(15)
$\sum_{i=1}^{b}\sum_{m\in\mathbb{Z}}\left(\operatorname{rk}(\mathcal{F}^{i})\cdot
P_{\mathcal{F}^{i}_{m}}-\operatorname{rk}(\mathcal{F}^{i}_{m})\cdot
P_{\mathcal{F}^{i}}\right)\leq 0$

We start by comparing our notion of Gieseker semistability with Ramanathan’s
notion of slope semistability for the underlying rational $G$-bundle.

######  Definition 10.3 ([Ram79]).

A rational $G$-bundle on a polarized smooth projective variety $X$ is a
$G$-bundle $\mathcal{G}$ on a big open set $U\subset X$. It is called slope
stable (respectively, semistable) if for all reductions of structure group
$\mathcal{G}_{P}\to U^{\prime}\subset U$ to a proper parabolic subgroup
$P\subset G$ on a big open subset $U^{\prime}$ and all dominant characters
$\chi$ of $P$, the associated line bundle $\chi_{*}(\mathcal{G}_{P})$
satisfies

$\deg\chi_{*}(\mathcal{G}_{P})<0\quad\text{(respectively, $\leq$)}$

Here the degree is defined with respect to the fixed polarization. A rational
$G$-bundle is called slope unstable if it is not slope semistable.

######  Proposition 10.4.

Let $K\supset k$ be a field extension. Let $(\mathcal{F}^{\bullet},\sigma)$ be
a $\rho$-sheaf on $X_{K}$. Suppose that $U$ is a big open subset of $X_{K}$
such that the tuple $\mathcal{F}^{\bullet}|_{U}$ is locally-free. Let
$\mathcal{G}$ denote the $G$-bundle on $U$ corresponding to
$(\mathcal{F}^{\bullet},\sigma)$. If $(\mathcal{F}^{\bullet},\sigma)$ is
Gieseker semistable, then the rational $G$-bundle $\mathcal{G}$ is slope
semistable in the sense of [Ram79].

###### Proof.

We will freely use the notation from the proof of Proposition 4.26 and assume,
after replacing $K$ with a field extension, that $K$ is algebraically closed.
Suppose that the rational $G$-bundle $\mathcal{G}$ is not slope semistable.
Then there exists a proper parabolic subgroup $P_{\lambda}\subset G$ and a
canonical (slope) $P_{\lambda}$-reduction of structure group
$\mathcal{G}_{P_{\lambda}}$ defined over a big open subset $Q$ contained in
$U$, as in [AAB02]. After replacing $U$ with the smaller subset $Q$, we can
assume without loss of generality that $\mathcal{G}_{P_{\lambda}}$ is defined
over $U$.

Let $\mathcal{G}_{L_{\lambda}}$ be the $L_{\lambda}$-bundle associated to the
$P_{\lambda}$-bundle $\mathcal{G}_{\lambda}$ via the quotient
$P_{\lambda}\twoheadrightarrow L_{\lambda}$. Let $Z_{\lambda}$ denote the
center of $P_{\lambda}$. The rational $L_{\lambda}$-bundle
$\mathcal{G}_{L_{\lambda}}$ determines a linear functional
$\psi:X^{*}(Z_{\lambda})_{\mathbb{R}}\rightarrow\mathbb{R}$ as in the proof of
Lemma 4.25. Since $\psi$ actually lies in $X_{*}(Z_{\lambda})_{\mathbb{Q}}$,
there is a multiple $N\psi$ that lies in the lattice of cocharacters
$X_{*}(Z_{\lambda})$. By the properties of the canonical slope reduction
[AAB02], $\psi$ lies in the interior of the cone of $P_{\lambda}$-dominant
coweights. It follows that $P_{\lambda}=P_{(N\psi)}$. Therefore
$f=(N\psi,\mathcal{G}_{P_{\lambda}})$ is a nondegenerate weighted parabolic
reduction of $(\mathcal{F},\sigma)$.

The group $G$ acts on each vector space $V^{j}$ via the homomorphism
$\rho_{j}$. For each $j$, we denote by $(\chi_{i,j})_{i=1}^{r_{j}}$ the tuple
of weights of the split maximal torus $T$ acting on $V^{j}$ (with possible
repetitions). Recall the function $\nu_{d-1}$ on the dominant cone
$\mathcal{C}_{\lambda}$ described in Lemma 4.23 (4)). In this case the formula
(4) includes an extra summation

(16)
$\nu_{d-1}(\delta)=\frac{1}{\sqrt{A_{d}}(d-1)!}\,\cdot\,\frac{\sum_{j=1}^{b}\sum_{i=1}^{r_{j}}\langle\delta,\chi^{\lambda}_{i,j}\rangle\cdot(c_{i,j}-c_{j})}{\sqrt{\sum_{j=1}^{b}\sum_{i=1}^{r_{j}}\left(\langle\delta,\chi^{\lambda}_{i,j}\rangle\right)^{2}}}$

where we define $c_{i,j}=\langle\psi,\chi_{i,j}^{\lambda}\rangle$ and
$c_{j}=\frac{1}{r_{j}}\langle\psi,\sum_{i=1}^{r_{j}}\chi_{i,j}^{\lambda}\rangle$.
Expression (16) for $\delta=N\psi$ yields

$\displaystyle\nu_{d-1}(N\psi)=\frac{1}{\sqrt{A_{d}}(d-1)!}\,\cdot\,\frac{\sum_{j=1}^{b}\sum_{i=1}^{r_{j}}\langle
N\psi,\chi^{\lambda}_{i,j}\rangle\cdot\left(\langle\psi,\chi_{i,j}^{\lambda}\rangle-\frac{1}{r_{j}}\sum_{l=1}^{r_{j}}\langle\psi,\chi_{i,l}^{\lambda}\rangle\right)}{\sqrt{\sum_{j=1}^{b}\sum_{i=1}^{r_{j}}\left(\langle
N\psi,\chi^{\lambda}_{i,j}\rangle\right)^{2}}}=$ (17)
$\frac{1}{\sqrt{A_{d}}(d-1)!}\,\cdot\,\frac{\sum_{j=1}^{b}\left(\sum_{i=1}^{r_{j}}\left(\langle\psi,\chi^{\lambda}_{i,j}\rangle\right)^{2}\right)-\frac{1}{r_{j}}\left(\sum_{i=1}^{r_{j}}\langle\psi,\chi_{i,j}^{\lambda}\rangle\right)\cdot\left(\sum_{l=1}^{r_{j}}\langle\psi,\chi_{l,j}^{\lambda}\rangle\right)}{\sqrt{\sum_{j=1}^{b}\sum_{i=1}^{r_{j}}\left(\langle
N\psi,\chi^{\lambda}_{i,j}\rangle\right)^{2}}}$

We shall show that the numerator of (17) is positive. This will contradict
Gieseker semistability of $(\mathcal{F},\sigma)$. For each index $j$, let
$\vec{v}_{j}\vcentcolon=[\langle\psi,\chi_{i}^{\lambda}\rangle]_{i=1}^{r_{j}}\in\mathbb{R}^{r_{j}}$
and set $\vec{1}_{j}\in\mathbb{R}^{r_{j}}$ to be the vector with all
coordinates equal to $1$. The numerator in (17) is

$\displaystyle\sum_{j=1}^{b}\left(\sum_{i=1}^{r_{j}}\left(\langle\psi,\chi^{\lambda}_{i,j}\rangle\right)^{2}\right)-\frac{1}{r_{j}}\left(\sum_{i=1}^{r_{j}}\langle\psi,\chi_{i,j}^{\lambda}\rangle\right)^{2}=\sum_{j=1}^{b}\frac{1}{r_{j}}\left[\left(\vec{1}_{j}\cdot_{\mathbb{R}}\vec{1}_{j}\right)\left(\vec{v}_{j}\cdot_{\mathbb{R}}\vec{v}_{j}\right)-\left(\vec{1}_{j}\cdot_{\mathbb{R}}\vec{v}_{j}\right)^{2}\right]$

where $\cdot_{\mathbb{R}}$ denotes the standard inner product on
$\mathbb{R}^{r}$. The Cauchy-Schwarz inequality implies that the quantity
above is always nonnegative and, moreover, it is strictly positive if there
exists some index $j$ such that $\vec{1}_{j}$ and $\vec{v}_{j}$ are not scalar
multiples of each other. Therefore, $\nu_{d-1}(N\psi)>0$ whenever
$\langle\psi,\chi_{i,j}^{\lambda}\rangle\neq\langle\psi,\chi_{l,j}^{\lambda}\rangle$
for some $j$ and $i\neq l$. We are left to show that there are $j$ and two
indexes $i\neq l$ satisfying this condition.

Let $\\{\zeta_{l,j}\\}$ denote the subset of $\\{\chi_{i,j}\\}$ consisting of
$B$-highest weights for an irreducible component of the representation
$\rho_{j}$. Each $\zeta_{l,j}$ can be uniquely expressed as a linear
combination
$\sum_{i=1}^{h}x_{z_{i}}z_{i}^{\vee}+\sum_{\alpha\in\Delta}x_{\alpha}\alpha$,
where all coefficients $x_{\alpha}$ are nonnegative. Since the parabolic
$P_{\lambda}$ is proper, we can choose some $\alpha\in I_{P_{\lambda}}$. Since
$\bigoplus\rho_{j}$ is faithful, at least one of the $\zeta_{l,j}$ must have a
nonzero coefficient $x_{\alpha}>0$; we fix such $l,j$. The restriction
$\chi_{l,j}^{\lambda}\in X^{*}(Z_{\lambda})$ is a $P_{\lambda}$-dominant
character such that the coefficient of $\alpha$ is not $0$. Since $\psi$ is in
the interior of the the $P_{\lambda}$-dominant cone in
$X_{*}(Z_{\lambda})_{\mathbb{R}}$, it follows that
$\langle\psi,\chi_{l,j}^{\lambda}\rangle>0$.

Let $W$ denote the Weyl group of $G$ permutting the set of characters
$\\{\chi_{i,j}\\}_{i=1}^{r_{j}}$ of the representation $\rho_{j}$. Let
$W_{P_{\lambda}}$ denote the subgroup of $W$ generated by the reflections of
the simple roots in $I_{P_{\lambda}}$. $W_{P_{\lambda}}$ acts on the
characters $X^{*}(Z_{\lambda})$, and the action is transitive on the set of
Weyl chambers in $X^{*}(Z_{\lambda})_{\mathbb{R}}$. The set of restrictions
$\\{\chi_{i,j}^{\lambda}\\}_{i=1}^{r_{j}}\subset X^{*}(Z_{\lambda})$ is
preserved by $W_{P_{\lambda}}$. The character $\chi_{l,j}^{\lambda}$ is not
fixed by the action of $W_{\lambda}$, because the coefficient of $\alpha$ is
not $0$. Therefore, there exists some $w\in W_{P_{\lambda}}$ such that
$\chi_{i,j}^{\lambda}\vcentcolon=w\cdot\chi_{l,j}^{\lambda}$ is contained in
the the $P_{\lambda}$-anti-dominant cone of $X^{*}(Z_{\lambda})_{\mathbb{R}}$.
This implies that $\langle\psi,\chi_{i,j}^{\lambda}\rangle<0$. Since
$\langle\psi,\chi_{i,j}^{\lambda}\rangle<0<\langle\psi,\chi_{l,j}^{\lambda}\rangle$,
we conclude that
$\langle\psi,\chi_{i}^{\lambda}\rangle\neq\langle\psi,\chi_{j}^{\lambda}\rangle$,
as desired. ∎

Example 10.10 below shows that the converse is not true even if we assume that
the underlying rational $G$-bundle is stable in the sense of Ramanathan. This
seems quite restrictive. In order to obtain a reasonable notion of stability,
it will be useful to impose the following condition on our representation
$\rho$.

######  Definition 10.5.

We say that a homomorphism of groups $\rho:G_{1}\to G_{2}$ is central if the
image of the identity component of the center of $G_{1}$ is in the center of
$G_{2}$.

Unless otherwise mentioned, for the rest of this subsection we will assume
that $\rho$ is central. This will allow us to have a better behaved notion of
stability which can be compared to previous notions in the literature. First,
we note that under this additional condition we can check semistability only
using cocharacters of the derived subgroup of $G$.

######  Proposition 10.6.

Let $K\supset k$ be an algebraically closed field, and suppose that $\rho$ is
central. Then in the semistability condition of Definition 10.2 it is enough
to look at cocharacters
$\lambda:(\mathbb{G}_{m})_{K}\longrightarrow[G_{K},G_{K}]$ whose image is in
the derived subgroup $[G_{K},G_{K}]\subset G_{K}.$

###### Proof.

Let $Z\subset G_{K}$ be the center and $[G_{K},G_{K}]\subset G_{K}$ be the
commutator subgroup. Consider the exact sequence

$1\longrightarrow F\longrightarrow Z\times[G_{K},G_{K}]\longrightarrow
G_{K}\longrightarrow 1$

where $F=Z\cap[G_{K},G_{K}]$ is a finite group (it is the center of the
semisimple group $[G_{K},G_{K}]$). Let $\lambda:(\mathbb{G}_{m})_{K}\to G_{K}$
be a cocharacter. Since $F$ is finite, there is an integer $N$ such that
$\lambda^{N}$ lifts to $Z\times[G_{K},G_{K}]$. If we replace $\lambda$ by
$\lambda^{N}$ then the parabolic subgroup $\mathcal{G}_{\lambda}$ associated
to $\lambda$ does not change. The only change in the associated filtration
$(\mathcal{F}^{i}_{m})_{m\in\mathbb{Z}}$ is that the indexes are multiplied by
$N$, and the expression (15) does not change. Therefore, to check
(semi)stability we can assume that the cocharacter $\lambda$ lifts to the
product $Z\times[G_{K},G_{K}]$. Assume $\lambda$ lifts, and write
$\lambda=(\lambda_{1},\lambda_{2})$ where $\lambda_{1}$ is the lift to $Z$ and
$\lambda_{2}$ is the lift to $[G_{K},G_{K}]$.

We claim that the filtrations associated to $(\lambda_{1},\lambda_{2})$ and
$(1,\lambda_{2})$ on each $V^{i}$ only differ by a constant shift in the
indexes. Indeed, since $\rho$ is central, the image of the composition
$\rho_{i}\circ\lambda_{1}$ lies in the center of $\operatorname{GL}(V^{i})$,
so $(\mathbb{G}_{m})_{K}$ acts on $V^{i}$, via $\rho_{i}\circ(\lambda_{1},1)$,
with a fixed weight $m_{i}$. If $v\in V^{i}$ is a vector where
$(\mathbb{G}_{m})_{K}$ acts with weight $m$ via
$\rho_{i}\circ(\lambda_{1},\lambda_{2})$, then the action, via
$\rho_{i}\circ(1,\lambda_{2})$, has weight $m-m_{i}$. Therefore, the
filtrations $(\mathcal{F}^{i}_{m})_{m\in\mathbb{Z}}$ induced by
$\rho_{i}\circ(1,\lambda_{2})$ and $\rho_{i}\circ(\lambda_{1},\lambda_{2})$
just differ by a shift by $m_{i}$, as claimed.

Finally, since the sum in (15) does not change by a shift by $m_{i}$ on each
$V^{i}$, we may assume that $\lambda_{1}$ is equal to the constant $1$. In
other words, we may assume that $\lambda$ factors through $[G_{K},G_{K}]$. ∎

Proposition 10.6 shows that the following definition is consistent with our
earlier notions.

######  Definition 10.7.

Suppose that $\rho$ is central. Let $K\supset k$ be an algebraically closed
field. A principal $\rho$-sheaf on $X_{K}$ is Gieseker stable (respectively,
Gieseker semistable) if for all nontrivial cocharacters
$\lambda:(\mathbb{G}_{m})_{K}\longrightarrow[G_{K},G_{K}]$ and all weighted
parabolic reductions $(\lambda,\mathcal{G}_{\lambda})$ with associated
underlying filtrations $(\mathcal{F}^{i}_{m})_{m\in\mathbb{Z}}$ of
$\mathcal{F}^{i}$, (as in Definition 3.15) we have

$\sum_{i=1}^{b}\sum_{m\in\mathbb{Z}}\left(\operatorname{rk}(\mathcal{F}^{i})\cdot
P_{\mathcal{F}^{i}_{m}}-\operatorname{rk}(\mathcal{F}^{i}_{m})\cdot
P_{\mathcal{F}^{i}_{m}}\right)<0\quad\quad(\text{respectively, $\leq$})$

A principal $\rho$-sheaf is slope stable (respectively, semistable) if we
replace the numerical condition with

$\sum_{i=1}^{b}\sum_{m\in\mathbb{Z}}\left(\operatorname{rk}(\mathcal{F}^{i})\cdot\deg{\mathcal{F}^{i}_{m}}-\operatorname{rk}(\mathcal{F}^{i}_{m})\cdot\deg{\mathcal{F}^{i}}\right)<0\quad\quad(\text{respectively,
$\leq$}).$

We say that a principal $\rho$-sheaf is Gieseker unstable (resp. slope
unstable) if it is not Gieseker semistable (resp. it is not slope semistable).

We immediately obtain that slope stable implies Gieseker stable, and Gieseker
semistable implies slope semistable. We proceed now to compare with other
notions of stability in the literature.

In [GLSS10, §1.1, p. 282] it is assumed that the field is algebraically
closed, $G$ is an arbitrary nonconnected reductive group, and the homomorphism
$\rho$ is assumed to be central. Furthermore, [GLSS10, §1.1, p. 282] requires
that the image of $\rho$ lies in the special group of
$V=V^{1}\oplus\cdots\oplus V^{b}$ as follows:

$\rho:G\longrightarrow\left(\operatorname{GL}(V^{1})\times\cdots\times\operatorname{GL}(V^{b})\right)\cap\operatorname{SL}(V)$

But this is not an essential condition. If we are given a representation
$\rho=(\rho_{1},\ldots,\rho_{b})$ we can add one more summand $V^{b+1}=k$ and
the representation

$\rho^{\prime}=(\rho_{1},\ldots,\rho_{b},(\det(\rho_{1})\cdots\det(\rho_{b}))^{-1})$

will automatically land in the special group. A $\rho^{\prime}$-sheaf is
$\rho$-sheaf together with an isomorphism
$\det(\mathcal{F}^{1})\otimes\cdots\otimes\det(\mathcal{F}^{b})\cong(\mathcal{F}^{b+1})^{\vee}$.
The $\rho^{\prime}$-sheaf is (semi)stable in the sense of [GLSS10] if and only
if the corresponding $\rho$-sheaf is (semi)stable.

Therefore, a principal $\rho$-sheaf is the same thing as a singular principal
$\rho^{\vee}$-bundle in [GLSS10] and our definition of (semi)stability
coincides with the definition in [GLSS10, §1.1, p. 282] (recall that they
assume that $\rho$ is central, so we can use Proposition 10.6).

Therefore, all the comparisons that are made in [GLSS10, §6] also hold for us.
If we consider orthogonal or symplectic groups with the standard
representation, a principal $\rho$-sheaf is the same thing as an orthogonal or
symplectic sheaf as defined in [GS03], and the conditions of (semi)stability
defined in this article coincide with those in [GS03].

[GS05] uses the adjoint representation
$\rho:G\to\operatorname{GL}(\mathfrak{g}^{\prime})$ into the semisimple part
$\mathfrak{g}^{\prime}\subset\mathfrak{g}$ of the Lie algebra (note that the
adjoint representation is trivial on the center $\mathfrak{z}$ of the Lie
algebra). If the center of $G$ is trivial this representation is faithful.
Then we can apply the techniques of this article. A principal $\rho$-sheaf is
the same thing as a principal $G$-sheaf as defined in [GS05], and the
conditions of stability and semistability coincide with those in [GS05].

######  Lemma 10.8 (Lemma 2.5.4 and Remark 2.5.5 in [GLSS10]).

A rational principal bundle $\mathcal{G}$ defined over $U$ is slope stable
(respectively, slope semistable) if for every big open subset
$U^{\prime}\subset U$, every nontrivial cocharacter
$\lambda:(\mathbb{G}_{m})_{K}\longrightarrow[G_{K},G_{K}]\subset G_{K}$ and
every reduction $\mathcal{G}_{\lambda}\to U^{\prime}\subset U$ to the
parabolic subgroup $P_{\lambda}$ defined by $\lambda$, we have

$\sum_{i=1}^{b}\sum_{m\in\mathbb{Z}}(\operatorname{rk}(\mathcal{F})\cdot\deg{\mathcal{F}^{i}_{m}}-\operatorname{rk}(\mathcal{F}_{i})\cdot\deg{\mathcal{F}^{i}})<0\quad\quad\text{(respectively,
$\leq$)}$

where $\mathcal{F}^{i}=(\rho_{i})_{*}\mathcal{G}$ and
$(\mathcal{F}^{i}_{m})_{m\in\mathbb{Z}}$ is the filtration associated to the
weighted parabolic reduction. ∎

The following corollary follows immediately.

######  Corollary 10.9.

If $\rho$ is central, then a principal $\rho$-sheaf is slope (semi)stable if
and only if the underlying rational principal bundle is slope (semi)stable in
the sense of Ramanathan (Definition 10.3). ∎

If $\dim X=1$, then the only big open subset of $X$ is $X$ itself. It follows
that a principal $\rho$-sheaf on a curve is the same thing as a principal
$G$-bundle, and the notion of Gieseker (semi)stability coincides with the
notion introduced by Ramanathan in [Ram75].

If the representation $\rho$ is not central, then our notion of Gieseker
semistability can be quite restrictive. This is illustrated by the next
example.

######  Example 10.10.

We construct a principal $\rho$-sheaf, with $\rho$ not central (Definition
10.5), which is Gieseker unstable, but the underlying rational principal
bundle is stable in the sense of Ramanathan (as in Lemma 10.8).

Let $W$ be a vector space over $k$. Set $V=W\oplus W$ and
$G=\operatorname{GL}(W)\times\operatorname{GL}(W)$. We define
$\rho:G\rightarrow\operatorname{GL}(V)$ to be the faithful representation
given by taking the direct sum of automorphisms

$\rho(A,B)=\left(\begin{array}[]{cc}A&0\\\ 0&B\\\ \end{array}\right)$

Note that, in the notation of Definition 10.2, we are taking $b=1$, and then
$\rho$ is not central.

Let $(\mathcal{E},\mathcal{E}^{\prime})$ be a pair of slope stable vector
bundles of rank $\dim W$ on $X$ and with
$d=\deg\mathcal{E}>\deg\mathcal{E}^{\prime}=d^{\prime}$. This pair defines a
principal $G$-bundle $P$ on $X$. In particular, it defines a principal
$\rho$-sheaf. It is stable in the sense of Ramanathan (Definition 10.3 or,
equivalently, Lemma 10.8 for the identity representation of
$\operatorname{GL}(W)\times\operatorname{GL}(W)$). However it is not Gieseker
semistable in the sense of Definition 10.2. Indeed, consider the central
cocharacter $\lambda$ of $G$ defined by
$\lambda(t)=t^{d}\operatorname{Id}_{W}\oplus
t^{d^{\prime}}\operatorname{Id}_{W}$ (note that the image is not in the
subgroup $[G,G]$). The corresponding parabolic $P_{\lambda}$ is the whole
group $G$. We have a weighted parabolic reduction
$f=(\lambda,\mathcal{G}_{\lambda})$, where $\mathcal{G}_{\lambda}$ is the
trivial parabolic reduction over $X$. The underlying associated filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ of $\mathcal{F}$ is given by

$\mathcal{F}_{m}=\left\\{\begin{array}[]{ll}\mathcal{E}\oplus\mathcal{E}^{\prime}&\text{if
$m\leq d^{\prime}$}\\\ \mathcal{E}&\text{if $d^{\prime}<m\leq d$}\\\
0&\text{if $d<m$}\\\ \end{array}\right.$

We have

$\operatorname{rk}(\mathcal{E}\oplus\mathcal{E}^{\prime})\cdot\deg(\mathcal{E})-\operatorname{rk}(\mathcal{E})\cdot\deg(\mathcal{E}\oplus\mathcal{E}^{\prime})=(d-d^{\prime})\dim
W>0$

and hence the numerical condition in the Gieseker semistability is not
satisfied.

###  10.2. Comparison of the leading term filtration with previous notions in
the literature

We end this section by comparing the leading term HN filtration with the
canonical (slope) parabolic reductions for $G$-bundles described in [AAB02,
GN18].

######  Proposition 10.11.

Suppose that the representation $\rho$ is central. Let $K\supset k$ be a field
extension such that $G_{K}$ is split. Let $(\mathcal{F}^{\bullet},\sigma)$ be
a $\rho$-sheaf on $X_{K}$ and choose a big open subset $U\subset X_{K}$ such
that $\mathcal{F}|_{U}$ is locally-free. Let $\mathcal{G}$ denote the rational
$G$-bundle on $U$ corresponding to $(\mathcal{F}^{\bullet},\sigma)$ and
suppose that $\mathcal{G}$ is (slope) unstable in the sense of Ramanathan. If
$f=(\lambda,\mathcal{G}_{\lambda})$ is the leading term HN weighted parabolic
reduction of $(\mathcal{F}^{\bullet},\sigma)$, then $\mathcal{G}_{\lambda}$ is
the canonical (slope) parabolic reduction as defined in [AAB02].

###### Proof.

We will use the setup and notation from Propositions 4.26 and 10.4. The group
$G$ acts on each vector space $V^{j}$ via the homomorphism $\rho_{j}$. For
each $j$, we denote by $(\chi_{i,j})_{i=1}^{r_{j}}$ the tuple of weights of
the split maximal torus $T$ acting on $V^{j}$ (with possible repetitions).
Since the representation $\rho$ is faithful, the collection of all the weights
$(\chi_{i,j})$ determines a positive definite inner product $\cdot_{\chi}$ on
$X_{*}(T)_{\mathbb{R}}$ given by

$\delta\cdot_{\chi}\gamma\vcentcolon=\sum_{j=1}^{b}\sum_{i=1}^{r_{j}}\langle\delta,\chi_{i,j}\rangle\cdot\langle\gamma,\chi_{i,j}\rangle$

which is invariant under the action of the Weyl group $W$. We can restrict
this inner product to $X_{*}(Z_{\lambda})_{\mathbb{R}}$ in order to obtain a
$W_{P_{\lambda}}$-invariant positive definite inner product.

Recall that we have

(18)
$\nu_{d-1}(\delta)=\frac{1}{\sqrt{A_{d}}(d-1)!}\,\cdot\,\frac{\sum_{j=1}^{b}\sum_{i=1}^{r_{j}}\langle\delta,\chi^{\lambda}_{i,j}\rangle\cdot(c_{i,j}-c_{j})}{\sqrt{\sum_{j=1}^{b}\sum_{i=1}^{r_{j}}\left(\langle\delta,\chi^{\lambda}_{i,j}\rangle\right)^{2}}}$

where we define $c_{i,j}=\langle\psi,\chi_{i,j}^{\lambda}\rangle$ and
$c_{j}=\frac{1}{r_{j}}\langle\psi,\sum_{i=1}^{r_{j}}\chi_{i,j}^{\lambda}\rangle$.
Set $\det_{j}=\det(V^{j})\circ\rho_{j}=\sum_{i=1}^{r_{j}}\chi_{i,j}$. Rewrite
(18) as

$\displaystyle\nu_{d-1}(\delta)=\frac{1}{\sqrt{A_{d}}(d-1)!}\,\cdot\,\frac{\sum_{j=1}^{b}\sum_{i=1}^{r_{j}}\langle\delta,\chi^{\lambda}_{i,j}\rangle\cdot\langle\psi,\chi_{i,j}^{\lambda}\rangle-\frac{1}{r_{j}}\langle\delta,\det_{j}\rangle\cdot\langle\psi,\det_{j}\rangle}{\sqrt{\sum_{j=1}^{b}\sum_{i=1}^{r_{j}}\left(\langle\delta,\chi^{\lambda}_{i,j}\rangle\right)^{2}}}=$
(19)
$\frac{1}{\sqrt{A_{d}}(d-1)!}\,\cdot\,\frac{\delta\cdot_{\chi}\psi-\sum_{j=1}^{b}\frac{1}{r_{j}}\langle\delta,\det_{j}\rangle\cdot\langle\psi,\det_{j}\rangle}{\sqrt{\delta\cdot_{\chi}\delta}}$

Recall that

$X_{*}(Z_{\lambda})_{\mathbb{R}}=\bigoplus_{l=1}^{h}\mathbb{R}z_{l}\oplus\bigoplus_{n\in
J_{P_{\lambda}}}\mathbb{R}\omega_{n}^{\vee}$

Choose a pair $z_{l}$ and $\omega_{n}^{\vee}$. Let $s_{n}$ denote the simple
reflection of $X_{*}(T)_{\mathbb{R}}$ determined by the root $\alpha_{n}$.
Since $\cdot_{\chi}$ is $W_{\lambda}$-invariant, we have

$z_{l}\cdot_{\chi}\omega_{n}^{\vee}=(s_{n}z_{l})\cdot_{\chi}(s_{n}\omega_{n}^{\vee})=z_{l}\cdot_{\chi}(-\omega_{n}^{\vee}).$

This shows that $z_{l}\cdot_{\chi}\omega_{n}^{\vee}=0$, and so the central
cocharacters $X_{*}(Z)_{\mathbb{R}}=\bigoplus_{i=1}^{h}\mathbb{R}z_{i}$ are
orthogonal to the fundamental coweights. Therefore the projection
$\pi_{Z}:X_{*}(Z_{\lambda})_{\mathbb{R}}\rightarrow\bigoplus_{n\in
J_{P_{\lambda}}}\mathbb{R}\omega_{q}^{\vee}$ is an orthogonal projection with
respect to $\cdot_{\chi}$.

By assumption, the central torus $Z$ maps to the center of
$\prod_{j=1}^{b}\operatorname{GL}(V^{j})$. For every index $l$, since
$\rho_{j}\circ z_{l}$ lands in the center of $\operatorname{GL}(V^{j})$, we
must have $\langle z_{l},\chi_{i,j}\rangle=\langle z_{l},\chi_{p,j}\rangle$
for any $i,p$. It follows that $\langle z_{l},\det_{j}\rangle=r_{j}\langle
z_{l},\chi_{1,j}\rangle$ for the fixed index $i=1$. This implies that

$\delta\cdot_{\chi}\psi-\sum_{j=1}^{b}\frac{1}{r_{j}}\langle\delta,{\det}_{j}\rangle\cdot\langle\psi,{\det}_{j}\rangle=\pi_{Z}(\delta)\cdot_{\chi}\pi_{Z}(\psi)$

Therefore, (19) turns out to be:

(20)
$\nu_{d-1}(\delta)=\frac{1}{\sqrt{A_{d}}(d-1)!}\,\cdot\,\frac{\pi_{Z}(\delta)\cdot_{\chi}\pi_{Z}(\psi)}{\sqrt{\delta\cdot_{\chi}\delta}}$

By Corollary 10.9, the $\rho$-sheaf $(\mathcal{F},\sigma)$ is unstable. Using
the proof of Proposition 10.6 we can conclude that the image of $\lambda$ is
not contained in the center of $G$. Therefore the parabolic $P_{\lambda}$ is a
proper subgroup of $G$. We denote by $f=(\lambda,\mathcal{G}_{\lambda})$ the
leading term HN weighted parabolic reduction of $(\mathcal{F},\sigma)$. To
prove that $\mathcal{G}_{\lambda}$ is the canonical parabolic reduction of the
rational $G$-bundle $\mathcal{G}$, we need to show the following (cf. [AAB02,
Thm. 6]):

  1. (i)

The linear functional $\psi\in X_{*}(Z_{\lambda})_{\mathbb{R}}$ lies in the
interior $\operatorname{Int}(\mathcal{C}_{\lambda})$ of the dominant cone
$\mathcal{C}_{\lambda}$.

  2. (ii)

The rational $L_{\lambda}$-bundle $\mathcal{G}_{L_{\lambda}}$ is slope
semistable.

We proceed with the proof of these statements.

(i) Since $f=(\lambda,\mathcal{G}_{\lambda})$ maximizes $\nu(f)$ in the
interior of $\mathcal{C}_{\lambda}$, we know that $\lambda$ is a critical
point of $\nu_{d-1}$. It follows from Cauchy-Schwarz that this critical point
is a scalar multiple of $\pi_{Z}(\psi)$. We must have
$\lambda=Q\cdot\pi_{Z}(\psi)$ for some constant $Q>0$, because $\nu(f)>0$ and
then $\nu_{d-1}(\lambda)\geq 0$. Since
$\lambda\in\operatorname{Int}(\mathcal{C}_{\lambda})$ by definition, it
follows that $\pi_{Z}(\psi)\in\operatorname{Int}(\mathcal{C}_{\lambda})$.
Therefore $\psi\in\operatorname{Int}(\mathcal{C}_{\lambda})$, as desired.

(ii) Suppose for the sake of contradiction that $\mathcal{G}_{\lambda}$ is not
slope semistable. Let $\mathcal{G}_{\overline{P}}$ be the canonical slope
parabolic reduction of $\mathcal{G}_{L_{\lambda}}$ for some proper parabolic
subgroup $\overline{P}\subset L_{\lambda}$, as in [AAB02]. This reduction is
defined over a big open subset. After shrinking the original open subset $U$,
we can assume that everything is defined over $U$.

Let $P$ denote the inverse image of $\overline{P}$ under the canonical
quotient map $P_{\lambda}\twoheadrightarrow L_{\lambda}$. Note that $P$ is a
parabolic subgroup of $G$ (c.f. second paragraph of the proof of [BH04, Prop.
3.1]). Let $\iota:P\hookrightarrow P_{\lambda}$ denote the inclusion and let
$q:P\twoheadrightarrow\overline{P}$ denote the quotient map. By [BH04, Lem.
2.1], there exists a unique $P$-parabolic reduction $\mathcal{G}_{P}$ of
$\mathcal{G}$ such that $\iota_{*}(\mathcal{G}_{P})=\mathcal{G}_{\lambda}$ and
$q_{*}(\mathcal{G}_{P})=\mathcal{G}_{\overline{P}}$. Let $L$ be the unique
Levi subgroup of $P$ containing the maximal torus $T\subset P_{\lambda}$, and
let $Z_{L}$ denote the maximal central torus of $L$. The canonical inclusion
$Z_{\lambda}\subset Z_{L}$ induces an inclusion
$X_{*}(Z_{\lambda})_{\mathbb{R}}\subset X_{*}(Z_{L})_{\mathbb{R}}$.

Since $\lambda=Q\cdot\pi_{Z}(\psi)$ for $Q>0$, (20) is given by

$\nu_{d-1}(\lambda)=\frac{1}{\sqrt{A_{d}}(d-1)!}\cdot\sqrt{\pi_{Z}(\psi)\cdot_{\chi}\pi_{Z}(\psi)}.$

On the other hand, let

$\psi_{L}=z+\sum_{n\in J_{P}}a_{n}\omega_{n}^{\vee}\in
X_{*}(Z_{L})_{\mathbb{R}}=X_{*}(Z)_{\mathbb{R}}\oplus\bigoplus_{n\in
J_{P}}\mathbb{R}\omega_{n}^{\vee}$

be the linear functional corresponding to the $L$-bundle $\mathcal{G}_{L}$
defined as in the proof of Lemma 4.25. Here $z\in X_{*}(Z)_{\mathbb{R}}$ and
$a_{m}\in\mathbb{R}$.

By construction, $\mathcal{G}_{L_{\lambda}}$ is isomorphic to the
$L_{\lambda}$-bundle obtained from $\mathcal{G}_{L}$ by extension of structure
group via the inclusion $L\subset L_{\lambda}$. This implies that
$\psi=z+\sum_{n\in J_{P_{\lambda}}}a_{n}\omega_{n}^{\vee}$, hence

(21) $\psi_{L}=\psi+\sum_{n\in J_{P}\setminus
J_{P_{\lambda}}}a_{n}\omega_{n}^{\vee}$

In particular we have $a_{n}>0$ for all $q\in J_{P_{\lambda}}$, because
$\psi\in\operatorname{Int}(\mathcal{C}_{\lambda})$.

We know that the bundle $q_{*}(\mathcal{G}_{\lambda})$ is the canonical slope
parabolic reduction of $\mathcal{G}_{L_{\lambda}}$. This implies that
$a_{n}>0$ for all the other indexes $n\in J_{P}\setminus J_{P_{\lambda}}$. We
conclude that $\psi_{L}$ is in the interior of the $P$-dominant cone
$\mathcal{C}_{P}$. This shows that there exists some positive integer $N>0$
such that the tuple $f^{\prime}=(N\cdot\pi_{Z}(\psi_{L}),\mathcal{G}_{P})$ is
a well-defined weighted parabolic reduction of $(\mathcal{F},\sigma)$. The
version of (20) for this parabolic reduction yields

$\nu_{d-1}(f^{\prime})=\frac{1}{\sqrt{A_{d}}(d-1)!}\cdot\sqrt{\pi_{Z}(\psi_{L})\cdot_{\chi}\pi_{Z}(\psi_{L})}.$

Equation (21) implies $\pi_{Z}(\psi_{L})=\pi_{Z}(\psi)+\sum_{n\in
J_{P}\setminus J_{P_{\lambda}}}a_{n}\omega_{n}^{\vee}$, therefore:

$\displaystyle\pi_{Z}(\psi_{L})\cdot_{\chi}\pi_{Z}(\psi_{L})=\pi_{Z}(\psi)\cdot_{\chi}\pi_{Z}(\psi)+2\pi_{Z}(\psi)\cdot_{\chi}\left(\sum_{n\in
J_{P}\setminus J_{P_{\lambda}}}a_{n}\omega_{n}^{\vee}\right)+$ (22)
$\displaystyle\left(\sum_{n\in J_{P}\setminus
J_{P_{\lambda}}}a_{n}\omega_{n}^{\vee}\right)\cdot_{\chi}\left(\sum_{n\in
J_{P}\setminus J_{P_{\lambda}}}a_{n}\omega_{n}^{\vee}\right)$

Note that $\cdot_{\chi}$ is a Weyl group invariant positive definite inner
product on $X_{*}(T)_{\mathbb{R}}$. In particular we must have
$\omega_{l}^{\vee}\cdot_{\chi}\omega_{n}^{\vee}\geq 0$ for all $l,n$. Since
$a_{n}>0$ for all $n\in J_{P}$, it follows that

$\displaystyle\pi_{Z}(\psi)\cdot_{\chi}\left(\sum_{n\in J_{P}\setminus
J_{P_{\lambda}}}a_{n}\omega_{n}^{\vee}\right)=\left(\sum_{l\in
J_{P_{\lambda}}}a_{l}\omega_{l}^{\vee}\right)\cdot_{\chi}\left(\sum_{n\in
J_{P}\setminus J_{P_{\lambda}}}a_{n}\omega_{n}^{\vee}\right)\geq 0$

On the other hand, since $\cdot_{\chi}$ is positive definite we have

$\left(\sum_{n\in J_{P}\setminus
J_{P_{\lambda}}}a_{n}\omega_{n}^{\vee}\right)\cdot_{\chi}\left(\sum_{n\in
J_{P}\setminus J_{P_{\lambda}}}a_{n}\omega_{n}^{\vee}\right)>0.$

Plugging the last two inequalities in (22) we get
$\pi_{Z}(\psi_{L})\cdot_{\chi}\pi_{Z}(\psi_{L})>\pi_{Z}(\psi)\cdot_{\chi}\pi_{Z}(\psi)$.
Therefore $\nu_{d-1}(f^{\prime})>\nu_{d-1}(\lambda)=\nu_{d-1}(f)$. Hence, the
leading coefficient of $\nu(f^{\prime})$ is strictly bigger than the leading
coefficient of $\nu(f)$, thus contradicting the maximality of $\nu(f)$. ∎

######  Remark 10.12.

When the representation $\rho$ is central, Proposition 10.11 shows that the
leading term HN $\Theta$-stratification of $\operatorname{Bun}_{\rho}(X)$ is a
refinement of the slope stratification for the stack of $G$-bundles defined in
[GN18, GN16] in the split case. This in turn implies that the finer
stratification induced by GHN filtrations is also a refinement of [GN18,
GN16].

##  Appendix A Gieseker-Harder-Narasimhan filtrations in positive
characteristic

Many of the arguments in this article do not use the hypothesis that the
characteristic of $k$ is $0$. In this appendix we explain what needs to be
modified in order to define Gieseker-Harder-Narasimhan filtrations in positive
characteristic. We can show that $\nu$ defines a weak $\Theta$-stratification
on $\text{Bun}_{\rho}(X)$ (as in [HLHJ21, Defn. 2.23]). This allows us to
define the GHN filtration of a $\rho$-sheaf after maybe passing to a purely
inseparable extension of the ground field.

Fix an arbitrary field $k$. We use the same setup and notation as in the
characteristic $0$ case ($X$ is a polarized smooth projective geometrically
irreducible $k$-variety, $G$ is a connected reductive group over $k$, and
$\rho$ is a faithful representation of $G$). Note that everything in
Subsection 2.2 applies to general $k$.

The construction of $\text{Red}_{G}(\mathcal{F})$ in Subsection 2.3 can be
done in a similar way in arbitrary characteristic. The proofs of all the
results in Subsection 2.3 remain valid, with the only exception that in Lemma
2.16 it is no longer clear that the natural base-change morphism
$\text{Red}_{G}(\mathcal{F})\times_{X_{S}}X_{T}\to\text{Red}_{G}(\mathcal{F}|_{X_{T}})$
is necessarily an isomorphism. This is because it is no longer true that
formation of $G$-invariants commutes with arbitrary base-change. We note,
however, that for any big open $U\subset X_{S}$ such that $\mathcal{F}$ is
locally free, the restriction
$\text{Red}_{G}(\mathcal{F})\times_{X_{S}}U_{T}\to\text{Red}_{G}(\mathcal{F}|_{X_{T}})_{U_{T}}$
is an isomorphism. This is because both sides are naturally identified with
the quotient $\mathcal{P}|_{U_{T}}/G$ by Proposition 2.17. This is used to
show the following lemma.

######  Lemma A.1.

Let $S$ be a $k$-scheme, and let $\mathcal{F}$ be a relative torsion-free
sheaf of rank $r$ on $X_{S}$. For any $T\to S$, there is a natural bijection
between the following two sets

  1. (A)

Sections of the structure morphism $\text{Red}_{G}(\mathcal{F}|_{X_{T}})\to
X_{T}$.

  2. (B)

Sections of the structure morphism
$\text{Red}_{G}(\mathcal{F})\times_{X_{S}}X_{T}\to X_{T}$.

###### Proof.

Let $U\subset X_{S}$ denote a big open subset such that $\mathcal{F}|_{U}$ is
locally-free. By Lemma 2.2 (c), sections in the set (A) are in correspondence
with sections $U_{T}\to\text{Red}_{G}(\mathcal{F}|_{X_{T}})_{U_{T}}$ defined
on the big open $U_{T}\subset X_{T}$. Since the base-change morphism
$\text{Red}_{G}(\mathcal{F})\times_{X_{S}}X_{T}\to\text{Red}_{G}(\mathcal{F}|_{X_{T}})$
restricts to an isomorphism over the open set $U_{T}$, these sections are in
bijection with sections
$U_{T}\to\text{Red}_{G}(\mathcal{F})\times_{X_{S}}U_{T}$. Another application
of Lemma 2.2 (c) shows that all such sections extend uniquely to a morphism
$X_{T}\to\text{Red}_{G}(\mathcal{F})\times_{X_{S}}X_{T}$ as in (B). ∎

In other words, even though the schemes
$\text{Red}_{G}(\mathcal{F})\times_{X_{S}}X_{T}$ and
$\text{Red}_{G}(\mathcal{F}|_{X_{T}})$ are not necessarily isomorphic, they
have the same sections over $X_{T}$ because they are isomorphic over a big
open subset.

The notion of $\rho$-sheaf and the stack $\text{Bun}_{\rho}(X)$ can be defined
in the same way for arbitrary characteristic.

######  Definition A.2.

A $\rho$-sheaf is a pair $(\mathcal{F},\sigma)$ where

  1. (1)

$\mathcal{F}$ is a family of torsion-free sheaves of rank $r$ on $X_{S}$.

  2. (2)

$\sigma$ is a section of the structure morphism
$\operatorname{Red}_{G}(\mathcal{F})\rightarrow X_{S}$.

The stack $\operatorname{Bun}_{\rho}(X)$ is the pseudofunctor from
$\left(\operatorname{Aff}_{k}\right)^{op}$ into groupoids that sends an affine
scheme $T\in\operatorname{Aff}_{k}$ to

$\operatorname{Bun}_{\rho}(X)\,(T)=\left[\;\;\text{groupoid of $\rho$-sheaves
$(\mathcal{F},\sigma)$ on $X_{S}$}\;\;\;\right].$

Once we know Lemma A.1, everything in Section 3 applies verbatim. Similarly,
the definitions and arguments in Subsections 4.1, 4.2, 4.3 and 4.4 apply
without a change. Hence we have the following result in arbitrary
characteristic

######  Proposition A.3.

The polynomial numerical invariant $\nu$ on the stack $\text{Bun}_{\rho}(X)$
is strictly $\Theta$-monotone, strictly $S$-monotone, and satisfies the HN
boundedness condition. ∎

Proposition A.3 yields the necessary hypotheses to apply [HLHJ21, Thm. 2.26]
(or [HL14, Thm. B]). This yields the following theorem (see [HLHJ21, Defn.
2.23] for a definition of weak $\Theta$-stratification).

######  Theorem A.4.

The polynomial numerical invariant $\nu$ defines a weak
$\Theta$-stratification on the stack $\text{Bun}_{\rho}(X)$. ∎

Note that in this case we only get a weak $\Theta$-stratification. In
particular, the corresponding notion of leading term HN filtration of a
$\rho$-sheaf $(\mathcal{F},\sigma)$ is not necessarily defined over the field
$K$ of definition of $(\mathcal{F},\sigma)$; one might need to allow to pass
to a finite purely inseparable extension of $K$. This is to be expected for
small characteristics, because of the failure of Behrend’s conjecture even in
the case when $X$ is a curve [Hei08a]. We note that it is still the case that
the leading HN filtration is unique up to scaling, even though it might not be
defined over the ground field.

The analogous definition of relative leading HN filtrations (Definition 4.32)
applies in arbitrary characteristic. However, since we are working with a weak
$\Theta$-stratification, the universal stratification $H\to S$ is no longer a
locally closed immersion. Instead, we get that the universal morphism $H\to S$
parametrizing relative leading term HN filtrations is surjective, radicial and
of finite type. Each connected component of $H$ is finite over its image.

As long as the characteristic of $k$ is not too small, the proofs in Section 5
can still be applied to show Theorem 5.7. More precisely, we just need that
condition (2) in Proposition 5.2 holds.

######  Theorem A.5.

Choose an algebraic closure $\overline{k}$ of $k$. Let $Z\subset
G_{\overline{k}}$ denote the maximal central torus of the base-change
$G_{\overline{k}}$. Suppose that the characteristic of $k$ does not divide the
order of the kernel of the simply-connected cover
$\widetilde{G_{\overline{k}}/Z}\to G_{\overline{k}}/Z$ of the semisimple group
$G_{\overline{k}}/Z$. Then the morphism
$\text{Bun}_{\rho}(X)\to\operatorname{Spec}(k)$ satisfies the existence part
of the valuative criterion for properness [Sta21, Tag 0CLK] in the case of
complete discrete valuation rings (as in Theorem 5.7). ∎

The definitions and constructions in Sections 7 and 8 apply in a similar way.
The only change is that the GHN filtration of a $\rho$-sheaf
$(\mathcal{F},\sigma)$ need not be defined over the field $K$ of definition,
but again one might need to pass to a finite purely inseparable extension. The
analogue of Theorem 8.18 still holds; the only thing that needs to be modified
is to replace the locally closed stratification $H\to S$ with a surjective
radicial morphism of finite type.

Finally, we note that the proof of Proposition 10.11 applies to arbitrary
characteristic. This means that if the representation $\rho$ is central, then
the stratification of $\text{Bun}_{\rho}(X)$ by radicial morphisms induced by
our Gieseker-Harder-Narasimhan filtrations refines the theory developed in
[GN16] in arbitrary characteristic.

######  Remark A.6.

The attentive reader might have noticed that we did not mention Section 6 in
this appendix. Unfortunately the results in [HL14, AHLH19] that we are using
to build moduli spaces do not apply in arbitrary characteristic.

######  Remark A.7.

Using the same techniques, one could generalize the results in this appendix
to the case when $X\to S$ is a smooth projective morphism with geometrically
connected fibers over an arbitrary base-scheme $S$, $G\to S$ is a reductive
group scheme with connected fibers, and $\rho$ is a faithful representation
$\rho:G\hookrightarrow\prod_{i}GL(V^{i})$ for some vector bundles $V^{i}$ on
$S$. We have chosen not to pursue this level of generality in this paper.

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