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

In particular, the function $\nu_{d-1}$ can be naturally interpreted as a
scale-invariant continuous function on the whole cone $\mathcal{C}_{\lambda}$
of dominant coweights with real coefficients.

###### Proof.

For any $\delta\in\mathcal{C}_{\lambda}\cap X_{*}(T)$, denote by $L_{\delta}$
the unique Levi subgroup of the parabolic $P_{\delta}$ containing the maximal
torus $T$. Note that $L_{\lambda}\subset L_{\delta}$. Denote by $Z_{\delta}$
the maximal central torus of $L_{\delta}$.

Let $\mathcal{G}_{L_{\delta}}$ be the $L_{\delta}$–bundle obtained from
$\mathcal{G}_{\delta}$ via the quotient $P_{\delta}\twoheadrightarrow
L_{\delta}$. Observe that $\mathcal{G}_{L_{\delta}}$ is obtained from
$\mathcal{G}_{L_{\lambda}}$ by extending the structure group via
$L_{\lambda}\hookrightarrow L_{\delta}$. Let $\mathcal{E}_{U}^{\delta}$ be the
vector bundle on $U$ obtained by extending the structure group via the
inclusion $L_{\delta}\hookrightarrow G\xrightarrow{\rho}\operatorname{GL}(V)$.
For all $\delta$ we have a canonical identification
$\mathcal{E}_{U}^{\delta}\cong\mathcal{E}^{\lambda}_{U}$. Hence we will denote
all such $\mathcal{E}^{\delta}_{U}$ by the same notation $\mathcal{E}_{U}$.
Let $\mathcal{E}$ be the reflexive sheaf on $X$ obtained by pushing forward
$\mathcal{E}_{U}$ under the open inclusion $U\hookrightarrow X$. Let
$(\mathcal{F}^{\delta}_{m})_{m\in\mathbb{Z}}$ be the underlying filtration of
$\mathcal{F}$ associated to the weighted parabolic reduction
$(\delta,\mathcal{G}_{\delta})$. The description of the associated graded at
the end of Subsection 3.3 shows that there is an identification
$\left(\bigoplus_{m\in\mathbb{Z}}\mathcal{F}^{\delta}_{m}/\mathcal{F}^{\delta}_{m+1}\right)|_{U}\xrightarrow{\sim}\mathcal{E}_{U}$
of associated graded sheaves on $U$.

This induces a natural identification
$\bigoplus_{m\in\mathbb{Z}}(\mathcal{F}^{\delta}_{m}/\mathcal{F}^{\delta}_{m+1})^{\vee\vee}\xrightarrow{\sim}\mathcal{E}$
of reflexive sheaves on $X$. We will describe a $\mathbb{Z}$-grading on
$\mathcal{E}$ determined by the action of the cocharacter $\delta$. The
inclusion $Z_{\lambda}\hookrightarrow T$ induces an inclusion
$X_{\ast}(Z_{\lambda})\hookrightarrow X_{\ast}(T)$. Recall that
$\chi_{i}^{\lambda}$ denotes the restriction of each character $\chi_{i}$ to
$X_{\ast}(Z_{\lambda})$. We have a $X^{*}(Z_{\lambda})$-grading on the vector
space $V$ with weight decomposition:

$V=\bigoplus_{\\{[\chi^{\lambda}_{i}]\\}}V_{\chi^{\lambda}_{i}}$

For any representative $\chi_{i}^{\lambda}$ of $[\chi^{\lambda}_{i}]$ we have
$V_{\chi^{\lambda}_{i}}=\bigoplus_{\\{j\,\mid\,\chi^{\lambda}_{j}=\chi^{\lambda}_{i}\\}}V_{\chi^{\lambda}_{j}}$.
Hence
$\dim(V_{\chi^{\lambda}_{i}})=\\#|\,\\{j\;\mid\;\chi^{\lambda}_{j}=\chi^{\lambda}_{i}\,\\}\,|$.
Since $Z_{\lambda}$ is central in $L_{\lambda}$, the action of $L_{\lambda}$
on $V$ preserves each $Z_{\lambda}$-weight space $V_{\chi^{\lambda}_{i}}$,
hence each $V_{\chi_{i}^{\lambda}}$ is a representation of $L_{\lambda}$. This
way we get the grading

$\mathcal{E}_{U}=\bigoplus_{\\{[\chi^{\lambda}_{i}]\\}}\mathcal{E}_{U,\chi^{\lambda}_{i}}$

where $\mathcal{E}_{U,\chi^{\lambda}_{i}}$ is the vector bundle
$\mathcal{G}_{L_{\lambda}}\times^{L_{\lambda}}V_{\chi_{i}^{\lambda}}$
associated to the representation $V_{\chi_{i}^{\lambda}}$ and $Z_{\lambda}$
acts with weight $\chi^{\lambda}_{i}$ on $\mathcal{E}_{U,\chi^{\lambda}_{i}}$.
Note that
$\operatorname{rk}(\mathcal{E}_{U,\chi^{\lambda}_{i}})=\dim(V_{\chi_{i}^{\lambda}})$.
This grading extends uniquely to a grading of the reflexive extension

$\bigoplus_{m\in\mathbb{Z}}(\mathcal{F}^{\delta}_{m}/\mathcal{F}^{\delta}_{m+1})^{\vee\vee}\cong\mathcal{E}=\bigoplus_{\\{[\chi^{\lambda}_{i}]\\}}\mathcal{E}_{\chi^{\lambda}_{i}}$

The cocharacter $\delta$ acts on $\mathcal{E}_{U}$ via its natural inclusion
in $Z_{\delta}$. Using $Z_{\delta}\hookrightarrow Z_{\lambda}$, we can view
$\delta$ as a cocharacter of $Z_{\lambda}$. It follows from our discussion
that $\delta$ acts with weight $\langle\delta,\chi^{\lambda}_{i}\rangle$ on
$\mathcal{E}_{U,\chi^{\lambda}_{i}}$. This shows that

$(\mathcal{F}_{m}^{\delta}/\mathcal{F}_{m+1}^{\delta})|_{U}=\bigoplus_{\\{[\chi^{\lambda}_{i}]\;\mid\;\langle\delta,\chi^{\lambda}_{i}\rangle=m\\}}\mathcal{E}_{U,\chi^{\lambda}_{i}}$

and therefore

(5)
$(\mathcal{F}_{m}^{\delta}/\mathcal{F}_{m+1}^{\delta})^{\vee\vee}=\bigoplus_{\\{[\chi^{\lambda}_{i}]\;\mid\;\langle\delta,\chi^{\lambda}_{i}\rangle=m\\}}\mathcal{E}_{\chi^{\lambda}_{i}}$

Observe that we have equality of slopes
$\widehat{\mu}_{d-1}\left(\mathcal{F}_{m}^{\delta}/\mathcal{F}_{m+1}^{\delta}\right)=\widehat{\mu}_{d-1}\left((\mathcal{F}_{m}^{\delta}/\mathcal{F}_{m+1}^{\delta})^{\vee\vee}\right)$.
Taking the leading coefficient in the formula (3) in Proposition 4.4, we get

(6)
$\nu_{d-1}(\delta)=\frac{\sqrt{A_{d}}}{(d-1)!}\,\cdot\,\frac{\sum\limits_{m\in\mathbb{Z}}m\cdot\left(\widehat{\mu}_{d-1}(\mathcal{F}^{\delta}_{m}/\mathcal{F}^{\delta}_{m+1})-\widehat{\mu}_{d-1}(\mathcal{F})\right)\cdot\operatorname{rk}(\mathcal{F}^{\delta}_{m}/\mathcal{F}^{\delta}_{m+1})}{\sqrt{\sum\limits_{m\in\mathbb{Z}}m^{2}\cdot\operatorname{rk}(\mathcal{F}^{\delta}_{m}/\mathcal{F}^{\delta}_{m+1})}}$

Using the description of the quotients
$\mathcal{F}^{\delta}_{m}/\mathcal{F}^{\delta}_{m+1}$ in terms of the
$\mathcal{E}_{\chi^{\lambda}_{i}}$, we can rewrite $(\ref{eqn_2})$ as:

$\displaystyle\nu_{d-1}(\delta)=\frac{\sqrt{A_{d}}}{(d-1)!}\,\cdot\,\frac{\sum\limits_{m\in\mathbb{Z}}\sum\limits_{\\{[\chi_{i}^{\lambda}]\;\mid\;\langle\delta,\chi_{i}^{\lambda}\rangle=m\\}}m\cdot\left(\widehat{\mu}_{d-1}(\mathcal{E}_{\chi^{\lambda}_{i}})-\widehat{\mu}_{d-1}(\mathcal{F})\right)\cdot\\#|\,\\{j\;\mid\;\chi^{\lambda}_{j}=\chi^{\lambda}_{i}\,\\}\,|}{\sqrt{\sum_{m\in\mathbb{Z}}m^{2}\cdot\\#|\,\\{j\;\mid\;\langle\delta,\chi^{\lambda}_{i}\rangle=m\,\\}\,|}}$
(7)
$=\frac{\sqrt{A_{d}}}{(d-1)!}\,\cdot\,\frac{\sum\limits_{i=1}^{r}\langle\delta,\chi^{\lambda}_{i}\rangle\cdot\left(\widehat{\mu}_{d-1}(\mathcal{E}_{\chi^{\lambda}_{i}})-\widehat{\mu}_{d-1}(\mathcal{F})\right)}{\sqrt{\sum_{i=1}^{r}\left(\langle\delta,\chi^{\lambda}_{i}\rangle\right)^{2}}}$

Observe that in the sum above we are indexing over all $\chi_{i}^{\lambda}$
with repetitions. This is done in order to account for the ranks.

We can further rewrite this by using the Hirzebruch-Riemann-Roch theorem
[Ful98, Ch. 15]. Let $H$ denote the hyperplane class on $X$ corresponding to
the line bundle $\mathcal{O}(1)$. Let $t_{1}$ be the degree $1$ term of the
Todd class of $X$. For any torsion-free sheaf $\mathcal{A}$ on $X$ with first
Chern class $c_{1}(\mathcal{A})$, Hirzebruch-Riemann-Roch implies that

$\widehat{\mu}_{d-1}(\mathcal{A})=\frac{1}{A_{d}\,\operatorname{rk}(\mathcal{A})}c_{1}(\mathcal{A})\cdot
H^{d-1}+\frac{1}{A_{d}}t_{1}\cdot H^{d-1}$

Using this formula for $\mathcal{E}_{\chi_{i}^{\lambda}}$ and $\mathcal{F}$
yields

$\widehat{\mu}_{d-1}(\mathcal{E}_{\chi^{\lambda}_{i}})-\widehat{\mu}_{d-1}(\mathcal{F}|_{X_{s}})=\frac{1}{A_{d}}\cdot\left(\frac{c_{1}(\mathcal{E}_{\chi^{\lambda}_{i}})\cdot
H^{d-1}}{\operatorname{rk}(\mathcal{E}_{\chi^{\lambda}_{i}})}-\frac{c_{1}(\mathcal{F})\cdot
H^{d-1}}{r}\right)$

Using $c_{i}=\frac{c_{1}(\mathcal{E}_{\chi^{\lambda}_{i}})\cdot
H^{d-1}}{\operatorname{rk}(\mathcal{E}_{\chi^{\lambda}_{i}})}$ and
$c=\frac{c_{1}(\mathcal{F})\cdot H^{d-1}}{r}$, we can rewrite (7) as:

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

∎

For each $\delta\in X_{*}(T)\cap\mathcal{C}_{\lambda}$, the inclusion
$Z_{\delta}\hookrightarrow T$ induces an identification

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

where the last inclusion comes from $Z_{\delta}\subset Z_{\lambda}$. Let us
denote by $\chi^{\delta}_{i}$ the restriction of each character $\chi_{i}$ to
$X_{*}(Z_{\delta})_{\mathbb{R}}$. In coordinates of the basis
$\mathcal{B}^{\vee}$, the tuple $(\chi_{i}^{\delta})_{i=1}^{r}$ is the matrix
$M_{\delta}$ obtained from $M$ by deleting the rows corresponding to
$\Delta\setminus I_{P_{\delta}}$. Note that the set of all such possible
matrices $M_{\delta}$ as we vary $\delta$ is finite, because there are only
finitely many possibilities for the set of indexes $J_{P_{\delta}}$.

As a consequence of Lemma 4.23 we get the following immediate corollary.

######  Corollary 4.24.

Set $\vec{a}=[c_{i}]_{i=1}^{r}$ to be the row vector of size $r$ with the
$i^{th}$ element given by $c_{i}$, and let $\vec{1}$ denote the row vector of
size $r$ such that all entries are $1$. Let $\vec{x}=[x_{i}]$ be the column
vector of the coordinates of $\delta\in\mathcal{C}_{\lambda}$ in the basis
$\mathcal{B}$. Then we can express $\nu_{d-1}$ in terms of the matrix
$M_{\lambda}$ as follows.

(8)
$\nu_{d-1}(\vec{x})=\frac{1}{\sqrt{A_{d}}(d-1)!}\,\cdot\,\frac{(\vec{a}-c\vec{1})M_{\lambda}\vec{x}}{\sqrt{\vec{x}^{T}M_{\lambda}^{T}M_{\lambda}\vec{x}}}$

∎

######  Lemma 4.25.

There exists a linear functional
$\psi:X^{*}(Z_{\lambda})_{\mathbb{R}}\rightarrow\mathbb{R}$ such that
$c_{i}=\psi(\chi_{i}^{\lambda})$ for all $i$.

###### Proof.

Recall that $j:U\hookrightarrow X_{s}$ is a big open set and $X_{s}$ is
smooth. The pushforward $j_{*}$ induces an isomorphism of Picard groups
$j_{*}:\operatorname{Pic}(U)\rightarrow\operatorname{Pic}(X_{s})$. Let
$X^{*}(L_{\lambda})$ be the group of characters of the Levi subgroup
$L_{\lambda}$. The $L_{\lambda}$-bundle $\mathcal{G}_{L_{\lambda}}$ on $U$
induces a homomorphism $X^{*}(L_{\lambda})\rightarrow\operatorname{Pic}(U)$
where any character $\chi\in X^{*}(L_{\lambda})$ is mapped to the associated
line bundle $\mathcal{G}_{L_{\lambda}}\times^{L_{\lambda}}\chi$. By post-
composing with $j_{\ast}$ we get a homomorphism

$X^{*}(L_{\lambda})\rightarrow\operatorname{Pic}(U)\rightarrow\operatorname{Pic}(X_{s})$

On the other hand, the first Chern class induces a homomorphism

$\operatorname{Pic}(X_{s})\rightarrow\mathbb{Z},\;\;\mathcal{L}\mapsto
c_{1}(\mathcal{L})\cdot H^{d-1}\in\mathbb{Z}.$

We compose this with the previous map to get a homomorphism

$\varphi:X^{*}(L_{\lambda})\rightarrow\mathbb{Z},\;\;\chi\mapsto
c_{1}\left(j_{*}(\mathcal{G}_{L_{\lambda}}\times^{L_{\lambda}}\chi)\right)\cdot
H^{d-1}$

Tensor with $\mathbb{R}$ to obtain a linear functional
$\varphi_{\mathbb{R}}:X^{*}(L_{\lambda})_{\mathbb{R}}\rightarrow\mathbb{R}$.

The inclusion $Z_{\lambda}\hookrightarrow L_{\lambda}$ yields a restriction
map on characters
$\operatorname{res}:X^{*}(L_{\lambda})_{\mathbb{R}}\rightarrow
X^{*}(Z_{\lambda})_{\mathbb{R}}$. This morphism $\operatorname{res}$ is an
isomorphism since the inclusion induces an isogeny between $Z_{\lambda}$ and
$L_{\lambda}/\,[L_{\lambda},L_{\lambda}]$. We set $\psi$ to be the linear
functional

$\psi=\varphi_{\mathbb{R}}\circ(\operatorname{res})^{-1}:X^{*}(Z_{\lambda})\rightarrow\mathbb{R},\;\;\;\;\;\overline{\chi}\mapsto\varphi_{\mathbb{R}}(\chi)=c_{1}\left(j_{*}(\,\mathcal{G}_{L_{\lambda}}\times^{L_{\lambda}}\chi)\right)\cdot
H^{d-1}$

for a character $\overline{\chi}$ of $Z_{\lambda}$ that comes from the
restriction of a character $\chi$ of $L_{\lambda}$. We are left to prove that
$\psi(\chi_{i}^{\lambda})=c_{i}$ for all $i$.

By definition $\mathcal{E}_{\chi_{i}^{\lambda}}|_{U}$ is the associated vector
bundle $\mathcal{G}_{L_{\lambda}}\times^{L_{\lambda}}V_{\chi^{\lambda}_{i}}$.
Hence we have
$\det(\mathcal{E}_{\chi^{\lambda}_{i}}|_{U})=\mathcal{G}_{\lambda}\times^{L_{\lambda}}\det(V_{\chi_{i}^{\lambda}})$.
Therefore:

$\displaystyle c_{i}=\frac{c_{1}(\mathcal{E}_{\chi^{\lambda}_{i}})\cdot
H^{d-1}}{\operatorname{rk}(\mathcal{E}_{\chi^{\lambda}_{i}})}=\frac{c_{1}(\det(\mathcal{E}_{\chi^{\lambda}_{i}}))\cdot
H^{d-1}}{\operatorname{rk}(\mathcal{E}_{\chi_{i}^{\lambda}})}=\frac{c_{1}(j_{*}\det(\mathcal{E}_{\chi^{\lambda}_{i}}|_{U}))\cdot
H^{d-1}}{\operatorname{rk}(\mathcal{E}_{\chi_{i}^{\lambda}})}=\frac{\varphi_{\mathbb{R}}\left(\det(V_{\chi_{i}^{\lambda}})\right)}{\operatorname{rk}(\mathcal{E}_{\chi_{i}^{\lambda}})}$

As a representation of $Z_{\lambda}$, the weight space
$V_{\chi_{i}^{\lambda}}$ is a direct sum of
$\operatorname{rk}(\mathcal{E}_{\chi_{i}^{\lambda}})$-many copies of the
character $\chi^{\lambda}_{i}$. Hence
$\det(V_{\chi_{i}^{\lambda}})=\operatorname{rk}(\mathcal{E}_{\chi_{i}^{\lambda}})\cdot\chi_{i}^{\lambda}$
as a $Z_{\lambda}$-character. This means

$c_{i}=\frac{\varphi_{\mathbb{R}}\left(\det(V_{\chi_{i}^{\lambda}})\right)}{\operatorname{rk}(\mathcal{E}_{\chi_{i}^{\lambda}})}=\frac{\psi\left(\operatorname{rk}(\mathcal{E}_{\chi_{i}^{\lambda}})\cdot\chi_{i}^{\lambda}\right)}{\operatorname{rk}(\mathcal{E}_{\chi_{i}^{\lambda}})}=\psi(\chi_{i}^{\lambda})$

∎

After all of our preliminary work, we are now ready to prove the main
proposition of this subsection.

######  Proposition 4.26.

The polynomial numerical invariant $\nu$ on the stack
$\operatorname{Bun}_{\rho}(X)$ satisfies the HN boundedness condition.

###### Proof.

We shall prove property (*) in Lemma 4.18. Fix an algebraically closed
extension $K\supset k$ and a scheme $S$ of finite type over $K$. After
replacing $k$ with the extension $K$, we can assume that the ground field $k$
is algebraically closed. Fix a family
$g:S\rightarrow\operatorname{Bun}_{\rho}(X)$ represented by a $\rho$-sheaf
$(\mathcal{F},\sigma)$ on $X_{S}$.

Choose a closed point $s\in S$. Let $f=(\lambda,\mathcal{G}_{\lambda})$ be a
nondegenerate weighted reduction of $(\mathcal{F}|_{X_{s}},\sigma|_{X_{s}})$.
Let $U$ be a big open subset of $X_{s}$ where $\mathcal{F}|_{X_{s}}$ is
locally-free and $(\lambda,\mathcal{G}_{\lambda})$ is defined. Suppose that
$\nu(f)>0$. The first step will be to construct another weighted parabolic
reduction $f^{\prime}$ that satisfies $\nu(f^{\prime})\geq\nu(f)$.

Let $P_{\lambda}\subset G$ the parabolic subgroup associated to the
cocharacter $\lambda$. Note that the discussion and lemmas before the
proposition apply to the weighted parabolic reduction
$(\lambda,\mathcal{G}_{\lambda})$ of $(\mathcal{F}|_{X_{s}},\sigma|_{X_{s}})$.
The scale-invariant continuous function $\nu_{d-1}$ (see (4) in Lemma 4.23)
attains a maximal value $\nu_{d-1}(\delta^{*})$ for some $\delta^{*}$ on the
domain $\mathcal{C}_{\lambda}$ (by restricting to the compact intersection of
$\mathcal{C}_{\lambda}$ with the unit sphere). We can assume without loss of
generality that the maximum $\delta^{*}$ is in the interior of the cone
$\mathcal{C}_{\lambda}$. Otherwise, let $F$ be the unique facet of
$\mathcal{C}_{\lambda}$ such that $\delta^{*}\in\operatorname{Int}(F)$. This
facet $F$ is the smaller cone of dominant coweights of a parabolic subgroup
$P_{\delta^{*}}\supset P_{\lambda}$. Then we can replace $\lambda$ with a
$\delta^{*}$ and replace the parabolic reduction
$(\lambda,\mathcal{G}_{\lambda})$ with $(\delta^{*},\mathcal{G}_{\delta^{*}})$
for the bigger parabolic.

Since the maximum $\delta^{*}$ is in the interior of $\mathcal{C}_{\lambda}$,
it must be a critical point of the function
$\nu_{d-1}:\mathcal{C}_{\lambda}\rightarrow\mathbb{R}$ as in (8) in Corollary
4.24. If $\nu_{d-1}$ is not identically $0$ (equivalently
$(\vec{a}-c\vec{1})M_{\lambda}\neq 0$ in (8)), the computation of the gradient
yields the coordinates $\vec{z}$ of a critical point, up to scaling:

(9)
$\vec{z}=(M_{\lambda}^{T}M_{\lambda})^{-1}M_{\lambda}^{T}(\vec{a}^{T}-c\vec{1}^{T})$

We consider three cases to construct a modified weighted parabolic reduction
$f^{\prime}$:

  1. (C1)

Suppose that $\nu_{d-1}(\lambda)\neq\nu_{d-1}(\delta^{*})$. Then $\nu_{d-1}$
is not identically $0$. Since all of the matrices and vectors in (9) have
rational coordinates, we can scale $\delta^{*}$ to have rational coordinates.
After further scaling, we can assume that $\delta^{*}$ is a cocharacter in
$\mathcal{C}_{\lambda}\cap X_{*}(T)$. It has coordinates

$\vec{y}=q\vec{z}=q(M_{\lambda}^{T}M_{\lambda})^{-1}M_{\lambda}^{T}(\vec{a}^{T}-c\vec{1}^{T})$

for $q\in\mathbb{Q}_{>0}$ (the negative multiples correspond to minimal
critical points). Set the modified weighted parabolic reduction to be
$f^{\prime}=(\delta^{*},\mathcal{G}_{\delta^{*}})$ with
$\nu_{d-1}(\delta^{*})>\nu_{d-1}(\lambda)$. By construction we have
$\nu(f^{\prime})>\nu(f)$.

  2. (C2)

Suppose that $\nu_{d-1}(\lambda)=\nu_{d-1}(\delta^{*})$ and the function
$\nu_{d-1}$ is not identically $0$. Then $\lambda$ is a critical point for
$\nu_{d-1}$ in the interior of the cone $\mathcal{C}_{\lambda}$ with
coordinates $\vec{y}=q\vec{z}$. In this case we define the weighted parabolic
reduction $f^{\prime}$ to be the original $f=(\lambda,\mathcal{G}_{\lambda})$
and set $\delta^{*}=\lambda$.

  3. (C3)

Suppose that $\nu_{d-1}(\lambda)=\nu_{d-1}(\delta^{*})$ and $\nu_{d-1}$ is
identically $0$. In this case $\vec{z}=0$. We set $f^{\prime}=f$ and denote
$\delta^{*}=\lambda$.

Let $f^{\prime}=(\delta^{*},\mathcal{G}_{\delta^{*}})$ with maximizing
$\delta^{*}$ as explained for each case above. Let
$(\mathcal{F}_{m}^{\delta^{*}})_{m\in\mathbb{Z}}$ be its corresponding
filtration with associated graded
$\bigoplus_{m\in\mathbb{Z}}\mathcal{F}^{\delta^{*}}_{m}/\mathcal{F}^{\delta^{*}}_{m+1}$.
We are left to find a uniform lower bound for the slopes
$\hat{\mu}_{d-1}(\mathcal{F}^{\delta^{*}}_{m}/\mathcal{F}^{\delta^{*}}_{m+1})$.
Since the slope of
$\mathcal{F}^{\delta^{*}}_{m}/\mathcal{F}^{\delta^{*}}_{m+1}$ is the same as
the slope of its reflexive hull
$(\mathcal{F}^{\delta^{*}}_{m}/\mathcal{F}^{\delta^{*}}_{m+1})^{\vee\vee}$,
this is equivalent to giving a uniform lower bound for

$\widehat{\mu}_{d-1}\left((\mathcal{F}_{m}^{\delta^{*}}/\mathcal{F}_{m+1}^{\delta^{*}})^{\vee\vee}\right)=\widehat{\mu}_{d-1}\left(\bigoplus_{\\{[\chi^{\lambda}_{i}]\;\mid\;\langle\delta^{*},\chi^{\lambda}_{i}\rangle=m\\}}\mathcal{E}_{\chi^{\lambda}_{i}}\right)$

(See Equation (5) in the proof of Lemma 4.23). By additivity of degrees and
ranks, it suffices to give a uniform lower bound for the slope
$\widehat{\mu}(\mathcal{E}_{\chi_{i}^{\lambda}})$ of each direct summand
above. Hirzebruch-Riemann-Roch implies that

$\widehat{\mu}_{d-1}(\mathcal{E}_{\chi_{i}^{\lambda}})=\frac{1}{A_{d}}\left(c_{i}+t_{1}\cdot
H^{d-1}\right)\;,$

Therefore it suffices to give a lower bound for the numbers $c_{i}$. Denote
the coordinates in $\mathcal{B}^{\vee}$ of the linear functional $\psi$ of
Lemma 4.25 by the column vector $\vec{\psi}$. Observe that
$\psi(\chi_{i}^{\lambda})=c_{i}$ translates into
$\vec{a}^{T}=M_{\lambda}\vec{\psi}$. We shall prove uniform bounds for the
vector $\vec{a}=[c_{i}]$. In cases (C1) and (C2) the coordinates $\vec{y}$ of
$\delta^{*}$ are given by

$\vec{y}=q\vec{z}=q(M_{\lambda}^{T}M_{\lambda})^{-1}M_{\lambda}^{T}(\vec{a}^{T}-c\vec{1})$

We can plug-in $\vec{a}^{T}=M_{\lambda}\vec{\psi}$ to obtain the following:

$\vec{y}=q(M_{\lambda}^{T}M_{\lambda})^{-1}M_{\lambda}^{T}M_{\lambda}\vec{\psi}-qc(M_{\lambda}^{T}M_{\lambda})^{-1}M_{\lambda}^{T}\vec{1}^{T}=q\vec{\psi}-qc(M_{\lambda}^{T}M_{\lambda})^{-1}M_{\lambda}^{T}\vec{1}^{T}$

Multiply both sides by $M_{\lambda}$ to get

(10)
$\vec{a}^{T}=M_{\lambda}\vec{\psi}=\frac{1}{q}M_{\lambda}\vec{y}+cM_{\lambda}(M_{\lambda}^{T}M_{\lambda})^{-1}M_{\lambda}^{T}\vec{1}^{T}.$

Since the family $\mathcal{F}$ is bounded, we see that
$c=\frac{c_{1}(\mathcal{F}|_{X_{s}})\cdot H^{d-1}}{r}$ admits uniform upper
and lower bounds that do not depend on $s$. Also we know that there are
finitely many possibilities for the matrix $M_{\lambda}$, one for each
possible choice of subset $I_{P_{\lambda}}$ of simple roots corresponding to
the parabolic $P_{\lambda}$. This gives uniform upper and lower bounds
$C_{max}$ and $C_{min}$ for the coefficients of matrix of the second term
$cM_{\lambda}(M_{\lambda}^{T}M_{\lambda})^{-1}M_{\lambda}^{T}\vec{1}^{T}$ in
(10), independent of $s$ and the weighted parabolic reduction $f$.

The $i^{th}$ coefficient of the row vector $M_{\lambda}\vec{y}$ is the weight
$\langle\delta^{*},\chi_{i}^{\lambda}\rangle$ of the sheaf
$\mathcal{E}_{\chi_{i}^{\lambda}}$. From (10) we get for each index $i$:

$C_{min}+\frac{1}{q}\operatorname{wgt}(\mathcal{E}_{\chi_{i}^{\lambda}})\leq
c_{i}\leq
C_{max}+\frac{1}{q}\operatorname{wgt}(\mathcal{E}_{\chi_{i}^{\lambda}})$

We can add $t_{1}\cdot H^{d-1}$ to get

(11)
$C_{min}+\frac{1}{q}\operatorname{wgt}(\mathcal{E}_{\chi_{i}^{\lambda}})+t_{1}\cdot
H^{d-1}\leq
A_{d}\cdot\widehat{\mu}_{d-1}(\mathcal{E}_{\chi_{i}^{\lambda}})\leq
C_{max}+\frac{1}{q}\operatorname{wgt}(\mathcal{E}_{\chi_{i}^{\lambda}})+t_{1}\cdot
H^{d-1}$

Let $\operatorname{wgt}(\mathcal{E}_{\chi_{i_{min}}^{\lambda}})$ be the
smallest weight among all of the $\mathcal{E}_{\chi_{i}^{\lambda}}$. Since
$q>0$, we obtain

$\displaystyle\frac{1}{A_{d}}C_{min}+\frac{1}{qA_{d}}\operatorname{wgt}(\mathcal{E}_{\chi_{i_{min}}^{\lambda}})+\frac{1}{A_{d}}t_{1}\cdot
H^{d-1}\leq\frac{1}{A_{d}}C_{min}+\frac{1}{qA_{d}}\operatorname{wgt}(\mathcal{E}_{\chi_{i}^{\lambda}})+\frac{1}{A_{d}}t_{1}\cdot
H^{d-1}\leq\widehat{\mu}_{d-1}(\mathcal{E}_{\chi_{i}^{\lambda}})$

for each $i$. So it suffices to give a uniform lower bound for
$\frac{1}{qA_{d}}\operatorname{wgt}(\mathcal{E}_{\chi_{i_{min}}^{\lambda}})$.
Replace $\mathcal{E}_{\chi_{i}^{\lambda}}$ by
$\mathcal{E}_{\chi_{i_{min}}^{\lambda}}$ in the upper inequality of (11) to
get

$\widehat{\mu}_{d-1}(\mathcal{E}_{\chi_{i_{min}}^{\lambda}})-\frac{1}{A_{d}}C_{max}-\frac{1}{A_{d}}t_{1}\cdot
H^{d-1}\leq\frac{1}{qA_{d}}\operatorname{wgt}(\mathcal{E}_{\chi_{i_{min}}^{\lambda}})\;.$

Therefore we just need to give a uniform lower bound for
$\widehat{\mu}_{d-1}(\mathcal{E}_{\chi_{i_{min}}^{\lambda}})$.

Set $m_{min}:=\langle\delta^{*},\chi_{i_{min}}^{\lambda}\rangle$. Note that
$\mathcal{E}_{\chi_{i_{min}}^{\lambda}}$ is a direct summand of the reflexive
hull of the graded piece
$\mathcal{F}^{\delta^{*}}_{m_{{min}}}/\mathcal{F}_{m_{min}+1}^{\delta^{*}}$ of
the filtration $(\mathcal{F}^{\delta^{*}}_{m})_{m\in\mathbb{Z}}$. We can think
of $\mathcal{E}_{\chi_{i_{min}}^{\lambda}}$ as a quotient of the reflexive
hull
$(\mathcal{F}^{\delta^{*}}_{m_{{min}}}/\mathcal{F}_{m_{min}+1}^{\delta^{*}})^{\vee\vee}$,
and there is a corresponding quotient sheaf
$\mathcal{F}^{\delta^{*}}_{m_{{min}}}/\mathcal{F}_{m_{min}+1}^{\delta^{*}}\twoheadrightarrow\mathcal{Q}$
with the same slope as $\mathcal{E}_{\chi_{i_{min}}^{\lambda}}$.

Since the weight of $\mathcal{E}_{\chi_{i_{min}}^{\lambda}}$ is minimal, it
follows that
$\mathcal{F}^{\delta^{*}}_{m_{{min}}}/\mathcal{F}_{m_{min}+1}^{\delta^{*}}$ is
the nontrivial graded piece with the smallest weight. This implies that
$\mathcal{F}^{\delta^{*}}_{m_{{min}}}/\mathcal{F}_{m_{min}+1}^{\delta^{*}}$ is
a quotient of $\mathcal{F}|_{X_{s}}$. Hence $\mathcal{Q}$ is also a quotient
of $\mathcal{F}|_{X_{s}}$. It follows that

$\widehat{\mu}_{d-1}(\mathcal{E}_{\chi_{i_{min}}^{\lambda}})=\widehat{\mu}_{d-1}(\mathcal{Q})\geq\widehat{\mu}_{d-1}(\mathcal{F}|_{X_{s}})_{min}\;,$

where $\widehat{\mu}_{d-1}(\mathcal{F}|_{X_{s}})_{min}$ is the smallest slope
appearing among the associated graded pieces of the Gieseker-Harder-Narasimhan
filtration of $\mathcal{F}|_{X_{s}}$ (c.f. [HL97, Def. 1.3.2]). The quantity
$\widehat{\mu}_{d-1}(\mathcal{F}|_{X_{s}})_{min}$ is uniformly bounded as we
vary $t\in T$, by constructibility of the locus of each fixed Harder-
Narasimhan type (see e.g. [Nit11]). This concludes the proof of cases (C1) and
(C2).

If we are in case (C3) we have $\vec{z}=0$ (see (9)). Then:

$(M_{\lambda}^{T}M_{\lambda})^{-1}M_{\lambda}^{T}\vec{a}^{T}=c(M_{\lambda}^{T}M_{\lambda})^{-1}M_{\lambda}^{T}\vec{1}^{T}$

Using $\vec{a}^{T}=M_{\lambda}\vec{\psi}$ we get
$\vec{a}^{T}=cM_{\lambda}(M_{\lambda}^{T}M_{\lambda})^{-1}M_{\lambda}^{T}\vec{1}^{T}$,
which is the second term in (10) and is bounded by our discussion of the cases
before. ∎

###  4.5. $\Theta$-stratification, semistability and leading term filtrations

After all of our work in this section, we can now construct a
$\Theta$-stratification on $\text{Bun}_{\rho}(X)$.

######  Theorem 4.27.

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

###### Proof.

[HLHJ21, Thm. 2.26] (or [HL14, Thm. B]) shows that it suffices to check that
$\nu$ satisfies HN boundedness and is strictly $\Theta$-monotone on the stack
$\operatorname{Bun}_{\rho}(X)$. This follows by Propositions 4.26 and 4.16. ∎

The following is a translation of the notion of $\Theta$-semistability from
[HL14, §2.1] for our numerical invariant $\nu$.

######  Definition 4.28.

Let $K\supset k$ be an algebraically closed field extension. Let
$(\mathcal{F},\sigma)$ be a $\rho$-sheaf on $X_{K}$. We say that
$(\mathcal{F},\sigma)$ is Gieseker semistable if for all weighted parabolic
reductions $(\lambda,\mathcal{G}_{\lambda})$ with associated underlying
filtrations $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ (as in Definition 3.15) we
have

$\sum_{m\in\mathbb{Z}}m\cdot(\overline{p}_{\mathcal{F}_{m}/\mathcal{F}_{m+1}}-\overline{p}_{\mathcal{F}})\cdot\operatorname{rk}(\mathcal{F}_{m}/\mathcal{F}_{m+1})\leq
0$

In general, if $K\supset k$ is any field extension, then we say that a
$\rho$-sheaf $(\mathcal{F},\sigma)$ on $X_{K}$ is Gieseker semistable if the
pullback $(\mathcal{F}|_{X_{\overline{K}}},\sigma|_{X_{\overline{K}}})$ is
semistable for any algebraic closure $\overline{K}\supset K$. A $\rho$-sheaf
$(\mathcal{F},\sigma)$ on $X_{K}$ is called unstable if it is not Gieseker
semistable.

The summation by parts argument in [HLHJ21, Prop. 4.1] shows that the
expression in Definition 4.28 can be rewritten as

(12)
$\sum_{m\in\mathbb{Z}}m\cdot(\overline{p}_{\mathcal{F}_{m}/\mathcal{F}_{m+1}}-\overline{p}_{\mathcal{F}})\cdot\operatorname{rk}(\mathcal{F}_{m}/\mathcal{F}_{m+1})=\sum_{m\in\mathbb{Z}}(\overline{p}_{\mathcal{F}_{m}}-\overline{p}_{\mathcal{F}})\cdot\operatorname{rk}(\mathcal{F}_{m})$

By Theorem 4.27, the polynomial numerical invariant $\nu$ defines a
$\Theta$-stratification on $\operatorname{Bun}_{\rho}(X)$. In particular the
locus of semistable $\rho$-sheaves is an open substack, which we will denote
by $\operatorname{Bun}_{\rho}(X)^{ss}$. Note that
$\operatorname{Bun}_{\rho}(X)^{ss}$ is a disjoint union of open and closed
substacks where the Hilbert polynomial of the underlying torsion-free sheaf is
constant. For every rational polynomial $P\in\mathbb{Q}[n]$, we will denote by
$\operatorname{Bun}_{\rho}(X)^{ss,P}$ the open and closed substack of
$\operatorname{Bun}_{\rho}(X)^{ss}$ parametrizing Gieseker semistable
$\rho$-sheaves $(\mathcal{F},\sigma)$ such that the Hilbert polynomial of
$\mathcal{F}$ is $P$.

We will next explain what it means to have a $\Theta$-stratification on
$\operatorname{Bun}_{\rho}(X)$ in terms of canonical filtrations.

######  Definition 4.29.

Suppose that $K$ is algebraically closed. Let $(\mathcal{F},\sigma)$ be an
unstable $\rho$-sheaf on $X_{K}$. A weighted parabolic reduction
$f=(\lambda,\mathcal{G}_{\lambda})$ is called canonical if the polynomial
invariant $\nu(f)$ is maximal among all weighted parabolic reductions of
$(\mathcal{F},\sigma)$. We say that the associated underlying filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ to $f=(\lambda,\mathcal{G}_{\lambda})$ is
a canonical filtration of $(\mathcal{F},\sigma)$.

We already know that a weighted parabolic reduction
$(\lambda,\mathcal{G}_{\lambda})$ determines a unique underlying filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$. By Proposition 3.9, the converse is
true. Namely, the underlying filtration $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$
uniquely determines the weighted parabolic reduction
$(\lambda,\mathcal{G}_{\lambda})$ over an open subset $U$ where
$\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m}/\mathcal{F}_{m+1}$ is locally-free.
Hence the data of a canonical weighted parabolic reduction is equivalent to
specifying its underlying canonical filtration.

######  Definition 4.30.

Let $K\supset k$ be an arbitrary field extension. Let $(\mathcal{F},\sigma)$
be an unstable $\rho$-sheaf on $X_{K}$. A $\mathbb{Z}$-weighted filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ of $\mathcal{F}$ is called a canonical
filtration if for any algebraic closure $\overline{K}\supset K$ we have that
$(\mathcal{F}_{m}|_{X_{\overline{K}}})_{m\in\mathbb{Z}}$ is a canonical
filtration of $(\mathcal{F}|_{X_{\overline{K}}},\sigma|_{X_{\overline{K}}})$

The fact that $\nu$ defines a $\Theta$-stratification on
$\operatorname{Bun}_{\rho}(X)$ implies that any field valued point of
$\operatorname{Bun}_{\rho}(X)$ admits a $\nu$-maximizing $\Theta$-filtration
that is unique up to scaling the weights. We can use our concrete descriptions
of $\Theta$-filtrations for $\operatorname{Bun}_{\rho}(X)$ to translate this
into the following.

######  Proposition 4.31.

Let $K\supset k$ be an arbitrary field extension. Let $(\mathcal{F},\sigma)$
be a $\rho$-sheaf on $X_{K}$. Then, $(\mathcal{F},\sigma)$ admits a canonical
filtration $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ which is uniquely determined
up to scaling all of the indexes by a constant rational number.

Furthermore, if the group $G_{K}$ is split, then there exists a (uniquely
determined) canonical weighted parabolic reduction
$(\lambda,\mathcal{G}_{\lambda})$ defined over $K$ such that its associated
underlying filtration is $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$. ∎

######  Definition 4.32.

Let $(\mathcal{F},\sigma)$ be field valued point of
$\operatorname{Bun}_{\rho}(X)$. Suppose that $(\mathcal{F},\sigma)$ is
unstable. We call the canonical filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ the leading term HN filtration of
$(\mathcal{F},\sigma)$. It is uniquely determined up to scaling the weights,
by Proposition 4.31.

On the other hand, if $(\mathcal{F},\sigma)$ is Gieseker semistable, then we
define the leading term HN filtration to be
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ where

$\mathcal{F}_{m}=\begin{cases}\mathcal{F}\text{ \; \; if m $\leq$ 0}\;\\\
0\;\;\;\;\;\;\;\text{otherwise}\;\end{cases}$

######  Remark 4.33.

Suppose that $f$ is the leading term HN filtration of $(\mathcal{F},\sigma)$.
If the polynomial $\nu(f)$ has degree $i$, then we say that $i$ is the leading
degree of $(\mathcal{F},\sigma)$. One should think of the leading term HN
filtration as an analogue of the slope filtration. Here the coefficient of
degree $i$ in the reduced Hilbert polynomial plays the role of the slope. This
can be made precise in the case of torsion-free sheaves, see Subsection 9.1.

The existence of a $\Theta$-stratification also implies that the notion of
leading term HN filtration behaves well in families. In order to explain this
precisely, we need to first define what is a filtration of a family of
$\rho$-sheaves.

Let $S$ be a $k$-scheme and let $(\mathcal{F},\sigma)$ be a $\rho$-sheaf on
$X_{S}$. A filtration of $\mathcal{F}$ is a $\mathbb{Z}$-weighted filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ of $\mathcal{F}$ satisfying the following
properties.

  1. (1)

$\mathcal{F}_{m+1}\subset\mathcal{F}_{m}$.

  2. (2)

$\mathcal{F}_{m}=0$ for $m\gg 0$ and $\mathcal{F}_{m}=\mathcal{F}$ for $m\ll
0$.

  3. (3)

$\mathcal{F}_{m}/\mathcal{F}_{m+1}$ is a $S$-flat family of torsion-free
sheaves.

An argument similar to Proposition 3.8 shows that the data of such filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ is equivalent to the data of a morphism
of stacks $\Theta_{S}\rightarrow\operatorname{Coh}_{r}^{tf}(X)$ such that the
restriction to $1_{S}$ recovers $\mathcal{F}$. This amounts to a torsion-free
sheaf $\widetilde{\mathcal{F}}$ on $X_{\Theta_{S}}$. Let $U$ be a big open
subset of $X_{S}$ where
$\bigoplus_{m\in\mathbb{Z}}\mathcal{F}_{m}/\mathcal{F}_{m+1}$ is locally-free.
Then, the restriction $\widetilde{\mathcal{F}}|_{U\times_{S}\Theta_{S}}$ is a
$\operatorname{GL}(V)$-bundle on $U\times_{S}\Theta_{S}$ and $\sigma$ induces
a $G$-reduction of structure group of the restriction
$\widetilde{\mathcal{F}}|_{U\times_{S}1_{S}}$.

######  Definition 4.34.

We say that $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ as above is a filtration of
$(\mathcal{F},\sigma)$ if the $G$-reduction of structure group (uniquely)
extends to $U\times_{S}\Theta_{S}$.

The uniqueness of the extension shows that this is a property, not a
structure. An argument similar to Proposition 3.12 implies that such a
filtration of $(\mathcal{F},\sigma)$ is equivalent to a morphism
$\Theta_{S}\rightarrow\operatorname{Bun}_{\rho}(X)$ such that the restriction
to $1_{S}$ recovers $(\mathcal{F},\sigma)$.

######  Definition 4.35.

Let $S$ be a $k$-scheme and let $(\mathcal{F},\sigma)$ be a $\rho$-sheaf on
$X_{S}$. We say that a filtration $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ of
$(\mathcal{F},\sigma)$ is a relative leading term HN filtration if for all
points $s\in S$ the restriction $(\mathcal{F}_{m}|_{X_{s}})_{m\in\mathbb{Z}}$
is the leading term HN filtration of $(\mathcal{F}|_{X_{s}},\sigma|_{X_{s}})$.

Suppose that $S$ is a connected scheme of finite type over a field extension
$K\supset k$ and assume that $G_{K}$ is split. In this case, every relative
filtration comes from a relative weighted parabolic reduction. Indeed, let
$(\mathcal{F},\sigma)$ be a $\rho$-sheaf on $X_{S}$, and let
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ be a relative filtration of
$(\mathcal{F},\sigma)$. Let $\widetilde{\mathcal{G}}$ denote the corresponding
$G$-bundle on $\Theta_{K}\times_{K}U$ for some big open subset $U\subset
X_{S}$. We denote by $\mathcal{G}$ the $G$-bundle on $U$ corresponding to
$(\mathcal{F},\sigma)$ (i.e. the restriction
$\widetilde{\mathcal{G}}|_{1\times S}$). Since $X$ is geometrically connected
and smooth, the fibers of $X_{S}\rightarrow S$ are integral, hence the fibers
of $U\rightarrow S$ are also integral. Since $S$ is connected, we conclude
that $U$ is a connected scheme of finite type over $K$. Proposition 3.14 shows
that $\widetilde{\mathcal{G}}$ arises from applying the Rees construction to a
weighted parabolic reduction $(\lambda,\mathcal{G}_{\lambda})$ of the
$G$-bundle $\mathcal{G}$ on $U$. We call $(\lambda,\mathcal{G}_{\lambda})$ a
relative weighted parabolic reduction of $(\mathcal{F},\sigma)$ and note that
the filtration $(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ is uniquely determined by
$(\lambda,\mathcal{G}_{\lambda})$. Then we say that the filtration
$(\mathcal{F}_{m})_{m\in\mathbb{Z}}$ comes from the relative weighted
parabolic reduction $(\lambda,\mathcal{G}_{\lambda})$.

Let $S$ be a $k$-scheme and $(\mathcal{F},\sigma)$ be a $\rho$-sheaf on
$X_{S}$. Let $H\rightarrow S$ be a locally closed stratification of $S$. This
means that $H$ is the disjoint union of locally closed subschemes of $S$ and
the morphism $H\rightarrow S$ is surjective.

######  Definition 4.36.

We say that the $S$-scheme $H$ is universal for leading term HN filtrations of
$(\mathcal{F},\sigma)$ if it represents the functor that classifies pairs
$\left(T,[(\mathcal{F}_{m})_{m\in\mathbb{Z}}]\right)$, where $T$ is a
$S$-scheme and $[(\mathcal{F}_{m})_{m\in\mathbb{Z}}]$ is an equivalence class
of relative leading term HN filtrations of
$(\mathcal{F}|_{X_{T}},\sigma|_{X_{T}})$ up to rational scaling the weights.

The fact that the numerical invariant $\nu$ induces a $\Theta$-stratification
implies the following property of the leading term HN filtration in families,
called a “schematic stratification” in [Nit11, GN14, GN18].

######  Proposition 4.37.

Relative leading term HN filtrations induce a stratification of
$\operatorname{Bun}_{\rho}(X)$ by locally closed substacks. Equivalently, for
any $k$-scheme $S$ and every $\rho$-sheaf $(\mathcal{F},\sigma)$ on $X_{S}$
there is a unique locally closed stratification $H\to S$ that is universal for
leading term HN filtrations of $(\mathcal{F},\sigma)$.

If, in addition, $S$ is locally of finite type over a field extension
$K\supset k$ such that $G_{K}$ is split, then the restriction of the universal
relative leading term HN filtration444 More precisely, we choose a
representative up to scaling.  to each connected component of the
stratification $H$ comes from a relative weighted parabolic reduction. ∎

##  5\. Completeness of the stack $\operatorname{Bun}_{\rho}(X)$

In this section we prove that the stack $\operatorname{Bun}_{\rho}(X)$
satisfies the existence part of the valuative criterion for properness [Sta21,
Tag 0CLK] in the case of complete discrete valuation rings (see Theorem 5.7).

The ideas that we use in this section (purely ramified extensions, Bruhat-Tits
buildings) already appear in [BS02, Section 6] (see [BP03] for positive
characteristic). Besides, V. Balaji has also considered extensions of families
of principal bundles in higher dimensions, in the context of Donaldson-
Uhlenbeck compactification [Bal07]. The case of positive characteristic for
curves has also been studied by J. Heinloth [Hei08b, Hei10]. For torsion free
sheaves, the valuative criteria was proved by S. Langton [Lan75].

Let $R$ be a complete discrete valuation $k$-algebra. We set
$K=\operatorname{Frac}(R)$, and choose a uniformizer $\pi$. By the Cohen
structure theorem, $R$ is isomorphic to the power series ring
$\kappa[[{\pi}]]$, where $\kappa$ is the residue field. Consider the base
change $X_{R}$. Let $\eta_{\kappa}$ denote the generic point of the special
fiber $X_{\kappa}$. Similarly, let $\eta_{K}$ be the generic point of the
generic fiber $X_{K}$. Note that the local $\mathcal{O}_{X_{R},\eta_{\kappa}}$
is a discrete valuation ring with generic point $\eta_{K}$ and special point
$\eta_{\kappa}$.

The following is a local characterization of when a $\rho$-sheaf extends.

######  Lemma 5.1.

Let $(\mathcal{F},\sigma)$ be a $\rho$-sheaf over $X_{K}$. Let $\mathcal{P}$
be the corresponding $G$-bundle over the generic fiber $\eta_{K}$. The
following statements are equivalent:

  1. (1)

There is a $\rho$-sheaf on $X_{R}$ that restricts to $(\mathcal{F},\sigma)$
over $X_{K}$.

  2. (2)

There is a $G$-bundle on
$\operatorname{Spec}(\mathcal{O}_{X_{R},\eta_{\kappa}})$ that restricts to
$\mathcal{P}$ over $\eta_{K}$.

###### Proof.

$(1)\Longrightarrow(2)$. Suppose that $(\widetilde{\mathcal{F}},\zeta)$ is a
$\rho$-sheaf on $X_{R}$ that extends $(\mathcal{F},\sigma)$. Let $U$ be a big
open subset of $X_{R}$ where $\widetilde{\mathcal{F}}$ is locally-free. Let
$\widetilde{\mathcal{P}}$ be the $G$-bundle on $U$ corresponding to the
restriction of $(\widetilde{\mathcal{F}},\zeta)$. Then
$\widetilde{\mathcal{P}}|_{\operatorname{Spec}(\mathcal{O}_{X_{R},\eta_{\kappa}})}$
is a $G$-bundle on $\operatorname{Spec}(\mathcal{O}_{X_{R},\eta_{\kappa}})$
that extends $\mathcal{P}$.

$(2)\Longrightarrow(1)$. Suppose that there is a $G$-bundle
$\widetilde{\mathcal{P}}$ on
$\operatorname{Spec}(\mathcal{O}_{X_{R},\eta_{\kappa}})$ that extends
$\mathcal{P}$. Let $\mathcal{A}$ be the corresponding
$\operatorname{GL}(V)$-bundle on
$\operatorname{Spec}(\mathcal{O}_{X_{R},\eta_{\kappa}})$ obtained via
extension of structure group using $\rho$. $\mathcal{A}$ is a vector bundle of
rank $r$ on $\operatorname{Spec}(\mathcal{O}_{X_{R},\eta_{\kappa}})$. By
construction we have a canonical identification
$\mathcal{A}|_{\eta_{K}}\xrightarrow{\sim}\mathcal{F}|_{\eta_{K}}$. Now
[Lan75, Prop. 6] implies that there is a (unique) family of torsion-free
sheaves $\widetilde{\mathcal{F}}$ on $X_{R}$ along with isomorphisms
$\widetilde{\mathcal{F}}|_{X_{K}}\xrightarrow{\sim}\mathcal{F}$ and
$\widetilde{\mathcal{F}}|_{\operatorname{Spec}(\mathcal{O}_{X_{R},\eta_{\kappa}})}\xrightarrow{\sim}\mathcal{A}$
that are compatible with the identification
$\mathcal{A}|_{\eta_{K}}\xrightarrow{\sim}\mathcal{F}|_{\eta_{K}}$.

We have a section
$s:\operatorname{Spec}(\mathcal{O}_{X_{R},\eta_{\kappa}})\rightarrow\operatorname{Red}_{G}(\widetilde{\mathcal{F}})|_{\operatorname{Spec}(\mathcal{O}_{X_{R},\eta_{\kappa}})}$
corresponding to the $G$-reduction $\widetilde{P}$. It identifies over the
point $\eta_{K}$ with the section
$\sigma:X_{K}\rightarrow\operatorname{Red}_{G}(\mathcal{F})$. Since
$\operatorname{Red}_{G}(\widetilde{\mathcal{F}})$ is of finite type over
$X_{R}$, we can spread the compatible pair $s$ and $\sigma$ to a section

$\widetilde{\sigma}:X_{R}\setminus
Z\rightarrow\operatorname{Red}_{G}(\widetilde{\mathcal{F}})$

defined away from a proper closed subscheme $Z\subsetneq X_{\kappa}$ of the
special fiber $X_{\kappa}$. Note that every point $z\in Z$ has codimension at
least 2 in $X_{R}$. Since $X_{R}$ is smooth over the regular scheme
$\operatorname{Spec}(R)$, it is itself regular [Sta21, Tag 07NF]. It follows
that for every $z\in Z$, the local ring $\mathcal{O}_{X_{R},z}$ has depth at
least 2. By the proof of Lemma 2.2 (c), the section $\widetilde{\sigma}$ on
$X_{R}\setminus Z$ uniquely extends to a section
$\zeta:X_{R}\rightarrow\operatorname{Red}_{G}(\widetilde{\mathcal{F}})$
defined over the whole $X_{R}$. The $\rho$-sheaf
$(\widetilde{\mathcal{F}},\zeta)$ restricts to $(\mathcal{F},\sigma)$, as
desired. ∎

We would like to show that the extension result in Lemma 5.1 (2) is true after
maybe passing to a finite extension $R^{\prime}\supset R$ of the discrete
valuation ring $R$. In order to do so, we prove the following general result
which could be of independent interest. 555V. Balaji pointed out to us that
Proposition 5.2 could also be deduced from [RR84, Proposition 3.12].

######  Proposition 5.2.

Let $\kappa$ be a perfect field. Let $G$ be a quasi-split connected reductive
group over $\kappa$. Suppose that the two following conditions are satisfied:

  1. (1)

The maximal central torus $Z\subset G$ is split over $\kappa$.

  2. (2)

Let $H$ denote the kernel of of the simply-connected cover
$\widetilde{G/Z}\rightarrow G/Z$ of the semisimple group $G/Z$. The
characteristic of $\kappa$ does not divide the order of $H$.

Set $\kappa[[{\pi}]]$ to be the formal power series ring. Let $\mathcal{P}$ be
a $G$-bundle on the fraction field $\operatorname{Spec}(\kappa(({\pi})))$.
Then, there exists some $m\geq 1$ such that the pullback of $\mathcal{P}$ to
$\kappa(({\pi^{\frac{1}{m}}}))$ extends to a $G$ bundle on
$\operatorname{Spec}(\kappa[[{\pi^{\frac{1}{m}}}]])$.

###### Proof.

Define $K\vcentcolon=\kappa(({\pi}))$ and $\mathcal{O}_{K}=\kappa[[{\pi}]]$.
We fix the choice of a separable closure $K_{s}$ of $K$. Set $K_{unr}$ to be
the maximal unramified extension of $K$ inside $K_{s}$. We will denote by
$\mathcal{O}_{K_{unr}}$ the ring of integers of $K_{unr}$. Let
$\Gamma_{K}:=\operatorname{Gal}(K_{s}/K)$ be the absolute Galois group of $K$.
We denote by $\Gamma_{unr}\vcentcolon=\operatorname{Gal}(K_{unr}/K)$ the
unramified Galois group.

We denote by $H^{1}(K,G)$ (resp. $H^{1}(\mathcal{O}_{K},G)$ ) the pointed set
of $G$-bundles on $\operatorname{Spec}(K)$ (resp.
$\operatorname{Spec}(\mathcal{O}_{K})$). We have a natural restriction
morphism $H^{1}(\mathcal{O}_{K},G)\rightarrow H^{1}(K,G)$. We are given an
element $\mathcal{P}\in H^{1}(K,G)$. Our goal is to show that, after maybe
replacing $K$ with an extension $K(\pi^{\frac{1}{m}})$, there exists some
$\widetilde{\mathcal{P}}\in H^{1}(\mathcal{O}_{K},G)$ that restricts to
$\mathcal{P}$.

Since $G$ is smooth, the base-change $\mathcal{P}_{K_{s}}$ is trivializable
over $\operatorname{Spec}(K_{s})$. The descent data for $\mathcal{P}$ yield a
class in the nonabelian Galois cohomology set $H^{1}(\Gamma_{K},G(K_{s}))$.
Since $G$ is connected and $\kappa$ is perfect, [BT87, 3.14 (3)] shows that
$H^{1}(\Gamma_{K},G(K_{s}))=H^{1}(\Gamma_{unr},K_{unr})$. So the torsor
$\mathcal{P}$ comes from an unramified cohomology class $\overline{c}\in
H^{1}(\Gamma_{unr},K_{unr})$. Notice that there is a natural action of
$\Gamma_{unr}$ on $G(\mathcal{O}_{K_{unr}})$. There is a map
$H^{1}(\Gamma_{unr},G(\mathcal{O}_{K_{unr}}))\rightarrow
H^{1}(\Gamma_{unr},G(K_{unr}))$ induced by the inclusion of
$\Gamma_{unr}$-groups $G(\mathcal{O}_{K_{unr}})\hookrightarrow G(K_{unr})$.
Our goal is to show that, after maybe replacing $K$ with
$K(\pi^{\frac{1}{m}})$, we have that $\overline{c}$ comes from a class in
$H^{1}(\Gamma_{unr},G(\mathcal{O}_{K_{unr}}))$.

Let us first suppose that $G$ is semisimple. There is a finite unramified
Galois extension $L\supset K$ contained in $K_{unr}$ such that
$\mathcal{P}_{L}$ is trivializable. Let $\mathcal{O}_{L}$ denote the ring of
integers of $L$. Set $\Gamma_{L/K}:=\operatorname{Gal}(L/K)$. This is a finite
quotient of $\Gamma_{unr}$. The descent data for $\mathcal{P}$ yield a class
$\overline{c}$ in the nonabelian Galois cohomology set
$H^{1}(\Gamma_{L/K},G(L))$.

Since $G_{K}$ is quasi-split, it follows that $G_{K}$ is residually quasi-
split [BT87, 2.2]. Therefore, we can use the description of unramified Galois
cohomology given in [BT87], we just need to be a bit careful if $G$ is not
simply-connected. We will denote by $G^{00}$ the subgroup of $G(K_{unr})$
generated by the $K_{unr}$-points of all parahoric subgroups of $G_{K_{unr}}$.
By [BT87, 1.8], $G^{00}$ admits the following description. Let $T$ denote a
maximal torus of $G$ over $\kappa$. We can think of $T_{K}$ as the centralizer
of a maximal split torus of $G_{K}$, since $G_{K}$ is quasi-split. Let
$j:\widetilde{G}\rightarrow G$ be a simply connected cover of $G$. Then, we
have $G^{00}=T(\mathcal{O}_{K_{unr}})\cdot j(\widetilde{G}(K_{unr}))$.

Choose a cocycle $c\in Z^{1}(\Gamma_{L/K},G(L))$ representing $\overline{c}$.
We will show that, after maybe replacing $K$ with some ramified extension
$K(\pi^{\frac{1}{m}})$, the cocycle $c$ takes values in the subgroup $G^{00}$.
Let $H$ denote the finite kernel of the isogeny $j:\widetilde{G}\rightarrow
G$. Note that $H$ is a finite product of groups $\mu_{j}$ of $j^{th}$ roots of
unity. By assumption (2), the characteristic of $\kappa$ does not divide any
of the orders of the $\mu_{j}$ appearing in $H$. After maybe enlarging the
unramified extension $L$, we can assume that all of these roots of unity are
contained in $L$. So $H_{L}$ is a finite constant étale group scheme. Since
$H$ is smooth, the following sequence of groups is exact.

$1\rightarrow H(K_{s})\rightarrow\widetilde{G}(K_{s})\rightarrow
G(K_{s})\rightarrow 1$

We can take $\operatorname{Gal}(K_{s}/L)$ invariants to get the following
exact sequence of pointed sets

$1\rightarrow H(L)\rightarrow\widetilde{G}(L)\rightarrow G(L)\rightarrow
H^{1}\left(\operatorname{Gal}(K_{s}/L),H(K_{s})\right)$

Hence the obstructions to lifting to $\widetilde{G}(L)$ the finitely many
elements in the image of the cocycle $c\in Z^{1}(\Gamma_{L/K},G(L))$ consist
of finitely many cohomology classes in
$H^{1}\left(\operatorname{Gal}(K_{s}/L),H(K_{s})\right)$. By Kummer theory,
these are killed after taking some cyclic extensions of $L$. After further
enlarging the unramified extension $L$, we can assume that these cyclic
extensions are totally ramified. Such cyclic extension must be of the form
$L(\pi_{L}^{\frac{1}{m}})$ for some uniformizer $\pi_{L}$ of $L$. Such
uniformizer must be of the form $\pi_{L}=u\pi$, where
$u\in\mathcal{O}_{L}^{\times}$. After replacing $L$ with $L(u^{\frac{1}{m}})$,
we can assume that the obstructions in
$H^{1}(\operatorname{Gal}(K_{s}/L),H(K_{s}))$ vanish on the extension
$L(\pi^{\frac{1}{m}})$ for some $m\geq 1$. So after replacing $K$ with some
sufficiently big ramified extension $K(\pi^{\frac{1}{m}})$, we can assume that
$c$ takes values in the group $j(\widetilde{G}(L))\subset G^{00}$. Hence
$\overline{c}$ is a class in the Galois cohomology set
$H^{1}\left(\Gamma_{L/K},G^{00}\cap G(L)\right)$. Since $G_{K}$ is residually-
quasisplit, the result [BT87, 3.15] and the construction of [BT87, 3.4 Lem. 2]
show that $\overline{c}$ is represented by a cocycle $z\in
Z^{1}(\Gamma_{L/K},P(\mathcal{O}_{L}))$ taking values in the group of
$\mathcal{O}_{L}$-points of a parahoric $P$ of $G_{K}$ defined over $K$. We
can without loss of generality take $P$ to be in the apartment corresponding
to a maximal split torus $S$ inside $T_{K}$, because $G(K)$ acts transitively
on apartments. The argument in [Lar95, Lem. 2.4] (see also [Ser96, Proposition
8] and [Gil94, Lemma I.1.3.2]) shows that there exists some $m\geq 1$ and some
$K$-rational cocharacter $\lambda:\mathbb{G}_{m}\to S$ such that
$\lambda(\pi^{\frac{1}{m}})\in G(K[\pi^{\frac{1}{m}}])$ conjugates the base-
changed parahoric $P_{\mathcal{O}_{K(\pi^{\frac{1}{m}})}}$ into the
hyperspecial parahoric $G_{\mathcal{O}_{K(\pi^{\frac{1}{m}})}}$. After
replacing $K$ with $K(\pi^{\frac{1}{m}})$, we can use the element
$\lambda(\pi^{\frac{1}{m}})$ to conjugate the cocycle $z$ into a cocycle
$z^{\prime}\in Z^{1}(\Gamma_{L/K},G(\mathcal{O}_{L}))$. This shows that
$\overline{c}$ comes from a cohomology class in
$H^{1}(\Gamma_{L/K},G(\mathcal{O}_{L}))$, and thus concludes the proof in the
case when $G$ is semisimple.

We now deal with the general case. Let $G$ be an arbitrary quasi-split
connected reductive group over $\kappa$. Let $Z$ denote the maximal central
torus of $G$. We have a short exact sequence of algebraic groups

$1\rightarrow Z\rightarrow G\rightarrow\overline{G}\rightarrow 1$

Here $\overline{G}=G/Z$ is a quasi-split semisimple group over $\kappa$
satisfying condition (2) of the proposition. By assumption $Z$ is split, and
so it is isomorphic to a finite product $(\mathbb{G}_{m})^{h}$ of copies of
$\mathbb{G}_{m}$. We have an exact sequence

$1\rightarrow Z(K_{unr})\rightarrow
G(K_{unr})\rightarrow\overline{G}(K_{unr})\rightarrow
H^{1}(\Gamma_{K_{s}/K_{unr}},Z(K_{s}))$

Hilbert’s Theorem 90 implies that $H^{1}(\Gamma_{K_{s}/K_{unr}},Z(K_{s}))=0$,
and so we have a short exact sequence of $\Gamma_{unr}$-groups

$1\rightarrow Z(K_{unr})\rightarrow
G(K_{unr})\rightarrow\overline{G}(K_{unr})\rightarrow 1$

The same thing holds for $\mathcal{O}_{K_{unr}}$-points. After taking
$\Gamma_{unr}$-invariants and replacing $Z$ with $(\mathbb{G}_{m})^{h}$, we
get a diagram of pointed sets

${H^{1}(\Gamma_{unr},\mathcal{O}_{K_{unr}}^{\times}))^{\oplus
h}}$${H^{1}(\Gamma_{unr},G(\mathcal{O}_{K_{unr}}))}$${H^{1}(\Gamma_{unr},\overline{G}(\mathcal{O}_{K_{unr}}))}$${H^{2}(\Gamma_{unr},\mathcal{O}_{K_{unr}}^{\times})^{\oplus
h}}$${H^{1}(\Gamma_{unr},K_{unr}^{\times})^{\oplus
h}}$${H^{1}(\Gamma_{unr},G(K_{unr}))}$${H^{1}(\Gamma_{unr},\overline{G}(K_{unr}))}$${H^{2}(\Gamma_{unr},K_{unr}^{\times}))^{\oplus
h}}$

After an application of Hilbert’s Theorem 90 to the first column, we get

${0}$${H^{1}(\Gamma_{unr},G(\mathcal{O}_{K_{unr}}))}$${H^{1}(\Gamma_{unr},\overline{G}(\mathcal{O}_{K_{unr}}))}$${H^{2}(\Gamma_{unr},\mathcal{O}_{K_{unr}}^{\times})^{\oplus
h}}$${0}$${H^{1}(\Gamma_{unr},G(K_{unr}))}$${H^{1}(\Gamma_{unr},\overline{G}(K_{unr}))}$${H^{2}(\Gamma_{unr},K_{unr}^{\times})^{\oplus
h}}$$\scriptstyle{f_{1}}$$\scriptstyle{g}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{3}}$

where the two horizontal rows are exact.

Let $\overline{c}\in H^{1}(\Gamma_{unr},G(K_{unr}))$ be our unramified Galois
cohomology class coming from $\mathcal{P}$. Our goal is to lift $\overline{c}$
to a class in $H^{1}(\Gamma_{unr},G(\mathcal{O}_{K_{unr}}))$. Consider the
image $f_{2}(\overline{c})\in H^{1}(\Gamma_{unr},\overline{G}(K_{unr}))$. By
the proof of the semisimple case, after maybe replacing $K$ with an extension
$K(\pi^{\frac{1}{m}})$ we can lift $f_{2}(\overline{c})$ to a cohomology class
$\overline{d}\in H^{1}(\Gamma_{unr},\overline{G}(\mathcal{O}_{K_{unr}}))$. We
now want to lift $\overline{d}$ to a class in
$H^{1}(\Gamma_{unr},G(\mathcal{O}_{K_{unr}}))$ whose image in
$H^{1}(\Gamma_{unr},G(\mathcal{O}_{K_{unr}}))$ is the original $\overline{c}$.
For this it suffices to show that $f_{1}(\overline{d})=0$.

We have $g(f_{1}(\overline{d}))=f_{3}(f_{2}(\overline{c}))=0$. Therefore, it
suffices to show that $g$ is injective. The morphism $g$ is the direct sum of
$h$ copies of the morphism
$H^{2}(\Gamma_{unr},\mathcal{O}_{K_{unr}}^{\times})\rightarrow
H^{2}(\Gamma_{unr},K_{unr}^{\times})$ coming from the inclusion of
$\Gamma_{unr}$-groups $\mathcal{O}_{K_{unr}}^{\times}\hookrightarrow
K_{unr}^{\times}$. These groups fit into a short exact sequence

$1\rightarrow\mathcal{O}_{K_{unr}}^{\times}\rightarrow
K_{unr}^{\times}\rightarrow\mathbb{Z}\rightarrow 1$

Here $\Gamma_{unr}$ acts trivially on $\mathbb{Z}$. We get an exact sequence
of abelian groups

(13) $H^{1}(\Gamma_{unr},\mathbb{Z})\rightarrow
H^{2}(\Gamma_{unr},\mathcal{O}_{K_{unr}}^{\times})\rightarrow
H^{2}(\Gamma_{unr},K_{unr}^{\times})$

Since $\Gamma_{unr}$ acts trivially on $\mathbb{Z}$, we have that
$H^{1}(\Gamma_{unr},\mathbb{Z})$ is the group of continuous homomorphisms from
$\Gamma_{unr}$ to $\mathbb{Z}$. Since $\Gamma_{unr}$ is profinite, the image
of such a homomorphism is finite. Therefore it must be $0$, because
$\mathbb{Z}$ is torsion-free. We conclude that
$H^{1}(\Gamma_{unr},\mathbb{Z})=0$. Hence (13) shows that $g$ is injective, as
desired. ∎

######  Remark 5.3.

The argument that allows us to go from semisimple to reductive groups is
similar to the argument of G. Faltings in [Fal93, Theorem II.4].

######  Remark 5.4.

It follows from the argument that $m\geq 1$ can be chosen independently of
$\mathcal{P}$. One can give a uniform upper bound that depends only on the
root data of $G/Z$.

######  Remark 5.5.

The argument above works verbatim for a general complete discrete valuation
ring $R$ with perfect residue field $\kappa$ (i.e. mixed characteristic). In
that case we can take $G$ to be a reductive group over $R$ with connected
fibers such that the generic fiber $G_{K}$ is residually quasi-split and the
analogous conditions (1) and (2) are satisfied.

In order to construct our generic $G$-bundle extension, we will make use of
formal gluing, also known as Beauville-Laszlo gluing [BL95]. Suppose that
$S\to\widehat{S}$ is a faithfully flat morphism of rings. Let $f\in S$ be a
nonzerodivisor. We denote by $\text{Glue}_{\text{Aff}}$ the category
consisting of triples $(\widehat{A},A_{f},\varphi)$, where

  1. (1)

$\widehat{A}$ is an affine scheme over $\operatorname{Spec}(\widehat{S})$.

  2. (2)

$A_{f}$ is an affine scheme over $\operatorname{Spec}(S_{f})$.

  3. (3)

$\varphi$ is an isomorphism
$\varphi:\widehat{A}|_{\operatorname{Spec}(\widehat{S}_{f})}\xrightarrow{\sim}A_{f}|_{\operatorname{Spec}(\widehat{S}_{f})}$.

A morphism $(\widehat{A},A_{f},\varphi)\to(\widehat{B},B_{f},\psi)$ consists
of a pair of morphisms $(g_{1}:\widehat{A}\to\widehat{B},\,g_{2}:A_{f}\to
B_{f})$ such that their base-changes to $\operatorname{Spec}(\widehat{S}_{f})$
fit into a commutative diagram

${\widehat{A}|_{\operatorname{Spec}(\widehat{S}_{f})}}$${A_{f}|_{\operatorname{Spec}(\widehat{S}_{f})}}$${\widehat{B}|_{\operatorname{Spec}(\widehat{S}_{f})}}$${B_{f}|_{\operatorname{Spec}(\widehat{S}_{f})}}$$\scriptstyle{\varphi}$$\scriptstyle{g_{1}|_{\operatorname{Spec}(\widehat{S}_{f})}}$$\scriptstyle{\psi}$$\scriptstyle{g_{2}|_{\operatorname{Spec}(\widehat{S}_{f})}}$

Given an affine scheme $A$ over $\operatorname{Spec}(S)$, we can form the two
pullbacks $A_{\operatorname{Spec}(\widehat{S})}$ and
$A_{\operatorname{Spec}(S_{f})}$. Note that there is a canonical isomorphism
$\varphi_{A,can}:\left(A_{\operatorname{Spec}(\widehat{S})}\right)|_{\operatorname{Spec}(\widehat{S}_{f})}\xrightarrow{\sim}\left(A_{\operatorname{Spec}(S_{f})}\right)|_{\operatorname{Spec}(\widehat{S}_{f})}$.
This defines a functor

$\text{Aff}_{can}:\text{Aff}_{\operatorname{Spec}(S)}\to\text{Glue}_{\text{Aff}},\;\;\;A\mapsto(A_{\operatorname{Spec}(\widehat{S})},A_{\operatorname{Spec}(S_{f})},\varphi_{A,can})$

Let $G$ be an affine flat group scheme over $\operatorname{Spec}(S)$. We can
similarly define a category $\text{Glue}_{G\text{Bun}}$ by replacing the tuple
of affine schemes with $G$-bundles and requiring that $\varphi$ is a morphism
of $G$-bundles. Denote by $BG(S)$ the groupoid of $G$-bundles over
$\operatorname{Spec}(S)$. The functor $\text{Aff}_{can}$ induces another
functor

$G\text{Bun}_{can}:BG(S)\to\text{Glue}_{G\text{Bun}},\;\;\;\mathcal{P}\mapsto(\mathcal{P}|_{\operatorname{Spec}(\widehat{S})},\mathcal{P}|_{\operatorname{Spec}(S_{f})},\varphi_{\mathcal{P},can})$

The following lemma is usually known in the literature as Beauville-Laszlo
gluing.

######  Lemma 5.6.

Suppose that the faithfully flat morphism $S\to\widehat{S}$ induces an
isomorphism $S/(f\cdot S)\xrightarrow{\sim}\widehat{S}/(f\cdot\widehat{S})$.
Then both of the functors $\text{Aff}_{can}$ and $G\text{Bun}_{can}$ described
above are equivalences of categories.

###### Proof.

We shall use [Sta21, Tag 05ES]. Since the gluing of modules preserves the
tensor product structure on modules (cf. [Sta21, Tag 05EU]), the equivalence
in [Sta21, Tag 05ES] induces an equivalence of monoid objects (i.e. algebras).
By applying the $\operatorname{Spec}$ functor we deduce that
$\text{Aff}_{can}$ is an equivalence, and that it is compatible with fiber
products relative to the base. This implies that $\text{Aff}_{S}$ induces an
equivalence of categories of affine schemes equipped with a right
$G$-action666One defines the corresponding gluing category similarly as in the
case for $G$-bundles above.. If $\mathcal{P}$ is a scheme equipped with a
right $G$-action, then by descent for the flat cover
$\operatorname{Spec}(\widehat{S})\to\operatorname{Spec}(S)$ we know that
$\mathcal{P}$ is a $G$-bundle if and only if
$\mathcal{P}_{\operatorname{Spec}(\widehat{S})}$ is a $G$-bundle. This shows
that the equivalence on affine schemes equipped with a right $G$-action
restricts to an equivalence $G\text{Bun}_{can}$ on the subcategories of
$G$-bundles. ∎

Now we are ready to prove the main theorem of this section.

######  Theorem 5.7.

The structure morphism
$\operatorname{Bun}_{\rho}(X)\rightarrow\operatorname{Spec}(k)$ satisfies the
existence part of the valuative criterion for properness [Sta21, Tag 0CLK] in
the case of complete discrete valuation rings. In other words, for all
complete discrete valuation rings $R$ with fraction field $K$ and all diagrams

${\operatorname{Spec}(K)}$${\operatorname{Bun}_{\rho}(X)}$${\operatorname{Spec}(R)}$${\operatorname{Spec}(k)}$$\scriptstyle{f}$

there is a finite extension $R^{\prime}\supset R$ of complete discrete
valuation rings and a morphism
$\operatorname{Spec}(R^{\prime})\rightarrow\operatorname{Bun}_{\rho}(X)$
fitting into a $2$-commutative diagram

${\operatorname{Spec}(K^{\prime})}$${\operatorname{Bun}_{\rho}(X)}$${\operatorname{Spec}(R^{\prime})}$${\operatorname{Spec}(k)}$$\scriptstyle{f_{K^{\prime}}}$

###### Proof.

We will use the notation from Lemma 5.1. By Lemma 5.1, it suffices to find a
finite extension $R^{\prime}\supset R$ of discrete valuation rings such that
the base-change $\mathcal{P}_{K^{\prime}}$ on the generic point
$\eta_{K^{\prime}}$ of
$\operatorname{Spec}(\mathcal{O}_{X_{R^{\prime}},\eta_{\kappa^{\prime}}})$
admits an extension to a $G$-bundle $\widetilde{\mathcal{P}}$ on the whole
$\operatorname{Spec}(\mathcal{O}_{X_{R^{\prime}},\eta_{\kappa^{\prime}}})$.

We can replace $k$ with $\kappa$ in order to assume that $\kappa$ is the
ground field. By the Cohen structure theorem, we have $R=\kappa[[{\pi}]]$.
After replacing $\kappa$ with a finite extension $\kappa^{\prime}$, we can
assume that $G$ is split. Set $S=\mathcal{O}_{X_{R},\eta_{\kappa}}$. Note that
$S\supset R$ is a discrete valuation ring with uniformizer $\pi\in R$. Let
$\widehat{S}$ denote the completion
$\widehat{\mathcal{O}}_{X_{R},\eta_{\kappa}}$. Let $\widehat{\mathcal{P}}$
denote the base change of $\mathcal{P}$ to
$\operatorname{Spec}(\operatorname{Frac}(\widehat{S}))$. We have that
$\widehat{S}=\kappa_{S}[[{\pi}]]$, where $\kappa_{S}\supset\kappa$ is the
residue field of $S$ (equivalently $\kappa_{S}$ is the fraction field of the
special fiber $X_{\kappa}$.

By formal gluing (Lemma 5.6 applied to $S\to\widehat{S}$ and $f=\pi$),
extending $\mathcal{P}$ to $\operatorname{Spec}(S)$ is equivalent to extending
$\widehat{\mathcal{P}}$ to $\operatorname{Spec}(\widehat{S})$. Indeed, such an
extension $(\widehat{\mathcal{P}})^{\sim}$ to
$\operatorname{Spec}(\widehat{S})$ yields a triple
$(\mathcal{P},(\widehat{\mathcal{P}})^{\sim},\varphi)$, which by Lemma 5.6
glues to an extension of $\mathcal{P}$ over $\operatorname{Spec}(S)$. Hence we
may replace $S$ with $\widehat{S}$.

Note that we are allowed to replace $\kappa[[{\pi}]]$ with a finite totally
ramified extension $\kappa[[{\pi^{\frac{1}{m}}}]]$ for some $m\geq 1$. This
replaces $\widehat{S}$ with $\kappa_{S}[[{\pi^{\frac{1}{m}}}]]$. The theorem
is therefore implied by Proposition 5.2 (by taking $\kappa=\kappa_{S}$ in that
proposition). ∎

Since the semistable locus $\operatorname{Bun}_{\rho}(X)^{ss}$ comes from a
$\Theta$-stratification, we automatically have semistable reduction for
$\rho$-sheaves by [AHLH19, §6]. We record this result as a separate theorem.

######  Theorem 5.8 (Semistable Reduction).

The substack $\operatorname{Bun}_{\rho}(X)^{ss}$ of Gieseker semistable points
satisfies the existence part of the valuative criterion for properness. ∎

The arguments in this section yield an interesting result in the case of
curves. Suppose that $X$ is a smooth projective curve. Then a $\rho$-sheaf is
the same thing as a $G$-bundle. The arguments above imply the following result
about extending $G$-bundles.

######  Corollary 5.9.

Let $X$ be a smooth projective geometrically connected curve over a field
$\kappa$ of arbitrary characteristic. Let $G$ be a quasi-split connected
reductive group over $\kappa$ satisfying the following two conditions:

  1. (1)

The maximal central torus of $G$ is split over $\kappa$.

  2. (2)

Let $H$ denote the kernel of of the simply-connected cover
$\widetilde{[G,G]}\rightarrow[G,G]$ of the derived subgroup of $G$. Then, the
characteristic of $\kappa$ does not divide the order of $H$.

Let $R$ be a complete discrete valuation ring over $\kappa$ with fraction
field $K$. Suppose that the residue field of $R$ is perfect. Let $\mathcal{P}$
be a $G$-bundle on the generic fiber $X_{K}$. Then, after maybe taking a root
of the uniformizer of $R$, we can extend $\mathcal{P}$ to a $G$-bundle on
$X_{R}$.

###### Proof.

This follows by applying the proof of Lemma 5.1, using formal gluing as in
Theorem 5.7 and then applying Proposition 5.2. ∎

##  6\. Moduli space for the semistable locus

###  6.1. The degree of a $\rho$-sheaf

Fix the choice of an algebraic closure $\overline{k}\supset k$. We denote by
$\Gamma_{k}$ the absolute Galois group
$\Gamma_{k}\vcentcolon=\operatorname{Gal}(\overline{k}/k)$. Let
$X^{*}(G_{\overline{k}})$ be the group of characters of the base-change
$G_{\overline{k}}$. Let $X^{*}(G_{\overline{k}})^{\vee}$ denote the dual group
$\operatorname{Hom}\left(X^{*}(G_{\overline{k}}),\,\mathbb{Z}\right)$. This
group admits a natural continuous action of the profinite absolute Galois
group $\Gamma_{k}$. We write
$\Lambda(G)\vcentcolon=\left(X^{*}(G_{\overline{k}})^{\vee}\right)_{\Gamma_{k}}$
to denote the quotient group of $\Gamma_{k}$-coinvariants of
$(X^{*}(G_{\overline{k}}))^{\vee}$. Note that $\Lambda(G)$ does not depend on
the choice of algebraic closure $\overline{k}$.

######  Example 6.1.

If the group $G$ is split over $k$, then $X^{*}(G_{\overline{k}})$ coincides
with the group $X^{*}(G)$ of characters defined over $k$. The action of the
Galois group $\Gamma_{k}$ on $X^{*}(G)$ is trivial. Therefore
$\Lambda(G)=X^{*}(G)^{\vee}$.

######  Example 6.2.

If the group $G$ is semisimple, then $\Lambda(G)=0$.

Let $K\supset k$ be an arbitrary algebraically closed field extension and let
$(\mathcal{F},\sigma)$ be a $\rho$-sheaf on $X_{K}$. Let $U$ be any big open
subset of $X_{K}$ where $\mathcal{F}$ is locally-free and denote by $j$ be the
open inclusion $j:U\hookrightarrow X_{K}$. If $\mathcal{G}$ is the
corresponding $G_{K}$-bundle on $U$, then for any character $\chi\in
X^{*}(G_{K})$ we can form the associated line bundle
$\mathcal{G}\times^{G_{K}}\chi$ over $U$. Since $X_{K}$ is smooth, the
pushforward $j_{*}(\mathcal{G}\times^{G_{K}}\chi)$ is a line bundle on $X_{K}$
whose first Chern class is
$c_{1}\left(j_{*}(\mathcal{G}\times^{G_{K}}\chi)\right)$. Let $H$ be the
hyperplane class on $X$ corresponding to the line bundle $\mathcal{O}_{X}(1)$
and define the group homomorphism (c.f Lemma 4.25):

$\psi:X^{*}(G_{K})\rightarrow\mathbb{Z},\;\;\psi(\chi)\vcentcolon=c_{1}\left(j_{*}(\mathcal{G}\times^{G_{K}}\chi)\right)\cdot
H^{d-1}.$

If we denote by $L$ the algebraic closure of $k$ in $K$, then base-change
induces a canonical identification
$X^{*}(G_{L})\xrightarrow{\sim}X^{*}(G_{K})$. Choose an isomorphism
$\overline{k}\xrightarrow{\sim}L$ over $k$. This yields an isomorphism of
abelian groups $X^{*}(G_{\overline{k}})\xrightarrow{\sim}X^{*}(G_{L})$ which
identifies $\psi$ with an element of $X^{*}(G_{\overline{k}})^{\vee}$. Since
any two choices of isomorphisms $\overline{k}\xrightarrow{\sim}L$ differ by an
element of the Galois group $\Gamma_{k}$, $\psi$ yields a well-defined element
of the quotient $\Lambda(G)$ that is independent of the choice of
$\overline{k}\xrightarrow{\sim}L$.

######  Definition 6.3.

Let $K\supset k$ be an algebraically closed field extension. Let
$(\mathcal{F},\sigma)$ be a $\rho$-sheaf on $X_{K}$. We define the degree
$\deg\,(\mathcal{F},\sigma)$ to be the element $\vartheta\in\Lambda(G)$
constructed above.

For an arbitrary field extension $K\supset k$, we set
$\deg\,(\mathcal{F},\sigma)$ to be
$\deg\,(\mathcal{F}|_{X_{\widetilde{K}}},\sigma|_{X_{\widetilde{K}}})$ for any
choice of an algebraically closed field extension $\widetilde{K}\supset K$.

######  Proposition 6.4.

Let $S$ be a $k$-scheme and let $(\mathcal{F},\sigma)$ be a $\rho$-sheaf on
$X_{S}$. Let $\deg:S\rightarrow\Lambda(G)$ denote the function defined by
$\deg(s)\vcentcolon=\deg\,(\mathcal{F}|_{X_{s}},\sigma|_{X_{s}})$. Then $\deg$
is locally constant on $S$.

###### Proof.

Since the stack $\operatorname{Bun}_{\rho}(X)$ is locally of finite type over
$k$, we can reduce to the case when $S$ is of finite type over $k$. Given that
the notion of degree is well-behaved under taking field extensions, we can
replace $k$ with an algebraic closure to assume that the ground field is
algebraically closed and, hence $\Gamma_{k}$ is trivial.

Let $U$ be a big open subset of $X_{S}$ where $\mathcal{F}$ is locally-free
and denote by $j:U\hookrightarrow X_{S}$ the open immersion. For each point
$s\in S$, let $j_{s}$ be the base-change $j_{s}:U_{s}\hookrightarrow X_{s}$.
Denote by $\mathcal{G}$ the corresponding $G$-bundle on $U$. Let $\chi\in
X^{*}(G)$ be a character of $G$ and denote by $\mathcal{L}$ the associated
line bundle $\mathcal{G}\times^{G}\chi$ on $U$. We want to show that the
function $f:S\rightarrow\mathbb{Z}$ given by

$f(s)\vcentcolon=c_{1}\left((j_{s})_{*}(\mathcal{L}|_{U_{s}})\right)\cdot
H^{d-1}$

is locally constant on $S$.

We first claim that $f$ is constructible on $S$. By Noetherian induction we
need to prove that if $S$ is integral then $f$ is constant on an open subset
of $S$. Assume that $S$ is integral. Since $k$ is perfect, there exists an
open dense subset of $S$ that is smooth over $k$. Therefore we can reduce to
the case when $S$ is a smooth variety. For each $s\in S$, consider the
following Cartesian diagram

${U_{s}}$${U}$${X_{s}}$${X_{S}}$$\scriptstyle{j_{s}}$$\scriptstyle{j}$

We get an induced base-change morphism of quasicoherent sheaves
$\varphi:j_{*}(\mathcal{L})|_{X_{s}}\rightarrow(j_{s})_{*}\left(\mathcal{L}|_{U_{s}}\right)$
where the right-hand side is a line bundle. Since $S$ is smooth over $k$ and
$X_{S}\rightarrow S$ is a smooth morphism, we conclude that $X_{S}$ is also
smooth over $k$, so $X_{S}$ is a regular scheme. The complement of the open
subset $U$ has codimension $\geq 2$ inside $X_{S}$, therefore the pushforward
$j_{*}(\mathcal{L})$ is a line bundle. It follows that
$(j_{*}(\mathcal{L})|_{X_{s}}$ is a line bundle, and so $\varphi$ is a
morphism of line bundles on $X_{s}$. Note that the restriction of $\varphi$ to
the big open subset $U_{s}\subset X_{s}$ is an isomorphism by construction. We
conclude that $\varphi$ must be an isomorphism of line bundles. This shows
that we can equivalently write $f$ as

$f(s)=c_{1}\left((j_{*}\mathcal{L})|_{X_{s}}\right)\cdot H^{d-1}$

Since $j_{*}\mathcal{L}$ is a line bundle its first Chern class is locally
constant in families, then $f$ is constant on the variety $S$. This concludes
the proof that $f$ is constructible.

In order to show that $f$ is locally constant, it suffices to check that $f$
is constant when the base $S$ is of the form $\operatorname{Spec}(R)$ for some
discrete valuation ring $R$. In this case $S$ is a regular integral scheme.
Since $X_{S}\rightarrow S$ is smooth, it follows that $X_{S}$ is regular
[Sta21, Tag 07NF] and we can apply the same argument as in the proof of
constructibility to see that $f$ is constant on $S$. ∎

As a consequence of Proposition 6.4, we have the following.

######  Corollary 6.5.

For every $\vartheta\in\Lambda(G)$, there exists an open and closed substack
$\operatorname{Bun}_{\rho}(X)_{\vartheta}\subset\operatorname{Bun}_{\rho}(X)$
parametrizing $\rho$-sheaves with degree $\vartheta$. Then

$\operatorname{Bun}_{\rho}(X)=\bigsqcup_{\vartheta\in\Lambda(G)}\operatorname{Bun}_{\rho}(X)_{\vartheta}$

###  6.2. Boundedness of the semistable locus

Let $P\in\mathbb{Q}[n]$ be a Hilbert polynomial and let
$\vartheta\in\Lambda(G)$. We define
$\operatorname{Bun}_{\rho}(X)^{ss,P}_{\vartheta}\vcentcolon=\operatorname{Bun}_{\rho}(X)^{ss,P}\cap\operatorname{Bun}_{\rho}(X)_{\vartheta}$.
By Corollary 6.5 we get a decomposition of the semistable locus

$\operatorname{Bun}_{\rho}(X)^{ss,P}=\bigsqcup_{\vartheta\in\Lambda(G)}\operatorname{Bun}_{\rho}(X)^{ss,P}_{\vartheta}$

The next proposition shows that each piece
$\operatorname{Bun}_{\rho}(X)^{ss,P}_{\vartheta}$ is bounded.

######  Proposition 6.6.

For all $\vartheta\in\Lambda(G)$ and all $P\in\mathbb{Q}[n]$, the stack
$\operatorname{Bun}_{\rho}(X)^{ss,P}_{\vartheta}$ is quasicompact.

###### Proof.

After replacing $k$ with a field extension, we can assume that $k$ is
algebraically closed. Since the forgetful morphism
$\operatorname{Forget}:\operatorname{Bun}_{\rho}(X)\rightarrow\operatorname{Coh}_{r}^{tf}(X)$
is of finite type, it is sufficient to show that the restriction
$\operatorname{Forget}|_{\operatorname{Bun}_{\rho}(X)^{ss,P}_{\vartheta}}$
factors through a quasicompact open substack of
$\operatorname{Coh}_{r}^{tf}(X)$. We shall show that the following set:

$\mathfrak{S}=\left\\{\;\operatorname{Forget}\left((\mathcal{F},\sigma)\right)\;\;\mid\;\;(\mathcal{F},\sigma)\in\operatorname{Bun}_{\rho}(X)^{ss,P}_{\vartheta}(k)\;\right\\}$

is a bounded family of torsion-free sheaves.

Let
$(\mathcal{F},\sigma)\in\operatorname{Bun}_{\rho}(X)^{ss,P}_{\vartheta}(k)$.
We claim that there exists a uniform upper bound $\mu_{max}$ (only depending
on $\vartheta$) such that for all torsion-free subsheaves
$\mathcal{V}\subset\mathcal{F}$ we have
$\widehat{\mu}_{d-1}(\mathcal{V})\leq\mu_{max}$. Since the Hilbert polynomial
$P$ of $\mathcal{F}$ is fixed, it will follow that the family $\mathfrak{S}$
is bounded by [HL97, Thm. 3.3.7].

We are left to show the claim. Since $G$ is a reductive group over $k$ of
characteristic $0$, $G$ is linearly reductive and the representation
$\rho:G\rightarrow\operatorname{GL}(V)$ decomposes into a direct sum of
irreducible representations $V=\bigoplus_{i=1}^{q}V_{i}$. If we denote
$H\vcentcolon=\prod_{i=1}^{q}\operatorname{GL}(V_{i})$, then $\rho$ factors
through a faithful homomorphism $\varphi:G\rightarrow H$. Since each
representation $V_{i}$ is irreducible, it follows that the image of $\varphi$
is not contained in any proper parabolic subgroup of $H$.

Let $U$ be a big open subset where $\mathcal{F}$ is locally-free and let
$\mathcal{G}$ be the corresponding $G$-bundle on $U$ which, by Proposition
10.4, is slope semistable in the sense of Ramanathan [Ram79]. By the analogue
of [BG03, Thm. 5.1] for rational principal bundles (as explained in [BG03,
§7]), the restriction of $\mathcal{G}$ to a general smooth complete
intersection curve $C\hookrightarrow X$ of degree sufficiently large is a
(slope) semistable $G$-bundle as in [Ram75]. Let
$\mathcal{H}\vcentcolon=\varphi_{*}(\mathcal{G})$ be the associated $H$-bundle
on $U$. Notice that $\varphi_{*}(\mathcal{G}|_{C})$ is the restriction
$\mathcal{H}|_{C}$. By [BH04, Prop. 5.3] $\varphi_{*}(\mathcal{G}|_{C})$ is
(slope) semistable on $C$, so the restriction of $\mathcal{H}$ to a general
smooth complete intersection curve $C$ of degree sufficiently large is slope
semistable in the sense of Ramanathan. By the argument in [BG03, Remark 5.2],
it follows that $\mathcal{H}$ is slope semistable.

This $H$-bundle $\mathcal{H}$ amounts to the data of a $q$-tuple
$(\mathcal{E}_{U,i})_{i=1}^{q}$ of vector bundles on $U$. Let $j$ denote the
open inclusion $j:U\hookrightarrow X$. For each $i$, define
$\mathcal{E}_{i}\vcentcolon=j_{*}(\mathcal{E}_{U,i})$. The fact that
$\mathcal{H}$ is semistable implies that each $\mathcal{E}_{i}$ is a slope
semistable sheaf: otherwise the slope filtration of $\mathcal{E}_{i}$ would
restrict over $U$ to a destabilizing filtration of the $\mathcal{E}_{U,i}$,
thus contradicting the fact that $\mathcal{H}$ is semistable.

For each $i$, $\det(V_{i})$ is a character of $G$ and, by the definition of
degree, $\vartheta(\det(V_{i}))=c_{1}(\mathcal{E}_{i})\cdot H^{d-1}$. By
Hizebruch-Riemann-Roch, the slope $\widehat{\mu}_{d-1}(\mathcal{E}_{i})$ is
given by

$\displaystyle\widehat{\mu}_{d-1}(\mathcal{E}_{i})=\frac{1}{A_{d}\,\operatorname{rk}(\mathcal{E}_{i})}c_{1}(\mathcal{E}_{i})\cdot
H^{d-1}+\frac{1}{A_{d}}t_{1}\cdot
H^{d-1}=\frac{1}{A_{d}\,\dim(V_{i})}\vartheta(\det(V_{i}))+\frac{1}{A_{d}}t_{1}\cdot
H^{d-1}$

Therefore $\widehat{\mu}_{d-1}(\mathcal{E}_{i})$ is completely determined by
$\vartheta$, and does not depend on $(\mathcal{F},\sigma)$. Denote by
$\mu_{max}$ the maximum of the slopes
$\widehat{\mu}_{d-1}(\mathcal{E}_{1}),\widehat{\mu}_{d-1}(\mathcal{E}_{2}),\ldots,\widehat{\mu}_{d-1}(\mathcal{E}_{q})$.

We have $\bigoplus_{i=1}^{q}\mathcal{E}_{i}=\mathcal{F}^{\vee\vee}$, where
$\mathcal{F}^{\vee\vee}$ denotes the reflexive hull of $\mathcal{F}$, because
both sides restrict to the same associated vector bundle
$\rho_{*}(\mathcal{G})$ on $U$. If we set
$\mathcal{F}_{j}\vcentcolon=\left(\bigoplus_{i=1}^{j}\mathcal{E}_{i}\right)\cap\mathcal{F}\subset\mathcal{F}^{\vee\vee}$,
this defines a filtration of $\mathcal{F}$ by saturated subsheaves:

$0=\mathcal{F}_{0}\subset\mathcal{F}_{1}\subset\cdots\subset\mathcal{F}_{q-1}\subset\mathcal{F}_{q}=\mathcal{F}$

For each $j$ we have
$(\mathcal{F}_{j}/\mathcal{F}_{j-1})|_{U}=\mathcal{E}_{U,j}$, and therefore
$(\mathcal{F}_{j}/\mathcal{F}_{j-1})^{\vee\vee}=\mathcal{E}_{j}$. This implies
that
$\widehat{\mu}_{d-1}(\mathcal{F}_{j}/\mathcal{F}_{j-1})=\widehat{\mu}_{d-1}(\mathcal{E}_{j})$.
Since $\mathcal{E}_{j}$ is slope semistable, it also follows that
$\mathcal{F}_{j}/\mathcal{F}_{j-1}$ is a slope semistable sheaf.

We have therefore found a filtration of $\mathcal{F}$ such that each graded
piece $\mathcal{F}_{j}/\mathcal{F}_{j-1}$ is a slope semistable sheaf such
that $\widehat{\mu}_{d-1}(\mathcal{F}_{j}/\mathcal{F}_{j-1})\leq\mu_{max}$.
This implies that subsheaves $\mathcal{V}\subset\mathcal{F}$ satisfy
$\widehat{\mu}_{d-1}(\mathcal{V})\leq\mu_{max}$, as desired. ∎

######  Remark 6.7.

If the representation $\rho$ is central (as in Definition 10.5), then the
degree $\vartheta$ of a $\rho$-sheaf $(\mathcal{F},\sigma)$ is determined by
the slope of $\mathcal{F}$. In particular, the degree $\vartheta$ is
determined by the Hilbert polynomial $P$ of $\mathcal{F}$. This shows that in
this case the notion of degree is superfluous and it suffices to specify the
Hilbert polynomial $P$ alone. Hence, Proposition 6.6 shows that
$\operatorname{Bun}_{\rho}(X)^{ss,P}$ is bounded for $\rho$ central.

###  6.3. Moduli space for the semistable locus

The work done in previous sections allows us to show that the stack of
semistable $\rho$-sheaves admits a proper good moduli space in the sense of
[Alp13].

######  Theorem 6.8.

For each choice of Hilbert polynomial $P\in\mathbb{Q}[n]$ and degree
$\vartheta\in\Lambda(G)$, the stack
$\operatorname{Bun}_{\rho}(X)^{ss,P}_{\vartheta}$ admits a proper good moduli
space $M_{\rho}(X)^{ss,P}_{\vartheta}$.

###### Proof.

It suffices to check the hypotheses of [HLHJ21, Thm. 2.26] (see also [HL14,
Thm. B]).

  1. (a)

Proposition 4.16 implies that the numerical invariant $\nu$ is strictly
$\Theta$-monotone and strictly $S$-monotone.

  2. (b)

By Proposition 6.6, the stack
$\operatorname{Bun}_{\rho}(X)^{ss,P}_{\vartheta}$ is bounded.

  3. (c)

By Theorem 5.7, the stack $\operatorname{Bun}_{\rho}(X)$ satisfies the
existence part of the valuative criterion of properness.

∎

If the group $G$ is semisimple, then the representation $\rho$ factors through
the subgroup $\operatorname{SL}(V)\subset\operatorname{GL}(V)$. In this case
the moduli space of semistable $\rho$-sheaves was already constructed in
[Sch02, GLSS08] using GIT. For semisimple groups the proof above provides an
alternative stack-theoretic approach which is intrinsic, in the sense that it
does not appeal to an embedding into an external GIT problem.

On the other hand, suppose that the representation $\rho$ is central. Then,
the degree $\vartheta$ is determined by the Hilbert polynomial $P$ and each
$\operatorname{Bun}_{\rho}(X)^{ss,P}$ admits a proper good moduli space
$M_{\rho}(X)^{ss,P}$. Let $\operatorname{Bun}_{G}(X)^{s}$ denote the stack of
slope stable $G$-bundles on $X$. By Corollary 10.9, the stack
$\operatorname{Bun}_{\rho}(X)^{ss}$ of Gieseker semistable $\rho$-sheaves
contains $\operatorname{Bun}_{G}(X)^{s}$ as an open substack. Therefore the
separated moduli space
$M_{\rho}(X)^{ss}=\bigsqcup_{P\in\mathbb{Q}[n]}M_{\rho}(X)^{ss,P}$ contains an
open subspace $M_{G}(X)^{s}$ which is a good moduli space for the stack
$\operatorname{Bun}_{G}(X)^{s}$.

##  7\. Generalization: $\rho$-sheaves for products of general linear groups

In this section we define the more general notion of $\rho$-sheaf valued in a
product of general linear groups. All of the arguments from previous sections
will apply in this more general context.

Example 10.10 shows that the notion of Gieseker stability could be quite
restrictive if we do not assume that the representation $\rho$ is central.
Since the dimension of the center of $\operatorname{GL}(V)$ is 1, the faithful
representation $\rho$ can only be central if the center of $G$ has dimension
at most $1$. In order to deal with reductive groups $G$ with large center, it
is convenient to allow the codomain of $\rho$ to be a product of general
linear groups. All of the previous results extend to this more general
context.

Fix a tuple $V^{\bullet}=(V^{i})_{i=1}^{b}$ of finite-dimensional vector
spaces over $k$ with $r_{i}\vcentcolon=\dim\,V^{i}$. Define
$\operatorname{GL}(V^{\bullet})\vcentcolon=\prod_{i=1}^{b}\operatorname{GL}(V^{i})$.
We fix once and for all a faithful representation
$\rho:G\rightarrow\operatorname{GL}(V^{\bullet})$, which induces
representations $\rho_{i}:G\rightarrow\operatorname{GL}(V^{i})$ by composing
with the natural projections
$\operatorname{GL}(V^{\bullet})\rightarrow\operatorname{GL}(V^{i})$.

Let $S$ be a $k$-scheme. Suppose that we are given a tuple
$\mathcal{F}^{\bullet}=(\mathcal{F}^{i})_{i=1}^{b}\in\prod_{i=1}^{b}\operatorname{Coh}_{r_{i}}^{tf}(X)\,(S)$
of families of torsion-free sheaves on $X_{S}$. We set

$E_{V^{i}}(\mathcal{F}^{i})\vcentcolon=\operatorname{Sym}^{\bullet}(\mathcal{F}^{i}\otimes_{k}(V^{i})^{\vee})\otimes_{\mathcal{O}_{X_{S}}}\operatorname{Sym}^{\bullet}(\det(\mathcal{F}^{i})^{\vee}\otimes_{k}\det(V^{i}))$

We also define
$H(\mathcal{F}^{i},V^{i})\vcentcolon=\underline{\operatorname{Spec}}_{X_{S}}(E_{V^{i}}(\mathcal{F}^{i}))$
and
$H(\mathcal{F}^{\bullet},V^{\bullet})\vcentcolon=\prod_{i=1}^{b}H(\mathcal{F}^{i},\,V^{i})\;$.
Each scheme $H(\mathcal{F}^{i},V^{i})$ admits a right action of the group $G$
via the representation $\rho_{i}$, as described in Subsection 2.3. Equip
$H(\mathcal{F}^{\bullet},V^{\bullet})$ with the corresponding diagonal action
of $G$ and form the relatively affine GIT quotient
$H(\mathcal{F}^{\bullet},V^{\bullet})/\\!/\\!\,G$ by taking the ring of
$G$-invariants.

Let $U$ be a big open subscheme such that $\mathcal{F}^{i}|_{U}$ is locally-
free for all $i$. The same construction as in Subsection 2.3 applies to yield
a divisor $\delta^{i}_{U}-1$ on $H(\mathcal{F}^{i},V^{i})|_{U}$. This divisor
is stable under the right action of $G$. We descend these divisors to get a
closed subscheme
$\prod_{i=1}^{b}\left(H(\mathcal{F}^{i},V^{i})|_{U}\right)_{\alpha^{i}_{U}=0}\hookrightarrow\left(H(\mathcal{F}^{\bullet},V^{\bullet})|_{U}\right)/\\!/\\!\,G$.
Define the scheme of $G$-reductions
$\operatorname{Red}_{G}(\mathcal{F}^{\bullet})$ to be the scheme-theoretic
closure of this subset in $H(\mathcal{F}^{\bullet},V^{\bullet})/\\!/\\!\,G$.

######  Definition 7.1.

A $\rho$-sheaf on $X_{S}$ is a pair
$\left(\mathcal{F}^{\bullet},\sigma\right)$, where
$\mathcal{F}^{\bullet}\in\prod_{i=1}^{b}\operatorname{Coh}_{r_{i}}^{tf}(X)\,(S)$
and $\sigma$ is a section
$\sigma:X_{S}\rightarrow\operatorname{Red}_{G}(\mathcal{F}^{\bullet})$.

The tuple $\mathcal{F}^{\bullet}|_{U}$ of locally-free sheaves defines a
$\operatorname{GL}(V^{\bullet})$-bundle $\mathcal{P}$ on $U$. The same
argument as in Subsection 2.3 shows that the data of a section
$\sigma:X_{S}\rightarrow\operatorname{Red}_{G}(\mathcal{F}^{\bullet})$ is
equivalent to a $G$-reduction of structure group for the
$\operatorname{GL}(V^{\bullet})$-bundle $\mathcal{P}$. In particular we obtain
a rational $G$-bundle $\mathcal{G}$ defined on $U$. This allows to define the
notion of degree $\text{deg}(\mathcal{F}^{\bullet},\sigma)$ of a $\rho$-sheaf
as in Definition 6.3.

We denote by $\operatorname{Bun}_{\rho}(X)$ the stack on groupoids that takes
an affine $k$-scheme $T$ to the groupoid of $\rho$-sheaves on $X_{T}$. There
is a natural forgetful morphism
$\operatorname{Forget}:\operatorname{Bun}_{\rho}(X)\rightarrow\prod_{i=1}^{b}\operatorname{Coh}_{r_{i}}^{tf}(X)$
defined by
$\operatorname{Forget}(\mathcal{F}^{\bullet},\sigma)=\mathcal{F}^{\bullet}$,
which is affine and of finite type by the same reasoning as in Proposition
3.6. Therefore $\operatorname{Bun}_{\rho}(X)$ is an algebraic stack with
affine diagonal and locally of finite type over $k$. All of the discussion for
filtrations in Subsection 3.3 applies in this more general context, the only
difference being that the underlying associated filtration is now a tuple of
filtrations $(\mathcal{F}^{\bullet}_{m})_{m\in\mathbb{Z}}$, where each
$(\mathcal{F}^{i}_{m})_{m\in\mathbb{Z}}$ is a filtration of the torsion-free
sheaf $\mathcal{F}^{i}$.

We define a family of line bundles $L_{n}$ on $\operatorname{Bun}_{\rho}(X)$
as follows. For each $i$, we have a family of line bundles $L_{n}^{i}$ on
$\operatorname{Coh}^{tf}_{r_{i}}(X)$ defined by the same formulas as in
Definition 4.1. Then, we define the box product
$L_{n}:=\boxtimes_{i=1}^{b}L^{i}_{n}$ on the stack
$\prod_{i=1}^{b}\operatorname{Coh}^{tf}_{r_{i}}(X)$, and use the morphism
$\operatorname{Forget}$ to pullback $L_{n}$ to a line bundle on
$\operatorname{Bun}_{\rho}(X)$, also denoted by $L_{n}$.

More explicitly, for every morphism
$f:T\rightarrow\operatorname{Bun}_{\rho}(X)$ represented by a $\rho$-sheaf
$(\mathcal{F}^{\bullet},\sigma)$ on $X_{T}$, we have

$f^{*}\,L_{n}=\left(\bigotimes_{i=1}^{b}f^{*}\,M^{i}_{n}\right)\otimes\left(\bigotimes_{i=1}^{b}(f^{*}\,b^{i}_{d})^{-\otimes\,\overline{p}_{\mathcal{F}^{i}}(n)}\right)$

where
$f^{*}\,M^{i}_{n}\vcentcolon=\det\,\left(R\pi_{T\,*}\left(\mathcal{F}^{i}(n)\right)\right)$
and
$b^{i}_{d}\vcentcolon=\bigotimes_{j=0}^{d}(M^{i}_{j})^{\otimes(-1)^{j}\binom{d}{j}}$.

We will use this family of line bundles $L_{n}$ to define a polynomial
numerical invariant on $\operatorname{Bun}_{\rho}(X)$. Let
$(\mathcal{F}^{\bullet},\sigma)$ be a $\rho$-sheaf on $X_{K}$ for some field
extension $K\supset k$. For every nondegenerate filtration $f$ with underlying
associated filtration $(\mathcal{F}^{\bullet}_{m})_{m\in\mathbb{Z}}$, we set
$\nu(f)$ to be the polynomial

$\nu(f)=\frac{\operatorname{wgt}(L_{n}|_{0})}{\sqrt{\sum_{i=1}^{b}\sum_{m\in\mathbb{Z}}m^{2}\cdot\operatorname{rk}(\mathcal{F}^{i}_{m}/\mathcal{F}^{i}_{m+1})}}$

The same computation as in Proposition 4.4 shows that this can be rewritten in
terms of the reduced Hilbert polynomials as follows:

$\nu(f)=\sqrt{A_{d}}\cdot\frac{\sum_{i=1}^{b}\sum_{m\in\mathbb{Z}}m\cdot(\overline{p}_{\mathcal{F}^{i}_{m}/\mathcal{F}^{i}_{m+1}}-\overline{p}_{\mathcal{F}^{i}})\cdot\operatorname{rk}(\mathcal{F}^{i}_{m}/\mathcal{F}^{i}_{m+1})}{\sqrt{\sum_{i=1}^{b}\sum_{m\in\mathbb{Z}}m^{2}\cdot\operatorname{rk}(\mathcal{F}^{i}_{m}/\mathcal{F}^{i}_{m+1})}}$

One can define a version of the affine Grassmannian in this more general
context. Let $S$ be a quasicompact $k$-scheme and let $\mathcal{F^{\bullet}}$
be a tuple of torsion-free sheaves in
$\prod_{i=1}^{b}\operatorname{Coh}_{r_{i}}^{tf}(X)\,(S)$. Let
$D\hookrightarrow X_{S}$ be an effective Cartier divisor that is flat over $S$
and set $Q=X_{S}\setminus D$. Assume that we have a section
$\sigma:Q\rightarrow\operatorname{Red}_{G}(\mathcal{F}^{\bullet})$ defined
over $Q$.

The affine Grassmannian
$\operatorname{Gr}_{X_{S},D,\mathcal{F}^{\bullet},\sigma}$ is the functor from
$\left(\operatorname{Aff}_{S}\right)^{op}$ into sets defined as follows. For
each affine scheme $T$ in $\operatorname{Aff}_{S}$,
$\operatorname{Gr}_{X_{S},D,\mathcal{F}^{\bullet},\sigma}(T)$ is the set of
equivalence classes of the following data

  1. (1)

A $\rho$-sheaf $(\mathcal{E}^{\bullet},\zeta)$ on $X_{T}$.

  2. (2)

A morphism
$\theta^{\bullet}:\mathcal{F}^{\bullet}|_{X_{T}}\to\mathcal{E}^{\bullet}$ of
tuples such that the restriction $\theta^{\bullet}|_{Q_{T}}$ is an
isomorphism. Moreover, we require that the following diagram induced by the
isomorphism $\theta^{\bullet}|_{Q_{T}}$ commutes.

${Q_{T}}$${\operatorname{Red}_{G}(\mathcal{E}^{\bullet}|_{Q_{T}})}$${\operatorname{Red}_{G}(\mathcal{F}^{\bullet}|_{Q_{T}})}$$\scriptstyle{\zeta|_{Q_{T}}}$$\scriptstyle{\sigma|_{Q_{T}}}$$\xrightarrow{\;\;\sim\;\;}$

There is a forgetful morphism
$\operatorname{Gr}_{{X_{S},D,\mathcal{F}^{\bullet},\sigma}}\rightarrow\prod_{i=1}^{b}\operatorname{Gr}_{X_{S},D,\mathcal{F}^{i}}$
that takes a point $(\mathcal{E}^{\bullet},\zeta,\theta^{\bullet})$ to the
tuple $(\mathcal{E}^{i},\theta^{i})_{i=1}^{b}$. The same proof as in
Proposition 4.9 shows that this forgetful morphism exhibits
$\operatorname{Gr}_{{X_{S},D,\mathcal{F}^{\bullet},\sigma}}$ as a closed ind-
subscheme of $\prod_{i=1}^{b}\operatorname{Gr}_{X_{S},D,\mathcal{F}^{i}}$.

Let $P^{\bullet}=(P^{i})_{i=1}^{b}$ be a tuple of rational polynomials. For
each $N\geq 0$ we define the subfunctor

$\operatorname{Gr}_{X_{S},D,\mathcal{F}^{\bullet},\sigma}^{\leq
N,P^{\bullet}}\vcentcolon=\operatorname{Gr}_{X_{S},D,\mathcal{F}^{\bullet},\sigma}\cap\prod_{i=1}^{b}\operatorname{Gr}_{X_{S},D,\mathcal{F}^{i}}^{\leq
N,P^{i}}$

which is a closed subscheme of
$\operatorname{Gr}_{X_{S},D,\mathcal{F}^{\bullet},\sigma}$ and it is
projective over $S$. There exists an integer $m\gg 0$ such that, for all
$n\geq m$, the restriction of the dual $L_{n}^{\vee}$ is $S$-ample on the
projective stratum
$\operatorname{Gr}_{X_{S},D,\mathcal{F}^{\bullet},\sigma}^{\leq
N,P^{\bullet}}$.

The proof of Proposition 4.16 applies without change in this more general
context. The argument of HN boundedness in Proposition 4.26 also applies, one
only needs to keep track of one extra index $1\leq i\leq b$ in all the
formulas. In particular, Theorem 4.27 holds and the numerical invariant $\nu$
induces a $\Theta$-stratification on the stack $\operatorname{Bun}_{\rho}(X)$.
All of the arguments in Subsection 4.5 and Section 5 apply without change.

The tuple of rational polynomials $P^{\bullet}=(P^{i})_{i=1}^{b}$ defines an
open substack
$\operatorname{Bun}_{\rho}(X)^{ss,P^{\bullet}}\subset\operatorname{Bun}_{\rho}(X)$
parametrizing Gieseker semistable $\rho$-sheaves
$(\mathcal{F}^{\bullet},\sigma)$ such that the Hilbert polynomial of
$\mathcal{F}^{i}$ is $P^{i}$. If we fix the degree $\vartheta$ of the
$\rho$-sheaf, we obtain an open and closed substack
$\operatorname{Bun}_{\rho}(X)^{ss,P^{\bullet}}_{\vartheta}\subset\operatorname{Bun}_{\rho}(X)^{ss,P^{\bullet}}$,
which is quasicompact by the same proof as in Proposition 6.6. It follows by
[HLHJ21, Thm. 2.26] that
$\operatorname{Bun}_{\rho}(X)^{ss,P^{\bullet}}_{\vartheta}$ admits a proper
good moduli space $M_{\rho}(X)^{ss,P^{\bullet}}_{\vartheta}$.

######  Theorem 7.2.

The polynomial numerical invariant $\nu$ induces a $\Theta$-stratification on
the stack $\operatorname{Bun}_{\rho}(X)$. In particular the leading term HN
filtration induces a stratification of $\operatorname{Bun}_{\rho}(X)$ by
locally closed susbtacks.

For each choice of degree $\vartheta\in\Lambda(G)$ and tuple of Hilbert
polynomials $P^{\bullet}$, the open substack
$\operatorname{Bun}_{\rho}(X)^{ss,P^{\bullet}}_{\vartheta}$ admits a proper
good moduli space $M_{\rho}(X)^{ss,P^{\bullet}}_{\vartheta}$. ∎

If the representation $\rho$ is central, then the degree $\vartheta$ is
determined by the tuple of Hilbert polynomials $P^{\bullet}$ and each
$\operatorname{Bun}_{\rho}(X)^{ss,P^{\bullet}}$ admits a proper good moduli
space $M_{\rho}(X)^{ss,P^{\bullet}}$. In this case, the stack
$\operatorname{Bun}_{\rho}(X)^{ss}$ of Gieseker semistable $\rho$-sheaves
contains the stack $\operatorname{Bun}_{G}(X)^{s}$ of slope stable $G$-bundles
as an open substack, by Corollary 10.9. Therefore the separated moduli space
$M_{\rho}(X)^{ss}=\bigsqcup_{P^{\bullet}\in(\mathbb{Q}[n])^{b}}M_{\rho}(X)^{ss,P^{\bullet}}$
contains an open subspace $M_{G}(X)^{s}$ which is a good moduli space for the
stack $\operatorname{Bun}_{G}(X)^{s}$. Hence $M_{\rho}(X)^{ss}$ is a locally
proper compactification of the moduli of slope stable $G$-bundles on $X$ (c.f.
[BG03, §7]).

Under the assumption that the representation $\rho$ is central, the
$\Theta$-stratification induced by the leading term HN filtration refines the
slope stratification of $\operatorname{Bun}_{G}(X)$ described in [GN18, GN16],
by Proposition 10.11.

##  8\. Gieseker-Harder-Narasimhan filtrations for $\rho$-sheaves

The example in Subsection 9.1 shows that the leading term HN filtration of a
$\rho$-sheaf is analogous to the leading term filtration of a torsion-free
sheaf as defined in [HLHJ21, §5.2]. In this subsection we describe a procedure
that iterates the leading term HN filtration in order to construct a finer
filtration for $\rho$-sheaves, called the Gieseker-Harder-Narasimhan
filtration (GHN filtration). We prove that this filtration induces a
stratification of $\operatorname{Bun}_{\rho}(X)$ by locally closed substacks.

To understand the notation and constructions in this section, it might help
the reader to look beforehand at the examples in Subsections 9.1 and 9.2.

###  8.1. Lexicographic filtrations

Fix a positive integer $q$. We equip the group $\mathbb{Z}^{q}$ with the
lexicographic order where two tuples
$\vec{m}=(m_{i})_{i=1}^{q},\vec{n}=(n_{i})_{i=1}^{q}\in\mathbb{Z}^{q}$ satisfy
$\vec{m}>\vec{n}$ if and only if there exists an index $j$ such that
$m_{j}>n_{j}$ and $m_{i}=n_{i}$ for all $i<j$. The lexicographic order endows
$\mathbb{Z}^{q}$ with the structure of a totally well-ordered abelian group.
For $\vec{m}=(m_{1},m_{2},\ldots,m_{q-1},m_{q})\in\mathbb{Z}^{q}$, we define
the successor by $\vec{m}+1:=(m_{1},m_{2},\ldots,m_{q-1},m_{q}+1)$.

Let $K\supset k$ be a field extension and let $\mathcal{F}$ be a torsion-free
sheaf on $X_{K}$.

######  Definition 8.1.

A lexicographic $\mathbb{Z}^{q}$-filtration of $\mathcal{F}$ is a sequence of
subsheaves $f=(\mathcal{F}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ indexed by
$\mathbb{Z}^{q}$ such that the following are satisfied.

  1. (1)

$\mathcal{F}_{\vec{m}}\subset\mathcal{F}_{\vec{n}}$ whenever
$\vec{m}>\vec{n}$.

  2. (2)

There exists an $M_{max}\in\mathbb{Z}^{q}$ such that $\mathcal{F}_{\vec{m}}=0$
for all $\vec{m}>M_{max}$, and there exists an $M_{min}\in\mathbb{Z}^{q}$ such
that $\mathcal{F}_{\vec{m}}=\mathcal{F}$ for all $\vec{m}<M_{min}$.

  3. (3)

For all $\vec{m}$, we have that $\mathcal{F}/\mathcal{F}_{\vec{m}}$ is
torsion-free.

The associated graded of a lexicographic $\mathbb{Z}^{q}$-filtration
$f=(\mathcal{F}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ is defined as
$\text{gr}(f)\vcentcolon=\bigoplus_{\vec{m}\in\mathbb{Z}^{q}}\mathcal{F}_{\vec{m}}/\mathcal{F}_{\vec{m}+1}$.
Conditions (2) and (3) in Definition 8.1 ensure that $\text{gr}(f)$ is a
torsion-free sheaf with the same rank as $\mathcal{F}$. Note that
$\text{gr}(f)$ is naturally equipped with a $\mathbb{Z}^{q}$-grading. We say
that $\vec{m}\in\mathbb{Z}^{q}$ is a jumping point of $f$ if the quotient
$\mathcal{F}_{\vec{m}}/\mathcal{F}_{\vec{m}+1}$ is not the zero sheaf. There
are finitely many jumping points.

######  Definition 8.2.

Let $f$ be a lexicographic $\mathbb{Z}^{q}$-filtration. Let
$\vec{p}=(p_{i})_{i=1}^{q}$ be a tuple of positive rational numbers. For all
coordinates $i$, we can scale the weights of $f$ at the $i^{th}$ coordinate by
the rational number $p_{i}$. Doing this for all $i$ yields a lexicographic
multi-weighted filtration denoted by $\vec{p}\cdot f$ that could possibly have
rational weights. If $\vec{p}\cdot f$ has integer weights, then we say that
the lexicographic $\mathbb{Z}^{q}$-filtration $\vec{p}\cdot f$ is obtained by
scaling the weights of $f$ by $\vec{p}$.

The scheme
$\mathbb{A}_{k}^{q}=\operatorname{Spec}(k[t_{1},t_{2},\ldots,t_{q}])$ admits
an action of $\mathbb{G}_{m}^{q}$ over $k$, where the $i^{th}$ coordinate of
$\mathbb{G}_{m}^{q}$ acts with weight $-1$ on $t_{i}$ and weight $0$ on the
other coordinates. Set
$\Theta^{q}_{k}\vcentcolon=\left[\,\mathbb{A}^{q}_{k}/\,\mathbb{G}_{m}^{q}\right]$
and denote by
$\Theta^{q}_{K}\vcentcolon=\left[\,\mathbb{A}^{q}_{K}/\,(\mathbb{G}_{m}^{q})_{K}\right]$
the base change. A multi-graded version of the Rees construction in
Proposition 3.8 can be applied to associate to any lexicographic
$\mathbb{Z}^{q}$-filtration a morphism of stacks
$\Theta^{q}_{K}\rightarrow\operatorname{Coh}^{tf}(X)$ in the following way.

######  Definition 8.3.

Given a lexicographic $\mathbb{Z}^{q}$-filtration
$f=(\mathcal{F}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ of $\mathcal{F}$, we
set multi-graded Rees construction $\widetilde{\mathcal{F}}_{f}$ to be the
$\mathbb{G}_{m}^{q}$-equivariant $\mathbb{A}^{q}_{K}$-flat sheaf on
$\mathbb{A}^{q}_{K}\times X$ defined by

$\widetilde{\mathcal{F}}_{f}=\bigoplus_{\vec{m}\in\mathbb{Z}^{q}}\left(\mathcal{F}_{\vec{m}}\cdot\prod_{i=1}^{q}t_{i}^{m_{i}}\right)$

Note that this is a coherent sheaf by condition (2) in Definition 8.1. If
$\vec{1}_{K}$ denotes the point $(1,1,\ldots,1)\in\mathbb{A}^{q}_{K}$, the
fiber at $\vec{1}_{K}\times X$ is isomorphic to the original sheaf
$\mathcal{F}$ and the fiber at $0_{K}\times X$ is the $\mathbb{Z}^{q}$-graded
sheaf $\text{gr}(f)$. The locus of points in $\mathbb{A}^{q}_{K}$ such that
the fiber of $\widetilde{\mathcal{F}}_{f}$ is torsion-free is open by the
slicing criterion for flatness [Sta21, Tag 046Y], and the fiber at the origin
$0_{K}$ is torsion-free by condition (3) in Definition 8.1. By
$\mathbb{G}^{q}_{m}$-equivariance, it follows that
$\widetilde{\mathcal{F}}_{f}$ is a family of torsion-free sheaves. So indeed
$\widetilde{\mathcal{F}}_{f}$ represents a morphism
$\Theta^{q}_{K}\rightarrow\operatorname{Coh}^{tf}(X)$.

More generally, a lexicographic $\mathbb{Z}^{q}$-filtration $f$ of a tuple
$\mathcal{F}^{\bullet}=(\mathcal{F}_{i})_{i=1}^{b}$ of torsion-free sheaves on
$X_{K}$ is defined to be a tuple $f=(f_{i})_{i=1}^{b}$ where $f_{i}$ is a
lexicographic $\mathbb{Z}^{q}$-filtration of $\mathcal{F}^{i}$. Here it also
makes sense to scale the weights by a tuple $\vec{p}=(p_{j})_{j=1}^{q}$ of
rational numbers; we just scale the weights of each $f_{i}$ by $\vec{p}$. If
$\mathcal{F}^{\bullet}$ is a point in
$\prod_{i=1}^{b}\operatorname{Coh}^{tf}_{r_{i}}(X)$, then the multi-graded
Rees construction applied to each component yields a morphism
$\Theta^{q}_{K}\rightarrow\prod_{i=1}^{b}\operatorname{Coh}^{tf}_{r_{i}}(X)$
represented by a tuple $\widetilde{\mathcal{F}^{\bullet}}_{f}$ such that the
fiber at $\vec{1}_{K}\times X$ recovers $\mathcal{F}^{\bullet}$.

Fix a tuple $V^{\bullet}$ of $k$-vector spaces and let
$\rho:G\rightarrow\operatorname{GL}(V^{\bullet})$ be faithful representation,
as in Section 7. Let $(\mathcal{F}^{\bullet},\sigma)$ be a $\rho$-sheaf on
$X_{K}$ and suppose that we are given a lexicographic
$\mathbb{Z}^{q}$-filtration $f$ of $\mathcal{F}^{\bullet}$. Applying the
multi-graded Rees construction, we obtain a homomorphism
$\Theta^{q}_{K}\rightarrow\prod_{i=1}^{b}\operatorname{Coh}^{tf}_{r_{i}}(X)$
represented by a tuple $\widetilde{\mathcal{F}^{\bullet}}_{f}$. If $U$ is a
big open subset of $X_{K}$ such that $\text{gr}(f)$ is locally-free, the
restriction $\widetilde{\mathcal{F}^{\bullet}}_{f}|_{\Theta^{q}_{K}\times U}$
is a tuple of locally-free sheaves, which we can view as a
$\operatorname{GL}(V^{\bullet})$-bundle on $\Theta^{q}_{K}\times_{K}U$. The
section $\sigma$ induces a $G$-reduction of structure group of the restriction
$\widetilde{\mathcal{F}^{\bullet}}_{f}|_{\vec{1}_{K}\times_{K}U}$.

######  Definition 8.4.

We say that $f$ is a lexicographic $\mathbb{Z}^{q}$-filtration of
$(\mathcal{F}^{\bullet},\sigma)$ if the $G$-reduction of
$\widetilde{\mathcal{F}^{\bullet}}_{f}|_{\vec{1}_{K}\times_{K}U}$ induced by
$\sigma$ extends to a $G$-reduction defined over $\Theta^{q}_{K}\times U$.

Since the $(\mathbb{G}_{m}^{q})_{K}$-orbit of $\vec{1}_{K}$ is scheme-
theoretically dense in $\mathbb{A}^{q}_{K}$, if such extension exists then it
must be unique. So, again, being a filtration of
$(\mathcal{F}^{\bullet},\sigma)$ is a property, not a structure.

###  8.2. Multi-weighted parabolic reductions

For this subsection we fix a field extension $K\supset k$.

Let $H$ be a reductive group over $K$ and let
$\vec{\lambda}=(\lambda_{i})_{i=1}^{q}$ be a tuple of $q$ cocharacters of $H$.
The first cocharacter $\lambda_{1}$ defines a parabolic subgroup
$P_{\lambda_{1}}$ with corresponding Levi subgroup $L_{\lambda_{1}}$. We
denote by $q_{\lambda_{1}}:P_{\lambda_{1}}\rightarrow L_{\lambda_{1}}$ the
quotient morphism. Suppose that the cocharacter
$\lambda_{2}:\mathbb{G}_{m}\rightarrow H$ factors through the subgroup
$L_{\lambda_{1}}$ and let $\overline{\lambda_{2}}:\mathbb{G}_{m}\rightarrow
L_{\lambda_{1}}$ be the corresponding cocharacter. This induces a parabolic
subgroup $P_{\overline{\lambda_{2}}}\subset L_{\lambda_{1}}$ with
corresponding Levi subgroup $L_{\overline{\lambda_{2}}}$. The preimage
$P_{(\lambda_{1},\lambda_{2})}:=(q_{\lambda_{1}})^{-1}(P_{\overline{\lambda_{2}}})$
is a parabolic subgroup of $H$ contained in $P_{\lambda_{1}}$ (c.f. second
paragraph of the proof of [BH04, Prop. 3.1]). We set
$L_{(\lambda_{1},\lambda_{2})}\vcentcolon=L_{\overline{\lambda_{2}}}$, the
joint centralizer of $\lambda_{1}$ and $\lambda_{2}$. By definition,
$L_{(\lambda_{1},\lambda_{2})}$ is a Levi subgroup of
$P_{(\lambda_{1},\lambda_{2})}$; we denote by
$q_{(\lambda_{1},\lambda_{2})}:P_{(\lambda_{1},\lambda_{2})}\rightarrow
L_{(\lambda_{1},\lambda_{2})}$ the quotient morphism. If the cocharacter
$\lambda_{3}$ factors through the $L_{(\lambda_{1},\lambda_{2})}$, then we can
similarly use $q_{(\lambda_{1},\lambda_{2})}$ to define a smaller parabolic
subgroup $P_{(\lambda_{1},\lambda_{2},\lambda_{3})}$ with Levi subgroup
$L_{(\lambda_{1},\lambda_{2},\lambda_{3})}$ given by the joint centralizer of
the cocharacters $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$, and iterate
this process.

######  Definition 8.5.

We say that the tuple $\vec{\lambda}$ is nested if at the $i^{th}$ step of the
procedure described above we have that $\lambda_{i}$ factors through the
subgroup $L_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{i-1})}$.

For a nested tuple $\vec{\lambda}=(\lambda_{1},\ldots,\lambda_{q})$, this
procedure terminates at the $q^{th}$ step and yields a parabolic subgroup
$P_{\vec{\lambda}}$ of $H$ with Levi subgroup $L_{\vec{\lambda}}$. Furthermore
these groups fit into nested chains of subgroups:

$P_{\vec{\lambda}}\subset
P_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{q-1})}\subset\cdots\subset
P_{(\lambda_{1},\lambda_{2})}\subset P_{\lambda_{1}}\subset H$
$L_{\vec{\lambda}}\subset
L_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{q-1})}\subset\cdots\subset
L_{(\lambda_{1},\lambda_{2})}\subset L_{\lambda_{1}}\subset H$

######  Definition 8.6.

Let $Y$ be a $K$-scheme. Let $\mathcal{H}$ be a $H$-bundle on $Y$. A
$q$-weighted parabolic reduction of $\mathcal{H}$ consists of a pair
$(\vec{\lambda},\mathcal{H}_{\vec{\lambda}})$, where $\vec{\lambda}$ is a
tuple of nested cocharacters of $H$ and $\mathcal{H}$ is a reduction of
structure group of $\mathcal{H}$ to the subgroup $P_{\vec{\lambda}}$.

Let $\theta:H\rightarrow F$ be a homomorphism of reductive groups over $K$.
Let $\vec{\lambda}=(\lambda_{i})_{i=1}^{q}$ be a nested tuple of cocharacters
of $H$. We set $\theta_{*}(\vec{\lambda})=(\theta\circ\lambda_{i})_{i=1}^{q}$,
which can be checked to be a nested tuple of cocharacters of $F$. The
homomorphism $\theta$ restricts to a well-defined homomorphism
$\theta:P_{\vec{\lambda}}\rightarrow P_{\theta_{*}(\vec{\lambda})}$. Let $Y$
be a $K$-scheme and let $\mathcal{H}$ be a $H$-bundle on $Y$. Given a
$q$-weighted parabolic reduction $(\vec{\lambda},\mathcal{H}_{\vec{\lambda}})$
of $H$, we can associate a $q$-weighted parabolic reduction
$(\theta_{*}(\vec{\lambda}),\theta_{*}\mathcal{H}_{\vec{\lambda}})$ of
$\theta_{*}\mathcal{H}$. Here $\theta_{*}\mathcal{H}_{\vec{\lambda}}$ is the
parabolic reduction obtained from $\mathcal{H}_{\vec{\lambda}}$ via the
restriction $\theta:P_{\vec{\lambda}}\rightarrow
P_{\theta_{*}(\vec{\lambda})}$.

We now return to our original group $G_{K}$. Suppose that we are given a
$G_{K}$-bundle $\mathcal{G}$ on $Y$. Given a $q$-weighted parabolic reduction
$(\vec{\lambda},\mathcal{G}_{\vec{\lambda}})$ as above, we shall construct a
$G_{K}$-bundle on $\Theta^{q}_{K}\times_{K}Y$ by iterating the Rees
construction.

Take the last cocharacter $\lambda_{q}$ of $\vec{\lambda}$ and consider the
$\mathbb{G}_{m}$-equivariant $P_{\vec{\lambda}}$-bundle
$\widetilde{\mathcal{G}}^{1}$ on $\mathbb{A}^{1}_{K}\times_{K}Y$ given by the
Rees construction
$\operatorname{Rees}(\mathcal{G}_{\vec{\lambda}},\lambda_{q})$. Using
$P_{\vec{\lambda}}\subset P_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{q-1})}$,
we extend the structure group of $\widetilde{\mathcal{G}}^{1}$ to the bigger
parabolic $P_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{q-1})}$. Given that the
subsequent cocharacter $\lambda_{q-1}$ factors through
$P_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{q-1})}$, we apply the Rees
construction to the
$P_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{q-1})}$-bundle
$\widetilde{\mathcal{G}}^{1}$ on $\mathbb{A}^{1}_{K}\times_{K}Y$ using the
cocharacter $\lambda_{q-1}$. This way we obtain the
$\mathbb{G}_{m}^{2}$-equivariant
$P_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{q-1})}$-bundle
$\widetilde{\mathcal{G}}^{2}\vcentcolon=\operatorname{Rees}(\mathcal{G}^{1},\lambda_{q-1})$
on $\mathbb{A}^{2}_{K}\times_{K}Y$. We iterate this procedure: at the $i^{th}$
step we have a $P_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{q-i})}$-bundle
$\widetilde{\mathcal{G}}^{i}$ on $\mathbb{A}^{i}_{K}\times_{K}Y$ and we apply
the Rees construction using the cocharacter $\lambda_{q-i}$ to get the
$\mathbb{G}_{m}^{i+1}$-equivariant
$P_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{q-i-1})}$-bundle
$\mathcal{G}^{i+1}\vcentcolon=\operatorname{Rees}(\mathcal{G}^{i},\lambda_{q-i})$
on $\mathbb{A}^{i+1}_{K}\times_{K}Y$.

######  Definition 8.7.

We denote by $\operatorname{Rees}(\mathcal{G}_{\vec{\lambda}},\vec{\lambda})$
the $P_{\lambda_{1}}$-bundle on $\Theta^{q}_{K}\times_{K}Y$ obtained at the
$q^{th}$ step of the procedure described above.

By construction, the restriction
$\operatorname{Rees}(\mathcal{G}_{\vec{\lambda}},\vec{\lambda})|_{\vec{1}_{K}\times_{K}Y}$
is canonically isomorphic to the $P_{\lambda_{1}}$-bundle obtained via
$\mathcal{G}_{\vec{\lambda}}$ using the inclusion $P_{\vec{\lambda}}\subset
P_{\lambda_{1}}$. We can extend the structure group from $P_{\lambda_{1}}$ to
$G_{K}$ in order to obtain a $G_{K}$-bundle
$\operatorname{Rees}(\mathcal{G}_{\vec{\lambda}},\vec{\lambda})\times^{P_{\lambda_{1}}}G_{K}$
on $\Theta^{q}_{K}\times_{K}Y$. The restriction to $\vec{1}_{K}\times_{K}Y$ is
canonically isomorphic to the original $G_{K}$-bundle $\mathcal{G}$.

######  Example 8.8.

If the group is a general linear group $\operatorname{GL}(V)$, this process
can be described more concretely as follows. Let $\mathcal{E}$ be a
$\operatorname{GL}(V)_{K}$-bundle on $Y$. We can think of $\mathcal{E}$ as a
vector bundle of constant rank. Let
$(\vec{\lambda},\mathcal{E}_{\vec{\lambda}})$ be a $q$-weighted parabolic
reduction of $\mathcal{E}$. For each index $i$, we set
$\mathcal{E}_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{i})}$ to be the
$P_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{i})}$-parabolic reduction of
$\mathcal{E}$ obtained from $\mathcal{E}_{\vec{\lambda}}$ by extending the
structure group $P_{\vec{\lambda}}\subset
P_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{i})}$. The pair
$\left(\lambda_{i},\mathcal{E}_{(\lambda_{1},\lambda_{2},\ldots,\lambda_{i})}\right)$
amounts to $\mathbb{Z}$-weighted filtration
$(\mathcal{E}^{(i)}_{m})_{m\in\mathbb{Z}}$ of $\mathcal{E}$. We use this to
define a lexicographic $\mathbb{Z}^{q}$-filtration
$f=(\mathcal{E}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ of $\mathcal{E}$ by
vector subbundles where for a tuple $\vec{m}=(m_{i})_{i=1}^{q}$, we set
$\mathcal{E}_{\vec{m}}=\bigcap_{i=1}^{q}\mathcal{E}^{(i)}_{m_{i}}$. Then we
apply the multi-graded Rees construction to $f$ in order to obtain a vector
bundle $\widetilde{\mathcal{E}}_{f}$ on $\Theta^{q}_{K}\times_{K}Y$. The
$\operatorname{GL}(V)_{K}$-bundle $\widetilde{\mathcal{E}}_{f}$ is precisely
$\operatorname{Rees}(\mathcal{G}_{\vec{\lambda}},\vec{\lambda})\times^{P_{\lambda_{1}}}\operatorname{GL}(V)_{K}$.
A similar description holds for $\operatorname{GL}(V^{\bullet})$-bundles, by
applying this construction to each component of a tuple
$\mathcal{E}^{\bullet}$.

Now let $(\mathcal{F}^{\bullet},\sigma)$ be a $\rho$-sheaf over $X_{K}$ and
let $f=(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ be a
lexicographic $\mathbb{Z}^{q}$-filtration of $\mathcal{F}^{\bullet}$. Let $U$
be a big open subset of $X_{K}$ where the tuple $\text{gr}(f)$ is locally-
free. Then the restriction
$(\widetilde{\mathcal{F}^{\bullet}}_{f})|_{\Theta^{q}_{K}\times_{K}U}$ is
locally-free, so we can view it as a $\operatorname{GL}(V^{\bullet})$-bundle.
Since $(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ is a
filtration of the $\rho$-sheaf, this $\operatorname{GL}(V^{\bullet})$ bundle
comes equipped with a canonical reduction of structure group
$\widetilde{\mathcal{G}}$ to $G_{K}$. Let $\mathcal{G}$ be the $G_{K}$-bundle
on $U$ corresponding to the $\rho$-sheaf $(\mathcal{F}^{\bullet},\sigma)$
(equivalently
$\mathcal{G}=\widetilde{\mathcal{G}}|_{\vec{1}_{K}\times_{K}U}$).

######  Definition 8.9.

We say that the lexicographic filtration
$f=(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ comes from a
$q$-weighted parabolic reduction $(\vec{\lambda},\mathcal{G}_{\vec{\lambda}})$
of $\mathcal{G}$ if
$\widetilde{\mathcal{G}}=\operatorname{Rees}(\mathcal{G}_{\vec{\lambda}},\vec{\lambda})\times^{P_{\lambda_{1}}}G_{K}$.

More explicitly, if $j:U\hookrightarrow X_{K}$ is a big open subset where the
tuple $\mathcal{F}^{\bullet}$ is locally-free, a $q$-weighted parabolic
reduction $(\vec{\lambda},\mathcal{G}_{\lambda})$ induces a $q$-weighted
parabolic reduction
$(\rho_{*}(\vec{\lambda}),\rho_{*}\mathcal{G}_{\vec{\lambda}})$ of the
$\operatorname{GL}(V^{\bullet})_{K}$-bundle
$\mathcal{F}^{\bullet}|_{U}:=\mathcal{E}^{\bullet}$. We define a lexicographic
$\mathbb{Z}^{q}$-filtration
$(\mathcal{E}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ of
$\mathcal{E}^{\bullet}$ associated to the $q$-weighted parabolic reduction
$(\rho_{*}(\vec{\lambda}),\rho_{*}\mathcal{G}_{\vec{\lambda}})$ and, for each
$\vec{m}\in\mathbb{Z}^{q}$, set
$\widetilde{\mathcal{E}}^{\bullet}_{\vec{m}}\vcentcolon=j_{*}(\mathcal{E}^{\bullet}_{\vec{m}})\cap\mathcal{F}^{\bullet}$
(the intersection being taken for each coordinate of the tuple). This defines
a lexicographic $\mathbb{Z}^{q}$-filtration of $\mathcal{F}^{\bullet}$, and
$(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}=(\widetilde{\mathcal{E}}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$
exactly when $(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$
comes from the $q$-weighted parabolic reduction
$(\vec{\lambda},\mathcal{G}_{\lambda})$.

###  8.3. The GHN filtration for $\rho$-sheaves

Fix a tuple $V^{\bullet}=(V^{i})_{i=1}^{b}$ of vector spaces over $k$ and let
$\rho:G\rightarrow\operatorname{GL}(V^{\bullet})$ be a faithful
representation. In this subsection, we recursively apply the leading term HN
filtration to the associated graded Levi sheaves in order to obtain a sequence
of multi-weighted parabolic reductions of $(\mathcal{F}^{\bullet},\sigma)$.

Let $K\supset k$ be a field extension such that $G_{K}$ is split and let
$(\mathcal{F}^{\bullet},\sigma)$ be an unstable $\rho$-sheaf on $X_{K}$. Let
us proceed to describe the recursive construction of the GHN filtration of
$(\mathcal{F}^{\bullet},\sigma)$.

First Step: Let $f_{can,1}:=(\mathcal{F}^{\bullet}_{m})_{m\in\mathbb{Z}}$ be
the leading term HN filtration of $(\mathcal{F}^{\bullet},\sigma)$ and let
$(\lambda_{1},\mathcal{G}_{\lambda_{1}})$ be the corresponding weighted
parabolic reduction. The cocharacter $\rho\circ\lambda_{1}$ induces a grading
on each vector space $V_{i}$. This induces a nontrivial direct sum
decomposition of the tuple $V^{\bullet}$. Indeed, if the decomposition was
trivial then $\rho\circ\lambda_{1}$ would be a cocharacter of the center of
$\operatorname{GL}(V^{\bullet})$ and using the same reasoning as in
Proposition 10.6 we would get $\nu(f|_{can,1})=0$, contradicting the fact that
$f|_{can,1}$ is destabilizing.

We can think of this direct sum decomposition as a finer tuple
$V_{\lambda_{1}}^{\bullet}$ of direct summands, where we have increased the
number of indexes. The group $\operatorname{GL}(V^{\bullet}_{\lambda_{1}})$ is
the Levi subgroup of $\operatorname{GL}(V^{\bullet})$ determined by the
cocharacter $\rho\circ\lambda_{1}$ which, by definition, contains the image of
the Levi $L_{\lambda_{1}}\subset G$ under $\rho$. We denote by
$\rho_{\lambda_{1}}:L_{\lambda_{1}}\rightarrow\operatorname{GL}(V^{\bullet}_{\lambda_{1}})$
the restriction to $L_{\lambda_{1}}$ of the homomorphism $\rho$.

The description of the associated graded
$f_{can,1}|_{0}=\left(\bigoplus_{m\in\mathbb{Z}}\mathcal{F}^{\bullet}_{m}/\mathcal{F}^{\bullet}_{m+1},\text{gr}(\sigma)\right)$
given at the end of Subsection 3.3 shows that we can canonically view
$f_{can,1}|_{0}$ as a $\rho_{\lambda_{1}}$-sheaf for the Levi
$L_{\lambda_{1}}$. We call this the $1^{st}$ associated graded Levi sheaf and
denote it by
$\text{gr}_{1}(\mathcal{F}^{\bullet},\sigma):=(\text{gr}_{1}(\mathcal{F}^{\bullet}),\text{gr}_{1}(\sigma))$.

Second Step: It might happen that the $\rho_{\lambda_{1}}$-sheaf
$\text{gr}_{1}(\mathcal{F}^{\bullet},\sigma)$ is unstable (see Subsection 9.1
below). If that is the case, we can consider the leading term HN filtration
$f_{can,2}=(\text{gr}_{1}(\mathcal{F}^{\bullet})_{m})_{m\in\mathbb{Z}}$ of the
$\rho_{\lambda_{1}}$-sheaf $\text{gr}_{1}(\mathcal{F}^{\bullet},\sigma)$. This
is by definition a $\mathbb{Z}$-filtration of
$\text{gr}_{1}(\mathcal{F}^{\bullet})$. Note that $(L_{\lambda_{1}})_{K}$ must
also be split, and therefore there exists a weighted parabolic reduction
$(\lambda_{2},\mathcal{G}_{\lambda_{2}})$ of
$\text{gr}_{1}(\mathcal{F}^{\bullet},\sigma)$ that induces the leading term HN
filtration $(\text{gr}_{1}(\mathcal{F}^{\bullet})_{m})_{m\in\mathbb{Z}}$.

By construction we can think of $(\lambda_{1},\lambda_{2})$ as a nested pair.
We set $L_{(\lambda_{1},\lambda_{2})}$ to be the Levi subgroup of
$L_{\lambda_{1}}$ centralizing $\lambda_{2}$ and denote by
$P_{(\lambda_{1},\lambda_{2})}$ the preimage under
$q_{\lambda_{1}}:P_{\lambda_{1}}\rightarrow L_{\lambda_{1}}$ of
$P_{\lambda_{2}}$. By [BH04, Lem. 2.1], the $P_{\lambda_{2}}$-parabolic
reduction $\mathcal{G}_{\lambda_{2}}$ lifts to a unique
$P_{(\lambda_{1},\lambda_{2})}$-parabolic reduction
$\mathcal{G}_{(\lambda_{1},\lambda_{2})}$ that recovers the
$P_{\lambda_{1}}$-reduction $\mathcal{G}_{\lambda_{1}}$ under the extension of
groups $P_{(\lambda_{1},\lambda_{2})}\subset P_{\lambda_{1}}$.

Each piece in the $\mathbb{Z}$-filtration
$(\text{gr}_{1}(\mathcal{F}^{\bullet}_{m}))_{m\in\mathbb{Z}}$ of the
associated graded
$\text{gr}_{1}(\mathcal{F}^{\bullet})=\bigoplus_{m\in\mathbb{Z}}\mathcal{F}^{\bullet}_{m}/\mathcal{F}_{m+1}^{\bullet}$
can be lifted to the original $\mathcal{F}^{\bullet}$ by taking the preimages
in each subquotient. This way we obtain a lexicographic
$\mathbb{Z}^{2}$-filtration
$(\mathcal{F}^{\bullet}_{(m_{1},m_{2})})_{(m_{1},m_{2})\in\mathbb{Z}^{2}}$ of
the original tuple $\mathcal{F}^{\bullet}$. This is the lexicographic
filtration coming from the 2-weighted parabolic reduction
$\left((\lambda_{1},\lambda_{2}),\mathcal{G}_{(\lambda_{1},\lambda_{2})}\right)$.
In particular
$(\mathcal{F}^{\bullet}_{(m_{1},m_{2})})_{(m_{1},m_{2})\in\mathbb{Z}^{2}}$ is
actually a lexicographic $\mathbb{Z}^{2}$-filtration of
$(\mathcal{F}^{\bullet},\sigma)$.

The cocharacter $\lambda_{2}$ induces a nontrivial direct sum decomposition of
the tuple $V^{\bullet}_{\lambda_{1}}$. We can think of this decomposition as a
(strictly) finer tuple $V^{\bullet}_{(\lambda_{1},\lambda_{2})}$ of direct
summands. The restriction of $\rho$ to the Levi subgroup
$L_{(\lambda_{1},\lambda_{2})}$ factors through
$\operatorname{GL}(V^{\bullet}_{(\lambda_{1},\lambda_{2})})$, which is a
smaller Levi subgroup of $\operatorname{GL}(V^{\bullet})$ contained in the
Levi $\operatorname{GL}(V^{\bullet}_{\lambda_{1}})$. We denote this
restriction by
$\rho_{(\lambda_{1},\lambda_{2})}:L_{(\lambda_{1},\lambda_{2})}\rightarrow\operatorname{GL}(V^{\bullet}_{(\lambda_{1},\lambda_{2})})$.
The associated graded $f_{can,2}|_{0}$ acquires canonically the structure of a
$\rho_{(\lambda_{1},\lambda_{2})}$-sheaf, denoted by
$\text{gr}_{2}(\mathcal{F}^{\bullet},\sigma):=(\text{gr}_{2}(\mathcal{F}^{\bullet}),\text{gr}_{2}(\sigma))$.
We call this the $2^{nd}$ associated graded Levi sheaf. We have that
$\text{gr}_{2}(\mathcal{F}^{\bullet})$ is the associated graded for the
lexicographic $\mathbb{Z}^{2}$-filtration
$(\mathcal{F}^{\bullet}_{(m_{1},m_{2})})_{(m_{1},m_{2})\in\mathbb{Z}^{2}}$
that we have defined. Moreover, its corresponding
$L_{(\lambda_{1},\lambda_{2})}$-rational bundle is the same as the one
obtained from the $P_{(\lambda_{1},\lambda_{2})}$-parabolic reduction
$\mathcal{G}_{(\lambda_{1},\lambda_{2})}$ via the Levi quotient map
$P_{(\lambda_{1},\lambda_{2})}\rightarrow L_{(\lambda_{1},\lambda_{2})}$. So
we can view $\text{gr}_{2}(\mathcal{F}^{\bullet})$ as the associated graded of
the 2-weighted parabolic reduction
$\left((\lambda_{1},\lambda_{2}),\mathcal{G}_{(\lambda_{1},\lambda_{2})}\right)$
777We entrust the reader with the precise formulation of what this means..

Recursion: If $\text{gr}_{2}(\mathcal{F}^{\bullet},\sigma)$ is not Gieseker
semistable, then we can iterate this construction and consider the leading
term HN filtration $f_{can,3}$, and so on. At the $j^{th}$ step of this
iteration we have a $j$-weighted parabolic reduction
$((\lambda_{1},\ldots,\lambda_{j}),\mathcal{G}_{(\lambda_{1},\ldots,\lambda_{j})})$
of $(\mathcal{F}^{\bullet},\sigma)$. The associated underlying lexicographic
$\mathbb{Z}^{j}$-filtration
$(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{j}}$ coming from
$((\lambda_{1},\ldots,\lambda_{j}),\mathcal{G}_{(\lambda_{1},\ldots,\lambda_{j}}))$
is uniquely determined up to scaling the weights. Recall that
$(\lambda_{1},\ldots,\lambda_{j})$ determines a Levi subgroup
$L_{(\lambda_{1},\ldots,\lambda_{j})}$ that centralizes all the cocharacters
in the tuple. We have an induced direct sum decomposition
$V^{\bullet}_{(\lambda_{1},\ldots,\lambda_{j})}$, so that the representation
$\rho$ restricted to $L_{(\lambda_{1},\ldots,\lambda_{j})}$ factors as
$\rho_{(\lambda_{1},\ldots,\lambda_{j})}:L_{(\lambda_{1},\ldots,\lambda_{j})}\rightarrow\operatorname{GL}(V^{\bullet}_{(\lambda_{1},\ldots,\lambda_{j})})$.
We also have a $j^{th}$ associated graded Levi
$\rho_{(\lambda_{1},\ldots,\lambda_{j})}$-sheaf
$(\text{gr}_{j}(\mathcal{F}^{\bullet}),\text{gr}_{j}(\sigma))$.

Termination: At each step of the process, the nontrivial direct sum
decomposition of $V^{\bullet}$ is finer than in the previous step. Since the
vector spaces $V^{i}$ are finite dimensional, this process must terminate
eventually. Suppose that it terminates at the $q^{th}$ step to yield a
$q$-weighted parabolic reduction
$\left(\vec{\lambda},\mathcal{G}_{\vec{\lambda}}\right)$. We call this a
Gieseker-Harder-Narasimhan (multi-weighted) parabolic reduction of
$(\mathcal{F}^{\bullet},\sigma)$. The corresponding $q^{th}$ associated graded
Levi $\rho_{\vec{\lambda}}$-sheaf
$(\text{gr}_{q}(\mathcal{F}^{\bullet}),\text{gr}_{q}(\sigma))$ is called the
Gieseker-Harder-Narasimhan associated Levi of
$(\mathcal{F}^{\bullet},\sigma)$. By construction,
$(\text{gr}_{q}(\mathcal{F}^{\bullet}),\text{gr}_{q}(\sigma))$ is Gieseker
semistable. We call the lexicographic $\mathbb{Z}^{q}$-filtration
$(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ the (multi-
weighted) Gieseker-Harder-Narasimhan filtration of the $\rho$-sheaf
$(\mathcal{F}^{\bullet},\sigma)$. This filtration is uniquely determined up to
scaling the weights by a rational tuple $\vec{p}$, because of the uniqueness
of the leading term HN filtration up to scaling at each step of the process.

This construction is recorded in the following definition.

######  Definition 8.10.

Suppose that the field extension $K\supset k$ is algebraically closed and let
$(\mathcal{F}^{\bullet},\sigma)$ be a $\rho$-sheaf on $X_{K}$. The Gieseker-
Harder-Narasimhan filtration of $(\mathcal{F}^{\bullet},\sigma)$ is the unique
(up to scaling of the weights) lexicographic $\mathbb{Z}^{q}$-filtration
$(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ of
$(\mathcal{F}^{\bullet},\sigma)$ constructed above. Its Gieseker-Harder-
Narasimhan associated Levi $\rho_{\vec{\lambda}}$-sheaf, as described above,
is Gieseker semistable.

If $K\supset k$ is not algebraically closed, then by uniqueness we can descend
the GHN filtration of the base change to any algebraic closure of $K$.
Therefore we obtain a unique (up to scaling the weights) lexicographic
$\mathbb{Z}^{q}$-filtration
$(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$, which we call
the GHN filtration of $(\mathcal{F}^{\bullet},\sigma)$.

######  Remark 8.11.

In particular, when the $\rho$-sheaf $(\mathcal{F}^{\bullet},\sigma)$ is
Gieseker semistable, the GHN filtration constructed above produces a trivial
filtration as in Definition 4.32 .

######  Remark 8.12.

If $G_{K}$ is split then, by construction, the GHN filtration
$(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ comes from a
$q$-weighted parabolic reduction
$(\vec{\lambda},\mathcal{G}_{\vec{\lambda}})$.

######  Remark 8.13.

Suppose in addition that $\rho$ is central. Then by Proposition 10.11 the
rational $P_{\vec{\lambda}}$-parabolic reduction $\mathcal{G}_{\vec{\lambda}}$
is finer than the canonical slope parabolic reduction of the corresponding
rational $G$-bundle, as in [AAB02] or [BG03, §7]. More precisely, the
parabolic subgroup $P_{\vec{\lambda}}$ is contained in the canonical slope
parabolic subgroup $P$ and the rational $P$-parabolic reduction
$\mathcal{G}_{P}$ obtained from $\mathcal{G}_{\vec{\lambda}}$ via extension of
structure group $P_{\vec{\lambda}}\subset P$ recovers the canonical slope
reduction.

In the case of torsion-free sheaves (i.e. when $G=\operatorname{GL}(V)$ and
$\rho$ is the identity), this filtration recovers the classical Gieseker-
Harder-Narasimhan filtration. We prove this in Subsection 9.1, where we
describe the procedure above in the case of torsion-free sheaves.

###  8.4. The GHN filtration in families

In this subsection we define a notion of relative GHN filtrations. We prove
that relative GHN filtrations induce a stratification of
$\operatorname{Bun}_{\rho}(X)$ by locally closed substacks.

Let $S$ be a $k$-scheme. Let $(\mathcal{F}^{\bullet},\sigma)$ be a
$\rho$-sheaf on $X_{S}$.

######  Definition 8.14.

A relative lexicographic $\mathbb{Z}^{q}$-filtration of the underlying tuple
of sheaves $\mathcal{F}^{\bullet}=(\mathcal{F}_{i})_{i=1}^{b}$ is a sequence
of tuples of subsheaves
$(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ satisfying the
following conditions

  1. (1)

$\mathcal{F}^{\bullet}_{\vec{m}}\subset\mathcal{F}^{\bullet}_{\vec{n}}$
whenever $\vec{m}>\vec{n}$.

  2. (2)

There exists some $M_{max}\in\mathbb{Z}^{q}$ such that
$\mathcal{F}^{\bullet}_{\vec{m}}=0$ for all $\vec{m}>M_{max}$. Similarly,
there exists some $M_{min}\in\mathbb{Z}^{q}$ such that
$\mathcal{F}^{\bullet}_{\vec{m}}=\mathcal{F}^{\bullet}$ for all
$\vec{m}<M_{min}$.

  3. (3)

For all $\vec{m}$, we have that
$\mathcal{F}^{\bullet}/\mathcal{F}^{\bullet}_{\vec{m}}$ is a tuple of families
of torsion-free sheaves on $X_{S}$.

For any such filtration
$f=(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$, we can apply
the multi-graded Rees construction in order to obtain a tuple
$\widetilde{\mathcal{F}^{\bullet}}_{f}$ of families of torsion-free sheaves on
$\Theta^{q}\times X_{S}$. Let $U$ be a big open subset of $X_{S}$ where the
associated graded $\text{gr}(f)$ is locally-free. One can use
$\mathbb{G}_{m}^{q}$-equivariance to conclude that the restriction
$(\widetilde{\mathcal{F}^{\bullet}}_{f})|_{\Theta^{q}\times U}$ is locally-
free. In particular, it can be viewed as a $\operatorname{GL}(V^{\bullet})$
bundle. The section $\sigma$ gives a $G$-reduction of structure group of
$(\widetilde{\mathcal{F}^{\bullet}}_{f})|_{\vec{1}\times U}$.

######  Definition 8.15.

We say that $f=(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$
is a relative lexicographic $\mathbb{Z}^{q}$-filtration of the $\rho$-sheaf
$(\mathcal{F}^{\bullet},\sigma)$ if the $G$-reduction induced by $\sigma$
extends (uniquely) to $\Theta_{k}^{q}\times U$.

We can also define relative $q$-weighted reductions for
$(\mathcal{F}^{\bullet},\sigma)$ in a similar way as in the case when
$S=\operatorname{Spec}(K)$. One can say as well when a relative lexicographic
$\mathbb{Z}^{q}$-filtration of $(\mathcal{F}^{\bullet},\sigma)$ comes from a
$q$-weighted parabolic reduction $(\vec{\lambda},\mathcal{G}_{\lambda})$, we
omit the details.

######  Definition 8.16.

Let $S$ and $(\mathcal{F}^{\bullet},\sigma)$ be as above and let
$(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ be a
lexicographic $\mathbb{Z}^{q}$-filtration of $(\mathcal{F}^{\bullet},\sigma)$,
where we might allow a different choice of $q$ over each connected component
of $S$. We say that
$(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ is a GHN
filtration for $(\mathcal{F}^{\bullet},\sigma)$ if for all points $s\in S$ the
restriction
$(\mathcal{F}^{\bullet}_{\vec{m}}|_{X_{s}})_{\vec{m}\in\mathbb{Z}^{q}}$ is the
GHN filtration of $(\mathcal{F}^{\bullet}|_{X_{s}},\sigma|_{X_{s}})$.

######  Definition 8.17.

Let $H\rightarrow S$ be a locally closed stratification of $S$. We say that
the $S$-scheme $H$ is universal for GHN filtrations of
$(\mathcal{F}^{\bullet},\sigma)$ if it represents the functor that classifies
pairs
$\left(T,\left[(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}]\right]\right)$,
where $T$ is a $S$-scheme and
$\left[(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}\right]$ is
an equivalence class of relative GHN filtrations of
$(\mathcal{F}^{\bullet}|_{X_{T}},\sigma|_{X_{T}})$, up to scaling the weights.

######  Theorem 8.18.

The relative GHN filtrations induce a stratification of
$\operatorname{Bun}_{\rho}(X)$ by locally closed substacks. Equivalently, for
any $k$-scheme $S$ and every $\rho$-sheaf $(\mathcal{F}^{\bullet},\sigma)$
there exists a unique locally closed stratification $H$ of $S$ that is
universal for GHN filtrations of $(\mathcal{F}^{\bullet},\sigma)$.

If, in addition, $S$ is locally of finite type over a field extension
$K\supset k$ such that $G_{K}$ is split, then the restriction of the universal
relative GHN filtration888 More precisely, we choose a representative up to
scaling.  to each connected component of the stratification comes from a
relative multi-weighted parabolic reduction.

###### Proof.

Let $S$ be a $k$-scheme and let $(\mathcal{F}^{\bullet},\sigma)$ be a
$\rho$-sheaf on $X_{S}$. We will denote by
$\operatorname{RelGies}(\mathcal{F}^{\bullet},\sigma)$ the functor that
parametrizes pairs
$\left(T,[(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}]\right)$,
where $T$ is a $S$-scheme and
$[(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}]$ is an
equivalence class of relative GHN filtrations of
$(\mathcal{F}^{\bullet}|_{X_{T}},\sigma|_{X_{T}})$ up to scaling the weights.
We would like to show that
$\operatorname{RelGies}(\mathcal{F}^{\bullet},\sigma)$ is represented by a
locally closed stratification of $S$. Since locally closed immersions satisfy
étale descent, we can check this after base-changing to a field extension
$K\supset k$ such that $G_{K}$ is split. After replacing $k$ with $K$, we can
assume without loss of generality that $G$ is split. Moreover, since
$\operatorname{Bun}_{\rho}(X)$ is locally of finite type over $k$, we can
reduce to the case when $S$ is of finite type over $k$. We will construct $H$
recursively.

First Step: We denote by $H_{1}$ the locally closed stratification of $S$ that
is universal for leading term HN filtrations of
$(\mathcal{F}^{\bullet},\sigma)$ (Proposition 4.37). There is a natural
forgetful functor
$\operatorname{RelGies}(\mathcal{F}^{\bullet},\sigma)\rightarrow H_{1}$ given
by sending a relative GHN filtration
$(\mathcal{F}^{\bullet}_{\vec{m}})_{\vec{m}\in\mathbb{Z}^{q}}$ to the relative
leading term HN filtration $(\mathcal{F}^{\bullet}_{m})_{m\in\mathbb{Z}}$
defined by

$\mathcal{F}^{\bullet}_{m}\vcentcolon=\bigcup_{(m_{2},m_{3},\ldots,m_{q})\in\mathbb{Z}^{q-1}}\mathcal{F}_{(m,m_{2},\ldots,m_{q})}$

The fact that this is a relative leading term HN filtration follows from the
pointwise construction of the GHN filtration and the fact that pulling back to
a fiber commutes with taking unions.

Choose a representative $f_{1}$ for the universal leading term HN filtration
of $(\mathcal{F}^{\bullet}|_{X_{H_{1}}},\sigma|_{X_{H_{1}}})$. By the last
statement in Proposition 4.37, the restriction of $f_{1}$ to each component of
$H_{1}$ comes from a relative weighted parabolic reduction