Motivic Donaldson–Thomas Invariants of Parabolic Higgs Bundles

Motivic Donaldson–Thomas Invariants of Parabolic
Higgs Bundles and Parabolic Connections on a Curve††This paper is a
contribution to the Special Issue on Integrability, Geometry, Moduli in honor
of Motohico Mulase for his 65th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Mulase.html

Roman FEDOROV †, Alexander SOIBELMAN ‡ and Yan SOIBELMAN §

R. Fedorov, A. Soibelman and Y. Soibelman

† University of Pittsburgh, Pittsburgh, PA, USA<EMAIL_ADDRESS>
‡ Aarhus University, Aarhus, Denmark<EMAIL_ADDRESS>
§ Kansas State University, Manhattan, KS, USA<EMAIL_ADDRESS>
Received November 19, 2019, in final form July 10, 2020; Published online July
27, 2020

Let $X$ be a smooth projective curve over a field of characteristic zero and
let $D$ be a non-empty set of rational points of $X$. We calculate the motivic
classes of moduli stacks of semistable parabolic bundles with connections on
$(X,D)$ and motivic classes of moduli stacks of semistable parabolic Higgs
bundles on $(X,D)$. As a by-product we give a criteria for non-emptiness of
these moduli stacks, which can be viewed as a version of the Deligne–Simpson
problem.

parabolic Higgs bundles; parabolic bundles with connections; motivic classes;
Donaldson–Thomas invariants; Macdonald polynomials

14D23; 14N35; 14D20

##  1 Introduction and main results

###  1.1 Overview

Let $\mathbf{k}$ be a field of characteristic zero and $X$ be a smooth
geometrically connected projective curve over $\mathbf{k}$ (geometric
connectedness means that $X$ remains connected after the base change to an
algebraic closure of $\mathbf{k}$). In [12] we calculated the motivic classes
of moduli stacks of semistable Higgs bundles on $X$. These motivic classes are
closely related to Donaldson–Thomas invariants, see [22, 23]. In [12] we also
calculated the motivic classes of moduli stacks of vector bundles with
connections on $X$ by relating them to the motivic classes of stacks of
semistable Higgs bundles.

In this paper, we extend these results to the parabolic case. Some of our
results are parallel to the results of A. Mellit in the case of finite fields
(see [30]). One difference is that we fix the eigenvalues of the residues.
Another difference is that by working over a field of characteristic zero, we
are also able to treat bundles with connections. We also note that the
calculation of the motivic class requires subtler techniques, than the
calculation of the volume of the corresponding stack over a finite field.

###  1.2 Moduli stacks

Let us briefly describe the moduli stacks whose motivic classes we will be
interested in. There will be three classes of stacks.

####  1.2.1 Parabolic bundles with connections

Let $D\subset X(\mathbf{k})$ be a non-empty set of rational points of $X$. A
_parabolic bundle_ of type $(X,D)$ is a collection
$\mathbf{E}=(E,E_{\bullet,\bullet})$, where $E$ is a vector bundle over $X$
and $E_{x,\bullet}$ is a flag in its fiber $E_{x}$ for $x\in D$:

$E_{x}=E_{x,0}\supseteq E_{x,1}\supseteq\dots\supseteq
E_{x,l}\supseteq\cdots,\qquad E_{x,l}=0\quad\text{for}\ l\gg 0.$

A _connection_ on $E$ with _poles bounded by $D$_ is a morphism of sheaves of
abelian groups $\nabla\colon E\to E\otimes\Omega_{X}(D)$ satisfying Leibniz
rule. (Here $\Omega_{X}$ is the canonical line bundle on $X$.) In this case
for $x\in D$ one defines the residue of the connection
$\operatorname{res}_{x}\nabla\in\operatorname{End}(E_{x})$. Let
$\zeta=\zeta_{\bullet,\bullet}=(\zeta_{x,j})$ be a sequence of elements of
$\mathbf{k}$ indexed by $D\times\mathbb{Z}_{>0}$ such that $\zeta_{x,j}=0$ for
$j\gg 0$. Let $\mathcal{C}{onn}(X,D,\zeta)$ denote the moduli stack
parameterizing collections $(E,E_{\bullet,\bullet},\nabla)$, where
$(E,E_{\bullet,\bullet})$ is a parabolic bundle of type $(X,D)$, $\nabla$ is a
connection on $E$ with poles bounded by $D$ such that
$(\operatorname{res}_{x}\nabla-\zeta_{x,j}1)(E_{x,{j-1}})\subset E_{x,j}$ for
all $x\in D$ and $j>0$. We usually skip $X$ and $D$ from the notation as they
are fixed, denoting $\mathcal{C}{onn}(X,D,\zeta)$ simply by
$\mathcal{C}{onn}(\zeta)$. We call the points of $\mathcal{C}{onn}(\zeta)$
_parabolic bundles with connections of type $(X,D)$ with eigenvalues $\zeta$._

For a parabolic bundle $\mathbf{E}=(E,E_{\bullet,\bullet})$ define the _class_
of $\mathbf{E}$ as the following collection of integers:

$(\operatorname{rk}E,\dim{E_{x,j-1}}-\dim{E_{x,j}},\deg E)\in\mathbb{Z}_{\geq
0}\times\mathbb{Z}_{\geq 0}[D\times\mathbb{Z}_{>0}]\times\mathbb{Z}.$ (1.1)

We also set $\operatorname{rk}\mathbf{E}:=\operatorname{rk}E$.

The stack $\mathcal{C}{onn}(\zeta)$ decomposes according to the classes of
parabolic bundles; denote the component corresponding to parabolic bundles of
class $\gamma$ by $\mathcal{C}{onn}_{\gamma}(\zeta)$. We will see that this
stack is an Artin stack of finite type over $\mathbf{k}$. One of our main
results (see Section 1.4 and Theorem 8.8) is the calculation of the motivic
class of this stack.

####  1.2.2 Parabolic Higgs bundles

Let $\zeta$ be as above. A _parabolic Higgs bundle with eigenvalues $\zeta$_
is a triple $(E,E_{\bullet,\bullet},\Phi)$, where $(E,E_{\bullet,\bullet})$ is
a parabolic bundle of type $(X,D)$, $\Phi\colon E\to E\otimes\Omega_{X}(D)$ is
a morphism of $\mathcal{O}_{X}$-modules (called a _Higgs field on
$(E,E_{\bullet,\bullet})$_) such that for all $x\in D$ and $j>0$ we have

$(\Phi-\zeta_{x,j}1)(E_{x,{j-1}})\subset E_{x,j}\otimes\Omega_{X}(D)_{x}.$

Denote the category and the stack of such Higgs bundles by
$\mathcal{H}{iggs}(\zeta)$. Unfortunately, this stack is not of finite type
over $\mathbf{k}$, and in fact, has an infinite motivic volume. To resolve the
problem we endow the category with a stability structure. Let
$\sigma=\sigma_{\bullet,\bullet}$ be a sequence of real numbers indexed by
$D\times\mathbb{Z}_{>0}$. Let $\varkappa\in\mathbb{R}_{\geq 0}$. We define
_the $(\varkappa,\sigma)$-degree_ of a parabolic bundle
$\mathbf{E}=(E,E_{\bullet,\bullet})$ by

$\deg_{\varkappa,\sigma}\mathbf{E}:=\varkappa\deg E+\sum_{x\in
D}\sum_{j>0}\sigma_{x,j}(\dim E_{x,j-1}-\dim E_{x,j})\in\mathbb{R}.$

If $\mathbf{E}\neq 0$, we define the _$(\varkappa,\sigma)$ -slope_ of
$\mathbf{E}$ as
$\deg_{\varkappa,\sigma}\mathbf{E}/\operatorname{rk}\mathbf{E}$.

We say that a sequence $\sigma=\sigma_{\bullet,\bullet}$ of real numbers
indexed by $D\times\mathbb{Z}_{>0}$ is a _sequence of parabolic weights_ if
for all $x\in D$ we have

$\sigma_{x,1}\leq\sigma_{x,2}\leq\cdots$ (1.2)

and for all $x$ and $j$ we have $\sigma_{x,j}\leq\sigma_{x,1}+1$. Let $\sigma$
be a sequence of parabolic weights. Let $\mathbf{E}=(E,E_{\bullet,\bullet})$
be a parabolic bundle. Let $F\subset E$ be a saturated vector subbundle (that
is, $E/F$ is torsion free). Set $F_{x,j}:=F_{x}\cap E_{x,j}$. Then
$\mathbf{F}:=(F,F_{\bullet,\bullet})$ is a parabolic bundle. We say that a
Higgs bundle $(\mathbf{E},\Phi)$ is _$\sigma$ -semistable_, if for all
saturated subbundles $F$ of $E$ preserved by $\Phi$ the $(1,\sigma)$-slope of
the corresponding parabolic bundle $\mathbf{F}$ is less than or equal to that
of $\mathbf{E}$. We have an open substack
$\mathcal{H}{iggs}_{\gamma}^{\sigma-{\rm ss}}(\zeta)$ of
$\mathcal{H}{iggs}_{\gamma}(\zeta)$ classifying $\sigma$-semistable parabolic
Higgs bundles. This stack is of finite type over $\mathbf{k}$; we will
calculate its motivic class (see Section 1.4 and Theorem 7.5).

We note that condition (1.2) is imposed on $\sigma$ to ensure that we have
Harder–Narasimhan filtrations for parabolic Higgs bundles. We also note that,
scaling $\varkappa$ and $\sigma$ by the same positive real number scales all
the slopes by the same number. This is why we restrict to the case
$\varkappa=1$ above (see Remark 3.10 for more details).

####  1.2.3 Semistable parabolic bundles with connections

We can also impose stability conditions on parabolic bundles with connections.
Moreover, for non-resonant connections we can work with more general stability
conditions, than those for Higgs bundles defined in the previous paragraph. A
sequence $\zeta$ as above is called _non-resonant_ if for all $x\in X$ and all
$i,j>0$ we have $\zeta_{x,i}-\zeta_{x,j}\notin\mathbb{Z}_{\neq 0}$. Take
$\varkappa\in\mathbb{R}_{\geq 0}$ and a sequence $\sigma$ of real numbers
indexed by $D\times\mathbb{Z}_{>0}$ and satisfying condition (1.2).

Assume that $\zeta$ is non-resonant. We define
$(\varkappa,\sigma)$-semistability of parabolic bundles with connections
similarly to semistability of Higgs bundles but using the
$(\varkappa,\sigma)$-slope. Denote the corresponding moduli stack by
$\mathcal{C}{onn}_{\gamma}^{(\varkappa,\sigma)-{\rm ss}}(\zeta)$; this is an
open substack of $\mathcal{C}{onn}_{\gamma}(\zeta)$. If $\zeta$ is resonant,
then $\sigma$ has to satisfy some additional conditions (see Proposition 8.12
and Remark 8.14).

###  1.3 Motivic Donaldson–Thomas invariants

Our formulas for motivic classes of the moduli stacks above are all given in
terms of certain motivic classes $\overline{B}_{\gamma}$ called _motivic
Donaldson–Thomas invariants_ (see Section 1.5 for this terminology), which we
are going to define. First of all, we recall that in [12, Section 2] we
defined (following earlier works [11, Section 1], [19], and [22]) the ring of
motivic classes of Artin stacks denoted $\operatorname{Mot}(\mathbf{k})$. We
also defined its dimensional completion
$\operatorname{\overline{Mot}}(\mathbf{k})$. For an Artin stack $\mathcal{S}$
of finite type over $\mathbf{k}$ we have its motivic class
$[\mathcal{S}]\in\operatorname{Mot}(\mathbf{k})$. We denote its image in
$\operatorname{\overline{Mot}}(\mathbf{k})$ by the same symbol.

For a curve $X$ and a partition $\lambda$ we defined the series
$J_{\lambda}^{\rm mot}(z),H_{\lambda}^{\rm
mot}(z)\in\operatorname{\overline{Mot}}(\mathbf{k})[[z]]$ in [12, Section
1.3.2]. The definitions (especially of $H_{\lambda}^{\rm mot}(z)$) are
somewhat long, so we will not recall them here inviting the reader to look
into [12]. We only note that $J_{\lambda}^{\rm mot}(z)$ and $H_{\lambda}^{\rm
mot}(z)$ are defined in terms of the motivic zeta-function of $X$ (cf. (3.3)
below). In particular, they only depend on $X$ but not on $D$. In this paper,
we will denote them by $J_{\lambda,X}^{\rm mot}(z)$ and $H_{\lambda,X}^{\rm
mot}(z)$ respectively to emphasize that they depend on the curve $X$ and to
ensure that they are not confused with motivic modified Macdonald polynomials
$\tilde{H}_{\lambda}^{\rm mot}(w_{\bullet};z)$ and with motivic
Hall–Littlewood polynomials $H_{\lambda}^{\rm mot}(w_{\bullet})$ defined
below.

The modified Macdonald polynomials $\tilde{H}_{\lambda}(w_{\bullet};q,z)$ are
symmetric functions in variables $w_{\bullet}=(w_{1},w_{2},\dots)$ with
coefficients in $\mathbb{Z}[q,z]$. In [30, Definition 2.5] the modified
Macdonald polynomials are defined as symmetric functions with coefficients in
$\mathbb{Q}[q,z]$ but it is well-known that the coefficients are integers
(see, e.g., [14] and references therein). Note that, formally speaking,
symmetric functions are not polynomials (they become polynomials upon plugging
in $w_{N+1}=w_{N+2}=\dots=0$). Let
$\mathbb{L}=\big{[}\mathbb{A}_{\mathbf{k}}^{1}\big{]}$ be the motivic class of
the affine line. We denote by $\tilde{H}_{\lambda}^{\rm mot}(w_{\bullet};z)$
the symmetric function with coefficients in
$\operatorname{Mot}(\mathbf{k})[z]$ obtained from
$\tilde{H}_{\lambda}(w_{\bullet};q,z)$ by substituting $\mathbb{L}$ for $q$.
We denote their images in the ring of symmetric functions with coefficients in
$\operatorname{\overline{Mot}}(\mathbf{k})[z]$ by $\tilde{H}_{\lambda}^{\rm
mot}(w_{\bullet};z)$ as well.

Let $\Gamma_{+}$ denote the commutative monoid of sequences
$(r,r_{\bullet,\bullet},d)$, where $r$ is a nonnegative integer,
$r_{\bullet,\bullet}$ is a sequence of nonnegative integers indexed by
$D\times\mathbb{Z}_{>0}$, $d$ is an integer, subject to the following
conditions:

  1. (i)

For all $x\in D$ we have $\sum\limits_{j=1}^{\infty}r_{x,j}=r$. In particular,
$r_{x,j}=0$ for $j$ large enough.

  2. (ii)

If $r=0$, then $d=0$ (and so $r_{x,j}=0$ for all $x$ and $j$).

The operation on $\Gamma_{+}$ is the componentwise addition. For
$\gamma=(r,r_{\bullet,\bullet},d)\in\Gamma_{+}$ we set
$\operatorname{rk}\gamma=r$. The significance of the monoid $\Gamma_{+}$ is
that the class of a parabolic bundle $\mathbf{E}$ defined by (1.1) is an
element of $\Gamma_{+}$. We also need a submonoid
$\Gamma_{+}^{\prime}\subset\Gamma_{+}$ given by $d\leq 0$. Consider the
completed monoid ring $\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}^{\prime}]]$,
we write its elements as
$\sum\limits_{\gamma\in\Gamma_{+}^{\prime}}A_{\gamma}e_{\gamma}$, where
$A_{\gamma}\in\operatorname{Mot}(\mathbf{k})$, $e_{\gamma}$ are basis vectors.
It is convenient to identify $e_{\gamma}$ with a monomial

$w^{r}\prod_{x\in D}\prod_{j=1}^{\infty}w_{x,j}^{r_{x,j}}z^{d},$

where $w_{\bullet,\bullet}=(w_{x,j})$ is a sequence of variables indexed by
$D\times\mathbb{Z}_{>0}$. Then we identify
$\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}^{\prime}]]$ with a subring of
$\operatorname{Mot}(\mathbf{k})\big{[}\big{[}w,w_{\bullet,\bullet},z^{-1}\big{]}\big{]}$.
Similarly, we consider the completed monoid ring
$\operatorname{\overline{Mot}}(\mathbf{k})[[\Gamma_{+}^{\prime}]]$. We note
that these rings are closely related to completed quantum tori considered in
[22, 23]. In our case, they are commutative, essentially because we are
working with 2-dimensional Calabi–Yau categories; see Sections 1.5 and 3.3 for
more details.

Finally, we need the notion of _plethystic exponent and logarithm_. Let
$\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}^{\prime}]]^{0}$ denote the subset
of $\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}^{\prime}]]$ consisting of
elements with zero constant terms. Then we have a bijection

$\operatorname{Exp}:\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}^{\prime}]]^{0}\to
1+\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}^{\prime}]]^{0}$

called the plethystic exponent. We refer the reader to Section 3.4 for the
definition. Let the plethystic logarithm $\operatorname{Log}$ be the inverse
bijection. Let us write

$\mathbb{L}\cdot\operatorname{Log}\left(\sum_{\lambda}w^{|\lambda|}J_{\lambda,X}^{\rm
mot}\big{(}z^{-1}\big{)}H_{\lambda,X}^{\rm mot}\big{(}z^{-1}\big{)}\prod_{x\in
D}\tilde{H}_{\lambda}^{\rm
mot}\big{(}w_{x,\bullet};z^{-1}\big{)}\right)=\sum_{\gamma\in\Gamma^{\prime}_{+}}\overline{B}_{\gamma}e_{\gamma},$

where the sum in the LHS is over all partitions. We call the elements
$\overline{B}_{\gamma}$ the _Donaldson–Thomas invariants_. Note that
$\overline{B}_{0}=0$.

When $X=\mathbb{P}^{1}_{\mathbf{k}}$, we can define motivic Donaldson–Thomas
invariants $B_{\gamma}$ by a simpler formula valid in
$\operatorname{Mot}(\mathbf{k})$:

$\mathbb{L}\cdot\operatorname{Log}\left(\sum_{\lambda}\frac{w^{|\lambda|}\prod\limits_{x\in
D}\tilde{H}_{\lambda}^{\rm
mot}\big{(}w_{x,\bullet};z^{-1}\big{)}}{\prod\limits_{h\in\operatorname{Hook}(\lambda)}\big{(}\mathbb{L}^{a(h)}-z^{-l(h)-1}\big{)}\big{(}\mathbb{L}^{a(h)+1}-z^{-l(h)}\big{)}}\right)=\sum_{\gamma\in\Gamma^{\prime}_{+}}B_{\gamma}e_{\gamma},$
(1.3)

where $\operatorname{Hook}(\lambda)$ stands for the set of hooks of $\lambda$,
$a(h)$ and $l(h)$ stand for the armlength and the leglength of the hook $h$
respectively. We show that for $X=\mathbb{P}^{1}$ the images of $B_{\gamma}$
in $\operatorname{\overline{Mot}}(\mathbf{k})$ are equal to
$\overline{B}_{\gamma}$.

###  1.4 Explicit formulas

The following explicit formulas for the motivic classes are parts of Theorem
8.8, Theorem 7.5, and Theorem 8.16 respectively. Let
$\gamma=(r,r_{\bullet,\bullet},d)\in\Gamma_{+}$, $\gamma\neq 0$. Let $\zeta$
be as in Section 1.2.1. For $\varkappa\in\mathbf{k}$ we define the
$(\varkappa,\zeta)$-degree and the $(\varkappa,\zeta)$-slope of parabolic
bundles similarly to $(\varkappa,\sigma)$-degree and
$(\varkappa,\sigma)$-slope defined in Section 1.2.3; the only difference is
that the $(\varkappa,\zeta)$-degree and $(\varkappa,\zeta)$-slope take values
in $\mathbf{k}$.

For each $\tau\in\mathbf{k}$, define the elements
$C_{\gamma}(\zeta)\in\operatorname{\overline{Mot}}(\mathbf{k})$, where
$\gamma$ ranges over elements of $\Gamma_{+}^{\prime}$ such that $\gamma=0$ or
the $(1,\zeta)$-slope of $\gamma$ is $\tau$, by the following formula

$\sum_{\begin{subarray}{c}\gamma\in\Gamma_{+}^{\prime}\\\
\deg_{1,\zeta}\gamma=\tau\operatorname{rk}\gamma\end{subarray}}\mathbb{L}^{-\chi(\gamma)}C_{\gamma}(\zeta)e_{\gamma}=\operatorname{Exp}\left(\sum_{\begin{subarray}{c}\gamma\in\Gamma_{+}^{\prime}\\\
\deg_{1,\zeta}\gamma=\tau\operatorname{rk}\gamma\end{subarray}}\overline{B}_{\gamma}e_{\gamma}\right),$

where $\chi(\gamma):=(g-1)r^{2}+\sum\limits_{x\in
D}\sum\limits_{j<j^{\prime}}r_{x,j}r_{x,j^{\prime}}$, $g$ is the genus of $X$.
Let $\gamma\in\Gamma_{+}$ be such that $\deg_{1,\zeta}\gamma=0$. Then,
according to Theorem 8.8, the stack $\mathcal{C}{onn}_{\gamma}(\zeta)$ is of
finite type over $\mathbf{k}$ and we have in
$\operatorname{\overline{Mot}}(\mathbf{k})$

$[\mathcal{C}{onn}_{\gamma}(\zeta)]=C_{(r,r_{\bullet,\bullet},d-Nr)}(\zeta),$

whenever $N$ is large enough. If $\deg_{1,\zeta}\gamma\neq 0$, then the stack
$\mathcal{C}{onn}_{\gamma}(\zeta)$ is empty.

Next, assume that $\zeta$ and $\sigma$ are as in Section 1.2.2, for
$\tau\in\mathbb{R}$ define the elements
$H_{\gamma}(\zeta,\sigma)\in\operatorname{\overline{Mot}}(\mathbf{k})$ by the
following formula

$\sum_{\begin{subarray}{c}\gamma\in\Gamma_{+}^{\prime}\\\
\deg_{0,\zeta}\gamma=0\\\
\deg_{1,\sigma}\gamma=\tau\operatorname{rk}\gamma\end{subarray}}\mathbb{L}^{-\chi(\gamma)}H_{\gamma}(\zeta,\sigma)e_{\gamma}=\operatorname{Exp}\left(\sum_{\begin{subarray}{c}\gamma\in\Gamma_{+}^{\prime}\\\
\deg_{0,\zeta}\gamma=0\\\
\deg_{1,\sigma}\gamma=\tau\operatorname{rk}\gamma\end{subarray}}\overline{B}_{\gamma}e_{\gamma}\right).$

Assume that $\deg_{0,\zeta}\gamma\\!=\\!0$, where
$\gamma\in\Gamma_{+}^{\prime}$. Then, according to Theorem 7.5, the stack
$\mathcal{H}{iggs}_{\gamma}^{\sigma-{\rm ss}}(\zeta)$ is of finite type over
$\mathbf{k}$ and we have

$\big{[}\mathcal{H}{iggs}_{\gamma}^{\sigma-{\rm
ss}}(\zeta)\big{]}=H_{(r,r_{\bullet,\bullet},d-Nr)}(\zeta,\sigma),$

whenever $N$ is large enough. If $\deg_{0,\zeta}\gamma\neq 0$, then the stack
$\mathcal{H}{iggs}_{\gamma}^{\sigma-{\rm ss}}(\zeta)$ is empty.

Finally, assume that $\zeta$ and $(\varkappa,\sigma)$ are as in Section 1.2.3.
For $\tau\in\mathbf{k}$, $\tau^{\prime}\in\mathbb{R}$ define the elements
$C_{\gamma}(\zeta,\varkappa,\sigma)\in\operatorname{\overline{Mot}}(\mathbf{k})$
by the following formula

$\sum_{\begin{subarray}{c}\gamma\in\Gamma_{+}^{\prime}\\\
\deg_{1,\zeta}\gamma=\tau\operatorname{rk}\gamma\\\
\deg_{\varkappa,\sigma}\gamma=\tau^{\prime}\operatorname{rk}\gamma\end{subarray}}\mathbb{L}^{-\chi(\gamma)}C_{\gamma}(\zeta,\varkappa,\sigma)e_{\gamma}=\operatorname{Exp}\left(\sum_{\begin{subarray}{c}\gamma\in\Gamma_{+}^{\prime}\\\
\deg_{1,\zeta}\gamma=\tau\operatorname{rk}\gamma\\\
\deg_{\varkappa,\sigma}\gamma=\tau^{\prime}\operatorname{rk}\gamma\end{subarray}}\overline{B}_{\gamma}e_{\gamma}\right).$

Let $\gamma\in\Gamma_{+}$ be such that $\deg_{1,\zeta}\gamma=0$. Then,
according to Theorem 8.16, we have

$\big{[}\mathcal{C}{onn}_{\gamma}^{(\varkappa,\sigma)-{\rm
ss}}(\zeta)\big{]}=C_{(r,r_{\bullet,\bullet},d-Nr)}(\zeta,\varkappa,\sigma),$

whenever $N$ is large enough. If $\deg_{1,\zeta}\gamma\neq 0$, then the stack
$\mathcal{C}{onn}_{\gamma}^{(\varkappa,\sigma)-{\rm ss}}(\zeta)$ is empty.

If $X=\mathbb{P}^{1}$, we get similar results valid in
$\operatorname{Mot}(\mathbf{k})$, by replacing $\overline{B}_{\gamma}$ with
$B_{\gamma}$ defined by a simpler formula (1.3).

######  Remark 1.1.

We note that each of the above motivic classes depends only on finitely many
DT-invariants. Indeed, $[\mathcal{C}{onn}_{\gamma}(\zeta)]$,
$\big{[}\mathcal{H}{iggs}_{\gamma}^{\sigma-{\rm ss}}(\zeta)\big{]}$, and
$\big{[}\mathcal{C}{onn}_{\gamma}^{(\varkappa,\sigma)-{\rm ss}}(\zeta)\big{]}$
depend only on $\overline{B}_{\gamma^{\prime}}$ with
$\operatorname{rk}\gamma^{\prime}\leq\operatorname{rk}\gamma$ and for a given
$\gamma$ there are only finitely many such
$\gamma^{\prime}\in\Gamma^{\prime}_{+}$.

######  Remark 1.2.

We note also that all the stacks whose motivic classes we are calculating are
of finite type over $\mathbf{k}$, so their motivic classes are defined in
$\operatorname{Mot}(\mathbf{k})$. However, we can only calculate their motivic
classes in $\operatorname{\overline{Mot}}(\mathbf{k})$ except when
$X=\mathbb{P}^{1}$. The reason is that, our calculation is based on the
calculation of motivic classes of stacks of vector bundles on $X$ (without
parabolic structures) with nilpotent endomorphisms. This calculation is
performed in [12] and is, in turn, based on the motivic analogue of Harder’s
residue formula (see [12, Theorem 1.5.1 and Section 4] and [15, Theorem
2.2.3]). This formula, which is essentially saying that “all vector bundles
have essentially the same motivic number of Borel reductions” involves some
limiting process and is, therefore, only valid in the completed ring
$\operatorname{\overline{Mot}}(\mathbf{k})$.

###  1.5 Aftermath

In Section 1.3 we defined the classes $\overline{B}_{\gamma}$. These classes
should be thought of as the Donaldson–Thomas invariants of the stack
$\mathcal{H}{iggs}(0)$ of parabolic Higgs bundles with nilpotent residues.
Note that this stack is the cotangent bundle of $\mathcal{B}{un}^{\rm
par}(X,D)$, while the stacks $\mathcal{H}{iggs}(\zeta)$ and
$\mathcal{C}{onn}(\zeta)$ are _twisted cotangent bundles_. We emphasize that
$\overline{B}_{\gamma}$ do not depend on $\zeta$, $\varkappa$, and $\sigma$.
The meaning of the formulas in Section 1.4 is that the Donaldson–Thomas
invariants of these twisted cotangent bundles are obtained by restricting the
range of $\gamma$ to the submonoid $\deg_{0,\zeta}\gamma=0$ in the case of
$\mathcal{H}{iggs}(\zeta)$ and to the submonoid $\deg_{1,\zeta}\gamma=0$ in
the case of $\mathcal{C}{onn}(\zeta)$.

Another feature of the formulas is that the motivic classes of the stacks
depend on _equations_ satisfied by $\varkappa$ and $\sigma$ rather than on
inequalities. In other words, there is no _wall-crossing_ in our case. This is
not very surprising, as the category $\mathcal{H}{iggs}(0)$, being a cotangent
bundle of $\mathcal{B}{un}^{\rm par}(X,D)$, is a 2-dimensional Calabi–Yau
category, cf. [36].

One can speculate that similar results should be valid for the twisted
cotangent stacks to the moduli stack of objects of any reasonable
1-dimensional category. Note that such cotangent stacks were studied by G.
Dobrovolska, V. Ginzburg, and R. Travkin in [9].

Another example of such a twisted cotangent stack is the category of vector
bundles with irregular connections and appropriate level structures. This
example is certainly more complicated as the corresponding abelian category
has infinite homological dimension. We hope to return to this question in
subsequent publications.

The formulas in Section 1.4 are explicit but complicated. However, one can see
that the motivic classes under considerations belong to the sub-$\lambda$-ring
of $\operatorname{\overline{Mot}}(\mathbf{k})$ generated by $\mathbb{L}$, $X$,
and the inverses of $\mathbb{L}^{i}-1$ for $i\geq 1$. We note that this ring
is probably _strictly larger_ , than the subring of
$\operatorname{\overline{Mot}}(\mathbf{k})$ generated by $\mathbb{L}$, the
symmetric powers $X^{(i)}$, and the inverses of $\mathbb{L}^{i}-1$ for $i\geq
1$. The reason is that $\operatorname{\overline{Mot}}(\mathbf{k})$ is unlikely
to be a special $\lambda$-ring, see Section 1.7. On the other hand, if
$X=\mathbb{P}^{1}$, then all our motivic classes are rational functions in
$\mathbb{L}$ with denominators being products of $\mathbb{L}^{i}-1$ for $i\geq
1$.

###  1.6 Other results

It is clear from the above formulas that we have a lot of equalities between
different motivic classes of Higgs bundles and bundles with connections. In
particular, we show in Propositions 9.1 and 9.2 that every motivic class of
the form $\big{[}\mathcal{H}{iggs}_{\gamma}^{\sigma-{\rm ss}}(\zeta)\big{]}$
or $\big{[}\mathcal{C}{onn}_{\gamma}^{(\varkappa,\sigma)-{\rm
ss}}(\zeta)\big{]}$ is equal to some motivic class of the form
$[\mathcal{C}{onn}_{\gamma}(\zeta)]$, provided that $\mathbf{k}$ is not a
finite extension of $\mathbb{Q}$. As a consequence, we derive from results of
Crawley-Boevey [8] a criterion of non-emptiness of our moduli stacks. It is
not difficult to see that if $X\neq\mathbb{P}^{1}_{\mathbf{k}}$, then the
stack $\big{[}\mathcal{H}{iggs}_{\gamma}^{\sigma-{\rm ss}}(\zeta)\big{]}$ is
non-empty if and only if $\deg_{0,\zeta}\gamma=0$, while the stack
$\big{[}\mathcal{C}{onn}_{\gamma}^{(\varkappa,\sigma)-{\rm ss}}(\zeta)\big{]}$
is non-empty if and only if $\deg_{1,\zeta}\gamma=0$. For
$X=\mathbb{P}^{1}_{\mathbf{k}}$ the question is much more subtle and is
related to the so-called Deligne–Simpson problem. This problem was originally
stated for $\mathbf{k}=\mathbb{C}$ in [40]. It may be reformulated for an
arbitrary algebraically closed field $\mathbf{k}$ of characteristic $0$ as
follows: given a sequence of $\mathfrak{gl}_{r}$-conjugacy classes
$C_{\bullet}$ indexed by $D$, does there exist a pair $(E,\nabla)$ consisting
of a rank $r$ vector bundle and a connection $\nabla$ on $E$ with poles
bounded by $D$ such that $\operatorname{Res}_{x}\nabla\in C_{x}$ for all $x\in
D$? Bundles with connections $(E,\nabla)$ parameterized by
$\mathcal{C}{onn}_{\gamma}\big{(}\mathbb{P}^{1}_{\mathbf{k}},D,\zeta\big{)}$
are exactly bundles with connections such that each residue
$\operatorname{Res}_{x}\nabla$ lies in the closure of a conjugacy class
determined by $\gamma$ and $\zeta$ (see [7]). If this conjugacy class is
semisimple for each $x\in D$, then the elements of
$\mathcal{C}{onn}_{\gamma}\big{(}\mathbb{P}^{1}_{\mathbf{k}},D,\zeta\big{)}$
are the solutions of the corresponding Deligne–Simpson problem. For a
comprehensive survey of the Deligne–Simpson problem, see [7, 24, 42].

We note that if $\mathbf{k}$ is not algebraically closed, one can ask a
subtler question of whether there is a $\mathbf{k}$-rational point in
$\big{[}\mathcal{H}{iggs}_{\gamma}^{\sigma-{\rm ss}}(\zeta)\big{]}$ or
$\big{[}\mathcal{C}{onn}_{\gamma}^{(\varkappa,\sigma)-{\rm
ss}}(\zeta)\big{]}$. We do not know the answer to this question. A somewhat
similar question for moduli spaces of quiver representations is considered by
V. Hoskins and F. Schaffhauser in [18].

At this point, we would like to emphasize that working with not necessarily
algebraically closed fields is inevitable in the motivic setup: even if one is
only interested in the case $\mathbf{k}=\mathbb{C}$, one still has to consider
all fields of characteristic zero, see Remark 2.2 below.

One of the motivations for this work is the non-abelian Hodge theory of C.
Simpson (see [39]). In this paper, Simpson constructs an equivalence between a
category of parabolic bundles with connections and a category of Higgs
bundles. In Proposition 9.4, we show that the corresponding stacks have equal
motivic classes. We note that neither statement can be derived from the other
(cf. [12, Remark 1.2.2]). We note also, that it is clear from our results that
there are many more equalities of motivic classes, than those that one can
guess from the non-abelian Hodge theory. We remark that V. Hoskins and S.
Pepin Lehalleur have shown in [17, Theorem 4.2] that the Voevodsky motives of
the coarse moduli spaces of bundles with connections and of Higgs bundles are
equal in the case when the rank and the degree are coprime. One can ask
whether this can be upgraded to the parabolic situation.

###  1.7 Other relations with previous work

We have already noted that our results are closely related with the results of
Mellit [30]. One difference is that Mellit counts the weighted number of
points over a finite field, while we work over a field of characteristic zero
and calculate motivic classes. Mellit counts the volumes of moduli stacks of
Higgs bundles but is not considering bundles with connections. Another
difference is that Mellit is not fixing the eigenvalues of Higgs fields.

On the other hand, Mellit’s answers are simpler as they do not involve
Schiffmann’s polynomials $H_{\lambda}$. In fact, Mellit’s simplification of
Schiffmann’s formula from [37] has nothing to do with parabolic structures.
This simplification is the content of Mellit’s papers [29, 30, 31]. We believe
that this simplification _does not go through_ in the motivic case because
Mellit is using the fact that the $\lambda$-ring structure on symmetric
functions is special (which, roughly speaking, means that
$\lambda_{\bullet}(xy)$ and $\lambda_{\bullet}(\lambda_{\bullet}(x))$ can be
expressed in terms of $\lambda_{\bullet}(x)$ and $\lambda_{\bullet}(y)$). It
is known (see [26]) that the Grothendieck $\lambda$-ring of varieties is not a
special $\lambda$-ring. We do not know whether the Grothendieck $\lambda$-ring
of _stacks_ $\operatorname{Mot}(\mathbf{k})$ and its completion
$\operatorname{\overline{Mot}}(\mathbf{k})$ are special. In any case, if one
replaces $\operatorname{\overline{Mot}}(\mathbf{k})$ by some its quotient that
is a special $\lambda$-ring, then one expects that the Mellit’s
simplifications are valid in this quotient. Examples of such quotients are the
Grothendieck ring of the category of Chow motives and the maximal special
quotient of $\operatorname{\overline{Mot}}(\mathbf{k})$ (see [26]).

The paper [6], although conjectural, contains an alternative approach to the
problem via upgrading the computation of the motivic class of Higgs bundles to
the problem about motivic Pandharipande–Thomas invariants on the non-compact
Calabi–Yau 3-fold associated with the spectral curve.

Note also that Mozgovoy and Schiffmann in [34] consider Higgs bundles with a
twist by an arbitrary line bundle of degree at least $2g-2$, where $g$ is the
genus of $X$. However, they do not consider parabolic structures and do not
fix eigenvalues.

Finally we note that the general philosophy of Donaldson–Thomas invariants and
the approach via motivic and cohomological Hall algebras (see [22, 23]) are
applicable to our situation. For more details about the approach that uses
motivic Hall algebras we refer the reader to [12, Section 1.6, Remark 3.6.3].

###  1.8 Organization of the article

In Section 3 we define the category $\mathcal{B}{un}^{\rm par}(X,D)$ of
parabolic bundles and its graded stack of objects denoted by the same letter.
Most of our stacks below will be stacks over $\mathcal{B}{un}^{\rm par}(X,D)$.

In Section 4 we study the stack of bundles with endomorphisms. This stack is
the main intermediate object in our calculations. First, we calculate the
motivic classes of stacks of parabolic bundles with nilpotent endomorphisms
with fixed generic type. The calculation is based on Theorem 4.4, saying that
these motivic classes are products of motivic classes of similar stacks
without parabolic structures and of “local stacks” independent of the curve.
This is a motivic analogue of [30, Theorem 5.6]. However, the proof in the
motivic case is significantly more involved and, hopefully, more conceptual.

The motivic classes of stacks of vector bundles (without parabolic structures)
with nilpotent endomorphisms are calculated in the proof of [12, Theorem
1.4.1]. Since the motivic classes of “local stacks” are independent of the
curve, it is enough to calculate them for $X=\mathbb{P}^{1}$; we do this using
the ideas and results of Mellit [30].

Then we use the formalism of plethystic powers to calculate the motivic
classes of stacks of parabolic bundles with arbitrary endomorphisms.

In Section 5 we study parabolic Higgs bundles with fixed eigenvalues. If the
eigenvalues are equal to zero, then the endomorphisms of a given parabolic
bundle and the Higgs fields on this bundle are parameterized by vector spaces
whose dimensions differ by some Euler characteristic. Thus, it is easy to
relate the motivic classes of the two stacks. If the eigenvalues are not zero,
then not every parabolic bundle admits a Higgs fields with these eigenvalues.
We give a criterion for existence of such a Higgs field in Lemma 5.3. This
allows us to express the motivic class of parabolic Higgs bundles with fixed
eigenvalues in terms of the motivic class of the so-called isoslopy parabolic
bundles with endomorphisms (see Proposition 5.4).

The motivic class of isoslopy parabolic bundles with endomorphisms is derived
from the results of Section 4 with the help of a factorization formula (see
Proposition 5.6). This is analogous to [12, Proposition 3.5.1].

In Section 6 we use a version of Kontsevich–Soibelman factorization formula to
calculate the motivic classes of stacks of semistable Higgs bundles. These
depend on two sets of parameters: the eigenvalues and the stability condition.
Somewhat surprisingly, these two sets come symmetrically in the answer.

Up to Section 7 we work with nonpositive vector bundles, that is, vector
bundles having no subbundle of positive degree. Without stability this
restriction is inevitable as otherwise the moduli stacks would have infinite
motivic volume. With a stability condition we can drop this technical
restriction; the motivic classes of semistable parabolic Higgs bundles whose
underlying vector bundles are not necessarily nonpositive are calculated in
Section 7.

In Section 8 we study the moduli stack of bundles with connections – with or
without stability condition. The strategy for connections is similar to that
for Higgs bundles except that the corresponding stacks are of finite type over
$\mathbf{k}$ even without stability conditions. So we first calculate the
motivic classes of bundles with connections with given eigenvalues without
stability conditions and without nonpositivity assumptions and then use a
version of Kontsevich–Soibelman factorization formula to calculate the motivic
classes of stacks of semistable bundles with connections.

In Section 9 we give a precise criterion for non-emptiness of moduli stacks of
Higgs bundles, bundles with connections, or semistable bundles with
connections. The idea is that such a stack is non-empty if and only if its
motivic class is non-zero. Using our explicit formulas we re-write each of our
motivic classes as the motivic class of a stack of bundles with connections
(without stability conditions). The non-emptiness of such a stack is decided
using Lemma 8.2 and Crawley-Boevey’s result [8, p. 1334, Corollary].

##  2 Preliminaries

###  2.1 Conventions

We denote by $\mathbf{k}$ a field of characteristic zero. We denote by $X$ a
smooth projective geometrically connected curve over $\mathbf{k}$ (recall that
geometric connectedness means that $X$ remains connected after the base change
to an algebraic closure of $\mathbf{k}$). We denote by $D$ a set of
$\mathbf{k}$-rational points of $X$ and by $\deg D$ the number of elements of
$D$.

If $E$ is a vector space or a vector bundle, we denote by $E^{\vee}$ the dual
vector space (resp. vector bundle). We identify vector bundles with their
sheaves of sections. If $F$ is a coherent sheaf, we denote by
$\mathcal{E}{nd}(F)$ the sheaf of its endomorphisms, we have
$\mathcal{E}{nd}(F)=F^{\vee}\otimes F$ if $F$ is a vector bundle.

####  2.1.1 Partitions and nilpotent matrices

By a partition we mean a non-increasing sequence of integers
$\lambda=\lambda_{1}\geq\lambda_{2}\geq\cdots$, where $\lambda_{l}=0$ for
$l\gg 0$. Set $|\lambda|:=\sum_{i}\lambda_{i}$. For partitions $\lambda$ and
$\mu$ we set
$\langle\lambda,\mu\rangle=\sum_{i}\lambda^{\prime}_{i}\mu^{\prime}_{i}$,
where $\lambda^{\prime}$ and $\mu^{\prime}$ are the conjugate partitions.

We denote by $\mathfrak{gl}_{r,\mathbf{k}}$ or simply by $\mathfrak{gl}_{r}$
the set of $r\times r$ matrices with entries in $\mathbf{k}$. We say that a
nilpotent matrix
$n\in\mathfrak{gl}_{|\lambda|}=\mathfrak{gl}_{|\lambda|,\mathbf{k}}$ is of
type $\lambda$, if for all $i\geq 1$ we have
$\dim\operatorname{Ker}n^{i}-\dim\operatorname{Ker}n^{i-1}=\lambda_{i}$. For
each partition $\lambda$ choose a nilpotent matrix $n_{\lambda}$ of type
$\lambda$. For concreteness, we can take for $n_{\lambda}$ the direct sum of
nilpotent Jordan blocks, where the number of blocks of size $i\times i$ is
equal to $\lambda_{i}-\lambda_{i+1}$.

A sequence $(w_{1},\dots,w_{l},\dots)$ we denote by $w_{\bullet}$.

####  2.1.2 Stacks

We will be working with stacks. All our stacks will have affine stabilizers.
Our stacks will be Artin stacks locally of finite type over a field except in
Section 4.3, where we will have to work with stacks whose points have
stabilizers of infinite type. For a stack $\mathcal{S}$ we often abuse
notation by writing $s\in\mathcal{S}$ to mean that $s$ is an object of the
groupoid $\mathcal{S}(\mathbf{k})$, or an object of the groupoid
$\mathcal{S}(K)$, where $K$ is an extension of $\mathbf{k}$. Following [27,
Chapter 5] we say that a $K^{\prime}$-point $\xi^{\prime}$ of $\mathcal{S}$ is
_equivalent_ to a $K^{\prime\prime}$-point $\xi^{\prime\prime}$ of
$\mathcal{S}$ if there is an extension $K\supset\mathbf{k}$ and
$\mathbf{k}$-embeddings $K^{\prime}\hookrightarrow K$,
$K^{\prime\prime}\hookrightarrow K$ such that $\xi^{\prime}_{K}$ is isomorphic
to $\xi^{\prime\prime}_{K}$ (as an object of $\mathcal{S}(K)$). The
corresponding equivalence classes are called _points of $\mathcal{S}$_; the
set of points is denoted by $|\mathcal{S}|$. See [12, Section 2] for more
details.

We write “morphism of stacks” to mean “1-morphism of stacks”. We write “of
finite type” to mean “of finite type over $\mathbf{k}$”.

###  2.2 Motivic functions and motivic classes

Recall that in [12, Section 2] we defined (following [11, Section 1], [19],
and [22]) the ring of motivic classes of Artin stacks denoted
$\operatorname{Mot}(\mathbf{k})$.

More generally, for an Artin stack $\mathcal{X}$ locally of finite type over
$\mathbf{k}$, we defined the $\operatorname{Mot}(\mathbf{k})$-module of
motivic functions on $\mathcal{X}$ denoted $\operatorname{Mot}(\mathcal{X})$.
For a morphism $f\colon\mathcal{X}\to\mathcal{Y}$ we have the pullback
homomorphism
$f^{*}\colon\operatorname{Mot}(\mathcal{Y})\to\operatorname{Mot}(\mathcal{X})$.
The pushforward homomorphism
$f_{!}\colon\operatorname{Mot}(\mathcal{X})\to\operatorname{Mot}(\mathcal{Y})$
is defined when $f$ is of finite type. We also defined the ring of completed
motivic classes, denoted $\operatorname{\overline{Mot}}(\mathbf{k})$, and
$\operatorname{\overline{Mot}}(\mathbf{k})$-modules of completed motivic
functions $\operatorname{\overline{Mot}}(\mathcal{X})$ with a (probably non-
injective) morphism
$\operatorname{Mot}(\mathcal{X})\to\operatorname{\overline{Mot}}(\mathcal{X})$.
We also defined the pullbacks and the pushforwards of completed motivic
functions.

We usually work with $\operatorname{Mot}(\mathbf{k})$ but our final results
are formulated in $\operatorname{\overline{Mot}}(\mathbf{k})$.

We defined the notion of a constructible subset of a stack. If
$\mathcal{X}\to\mathcal{Y}$ is a morphism of finite type, and
$\mathcal{S}\subset\mathcal{X}$ is a constructible subset, we defined the
motivic function
$[\mathcal{S}\to\mathcal{Y}]\in\operatorname{Mot}(\mathcal{Y})$. Recall [12,
Proposition 2.6.1]:

######  Proposition 2.1.

Assume that we are given $A,B\in\operatorname{Mot}(\mathcal{X})$ are such that
for all field extensions $K\supset\mathbf{k}$ and for all
$\mathbf{k}$-morphisms $\xi\colon\operatorname{Spec}K\to\mathcal{X}$ we have
$\xi^{*}A=\xi^{*}B$. Then $A=B$.

######  Remark 2.2.

The previous proposition is one of the reasons we have to work with arbitrary
fields. Indeed, even if we start with $\mathbf{k}=\mathbb{C}$, to be able to
apply the proposition we have to consider all finitely generated extensions of
$\mathbb{C}$; see, for example, Section 4.3.5.

In Section 9 we will need the following proposition.

######  Proposition 2.3.

An Artin stack of finite type over $\mathbf{k}$ is non-empty if and only if
its motivic class in $\operatorname{\overline{Mot}}(\mathbf{k})$ is not equal
to zero.

###### Proof.

The ‘if’ direction is obvious. For the other direction assume for a
contradiction that $\mathcal{S}$ is a non-empty Artin stack of finite type
over $\mathbf{k}$ such that
$[\mathcal{S}]=0\in\operatorname{\overline{Mot}}(\mathbf{k})$. This means that
for all $m\in\mathbb{Z}$ we have $[\mathcal{S}]\in
F^{m}\operatorname{Mot}(\mathbf{k})$, where $F^{\bullet}$ is the dimensional
filtration on $\operatorname{Mot}(\mathbf{k})$. According to [25, Propositions
3.5.6 and 3.5.9] every Artin stack of finite type with affine stabilizers has
a stratification by global quotients of the form $T/{\rm GL}_{n}$, where $T$
is a scheme. Thus, replacing $\mathcal{S}$ with a stratification and clearing
the denominators, we may assume that $\mathcal{S}$ is a disjoint union of
integral affine schemes. Recall from [12, Section 2.5] that
$\operatorname{Mot}(\mathbf{k})$ is the localization of the K-ring of
varieties $\operatorname{Mot}_{\rm var}(\mathbf{k})$ with respect to the
multiplicative set generated by $\mathbb{L}$ and $\mathbb{L}^{i}-1$, where
$i>0$. Thus, multiplying $\mathcal{S}$ by a certain product of these elements,
we may assume that the class of $\mathcal{S}$ in $\operatorname{Mot}_{\rm
var}(\mathbf{k})$ belongs to the subgroup $F^{m-1}\operatorname{Mot}_{\rm
var}(\mathbf{k})$ generated by the classes of the varieties of dimension at
most $m-1$, where $m=\dim\mathcal{S}$. Compactifying each top-dimensional
connected component of $\mathcal{S}$ and taking the resolution of
singularities, we may assume that $\mathcal{S}$ is the disjoint union of
smooth projective $\mathbf{k}$-varieties.

Recall that the Hodge–Deligne polynomial of a smooth projective variety $Y$ is

$\sum_{p,q=0}^{\dim Y}(-1)^{p+q}h^{p,q}(Y)u^{p}v^{q},$

where $h^{p,q}=\dim H^{q}(Y,\wedge^{p}\Omega_{Y})$. This extends uniquely to a
homomorphism $E\colon\operatorname{Mot}_{\rm
var}(\mathbf{k})\to\mathbb{Z}[u,v]$. Clearly, $E([Y])$ has degree $2m$, if $Y$
is a smooth projective variety of dimension $m$. On the other hand, $E([Y])$
has degree at most $2m-2$, if $Y$ is any variety of dimension at most $m-1$.
We see that, on the one hand $E(\mathcal{S})$ has degree $2m$, on the other
hand it has degree $2m-2$. We come to contradiction. ∎

###  2.3 Principal bundles and special groups

Let $H$ be an algebraic group of finite type over $\mathbf{k}$. Recall that a
_principal $H$-bundle_ over a $\mathbf{k}$-stack $\mathcal{B}$ is a stack $E$
together with a schematic smooth surjective morphism of finite type
$E\to\mathcal{B}$ and an action $a\colon H\times_{\mathbf{k}}E\to E$ such that
$H$ acts simply transitively on the fibers of $E\to\mathcal{B}$. More
precisely, the simple transitivity means that the morphism $(a,p_{2})\colon
H\times E\to E\times_{\mathcal{B}}E$ is an isomorphism, where $p_{2}\colon
H\times E\to E$ is the projection. A principal bundle is _trivial_ if there is
an isomorphism $E\approx H\times\mathcal{B}$ compatible with the action and
the projection to $\mathcal{B}$. If $E$ is a principal $H$-bundle over
$\mathcal{B}$ and $\mathcal{B}^{\prime}\to\mathcal{B}$ is a morphism, then one
gets an induced principal $H$-bundle
$E\times_{\mathcal{B}}\mathcal{B}^{\prime}$ over $\mathcal{B}^{\prime}$. The
group $H$ is called _special_ if every principal $H$-bundle $E$ over a scheme
$B$ of finite type over $\mathbf{k}$ is locally trivial over $B$ in the
Zariski topology.

The following lemma is standard, see, e.g., [3, Section 2.3].

######  Lemma 2.4.

Let $H$ be a special group and $E\to\mathcal{B}$ be a principal $H$-bundle,
where $\mathcal{B}$ is an Artin stack of finite type over $\mathbf{k}$. Then
in $\operatorname{Mot}(\mathbf{k})$ we have $[E]=[H][\mathcal{B}]$.

###### Proof.

The case when $\mathcal{B}$ is a scheme is easily proved by Noetherian
induction. If $\mathcal{B}$ is a stack, then, using again [25, Propositions
3.5.6 and 3.5.9], we may assume that $\mathcal{B}$ is a global quotient:
$\mathcal{B}=S/{\rm GL}_{n}$, where $S$ is a scheme. Then we have the
cartesian diagram

$\begin{CD}E^{\prime}@>{}>{}>E\\\ @V{}V{}V@V{}V{}V\\\
S@>{}>{}>\mathcal{B},\end{CD}$

where $E^{\prime}=E\times_{\mathcal{B}}S$ and $E=E^{\prime}/{\rm GL}_{n}$.
Applying [12, Corollary 2.2.2] with
$\mathcal{S}=\operatorname{Spec}\mathbf{k}$, we get $[E^{\prime}]=[E][{\rm
GL}_{n}]$ and $S=[\mathcal{B}][{\rm GL}_{n}]$. Next, $E^{\prime}$ is a
principal $H$-bundle over the scheme $S$ so $[E^{\prime}]=[H][S]$. Combining
these equations we easily get the required statement. ∎

Recall that a $\mathbf{k}$-group $U$ is _unipotent_ if it can be embedded into
a group of strictly upper triangular matrices. Every unipotent subgroup is
obtained from the additive group of the 1-dimensional vector space by iterated
extensions.

######  Lemma 2.5.

Let $H$ be an algebraic group of finite type over $\mathbf{k}$ and let $U$ be
a unipotent subgroup. Assume that $H/U$ is special. Then $H$ is special. In
particular, every unipotent group is special.

###### Proof.

Assume first that $H$ is a unipotent group. We claim that every principal
$H$-bundle over an affine scheme is trivial. We prove this by induction on
$\dim H$. If $\dim H=1$, then $H$ is the additive group and the principal
$H$-bundles over a scheme $B$ are classified by the coherent cohomology group
$H^{1}(B,\mathcal{O}_{B})$, which vanishes as soon as $B$ is affine. If $\dim
H>1$, then there is a subgroup $H^{\prime}\subset H$ such that $\dim
H^{\prime}<\dim H$ and $\dim(H/H^{\prime})<\dim H$. The groups $H^{\prime}$
and $H/H^{\prime}$ are unipotent. Recall that the principal $H$-bundles over
$B$ are classified by the first non-abelian étale cohomology group
$H_{\text{\'{e}t}}^{1}(B,H)$. Now the statement follows from the exact
sequence $H_{\text{\'{e}t}}^{1}(B,H^{\prime})\to H_{\text{\'{e}t}}^{1}(B,H)\to
H_{\text{\'{e}t}}^{1}(B,H/H^{\prime})$. In particular, every unipotent group
is special.

Now assume that $H/U$ is special, where $U$ is unipotent, and consider, for an
affine $B$, the exact sequence $1=H_{\text{\'{e}t}}^{1}(B,U)\to
H_{\text{\'{e}t}}^{1}(B,H)\to H_{\text{\'{e}t}}^{1}(B,H/U)$. We see that a
principal $H$-bundle is trivial over an affine scheme if the induced principal
$H/U$-bundle is trivial. The lemma follows. ∎

######  Lemma 2.6.

Let $Z_{\lambda}$ be the centralizer of $n_{\lambda}$ in ${\rm
GL}_{|\lambda|}$. Then $Z_{\lambda}$ is a special group.

###### Proof.

It is well-known that the quotient of $Z_{\lambda}$ by its unipotent radical
is the product of ${\rm GL}_{r_{i}}$ for some $r_{i}\in\mathbb{Z}_{>0}$. It
remains to note that ${\rm GL}_{r_{i}}$ are special groups and the product of
special groups is special. ∎

##  3 Parabolic bundles

###  3.1 Definitions and notations

Recall that $\mathbf{k}$ denotes a field of characteristic zero, $X$ stands
for a smooth projective geometrically connected curve over $\mathbf{k}$, and
$D$ is a set of rational points of $X$. We often assume that
$D\neq\varnothing$; in this case $X$ has a divisor of degree one defined over
$\mathbf{k}$. We will often have to consider sequences indexed by
$D\times\mathbb{Z}_{>0}$ or by $D\times\mathbb{Z}_{\geq 0}$. A typical
notation will be $r_{\bullet,\bullet}$. If $x\in D$, then $r_{x,\bullet}$
stands for the sequence $r_{x,1},r_{x,2},\dots$ (or $r_{x,0},r_{x,1},\dots$).

The monoid of all sequences $r_{\bullet,\bullet}$ indexed by
$D\times\mathbb{Z}_{>0}$ with terms $r_{x,j}$ in a commutative monoid $S$ and
such that $r_{x,j}=0$ for $j\gg 0$ (that is, functions on
$D\times\mathbb{Z}_{>0}$ with finite support) will be denoted by
$S[D\times\mathbb{Z}_{>0}]$.

######  Definition 3.1.

A _parabolic bundle_ of type $(X,D)$ is a collection
$(E,E_{\bullet,\bullet})$, where $E$ is a vector bundle over $X$ and
$E_{x,\bullet}$ is a flag in $E_{x}$ for $x\in D$:

$E_{x}=E_{x,0}\supseteq E_{x,1}\supseteq\dots\supseteq
E_{x,l}\supseteq\cdots,\qquad E_{x,l}=0\quad\text{for}\ l\gg 0.$

We have the category $\mathcal{B}{un}^{\rm par}(X,D)$ of parabolic bundles. We
sometimes denote a parabolic bundle by a single boldface letter:
$\mathbf{E}=(E,E_{\bullet,\bullet})$. The morphism from $\mathbf{E}$ to
$\mathbf{E}^{\prime}$ is a morphism $\varphi\colon E\to E^{\prime}$ such that
for all $x\in D$ and $j\geq 0$ we have $\varphi(E_{x,j})\subset
E^{\prime}_{x,j}$. This category is an additive $\mathbf{k}$-linear category.
The direct sum of $(E,E_{\bullet,\bullet})$ and
$(E^{\prime},E^{\prime}_{\bullet,\bullet})$ is $(E\oplus
E^{\prime},E_{\bullet,\bullet}\oplus E^{\prime}_{\bullet,\bullet})$. We note
that the decomposition of a parabolic bundle into a direct sum of
indecomposable parabolic bundles is unique up to isomorphism, while the
isotypic summands are unique; the proof is similar to [1, Theorem 3] (see also
[12, Proposition 3.1.2]).

We often skip $X$ and $D$ from the notation, writing $\mathcal{B}{un}^{\rm
par}$ instead of $\mathcal{B}{un}^{\rm par}(X,D)$.

Abusing notation, we denote by $\mathcal{B}{un}^{\rm par}$ the stack of
objects of the category $\mathcal{B}{un}^{\rm par}$. Precisely, if $S$ is a
$\mathbf{k}$-scheme, then $\mathcal{B}{un}^{\rm par}(S)$ is the groupoid of
collections $(E,E_{\bullet,\bullet})$, where $E$ is a vector bundle over
$S\times_{\mathbf{k}}X$, $E_{x,\bullet}$ is a filtration by vector subbundles
of the restriction of $E$ to $S\times_{\mathbf{k}}x$ for $x\in D$. Here by a
subbundle of $E|_{S\times_{\mathbf{k}}x}$ we mean a subsheaf that splits off
as a direct summand Zariski locally over $S$.

###  3.2 Monoids $\boldsymbol{\Gamma_{+}}$ and
$\boldsymbol{\Gamma^{\prime}_{+}}$

Consider the free abelian group
$\mathbb{Z}\times\mathbb{Z}[D\times\mathbb{Z}_{>0}]\times\mathbb{Z}$ and its
subgroup $\Gamma$ consisting of $(r,r_{\bullet,\bullet},d)$ such that for all
$x\in D$ we have $\sum_{j=1}^{\infty}r_{x,j}=r$.

Let $\Gamma_{+}\subset\Gamma$ be the monoid of sequences
$(r,r_{\bullet,\bullet},d)$ such that

  1. (i)

$r\geq 0$ and for all $x\in D$ and $j>0$ we have $r_{x,j}\geq 0$;

  2. (ii)

if $r=0$, then $d=0$.

Note that it follows from these conditions that $r=0$ implies that
$(r,r_{\bullet,\bullet},d)$ is the zero sequence; we denote it by 0. Define
the _class function_ :

$\operatorname{cl}\colon\ \mathcal{B}{un}^{\rm
par}\to\Gamma_{+},\qquad(E,E_{\bullet,\bullet})\mapsto(\operatorname{rk}E,\dim
E_{x,j-1}-\dim E_{x,j},\deg E).$

For $\gamma=(r,r_{\bullet,\bullet},d)\in\Gamma$ we set
$\operatorname{rk}\gamma:=r$. For a parabolic bundle $\mathbf{E}$ we set
$\operatorname{rk}\mathbf{E}:=\operatorname{rk}\operatorname{cl}(\mathbf{E})$.

For $\gamma\in\Gamma_{+}$, we denote by $\mathcal{B}{un}^{\rm par}_{\gamma}$
the stack of objects of class $\gamma$; this is an open and closed substack of
$\mathcal{B}{un}^{\rm par}$. It is often convenient to think of
$\mathcal{B}{un}^{\rm
par}=\bigsqcup\limits_{\gamma\in\Gamma_{+}}\mathcal{B}{un}_{\gamma}^{\rm par}$
as a $\Gamma_{+}$-graded stack. Note that $\mathcal{B}{un}^{\rm par}_{0}$ has
a single object: the parabolic bundle of rank zero.

Let $\gamma=(r,r_{\bullet,\bullet},d)\in\Gamma_{+}$. The projection
$\mathcal{B}{un}_{\gamma}^{\rm par}(X,D)\to\mathcal{B}{un}_{r,d}(X)$ to the
stack of rank $r$ degree $d$ vector bundles on $X$ is schematic and of finite
type (in fact, projective). Thus $\mathcal{B}{un}_{\gamma}^{\rm par}(X,D)$ and
$\mathcal{B}{un}^{\rm par}(X,D)$ are Artin stacks locally of finite type.

Let us call a vector bundle on $X$ _nonpositive_ if it does not have a
subbundle of positive degree. Recall that in [12, Section 3.2]) we called a
vector bundle on $X$ HN-nonnegative, if its Harder–Narasimhan spectrum is
nonnegative. by [12, Lemma 3.2.1(i)] there is an open substack
$\mathcal{B}{un}^{+}(X)\subset\mathcal{B}{un}(X)$ classifying HN-nonnegative
vector bundles. Moreover, by [12, Lemma 3.2.1(iii)] the substack of
$\mathcal{B}{un}^{+}(X)$ corresponding to vector bundles of rank $r$ and
degree $d$ is of finite type (this substack was denoted by
$\mathcal{B}{un}_{r,d}^{\geq 0}(X)$ in loc. cit.).

By [12, Lemma 3.2.1(ii)] a bundle $E$ is HN-nonnegative if and only if it has
no quotient bundles of negative degree. Thus a vector bundle $E$ is
nonpositive if and only if its dual $E^{\vee}$ is HN-nonnegative. The
following lemma is now clear.

######  Lemma 3.2.

There is an open substack $\mathcal{B}{un}^{-}(X)$ of $\mathcal{B}{un}(X)$
classifying nonpositive vector bundles. The assignment $E\mapsto E^{\vee}$ is
an isomorphism between $\mathcal{B}{un}^{-}(X)$ and $\mathcal{B}{un}^{+}(X)$.
The component of $\mathcal{B}{un}^{-}(X)$ corresponding to vector bundles of
fixed degree and rank is of finite type.

Set

$\mathcal{B}{un}^{{\rm par},-}=\mathcal{B}{un}^{{\rm
par},-}(X,D):=\mathcal{B}{un}^{\rm
par}(X,D)\times_{\mathcal{B}{un}(X)}\mathcal{B}{un}^{-}(X).$

In other words, $\mathcal{B}{un}^{{\rm par},-}$ is the stack (and the
category) of parabolic bundles on $X$ whose underlying vector bundle is
nonpositive. Set also

$\mathcal{B}{un}^{{\rm par},-}_{\gamma}=\mathcal{B}{un}^{{\rm
par},-}_{\gamma}(X,D):=\mathcal{B}{un}^{\rm
par}_{\gamma}(X,D)\times_{\mathcal{B}{un}(X)}\mathcal{B}{un}^{-}(X).$

######  Lemma 3.3.

For all $\gamma\in\Gamma_{+}$ the stack $\mathcal{B}{un}^{{\rm
par},-}_{\gamma}(X,D)$ is an Artin stack of finite type.

###### Proof.

Follows from Lemma 3.2. ∎

Let $\Gamma^{\prime}_{+}$ be the submonoid of $\Gamma_{+}$ consisting of
sequences with $d\leq 0$. Clearly, $\mathcal{B}{un}_{\gamma}^{{\rm
par},-}\neq\varnothing$ only if $\gamma\in\Gamma^{\prime}_{+}$.

###  3.3 Categories over $\boldsymbol{\mathcal{B}{un}^{\rm par}}$ and
$\boldsymbol{\Gamma_{+}}$-graded stacks

We will consider below many categories with a forgetful functor to
$\mathcal{B}{un}^{\rm par}$ (e.g., the category of parabolic Higgs bundles).
Let $\mathcal{C}$ be such a category and denote by $\mathcal{C}$ its stack of
objects as well. Assume that the morphism $\mathcal{C}\to\mathcal{B}{un}^{\rm
par}$ is of finite type. Define the stacks

$\displaystyle\mathcal{C}_{\gamma}:=\mathcal{C}\times_{\mathcal{B}{un}^{\rm
par}}\mathcal{B}{un}^{\rm
par}_{\gamma},\qquad\\!\\!\mathcal{C}^{-}:=\mathcal{C}\times_{\mathcal{B}{un}^{\rm
par}}\mathcal{B}{un}^{{\rm
par},-},\qquad\\!\\!\mathcal{C}_{\gamma}^{-}:=\mathcal{C}\times_{\mathcal{B}{un}^{\rm
par}}\mathcal{B}{un}^{{\rm par},-}_{\gamma}.\\!\\!$ (3.1)

By Lemma 3.3, the stack $\mathcal{C}_{\gamma}^{-}$ is of finite type. The
stack $\mathcal{C}=\bigsqcup\limits_{\gamma\in\Gamma_{+}}\mathcal{C}_{\gamma}$
is $\Gamma_{+}$-graded, while the stack
$\mathcal{C}^{-}=\bigsqcup\limits_{\gamma\in\Gamma_{+}}\mathcal{C}^{-}_{\gamma}$
is $\Gamma^{\prime}_{+}$-graded. Moreover, the stack $\mathcal{C}^{-}$ is of
finite type as a graded stack, that is, its graded components are stacks of
finite type.

The group ring $\operatorname{Mot}(\mathbf{k})[\Gamma_{+}]$ is closely related
to _quantum tori_ (cf. [22]). This has a natural basis $e_{\gamma}$, where
$\gamma$ ranges over $\Gamma_{+}$; the multiplication is given by
$e_{\gamma}e_{\gamma^{\prime}}=e_{\gamma+\gamma^{\prime}}$. The reason for the
multiplication in the quantum torus to be commutative is that we are actually
working with 2-dimensional Calabi–Yau categories, and hence the
skewsymmetrization of the Euler form vanishes (cf. [36]).

Let $\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}]]$ be the completion of
$\operatorname{Mot}(\mathbf{k})[\Gamma_{+}]$ (this can be viewed as the group
of $\operatorname{Mot}(\mathbf{k})$-valued functions on $\Gamma_{+}$).

If $\mathcal{D}$ is a $\Gamma_{+}$-graded stack of finite type, we consider
the generating series

$[\mathcal{D}]:=\sum_{\gamma\in\Gamma_{+}}[\mathcal{D}_{\gamma}]e_{\gamma}\in\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}]].$
(3.2)

We call $[\mathcal{D}]$ the _graded motivic class_ of the stack $\mathcal{D}$.
Recall that in [22] the motivic Donaldson–Thomas series was defined as an
element of the completed motivic quantum torus. In our case, it associates to
a $\Gamma_{+}$-graded stack $\mathcal{D}$ an infinite series in the
commutative motivic quantum torus corresponding to the monoid $\Gamma_{+}$
endowed with the trivial bilinear form. Thus, it coincides with
$[\mathcal{D}]$.

Sometimes it is convenient to write $e_{\gamma}$ explicitly as

$e_{\gamma}=w^{r}\prod_{x\in
D}\prod_{j=1}^{\infty}w_{x,j}^{r_{x,j}}z^{d}\in\mathbb{Z}\big{[}\big{[}w,w_{\bullet,\bullet},z,z^{-1}\big{]}\big{]}\subset\operatorname{Mot}(\mathbf{k})\big{[}\big{[}w,w_{\bullet,\bullet},z,z^{-1}\big{]}\big{]},$

where $\gamma=(r,r_{\bullet,\bullet},d)$. Here $w=w_{\bullet,\bullet}$ stands
for the collection of variables

$w_{x,\bullet}:=(w_{x,1},w_{x,2},\dots,w_{x,j},\dots),\qquad x\in D.$

The variables $w$, $z$, and $w_{x,j}$ for $x\in D$, $j\geq 1$ are commuting
variables.

######  Remark 3.4.

Note that we do not fix the lengths of flags. Let us fix a function $l\colon
D\to\mathbb{Z}_{>0}$ and consider only flags of length at most $l(x)$ at $x$.
Let $\Gamma_{+,l}$ be the submonoid of $\Gamma_{+}$ consisting of sequences
$(r,r_{\bullet,\bullet},d)$ such that $r_{x,j}=0$ whenever $j>l(x)$. We have
the obvious projection $\Gamma_{+}\to\Gamma_{+,l}$, which induces a
homomorphism
$\Pi_{l}\colon\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}]]\to\operatorname{Mot}(\mathbf{k})[[\Gamma_{+,l}]]$.
Explicitly, this is just setting $w_{x,j}=0$ whenever $j>l(x)$. Let
$\mathcal{D}$ be as in (3.2), then
$\Pi_{l}[\mathcal{D}]=\sum\limits_{\gamma\in\Gamma_{+,l}}[\mathcal{D}_{\gamma}]e_{\gamma}\in\operatorname{Mot}(\mathbf{k})[[\Gamma_{+,l}]]$
is the graded motivic class of the substack of $\mathcal{D}$ where the lengths
of the flags are bounded by $l$. We see that the difference between fixing the
length of flags and allowing flags of arbitrary lengths corresponds on the
quantum torus side to the difference between polynomials in infinite number of
variables and polynomials in finite number of variables.

In our applications, $[\mathcal{D}]$ will be symmetric in each sequence of
variables $w_{x,\bullet}$ (cf. Remark 4.2). In this case, the difference
between fixing and not fixing the lengths corresponds on the side of motivic
classes to the difference between symmetric polynomials and symmetric
functions, cf. [28, Chapter 1, Section 2].

We emphasize that $\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}]]$ is not a
ring. However,
$\operatorname{Mot}(\mathbf{k})[[\Gamma^{\prime}_{+}]]\subset\operatorname{Mot}(\mathbf{k})[[w,w_{\bullet,\bullet},z^{-1}]]$
is a ring and
$[\mathcal{C}^{-}]\in\operatorname{Mot}(\mathbf{k})[[\Gamma^{\prime}_{+}]]$
whenever $\mathcal{C}$ is a stack of finite type over $\mathcal{B}{un}^{\rm
par}$. This is in accordance to the general theory in [22], where one fixes a
strict sector in $\mathbb{R}^{2}$ in order to have well-defined
Donaldson–Thomas invariants. We can also replace
$\operatorname{Mot}(\mathbf{k})$ with
$\operatorname{\overline{Mot}}(\mathbf{k})$ in all the above constructions.

###  3.4 Motivic zeta-functions and plethystic operations

Following [21], for a variety $Y$ define its _motivic zeta-funcion_ by

$\zeta_{Y}(z):=\sum_{n=0}^{\infty}\big{[}Y^{(n)}\big{]}z^{n}\in\operatorname{Mot}(\mathbf{k})[[z]],$
(3.3)

where $Y^{(n)}=Y^{n}/\Sigma_{n}$ is the $n$-th symmetric power of $Y$
($\Sigma_{n}$ denotes the group of permutations). Consider the group
$(1+z\operatorname{Mot}(\mathbf{k})[[z]])^{\times}$, where the group operation
is multiplication. According to [10, Theorem 2.3] $\zeta$ can be uniquely
extended to a homomorphism

$\zeta\colon\
\operatorname{Mot}(\mathbf{k})\to(1+z\operatorname{Mot}(\mathbf{k})[[z]])^{\times}\colon\
A\mapsto\zeta_{A}(z)$

such that we have
$\zeta_{\mathbb{L}^{n}A}(z)=\zeta_{A}\big{(}\mathbb{L}^{n}z\big{)}$ for all
$n\in\mathbb{Z}$ and $A\in\operatorname{Mot}(\mathbf{k})$. Clearly,

$\zeta_{A}(z)\equiv 1+Ax\pmod{z^{2}}.$

Thus we have equipped $\operatorname{Mot}(\mathbf{k})$ with a $\lambda$-ring
structure. Note that $\operatorname{Mot}(\mathbf{k})$ is _not_ a special
$\lambda$-ring, in particular, $\zeta(AB)$ cannot be expressed in terms of
$\zeta(A)$ and $\zeta(B)$ (so some authors would call this a
pre-$\lambda$-ring structure).

According to loc. cit., this homomorphism $\zeta$ is continuous with respect
to the dimensional filtration on $\operatorname{Mot}(\mathbf{k})$, so it
extends to a homomorphism

$\zeta\colon\
\operatorname{\overline{Mot}}(\mathbf{k})\to\big{(}1+z\operatorname{\overline{Mot}}(\mathbf{k})[[z]]\big{)}^{\times},$

which coincides with the one constructed in [12, Section 1.3.1].

Let $\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}^{\prime}]]^{0}$ stand for the
series without constant term. We define the _plethystic exponent_
$\operatorname{Exp}\colon\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}^{\prime}]]^{0}\to(1+\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}^{\prime}]]^{0})^{\times}$
by

$\operatorname{Exp}\left(\sum_{\gamma\in\Gamma_{+}^{\prime}}A_{\gamma}e_{\gamma}\right)=\prod_{\gamma\in\Gamma_{+}^{\prime}}\operatorname{Exp}(A_{\gamma}e_{\gamma})=\prod_{\gamma\in\Gamma_{+}^{\prime}}\zeta_{A_{\gamma}}(e_{\gamma}).$

One shows easily that this is an isomorphism of abelian groups. Denote the
inverse isomorphism by $\operatorname{Log}$ (the _plethystic logarithm_).
Finally, we define the _plethystic power_ by

$\operatorname{\mathrm{Pow}}\colon\
\big{(}1+\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}^{\prime}]]^{0}\big{)}\times\operatorname{Mot}(\mathbf{k})\to
1+\operatorname{Mot}(\mathbf{k})[[\Gamma_{+}^{\prime}]]^{0}\colon\
(f,A)\mapsto\operatorname{Exp}(A\operatorname{Log}(f)).$

We note that we can similarly define $\operatorname{Exp}$,
$\operatorname{Log}$, and $\operatorname{\mathrm{Pow}}$ for the completed ring
$\operatorname{\overline{Mot}}(\mathbf{k})$, which coincide with the
operations defined in [12] when $D=\varnothing$.

###  3.5 Parabolic subbundles and quotient bundles

Let $\mathbf{E}=(E,E_{\bullet,\bullet})$ be a parabolic bundle of type
$(X,D)$. We say that
$\mathbf{E}^{\prime}=(E^{\prime},E^{\prime}_{\bullet,\bullet})$ is a _strict_
parabolic subbundle of $\mathbf{E}$ if $E^{\prime}$ is a saturated subbundle
of $E$ (that is, $E/E^{\prime}$ is torsion free) and for all $x$ and $j$ we
have $E^{\prime}_{x,j}=E_{x,j}\cap E^{\prime}_{x}$. Note that strict parabolic
subbundles of $\mathbf{E}=(E,E_{\bullet,\bullet})$ are in bijective
correspondence with saturated subbundles of $E$. Let
$\mathbf{E}^{\prime}=(E^{\prime},E^{\prime}_{\bullet,\bullet})$ be a strict
parabolic subbundle of $\mathbf{E}=(E,E_{\bullet,\bullet})$; set
$E^{\prime\prime}:=E/E^{\prime}$. Then we have a parabolic structure on
$E^{\prime\prime}$ given by
$E^{\prime\prime}_{x,j}:=E_{x,j}/E^{\prime}_{x,j}$. We call the parabolic
bundle
$\mathbf{E}/\mathbf{E}^{\prime}:=(E^{\prime\prime},E^{\prime\prime}_{\bullet,\bullet})$
the _quotient parabolic bundle_ of $\mathbf{E}$. Thus, the quotient parabolic
bundles of $\mathbf{E}$ are also in bijective correspondence with saturated
subbundles of $E$. Finally, in the above situation, we say that

$0\to\mathbf{E}^{\prime}\to\mathbf{E}\to\mathbf{E}/\mathbf{E}^{\prime}\to 0$
(3.4)

is a _short exact sequence_. We also say that _$\mathbf{E}$ is an extension of
$\mathbf{E}/\mathbf{E}^{\prime}$ by $\mathbf{E}^{\prime}$._ It is clear that
in this case we have
$\operatorname{cl}(\mathbf{E})=\operatorname{cl}(\mathbf{E}^{\prime})+\operatorname{cl}(\mathbf{E}/\mathbf{E}^{\prime})$.
One can use the short exact sequences above to define the group
$K_{0}(\mathcal{B}{un}^{\rm par})$; the class function $\operatorname{cl}$
extends to $\operatorname{cl}\colon K_{0}(\mathcal{B}{un}^{\rm
par})\to\Gamma$.

######  Remark 3.5.

The category $\mathcal{B}{un}^{\rm par}$ is not abelian. It can be extended to
an abelian category by viewing vector bundles with flags as coherent sheaves
on orbifold curves. Then the abelian category is the category of coherent
sheaves on this orbifold. This extension will not be used in the current
paper. On the other hand, if we define short exact sequences in
$\mathcal{B}{un}^{\rm par}$ as sequences isomorphic to some sequence of the
form (3.4), then $\mathcal{B}{un}^{\rm par}$ becomes an exact category in the
sense of Quillen.

Let $\varphi\colon E\to F$ be a morphism of vector bundles on $X$. We say that
$\varphi$ is _generically an isomorphism_ if it is an isomorphism at the
generic point of $X$. Equivalently, $\varphi$ is an isomorphism over a non-
empty Zariski open subset of $X$. Another reformulation is that $\varphi$ is
injective and $F/\varphi(E)$ is a torsion sheaf. Sometimes one says in this
situation that $E$ is a lower modification of $F$.

######  Definition 3.6.

We say that a morphism of parabolic bundles
$\varphi\colon(E,E_{\bullet,\bullet})\to(F,F_{\bullet,\bullet})$ is
_generically an isomorphism_ if the underlying morphism $E\to F$ is
generically an isomorphism.

######  Lemma 3.7.

Let $\varphi\colon\mathbf{E}\to\mathbf{F}$ be a morphism of parabolic bundles.
Then there are strict parabolic subbundles
$\mathbf{E}^{\prime}\subset\mathbf{E}$ and
$\mathbf{F}^{\prime}\subset\mathbf{F}$ such that $\varphi$ can be decomposed
as

$\mathbf{E}\xrightarrow{\varphi_{1}}\mathbf{E}/\mathbf{E}^{\prime}\xrightarrow{\varphi_{2}}\mathbf{F}^{\prime}\xrightarrow{\varphi_{3}}\mathbf{F},$

where $\varphi_{1}$ is the canonical projection, $\varphi_{2}$ is generically
an isomorphism, $\varphi_{3}$ is the canonical embedding.

###### Proof.

Write $\mathbf{E}=(E,E_{\bullet,\bullet})$,
$\mathbf{F}=(F,F_{\bullet,\bullet})$, let $\varphi^{\prime}\colon E\to F$ be
the underlying morphism of vector bundles. Note that
$\operatorname{Ker}\varphi^{\prime}$ is a vector subbundle of $E$. Indeed,
$E/\operatorname{Ker}\varphi^{\prime}$ is isomorphic to a subsheaf of $F$, so
it is torsion free. Let $\mathbf{E}^{\prime}$ be the strict subbundle of
$\mathbf{E}$ whose underlying vector bundle is
$\operatorname{Ker}(\varphi^{\prime})$. Let $F^{\prime}$ be the saturation of
the image of $\varphi^{\prime}$ (that is, $F^{\prime}$ is the unique saturated
vector subbundle of $F$ containing $\varphi^{\prime}(E)$ such that the
quotient $F^{\prime}/\varphi^{\prime}(E)$ is a torsion sheaf). Let
$\mathbf{F}^{\prime}$ be the strict subbundle of $\mathbf{F}$ whose underlying
vector bundle is $F^{\prime}$. Now the existence of the decomposition is
clear. ∎

###  3.6 Generalized degrees and slopes

Let $A$ be a $\mathbb{Q}$-vector space (in applications it will be
$\mathbf{k}$ or $\mathbb{R}$). Let $\varkappa\in A$,
$\zeta=\zeta_{\bullet,\bullet}\in A[D\times\mathbb{Z}_{>0}]$. Then we define
the homomorphism $\deg_{\varkappa,\sigma}\colon\Gamma\to A$ by

$\deg_{\varkappa,\zeta}(r,r_{\bullet,\bullet},d)=\varkappa d+\sum_{x\in
D}\sum_{j>0}\zeta_{x,j}r_{x,j}.$

If $\operatorname{rk}\gamma\neq 0$, we define the $(\varkappa,\zeta)$-slope of
$\gamma$ by $\deg_{\varkappa,\zeta}\gamma/\operatorname{rk}\gamma$. We write
$\deg_{\varkappa,\zeta}\mathbf{E}$ for
$\deg_{\varkappa,\zeta}\operatorname{cl}(\mathbf{E})$ and call
$\deg_{\varkappa,\zeta}\mathbf{E}/\operatorname{rk}\mathbf{E}$ _the
$(\varkappa,\sigma)$-slope of $\mathbf{E}$_.

We say that a parabolic bundle $\mathbf{E}$ is _$(\varkappa,\zeta)$ -isoslopy_
if the $(\varkappa,\zeta)$-slope of any direct summand of $\mathbf{E}$ is
equal to the $(\varkappa,\zeta)$-slope of $\mathbf{E}$.

We remark that it is common to write $\zeta\star\gamma$ for
$\deg_{0,\zeta}\gamma$ and $\deg_{\zeta}\gamma$ for $\deg_{1,\zeta}\gamma$ but
we prefer a uniform notation. We also remark that for an exact sequence (3.4)
we have
$\deg_{\varkappa,\zeta}\mathbf{E}=\deg_{\varkappa,\zeta}\mathbf{E}^{\prime}+\deg_{\varkappa,\zeta}(\mathbf{E}/\mathbf{E}^{\prime})$.

###  3.7 Parabolic weights and stability conditions

The following definition should be compared to [30, Definition 6.9].

######  Definition 3.8.

We say that a sequence $\sigma=\sigma_{\bullet,\bullet}$ of real numbers
indexed by $D\times\mathbb{Z}_{>0}$ is a  _sequence of parabolic weights_ if
for all $x\in D$ we have

$\sigma_{x,1}\leq\sigma_{x,2}\leq\cdots$ (3.5)

and for all $x$ and $j$ we have $\sigma_{x,j}\leq\sigma_{x,1}+1$.

To every sequence of parabolic weights we will associate a notion of stability
on parabolic bundles in Definition 3.9 below. Thus we denote the set of all
sequences of parabolic weights by
$\operatorname{Stab}=\operatorname{Stab}(X,D)$.

Fix $\sigma\in\operatorname{Stab}$.

######  Definition 3.9.

A parabolic bundle $\mathbf{E}$ is _$\sigma$ -semistable_ if for all strict
parabolic subbundles $\mathbf{E}^{\prime}\subset\mathbf{E}$ we have

$\frac{\deg_{1,\sigma}\mathbf{E}^{\prime}}{\operatorname{rk}\mathbf{E}^{\prime}}\leq\frac{\deg_{1,\sigma}\mathbf{E}}{\operatorname{rk}\mathbf{E}}.$
(3.6)

######  Remark 3.10.

We can similarly define semistability for any $\varkappa>0$ replacing the
condition $\sigma_{x,j}\leq\sigma_{x,1}+1$ with
$\sigma_{x,j}\leq\sigma_{x,1}+\varkappa$. However, scaling $\varkappa$ and
$\sigma$ by the same positive real number scales all the slopes by the same
number, so we would get the same notion of semistability. This is why we
restrict to the case $\varkappa=1$ above. The case $\varkappa=0$ would not
yield stacks of finite type; however, the possibility of taking $\varkappa=0$
will be useful below, when we work with connections (see Section 8.3).

######  Proposition 3.11.

Let $\mathbf{E}\in\mathcal{B}{un}^{\rm par}$ be a parabolic bundle. Then there
is a unique filtration
$0=\mathbf{E}_{0}\subset\mathbf{E}_{1}\subset\dots\subset\mathbf{E}_{m}=\mathbf{E}$
by strict parabolic subbundles such that all the quotients
$\mathbf{E}_{i}/\mathbf{E}_{i-1}$ are $\sigma$-semistable and we have
$\tau_{1}>\dots>\tau_{m}$, where $\tau_{i}$ is the $(1,\sigma)$-slope of
$\mathbf{E}_{i}/\mathbf{E}_{i-1}$.

###### Proof.

We start with a Lemma.

######  Lemma 3.12.

Let the morphism $\mathbf{E}\to\mathbf{F}$ be generically an isomorphism. Then
$\deg_{1,\sigma}\mathbf{E}\leq\deg_{1,\sigma}\mathbf{F}$.

###### Proof.

Write $\mathbf{E}=(E,E_{\bullet,\bullet})$ and
$\mathbf{F}=(F,F_{\bullet,\bullet})$. Let $\varphi\colon E\to F$ be the
underlying morphism of vector bundles. For $x\in D$ let $d_{x}$ denote the
dimension of the kernel of $\varphi_{x}$. Then

$\deg E=\deg F-\operatorname{length}(F/\varphi(E))\leq\deg F-\sum_{x\in
D}d_{x}.$

On the other hand, for all $x\in D$ and $i>0$ we have $\dim E_{x,i}\leq\dim
F_{x,i}+d_{x}$. Hence

$\displaystyle\deg_{1,\sigma}\mathbf{E}$ $\displaystyle=\deg
E+\sum_{x,j>0}\sigma_{x,j}(\dim E_{x,j-1}-\dim E_{x,j})$ $\displaystyle=\deg
E+\sum_{x\in
D}\left(\sigma_{x,1}\operatorname{rk}E+\sum_{i>0}(\sigma_{x,i+1}-\sigma_{x,i})\dim
E_{x,i}\right)$ $\displaystyle\leq\deg F-\sum_{x\in D}d_{x}+\sum_{x\in
D}\left(\sigma_{x,1}\operatorname{rk}F+\sum_{i>0}(\sigma_{x,i+1}-\sigma_{x,i})(\dim
F_{x,i}+d_{x})\right)$ $\displaystyle=\deg_{1,\sigma}\mathbf{F}+\sum_{x\in
D}d_{x}\left(-1+\sum_{i>0}(\sigma_{x,i+1}-\sigma_{x,i})\right)\leq\deg_{1,\sigma}\mathbf{F}.$

Lemma 3.12 is proved. ∎

The rest of the proof of Proposition 3.11 is completely analogous to the proof
of [16, Section 1.3] in view of Lemma 3.7. ∎

In the situation of Proposition 3.11(i) we say that the filtration is the
_Harder–Narasimhan filtration_ of $E$ (or _HN-filtration_ for short) and
$\tau_{1}>\dots>\tau_{m}$ is the $\sigma$-HN spectrum of $\mathbf{E}$. We
define $\mathcal{B}{un}^{{\rm par},\leq\tau}$ and $\mathcal{B}{un}^{{\rm
par},\geq\tau}$ as full subcategories of $\mathcal{B}{un}^{\rm par}$ whose
objects are parabolic bundles with the $\sigma$-HN spectrum contained in
$(-\infty,\tau]$ (resp. $[\tau,\infty)$). We emphasize that the categories
$\mathcal{B}{un}^{{\rm par},\leq 0}$ and $\mathcal{B}{un}^{{\rm par},-}$
should not be confused with each other: they coincide only if $\sigma=0$. The
following lemma is standard.

######  Lemma 3.13.

Let $\mathbf{E}$ be an object of $\mathcal{B}{un}^{{\rm par},\leq\tau}$ and
$\mathbf{E}^{\prime}$ be an object of $\mathcal{B}{un}^{{\rm
par},\geq\tau^{\prime}}$, where $\tau<\tau^{\prime}$. Then
$\operatorname{Hom}_{\mathcal{B}{un}^{\rm
par}}(\mathbf{E}^{\prime},\mathbf{E})=0$.

##  4 Parabolic pairs

###  4.1 Parabolic pairs and their generic Jordan types

The notion of parabolic pair, interesting by itself, will be used as a
technical tool for studying parabolic Higgs bundles in Section 5 and parabolic
bundles with connections in Section 8. Our main results in this section are
Theorem 4.19 and Corollary 4.21. They give explicit answers for graded motivic
classes of stacks of nilpotent parabolic pairs and parabolic pairs
respectively. We will also give a simplified answer in the case
$X=\mathbb{P}^{1}$ in Section 4.7.

######  Definition 4.1.

A _parabolic pair_ $(\mathbf{E},\Psi)$ consists of a parabolic bundle

$\mathbf{E}=(E,E_{\bullet,\bullet})\in\mathcal{B}{un}^{\rm par}(X,D)$

and an endomorphism $\Psi$ of $\mathbf{E}$ (that is, an endomorphism of $E$
preserving each $E_{x,j}$). If $\Psi$ is nilpotent we will speak about
_nilpotent parabolic pairs_.

Parabolic pairs as well as nilpotent parabolic pairs form an additive
$\mathbf{k}$-linear category denoted $\mathcal{P}{air}=\mathcal{P}{air}(X,D)$
(resp. $\mathcal{P}{air}^{\rm nilp}=\mathcal{P}{air}^{\rm nilp}(X,D)$). Again,
we abuse notation by denoting the stacks of objects by the same symbols. We
define $\mathcal{P}{air}_{\gamma}$, $\mathcal{P}{air}_{\gamma}^{\rm nilp}$,
$\mathcal{P}{air}_{\gamma}^{-}$, $\mathcal{P}{air}_{\gamma}^{{\rm nilp},-}$
etc. following the general construction (3.1) of Section 3.3.

The forgetful morphisms $\mathcal{P}{air}\to\mathcal{B}{un}^{\rm par}$ and
$\mathcal{P}{air}^{\rm nilp}\to\mathcal{B}{un}^{\rm par}$ are schematic and of
finite type. In particular, $\mathcal{P}{air}^{-}$ and $\mathcal{P}{air}^{{\rm
nilp},-}$ are $\Gamma^{\prime}_{+}$-graded Artin stacks of finite type (in the
graded sense).

Let $K\supset\mathbf{k}$ be an extension and
$(E,E_{\bullet,\bullet},\Psi)\in\mathcal{P}{air}^{\rm nilp}(K)$. If we
trivialize $E$ at the generic point of
$X_{K}=X\times_{\mathbf{k}}\operatorname{Spec}K$, $\Psi$ becomes a
$\operatorname{rk}E\times\operatorname{rk}E$ nilpotent matrix. Its Jordan type
is a partition $\lambda$ of $\operatorname{rk}E$. Thus we get a locally closed
stratification of $\mathcal{P}{air}^{\rm nilp}$ according to the generic
Jordan type of the nilpotent endomorphism

$\mathcal{P}{air}^{\rm nilp}(X,D)=\bigsqcup_{\lambda}\mathcal{P}{air}^{\rm
nilp}(X,D,\lambda),$

where the disjoint union is over all partitions. In other words,
$\mathcal{P}{air}^{\rm nilp}(X,D,\lambda)$ classifies nilpotent parabolic
pairs such that the endomorphism is generically conjugate to $n_{\lambda}$
(that is, conjugate to $n_{\lambda}$ at the generic point of $X$, or,
equivalently, at each point of a non-empty Zariski open subset of $X$). We
remark that any endomorphism generically conjugate to $n_{\lambda}$ is
necessarily nilpotent.

Again, we define the $\Gamma^{\prime}_{+}$-graded stacks
$\mathcal{P}{air}^{{\rm nilp},-}(X,D,\lambda)$ using the general formalism
(3.1) of Section 3.3.

###  4.2 Motivic classes of parabolic bundles with nilpotent endomorphisms

Our goal in this section is to calculate the graded motivic class (that is,
the motivic Donaldson–Thomas series, cf. [22])

$\displaystyle\big{[}\mathcal{P}{air}^{{\rm nilp},-}(X,D,\lambda)\big{]}$
$\displaystyle=\sum_{\gamma\in\Gamma^{\prime}_{+}}\big{[}\mathcal{P}{air}_{\gamma}^{{\rm
nilp},-}(X,D,\lambda)\big{]}e_{\gamma}$
$\displaystyle=w^{|\lambda|}\sum_{\gamma=(r,r_{\bullet,\bullet},d)\in\Gamma^{\prime}_{+}}\big{[}\mathcal{P}{air}_{\gamma}^{{\rm
nilp},-}(X,D,\lambda)\big{]}\prod_{x,j}w_{x,j}^{r_{x,j}}z^{d}.$ (4.1)

The partition $\lambda$ is fixed until the end of Section 4.3. This graded
motivic class is calculated as follows. First, in Section 4.3 we write this
graded motivic class as the product of a term that is independent of the
parabolic structures, and the “local” terms independent of the curve. The
first term has been calculated in [12]. Since the local terms are independent
of the curve, it is enough to calculate them when $X=\mathbb{P}^{1}$. More
precisely, we will work with $\mathbb{P}^{1}$ and two points with parabolic
structures (that is, $D=\\{0,\infty\\}$) but we will calculate the sum over
all partitions (Section 4.4). This part is very similar to [30, Section 5.4].
In Section 4.6 we give the explicit answer for the graded motivic classes
under consideration. Using the formalism of plethystic powers we then easily
calculate the class of parabolic bundles with not necessarily nilpotent
endomorphisms.

######  Remark 4.2.

We will see in Theorem 4.19 that (4.1) is a symmetric function in
$w_{x,\bullet}$ for each $x\in D$. This can be explained as follows. Note that
the “Weyl group” $W:=\prod\limits_{x\in D}\Sigma_{\infty}$ acts on
$\Gamma_{+}$ and $\Gamma^{\prime}_{+}$ in the obvious way (here
$\Sigma_{\infty}$ is the inductive limit of the permutation groups
$\Sigma_{l}$). Using the commutativity of the motivic Hall algebra of the
category of representations of the Jordan quiver (the quiver with one vertex
and one loop), one can easily show that the motivic classes
$\big{[}\mathcal{P}{air}_{\gamma}^{{\rm nilp},-}(X,D,\lambda)\big{]}$ are
$W$-invariant. Thus, we can re-write (4.1) as

$w^{|\lambda|}\mathop{\sum\nolimits^{\prime}}\limits_{\gamma=(r,r_{\bullet,\bullet},d)\in\Gamma^{\prime}_{+}}\big{[}\mathcal{P}{air}_{\gamma}^{{\rm
nilp},-}(X,D,\lambda)\big{]}\prod_{x\in
D}m_{r_{x,\bullet}}(w_{x,\bullet})z^{d},$

where the summation is only over $r_{\bullet,\bullet}$ such that we have
$r_{x,j}\geq r_{x,j+1}$ for all $x$ and $j$ (that is, $r_{x,\bullet}$ is a
partition of $|\lambda|$). Here, for a partition $\mu$, $m_{\mu}$ is the
symmetric function equal to the sum of all monomials whose ordered list of
exponents is $\mu$.

###  4.3 Factorization of graded motivic classes of stacks
of nilpotent parabolic pairs

In this section we factorize (4.1) as the product of the global part
(depending only on $X$ but not on $D$) and the local parts corresponding to
points of $D$ (but independent of $X$). This is a motivic version of [30,
Theorem 5.6]. We follow the same ideas, though some parts of Mellit’s proof do
not work in the motivic case and must be replaced by different arguments. On
the other hand, we were able to simplify some parts of Mellit’s proof, in
particular, by working with stacks.

Note that $\mathcal{P}{air}^{{\rm nilp},-}(X,\varnothing,\lambda)$ classifies
pairs $(E,\Psi)$, where $E$ is a nonpositive vector bundle on $X$, $\Psi$ is
an endomorphism of $E$ generically conjugate to $n_{\lambda}$ but there are no
parabolic structures.

######  Lemma 4.3.

The motivic class $\big{[}\mathcal{P}{air}_{\gamma}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\varnothing,\lambda\big{)}\big{]}\in
w^{|\lambda|}\operatorname{Mot}(\mathbf{k})\big{[}\big{[}z^{-1}\big{]}\big{]}$
is invertible.

###### Proof.

It is enough to show that the degree 0 part
$\big{[}\mathcal{P}{air}_{|\lambda|,0}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\varnothing,\lambda\big{)}\big{]}$ is invertible
in $\operatorname{Mot}(\mathbf{k})$. Note that a nonpositive vector bundle of
degree 0 on $\mathbb{P}^{1}$ is necessarily trivial. Thus, this degree zero
part classifies pairs $(E,\Psi)$, where $E$ is a trivial vector bundle and
$\Psi$ is a constant endomorphism conjugate to $n_{\lambda}$. It is easy to
see that this stack is isomorphic to the classifying stack of the centralizer
$Z_{\lambda}$ of $n_{\lambda}$. By Lemma 2.6 $Z_{\lambda}$ is special. Now it
follows from Lemma 2.4 or [12, Lemma 2.2.3] that
$\big{[}\mathcal{P}{air}_{|\lambda|,0}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\varnothing,\lambda\big{)}\big{]}=1/[Z_{\lambda}]$.
∎

We need some notation. Note that $\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\infty,\lambda\big{)}\big{]}\in
w^{|\lambda|}\operatorname{Mot}(\mathbf{k})\big{[}\big{[}w_{\infty,\bullet},z^{-1}\big{]}\big{]}$.
For $x\in D$ let

$\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\infty,\lambda\big{)}\big{]}_{x}\in
w^{|\lambda|}\operatorname{Mot}(\mathbf{k})\big{[}\big{[}w_{x,\bullet},z^{-1}\big{]}\big{]}$

denote the result of replacing $w_{\infty,\bullet}$ by $w_{x,\bullet}$ in this
series.

######  Theorem 4.4.

We have in $\operatorname{Mot}(\mathbf{k})[[\Gamma^{\prime}_{+}]]$.

$\big{[}\mathcal{P}{air}^{{\rm
nilp},-}(X,D,\lambda)\big{]}=\big{[}\mathcal{P}{air}^{{\rm
nilp},-}(X,\varnothing,\lambda)\big{]}\prod_{x\in
D}\frac{\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\infty,\lambda\big{)}\big{]}_{x}}{\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\varnothing,\lambda\big{)}\big{]}}.$

The proof of the theorem occupies the rest of Section 4.3; it is based on the
local study of stacks in the formal neighborhood of $D$.

The main idea of the proof is very simple. Let us assume that $D=\\{x\\}$ is a
single rational point of $X$. In Section 4.3.2 we will define the stack
$\mathcal{P}{air}^{{\rm loc},{\rm fl}}$ classifying triples
$(F,\Phi,F_{\bullet})$, where $F$ is a rank $|\lambda|$ vector bundle over the
formal completion of $X$ at $x$, $\Phi$ is a nilpotent endomorphism of $F$
generically conjugate to $n_{\lambda}$, $F_{\bullet}$ is a flag in $F_{x}$
preserved by $\Phi(x)$. We have an obvious restriction morphism
$\mathcal{P}{air}^{{\rm nilp},-}(X,x,\lambda)\to\mathcal{P}{air}^{{\rm
loc},{\rm fl}}$. We will see in Lemma 4.12 that this restriction morphism has
constant fiber. Thus, one is tempted to write the graded motivic class
$\big{[}\mathcal{P}{air}^{{\rm nilp},-}(X,x,\lambda)\big{]}$ as the product of
the graded motivic class of this fiber and of $\mathcal{P}{air}^{{\rm
loc},{\rm fl}}$. This would quickly lead to the proof of the theorem.
Unfortunately, $\mathcal{P}{air}^{{\rm loc},{\rm fl}}$ is not an Artin stack
as its points have inertia groups of infinite type, so its motivic class does
not make sense. The major part of the proof consists of going around this
problem.

Let us give the overview of the proof. In Section 4.3.1 we define and study
the schemes of jets into $\mathfrak{gl}_{|\lambda|}$. In Section 4.3.2 we
study the local stacks; they are essentially the quotients of the schemes of
jets by the group of jets of ${\rm GL}_{|\lambda|}$. In Section 4.3.3 we re-
write the theorem as a statement about motivic classes of graded components.
In Section 4.3.4 we study the fibers of the localization map; this is the main
part of the proof. We complete the proof in Section 4.3.5.

####  4.3.1 Jets

We will denote the non-archimedean local field $\mathbf{k}((t))$ by
$\mathbb{K}$ and its ring of integers $\mathbf{k}[[t]]$ by $\mathbb{O}$. The
order of pole at $t=0$ gives rise to a valuation map
$\operatorname{val}\colon\mathbb{K}\to\mathbb{Z}\cup\\{-\infty\\}$, where
$\operatorname{val}(0)=-\infty$. Clearly, $\operatorname{val}$ extends to
$\mathfrak{gl}_{r,\mathbb{K}}$ as the maximum of valuations of all matrix
elements. Let $J(X)$ denote the jet scheme of a scheme $X$ (this is a scheme
of infinite type), and let $J_{N}(X)$ denote the scheme of order $N-1$ jets.
In particular, $J_{1}(X)=X$.

For an algebraic group $G$ of finite type over $\mathbf{k}$ we have the jet
group $G_{\mathbb{O}}:=J(G)$ and the jet group of finite type $J_{N}(G)$. The
$N$-th congruence subgroup $G^{(N)}$ is the kernel of the projection
$G_{\mathbb{O}}\to J_{N}(G)$. We also have the ind-group of loops
$G_{\mathbb{K}}$ containing $G_{\mathbb{O}}$. Let
$\Delta:=\operatorname{Spec}\mathbb{O}$ be the formal disc and
$\mathring{\Delta}:=\operatorname{Spec}\mathbb{K}$ be its generic point (the
punctured formal disc). Also set
$\Delta_{N}:=\operatorname{Spec}k[[t]]/t^{N}$. The groups ${\rm
GL}_{r,\mathbb{O}}$, ${\rm GL}_{r,\mathbb{K}}$, and $J_{N}({\rm GL}_{r})$ are
the groups of automorphisms of the trivial vector bundles on $\Delta$,
$\mathring{\Delta}$, and $\Delta_{N}$ respectively. For more details on the
jet and loop groups we refer the reader to [43].

Set $r=|\lambda|$. Consider the orbit stratification of $\mathfrak{gl}_{r}$
under the adjoint action of ${\rm GL}_{r}$:
$\mathfrak{gl}_{r}=\bigsqcup\limits_{\mu\vdash r}\mathcal{O}_{\mu}$, where
$\mathcal{O}_{\mu}$ is the adjoint orbit containing $n_{\mu}$. Let
$\overline{\mathcal{O}}_{\mu}$ denote the Zariski closure of
$\mathcal{O}_{\mu}$. Set

$J(\lambda):=J\big{(}\overline{\mathcal{O}}_{\lambda}\big{)}-\bigcup_{\mathcal{O}_{\mu}\subset\overline{\mathcal{O}}_{\lambda}-\mathcal{O}_{\lambda}}J\big{(}\overline{\mathcal{O}}_{\mu}\big{)}.$

Note that $J(\lambda)$ parameterizes morphisms $\Delta\to\mathfrak{gl}_{r}$
such that the image of the generic point of $\Delta$ is in
$\mathcal{O}_{\lambda}$, that is, jets that are generically conjugate to
$n_{\lambda}$.

######  Definition 4.5.

We say that a loop $g\in{\rm GL}_{r,\mathbb{K}}(\mathbf{k})={\rm
GL}_{r}(\mathbb{K})$ is _kernel-strict_ if
$g^{-1}n_{\lambda}g\in\mathfrak{gl}_{r,\mathbb{O}}$ and $g^{-1}$ induces an
isomorphism between the $\mathbb{O}$-modules
$\operatorname{Ker}n_{\lambda}\otimes_{\mathbf{k}}\mathbb{O}$ and
$\operatorname{Ker}\big{(}g^{-1}n_{\lambda}g\big{)}$.

######  Remark 4.6.

Note that our definition is a little different from [30, Definition 3.8].
Mellit’s definition of kernel-strictness depends also on a choice of a matrix
$\theta$. In terminology of Mellit our $g$ is kernel-strict for
$\theta=g^{-1}n_{\lambda}g$. Note also that the results of Mellit we are using
here and below are formulated over finite fields but are valid over any field,
proofs being the same.

Let $\Phi$ be a $\mathbf{k}$-point of $J(\lambda)$, then there is a kernel-
strict $g\in{\rm GL}_{r,\mathbb{K}}(\mathbf{k})={\rm GL}_{r}(\mathbb{K})$ such
that $\Phi=g^{-1}n_{\lambda}g$. Set $\deg\Phi:=\operatorname{val}(\det g)$.
The existence of such $g$ and independence of the degree on the choice of $g$
is proved in [30, Lemma 3.7]. It follows also from loc. cit. that the degree
is nonnegative.

If $K\supset\mathbf{k}$ is a field extension, we similarly define the degree
of a $K$-point of $J(\lambda)$. The degree is compatible with field
extensions. Thus we get a stratification of the set of points of $J(\lambda)$:
$|J(\lambda)|=\bigsqcup\limits_{d\geq 0}J_{d}(\lambda)$. Let $\pi_{N}\colon
J(\lambda)\to J_{N}\big{(}\overline{\mathcal{O}}_{\lambda}\big{)}$ be the
truncation map. Set $J_{d,N}(\lambda):=\pi_{N}(J_{d}(\lambda))$.

######  Proposition 4.7.

For a nonnegative integer $d$ and a partition $\lambda$ there is a positive
integer $N(d,\lambda)$ such that for $N>N(d,\lambda)$ we have

  1. $(i)$

For all $\Phi$ in $J_{d}(\lambda)$ there is a kernel-strict $g$ with
$\operatorname{val}(g)<N/2$, $\operatorname{val}\big{(}g^{-1}\big{)}<N/2$ such
that $g\Phi g^{-1}=n_{\lambda}$.

  2. $(ii)$

$J_{d}(\lambda)=\pi_{N}^{-1}(J_{d,N}(\lambda))$.

  3. $(iii)$

Any two points in the same fiber of the projection $J_{d}(\lambda)\to
J_{d,N}(\lambda)$ are conjugate by an element of ${\rm GL}_{r,\mathbb{O}}$.

###### Proof.

By [30, Lemma 3.7] there is $N_{0}\geq 1$ (depending on $\lambda$ and $d$)
such that for all $\Phi\in J_{d}(\lambda)$ there is a kernel-strict $g$ with
$\operatorname{val}(g)<N_{0}$ such that $g\Phi g^{-1}=n_{\lambda}$. Then
$\operatorname{val}(\det g)=d$. Set $N_{1}:=rN_{0}+d$. Then by Cramer’s rule
$\operatorname{val}\big{(}g^{-1}\big{)}<N_{1}$. We also have
$\operatorname{val}(g)<N_{1}$. Take $N(d,\lambda):=4N_{1}$. With this choice
of $N(d,\lambda)$ part (i) of the proposition is clear.

Note that (ii) is saying that the degree of an infinite jet depends only on
its $N$-th truncation if $N>4N_{1}$. Thus to prove (ii) we need to show that
if $\Phi\in J_{d}(\lambda)$ and $\Phi^{\prime}\in J(\lambda)$ is such that
$\Phi^{\prime}\equiv\Phi\pmod{z^{4N_{1}}}$, then $\Phi^{\prime}\in
J_{d}(\lambda)$. Choose a kernel-strict $g$ with
$\operatorname{val}(g)<N_{1}$, $\operatorname{val}\big{(}g^{-1}\big{)}<N_{1}$
such that $g\Phi g^{-1}=n_{\lambda}$. Then $g\Phi^{\prime}g^{-1}\equiv
n_{\lambda}\pmod{z^{2N_{1}}}$. We need a lemma.

######  Lemma 4.8.

There is $g^{\prime}\in{\rm GL}_{r}^{(2N_{1})}$ such that
$g^{\prime}g\Phi^{\prime}g^{-1}(g^{\prime})^{-1}=n_{\lambda}$.

###### Proof.

Set $\Phi^{\prime\prime}:=g\Phi^{\prime}g^{-1}$ and $N_{2}:=2N_{1}$. Write
$\Phi^{\prime\prime}\equiv
n_{\lambda}+\Phi_{N_{2}}z^{N_{2}}\pmod{z^{N_{2}+1}}$. Since
$\Phi^{\prime\prime}\in J\big{(}\overline{\mathcal{O}}_{\lambda}\big{)}$,
$\Phi_{N_{2}}$ belongs to the tangent space to
$\overline{\mathcal{O}}_{\lambda}$ at $n_{\lambda}$, which is naturally
identified with $[n_{\lambda},\mathfrak{gl}_{r}]$. Thus there is
$g_{{}_{N_{2}}}\in\mathfrak{gl}_{r}$ such that
$[n_{\lambda},g_{{}_{N_{2}}}]=\Phi_{N_{2}}$. Then
$\big{(}1+g_{{}_{N_{2}}}z^{N_{2}}\big{)}\Phi^{\prime\prime}\big{(}1+g_{{}_{N_{2}}}z^{N_{2}}\big{)}^{-1}\equiv
n_{\lambda}\pmod{z^{N_{2}+1}}$.

Repeating this process we find
$g_{{}_{N_{2}+1}},g_{{}_{N_{2}+2}},\ldots\in\mathfrak{gl}_{r}$ such that for
$j>0$ we have

$\displaystyle\big{(}1+g_{{}_{N_{2}+j}}z^{N_{2}+j}\big{)}\cdots\big{(}1+g_{{}_{N_{2}}}z^{N_{2}}\big{)}\Phi^{\prime\prime}\big{(}1+g_{{}_{N_{2}}}z^{N_{2}}\big{)}^{-1}\cdots\big{(}1+g_{{}_{N_{2}+j}}z^{N_{2}+j}\big{)}^{-1}$
$\displaystyle\qquad{}\equiv n_{\lambda}\pmod{z^{N_{2}+j+1}}.$

It remains to take
$g^{\prime}=\prod\limits_{j=0}^{\infty}\big{(}1+g_{{}_{N_{2}+j}}z^{N_{2}+j}\big{)}$.
Lemma 4.8 is proved. ∎

We return to the proof Proposition 4.7. Note that $g^{-1}g^{\prime}g\in{\rm
GL}_{r,\mathbb{O}}$. It is easy to see that the set of kernel-strict loops is
invariant under the multiplication by points of ${\rm GL}_{r,\mathbb{O}}$ on
the right, so $g^{\prime}g=g\big{(}g^{-1}g^{\prime}g\big{)}$ is kernel-strict.
Clearly, $\operatorname{val}(\det(g^{\prime}g))=\operatorname{val}(\det g)=d$,
so (ii) follows. Further, $g^{-1}g^{\prime}g$ conjugates $\Phi^{\prime}$ to
$\Phi$, so (iii) follows as well. Proposition 4.7 is proved. ∎

For every $d\geq 0$ we fix $N(d,\lambda)$ satisfying the conditions of the
above proposition.

######  Definition 4.9.

For $N>N(d,\lambda)$ we call any jet in $J_{d,N}(\lambda)$ _stabilized_ and
call $d$ its _degree_.

According to the proposition, the degree of a stabilized jet is well-defined
and any two lifts of a stabilized jet to an infinite jet are conjugate by an
element of ${\rm GL}_{r,\mathbb{O}}$. Note that for every jet $\Phi\in
J_{d}(\lambda)$ its truncation $\pi_{N}(\Phi)$ is stabilized for $N$ large
enough.

####  4.3.2 Local stacks

Consider the quotient stack $\mathcal{P}{air}^{\rm loc}=\mathcal{P}{air}^{\rm
loc}(\lambda):=J(\lambda)/{\rm GL}_{r,\mathbb{O}}$, where ${\rm
GL}_{r,\mathbb{O}}$ acts by conjugation. We skip $\lambda$ from the notation
as it is fixed until the end of Section 4.3. Since the degree function on
$J(\lambda)$ is ${\rm GL}_{r,\mathbb{O}}$-invariant, we get the degree
function on the points of $\mathcal{P}{air}^{\rm loc}$.

Note that $\mathcal{P}{air}^{\rm loc}$ classifies pairs $(F,\Phi)$, where $F$
is a rank $r=|\lambda|$ vector bundle over $\Delta$, $\Phi$ is a nilpotent
endomorphism of $F$ generically conjugate to $n_{\lambda}$. This follows from
the fact that every vector bundle on $\Delta$ is trivial. It also follows that
every $K$-point of $\mathcal{P}{air}^{\rm loc}$ is isomorphic to a point of
the form $(\mathbb{O}^{r},\Phi)$, where $\Phi\in\mathfrak{gl}_{r,\mathbb{O}}$.

We emphasize that $\mathcal{P}{air}^{\rm loc}$ is not an Artin stack (its
isotropy groups are not of finite type).

Define $\mathcal{P}{air}^{\rm
loc}_{N}:=J_{N}\big{(}\overline{\mathcal{O}}_{\lambda}\big{)}/J_{N}({\rm
GL}_{r})$; this is an Artin stack of finite type. The points of
$\mathcal{P}{air}^{\rm loc}_{N}$ are the pairs $(F,\Phi)$ where $F$ is a
vector bundle on $\Delta_{N}$, $\Phi$ is an endomorphism of $F$ such that if
we trivialize $F$, $\Phi$ becomes a jet with values in
$\overline{\mathcal{O}}_{\lambda}$. We say that $(F,\Phi)$ is _stabilized_ if
$\Phi$ is stabilized in the sense of Definition 4.9. Note that this does not
depend on the trivialization of $F$. If $(F,\Phi)$ is stabilized, then we have
a well-defined notion of the degree of $(F,\Phi)$. Explicitly, we can lift
$(F,\Phi)$ to a point $(\mathbb{O}^{r},\overline{\Phi})$ of
$\mathcal{P}{air}^{\rm loc}$, and the degree of $(F,\Phi)$ is equal to the
degree of $\overline{\Phi}\in\mathfrak{gl}_{r,\mathbb{O}}$.

We define the stack $\mathcal{P}{air}^{{\rm loc},{\rm fl}}$ as the stack
classifying triples $(F,\Phi,F_{\bullet})$, where $(F,\Phi)$ is a point of
$\mathcal{P}{air}^{\rm loc}$, $F_{\bullet}$ is a flag in the fiber $F_{0}$
preserved by $\Phi(0)$. We define the stack $\mathcal{P}{air}^{{\rm loc},{\rm
fl}}_{N}$ as the stack classifying triples $(F,\Phi,F_{\bullet})$, where
$(F,\Phi)$ is a point of $\mathcal{P}{air}^{\rm loc}_{N}$, $F_{\bullet}$ is a
flag in $F_{0}$ preserved by $\Phi(0)$.

####  4.3.3 Preparation for the proof of Theorem 4.4

We will assume that $D=x$ is a single rational point of $X$. This will
unburden the notation; the general case is proved similarly. Thus we want to
prove that

$\big{[}\mathcal{P}{air}^{{\rm
nilp},-}(X,x,\lambda)\big{]}\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\varnothing,\lambda\big{)}\big{]}=\big{[}\mathcal{P}{air}^{{\rm
nilp},-}(X,\varnothing,\lambda)\big{]}\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\infty,\lambda\big{)}\big{]}_{x}.$

Equating the graded components, we see that this reduces to the following
proposition.

######  Proposition 4.10.

Let $d$ be a nonpositive integer, $r_{\bullet}=(r_{1},r_{2},\dots)$ be a
sequence of nonnegative integers such that $\sum_{i}r_{i}=r=|\lambda|$. Then
we have in $\operatorname{Mot}(\mathbf{k})$:

$\displaystyle\sum_{d^{\prime}+d^{\prime\prime}=d}\big{[}\mathcal{P}{air}^{{\rm
nilp},-}_{r,r_{\bullet},d^{\prime}}(X,x,\lambda)\times\mathcal{P}{air}^{{\rm
nilp},-}_{r,d^{\prime\prime}}\big{(}\mathbb{P}^{1},\varnothing,\lambda\big{)}\big{]}$
$\displaystyle\qquad{}=\sum_{d^{\prime}+d^{\prime\prime}=d}\big{[}\mathcal{P}{air}^{{\rm
nilp},-}_{r,d^{\prime}}(X,\varnothing,\lambda)\times\mathcal{P}{air}^{{\rm
nilp},-}_{r,r_{\bullet},d^{\prime\prime}}\big{(}\mathbb{P}^{1},\infty,\lambda\big{)}\big{]}.$

This proposition will be proved in Section 4.3.5. We emphasize that the sum is
over all $d^{\prime},d^{\prime\prime}\in\mathbb{Z}$ with
$d^{\prime}+d^{\prime\prime}=d$ but the terms are non-zero only if
$d^{\prime},d^{\prime\prime}\in[d,0]$. We note that the RHS is manifestly
independent of $x$. Thus, the LHS is independent of $x$ as well.

####  4.3.4 The restriction to the formal neighborhood of $\boldsymbol{x}$

We keep the simplifying assumption that $D=\\{x\\}$ is a single point; we
write $r_{\bullet}$ instead of $r_{x,\bullet}$. Fix
$\gamma=(r,r_{\bullet},d)\in\Gamma_{+}^{\prime}$. For $x\in X$ let
$\mathcal{O}_{X,x}$ be the local ring of $x$ and $\hat{\mathcal{O}}_{X,x}$ be
its formal completion. Set
$\Delta_{x}:=\operatorname{Spec}\hat{\mathcal{O}}_{X,x}$. Choose a formal
coordinate at $x$, use it to identify $\Delta_{x}$ with $\Delta$ and the
$N$-th infinitesimal neighborhood $\Delta_{x,N}$ of $x$ with $\Delta_{N}$.
Consider the restriction morphism

$\operatorname{loc}_{x}^{\rm fl}\colon\
\mathcal{P}{air}_{r,r_{\bullet},d}^{{\rm
nilp},-}(X,x,\lambda)\to\mathcal{P}{air}^{{\rm loc},{\rm
fl}}_{r_{\bullet}},\qquad(E,\Psi,E_{x,\bullet})\mapsto(E|_{\Delta_{x}},\Psi|_{\Delta_{x}},E_{x,\bullet}).$
(4.2)

Similarly we have a morphism

$\operatorname{loc}_{x}\colon\ \mathcal{P}{air}_{r,d}^{{\rm
nilp},-}(X,\varnothing,\lambda)\to\mathcal{P}{air}^{\rm
loc},\qquad(E,\Psi)\mapsto(E|_{\Delta_{x}},\Psi|_{\Delta_{x}}).$ (4.3)

Our nearest goal is to describe the fibers of these morphisms. For a
nonpositive integer $e$, let ${\mathcal{F}}{ib}_{e}(X,x)$ denote the open
substack of $\mathcal{P}{air}^{{\rm nilp},-}_{r,e}(X,\varnothing,\lambda)$
consisting of $(E,\Psi)$ such that $\Psi$ is conjugate to $n_{\lambda}$ at
$x$. Let $\overline{{\mathcal{F}}{ib}}_{e}(X,x)$ denote the stack of triples
$(E,\Psi,s)$, where $(E,\Psi)$ is a point of ${\mathcal{F}}{ib}_{e}(X,x)$, $s$
is a trivialization of $E$ over $\Delta_{x}$ such that $\Psi=n_{\lambda}$ in
this trivialization. Recall that in Section 4.3.2 we defined the notion of
degree for the points of $\mathcal{P}{air}^{\rm loc}$.

######  Lemma 4.11.

  1. $(i)$

The fiber of $\operatorname{loc}_{x}^{\rm fl}$ over $(F,\Phi,F_{\bullet})$ is
isomorphic to $\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)$, where $e$ is the
degree of $(F,\Phi)$.

  2. $(ii)$

Similarly, the fiber of $\operatorname{loc}_{x}$ over $(F,\Phi)$ is isomorphic
to $\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)$, where $e$ is the degree of
$(F,\Phi)$.

###### Proof.

We prove (ii) first. Fix a trivialization of $F$ on the formal disc $\Delta$.
Then $\Phi$ becomes an element of $\mathfrak{gl}_{r,\mathbb{K}}$ and we choose
a kernel-strict $g$ such that $g\Phi g^{-1}=n_{\lambda}$. Then
$\operatorname{val}(\det g)=e$.

Denote the fiber under consideration by $\overline{{\mathcal{F}}{ib}}$. The
fiber can be described as the stack of triples $(E,\Psi,s)$, where $E$ is a
nonpositive vector bundle, $\Psi$ is an endomorphism, $s$ is the
trivialization of $E$ over $\Delta_{x}$ such that in this trivialization we
have $\Psi|_{\Delta_{x}}=\Phi$. Note that such $\Psi$ is automatically
conjugate to $n_{\lambda}$ at the generic point of $X$.

If $(E,\Psi,s)$ is a point of $\overline{{\mathcal{F}}{ib}}$, then $E|_{X-x}$
is trivialized over the punctured disc $\mathring{\Delta}_{x}$, and we use the
$g$ chosen above to glue $E|_{X-x}$ with the trivial bundle
$\mathbf{k}^{r}\times\Delta_{x}$ on $\mathring{\Delta}_{x}$ (we recall that
$g$ can be viewed as an automorphism of the trivial vector bundle on
$\mathring{\Delta}_{x}$). We obtain a new vector bundle $E^{\prime}$ on $X$
with an isomorphism $E^{\prime}|_{X-x}\simeq E|_{X-x}$ and a trivialization
over $\Delta_{x}$. Thus $\Psi$ gives rise to an endomorphism $\Psi^{\prime}$
of $E^{\prime}|_{X-x}$. It is easy to derive from the definition of $g$ that
in the given trivialization we have
$\Psi^{\prime}|_{\mathring{\Delta}_{x}}=n_{\lambda}$. Thus $\Psi^{\prime}$
extends to $x$ and, moreover, in the given trivialization of $E^{\prime}$ over
$\Delta_{x}$ we have $\Psi^{\prime}|_{\Delta_{x}}=n_{\lambda}$.

Note that $E^{\prime}$ is nonpositive. Indeed, $\operatorname{Ker}\Psi$ is
nonpositive as a subbundle of $E$. Since $g$ is kernel-strict, the isomorphism
between $\operatorname{Ker}\Psi$ and $\operatorname{Ker}\Psi^{\prime}$ extends
from $X-x$ to $X$. Thus $\operatorname{Ker}\Psi^{\prime}$ is also nonpositive.
But by [30, Proposition 5.3] this implies that $E^{\prime}$ is nonpositive as
well.

Next, we have an isomorphism between $\wedge^{r}E$ and $\wedge^{r}E^{\prime}$
over $X-x$, and it has a zero of order $\operatorname{val}(\det g)$ at $x$.
Thus $\deg
E^{\prime}=\deg\wedge^{r}E^{\prime}=\deg\wedge^{r}E+\operatorname{val}(\det
g)=\deg E+e=d+e$.

We have constructed a morphism
$\overline{{\mathcal{F}}{ib}}\to\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)$.
Conversely, given a point $(E^{\prime},\Psi^{\prime},s^{\prime})$ of
$\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)$, we use $g$ and $s^{\prime}$ to
construct a new bundle $E$ with an isomorphism to $E^{\prime}$ over $X-x$ and
a trivialization over $\Delta_{x}$. Then $\Psi^{\prime}|_{X-x}$ give rise to
an endomorphism of $E|_{X-x}$ and we check that it extends to $x$ and,
moreover, in the trivialization of $E$ over $\Delta_{x}$ we have
$\Psi|_{\Delta_{x}}=\Phi$. Now it is easy to see that the two constructions
are inverse to each other. This proves (ii).

Now (i) follows from the cartesian diagram

$\begin{CD}\mathcal{P}{air}_{r,r_{\bullet},d}^{{\rm
nilp},-}(X,x,\lambda)@>{\operatorname{loc}_{x}^{\rm
fl}}>{}>\mathcal{P}{air}^{{\rm loc},{\rm fl}}_{r_{\bullet}}\\\
@V{}V{}V@V{}V{}V\\\ \mathcal{P}{air}_{r,d}^{{\rm
nilp},-}(X,\varnothing,\lambda)@>{\operatorname{loc}_{x}}>{}>\mathcal{P}{air}^{\rm
loc},\end{CD}$

where the vertical arrows correspond to forgetting the flags. ∎

Consider now the compositions of (4.2) and (4.3) with restrictions to the
$N$-th infinitesimal neighborhood of $x$.

$\operatorname{loc}_{x,N}^{\rm fl}\colon\
\mathcal{P}{air}_{r,r_{\bullet},d}^{{\rm
nilp},-}(X,x,\lambda)\to\mathcal{P}{air}^{{\rm loc},{\rm
fl}}_{r_{\bullet},N},\qquad(E,\Psi,E_{x,\bullet})\mapsto(E|_{\Delta_{x,N}},\Psi|_{\Delta_{x,N}},E_{x,\bullet}).$
(4.4)

Similarly we have a morphism

$\operatorname{loc}_{x,N}\colon\ \mathcal{P}{air}_{r,d}^{{\rm
nilp},-}(X,\varnothing,\lambda)\to\mathcal{P}{air}^{\rm
loc}_{N},\qquad(E,\Psi)\mapsto(E|_{\Delta_{x,N}},\Psi|_{\Delta_{x,N}}).$ (4.5)

Let $(F,\Phi,F_{\bullet})$ be a point of $\mathcal{P}{air}^{{\rm loc},{\rm
fl}}_{r_{\bullet},N}$ and assume that $(F,\Phi)$ is stabilized. By Definition
4.9 we can find $\Phi^{\prime}\in\mathfrak{gl}_{r,\mathbb{O}}$ such that
$(\mathbb{O}^{r},\Phi^{\prime})$ lifts $(F,\Phi)$ and $N>N(e,\lambda)$, where
$e$ is the degree of $\Phi^{\prime}$ and $N(e,\lambda)$ is the integer number
from Proposition 4.7, which was fixed just before Definition 4.9. Choose a
kernel-strict $g\in\mathfrak{gl}_{r,\mathbb{K}}$ such that
$g\Phi^{\prime}g^{-1}=n_{\lambda}$. Denote by $Z^{(N)}_{g}$ the intersection
$Z_{\mathbb{O}}\cap\big{(}g^{-1}{\rm GL}^{(N)}g\big{)}\subset{\rm
GL}_{r,\mathbb{K}}$, where $Z=Z_{\lambda}$ is the centralizer of $n_{\lambda}$
in ${\rm GL}_{|\lambda|}$ (cf. Lemma 2.6). Recall that by Proposition 4.7(i)
we may assume that the orders of the poles of $g$ and $g^{-1}$ are less than
$N/2$ so we have $Z^{(N)}_{g}\subset Z^{(1)}$. This is a pro-unipotent group.
Clearly, $Z_{\mathbb{O}}$ (and thus $Z^{(N)}_{g}$ as well) acts on
$\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)$ by changing the trivialization of
$E$ on $\Delta_{x}$.

######  Lemma 4.12.

  1. $(i)$

Let $(F,\Phi,F_{\bullet})\in\mathcal{P}{air}_{r_{\bullet},N}^{{\rm loc},{\rm
fl}}$ be such that $(F,\Phi)$ is stabilized, choose $g\in{\rm
GL}_{r,\mathbb{K}}$ as in the previous paragraph. Then the fiber of
$\operatorname{loc}_{x,N}^{\rm fl}$ over $(F,\Phi,F_{\bullet})$ is isomorphic
to $\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)/Z^{(N)}_{g}$, where $e$ is the
degree of $(F,\Phi)$.

  2. $(ii)$

Similarly, the fiber of $\operatorname{loc}_{x,N}$ over $(F,\Phi)$ is
isomorphic to $\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)/Z^{(N)}_{g}$.

###### Proof.

Let us prove (ii), the proof of (i) is completely analogous. Denote the fiber
under consideration by ${\mathcal{F}}{ib}$ and let
$\overline{{\mathcal{F}}{ib}}$ be the fiber of $\operatorname{loc}_{x}$ over
$(\mathbb{O}^{r},\Phi^{\prime})$. Then we have a restriction morphism
$\overline{{\mathcal{F}}{ib}}\to{\mathcal{F}}{ib}$. It follows from the
stability of $(F,\Phi)$ and Proposition 4.7(iii) that this morphism is
surjective. On the other hand, it is easy to see that two points of
$\overline{{\mathcal{F}}{ib}}$ map to the same point of ${\mathcal{F}}{ib}$ if
and only if they differ by the action of an element of
$gZ_{\mathbb{O}}g^{-1}\cap{\rm GL}^{(N)}$. On the other hand, according to
Lemma 4.11,
$\overline{{\mathcal{F}}{ib}}\simeq\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)$.
One checks that under this isomorphism, the action of
$gZ_{\mathbb{O}}g^{-1}\cap{\rm GL}^{(N)}$ on $\overline{{\mathcal{F}}{ib}}$
corresponds to the action of $Z^{(N)}_{g}$ on
$\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)$. ∎

We need to calculate the motivic class of this fiber. Set
$Z_{g}:=Z_{\mathbb{O}}/Z^{(N)}_{g}$; this is a group of finite type.

######  Lemma 4.13.

We have in $\operatorname{Mot}(\mathbf{k})$

$\big{[}\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)/Z^{(N)}_{g}\big{]}=[{\mathcal{F}}{ib}_{d+e}(X,x)]/[Z_{g}].$

###### Proof.

For large $M$ we have $Z^{(M)}\subset Z_{g}^{(N)}$ and this subgroup is
normal.

_Claim._ The groups $Z^{(N)}_{g}/Z^{(M)}$, $Z_{\mathbb{O}}/Z^{(M)}$, and
$Z_{\mathbb{O}}/Z^{(N)}_{g}$ are special.

_Proof of the claim._ Recall that $Z_{g}^{(N)}\subset Z^{(1)}$. The group
$Z^{(N)}_{g}/Z^{(M)}$ is special, since every unipotent group is special by
Lemma 2.5. Next, the quotient of $Z_{\mathbb{O}}/Z^{(M)}$ by the unipotent
subgroup $Z^{(1)}/Z^{(M)}$ is equal to $Z$ and the statement follows from
Lemmas 2.5 and 2.6. A similar argument shows that $Z_{\mathbb{O}}/Z^{(N)}_{g}$
is special.∎

We continue with the proof of Lemma 4.13. Next,
$\overline{\mathcal{F}}{ib}_{d+e}(X,x)/Z^{(M)}$ is a
$Z_{\mathbb{O}}/Z^{(M)}$-principal bundle over ${\mathcal{F}}{ib}_{d+e}(X,x)$.
Since $Z_{\mathbb{O}}/Z^{(M)}$ is a special group, we get by Lemma 2.4

$\big{[}\overline{\mathcal{F}}{ib}_{d+e}(X,x)/Z^{(M)}\big{]}=\big{[}Z_{\mathbb{O}}/Z^{(M)}\big{]}[{\mathcal{F}}{ib}_{d+e}(X,x)].$

Similarly, $\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)/Z^{(M)}$ is a
$Z^{(N)}_{g}/Z^{(M)}$-principal bundle over
$\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)/Z^{(N)}_{g}$ and we get

$\big{[}\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)/Z^{(M)}\big{]}=\big{[}Z^{(N)}_{g}/Z^{(M)}\big{]}\big{[}\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)/Z^{(N)}_{g}\big{]}.$

Finally, $Z_{\mathbb{O}}/Z^{(M)}$ is a $Z^{(N)}_{g}/Z^{(M)}$-principal bundle
over $Z_{g}$ and we have

$\big{[}Z_{\mathbb{O}}/Z^{(M)}\big{]}=\big{[}Z^{(N)}_{g}/Z^{(M)}\big{]}[Z_{g}].$

The lemma follows from these three equations. ∎

######  Lemma 4.14.

Let $N$ be an integer larger than $N(j,\lambda)$ for all $j=0,\dots,-d$.
Assume that the fiber of (4.5) over $(F,\Phi)$ is non-empty. Then $(F,\Phi)$
is stabilized. A similar statement holds for the fibers of (4.4).

###### Proof.

We prove the statement about the fibers of (4.5), the other statement being
analogous. Let $(E,\Psi)$ be a point of the fiber, then the fiber of
$\operatorname{loc}_{x}$ over $(E|_{\Delta_{x}},\Psi|_{\Delta_{x}})$ is non-
empty. Then by Lemma 4.11 this fiber is isomorphic to
$\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)$, where $e$ is the degree of
$(E|_{\Delta_{x}},\Psi|_{\Delta_{x}})$ (recall that $e\geq 0$). Since
$\overline{{\mathcal{F}}{ib}}_{d+e}(X,x)$ classifies nonpositive vector
bundles, we get $d+e\leq 0$. Thus $N>N(e,\lambda)$ and we see that $(F,\Phi)$
is stabilized. ∎

####  4.3.5 Proof of Proposition 4.10

Let us take an integer $N$ larger than $N(j,\lambda)$ for all $j=0,\dots,-d$.
It is enough to show that the motivic functions

$A:=\sum_{d^{\prime}+d^{\prime\prime}=d}\big{[}\mathcal{P}{air}^{{\rm
nilp},-}_{r,r_{\bullet},d^{\prime}}(X,x,\lambda)\times\mathcal{P}{air}^{{\rm
nilp},-}_{r,d^{\prime\prime}}\big{(}\mathbb{P}^{1},\varnothing,\lambda\big{)}\to\mathcal{P}{air}^{{\rm
loc},{\rm fl}}_{r_{\bullet},N}\times\mathcal{P}{air}^{\rm loc}_{N}\big{]}$

and

$B:=\sum_{d^{\prime}+d^{\prime\prime}=d}\big{[}\mathcal{P}{air}^{{\rm
nilp},-}_{r,r_{\bullet},d^{\prime}}\big{(}\mathbb{P}^{1},\infty,\lambda\big{)}\times\mathcal{P}{air}^{{\rm
nilp},-}_{r,d^{\prime\prime}}(X,\varnothing,\lambda)\to\mathcal{P}{air}^{{\rm
loc},{\rm fl}}_{r_{\bullet},N}\times\mathcal{P}{air}^{\rm loc}_{N}\big{]}$

are equal. The morphisms are $\operatorname{loc}_{x,N}^{\rm
fl}\times\operatorname{loc}_{\infty,N}$ and
$\operatorname{loc}_{\infty,N}^{\rm fl}\times\operatorname{loc}_{x,N}$
respectively.

Let $K\supset\mathbf{k}$ be an extension and let $\xi$ be a $K$-point of the
stack $\mathcal{P}{air}^{{\rm loc},{\rm
fl}}_{r_{\bullet},N}\times\mathcal{P}{air}^{\rm loc}_{r,N}$ represented by
$((F,\Phi,F_{\bullet}),(F^{\prime},\Phi^{\prime}))$. By Proposition 2.1 it is
enough to show that $\xi^{*}A=\xi^{*}B$. Using base change, we may assume that
$K=\mathbf{k}$. According to Lemma 4.14, these motivic functions are zero
unless $(F,\Phi)$ and $(F^{\prime},\Phi^{\prime})$ are stabilized.

Let us lift $(F,\Phi)$ and $(F^{\prime},\Phi^{\prime})$ to
$(\mathbb{O}^{r},\overline{\Phi})$ and
$(\mathbb{O}^{r},\overline{\Phi}^{\prime})$, where
$\Phi^{\prime},\overline{\Phi}^{\prime}\in\mathfrak{gl}_{r,\mathbb{O}}$.
Choose kernel-strict $g,g^{\prime}\in{\rm GL}_{r,\mathbb{K}}$ such that $g$,
$g^{-1}$, $g^{\prime}$, and $(g^{\prime})^{-1}$ have poles of order less than
$N/2$ and such that
$g\overline{\Phi}g^{-1}=g^{\prime}\overline{\Phi}^{\prime}(g^{\prime})^{-1}=n_{\lambda}$.

Then, according to Lemmas 4.12 and 4.13 (applied to $X$ and $\mathbb{P}^{1}$)
we get

$\displaystyle\xi^{*}A$
$\displaystyle=\frac{\sum\limits_{d^{\prime}+d^{\prime\prime}=d}[{\mathcal{F}}{ib}_{d^{\prime}+e}(X,x)][{\mathcal{F}}{ib}_{d^{\prime\prime}+e^{\prime}}(\mathbb{P}^{1},\infty)]}{[Z_{g}][Z_{g^{\prime}}]}$
$\displaystyle=\frac{\sum\limits_{d^{\prime}+d^{\prime\prime}=d+e+e^{\prime}}[{\mathcal{F}}{ib}_{d^{\prime}}(X,x)][{\mathcal{F}}{ib}_{d^{\prime\prime}}(\mathbb{P}^{1},\infty)]}{[Z_{g}][Z_{g^{\prime}}]}.$

Similarly,

$\displaystyle\xi^{*}B$
$\displaystyle=\frac{\sum\limits_{d^{\prime}+d^{\prime\prime}=d}[{\mathcal{F}}{ib}_{d^{\prime}+e}(\mathbb{P}^{1},\infty)][{\mathcal{F}}{ib}_{d^{\prime\prime}+e^{\prime}}(X,x)]}{[Z_{g}][Z_{g^{\prime}}]}$
$\displaystyle=\frac{\sum\limits_{d^{\prime}+d^{\prime\prime}=d+e+e^{\prime}}[{\mathcal{F}}{ib}_{d^{\prime}}(\mathbb{P}^{1},\infty)][{\mathcal{F}}{ib}_{d^{\prime\prime}}(X,x)]}{[Z_{g}][Z_{g^{\prime}}]}.$

We see that $\xi^{*}A=\xi^{*}B$. This completes the proof of Proposition 4.10
and thus the proof of Theorem 4.4.∎

######  Remark 4.15.

We emphasize that we only worked with motivic classes of Artin stacks of
finite type. It seems plausible that one can define motivic classes of stacks
like $\big{[}\,\overline{{\mathcal{F}}{ib}}_{d}(X,x)\big{]}$ using some ideas
of motivic integration. This would significantly simplify our argument.
Unfortunately, we were not able to develop such a formalism.

###  4.4 Case of $\boldsymbol{\mathbb{P}^{1}}$ and two points

Consider the case when $X=\mathbb{P}^{1}$, $D=\\{0,\infty\\}$.

######  Proposition 4.16.

We have in
$\operatorname{Mot}(\mathbf{k})[[\Gamma^{\prime}_{+}]]\subset\operatorname{Mot}(\mathbf{k})\big{[}\big{[}w,w_{0,\bullet},w_{\infty,\bullet},z^{-1}\big{]}\big{]}$

$\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\\{0,\infty\\}\big{)}\big{]}=\operatorname{Exp}\left(w\frac{\sum_{j=1}^{\infty}\sum_{j^{\prime}=1}^{\infty}w_{0,j}w_{\infty,j^{\prime}}}{(\mathbb{L}-1)(1-z^{-1})}\right),$
(4.6)

where $\operatorname{Exp}$ is the plethystic exponent defined in Section 3.4.

###### Proof.

We note that the proof in [30, Section 5.4] goes through in the motivic case
as well. The only difference is that Mellit uses the Hall algebra of the
Jordan quiver (that is, the Hall algebra of the category of vector spaces with
nilpotent endomorphisms); this Hall algebra has to be replaced with the
similar motivic Hall algebra in our case.

Let us give more details. In [12, Section 5], to any smooth projective
geometrically connected curve over $\mathbf{k}$ we associated the Hall algebra
of the category of coherent sheaves on this curve, denoted by $\mathcal{H}$.
Let us take the curve to be $\mathbb{P}^{1}_{\mathbf{k}}$ and let
$\mathcal{H}_{0}$ be the subalgebra of torsion sheaves supported at
$0\in\mathbb{P}^{1}_{\mathbf{k}}$. Since such sheaves are identified with
finite dimensional representations of the Jordan quiver, we can view
$\mathcal{H}_{0}$ as the Hall algebra of the category of such representations.
Of course, we could have taken any other curve and any rational point.

Following Mellit, we re-write the LHS of (4.6) in terms of products of certain
elements of this algebra. This part of Mellit’s proof is geometric, so it is
easily carried to the motivic case. The rest of the proof is a calculation in
this Hall algebra; the necessary identities in the Hall algebra are easily
derived from results of [12, Section 5]. ∎

######  Remark 4.17.

Note that the RHS of (4.6) can be written without plethystic exponents as
follows (cf. [12, Lemma 5.7.3])

$\prod_{d=0}^{-\infty}\prod_{j=1}^{\infty}\prod_{j^{\prime}=1}^{\infty}\left(1+\sum_{i\geq
1}\frac{\mathbb{L}^{i(i-1)}}{\big{(}\mathbb{L}^{i}-1\big{)}\cdots\big{(}\mathbb{L}^{i}-\mathbb{L}^{i-1}\big{)}}z^{id}w^{i}w_{0,j}^{i}w_{\infty,j^{\prime}}^{i}\right).$

###  4.5 Motivic modified Macdonald polynomials

For a commutative unital ring $A$, let $\operatorname{Sym}_{A}[w_{\bullet}]$
be the ring of symmetric functions with coefficients in $A$ in variables
$w_{\bullet}$. In this section we define axiomatically the images of modified
Macdonald polynomials in
$\operatorname{Sym}_{\operatorname{Mot}(\mathbf{k})[[z]]}[w_{\bullet}]$.

Consider the modified Macdonald polynomials
$\tilde{H}_{\lambda}(w_{\bullet};q,z)\in\operatorname{Sym}_{\mathbb{Z}[q,z]}[w_{\bullet}]$.
For a definition see, for example, [30, Definition 2.5]. It is not clear from
this definition that the coefficients of
$\tilde{H}_{\lambda}(w_{\bullet};q,z)$ are integers, but this is well-known
(see, e.g., [14] and references therein). Note that $\tilde{H}_{\lambda}$ is a
symmetric function so, formally speaking, it is not a polynomial.

We denote by $\tilde{H}^{\rm
mot}_{\lambda}(w_{\bullet};z)\in\operatorname{Sym}_{\operatorname{Mot}(\mathbf{k})[z]}[w_{\bullet}]$
the image of the corresponding modified Macdonald polynomial under the
homomorphism
$\operatorname{Sym}_{\mathbb{Z}[q,z]}[w_{\bullet}]\to\operatorname{Sym}_{\operatorname{Mot}(\mathbf{k})[z]}[w_{\bullet}]$
sending $q$ to $\mathbb{L}$; we call these images _motivic modified Macdonald
polynomials_.

Define the motivic Hall–Littlewood polynomials as the specialization

$H_{\lambda}^{\rm mot}(w_{\bullet}):=\tilde{H}_{\lambda}^{\rm
mot}(w_{\bullet};0).$

Thus $H_{\lambda}^{\rm mot}$ is the image of the usual Hall–Littlewood
polynomial under the homomorphism
$\mathbb{Z}[q,w_{\bullet}]\to\operatorname{Mot}(\mathbf{k})[w_{\bullet}]$
sending $q$ to $\mathbb{L}$. The motivic Hall–Littlewood polynomials can be
interpreted as follows: let $\operatorname{Fl}_{\lambda}$ stand for the scheme
of all flags in $\mathbf{k}^{|\lambda|}$ preserved by $n_{\lambda}$. Then
$\operatorname{Fl}_{\lambda}$ is graded by the type of the flag. It is not
difficult to check that we have

$H^{\rm
mot}_{\lambda}(w_{\bullet})=[\operatorname{Fl}_{\lambda}]\in\operatorname{Mot}(\mathbf{k})[w_{\bullet}]$

(cf. [30, Theorem 2.12, Corollary 2.13]).

It follows from [28, Chapter 3, equation (2.7)] that $H_{\lambda}^{\rm mot}$
form a basis of the $\operatorname{Mot}(\mathbf{k})$-module
$\operatorname{Sym}_{\operatorname{Mot}(\mathbf{k})}[w_{\bullet}]$. Thus
$\tilde{H}_{\lambda}^{\rm mot}$ also form a basis of
$\operatorname{Sym}_{\operatorname{Mot}(\mathbf{k})[[z]]}[w_{\bullet}]$.

######  Proposition 4.18.

  1. $(a)$

The motivic modified Macdonald polynomials $\tilde{H}_{\lambda}^{\rm mot}$
satisfy the following properties

    1. $(i)$

$\operatorname{Exp}\left(\frac{\sum\limits_{j=1}^{\infty}\sum\limits_{j^{\prime}=1}^{\infty}w_{0,j}w_{\infty,j^{\prime}}}{(\mathbb{L}-1)(1-z)}\right)=\sum_{\lambda}a_{\lambda}(z)\tilde{H}_{\lambda}^{\rm
mot}(w_{0,\bullet};z)\tilde{H}_{\lambda}^{\rm mot}(w_{\infty,\bullet};z),$
(4.7)

where $a_{\lambda}$ are invertible elements of
$\operatorname{Mot}(\mathbf{k})[[z]]$.

    2. $(ii)$

$\tilde{H}_{\lambda}^{\rm
mot}=\sum\limits_{\mu\colon\mu^{\prime}\prec\lambda^{\prime}}b_{\lambda\mu}H_{\mu}^{\rm
mot}$, where $b_{\lambda\mu}$ are some elements of
$\operatorname{Mot}(\mathbf{k})[[z]]$ such that $b_{\lambda\lambda}$ are
invertible. $($Here $\mu^{\prime}$ and $\lambda^{\prime}$ stand for the
conjugate partitions, $\prec$ is the usual order on partitions.$)$

    3. $(iii)$

$\tilde{H}^{\rm mot}_{\lambda}(1_{\bullet};z)=1$, where $1_{\bullet}$ stands
for the sequence $(1,0,\dots,0,\dots)$.

    4. $(iv)$

$\tilde{H}^{\rm mot}_{\lambda}$ is homogeneous in $w_{\bullet}$ of degree
$|\lambda|$.

  2. $(b)$

The motivic modified Macdonald polynomials are uniquely determined by
properties $(i)$–$(iv)$.

  3. $(c)$

Additionally we have

$a_{\lambda}(z)=\frac{1}{\prod\limits_{h\in\operatorname{Hook}(\lambda)}\big{(}\mathbb{L}^{a(h)}-z^{l(h)+1}\big{)}\big{(}\mathbb{L}^{a(h)+1}-z^{l(h)}\big{)}},$
(4.8)

where $\operatorname{Hook}(\lambda)$ stands for the set of hooks of $\lambda$,
$a(h)$ and $l(h)$ stand for the armlength and the leglength of the hook $h$
respectively.

###### Proof.

(a) It is enough to check the corresponding properties for the usual modified
Macdonald polynomials
$\tilde{H}_{\lambda}\in\operatorname{Sym}_{\mathbb{Z}[q,z]}[w_{\bullet}]$ and
Hall–Littlewood polynomials
$H_{\lambda}\in\operatorname{Sym}_{\mathbb{Z}[q]}[w_{\bullet}]$. To prove
property (ii) we note first that according to [30, Definition 2.5] we have

$\tilde{H}_{\lambda}[(q-1)w_{\bullet}]=\sum_{\mu\colon\mu^{\prime}\prec\lambda^{\prime}}c_{\lambda\mu}(q,z)m_{\mu^{\prime}}(w_{\bullet}),$

where $[(q-1)w_{\bullet}]$ stands for the plethystic action as in [30, Section
2.1]. Recalling that the Hall–Littlewood polynomials are $z=0$ specializations
of the modified Macdonald polynomials, we get

$H_{\lambda}[(q-1)w_{\bullet}]=\sum_{\mu\colon\mu^{\prime}\prec\lambda^{\prime}}c_{\lambda\mu}(q,0)m_{\mu^{\prime}}(w_{\bullet}).$

Now it is easy to see that an analogue of property (ii) holds in
$\operatorname{Sym}_{\mathbb{Q}[q,z]}(w_{\bullet})$: we can write
$\tilde{H}_{\lambda}=\sum\limits_{\mu\colon\mu^{\prime}\prec\lambda^{\prime}}b^{\prime}_{\lambda\mu}H_{\mu}$,
where $b^{\prime}_{\lambda\mu}$ are some elements of $\mathbb{Q}[q,z]$. Next,
$H_{\lambda}$ form a basis in
$\operatorname{Sym}_{\mathbb{Z}[q]}(w_{\bullet})$ (by [28, Chapter 3, equation
(2.7)]), so $\tilde{H}_{\lambda}$ form a basis in
$\operatorname{Sym}_{\mathbb{Z}[q][[w]]}(w_{\bullet})$. It follows that
$b^{\prime}_{\lambda\mu}\in\mathbb{Z}[w][[z]]$ and
$b^{\prime}_{\lambda\lambda}$ are invertible in this ring. Now property (ii)
follows.

It is sufficient to prove properties (iii), and (iv) in
$\operatorname{Sym}_{\mathbb{Q}(q,z)}[w_{\bullet}]$. Property (iii) is clear
from [30, Definition 2.5]. Property (iv) follows, for example, from the
definition of $\tilde{H}$ given in [13].

We first prove an analogue of property (i) in
$\operatorname{Sym}_{\mathbb{Q}(q,z)}[w_{\bullet}]$. Recall from loc. cit.
that $\operatorname{Sym}_{\mathbb{Q}(q,z)}[w_{\bullet}]$ carries the
$q,z$-scalar product $(\cdot,\cdot)_{q,z}$. for which

$\operatorname{Exp}\left(\frac{\sum\limits_{j=1}^{\infty}\sum\limits_{j^{\prime}=1}^{\infty}w_{0,j}w_{\infty,j^{\prime}}}{(q-1)(1-z)}\right)$

is the reproducing kernel. This means that if $f_{\lambda}(w_{\bullet};q,z)$
is any graded $\mathbb{Q}(q,z)$-basis in
$\operatorname{Sym}_{\mathbb{Q}(q,z)}[w_{\bullet}]$ indexed by partitions and
$f^{\vee}_{\lambda}(w_{\bullet};q,z)$ is the dual basis with respect to
$(\cdot,\cdot)_{q,z}$, then

$\operatorname{Exp}\left(\frac{\sum\limits_{j=1}^{\infty}\sum\limits_{j^{\prime}=1}^{\infty}w_{0,j}w_{\infty,j^{\prime}}}{(q-1)(1-z)}\right)=\sum_{\lambda}f_{\lambda}(w_{0,\bullet};q,z)f^{\vee}_{\lambda}(w_{\infty,\bullet};q,z).$
(4.9)

Next, by property (iv) the basis $H_{\lambda}(w_{\bullet};q,z)$ is a graded
basis. Thus, by [30, Proposition 2.7] the dual of
$H_{\lambda}(w_{\bullet};q,z)$ is equal to
$a_{\lambda}(q,z)H_{\lambda}(w_{\bullet};q,z)$ for some
$a_{\lambda}(q,z)\in\mathbb{Q}(q,z)$. Further, in [30, Section 2.4] it is
shown that

$a_{\lambda}(q,z)=\frac{1}{\prod\limits_{h\in\operatorname{Hook}(\lambda)}\big{(}q^{a(h)}-z^{l(h)+1}\big{)}\big{(}q^{a(h)+1}-z^{l(h)}\big{)}}.$
(4.10)

It is clear from this formula that $a_{\lambda}(q,z)\in\mathbb{Z}(q)[[z]]$.
Now property (i) follows from (4.9).

The condition in part (c) follows from (4.10).

Now we prove part (b). Since $\tilde{H}_{\lambda}^{\rm mot}$ form a basis of
$\operatorname{Sym}_{\operatorname{Mot}(\mathbf{k})[[z]]}[w_{\bullet}]$, there
is a unique $\operatorname{Mot}(\mathbf{k})[[z]]$-linear scalar product on
$\operatorname{Sym}_{\operatorname{Mot}(\mathbf{k})[[z]]}[w_{\bullet}]$ such
that $\langle\tilde{H}_{\lambda}^{\rm mot},\tilde{H}_{\mu}^{\rm
mot}\rangle=\delta_{\lambda\mu}\frac{1}{a_{\lambda}}$. (This is the scalar
product such that the LHS of (4.7) is the reproducing kernel for this
product).

Let $H^{\prime}_{\lambda}=H^{\prime}_{\lambda}(w_{\bullet};z)$ be symmetric
functions satisfying conditions of part (a), we need to show that
$H^{\prime}_{\lambda}=\tilde{H}_{\lambda}^{\rm mot}$. Applying condition (ii),
we see that we can write

$H^{\prime}_{\lambda}=\sum_{\mu\colon\mu^{\prime}\prec\lambda^{\prime}}c_{\lambda\mu}\tilde{H}_{\mu}^{\rm
mot},$ (4.11)

where $c_{\lambda\lambda}$ is invertible. Condition (i) shows that we have

$\sum_{\lambda}a_{\lambda}(z)\tilde{H}_{\lambda}^{\rm
mot}(w_{0,\bullet};z)\tilde{H}_{\lambda}^{\rm
mot}(w_{\infty,\bullet};z)=\sum_{\lambda}a^{\prime}_{\lambda}(z)H^{\prime}_{\lambda}(w_{0,\bullet};z)H^{\prime}_{\lambda}(w_{\infty,\bullet};z),$

with invertible $a^{\prime}_{\lambda}(z)$. Recalling that
$H^{\prime}_{\lambda}$ form a graded basis in
$\operatorname{Sym}_{\operatorname{Mot}(\mathbf{k})[[z]]}[w_{\bullet}]$, we
see that $\langle
H^{\prime}_{\lambda},H^{\prime}_{\mu}\rangle=\delta_{\lambda\mu}\frac{1}{a^{\prime}_{\lambda}}$.
Indeed, $H_{\lambda}^{\rm mot}$ and $a_{\lambda}H_{\lambda}^{\rm mot}$ are
dual basis for the scalar product, so the above equality shows that
$H^{\prime}_{\lambda}$ and $a^{\prime}_{\lambda}H^{\prime}_{\lambda}$ are dual
basis as well.

We prove that $H^{\prime}_{\lambda}=\tilde{H}_{\lambda}^{\rm mot}$ by
induction on the conjugate partition $\lambda^{\prime}$ with respect to
$\prec$. Thus we assume that $H^{\prime}_{\mu}=\tilde{H}_{\mu}^{\rm mot}$
whenever $\mu^{\prime}\prec\lambda^{\prime}$. Taking the scalar product of
(4.11) with $\tilde{H}_{\mu}^{\rm mot}=H^{\prime}_{\mu}$, we see that
$c_{\lambda\mu}\langle\tilde{H}_{\mu}^{\rm mot},\tilde{H}_{\mu}^{\rm
mot}\rangle=0$ whenever $\mu\neq\lambda$. We see that $c_{\lambda\mu}=0$ so
that $H^{\prime}_{\lambda}=c_{\lambda\lambda}\tilde{H}_{\lambda}^{\rm mot}$.
Now condition (iii) implies that $c_{\lambda\lambda}=1$. ∎

###  4.6 Explicit formulas for the graded motivic classes of nilpotent pairs

Now we are ready to give the precise formula for
$\big{[}\mathcal{P}{air}^{{\rm nilp},-}(X,D,\lambda)\big{]}$. Recall that for
a partition $\lambda$ we defined $J_{\lambda}^{\rm mot}(z),H_{\lambda}^{\rm
mot}(z)\in\operatorname{\overline{Mot}}(\mathbf{k})[[z]]$ in [12, Section
1.3.2]. In this paper, we will denote them by $J_{\lambda,X}^{\rm mot}(z)$ and
$H_{\lambda,X}^{\rm mot}(z)$ respectively to emphasize that they depend on the
curve $X$ and to ensure that they are not confused with motivic modified
Macdonald polynomials $\tilde{H}_{\lambda}^{\rm mot}(w_{\bullet};z)$ and with
motivic Hall–Littlewood polynomials $H_{\lambda}^{\rm mot}(w_{\bullet})$.
Denote by $g$ the genus of $X$.

######  Theorem 4.19.

We have in $\operatorname{\overline{Mot}}(\mathbf{k})[[\Gamma^{\prime}_{+}]]$

$\big{[}\mathcal{P}{air}^{{\rm
nilp},-}(X,D,\lambda)\big{]}=w^{|\lambda|}\mathbb{L}^{(g-1)\langle\lambda,\lambda\rangle}J_{\lambda,X}^{\rm
mot}\big{(}z^{-1}\big{)}H_{\lambda,X}^{\rm mot}\big{(}z^{-1}\big{)}\prod_{x\in
D}\tilde{H}_{\lambda}^{\rm mot}\big{(}w_{x,\bullet};z^{-1}\big{)}.$

###### Proof.

According to [12, Theorem 1.4.1], we have

$\sum_{\lambda}\big{[}\mathcal{P}{air}^{{\rm
nilp},+}(X,\varnothing,\lambda)\big{]}=w^{|\lambda|}\sum_{\lambda}\mathbb{L}^{(g-1)\langle\lambda,\lambda\rangle}J^{\rm
mot}_{\lambda,X}(z)H^{\rm mot}_{\lambda,X}(z),$

where the superscript “$+$” stands for HN-nonnegative vector bundles (see
Section 3.2 and [12, Section 3.2] for the definition). Inspecting the proof,
we see that for each $\lambda$ the summands are equal:

$\big{[}\mathcal{P}{air}^{{\rm
nilp},+}(X,\varnothing,\lambda)\big{]}=w^{|\lambda|}\mathbb{L}^{(g-1)\langle\lambda,\lambda\rangle}J_{\lambda,X}^{\rm
mot}(z)H_{\lambda,X}^{\rm mot}(z).$

By Lemma 3.2 we get an isomorphism of stacks

$\mathcal{P}{air}^{{\rm
nilp},+}_{r,d}(X,\varnothing,\lambda)\simeq\mathcal{P}{air}^{{\rm
nilp},-}_{r,-d}(X,\varnothing,\lambda).$

Thus

$\big{[}\mathcal{P}{air}^{{\rm
nilp},-}(X,\varnothing,\lambda)\big{]}=w^{|\lambda|}\mathbb{L}^{(g-1)\langle\lambda,\lambda\rangle}J_{\lambda,X}^{\rm
mot}\big{(}z^{-1}\big{)}H_{\lambda,X}^{\rm mot}\big{(}z^{-1}\big{)}.$

To be able to apply Theorem 4.4, we need the following lemma.

######  Lemma 4.20.

We have in $\operatorname{Mot}(\mathbf{k})[[w_{\infty,\bullet},z^{-1}]]$

$\frac{\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\infty,\lambda\big{)}\big{]}}{\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\varnothing,\lambda\big{)}\big{]}}=\tilde{H}_{\lambda}^{\rm
mot}\big{(}w_{\infty,\bullet};z^{-1}\big{)}.$

###### Proof.

Our proof is similar to that of [30, Theorem 5.5]. Let
$H^{\prime}_{\lambda}\in\operatorname{Mot}(\mathbf{k})[[w_{\bullet},z]]$ be
the series such that

$\frac{\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\infty,\lambda\big{)}\big{]}}{\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\varnothing,\lambda\big{)}\big{]}}=H^{\prime}_{\lambda}\big{(}w_{\infty,\bullet};z^{-1}\big{)}.$

Denote by $\mathcal{C}_{\lambda\mu}$ the stack classifying pairs $(E,\Psi)$,
where $E$ is a nonpositive vector bundle of rank $|\lambda|$ on
$\mathbb{P}^{1}$, $\Psi$ is an endomorphism of $E$ generically conjugate to
$n_{\lambda}$ and conjugate to $n_{\mu}$ at $x=\infty$. Then
$\mathcal{C}_{\lambda\mu}$ is graded by the degree of $E$, and we have
$[\mathcal{C}_{\lambda\mu}]\in\operatorname{Mot}(\mathbf{k})\big{[}\big{[}z^{-1}\big{]}\big{]}$.
Clearly, we have

$H^{\prime}_{\lambda}\big{(}w_{\bullet};z^{-1}\big{)}=\sum_{\mu}\frac{[\mathcal{C}_{\lambda\mu}]}{\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\varnothing,\lambda\big{)}\big{]}}H_{\mu}^{\rm
mot}(w_{\bullet}),$ (4.12)

where $H_{\mu}^{\rm mot}$ are the motivic Hall–Littlewood polynomials. Note
that $C_{\lambda\mu}=\varnothing$ unless $\mu^{\prime}\prec\lambda^{\prime}$
because for all $i$ the dimension of the fiber of $\operatorname{Ker}\Psi^{i}$
is semicontinuous on $\mathbb{P}^{1}$. Now it is easy to see that
$H^{\prime}_{\lambda}$ are symmetric functions with coefficients in
$\operatorname{Mot}(\mathbf{k})[[z]]$. We will use Proposition 4.18(b) to show
that for all $\lambda$ we have $H^{\prime}_{\lambda}=\tilde{H}_{\lambda}^{\rm
mot}$.

To show that $H^{\prime}_{\lambda}$ satisfy property (ii) of Proposition
4.18(a) it remains to show that $[\mathcal{C}_{\lambda\lambda}]$ is
invertible. This is completely similar to the proof of Lemma 4.3.

To show that $H^{\prime}_{\lambda}$ satisfy condition (i) of Proposition
4.18(a), we note that combining Theorem 4.4 and Proposition 4.16, we get

$\operatorname{Exp}\left(\frac{\sum\limits_{j=1}^{\infty}w_{0,j}w_{\infty,j}}{(\mathbb{L}-1)(1-z)}\right)=\sum_{\lambda}a^{\prime}_{\lambda}(z)H^{\prime}_{\lambda}(w_{0,\bullet};z)H^{\prime}_{\lambda}(w_{\infty,\bullet};z),$

where $a^{\prime}_{\lambda}(z^{-1})=\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\varnothing,\lambda\big{)}\big{]}$ and the
statement follows.

Condition (iii) of Proposition 4.18(a) is obvious. Condition (iv) follows from
(4.12). We note also for the future use that it is clear from the argument
that we have

$\big{[}\mathcal{P}{air}^{{\rm
nilp},-}\big{(}\mathbb{P}^{1},\varnothing,\lambda\big{)}\big{]}=a^{\prime}_{\lambda}\big{(}z^{-1}\big{)}=a_{\lambda}\big{(}z^{-1}\big{)},$
(4.13)

where $a_{\lambda}(z)$ is given by (4.8). The proof of Lemma 4.20 is complete.
∎

Now Theorem 4.4 completes the proof of Theorem 4.19. ∎

######  Corollary 4.21.

We have in $\operatorname{\overline{Mot}}(\mathbf{k})[[\Gamma^{\prime}_{+}]]$

$\displaystyle\big{[}\mathcal{P}{air}^{-}(X,D)\big{]}=\operatorname{\mathrm{Pow}}\left(\sum_{\lambda}w^{|\lambda|}\mathbb{L}^{(g-1)\langle\lambda,\lambda\rangle}J_{\lambda,X}^{\rm
mot}\big{(}z^{-1}\big{)}H_{\lambda,X}^{\rm mot}\big{(}z^{-1}\big{)}\prod_{x\in
D}\tilde{H}_{\lambda}^{\rm