# The parabolic Verlinde formula: iterated residues and wall-crossings

Andras Szenes  Section de mathématiques, Université de Genève
<EMAIL_ADDRESS>and  Olga Trapeznikova  Section de mathématiques,
Université de Genève<EMAIL_ADDRESS>
###### Abstract.

We give a new proof for the parabolic Verlinde formula in all ranks based on a
comparison of wall-crossings in Geometric Invariant Theory and certain
iterated residue functionals. On the way, we develop a tautological variant of
Hecke correspondences, calculate the Hilbert polynomials of the moduli spaces,
and present a new, transparent, local approach to the $\rho$-shift problem of
the theory.

## 1\. Introduction

### 1.1. The Verlinde formula

The Verlinde formula is a strikingly beautiful statement in Enumerative
Geometry motivated by quantum physics [Ver]. Our focus in this paper will be
the more difficult, parabolic variant, which we briefly describe below.

Let $C$ be a smooth, complex projective curve of genus $g\geq 1$, and fix an
auxiliary point $p\in C$. We will call a vector
$c=(c_{1}>c_{2}>\dots>c_{r})\in\mathbb{R}^{r}$ satisfying $\sum c_{i}=0$ and
$c_{1}-c_{r}<1$ regular if no nontrivial subset of its coordinates sums to an
integer. For such a $c\in\mathbb{R}^{r}$, there exists a smooth projective
moduli space $P_{0}(c)$ ([Se, MS, B]), whose points are in one-to-one
correspondence with the equivalence classes of pairs $(W,F_{*})$, where $W\to
C$ is a vector bundle of rank $r$ on $C$ with trivial determinant, $F_{*}$ is
a full flag of the fiber $W_{p}$, and the pair satisfies a certain parabolic
stability condition depending on $c$ (cf. §2.1). This condition roughly states
that for a proper subbundle $W^{\prime}\subset W$, the degree
$\deg(W^{\prime})$ is strictly smaller than the sum of a subset of the
coordinates of $c$ depending on the position of $W^{\prime}_{p}$ with respect
to $F_{*}$.

There is a natural way to associate to a positive integer $k$ and an integer
vector $\lambda\in\mathbb{Z}^{r}$ satisfying $\lambda_{1}+\dots+\lambda_{r}=0$
a line bundle $\mathcal{L}({k;\lambda})$ on $P_{0}(c)$, in such a way that if
$c=\lambda/k$, then $\mathcal{L}({k;\lambda})$ is ample. The parabolic
Verlinde formula is the following expression for the Euler characteristic of
the ample line bundle $\mathcal{L}({k;\lambda})$: assume $c=\lambda/k$ is
regular; then

(1)
$\chi(P_{0}(c),\mathcal{L}({k;\lambda}))=N_{r,k}\cdot\sum\frac{(-i)^{r\choose
2}\exp(2\pi i\,\widehat{\lambda}\cdot
x)}{\prod_{i<j}\left(2\sin\pi(x_{i}-x_{j})\right)^{2g-1}}$

where $N_{r,k}=r(r(k+r)^{r-1})^{g-1}$,
$\widehat{\lambda}=\lambda+\frac{1}{2}(r-1,r-3,\dots,1-r)$, and the sum is
taken over the finite set of those points in the interior of the
parallelopiped

$\\{x=(x_{1},x_{2},\dots,x_{r}=0)|\;0<x_{i}-x_{i+1}<1\text{ for
}i=1,\dots,r-1\\}$

which satisfy the conditions

  * •

$(k+r)x\in\mathbb{Z}^{r}$

  * •

$x_{i}-x_{j}\notin\mathbb{Z}$ for $1\leq i<j<r$.

We note that this finite set is a set of lattice points in the interior of
$(r-1)!$ identical simplices (cf. the rhombus on Figure 1).

(0,1,-1)(1,-1,0)(0,0,0)(2,2,-4)(4,-2,-2)

(0,0,0)(0,1,1)(1,1,0)(1,2,1)

Figure 1. The set of $\lambda$s (left), and the finite set from (1) (right)
for $k=6$, $r=3$.

######  Remark 1.1.

Equality (1) remains valid in greater generality, for certain cases when
$\lambda/k$ is non-regular. This slightly more technical statement will be
given in Theorems 4.7 and 4.8.

Notation: We denote the discrete sum in (1) depending on $k,\lambda,r$ and $g$
by $\mathrm{Ver}(k,\lambda)$. In what follows, the shift
$\frac{1}{2}(r-1,r-3,\dots,1-r)$ will be denoted by $\rho$, and thus we have
$\widehat{\lambda}=\lambda+\rho$.

Equality (1), the parabolic Verlinde formula has attracted a lot of attention
over the years, and there is a number of different proofs. There is a
generalization of this formula associated to a simply connected compact Lie
group, and the form presented here corresponds to the case of the group
$SU(r)$.

In this article, we give a novel proof of this result, which stands out with
its technical simplicity. We believe the methods and ideas described in the
paper will have other applications in Geometric Invariant Theory and the study
of moduli spaces.

Below, we give a quick sketch of the strategy of the proof, treating the
example of the case of rank 3 in §1.2-4. Next, we give a short guide to the
contents of the paper in §1.5.

Our work has close relationship with several earlier approaches, and we
describe these links in §1.6

Acknowledgments. The authors gratefully acknowledge the help and insights of
Michael Thaddeus at several stages of this work, as well as the advice and
encouragement of Tamás Hausel and Michèle Vergne. We had useful discussions
with Eckhard Meinrenken, Gábor Tardos and Chris Woodward. This research was
supported by SNF grant 175799, and the NCCR SwissMAP.

### 1.2. The residue formula

The proof is based on 3 ideas. We will follow the arguments below for the case
$r=3$. We fix thus an integer $k>1$ and an integer vector
$\lambda=(\lambda_{1}>\lambda_{2}>\lambda_{3})$, such that
$\lambda_{1}+\lambda_{2}+\lambda_{3}=0$ and $\lambda_{1}-\lambda_{r}<k$.

We start with the study of the right hand side of (1), which, for $r=3$, may
be written in the following somewhat simplified form

$\mathrm{Ver}(k,\lambda)=N_{3,k}\sum_{0<n_{2}<n_{1}<k+3}\frac{2i\sin
2\pi\frac{(\lambda_{1}+1)n_{1}+\lambda_{2}n_{2}}{k+3}}{\left(8\sin\pi\frac{n_{1}-n_{2}}{k+3}\sin\pi\frac{n_{2}}{k+3}\sin\pi\frac{n_{1}}{k+3}\right)^{2g-1}},$

where $n_{1},n_{2}$ are integers. Using Theorem 4.7 and Remark 4.6 one can
show that

$\mathrm{Ver}(k,\lambda)=\begin{cases}p_{+}(k;\lambda),\;\lambda_{2}>0,\\\
p_{-}(k;\lambda),\;\lambda_{2}<0,\end{cases}$

where $p_{+}$ and $p_{-}$ are two polynomials, given by the right hand sides
of the expressions of Example 4 on page 4. We note two properties of
$p_{\pm}(k,\lambda)$:

  1. A.

The wall-crossing difference $p_{-}-p_{+}$ has a relatively simple form (cf.
Example 5 with $\lambda_{1}+\lambda_{3}$ replaced by $-\lambda_{2}$):

$p_{-}(k;\lambda)-p_{+}(k;\lambda)=\operatornamewithlimits{Res}_{y=0}\operatornamewithlimits{Res}_{x=0}\frac{(-3(k+3)^{2})^{g}\cdot
e^{(\lambda_{1}+1)x-\lambda_{2}y}}{(1-e^{x(k+3)})w_{\Phi}(x,y)^{2g-1}}dxdy,$

where $w_{\Phi}(x,y)=2\mathrm{sinh}(\frac{x}{2})\cdot
2\mathrm{sinh}(\frac{y}{2})\cdot 2\mathrm{sinh}(\frac{x+y}{2})$.

  2. B.

An easy calculation via substitution shows that for any permutation on 3
elements $\sigma\in\Sigma_{3}$, our polynomials have the following symmetries:

(2)
$p_{+}(k;\sigma\cdot\lambda+\theta_{1}[k])=(-1)^{\sigma}p_{+}(k;\lambda+\theta_{1}[k])$

and

(3)
$p_{-}(k;\sigma\cdot\lambda+\theta_{-1}[k])=(-1)^{\sigma}p_{-}(k;\lambda+\theta_{-1}[k]),$

where

$\theta_{1}[k]=\frac{k}{3}(1,1,-2)+(0,1,-1)\quad\theta_{-1}[k]=\frac{k}{3}(2,-1,-1)+(1,-1,0).$

### 1.3. Wall-crossings in moduli spaces

Now consider the left hand side of (1). It is easy to check that the set of
isomorphism classes of parabolic bundles in $P_{0}(c)$ remains unchanged as
long as $c_{2}$ does not change sign. Hence, effectively, we have two moduli
spaces $P_{0}(>)$ and $P_{0}(<)$, corresponding to the two chambers separated
by the red ($c_{2}=0$) line in Figure 1. Introduce the notation

$q_{+}(k;\lambda)=\chi(P_{0}(>),\mathcal{L}({k;\lambda})),\quad
q_{-}(k;\lambda)=\chi(P_{0}(<),\mathcal{L}({k;\lambda}))$

for the generalized Hilbert polynomials of these two spaces.

In §5, we derive a simple formula (47) for the wall-crossing difference in
Geometric Invariant Theory. The formula has the form of a residue of an
equivariant integral, taken with respect to the equivariant parameter. In our
case, the space on which we integrate is the space of rank-3 parabolic bundles
which split into a direct sum of a rank-2 and a rank-1 bundle. This
equivariant integral may be evaluated using induction on the rank (cf. the
detailed calculation in Example 6 on page 6), and the result is

(4)
$q_{-}(k;\lambda)-q_{+}(k;\lambda)=\operatornamewithlimits{Res}_{u=0}\operatornamewithlimits{Res}_{z=0}\frac{(-3(k+3)^{2})^{g}\cdot
e^{\lambda_{1}z+\lambda_{2}u+z}}{-w_{\Phi}(z,-u)^{2g-1}(1-e^{(k+3)z})}dzdu,$

where $u$ plays the role of the equivariant parameter, the generator of
$H^{*}_{\mathbb{C}^{*}}(\mathrm{pt})$. This iterated residue coincides with
the expression above after changing $(z,u)$ to $(x,-y)$, and thus we have

(5) $p_{+}-p_{-}=q_{+}-q_{-}.$

### 1.4. Hecke correspondences, Serre duality and the symmetry argument

Hecke correspondences between moduli spaces of bundles of different degrees
were introduced by Narasimhan and Ramanan in [NR]. In §7 of our paper, we
describe a "tautological" variant of this construction, which identifies the
same space with several moduli spaces of parabolic bundles with different
degrees and weights. Using this construction we can fiber our two moduli
spaces, $P_{0}(>)$ and $P_{0}(<)$ over the moduli spaces of stable bundles
(without parabolic structure) of degrees 1 and $-1$:

$\mathrm{Flag}_{3}\to P_{0}(<)\to N_{-1},\quad N_{1}\leftarrow
P_{0}(>)\leftarrow\mathrm{Flag}_{3},$

where the fibers are full flags of 3-dimensional vector spaces. Serre duality
applied to a $\mathrm{Flag}_{3}$-bundle implies a $\Sigma_{3}$-antisymmetry of
the Euler characteristics of line bundles on this space, and after careful
identification of these bundles, we derive the same symmetry properties for
the functions $q_{\pm}$ as we did for the polynomials $p_{\pm}$:
$q_{+}(k;\lambda)$ satisfies (2), while $q_{-}(k;\lambda)$ satisfies (3).

The final argument is elegant: we can rearrange equation (5) describing the
equality of wall-crossings as

$p_{+}-q_{+}=p_{-}-q_{-},$

and we introduce the notation $\Theta(k;\lambda)$ for this polynomial. Then
$\Theta$ satisfies both (2) and (3), and thus it is anti-invariant with
respect to an affine Weyl group action in the plane for each fixed $k$. This
implies that $\Theta(k;\lambda)$ vanishes and this completes the proof.

### 1.5. Contents of the paper

There are a number of complications which arise when $r>3$. We will highlight
these in this section, and also give a brief guide to the contents of the
paper.

We start with a quick introduction into the theory of parabolic bundles in §2.
Here we describe the line bundles we are considering, as well as the chamber
structure of the space of parabolic weights induced by the stability
condition. The combinatorics of the iterated residue formulas mentioned in
§1.2 above is considerably more complicated in the higher rank case, and is
best treated using the notion of diagonal bases of hyperplane arrangements
introduced in [Sz1]; we review this construction in the special case of the
$A_{r}$ root arrangement in §3.

Using this notion, in §4, we present a residue formula for the Verlinde sums
on the right hand side of (1) obtained in [Sz2] (Theorems 4.4 and 4.7). It
turns out that because of a standard $\rho$-shift type effect in the theory,
this residue formula does not have a manifestly polynomial form on our
chambers, and thus, we formulate our main result, Theorem 4.8 in two parts: in
part I. we state the equality of the Euler characteristics of line bundles
with a modified residue formula, which is manifestly polynomial on our
chambers, and in part II. we state the equality of the modified formula with
the original residue formula from [Sz2]. Part II. is proved in §10, while the
the proof of part I. takes up the rest of the paper.

At the end of §4, we present our wallcrossing formula for Verlinde sums in
Proposition 4.16, which uses in an essential manner the yoga of diagonal bases
(cf. property A. above for the case of $r=3$).

The geometric part of our work starts in §5, where we derive a simple general
result, formula (47), for wallcrossings in GIT. We apply this result to
parabolic moduli spaces in §6, and, using induction on the rank, obtain
Theorem 6.11, the higher rank version of formula (4) above.

It is downhill from here: in §7 we describe the tautological Hecke
correspondences we need in several places in the paper, and in §8 we derive
the Weyl-symmetries of the polynomials $q_{\pm}$, and finish the proof along
the lines sketched above.

We are essentially done, but we hit a snag when checking the beginning of our
induction on the rank: our argument does not work for $r=2$. Roughly, the
reason for this is that we need our simplex of parabolic weights to have at
least 2 regular vertices, and for $r=2$, we have only 1. The way out is to
consider the moduli space with two punctures and then all the pieces fall in
place. This argument is carried out in §9.

### 1.6. Historical remarks

There is a long list of proofs of the Verlinde formula, and we cannot do
justice to all the approaches in this short introduction. We will thus focus
on the historical lineage of our paper, and the works that are closest in
spirit to what we do (cf. [S] for a more comprehensive overview).

The proofs of the Verlinde formula fall in two categories: proofs of the
fusion rules and proofs that find some interpretation of the "Fourier
transformed" discrete sum on the right hand side of (1); our work belongs to
this second group. Another line of division concerns the model which one uses
for the moduli spaces: via the Narasimhan-Seshadri correspondence, the moduli
spaces of vector bundles may equally be presented as symplectic manifolds of
certain types of flat connections on punctured Riemann surfaces, and this
opens the way of using the methods of symplectic geometry. While these
symplectic approaches lead to results equivalent to the ones coming out of the
algebro-geometric setup, the fields of applications of the two approaches seem
to be very different.

The idea of proving the Verlinde formula via wall crossings appeared in the
seminal paper of Michael Thaddeus [Th2]. He used a geometric approach and
managed to prove the Verlinde formula in rank 2 by crossing walls in the
moduli of stable pairs. The master space construction, which plays a central
role in our paper, first appeared in his works as well [Th1]. In a sense, our
paper may be thought of as the completion of his program.

A paper closely related to our work is that of Jeffrey and Kirwan [JK], who
also use the residue calculus introduced in [Sz1, Sz2]. This paper approaches
the problem from a symplectic/cohomological point of view, and has a somewhat
different angle form ours. The use of iterated residues is not quite as
consistent in [JK] as in our work, and the parabolic case was not resolved
from this point of view [J]. The geometric model used to represent the moduli
spaces as quotients is rather complicated.

In a comprehensive paper [BL] covering the case of all compact groups, Bismut,
Laborie used a differential–geometric approach to find the generating function
for the parabolic Verlinde formula. This work was the motivation for the
residue formula in [Sz2], which is also used in the present paper.

In a remarkable series of papers of Alekseev, Meinrenken and Woodward [AMW],
again, approaching the subject from the symplectic point of view, gave a
direct proof of (1), using reduction in infinite dimensions. A general
approach related to twisted K-theory was introduced by Meinrenken in [M]. We
should also mention recent work by Loizides and Meinrenken in [LM], which
employs the residue techniques of [Sz2].

Finally, we drew motivation from the paper of Teleman and Woodward [TW], where
the Verlinde formula is put in the framework of localization in K-theory of
stacks. This very impressive work is probably accessible to a small number of
experts only. In the present article, we demonstrate, in particular, that the
sophisticated tools employed in [TW], at least in this instance, may be
replaced by a simple combinatorial device.

In summary, the virtues of this article are:

  * •

A proof of the parabolic Verlinde formula which needs as background only the
basics of GIT.

  * •

The discrete sum, and the generating function giving the coefficients of the
Hilbert polynomial are treated at the same time, and the $\rho$-shift is dealt
with explicitly.

  * •

A few technical innovations such that an efficient wall-crossing formula in
GIT (Theorem 5.6) and the tautological Hecke correspondences keep the
arguments simple, and the technical difficulties related to infinite
dimensional quotients or singularities, in our approach, are absorbed by a
combinatorial device: the theory of iterated residues.

## 2\. Parabolic bundles

### 2.1. Definitions

Let $C$ be a smooth complex projective curve of genus $g\geq 2$, and fix a
point $p\in{C}$.

  * •

A parabolic bundle on $C$ is a vector bundle $W$ of rank $r$ with a full flag
$F_{*}$ in the fiber over $p$:

$W_{p}=F_{r}\supsetneq...\supsetneq F_{1}\supsetneq F_{0}=0$

and parabolic weights $c=(c_{1},...,c_{r})$ assigned to
$F_{r},F_{r-1},...,F_{1}$, satisfying the conditions

$c_{1}>c_{2}>...>c_{r}\text{ and }c_{1}-c_{r}<1.$

  * •

The parabolic degree111For technical reasons, we have chosen a sign convention
opposite to that in the majority of treatments in the literature. and the
parabolic slope of $W$ are defined as

$\textit{pardeg}(W)=\mathrm{deg}(W)-\sum_{i=1}^{r}c_{i};\,\,\,\,\,\textit{parslope}(W)=\frac{\textit{pardeg}(W)}{\mathrm{rank}(W)}.$

  * •

A morphism $f:W\to W^{\prime}$ of parabolic bundles is a morphism of vector
bundles satisfying $f_{p}(F_{i})\subset{F^{\prime}_{j-1}}$ if
$c_{r-i+1}<c^{\prime}_{r-j+1}$. In particular, an endomorphism of a parabolic
bundle $W$ is a vector bundle endomorphism preserving the flag $F_{*}$.

  * •

Denote by $\mathrm{ParHom}(W,W^{\prime})$ the sheaf of parabolic morphisms
from $W$ to $W^{\prime}$. Then there is a short exact sequence of sheaves

(6) $0\to\mathrm{ParHom}(W,W^{\prime})\to\mathrm{Hom}(W,W^{\prime})\to
T_{p}\to{0},$

where $T_{p}$ is a torsion sheaf supported at $p$. The rank of $T_{p}$ is the
number of pairs $(i,j)$, s.t. $c_{i}<c^{\prime}_{j}$ (cf. [BH]).

If $W^{\prime}\subset{W}$ is a subbundle of $W$, then both $W^{\prime}$ and
the quotient $W/W^{\prime}$ inherit a parabolic structure from $W$ in a
natural way (cf. [MS], definition 1.7).

  * •

A parabolic bundle $W$ is stable of weight $c$, if any proper subbundle
$W^{\prime}\subset{W}$ satisfies
$\textit{parslope}(W^{\prime})<\textit{parslope}(W)$; and $W$ is semistable of
weight $c$, if the inequality is not strict.

######  Remark 2.1.

Note that the parabolic stability condition depends on the parabolic weights
only up to adding the same constant to all weights $c_{i}$.

### 2.2. Construction of the moduli spaces

We start with a quick review of the construction of Mehta and Seshadri [MS] of
the moduli space of stable parabolic bundles. It follows from Remark 2.1 that,
without loss of generality, we can assume that the parabolic weights of a
rank-$r$ degree-$d$ bundle belong to the simplex

$\Delta_{d}=\left\\{(c_{1},c_{2},...,c_{r})\,|\,c_{1}>{c_{2}}>...>{c_{r}},\,c_{1}-c_{r}<1,\,\sum_{i}c_{i}=d\right\\}.$

######  Definition 2.2.

We will call a vector $c=(c_{1},\dots,c_{r})\in\mathbb{R}^{r}$ such that
$\sum_{i}c_{i}\in\mathbb{Z}$ regular if for any nontrivial subset
$\Psi\subset\\{1,2,\dots,r\\}$, we have
$\sum_{i\in\Psi}c_{i}\notin\mathbb{Z}$.

Now choose an integer $d\gg 0$ such that $H^{1}(W)=0$ and $W$ is generated by
global sections for any rank-$r$ degree-$d$ semistable parabolic bundle $W$ of
parabolic degree $0$. Put $\chi=r(1-g)+d$ and consider the

  * •

Groethendieck quot scheme $Quot(\chi,r)$ ([G]) parametrizing quotients
$\mathcal{O}^{\chi}\twoheadrightarrow W$, where $W$ is a coherent sheaf of
degree $d$ and rank $r$.

  * •

This space is endowed with a universal bundle $UQ$, and a genericly free
action of the group $G=PSL(\chi)$, which does not, however, lift to $UQ$.

  * •

Let $\mathrm{LFQuot}\subset Quot(\chi,r)$ be the open subscheme consisting of
locally free quotients $W$, such that the induced map
$H^{0}(\mathcal{O}^{\chi})\to H^{0}(W)$ is an isomorphism.

  * •

Denote by $XQ$ the total space of the flag bundle $\mathrm{Flag}({UQ}_{p})$ on
$\mathrm{LFQuot}\times{p}$. This space is endowed with the flag of vector
bundles $Fl_{1}\subset\dots\subset Fl_{r-1}\subset Fl_{r}=UQ_{p}$.

  * •

Let $k\in\mathbb{Z}$ and $(\lambda_{1},...,\lambda_{r})\in\mathbb{Z}^{r}$,
such that $\sum_{i=1}^{r}\lambda_{i}=kd$, and consider the line bundle

$L(k;\lambda)=\mathrm{det}({UQ}_{p})^{k(1-g)}\otimes
det(\pi_{*}UQ)^{-k}\otimes({Fl}_{r}/{Fl}_{r-1})^{\lambda_{1}}\otimes...\otimes({Fl}_{1})^{\lambda_{r}}$

on $XQ$, which does carry a $G$-linearization (lift of the $G$-action from
$XQ$).

  * •

Finally, assume $c\in\Delta_{d}$ is regular (cf. Definition 2.2 above) and
define $\widetilde{P}_{d}(c)$, the moduli space of stable parabolic weight-$c$
vector bundles on $C$ as the GIT quotient ${XQ\sslash^{c}G}$ of $XQ$ with
respect to any linearization $L(k;\lambda)$, such that $\lambda/k=c$.

######  Theorem 2.3 ([Se]).

Assume that $c\in\Delta_{d}$ is a regular weight vector. Then the moduli space
$\widetilde{P}_{d}(c)$ is a smooth projective variety of dimension
$r^{2}(g-1)+\binom{r}{2}+1$, whose points are in one-to-one correspondence
with the set of isomorphism classes of stable parabolic bundles of weight $c$
(cf. §2.1).

######  Remark 2.4.

Via the determinant map, the moduli space $\widetilde{P}_{d}(c)$ fibers over
the Jacobian of degree-$d$ line bundles with isomorphic fibers, and in this
paper, we will focus on the moduli space

$P_{d}(c)=\\{W\in\widetilde{P}_{d}(c)|\;\det W\simeq\mathcal{O}(d{p})\\},$

which is smooth, projective and has dimension $(r^{2}-1)(g-1)+\binom{r}{2}$.

######  Remark 2.5.

Note that tensoring with the line bundle $\mathcal{O}(mp)$ induces an
isomorphism: $\otimes\mathcal{O}(mp):P_{d}(c)\to P_{d+rm}(c)$, so the moduli
spaces $P_{d}(c)$, essentially, depend only on $d$ modulo $r$.

### 2.3. The Picard group of $P_{d}(c)$

For a regular $c\in\Delta_{d}$, there exist universal bundles $U$ over
$P_{d}(c)\times{C}$ endowed with a flag
$\mathcal{F}_{1}\subset\dots\subset\mathcal{F}_{r-1}\subset\mathcal{F}_{r}=U_{p}$,
and satisfying the obvious tautological properties. In general, such universal
bundles $U$, and hence the flag line bundles
$\mathcal{F}_{i+1}/\mathcal{F}_{i}$ are unique only up to tensoring by the
pull-back of a line bundle from $P_{d}(c)$. Nevertheless, we have the
following statement, which is easy to verify.

######  Lemma 2.6.

For $k\in\mathbb{Z}$ and
$\lambda=(\lambda_{1},...,\lambda_{r})\in\mathbb{Z}^{r}$, such that
$\sum_{i=1}^{r}\lambda_{i}=kd$, the line bundle

$\mathcal{L}_{d}(k;{\lambda})=\mathrm{det}(U_{p})^{k(1-g)}\otimes\mathrm{det}(\pi_{*}U)^{-k}\otimes(\mathcal{F}_{r}/\mathcal{F}_{r-1})^{\lambda_{1}}\otimes...\otimes(\mathcal{F}_{1})^{\lambda_{r}}$

on $P_{d}(c)$ is independent of the choice of the universal bundle $U$.

######  Remark 2.7.

The line bundle $L(k;\lambda)$ defined in §2.2 descends to the line bundle
$\mathcal{L}_{d}(k;\lambda)$ on the GIT quotient $P_{d}(c)$.

Notation: We will say that $U$ is normalized if the line subbundle
$\mathcal{F}_{1}\subset U_{p}$ is trivial. The parameter $k$ is often called
the level.

Let $\omega\in H^{2}(C)$ be the fundamental class of our curve $C$, and
$e_{1},...,e_{2g}$ a basis of $H^{1}(C)$, such that $e_{i}e_{i+g}=\omega$ for
$1\leq i\leq g$, and all other intersection numbers $e_{i}e_{j}$ equal 0. For
a class $\delta\in H^{*}(P\times C)$ of a product, we introduce the following
notation for its Künneth components (cf. [W]):

(7) $\delta=\delta_{(0)}\otimes 1+\sum_{i}\delta_{(e_{i})}\otimes
e_{i}+\delta_{(2)}\otimes\omega\in\bigoplus_{i=0}^{2}H^{*-i}(P)\otimes
H^{i}(C).$

Later, we will need the following formula, which can be proved by a
straightforward calculation.

######  Lemma 2.8.

$2c_{1}(\mathcal{L}_{d}(r;d,...,d))=c_{2}(\mathrm{End}_{0}(U_{d}))_{(2)},$
where $\mathrm{End}_{0}$ stands for traceless endomorphisms.

### 2.4. Walls and chambers

The central question we address in this paper is how the moduli space of
stable parabolic bundles depends on the choice of parabolic weights. Let $W$
be a vector bundle of degree $d$ with a fixed full flag $\mathrm{F}_{*}$ of
the fiber $W_{p}$, and let us try to determine the structure of the set of
parabolic weights $c\in\Delta_{d}$ for which $W$ is stable. Clearly, for this
we need to study the set of parabolic weights $c=(c_{1},c_{2},\dots c_{r})$
for which one can find a proper subbundle $W^{\prime}\subset W$ such that

(8) $\textit{parslope}(W^{\prime})=\textit{parslope}(W)=0.$

A subbundle $W^{\prime}\subset W$ determines a short exact sequence of
parabolic bundles

$0\to W^{\prime}\to W\to W^{\prime\prime}\to 0$

and the position of $W^{\prime}_{p}$ with respect to $\mathrm{F}_{*}$ gives
rise to a nontrivial partition of the set $\\{1,2,\dots,r\\}$ into two sets,
$\Pi^{\prime}$ and $\Pi^{\prime\prime}$ (cf. [MS], definition 1.7); the
parabolic weights of $W^{\prime}$ and $W^{\prime\prime}$ are then
$c^{\prime}=(c_{i})_{i\in\Pi^{\prime}}$ and
$c^{\prime\prime}=(c_{i})_{i\in\Pi^{\prime\prime}}$, correspondingly. The
slope condition (8) translates into a pair of equivalent equalities:

(9) $d^{\prime}=\sum_{i\in\Pi^{\prime\prime}}{c_{i}},\quad
d^{\prime\prime}=\sum_{i\in\Pi^{\prime\prime}}{c_{i}},$

where $d^{\prime}$, $d^{\prime\prime}=d-d^{\prime}$ are the degrees of
$W^{\prime}$ and $W^{\prime\prime}$, respectively. This means that the
critical values of $c\in\Delta_{d}$ for which (8) is possible lie on the union
of affine hyperplanes (or walls) defined by the equations

$\sum_{i\in\Pi^{\prime}}{c_{i}}=l,\;\text{where }l\in\mathbb{Z}\text{, and
}\Pi^{\prime}\subset\\{1,2,\dots,r\\}\text{ nontrivial.}$

As only finitely many of these walls intersect the simplex $\Delta_{d}$, their
complement is a finite union of open polyhedral chambers. It is easy to verify
that as we vary $c$ inside one of these chambers, the stability condition, and
thus the moduli space $P_{d}(c)$ does not change.

######  Example 1.

Consider the case of rank-3 degree-0 stable parabolic bundles with parabolic
weights $c=(c_{1},c_{2},c_{3})\in\Delta_{0}$. The set $\Delta_{0}$ is an open
triangle with vertices $(0,0,0),(\frac{2}{3},-\frac{1}{3},-\frac{1}{3})$ and
$(\frac{1}{3},\frac{1}{3},-\frac{2}{3})$ (cf. Figure 2), and there exist only
two essentially different stability conditions. The wall separating the two
regimes is given by the condition $c_{2}=0$. We write $P_{0}(>)$ for the
moduli space $P_{0}(c_{1},c_{2},c_{3})$ with $c_{2}>0$, and $P_{0}(<)$ for
$P_{0}(c_{1},c_{2},c_{3})$ with $c_{2}<0$.

$(0,0,0)$$(\frac{2}{3},-\frac{1}{3},-\frac{1}{3})$$(\frac{1}{3},\frac{1}{3},-\frac{2}{3})$$P_{0}(>)$$P_{0}(<)$
Figure 2. The space of admissible parabolic weights for rank $r=3$.

## 3\. Wall-crossing in the Verlinde formula

A key component of our approach is the notion of diagonal basis and the
associated generalized Bernoulli polynomials introduced for general hyperplane
arrangements in [Sz1]. Using this formalism, we will be able to formulate our
main result, Theorem 4.8.

### 3.1. Notation

We begin by setting up some extra notation for the space of parabolic weights
introduced in §2.1.

  * •

Let $V=\mathbb{R}^{r}/\mathbb{R}(1,1,\dots,1)$ be the $r-1$-dimensional vector
space, obtained as the quotient of $\mathbb{R}^{r}$. The dual space $V^{*}$ is
then naturally represented as

$V^{*}=\\{a=(a_{1},\dots,a_{r})\in\mathbb{R}^{r}|\;a_{1}+\dots+a_{r}=0\\}.$

Let $x_{1},x_{2},\dots,x_{r}$ be the coordinates on $\mathbb{R}^{r}$; given
$a\in V^{*}$, we will write $\langle a,x\rangle$ for the linear function
$\sum_{i}a_{i}x_{i}$ on $V$. We will sometimes identify this linear function
with the vector $a$ itself.

  * •

The vector space $V^{*}$ is endowed with a lattice $\Lambda$ of full rank:

$\Lambda=\\{\lambda=(\lambda_{1},\dots,\lambda_{r})\in\mathbb{Z}^{r}|\;\lambda_{1}+\dots+\lambda_{r}=0\\}.$

In particular, for $1\leq i\neq j\leq r$, we can define the element
$\alpha^{ij}=x_{i}-x_{j}$ in $\Lambda$.

  * •

Our arrangement is the set of hyperplanes $\\{x_{i}=x_{j}\\}\subset V$, $1\leq
i<j\leq r$. It will be convenient for us to think about this set as the set of
roots of the $A_{r-1}$ root system with the opposite roots identified:

${\Phi}=\\{\ \pm\alpha^{ij}|\,1\leq i<j\leq r\\}.$

Note that $V^{*}$ carries a natural action of the permutation group
$\Sigma_{r}$, permuting the coordinates $x_{j},\,j=1,\dots,r$, and this action
restricts to an action on $\Phi$ as well.

  * •

The basic object of the theory is an ordered linear basis $\mathbf{B}$ of
$V^{*}$ consisting of the elements of $\Phi$. Let us denote the set of these
objects by $\mathcal{B}$:

$\mathcal{B}=\left\\{\mathbf{B}=\left(\beta^{[1]},\dots,\beta^{[r-1]}\right)\in\Phi^{r-1}|\;\mathbf{B}\text{
-- basis of }V^{*}\right\\}$

  * •

For $\mathbf{B}\in\mathcal{B}$, we will write $\mathrm{Fl}(\mathbf{B})$ for
the full flag

$\left[V^{*}=\langle\beta^{[1]},\beta^{[2]},\dots,\beta^{[r-1]}\rangle_{\mathrm{lin}},\dots,\langle\beta^{[r-1]},\beta^{[r-2]}\rangle_{\mathrm{lin}},\langle\beta^{[r-1]}\rangle_{\mathrm{lin}}\right],$

where $\langle\cdot\rangle_{\mathrm{lin}}$ stands for linear span.

### 3.2. Diagonal bases

######  Definition 3.1.

  * •

For $\tau\in\Sigma_{r-1}$ and $\mathbf{B}\in\mathcal{B}$, we will write
$\mathbf{B}\circlearrowleft\tau$ for the permuted sequence
$(\beta^{[\tau(1)]},\beta^{[\tau(2)]},\dots,\beta^{[\tau(r-1)]})$.

  * •

For two elements $\mathbf{B},\mathbf{C}\in\mathcal{B}$ we will write
$\mathbf{B}\dashv\mathbf{C}$ if for any $\tau\in\Sigma_{r-1}$, we have
$\mathrm{Fl}(\mathbf{B}\circlearrowleft\tau)\neq\mathrm{Fl}(\mathbf{C})$.

  * •

A subset $\mathcal{D}\subset\mathcal{B}$ of $(r-1)!$ elements is called a
diagonal basis if for any two different elements
$\mathbf{B},\mathbf{C}\in\mathcal{D}$, we have $\mathbf{B}\dashv\mathbf{C}$.

######  Remark 3.2.

This definition is motivated by a construction [Sz1], which associates to each
diagonal basis $\mathcal{D}$ a pair of dual bases of the middle homology and
the cohomology of the complexified hyperplane arrangement on
$V\otimes_{\mathbb{R}}\mathbb{C}$ defined by $\Phi$. The dimension of these
(co)homology spaces is $(r-1)!$.

### 3.3. Combinatorial interpretation

This notion has the following purely combinatorial form.

  * •

We can think of $\Phi$ as the edges of the complete graph on $r$ vertices.

  * •

Then the set $\mathcal{B}$ may be thought of as the set of spanning trees of
this graph with edges enumerated from $1$ to $r-1$. We will introduce the
notation

$\mathbf{B}\mapsto\mathrm{Tree}(\mathbf{B})$

for this ordered tree.

  * •

In this language, the flag $\mathrm{Fl}(\mathbf{B})$ corresponds to a sequence
of $r$ nested partitions of the vertices (starting with the total partition
into 1-element sets and ending with the trivial partition) associated to
$\mathrm{Tree}(\mathbf{B})$, the $j$th partition being the one induced by the
first $j-1$ edges. For example, the ordered tree $[(2,4)(1,3),(1,2)]$ induces
the same sequence of partitions as $[(1,4),(2,3),(1,2)]$ (see Figure 3)

  * •

A diagonal basis $\mathcal{D}$ is then a set of $(r-1)!$ ordered trees such
that the $(r-1)!$ partition sequences obtained by reordering the edges of any
one of the ordered trees are different from $(r-1)!-1$ sequences of partitions
obtained from the remaining elements of $\mathcal{D}$.

1231243

1234123412341234

Figure 3. $\mathbf{B}=(\alpha^{1,3},\alpha^{1,2},\alpha^{3,4})$

### 3.4. Examples

There are essentially 2 known constructions of diagonal bases [Sz1].

The Hamiltonian basis. For each permutation $\sigma\in\Sigma_{r}$, we can
define

(10)
$\sigma(\mathbf{B})=(\alpha^{\sigma(r-1),\sigma(r)},\alpha^{\sigma(r-2),\sigma(r-1)},\dots,\alpha^{\sigma(1),\sigma(2)})\in\mathcal{B}.$

The set
$\mathcal{H}_{m}=\\{\sigma(\mathbf{B})|\;\sigma\in\Sigma_{r},\,\sigma(1)=m\\}$
is then a diagonal basis. In the combinatorial description, this diagonal
basis corresponds to the set of Hamiltonian paths starting at vertex $m$, and
endowed with the reversed natural ordering of edges.

######  Example 2.

Here are some examples of Hamiltonian bases:

  * •

for $r=3$:
$\mathcal{H}_{1}=\\{(\alpha^{2,3},\alpha^{1,2}),(\alpha^{3,2},\alpha^{1,3})\\}$,

  * •

and for $r=4$:

$\mathcal{H}_{1}=\\{(\alpha^{3,4},\alpha^{2,3},\alpha^{1,2}),(\alpha^{2,4},\alpha^{3,2},\alpha^{1,3}),(\alpha^{4,3},\alpha^{2,4},\alpha^{1,2}),\\\
(\alpha^{3,2},\alpha^{4,3},\alpha^{1,4}),(\alpha^{4,2},\alpha^{3,4},\alpha^{1,3}),(\alpha^{2,3},\alpha^{4,2},\alpha^{1,4})\\}$

II.  The no-broken-circuit bases. Let $\upsilon:\\{1,\dots,r(r-1)/2\\}\to\Phi$
be a total ordering, which we will represent as an order relation
$\overset{\upsilon}{<}$ on $\Phi$. To this ordering, one can associate the
following, so called noncommutative no-broken-circuit diagonal basis [Sz1]:

$\mathcal{D}[\upsilon]=\left\\{\left(\beta^{[1]},\dots,\beta^{[r-1]}\right)\in\mathcal{B}\left|\quad\beta^{[1]}\overset{\upsilon}{<}\dots\overset{\upsilon}{<}\beta^{[r-1]},\right.\,\text{and}\right.\\\
\left.\alpha^{ij}\overset{\upsilon}{<}\beta^{[m]}\Rightarrow(\alpha^{ij},\beta^{[m]},\dots,\beta^{[r-1]})\text{
linearly independent}\right\\}.$

######  Example 3.

Let
$\alpha^{1,3}\overset{\upsilon}{<}\alpha^{1,4}\overset{\upsilon}{<}\alpha^{2,3}\overset{\upsilon}{<}\alpha^{2,4}\overset{\upsilon}{<}\alpha^{1,2}\overset{\upsilon}{<}\alpha^{3,4}$
be the ordering of the positive roots for rank $r=4$. Then

$\mathcal{D}[\upsilon]=\\{(\alpha^{1,3},\alpha^{1,2},\alpha^{3,4}),(\alpha^{1,3},\alpha^{1,4},\alpha^{2,3}),(\alpha^{1,3},\alpha^{1,4},\alpha^{2,4}),\\\
(\alpha^{1,3},\alpha^{1,4},\alpha^{1,2}),(\alpha^{1,3},\alpha^{2,3},\alpha^{3,4}),(\alpha^{1,3},\alpha^{2,3},\alpha^{2,4})\\}$

is the corresponding no-broken-circuit diagonal basis.

######  Remark 3.3.

The hyperplane arrangement induced by $\Phi$ is invariant under the natural
action of $\Sigma_{r}$ on the vector space $V$. It follows easily from the
definition that if $\mathcal{D}$ is a diagonal basis and $\sigma\in\Sigma_{r}$
is a permutation, then $\sigma(\mathcal{D})$ is also a diagonal basis.

## 4\. The residue formula and the main result

In this section, we recall the residue formula from [Sz1] for
$Ver(k,\lambda)$, the discrete Verlinde sum on the right hand side of (1). The
key feature of this formula is that it exposes the piecewise polynomial nature
of $Ver(k,\lambda)$, which is key for our wall-crossing analysis. While the
objects are relatively simple, the formalism is heavy with notation, so we
begin by describing the 1-dimensional case.

### 4.1. The residue formula in dimension 1

The story begins with the Fourier series

(11) $\frac{1}{(2\pi i)^{m}}\sum_{n\in\mathbb{Z}\setminus{0}}\frac{\exp(2\pi
ian)}{n^{m}}$

for $m\geq 2$, which is a periodic, piecewise polynomial function given by the
formula

$\operatornamewithlimits{Res}_{x=0}\frac{\exp(\\{a\\}x)}{1-\exp(x)}\frac{dx}{x^{m}},$

where $\\{a\\}$ is the fractional part of the real number $a$. The polynomial
functions thus obtained on the interval $[0,1]$ are called Bernoulli
polynomials. The polynomial on the interval containing the real number
$c\in\mathbb{R}\setminus\mathbb{Z}$ is given by

$\operatornamewithlimits{Res}_{x=0}\frac{\exp((a-[c])x)}{1-\exp(x)}\frac{dx}{x^{m}},$

where $[c]$ is the integer part of $c$.

Now we pass to a trigonometric version of this formula, calculating finite
sums of values of rational trigonometric functions over rational points with
denominators equal to an integer $k$.

We replace thus the rational function $x^{-m}$ by the (hyperbolic)
trigonometric function $f(x)=(2\sinh(x/2))^{-2m}$, and introduce an integer
parameter $\lambda$ related to $a$ via $ka=\lambda$. We consider the sum of
values of the function $f$ over a finite set of rational points in analogy
with (11):

$\sum_{n=1}^{k-1}\frac{exp(2\pi i\lambda n/k)}{(2\sin(\pi n/k))^{2m}},$

where $\lambda,k\in\mathbb{Z}$. This sum is again periodic in $\lambda\mod k$,
and for $m\geq 2$ we can evaluate it via the residue theorem as

$(-1)^{m}\operatornamewithlimits{Res}_{z=1}\frac{z^{k\\{\lambda/k\\}}}{(z^{1/2}-z^{-1/2})^{2m}}\cdot\frac{k\,dz}{z(1-z^{k})}\overset{z=\exp(x/k)}{=}\\\
(-1)^{m}\operatornamewithlimits{Res}_{x=0}\frac{\exp(\\{\lambda/k\\}\cdot
x)}{1-\exp(x)}\cdot f(x/k)\,dx.$

Again, this is a piecewise polynomial function in the pair $(k,\lambda)$,
which is polynomial in the cones bounded by the lines $\lambda=qk$,
$q\in\mathbb{Z}$.

Note that in these calculations, a key role is played by the Bernoulli
operator:

(12) $f\mapsto\mathrm{Ber}[f](a)=\frac{f(x)\exp(ax)\,dx}{1-\exp(x)},$

which transforms meromorphic functions in the variable $x$ into polynomials in
$a$, and plays the role of a generalized Fourier operator.

### 4.2. The multidimensional case

Now we return to the setup of §3 with the vector space $V$ endowed with the
hyperplane arrangement $\Phi$. We introduce the notation $\mathcal{F}_{\Phi}$
for the space of meromorphic functions defined in a neighborhood of $0$ in
$V\otimes_{\mathbb{R}}\mathbb{C}$ with poles on the union of hyperplanes

$\bigcup_{1\leq i<j\leq r}\\{x|\;\langle\alpha^{ij},x\rangle=0\\}.$

In particular, the function

$w_{\Phi}=\prod_{i<j}\left(2\mathrm{sinh}(\pi(x_{i}-x_{j}))\right)$

is an element of $\mathcal{F}_{\Phi}$.

To write down our residue formula, we need a multidimensional generalization
of the notions of integer and fractional parts. Given a basis
$\mathbf{B}=(\beta^{[1]},\dots,\beta^{[r-1]})\in\mathcal{B}$ of $V^{*}$, and
an element $a\in V^{*}$, we define $[a]_{\mathbf{B}}$ and
$\\{a\\}_{\mathbf{B}}$ to be the unique elements of $V^{*}$ satisfying

  * •

$[a]_{\mathbf{B}}=a-\\{a\\}_{\mathbf{B}}\in\Lambda$, and

  * •

$\\{a\\}_{\mathbf{B}}\in\sum_{j=1}^{r-1}[0,1)\beta^{[j]}.$

This notion naturally induces a chamber structure on $V^{*}$: we will call
$a\in V^{*}$ regular if $a$ is a point of continuity for the functions
$a\mapsto[a]_{\mathbf{B}},\\{a\\}_{\mathbf{B}}$ for all
$\mathbf{B}\in\mathcal{B}$, i.e. when
$\\{a\\}_{\mathbf{B}}\in\sum_{j=1}^{r-1}(0,1)\beta^{[j]}$. Now, for regular
$a$ and $b$ we define the equivalence relation

(13) $a\sim b\text{ when
}[a]_{\mathbf{B}}=[b]_{\mathbf{B}}\quad\forall\mathbf{B}\in\mathcal{B}.$

The equivalence classes for this relation form a $\Lambda$-periodic system of
chambers in $V^{*}$.

Convention: We will think of a partition $\Pi$ of $\\{1,2,\dots,r\\}$ into two
nonempty sets as an ordered partition $\Pi=(\Pi^{\prime},\Pi^{\prime\prime})$
such that $r\in\Pi^{\prime\prime}$, and we will call these objects nontrivial
partitions for short.

######  Lemma 4.1.

The equivalence classes of the relation $\sim$ are precisely the chambers in
$V^{*}$ created by the walls parameterized by a nontrivial partition
$\Pi=(\Pi^{\prime},\Pi^{\prime\prime})$ of the first $r$ positive integers,
and an integer $l$:

(14) $S_{\Pi,l}=\\{c\in V^{*}|\;\sum_{j\in\Pi^{\prime}}c_{j}=l\\}$

######  Remark 4.2.

Note that the walls given in (14) are precisely the same as the ones given in
(9) for the case $d=0$, where they play the role of walls separating the
chambers of parabolic weights $c$ in which the parabolic moduli spaces
$P_{0}(c)$ are naturally the same. This "coincidence" is precisely what we
need for our comparative wall-crossing strategy. There is a small
terminological issue here: the "chambers" in §2.4 are the intersections of the
equivalence classes of $\sim$ defined above with the open simplex
$\Delta_{0}\overset{\mathrm{def}}{=}\Delta$, where the parabolic weights live
(cf. Figures 2 and 4). We will use the term "chamber" in both cases if this
causes no confusion.

$S_{(\\{2\\},\\{1,3\\}),0}$$S_{(\\{1,2\\},\\{3\\}),1}$$S_{(\\{1\\},\\{2,3\\}),1}$(0,0,0)(0,1,-1)(1,-1,0)(1,0,-1)$P_{0}(>)$$P_{0}(<)$$S_{(\\{1\\},\\{2,3\\}),0}$$S_{(\\{1,2\\},\\{3\\}),0}$
Figure 4. Chambers for rank $r=3$.

Each element $\mathbf{B}=(\beta^{[1]},\dots,\beta^{[r-1]})\in\mathcal{B}$
defines an iterated version of the Bernoulli operator (12) on the space of
functions $\mathcal{F}_{\Phi}$: interpreting the elements $a,\beta^{[j]}\in
V^{*}$ as linear functions on $V$, we define

(15)
$\operatornamewithlimits{iBer}_{\mathbf{B}}\left[f(x)\right](a)=\frac{1}{(2\pi
i)^{r-1}}\int\displaylimits_{Z(\mathbf{B})}\frac{f(x)\exp\langle
a,x\rangle\;d\langle\beta^{[1]},x\rangle\wedge\dots\wedge
d\langle\beta^{[r-1]},x\rangle\;}{(1-exp(\langle\beta^{[1]},x\rangle))\;\dots(1-exp(\langle\beta^{[r-1]},x\rangle))\;},$

where the naturally oriented cycle $Z_{\mathbf{B}}$ is defined by

$Z_{\mathbf{B}}=\\{v\in
V\otimes_{\mathbb{R}}\mathbb{C}:\,|\langle\beta^{[j]},x\rangle|\;=\varepsilon_{j},\,j=\,\dots,r-1\\}\subset
V\otimes_{\mathbb{R}}\mathbb{C}\setminus\\{w_{\Phi}(x)=0\\},$

with real constants $\varepsilon_{j}$ satisfying
$0\leq\varepsilon_{r-1}\ll\dots\ll\varepsilon_{1}$. Thus again,
$\operatornamewithlimits{iBer}_{\mathbf{B}}$ is a linear operator associating
to a function in $\mathcal{F}_{\Phi}$ a polynomial on $V^{*}$.

######  Remark 4.3.

Let us make a small remark about the computational aspects of the operator
$\operatornamewithlimits{iBer}_{\textbf{B}}$. Denoting the coordinate
$\langle\beta^{[j]},x\rangle$ by $y_{j}$, $j=1,\dots,r-1$, and writing $f$ and
$a$ in these coordinates: $f(x)=\hat{f}(y)$, $\langle
a,x\rangle=\langle\hat{a},y\rangle$, we can rewrite (15) as

$\operatornamewithlimits{iBer}_{\mathbf{B}}\left[f(x)\right](a)=\operatornamewithlimits{Res}_{y_{1}=0}\dots\operatornamewithlimits{Res}_{y_{r-1}=0}\frac{\hat{f}(y)\exp\langle\hat{a},y\rangle\;dy_{1}\wedge\dots\wedge
dy_{r-1}}{(1-exp(y_{1}))\;\dots(1-exp(y_{r-1}))\;},$

where iterating the residues here means that we keep the variables with lower
indices as unknown constants, and then use geometric series expansions of the
type

$\frac{1}{1-exp(y_{1}-y_{2})}=\frac{y_{1}-y_{2}}{1-exp(y_{1}-y_{2})}\cdot\frac{1}{y_{1}-y_{2}}=\frac{y_{1}-y_{2}}{1-exp(y_{1}-y_{2})}\cdot\sum_{n=0}^{\infty}\frac{y_{2}^{n}}{y_{1}^{n+1}}.$

### 4.3. Invariance of diagonal bases and the main results

Diagonal bases have the following key invariance property.

######  Theorem 4.4 ([Sz1]).

Let $f\in\mathcal{F}_{\Phi}$, and $c\in V^{*}$ be regular; let $\mathcal{D}$
be a diagonal basis of $\Phi$. Then the functional (cf. (15) above)

$f\mapsto\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[f(x)](a-[c]_{\mathbf{B}})$

transforming a meromorphic function $f\in\mathcal{F}_{\Phi}$ into a polynomial
in the variable $a\in V^{*}$ is independent of the choice of the diagonal
basis $\mathcal{D}$. In particular, for regular $a\in V^{*}$, the functional

(16)
$f\mapsto\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[f(x)](\\{a\\}_{\mathbf{B}})$

transforms $f$ into a well-defined piecewise polynomial function on $V^{*}$,
which is polynomial in each chamber.

As this functional is invariantly defined, it is not surprising that it is
equivariant with respect to the symmetries of our hyperplane arrangement. For
$\sigma\in\Sigma_{r}$, we define, as usual

(17) $\sigma\cdot f(x)=f(\sigma^{-1}x).$

This convention is consistent with (10).

######  Lemma 4.5.

Let $f\in\mathcal{F}_{\Phi}$, and $\sigma\in\Sigma_{r}$, and pick any diagonal
basis $\mathcal{D}$. Then

$\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[f(x)](\sigma\cdot
a-[\sigma\cdot
c]_{\mathbf{B}})=\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[\sigma^{-1}\cdot
f(x)](a-[c]_{\mathbf{B}})$

###### Proof.

Indeed, it is sufficient to note that

$\langle\sigma\cdot a-[\sigma\cdot
c]_{\mathbf{B}},x\rangle=\langle\sigma\cdot(a-[c]_{\mathbf{B}}),x\rangle=\langle
a-[c]_{\mathbf{B}},\sigma^{-1}(x)\rangle,$

perform the linear substitution $x=\sigma(y)$, and conclude that

$\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[f(x)](\sigma\cdot
a-[\sigma\cdot
c]_{\mathbf{B}})=\sum_{\mathbf{B}\in\sigma^{-1}(\mathcal{D})}\operatornamewithlimits{iBer}_{\mathbf{B}}[\sigma^{-1}\cdot
f(x)](a-[c]_{\mathbf{B}}).$

Now the statement follows from the fact that $\sigma\in\Sigma$ takes a
diagonal basis to another diagonal basis (cf. Remark 3.3). ∎

######  Remark 4.6.

By picking the Hamiltonian diagonal basis
$\mathcal{H}_{1}=\\{\sigma\cdot\mathbf{B}_{0}|\;\sigma\in\mathrm{Stab}(1,\Sigma_{r})\\}$,
we can turn the argument in the proof above around, and obtain the following
formula:

$\sum_{\mathbf{B}\in\mathcal{H}_{1}}\operatornamewithlimits{iBer}_{\mathbf{B}}[f(x)](a-[c]_{\mathbf{B}})=\sum_{\sigma\in\mathrm{Stab}(1,\Sigma_{r})}\operatornamewithlimits{iBer}_{\mathbf{B}_{0}}[\sigma\cdot
f(x)](\sigma\cdot a-[\sigma\cdot c]_{\mathbf{B}})=\\\
\operatornamewithlimits{Res}_{y_{1}=0}\dots\operatornamewithlimits{Res}_{y_{r-1}=0}\sum_{\sigma\in\mathrm{Stab}(1,\Sigma_{r})}\frac{\sigma\cdot
f(y)\exp\langle\sigma\cdot a-[\sigma\cdot
c]_{\mathbf{B}},y\rangle\;dy_{1}\wedge\dots\wedge
dy_{r-1}}{(1-exp(y_{1}))\;\dots(1-exp(y_{r-1}))\;},$

where

${\bf{B_{0}}}=(y_{1}=x_{r-1}-x_{r},\dots,y_{r-2}=x_{2}-x_{3},y_{r-1}=x_{1}-x_{2})\in\mathcal{B}.$

Now we are ready to write down the residue formula for the Verlinde sums
proved in [Sz2, Theorem 4.2]. Recall that we denoted by
$\mathrm{Ver}(k,\lambda)$ the finite sum on the right hand side of (1).

######  Theorem 4.7.

Let $g\geq 1$, $k\in\mathbb{Z}^{>0}$, $\lambda\in\Lambda$, and let
$\mathcal{D}$ be any diagonal basis of $\Phi$. Introducing the notation
$\widehat{k}=k+r$, and $\widehat{\lambda}=\lambda+\rho$, we have

(18)
$\mathrm{Ver}(k,\lambda)=\tilde{N}_{r,k}\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}\left[w_{\Phi}^{1-2g}(x/\widehat{k})\right]\left(\widehat{\lambda}/\widehat{k}-[\,\widehat{\k{c}}\,]_{\mathbf{B}}\right),$

where $\tilde{N}_{r,k}=(-1)^{{r\choose 2}(g-1)}N_{r,k}$ (cf. (1)) and
$\widehat{\k{c}}\in V^{*}$ is a regular point in a chamber that contains
$\widehat{\lambda}/\widehat{k}$ in its closure.

Now, if we look at our main goal (1): proving the equality

(19)
$\mathrm{Ver}(k,\lambda)=\chi(P_{0}(\lambda/k),\mathcal{L}_{0}({k;\lambda})),$

then we discover a rather embarrassing mismatch. Both sides are piecewise
polynomial functions, however,

  * •

according to the HRR theorem,
$\chi(P(\lambda/k),\mathcal{L}_{0}({k;\lambda}))$ is polynomial on the cones
over the the equivalence classes (cf. (13)) of $\lambda/k$, while

  * •

according to (18), $\mathrm{Ver}(k,\lambda)$ is polynomial on the cones over
the equivalence classes of $\widehat{\lambda}/\widehat{k}$,

and these conic partitions of $\\{(k,\lambda)|\;\lambda/k\in\Delta\\}$ could
clearly be different (cf. Figure 5 for a sketch of this problem).

$\lambda$$k$$1$$-r$$-r+1$$(0,0)$$\lambda$$\widehat{\lambda}$$\frac{\lambda}{k}$$\frac{\widehat{\lambda}}{\widehat{k}}$
Figure 5. $\lambda/k$ is in the orange chamber, while
${\widehat{\lambda}}/{\widehat{k}}$ is in the green chamber.

Thus for (1) to be true, some miracle needs to occur, and these miracles are
well-known in the area of "quantization commutes with reduction" [MS, V, SzV].
We will return to this problem in §10, but for now, we will be satisfied to
use (18) to write down a (conjectural for the moment) formula for
$\chi(P_{0}(\lambda/k),\mathcal{L}_{0}({k;\lambda}))$, which is manifestly
polynomial on the cones where $\lambda/k$ is in a fixed equivalence class.

Let us fix a regular $c\in\Delta$ marking a particular chamber in $\Delta$.
The two cones $\\{(k;\lambda)|\,\lambda/k\sim c\\}$ and
$\\{(k;\lambda)|\,\widehat{\lambda}/\widehat{k}\sim c\\}$ intersect along an
open cone (this cone is shaded in orange on Figure 5), and on this
intersection, the expression

(20)
$\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}\left[w_{\Phi}^{1-2g}(x/\widehat{k})\right]\left(\widehat{\lambda}/\widehat{k}-[\lambda/k]_{\mathbf{B}}\right)$

coincides with the right hand side of (18). As (20) is manifestly polynomial
on each cone where $\lambda/k$ is in a particular chamber in $\Delta$, this
expression will be then our main candidate for
$\chi(P_{0}(\lambda/k),\mathcal{L}_{0}({k;\lambda}))$.

Our plan is thus to split the proof of (19) into three parts: the first is
equality (18), and the other two are given in our main theorem below. We
formulated all our statements in a manner that allows us to treat the cases
when $\lambda/k$ or $\widehat{\lambda}/\widehat{k}$ are on a boundary
separating two of our chambers in $\Delta$.

######  Theorem 4.8.

Let $\lambda\in\Lambda$ and $k\in\mathbb{Z}^{>0}$ be such that
$\lambda/k\in\Delta$. Let c̨ and $\widehat{\k{c}}\in\Delta$ be regular
elements, specifying two chambers in $\Delta$, which contain $\lambda/k$ and
$\widehat{\lambda}/\widehat{k}$ in their closures, correspondingly. Then for
any diagonal basis $\mathcal{D}$, the following two equalities hold:

(I.)
$\chi(P_{0}({\k{c}}),\mathcal{L}({k;\lambda}))=\tilde{N}_{r,k}\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}\left[w_{\Phi}^{1-2g}(x/\widehat{k})\right]\left(\widehat{\lambda}/\widehat{k}-[{\k{c}}]_{\mathbf{B}}\right),$

and

(II.)
$\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}\left[w_{\Phi}^{1-2g}(x/\widehat{k})\right]\left(\widehat{\lambda}/\widehat{k}-[{\k{c}}]_{\mathbf{B}}\right)=\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}\left[w_{\Phi}^{1-2g}(x/\widehat{k})\right]\left(\widehat{\lambda}/\widehat{k}-[\,\widehat{\k{c}}\,]_{\mathbf{B}}\right).$

######  Remark 4.9.

Part I. of the theorem implies that if $\lambda/k\in\Delta$ is not regular,
then

$\chi(P_{0}({c^{+}}),\mathcal{L}({k;\lambda}))=\chi(P_{0}({c^{-}}),\mathcal{L}({k;\lambda})),$

for regular $c^{\pm}\in\Delta$ in two neighboring chambers that contain
$\lambda/k$ in their closure (cf. Proposition 10.1 and Remark 10.4).

Before we proceed, we formulate a mild generalization of part I. of our
theorem. As observed above, if we fix a generic $c\in\Delta$, and vary
$(\lambda,k)$ in such a way that $\lambda/k\sim c$, then both sides of the
equality (I.) are manifestly polynomial, and thus we can extend the validity
of this equality as follows.

######  Corollary 4.10.

Let $c\in\Delta$ be a regular element, which thus specifies a chamber in
$\Delta$ and a parabolic moduli space $P_{0}(c)$ as well. Then for a diagonal
basis $\mathcal{D}$, an arbitrary weight $\lambda\in\Lambda$, and a positive
integer $k$, we have

(21)
$\chi(P_{0}(c),\mathcal{L}(k;\lambda))=\tilde{N}_{r,k}\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[w_{\Phi}^{1-2g}(x/\widehat{k})](\widehat{\lambda}/\widehat{k}-[c]_{\mathbf{B}}).$

######  Example 4.

Let us write down these formulas in case of $r=3$ explicitly. Let
$\mathcal{D}$ be the diagonal basis from Example 2; then using Remark 4.6, we
obtain

$\chi(P_{0}(<),\mathcal{L}(k;\lambda))=(-1)^{g-1}(3(k+3)^{2})^{g}\\\
\operatornamewithlimits{Res}_{y=0}\operatornamewithlimits{Res}_{x=0}\frac{e^{\lambda_{1}x+(\lambda_{1}+\lambda_{2})y+x+y}-e^{\lambda_{1}x+(\lambda_{1}+\lambda_{3})y+x}}{(1-e^{x(k+3)})(1-e^{y(k+3)})w_{\Phi}(x,y)^{2g-1}}dxdy$

and

$\chi(P_{0}(>),\mathcal{L}(k;\lambda))=(-1)^{g-1}(3(k+3)^{2})^{g}\\\
\operatornamewithlimits{Res}_{y=0}\operatornamewithlimits{Res}_{x=0}\frac{e^{\lambda_{1}x+(\lambda_{1}+\lambda_{2})y+x+y}-e^{\lambda_{1}x+(\lambda_{1}+\lambda_{3})y+x+y(k+3)}}{(1-e^{x(k+3)})(1-e^{y(k+3)})w_{\Phi}(x,y)^{2g-1}}\;dxdy,$

where
$w_{\Phi}(x,y)=2\mathrm{sinh}(\frac{x}{2})2\mathrm{sinh}(\frac{y}{2})2\mathrm{sinh}(\frac{x+y}{2})$.

### 4.4. The walls

Our first step is to identify the wall-crossing terms of the residue formula
(21), which originate in the discontinuities of the function
$c\mapsto\\{c\\}_{\mathbf{B}}$. These discontinuities occur on "walls": the
affine hyperplanes (14). The following is straightforward:

######  Lemma 4.11.

Let $S_{\Pi,l}$ be the wall defined by (14), and
$\mathbf{B}=(\beta^{[1]},\dots,\beta^{[r-1]})\in\mathcal{B}$ an ordered basis
of $V^{*}$. Then, as a function of $c$, the fractional part function
$\\{c\\}_{\mathbf{B}}$ has a discontinuity on the wall $S_{\Pi,l}$ exactly
when $\mathrm{Tree}(\mathbf{B})$ (cf. page • ‣ 3.3) is a union of a tree on
$\Pi^{\prime}$, a tree on $\Pi^{\prime\prime}$ (the enumeration of the edges
is irrelevant here) and a single edge (which we will call the link) connecting
$\Pi^{\prime}$ and $\Pi^{\prime\prime}$.

Notation: We will denote the element of $\mathbf{B}$ corresponding to this
edge by $\beta_{\mathrm{link}}$; this vector thus depends on $\mathbf{B}$ and
the partition $\Pi$.

Now choose two regular elements $c^{+},c^{-}\in V^{*}$ in two neighboring
chambers separated by the wall $S_{\Pi,l}$, in such a way that

(22) $[c^{+}_{\Pi^{\prime}}]=l\text{ \, and \, }[c^{-}_{\Pi^{\prime}}]=l-1,$

where

$c_{\Pi^{\prime}}\overset{\mathrm{def}}{=}\sum_{i\in\Pi^{\prime}}c_{i},$

and, as usual, $[q]$ stands for the integer part of the real number $q$. Now
introduce the notation

$p_{\pm}(k;\lambda)=\tilde{N}_{r,k}\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[w_{\Phi}^{1-2g}(x/\widehat{k})](\widehat{\lambda}/\widehat{k}-[c^{\pm}]_{\mathbf{B}})$

for the two polynomial functions in $(k,\lambda)$ corresponding to $c^{+}$ and
$c^{-}$, respectively. We define the  wall-crossing term in our residue
formula (21) as the difference between these two polynomials:

$p_{+}(k;\lambda)-p_{-}(k;\lambda).$

Using Lemma 4.1 and (22), we obtain the following simple residue formula for
this difference.

######  Lemma 4.12.

Let $(\Pi,l)$, $c^{+}$ and $c^{-}$ be as above, and let us fix a diagonal
basis $\mathcal{D}\subset\mathcal{B}$. Denote by $\mathcal{D}|\Pi$ the subset
of those elements of $\mathcal{D}$, which satisfy the condition described in
Lemma 4.11. Then

(23) $p_{+}(k,\lambda)-p_{-}(k,\lambda)=\\\
\tilde{N}_{r,k}\sum_{\mathbf{B}\in\mathcal{D}|\Pi}\operatornamewithlimits{iBer}_{B}\left[(1-\exp(\beta_{\mathrm{link}}(x)))w_{\Phi}^{1-2g}(x/\widehat{k})\right]\left(\lambda/\widehat{k}-[c^{+}]_{\mathbf{B}}\right),$

where $\beta_{\mathrm{link}}$ is the "link" element of $\mathbf{B}$ (depending
on $\Pi$ and $\mathbf{B}$) defined after Lemma 4.11.

######  Remark 4.13.

Note that the multiplication by $1-\exp(\beta_{\mathrm{link}}(x))$ in (23) has
the effect of canceling one of the factors in the denominator in the
definition (15) of the operation $\operatornamewithlimits{iBer}$.

######  Example 5.

Calculating the difference of two polynomials from Example 4, we get the wall-
crossing term for rank 3 case:

$\begin{split}p_{-}(k;\lambda)-p_{+}(k;\lambda)=(-3(k+3)^{2})^{g}\operatornamewithlimits{Res}_{y=0}\operatornamewithlimits{Res}_{x=0}\frac{e^{\lambda_{1}x+(\lambda_{1}+\lambda_{3})y+x}}{(1-e^{x(k+3)})w_{\Phi}(x,y)^{2g-1}}dxdy.\end{split}$

### 4.5. Wall-crossing and diagonal bases

Now we pass to the study of the combinatorial object $\mathcal{D}|\Pi$ defined
in Lemma 4.12. One thing we will discover is that even though each diagonal
basis consists of $(r-1)!$ elements and the right hand side of (23) does not
depend on the choice of $\mathcal{D}$, the number of elements in
$\mathcal{D}|\Pi$ might vary with $\mathcal{D}$.

First we look at the case of the Hamiltonian basis $\mathcal{H}_{1}$. Form now
on, we will use the notation $|\Pi^{\prime}|=r^{\prime}$ and
$|\Pi^{\prime\prime}|=r^{\prime\prime}$ for a nontrivial partition
$\Pi=(\Pi^{\prime},\Pi^{\prime\prime})$, (recall the convention
$r\in\Pi^{\prime\prime}$). The following statement is easy to verify.

######  Lemma 4.14.

Let $\Pi=(\Pi^{\prime},\Pi^{\prime\prime})$ be a nontrivial partition, such
that $1\in\Pi^{\prime}$ (the other case is analogous). Then

$\mathcal{H}_{1}|\Pi=\\{\sigma(\mathbf{B})|\;\sigma(1)=1,\text{ and
}\sigma(\Pi^{\prime})\in\Pi^{\prime}\\}.$

In particular, $|\mathcal{H}_{1}|\Pi|=(r^{\prime}-1)!\cdot r^{\prime\prime}!.$

It turns out that for our geometric applications, instead of
$\mathcal{H}_{1}$, we will need to choose a particular nbc-basis, where the
ordering is chosen to be consistent with $\Pi$.

To simplify our terminology, we will use the language of graphs and edges
introduced in §3.3, and we will think of $\alpha^{ij}\in\Phi$ as an edge in
the complete graph on $r$ vertices. To define the ordering $\upsilon$, we need
to choose an edge between $\Pi^{\prime},\Pi^{\prime\prime}$; the choice is
immaterial, but for simplicity we settle for
$m\overset{\mathrm{def}}{=}\mathrm{max}\\{i\in\Pi^{\prime}\\}$ and
$r\in\Pi^{\prime\prime}$, and set $\beta_{\mathrm{link}}=\alpha^{m,r}$ to be
the smallest element according to $\upsilon$.

The $\upsilon$-ordered list of edges thus starts with $\beta_{\mathrm{link}}$,
and then continues with the remaining $r^{\prime}\cdot r^{\prime\prime}-1$
edges connecting $\Pi^{\prime}$ and $\Pi^{\prime\prime}$. Next we list the
$r^{\prime}(r^{\prime}-1)/2$ edges connecting vertices in $\Pi^{\prime}$ in
any order, and finally, we list the remaining edges, those connecting vertices
in $\Pi^{\prime\prime}$.

Notation: We introduce the natural notation $\Phi^{\prime}$ and
$\Phi^{\prime\prime}$ for the $A_{r^{\prime}}$ and $A_{r^{\prime\prime}}$ root
systems corresponding to $\Pi^{\prime}$ and $\Pi^{\prime\prime}$, and we
denote by $\mathcal{D}[\upsilon]$, $\mathcal{D}^{\prime}[\upsilon]$ and
$\mathcal{D}^{\prime\prime}[\upsilon]$, the diagonal nbc-bases induced by the
ordering $\upsilon$ on $\Phi$, $\Phi^{\prime}$ and $\Phi^{\prime\prime}$,
respectively.

######  Lemma 4.15.

Given elements $\mathbf{B}^{\prime}\in\mathcal{D}^{\prime}[\upsilon]$ and
$\mathbf{B}^{\prime\prime}\in\mathcal{D}^{\prime\prime}[\upsilon]$, we can
define an element of $\mathcal{D}[\upsilon]$ as follows: we start with
$\beta_{\mathrm{link}}$, then append $\mathbf{B}^{\prime}$, and then continue
with $\mathbf{B}^{\prime\prime}$. This construction creates a one-to-one
correspondence

(24)
$\mathcal{D}^{\prime}[\upsilon]\times\mathcal{D}^{\prime\prime}[\upsilon]\to\mathcal{D}[\upsilon]|\pi;$

in particular,
$|\mathcal{D}[\upsilon]|\pi|=(r^{\prime}-1)!\cdot(r^{\prime\prime}-1)!$.

Finally, putting Lemmas 4.12 and 4.15 together, we arrive at the following
elegant statement:

######  Proposition 4.16.

Let $(\Pi,l)$, $c^{+}$ and $c^{-}$ be as in Lemma 4.12, and let
$\mathcal{D}^{\prime}$ and $\mathcal{D}^{\prime\prime}$ be diagonal bases of
$\Phi^{\prime}$ and $\Phi^{\prime\prime}$ correspondingly. Then

(25) $p_{+}(k;\lambda)-p_{-}(k;\lambda)=(k+r)\tilde{N}_{r,k}\cdot\\\
\sum_{\mathbf{B}^{\prime}\in\mathcal{D}^{\prime}}\sum_{\mathbf{B}^{\prime\prime}\in\mathcal{D}^{\prime\prime}}\operatornamewithlimits{Res}_{\beta_{\mathrm{link}}=0}\operatornamewithlimits{iBer}_{\mathbf{B}^{\prime}}\operatornamewithlimits{iBer}_{\mathbf{B}^{\prime\prime}}\left[w_{\Phi}^{1-2g}(x/\widehat{k})\right]\left(\widehat{\lambda}/\widehat{k}-[c^{+}]_{\mathbf{B}}\right)\,d\beta_{\mathrm{link}},$

where
$\operatornamewithlimits{Res}_{\beta_{\mathrm{link}}=0}\operatornamewithlimits{iBer}_{B^{\prime}}\operatornamewithlimits{iBer}_{B^{\prime\prime}}\,d\beta_{\mathrm{link}}$
is simply $\operatornamewithlimits{iBer}_{\mathbf{B}}$ (cf (15)) with
$\mathbf{B}$ obtained by appending $\mathbf{B}^{\prime}$, and then
$\mathbf{B}^{\prime\prime}$ to $\beta_{\mathrm{link}}$, and with the factor
$(1-\exp\langle\beta_{\mathrm{link}},x\rangle)$ removed from the denominator.

######  Remark 4.17.

The expression

$\operatornamewithlimits{Res}_{\beta_{\mathrm{link}}=0}\operatornamewithlimits{iBer}_{\mathbf{B}^{\prime}}\operatornamewithlimits{iBer}_{\mathbf{B}^{\prime\prime}}\left[w_{\Phi}^{1-2g}(x/\widehat{k})\right]\left(\widehat{\lambda}/\widehat{k}-[c^{+}]_{\mathbf{B}}\right)\,d\beta_{\mathrm{link}}$

may equally be interpreted as follows. We write

$\Lambda\ni\widehat{\lambda}/\widehat{k}-[c^{+}]_{\mathbf{B}}=m_{\mathrm{link}}\beta_{\mathrm{link}}+n^{\prime}+n^{\prime\prime}$

according to the splitting of $\mathbf{B}$, think of $w(x/\widehat{k})$ as a
function in $\mathcal{F}_{\Phi^{\prime\prime}}$ with some fixed values of the
parameters from $\mathbf{B}^{\prime}$ and $\beta_{\mathrm{link}}$, and then
calculate

$\operatornamewithlimits{iBer}_{\mathbf{B}^{\prime\prime}}[w_{\Phi}^{1-2g}(x/\widehat{k})](n^{\prime\prime}).$

The result will be a rational function $Q$ in the variables from
$\mathbf{B}^{\prime}$ and $\beta_{\mathrm{link}}$, and we proceed to calculate
$\operatornamewithlimits{iBer}_{\mathbf{B}^{\prime}}[Q](n^{\prime})$ to obtain
a function $F$ in the variable $\beta_{\mathrm{link}}$, and finally the answer
is
$\operatornamewithlimits{Res}_{\beta_{\mathrm{link}}=0}\exp(\beta_{\mathrm{link}})F(\beta_{\mathrm{link}})d\beta_{\mathrm{link}}$.

We observe that since the trees $\mathrm{Tree}({\mathop{\mathchoice{\mbox{\rm
B}}{\mbox{\rm B}}{\mbox{\rm\scriptsize B}}{\mbox{\rm\tiny
B}}}\nolimits}^{\prime})$ and $\mathrm{Tree}({\mathop{\mathchoice{\mbox{\rm
B}}{\mbox{\rm B}}{\mbox{\rm\scriptsize B}}{\mbox{\rm\tiny
B}}}\nolimits}^{\prime\prime})$ are disjoint, the order of the application of
the operations $\operatornamewithlimits{iBer}_{B^{\prime}}$ and
$\operatornamewithlimits{iBer}_{B^{\prime\prime}}$ is immaterial.

## 5\. Wall crossing in master space

Master spaces were introduced by Thaddeus in [Th1] in order to understand GIT
quotients when varying linearizations. Following his footsteps, in this
section, we describe a simple but very effective method to control the changes
in the Euler characteristics of line bundles when crossing a wall in the space
of linearizations.

### 5.1. Wall-crossing and holomorphic Euler characteristics

We begin by recalling the basic notions of Geometric Invariant Theory.

Let $X$ be a smooth projective variety over $\mathbb{C}$, and $G$ a reductive
group acting on $X$. A linearization of this action is a line bundle $L$ on
$X$ with a lifting of the $G$-action to a linear action on $L$. An ample
linearization is $G$-effective, if $L^{n}$ has a nonzero $G$-invariant section
for some $n>0$; the space of such linearizations is called the $G$-effective
ample cone; we denote this cone by $\mathrm{Cone}_{G}(X)$.

For $L\in\mathrm{Cone}_{G}(X)$, we define the invariant-theoretic quotient
$X\sslash^{L}G$ as the Proj of the graded ring of invariant sections of the
powers of $L$:

$M_{L}=\mathrm{Proj}\bigoplus_{n}H^{0}(X,L^{n})^{G}.$

According to Mumford’s Geometric Invariant Theory [MF], there is a partition
of $X$ depending on $L$:

(26) $X=X^{\mathrm{s}}[L]\cup X^{\mathrm{sss}}[L]\cup X^{\mathrm{us}}[L]$

into the set of stable, strictly semistable, and unstable points, such that
there is a surjective map $(X^{\mathrm{s}}[L]\cup X^{\mathrm{sss}}[L])/G\to
M_{L}$, which is a bijection if $X^{\mathrm{sss}}[L]$ is empty, and the
quotient $X^{\mathrm{s}}[L]/G$ is a smooth orbifold.

In [DH], Dolgachev and Hu studied the dependence of the GIT quotient
$M_{L}=X\sslash^{L}G$ on $L$. They showed that $\mathrm{Cone}_{G}(X)$ is
divided by hyperplanes, called walls, into finitely many convex chambers, such
that when $L$ varies within a chamber, the partition (26) and thus the GIT
quotient $M_{L}$ remains unchanged. Moreover, an ample effective linearization
lies on a wall precisely when it possesses a strictly semistable point.

Now let us consider two neighboring chambers, with smooth GIT quotients
$M_{+}$ and $M_{-}$. We pick an arbitrary linearization $\mathcal{L}$ of the
$G$-action on $X$, which descends to $M_{+}$ and $M_{-}$. This last condition
means that if $S\subset G$ is the stabilizer of a generic point in $X$, then
$S$ acts trivially on the fibers of $\mathcal{L}$. We will call such
linearizations descending.

Thus, given such a descending linearization $\mathcal{L}$ of the $G$-action on
$X$, we obtained two line bundles: one on $M_{+}$ and one on $M_{-}$, which,
by abuse of notation, we will denote by the same letter $\mathcal{L}$. Via
taking Chern classes, this construction creates a correspondence between
classes in $H^{2}(M_{+},\mathbb{Z})$ and $H^{2}(M_{-},\mathbb{Z})$, which we
will assume to be an isomorphism of free $\mathbb{Z}$-modules. We will thus
identify these lattices, and introduce the notation $\Gamma$ for them:

$\Gamma=H^{2}(M_{+},\mathbb{Z})\simeq H^{2}(M_{-},\mathbb{Z}).$

The walls mentioned above can be thought of as hyperplanes in
$\Gamma_{\mathbb{R}}=\Gamma\otimes_{\mathbb{Z}}\mathbb{R}$.

Our goal in this section is to compare the holomorphic Euler characteristics
$\chi(M_{+},\mathcal{L})$ and $\chi(M_{-},\mathcal{L})$, which are given by
the Hirzebruch-Riemann-Roch theorem:

$\chi(M_{\pm},\mathcal{L})=\int_{M_{\pm}}\exp(c_{1}(\mathcal{L}))\mathrm{Todd}(M_{\pm}).$

As this expression is manifestly polynomial in $c_{1}(\mathcal{L})$, we obtain
thus two polynomials on $\Gamma$, and our goal is to calculate their
difference, the wall-crossing term

(27) $\chi(M_{+},\mathcal{L})-\chi(M_{-},\mathcal{L}).$

### 5.2. The master space construction

To simplify our setup, we will make some additional assumptions.

######  Assumptions 5.1.

  1. (1)

The generic stabilizer of $X$ is trivial.

  2. (2)

Let $L_{+}$ and $L_{-}$ be two ample linearizations of the $G$-action on $X$
from the adjacent chambers corresponding to the quotients $M_{+}$ and $M_{-}$.
Without loss of generality, we can assume that the linearization
$L_{0}=L_{+}\otimes{L}_{-}$ lies on the single wall separating the two
chambers, and that the interval connecting $c_{1}(L_{+})$ and $c_{1}(L_{-})$
in $\Gamma_{\mathbb{R}}=\Gamma\otimes_{\mathbb{Z}}\mathbb{R}$ does not
intersect any other walls.

  3. (3)

Let $X^{0}$ be the set of those semistable points $x\in
X^{\mathrm{ss}}[L_{0}]$ which are not stable for $L_{\pm}$:

$X^{0}:=X^{\mathrm{ss}}[L_{0}]\setminus(X^{s}[L_{+}]\cup X^{s}[L_{-}])$

We assume that $X^{0}$ is smooth, and that for $x\in X^{0}$ the stabilizer
subgroup $G_{x}\subset G$ is isomorphic to $\mathbb{C}^{*}$.

  4. (4)

Assume that there is a linearization ${\vec{L}}$ of the $G$-action on $X$ such
that $L_{+}=L_{-}\otimes{\vec{L}}^{n}$ for some positive integer $n$, and such
that for each $x\in X^{0}$, the stabilizer subgroup $G_{x}$ acts freely on
$L_{x}\setminus{0}$.

Now we introduce the master space  construction of Thaddeus [Th1]. Consider
the variety $Y=\mathbb{P}(\mathcal{O}\oplus{{\vec{L}}})$, which is a
$\mathbb{P}^{1}$-bundle over $X$ endowed with the additional
$\mathbb{C}^{*}$-action $(1,t^{-1})$. As $Y$ is a projectivization of a vector
bundle on $X$, it comes equipped with $\mathcal{O}(1)$, which is the standard
${G}\times\mathbb{C}^{*}$-equivariant line bundle. To simplify our notation,
we will denote the same way the linearizations of the $G$-action on $X$ and
their pull-backs (with tautological $G$-action) to $Y$.

The master space $Z$ then is the GIT quotient of $Y$ with respect to the
linearization $L_{-}(n)=L_{-}\otimes\mathcal{O}(n)$:

$Z=Y\sslash^{L_{-}(n)}{{G}},$

which inherits a $\mathbb{C}^{*}$-action from $Y$. Some additional notation:

  * •

We will denote this copy of $\mathbb{C}^{*}$ by $T$,

  * •

the projection $Y\to X$ by $\pi$, and the quotient map $Y^{s}\to Z$ by $\psi$.

  * •

Introduce the notation $Y(0:\cdot)$ and $Y(\cdot:0)$ for the two copies of $X$
in $Y$, corresponding to the two poles of the projective line; then $Y$ is
partitioned into 3 sets:

$Y=Y(0:\cdot)\sqcup Y(\cdot:0)\sqcup{\vec{L}}^{\circ},$

where ${\vec{L}}^{\circ}$ is the line bundle ${\vec{L}}$ with the zero-section
removed. We will write $\pi_{\circ}$ for the restriction of $\pi$ to
${\vec{L}}^{\circ}$. We can collect our maps on the following diagram.

${{\vec{L}}^{\circ}}$${Y=\mathbb{P}(\mathcal{O}\oplus{\vec{L}})\supset
Y^{\mathrm{s}}}$${Z}$${X}$$\scriptstyle{\pi_{\circ}}$$\scriptstyle{\pi}$$\scriptstyle{\psi}$

######  Proposition 5.2.

  1. (1)

There are embeddings

$\iota_{-}:M_{-}\to Z\quad\text{and}\quad\iota_{+}:M_{+}\to Z$

obtained as the quotients $Y^{\mathrm{s}}\cap Y(\cdot:0)/G$ and
$Y^{s}\cap Y(0:\cdot)/G$, correspondingly.

  2. (2)

The strictly semistable locus of $Y$ with respect to the linearization
$L_{-}(n)$ is empty, and the GIT quotient $Z=Y^{\mathrm{s}}/G$ is smooth.

  3. (3)

There is an embedding $\iota_{0}:X^{0}/G\to Z$, obtained via
$\psi({\pi_{\circ}}^{-1}(X^{0}))$. We denote the image of $\iota_{0}$ by
$Z^{0}$.

  4. (4)

The fixed point locus $Z^{T}$ is the disjoint union of $\iota_{+}(M_{+})$,
$\iota_{-}(M_{-})$, and $Z^{0}$.

###### Proof.

(1)-(3) follow from [Th1, 4.2, 4.3]. To prove (4), first note that
$Y(\cdot:0)$ and $Y(0:\cdot)$ are fixed by $T$, so we immediately obtain that
$M_{\pm}\subset Z$ are fixed components. Also the $G$-action on $Y$ commutes
with the $T$-action, so a point $\psi(y)\in{\psi(\pi_{\circ}^{-1}(X))}$ is
fixed by $T$ if and only if the $T$-orbit
$T\cdot{y}\subset\pi_{\circ}^{-1}(X)$ is contained in the $G$-orbit
$G\cdot{y}\subset\pi_{\circ}^{-1}(X)$. Since
$T\cdot{y}\subset\pi_{\circ}^{-1}(x)$ for some $x\in{X}$, we need
$y\in\pi_{\circ}^{-1}(X^{0})$. Moreover, for any
$y\in\pi_{\circ}^{-1}(x)\subset{\pi_{\circ}^{-1}(X^{0})}$,
$T\cdot{y}=\pi_{\circ}^{-1}(x)=G_{x}\cdot{y}$, so a point
$\psi(y)\in{\psi(\pi_{\circ}^{-1}(X))}$ is fixed by $T$ if and only if
$\psi(y)\in{\psi(\pi_{\circ}^{-1}(X^{0}))=Z^{0}}$. ∎

Construction: Given a $G$-equivariant vector bundle $E$ on $X$, we can
construct a $T$-equivariant vector bundle $\zeta(E)\to Z$ on $Z$ by first
pulling $E$ back from $X$ to $Y$, and endowing the resulting bundle $\pi^{*}E$
with the trivial action of $T$, and the action of $G$ pulled back from $X$. We
then obtain $\zeta(E)\to Z$ by descending $\pi^{*}E$ to $Z$. Then it is easy
to verify the following.

######  Lemma 5.3.

The restriction of the line bundle $\zeta({\vec{L}})$ to $Z^{0}$ is trivial
with $T$-weight 1.

Before we formulate our wall-crossing formula, we need one more ingredient:
the identification of the normal bundles of the fixed point components of $Z$.

######  Lemma 5.4.

  1. (1)

The normal bundle on the component $M_{+}$ of $Z^{T}$ is
$\zeta({\vec{L}}^{-1})\big{|}_{M_{+}}$, and the normal bundle of $M_{-}$ is
$\zeta({\vec{L}})\big{|}_{M_{-}}$.

  2. (2)

The normal bundle $N_{Z^{0}}$ of $Z^{0}=X^{0}/G\subset Z$ may be described as
the descent of the normal bundle $N_{X^{0}}$ of $X^{0}\subset X$. The weights
of the action may be computed by fixing $x\in X^{0}$, identifying the
stabilizer subgroup $G_{x}\subset G$ with $T$ via its action on the fiber
${\vec{L}}_{x}$, and then considering the action of $G_{x}$ on $N_{X^{0}}$.

######  Definition 5.5.

Given a $T$-vector bundle $V$ on a manifold on which $T$ acts trivially, the
$T$-equivariant K-theoretical Euler class of $V^{*}$, which we denote by
$E_{t}(V)$, may be described as follows: let $x_{1},\dots,x_{n}$ be the Chern
roots of $V$, and $l_{1},\dots l_{n}\in\mathbb{Z}$ be the corresponding
$T$-weights. Then

$E_{t}(V)=\prod_{j=1}^{n}\left(1-t^{-l_{j}}\exp(-x_{j})\right).$

Now we are ready to write down our wall-crossing formula for (27). A key role
will be played by the following notion: given a rational differential 1-form
on the Riemann sphere, let us denote taking the sum of residues at 0 and at
infinity by $\mu\mapsto\operatornamewithlimits{Res}_{t=0,\infty}\mu$:

$\operatornamewithlimits{Res}_{t=0,\infty}\overset{\mathrm{def}}{=}\operatornamewithlimits{Res}_{t=0}+\operatornamewithlimits{Res}_{t=\infty}.$

######  Theorem 5.6.

Let $\mathcal{L}$ be a linearization of the $G$-action on $X$, and denote, as
above, by $\zeta(\mathcal{L})$ the $T$-equivariant line bundle on $Z$ obtained
by pull-back to $Y$ and descent to $Z$. If Assumptions 5.1 hold, then

(28)
$\chi(M_{+},\mathcal{L})-\chi(M_{-},\mathcal{L})=\operatornamewithlimits{Res}_{t=0,\infty}\int_{Z^{0}}\frac{\mathrm{ch}_{t}(\zeta(\mathcal{L})\big{|}_{Z^{0}})}{E_{t}(N_{Z^{0}})}\mathrm{Todd}(Z^{0})\frac{dt}{t},$

where $N_{Z^{0}}$ is the $T$-equivariant bundle on $Z^{0}$ described in Lemma
5.4, $\mathrm{ch}_{t}$ is the $T$-equivariant Chern character, and
$E_{t}(N_{Z^{0}})$ is the K-theoretical Euler class of $N^{*}_{Z^{0}}$.

###### Proof of Theorem 5.6.

The Atiyah-Bott fixed-point formula [AB1] applied to the line bundle
$\mathcal{L}$ on our master space $Z$ yields

(29) $\chi_{t}(Z,\mathcal{L})=\sum_{F\subset
Z^{T}}\int_{F}\frac{\mathrm{ch}_{t}(\zeta(\mathcal{L})\big{|}_{F})}{E_{t}(N_{F})}\mathrm{Todd}(F),$

where the sum is taken over the connected components of the fixed point locus
$Z^{T}$.

In Proposition 5.2, we identified these components as $M_{+},M_{-}$ and
${Z^{0}}$. According to Lemma 5.4, for $M_{-}$, the normal bundle is simply
$\zeta({\vec{L}})$, and thus the contribution of $M_{-}$ is equal to

$\int_{M_{-}}\frac{\mathrm{ch}(\mathcal{L})\mathrm{Todd}(M_{-})}{1-t\exp(-c_{1}({\vec{L}}))}.$

A similar calculation gives the contribution of $M_{+}$ as

$\int_{M_{+}}\frac{\mathrm{ch}(\mathcal{L})\mathrm{Todd}(M_{+})}{1-t^{-1}\exp(c_{1}({\vec{L}}))}.$

We observe that $\chi_{t}(Z,\mathcal{L})$ is a Laurent polynomial in $t$ since
it is the alternating sum of $T$-characters of finite dimensional vector
spaces. Thus, as a function of $t$, $\chi_{t}(Z,\mathcal{L})$ has poles only
at $t=0,\infty$, and by the Residue Theorem, we have

$\operatornamewithlimits{Res}_{t=0,\infty}\chi_{t}(Z,\mathcal{L})\frac{dt}{t}=0.$

On the other hand, since

$\operatornamewithlimits{Res}_{t=0,\infty}\frac{A}{1-t^{-1}B}\frac{dt}{t}=-A\;\text{
and }\;\operatornamewithlimits{Res}_{t=0,\infty}\frac{A}{1-tB}\frac{dt}{t}=A,$

we have

$\operatornamewithlimits{Res}_{t=0,\infty}\int_{M_{-}}\frac{\mathrm{ch}(\mathcal{L})\mathrm{Todd}(M_{-})}{1-t\exp(-c_{1}({\vec{L}}))}\frac{dt}{t}=\chi(M_{-},L)\text{
and }\\\
\operatornamewithlimits{Res}_{t=0,\infty}\int_{M_{+}}\frac{\mathrm{ch}(\mathcal{L})\mathrm{Todd}(M_{+})}{1-t^{-1}\exp(c_{1}({\vec{L}}))}\frac{dt}{t}=-\chi(M_{+},L).$

Now, applying the functional $\operatornamewithlimits{Res}_{t=0,\infty}$ to
the two sides of (29) multiplied by $dt/t$ gives us the desired result (47).

∎

## 6\. Wallcrossings in parabolic moduli spaces

In this section we apply Theorem 5.6 to wall crossings in the moduli space of
parabolic bundles.

From now on, we assume that $d=0$, and we write $\Delta$ for the corresponding
set of admissible parabolic weights $\Delta_{0}$. Recall from Section 2.2 that
for regular $c\in\Delta$, the moduli space of stable parabolic bundles
$P_{0}(c)$ is the GIT quotient $XQ\sslash^{c}PSL(\chi),$ where $XQ$ is the
subspace of the total space of the flag bundle over the Quot scheme. Let us
fix a partition $\Pi=(\Pi^{\prime},\Pi^{\prime\prime})$ and an integer $l$,
and introduce the notation $\Delta^{\prime}_{l}$ and
$\Delta^{\prime\prime}_{-l}$ for the simplices of parabolic weights of
$\Pi^{\prime}$ and $\Pi^{\prime\prime}$. Let $\phi\in\Sigma_{r}$ be the unique
permutation which sends $\\{1,...,r^{\prime}\\}$ to $\Pi^{\prime}$ preserving
the order of first $r^{\prime}$ and the last $r^{\prime\prime}$ elements. We
choose $c^{0}=(c^{0}_{1},...,c^{0}_{r})\in{S}_{\pi,l}$ and two regular
elements $c^{+},c^{-}\in\Delta$ in two neighboring chambers separated by the
wall $S_{\Pi,l}$, such that

$c^{\pm}=c^{0}\pm\epsilon(...,0,1,0,...,0,-1)$

for some positive $\epsilon\in\mathbb{Q}$, where $1$ and $-1$ are on the
$\phi(r^{\prime})^{\text{th}}$ and $r^{\text{th}}$ places, respectively. Let

$c^{\prime}=\sum_{i\in\Pi^{\prime}}c^{0}_{i}x_{i}\in\Delta^{\prime}_{l}\text{\,\,
and
\,\,}c^{\prime\prime}=\sum_{i\in\Pi^{\prime\prime}}c^{0}_{i}x_{i}\in\Delta^{\prime\prime}_{-l}.$

For $(k,\lambda)\in\mathbb{Z}\times\Lambda$, consider the polynomials

$q_{\pm}(k,\lambda)=\chi(P_{0}(c^{\pm}),\mathcal{L}_{0}(k;\lambda)).$

Our goal is to calculate the difference of these two polynomials.

Notation: To simplify our notation, from now on, we omit the index $t$ from
the symbols for equivariant characteristic classes.

### 6.1. The master space construction

We construct the master space $Z$ from §5.2 using the following data:

  * •

a smooth variety $X=XQ$ (cf. §2.2);

  * •

linearizations $L^{\pm}=L(k;\lambda^{\pm})$ of the $G$-action on $X$ (cf.
§2.2), such that $\lambda^{\pm}/k=c^{\pm}$;

  * •

the linearization ${\vec{L}}=L(0;x_{\phi(r^{\prime})}-x_{r})$ of the
$G$-action on $X$.

The following statement is easy to verify.

######  Lemma 6.1.

([BH, §3.2]) The subset $X^{0}\subset{X}$ is the set of points representing
vector bundles $W$ on $C$, such that $W$ splits as a direct sum
$W^{\prime}\oplus{W^{\prime\prime}}$, where $W^{\prime}$ and
$W^{\prime\prime}$ are, respectively, $c^{\prime}$ and
$c^{\prime\prime}$-stable parabolic bundles. Therefore, we have the following
description of the locus $Z^{0}$:

$Z^{0}=\\{W=W^{\prime}\oplus
W^{\prime\prime}\,|\,W^{\prime}\in\widetilde{P}_{l}(c^{\prime});\,W^{\prime\prime}\in\widetilde{P}_{-l}(c^{\prime\prime});\,\,det(W)\simeq\mathcal{O}\\}.$

######  Remark 6.2.

Note that $Z^{0}$ is fibered over $Jac^{l}$ with fibre
$P_{l}(c^{\prime})\times{P_{-l}(c^{\prime\prime})}$ by the determinant map
$\widetilde{P}_{l}(c^{\prime})\to Jac^{l}$ and

$H^{*}(Z^{0},\mathbb{Q})\simeq
H^{*}(P_{l}(c^{\prime})\times{P_{-l}(c^{\prime\prime})},\mathbb{Q})\otimes
H^{*}(Jac^{l},\mathbb{Q}).$

######  Remark 6.3.

If the rank of the vector bundle $W\in\widetilde{P}_{l}(c)$ is $1$, then $c=l$
and $\widetilde{P}_{l}(l)$ is isomorphic to $Jac^{l}$, while ${P}_{l}(l)$ is a
point.

Now we need to verify the hypotheses of Theorem 5.6. First note that in our
present construction $X$ is not projective, however, it contains all
semisimple points of the Quot scheme for all possible polarizations, and hence
the missing points of the Quot scheme have no effect on any of our
constructions.

Assumptions 5.1 (1)-(2) are trivially satisfied, so we study the action of the
stabilizer $G_{x}\subset{SL_{\chi}}$ of point $x\in{X}$ on the fiber
${\vec{L}}_{x}\setminus{0}$.

  * •

For a general point $x\in X$ the stabilizer of $x$ is the center
$\mathbb{Z}_{\chi}\subset SL(\chi)$, which acts trivially on the fiber
${\vec{L}}_{x}\setminus{0}$.

  * •

For $x\in X^{0}$, any element of the stabilizer of $x$ induces an automorphism
of the corresponding vector bundle $W=W^{\prime}\oplus{W^{\prime\prime}}$, so
$G_{x}\simeq\mathbb{C}^{*}\times\mathbb{C}^{*}\subset GL_{\chi}$. Then
$(t_{1},t_{2})\in{G}_{x}$ is in $SL_{\chi}$ if and only if
$t_{1}^{\chi^{\prime}}t_{2}^{\chi^{\prime\prime}}=1$, where
$\chi^{\prime}=\chi(W^{\prime})$ and
$\chi^{\prime\prime}=\chi(W^{\prime\prime})$. Note that $(t_{1},t_{2})$ acts
on ${\vec{L}}_{x}$ as $t_{1}t_{2}^{-1}$, and we need $t_{1}=t_{2}$ (hence
$t_{1}^{\chi}=1$) for this action to be trivial, so the stabilizer of any
point in ${\vec{L}}_{x}\setminus{0}$ is the center $\mathbb{Z}_{\chi}\subset
SL_{\chi}$.

Then the action of $G={P}SL(\chi)$ is free on $Y\setminus(Y(0:\cdot)\cup
Y(\cdot:0))$ and the action of $G_{x}\subset PSL(\chi)$ on
${\vec{L}}_{x}\setminus{0}$ induces an isomorphism
$G_{x}\simeq\mathbb{C}^{*}\simeq T$.

Now by Theorem 5.6, the wall-crossing polynomial
$q_{-}(k;\lambda)-q_{+}(k;\lambda)$ is equal to

(30)
$\begin{split}\operatornamewithlimits{Res}_{t=0,\infty}\int_{Z^{0}}\frac{ch(\mathcal{L}_{0}(k;{\lambda})\big{|}_{Z^{0}})}{E(N_{Z^{0}})}\mathrm{Todd}(Z^{0})\,\frac{dt}{t}.\end{split}$

Note that in our case, the $T$-action on $Z$ is free outside the fixed locus
$Z^{T}$, so as a function in $t\in T$, the integral in (30) may have poles
only at $t=0,1,\infty$. Then, using the Residue Theorem and substituting
$t=e^{u}$, we conclude that (30) equals

(31)
$\begin{split}-\operatornamewithlimits{Res}_{u=0}\int_{Z^{0}}\frac{ch(\mathcal{L}_{0}(k;{\lambda})\big{|}_{Z^{0}})}{E(N_{Z^{0}})}\mathrm{Todd}(Z^{0})\,du,\end{split}$

and thus our goal is to calculate this integral.

Our first step is to identify the characteristic classes under the integral
sign (cf. Proposition 6.9 for the result).

We start with the study of the restriction of the line bundle
$\mathcal{L}_{0}(k;{\lambda})$ to the fixed locus $Z^{0}\subset Z$. Let
$\mathcal{J}$ be the Poincare bundle over $Jac\times C$, such that
$c_{1}(\mathcal{J})_{(0)}=0$; define $\eta\in H^{2}(Jac)$ by
$(\sum_{i}c_{1}(\mathcal{J})_{(e_{i})}\otimes e_{i})^{2}=-2\eta\otimes\omega$
(cf. §2.3), then (cf. [Z]) for any $m\in\mathbb{Z}$

(32) $\int_{Jac}e^{\eta m}=m^{g}.$

Recall that for a parabolic weight $c=(c_{1},...,c_{r})\in\Delta$ we have set
$c_{\Pi^{\prime}}=\sum_{i\in\Pi^{\prime}}c_{i}.$

######  Lemma 6.4.

Let $\lambda=(\lambda_{1},...,\lambda_{r})\in\Lambda$, $k\in\mathbb{Z}^{>0}$
and let $\Pi=(\Pi^{\prime},\Pi^{\prime\prime})$ be a nontrivial partition with
$r\in\Pi^{\prime\prime}$. Let

$\lambda^{\prime}=\sum_{i\in\Pi^{\prime}}\lambda_{i}x_{i}\text{\,\, and
\,\,}\lambda^{\prime\prime}=\sum_{i\in\Pi^{\prime\prime}}\lambda_{i}x_{i},$

and define $\delta$ by $(\lambda/k)_{\Pi^{\prime}}=l+\delta$. Then

$\begin{split}ch&(\mathcal{L}_{0}(k;{\lambda})\big{|}_{Z^{0}})=\,e^{k{\delta}u}exp\left(\frac{\eta
k}{r^{\prime}}+\frac{\eta k}{r^{\prime\prime}}\right)\cdot\\\
&ch(\mathcal{L}_{l}(k;\lambda_{1}^{\prime},...,\lambda^{\prime}_{r^{\prime}}-k\delta)\boxtimes\mathcal{L}_{-l}(k;\lambda_{1}^{\prime\prime},...,\lambda_{r^{\prime\prime}}^{\prime\prime}+k\delta)),\end{split}$

where $\boxtimes$ denotes the external tensor product of line bundles on
$P_{l}(c^{\prime})\times{P_{-l}(c^{\prime\prime})}$.

######  Lemma 6.5.

Denote by $\widetilde{U}^{\prime}$ and $\widetilde{U}^{\prime\prime}$ the
universal bundles over $\widetilde{P}_{l}(c^{\prime})\times{C}$ and
$\widetilde{P}_{-l}(c^{\prime\prime})\times{C}$ with the standard
normalization (cf. §2.3 ), and denote by $\pi$ projections along $C$. Then the
equivariant normal bundle to the fixed locus $Z^{0}\subset{Z}$ is

${N_{Z^{0}}}=R_{T}^{1}\pi_{*}(ParHom(\widetilde{U}^{\prime},\widetilde{U}^{\prime\prime}))\oplus
R_{T}^{1}\pi_{*}(ParHom(\widetilde{U}^{\prime\prime},\widetilde{U}^{\prime})),$

where $T\simeq\mathbb{C}^{*}$-action has weights $(-1,1)$.

### 6.2. Calculation of the characteristic classes of ${N_{Z^{0}}}$

Before we calculate the equivariant K-theoretical Euler class of the conormal
bundle $N^{*}_{Z^{0}}$, we need to introduce some notations. Recall that for
$1\leq i,j\leq r$, the differences $x_{i}-x_{j}\in V^{*}$ are linear functions
on $V$, and the function $x_{i}-x_{j}$ corresponds to the linearization
$L_{0}(0;x_{i}-x_{j})$ on $X$, which descends to the line bundle
$\mathcal{L}_{0}(0;x_{i}-x_{j})$ on the moduli space $P_{0}(c)$ (cf. §2.2). As
in §5.2, we denote by $\zeta(L_{0}(0;x_{i}-x_{j})$ the line bundle on $Z$
obtained by the pullback and then descent. This way, we obtain a
correspondence between the linear functions $x_{i}-x_{j}$ and the
$T$-equivariant line bundles on $Z$.

As $Z^{0}$ is a connected component of the fixed locus of the $T$\- action on
$Z$, and thus its equivariant cohomology factors: $H^{*}_{T}(Z^{0})\simeq
H^{*}(Z^{0})\otimes\mathbb{C}[u]$. In particular, there are canonical
embeddings $H^{*}(Z^{0})\hookrightarrow H^{*}_{T}(Z^{0})$ and
$\mathbb{C}[u]\hookrightarrow H^{*}_{T}(Z^{0})$.

Recall the definition of the permutation $\phi\in\Sigma_{r}$ given at the
beginning of this chapter: $\phi$ takes the first $r^{\prime}$ numbers to
$\Pi^{\prime}$, preserving the order of the first $r^{\prime}$ and the last
$r^{\prime\prime}$ elements. We introduce the symbols

(33)
$\begin{split}&z^{\prime}_{i}-z^{\prime}_{j}=c_{1}(\zeta(L_{0}(0;x_{\phi(i)}-x_{\phi(j)}))\big{|}_{Z^{0}}),\,\,(1\leq
i,j\leq r^{\prime})\\\
&z^{\prime\prime}_{i}-z^{\prime\prime}_{j}=c_{1}(\zeta({L}_{0}(0;x_{\phi(r^{\prime}+i)}-x_{\phi(r^{\prime}+j)}))\big{|}_{Z^{0}}),\,\,(1\leq
i,j\leq r^{\prime\prime})\\\
&u=(z^{\prime}_{r^{\prime}}-z^{\prime\prime}_{r})=c_{1}(\zeta({L}_{0}(0;x_{\phi(r^{\prime})}-x_{r}))\big{|}_{Z^{0}})\end{split}$

for the equivariant cohomology classes in $H^{2}_{T}(Z^{0})$. The last
equalities are consistent with Lemma 5.3.

######  Remark 6.6.

Note that (cf. Remark 6.2)

$z_{i}^{\prime}-z_{j}^{\prime}=c_{1}(\mathcal{F}^{\prime}_{r-i+1}/\mathcal{F}^{\prime}_{r-i}\otimes(\mathcal{F}^{\prime}_{r-j+1}/\mathcal{F}^{\prime}_{r-j})^{*})\in
H^{2}(P_{l}(c^{\prime})),$
$z^{\prime\prime}_{i}-z^{\prime\prime}_{j}=c_{1}(\mathcal{F}^{\prime\prime}_{r-i+1}/\mathcal{F}^{\prime\prime}_{r-i}\otimes(\mathcal{F}^{\prime\prime}_{r-j+1}/\mathcal{F}^{\prime\prime}_{r-j})^{*})\in
H^{2}(P_{-l}(c^{\prime\prime})),$

where $\mathcal{F}^{\prime}_{i}$ and $\mathcal{F}^{\prime\prime}_{i}$ are the
flag bundles (cf. §2.3) on $P_{0}(c^{\prime})$ and $P_{0}(c^{\prime\prime})$,
correspondingly.

Taking into account these identifications, functions on $V$ give rise to
equivariant cohomology classes on $Z^{0}$. To make the splitting
$H^{*}_{T}(Z^{0})\simeq H^{*}(Z^{0})\otimes\mathbb{C}[u]$, explicit, however,
we will write these classes in the form $f_{u}(z^{\prime},z^{\prime\prime})$,
thinking of them as functions of the differences of the $z_{i}^{\prime}$s and
the differences of the $z_{i}^{\prime\prime}$s, depending on the parameter
$u$. With this convention, we introduce

$\begin{split}&w_{u}^{\times}(z^{\prime},z^{\prime\prime})=\prod_{\begin{subarray}{c}i,j\\\
\phi(i)<\phi(r^{\prime}+j)\end{subarray}}2\mathrm{sinh}(z^{\prime}_{i}-z^{\prime\prime}_{j})\prod_{\begin{subarray}{c}i,j\\\
\phi(r^{\prime}+j)<\phi(i)\end{subarray}}2\mathrm{sinh}(z^{\prime\prime}_{j}-z^{\prime}_{i}),\\\
&\rho_{u}^{\times}(z^{\prime},z^{\prime\prime})=\frac{1}{2}\sum_{\begin{subarray}{c}i,j\\\
\phi(i)<\phi(r^{\prime}+j)\end{subarray}}(z^{\prime}_{i}-z^{\prime\prime}_{j})+\frac{1}{2}\sum_{\begin{subarray}{c}i,j\\\
\phi(r^{\prime}+j)<\phi(i)\end{subarray}}(z^{\prime\prime}_{j}-z^{\prime}_{i}),\\\
\end{split}$

where according to (33),

$z^{\prime}_{i}-z_{j}^{\prime\prime}=(z^{\prime}_{i}-z^{\prime}_{r^{\prime}})+u-(z^{\prime\prime}_{j}-z^{\prime\prime}_{r})=c_{1}(\zeta(\mathcal{L}_{0}(0;x_{\phi(i)}-x_{\phi(r^{\prime}+j)}))\big{|}_{Z^{0}})\in
H^{2}_{T}(Z^{0}).$

Now we are ready to write down our formula for the K-theoretical Euler class
$E(N_{Z^{0}})$ (cf. definition 5.5 with $t=e^{u}$).

######  Proposition 6.7.

$\begin{split}E(N_{Z^{0}})^{-1}=&\,(-1)^{lr+r^{\prime}r^{\prime\prime}(g-1)}e^{-rlu}exp\left(\frac{\eta
r}{r^{\prime}}+\frac{\eta
r}{r^{\prime\prime}}\right)w_{u}^{\times}(z^{\prime},z^{\prime\prime})^{1-2g}exp(\rho_{u}^{\times}(z^{\prime},z^{\prime\prime}))\\\
&ch(\mathcal{L}_{l}(r^{\prime\prime};-l,...,-l,-l+rl)\boxtimes\mathcal{L}_{-l}(r^{\prime};l,...,l,l-rl)).\\\
\end{split}$

###### Proof.

It follows from the short exact sequence (6) for parabolic morphisms that

$ch(-\pi_{!}(ParHom(\widetilde{U}^{\prime\prime},\widetilde{U}^{\prime}))=-ch(\pi_{!}(Hom(\widetilde{U}^{\prime\prime},\widetilde{U}^{\prime})))+\sum_{\begin{subarray}{c}i,j\\\
\phi(i)<\phi(r^{\prime}+j)\end{subarray}}e^{z^{\prime}_{i}-z^{\prime\prime}_{j}}$

and

$ch(-\pi_{!}(ParHom(\widetilde{U}^{\prime},\widetilde{U}^{\prime\prime}))=-ch(\pi_{!}(Hom(\widetilde{U}^{\prime},\widetilde{U}^{\prime\prime}))+\sum_{\begin{subarray}{c}i,j\\\
\phi(r^{\prime}+j)<\phi(i)\end{subarray}}e^{z^{\prime\prime}_{j}-z^{\prime}_{i}},$

so by Lemma 6.5

(34)
$\begin{split}&ch(N_{Z^{0}})=ch(-\pi_{!}(Hom(\widetilde{U}^{\prime},\widetilde{U}^{\prime\prime}))\oplus-\pi_{!}(Hom(\widetilde{U}^{\prime},\widetilde{U}^{\prime\prime})^{*}))\\\
&+\sum_{\begin{subarray}{c}i,j\\\
\phi(i)<\phi(r^{\prime}+j)\end{subarray}}e^{z^{\prime}_{i}-z^{\prime\prime}_{j}}+\sum_{\begin{subarray}{c}i,j\\\
\phi(r^{\prime}+j)<\phi(i)\end{subarray}}e^{z^{\prime\prime}_{j}-z^{\prime}_{i}}.\end{split}$

######  Lemma 6.8.

We denote by $[f(x)]^{W}$ the multiplicative class of the vector bundle $W$
given by the function $f(x)$ in Chern roots of $W$. Let $S$ be a vector bundle
on $P\times C$ with $T$-weight 1, and $\pi:P\times C\to P$ projection along
the curve, then

$E(-\pi_{!}S\oplus-\pi_{!}S^{*})^{-1}=(-1)^{\mathrm{rk}(-\pi_{!}S)}\frac{exp(-ch_{2}(S)_{(2)})}{[(2\mathrm{sinh}(x/2))^{2g-2}]^{S_{p}}}.$

###### Proof.

Note that

$E(-\pi_{!}S)^{-1}=\left[\frac{1}{1-t^{-1}e^{-x}}\right]^{-\pi_{!}S}=\left[\frac{-te^{x}}{1-te^{x}}\right]^{-\pi_{!}S}$

and

$E(-\pi_{!}S^{*})^{-1}=\left[\frac{1}{1-te^{-x}}\right]^{-\pi_{!}S^{*}}=\left[\frac{1}{1-te^{x}}\right]^{(-\pi_{!}S^{*})^{*}}.$

Applying Serre duality and the Grothendieck-Riemann-Roch Theorem we obtain

$ch(-\pi_{!}S)+ch((-\pi_{!}S^{*})^{*})=ch(-\pi_{!}S)+ch(\pi_{!}(S\otimes
K_{C}))=\\\ ch(-\pi_{!}S)+\pi_{*}(ch(S\otimes{K_{C}})\mathrm{Todd}(C))=\\\
ch(-\pi_{!}S)+ch(\pi_{!}S)+(2g-2)ch(S_{p})=(2g-2)ch(S_{p}),$

where $K_{C}$ is the canonical sheaf on the curve $C$, hence

$\left[\frac{1}{1-te^{x}}\right]^{-\pi_{!}S\oplus(-\pi_{!}S^{*})^{*}}=\left[\frac{1}{(1-te^{x})^{2g-2}}\right]^{S_{p}}=\frac{exp(-c_{1}(S_{p})(g-1))}{[(2\mathrm{sinh}(x/2))^{2g-2}]^{S_{p}}}.$

Since

$[-te^{x}]^{-\pi_{!}S}=(-1)^{\mathrm{rk}(-\pi_{!}S)}exp(c_{1}(-\pi_{!}S))$

and by the Grothendieck-Riemann-Roch theorem

$ch_{1}(-\pi_{!}S)=ch_{1}(S_{p})(g-1)-ch_{2}(S)_{(2)},$

we conclude that

$[-te^{x}]^{-\pi_{!}S}=(-1)^{\mathrm{rk}(-\pi_{!}S)}exp(c_{1}(S_{p})(g-1))exp(-ch_{2}(S)_{(2)}),$

which finishes the proof of Lemma 6.8. ∎

Note that the last two terms in (34) are the sums of Chern characters of line
bundles, so they contribute the multiplicative factor

$\frac{exp(\rho_{u}^{\times}(z^{\prime},z^{\prime\prime}))}{w_{u}^{\times}(z^{\prime},z^{\prime\prime})}$

to the equivariant class $E(N_{Z^{0}})^{-1}$; and using Lemma 6.8 with
$S=Hom(\widetilde{U}^{\prime},\widetilde{U}^{\prime\prime})$, we obtain that
the inverse of the K-theoretical Euler class of the first term in (34) is

$(-1)^{lr+r^{\prime}r^{\prime\prime}(g-1)}w_{u}^{\times}(z^{\prime},z^{\prime\prime})^{2-2g}exp(-ch_{2}(Hom(\widetilde{U}^{\prime},\widetilde{U}^{\prime\prime}))_{(2)}).$

Note that

$-ch_{2}(Hom(\widetilde{U}^{\prime},\widetilde{U}^{\prime\prime}))_{(2)}=\frac{1}{2}c_{2}(\mathrm{End}_{0}(\widetilde{U}^{\prime}\oplus\widetilde{U}^{\prime\prime}))_{(2)}\\\
-\frac{1}{2}c_{2}(\mathrm{End}_{0}(\widetilde{U}^{\prime}))_{(2)}-\frac{1}{2}c_{2}(\mathrm{End}_{0}(\widetilde{U}^{\prime\prime}))_{(2)}\\\
=c_{1}\left(\mathcal{L}^{r}_{0}\big{|}_{Z^{0}}\otimes\mathcal{L}_{l}(-r^{\prime};-l,...,-l)\boxtimes\mathcal{L}_{-l}(-r^{\prime\prime};l,...,l)\right).$

The latter equality follows from Lemma 2.8. Finally, using Lemma 6.4 to
calculate the Chern character of $\mathcal{L}^{r}_{0}\big{|}_{Z^{0}}$, we
obtain the formula for the class $E(N_{Z^{0}})^{-1}$, and the proof of the
Lemma is complete. ∎

### 6.3. The wall-crossing formula

Putting Lemma 6.4 and Proposition 6.7 together, we obtain the following.

######  Proposition 6.9.

The wall-crossing term (31) is equal to

$K\operatornamewithlimits{Res}_{u=0}e^{(k\delta-
rl)u}\int\limits_{P_{l}(c^{\prime})\times
P_{-l}(c^{\prime\prime})}\big{[}(w_{u}^{\times}(z^{\prime},z^{\prime\prime}))^{1-2g}exp({\rho_{u}^{\times}(z^{\prime},z^{\prime\prime})})\cdot\\\
ch(\mathcal{L}_{l}(k+r^{\prime\prime};\lambda_{1}^{\prime}-l,...,\lambda^{\prime}_{r^{\prime}-1}-l,\lambda^{\prime}_{r^{\prime}}-l-k{\delta}+rl)\boxtimes\\\
\mathcal{L}_{-l}(k+r^{\prime};\lambda^{\prime\prime}_{1}+l,...,\lambda^{\prime\prime}_{r^{\prime\prime}-1}+l,\lambda^{\prime\prime}_{r^{\prime\prime}}+l+k\delta-
rl))\mathrm{Todd}(P_{l}(c^{\prime})\times
P_{-l}(c^{\prime\prime}))\big{]}\,du,$

where $\delta$ is a parameter depending on $\lambda$ and the wall $S_{\Pi,l}$
(cf. Lemma 6.4) and $K$ is the constant
$(-1)^{lr+r^{\prime}r^{\prime\prime}(g-1)}\frac{(r(k+r))^{g}}{(r^{\prime}r^{\prime\prime})^{g}}$.

Now all that is left to do is to perform the integral, using an induction on
the rank based on Corollary 4.10. We will begin with the case $l=0$, as it is
simpler. For $l=0$, the integral from Proposition 6.9 has the form

(35) $\int\limits_{P_{0}(c^{\prime})\times
P_{0}(c^{\prime\prime})}\big{[}w_{u}^{\times}(z^{\prime},z^{\prime\prime})^{1-2g}e^{\rho_{u}^{\times}(z^{\prime},z^{\prime\prime})}\mathrm{Todd}(P_{0}(c^{\prime}))\mathrm{Todd}(P_{0}(c^{\prime\prime}))\\\
ch(\mathcal{L}_{0}(k+r^{\prime\prime};\lambda_{1}^{\prime},...,\lambda^{\prime}_{r^{\prime}-1},\lambda^{\prime}_{r^{\prime}}-k{\delta})\boxtimes\mathcal{L}_{0}(k+r^{\prime};\lambda^{\prime\prime}_{1},...,\lambda^{\prime\prime}_{r^{\prime\prime}-1},\lambda^{\prime\prime}_{r^{\prime\prime}}+k{\delta}))\big{]}.$

The inductive hypothesis (21) maybe cast in the following form

(36)
$\int\limits_{P_{0}(c)}ch(\mathcal{L}_{0}(k;\lambda))\mathrm{Todd}(P_{0}(c))=\\\
\tilde{N}_{r,k}\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[\exp\langle\lambda,x/\widehat{k}\rangle\cdot
w_{\Phi}(x/\widehat{k})^{1-2g}](\rho/\widehat{k}-[c]_{\mathbf{B}}).$

Now let us fix $k$, and allow to vary $\lambda$. We can extend this equality
by linearity to arbitrary linear combinations of Chern characters of line
bundles of the form

$\sum_{i}ch(\mathcal{L}_{0}(k;\lambda^{i}))=ch(\mathcal{L}_{0}(k;0))\cdot\sum_{i}ch(\mathcal{L}_{0}(0;\lambda^{i})).$

Since any polynomial on $V$, up to a fixed degree may be represented as a
linear combination of exponential functions of the form
$\exp\langle\lambda,x/\widehat{k}\rangle$, formula (36) may be generalized in
the following way.

######  Lemma 6.10.

Let $G(x)$ be a formal power series on $V$, and denote by $G(z)$ the
characteristic class in $H^{*}(P_{0}(c))$ obtained by the identification of
functions on $V$ and cohomology classes of $P_{0}(c)$, described before the
equation (33). Then we have

(37)
$\int\limits_{P_{0}(c)}ch(\mathcal{L}_{0}(k;0))G(z)\mathrm{Todd}(P_{0}(c))=\tilde{N}_{r,k}\cdot\\\
\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[G(x/\widehat{k})\cdot
w_{\Phi}^{1-2g}(x/\widehat{k})](\rho/\widehat{k}-[c]_{\mathbf{B}}).$

Finally, let $\mathcal{D}^{\prime}$ and $\mathcal{D}^{\prime\prime}$ be
Hamiltonian bases (cf. §4.5). Since

$w_{\Phi^{\prime}}(x/\widehat{k})w_{\Phi^{\prime\prime}}(x/\widehat{k})w^{\times}_{u}(x/\widehat{k})=w_{\Phi}(x/\widehat{k}),$
$\rho^{\prime}(x/\widehat{k})\rho^{\prime\prime}(x/\widehat{k})\rho^{\times}_{u}(x/\widehat{k})=\rho(x/\widehat{k}),$

where $w_{\Phi^{\prime}},w_{\Phi^{\prime\prime}}$ and
$\rho^{\prime},\rho^{\prime\prime}$ are naturally defined for the root systems
$\Phi^{\prime}$ and $\Phi^{\prime\prime}$ (cf. §4.5), the integral (35) is
equal to

$\tilde{N}_{r^{\prime},k+r^{\prime\prime}}\tilde{N}_{r^{\prime\prime},k+r^{\prime}}\sum_{\textbf{B}^{\prime}\in\mathcal{D}^{\prime}}\sum_{\textbf{B}^{\prime\prime}\in\mathcal{D}^{\prime\prime}}\operatornamewithlimits{iBer}_{\textbf{B}^{\prime}}\operatornamewithlimits{iBer}_{\textbf{B}^{\prime\prime}}[w_{\Phi}(x/\widehat{k})^{1-2g}e^{\rho(x/\widehat{k})}]\\\
((\lambda_{1}^{\prime},...,\lambda^{\prime}_{r^{\prime}-1},\lambda^{\prime}_{r^{\prime}}-k{\delta})/\widehat{k}-[c^{\prime}]_{\textbf{B}^{\prime}}+(\lambda^{\prime\prime}_{1},...,\lambda^{\prime\prime}_{r^{\prime\prime}-1},\lambda^{\prime\prime}_{r^{\prime\prime}}+k\delta)/\widehat{k}-[c^{\prime\prime}]_{\textbf{B}^{\prime\prime}}).$

Identifying $u$ (cf. (33)) with the "link" element of the diagonal basis
$\mathcal{D}=(\alpha^{\phi(r^{\prime}),r}\,\mathcal{D}^{\prime}\,\mathcal{D}^{\prime\prime})$
(cf. §4.5), and moving the factor $e^{k\delta u}$ from Proposition 6.9 inside
the argument of $\operatornamewithlimits{iBer}$, we obtain the proof of the
following theorem for $l=0$.

######  Theorem 6.11.

Let $c^{\pm}\in\Delta$ be in the neighbouring chambers; then the wall-crossing
term

$\chi(P_{0}(c^{+}),\mathcal{L}_{0}(k;\lambda))-\chi(P_{0}(c^{-}),\mathcal{L}_{0}(k;\lambda))$

is equal to

$\begin{split}&(k+r)\tilde{N}_{r,k}\sum_{\textbf{B}^{\prime}\in\mathcal{D}^{\prime}}\sum_{\textbf{B}^{\prime\prime}\in\mathcal{D}^{\prime\prime}}\operatornamewithlimits{Res}_{\alpha^{\phi(r^{\prime}),r}=0}\operatornamewithlimits{iBer}_{\textbf{B}^{\prime}}\operatornamewithlimits{iBer}_{\textbf{B}^{\prime\prime}}[w_{\Phi}(x/\widehat{k})^{1-2g}](\widehat{\lambda}/\widehat{k}-[c^{+}]_{\textbf{B}})\,d\alpha^{\phi(r^{\prime}),r},\end{split}$

where $\mathcal{D}^{\prime}$ and $\mathcal{D}^{\prime\prime}$ are the diagonal
bases of $\Phi^{\prime}$ and $\Phi^{\prime\prime}$ (cf. §4.5) correspondingly.

######  Remark 6.12.

Note that this wall-crossing term coincides with the one from Proposition
4.16.

######  Example 6.

It follows from Example 1 that in case of rank 3, the permutation
$\phi\in\Sigma_{3}$ sends $(1,2,3)$ to $(1,3,2)$. Then
$u=c_{1}(\mathcal{F}_{1}^{\prime}\otimes{\mathcal{F}^{\prime\prime}_{1}}^{*})$
and let
$z=z^{\prime\prime}_{1}-z^{\prime\prime}_{2}=c_{1}(\mathcal{F}_{2}^{\prime\prime}/\mathcal{F}_{1}^{\prime\prime}\otimes{\mathcal{F}_{1}^{\prime\prime}}^{*})$.
Then the inverse of the K-theoretical Euler class of the conormal bundle is
(cf. Proposition 6.7)

$ch(\mathcal{L})e^{\frac{9\eta}{2}}e^{\frac{z}{2}}\left(2\mathrm{sinh}\left(\frac{u}{2}\right)2\mathrm{sinh}\left(\frac{z-u}{2}\right)\right)^{1-2g},$

where $\mathcal{L}=\mathcal{L}_{0}(2;0,0)$ is a line bundle on the moduli
space $P_{0}$ of rank-2 degree-0 stable parabolic bundles. The Chern character
of the restriction of the line bundle
$\mathcal{L}_{0}(k;\lambda_{1},\lambda_{2},\lambda_{3})$ to $\Sigma$ is

$e^{\frac{3k\eta}{2}}ch(\mathcal{L}_{0}^{k})e^{\lambda_{1}z+\lambda_{2}u}.$

Hence the wall-crossing term

$\chi(P_{0}(<),\mathcal{L}_{0}(k,\lambda))-\chi(P_{0}(>),\mathcal{L}_{0}(k,\lambda))$

is equal to

$-\left(\frac{3(k+3)}{2}\right)^{g}\operatornamewithlimits{Res}_{u=0}\frac{e^{\lambda_{2}u}}{(2\mathrm{sinh}(\frac{u}{2}))^{2g-1}}\cdot\int\limits_{P_{0}}\frac{ch(\mathcal{L}_{0}(k+1;\lambda_{1}+\frac{1}{2},-\lambda_{1}-\frac{1}{2}))}{(2\mathrm{sinh}(\frac{z-u}{2}))^{2g-1}}\mathrm{Todd}(P_{0})du.$

The integral is the Euler charactersitics of a line bundle on a moduli space
of degree-$0$ rank-$2$ stable parabolic bundles, so we can calculate it using
the induction by rank. It is equal to

$(-1)^{g-1}(2(k+3))^{g}\operatornamewithlimits{Res}_{z=0}\frac{e^{(\lambda_{1}+1)z}}{(2\mathrm{sinh}(\frac{z-u}{2})2\mathrm{sinh}(\frac{z}{2}))^{2g-1}(1-e^{(k+3)z})}dz,$

so the wall-crossing term is

$(-3(k+3)^{2})^{g}\operatornamewithlimits{Res}_{u=0}\operatornamewithlimits{Res}_{z=0}\frac{e^{\lambda_{1}z+\lambda_{2}u+z}}{\tilde{w}_{\phi}(z,u)^{2g-1}(1-e^{(k+3)z})}dzdu,$

where
$\tilde{w}_{\phi}(z,u)=2\mathrm{sinh}(\frac{z-u}{2})2\mathrm{sinh}(\frac{u}{2})2\mathrm{sinh}(\frac{z}{2})$.
Note that this is exactly the same polynomial as in Example 5 after changing
$(z,u)$ to $(x,-y)$.

## 7\. Tautological Hecke correspondences

If $l\neq 0$, then we need one more step in our proof, which uses the Hecke
correspondence to calculate the wall-crossing term (31).

### 7.1. The Hecke correspondence

Given a rank-$r$ degree-$d$ vector bundle $W$ with a full flag $0\subsetneq
F_{1}\subsetneq...\subsetneq F_{r}=W_{p}$ at $p$, one can obtain a rank-$r$
degree-$d-1$ vector bundle $W^{\prime}$ with a full flag $0\subsetneq
G_{1}\subsetneq...\subsetneq G_{r}=W^{\prime}_{p}$ using the tautological
Hecke correspondence construction as follows.

The evaluation map $W\to W_{p}$ induces the short exact sequence of the
associated sheaves of sections

(38) $0\to\mathcal{W}^{\prime}\overset{\tilde{\alpha}}{\to}\mathcal{W}\to
W_{p}/F_{r-1}\to 0$

on curve $C$. Since $\mathcal{W}^{\prime}$ is a kernel of $\tilde{\alpha}$, it
is a locally free sheaf, thus gives a rank-$r$ vector bundle $W^{\prime}$ over
$C$ with $det(W^{\prime})\simeq det(W)\otimes\mathcal{O}(-p)$. The image of
the associated morphism of vector bundles $\alpha$ at the point $p$ is
$F_{r-1}\subset{W}_{p}$, so $\alpha_{p}:W^{\prime}_{p}\to W_{p}$ has a one-
dimensional kernel $G_{1}\subset W^{\prime}_{p}$. Moreover, compositions of
$\alpha_{p}$ with the quotient morphisms $F_{r-1}\to F_{r-1}/F_{i}$ induce a
full flag of the corresponding kernels $G_{1}\subsetneq...\subsetneq
G_{r-1}\subsetneq G_{r}=W^{\prime}_{p}$ in $W^{\prime}_{p}$.

Denote this operator between the sets of isomorphism classes of degree-$d$ and
$d-1$ vector bundles with a flag at $p$ by

$\mathcal{H}:(W,F_{*})\mapsto(W^{\prime},G_{*}).$

Similarly, for any $m\geq 0$, one can define the operator $\mathcal{H}^{m}$
between the sets of isomorphism classes of degree-$d$ and $d-m$ vector bundles
with a flag at the point $p$ by iterating the above construction $m$ times.
Clearly, these maps are independent of the parabolic weights.

######  Proposition 7.1.

Let $c\in\Delta$ be a regular (cf. page 2.2) point. Then the operator
$\mathcal{H}$ induces an isomorphism between the moduli spaces
$P_{d}(c_{1},...,c_{r})$ and $P_{d-1}(c_{2},...,c_{r},c_{1}-1)$.

###### Proof.

First, we need to show that if $W\in P_{d}(c_{1},...,c_{r})$ is a parabolic
stable bundle with parabolic weights $(c_{1},...,c_{r})$, then $W^{\prime}$,
its image under the Hecke operator $\mathcal{H}$, is parabolic stable with
respect to parabolic weights $(c_{2},...,c_{r},c_{1}-1)$. For this, consider
the subbundle $V^{\prime}\subset{W^{\prime}}$ and let
$\alpha(V^{\prime})=V\subset{W}$ (cf. (38)) be its image. Since $W$ is
parabolic stable,

$\textit{par}\textit{slope}(V)<\textit{par}\textit{slope}(W)=\textit{par}\textit{slope}(W^{\prime}).$

We need to prove that
$\textit{par}\textit{slope}(V^{\prime})<\textit{par}\textit{slope}(W^{\prime})$.
There are two possible cases:

  * •

If $\alpha$ maps $V^{\prime}$ to $V$ isomorphically, then
$deg(V^{\prime})=deg(V)$ and $V_{p}\subset{F_{r-1}}$, hence
$\textit{par}\textit{slope}(V^{\prime})=\textit{par}\textit{slope}(V)<\textit{par}\textit{slope}(W^{\prime})$.

  * •

Otherwise, $deg(V^{\prime})=deg(V)-1$, and $V_{p}$ is not contained in
${F_{r-1}}$, so one of the parabolic weights of $V^{\prime}$ is $c_{1}-1$.
Then, as in the previous case,
$\textit{par}\textit{slope}(V^{\prime})=\textit{par}\textit{slope}(V)$, and
the result follows.

To show that the map $\mathcal{H}$ is an isomorphism, note that
$\mathcal{H}^{r}$ maps

(39) $P_{d}(c_{1},c_{2},...,c_{r})\to P_{d-r}(c_{1}-1,c_{2}-1,,...,c_{r}-1).$

It is easy to check that given $\mathcal{W}$ and iterating the associated
morphism of locally free sheaves of sections (38) $r$ times, we obtain a
subsheaf $\mathcal{W}^{\prime}\subset\mathcal{W}$ of sections of $W$ which
vanishes at the point $p$. So the map (39) is just tensoring by
$\mathcal{O}(-p)$, and hence it is an isomorphism. ∎

Now we can define an operator $\mathcal{H}^{m}$ for any $m\in\mathbb{Z}$,
taking the inverse map if necessary. We will need the following statement,
which follows from Proposition 7.1 and the construction of $\mathcal{H}^{m}$.

######  Corollary 7.2.

Let $m\geq 0$. Then under the isomorphism $\mathcal{H}^{m}$ the line bundle
$\mathcal{L}_{d}(k;\lambda_{1},...,\lambda_{r})$ corresponds to the line
bundle
$\mathcal{L}_{d-m}(k;\lambda_{r-m+1},...,\lambda_{r},\lambda_{1}-k,...,\lambda_{r-m}-k)$.

### 7.2. The effect of the Hecke correspondence on the integral

Recall that our goal is to calculate the wall-crossing term from Proposition
6.9. For simplicity, we assume that $l$ is positive (the other case is
analogous). We apply the Hecke operators $\mathcal{H}^{l}$ and
$\mathcal{H}^{-l}$ to the moduli spaces $P_{l}(c^{\prime})$ and
$P_{-l}(c^{\prime\prime})$ to obtain

$P_{0}^{\prime}=P_{0}(c^{\prime}_{l+1},...,c^{\prime}_{r^{\prime}},c^{\prime}_{1}-1,...,c^{\prime}_{l}-1)\simeq
P_{l}(c^{\prime})\text{\, and \,}$
$P_{0}^{\prime\prime}=P_{0}(c^{\prime\prime}_{r^{\prime\prime}-l+1}+1,...,c^{\prime\prime}_{r^{\prime\prime}}+1,c^{\prime\prime}_{1},...,c^{\prime\prime}_{r^{\prime\prime}-l})\simeq
P_{-l}(c^{\prime\prime}).$

Recall (cf. page • ‣ 3.1) that there is a natural action of the group
$\Sigma_{r}$ on $V^{*}$, and hence (cf. page 33) on
$H^{2}(P_{l}(c^{\prime})\times{P_{-l}(c^{\prime\prime})})$. Let
$\tau^{\prime}\in\Sigma_{r^{\prime}}$ and
$\tau^{\prime\prime}\in\Sigma_{r^{\prime\prime}}$ be the cyclic permutations
defined by

$\tau^{\prime}\cdot(c^{\prime}_{1}-1,...,c^{\prime}_{l}-1,c^{\prime}_{l+1},...,c^{\prime}_{r^{\prime}})=(c^{\prime}_{l+1},...,c^{\prime}_{r^{\prime}},c^{\prime}_{1}-1,...,c^{\prime}_{l}-1)$

and

$\tau^{\prime\prime}\cdot(c^{\prime\prime}_{1},...,c^{\prime\prime}_{r^{\prime\prime}-l},c^{\prime\prime}_{r^{\prime\prime}-l+1}+1,...,c^{\prime\prime}_{r}+1)=(c^{\prime\prime}_{r^{\prime\prime}-l+1}+1,...,c^{\prime\prime}_{r^{\prime\prime}}+1,c^{\prime\prime}_{1},...,c^{\prime\prime}_{r^{\prime\prime}-l}).$

And set
$\tau=(\tau^{\prime},\tau^{\prime\prime})\in\Sigma_{r^{\prime}}\times\Sigma_{r^{\prime\prime}}\subset\Sigma_{r}$.
Note that

$\tau^{\prime}\cdot(-l+r^{\prime},...,-l+r^{\prime},-l,...,-l)=\tau^{\prime}\cdot\rho^{\prime}-\rho^{\prime}$

and

$\tau^{\prime\prime}\cdot(l,...,l,l-r^{\prime\prime},...,l-r^{\prime\prime})=\tau^{\prime\prime}\cdot\rho^{\prime\prime}-\rho^{\prime\prime},$

so applying the Hecke operator $\mathcal{H}^{l}\times\mathcal{H}^{-l}$ to the
wall-crossing term from Proposition 6.9 and using Corollary 7.2, we obtain
that the wall-crossing term (31) is equal to

(40) $K\operatornamewithlimits{Res}_{u=0}e^{(k\delta-
rl)u}\int_{P_{0}^{\prime}\times P^{\prime\prime}_{0}}\big{(}\tau\cdot
w_{u}^{\times}(z^{\prime},z^{\prime\prime})^{1-2g}e^{\tau\cdot\rho_{u}^{\times}(z^{\prime},z^{\prime\prime})}\\\
ch(\mathcal{L}_{0}(k+r^{\prime\prime};\tau^{\prime}\cdot(\lambda^{\prime}_{1}-\widehat{k},...,\lambda^{\prime}_{l}-\widehat{k},\lambda^{\prime}_{l+1},...,\lambda^{\prime}_{r^{\prime}-1},\lambda^{\prime}_{r^{\prime}}-k\delta+rl))\\\
ch(\mathcal{L}_{0}(k+r^{\prime};\tau^{\prime\prime}\cdot(\lambda^{\prime\prime}_{1},...,\lambda^{\prime\prime}_{r^{\prime}-l},\lambda^{\prime\prime}_{r^{\prime\prime}-l+1}+\widehat{k},...,\lambda^{\prime\prime}_{r^{\prime\prime}}+\widehat{k}+k\delta-
rl))\\\
e^{\tau^{\prime\prime}\cdot\rho^{\prime\prime}(z^{\prime},z^{\prime\prime})-\rho^{\prime\prime}(z^{\prime},z^{\prime\prime})}e^{\tau^{\prime}\cdot\rho^{\prime}(z^{\prime},z^{\prime\prime})-\rho^{\prime}(z^{\prime},z^{\prime\prime})}\mathrm{Todd}(P_{0}^{\prime})\mathrm{Todd}(P_{0}^{\prime\prime})\big{)}\,du.$

As in §6.3, according to Lemma 6.10, we can calculate this integral using the
induction on rank. Let $\mathcal{D}^{\prime}$ and $\mathcal{D}^{\prime\prime}$
be two Hamiltonian diagonal bases. Then $\tau^{\prime}(\mathcal{D}^{\prime})$
and $\tau^{\prime\prime}(\mathcal{D}^{\prime\prime})$ are also Hamiltonian
diagonal bases (cf. Remark 3.3) and the integral in (40) is equal to

(41)
$(-1)^{lr}\tilde{N}_{r^{\prime},k+r^{\prime\prime}}\tilde{N}_{r^{\prime\prime},k+r^{\prime}}\sum_{\textbf{B}^{\prime}\in\tau^{\prime}(\mathcal{D}^{\prime})}\sum_{\textbf{B}^{\prime\prime}\in\tau^{\prime\prime}(\mathcal{D}^{\prime\prime})}\\\
\operatornamewithlimits{iBer}_{\textbf{B}^{\prime}}\operatornamewithlimits{iBer}_{\textbf{B}^{\prime\prime}}[\tau\cdot
w^{\times}_{u}(x/\widehat{k})^{1-2g}(w_{\Phi^{\prime}}(x/\widehat{k})w_{\Phi^{\prime\prime}}(x/\widehat{k}))^{1-2g}e^{\tau\cdot\rho(x/\widehat{k})}]\\\
(\tau^{\prime}\cdot(\lambda^{\prime}_{1}-\widehat{k},...,\lambda^{\prime}_{l}-\widehat{k},\lambda^{\prime}_{l+1},...,\lambda^{\prime}_{r^{\prime}-1},\lambda^{\prime}_{r^{\prime}}-k{\delta}+rl)/\widehat{k}-\\\
[\tau^{\prime}\cdot(c^{\prime}_{1}-1,...,c^{\prime}_{l}-1,c^{\prime}_{l+1},...,c^{\prime}_{r^{\prime}})]_{\textbf{B}^{\prime}}+\\\
\tau^{\prime\prime}\cdot(\lambda^{\prime\prime}_{1},...,\lambda^{\prime\prime}_{r^{\prime}-l},\lambda^{\prime\prime}_{r^{\prime\prime}-l+1}+\widehat{k},...,\lambda^{\prime\prime}_{r^{\prime\prime}-1}+\widehat{k},\lambda^{\prime\prime}_{r^{\prime\prime}}+\widehat{k}+k\delta-
rl)/\widehat{k}-\\\
[\tau^{\prime\prime}\cdot(c_{1}^{\prime\prime},...,c^{\prime\prime}_{r^{\prime}-l},c^{\prime\prime}_{r^{\prime\prime}-l+1}+1,...,c^{\prime\prime}_{r^{\prime\prime}}+1)]_{\textbf{B}^{\prime\prime}}).$

To arrive at Theorem 6.11, we need to make additional transformations of
formula (41): first, we shift $\lambda^{\prime}$ and $\lambda^{\prime\prime}$
, and then we apply Lemma 4.5 to eliminate the cyclic permutation $\tau$.

Note that given an ordered basis $\textbf{B}\in\mathcal{B}$ and an element
$v\in V^{*}$ such that $\\{v\\}_{\textbf{B}}=0$, for any weight
$\lambda\in{\Lambda}$ and positive integer $k$ one have

(42)
$({\lambda}+\widehat{k}v)/\widehat{k}-[c+v]_{\textbf{B}}={\lambda}/\widehat{k}-[c]_{\textbf{B}}.$

In particular, to perform the shift of $\lambda^{\prime}$ in (41), we use the
following equality for any $\mathbf{B}^{\prime}\in\mathcal{D}^{\prime}$ :

(43)
$(\lambda^{\prime}_{1}-\widehat{k},...,\lambda^{\prime}_{l}-\widehat{k},\lambda^{\prime}_{l+1},...,\lambda^{\prime}_{r^{\prime}-1},\lambda^{\prime}_{r^{\prime}}-k{\delta}+rl)/\widehat{k}-\\\
[(c^{\prime}_{1},...,c^{\prime}_{r^{\prime}-1},c_{r^{\prime}}^{\prime}-l)-(1,...,1,0,...0,-l)]_{\mathbf{B}^{\prime}}=\\\
(\lambda^{\prime}_{1},...,\lambda^{\prime}_{r^{\prime}-1},\lambda^{\prime}_{r^{\prime}}-k{\delta}+rl-l\widehat{k})/\widehat{k}-[(c^{\prime}_{1},...,c^{\prime}_{r^{\prime}-1},c^{\prime}_{r^{\prime}}-l)]_{\mathbf{B}^{\prime}},$

which clearly remains true after changing $\mathcal{D}^{\prime}$ to
$\tau^{\prime}(\mathcal{D}^{\prime})$ and applying $\tau^{\prime}$ to both
sides of the equation. Similarly, shifting the last terms of (41) by
$\tau^{\prime\prime}(0,...,0,-1,...-1,-1+l)$, we can rewrite (41) as

$(-1)^{lr}\tilde{N}_{r^{\prime},k+r^{\prime\prime}}\tilde{N}_{r^{\prime\prime},k+r^{\prime}}\sum_{\textbf{B}^{\prime}\in\tau^{\prime}(\mathcal{D}^{\prime})}\sum_{\textbf{B}^{\prime\prime}\in\tau^{\prime\prime}(\mathcal{D}^{\prime\prime})}\operatornamewithlimits{iBer}_{\textbf{B}^{\prime}}\operatornamewithlimits{iBer}_{\textbf{B}^{\prime\prime}}\\\
[\tau\cdot
w^{\times}_{u}(x/\widehat{k})^{1-2g}(w_{\Phi^{\prime}}(x/\widehat{k})w_{\Phi^{\prime\prime}}(x/\widehat{k}))^{1-2g}e^{\tau\cdot\rho(x/\widehat{k})}]\\\
(\tau^{\prime}\cdot(\lambda^{\prime}_{1},...,\lambda^{\prime}_{r^{\prime}-1},\lambda^{\prime}_{r^{\prime}}-k{\delta}+rl-l\widehat{k})/\widehat{k}-[\tau^{\prime}\cdot(c^{\prime}_{1},...,c^{\prime}_{r^{\prime}-1},c^{\prime}_{r^{\prime}}-l)]_{\textbf{B}^{\prime}}+\\\
\tau^{\prime\prime}\cdot(\lambda^{\prime\prime}_{1},...,\lambda^{\prime\prime}_{r^{\prime\prime}-1},\lambda^{\prime\prime}_{r^{\prime\prime}}+k\delta-
rl+l\widehat{k})/\widehat{k}-[\tau^{\prime\prime}\cdot(c^{\prime\prime}_{1},...,c^{\prime\prime}_{r^{\prime\prime}-1},c^{\prime\prime}_{r"}+l)]_{\textbf{B}^{\prime\prime}}).$

Finally, identifying $u$ (cf. (33)) with the "link" element of the diagonal
basis
$\tau(\mathcal{D})=(\alpha^{\tau\phi(r^{\prime}),\tau(r)}\,\tau^{\prime}(\mathcal{D}^{\prime})\,\tau^{\prime\prime}(\mathcal{D}^{\prime\prime}))$
(cf. §4.5) and

  * •

moving the factor $e^{(k\delta-rl)u}$ from (40) inside the argument of
$\operatornamewithlimits{iBer}_{\textbf{B}}$, where
$\textbf{B}=(\alpha^{\tau\phi(r^{\prime}),\tau(r)}\,{\textbf{B}}^{\prime}\,{\textbf{B}}^{\prime\prime})$,

  * •

applying (42) with
$\textbf{B}=(\alpha^{\tau\phi(r^{\prime}),\tau(r)}\,{\textbf{B}}^{\prime}\,{\textbf{B}}^{\prime\prime})$
and $v=l\alpha^{\tau\phi(r^{\prime}),\tau(r)}$,

  * •

applying Lemma 4.5,

  * •

and using the fact that

$\tau^{-1}\cdot(w_{\Phi^{\prime}}(x/\widehat{k})w_{\Phi^{\prime\prime}}(x/\widehat{k}))=(-1)^{lr}w_{\Phi^{\prime}}(x/\widehat{k})w_{\Phi^{\prime\prime}}(x/\widehat{k}),$

we obtain the formula of Theorem 6.11 for arbitrary $l\in\mathbb{Z}$.

## 8\. Affine Weyl symmetry and the proof of part I of Theorem 4.8

In this section, we prove certain symmetry properties of our Hilbert
polynomials on the left hand side of (1), and we finish the proof of part I of
Theorem 4.8. We start with the basic instance of symmetry of Hilbert
polynomials: relative Serre duality.

### 8.1. Serre duality

######  Proposition 8.1.

Let $\mathcal{E}\to{X}$ be a rank 2 vector bundle over a smooth variety $X$,
$\pi:Y=\mathbb{P}(\mathcal{E})\to X$ its projectivization and $\omega_{X/Y}$
the relative cotangent line bundle. Then

$\chi(Y,\pi^{*}\mathcal{L}\otimes\omega_{X/Y}^{m})=-\chi(Y,\pi^{*}\mathcal{L}\otimes\omega_{X/Y}^{-m+1})$

for any line bundle $\mathcal{L}\in Pic(X)$.

###### Proof.

It follows from the short exact sequence

$0\to\Omega_{X/Y}\otimes\mathcal{O}_{X}\to\pi^{*}\mathcal{E}(-1)\to\mathcal{O}_{X}\to
0,$

that

$\omega_{X/Y}=\wedge^{2}(\pi^{*}\mathcal{E}(-1))=\pi^{*}(\wedge^{2}\mathcal{E})\otimes\mathcal{O}(-2).$

By Serre duality for families of curves [H, Ch III, §7-8] we have then

$\chi(Y,\pi^{*}(\mathcal{L}\otimes(\wedge^{2}\mathcal{E})^{m})\otimes\mathcal{O}(-2m))=\\\
-\chi(Y,\pi^{*}(\mathcal{L}\otimes(\wedge^{2}\mathcal{E})^{m})\otimes\pi^{*}(\wedge^{2}\mathcal{E})^{-2m+1}\otimes\mathcal{O}(2m-2)).$

∎

Now we can iterate this statement to the case of flag bundles.

######  Proposition 8.2.

Let $\pi:Y=\mathrm{Flag}(\mathcal{E})\to X$ be a rank $r$ flag bundle over
$X$. Let $\mathcal{L}$ be a line bundle on $X$, and $\mathcal{F}_{1}$,
$\mathcal{F}_{2}/\mathcal{F}_{1},...,\mathcal{F}_{r}/\mathcal{F}_{r-1}$ the
standard flag line bundles on $Y$. For $k\in\mathbb{Z}$ and
$\lambda=(\lambda_{1},...,\lambda_{r})\in\Lambda$ denote by

$\mathcal{L}{(k;\lambda)}=(\pi^{*}\mathcal{L})^{k}\otimes(\mathcal{F}_{r}/\mathcal{F}_{r-1})^{\lambda_{1}}\otimes(\mathcal{F}_{r-1}/\mathcal{F}_{r-2})^{\lambda_{2}}\otimes...\otimes\mathcal{F}_{1}^{\lambda_{r}}.$

Consider the polynomial

$q(k;\lambda_{1},\lambda_{2},...,\lambda_{r})=\chi(Y,\mathcal{L}{(k;\lambda_{1},\lambda_{2},...,\lambda_{r})})$

in $(k,\lambda)\in\mathbb{Z}\times\Lambda$ and extend this definition to
$\mathbb{R}\times V^{*}$. Then $q(k;\lambda-\rho)$ is anti-invariant under the
permutations of $\lambda_{1},\lambda_{2},...,\lambda_{r}$.

### 8.2. The Weyl anti-symmetry of the functions $q_{1}$ and $q_{-1}$

Armed with this statement, we are ready to take on the symmetries of the
Hilbert polynomial of our parabolic moduli spaces. We note that the two sets
$\Delta_{\pm 1}$ of weights for degree-${\pm 1}$ stable parabolic bundles are
simplices with one of their vertices at $(\frac{1}{r},...,\frac{1}{r})$ and
$(\frac{-1}{r},...,\frac{-1}{r})$, correspondingly (cf. §2.2).

Denote by $N_{\pm 1}$ the moduli spaces of rank-$r$ degree-$\pm 1$ stable
vector bundles and by $UN$ any universal bundle over $N_{\pm 1}\times C$ (cf.
e.g [AB2]).

######  Lemma 8.3.

Let $c=(c_{1},...,c_{r})$ be a parabolic weight from the chamber in
$\Delta_{1}$, which has as one of its vertices the (regular) point
$(\frac{1}{r},...,\frac{1}{r})$. Then the moduli space $P_{1}(c)$ of rank-$r$
degree-$1$ stable parabolic bundles is isomorphic to the flag bundle
$\mathrm{Flag}({UN}_{p})$ over $N_{1}$. An analogous statement holds in the
case of degree $-1$ and the point
$(\frac{-1}{r},...,\frac{-1}{r})\in\Delta_{-1}$.

###### Proof.

A simple calculation shows that the point $(c_{1},...,c_{r})\in\Delta_{1}$,
such that all $c_{i}>0$, lies inside the chamber in $\Delta_{1}$ with the
vertex $(\frac{1}{r},...,\frac{1}{r})$. Hence it is enough to prove the first
statement for the moduli space $P_{1}(c_{1},...,c_{r})$ with positive
parabolic weights.

Moreover, it is sufficient to show that if $(W,F_{*})$ is a parabolic stable
vector bundle which represents a point in $P_{1}(c_{1},...,c_{r})$, then $W$
is stable as an ordinary bundle. Assume that $W$ admits a proper subbundle
$W^{\prime}$ with
$\textit{slope}(W^{\prime})\geq\textit{slope}(W)=\frac{1}{r}$, then
$\mathrm{deg}(W^{\prime})\geq 1$. Since all parabolic weights of $W$ are
positive, this implies that
$\textit{parslope}(W^{\prime})>0=\textit{parslope}(W)$, and therefore $W$ is
parabolic unstable. The proof for degree-($-1$) bundles is analogous. ∎

Denote the moduli spaces described above by $P_{1}(>)$ and $P_{-1}(<)$,
correspondingly, and their images under the Hecke isomorphisms $\mathcal{H}$
and $\mathcal{H}^{-1}$ by $P_{0}(>)$ and $P_{0}(<)$.

The following statement is straightforward (cf. Lemma 2.8).

######  Lemma 8.4.

The line bundles $\mathcal{L}_{1}(r;1,...,1)$ and
$\mathcal{L}_{-1}(r;-1,...,-1)$ on $P_{1}(>)$ and $P_{-1}(<)$ defined in Lemma
2.6 may be obtained as pullbacks of the ample generators of the Picard groups
$\mathrm{Pic}(N_{\pm 1})$.

######  Example 7.

In case of rank-3 parabolic bundles the moduli space
$P_{1}(c_{1},c_{2},c_{3})$ with $2c_{3}>c_{1}+c_{2}-1$ is a flag bundle over
$N_{1}$ and it is isomorphic to the moduli space $P_{0}(>)$ from Example 1,
while the moduli space $P_{-1}(c_{1},c_{2},c_{3})$ with $2c_{1}<c_{2}+c_{3}+1$
is a flag bundle over $N_{-1}$ and it is isomorphic to $P_{0}(<)$.

Now we establish the Weyl anti-symmetry of the polynomials

$q_{-1}(k;\lambda_{1},...,\lambda_{r})=\chi(P_{0}(<),\mathcal{L}_{0}(k;\lambda_{1},...,\lambda_{r}))$

and

$q_{1}(k;\lambda_{1},...,\lambda_{r})=\chi(P_{0}(>),\mathcal{L}_{0}(k;\lambda_{1},...,\lambda_{r}))$

defined on $\mathbb{R}\times\Lambda$, as in Proposition 8.2. Let
$\tau\in\Sigma_{r}$ be the cyclic permutation, such that
$\tau\cdot(c_{1},...,c_{r})=(c_{2},...,c_{r},c_{1})$, and consider two points
in $V^{*}$:

$\theta_{1}[k]=\frac{k+r}{r}\cdot(1,1,\dots,1)-(k+r)x_{r}-\rho=\tau\cdot(\frac{k}{r}-k,\frac{k}{r},...,\frac{k}{r})-\tau\cdot(\rho)=\\\
\left(\frac{k}{r}-\frac{r-1}{2}+1,\frac{k}{r}-\frac{r-1}{2}+2,....,\frac{k}{r}-\frac{r-1}{2}+r-1,-k+\frac{k}{r}-\frac{r-1}{2}\right)$

and

$\theta_{-1}[k]=-\frac{k+r}{r}\cdot(1,1,\dots,1)+(k+r)x_{1}-\rho=\\\
\tau^{-1}\cdot(-\frac{k}{r},...,-\frac{k}{r},-\frac{k}{r}+k)-\tau^{-1}\cdot(\rho)=\\\
\left(k-\frac{k}{r}+\frac{r-1}{2},-\frac{k}{r}-\frac{r-1}{2},-\frac{k}{r}-\frac{r-1}{2}+1,...,-\frac{k}{r}-\frac{r-1}{2}+r-2\right).$

######  Proposition 8.5.

The polynomials $q_{1}(k;\lambda+\theta_{1}[k])$ and
$q_{-1}(k;\lambda+\theta_{-1}[k])$ are anti-invariant under the action of the
Weyl group by permutations of $\lambda_{1},...,\lambda_{r}$.

###### Proof.

Recall that the moduli space $P_{0}(>)$ is isomorphic to the flag bundle
$P_{1}(>)$ over $N_{1}$ under the Hecke isomorphism $\mathcal{H}^{-1}$. Then
using Corollary 7.2, Proposition 8.2 and Lemma 8.4, for any permutation
$\sigma\in\Sigma_{r}$ we obtain

$q_{1}(k;\sigma\cdot\lambda+\theta_{1}[k])\overset{\mathrm{def}}{=}\chi(P_{0}(>),\mathcal{L}_{0}(k;\sigma\cdot\lambda+\theta_{1}[k]))\overset{{\ref{LineHecke}}}{=}\\\
\chi(P_{1}(>),\mathcal{L}_{1}(k;\tau^{-1}\cdot\sigma\cdot\lambda+(\frac{k}{r},...,\frac{k}{r})-\rho))\overset{{\ref{Serre}},\ref{PicN}}{=}\\\
(-1)^{\sigma}\chi(P_{1}(>),\mathcal{L}_{1}(k;\tau^{-1}\cdot\lambda+(\frac{k}{r},...,\frac{k}{r})-\rho))\overset{{\ref{LineHecke}}}{=}\\\
(-1)^{\sigma}\chi(P_{0}(>),\mathcal{L}_{0}(k;\lambda+\theta_{1}[k]))\overset{\mathrm{def}}{=}(-1)^{\sigma}q_{1}(k;\lambda+\theta_{1}[k]).$

The proof for $q_{-1}$ is similar. ∎

The two group actions in Proposition 8.5 may be combined in the following
manner. For $k\geq 0$, we define an action of the affine Weyl group
$\Sigma\rtimes\Lambda$ on $\Lambda\times\mathbb{Z}$, which acts trivially on
the second factor, the level, and the action at level $k$ is given by setting

$\sigma.\lambda=\sigma\cdot(\lambda+\rho)-\rho\text{\,\,\, and
\,\,\,}\gamma.\lambda=\lambda+(k+r)\gamma\text{ for
}\sigma\in\Sigma,\,\gamma\in\Lambda.$

We denote the resulting group of affine-linear transformations of $V^{*}$ by
$\widetilde{\Sigma}[{k}]$, and note that the action is defined in such a way
that

(44) $\sigma.\lambda+\rho=\sigma\cdot(\lambda+\rho)\text{ and
}(\gamma.\lambda+\rho)/\widehat{k}=\gamma+(\lambda+\rho)/\widehat{k}$

It is easy to verify that the stabilizer subgroup

$\Sigma_{r}^{+}\overset{\mathrm{def}}{=}\mathrm{Stab}(\theta_{1}[k],\widetilde{\Sigma}[{k}])\subset\widetilde{\Sigma}[{k}]$

is generated by the transpositions $s_{i,i+1},\;1\leq i\leq r-2$ and the
reflection $\alpha^{r-1,r}\circ s_{r-1,r};$

similarly,

$\Sigma_{r}^{-}\overset{\mathrm{def}}{=}\mathrm{Stab}(\theta_{-1}[k],\widetilde{\Sigma}[{k}])\subset\widetilde{\Sigma}[{k}]$

is generated by $s_{i,i+1}$, $2\leq i\leq r-1$ and the reflection
$\alpha^{1,2}\circ s_{1,2}$.

Then Proposition 8.5 maybe recast in the following form: the polynomial
$q_{1}(k;\lambda)$ is anti-invariant with respect to the copy $\Sigma_{r}^{+}$
of the symmetric group $\Sigma_{r}$, while $q_{-1}(k;\lambda)$ is anti-
invariant with respect to the copy $\Sigma_{r}^{-}$ of the symmetric group
$\Sigma_{r}$.

The following statement is straightforward:

######  Lemma 8.6.

Both subgroups $\Sigma_{r}^{\pm}$ are isomorphic to $\Sigma_{r}$ and for
$r>2$, the two subgroups generate the affine Weyl group
$\widetilde{\Sigma}[{k}]$.

### 8.3. The Weyl anti-symmetry of the polynomials $p_{1}$ and $p_{-1}$

Following (21), we define the two polynomials

$p_{\pm
1}(k;\lambda)=\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[w_{\Phi}^{1-2g}(x/\widehat{k})](\widehat{\lambda}/\widehat{k}-[\theta_{\pm
1}]_{\mathbf{B}}),$

where $\theta_{1}=\frac{1}{r}\cdot(1,1,\dots,1)-x_{r}$, and
$\theta_{-1}=-\frac{1}{r}\cdot(1,1,\dots,1)+x_{1}$.

######  Proposition 8.7.

The polynomial $p_{1}(k;\lambda)$ is anti-invariant with respect to
$\Sigma_{r}^{+}$, and $p_{-1}(k;\lambda)$ is anti-invariant with respect to
$\Sigma_{r}^{-}$.

###### Proof.

We recall that the points $\theta_{\pm 1}[k]$ are the fixed points of the
actions of $\Sigma^{\pm}$, and clearly $\lim_{k\to\infty}\theta_{\pm
1}[k]/k=\theta_{\pm 1}$. This means, that we can fix a small open ball
$D\subset V^{*}$ centered at $\theta_{1}$ such that

(45) $\lambda/k\in
D\Longrightarrow\forall\sigma\in\Sigma^{+}:\;(\sigma.\lambda+\rho)/\widehat{k}\sim\theta_{1}.$

Then for $\lambda/k\in D$ we have

$p_{1}(k;\lambda)=\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[w_{\Phi}^{1-2g}(x/\widehat{k})](\\{\widehat{\lambda}/\widehat{k}\\}_{\mathbf{B}}).$

Now, let us consider a generator of $\Sigma^{+}$ of the type
$\sigma=s_{i,i+1}$, $1\leq i\leq r-2$. Using (44), and Lemma 4.5, and the fact
that $\sigma\cdot w_{\Phi}=-w_{\Phi}$ we obtain

$p_{1}(k;\sigma.\lambda)=\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[w_{\Phi}^{1-2g}(x/\widehat{k})](\sigma\cdot\\{\widehat{\lambda}/\widehat{k}\\}_{\mathbf{B}})=\\\
\sum_{\mathbf{B}\in\mathcal{D}}\operatornamewithlimits{iBer}_{\mathbf{B}}[(-w_{\phi})^{1-2g}(x/\widehat{k})](\\{\widehat{\lambda}/\widehat{k}\\}_{\mathbf{B}})=-p_{1}(k;\lambda)$

The case of the last generator $\alpha^{r-1,r}\circ s_{r-1,r}$ is similar, but
after the substitution, we need to use the equality
$\\{\alpha^{r-1,r}+\widehat{\lambda}/\widehat{k}\\}_{\mathbf{B}}=\\{\widehat{\lambda}/\widehat{k}\\}_{\mathbf{B}}$
to obtain
$p_{1}(k;\widehat{k}\alpha^{r-1,r}+s_{r-1,r}.\lambda)=-p_{1}(k;\lambda)$. ∎

### 8.4. Proof of part I. of Theorem 4.8

Recall that in Lemma 4.1, we introduced a chamber structure on $\Delta\subset
V^{*}$ created by the walls $S_{\Pi,l}$, where
$\Pi=(\Pi^{\prime},\Pi^{\prime\prime})$ is a nontrivial partition, and
$l\in\mathbb{Z}$. Before we proceed, we introduce some extra notation. Denote
by

$\widecheck{\Delta}=\\{(k;a)|\,a/k\in\Delta\\}\subset\mathbb{R}^{>0}\times
V^{*}$

the cone over $\Delta\subset V^{*}$, and let

$\widecheck{\Delta}^{\mathrm{reg}}=\\{(k;a)|\,a/k\in\Delta\,\text{is
regular}\\}\subset\widecheck{\Delta}$

be the set of its regular points. Denote by
$\widecheck{S}_{\Pi,l}\subset\widecheck{\Delta}$ the cone over the wall
$S_{\Pi,l}\subset\Delta$; then $\widecheck{\Delta}^{\mathrm{reg}}$ is the
complement of the union of walls $\widecheck{S}_{\Pi,l}$ in
$\widecheck{\Delta}$. Finally, denote by
$\widecheck{\Delta}^{\mathrm{reg}}_{\Lambda}$ the intersection of the lattice
$\mathbb{Z}^{>0}\times\Lambda$ with $\widecheck{\Delta}^{\mathrm{reg}}$.

By substituting $\textit{\k{c}}=\lambda/k$, we can consider the left-hand side
and the right-hand side of formula I. of Theorem 4.8 as functions in
$(k,\lambda)\in\widecheck{\Delta}^{\mathrm{reg}}_{\Lambda}$. We denote by
$q(k;\lambda)$ and $p(k;\lambda)$ the left-hand side and the right-hand side,
correspondingly.

We showed that $q(k;\lambda)$ and $p(k;\lambda)$ are polynomials on the cone
over each chamber in $\Delta$ (cf. Theorem 4.4, §2.4). We proved that the
wall-crossing terms, i.e. the differences between polynomials on neighbouring
chambers, for $q(k;\lambda)$ (cf. Theorem 6.11) and for $p(k;\lambda)$ (cf.
Proposition 4.16) coincide, hence there exists a polynomial
$\Theta(k;\lambda)$ on $\mathbb{Z}^{>0}\times\Lambda$, such that the
restriction of $\Theta(k;\lambda)$ to
$\widecheck{\Delta}^{\mathrm{reg}}_{\Lambda}$ is equal to the difference
$p(k;\lambda)-q(k;\lambda)$.

Now for $r>2$, we can conclude that

$\Theta(k;\lambda)=p_{1}(k;\lambda)-q_{1}(k;\lambda)=p_{-1}(k;\lambda)-q_{-1}(k;\lambda),$

where $p_{\pm 1}(k;\lambda)$ and $q_{\pm 1}(k;\lambda)$ are the restrictions
of $p(k;\lambda)$ and $q(k;\lambda)$ to two specific chambers defined in §8.3
and §8.2. Then, according to Propositions 8.5 and 8.7, the polynomial
$\Theta(k;\lambda)$ is anti-invariant with respect to the action of the
subgroups $\Sigma^{\pm}_{r}$, and hence by Lemma 8.6, it is anti-invariant
under the action of the entire affine Weyl group $\widetilde{\Sigma}[{k}]$. It
is easy to see that any such polynomial function has to vanish, and thus
$p(k;\lambda)=q(k;\lambda)$, and this completes the proof of part I. of
Theorem 4.8 for the case when $\lambda/k\in\Delta$ is regular.

As in Corollary 4.10, we can extend $p(k;\lambda)$ from the interior of each
chamber to its boundary by polynomiality. Clearly, to prove part I. of Theorem
4.8 for the cases when $\lambda/k$ is not regular, it is sufficient to show,
that these extensions from the chambers containing $\lambda/k$ in their
closure give the same value on $(k;\lambda)$. It follows from Remark 10.4,
that this is the case, and this completes the proof of part I. of Theorem 4.8
(cf. Remark 4.9).

## 9\. Rank 2, two points

Unfortunately, the argument above does not work for $r=2$, because, in this
case, $\theta_{1}[k]=\theta_{-1}[k]$, the groups $\Sigma_{r}^{-}$ and
$\Sigma_{r}^{+}$ coincide, and thus they do not generate the entire affine
Weyl group. The way out is to pass to the 2-punctured case.

### 9.1. Wall-crossing

We will thus fix two points: $p,s\in C$, and study the moduli space of rank-2,
stable parabolic bundles $W$ with fixed determinant isomorphic to
$\mathcal{O}(pd)$, with parabolic structure given by a line $F_{1}\subset
W_{p}$ with weight $(c,-c)$, and a line $G_{1}\subset W_{s}$ with weight
$(a,-a)$.

Now we need to repeat the analysis of our work so far in this somewhat simpler
case; some details thus will be omitted.

Set $d=0$; then the space of admissible weights (cf. Figure 6) is a square

$\Box=\\{(c,a)\,|\,1>2c>0,\,1>2a>0,\,\\},$

which has two adjacent chambers defined by the conditions

$c>a\text{ \, and \, }c<a.$

Denote the corresponding moduli spaces by $P_{0}(c>a)$ and $P_{0}(c<a)$.

$P_{0}(c<a)$$P_{0}(c>a)$$(0,0)$$(\frac{1}{2},0)$$(0,\frac{1}{2})$$(\frac{1}{2},\frac{1}{2})$
Figure 6. The space of admissible weights in the case of rank $r=2$, two
points.

Again, we have universal bundles over $P_{0}(c>a)\times C$ and
$P_{0}(c<a)\times C$, which we will denote by the same symbol $U$; this bundle
is endowed with two flags, $\mathcal{F}_{1}\subset\mathcal{F}_{2}=U_{p}$ and
$\mathcal{G}_{1}\subset\mathcal{G}_{2}=U_{s}$. For $\mu,\lambda\in\mathbb{Z}$,
we introduce the line bundle

$\begin{split}\mathcal{L}(k;\lambda,{\mu})=&\mathrm{det}({U}_{p})^{k(1-g)}\otimes\mathrm{det}(\pi_{*}(U))^{-k}\\\
&\otimes(\mathcal{F}_{2}/\mathcal{F}_{1})^{\lambda}\otimes(\mathcal{F}_{1})^{-\lambda}\otimes(\mathcal{G}_{2}/\mathcal{G}_{1})^{\mu}\otimes(\mathcal{G}_{1})^{-\mu}.\end{split}$

We repeat the construction of the master space from Section 5.1, choosing a
point $(c^{0},c^{0})$ on the wall and two points

$(c,a)^{\pm}=(c^{0},c^{0})\pm\epsilon(1,0)\in\Box,\,\,\,\epsilon\in\mathbb{Q}_{>0}$

from the adjacent chambers. We can identify the fixed point set $Z^{0}$ as
follows.

######  Lemma 9.1.

The locus $Z^{0}$ defined in Proposition 5.2 is

$Z^{0}\simeq Jac^{0}\simeq\\{V=L\oplus{L^{-1}}\,|\,L_{p}\subset
F_{1},L_{s}^{-1}\subset G_{1}\\}.$

As in §6.1, denote by $\mathcal{J}$ the universal bundle over $Jac^{0}\times
C$ normalized in such a way that $c_{1}(\mathcal{J})_{(0)}=0$ (cf. (7)).
Define

$\eta\in H^{2}(Jac)\;\text{\,\, by\, \,