The parabolic Verlinde formula: iterated residues and wall-crossings

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.


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 ě 1, and fix an auxiliary point p P C. We will call a vector c " pc 1 ą c 2 ą ¨¨¨ą c r q P R r satisfying ř c i " 0 and c 1 ´cr ă 1 regular if no nontrivial subset of its coordinates sums to an integer.For such a c P R r , there exists a smooth projective moduli space P 0 pcq ( [Se, MS, B]), whose points are in one-to-one correspondence with the equivalence classes of pairs pW, F ˚q, where W Ñ 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 1 Ă W, the degree degpW 1 q is strictly smaller than the sum of a subset of the coordinates of c depending on the position of W 1 p with respect to F ˚.
There is a natural way to associate to a positive integer k and an integer vector λ P Z r satisfying λ 1 `¨¨¨`λ r " 0 a line bundle Lpk; λq on P 0 pcq, in such a way that if c " λ{k, then Lpk; λq is ample.The parabolic Verlinde formula is the following expression for the Euler characteristic of the ample line bundle Lpk; λq: assume c " λ{k is regular; then (1) χpP 0 pcq, Lpk; λqq " N r,k ¨ÿ p´iq p r 2 q expp2πi p λ ¨xq ś iăj `2 sin πpx i ´xj q ˘2g´1 where N r,k " rprpk `rq r´1 q g´1 , p λ " λ `1 2 pr ´1, r ´3, . . ., 1 ´rq, and the sum is taken over the finite set of those points in the interior of the parallelopiped tx " px 1 , x 2 , . . ., x r " 0q| 0 ă x i ´xi`1 ă 1 for i " 1, . . ., r ´1u which satisfy the conditions ‚ pk `rqx P Z r ‚ x i ´xj R Z for 1 ď i ă j ă r. 1 We note that this finite set is a set of lattice points in the interior of pr ´1q! identical simplices (cf. the rhombus on Figure 1).
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 Verpk, λq " N 3,k ÿ 0ăn 2 ăn 1 ăk`3 2i sin 2π where n 1 , n 2 are integers.Using Theorem 4.7 and Remark 4.6 one can show that Verpk, λq " # p `pk; λq, λ 2 ą 0, p ´pk; λq, λ 2 ă 0, where p `and p ´are two polynomials, given by the right hand sides of the expressions of Example 4 on page 17.We note two properties of p ˘pk, λq: A. The wall-crossing difference p ´´p `has a relatively simple form (cf. Example 5 with λ 1 `λ3 replaced by ´λ2 ): p ´pk; λq ´p`p k; λq " Res y"0

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 pcq remains unchanged as long as c 2 does not change sign.Hence, effectively, we have two moduli spaces P 0 pąq and P 0 păq, corresponding to the two chambers separated by the red (c 2 " 0) line in Figure 1.Introduce the notation q `pk; λq " χpP 0 pąq, Lpk; λqq, q ´pk; λq " χpP 0 păq, Lpk; λqq 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 31), and the result is (4) q ´pk; λq ´q`p k; λq " Res u"0

Res
z"0 p´3pk `3q 2 q g ¨eλ 1 z`λ 2 u`z ´wΦ pz, ´uq 2g´1 p1 ´epk`3qz q dzdu, where u plays the role of the equivariant parameter, the generator of H C˚p ptq.This iterated residue coincides with the expression above after changing pz, uq to px, ´yq, 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 pąq and P 0 păq over the moduli spaces of stable bundles (without parabolic structure) of degrees 1 and ´1: where the fibers are full flags of 3-dimensional vector spaces.Serre duality applied to a Flag 3 -bundle implies a Σ 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 ˘as we did for the polynomials p ˘: q `pk; λq satisfies (2), while q ´pk; λq 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 Θpk; λq for this polynomial.Then Θ 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 Θpk; λq vanishes and this completes the proof.

Contents of the paper
There are a number of complications which arise when r ą 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 ρ-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 Weylsymmetries of the polynomials q ˘, 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.

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 ρ-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.

Definitions
Let C be a smooth complex projective curve of genus g ě 2, and fix a point p P C.
‚ A parabolic bundle on C is a vector bundle W of rank r with a full flag F ˚in the fiber over p: and parabolic weights c " pc 1 , ..., c r q assigned to F r , F r´1 , ..., F 1 , satisfying the conditions c 1 ą c 2 ą ... ą c r and c 1 ´cr ă 1. ‚ The parabolic degree 1 and the parabolic slope of W are defined as pardegpWq " degpWq ´r ÿ i"1 In particular, an endomorphism of a parabolic bundle W is a vector bundle endomorphism preserving the flag F ˚.
1 For technical reasons, we have chosen a sign convention opposite to that in the majority of treatments in the literature.
‚ Denote by ParHompW, W 1 q the sheaf of parabolic morphisms from W to W 1 .Then there is a short exact sequence of sheaves where T p is a torsion sheaf supported at p.The rank of T p is the number of pairs pi, jq, s.t.c i ă c 1 j (cf.[BH]).If W 1 Ă W is a subbundle of W, then both W 1 and the quotient W{W 1 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 1 Ă W satisfies parslopepW 1 q ă parslopepWq; 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 .

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 Definition 2.2.We will call a vector c " pc 1 , . . ., c r q P R r such that ř i c i P Z regular if for any nontrivial subset Ψ Ă t1, 2, . . ., ru, we have ř iPΨ c i R Z. Now choose an integer d " 0 such that H 1 pWq " 0 and W is generated by global sections for any rank-r degree-d semistable parabolic bundle W of parabolic degree 0. Put χ " rp1 ´gq `d and consider the ‚ Groethendieck quot scheme Quotpχ, rq ( [G]) parametrizing quotients O χ ։ 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 " PSLpχq, which does not, however, lift to UQ. ‚ Let LFQuot Ă Quotpχ, rq be the open subscheme consisting of locally free quotients W, such that the induced map H 0 pO χ q Ñ H 0 pWq is an isomorphism.‚ Denote by XQ the total space of the flag bundle FlagpUQ p q on LFQuot ˆp.
This space is endowed with the flag of vector bundles Fl 1 Ă ¨¨¨Ă Fl r´1 Ă Fl r " UQ p .‚ Let k P Z and pλ 1 , ..., λ r q P Z r , such that ř r i"1 λ i " kd, and consider the line bundle Lpk; λq " detpUQ p q kp1´gq b detpπ ˚UQq ´k b pFl r {Fl r´1 q λ 1 b ... b pFl 1 q λ r on XQ, which does carry a G-linearization (lift of the G-action from XQ). ‚ Finally, assume c P ∆ d is regular (cf.Definition 2.2 above) and define r P d pcq, the moduli space of stable parabolic weight-c vector bundles on C as the GIT quotient XQ c G of XQ with respect to any linearization Lpk; λq, such that λ{k " c.

Theorem 2.3 ([Se]
). Assume that c P ∆ d is a regular weight vector.Then the moduli space r P d pcq is a smooth projective variety of dimension r 2 pg ´1q ``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 r P d pcq fibers over the Jacobian of degree-d line bundles with isomorphic fibers, and in this paper, we will focus on the moduli space which is smooth, projective and has dimension pr 2 ´1qpg ´1q ``r 2 ˘.
Remark 2.5.Note that tensoring with the line bundle Opmpq induces an isomorphism: bOpmpq : P d pcq Ñ P d`rm pcq, so the moduli spaces P d pcq, essentially, depend only on d modulo r.

The Picard group of P d pcq
For a regular c P ∆ d , there exist universal bundles U over P d pcq ˆC endowed with a flag F 1 Ă ¨¨¨Ă F r´1 Ă F r " U p , and satisfying the obvious tautological properties.In general, such universal bundles U, and hence the flag line bundles F i`1 {F i are unique only up to tensoring by the pull-back of a line bundle from P d pcq.Nevertheless, we have the following statement, which is easy to verify.Lemma 2.6.For k P Z and λ " pλ 1 , ..., λ r q P Z r , such that ř r i"1 λ i " kd, the line bundle L d pk; λq " detpU p q kp1´gq b detpπ ˚Uq ´k b pF r {F r´1 q λ 1 b ... b pF 1 q λ r on P d pcq is independent of the choice of the universal bundle U.
Remark 2.7.The line bundle Lpk; λq defined in §2.2 descends to the line bundle L d pk; λq on the GIT quotient P d pcq.
Notation: We will say that U is normalized if the line subbundle F 1 Ă U p is trivial.The parameter k is often called the level.
Let ω P H 2 pCq be the fundamental class of our curve C, and e 1 , ..., e 2g a basis of H 1 pCq, such that e i e i`g " ω for 1 ď i ď g, and all other intersection numbers e i e j equal 0. For a class δ P H ˚pP ˆCq of a product, we introduce the following notation for its Künneth components (cf.[W]): Later, we will need the following formula, which can be proved by a straightforward calculation.

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 F ˚of the fiber W p , and let us try to determine the structure of the set of parabolic weights c P ∆ d for which W is stable.
Clearly, for this we need to study the set of parabolic weights c " pc 1 , c 2 , . . .c r q for which one can find a proper subbundle W 1 Ă W such that (8) parslopepW 1 q " parslopepWq " 0.
A subbundle W 1 Ă W determines a short exact sequence of parabolic bundles and the position of W 1 p with respect to F ˚gives rise to a nontrivial partition of the set t1, 2, . . ., ru into two sets, Π 1 and Π 2 (cf.[MS], definition 1.7); the parabolic weights of W 1 and W 2 are then c 1 " pc i q iPΠ 1 and c 2 " pc i q iPΠ 2 , correspondingly.The slope condition (8) translates into a pair of equivalent equalities: where d 1 , d 2 " d ´d1 are the degrees of W 1 and W 2 , respectively.This means that the critical values of c P ∆ d for which (8) is possible lie on the union of affine hyperplanes (or walls) defined by the equations ÿ iPΠ 1 c i " l, where l P Z, and Π 1 Ă t1, 2, . . ., ru nontrivial.
As only finitely many of these walls intersect the simplex ∆ 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 pcq does not change.

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.

Notation
We begin by setting up some extra notation for the space of parabolic weights introduced in §2.1.
‚ Let V " R r {Rp1, 1, . . ., 1q be the r ´1-dimensional vector space, obtained as the quotient of R r .The dual space V ˚is then naturally represented as V ˚" ta " pa 1 , . . ., a r q P R r | a 1 `¨¨¨`a r " 0u.
Let x 1 , x 2 , . . ., x r be the coordinates on R r ; given a P V ˚, we will write xa, xy for the linear function ř 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 Λ of full rank: In particular, for 1 ď i ‰ j ď r, we can define the element α ij " x i ´xj in Λ. ‚ Our arrangement is the set of hyperplanes tx i " x j u Ă V, 1 ď i ă j ď 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: Note that V ˚carries a natural action of the permutation group Σ r , permuting the coordinates x j , j " 1, . . ., r, and this action restricts to an action on Φ as well.‚ The basic object of the theory is an ordered linear basis B of V ˚consisting of the elements of Φ.Let us denote the set of these objects by B: ‚ For B P B, we will write FlpBq for the full flag " V ˚" xβ r1s , β r2s , . . ., β rr´1s y lin , . . ., xβ rr´1s , β rr´2s y lin , xβ rr´1s y lin ı , where x¨y lin stands for linear span.

Diagonal bases
Definition 3.1.‚ For τ P Σ r´1 and B P B, we will write B ö τ for the permuted sequence pβ rτp1qs , β rτp2qs , . . ., β rτpr´1qs q. ‚ For two elements B, C P B we will write B % C if for any τ P Σ r´1 , we have FlpB ö τq ‰ FlpCq.‚ A subset D Ă B of pr ´1q! elements is called a diagonal basis if for any two different elements B, C P D, we have B % C.
Remark 3.2.This definition is motivated by a construction [Sz1], which associates to each diagonal basis D a pair of dual bases of the middle homology and the cohomology of the complexified hyperplane arrangement on V b R C defined by Φ.The dimension of these (co)homology spaces is pr ´1q!.

Combinatorial interpretation
This notion has the following purely combinatorial form.
‚ We can think of Φ as the edges of the complete graph on r vertices.
‚ Then the set 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 for this ordered tree.‚ In this language, the flag FlpBq 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 TreepBq, the jth partition being the one induced by the first j ´1 edges.For example, the ordered tree rp2, 4qp1, 3q, p1, 2qs induces the same sequence of partitions as rp1, 4q, p2, 3q, p1, 2qs (see Figure 3.3) ‚ A diagonal basis D is then a set of pr ´1q! ordered trees such that the pr ´1q! partition sequences obtained by reordering the edges of any one of the ordered trees are different from pr ´1q! ´1 sequences of partitions obtained from the remaining elements of D.
The set H m " tσpBq| σ P Σ r , σp1q " mu 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.
Remark 3.3.The hyperplane arrangement induced by Φ is invariant under the natural action of Σ r on the vector space V.It follows easily from the definition that if D is a diagonal basis and σ P Σ r is a permutation, then σpDq is also a diagonal basis.

The residue formula and the main result
In this section, we recall the residue formula from [Sz1] for Verpk, λq, 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 Verpk, λq, 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.

The residue formula in dimension 1
The story begins with the Fourier series for m ě 2, which is a periodic, piecewise polynomial function given by the formula Res where tau is the fractional part of the real number a.The polynomial functions thus obtained on the interval r0, 1s are called Bernoulli polynomials.The polynomial on the interval containing the real number c P RzZ is given by Res where rcs 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 fpxq " p2 sinhpx{2qq ´2m , and introduce an integer parameter λ related to a via ka " λ.We consider the sum of values of the function f over a finite set of rational points in analogy with (11): where λ, k P Z.This sum is again periodic in λ mod k, and for m ě 2 we can evaluate it via the residue theorem as Again, this is a piecewise polynomial function in the pair pk, λq, which is polynomial in the cones bounded by the lines λ " qk, q P Z.
Note that in these calculations, a key role is played by the Bernoulli operator: which transforms meromorphic functions in the variable x into polynomials in a, and plays the role of a generalized Fourier operator.

The multidimensional case
Now we return to the setup of §3 with the vector space V endowed with the hyperplane arrangement Φ.We introduce the notation F Φ for the space of meromorphic functions defined in a neighborhood of 0 in V b R C with poles on the union of hyperplanes ď 1ďiăjďr tx| xα ij , xy " 0u.
In particular, the function To write down our residue formula, we need a multidimensional generalization of the notions of integer and fractional parts.Given a basis B " pβ r1s , . . ., β rr´1s q P B of V ˚, and an element a P V ˚, we define ras B and tau B to be the unique elements of V ˚satisfying ‚ ras B " a ´tau B P Λ, and ‚ tau B P ř r´1 j"1 r0, 1qβ rjs .This notion naturally induces a chamber structure on V ˚: we will call a P V regular if a is a point of continuity for the functions a Þ Ñ ras B , tau B for all B P B, i.e. when tau B P ř r´1 j"1 p0, 1qβ rjs .Now, for regular a and b we define the equivalence relation (13) a " b when ras B " rbs B @B P B.
The equivalence classes for this relation form a Λ-periodic system of chambers in V ˚.
Convention: We will think of a partition Π of t1, 2, . . ., ru into two nonempty sets as an ordered partition Π " pΠ 1 , Π 2 q such that r P Π 2 , and we will call these objects nontrivial partitions for short.
Lemma 4.1.The equivalence classes of the relation " are precisely the chambers in V created by the walls parameterized by a nontrivial partition Π " pΠ 1 , Π 2 q of the first r positive integers, and an integer 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 pcq 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 " defined above with the open simplex ∆ 0 def " ∆, where the parabolic weights live (cf.Fig- ures 2 and 4).We will use the term "chamber" in both cases if this causes no confusion.
(1,0,-1) where the naturally oriented cycle Z B is defined by 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

Invariance of diagonal bases and the main results
Diagonal bases have the following key invariance property.
Theorem 4.4 ([Sz1]).Let f P F Φ , and c P V ˚be regular; let D be a diagonal basis of Φ.
Then the functional (cf.(15) above) transforming a meromorphic function f P F Φ into a polynomial in the variable a P V ˚is independent of the choice of the diagonal basis D. In particular, for regular a P V ˚, the functional 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 σ P Σ r , we define, as usual (17) σ ¨fpxq " fpσ ´1xq.
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 Verpk, λq the finite sum on the right hand side of (1).
Theorem 4.7.Let g ě 1, k P Z ą0 , λ P Λ, and let D be any diagonal basis of Φ. Introducing the notation p k " k `r, and p λ " λ `ρ, we have where Ñr,k " p´1q p r 2 qpg´1q N r,k (cf.(1)) and p c ˛P V ˚is a regular point in a chamber that contains p λ{ p k in its closure.Now, if we look at our main goal (1): proving the equality (19) Verpk, λq " χpP 0 pλ{kq, L 0 pk; λqq, then we discover a rather embarrassing mismatch.Both sides are piecewise polynomial functions, however, ‚ according to the HRR theorem, χpPpλ{kq, L 0 pk; λqq is polynomial on the cones over the the equivalence classes (cf.( 13)) of λ{k, while ‚ according to (18), Verpk, λq is polynomial on the cones over the equivalence classes of p λ{ p k, and these conic partitions of tpk, λq| λ{k P ∆u could clearly be different (cf. Figure 5 for a sketch of this problem).Thus for (1) to be true, some miracle needs to occur, and these miracles are wellknown 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 χpP 0 pλ{kq, L 0 pk; λqq, which is manifestly polynomial on the cones where λ{k is in a fixed equivalence class.
Let us fix a regular c P ∆ marking a particular chamber in ∆.The two cones tpk; λq| λ{k " cu and tpk; λq| p λ{ p k " cu intersect along an open cone (this cone is shaded in orange on Figure 5), and on this intersection, the expression with the right hand side of (18).As ( 20) is manifestly polynomial on each cone where λ{k is in a particular chamber in ∆, this expression will be then our main candidate for χpP 0 pλ{kq, L 0 pk; λqq.
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 λ{k or p λ{ p k are on a boundary separating two of our chambers in ∆.Remark 4.9.Part I. of the theorem implies that if λ{k P ∆ is not regular, then χpP 0 pc `q, Lpk; λqq " χpP 0 pc ´q, Lpk; λqq, for regular c ˘P ∆ in two neighboring chambers that contain λ{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 P ∆, and vary pλ, kq in such a way that λ{k " 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 P ∆ be a regular element, which thus specifies a chamber in ∆ and a parabolic moduli space P 0 pcq as well.Then for a diagonal basis D, an arbitrary weight λ P Λ, and a positive integer k, we have Example 4. Let us write down these formulas in case of r " 3 explicitly.Let D be the diagonal basis from Example 2; then using Remark 4.6, we obtain χpP 0 păq, Lpk; λqq " p´1q g´1 p3pk `3q 2 q g Res y"0

Res
x"0 e λ 1 x`pλ 1 `λ2 qy`x`y ´eλ 1 x`pλ 1 `λ3 qy`x`ypk`3q p1 ´expk`3q qp1 ´eypk`3q qw Φ px, yq 2g´1 dxdy, where w Φ px, yq " 2sinhp x 2 q2sinhp y 2 q2sinhp x`y 2 q. 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 Þ Ñ tcu B .These discontinuities occur on "walls": the affine hyperplanes ( 14).The following is straightforward: Lemma 4.11.Let S Π,l be the wall defined by (14), and B " pβ r1s , . . ., β rr´1s q P B an ordered basis of V ˚.Then, as a function of c, the fractional part function tcu B has a discontinuity on the wall S Π,l exactly when TreepBq (cf.page 11) is a union of a tree on Π 1 , a tree on Π 2 (the enumeration of the edges is irrelevant here) and a single edge (which we will call the link) connecting Π 1 and Π 2 .
Notation: We will denote the element of B corresponding to this edge by β link ; this vector thus depends on B and the partition Π.
Now choose two regular elements c `, c ´P V ˚in two neighboring chambers separated by the wall S Π,l , in such a way that (22) rc Π1 s " l and rc Π1 s " l ´1, where and, as usual, rqs stands for the integer part of the real number q.Now introduce the notation for the two polynomial functions in pk, λq 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 `pk; λq ´p´p k; λq.
Using Lemma 4.1 and ( 22), we obtain the following simple residue formula for this difference.p ´pk; λq ´p`p k; λq " p´3pk `3q 2 q g Res y"0

Wall-crossing and diagonal bases
Now we pass to the study of the combinatorial object D|Π defined in Lemma 4.12.One thing we will discover is that even though each diagonal basis consists of pr ´1q! elements and the right hand side of ( 23) does not depend on the choice of D, the number of elements in D|Π might vary with D.
It turns out that for our geometric applications, instead of H 1 , we will need to choose a particular nbc-basis, where the ordering is chosen to be consistent with Π.
To simplify our terminology, we will use the language of graphs and edges introduced in §3.3, and we will think of α ij P Φ as an edge in the complete graph on r vertices.To define the ordering υ, we need to choose an edge between Π 1 , Π 2 ; the choice is immaterial, but for simplicity we settle for m def " maxti P Π 1 u and r P Π 2 , and set β link " α m,r to be the smallest element according to υ.
The υ-ordered list of edges thus starts with β link , and then continues with the remaining r 1 ¨r2 ´1 edges connecting Π 1 and Π 2 .Next we list the r 1 pr 1 ´1q{2 edges connecting vertices in Π 1 in any order, and finally, we list the remaining edges, those connecting vertices in Π 2 .Notation: We introduce the natural notation Φ 1 and Φ 2 for the A r 1 and A r 2 root systems corresponding to Π 1 and Π 2 , and we denote by Drυs, D 1 rυs and D 2 rυs, the diagonal nbc-bases induced by the ordering υ on Φ, Φ 1 and Φ 2 , respectively.Lemma 4.15.Given elements B 1 P D 1 rυs and B 2 P D 2 rυs, we can define an element of Drυs as follows: we start with β link , then append B 1 , and then continue with B 2 .This construction creates a one-to-one correspondence (24) D 1 rυs ˆD2 rυs Ñ Drυs|π; in particular, |Drυs|π| " pr 1 ´1q! ¨pr 2 ´1q!.
Finally, putting Lemmas 4.12 and 4.15 together, we arrive at the following elegant statement: Proposition 4.16.Let pΠ, lq, c `and c ´be as in Lemma 4.12, and let D 1 and D 2 be diagonal bases of Φ 1 and Φ 2 correspondingly.Then (25) p `pk; λq ´p´p k; λq " pk `rq Ñr,k ÿ The result will be a rational function Q in the variables from B 1 and β link , and we proceed to calculate iBer B 1 rQspn 1 q to obtain a function F in the variable β link , and finally the answer is Res β link "0 exppβ link qFpβ link qdβ link .We observe that since the trees TreepB 1 q and TreepB 2 q are disjoint, the order of the application of the operations iBer B 1 and iBer B 2 is immaterial.

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.

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 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 ą 0; the space of such linearizations is called the G-effective ample cone; we denote this cone by Cone G pXq.
For L P Cone G pXq, we define the invariant-theoretic quotient X L G as the Proj of the graded ring of invariant sections of the powers of L: According to Mumford's Geometric Invariant Theory [MF], there is a partition of X depending on L: (26) X " X s rLs Y X sss rLs Y X us rLs into the set of stable, strictly semistable, and unstable points, such that there is a surjective map pX s rLs Y X sss rLsq{G Ñ M L , which is a bijection if X sss rLs is empty, and the quotient X s rLs{G is a smooth orbifold.
In [DH], Dolgachev and Hu studied the dependence of the GIT quotient M L " X L G on L. They showed that Cone G pXq 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 L of the G-action on X, which descends to M `and M ´.This last condition means that if S Ă G is the stabilizer of a generic point in X, then S acts trivially on the fibers of L. We will call such linearizations descending.
Thus, given such a descending linearization 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 L. Via taking Chern classes, this construction creates a correspondence between classes in H 2 pM `, Zq and H 2 pM ´, Zq, which we will assume to be an isomorphism of free Z-modules.We will thus identify these lattices, and introduce the notation Γ for them: The walls mentioned above can be thought of as hyperplanes in Γ R " Γ b Z R.
Our goal in this section is to compare the holomorphic Euler characteristics χpM `, Lq and χpM ´, Lq, which are given by the Hirzebruch-Riemann-Roch theorem: As this expression is manifestly polynomial in c 1 pLq, we obtain thus two polynomials on Γ , and our goal is to calculate their difference, the wall-crossing term (27) χpM `, Lq ´χpM ´, Lq.

The master space construction
To simplify our setup, we will make some additional assumptions.
(1) The generic stabilizer of X is trivial.(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 `b L ´lies on the single wall separating the two chambers, and that the interval connecting c 1 pL `q and c 1 pL ´q in Γ R " Γ b Z R does not intersect any other walls.
(3) Let X 0 be the set of those semistable points x P X ss rL 0 s which are not stable for L ˘: X 0 :" X ss rL 0 szpX s rL `s Y X s rL ´sq We assume that X 0 is smooth, and that for x P X 0 the stabilizer subgroup (4) Assume that there is a linearization L of the G-action on X such that L `" L ´b L n for some positive integer n, and such that for each x P X 0 , the stabilizer subgroup G x acts freely on L x z0.
Now we introduce the master space construction of Thaddeus [Th1].Consider the variety Y " PpO ' Lq, which is a P 1 -bundle over X endowed with the additional C ˚-action p1, t ´1q.As Y is a projectivization of a vector bundle on X, it comes equipped with Op1q, which is the standard G ˆC˚equivariant line bundle.To simplify our notation, we will denote the same way the linearizations of the Gaction 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 ´pnq " L ´b Opnq: which inherits a C ˚-action from Y. Some additional notation: ‚ We will denote this copy of C ˚by T , ‚ the projection Y Ñ X by π, and the quotient map Y s Ñ Z by ψ. ‚ Introduce the notation Yp0 : ¨q and Yp¨: 0q for the two copies of X in Y, corresponding to the two poles of the projective line; then Y is partitioned into 3 sets: where L ˝is the line bundle L with the zero-section removed.We will write π ˝for the restriction of π to L ˝.We can collect our maps on the following diagram.
(1) There are embeddings obtained as the quotients Y s X Yp¨: 0q{G and Y s X Yp0 : ¨q{G, correspondingly.
(2) The strictly semistable locus of Y with respect to the linearization L ´pnq is empty, and the GIT quotient Z " Y s {G is smooth.(3) There is an embedding ι 0 : X 0 {G Ñ Z, obtained via ψpπ ˝´1 pX 0 qq.We denote the image of ι 0 by Z 0 .(4) The fixed point locus Z T is the disjoint union of ι `pM `q, ι ´pM ´q, and Z 0 .
Proof.(1)-(3) follow from [Th1,4.2,4.3].To prove (4), first note that Yp¨: 0q and Yp0 : ¨q are fixed by T , so we immediately obtain that M ˘Ă Z are fixed components.Also the G-action on Y commutes with the T -action, so a point ψpyq P ψpπ ´1 ˝pXqq is fixed by T if and only if the T -orbit T ¨y Ă π ´1 ˝pXq is contained in the G-orbit G ¨y Ă π ´1 ˝pXq.Since T ¨y Ă π ´1 ˝pxq for some x P X, we need y P π ´1 ˝pX 0 q.Moreover, for any y P π ´1 ˝pxq Ă π ´1 ˝pX 0 q, T ¨y " π ´1 ˝pxq " G x ¨y, so a point ψpyq P ψpπ ´1 ˝pXqq is fixed by T if and only if ψpyq P ψpπ ´1 ˝pX 0 qq " Z 0 .
Construction: Given a G-equivariant vector bundle E on X, we can construct a T -equivariant vector bundle ζpEq Ñ Z on Z by first pulling E back from X to Y, and endowing the resulting bundle π ˚E with the trivial action of T , and the action of G pulled back from X.We then obtain ζpEq Ñ Z by descending π ˚E to Z. Then it is easy to verify the following.
Lemma 5.3.The restriction of the line bundle ζp Lq 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.
(1) The normal bundle on the component M `of Z T is ζp L ´1q ˇˇM `, and the normal bundle of M ´is ζp Lq ˇˇM ´.
(2) The normal bundle N Z 0 of Z 0 " X 0 {G Ă Z may be described as the descent of the normal bundle N X 0 of X 0 Ă X.The weights of the action may be computed by fixing x P X 0 , identifying the stabilizer subgroup G x Ă G with T via its action on the fiber 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 pVq, may be described as follows: let x 1 , . . ., x n be the Chern roots of V, and l 1 , . . .l n P Z be the corresponding T -weights.Then 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 µ Þ Ñ Res t"0,8 µ: . Theorem 5.6.Let L be a linearization of the G-action on X, and denote, as above, by ζpLq the T -equivariant line bundle on Z obtained by pull-back to Y and descent to Z.If Assumptions 5.1 hold, then where N Z 0 is the T -equivariant bundle on Z 0 described in Lemma 5.4, ch t is the Tequivariant Chern character, and E t pN Z 0 q is the K-theoretical Euler class of N Z0 .
Proof of Theorem 5.6.The Atiyah-Bott fixed-point formula [AB1] applied to the line bundle L on our master space Z yields (29) χ t pZ, Lq " 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 ζp Lq, and thus the contribution of M ´is equal to ż 1 ´t expp´c 1 p Lqq .
A similar calculation gives the contribution of M `as ż We observe that χ t pZ, Lq 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, χ t pZ, Lq has poles only at t " 0, 8, and by the Residue Theorem, we have Now, applying the functional Res t"0,8 to the two sides of (29) multiplied by dt{t gives us the desired result (47).

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 ∆ for the corresponding set of admissible parabolic weights ∆ 0 .Recall from Section 2.2 that for regular c P ∆, the moduli space of stable parabolic bundles P 0 pcq is the GIT quotient XQ c PSLpχq, where XQ is the subspace of the total space of the flag bundle over the Quot scheme.Let us fix a partition Π " pΠ 1 , Π 2 q and an integer l, and introduce the notation ∆ 1 l and ∆ 2 ´l for the simplices of parabolic weights of Π 1 and Π 2 .Let φ P Σ r be the unique permutation which sends t1, ..., r 1 u to Π 1 preserving the order of first r 1 and the last r 2 elements.We choose c 0 " pc 0 1 , ..., c 0 r q P S π,l and two regular elements c `, c ´P ∆ in two neighboring chambers separated by the wall S Π,l , such that c ˘" c 0 ˘ǫp..., 0, 1, 0, ..., 0, ´1q for some positive ǫ P Q, where 1 and ´1 are on the φpr 1 q th and r th places, respectively.Let For pk, λq P Z ˆΛ, consider the polynomials q ˘pk, λq " χpP 0 pc ˘q, L 0 pk; λqq.
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.

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 ˘" Lpk; λ ˘q of the G-action on X (cf.§2.2), such that λ ˘{k " c ˘; ‚ the linearization L " Lp0; x φpr 1 q ´xr q of the G-action on X.The following statement is easy to verify.Lemma 6.1.([BH,§3.2])The subset X 0 Ă X is the set of points representing vector bundles W on C, such that W splits as a direct sum W 1 ' W 2 , where W 1 and W 2 are, respectively, c 1 and c 2 -stable parabolic bundles.Therefore, we have the following description of the locus Z 0 : Z 0 " tW " W 1 ' W 2 | W 1 P r P l pc 1 q; W 2 P r P ´lpc 2 q; detpWq » Ou.
Remark 6.2.Note that Z 0 is fibered over Jac l with fibre P l pc 1 q ˆP´l pc 2 q by the determinant map r P l pc 1 q Ñ Jac l and H ˚pZ 0 , Qq » H ˚pP l pc 1 q ˆP´l pc 2 q, Qq b H ˚pJac l , Qq.
Remark 6.3.If the rank of the vector bundle W P r P l pcq is 1, then c " l and r P l plq is isomorphic to Jac l , while P l plq 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 Ă SL χ of point x P X on the fiber L x z0.
‚ For a general point x P X the stabilizer of x is the center Z χ Ă SLpχq, which acts trivially on the fiber L x z0.‚ For x P X 0 , any element of the stabilizer of x induces an automorphism of the corresponding vector bundle Then pt 1 , t 2 q P G x is in SL χ if and only if t χ 1 1 t χ 2 2 " 1, where χ 1 " χpW 1 q and χ 2 " χpW 2 q.Note that pt 1 , t 2 q acts on L x as t 1 t ´1 2 , and we need t 1 " t 2 (hence t χ 1 " 1) for this action to be trivial, so the stabilizer of any point in Then the action of G " PSLpχq is free on YzpYp0 : ¨q Y Yp¨: 0qq and the action of Now by Theorem 5.6, the wall-crossing polynomial q ´pk; λq ´q`p k; λq is equal to Note that in our case, the T -action on Z is free outside the fixed locus Z T , so as a function in t P T , the integral in (30) may have poles only at t " 0, 1, 8.Then, using the Residue Theorem and substituting t " e u , we conclude that (30) equals 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 L 0 pk; λq to the fixed locus Z 0 Ă Z.Let J be the Poincare bundle over Jac ˆC, such that c 1 pJq p0q " 0; define η P H 2 pJacq by p ř i c 1 pJq pe i q b e i q 2 " ´2η b ω (cf.§2.3), then (cf.[Z]) for any m P Z (32) ż Jac e ηm " m g .
Recall that for a parabolic weight c " pc 1 , ..., c r q P ∆ we have set c Π 1 " ř iPΠ 1 c i .
Lemma 6.4.Let λ " pλ 1 , ..., λ r q P Λ, k P Z ą0 and let Π " pΠ 1 , Π 2 q be a nontrivial partition with r P Π 2 .Let and define δ by pλ{kq Π 1 " l `δ.Then chpL 0 pk; λq ˇˇZ 0 q " e kδu exp ˆηk , where b denotes the external tensor product of line bundles on P l pc 1 q ˆP´l pc 2 q.Lemma 6.5.Denote by r U 1 and r U 2 the universal bundles over r P l pc 1 q ˆC and r P ´lpc 2 q ˆC with the standard normalization (cf.§2.3 ), and denote by π projections along C. Then the equivariant normal bundle to the fixed locus Z 0 Ă Z is where T » C ˚-action has weights p´1, 1q.

Calculation of the characteristic classes of N Z 0
Before we calculate the equivariant K-theoretical Euler class of the conormal bundle N Z0 , we need to introduce some notations.Recall that for 1 ď i, j ď r, the differences x i ´xj P V ˚are linear functions on V, and the function x i ´xj corresponds to the linearization L 0 p0; x i ´xj q on X, which descends to the line bundle L 0 p0; x i ´xj q on the moduli space P 0 pcq (cf.§2.2).As in §5.2, we denote by ζpL 0 p0; x i ´xj q the line bundle on Z obtained by the pullback and then descent.This way, we obtain a correspondence between the linear functions x i ´xj 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 pZ 0 q » H ˚pZ 0 q b Crus.In particular, there are canonical embeddings H ˚pZ 0 q ã Ñ H T pZ 0 q and Crus ã Ñ H T pZ 0 q.
Recall the definition of the permutation φ P Σ r given at the beginning of this chapter: φ takes the first r 1 numbers to Π 1 , preserving the order of the first r 1 and the last r 2 elements.We introduce the symbols j " c 1 pζpL 0 p0; x φpiq ´xφpjq qq ˇˇZ 0 q, p1 ď i, j ď r 1 q z 2 i ´z2 j " c 1 pζpL 0 p0; x φpr 1 `iq ´xφpr 1 `jq qq ˇˇZ 0 q, p1 ď i, j ď r 2 q u " pz 1 r 1 ´z2 r q " c 1 pζpL 0 p0; x φpr 1 q ´xr qq ˇˇZ 0 q (33) for the equivariant cohomology classes in H 2 T pZ 0 q.The last equalities are consistent with Lemma 5.3.Remark 6.6.Note that (cf.Remark 6.2) 1 {F 2 r´j q ˚q P H 2 pP ´lpc 2 qq, where F 1 i and F 2 i are the flag bundles (cf.§2.3) on P 0 pc 1 q and P 0 pc 2 q, correspondingly.
Taking into account these identifications, functions on V give rise to equivariant cohomology classes on Z 0 .To make the splitting H T pZ 0 q » H ˚pZ 0 q b Crus, explicit, however, we will write these classes in the form f u pz 1 , z 2 q, thinking of them as functions of the differences of the z 1 i s and the differences of the z 2 i s, depending on the parameter u.With this convention, we introduce where according to (33), r q " c 1 pζpL 0 p0; x φpiq ´xφpr 1 `jq qq ˇˇZ 0 q P H 2 T pZ 0 q.Now we are ready to write down our formula for the K-theoretical Euler class EpN Z 0 q (cf.definition 5.5 with t " e u ).Proposition 6.7.EpN Z 0 q ´1 " p´1q lr`r 1 r 2 pg´1q e ´rlu exp ´ηr r 1 `ηr r 2 ¯wû pz 1 , z 2 q 1´2g exppρ û pz 1 , z 2 qq chpL l pr 2 ; ´l, ..., ´l, ´l `rlq b L ´lpr 1 ; l, ..., l, l ´rlqq.
Proof.It follows from the short exact sequence (6) for parabolic morphisms that so by Lemma 6.5 Lemma 6.8.We denote by rfpxqs W the multiplicative class of the vector bundle W given by the function fpxq in Chern roots of W. Let S be a vector bundle on P ˆC with T -weight 1, and π : P ˆC Ñ P projection along the curve, then Ep´π !S ' ´π! S ˚q´1 " p´1q rkp´π !Sq expp´ch 2 pSq p2q q rp2sinhpx{2qq 2g´2 s S p .
Note that the last two terms in (34) are the sums of Chern characters of line bundles, so they contribute the multiplicative factor to the equivariant class EpN Z 0 q ´1; and using Lemma 6.8 with S " Homp r U 1 , r U 2 q, we obtain that the inverse of the K-theoretical Euler class of the first term in ( 34) is p´1q lr`r 1 r 2 pg´1q w û pz 1 , z 2 q 2´2g expp´ch 2 pHomp r U 1 , r U 2 qq p2q q.
The latter equality follows from Lemma 2.8.Finally, using Lemma 6.4 to calculate the Chern character of L r 0 ˇˇZ 0 , we obtain the formula for the class EpN Z 0 q ´1, and the proof of the Lemma is complete.
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 p´1q g´1 p2pk `3qq g Res z"0 e pλ 1 `1qz p2sinhp z´u 2 q2sinhp z 2 qq 2g´1 p1 ´epk`3qz q dz, so the wall-crossing term is p´3pk `3q 2 q g Res u"0

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

The Hecke correspondence
Given a rank-r degree-d vector bundle W with a full flag 0 Ĺ F 1 Ĺ ... Ĺ F r " W p at p, one can obtain a rank-r degree-d ´1 vector bundle W 1 with a full flag 0 Ĺ G 1 Ĺ ... Ĺ G r " W 1 p using the tautological Hecke correspondence construction as follows.
The evaluation map W Ñ W p induces the short exact sequence of the associated sheaves of sections is a kernel of α, it is a locally free sheaf, thus gives a rank-r vector bundle W 1 over C with detpW 1 q » detpWq b Op´pq.The image of the associated morphism of vector bundles α at the point p is F r´1 Ă W p , so Moreover, compositions of α p with the quotient morphisms F r´1 Ñ F r´1 {F i induce a full flag of the corresponding kernels G 1 Ĺ ... Ĺ G r´1 Ĺ G r " W 1 p in W 1 p .Denote this operator between the sets of isomorphism classes of degree-d and d ´1 vector bundles with a flag at p by Similarly, for any m ě 0, one can define the operator 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 P ∆ be a regular (cf.page 7) point.Then the operator H induces an isomorphism between the moduli spaces P d pc 1 , ..., c r q and P d´1 pc 2 , ..., c r , c 1 ´1q.
Proof.First, we need to show that if W P P d pc 1 , ..., c r q is a parabolic stable bundle with parabolic weights pc 1 , ..., c r q, then W 1 , its image under the Hecke operator H, is parabolic stable with respect to parabolic weights pc 2 , ..., c r , c 1 ´1q.For this, consider the subbundle V 1 Ă W 1 and let αpV 1 q " V Ă W (cf. ( 38)) be its image.Since W is parabolic stable, parslopepVq ă parslopepWq " parslopepW 1 q.
We need to prove that parslopepV 1 q ă parslopepW 1 q.There are two possible cases: ‚ If α maps V 1 to V isomorphically, then degpV 1 q " degpVq and V p Ă F r´1 , hence parslopepV 1 q " parslopepVq ă parslopepW 1 q.‚ Otherwise, degpV 1 q " degpVq ´1, and V p is not contained in F r´1 , so one of the parabolic weights of V 1 is c 1 ´1.Then, as in the previous case, parslopepV 1 q " parslopepVq, and the result follows.
It is easy to check that given W and iterating the associated morphism of locally free sheaves of sections (38) r times, we obtain a subsheaf W 1 Ă W of sections of W which vanishes at the point p.So the map (39) is just tensoring by Op´pq, and hence it is an isomorphism.Now we can define an operator H m for any m P Z, taking the inverse map if necessary.We will need the following statement, which follows from Proposition 7.1 and the construction of H m .Corollary 7.2.Let m ě 0. Then under the isomorphism H m the line bundle L d pk; λ 1 , ..., λ r q corresponds to the line bundle L d´m pk; λ r´m`1 , ..., λ r , λ 1 ´k, ..., λ r´m ´kq.

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 H l and H ´l to the moduli spaces P l pc 1 q and P ´lpc 2 q to obtain P 1 0 " P 0 pc 1 l`1 , ..., c 1 r 1 , c 1 1 ´1, ..., c 1 l ´1q » P l pc 1 q and P 2 0 " P 0 pc 2 r 2 ´l`1 `1, ..., c 2 r 2 `1, c 2 1 , ..., c 2 r 2 ´lq » P ´lpc 2 q.
As in §6.3, according to Lemma 6.10, we can calculate this integral using the induction on rank.Let D 1 and D 2 be two Hamiltonian diagonal bases.Then τ 1 pD 1 q and τ 2 pD 2 q are also Hamiltonian diagonal bases (cf.Remark 3.3) and the integral in ( 40) is equal to To arrive at Theorem 6.11, we need to make additional transformations of formula (41): first, we shift λ 1 and λ 2 , and then we apply Lemma 4.5 to eliminate the cyclic permutation τ.
Note that given an ordered basis B P B and an element v P V ˚such that tvu B " 0, for any weight λ P Λ and positive integer k one have (42) pλ `p kvq{ p k ´rc `vs B " λ{ p k ´rcs B .
Now we can iterate this statement to the case of flag bundles.

´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 ∆ ˘1 of weights for degree-˘1 stable parabolic bundles are simplices with one of their vertices at p 1 r , ..., 1 r q and p ´1 r , ..., ´1 r q, correspondingly (cf.§2.2).Denote by N ˘1 the moduli spaces of rank-r degree-˘1 stable vector bundles and by UN any universal bundle over N ˘1 ˆC (cf.e.g [AB2]).
Lemma 8.3.Let c " pc 1 , ..., c r q be a parabolic weight from the chamber in ∆ 1 , which has as one of its vertices the (regular) point p 1 r , ..., 1 r q.Then the moduli space P 1 pcq of rank-r degree-1 stable parabolic bundles is isomorphic to the flag bundle FlagpUN p q over N 1 .An analogous statement holds in the case of degree ´1 and the point p ´1 r , ..., ´1 r q P ∆ ´1.
Proof.A simple calculation shows that the point pc 1 , ..., c r q P ∆ 1 , such that all c i ą 0, lies inside the chamber in ∆ 1 with the vertex p 1 r , ..., 1 r q.Hence it is enough to prove the first statement for the moduli space P 1 pc 1 , ..., c r q with positive parabolic weights.
Moreover, it is sufficient to show that if pW, F ˚q is a parabolic stable vector bundle which represents a point in P 1 pc 1 , ..., c r q, then W is stable as an ordinary bundle.Assume that W admits a proper subbundle W 1 with slopepW 1 q ě slopepWq " 1 r , then degpW 1 q ě 1.Since all parabolic weights of W are positive, this implies that parslopepW 1 q ą 0 " parslopepWq, and therefore W is parabolic unstable.The proof for degree-(´1) bundles is analogous.Denote the moduli spaces described above by P 1 pąq and P ´1păq, correspondingly, and their images under the Hecke isomorphisms H and H ´1 by P 0 pąq and P 0 păq.
The following statement is straightforward (cf.Lemma 2.8).
The proof for q ´1 is similar.
The two group actions in Proposition 8.5 may be combined in the following manner.For k ě 0, we define an action of the affine Weyl group Σ ¸Λ on Λ ˆZ, which acts trivially on the second factor, the level, and the action at level k is given by setting σ.λ " σ ¨pλ `ρq ´ρ and γ.λ " λ `pk `rqγ for σ P Σ, γ P Λ.
We denote the resulting group of affine-linear transformations of V ˚by r Σrks, and note that the action is defined in such a way that (44) σ.λ `ρ " σ ¨pλ `ρq and pγ.λ `ρq{ p k " γ `pλ `ρq{ p k 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 l " tpc, aq | 1 ą 2c ą 0, 1 ą 2a ą 0, u, which has two adjacent chambers defined by the conditions c ą a and c ă a.
Denote the corresponding moduli spaces by P 0 pc ą aq and P 0 pc ă aq.
Again, we have universal bundles over P 0 pc ą aq ˆC and P 0 pc ă aq ˆC, which we will denote by the same symbol U; this bundle is endowed with two flags, F 1 Ă F 2 " U p and G 1 Ă G 2 " U s .For µ, λ P Z, we introduce the line bundle Lpk; λ, µq "detpU p q kp1´gq b detpπ ˚pUqq ´k b pF 2 {F 1 q λ b pF 1 q ´λ b pG 2 {G 1 q µ b pG 1 q ´µ.
We repeat the construction of the master space from Section 5.1, choosing a point pc 0 , c 0 q on the wall and two points pc, aq ˘" pc 0 , c 0 q ˘ǫp1, 0q P l, ǫ P Q ą0 from the adjacent chambers.We can identify the fixed point set Z 0 as follows.
and, applying Theorem 5.6, we obtain the following expression for their difference.

Symmetry
Denote by P ´1pc ą aq the image of the moduli space P 0 pc ą aq under the Hecke isomorphism H (cf. §7) at the point p and by P ´1pc ă aq the image of the moduli space P 0 pc ă aq under the Hecke isomorphism H at the point s.
We have the following analogue of Lemma 8.3.
Lemma 9.3.Denote by N ´1 the moduli space of rank-2 degree-(´1) stable bundles on C and by UN any universal bundle over N ´1 ˆC.Then the moduli spaces P ´1pc ą aq and P ´1pc ă aq are isomorphic to the bundle PpUN p q ˆPpUN s q over N ´1.
Proof.It is a simple exercise to show that r h ą pk; λ, µq and r h ă pk; λ, µq satisfy the identities appearing in Lemmas 9.2 and 9.5, and hence the polynomial Θpk; λ, µq " h ą pk; λ, µq ´r h ą pk; λ, µq " h ă pk; λ, µq ´r h ă pk; λ, µq satisfies all four Σ 2 -symmetries listed in Lemma 9.5.These groups together generate a double action of the affine Weyl group r Σ in λ and µ separately, and this implies the vanishing of Θ.
As P 0 pc ą aq is a P 1 -bundle over the moduli space of rank-2 degree-0 stable parabolic bundles P 0 pc, ´cq, substituting µ " 0 in r h ą , we obtain the Verlinde formula for rank 2.
10.The combinatorics of the rQ, Rs " 0 In this section, we give a proof of the second part of Theorem 4.8.Let λ{k P ∆, and fix a regular element c ˛P ∆ in a chamber containing λ{k in its closure, and another regular element p c ˛P ∆ containing p λ{ p k in its closure.Our goal is to prove the the equality p c ˛pk; λq " p p c ˛pk; λq, where we define for a regular c P ∆ and diagonal basis D. This is a subtle statement, which is a combinatorial-geometric projection of the idea of quantization commutes with reduction (or rQ, Rs " 0 for short, cf.[MS, SzV]).If λ{k " p λ{ p k, i.e. when λ{k and p λ{ p k are regular elements in the same chamber in ∆, then p c ˛pk; λq " p p c ˛pk; λq is a tautology.We assume thus that this is not the case, and denote by Spk, λq the set of walls separating c ˛and p c ˛, or containing either λ{k or p λ{ p k or both.Equivalently, the wall S Π,l belongs to Spk, λq if pλ{kq Π 1 ě l ě p p λ{ p kq Π 1 or pλ{kq Π 1 ď l ď p p λ{ p kq Π 1 , where c Π 1 " ř iPΠ 1 c i for an element c " pc 1 , ..., c r q P V ˚.Clearly, there is a path in ∆ connecting c ˛and p c ˛, which intersects only walls from Spk, λq in a generic points.Then to prove the equality p c ˛pk; λq " p p c ˛pk; λq, it is enough to show the following, at first sight somewhat surprising fact.
Remark 10.4.Note that if λ{k P ∆ is non-regular, then it belongs to some wall from the set Spk, λq.Hence proposition 10.1 implies that the right-hand side of formula (I.) of Theorem 4.8 is a well-defined function on the cone over ∆: tpk, λq P Z ą0 ˆΛ| λ{k P ∆u.

Figure 4 .
Figure 4. Chambers for rank r " 3.Each element B " pβ r1s , . . ., β rr´1s q P B defines an iterated version of the Bernoulli operator (12) on the space of functions F Φ : interpreting the elements a, β rjs P V ås

Lemma 4. 5 .
Let f P F Φ , and σ P Σ r , and pick any diagonal basis D. Then ÿ BPD iBer B rfpxqspσ ¨a ´rσ ¨cs B q " ÿ BPD iBer B rσ ´1 ¨fpxqspa ´rcs B q

Figure 5 .
Figure5.λ{k is in the orange chamber, while p λ{ p k is in the green chamber.

Lemma 4. 12 .
Let pΠ, lq, c `and c ´be as above, and let us fix a diagonal basis D Ă B. Denote by D|Π the subset of those elements of D, which satisfy the condition described in Lemma 4.11.Then (23) p `pk, λq ´p´p k, λq " β link is the "link" element of B (depending on Π and B) defined after Lemma 4.11.Remark 4.13.Note that the multiplication by 1 ´exppβ link pxqq in (23) has the effect of canceling one of the factors in the denominator in the definition (15) of the operation iBer.Example 5. Calculating the difference of two polynomials from Example 4, we get the wall-crossing term for rank 3 case:

Lemma 9. 4 .
Under the Hecke isomorphism H at p, the line bundle Lp2k; λ, µq on P 0 pc ą aq corresponds to the line bundle L k ´1 b Trps λ´k b Trss µ on P ´1pc ą aq.Under the Hecke isomorphism H at the point s, Lp2k; λ, µq on P 0 pc ă aq corresponds to the line bundle L k ´1 b Trps λ b Trss µ´k on P ´1pc ă aq.As in §8.2, applying Serre duality for families of curves (cf.Proposition 8.2)

Φ
px{ p kqsp p λ{ p k ´rcs B q By picking the Hamiltonian diagonal basis H 1 " tσ ¨B0 | σ P Stabp1, Σ r qu, we can turn the argument in the proof above around, and obtain the following formula: ÿ B , xy " xσ ¨pa ´rcs B q, xy " xa ´rcs B , σ ´1pxqy, perform the linear substitution x " σpyq, and conclude that ÿ BPD iBer B rfpxqspσ ¨a ´rσ ¨cs B q " ÿ B rσ ´1 ¨fpxqspa ´rcs B q.Now the statement follows from the fact that σ P Σ takes a diagonal basis to another diagonal basis (cf.Remark 3.3).
´rc `sB ¯dβ link , where Res β link "0 iBer B 1 iBer B 2 dβ link is simply iBer B (cf (15)) with B obtained by appending B 1 , and then B 2 to β link , and with the factor p1 ´expxβ link , xyq removed from the denominator.
Φpx{ p kq ı ´p λ{ p k ´rc `sB ¯dβ link may equally be interpreted as follows.We writeΛ Q p λ{ p k ´rc `sB " m link β link `n1 `n2according to the splitting of B, think of wpx{ p kq as a function in F Φ 2 with some fixed values of the parameters from B 1 and β link , and then calculate iBer B 2 rw 1´2g Φ px{ p kqspn 2 q.