Motivic Donaldson-Thomas invariants of parabolic Higgs bundles and parabolic connections on a curve

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


Introduction and main results
1.1. Let k be a field of characteristic zero and X be a smooth geometrically connected projective curve over k (geometric connectedness means that X remains connected after the base change to an algebraic closure of k). In [FSS] we calculated the motivic classes of moduli stacks of semistable Higgs bundles on X. These motivic classes are closely related to Donaldson-Thomas invariants, see [KS1,KS2]. In [FSS] we also calculated the motivic classes of moduli stacks of vector bundles with connections on X by relating them to the semistable Higgs bundles.
In this paper, we extend these results to the parabolic case. Some of our results are parallel to the results of A. Mellit in the case of finite fields (see [Mel1]). One difference is that we fix the eigenvalues of the residues. Another difference is that by working over a field of characteristic zero, we are also able to treat bundles with connections. We also note that the calculation of the motivic class requires subtler techniques, than the calculation of the volume of a corresponding stack over a finite field.
1.2. Moduli Stacks. Let us briefly describe the moduli stacks whose motivic classes we will be interested in. There will be three classes of stacks.
1.2.1. Parabolic bundles with connections. Let D ⊂ X(k) be a non-empty set of rational points of X. A parabolic bundle of type (X, D) is a collection E = (E, E •,• ), where E is a vector bundle over X and E x,• is a flag in its fiber E x for x ∈ D: E x = E x,0 ⊇ E x,1 ⊇ . . . ⊇ E x,l ⊇ . . . , E x,l = 0 for l ≫ 0. A connection on E with poles bounded by D is a morphism of sheaves of abelian groups ∇ : E → E ⊗ Ω X (D) satisfying Leibniz rule. In this case for x ∈ D one defines the residue of the connection res x ∇ ∈ End(E x ). Let ζ = ζ •,• = (ζ x,j ) be a sequence of elements of k indexed by D × Z >0 such that ζ x,j = 0 for j ≫ 0. Let Conn(X, D, ζ) denote the moduli stack parameterizing collections (E, E •,• , ∇), where (E, E •,• ) is a parabolic bundle of type (X, D), ∇ is a connection on E with poles bounded by D such that (res x ∇−ζ x,j 1)(E x,j−1 ) ⊂ E x,j for all x ∈ D and j > 0. We usually skip X and D from the notation as they are fixed, denoting Conn(X, D, ζ) simply by Conn(ζ). We call the points of Conn(ζ) parabolic bundles with connections of type (X, D) with eigenvalues ζ.
For a parabolic bundle E = (E, E •,• ) define the class of E as the following collections of integers: (1) We also set rk E := rk E. The stack Conn(ζ) decomposes according to the classes of parabolic bundles; denote the component corresponding to parabolic bundles of class γ by Conn γ (ζ). We will see that this stack is an Artin stack of finite type. One of our main results (see Section 1.4 and Theorem 8.2.5) is the calculation of the motivic class of this stack.
1.2.2. Parabolic Higgs bundles. Let ζ be as above. A parabolic Higgs bundle with eigenvalues ζ is a triple (E, E •,• , Φ), where (E, E •,• ) is a parabolic bundle of type (X, D), Φ : E → E ⊗ Ω X (D) is a morphism of O X -modules (called a Higgs field on (E, E •,• )) such that for all x ∈ D and j > 0 we have (Φ − ζ x,j 1)(E x,j−1 ) ⊂ E x,j ⊗ Ω X (D) x . Denote the category and the stack of such Higgs bundles by Higgs(ζ). Unfortunately, this stack is not of finite type, and in fact, has an infinite motivic volume. To resolve the problem we endow the category with a stability structure. Let σ = σ •,• be a sequence of real numbers indexed by D × Z >0 . Let κ ∈ R. We define the (κ, σ)-degree of a parabolic bundle E = (E, E •,• ) by If E = 0, we define the (κ, σ)-slope of E as deg κ,σ E/ rk E.
We say that a sequence σ = σ •,• of real numbers indexed by D × Z >0 is a sequence of parabolic weights if for all x ∈ D we have (2) σ x,1 ≤ σ x,2 ≤ . . . and for all x and j we have σ x,j ≤ σ x,1 + 1. Let σ be a sequence of parabolic weights. Let E = (E, E •,• ) be a parabolic bundle. Let F ⊂ E be a vector subbundle. Set F x,j := F x ∩ E x,j . Then F := (F, F •,• ) is a parabolic bundle. We say that a Higgs bundle (E, Φ) is σ-semistable, if for all subbundles F of E preserved by Φ the (1, σ)-slope of the corresponding parabolic bundle F is less or equal than that of E.
We have an open substack Higgs σ−ss γ (ζ) of Higgs γ (ζ) classifying σ-semistable parabolic Higgs bundles. This stack is of finite type; we will calculate its motivic class (see Section 1.4 and Theorem 7.1.5).
We note that condition (2) is imposed on σ to ensure that we have Harder-Narasimhan filtrations for parabolic Higgs bundles.
1.2.3. Semistable parabolic bundles with connections. We can also impose stability conditions on parabolic bundles with connections. Moreover, for non-resonant connections, we can work with stability conditions more general than those for Higgs bundles defined in the previous paragraph. A sequence ζ as above is called nonresonant if for all x ∈ X and all i, j > 0 we have ζ x,i − ζ x,j / ∈ Z =0 . Take κ ∈ R ≥0 and a sequence σ of real numbers indexed by D × Z >0 and satisfying condition (2).
1.3. Motivic Donaldson-Thomas invariants. Our formulas for motivic classes of the moduli stacks above are all given in terms of certain motivic classes B γ called motivic Donaldson-Thomas invariants (see Section 1.5 for this terminology), which we are going to define. First of all, we recall that in [FSS,Sect. 2] we defined (following earlier works [Eke1,Sect. 1], [Joy], and [KS1]) the ring of motivic classes of Artin stacks denoted Mot(k). We also defined its dimensional completion Mot(k). For an Artin stack S of finite type we have its motivic class [S] ∈ Mot(k). We denote its image in Mot(k) by the same symbol.
For a curve X and a partition λ we defined the series J mot λ (z), H mot λ (z) ∈ Mot(k) [[z]] in [FSS,Sect. 1.3.2]. The definitions (especially of H mot λ (z)) are somewhat long, so we will not recall them here inviting the reader to look into [FSS]. We only note that J mot λ (z) and H mot λ (z) are defined in terms of the motivic zeta-function of X (cf. (6) below). In particular, they only depend on X but not on D. In this paper, we will denote them by J mot λ,X (z) and H mot λ,X (z) respectively to emphasize that they depend on the curve X and to ensure that they are not confused with motivic modified Macdonald polynomialsH mot λ (w • ; z) and with motivic Hall-Littlewood polynomials H mot λ (w • ) defined below. The modified Macdonald polynomialsH λ (w • ; q, z) are defined e.g. by [Mel1,Def. 2.5]. These are symmetric functions in variables w • = (w 1 , w 2 , . . .) with coefficients in Z [q, z]. It is well-known that the coefficients are integers (see e.g. [HHL] and references therein). Note that, formally speaking, symmetric functions are not polynomials (they become polynomials upon plugging in w N +1 = w N +2 = . . . = 0). Let L = [A 1 k ] be the motivic class of the affine line. We denote byH mot λ (w • ; z) the symmetric function with coefficients in Mot(k) [z] or Mot(k)[z] obtained fromH λ (w • ; q, z) by substituting L for q.
Let Γ + denote the commutative monoid of sequences (r, r •,• , d), where r is a nonnegative integer, r •,• is a sequence of nonnegative integers indexed by D × Z >0 , d is an integer, subject to the following conditions: (i) For all x ∈ D we have ∞ j=1 r x,j = r. In particular, r x,j = 0 for j large enough. (ii) if r = 0, then d = 0 (and so r x,j = 0 for all x and j). The operation on Γ + is the componentwise addition. For γ = (r, r •,• , d) ∈ Γ + we set rk γ = r. The significance of the monoid Γ + is that the class of a parabolic bundle E defined by (1) is an element of Γ + . We also need a submonoid Γ ′ + ⊂ Γ + given by d ≤ 0. Consider the completed monoid ring Mot(k)[[Γ ′ + ]], we write its elements as γ∈Γ ′ + A γ e γ , where A γ ∈ Mot(k), e γ are basis vectors. It is convenient to identify e γ with a monomial ]. We note that these rings are instances of completed quantum tori considered in [KS1,KS2]. In our case, they are commutative.
Finally, we need the notion of plethystic exponent and logarithm.
where the sum in the LHS is over all partitions. We call the elements B γ the Donaldson-Thomas invariants.
Note that B 0 = 0. When X = P 1 k , we can define motivic Donaldson-Thomas invariants B γ by a simpler formula valid in Mot(k): where Hook(λ) stands for the set of hooks of λ, a(h) and l(h) stand for the armlength and the leglength of the hook h respectively. We show that for X = P 1 the images of B γ in Mot(k) are equal to B γ .
Next, assume that ζ and σ are as in Section 1.2.2, for τ ∈ R define the elements H γ (ζ, σ) ∈ Mot(k) by the following formula Then, according to Theorem 7.1.5, the stack Higgs σ−ss γ (ζ) is of finite type and we have Finally, assume that ζ and (κ, σ) are as in Section 1.2.3. For τ ∈ k, τ ′ ∈ R define the elements C γ (ζ, κ, σ) ∈ Mot(k) by the following formula.
If X = P 1 , we get similar results valid in Mot(k), by replacing B γ with B γ defined by a simpler formula (3).
Remark 1.4.1. We note that all the stacks whose motivic classes we are calculating are of finite type, so their motivic classes are defined in Mot(k). However, we can only calculate their motivic classes in Mot(k) except when X = P 1 .
1.5. Aftermath. In Section 1.3 we defined the classes B γ . These classes should be thought of as the Donaldson-Thomas invariants of the stack Higgs(0) of parabolic Higgs bundles with nilpotent residues. Note that this stack is the cotangent bundle of Bun par (X, D), while the stacks Higgs(ζ) and Conn(ζ) are twisted cotangent bundles. We emphasize that B γ do not depend on ζ, κ, and σ. The meaning of the formulas in Section 1.4 is that the Donaldson-Thomas invariants of these twisted cotangent bundles are obtained by restricting the range of γ to the submonoid deg 0,ζ γ = 0 in the case of Higgs(ζ) and to the submonoid deg 1,ζ γ = 0 in the case of Conn(ζ).
Another feature of the formulas is that the motivic classes of the stacks depend on equations satisfied by κ and σ rather than on inequalities. In other words, there is no wall-crossing in our case. This is not very surprising, as the category Higgs(0), being a cotangent bundle of Bun par (X, D), is a 2-dimensional Calabi-Yau category.
One can speculate that similar results should be valid for the twisted cotangent stacks to the moduli stack of objects of any reasonable 1-dimensional category. A result in this direction is obtained by G. Dobrovolska, V. Ginzburg, and R. Travkin (see [DGT]).
Another example of such a twisted cotangent stack is the category of vector bundles with irregular connections and appropriate level structures. We hope to return to this question in subsequent publications.
1.6. Other results. It is clear from the above formulas that we have a lot of equalities between different motivic classes of Higgs bundles and bundles with connections. In particular, we show in Propositions 9.1.1 and 9.1.2 that every motivic class of the form [Higgs σ−ss γ (ζ)] or [Conn (κ,σ)−ss γ (ζ)] is equal to some motivic class of the form [Conn γ (ζ)]. As a consequence, we derive from results of Crawley-Boevey [CB2] a criterion for when a moduli stack is non-empty. It is not difficult to see that if X = P 1 k , then the stacks of semistable Higgs bundles are always non-empty while the stacks of semistable bundles with connections are non-empty if and only if deg 1,ζ γ = 0. For X = P 1 k the question is much more subtle and is related to the so-called Deligne-Simpson problem.
This problem was originally stated for k = C in [Sim3]. It may be reformulated for an arbitrary algebraically closed fields k of characteristic 0 as follows: given a sequence of gl r -conjugacy classes C • indexed by D, does there exist a pair (E, ∇) consisting of a rank r vector bundle and a connection ∇ on E with poles bounded by D such that Res x ∇ ∈ C x for all x ∈ D? Connections ∇ coming from Conn γ (P 1 k , D, ζ) are exactly the connections such that each residue Res x ∇ lies in the closure of a conjugacy class determined by γ and ζ (see [CB1]). If this conjugacy class is semisimple for each x ∈ D, then the elements of Conn γ (P 1 k , D, ζ) are the solutions of the corresponding Deligne-Simpson problem. For a more comprehensive survey of the Deligne-Simpson problem, see [CB1,Kos,Sim1].
We note that if k is not algebraically closed, one can ask a subtler question of whether there is a k-rational point in [Higgs σ−ss γ (ζ)] or [Conn (κ,σ)−ss γ (ζ)]. We do not know the answer to this question. A somewhat similar question for moduli spaces of quiver representations is considered by V. Hoskins and F. Schaffhauser in [HS].
We would like to emphasize that even if one is only interested in the case k = C, one still has to consider all fields of characteristic zero, see Remark 2.2.2 below.
One of the motivations for this work is the non-abelian Hodge theory of C. Simpson (see [Sim2]). In this paper, Simpson constructs a diffeomorphism between a moduli space of parabolic bundles with connections and a moduli space of Higgs bundles. In Proposition 9.2.1, we show that the corresponding stacks have equal motivic classes. We note that neither statement can be derived from the other (cf. [FSS,Rem. 1.2.2]). We note also, that it is clear from our results that there are many more equalities of motivic classes, than those that one can guess from the non-abelian Hodge theory. We remark that V. Hoskins and P. L. Simon have shown in [HL,Thm. 4.2] that the Voevodsky motives of the coarse moduli spaces of bundles with connections and of Higgs bundles are equal in the case when the rank and the degree are coprime. One can ask whether this can be upgraded to our situation. 1.7. Other relations with previous work. We note that our results are closely related with the results of Mellit [Mel1]. One difference is that Mellit counts the weighted number of points over a finite field, while we work over a field of characteristic zero and calculate motivic classes. Mellit counts the volumes of moduli stacks of Higgs bundles but is not considering bundles with connections. Another difference is that Mellit is not fixing the eigenvalues of Higgs fields.
On the other hand, Mellit's answers are simpler as they do not involve Schiffmann's polynomials H mot λ,X . In fact, Mellit's simplification of Schiffmann's formula has nothing to do with parabolic structures. This simplification is the content of Mellit's papers [Mel1,Mel2]. In these papers he simplifies the formulas of [Sch]. We believe that this simplification does not go through in the motivic case because Mellit is using the fact that the λ-ring structure on symmetric functions is special (which, roughly speaking, means that λ • (xy) and λ • (λ • (x)) can be expressed in terms of λ • (x) and λ • (y)). It is known (see [LL]) that the Grothendieck λ-ring of varieties is not a special λ-ring. However, if one replaces Mot(k) by its quotient that is a special λ-ring, then one expects that the Mellit's simplifications are valid in this quotient. In particular, one can work with the Grothendieck ring of the category of Chow motives or with the maximal special quotient (see [LL]).
One of the main technical statements in our paper is Theorem 4.3.2, which is a motivic analogue of [Mel1,Thm. 5.6]. While we generally follow the strategy of Mellit, the motivic case required introduction of new ideas.
The paper [CDDP], although conjectural, contains an alternative approach to the problem via upgrading the computation of the motivic class of Higgs bundles to the problem about motivic Pandharipande-Thomas invariants on the non-compact Calabi-Yau 3-fold associated with the spectral curve.
Note also that Mozgovoy and Schiffmann in [MS] consider Higgs bundles with a twist by an arbitrary line bundle of degree at least 2g − 2, where g is the genus of X. However, they do not consider parabolic structures and do not fix eigenvalues.
Finally we note that the general philosophy of Donaldson-Thomas invariants and the approach via motivic and cohomological Hall algebras (see [KS1,KS2]) are applicable to our situation. For more details about the approach that uses motivic Hall algebras we refer the reader to [FSS,Sect. 1.6,Rem. 3.6.3].
1.8. Organization of the article. In Section 3 we define the category Bun par (X, D) of parabolic bundles and its graded stack of objects denoted by the same letter. Most of our stacks below will be stacks over Bun par (X, D).
In Section 4 we study the stack of bundles with endomorphisms. First, we calculate the motivic classes of stacks of parabolic bundles with nilpotent endomorphisms combining the motivic version of [Mel1,Thm. 5.6] and our results [FSS]. Then we use the formalism of plethystic powers to calculate the motivic classes of stacks of parabolic bundles with arbitrary endomorphisms.
In Section 5 we study parabolic Higgs bundles with fixed eigenvalues. We reduce the calculation of their motivic classes to the results of Section 4 using the notion of isoslopy parabolic bundle and a factorization formula (see Proposition 5.4.3).
In Section 6 we use a version of Kontsevich-Soibelman factorization formula to calculate the motivic classes of stacks of semistable Higgs bundles. These depend on two sets of parameters: the eigenvalues and the stability condition. Somewhat surprisingly, these two sets come symmetrically in the answer.
Up to Section 7 we work with nonpositive vector bundles, that is, vector bundles having no subbundle of positive degree. Without stability this restriction is inevitable as otherwise the moduli stacks would have infinite motivic volume. With a stability condition we can drop this technical restriction; the motivic classes of semistable parabolic Higgs bundles whose underlying vector bundles are not necessarily nonpositive are calculated in Section 7.
In Section 8 we study the moduli stack of bundles with connections-with or without stability condition. In Section 9 we give a precise criterion for when moduli stacks of Higgs bundles, bundles with connections, or semistable bundles with connections are non-empty.
1.9. Acknowledgements. We thank E. Diaconescu, J. Heinloth, O. Schiffmann, and especially A. Mellit for useful discussions and correspondence. A part of this work was done while R.F. was visiting Max Planck Institute of Mathematics in Bonn, and a part when he was visiting A. Mellit at the University of Vienna. The work of R.F. was partially supported by NSF grant DMS-1406532. A.S. and Y.S. thank IHES for excellent research conditions and hospitality. The work of Y.S. was partially supported by NSF grants and Munson-Simu Faculty Award at Kansas State University.

Preliminaries
2.1. Conventions. We denote by k a field of characteristic zero. We denote by X a smooth projective geometrically connected curve over k (recall that geometric connectedness means that X remains connected after the base change to an algebraic closure of k). We denote by D a set of k-rational points of X and by deg D the number of elements of D.
If E is a vector space or a vector bundle, we denote by E ∨ the dual vector space (resp. vector bundle). We identify vector bundles with their sheaves of sections. If F is a coherent sheaf, we denote by End (F ) the sheaf of its endomorphisms, we have End (F ) = F ∨ ⊗ F if F is a vector bundle.
We denote by gl r,k or simply by gl r the set of r × r matrices with entries in k. We say that a nilpotent matrix n ∈ gl |λ| = gl |λ|,k is of type λ, if for all i ≥ 1 we have dim Ker n i − dim Ker n i−1 = λ i . For each partition λ choose a nilpotent matrix n λ of type λ. For concreteness, we can take for n λ the direct sum of nilpotent Jordan blocks, where the number of blocks of size i × i is equal to λ i − λ i+1 .
2.1.2. Stacks. We will be working with stacks. All our stacks will have affine stabilizers. Our stacks will be Artin stacks locally of finite type except in Section 4.3, where we will have to work with stacks whose points have stabilizers of infinite type. For a stack S we often abuse notation by writing s ∈ S to mean that s is an object of the groupoid S(k), or an object of the groupoid S(K), where K is an extension of k. Following [LMB,Ch. 5] we say that a K ′ -point ξ ′ of S is equivalent to a K ′′ -point ξ ′′ of S if there is an extension K ⊃ k and k-embeddings K ′ ֒→ K, K ′′ ֒→ K such that ξ ′ K is isomorphic to ξ ′′ K (as an object of S(K)). The corresponding equivalence classes are called points of S; the set of points is denoted by |S|. See [FSS,Sect. 2] for more details.
We write "morphism of stacks" to mean "1-morphism of stacks".

Motivic functions and motivic classes.
Recall that in [FSS,Sect. 2] we defined (following [Eke1, Sect. 1], [Joy], and [KS1]) the ring of motivic classes of Artin stacks denoted Mot(k). More generally, for an Artin stack X locally of finite type over k, we defined the Mot(k)-module of motivic functions on X denoted Mot(X ). For a morphism f : X → Y we have the pullback homomorphism f * : Mot(Y) → Mot(X ). The pushforward homomorphism f ! : Mot(X ) → Mot(Y) is defined when f is of finite type. We also defined the ring of completed motivic classes, denoted Mot(k), and Mot(k)-modules of completed motivic functions Mot(X ) with a (probably non-injective) morphism Mot(X ) → Mot(X ). We also defined the pullbacks and the pushforwards of completed motivic functions.
We usually work with Mot(k) but our final results are formulated in Mot(k).
We defined the notion of a constructible subset of a stack. If X → Y is a morphism of finite type, and S ⊂ X is a constructible subset, we defined the motivic function [S → Y] ∈ Mot(Y). Recall [FSS,Prop. 2.6.1]: Proposition 2.2.1. Assume that we are given A, B ∈ Mot(X ) are such that for all fields K and for all morphisms ξ : Spec K → X we have ξ * A = ξ * B. Then A = B.
Remark 2.2.2. The previous proposition is one of the reasons we have to work with arbitrary fields. Indeed, even if we start with k = C, to be able to apply the proposition we have to consider all finitely generated extensions of C; see, for example, Section 4.3.17.
In Section 9 we will need the following proposition. Proof. The 'if' direction is obvious. For the other direction assume for a contradiction that S is a non-empty Artin stack of finite type over k such that [S] = 0 ∈ Mot(k). This means that for all m ∈ Z we have [S] ∈ F m Mot(k), where F • is the dimensional filtration on Mot(k). According to [Kre,Prop. 3.5.6,Prop. 3.5.9] every Artin stack of finite type with affine stabilizers has a stratification by global quotients of the form T /GL n , where T is a scheme. Thus, replacing S with a stratification and clearing the denominators, we may assume that S is a disjoint union of integral affine schemes. Recall from [FSS,Sect. 2.5] that Mot(k) is the localization of the K-ring of varieties Mot var (k) with respect to the multiplicative set generated by L and L i − 1, where i > 0. Thus, multiplying S by a certain product of these elements, we may assume that the class of S in Mot var (k) belongs to the subgroup F m−1 Mot var (k) generated by the classes of the varieties of dimension at most m − 1, where m = dim S. Compactifying each top-dimensional connected component of S and taking the resolution of singularities, we may assume that S is the disjoint union of smooth projective k-varieties.
Recall that the Hodge-Deligne polynomial of a smooth projective variety X is dim X p,q=0 (−1) p+q h p,q (X)u p v q , where h p,q = dim H q (X, ∧ p Ω X ). This extends uniquely to a homomorphism E : Mot var (k) → Z [u, v]. Clearly, E([X]) has degree 2m, if X is a smooth projective variety of dimension m. On the other hand, E([X]) has degree at most 2m − 2, if X is any variety of dimension at most m − 1. We see that, on the one hand E(S) has degree 2m, on the other hand it has degree 2m − 2. We come to contradiction.
2.3. Principal bundles and special groups. Let H be an algebraic group of finite type over k. Recall that a principal H-bundle over a k-stack B is a stack E together with a schematic smooth surjective morphism of finite type E → B and an action a : H × k E → E such that H acts simply transitively on the fibers of E → B. More precisely, the simple transitivity means that the morphism (a, p 2 ) : compatible with the action and the projection to B. If E is a principal H-bundle over B and B ′ → B is a morphism, then one gets an induced principal H-bundle E × B B ′ over B ′ . The group H is called special if every principal H-bundle E over a scheme B of finite type over k is locally trivial over B in the Zariski topology.
The following lemma is standard, see e.g. [BD,Sect. 2 Proof. The case when B is a scheme is easily proved by Noetherian induction. If B is a stack, then, using again [Kre,Prop. 3.5.6,Prop. 3.5.9], we may assume that B is a global quotient: B = S/GL n , where S is a scheme. Then we have the cartesian diagram Recall that a k-group U is unipotent if it can be embedded into a group of strictly upper triangular matrices. Every unipotent subgroup is obtained from the additive group of the 1-dimensional vector space by iterated extensions.
Lemma 2.3.2. Let H be an algebraic group of finite type over k and let U be a unipotent subgroup. Assume that H/U is special. Then H is special. In particular, every unipotent group is special.
Proof. Assume first that H is a unipotent group. We claim that every principal H-bundle over an affine scheme is trivial. We prove this by induction on dim H.
We see that a principal H-bundle is trivial over an affine scheme if the induced principal H/U -bundle is trivial. The lemma follows.
Proof. It is well-known that the quotient of Z λ by its unipotent radical is the product of GL ri for some r i ∈ Z >0 . It remains to note that GL ri are special groups and the product of special groups is special.

Parabolic bundles
3.1. Recall that k denotes a field of characteristic zero, X stands for a smooth projective geometrically connected curve over k, and D is a set of rational points of X. We often assume that D = ∅; in this case X has a divisor of degree one defined over k. We will often have to consider sequences indexed D × Z >0 or D × Z ≥0 . A typical notation will be r •,• . If x ∈ D, then r x,• stands for the sequence r x,1 , r x,2 , . . . (or r x,0 , r x,1 , . . . ).
The monoid of all sequences r •,• indexed by D × Z >0 with terms r x,j in a commutative monoid S and such that r x,j = 0 for j ≫ 0 (that is, functions on D × Z >0 with finite support) will be denoted by S[D × Z >0 ].
We have the category Bun par (X, D) of parabolic bundles. We sometimes denote a parabolic bundle by a single boldface letter: . We note that the decomposition of a parabolic bundle into a direct sum of indecomposable vector bundles is unique up to isomorphism, while the isotypic summands are unique; the proof is similar to [Ati1,Thm. 3] (see also [FSS,Prop. 3

.1.2]).
We often skip X and D from the notation, writing Bun par instead of Bun par (X, D).
Abusing notation, we denote by Bun par the stack of objects of the category Bun par . Precisely, if S is a k-scheme, then Bun par (S) is the groupoid of collections (E, E •,• ), where E is a vector bundle over S × k X, E x,• is a filtration by vector subbundles of the restriction of E to S × k x for x ∈ D.

3.2.
Monoids Γ + and Γ ′ + . Consider the free abelian group Z × Z[D × Z >0 ] × Z and its subgroup Γ consisting of (r, r •,• , d) such that for all x ∈ D we have ∞ j=1 r x,j = r. Let Γ + ⊂ Γ be the monoid of sequences (r, r •,• , d) such that (i) r ≥ 0 and for all x ∈ D and j > 0 we have r x,j ≥ 0; (ii) if r = 0, then d = 0. Note that it follows from these conditions that r = 0 implies that (r, r •,• , d) is the zero sequence; we denote it by 0. Define the class function: For γ = (r, r •,• , d) ∈ Γ we set rk γ := r. For a parabolic bundle E we set rk E := rk cl(E). For γ ∈ Γ + , we denote by Bun par γ the stack of objects of class γ; this is an open and closed substack of Bun par . It is often convenient to think of Bun par = γ∈Γ+ Bun par γ as a Γ + -graded stack. Note that Bun par 0 has a single object: the vector bundle of rank zero.
Let γ = (r, r •,• , d) ∈ Γ + . The projection Bun par γ (X, D) → Bun r,d (X) to the stack of rank r degree d vector bundles on X is schematic and of finite type (in fact, projective). Thus Bun par γ (X, D) and Bun par (X, D) are Artin stacks locally of finite type.
Let us call a vector bundle on X nonpositive if it does not have a subbundle of positive degree. Clearly, a vector bundle E is nonpositive if and only if its dual E ∨ has no quotient vector bundles of negative degree. By [FSS,Lemma 3.2.1] a bundle E is nonpositive if and only if its dual E ∨ is HN-nonnegative in the sense that its Harder-Narasimhan spectrum is nonnegative (cf. [FSS,Sect. 3.2]). Now it follows from loc. cit. that there is an open substack Bun − (X) ⊂ Bun(X) classifying nonpositive vector bundles. Set In other words, Bun par,− is the stack (and the category) of parabolic bundles on X whose underlying vector bundle is nonpositive. Set also  Let Γ ′ + be the submonoid of Γ + consisting of sequences with d ≤ 0. Clearly, Bun par,− γ = ∅ only if γ ∈ Γ ′ + . 3.3. Categories over Bun par and Γ + -graded stacks. We will consider below many categories with a forgetful functor to Bun par (e.g., the category of parabolic Higgs bundles). Let C be such a category and denote by C its stack of objects as well. Assume that the morphism C → Bun par is of finite type. Define the stacks Moreover, the stack C − is of finite type as a graded stack, that is, its graded components are stacks of finite type.
The group ring Mot(k)[Γ + ] is an instance of a quantum torus (cf. [KS1]). This has a natural basis e γ , where γ ranges over Γ + ; the multiplication is given by e γ e γ ′ = e γ+γ ′ . The reason that multiplication is commutative is that we are actually working with 2-dimensional Calabi-Yau categories. Let Mot(k)[[Γ + ]] be the completion of Mot(k)[Γ + ] (this can be viewed as the group of Z-valued functions on Γ + ).
If D is a Γ + -graded stack of finite type, we define its motivic class (also known as motivic Donaldson-Thomas series) as Of course, this can also be thought of as the generating function for the motivic classes of stacks D γ . Sometimes it is convenient to write e γ explicitly as The variables w, z, and w x,j for x ∈ D, j ≥ 1 are commuting variables.
Remark 3.3.1. Note that we do not fix the lengths of flags. Let us fix a function l : D → Z >0 and consider only flags of length at most l(x) at x. Let Γ +,l be the submonoid of Γ + consisting of sequences (r, r •,• , d) such that r x,j = 0 whenever j > l(x). We have the obvious projection Γ + → Γ +,l , which induces a homomorphism is the motivic class of the substack of D where the lengths of the flags are bounded by l. We see that the difference between fixing the length of flags and allowing flags of arbitrary lengths corresponds on the quantum torus side to the difference between polynomials in infinite number of variables and polynomials in finite number of variables.
In our applications, [D] will be symmetric in each sequence of variables w x,• (cf. Remark 4.2.1). In this case, the difference between fixing and not fixing the lengths corresponds on the side of motivic classes to the difference between symmetric polynomials and symmetric functions, cf. [Mac,Ch. 1,Sect. 2].
] whenever C is a stack of finite type over Bun par . This is in accordance to the general theory in [KS1], where one fixes a strict sector in R 2 in order to have well-defined Donaldson-Thomas invariants. We can also replace Mot(k) with Mot(k) in all the above constructions.
3.4. Motivic zeta-functions and plethystic operations. Following [Kap], for a variety Y define its motivic zeta-funcion by where Y (n) = Y n /Σ n is the n-th symmetric power of Y (Σ n denotes the group of permutations). Consider the group (1 + z Mot(k) [[z]]) × , where the group operation is multiplication. According to [Eke2,Thm .2.3] ζ can be uniquely extended to a homomorphism such that we have the identity ζ L n A (z) = ζ A (L n z) for all n ∈ Z and A ∈ Mot(k). In other words, we equip Mot(k) with a lambda-ring structure. Note that Mot(k) is not a special lambda-ring, in particular, ζ(AB) cannot be expressed in terms of ζ(A) and ζ(B) (so some authors would call this a pre-lambda-ring structure). According to loc. cit., this homomorphism ζ is continuous with respect to the dimensional filtration on Mot(k), so it extends to a homomorphism which coincides with the one constructed in [FSS,Sect. 1 ] 0 stand for the series without constant term. We define the plethystic exponent Exp : One shows easily that this is an isomorphism of abelian groups. Denote the inverse isomorphism by Log (the plethystic logarithm). Finally, we define the plethystic power by We note that we can similarly define Exp, Log and Pow for the completed ring Mot(k), which coincide with the operations defined in [FSS] when D = ∅.
Thus, the quotient parabolic bundles of E are also in bijective correspondence with saturated subbundles of E. Finally, in the above situation, we say that One can use the short exact sequences above to define the group K 0 (Bun par ); the class function cl extends to cl : K 0 (Bun par ) → Γ.
Remark 3.5.1. The category Bun par is not abelian. It can be extended to an abelian category by viewing vector bundles with flags as coherent sheaves on orbifold curves. Then the abelian category is the category of coherent sheaves on this orbifold. This extension will not be used in the current paper. On the other hand, if we define short exact sequences in Bun par as sequences isomorphic to some sequence of the form (7), then Bun par becomes an exact category in the sense of Quillen.
Let ϕ : E → F be a morphism of vector bundles on X. We say that ϕ is generically an isomorphism if it is an isomorphism at the generic point of X. Equivalently, ϕ is an isomorphism over a non-empty Zariski open subset of X. Another reformulation is that ϕ is injective and F/ϕ(E) is a torsion sheaf. Sometimes one says in this situation that E is a lower modification of F . Definition 3.5.2. We say that a morphism of parabolic bundles ϕ : Lemma 3.5.3. Let ϕ : E → F be a morphism of parabolic bundles. Then there are strict parabolic subbundles E ′ ⊂ E and F ′ ⊂ F such that ϕ can be decomposed as where ϕ 1 is the canonical projection, ϕ 2 is generically an isomorphism, ϕ 3 is the canonical embedding.
, let ϕ ′ : E → F be the underlying morphism of vector bundles. Note that Ker ϕ ′ is a vector subbundle of E. Indeed, E/ Ker ϕ ′ is isomorphic to a subsheaf of F , so it is torsion free. Let E ′ be the strict subbundle of E whose underlying vector bundle is Ker(ϕ ′ ). Let F ′ be the saturation of the image of ϕ ′ (that is, F ′ is the unique saturated vector subbundle of F containing ϕ ′ (E) such that the quotient F ′ /ϕ ′ (E) is a torsion sheaf). Let F ′ be the strict subbundle of F whose underlying vector bundle is F ′ . Now the existence of the decomposition is clear.
3.6. Generalized degrees and slopes. Let A be a Q-vector space (in applications it will be k or R). Let We define the (κ, ζ)-slope of γ by deg κ,ζ γ/ rk γ. We write deg κ,ζ E for deg κ,ζ cl(E) and call We remark that it is common to write ζ ⋆ γ for deg 0,ζ γ and deg ζ γ for deg 1,ζ γ but we prefer a uniform notation. We also remark that for an exact sequence (7) 3.7. Parabolic weights and stability conditions. The following definition should be compared to [Mel1,Def. 6.9].
Definition 3.7.1. We say that a sequence σ = σ •,• of real numbers indexed by D × Z >0 is a sequence of parabolic weights if for all x ∈ D we have and for all x and j we have σ x,j ≤ σ x,1 + 1.
To every sequence of parabolic weights we will associate a notion of stability on parabolic bundles in Definition 3.7.2 below. Thus we denote the set of all sequences of parabolic weights by Stab = Stab(X, D).
Proposition 3.7.3. Let E ∈ Bun par be a parabolic bundle. Then there is a unique filtration 0 = E 0 ⊂ E 1 ⊂ . . . ⊂ E m = E by strict parabolic subbundles such that all the quotients E i /E i−1 are σ-semistable and we have Proof. We start with a Lemma.
On the other hand, for all x ∈ D and The rest of the proof is completely analogous to the proof of [HN,Sect. 1.3] in view of Lemma 3.5.3.
In the situation of Proposition 3.7.3(i) we say that the filtration is the Harder-Narasimhan filtration of E (or HN-filtration for short) and τ 1 > . . . > τ m is the σ-HN spectrum of E. We define Bun par,≤τ and Bun par,≥τ as full subcategories of Bun par whose objects are parabolic bundles with σ-HN spectrum contained in (−∞, τ ] (resp. [τ, ∞)). We emphasize that the categories Bun par,≤0 and Bun par,− should not be confused with each other: they coincide only if σ = 0. The following lemma is standard.

Parabolic pairs
4.1. The notion of parabolic pair, interesting by itself, will be used as a technical tool for studying parabolic Higgs bundles in Section 5 and parabolic bundles with connections in Section 8. Our main results in this section are Theorem 4.6.1 and Corollary 4.6.3. They give explicit answers for motivic classes of stacks of nilpotent parabolic pairs and parabolic pairs respectively. We will also give a simplified answer in the case X = P 1 in Section 4.7.
Definition 4.1.1. A parabolic pair (E, Ψ) consists of a parabolic bundle E = (E, E •,• ) ∈ Bun par (X, D) and its endomorphism Ψ (that is, an endomorphism of E preserving each E x,j ). If Ψ is nilpotent we will speak about nilpotent parabolic pairs.
Parabolic pairs as well as nilpotent parabolic pairs form an additive k-linear category denoted Pair = Pair(X, D) (resp. Pair nilp = Pair nilp (X, D)). Again, we abuse notation by denoting the stacks of objects by the same symbols. We define Pair γ , Pair nilp γ , Pair − γ , Pair nilp,− γ etc. following the general construction (4) of Section 3.3.
The forgetful morphisms Pair → Bun par and Pair nilp → Bun par are schematic and of finite types. In particular, Pair − and Pair nilp,− are Γ ′ + -graded Artin stacks of finite type (in the graded sense). Let K ⊃ k be an extension and (E, E •,• , Ψ) ∈ Pair nilp (K). If we trivialize E at the generic point of X K = X × k Spec K, Ψ becomes a rk E × rk E nilpotent matrix. Its Jordan type is a partition λ of rk E. Thus we get a locally closed stratification of Pair nilp according to the generic Jordan type of the nilpotent endomorphism where the disjoint union is over all partitions. In other words, Pair nilp (X, D, λ) classifies nilpotent parabolic pairs such that the endomorphism is generically conjugate to n λ (that is, conjugate to n λ at the generic point of X, or, equivalently, at each point of a non-empty Zariski open subset of X). We remark that any endomorphism generically conjugate to n λ is necessarily nilpotent. Again, using the general formalism (4) of Section 3.3 we define the Γ ′ + -graded stacks Pair nilp,− (X, D, λ) 4.2. Motivic classes of parabolic bundles with nilpotent endomorphisms. Our goal in this section is to calculate the graded motivic class (that is, the motivic Donaldson-Thomas series, cf. [KS1]) The partition λ is fixed through the end of Section 4.3. This motivic class is calculated as follows. First, in Section 4.3 we write this motivic class as the product of a term that is independent of the parabolic structures, and the "local" terms independent of the curve. The first term has been calculated in [FSS]. The terms corresponding to parabolic structures are enough to calculate when X = P 1 . More precisely, we will work with P 1 and two points with parabolic structures (that is, D = {0, ∞}) but we will calculate the sum over all partitions (Section 4.4). This part is very similar to [Mel1,Sect. 5.4]. In Section 4.6 we give the explicit answer for the motivic classes under consideration. Using the formalism of plethystic powers we then easily calculate the class of parabolic bundles with not necessarily nilpotent endomorphisms.
Remark 4.2.1. We will see in Theorem 4.6.1 that (10) is a symmetric function in w x,• for each x ∈ D. This can be explained as follows. Note that the "Weyl group" W := x∈D Σ ∞ acts on Γ + and Γ ′ + in the obvious way (here Σ ∞ is the inductive limit of the permutation groups Σ l ). Using the commutativity of the motivic Hall algebra of the category of representations of the Jordan quiver (the quiver with one vertex and one loop), one can easily show that the motivic classes [Pair nilp,− γ (X, D, λ)] are W -invariant. Thus, we can re-write (10) as where the summation is only over r •,• such that we have r x,j ≥ r x,j+1 for all x and j (that is, r x,• is a partition of |λ|). Here, for a partition µ, m µ is the symmetric function equal to the sum of all monomials whose ordered list of exponents is µ.

4.3.
Factorization of graded motivic classes of stacks of nilpotent parabolic pairs. In this section we factorize (10) as the product of the global part (depending only on X but not on D) and the local parts corresponding to points of D (but independent of X). This is a motivic version of [Mel1,Theorem 5.6]. We follow the same ideas, though some parts of Mellit's proof do not work in the motivic case and must be replaced by different arguments. On the other hand, we were able to simplify some parts of Mellit's proof, in particular, by working with stacks.
Note that Pair nilp,− (X, ∅, λ) classifies pairs (E, Ψ), where E is a nonpositive vector bundle on X, Ψ is an endomorphism of E generically conjugate to n λ but there are no parabolic structures. Proof. It is enough to show that the degree 0 part [Pair nilp,− |λ|,0 (P 1 , ∅, λ)] is invertible in Mot(k). Note that a nonnegative vector bundle of degree 0 on P 1 is necessarily trivial. Thus, this degree zero part classifies pairs (E, Ψ), where E is a trivial vector bundle and Ψ is a constant endomorphism conjugate to n λ . It is easy to see that this stack is isomorphic to the classifying stack of the centralizer Z λ of n λ . By Lemma 2.3.3 Z λ is special. Now it follows from Lemma 2.3.1 or [FSS,Lemma 2.2.3] that [Pair nilp,− |λ|,0 (P 1 , ∅, λ)] = 1/[Z λ ].
We need some notation. Note that [Pair nilp,− (P 1 , denote the result of replacing w ∞,• by w x,• in this series.
The proof of the theorem occupies the rest of Section 4.3; it is based on the local study of stacks in the formal neighborhood of D.
The main idea of the proof is very simple. Let us assume that D = {x} is a single rational point of X. In Section 4.3.9 we will define the stack Pair loc,f l classifying triples (F, Φ, F • ), where F is a rank |λ| vector bundle over the formal completion of X at x, Φ is a nilpotent endomorphism of F generically conjugate to n λ , F • is a flag in F x preserved by Φ(x). We have an obvious restriction morphism Pair nilp,− (X, x, λ) → Pair loc,f l . We will see in Lemma 4.3.14 that this restriction morphism has constant fiber. Thus, one is tempted to write the graded motivic class [Pair nilp,− (X, x, λ)] as the product of the motivic class of this fiber and of Pair loc,f l . This would quickly lead to the proof of the theorem. Unfortunately, Pair loc,f l is not an Artin stack as its points have inertia groups of infinite type, so its motivic class does not make sense. The major part of the proof consists of going around this problem.
Let us give the overview of the proof. In Section 4.3.3 we define and study the schemes of jets into gl |λ| . In Section 4.3.9 we study the local stacks; they are essentially the quotients of the schemes of jets by GL |λ| . In Section 4.3.10 we re-write the theorem as a statement about motivic classes of graded components. In Section 4.3.12 we study the fibers of the localization map; this is the main part of the proof. We complete the proof in Section 4.3.17. 4.3.3. Jets. We will denote the non-archimedean local field k((t)) by K and its ring of integers k[[t]] by O. The order of pole at z = 0 gives rise to a valuation map val : K → Z ∪ {−∞}, where val(0) = −∞. Clearly, val extends to gl r,K as the maximum of valuations of all matrix elements. Let J(X) denote the jet scheme of a scheme X (this is a scheme of infinite type), and let J N (X) denote the scheme of order N − 1 jets. In particular, J 1 (X) = X.
For an algebraic group G of finite type over k we have the jet group G O := J(G) and the jet group of finite type J N (G). The N -th congruence subgroup G (N ) is the kernel of the projection G O → J N (G). We also have the ind-group of loops G K containing G O . Let ∆ := Spec O be the formal disc and∆ := Spec K be its generic point (the punctured formal disc). Also set ∆ N := Spec k[[t]]/t N . The groups GL r,O , GL r,K , and J N (GL r ) are the groups of automorphisms of the trivial vector bundles on ∆,∆, and ∆ N respectively. For more details on the jet and loop groups we refer the reader to [Sor].
Set r = |λ|. Consider the orbit stratification of gl r under the adjoint action of GL r : Note that J(λ) parameterizes morphisms ∆ → gl r such that the image of the generic point of ∆ is in O λ , that is, jets that are generically conjugate to n λ . Definition 4.3.4. We say that a loop g ∈ GL r,K (k) = GL r (K) is kernel-strict if g −1 n λ g ∈ gl r,O and g induces an isomorphism between the O-modules Ker n λ ⊗ k O and Ker(g −1 n λ g).
Remark 4.3.5. Note that our definition is a little different from [Mel1,Def. 3.8]. Mellit's definition of kernelstrictness depends also on a choice of a matrix θ. In terminology of Mellit our g is kernel-strict for θ = g −1 n λ g. Note also that the results of Mellit we are using here and below are formulated over finite fields but are valid over any field, proofs being the same.
Let Φ be a k-point of J(λ), then there is a kernel-strict g ∈ GL r,K (k) = GL r (K) such that Φ = g −1 n λ g. Set deg Φ := val(det g). The existence of such g and independence of the degree on the choice of g is proved in [Mel1,Lm. 3.7]. It follows also from loc. cit. that the degree is nonnegative.
If K ⊃ k is a field extension, we similarly define the degree of a K-point of J(λ). The degree is compatible with field extensions. Thus we get a stratification of the set of points of J(λ):  Proof. By [Mel1,Lm. 3.7] there is N 0 ≥ 1 (depending on λ and d) such that for all Φ ∈ J d (λ) there is a kernel-strict g with val(g) < N 0 such that gΦg −1 = n λ . Then val(det g) = d. Set N 1 := rN 0 + d. Then by Cramer's rule val(g −1 ) < N 1 . We also have val(g) < N 1 . Take N (d, λ) := 4N 1 . With this choice of N (d, λ) part (i) of the proposition is clear. To . Choose a kernel-strict g with val(g) < N 1 , val(g −1 ) < N 1 such that gΦg −1 = n λ . Then gΦ ′ g −1 ≡ n λ (mod z 2N1 ).
Note that g −1 g ′ g ∈ GL r,O . It is easy to see that the set of kernel-strict loops is invariant under the multiplication by points of GL r,O on the right, so g ′ g = g(g −1 g ′ g) is kernel-strict. Clearly, val(det(g ′ g)) = val(det g) = d, so (i) follows. Further, g −1 g ′ g conjugates Φ ′ to Φ, so (iii) follows as well.
For every d ≥ 0 we fix N (d, λ) satisfying the conditions of the above proposition. According to the proposition, the degree of a stabilized jet is well-defined and any two lifts of a stabilized jet to an infinite jet are conjugate by an element of GL r,O . Note that for every jet Φ ∈ J d (λ) its truncation π N (Φ) is stabilized for N large enough. 4.3.9. Local stacks. Consider the quotient stack Pair loc = Pair loc (λ) := J(λ)/GL r,O , where GL r,O acts by conjugation. We skip λ from the notation as it is fixed through the end of Section 4.3. Since the degree function on J(λ) is GL r,O -invariant, we get the degree function on the points of Pair loc .
Note that Pair loc classifies pairs (F, Φ), where F is a rank r = |λ| vector bundle over ∆, Φ is a nilpotent endomorphism of F generically conjugate to n λ . This follows from the fact that every vector bundle on ∆ is trivial. It also follows that every K-point of Pair loc is isomorphic to a point of the form (O r , Φ), where Φ ∈ gl r,O .
We emphasize that Pair loc is not an Artin stack (its isotropy groups are not of finite type). Define Pair loc N := J N (O λ )/J N (GL r ); this is an Artin stack of finite type. The points of Pair loc N are the pairs (F, Φ) where F is a vector bundle on ∆ N , Φ is an endomorphism of F such that if we trivialize F , Φ becomes a jet with values in O λ . We say that (F, Φ) is stabilized if Φ is stabilized in the sense of Definition 4.3.8. Note that this does not depend on the trivialization of F . If (F, Φ) is stabilized, then we have a well-defined notion of the degree of (F, Φ). Explicitly, we can lift (F, Φ) to a point (O r , Φ) of Pair loc , and the degree of (F, Φ) is equal to the degree of Φ ∈ gl r,O .
We define the stack Pair loc,f l as the stack classifying triples (F, Φ, F • ), where (F, Φ) is a point of Pair loc , F • is a flag in F 0 preserved by Φ(0). We define the stack Pair loc,f l N as the stack classifying triples (F, Φ, F • ), where (F, Φ) is a point of Pair loc N , F • is a flag in F 0 preserved by Φ(0).

4.3.10.
Preparation for the proof of Theorem 4.3.2. We will assume that D = x is a single rational point of X. This will unburden the notation; the general case is proved similarly. Thus we want to prove that Equating the graded components, we see that this reduces to the following proposition.
Proposition 4.3.11. Let d be a nonpositive integer, r • = (r 1 , r 2 , . . .) be a sequence of nonnegative integers such that i r i = r = |λ|. Then we have in Mot(k): This proposition will be proved in Section 4.3.17. We emphasize that the sum is over all d ′ , d ′′ ∈ Z with d ′ + d ′′ = d but the terms are non-zero only if d ′ , d ′′ ∈ [d, 0]. We note that the RHS is manifestly independent of x. Thus, the LHS is independent of x as well.
4.3.12. The restriction to the formal neighborhood of x. We keep the simplifying assumption that D = {x} is a single point; we write r • instead of r x,• . Fix γ = (r, r • , d) ∈ Γ ′ + . For x ∈ X let O X,x be the local ring of x and O X,x be its formal completion. Set ∆ x := SpecÔ X,x . Choose a formal coordinate at x, use it to identify ∆ x with ∆ and the N -th infinitesimal neighborhood ∆ x,N of x with ∆ N . Consider the restriction morphism Similarly we have a morphism Our nearest goal is to describe the fibers of these morphisms. For a nonpositive integer e, let F ib e (X, x) denote the open substack of Pair nilp,− r,e (X, ∅, λ) consisting of (E, Ψ) such that Ψ is conjugate to n λ at x. Let F ib e (X, x) denote the stack of triples (E, Ψ, s), where (E, Ψ) is a point of F ib e (X, x), s is a trivialization of E over ∆ x such that Ψ = n λ in this trivialization.
, where e is the degree of (F, Φ).
(ii) Similarly, the fiber of loc x over (F, Φ) is isomorphic to F ib d+e (X, x), where e is the degree of (F, Φ).
Proof. We prove (ii) first. Fix a trivialization of F on the formal disc ∆. Then Φ becomes an element of gl r,K and we choose a kernel-strict g such that gΦg −1 = n λ . Then val(det g) = e. Denote the fiber under consideration by F ib. The fiber can be described as the stack of triples (E, Ψ, s), where E is a nonpositive vector bundle, Ψ is an endomorphism, s is the trivialization of E over ∆ x such that in this trivialization we have Ψ| ∆x = Φ. Note that such Ψ is automatically conjugate to n λ at the generic point of X.
If (E, Ψ, s) is a point of F ib, then E| X−x is trivialized over the punctured disc∆ x , and we use the chosen above g to glue E| X−x with the trivial bundle k r ×∆ x on∆ x (we recall that g can be viewed as an automorphism of the trivial vector bundle on∆ x ). We obtain a new vector bundle E ′ with an isomorphism E ′ | X−x ≃ E| X−x and a trivialization over ∆ x . Thus Ψ gives rise to an endomorphism Ψ ′ of E ′ | X−x . It is easy to derive from the definition of g that in the given trivialization we have Ψ ′ |∆ x = n λ . Thus Ψ ′ extends to x and, moreover, in the given trivialization of E ′ over ∆ x we have Ψ ′ | ∆x = n λ .
Note that E ′ is nonpositive. Indeed, Ker Ψ is nonpositive as a subbundle of E. Since g is kernel-strict, the isomorphism between Ker Ψ and Ker Ψ ′ extends from X − x to X. Thus Ker Ψ ′ is also nonpositive. But by [Mel1,Prop 5.3] this implies that E ′ is nonpositive as well.
Next, we have an isomorphism between ∧ r E and ∧ r E ′ over X − x, and it has a zero of order val(det g) at x.
We have constructed a morphism F ib → F ib d+e (X, x). Conversely, given a point (E ′ , Ψ ′ , s ′ ) of F ib d+e (X, x), we use g and s ′ to construct a new bundle E with an isomorphism to E ′ over X − x and a trivialization over ∆ x . Then Ψ ′ | X−x give rise to an endomorphism of E| X−x and we check that it extends to x and, moreover, in the trivialization of E over ∆ x we have Ψ| ∆x = Φ. Now it is easy to see that the two constructions are inverse to each other. This proves (ii). Now (i) follows from the cartesian diagram where the vertical arrows correspond to forgetting the flags.
Consider now the compositions of (13) and (14) with restrictions to the N -th infinitesimal neighborhood of x. Let (F, Φ, F • ) be a point of Pair loc,f l r•,N and assume that (F, Φ) is stabilized. By Definition 4.3.8 we can find Φ ′ ∈ gl r,O such that (O r , Φ ′ ) lifts (F, Φ) and N > N (e, λ), where e is the degree of Φ ′ and N (e, λ) is from Proposition 4.3.6. Choose a kernel-strict g ∈ gl r,K such that gΦ ′ g −1 = n λ . Denote by Z (N ) g the intersection Z O ∩ (g −1 GL (N ) g) ⊂ GL r,K , where Z is the centralizer of n λ in GL |λ| . Recall that by Proposition 4.3.6(i) we may assume that the orders of the poles of g and g −1 are less than N/2 so we have Z (N ) g ⊂ Z (1) . This is a pro-unipotent group. Clearly, Z O (and thus Z (N ) g as well) acts on F ib d+e (X, x) by changing the trivialization of E on ∆ x . Lemma 4.3.14. (i) Let (F, Φ, F • ) ∈ Pair loc,f l r•,N be such that (F, Φ) is stabilized, choose g ∈ GL r,K as in the previous paragraph. Then the fiber of loc f l x,N over , where e is the degree of (F, Φ).
(ii) Similarly, the fiber of loc x,N over (F, Φ) is isomorphic to F ib d+e (X, x)/Z Proof. Let us prove (i), the proof of (ii) is completely analogous. Denote the fiber under consideration by F ib and let F ib be the fiber of loc f l x over (O r , Φ ′ ). Then we have a restriction morphism F ib → F ib. It follows from stability of (F, Φ, F • ) and Proposition 4.3.6(iii) that this morphism is surjective. On the other hand, it is easy to see that two points of F ib map to the same point of F ib if and only if they differ by an action of an element of gZ O g −1 ∩ GL (N ) . On the other hand, according to Lemma 4.3.13, F ib ≃ F ib d+e (X, x). One checks that under this isomorphism, the action of gZ O g −1 ∩ GL (N ) on F ib corresponds to the action of Z (N ) g on F ib d+e (X, x).
We need to calculate the motivic class of this fiber. Set Z g := Z O /Z (N ) g ; this is a group of finite type.
The lemma follows from these three equations.
Lemma 4.3.16. Let N be an integer larger than N (j, λ) for all j = 0, . . . , −d. Assume that the fiber of (16) over (F, Φ) is non-empty. Then (F, Φ) is stabilized. A similar statement holds for the fibers of (15).
Proof. We prove the statement about the fibers of (16), the other statement being analogous. Let (E, Ψ) be a point of the fiber, then the fiber of loc x over (E| ∆x , Ψ| ∆x ) is non-empty. Then by Lemma 4.3.13 this fiber is isomorphic to F ib d+e (X, x), where e is the degree of (E| ∆x , Ψ| ∆x ). Since F ib d+e (X, x) classifies nonpositive vector bundles, we get d + e ≤ 0. Thus N > N (e, λ) and we see that (F, Φ) is stabilized. are equal. The morphisms are loc f l x,N × loc ∞,N and loc f l ∞,N × loc x,N respectively. Let K ⊃ k be an extension and let ξ be a K-point of Pair loc,f l r•,N × Pair loc r,N represented by ((F, Φ, F • ), (F ′ , Φ ′ )). By Proposition 2.2.1 it is enough to show that ξ * A = ξ * B. Using base change, we may assume that K = k. According to Lemma 4.3.16, these motivic functions are zero unless (F, Φ) and (F ′ , Φ ′ ) are stabilized.
Let us lift (F, Φ) and Choose kernel-strict g, g ′ ∈ GL r,K such that g, g −1 , g ′ , and (g ′ ) −1 have poles of order less than N/2 and such that Then, according to Lemmas 4.3.14 and 4.3.15 (applied to X and P 1 ) we get Similarly, We see that ξ * A = ξ * B. This completes the proof of Proposition 4.3.11 and thus the proof of Theorem 4.3.2.
where Exp is the plethystic exponent defined in Section 3.4.
Proof. We note that the proof in [Mel1,Sect. 5.4] goes through in the motivic case as well. The only difference is that Mellit uses the Hall algebra of the Jordan quiver (that is, the Hall algebra of the category of vector spaces with nilpotent endomorphisms); this Hall algebra has to be replaced with the similar motivic Hall algebra in our case. Note that this motivic Hall algebra is just a subalgebra of the Hall algebra constructed in [FSS,Sect. 5] corresponding to torsion sheaves with one point support. Thus, most of the results on motivic Hall algebras needed for the calculation can be found in loc. cit.
where Hook(λ) stands for the set of hooks of λ, a(h) and l(h) stand for the armlength and the leglength of the hook h respectively.   [Mel1,Def. 2.5]. Property (iv) follows, for example, from the definition ofH given in [GH].
The modified Macdonald polynomials are shown to satisfy the condition in part (c) in [Mel1,Sect. 2.4]. Now we prove part (b  . Now condition (iii) implies that c λλ = 1. 4.6. Explicit formulas for the motivic classes of nilpotent pairs. Now we are ready to give the precise formula for [Pair nilp,− (X, D, λ)]. Recall that for a partition λ we defined J mot [FSS,Sect. 1.3.2]. In this paper, we will denote them by J mot λ,X (z) and H mot λ,X (z) respectively to emphasize that they depend on the curve X and to ensure that they are not confused with motivic modified Macdonald polynomialsH mot λ (w • ; z) and with motivic Hall-Littlewood polynomials H mot λ (w • ).
Inspecting the proof, we see that for each λ the summands are equal:

Thus
[Pair nilp,− (X, ∅, λ)] = w |λ| L (g−1) λ,λ J mot λ,X (z −1 )H mot λ,X (z −1 ). To be able to apply Theorem 4.3.2, we need the following lemma. Proof. Our proof is similar to that of [Mel1,Th. 5.5]. Let H ′ λ ∈ Mot(k) [[w • , z]] be the series such that [Pair nilp,− (P 1 ,∞,λ)] [Pair nilp,− (P 1 ,∅,λ)] = H ′ λ (w ∞,• ; z −1 ). Denote by C λµ the stack classifying pairs (E, Ψ), where E is a nonpositive vector bundle of rank |λ| on P 1 , Ψ is an endomorphism of E generically conjugate to n λ and conjugate to n µ at x = ∞. Then C λµ is graded by the degree of E, and we have [ where H mot µ are the motivic Hall-Littlewood polynomials. Note that C λµ = ∅ unless µ ′ ≺ λ ′ because for all i the dimension of the fiber of Ker Ψ i is semicontinuous on P 1 . Now it is easy to see that H ′ λ are symmetric functions with coefficients in Mot(k) [[z]]. We will use Proposition 4.5.2(b) to show that for all λ we have H ′ λ =H mot λ . To show that H ′ λ satisfy property (ii) of Proposition 4.5.2(a) it remains to show that [C λλ ] is invertible. This is completely similar to the proof of Lemma 4.3.1.
Proof. The argument is similar to the proof of [FSS,Prop 3.8.1]. In more details, let (E, Ψ, E •,• ) be a K-point of Pair − (X, D). According to [FSS,Lm. 3.8.3], we can uniquely decompose is the residue field of x i , and R k(xi)/K is the pushforward functor. It follows easily from the proof of [FSS,Lm. 3.8.3], that we can write uniquely [Pair nilp,− (P 1 , D, λ)] = w |λ| x∈DH λ (w x, where Hook(λ) stands for the set of hooks of λ, a(h) and l(h) stand for the armlength and the leglength of the hook h respectively. Arguing as in Corollary 4.6.3, we get in Mot(k) , L .

Parabolic Higgs bundles with fixed eigenvalues
5.1. Let X and D be as above. Our goal in this section is to calculate the motivic Donaldson-Thomas series of the category of parabolic Higgs bundles. More precisely, we calculate the motivic classes of the moduli stacks of Higgs bundles with fixed eigenvalues and with nonpositive underlying vector bundles. Our argument is similar to [FSS,]. The main result is Corollary 5.4.6. We denote by Ω X the canonical line bundle on X.
We denote the category (and the Artin stack) of parabolic Higgs bundles by Higgs = Higgs(X, D). We define the Γ ′ + -graded stack Higgs − = Higgs − (X, D) following the general formalism of Section 3.3, that is, Higgs − is the open substack of Higgs corresponding to Higgs bundles with nonpositive underlying vector bundle. Clearly, this stack is of finite type in the graded sense (that is, the graded components are of finite type). In this section X and D are fixed, so we skip them from the notation.

5.2.
Existence of Higgs fields with prescribed residues. To determine a criterion for the existence of a Higgs bundle with prescribed residues, we use an approach similar to [Ati2,Mih1,Mih2]. Let Φ : E → E⊗Ω X (D) be a morphism. In this case, for all x ∈ D we have a residue Res x Φ ∈ End(E x ).
Proposition 5.2.1. Let E be a vector bundle on X and let for x ∈ D, ρ x ∈ End(E x ) be an endomorphism of the fiber of E at x. There exists a Higgs field Φ : for all ϕ ∈ End(E), where tr stands for the trace.
Proof. Consider the short exact sequence of sheaves where K X is the constant sheaf corresponding to the function field of X. Let Ω K be the constant sheaf of meromorphic differential forms on X. We can obtain a new short exact sequence of sheaves: by taking the tensor product of the first sequence with End (E) ⊗ Ω X . Note that the middle term in this sequence is a constant sheaf, while the last term is an (infinite) direct sum of skyscraper sheaves. That is, , where (Ω K /Ω X ) x is the vector space of polar parts at x of meromorphic 1-forms, the summation is taken over all closed points of X, and i x : x → X is the inclusion. Passing to the long exact sequence for cohomology we obtain the following exact sequence of vector spaces: This implies that H 1 (X, End(E) ⊗ Ω X ) may be presented as the quotient of x∈X End(E x ) ⊗ (Ω K /Ω X ) x by the image of End(E) ⊗ Ω K (compare with the adelic description of cohomology given in [Ser,Ch. 2.,Sect. 5]). Further note that the required Higgs field Φ always exists locally, defined as Φ Under the above presentation of H 1 (X, End (E) ⊗ Ω X ), the local solutions Φ x define a cohomology class a(E, D, ρ • ) ∈ H 1 (X, End (E) ⊗ Ω X ). Moreover, it follows from the exact sequence that a(E, D, ρ • ) = 0 if and only if Φ can be defined globally.
Serre duality defines a bilinear pairing H 1 (X, End (E) ⊗ Ω X ) × End(E) → k. Using the above presentation for H 1 (X, End (E) ⊗ Ω X ) this pairing may be evaluated on a(E, D, ρ • ) as Since the pairing is perfect, x∈D tr(ρ x ϕ x ) = 0 for all ϕ ∈ End(E) if and only if a(E, D, ρ • ) = 0. The proof is complete.

Parabolic Higgs bundles with fixed eigenvalues.
Recall that k[D × Z >0 ] is the set of all sequences ζ = ζ •,• = (ζ x,j ) indexed by D × Z >0 such that ζ x,j = 0 for j ≫ 0. * For ζ ∈ k[D × Z >0 ] let Higgs(ζ) = Higgs(X, D, ζ) denote the full subcategory of Higgs (and its stack of objects) corresponding to collections x for all x ∈ D and j > 0. Again, the Γ ′ + -graded stack Higgs − (ζ) is defined following the formalism of Section 3.3.

Parabolic pairs with isoslopy underlying parabolic bundles. Recall that for
Let Pair Let K ⊃ k be a field extension. Let ξ : Spec K → Bun par,− γ be a point represented by a parabolic bundle E = (E, E •,• ). In view of Proposition 2.2.1, we only need to check that the ξ-pullbacks of (26) are equal. If E is not (0, ζ)-isoslopy, then, by Lemma 5.3.1, the pullbacks are equal to zero, so we assume that E is (0, ζ)-isoslopy.
Let Higgs(E, ζ) denote the space of Higgs fields on E with eigenvalues ζ (that is, the E-fiber of the projection Higgs(ζ) → Bun par ). By Lemma 5.3.1, Higgs(E, ζ) is non-empty, so it is a torsor over the vector space Higgs(E, 0). Thus, On the other hand, we have ξ * [Pair (0,ζ)−iso,− γ → Bun par,− γ ] = L dim End(E) . It remains to prove the following lemma. * According to our convention we should denote ζ by ζ•,• but it does not look nice in the formulas. Proof. Write E = (E, E •,• ). Let End (E) ⊂ End(E) be the subsheaf of endomorphisms preserving flags. One checks easily that the trace pairing gives an isomorphism between the dual sheaf End (E) ∨ ⊗Ω X and Higgs(E, 0), where Higgs(E, 0) stands for the sheaf of Higgs fields on E with zero eigenvalues. Thus by Riemann-Roch Theorem we have It remains to calculate deg End (E). For x ∈ D consider the fiber E x , its ring of endomorphisms End(E x ), its subspace V x of endomorphisms preserving the flag E x,• , and the quotient W x := End(E x )/V x . Further, consider the torsion sheaf W := ⊕ x∈D (i x ) * W x , where, as before, i x : x → X is the inclusion. We have an exact sequence and the lemma follows.
The lemma completes the proof of Proposition 5.4.1.
We note that the product makes sense because for a γ ∈ Γ ′ + there are only finitely many ways to write γ as the sum of elements of Γ ′ + . Also, the order of the multiples is irrelevant, since we are working with a commutative quantum torus.
Proof. The proof is almost the same as the proof of [FSS,Lm. 3.5.3 ] (see also [FSS,Prop. 3.5.1]).
We need some notation. Let us write where Log is the plethystic logarithm defined in Section 3.4, the summation is over all partitions. We note that B γ are W -invariant, where W = x∈D Σ ∞ (cf. Remark 4.2.1). Note also that B 0 = 0.
Definition 5.4.4. The motivic classes B γ ∈ Mot(k) are called motivic Donaldson-Thomas invariants of the pair (X, D).
where Exp is the plethystic exponent defined in Section 3.4.
Proof. First of all, using Corollary 4.6.3 and properties of plethystic operations, we get Now, it remains to use Proposition 5.4.3 and equate the slopes (cf. [FSS,Lm. 3.7.1]).
Corollary 5.4.6. We have in where Exp is the plethystic exponent defined in Section 3.4.
5.5. Case of P 1 . Assume now that X = P 1 . Then we have a simpler result. Moreover, it is more precise in the sense that we get an answer in Mot(k) rather than in Mot(k). Define the Donaldson-Thomas invariants B γ ∈ Mot(k) by We have precisely the same formula as in Corollary 5.4.6, where B γ is replaced with B γ (thus, the formula is valid in Mot(k)). The proof is the same as of Corollary 5.4.6 except that one uses (25) instead of Corollary 4.6.3. Comparing Corollary 4.6.3 with (25), we see that the images of B γ in Mot(k) are equal to B γ . 6. Stability conditions for Higgs bundles 6.1. Harder-Narasimhan filtration. Recall that in Section 3.7 we defined the set Stab of sequences of parabolic weights. To every sequence of parabolic weights we associated a stability condition on parabolic bundles in Definition 3.7.1. We want to extend this to Higgs bundles and to calculate the motivic classes of stacks of semistable parabolic Higgs bundles with nonpositive underlying vector bundles. Let X and D be as before and let σ ∈ Stab. (9) is satisfied for all strict subbundles preserved by Φ.
6.2. Kontsevich-Soibelman factorization formula. The general formalism of [KS1] implies the following factorization formula valid in Mot(k)[[Γ ′ + ]]. One can also give a direct proof along the lines of the proof of [FSS,Prop .3.6 Now, taking the plethystic logarithms of both sides and using Corollary 5.4.6, we get the following statement. 7. Stabilization 7.1. Let X and D be as before. We will be assuming that D = ∅. Note that this implies that X has a k-rational divisor of degree one. Set δ := max(2g − 2 + deg D, 0). Let σ ∈ Stab. Our goal in this section is to calculate the motivic class of the moduli stack of σ-semistable parabolic Higgs bundles without nonnegativity assumption. The main result in this section is Theorem 7.1.5. We start with the following analogue of [MS,Lm. 3.1].
Next, we have an analogue of [MS,Lm. 3.2].
Lemma 7.1.2. Let (E, Φ) be a σ-semistable Higgs bundle. Assume that deg 1,σ E < − r(r−1) 2 δ, where r = rk E. Then E ∈ Bun par,≤0 . † Note that all but countably many multiples are equal to one. We can understand the countable product as a clockwise product as in [KS1,KS2]. Note, however, that this is a product in a commutative ring.
Proof. Let τ 1 > τ 2 > . . . > τ m be the σ-HN-spectrum of E. Denote by r i the jumps of the ranks of σ-HNfiltration. By Lemma 7.1.1 we have and the statement follows.
Now we can formulate our first main result.
(ii) The stack Higgs σ−ss γ (ζ) is of finite type and we have in Mot(k) whenever N is large enough (it suffices to take N > |σ| + r−1 2 δ + d/r.) Proof. For part (ii) combine Corollary 7.1.4 with Proposition 6.2.1. Part (i) is clear from part (ii).
An immediate corollary of the above theorem and formula (31) is the following curious observation.

Motivic classes of parabolic connections
8.1. Let X and D be as above. Our goal in this section is to calculate the motivic classes of the moduli stacks of parabolic bundles with connections with prescribed eigenvalues of residues. In Section 8.3 we put stability conditions on these moduli stacks and calculate the motivic classes of substacks of semistable parabolic bundles with connections. Our argument is similar to the argument for Higgs bundles.
Let E be a vector bundle on X. A connection on E with poles bounded by D is a morphism of sheaves of abelian groups ∇ : E → E ⊗ Ω X (D) satisfying Leibniz rule. In this case for x ∈ D one defines the residue of the connection res x ∇ ∈ End(E x ).
We denote the category (and the Artin stack) of parabolic connections by Conn = Conn(X, D). In this section X and D are fixed, so we skip them from the notation.
Proof. The proof is completely analogous to the proof of Proposition 5.4.1 with Lemma 5.3.1 replaced by Lemma 8.1.2. 8.2. Stabilization of isoslopy parabolic bundles. As in Section 7 we will be assuming that D = ∅. Recall that this implies that X a k-rational divisor of degree one. As before, set δ := max(2g − 2 + deg D, 0).
Recall that every vector bundle E on X has a unique HN-filtration and the slopes of the quotients form a sequence called the HN-spectrum of E. We start with the following analogue of [MS,Lm. 4.1].
Proof. One shows that the extensions of a parabolic bundle E ′′ by a parabolic bundle E ′ (in the sense of Section 3.5) are classified by a vector space Ext 1 (E ′′ , E ′ ) dual to Hom(E ′ , E ′′ (Ω X (D)). Let 0 = E 0 ⊂ E 1 ⊂ . . . ⊂ E m = E be the Harder-Narasimhan filtration of E. Let E i be the strict parabolic subbundle with the underlying vector bundle E i . We have an exact sequence 0 → E i → E → E/E i → 0. Note that by the assumption the Harder-Narasimhan spectrum of E i is contained in [τ i , ∞), while the Harder-Narasimhan spectrum of (E/E i )(Ω X (D)) is contained in (−∞, τ i ). It follows that Hom E i , (E/E i )(Ω X (D)) = 0. Thus Next, we have an analogue of [MS,Cor. 4.2] whose proof is similar to loc. cit. and to that of Lemma 7.1.2.
Fix ζ ∈ k[D × Z >0 ]. Let | • | be any norm on the Q-vector subspace of k generated by the components of ζ. If k is embedded into C, we can take the usual absolute value for | • |. We set |ζ| := x∈D (max i |ζ x,i |). We have an analogue of [FSS,Lm. 3.2.3(i)] (cf. also Lemma 7.1.3). . Assume for a contradiction that E / ∈ Bun par,− . By Lemma 8.2.2, E is decomposable. Let E ′ be an indecomposable summand of E such that E ′ / ∈ Bun par,− . By the definition of isoslopy bundles, we have and Lemma 8.2.2 gives a contradiction.
Proof. Since X has a divisor of degree one, it has a line bundle of degree N . Tensorisation with this line bundle gives Pair (1,ζ)−iso γ
8.2.6. Case of P 1 . If X = P 1 , we obtain simpler and more precise results. Define B γ ∈ Mot(k) by (28). Define C γ (ζ) ∈ Mot(k) by the same formula (33) but with B γ replaced by B γ . Then Theorem 8.2.5 holds in Mot(k). 8.3. Stability conditions for bundles with connections. Recall that in Definition 3.7.1 we defined the notion of a sequence of parabolic weights. For non-resonant connections one can work with more general sequences of parabolic weights. Let us give the definitions. Definition 8.3.1. We say that ζ = ζ •,• ∈ k[D × Z >0 ] is non-resonant if for all x ∈ X and all i, j > 0 we have The importance of this definition is in the following lemma.
Lemma 8.3.2. Let ζ ∈ k[D×Z >0 ] be non-resonant and let ϕ be a morphism in Conn(ζ) such that the underlying morphism of vector bundles is generically an isomorphism. Then the underlying morphism of vector bundles is an isomorphism.
Proof. One easily reduces to the case k = C. Take x ∈ D. Since ζ is non-resonant, one can find a subset Ω of C containing {ζ x,j |j > 0} and such that the exponential function induces a bijection between Ω and C − 0. Then it is well-known that every regular connection on the punctured formal disc has a unique extension to the puncture such that the eigenvalues of the residues are in Ω. The statement follows.
Define the space of extended sequences of parabolic weights Stab ′ as the set of pairs (κ, σ), where κ ∈ R ≥0 , σ = σ •,• is a sequence of real numbers, indexed by D × Z >0 , such that for all x ∈ D we have (8).
Definition 8.3.3. Let (κ, σ) ∈ Stab ′ . A parabolic connection (E, ∇) is (κ, σ)-semistable if for all strict parabolic subbundles E ′ ⊂ E preserved by ∇ we have Proof. Write E = (E, E •,• ) and F = (F, F •,• ). Let ϕ : E → F be the underlying morphism of vector bundles. By Lemma 8.3.2, ϕ is an isomorphism. Thus dim E x,j ≤ dim F x,j for all x and j. Therefore In the resonant case, the proof is completely analogous to the proof of [HN,Sect. 1.3] in view of Lemma 3.7.4. (Cf. Propositions 3.7.3 and 6.1.2).
Remark 8.3.6. More generally, If ζ is not non-resonant, one can work with any (κ, σ) ∈ Stab ′ such that σ x,j − σ x,1 ≤ κ for all x and j. However, the notion of stability does not change if we scale (κ, σ). Thus, we can always assume that κ = 1, in which case σ ∈ Stab, or κ = 0, in which case σ = 0. The latter case corresponds to the trivial stability condition; the corresponding motivic class has been calculated in Theorem 8.2.5.
Similarly to Proposition 6.2.1 the Kontsevich-Soibelman factorization formula implies the following proposition.
We note that if κ = 0, and σ = 0, then this theorem is essentially Theorem 8.2.5.
It remains to show that the respective terms in the sums are equal. Clearly, we have N > 3|ζ| + (rk β i − 1)δ/2 and the statement follows from the induction hypothesis.
As usual, if X = P 1 we obtain similar formulas valid in Mot(k) by replacing B γ with B γ in (36). 9. Equalities of motivic classes and non-emptiness of moduli stacks 9.1. The main goal of this section is to give a criterion for when stacks Conn (κ,σ)−ss γ (ζ) or Higgs σ−ss γ (ζ) are non-empty. For stacks Conn γ (ζ) such a criterion follows easily from [CB1,CB2]. We reduce the case, when stability condition is present, to the case of Conn γ (ζ) using some equalities between motivic classes and Proposition 2.2.3.
It follows from Theorem 8.3.8 that the motivic class of Conn (κ,σ)−ss γ (ζ) depends only on the submonoid of Γ + given by the equations deg 1,ζ γ ′ = 0, deg κ,σ γ ′ = τ rk γ ′ , where τ = deg κ,σ γ/ rk γ. Using this fact, one can give a lot of examples of seemingly unrelated moduli stacks Conn (κ,σ)−ss γ (ζ) having the same motivic class. Note that replacing γ with γ − N (rk γ)1 shifts the (κ, σ)-slopes by κN . An analogous statement for moduli spaces of parabolic Higgs bundles is the content of Corollary 7.1.6 above. Finally, we can get a lot of equalities between motivic classes of parabolic Higgs bundles and motivic classes of connections. In the following proposition, we show that motivic classes of the form [Conn γ (ζ)] are universal.

Moreover, we have [Conn
Similarly we have the following proposition whose proof is also similar.
We emphasize that this algebraic result does not follow from the smooth isomorphism of Simpson. Nor the isomorphism of Simpson can be derived from the above result. We would also like to mention [HL,Thm. 4.2], where a stronger statement is proved in the case when parabolic structures are absent and the rank and degree are coprime. 9.3. Indecomposable parabolic bundles and non-emptiness of moduli stacks. 9.3.1. Indecomposable parabolic bundles. Here we recall results of [CB2]. Recall that X is a smooth projective curve of genus g, D ⊂ X(k) is a non-empty set. Let γ ∈ Γ + . We would like to know whether there exists an indecomposable parabolic bundle of class γ. The following simple statement is noted in [CB1,Introduction].
Lemma 9.3.2. Assume that k is algebraically closed. If g > 0, then for all γ ∈ Γ + there is an indecomposable parabolic bundle of class γ.
Proof. It is well-known that there is an indecomposable vector bundle on X of rank rk γ. Now one extends it arbitrarily to a parabolic bundle of class γ.
Next, let X = P 1 . Fix γ = (r, r •,• , d) ∈ Γ ′ and choose a sequence w • indexed by D such that r x,j = 0 for j ≥ w x . Consider the star-shaped graph G w• with vertices v * and v x,j where x ∈ D, j is between 1 and w x − 1. The vertex v * is connected to all the vertices of the form v x,1 , the vertex v x,i is connected to v x,i±1 (see the picture) .

Star-shaped graph
Consider the Kac-Moody Lie algebra g w• associated to this graph. Let Λ w• be the root lattice of g w• ; we identify it with the free abelian group generated by the set of vertices. Then γ gives rise to an element of Λ w• given by Now [CB2, p. 1334, Corollary] can be re-formulated as follows.
Proposition 9.3.3. In the above notation, there is a non-zero indecomposable parabolic bundle E ∈ Bun par γ if and only if ρ γ,w• is a root of g w• .
We see that ρ γ,w• does not depend on d. Thus if there is an indecomposable parabolic bundle E with cl(E) = (r, r •,• , d), then for any d ′ there is an indecomposable parabolic bundle of class (r, r •,• , d ′ ). Secondly, we see that the property of ρ γ,w• being a root does not depend on the choice of w • as long as the components of w • are large enough. By a slight abuse of terminology we say that γ is a root in this case.
Remark 9.3.4. In fact, one can consider an infinite star-shaped graph G D with deg D infinite rays, and the corresponding Kac-Moody Lie algebra g D , which is the inductive limit of g w• . Then we have a homomorphism