The pure cohomology of multiplicative quiver varieties

To a quiver $Q$ and choices of nonzero scalars $q_i$, non-negative integers $\alpha_i$, and integers $\theta_i$ labeling each vertex $i$, Crawley-Boevey--Shaw associate a"multiplicative quiver variety"$\mathcal{M}_\theta^q(\alpha)$, a trigonometric analogue of the Nakajima quiver variety associated to $Q$, $\alpha$, and $\theta$. We prove that the pure cohomology, in the Hodge-theoretic sense, of the stable locus $\mathcal{M}_\theta^q(\alpha)^s$ is generated as a $\mathbb{Q}$-algebra by the tautological characteristic classes. In particular, the pure cohomology of genus $g$ twisted character varieties of $GL_n$ is generated by tautological classes.


Introduction
Let Q = (I, Ω) be a quiver. Fix a vector q ∈ (C × ) I . Associated to these data is a noncommutative algebra Λ q , the multiplicative preprojective algebra [CBS] of Q with parameter q. Letting α ∈ Z I ≥0 be a dimension vector for Q and choosing a stability condition θ ∈ Z I , we get a moduli space M q θ (α) of θ-semistable representations of Λ q with dimension vector α, called a multiplicative quiver variety, investigated in [CBS, Ya] (and both investigated and substantially generalized in [Bo]). Multiplicative quiver varieties provide concrete realizations of character varieties and related spaces: see [BY, BK, ST] among others.
1.1. Results. As for its cousins, the Nakajima quiver varieties, the multiplicative quiver variety M q θ (α) is defined as a GIT quotient (at a character χ θ : G → G m ) of an affine algebraic variety Rep(Λ q , α) by the group G = i GL(α i ) /∆(G m ), a product of general linear groups modulo the diagonal copy of G m ; when it is a free quotient, this endows M q θ (α) with a map M q θ (α) → BG.  (2) In particular, if M q θ (α) = M q θ (α) s and M q θ (α) is connected, then H * (BG, Q) → H * M q θ (α), Q surjects onto P H * M q θ (α) .
In light of Theorem 1.2 of [MN], Theorem 1.1 is nicely consonant with Hausel's "purity conjecture" (cf. [Ha] as well as [HLV,Theorem 1.3.1 and Corollary 1.3.2], and the discussion around Conjecture 1.1.3 of [HWW]), which predicts that when M q θ (α) = M q θ (α) s , one should have an isomorphism P H * (M q θ (α) s ) ∼ = H * M θ (α) s , Q , where M θ (α) s denotes the corresponding Nakajima quiver variety. In the special case in which Q is a quiver with a single node and g ≥ 1 loops, the dimension vector is α = n, and q ∈ C × is a primitive nth root of unity, the multiplicative quiver variety M q θ (α) is identified with the GL n -character variety Char(Σ g , GL n , q Id) of a genus g surface with a single puncture with residue q Id, sometimes called a genus g twisted character variety [HR]. We obtain: Corollary 1.2. The pure cohomology P H * Char(Σ g , GL n , q Id) is generated by tautological classes. Corollary 1.2 has already appeared in [Sh], where it was deduced, via the non-abelian Hodge theorem, from Markman's theorem [Ma] that the cohomology of the moduli space of GL n -Higgs bundles of degree 1 on a smooth projective genus g curve is generated by tautological classes. A novelty of our result, compared to [Sh], is that we avoid invoking non-abelian Hodge theory: instead, we deduce Corollary 1.2 (as well as Theorem 1.1) via a more direct and concrete method that invokes only basic facts of ordinary mixed Hodge theory as in [TdH-II]. 1 . Theorem 1.1 has the following slightly different but equivalent formulation. Choose a subgroup S ⊂ i GL(α i ) whose projection S → G is a finite covering. Then one can form the stack quotient Rep(Λ q , α) θ -s /S, which comes with a morphism π : Rep(Λ q , α) θ -s /S → M q θ (α) s that is a gerbe, in fact a torsor over the commutative group stack BH where H = ker(S → G). We have an isomorphism H * (BS, Q) ∼ = H * (BG, Q) and π induces an isomorphism H * M q θ (α) s , Q ∼ = H * Rep(Λ q , α) θ -s /S, Q . Thus Theorem 1.1 can be restated as: For each connected open substack U ⊆ Rep(Λ q , α) θ -s /S, the pure cohomology P H * (U ) is generated as a Q-algebra by the Chern classes of tautological bundles Rep(Λ q , α) θ -s × S V associated to finite-dimensional representations V of S.
It is Theorem 1.3 that we prove directly: the tautological bundles Rep(Λ q , α) θ -s × S V that appear naturally and geometrically in our proof do not themselves descend to the multiplicative quiver variety in general, so it is more convenient to work on the Deligne-Mumford stack Rep(Λ q , α) θ -s /S.
Unlike the situation of quiver varieties in [MN], we know of no obvious generalizations of Theorems 1.1 and 1.3 to other even-oriented cohomology theories (such as topological K-theory or elliptic cohomology). However, we do obtain the following analogue of Theorem 1.6 of [MN]. (2) There is a finite list of tautological bundles from which every object of D b coh (M) is obtained by finitely many applications of (i) direct sum, (ii) cohomological shift, and (iii) cone.
As for the analogous result in [MN], we emphasize that Theorem 1.4(2) is not simply a formal consequence of Theorem 1.4(1), since we do not include taking direct summands (i.e., retracts) among the operations (i)-(iii). It would be interesting to know generators for D b coh (M) for more general dimension vectors α than in Theorem 1.4.

1.2.
Method of Proof. The proof of Theorem 1.3 is broadly similar to the proof used in [MN] to establish that tautological classes generate the cohomology of Nakajima quiver varieties.
A main part of the proof consists in producing a suitable modular compactification of the multiplicative quiver variety (or rather its Deligne-Mumford stack analogue). One major difference from the Nakajima quiver variety case arises already at this stage: one frequently relies on q being an appropriate tuple of primitive roots of unity to deduce that M q θ (α) parameterizes only stable representations, independently of the choice of θ; whereas in [MN], we assumed, without significant loss of generality, that θ was a generic stability condition. We note that such a genericity assumption here would exclude the possibility of applications to the character variety Char(Σ g , GL n , q Id); hence we avoid it. Instead we identify a compactification by a "projective Artin stack" M, a quotient of a quasiprojective scheme by a reductive group whose coarse moduli space is a projective scheme. Known techniques [Ki, ER] allow us to replace the Artin stack compactification by a projective Deligne-Mumford stack at no cost to the validity of our approach.
The second stage is to identify a complex on M q θ (α)×M that, roughly speaking, resolves the graph of the embedding M q θ (α) ֒→ M. Again, while this is morally similar to [MN], the actual construction and proofs are more complicated and subtle. This is essentially because our compactification of the Nakajima quiver variety relied on a graded 3-Calabi-Yau algebra, whereas the compactification of M q θ (α) uses an algebra, denoted by A in the body of this paper, that may (conjecturally) be what one might call a "relative 2g-Koszul algebra" in most cases but (as far as we know) is not known to be so. Fortunately it turns out that we can proceed as if the algebra A were known to have certain desired properties, carry out some constructions, and check by hand that the resulting complex behaves as hoped. Unfortunately, in the generality in which we work here (and again unlike [MN]), it seems one cannot expect the complex to actually provide a resolution of the structure sheaf of the graph of the embedding: instead, we rely on work of Markman [Ma] to show that an appropriate Chern class of the complex we built is the Poincaré dual of the fundamental class of the graph.
The final step is to deduce the theorem via usual integral transform arguments. In [MN], we used Nakajima's result that the (integral) cohomology of a quiver variety is generated by algebraic cycles, hence is surjected onto by the cohomology of any compactification. Such an assertion is not true of the multiplicative quiver varieties M q θ (α). Instead, what is always true is that the cohomology of any reasonable smooth compactification-which is always Hodge-theoretically pure-surjects onto the pure part of the cohomology of any open subset. This yields the assertion of the theorem, which in any case would be the best possible result, given that the cohomology H * (BG, Q) is pure; but its Hodge-theoretic nature also necessitates working with rational cohomology.
It is an interesting question to characterize the image of H * (BG, Z) in H * M q θ (α), Z .
1.3. Acknowledgments. We are grateful to Gwyn Bellamy, Ben Davison, Tamás Hausel, and Travis Schedler for helpful conversations, and to Donu Arapura and Ajneet Dhillon for help with references. The first author was supported by EPSRC programme grant EI/I033343/1 and a Fisher Visiting Professorship at the University of Illinois at Urbana-Champaign. The second author was supported by NSF grants DMS-1502125 and DMS-1802094 and a Simons Foundation fellowship. The authors are also grateful to the Department of Mathematics of the University of Notre Dame for its hospitality during part of the preparation of this paper.
1.4. Notation. Throughout, k denotes a field of characteristic 0. In Sections 1 and 6, k = C.

Quivers and Multiplicative Preprojective Algebras
2.1. Truncations of Graded Algebras. We will frequently use certain "truncations" of a Z ≥0graded algebra A in what follows. For a Z-graded vector space V and integer n, we write V ≥n = ⊕ m≥n V m , a vector space graded by {n, n + 1, . . . }. We note the vector space injection V ≥n → V that is the identity on the mth graded piece for m ≥ n. • a / / t(a) • . The double of Q is a quiver Q dbl = (I, H = Ω ⊔ Ω) with the same vertex set I as for Q and the set of arrows H = Ω ⊔ Ω where Ω is the arrow set of Q and Ω is a set equipped with a bijection to Ω, written Ω ∋ a ↔ a * ∈ Ω. We extend this bijection canonically to an involution on H = Ω ⊔ Ω, still written a → a * , and decree s(a * ) = t(a), t(a * ) = s(a). For each arrow a ∈ H we write Fix an integer N ≥ 1. The graded tripled quiver Q gtr associated to Q (cf. Section 4 of [MN]) is a quiver defined as follows. We give Q gtr the vertex set I gtr = I × [0, N ] where I is the vertex set of Q. If Ω is the edge set of Q and H = Ω ⊔ Ω the associated set of pairs of an edge together with an orientation, we give Q gtr the arrow set Thus, s(h, n) = (s(h), n) and t(h, n) = (t(h), n + 1); (2) for each i ∈ I, n ∈ [0, N − 1] we have arrows t (i,n) with s(t i,n ) = (i, n) and t(t (i,n) ) = (i, n + 1).
More discussion can be found in [MN].
2.3. Path Algebras. Let S = i Se i be a semisimple algebra with orthogonal system of idempotents {e i }. Suppose A is an algebra with homomorphism S → A. We say that x ∈ A has diagonal Peirce decomposition if x ∈ i∈I e i Ae i , or equivalently if it lies in the centralizer Z A (S).
Given a quiver Q, We let kQ denote the path algebra of the quiver. Thus, we have a finitedimensional semisimple k-algebra S = i∈I ke i with idempotents e i labelled by the vertices i ∈ I. We define an S-bimodule B = B(Q), with k-basis labelled by the arrows, and "arrows written leftto-right," so e i ae j = 0 unless i = s(a), j = t(a), and so that e s(a) ae t(a) = a. Then kQ = T S (B(Q)) (the tensor algebra).
It is natural to grade the path algebra kQ of any quiver Q = (I, H)-for example, kQ dbl -by taking the semisimple algebra S to lie in degree 0 and the arrows h ∈ H to lie in degree 1: this is the standard nonnegative grading on the tensor algebra. The algebra kQ dbl t thus is naturally bigraded, hence has total grading with deg(t) = 1. We can also grade kQ gtr by putting the semisimple algebra i∈I gtr ke i in degree 0 and the arrows in degree 1. We obtain a graded algebra homomorphism The graded algebra kQ gtr has the property kQ gtr ≥N +1 = 0, so we obtain a homomorphism This equivalence identifies representations of kQ dbl [t] [0,N ] with representations of the quotient kQ gtr /J, where J denotes the two-sided ideal 2.4. Universal Localizations. We briefly review some aspects of universal localizations that may be unfamiliar to the reader, using Chapter 4 of [Sch] as our reference; see also [Co]. Suppose that R is a ring with 1 and Σ is a set of elements of R. Then there is a ring R Σ with a homomorphism R → R Σ that is universal with respect to the property that for every r ∈ Σ, r becomes invertible in R Σ . The ring R Σ is called the universal localization of R at Σ; an alternative notation that is sometimes preferable is Σ −1 R. The universal localization is constructed as follows: letting Σ −1 denote the set of symbols a −1 for a ∈ Σ, we define This has the universal property claimed. We will need the following properties, which follow immediately from the universal property.
(3) Given a two-sided ideal I ⊆ R, let Σ denote the image of Σ in R/I and I Σ denote the two-sided ideal in R Σ generated by I. Then (R/I) Σ ∼ = R Σ /I Σ .
2.5. Multiplicative Preprojective Algebras. We review the multiplicative preprojective algebra of a quiver Q as defined in [CBS]. Given a quiver Q with double Q dbl = (I, H), for each arrow a ∈ H of Q dbl , we define g a = 1 + aa * ∈ kQ dbl . Write L Q for the algebra obtained by universal localization of kQ dbl inverting Σ = {g a | a ∈ H}. Identify the tuple q ∈ (k × ) I with the element i∈I q i e i ∈ S. Crawley-Boevey and Shaw choose an ordering of the arrows in H and define ρ CBS = −→ a∈H g ǫ(a) a − q (the arrow over the product indicates that it is taken in the chosen order). It is proven in [CBS] that, up to isomorphism, the quotient algebra L Q /(ρ CBS ) does not depend on the choice of ordering. Thus, in this paper we specifically fix an ordering Ω = {a 1 , . . . , a g } on the arrows in Q, and let Definition 2.4. The associated multiplicative preprojective algebra is where ρ CBS is defined as in (2.3).
2.6. Homogenized Multiplicative Preprojective Algebras. A principal tool in this paper is a certain graded algebra A that "homogenizes" the multiplicative preprojective algebra Λ q of [CBS]. Here we construct the algebra A and collect some basic facts about A and its relation to the multiplicative preprojective algebra Λ q .
Thus, fix a quiver Q. We consider kQ dbl [t] = kQ dbl t /(ta − at | a ∈ kQ dbl ) as a nonnegatively graded algebra, with the generators a ∈ H, t all in degree 1, and S = ⊕ i∈I ke i in degree 0. We let for all a ∈ H.
Remark 2.5. Each G a has diagonal Peirce decomposition: more precisely, We note the obvious equalities Given q ∈ (k × ) I , we identify q with q := i∈I q i e i ∈ kQ dbl , a sum of idempotents in the path algebra (which thus also has diagonal Peirce decomposition). Analogously to [CBS], the algebra kQ dbl [t] admits a universal localization in which the elements G a , a ∈ H, and t are inverted: we write L t for this universal localization. The algebra L t contains ] 0 at the elements g a , a ∈ H, as in [CBS] and reviewed above. As above, fix an ordering Ω = {a 1 , . . . , a g } on the arrows in Q. Write The element ρ has diagonal Peirce decomposition, and so ρe i = e i ρ, and (ρ) = ({ρe i |i ∈ I}).
Proposition 2.7. Write Σ = {G a | a ∈ H} ∪ {t}. We have: (1) A is a graded algebra where a i , a * i and t have degree 1 (and S = i∈I ke i lies in degree 0). (2) The universal localization where Λ q (Q) =: Λ q denotes the multiplicative preprojective algebra of [CBS].

Representations and their Moduli
3.1. Representations of kQ dbl and kQ gtr . Fixing some N ≥ 2g, where g is the number of arrows in Q, we form the graded-tripled quiver Q gtr associated to Q as above. 2 Given a dimension vector α ∈ Z I ≥0 for the quiver Q dbl , we write α gtr ∈ Z for the dimension vector for kQ gtr for which α gtr i,n = α i for all n ∈ [0, N ]. We write Rep(kQ dbl , α) for the space of representations of kQ dbl with dimension vector α and G = i GL(α i ) for the automorphism group; thus Similarly we write Rep(kQ gtr , α gtr ) for the space of representations of kQ gtr with dimension vector α gtr , and G gtr for the automorphism group.
As in the construction of Section 4.3 of [MN], there is a natural "induction functor" from the category of representations of kQ dbl with dimension vector α to the category of representations of kQ gtr of dimension vector α gtr . The construction proceeds as follows. To a representation V of kQ dbl we may associate the Z ≥0 -graded vector space V [t], and let arrows h of Q dbl act as multiplication followed by shift-of-grading. This makes [0,N ] -module, and finally apply Lemma 2.2 to get a representation of kQ gtr : in fact, a representation of the quotient kQ gtr /J where J is as in (2.2).
More concretely, the above construction is the following. Suppose we have a representation to act by shift of Z-grading; and (3) defining each generator of kQ dbl [t] corresponding to h ∈ H to act as the composite The construction determines a morphism of algebraic varieties ("induction") Thus, given a representation (a h : V s(h) → V t(h) ) h∈H of kQ dbl on V , and (g i,n ) ∈ G gtr , we have Proposition 3.1. The map Ind of (3.1) defines a G gtr -equivariant open immersion of G gtr × G Rep(kQ dbl , α) in Rep(kQ gtr /J, α gtr ), whose image consists of those (h, n), t i,n for which:

Representations of A and
3.3. Representations of A and Λ q . We note: Recall from (2.7) that, letting Σ = {G a | a ∈ H} ∪ {t}, we have a graded algebra isomorphism

and thus a graded
In the opposite direction, we have a functor (Λ t ⊗ A −) 0 : A -Gr ≥0 → Λ q -Mod. We have: form an adjoint pair.
( It is immediate from the construction of Section 3.1 that: Proposition 3.4 (cf. Prop. 4.7 of [MN]). The morphism Ind of (3.1) restricts to an open immersion: Its image consists of those representations on which the elements t, G a act invertibly whenever their domain and target lie in the range [0, N ].

Corollary 3.5. The map Ind defines an open immersion of moduli stacks
3.5. Semistability and Stability. We next discuss (semi)stability of representations and the corresponding GIT quotients. For any quiver Q = (I, Ω) with dimension vector α ∈ Z I ≥0 , a GIT stability condition is given by θ ∈ Z I ≥0 satisfying i θ i α i = 0. The vector θ determines a character χ θ : i GL(α i ) → G m , χ(g i ) i∈I = i det(g i ) θi , and the condition i θ i α i = 0 guarantees that the diagonal copy ∆(G m ) of G m in i GL(α i ) lies in the kernel of χ; we require this because ∆(G m ) acts trivially on Rep(Q, α). Given dimension vectors β, α, we write β < α if β = α and β i ≤ α i for all i ∈ I.
We now turn to stability conditions for the doubled and tripled quivers Q dbl and Q gtr for a fixed quiver Q. Suppose θ is a stability condition for Q dbl and dimension vector α. We construct a stability condition θ gtr for Q gtr with dimension vector α gtr as follows. For a representation M of kQ gtr of dimension vector α gtr , we write δ i,n (M ) := dim(M i,n ); we will write θ gtr as a linear combination of the δ i,n . Also, we note that it suffices to construct a rational linear functional θ gtr , since any positive integer multiple of θ gtr evidently defines the same stable and semistable loci. We fix an ordering on the vertices of Q, identifying I = {1, . . . , r}. and a positive integer T ≫ 0. We define: Proposition 3.6. Suppose M = Ind(N ) for some representation N of kQ dbl with dimension vector α. Then M is semistable, respectively stable, with respect to θ gtr if and only if N is semistable, respectively stable, with respect to θ.
The proof is an easy adaptation of that of Proposition 4.12(4) of [MN].
We remark that the above construction does not match [MN]: there we chose to construct a stability θ gtr for Q gtr that would be nondegenerate if θ was, whereas here we ignore this possible requirement. While it would be possible to copy the construction of a stability θ gtr from [MN] and prove analogues of the statements of [MN], there are cases important to multiplicative quiver varieties in which it is not possible to find a stability condition for kQ dbl that is nondegenerate in the sense used in [MN]: for example, the case when Q has a single vertex and loops based at that vertex, with dimension vector α = n > 1. However, again for multiplicative quiver varieties, in some interesting cases the choice of the parameter q can guarantee that every semistable representation of Λ q is automatically stable (though not for numerical reasons, as nondegeneracy guarantees). Indeed, we say q = (q i ) i∈I ∈ (k × ) I is a primitive αth root of unity if q α := q αi i = 1 and q β = 1 for all 0 < β < α. We have: (1) Suppose that M is a representation of Λ q with dimension vector α. Then q α = 1.
(2) In particular, if q is a primitive αth root of 1, then every representation of Λ q of dimension vector α is θ-stable for every θ.
For example, if Q = ({ * }, E) where E has g loops at * , α = n, and q is a primitive nth root of 1, then every representation of Λ q of dimension n is stable for every θ; the corresponding moduli space of representations of Λ q is the character variety Char(Σ g , GL n , q Id) of the introduction.
Remark 3.8. It would be interesting to characterize those stability conditions θ gtr for kQ gtr with the property that there is a stability condition θ for kQ dbl so that if M = Ind(N ) then M is θ gtr -(semi)stable if and only if N is θ-(semi)stable.
However, one can make a choice of subgroup S ⊂ G that ensures that the quotient stack Rep(Λ q , α) θ−s /S is a Deligne-Mumford stack and that Rep(Λ q , α) θ−s /S → M q θ (α) s is a finite gerbe (indeed a principal BH-bundle for a finite abelian group H). Indeed, for example, we can choose any character ρ : G gtr → G m for which the composite with the diagonal embedding ρ • ∆ : G m → G m is nontrivial, hence surjective. Then S gtr := ker(ρ) has the property that G gtr = S gtr · ∆(G m ) and similarly letting S = G ∩ S gtr we have G = S · ∆(G m ). Moreover, since ∆(G m ) is the stabilizer of every point of Rep(Λ q , α) θ -s and H := ∆(G m ) ∩ S is finite, we get: Lemma 3.10. The quotient Rep(Λ q , α) θ -s /S is a Deligne-Mumford stack and the natural morphism Rep(Λ q , α) θ -s /S → M q θ (α) s is a torsor for the commutative group stack BH (in particular, is a finite gerbe over M q θ (α) s ).
Lemma 3.12. The stack Rep(Λ q , α) θ -s /S is smooth. The stack M st is integral and its coarse moduli space is a projective scheme. The natural morphism Rep(Λ q , α) θ -s /S ֒→ M st is an open immersion.
Proof. The smoothness of Rep(Λ q , α) θ -s /S is Theorem 1.10 of [CBS]. The remaining assertions are immediate.
We may apply the results of [Ki] or [ER] to Rep(kQ gtr , α gtr ) θ gtr -ss /S gtr and its closed substack M st to obtain a projective Deligne-Mumford stack (i.e., a Deligne-Mumford stack whose coarse space is a projective scheme) M spanned by the arrows, so that kQ dbl [t] is identified with the tensor algebra T S[t] (B). As in [CBS,p.190], the bimodule that is the target of the universal S-linear bimodule derivation of kQ dbl [t] satisfies [CBS,p. 190], for the universal localization L t we also get Ω L t with the obvious identification of the universal derivation δ Lt/S[t] . We write: A.
The module P 1 is evidently projective as a bimodule. Via the above description, we obtain a collection of bimodule basis elements

An Exact Sequence. We write
Write η i = e i ⊗ 1 = 1 ⊗ e i , i ∈ I, for the obvious bimodule generators of P 0 . Define graded bimodule maps by β(η a ) = aη s(a) − η t(a) a for arrows a of Q gtr , and where ∆ a = δ(G a ) (where δ denotes the universal derivation). It is then immediate that α(η i ) = e i · δ(ρ); in particular, letting θ : P 0 → (ρ)/(ρ 2 ) denote the map defined by θ(p ⊗ q) = pρq and writing φ for the isomorphism defined by (4.1), we have: Imitating the proof of Lemma 3.1 of [CBS] gives: where γ(p ⊗ q) = pq, is an exact sequence of Z-graded bimodules.
Proof. As in [Sch,Theorem 10.3], one gets an exact sequence As in [CBS], splicing this sequence and the defining sequence for Ω S[t] A and applying (4.4) gives a commutative diagram The vertical arrows φ, ψ are isomorphisms and θ is surjective, yielding the assertion.

Dual of the Map
We consider A e as a left A e -module where a ⊗ a ′ ∈ A e acts by We remark that A e naturally also has a right A e -module structure commuting with the left A e -action, where a ⊗ a ′ ∈ A e acts on the right by Given a finitely generated left A e -module, we form P ∨ = Hom A e (P, A e ), the dual over the enveloping algebra; by the above discussion, this module has a right A e -module structure, which we can identify with a left A e -module structure via the isomorphism We now want to calculate the dual α ∨ of the map α of (4.2) using the formula (4.3). Note that ∆(G a ) = aδ(a * ) + δ(a)a * = aη a * + η a a * .
We thus find from Formula (4.3) that the η a -component of α is given by and zero otherwise. Let {η ∨ a } denote the basis of P ∨ 1 dual to the basis {η a } of P 1 ; we note that (4.6) η ∨ a ∈ e t(a) P ∨ 1 e s(a) . It follows from the above formulas: Lemma 4.2. For all a ∈ Ω, we have Proof. The element D is a product of elements of diagonal Peirce type, hence itself is of diagonal Peirce type. Thus, using e s(a) η ∨

This completes the proof.
Suppose now that M is a graded right Λ t -module; then M = M ≥0 is a graded right A-submodule of M. For example, we could take M = Λ t itself, as in (2.7). We consider the map Remark 4.4. We note that, under the above hypothesis on M , for any product Q of elements G a , a ∈ H, of degree deg(Q), the elements Qt − deg(Q) and t deg(Q) Q −1 of Λ t give well defined operators of right multiplication on M that satisfy all relations in Λ t .
Proposition 4.5. Suppose that M = M ≥0 for a graded right Λ t -module M. Then for all m ∈ M and all i ∈ I and 1 ≤ j ≤ g, Proof. (1) We first prove that m G aj Dη ∨ i − Dη ∨ i G aj ∈ Im(1 M ⊗ α ∨ ) by (strong) induction on j. Base Case. j = 1. By Lemma 4.3, the assertion is true for i = s(a 1 ). From Lemma 4.2, we have . This completes the base case.

Induction
Step. Assume m G a k Dη ∨ i − Dη ∨ i G a k ∈ Im(1 M ⊗ α ∨ ) for all i ∈ I and k < j. Again, by Lemma 4.3, we have mG aj Dη ∨ i − mDη ∨ i G aj ∈ Im(1 M ⊗ α ∨ ) for i = s(a j ). Applying Lemma 4.2 gives where the last equality applies the inductive hypothesis. This completes the induction step, thus proving the assertion for the elements follows the analogous descending induction on j.
(2) Taking note of Remark 4.4, from (4.7) we have α ∨ (mG aj t −2 η aj ) = mG aj t −2 a * j R aj η ∨ s(aj ) L aj − mG aj t −2 qR a * j η ∨ t(aj ) L a * j a * j . Applying part (1) of the proposition to the right-hand side of this formula gives where the last equality uses (2.4); in particular this gives the first assertion of Part (2) of the proposition. The second assertion follows similarly.

Analysis of the Ext-Complex
5.1. The Complex (4.5) and the Hom-Functor. Let M, N be graded left A-modules such that M is finitely generated and projective as a k[t]-module. To the exact sequence we apply the functor Hom A (−, N ) to obtain an exact sequence We continue the sequence (5.1) using Thus, we would like to compute the cokernel of the map (5.2).
Proof. By projectivity, it suffices to check for P = A e , where it follows by adjunction.
Corollary 5.2. Under the hypotheses of Proposition 5.1, the cokernel of the map (5.2) is We note the following identities, which are immediate from adjunction: is the graded left A-module associated to a finite-dimensional left Λ q -module M . Then: N ). 5.2. The Ext-Complex. Fix N ≥ 2g. Let V be a finite-dimensional representation of Λ q of dimension vector α, and let V = V [t] be the corresponding graded A-module as in Section 3.2, and specifically as in Lemma 3.3. Suppose W is a Z ≥0 -graded A [0,N ] = A/A ≥N +1 -module, identified with a representation of Q gtr that has dimension vector α gtr . Thus τ [0,N ] V is also identified with a representation of Q gtr that has dimension vector α gtr .
Let P • denote the complex of (4.2). We consider the complex Hom A (P • ⊗ A V, W ). Since the sources and target of the Homs in this complex are graded A-modules, each Hom-space can be regarded as a graded vector space; we write for its degree 0 graded piece. As in [MN], using Lemma 5.3 we may identify Ext with: If,in addition,τ [0,N ] V is θ-stable and W is θ-semistable, both of dimension vector α gtr , then: (2) We have ker(∂ 0 ) = 0 unless τ [0,N ] V ∼ = W , in which case ker(∂ 0 ) ∼ = k.
Cyclically permuting, these conditions become It is immediate from the conditions (5.6) that on W 2g−m , m ≥ 2, we have that Φ * commutes with all a j and a * j , whereas for m = 1 we may write Φ * | W2g−1 = tt −2 Dφ * t and again Φ * commutes with a j , a * j . Thus Φ * defines an . This completes the proof.

Cohomology of Varieties and Stacks
In the remainder of the paper, the base field k is assumed to be C. Here as throughout the paper, we use H * (X) to denote cohomology with Q-coefficients, and H BM * (X) to denote Borel-Moore homology with Q-coefficients; if X is a smooth Deligne-Mumford stack, there is a canonical isomorphism H * (X) ∼ = H BM * (X).
6.1. Mixed Hodge Structure on the Cohomology of an Algebraic Stack. Suppose that X is an algebraic stack of finite type over C. It follows from Example 8.3.7 of [TdH-III] that the cohomology H * (X) comes equipped with a functorial mixed Hodge structure. The Gysin map satisfies the projection formula: for classes c ∈ H * (X), c ′ ∈ H * (Y ), we have Suppose X and Y are smooth Deligne-Mumford stacks and C ∈ H * (X ×Y ) is a cohomology class. By the Künneth theorem we have H * (X × Y ) ∼ = H * (X) ⊗ H * (Y ), and thus we may write C = x i ⊗ y i with x i ∈ H * (X), y i ∈ H * (Y ). The classes x i , y i are the Künneth components of C (with respect to X or Y respectively). Now suppose that f : X → Y is a representable morphism from a smooth Deligne-Mumford stack X to a smooth, proper Deligne-Mumford stack Y . The graph morphism X (1,f ) − −− → X × Y is not usually a closed immersion. Proposition 6.3 (cf. Proposition 2.1 of [MN]). The image of f * : H * (Y ) → H * (X) is contained in the span of the Künneth components of (1, f ) * [X] with respect to the left-hand factor X.
/ / Y for the projections. Write p * : Y → Spec(C) for the projection to a point; then (p X ) * exists since Y is proper. We have f * = (1, f ) * p * Y and (p X ) * (1, f ) * = id. Using the projection formula, then, we get This proves the claim.
Proof. Consider first the case of smooth quasi-projective varieties X • ⊂ X. Then, for any smooth projective compactification X of X, the image of H * (X) → H * (X • ) is independent of the choice of X: for example, by the Weak Factorization theorem, any two such X, X ′ are related by a sequence of blow-ups and blow-downs along smooth centers in the complement of X • , and the claimed independence follows from the usual formula for the cohomology of a blow-up. Since the image of H k (X) in H k (X • ) is W k H * K(X • ) by Corollaire 3.2.17 of [TdH-II], the claim follows in this case. We now consider the general case. By the assumptions, X and X • are (separated) quasi-projective smooth Deligne-Mumford stacks that are global quotients. By Theorem 1 of [KV], there exist a smooth quasi-projective scheme W and a finite flat LCI morphism W → X; the fiber product X • × X W → X • is then also finite, flat, and LCI. Using the commutative square and base change, we find: are surjective (indeed, q * q * and q • * (q • ) * are multiplication by the degree of q).
(2) Since the Gysin maps q • * , q * are morphisms of mixed Hodge structures by Proposition 6.2, is a complex of locally free sheaves on M of ranks r −1 , r 0 , r 1 respectively.
It follows that the scheme-theoretic support of H 1 (C ∨ ) is the reduced diagonal Γ.
It remains to check that the same is true of H 1 (C). To do that, we again start with τ [0,N ] V ǫ , τ [0,N ] V ′ ǫ as above, but consider them as graded A-modules (i.e., forgetting the k[ǫ]-module structure) and form the complex C. Assume without loss of generality that k ⊗ k[ǫ] τ [0,N ] V ǫ ∼ = k ⊗ k[ǫ] τ [0,N ] V ′ ǫ as graded A-modules. We have a short exact sequence of graded A-modules is nonzero, it is an isomorphism, since both its domain and target are stable of dimension vector α gtr ; in which case both (6.4) and its analogue for τ [0,N ] V ′ ǫ are split extensions. This means that the tangent vector to Rep(Λ q , α) θ -s /S × Rep(Λ q , α) θ -s /S determined by (V ǫ , V ′ ǫ ) is zero, and thus irrelevant to our analysis of the scheme-theoretic support of H 1 (C). Thus we may assume that the composite (6.5) is zero, and so the morphism φ is a homomorphism of 1-extensions. Now if φ(ǫτ [0,N ] V ǫ ) = 0, then again by stability it maps isomorphically onto ǫτ [0,N ] V ′ ǫ . Since (6.4) is nonsplit, it follows that φ is an isomorphism, implying that the tangent vector determined by (V ǫ , V ′ ǫ ) is tangent to Γ, and again irrelevant to our analysis of the scheme-theoretic support of H 1 (C). Finally then, we may assume that φ(ǫτ [0,N ] V ǫ ) = 0. It follows that φ factors through the quotient k ⊗ k[ǫ] τ [0,N ] V ǫ ; similarly its image lies in ǫτ [0,N ] V ′ ǫ . It follows that Hom A -Gr τ [0,N ] V ǫ , τ [0,N ] V ′ ǫ is scheme-theoretically supported over Spec(k) ⊂ Spec k[ǫ], and hence by Proposition 5.4(1) that the same is true of H 1 (C). Since this is true for every Spec k[ǫ] → Rep(Λ q , α) θ -s /S × Rep(Λ q , α) θ -s /S not tangent to Γ, we conclude that H 1 (C) has scheme-theoretic support equal to Γ, as required.
By Proposition 6.5, then, we conclude that [Γ] = c m (C). By Proposition 6.3, the Künneth components of c m (C) thus span the image of the restriction map H * Rep(Λ q , α) θ -s /S −→ H * Rep(Λ q , α) θ -s /S , which by Proposition 6.4 is exactly ⊕ m W m H m Rep(Λ q , α) θ -s /S . Since the Chern classes of C are polynomials in the Chern classes of the tautological bundles (see the proof of Proposition 2.4(ii) of [MN]), this completes the proof of Theorem 1.3, hence also of Theorem 1.1. 6.6. Proof of Theorem 1.4. The proof of Theorem 1.4 is essentially identical to that of Theorem 1.6 of [MN] (and we note that Theorem 1.4 holds whenever k is any field of characteristic zero and q ∈ k × ). Indeed, the assumption that there is a vertex i 0 ∈ I for which α i0 = 1 guarantees the following. First, we may take S = i =i0 GL(α i ), which acts freely on the stable locus: thus, M q θ (α) s is a fine moduli space for stable representations of Λ q . Second, exactly as in the proof of Theorem 1.6 of [MN], in the complex (5.3), there are direct sum decompositions L(V 0 , W 0 ) = Hom(V 0,i0 , W 0,i0 ) ⊕ ⊕ i =i0 Hom(V 0,i , W 0,i ) and L(V 0 , W 2g ) = Hom(V 0,i0 , W 2g,i0 ) ⊕ ⊕ i =i0 Hom(V 0,i , W 2g,i ) , so that the complex obtained by modifying (5.3) given by has no cohomology at the ends, and in the middle has cohomology H that is a rank m = codim(Γ) vector bundle. Moreover, the remaining map k = Hom(V 0,i0 , W 0,i0 ) → E(V 0 , W 1 ) defines a section s of H whose scheme-theoretic zero locus is Z(s) = Γ. The remainder of the proof now copies that of Theorem 1.6 of [MN].