BPS Lie algebras and the less perverse filtration on the preprojective CoHA
aa r X i v : . [ m a t h . R T ] A ug BPS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THEPREPROJECTIVE COHA
BEN DAVISON
Abstract.
We introduce a new perverse filtration on the Borel–Moore homology of the stackof representations of a preprojective algebra Π Q , by proving that the derived direct image ofthe dualizing mixed Hodge module along the morphism to the coarse moduli space is pure. Weshow that the zeroth piece of the resulting filtration on the preprojective CoHA is isomorphic tothe universal enveloping algebra of the associated BPS Lie algebra g Π Q , and that the sphericalLie subalgebra of this algebra contains half of the Kac–Moody Lie algebra associated to thereal subquiver of Q . Lifting g Π Q to a Lie algebra in the category of mixed Hodge modules onthe coarse moduli space of Π Q -modules, we prove that the intersection cohomology of spacesof semistable Π Q -modules provide “cuspidal cohomology” for g Π Q – a conjecturally completespace of simple hyperbolic roots for this Lie algebra. Introduction
Main results.
Let Q be the double of a quiver Q , and let Π Q := C Q/ h P a ∈ Q [ a, a ∗ ] i be thepreprojective algebra. Let S be a Serre subcategory of the category of C Q -modules. We set H A S Π Q := M d ∈ N Q H BM (cid:0) M S d (Π Q ) , Q (cid:1) ⊗ L − χ Q ( d , d ) , the (shifted) Borel–Moore homology of the stack M S (Π Q ) of finite-dimensional Π Q -modules whichare objects of S . Here L = H c ( A , Q ) is a Tate twist, which is introduced so that the object H A S Π Q carries an associative multiplication. The resulting algebra plays a key role in geometricrepresentation theory; it is the algebra of all conceivable raising operators on the cohomology ofNakajima’s quiver varieties, and so via several decades of work [38, 37, 18, 52, 48, 31]... containshalf of various quantum groups associated to Q .Let JH : M (Π Q ) → M (Π Q ) be the semisimplification morphism to the coarse moduli space ofΠ Q -modules. We study H A S Π Q via the richer object RA Π Q := M d ∈ N Q JH ∗ DQ M (Π Q ) ⊗ L − χ Q ( d , d ) , the derived direct image of the dualizing sheaf. The derived category of mixed Hodge moduleson M (Π Q ) is a tensor category via convolution, and RA Π Q is an algebra object in this category,from which we recover H A S Π Q by restricting to M S (Π Q ) and taking hypercohomology. Theorem A (Corollary 4.9 ) . There is an isomorphism (1) RA Π Q ∼ = M n ∈ Z ≥ H n ( RA Π Q )[ − n ] and H n ( RA Π Q ) is pure of weight n , i.e. RA Π Q is pure . As a result of the above decomposition, themixed Hodge structure H A S Π Q carries an ascending perverse filtration L ≤• H A S Π Q , starting in degreezero, which is respected by the algebra structure on H A S Π Q . Moreover, there is an isomorphism ofalgebras L ≤ H A S Π Q ∼ = U ( g S Π Q ) where g S Π Q is isomorphic to the BPS Lie algebra [14] determined by a quiver e Q , potential f W and Serre subcategory e S of the category of C e Q -modules defined in § In the interests of digestibility, in the introduction we state all results without reference to extra gauge groups G , stability conditions or slopes. The results in the main body incorporate these generalisations. The main geometric content of the theorem amounts to the statement that the derived directimage with compact support JH ! Q M (Π Q ) is pure, i.e. this complex satisfies the statement ofthe celebrated decomposition theorem of Beilinson, Bernstein, Deligne and Gabber [2], or moreprecisely, Saito’s version in the language of mixed Hodge modules [42, 45]. This is rather surprising,since the preconditions of that theorem are not met; M (Π Q ) is a highly singular stack, and p is notprojective. The algebraic content of the theorem is that the lowest piece of the perverse filtrationcan be expressed in terms of the BPS Lie algebras introduced by myself and Sven Meinhardt in[14], as part of a project to realise the cohomological Hall algebras defined by Kontsevich andSoibelman [28] as positive halves of generalised Yangians. This is also quite striking; the BPSLie algebra is defined by a quite different perverse filtration, on vanishing cycle cohomology of adifferent Calabi–Yau category.1.2. Cuspidal cohomology.
In general, the BPS Lie algebra g Π Q satisfies the condition on thedimensions of the cohomologically graded pieces X n ∈ Z dim( H n ( g Π Q , d )) q n/ = a Q, d ( q − )where the polynomials on the right hand side are the polynomials introduced introduced by VictorKac in [22], counting d -dimensional absolutely irreducible Q -representations over a finite field oforder q. A conjecture of Bozec and Schiffmann [6, Conj.1.3] states that the Kac polynomials onthe right hand side are the characteristic functions of the N Q -graded pieces of a cohomologicallygraded Borcherds algebra, and so it is natural to suspect that g Π Q itself is the positive halfof a cohomologically graded Borcherds algebra. In particular, g Π Q should be given by somecohomologically graded Cartan datum, including the data of (usually infinitely many) imaginarysimple roots.One of the motivations for pursuing a lift of the BPS Lie algebra to the category of mixedHodge modules is a question of Olivier Schiffmann [47]: is there any geometric description of theCartan datum, for example some algebraic variety M cusp , d (Π Q ) along with a natural embeddingΨ : H ( M cusp , d (Π Q ) , Q ) ֒ → g Π Q , d as the space of imaginary simple roots of weight d ? Such aconstruction would answer in the affirmative the complex geometric analogue of Conjecture 3.5 of[47].We can still make sense of the above conjecture in the absence of a proof that g Π Q is thepositive part of a Borcherds algebra. We do so via the special case S = C Q -mod of our moregeneral theorem on primitive generators: Theorem B.
Let d be such that there exists a simple d -dimensional Π Q -module, let ̟ ′ : M S d (Π Q ) ֒ →M d (Π Q ) be the inclusion, and set cu S Π Q , d := H (cid:0) M S d (Π Q ) , ̟ ′ ! IC M d (Π Q ) ( Q ) (cid:1) ⊗ L χ Q ( d , d ) . There is a canonical decomposition g S Π Q , d ∼ = cu S Π Q , d ⊕ l of mixed Hodge structures, such that the Lie bracket g S Π Q , d ′ ⊗ g S Π Q , d ′′ [ · , · ] −−→ g S Π Q , d for d ′ + d ′′ = d factors through the inclusion of l . In particular, the mixed Hodge structures cu S Π Q , d give a collection of canonical subspaces of generators for g S Π Q . The proof of the above theorem uses the construction of the new perverse filtration on H A S Π Q arising from Theorem A, and the resulting lift of the Lie algebra g Π Q to a Lie algebra object inthe category of pure Hodge modules on M (Π Q ). In particular, the decomposition into generatorsand non-generators in the BPS Lie algebra arises from the decomposition theorem for perversesheaves/mixed Hodge modules. We will adopt the convention throughout that where an expected S superscript is missing, we assume that S is the whole category C Q -mod. PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 3
We conjecture that aside from the known simple roots of N Q -degree 1 i for i a vertex of Q ,or d such that χ Q ( d , d ) = 0, these are all of the generators; see Conjecture 7.7 for the precisestatement.Since by the decomposition theorem there is a canonical embedding H (cid:0) M d (Π Q ) , IC M d (Π Q ) ( Q ) (cid:1) ⊂ H ( X, Q )where X → M d (Π Q ) is a semi-small resolution, Theorem B suggests that the answer to thequestion above is “yes”, and the above embedding provides a route towards [47, Conj.3.5]. Forexample, any symplectic resolution is a semi-small [23] resolution of singularities, and thus itscohomology contains “cuspidal” cohomology as a canonical summand.1.3. Comparison with the perverse filtration of [14] . We call the filtration introduced inTheorem A the less perverse filtration, in order to distinguish it from a different perverse filtration,that was introduced in joint work with Sven Meinhardt [14]. This is a perverse filtration on thecritical CoHA H A e S e Q, f W (as defined in [28, Sec.7]) that is the crucial part of the definition of theBPS Lie algebra in Theorem A. We recall some of the main facts regarding critical CoHAs, inorder to explain the relationship between the two filtrations.Let Q ′ be a symmetric quiver, i.e. we assume that for each pair of vertices i, j there are asmany arrows going from i to j as from j to i . Let W ′ ∈ C Q ′ be a potential, and let S ′ be a Serresubcategory of the category of C Q ′ -modules. We continue to denote by JH : M ( Q ′ ) → M ( Q ′ ) thesemisimplification map. We define RA Q ′ ,W ′ := M d ∈ N Q JH ∗ φ mon T r( W ′ ) Q M d ( Q ′ ) ⊗ L χ Q ′ ( d , d ) / . See § § RA Q ′ ,W ′ carries the structure of an algebra in the derived category of monodromic mixed Hodge moduleson M ( Q ′ ), and we obtain the algebra H A S ′ Q ′ ,W ′ by taking (exceptional) restriction and hyperco-homology of RA Q ′ ,W ′ .We may summarise the main results of [14] on BPS Lie algebras as follows; there is an isomor-phism of complexes of monodromic mixed Hodge modules RA Q ′ ,W ′ ∼ = M n ∈ Z ≥ H n ( RA Q ′ ,W ′ )[ − n ] , inducing a filtration P ≤• H A S ′ Q ′ ,W ′ beginning in degree one. The algebra Gr P H A S ′ Q ′ ,W ′ is super-commutative, so that P ≤ H A S ′ Q ′ ,W ′ is closed under the commutator Lie bracket, and is called the BPS Lie algebra , denoted g S ′ Q ′ ,W ′ .We define the tripled quiver e Q to be the quiver obtained from Q by adding a loop to eachvertex, and we define f W as in (21). We set e S to be the Serre subcategory containing those C e Q -modules for which the underlying C Q -module is an object of S . Then via the dimensionalreduction isomorphism [11, Thm.A.1] there is an isomorphism of algebras [40, 53] H A e S e Q, f W ∼ = H A S Π Q via which H A S Π Q inherits a perverse filtration, which we denote P ≤• H A S Π Q .Switching to the ordinary English meaning of the word, the filtration L ≤• H A S Π Q seems less per-verse than P ≤• H A S Π Q since it comes directly from the geometry of the map M (Π Q ) → M (Π Q ),rather than the more circuitous route of dimensional reduction, vanishing cycles, and the semisim-plification morphism M ( e Q ) → M ( e Q ) for the auxiliary quiver e Q . The two filtrations are ratherdifferent ; for instance, the BPS Lie algebra lives inside L ≤ H A S Π Q , while P ≤ H A S Π Q = 0. Ingeneral, perverse degrees with respect to the new filtration are lower than for the old one. It isfor these two reasons that we call the new filtration the less perverse filtration. See [3] for a comprehensive treatment of when we may expect to find such a resolution. As a consequence of this difference, there is value in considering them both simultaneously; see § BEN DAVISON
Halpern–Leistner’s conjecture.
Our purity theorem is independent from the statement(proved in [12]) that the mixed Hodge structure on H A Π Q is pure. We explain the particularutility of the purity statement of the current paper, with reference to a particular application: theproof of a conjecture of Halpern–Leistner [19]. The details will appear in forthcoming work withSjoerd Beentjes.Let X be a K3 surface, fix a generic ample class H ∈ NS( X ) Q , and fix a Hilbert polynomial P .Then there is a moduli stack C oh HP ( X ) of H -semistable coherent sheaves with Hilbert polynomial P , and Halpern–Leistner conjectures that the mixed Hodge structure on H BM ( C oh HP ( X ) , Q )is pure. The above-mentioned purity result of [12] encouraged this statement, while the purityresult of the current paper provides the means to prove it. The proof idea is easy to explain:locally, the morphism p : C oh HP ( X ) → C oh HP ( X ) to the coarse moduli space is modelled as themorphism M d (Π Q ) → M d (Π Q ) for some quiver Q , and so Theorem A tells us that the directimage p ! Q M d (Π Q ) is locally, and hence globally, pure. The result then follows from the fact thatthe direct image of a pure complex of mixed Hodge modules along a projective morphism is pure.1.5. The algebras U ( g C ) and U ( g Σ g ) . The construction and results of the present paper canbe applied in nonabelian Hodge theory, since they concern any category for which the moduli ofobjects is locally modeled by moduli stacks of modules for preprojective algebras.Let C be a smooth genus g complex projective curve, which for ease of exposition we assumeto be defined over Z , and let H iggs sst r, ( C ) denote the complex algebraic stack of semistable rank r degree zero Higgs bundles on C . By [36] there is an equality X r ≥ ,i,n ∈ Z dim( Gr Wn ( H BM − i ( H iggs sst r, ( C ) , Q )))( − i q n/ g − r T r (2) = Exp q / ,T X r ≥ Ω C,r, ( q / )(1 − q ) − T r where Ω C,r, ( q / ) = a C,r, ( q / , . . . , q / ) is a specialization of Schiffmann’s polynomial, countingabsolutely indecomposable vector bundles of rank r on C over F q . On the right hand side we havetaken the plethystic exponential, an operation which satisfies the identity(3) Exp X r,i ∈ Z > × Z ( − i dim g r,i ( q i/ ) T r = X r,i ∈ Z > × Z ( − i dim U ( g ) r,i ( q i/ ) T r for g any Z > × Z -graded Lie algebra with finite-dimensional graded pieces. We presume that thesecond grading agrees with the cohomological grading, so that the Koszul sign rule is in effectwith respect to it, e.g. [ a, b ] = ( − | a || b | +1 [ b, a ]for | a | and | b | the Z -degrees of a and b respectively. This explains the introduction of the signsin (3). Via a similar argument to the previous subsection, we may show that the Borel–Moorehomology of H iggs sst r, ( C ) is pure, so that the only terms that contribute on the left hand side of(2) have n = i .Putting all of these hints together, it is natural to conjecture (as in [47]) that there is some Liealgebra g C , and an isomorphism H Higgs C := M r ≥ H BM ( H iggs sst r, ( C ) , Q ) ⊗ L ( g − r ∼ = U q ( g C [ u ])where the right hand side is a deformation of the universal enveloping algebra of a current alge-bra for some Lie algebra g C , which should be a “curve” cousin of the Kac–Moody Lie algebrasassociated to quivers.This Lie algebra should be defined as the BPS Lie algebra associated to the noncompact Calabi–Yau threefold Y = Tot C ( ω C ⊕ O C ). Technically, this presents some well-known complications:stacks of coherent sheaves on Y do not have a global critical locus description, so that the definitionof vanishing cycle sheaves on them requires a certain amount of extra machinery (see [21, 4]). The PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 5 outcome of this paper is that there is a “less perverse” definition of U ( g C ) ready off the shelf,avoiding d critical structures, vanishing cycles etc.: we may define U ( g C ) := M r ≥ H ( H iggs sst r, ( C ) , τ ≤ p ∗ DQ H iggs sst r, ( C ) ) ⊗ L ( g − r where p : H iggs sst r, ( C ) → H iggs sst r, ( C ) is the morphism to the coarse moduli space, and the mul-tiplication is via the correspondences in the CoHA of Higgs sheaves as in [46, 34]. Similarly, wedefine U ( g nilp C ) := M r ≥ H ( H iggs sst r, ( C ) , g ∗ g ! τ ≤ p ∗ DQ H iggs sst r, ( C ) ) ⊗ L ( g − r ⊂ H Higgs , nilp C := M r ≥ H BM ( H iggs sst , nilp r, ( C ) , Q ) ⊗ L ( g − r where g : H iggs sst , nilp r, ( C ) → H iggs sst r, ( C ) is the inclusion of the locus for which the Higgs field isnilpotent, to define the correct enveloping algebra inside the CoHA of nilpotent Higgs bundles[46].Whenever the morphism p from the stack of objects in a category C to the coarse modulispace is locally modeled as the semisimplification morphism from the stack of representations ofa preprojective algebra, the definition of the enveloping algebra of the BPS Lie algebra for C isforced by Theorem A; we likewise define U ( g Σ g ) := M r ≥ H ( M Betti g,r , τ ≤ p ∗ DQ M Betti g,r ) ⊗ L ( g − r ⊂H Σ g := M r ≥ H BM ( M Betti g,r , Q ) ⊗ L ( g − r where p : M Betti g,r → M
Betti g,r is the semisimplification morphism from the moduli stack of r -dimensional π (Σ g )-modules to the coarse moduli space, for Σ g a genus g Riemann surface without boundary .The object H Σ g is the CoHA of representations of the stack of C [ π q (Σ g )]-modules defined in [10].We leave the detailed study of the algebras U ( g C ) and U ( g Σ g ), as well as a general treatment ofthe perverse filtration on CoHAs for 2CY categories [39, 25] to future work.1.6. Notation and conventions.
All schemes and stacks are defined over C , and assumed to belocally of finite type. All quivers are finite. All functors are derived.If X is a scheme or stack, and p : X → pt is the morphism to point, we often write H for thederived functor p ∗ , and H i for the i th cohomology of H , i.e. we abbreviate H ( F ) := H ( X , F ) H i ( F ) := H i ( X , F ) . For X an irreducible scheme or stack, we write H ( X, Q ) vir = H ( X, Q ) ⊗ L − dim( X ) / where thehalf Tate twist is as in § H (B C ∗ , Q ) vir := H (B C ∗ , Q ) ⊗ L / . We define H BM ( X , Q ) = H DQ X where D is the Verdier duality functor.If C is a triangulated category equipped with a t structure we write H ( F ) = M i ∈ Z H i ( F )[ − i ]when the right hand side exists in C .If V is a cohomologically graded vector space with finite-dimensional graded pieces, we define χ t ( V ) := X i ∈ Z ( − i dim( V i ) t i/ . This case is slightly different, since the moduli stack of π (Σ g )[ ω ]-modules is written as a global critical locus;see [10]. BEN DAVISON If V also carries a weight filtration W n V , we define the weight polynomial(4) χ wt ( V ) := X i,n ∈ Z ( − i dim( Gr W n ( V i )) t n/ . Acknowledgements.
During the writing of the paper, I was supported by the starter grant“Categorified Donaldson–Thomas theory” No. 759967 of the European Research Council. I wasalso supported by a Royal Society university research fellowship.I would like to thank Olivier Schiffmann for helpful conversations, and Tristan Bozec for pa-tiently explaining his work on crystals to me. Finally, I offer my heartfelt gratitude to Paul,Sophia, Sacha, Kristin and Nina, for their help and support throughout the writing of this paper.2.
Background on CoHAs
Monodromic mixed Hodge modules.
Mixed Hodge modules.
Let X be an algebraic variety. We define as in [44, 45] the category MHM ( X ) of mixed Hodge modules on X . There is an exact functor rat X : D b ( MHM ( X )) → D b ( Perv ( X ))and moreover the functor rat X : MHM ( X ) → Perv ( X ) is faithful. We will make light use of thelarger category of monodromic mixed Hodge modules MMHM ( X ) considered in [28, 14], which isdefined to be the Serre quotient B X / C X , of two full subcategories of MHM ( X × A ). Here, B X is the full subcategory containing those objects for which the cohomology mixed Hodge modulesare locally constant, away from the origin, when restricted to { x } × A for each x ∈ X . Thecategory C X is the full subcategory containing those F for which such restrictions have globallyconstant cohomology sheaves.The functor ( X × G m ֒ → X × A ) ! provides an equivalence of categories between MMHM ( X )and the full subcategory of mixed Hodge modules on X × G m containing those F satisfying thecondition that the restriction to each { x } × G m has locally constant cohomology sheaves. Write G for a quasi-inverse. We define the inclusion τ : X ֒ → X × G m by setting τ ( x ) = ( x, rat mon X = rat X ◦ τ ∗ [ − ◦ G : MMHM ( X ) → Perv ( X ) . Let z X : X ֒ → X × A be the inclusion of the zero section. Then z X, ∗ : MHM ( X ) → MMHM ( X )is an inclusion of tensor categories, where the tensor product on the target is the one describedbelow. We write MMHS := MMHM (pt). The category of polarizable mixed Hodge structuresis a full subcategory of
MMHS via z pt , ∗ .2.1.2. Six functors.
Excepting the definition of tensor products, the six functor formalism forcategories of monodromic mixed Hodge modules is induced in a straightforward way by that ofmixed Hodge modules, e.g. for f : X → Y a morphism of varieties we define f ∗ , f ! : D b ( MMHM ( X )) → D b ( MMHM ( Y ))to be the functors induced by( f × id A ) ∗ , ( f × id A ) ! : D b ( MHM ( X × A )) → D b ( MHM ( Y × A ))respectively. The functor D X : MHM ( X × A ) → MHM ( X × A ) op sends objects of C X toobjects of C op X , inducing the functor D mon X : MMHM ( X ) → MMHM ( X ) op . We may omit themon superscript when doing so is unlikely to cause confusion.If X and Y are schemes over S , and F ∈
Ob(
MMHM ( X )), G ∈
Ob(
MMHM ( Y )), then takingtheir external tensor product (as mixed Hodge modules) we obtain J ∈
Ob(
MHM ( Z × A )), where Z = X × S Y . We define F ⊠ S G := (id Z × +) ∗ J ∈
Ob(
MMHM ( Z )) . PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 7 If X is a monoid over S , i.e. there exist S -morphisms ν : X × S X → Xi : S → X satisfying the standard axioms, and F , G ∈
Ob( D b ( MMHM ( X ))) we define F ⊠ ν G := ν ∗ ( F ⊠ S G ) ∈ Ob( D b ( MMHM ( X ))) . This monoidal product is symmetric if ν is commutative, and is exact if ν is finite. If ν iscommutative, we define Sym ν ( F ) := M i ≥ Sym i ν ( F )where Sym i ν ( F ) is the S i -invariant part of F ⊠ ν . . . ⊠ ν F | {z } i times , and the S i -action is defined via the isomorphism(5) F ⊠ ν . . . ⊠ ν F | {z } i times ∼ = e ∗ F ⊠ . . . ⊠ F | {z } i times ! where e : X × S · · · × S X ֒ → X × · · · × X is the natural embedding. By [32] (see [14, Sec.3.2]), thetarget of (5) carries a natural S i -action. The functor z ∗ : MHM ( X ) → MMHM ( X ) from § MMHMs on stacks. If X is a connected locally finite type Artin stack we define the boundedderived category of monodromic mixed Hodge modules D b MMHM ( X ) as in [13]. Since all Artinstacks that we encounter for the rest of this paper will be global quotient stacks, and aside fromsome half Tate twists almost all monodromic mixed Hodge modules will be monodromy-free, thereader may think of this as the category of G -equivariant mixed Hodge modules described in [1].The category D b MMHM ( X ) admits a natural t structure for which the heart is the category MMHM ( X ) of monodromic mixed Hodge modules on X , which admits a faithful functor rat mon X to the category of perverse sheaves on X . If X ∼ = X/G is a global quotient stack, then up to acohomological shift by dim( G ) this is the category of G -equivariant perverse sheaves on X . Forfull generality and detail, we refer the reader to [13]. If X is not necessarily connected we define D lb MMHM ( X ) = Y X ′ ∈ π ( X ) D b MMHM ( X ) . Let X be a connected locally finite type Artin stack. We define the category D MMHM (cid:1) ( X )by setting the objects to be Z -tuples of objects F ≤ n ∈ D b MMHM ( X ) such that H m ( F ≤ n ) = 0for m > n , along with the data of isomorphisms τ ≤ n − F ≤ n ∼ = F ≤ n − . We define D MMHM (cid:0) ( X )in the analogous way, by considering tuples of objects F ≥ n along with isomorphisms τ ≥ n F ≥ n − ∼ = F ≥ n . If X is a disjoint union of locally finite type Artin stacks we define D MMHM (cid:1) ( X ) = Y X ′ ∈ π ( X ) D MMHM (cid:1) ( X ′ )and likewise for D MMHM (cid:0) ( X ). For f : X → Y a morphism of Artin stacks we define functors f ∗ : D MMHM (cid:1) ( X ) → D MMHM (cid:1) ( Y ) and f ! : D MMHM (cid:0) ( X ) → D MMHM (cid:0) ( Y ) in thenatural way (see [13]). The selling point of the categories introduced in this paragraph is thatthey give us a setting to talk about direct images of complexes of monodromic mixed Hodgemodules along non-representable morphisms of stacks without needing a full theory of unboundedderived categories of such objects.We define D lb MHM ( X ) , D MHM (cid:1) ( X ) etc. the same way, and consider these categories assubcategories of their monodromic counterparts via z X, ∗ . BEN DAVISON
Weight filtrations. If X is a scheme, an object F ∈
Ob(
MMHM ( X )) inherits a weightfiltration from its weight filtration in MHM ( X × A ), and is called pure of weight n if Gr Wi ( F ) = 0for i = n . For X a stack, an object F ∈ D
MMHM (cid:1) ( X ) is called pure if H i ( F ) is pure of weight i for every i . An object of D MMHM (cid:1) ( X ) or D MMHM (cid:0) ( X ) is called pure if its pullback alonga smooth atlas is pure. Via Saito’s results, if F ∈
Ob( D MMHM (cid:1) ( X )) is pure, then F ∼ = H ( F ).Furthermore, if p : X → Y is projective, then p ∗ F is pure.2.1.5. Intersection cohomology complexes.
Let X be a stack. Then Q X ∈ Ob( D b MMHM ( X ))is defined by the property that for all smooth morphisms q : X → X with X a scheme, q ∗ Q X ∼ = Q X ,the constant complex of mixed Hodge modules on X .Likewise, if X is irreducible we define IC X ( Q ) by the property that q ∗ IC X ( Q ) ∼ = IC X ( Q ), theintersection mixed Hodge module complex on X . Note that unless X is zero-dimensional, IC X ( Q )is not a mixed Hodge module, but rather a complex with cohomology concentrated in degree d = dim( X ). This complex is pure, i.e. its d th cohomology mixed Hodge module is pure of weight d . Consider the morphism s : A x x −−−−→ A . We define L / = cone( Q A → s ∗ Q A ) ∈ D b ( MMHM (pt)) . This complex has cohomology concentrated in degree 1, and is pure. Moreover there is an isomor-phism ( L / ) ⊗ ∼ = L , justifying the notation.We define(6) IC X := IC X ( Q ) ⊗ L − dim( X ) / . Since L / is pure, this is a pure monodromic mixed Hodge module.2.1.6. G-equivariant MMHMs.
Assume that we have fixed an algebraic group G , and let X = X/H be a global quotient stack, where an embedding G ⊂ H is understood. Examples relevant to thispaper will be X = M G, ζ -ss ( Q ) or X = M G, ζ -ss ( Q ), defined in § f IC X := IC X ⊗ L − dim( G ) / (7)The motivation for introducing the extra Tate twist in (7) alongside the one in § H = G . Thinking of the underlying complex of perverse sheaves for f IC X as a G -equivariant complex of perverse sheaves on X , the extra twist of (7) means that this complexis a genuine perverse sheaf (without shifting).Continuing in the same vein, we shift the natural t structure on D MMHM (cid:1) ( X ), so that forexample H G,i (cid:16) f IC M G, ζ -ss ( Q ) (cid:17) = 0 if and only if i = 0 , (8)where the cohomology functor is with respect to the shifted t structure. We denote by MMHM G ( X )the heart of this t structure (i.e. the shift by dim( G ) of the usual t structure), and τ G ≤• and τ G ≥• the truncation functors with respect to this t structure.2.1.7. Vanishing cycles.
Let X be an algebraic stack , and let f ∈ Γ ( X ) be a regular function onit. An integral part of Saito’s theory is the construction of a functor φ f [ −
1] :
MHM ( X ) → MHM ( X )lifting the usual vanishing cycle functor ϕ f [ −
1] :
Perv ( X ) → Perv ( X ) , in the sense that there is a natural equivalence rat X φ f ∼ = ϕ f rat X . There is a further lift φ mon f : MHM ( X ) → MMHM ( X ) We state all of Saito’s results for stacks, as opposed to schemes. The details of the extension to stacks can befound in [13].
PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 9 satisfying rat mon X φ mon f ∼ = ϕ f rat X , defined by φ mon f : MHM ( X ) → MMHM ( X ) F 7→ j ! φ f/u ( X × G m → X ) ∗ F where u is a coordinate for G m and j : X × G m → X × A is the natural inclusion. The vanishing cycle functor commutes with Verdier duality, i.e. by [43]there is a natural isomorphism of functors φ mon f D X ∼ = D mon X φ mon f : MHM ( X ) → MMHM ( X ) . Let g ∈ Γ ( Y ) be a regular function on the stack Y . Then there is a Thom–Sebastiani naturalisomorphism [41] φ mon f ⊠ φ mon g → (cid:0) φ mon f ⊞ g ( • ⊠ • ) (cid:1) f − (0) × g − (0) : MHM ( X ) × MHM ( Y ) → MMHM ( X × Y ) . Quivers and their representations.
In this section we fix notation regarding quiver rep-resentations.By a quiver Q we mean a pair of finite sets Q and Q (the arrows and vertices respectively)along with a pair of morphisms s, t : Q → Q taking each arrow to its source and target, respec-tively. We say that Q is symmetric if for every pair of vertices i, j ∈ Q there are as many arrows a satisfying s ( a ) = i and t ( a ) = j as there are arrows satisfying s ( a ) = j and t ( a ) = i .We refer to elements d ∈ N Q as dimension vectors . We define a bilinear form on the set ofdimension vectors by(9) χ Q ( d ′ , d ′′ ) = X i ∈ Q d ′ i d ′′ i − X a ∈ Q d ′ s ( a ) d ′′ t ( a ) . If Q is symmetric this form is symmetric. We define the form ( • , • ) Q on N Q via(10) ( d , d ′ ) Q = χ Q ( d , d ′ ) + χ Q ( d ′ , d ) . For K a field, we denote by KQ the free path algebra of Q over K . Recall that this algebracontains | Q | mutually orthogonal idempotents e i for i ∈ Q , the “lazy paths”. We define thedimension vector dim( ρ ) ∈ N Q of a KQ -representation via dim( ρ ) i = dim K ( e i · ρ ). If W ∈ C Q cyc is a linear combination of cyclic words in Q , we denote by Jac( Q, W ) the quotient of C Q by thetwo-sided ideal generated by the noncommutative derivatives ∂W/∂a for a ∈ Q , as defined in[17].2.2.1. Extra gauge group.
For each pair of (not necessarily distinct) vertices i, j fix a complexvector space V i,j with basis the arrows from i to j . SetG Q := Y i,j GL( V i,j ) . Then G Q acts on A d ( Q ) via the isomorphism A d ( Q ) ∼ = M i,j ∈ Q V i,j ⊗ Hom( C d i , C d j ) . We fix a complex algebraic group G , and fix a homomorphism G → G Q . We defineGL d := Y i ∈ Q GL d i gl d := Y i ∈ Q gl d i f GL d := GL d × G. Throughout the paper we fix C = H (B G, Q ) . Here we employ the standard abuse of notation, identifying vector spaces with their total spaces, consideredas algebraic varieties.
Stability conditions.
By a
King stability condition we mean a tuple ζ ∈ Q Q + . The slopeof a nonzero dimension vector d ∈ N Q is defined by µ ζ ( d ) = ζ · d P i ∈ Q d i , and we define the slope of a nonzero KQ -module by setting µ ζ ( ρ ) = µ ζ (dim( ρ )) . For θ ∈ Q we define Λ ζθ := { d ∈ N Q \ { } : µ ζ ( d ) = θ } ∪ { } . A KQ -module ρ is called ζ -stable if for all proper nonzero submodules ρ ′ ⊂ ρ we have µ ζ ( ρ ′ ) < µ ζ ( ρ ), and is ζ -semistable if the weak version of this inequality is satisfied. We denote by A ζ -ss d ( Q ) ⊂ A d ( Q ) := Y a ∈ Q Hom( C d s ( a ) , C d t ( a ) )the open subvariety of ζ -semistable C Q -modules.We set M G, ζ -ss d ( Q ) := A ζ -ss d ( Q ) / f GL d , where the quotient is the stack-theoretic quotient. If G is trivial this stack is isomorphic to thestack of ζ -semistable d -dimensional C Q -modules. In [27] King constructs M ζ -ss d ( Q ), the coarsemoduli space of ζ -semistable d -dimensional C Q -representations. We denote by M G, ζ -ss d ( Q ) thestack-theoretic quotient of this variety by the G -action. We denote by JH G : M G, ζ -ss ( Q ) → M G, ζ -ss ( Q )the natural map. If G is trivial, this is the morphism which, at the level of points, takes d -dimensional C Q -modules to their semisimplifications.Given an algebra A , presented as a quotient of a free path algebra C Q by some two-sided ideal R , we denote by M G, ζ -ss ( A ) the moduli stack of ζ -semistable A -modules, and by M G, ζ -ss d ( A ) thesubstack of d -dimensional A -modules. Similarly, we denote by M G, ζ -ss ( A ) the stack-theoreticquotient of the coarse moduli scheme by the G -action.2.2.3. Monoidal structure.
The stack M G, ζ -ss θ ( Q ) is a monoid in the category of stacks over B G ,via the morphism ⊕ G : M G, ζ -ss θ ( Q ) × B G M G, ζ -ss θ ( Q ) → M G, ζ -ss θ ( Q )taking a pair of ζ -polystable C Q -modules to their direct sum. This morphism is finite and com-mutative [33, Lem.2.1], and so the monoidal product F ⊠ ⊕ G G := ⊕ G ∗ ( F ⊠ B G G )for F , G ∈ D
MMHM (cid:1) ( X ) is biexact and symmetric.2.2.4. Subscript conventions.
Throughout the paper, if X is some object that admits a decomposi-tion with respect to dimension vectors d ∈ N Q , we denote by X d the subobject corresponding tothe dimension vector d . If F is a sheaf or mixed Hodge module defined on X , a stack that admitsa decomposition indexed by dimension vectors, we denote by F d its restriction to X d . Finally, if f : X → Y is a morphism preserving natural decompositions of X and Y indexed by dimensionvectors, we denote by f d : X d → Y d the induced morphism.If a stability condition ζ is fixed, we set X θ = ` d ∈ Λ ζθ X d , and extend the conventions of theprevious paragraph in the obvious way to objects admitting decompositions indexed by dimensionvectors, along with morphisms that preserve these decompositions. PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 11
Serre subcategories.
Throughout the paper, S will be used to denote a Serre subcategoryof the category of C Q -modules, i.e. S is a full subcategory such that if0 → ρ ′ → ρ → ρ ′′ → C Q -modules then ρ is an object of S if and only if ρ ′ and ρ ′′ are. Weassume that S admits a geometric definition, in the sense that there is an inclusion of stacks ̟ : M S ,G, ζ -ss ( Q ) ֒ → M G, ζ -ss ( Q )which at the level of complex points is the inclusion of the set of objects of S , and a correspondinginclusion ̟ ′ : M S ,G, ζ -ss ( Q ) ֒ → M G, ζ -ss ( Q )of coarse moduli spaces.If the definition of an object F S depends on a choice of some Serre subcategory S of the categoryof C Q ′ -modules, for some quiver Q ′ , we omit the superscript S as shorthand for the case in whichwe choose S to be the entire category of C Q ′ -modules.2.3. Critical CoHAs.
We set DT GQ,W, θ := ̟ ∗ ̟ ! φ mon Tr ( W ) f IC M G, ζ -ss ( Q ) RA S ,G, ζ Q,W, θ := JH G ∗ DT GQ,W, θ H A S ,G, ζ Q,W, θ := H (cid:16) M G, ζ -ss ( Q ) , ̟ ∗ ̟ ! φ mon Tr ( W ) f IC M G, ζ -ss ( Q ) (cid:17) . Assumption 2.1.
We will assume throughout that we have chosen
Q, W, θ , S , G, ζ so that H A S ,G, ζ Q,W, θ is a free C = H (B G, Q ) -module. The purity of H A S , ζ Q,W, θ is a sufficient, but not necessary condition for the assumption to hold;see [13] for an impure example for which the assumption holds.Given dimension vectors d ′ , d ′′ ∈ Λ ζθ with d = d ′ + d ′′ we define A ζ -ss d ′ , d ′′ ( Q ) ⊂ A ζ -ss d ( Q )to be the subset of linear maps preserving the Q -graded subspace C d ′ ⊂ C d , and we defineGL d ′ , d ′′ ⊂ GL d to be the subgroup preserving the same subspace. We define π ′ , π ′ , π ′ to be the natural morphismsfrom A ζ -ss d ′ , d ′′ ( Q ) to A ζ -ss d ′ ( Q ), A ζ -ss d ( Q ) and A ζ -ss d ′′ ( Q ) respectively. We define M G, ζ -ss d ′ , d ′′ ( Q ) := A ζ -ss d ′ , d ′′ ( Q ) / (GL d ′ , d ′′ × G ) . Finally we define M G, ζ -ss θ ( Q ) (2) to be the union of the stacks M G, ζ -ss d ′ , d ′′ ( Q ) across all d ′ , d ′′ ∈ Λ ζθ .Consider the commutative diagram(11) M G, ζ -ss θ ( Q ) (2) π × π u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ π ' ' ❖❖❖❖❖❖❖❖❖❖❖ M G, ζ -ss θ ( Q ) × B G M G, ζ -ss θ ( Q ) JH G × B G JH G (cid:15) (cid:15) M G, ζ -ss θ ( Q ) JH G (cid:15) (cid:15) M G, ζ -ss θ ( Q ) × B G M G, ζ -ss θ ( Q ) ⊕ G / / M G, ζ -ss θ ( Q )where π , π , π are induced by π ′ π ′ , π ′ respectively. Set A = M G, ζ -ss θ ( Q ) × B G M G, ζ -ss θ ( Q ) B = M G, ζ -ss θ ( Q ) (2) O = M G, ζ -ss θ ( Q ) . Via the Thom–Sebastiani isomorphism and the composition of appropriate Tate twists of themorphisms ̟ ′∗ ̟ ′ ! ( JH G × B G JH G ) ∗ φ mon Tr ( W ) (cid:16) Q A → ( π × π ) ∗ Q B (cid:17) and ̟ ′∗ ̟ ′ ! JH G ∗ φ mon Tr ( W ) D mon O (cid:16) Q O → π , ∗ Q B (cid:17) we define the morphism(12) ⋆ : RA S ,G, ζ Q,W ⊠ ⊕ G RA S ,G, ζ Q,W → RA S ,G, ζ Q,W i.e. a multiplication operation on RA S ,G, ζ Q,W . The proof that this operation is associative is standard,and is as in [28, Sec.2.3]. Applying H to this morphism we obtain a morphism H A S ,G, ζ Q,W ⊗ C H A S ,G, ζ Q,W → H A S ,G, ζ Q,W and we define the (associative) multiplication on H A S ,G, ζ Q,W by composing with the surjection H A S ,G, ζ Q,W ⊗ H A S ,G, ζ Q,W → H A S ,G, ζ Q,W ⊗ C H A S ,G, ζ Q,W . The PBW theorem.
We next recall some fundamental results for critical CoHAs from [14]. For ease of exposition we assume that Q is symmetric, though for generic stability conditionsall results are stated more generally in [14, 13].Firstly, there is an isomorphism JH G ∗ DT G, ζ Q,W, θ ∼ = H (cid:16) JH G ∗ DT G, ζ Q,W, θ (cid:17) and τ G ≤ (cid:16) JH G ∗ DT G, ζ Q,W, θ (cid:17) = 0. By base change we have(13) RA S ,G, ζ Q,W, θ ∼ = ̟ ′∗ ̟ ′ ! H (cid:16) JH G ∗ DT G, ζ Q,W, θ (cid:17) . Setting(14)
BPS S ,G, ζ Q,W, θ := ̟ ′∗ ̟ ′ ! τ ≤ JH G ∗ DT G, ζ Q,W, θ ⊗ L − / there is an isomorphism BPS S ,G, ζ Q,W, θ ∼ = ( ̟ ′∗ ̟ ′ ! φ mon T r ( W ) f IC M G, ζ -ss θ ( Q ) if M ζ -st θ ( Q ) = ∅ S ,G, ζ Q,W, θ := H (cid:16) M G, ζ -ss θ ( Q ) , BPS S ,G, ζ Q,W, θ (cid:17) . There is a natural action of H (B C ∗ , Q ) on RA S ,G, ζ Q,W, θ and this induces the morphism(15) BPS S ,G, ζ Q,W, θ ⊗ H (B C ∗ , Q ) vir → RA S ,G, ζ Q,W, θ . Given an algebra A and an A -bimodule L we defineT A ( L ) := M i ≥ L ⊗ A · · · ⊗ A L | {z } i times where the i = 0 summand is the A -bimodule A . Given an A -linear Lie algebra g , i.e. a Lie algebra g , with an A -action such that the Lie algebra map g ⊗ g → g factors through the surjection g ⊗ g → g ⊗ A g we define the A -linear universal enveloping algebra as the quotient algebraT A ( g ) / h a ⊗ b − b ⊗ a − [ a, b ] g i . Likewise, if N is an A -bimodule we define Sym A ( N ) := T A ( g ) / h a ⊗ b − b ⊗ a i . For the extension to the G -equivariant case considered here, we refer the reader to [13]. PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 13
The main structural result for the critical CoHA is the PBW theorem:
Theorem 2.2.
The morphism (16) Φ :
Sym ⊕ G (cid:16) BPS S ,G, ζ Q,W, θ ⊗ H (B C ∗ , Q ) vir (cid:17) → RA S ,G, ζ Q,W, θ obtained by combining (15) with the iterated CoHA multiplication map is an isomorphism in D MMHM (cid:1) ( M S ,G, ζ -ss θ ( Q )) . Moreover H (Φ) is an isomorphism of algebra objects in D MMHM (cid:1) ( M G, ζ -ss θ ( Q )) ,and H (Φ) is an isomorphism of Λ ζθ -graded monodromic mixed Hodge structures (17) Sym C (cid:16) BPS S ,G, ζ Q,W, θ ⊗ H (B C ∗ , Q ) vir (cid:17) → H A S ,G, ζ Q,W, θ . Remark 2.3.
Strictly speaking, the symmetric monoidal structure on D MMHM (cid:1) ( M S ,G, ζθ ( Q )) should be twisted by a sign depending on the Euler form of Q , in the definition of the domain of Φ .In this paper we only consider Hall algebras RA S ,G, ζ Q,W, θ and H A S ,G, ζ Q,W, θ for quivers Q satisfying thecondition that χ Q ( d ′ , d ′′ ) ∈ · Z for all d ′ , d ′′ ∈ N Q , and so we may omit this added complication(see [14, Sec.1.6] for details). By (13) the algebra H A S ,G, ζ Q,W, θ carries a filtration defined by P ≤ i H A S ,G, ζ Q,W, θ := H (cid:16) M G, ζ -ss θ ( Q ) , ̟ ′∗ ̟ ′ ! τ G ≤ i JH G ∗ DT S ,G, ζ Q,W, θ (cid:17) . We define g S ,G, ζ Q,W, θ := P ≤ H A S ,G, ζ Q,W, θ ∼ = H (cid:16) M S ,G, ζ -ss θ ( Q ) , ̟ ′∗ ̟ ′ ! H G, (cid:16) JH G ∗ DT S ,G, ζ Q,W, θ (cid:17)(cid:17) ∼ = BPS S ,G, ζ Q,W, θ ⊗ L / . By Theorem 2.2 the associated graded algebra Gr P H A S ,G, ζ Q,W, θ is supercommutative, and so g S ,G, ζ Q,W, θ is closed under the commutator bracket in H A S ,G, ζ Q,W, θ . The resulting Lie algebra is calledthe BPS Lie algebra [14].
Proposition 2.4.
The universal map τ : U C ( g S ,G, ζ Q,W, θ ) → H A S ,G, ζ Q,W, θ is an inclusion of algebras.Proof. The projection BPS S ,G, ζ Q,W, θ ⊗ H (B C ∗ , Q ) vir → g S ,G, ζ Q,W, θ induces a morphism π : Sym C (cid:16) BPS S ,G, ζ Q,W, θ ⊗ H (B C ∗ , Q ) vir (cid:17) → Sym C (cid:16) g S ,G, ζ Q,W, θ (cid:17) which is a left inverse to the morphism Sym C (cid:16) g S ,G, ζ Q,W, θ (cid:17) → Sym C (cid:16) BPS S ,G, ζ Q,W, θ ⊗ H (B C ∗ , Q ) vir (cid:17) induced by the inclusion g S ,G, ζ Q,W, θ ֒ → BPS S ,G, ζ Q,W, θ ⊗ H (B C ∗ , Q ) vir . We obtain the commutativediagram of Λ ζθ -graded cohomologically graded mixed Hodge structures U C ( g S ,G, ζ Q,W, θ ) τ / / H A S ,G, ζ Q,W, θ Φ − / / Sym C (cid:16) g S ,G, ζ Q,W, θ ⊗ H (B C ∗ , Q ) (cid:17) π (cid:15) (cid:15) Sym C (cid:16) g S ,G, ζ Q,W, θ (cid:17) PBW ∼ = O O = / / Sym C ( g S ,G, ζ Q,W, θ )so that τ is indeed injective. (cid:3) Preprojective CoHAs
The 2-dimensional approach.
Given a quiver Q we define the doubled quiver Q by setting Q = Q and Q = Q ` Q op1 , where Q op1 is the set { a ∗ : a ∈ Q } , and we set s ( a ∗ ) = t ( a ) t ( a ∗ ) = s ( a ) . We define the preprojective algebra as in the introduction:Π Q := C Q/ h X a ∈ Q [ a, a ∗ ] i . For each i, j ∈ Q let V i,j be the vector space with basis given by the set of arrows from i to j .We set GL ′ edge := Y i = j GL( V i,j ) × Y i Sp( V i,i )(18) GL edge := GL ′ edge × C ∗ ~ (19)where C ∗ ~ is a copy of C ∗ . Decomposing A d ( Q ) = Y i = j (cid:0) V i,j ⊗ Hom( C d s ( a ) , C d t ( a ) ) (cid:1) ∗ × (cid:0) V i,j ⊗ Hom( C d s ( a ) , C d t ( a ) ) (cid:1) × Y i (( V i,i ⊕ V ∗ i,i ) ⊗ Hom( C d i , C d i ))it follows that A d ( Q ) carries an action of GL ′ edge preserving the natural symplectic form. Welet C ∗ ~ act by scaling all of A d ( Q ), so that it acts with weight two on the symplectic form. Inthe following, we assume that the gauge group action G → G Q factors through the morphismGL edge → G Q that we have defined here.We denote by ⊕ G red : M G, ζ -ss θ ( Q ) × B G M G, ζ -ss θ ( Q ) → M G, ζ -ss θ ( Q )the morphism taking a pair of polystable C Q -modules to their direct sum.3.1.1. Serre subcategories.
Let S be a Serre subcategory of the category of C Q -modules As in [7]we may consider the examples(1) N is the full subcategory of C Q -modules ρ for which there is a flag of Q -graded subspaces0 ⊂ V . . . ⊂ V of the underlying vector space of ρ such that ρ ( a )( L i ) ⊂ L i − and ρ ( a ∗ )( L i ) ⊂ L i − for every a ∈ Q .(2) SN is the full subcategory of C Q -modules ρ for which there is a flag of Q -graded subspacesas above, satisfying the weaker condition that ρ ( a )( L i ) ⊂ L i − and ρ ( a ∗ )( L i ) ⊂ L i .(3) SSN is the full subcategory of C Q -modules satisfying the same conditions as for SN , butwith the added condition that each of the subquotients L i /L i − is supported at a singlevertex.Let ̟ red : M S ,G, ζ -ss ( Q ) → M G, ζ -ss ( Q ) ̟ ′ red : M S ,G, ζ -ss ( Q ) → M G, ζ -ss ( Q )denote the inclusion of the stack, or respectively the stack-theoretic quotient of the coarse modulispace, of modules in S . Fix a slope θ ∈ Q . We define DT S ,G, ζ Π Q , d := ̟ red , ∗ ̟ ! red ι ∗ ι ! Q M G, ζ -ss d ( Q ) ⊗ L χ e Q ( d , d ) / H A S ,G, ζ Π Q , d := H (cid:16) M G, ζ -ss d ( Q ) , DT S ,G, ζ Π Q , d (cid:17) H A S ,G, ζ Π Q , θ := M d ∈ Λ ζθ H A S ,G, ζ Π Q , d . Remark 3.1.
Since χ e Q ( · , · ) only takes even values, these are genuine mixed Hodge structures, asopposed to monodromic mixed Hodge structures. PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 15
By Verdier duality, there is an isomorphism H A S ,G, ζ Π Q , d ∼ = H BM ( M S ,G, ζ -ss d (Π Q ) , Q ) ⊗ L dim( G ) − χ Q ( d , d ) . Assumption 3.2.
We will always choose S so that H A S ,G, ζ Π Q , θ is free as a C -module. It is a consequence of purity that this assumption holds if we set S to be any of C Q -mod, N , SN or SSN — see [12], [49] as well as [13] for details and discussion.The Λ ζθ -graded mixed Hodge structure H A S ,G, ζ Π Q carries a Hall algebra structure introducedby Schiffmann and Vasserot in the case of the Jordan quiver [48]. It is defined in terms ofcorrespondences. Since the algebra defined this way is isomorphic to the critical CoHA introducedin § RA S ,G, ζ Π Q , d := JH G red ̟ red , ∗ ̟ ! red ι ∗ ι ! Q M G, ζ -ss d ( Q ) ⊗ L χ e Q ( d , d ) / RA S ,G, ζ Π Q , θ := M d ∈ Λ ζθ RA S ,G, ζ Π Q , d the correspondence diagrams that are used to define the Hall algebra structure on H A S ,G, ζ Π Q , θ canbe used to define an algebra structure on RA S ,G, ζ Π Q , θ with respect to the monoidal structure ⊠ ⊕ G red .Since this algebra object will again be isomorphic to the direct image of an algebra object RA e S ,G, ζ e Q, f W , θ for e S , e Q, f W chosen as in § The 3-dimensional description.
For the description of the preprojective CoHA in termsof vanishing cycles, we introduce a particular class of quivers with potential.We start with a quiver Q (not assumed to be symmetric). Then we define the tripled quiver e Q as in § Q , with an additional set of edge-loops Ω = { ω i : i ∈ Q } added to theset of arrows Q , where s ( ω i ) = t ( ω i ) = i . The quiver e Q is symmetric. We extend the action ofGL edge to an action on(20) A d ( e Q ) ∼ = A d ( Q ) × Y i ∈ Q gl i by letting GL ′ edge act trivially on Q i ∈ Q gl i and letting C ∗ ~ act with weight −
2. In what follows,we assume that the G -action, defined by G → G e Q , factors through the inclusion of GL edge .We fix(21) f W = X a ∈ Q [ a, a ∗ ] X i ∈ Q ω i . The function Tr( f W ) is GL edge -invariant, and thus induces a function T r( f W ) on M G ( e Q ).We denote by r : M G ( e Q ) → M G ( Q )the forgetful map taking a C e Q -module to its underlying C Q -module. This morphism is theprojection map from the total space of a vector bundle. The function T r ( f W ) has weight one withrespect to the function that scales the fibres.We denote by ι : M G (Π Q ) ֒ → M G ( Q )the inclusion of the substack of representations satisfying the preprojective algebra relations. Then ι is also the inclusion of the set of points x for which r − ( x ) ⊂ Tr ( W ) − (0). By the dimensionalreduction theorem [11, Thm.A.1] there is a natural isomorphism(22) ι ∗ ι ! → r ∗ φ mon Tr ( W ) r ∗ . Let S be a Serre subcategory of the category of C Q -modules. We denote by e S the Serresubcategory of the category of C e Q -modules ρ satisfying the condition that the underlying C Q -module of ρ is an object of S . As in § DT e S ,G, ζ e Q, f W , θ := ̟ ∗ ̟ ! φ mon T r ( f W ) f IC M G, ζ -ss ( e Q ) RA e S ,G, ζ e Q, f W , θ := JH G ∗ DT e S ,G, ζ e Q, f W , θ H A e S ,G, ζ e Q, f W , θ := H (cid:16) M G, ζ -ss ( e Q ) , DT e S ,G, ζ e Q, f W , θ (cid:17) where the last two objects carry algebra structures via the diagram of correspondences (11).3.2.1. Stability conditions and dimensional reduction.
Via the isomorphism (22) there is a naturalisomorphism r ∗ DT G e Q, f W ∼ = ι ∗ ι ! Q M G ( Q ) ⊗ L χ e Q ( d , d ) / . Applying ̟ red , ∗ ̟ ! red and base change to this isomorphism, gives a natural isomorphism(23) r ∗ DT e S ,G e Q, f W ∼ = ̟ red , ∗ ̟ ! red ι ∗ ι ! Q M G ( Q ) ⊗ L χ e Q ( d , d ) / . We would like to be able to incorporate stability conditions into isomorphism (23) but there is anobvious problem: far from being the projection from a total space of a vector bundle, the forgetfulmorphism from M G, ζ -ss d ( e Q ) → M G, ζ -ss d ( Q ) is not even defined! This is because the underlying C Q -module of a ζ -semistable C e Q -module may be unstable. On the way to resolving the problem,we define M G, ζ -ss ◦ d ( e Q ) := r − d ( M G, ζ -ss d ( Q )) . Then the morphism r ◦ d : M G, ζ -ss ◦ d ( e Q ) → M G, ζ -ss d ( Q ) obtained by restricting r d is the projectionfrom the total space of a vector bundle, as required in the statement of the dimensional reductiontheorem. We will resolve the above problem by use of the following helpful fact. Proposition 3.3. [12, Lem.6.5]
The critical locus of the function T r( f W ) on M ζ -ss ( e Q ) lies inside M G, ζ -ss ◦ ( e Q ) . As a consequence, the support of DT e S ,G, ζ e Q, f W , d is contained in M G, ζ -ss ◦ d ( e Q ) . The absolute CoHA.
Let κ : M G, ζ -ss ◦ d ( e Q ) ֒ → M G, ζ -ss d ( e Q )be the inclusion. We define DT ◦ , e S ,G, ζ e Q, f W , d := κ ∗ DT e S ,G, ζ e Q, f W , d . By dimensional reduction (22) there is an isomorphism(24) r ◦ d , ∗ DT ◦ , e S ,G, ζ e Q, f W , d ∼ = DT S ,G, ζ Π Q , d , and so there is an isomorphism of C-modules(25) H (cid:16) M G, ζ -ss ◦ θ ( e Q ) , DT ◦ , e S ,G, ζ e Q, f W , θ (cid:17) ∼ = H A S ,G, ζ Π Q , θ . On the other hand by Proposition 3.3 we deduce that there are isomorphisms H (cid:16) M G, ζ -ss ◦ θ ( e Q ) , DT ◦ , e S ,G, ζ e Q, f W , θ (cid:17) ∼ = H (cid:16) M G, ζ -ss θ ( e Q ) , DT e S ,G, ζ e Q, f W , θ (cid:17) (26) = H A e S ,G, ζ e Q, f W , θ . Combining (26) and (25) yields the following isomorphism of C-modules in the category of Λ ζθ -graded, cohomologically graded mixed Hodge structures:(27) H A S ,G, ζ Π Q , θ ∼ = H A e S ,G, ζ e Q, f W , θ . As such, H A S ,G, ζ Π Q , θ inherits a C-linear algebra structure from the algebra structure on H A S ,G, ζ e Q, f W , θ . PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 17
The relative CoHA.
In this section we lift the absolute CoHA constructed in § RA S ,G, ζ Π Q , θ in the category D MHM (cid:1) ( M G, ζ -ss θ ( Q )).We denote by κ ′ : M G, ζ -ss ◦ ( e Q ) → M G, ζ -ss ( e Q ) the inclusion of the subscheme of C e Q -modulesfor which the underlying C Q -module is ζ -semistable, and we denote by r ′ : M G, ζ -ss ◦ ( e Q ) → M G, ζ -ss ( Q )the forgetful morphism. To make it easier to keep track of them all, we arrange some of themorphisms introduced in this section into a commutative diagram : M ( e Q ) r / / M ( Q ) M ζ -ss ( e Q ) ?(cid:31) O O JH (cid:15) (cid:15) M ζ -ss ◦ ( e Q ) ? _ κ o o JH ◦ (cid:15) (cid:15) r ◦ / / M ζ -ss ( Q ) JH red (cid:15) (cid:15) ?(cid:31) O O M ζ -ss ( e Q ) M ζ -ss ◦ ( e Q ) ? _ κ ′ o o r ′ / / M ζ -ss ( Q ) . (28)By dimensional reduction there is a natural isomorphism(29) r ′∗ κ ′∗ RA S ,G, ζ e Q, f W , d ∼ = RA S ,G, ζ Π Q , d obtained via commutativity of the diagram (28). Since κ ′ and r ′ are morphisms of monoids, theobject RA S ,G, ζ Π Q , d inherits an algebra structure, as promised in § H ( M G, ζ -ss ◦ ( e Q ) , κ ′∗ RA S ,G, ζ e Q, f W ) ∼ = H ( M G, ζ -ss ( e Q ) , RA S ,G, ζ e Q, f W ) ∼ = H A S ,G, ζ Π Q , θ (the second is (27)). The Hall algebra structure on H A S ,G, ζ Π Q , θ comes from applying H to theHall algebra structure on RA S ,G, ζ Π Q , θ , i.e. RA S ,G, ζ Π Q , θ is a lift of the CoHA H A S ,G, ζ Π Q , θ to the category D MHM (cid:1) ( M G, ζ -ss θ ( Q )). 4. BPS sheaves on M ( Q )4.1. Generalities on the BPS sheaves for ( e Q, f W ) . Let Q be a quiver, then we define e Q and f W as in § ζ ∈ Q Q and a slope θ ∈ Q , as well as an extra gaugegroup G along with a homomorphism G → GL edge as in § G -invariantSerre subcategory S of the category of C Q -modules, satisfying Assumption 3.2, and define e S asin § e S satisfies Assumption 2.1 via the isomorphism (27).With this data fixed, we define the BPS sheaf BPS e S ,G, ζ e Q, f W , θ ∈ D lb MMHM ( M G, ζ -ss θ ( e Q ))as in (14). If S = C Q -mod, so e S = C e Q -mod, this is a G -equivariant (monodromic) mixed Hodgemodule, otherwise, it may be a complex of monodromic mixed Hodge modules with cohomologyin several degrees.4.1.1. The 2d BPS sheaf.
We define GL edge as in (19). We let GL edge act on A via the projectionto C ∗ ~ , and the weight -2 action of C ∗ ~ on A . The inclusion A → gl d t ( t · Id C d i ) i ∈ Q , along with the decomposition (20), induces a GL edge -equivariant inclusion A d ( Q ) × A ֒ → A d ( e Q ) . It is not hard to show that this is an open subscheme; we leave the proof to the reader. We indicate the version where G = { } . In general, there should be G superscripts everywhere. This induces the inclusion l : M G, ζ -ss ( Q ) × B G A ֒ → M G, ζ -ss ( e Q ) . We denote the projection by h : M G, ζ -ss ( Q ) × B G A → M G, ζ -ss ( Q ) . The following theorem is essentially proved in [12, Lem.4.1], though see [13] for the adjustmentsnecessary to incorporate the additional data of S , G, ζ . Theorem/Definition 4.1.
There exists an object (30)
BPS S ,G, ζ Π Q , θ ∈ D lb MMHM ( M G, ζ -ss θ ( Q )) along with an isomorphism (31) BPS e S ,G, ζ e Q, f W , θ ∼ = l ∗ ( h ∗ BPS S ,G, ζ Π Q , θ ⊗ L − / ) . In words, the theorem says that
BPS e S ,G, ζ e Q, f W , θ is supported on the locus containing those C e Q -modules for which all of the generalised eigenvalues of all of the operators ω i · are the samecomplex number t , and the sheaf does not depend on this complex number.By (31) there is an isomorphism of Λ ζθ -graded, cohomologically graded mixed Hodge structures(32) H (cid:16) M G, ζ -ss θ ( e Q ) , BPS e S ,G, ζ e Q, f W , θ (cid:17) ⊗ L / ∼ = H (cid:16) M G, ζ -ss θ ( Q ) , BPS S ,G, ζ Π Q , θ (cid:17) . Definition 4.2.
We define the Lie algebra g S ,G, ζ Π Q , θ := H (cid:16) M G, ζ -ss θ ( Q ) , BPS S ,G, ζ Π Q , θ (cid:17) . The Lie algebra structure is induced by isomorphism (32) and the Lie algebra structure on g S ,G, ζ Π Q , θ ∼ = g e S ,G, ζ e Q, f W , θ ∼ = H (cid:16) M G θ ( e Q ) , BPS S ,G, ζ e Q, f W , θ (cid:17) ⊗ L / . Combining with (27) and (17) there is a PBW isomorphism(33)
Sym B (cid:16) g S ,G, ζ Π Q , θ ⊗ H (B C ∗ , Q ) (cid:17) → H A S ,G, ζ Π Q , θ . Remark 4.3.
In contrast with (17) there is no half Tate twist in (33) , and all of the terms in (33) are defined as mixed Hodge structures without any monodromy.
We note that the image of l lies within M G, ζ -ss ◦ ( e Q ), and thus there is an isomorphism(34) r ′∗ κ ′∗ BPS e S ,G, ζ e Q, f W , θ ⊗ L / ∼ = BPS S ,G, ζ Π Q , θ and so, via (29) and the PBW theorem (16), an isomorphism(35) Sym ⊕ G red (cid:16) BPS S ,G, ζ Π Q , θ ⊗ H (B C ∗ , Q ) (cid:17) ∼ = RA S ,G, ζ Π Q , θ lifting (33).4.2. Restricted Kac polynomials.
In this section we assume that G is trivial, so we drop itfrom the notation. Also, we will work with the degenerate stability condition ζ = (0 , . . . ,
0) andslope θ = 0, so that we may drop ζ and θ from the notation too.We recall the connection between the BPS Lie algebra g S Π Q ∼ = H (cid:16) M ( e Q ) , BPS e S e Q, f W (cid:17) ⊗ L / . and (restricted) Kac polynomials. In the case in which S = C Q -mod, it is proved in [12] that g S Π Q is pure, of Tate type, and has vanishing even cohomology. Thus we have the equality ofpolynomials χ t ( g Π Q , d ) = χ wt ( g Π Q , d ) . (36)By [35] there is an equality(37) χ wt ( g Π Q , d ) = a Q, d ( t − ) PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 19 where a Q, d ( t ) is the Kac polynomial for Q , defined to be the polynomial such that if q = p r is a prime power, a Q, d ( q ) is the number of isomorphism classes of absolutely indecomposable d -dimensional C Q -modules. So combining (36) and (37) we deduce(38) χ t ( g Π Q , d ) = a Q, d ( t − ) . Similarly, by [12, Sec.7.2] the mixed Hodge structures on g SN Π Q , g SSN Π Q are pure, of Tate type. Ina little more detail, by purity of H A SN Π Q and H A SSN Π Q , proved in [49, Sec.4.3], along with the PBWtheorem (17), we deduce that g SN Π Q , d and g SSN Π Q , d are pure of Tate type, since they are subobjectsof pure mixed Hodge structures of Tate type.There are analogues of the Kac polynomials for these Serre subcategories. We recall from [7]that a representation of Q is called 1-nilpotent if there is a flag 0 = F ⊂ F ⊂ . . . ⊂ F r = C d i forevery i ∈ Q such that ρ ( a )( F n ) ⊂ F n − for every n , for every a an edge-loop at i . We define a SN Q, d ( t )to be the polynomial counting the isomorphism classes of absolutely indecomposable 1-nilpotent d -dimensional F q Q -modules, and a SSN Q, d ( t ) to be the analogous count of absolutely indecomposablenilpotent representations. Then by [7] (see also [12, Sec.7.2] for details on the passage to BPScohomology) there are identities χ wt ( g SN Π Q , d ) = a SN Q, d ( t )(39) χ wt ( g SSN Π Q , d ) = a SSN Q, d ( t ) . (40)On the other hand since the mixed Hodge structures on g SN Π Q , d and g SSN Π Q , d are pure, their weightpolynomials agree with their characteristic polynomials. So (39) and (40) yield χ t ( g SN Π Q , d ) = a SN Q, d ( t )(41) χ t ( g SSN Π Q , d ) = a SSN Q, d ( t )(42)respectively.4.2.1. Serre relation for BPS sheaves.
Let i, j ∈ Q be distinct elements of Q , and assume that Q has no edge-loops at i . Let(43) d = ( e + 1) · i + 1 j where e is the number of edges between i and j in the underlying graph of Q . In Proposition 4.5we prove a vanishing theorem for BPS sheaves, that strengthens the identity H ( g Π Q , d ) = 0resulting from the Serre relations in g Q (see § Proposition 4.4. [51, Lem.4.7]
Let Q ′ be a symmetric quiver, let W ′ ∈ C Q ′ cyc be a superpotential,let ζ ∈ Q Q be a stability condition, and let d ∈ N Q be a dimension vector. Let q d : M ζ -ss d ( Q ′ ) → M d ( Q ′ ) be the affinization map. Then there is an isomorphism q d , ∗ BPS ζ Q ′ ,W ′ , d ∼ = BPS Q ′ ,W ′ , d . Proposition 4.5.
Let ζ ∈ Q Q be an arbitrary stability condition, let S be arbitrary, and let d = ( e + 1) · i + 1 j with i , j as above. There is an identity in MHM ( M ζ -ss d ( Q ))(44) BPS S ,G, ζ Π Q , d = 0 . Proof.
Since
BPS S ,G, ζ Π Q , d = ̟ ′ red , ∗ ̟ ′ ! red BPS G, ζ Π Q , d it is sufficient to prove (44) under the assumptionthat S = C Q -mod. In addition, we may assume that G is trivial, since a G -equivariant perversesheaf is trivial if and only if the underlying perverse sheaf is.By Theorem/Definition 4.1, we may equivalently prove that BPS ζ e Q, f W , d = 0. There are threecases to consider:(1) ζ i < ζ j (2) ζ i = ζ j (3) ζ i > ζ j . The proofs for (1) and (3) are the same, while (2) follows from (1) and the identity
BPS e Q, f W , d ∼ = (cid:16) M ζ -ss d ( e Q ) → M d ( e Q ) (cid:17) ∗ BPS ζ e Q, f W , d , which is a special case of Proposition 4.4. So we concentrate on (1).We claim that(45) M ζ -ss d ( e Q ) ∩ crit( T r( f W )) = ∅ . We first note that a point in the left hand side of (45) represents a ζ -semistable Jac( e Q, f W )-module. By Proposition 3.3, the underlying Π Q -module of ρ is ζ -semistable. On the other hand,there are no ζ -semistable d -dimensional Π Q -modules ρ , as for such a ρ the subspace spanned by e i · ρ, b · ρ, . . . , b e − · ρ is a submodule, where b , . . . , b e − are the arrows in Q with source j andtarget i . This proves the claim.Now the proposition follows from the definition (14) and the equalitysupp (cid:16) DT ζ e Q, f W , θ (cid:17) = M ζ -ss d ( e Q ) ∩ crit( T r( f W ))which follows from the fact that for f a regular function on a smooth space X , φ f Q X is supportedon the critical locus of f . (cid:3) Purity of BPS sheaves.
Let X be a stack. We say that G ∈ D
MMHM (cid:1) ( X ) is pure belowif for all integers m < n (46) Gr m W ( H n G ) = 0 . Similarly, we say that G is pure above if (46) holds for all m > n . I.e. purity is the combinationof being pure above and pure below.For example, if X is a smooth variety then H ( X, Q ) is pure below (considered as a mixedHodge module on a point), while if X is projective, H ( X, Q ) is pure above. By Poincar´e duality,it follows that H c ( X, Q ) is pure above if X is smooth. From the long exact sequence in compactlysupported cohomology, and the fact that a variety can be stratified into smooth pieces, it followsthat H c ( X, Q ) is pure above for all varieties X . We will use the following generalisation of thisfact. Lemma 4.6.
Let X be a finite type stack. Let p : X → Y be a morphism of stacks. Then p ! Q X ispure above.Proof. Since p ! Q X only depends on the reduced structure of X , we may assume that X is reduced.We first claim that X can be written as a disjoint union X = S i ∈ I X i of locally closed smoothsubstacks, where I = { , . . . n } is ordered so that X i is open inside X ≤ i := [ j ≤ i X j . This follows from the fact that X sm is smooth and dense inside the reduced stack X , Noetherianinduction, and our assumption that X is of finite type.For q a morphism of varieties, q ! decreases weights. Since, for q : Z → Z ′ a morphism of stacks, q ! is still defined in terms of morphisms of varieties, it still decreases weights. Thus q ! Q Z is pureabove if Z is smooth.We define p i : X i → Y p ≤ i : X ≤ i → Y to be the restrictions of p . Under our assumptions on X , there are distinguished triangles p i, ! Q X i → p ≤ i, ! Q X ≤ i → p ≤ i − Q X ≤ i − . The first term term is pure above, the last term is pure above by induction on i , and so the middleterm is pure above, by the long exact sequence in cohomology. In particular, since p ≤ n = p , wededuce that p ! Q X is pure above. (cid:3) The purpose of this section is to prove the following purity theorem.
PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 21
Theorem 4.7.
The mixed Hodge module
BPS G, ζ Π Q ∈ MHM ( M G, ζ -ss ( Q )) is pure.Proof. Setting q : M ζ -ss ( Q ) → M G, ζ -ss ( Q )to be the quotient map, we have BPS ζ e Q, f W ∼ = q ∗ BPS G, ζ e Q, f W and so it is enough to prove that F = BPS ζ e Q, f W is pure, i.e. we can assume that G = { } . Recall that F ∼ = φ mon T r( W ) IC M ζ -ss ( e Q ) . Since φ mon T r( W ) commutes with Verdier duality, and IC M ζ -ss ( e Q ) ∼ = D M ζ -ss ( e Q ) IC M ζ -ss ( e Q ) , there is anisomorphism(47) D M ζ -ss ( e Q ) F ∼ = F . Now for X a variety and G , L objects of D MMHM (cid:1) ( X ) and MMHM ( X ) respectively, thereare isomorphisms H n ( D X G ) ∼ = D X H − n G D X ( Gr n W L ) ∼ = Gr − n W D X ( L )and so the existence of the isomorphism (47) implies that for m, n ∈ Z we have(48) Gr m W ( H n F ) = 0if and only if Gr − m W (cid:0) H − n F (cid:1) = 0 . In particular, F is pure below if and only if it is pure above. Since by (31) we may write F ∼ = l ∗ (cid:16) BPS ζ Π Q ⊠ IC A (cid:17) and IC A is pure, we deduce that the same symmetry of impurity holds for BPS ζ Π Q : ∗ BPS ζ Π Q is pure below if and only if it is pure above.We will complete the proof by showing that BPS ζ Π Q is pure below. We again consider the com-mutative diagram (28). By 24 there is an isomorphism r ◦∗ κ ∗ DT ζ e Q, f W ∼ = ι ∗ ι ! Q M ζ -ss ( Q ) ⊗ L χ e Q ( d , d ) / and thus an isomorphism r ′∗ JH ◦∗ κ ∗ DT ζ e Q, f W ∼ = JH red , ∗ ι ∗ ι ! Q M ζ -ss ( Q ) ⊗ L χ e Q ( d , d ) / . (49)Applying Verdier duality to the right hand side of (49), we get(50) D M ζ -ss ( Q ) JH red , ∗ ι ∗ ι ! Q M ζ -ss ( Q ) ⊗ L χ e Q ( d , d ) / ∼ = JH red , ! ι ! Q M ζ -ss (Π Q ) ⊗ L χ Q ( d , d ) . The Tate twist comes from the calculationsdim( M ζ -ss ( Q )) = − χ e Q ( d , d ) − d · d χ Q ( d , d ) = d · d + χ e Q ( d , d ) / . By Lemma 4.6 the isomorphic objects of (50) are pure above, and thus the objects of (49) arepure below. On the other hand, there are isomorphisms r ′∗ JH ◦∗ κ ∗ DT ζ e Q, f W ∼ = r ′∗ κ ′∗ JH ∗ DT ζ e Q, f W (51) ∼ = r ′∗ κ ′∗ Sym ⊕ (cid:16) BPS ζ e Q, f W ⊗ H (B C ∗ , Q ) vir (cid:17) ∼ = Sym ⊕ (cid:16) BPS ζ Π Q ⊗ H (B C ∗ , Q ) vir ⊗ L (cid:17) where we have used the PBW theorem (16), and the fact that r ′ and κ ′ are morphisms of monoidsto commute them past Sym ⊕ . Combining (49) and (51) there is an inclusion BPS ζ Π Q ⊗ L / ⊂ JH red , ∗ ι ∗ ι ! Q M ζ -ss ( Q ) ⊗ L χ e Q ( d , d ) / . We deduce that
BPS ζ Π Q is pure below, and thus also pure above by ( ∗ ). (cid:3) Corollary 4.8.
The BPS sheaf
BPS G, ζ e Q, f W ∈ MMHM G ( M ζ ( e Q )) is pure.Proof. This follows from Theorem 4.7 and the isomorphism (31). (cid:3)
The purity statement of Theorem A is a special case of the following corollary of Theorem 4.7:
Corollary 4.9.
The underlying objects of the relative CoHAs in M G, ζ -ss θ ( e Q ) and M G, ζ -ss θ ( Q ) ,i.e. RA G, ζ Π Q , θ ∈ D MHM (cid:1) ( M G, ζ -ss θ ( Q )) RA G, ζ e Q, f W , θ ∈ D MMHM (cid:1) ( M G, ζ -ss θ ( e Q )) respectively, are pure. In particular, applying Verdier duality to the first of these statements, andtaking the appropriate Tate twist, the complex of mixed Hodge modules JH red , ! Q M G, ζ -ss (Π Q ) (52) is pure.Proof. These purity statements follow from Theorem 4.7 and Corollary 4.8, respectively, via (35)and (16), respectively. (cid:3)
Remark 4.10.
Given that the morphism JH : M G, ζ -ss ( e Q ) → M G, ζ -ss ( e Q ) is approximated by proper maps (in the sense of [14] ) and thus sends pure monodromic mixedHodge modules to pure monodromic mixed Hodge modules (see [13] ), it might feel natural, inlight of Corollary 4.9, to conjecture that DT G, ζ e Q, f W is a pure monodromic mixed Hodge module on M G, ζ -ss ( e Q ) . However this statement turns out to be false. For example in the case of Q theJordan quiver, G = { } ζ = (0 , . . . , and d = 4 , impurity follows from the main result of [16] .It seems that purity goes no “higher” than BPS sheaves. The less perverse filtration
The Hall algebra in D MHM (cid:1) ( M G, ζ -ss ( Q )) . We consider the following diagram, wherethe top three rows are defined from diagram (11) (substituting e Q for Q there) by pulling backalong the open embeddings κ : M G, ζ -ss ◦ θ ( e Q ) ֒ → M G, ζ -ss θ ( e Q ) κ ′ : M G, ζ -ss ◦ θ ( e Q ) ֒ → M G, ζ -ss θ ( e Q ) . PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 23 (53) M G, ζ -ss ◦ θ ( e Q ) (2) π × π t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ π ' ' PPPPPPPPPPPP M G, ζ -ss ◦ θ ( e Q ) × B G M G, ζ -ss ◦ θ ( e Q ) JH ◦ × B G JH ◦ (cid:15) (cid:15) M G, ζ -ss ◦ θ ( e Q ) JH ◦ (cid:15) (cid:15) M G, ζ -ss ◦ θ ( e Q ) × B G M G, ζ -ss ◦ θ ( e Q ) r ′ × B G r ′ (cid:15) (cid:15) ⊕ G / / M G, ζ -ss ◦ θ ( e Q ) r ′ (cid:15) (cid:15) M G, ζ -ss θ ( Q ) × B G M G, ζ -ss θ ( Q ) ⊕ G red / / M G, ζ -ss ◦ θ ( Q ) . By base change there is a natural isomorphism RA S ,G, ζ -ssΠ Q , θ = r ′∗ κ ∗ RA e S ,G, ζ -ss e Q, f W , θ . Since the bottom square of (53) commutes, this complex of monodromic mixed Hodge modulesinherits an algebra structure in D MHM (cid:1) ( M S ,G, ζ -ss θ ( Q )), i.e. we obtain the morphism(54) ⊕ G red , ∗ (cid:16) RA S ,G, ζ -ssΠ Q , θ ⊠ B G RA S ,G, ζ -ssΠ Q , θ (cid:17) → RA S ,G, ζ -ssΠ Q , θ and applying the functor H to this morphism we recover the algebra structure on H A S ,G, ζ Π Q , θ .5.2. The relative Lie algebra in MHM G ( M ζ -ss θ ( Q )) . By Theorem 2.2 there is a split inclusion
BPS G, ζ e Q, f W , θ ⊗ L / → RA G, ζ e Q, f W , θ . The commutator Lie bracket provides a morphism[ · , · ] : (cid:16) BPS G, ζ e Q, f W , θ ⊗ L / (cid:17) ⊠ ⊕ G (cid:16) BPS G, ζ e Q, f W , θ ⊗ L / (cid:17) → T = τ G ≤ RA G, ζ e Q, f W , θ and by Theorem 2.2 again, the target T fits into a split triangle BPS G, ζ e Q, f W , θ ⊗ L / → T → H G, ( T ) . and the composition (cid:16) BPS G, ζ e Q, f W , θ ⊗ L / (cid:17) ⊠ ⊕ G (cid:16) BPS G, ζ e Q, f W , θ ⊗ L / (cid:17) → H G, ( T )is the zero morphism. The commutator Lie bracket thus induces a morphism(55) [ · , · ] : (cid:16) BPS G, ζ e Q, f W , θ ⊗ L / (cid:17) ⊠ ⊕ G (cid:16) BPS G, ζ e Q, f W , θ ⊗ L / (cid:17) → BPS G, ζ e Q, f W , θ ⊗ L / and applying r ′∗ κ ′∗ we obtain the Lie bracket(56) [ · , · ] : BPS G, ζ Π Q , θ ⊠ ⊕ G red BPS G, ζ Π Q , θ → BPS G, ζ Π Q , θ . This is a Lie algebra object inside the category
MHM G ( M ζ -ss θ ( Q )), from which we obtain g ζ Π Q , θ by applying H . Likewise, applying ̟ ′ red , ∗ ̟ ′ ! red to (56) we obtain a Lie algebra structure on theobject BPS S ,G, ζ Π Q , θ ∈ D lb MHM ( M ζ -ss θ ( Q )), which becomes the Lie algebra g S ,G, ζ Π Q , θ after applying H .5.3. Definition of the filtration.
By base change, the algebra morphism (54) is given by ap-plying ̟ ′ red , ∗ ̟ ′ ! red to ⊕ G red , ∗ (cid:16) RA G, ζ Π Q , θ ⊠ B G RA G, ζ Π Q , θ (cid:17) → RA G, ζ Π Q , θ , a morphism in D MHM (cid:1) ( M G, ζ -ss θ ( Q )). Furthermore by (35) there is an isomorphism RA G, ζ Π Q , θ ∼ = Sym ⊕ G red (cid:16) BPS G, ζ Π Q , θ ⊗ H (B C ∗ , Q ) (cid:17) . Since
BPS G, ζ Π Q , θ ∈ MHM G ( M ζ -ss θ ( Q )) there is an isomorphism H (cid:16) BPS G, ζ Π Q , θ ⊗ H (B C ∗ , Q ) (cid:17) ∼ = BPS G, ζ Π Q , θ ⊗ H (B C ∗ , Q ) (i.e. the right hand side is isomorphic to its total cohomology) and so also an isomorphism(57) Sym ⊕ G red (cid:16) BPS G, ζ Π Q , θ ⊗ H (B C ∗ , Q ) (cid:17) ∼ = H (cid:16) Sym ⊕ G red (cid:16) BPS G, ζ Π Q , θ ⊗ H (B C ∗ , Q ) (cid:17)(cid:17) . We could alternatively have deduced the existence of this isomorphism from the purity of the lefthand side of (57). It follows that for every p ∈ Z the morphism τ G ≤ p RA G, ζ Π Q , θ → RA G, ζ Π Q , θ has a left inverse α p , and so H ̟ ′ red , ∗ ̟ ′ ! red α p provides a left inverse to the morphism H (cid:16) M G, ζ -ss θ ( Q ) , τ G ≤ p ̟ ′ red , ∗ ̟ ′ ! red RA G, ζ Π Q , θ (cid:17) → H A S ,G, ζ Π Q , θ . Thus the objects L ≤ p H A S ,G, ζ Π Q , θ := H (cid:16) M G, ζ -ss θ ( Q ) , ̟ ′ red , ∗ ̟ ′ ! red τ G ≤ p RA G, ζ Π Q , θ (cid:17) provide an ascending filtration of H A S ,G, ζ Π Q , θ , the less perverse filtration .5.3.1. A warning.
A variant of [14, Warning 5.5] is in force here; if S is not the entire category C Q -mod, the perverse filtration that we have defined here may be quite different from the perversefiltration given by applying perverse truncation functors to JH red , ∗ DT S ,G, ζ Π Q , θ . For instance, let Q bethe Jordan quiver, with one loop, and consider the Serre subcategory SSN . Then one may easilyverify that(58) JH red , ∗ DT SSN Π Q , ∼ = i ∗ Q A ⊗ H (B C ∗ , Q )where i : A ֒ → A is the inclusion of a coordinate hyperplane. In particular, the zeroth perversecohomology of (58) is zero, while if instead we apply ̟ ′ red , ∗ ̟ ′ ! red to the zeroth cohomology of JH red , ∗ DT Π Q , ∼ = Q A ⊗ H (B C ∗ , Q ) ⊗ L − we get the (shifted) mixed Hodge module Q A , and we find L ≤ H A SSN Π Q , ∼ = H ( A , Q ) = 0 . This distinction between the two choices of filtration on H A SSN Π Q is crucial in § Deformed dimensional reduction.
We can generalise the results of this paper, incorpo-rating deformed potentials as introduced in [15]. We indicate how this goes in this section. Wewill not use this generalisation of the less perverse filtration, except in the statement of Corollary6.7 and the example of § W ∈ C Q cyc be a G -invariant linear combination of cyclic words in Q . We make theassumption that there is a grading of the arrows of e Q so that f W + W is quasihomogeneous ofpositive degree. Then in [15] it was shown that there is a natural isomorphism. r ∗ JH G ∗ DT G e Q, f W + W ∼ = φ mon T r( W ) JH G red , ∗ DT G Π Q . In particular, since φ mon T r( W ) is exact, and JH G red , ∗ DT G Π Q is pure by Theorem A, there is an isomor-phism r ∗ JH G ∗ DT G e Q, f W + W ∼ = H (cid:16) r ∗ JH G ∗ DT G e Q, f W + W (cid:17) and so H A G e Q, f W + W carries a less perverse filtration, defined in the same way as the less perversefiltration for H A G e Q, f W . As in § r ∗ BPS G e Q, f W + W ∼ = φ mon T r( W ) BPS G Π Q which recovers the BPS Lie algebra g G Π Q ,W := g G e Q, f W + W after applying H . This isomorphism is given in [13].
PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 25
For a final layer of generality, for S a Serre subcategory of C Q -mod and f W + W a quasiho-mogeneous potential as above, one may consider the object DT S ,G e Q, f W + W := ̟ ∗ ̟ ! DT G e Q, f W + W , andthe associated Hall algebra H A S ,G e Q, f W + W which carries a (less) perverse filtration defined by L ≤ i H A S ,G e Q, f W + W := H ̟ ′ red , ∗ ̟ ′ ! red τ G ≤ i r ∗ JH G ∗ DT G e Q, f W + W with 2d BPS sheaf BPS S ,G Π Q ,W = r ∗ BPS S ,G e Q, f W + W ⊗ L / ∼ = ̟ ′ red , ∗ ̟ ′ ! red φ mon T r( W ) BPS S ,G Π Q and associated BPS Lie algebra g S ,G Π Q ,W = H (cid:16) M ( Q ) , BPS S ,G Π Q ,W (cid:17) . Even at this maximal level ofgenerality, we find that the spherical Lie subalgebra is a Kac-Moody Lie algebra, by Corollary 6.76. The zeroth piece of the filtration
The subalgebra L ≤ H A S ,G, ζ Π Q , θ . By (35), the less perverse filtration on H A S ,G, ζ Π Q , θ begins indegree zero, and thus the subobject L ≤ H A S ,G, ζ Π Q , θ is closed under the CoHA multiplication. It turns out that this subalgebra has a very naturaldescription in terms of the BPS Lie algebra, completing the proof of Theorem A: Theorem 6.1.
There is an isomorphism of algebras L ≤ H A S ,G, ζ Π Q , θ ∼ = U B ( g G, S , ζ Π Q , θ ) . (59) Proof.
Applying τ G ≤ to the isomorphism (35) in the special case S = C Q -mod yields the isomor-phism(60) Sym ⊕ G red (cid:16) BPS G, ζ Π Q , θ (cid:17) ∼ = τ G ≤ (cid:16) RA G, ζ Π Q , θ (cid:17) defined via the relative CoHA multiplication. Now applying H ̟ ′ red , ∗ ̟ ′ ! red to (60) we obtain theisomorphism Sym B ( g S ,G, ζ Π Q , θ ) ∼ = −→ L ≤ H A S ,G, ζ Π Q , θ . By Proposition 2.4, the image of the induced embedding
Sym B ( g S ,G, ζ Π Q , θ ) ֒ → H A S ,G, ζ Π Q , θ is precisely the subalgebra U B ( g G, S , ζ Π Q , θ ). (cid:3) The perverse filtration P ≤• L ≤ H A S ,G, ζ Π Q , θ . Since via (27) there is an inclusion I : L ≤ H A S ,G, ζ Π Q , θ ֒ → H A e S ,G, ζ e Q, f W , θ and H A e S ,G, ζ e Q, f W , θ carries the “more” perverse filtration P ≤• H A e S ,G, ζ e Q, f W , θ defined in [14], we obtain aperverse filtration on L ≤ H A S ,G, ζ Π Q , θ itself, for which the i th piece is(61) L ≤ H A S ,G, ζ Π Q , θ ∩ I − (cid:16) P ≤ i H A e S ,G, ζ e Q, f W , θ (cid:17) . Writing L ≤ H A S ,G, ζ Π Q , θ ∼ = H ̟ ′∗ ̟ ′ ! Sym ⊕ G (cid:16) BPS G, ζ e Q, f W , θ ⊗ L / (cid:17) , we have that L ≤ H A S ,G, ζ Π Q , θ ∩ I − (cid:16) P ≤ n H A e S ,G, ζ e Q, f W , θ (cid:17) ∼ = H ̟ ′∗ ̟ ′ ! τ G ≤ n Sym ⊕ G (cid:16) BPS G, ζ e Q, f W , θ ⊗ L / (cid:17) ∼ = H ̟ ′∗ ̟ ′ ! n M i =0 (cid:16) Sym i ⊕ G (cid:16) BPS G, ζ e Q, f W , θ ⊗ L / (cid:17)(cid:17) ∼ = H n M i =0 (cid:16) Sym i ⊕ G (cid:16) BPS e S ,G, ζ e Q, f W , θ ⊗ L / (cid:17)(cid:17) ∼ = n M i =0 (cid:16) Sym iB (cid:16) g S ,G, ζ Π Q , θ (cid:17)(cid:17) . We deduce the following
Proposition 6.2.
Under the isomorphism (59) , the perverse filtration (61) is sent to the orderfiltration on the universal enveloping algebra U B ( g G, S , ζ Π Q , θ ) . Nakajima quiver varieties.
As preparation for the proof of Theorem 6.6 below, we recallsome fundamental results regarding the action of H A Π Q on the cohomology of Nakajima quivervarieties, recasting these results in terms of vanishing cycle cohomology along the way.6.2.1. Nakajima quiver varieties as critical loci.
Given a quiver Q and a dimension vector f ∈ N Q ,we define the quiver Q f by adding one vertex ∞ to the vertex set Q , and for each vertex i ∈ Q we add f i arrows a i, , . . . , a i, f i with source ∞ and target i .Given a dimension vector d ∈ N Q we denote by d + the dimension vector for Q f defined by • d + | Q = d • d + ∞ = 1.From the quiver Q f we form the quiver f Q f via the tripling construction of § i ∈ Q there are f i arrows a ∗ i, , . . . , a ∗ i, f i in ( f Q f ) with source i and target ∞ . We denote by f W f thecanonical cubic potential for f Q f . We form Q + by removing the loop ω ∞ from f Q f , and form W + from f W f by removing all paths containing ω ∞ . So in symbols W + = X a ∈ Q [ a, a ∗ ] + f i X m =1 a i,m a ∗ i,m ! X i ∈ Q ω i . We define the stability condition ζ ∈ Q Q +0 by setting ζ i = 0 for i ∈ Q and ζ ∞ = 1. Then a d + -dimensional C Q + -module ρ is ζ + -stable if and only if it is ζ + -semistable. This occurs if andonly if the vector space e ∞ · ρ ∼ = C generates ρ under the action of C Q + .We define the fine moduli space M f , d ( Q ) = A ζ + -ss d + ( Q + ) / GL d , which carries the function T r( W + ) d .Following Nakajima [38], we define M ( f , d ) ⊂ A ζ + -ss d + ( Q f ) / GL d to be intersection with theGL d -quotient of the zero set of the moment map A d ( Q ) × A d ( Q op ) × Y i ∈ Q ( C d i ) f i × Y i ∈ Q (( C d i ) f i ) ∗ → gl d ( A, A ∗ , I, J ) [ A, A ∗ ] + IJ.
We define the embedding ι : M ( f , d ) ֒ → M f , d ( Q ) by extending a C Q f -module to a C Q + module,setting the action of each of the ω i for i ∈ Q to be zero. If M ( f , d ) = ∅ then(62) dim( M ( f , d )) = 2 f · d − χ Q ( d , d ) . Proposition 6.3.
There is an equality of subschemes crit (cid:0) T r( W + ) d (cid:1) = M ( f , d ) . PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 27
Moreover, there is an isomorphism of mixed Hodge modules (63) φ mon T r( W + ) d IC M f , d ( Q ) ∼ = ι ∗ IC M ( f , d ) , so that, in particular, there is an isomorphism of mixed Hodge structures (64) H (cid:16) M f , d ( Q ) , φ mon T r( W + ) d IC M f , d ( Q ) (cid:17) ∼ = H ( M ( f , d ) , Q ) ⊗ L χ Q ( d , d ) − f · d . Proof.
The (Verdier dual of the) isomorphism (64) is constructed in [12, Thm.6.6] via dimensionalreduction, so we just need to prove the other parts of the proposition.Let c be an arrow from ∞ to i in Q + . Then ∂W + /∂c ∗ = ω i c∂W + /∂c = c ∗ ω i . For a an arrow in Q we have ∂W/∂a ∗ = ω t ( a ) a − aω s ( a ) ∂W/∂a = a ∗ ω t ( a ) − ω s ( a ) a ∗ . Putting these facts together, we have an isomorphism of algebras(65) Jac( Q + , W + ) ∼ = Π Q f [ ω ] / h e ∞ ωe ∞ = 0 i where ω = X i ∈ Q +0 ω i . We consider the fine moduli space N = A ζ + -ss d + ( f Q f ) / GL d . Then the critical locus of T r( f W f ) is identified with the total space of the tautological line bundleon M ( f , d ) for which the fibre over ρ is the space of endomorphisms of ρ . By (65), crit( T r( W + ) d )is the zero section, i.e. it is M ( f , d ).Since the (scheme-theoretic) critical locus of T r( W + ) d is smooth, by the holomorphic Bott–Morse lemma this function can be written analytically locally (on M ( f , d )) as T r( W + ) d = x + . . . + x e where e is the codimension of M ( f , d ) inside M f , d ( Q ), and so φ mon T r( W + ) d IC M f , d ( Q ) is analyticallylocally isomorphic to IC M ( f , d ) . In particular, as a perverse sheaf it is locally isomorphic to Q M f , d ( Q ) [2 f · d − χ Q ( d , d )], and is thus determined by its monodromy . Finally, in cohomologicaldegree 2 χ Q ( d , d ) − f · d the right hand side of (64) is a vector space spanned by the componentsof M ( f , d ), while the left hand side is a vector space spanned by the components of M ( f , d ) onwhich the monodromy of the underlying perverse sheaf of φ mon T r( W + ) d IC M f , d ( Q ) is trivial. Sincethese dimensions are the same, the monodromy is trivial, and isomorphism (63) follows. (cid:3) CoHA modules from framed representations.
There is a general construction, producingmodules for cohomological Hall algebras out of moduli spaces of framed quiver representations,see [50] for a related example and discussion. We recall the variant relevant to us.Let d ′ , d ′′ ∈ N Q be dimension vectors, with d = d ′ + d ′′ . We define A ζ + -ss d ′ , d ′′ + ( Q + ) ⊂ A ζ + -ss d + ( Q + )to be the subspace of C Q + -modules ρ such that the underlying C e Q -module of ρ preserves the Q -graded flag(66) 0 ⊂ C d ′ ⊂ C d These are indeed mixed Hodge modules, since by (62) there are an even number of half Tate twists in thedefinition (6) of the right hand side of (63). This is the monodromy around M f , d ( Q ) and is unrelated to the “monodromic” in “monodromic mixed Hodgemodule”. and for every arrow c ∗ with t ( c ∗ ) = ∞ we have ρ ( c ∗ )( C d ′ ) = 0. Given such a ρ we obtain a shortexact sequence(67) 0 → ρ ′ → ρ → ρ ′′ → ρ ′ ) = ( d ′ ,
0) and dim( ρ ′′ ) = d ′′ + . We set M f , d ′ , d ′′ ( Q ) := A ζ + -ss d ′ , d ′′ + ( Q + ) / GL d ′ , d ′′ where GL d ′ , d ′′ ⊂ GL d is the subgroup preserving the flag (66). There are morphisms π : M f , d ′ , d ′′ ( Q ) → M d ′ ( e Q ) π : M f , d ′ , d ′′ ( Q ) →M f , d ( Q ) π : M f , d ′ , d ′′ ( Q ) →M f , d ′′ ( Q )taking a point representing the short exact sequence (67) to ρ ′ , ρ, ρ ′′ respectively.Then in the correspondence diagram M f , d ′ , d ′′ ( Q ) π × π u u ❧❧❧❧❧❧❧❧❧❧❧❧❧ π & & ◆◆◆◆◆◆◆◆◆◆◆ M d ′ ( e Q ) × M f , d ′′ ( Q ) M f , d ( Q )the morphism π is a proper morphism between smooth varieties, so that as in § α : Q M d ′ ( e Q ) ×M f , d ′′ ( Q ) ⊗ L ♥ → ( π × π ) ∗ Q M f , d ′ , d ′′ ( Q ) ⊗ L ♥ β : π , ∗ Q M f , d ′ , d ′′ ( Q ) ⊗ L ♥ → Q M f , d ( Q ) ⊗ L χ e Q ( d , d ) − f · d where ♥ = χ e Q ( d ′ , d ′ ) + χ e Q ( d ′′ , d ′′ ) − f · d ′′ and β is the Verdier dual of Q M f , d ( Q ) ⊗ L χ e Q ( d , d ) − f · d → π , ∗ Q M f , d ′ , d ′′ ( Q ) ⊗ L χ e Q ( d , d ) − f · d . For d , f ∈ N Q we define H N ∗ f , d := H (cid:16) M f , d ( Q ) , φ mon T r( W + ) Q M f , d ( Q ) (cid:17) ⊗ L χ e Q ( d , d ) / − f · d and we define H N ∗ f := M d ∈ N Q H N ∗ f , d . Applying φ mon T r( W + ) and taking hypercohomology, via the Thom–Sebastiani isomorphism we obtaina morphism H ( φ mon T r( W + ) β ) ◦ H ( φ mon T r( W + ) α ) ◦ TS : H A e Q, f W , d ′ ⊗ H N ∗ f , d ′′ → H N ∗ f , d endowing H N ∗ f with the structure of a H A e Q, f W -module. By Proposition 6.3 there is an isomor-phism H N ∗ f , d ∼ = H ( M ( f , d ) , Q ) ⊗ L χ Q ( d , d ) − f · d . Here we have used the calculationdim( M f , d ( Q )) = − χ e Q ( d , d ) + 2 f · d along with (64). As such, we obtain an action of H A Π Q ∼ = H A e Q, f W on the cohomology of Nakajimaquiver varieties. PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 29
Remark 6.4.
Let M S ( f , d ) ⊂ M ( f , d ) be the subvariety for which the underlying Q -module of ρ lies in some Serre subcategory S . Applying exceptional restriction functors to the above morphismsof mixed Hodge modules, we may likewise define an action of H A S Π Q on M d ∈ N Q H BM ( M S ( f , d ) , Q ) ⊗ L f · d − χ Q ( d , d ) , although we will make no use of this generalisation here. Kac–Moody Lie algebras and quiver varieties.
For the rest of § Q hasno edge-loops. Let i ∈ Q be a vertex and let 1 i ∈ N Q be the basis vector for the vertex i . Thenthere is an isomorphism Ψ i : H A e Q, f W , i ∼ = H ( A / C ∗ , Q ) ∼ = Q [ u ]and we set(68) α i := Ψ − i (1) ∈ H A e Q, f W , i . The action of α i provides a morphism α i · : H N ∗ f , d → H N ∗ f , d +1 i . Consider the semisimplification map JH : M f , d ( Q ) → M d + ( Q f ). Following Lusztig [30] we con-sider the Lagrangian subvariety L ( f , d ) = JH − (0) . There is a contracting C ∗ action on M ( f , d ), contracting it onto the projective variety L ( f , d ),and so we obtain the first in the sequence of isomorphisms H ( M ( f , d ) , Q ) ∼ = H ( L ( f , d ) , Q ) ∼ = H c ( L ( f , d ) , Q ) ∼ = H BM ( L ( f , d ) , Q ) ∗ . Note that since L ( f , d ) is Lagrangian, its top degree compactly supported cohomology is in degreedim( M ( f , d )) = 2 f · d − χ Q ( d , d ) . Setting H N f , d := H BM ( L ( f , d ) , Q ) ⊗ L f · d − χ Q ( d , d ) H N f := M d ∈ N Q H N f , d we deduce that there is a H A Π Q action on H N f by lowering operators, for which ( α i · ) ∗ isthe lowering operator constructed by Nakajima. By the degree bound on the cohomology of H c ( L ( f , d ) , Q ), H i N f , d = ( i < Q · { top-dimensional components of L ( f , d ) } if i = 0 . The main theorem regarding the operators ( α i · ) ∗ is the following part of Nakajima’s work. Theorem 6.5. [37, 38]
There is an action of the Kac–Moody Lie algebra g on each H N f sendingthe generators f i for i ∈ Q to the operators ( α i · ) ∗ . With respect to this g Q action, the submodule H N f , d is the irreducible highest weight module with highest weight f . The original statement of Nakajima’s theorem does not involve any vanishing cycles, i.e. it onlyinvolves the right hand side of the isomorphism (64). Likewise, the correspondences considered in[37, 38] come from an action of the Borel–Moore homology of the stack of Π Q -representations, notfrom the critical cohomology of the stack of Π Q -representations. For the compatibility betweenthe two actions via the dimensional reduction isomorphisms (64) and (27) see e.g. [53, Sec.4]. The subalgebra L ≤ H A Π Q . In this section we concentrate on the case in which S = C Q -mod, G is trivial and ζ = (0 , . . . ,
0) is the degenerate stability condition (i.e. we essentiallydo not consider stability conditions). Note that by (38), the Lie algebra g Π Q is concentrated incohomological degrees less than or equal to zero, and so by (78) there is an isomorphism(69) L ≤ H A Π Q ∼ = U (cid:0) H ( g Π Q ) (cid:1) . We thus reduce the problem to calculating H ( g Π Q ). By (38) again, there is an equality(70) dim( H ( g Π Q , d )) = a Q, d (0) . If d i = 0 for some i ∈ Q for which there is an edge-loop b , there is a free action of F q on the setof absolutely indecomposable d -dimensional C Q -modules, defined by z · ρ ( a ) = ( ρ ( a ) + z · Id e i · ρ if a = bρ ( a ) otherwiseand thus a Q, d (0) = 0, and so H ( g Π Q , d ) = 0. It follows that if d i = 0 for any vertex i supportingan edge loop, then L ≤ H A ζ Π Q , d = 0.We define Q ′ , the real subquiver of Q , to be the full subquiver of Q containing those verticesof Q that do not support any edge-loops, along with all arrows between these vertices. From theabove considerations, we deduce that the morphism of algebras in D MHM (cid:1) ( M ( Q )) RA Π Q ′ → RA Π Q becomes an isomorphism after applying L ≤ H .Hausel’s (first) famous theorem regarding Kac polynomials [20] states that(71) a Q ′ , d (0) = dim( g Q ′ , d )where g Q ′ is the Kac–Moody Lie algebra associated to the quiver (without edge-loops) Q ′ . Wewill not recall the definition of g Q ′ , since in any case it is a special case of the Borcherds–Bozecalgebra (i.e. the case in which I im = ∅ ), which we recall in § H ( g Π Q , d )) = dim( g Q ′ , d )and from there to the obvious conjecture regarding the algebra L ≤ H A ζ Π Q , which we now prove. Theorem 6.6.
There is an isomorphism of algebras (73) U ( n − Q ′ ) ∼ = L ≤ H A ζ Π Q where n − Q ′ is the negative part of the Kac–Moody Lie algebra for the real subquiver of Q . Moreoverthe isomorphism restricts to an isomorphism with the BPS Lie algebra (74) n − Q ′ ∼ = H ( g Π Q ) under the isomorphism (69) .Proof. We construct the isomorphism (74), then the isomorphism (73) is constructed via (69).Consider the dimension vector e = 1 i , where i does not support any edge-loops. The coarsemoduli space M e ( Q ) is just a point, and so the less perverse filtration on H A Π Q ,e = H (B C ∗ , Q )is just the cohomological filtration. In particular, the element α i from (68) lies in less perversedegree 0. On the other hand, there is an isomorphism M e ( e Q ) ∼ = A and writing H A e Q, f W ,e = H ( A , IC A ) ⊗ H (B C ∗ , Q ) vir we see that α i has perverse degree 1, i.e. by definition it is an element of g Π Q ,e .We claim that there is a Lie algebra homomorphism Φ : n − Q ′ → H ( g Π Q ) sending f i to α i . Thealgebra n − Q ′ is the free Lie algebra generated by f i for i ∈ Q subject to the Serre relations:[ f i , · ] − (1 i , j ) Q ( f j ) = 0 PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 31 where ( · , · ) Q is the symmetrized Euler form (10). So to prove the claim we only need to prove thatthe elements α i satisfy the Serre relations. This follows from the stronger claim: for any distinctpair of vertices i, j ∈ Q if we set γ = 1 j + (1 − (1 i , j ))1 i then there is an equality g Π Q , γ = 0 . This follows from Proposition 4.5. Alternatively, this follows from (38) and the claim that a Q, γ ( t ) = 0. Since the Kac polynomial is independent of the orientation of Q this equality isclear: if all the arrows are directed from i to j , there are no indecomposable γ -dimensional KQ -modules for any field K .We next claim that the morphism n − Q ′ → H ( g Π Q ) is injective. This follows from Nakajima’stheorem, i.e. we have a commutative diagram n − Q ′ u & & ▼▼▼▼▼▼▼▼▼▼▼▼ Φ f i α i / / H ( g Π Q ) (cid:15) (cid:15) End Q ( L f H N ∗ f )via the module structure of each H N ∗ f , and the morphism u is injective since the representa-tion L f H N ∗ f is a faithful representation. Faithfulness follows, for example, from the Kac–Weylcharacter formula.By Hausel’s identity (71) the graded dimensions of the source and target of Φ are the same,and so Φ is an isomorphism. Comparing with (69), the induced morphism U ( n − Q ′ ) → L ≤ H A ζ Π Q is an isomorphism. (cid:3) Theorem 6.6 shows that Kac–Moody Lie algebras are a natural piece of the BPS Lie algebra inthe basic case in which we do not modify potentials, and do not restrict to a Serre subcategory.The next proposition shows that Kac–Moody Lie algebras are a somewhat universal feature of thecohomological Hall algebras that we are considering.
Corollary 6.7.
Let Q be a quiver, let Q ′ be the real subquiver of Q , let W ∈ C Q cyc be a potentialsuch that f W + W is quasihomogeneous, and let S be any Serre subcategory of C Q -mod containingeach of the 1-dimensional simple modules S i with dimension vector i , for i ∈ Q ′ . Then there isa N Q -graded inclusion of Lie algebras n − Q ′ ֒ → g S Π Q ,W with image the Lie subalgebra of g S Π Q ,W generated by the graded pieces g S Π Q ,W , i for i ∈ Q ′ .Proof. By the proof of Theorem 6.6 the subalgebra n − Q ′ ⊂ g Π Q is obtained by applying H to theLie subalgebra object G of BPS Π Q generated by the objects BPS Π Q , i for i ∈ Q ′ , i.e. n − Q ′ ∼ = H G .There is a decomposition BPS Π Q ∼ = G ⊕ L of mixed Hodge modules, by purity of
BPS Π Q , and hence the inclusion G ֒ → BPS Π Q splits in thecategory of mixed Hodge modules. Applying H ̟ ′∗ ̟ ′ ! φ mon W gives an inclusion of Lie algebras H ̟ ′∗ ̟ ′ ! φ mon W G ֒ → g S Π Q ,W . Each mixed Hodge module G d is supported at the origin of M d ( Q ), since G is generated by mixedHodge modules supported on the nilpotent locus. The Serre subcategory S contains all nilpotent C Q -modules supported on the subquiver Q ′ , since it is closed under extensions. As such thenatural morphisms ̟ ′∗ ̟ ′ ! G → G φ mon T r( W ) G → G , are isomorphisms, and ̟ ′∗ ̟ ′ ! φ mon T r( W ) G ∼ = G as a Lie algebra object in MHM ( M ( Q )), proving thecorollary. (cid:3) The subalgebra L ≤ H A SSN Π Q . Moving to the case of strongly semi-nilpotent Π Q -modules(see § L ≤ H A SSN Π Q , in order to compare with work of Bozec [5].Interestingly, we find that the BPS Lie algebra g SSN Π Q is not identified with Bozec’s Lie algebra g Q under the natural isomorphism between their two enveloping algebras, although the two Liealgebras are isomorphic.First we note that by (42) the Lie algebra g SSN Π Q is concentrated in cohomologically nonnegativedegrees. Applying H to (17), the morphism Sym ( H ( g SSN Π Q )) → H A SSN Π Q is an isomorphism, and thus there is an identity(75) U ( H ( g SSN Π Q )) = H A SSN Π Q . Comparing with (78) we deduce that L ≤ H A SSN Π Q = H A SSN Π Q and so for the rest of § H A SSN Π Q to denote this subalgebra.6.4.1. The Borcherds–Bozec algebra.
We write Q = I real a I im , where I real is the set of vertices that do not support an edge-loop, and I im is the set vertices thatdo. We furthermore decompose I im = I iso a I hyp where I iso is the set of vertices supporting exactly one edge-loop, and the vertices of I hyp supportmore than one.Out of the quiver Q we build the Borcherds–Bozec algebra g Q as follows. We set I ∞ = ( I real × { } ) a ( I im × Z > )and we extend the form (10) to a bilinear form on N I ∞ by setting((1 i ′ ,n ) , (1 j ′ ,m )) = mn (1 i ′ , j ′ ) Q and extending linearly. The Lie algebra g Q is a Borcherds algebra associated to a generalisedCartan datum for which the Cartan matrix is the form ( · , · ) expressed in the natural basis of N I ∞ .More explicitly, we define g Q to be the free Lie algebra generated over Q by h i ′ , e i , f i for i ′ ∈ Q and i ∈ I ∞ subject to the relations[ h i ′ , h j ′ ] =0[ h j ′ , e ( i ′ ,n ) ] = n (1 j ′ , i ′ ) Q · e ( i ′ ,n ) [ h j ′ , f ( i ′ ,n ) ] = − n (1 j ′ , i ′ ) Q · f ( i ′ ,n ) [ e j , · ] − ( j,i ) e i = [ f j , · ] − ( j,i ) f i =0 if j ∈ I real × { } , i = j [ e i , e j ] = [ f i , f j ] =0 if ( i, j ) = 0[ e i , f j ] = δ i,j nh i ′ if i = ( i ′ , n ) . The positive half n + Q has an especially quick presentation: it is the Lie algebra over Q freelygenerated by e i for i ∈ I ∞ , subject to the relations[ e i , · ] − ( i,j ) ( e j ) = 0 if i ∈ I real × { } , i = j (76) [ e i , e j ] = 0 if ( i, j ) = 0 . (77) PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 33
Lagrangian subvarieties.
Define Λ ( d ) = A SSN d ( Q ) ∩ µ − (0), the subvariety of A d ( Q ) param-eterising strictly semi-nilpotent Π Q -modules. By [5, Thm.1.15], this is a Lagrangian subvarietyof A d ( Q ). If d = n · i ′ for some i ′ ∈ Q , by [5, Thm.1.4] the irreducible components of Λ ( d ) areindexed by tuples ( n , . . . , n r ) such that P n s = n . Let I l be the two-sided ideal in C Q containingall those paths in Q containing at least l instances of arrows a ∈ Q . The tuple c correspondingto a component Λ ( d ) c is given by the successive dimensions of the subquotients in the filtration0 = I r · ρ ⊂ I r − · ρ ⊂ . . . ρ for ρ a module parameterised by a generic point on Λ ( d ) c . For example there is an equality Λ ( d ) ( n ) = A d ( Q op ) ⊂ A SSN d ( Q ) ∩ µ − (0) . To translate Bozec’s results into our setting we follow the arguments of [40, Sec.2]. Unpicking thedefinitions, we have H A SSN Π Q := M d ∈ N Q H BM ( Λ ( d ) / GL d , Q ) ⊗ L − χ Q ( d , d ) . Since Λ ( d ) is a Lagrangian subvariety of the 2( d · d − χ Q ( d , d ))-dimensional subvariety A d ( Q ),the irreducible components of Λ ( d ) / GL d are − χ Q ( d , d )-dimensional. It follows that H A SSN Π Q has a natural basis given by [ Λ ( d ) e ] where Λ ( d ) e are the irreducible components of Λ ( d ). Theorem 6.8. [5, Prop.1.18, Thm.3.34]
There is an isomorphism of algebras U ( n + Q ) → H A SSN Π Q which sends e i ′ ,n to [ Λ ( n · i ′ ) ( n ) ] . Combining with (75) we obtain an isomorphism(78) F : U ( n + Q ) ∼ = −→ U ( H ( g SSN Π Q )) . Corollary 6.9.
There exists an isomorphism of Lie algebras n + Q ∼ = H ( g SSN Π Q ) where the left hand side is the Borcherds–Bozec algebra of the full quiver Q .Proof. Let S be a minimal N Q -graded generating set of H ( g SSN Π Q ). Then F − ( S ) is a minimalgenerating set of U ( n + Q ) and so | S d | is the number of degree d generators of n + Q . We show thatthe elements of S satisfy the Serre relations.Let g, g ′ ∈ S be of degree m · i ′ and n · j ′ respectively. If (1 i ′ , j ′ ) Q = 0 then g and g ′ commute,i.e. they satisfy the Serre relation [ g, g ′ ] = 0, dealing with (77).We next consider (76). Assume that there are no edge-loops of Q at i ′ , so that up to a scalarmultiple g = e ( i ′ , . Write g ′ as a linear combination of monomials Q lr =1 e ( j ′ ,t r ) for t r ∈ N summingto n . Since [ e ( i ′ , , · ] is a derivation, the identity[ e ( i ′ , , g ′ ] − n (1 i ′ , j ′ ) Q = 0follows from the identities [ e ( i ′ , , e ( j ′ ,t r ) ] − t r (1 i ′ , j ′ ) Q = 0 . So the generators S satisfy the Serre relations and there is a surjection n + Q → H ( g SSN Π Q ), which isinjective since the graded pieces have the same dimensions. (cid:3) Although Corollary 6.9 establishes that they are abstractly isomorphic, we spend the rest of § difference between the two Lie subalgebras F ( n + Q ) and H ( g SSN Π Q ). Isotropic vertices.
Let i ′ ∈ I iso , and set d = n · i ′ . Let∆ n : A ֒ → M d ( e Q )be the inclusion, sending ( z , z , z ) ∈ A to the Jac( e Q, f W )-representation for which the action ofthe three arrows a, a ∗ , ω i ′ is scalar multiplication by z , z , z respectively. Then by [12, Thm.5.1]there is an isomorphism(79) BPS e Q, f W , d ∼ = ∆ n, ∗ Q A ⊗ L − / = ∆ n, ∗ IC A . Thus by (34) there is an isomorphism
BPS Π Q , d ∼ = ∆ red ,n, ∗ Q A ⊗ L − = ∆ red ,n, ∗ IC A where ∆ red ,n : A ֒ → M n · i ′ ( Q )is the inclusion taking z , z to the module ρ for which a and a ∗ act via multiplication by z and z respectively. Thus we find that BPS
SSN Π Q , d ∼ = ̟ ′ red , ∗ ̟ ′ ! red ∆ red ,n, ∗ IC A (80) ∼ =∆ SSN red ,n, ∗ Q A (81)where ∆ SSN red ,n, ∗ : A ֒ → M n · i ′ ( Q )takes z to the module ρ for which a ∗ acts via multiplication by z and a acts via the zero map. By(35) we deduce the following proposition. Proposition 6.10.
There is an isomorphism in D MHM (cid:1) ( M ( Q ))(82) M n ≥ RA SSN Π Q ,n · i ′ ∼ = Sym ⊕ red M n ≥ ∆ SSN red ,n, ∗ Q A ⊗ H (B C ∗ , Q ) . In particular, L n ≥ RA SSN Π Q ,n · i ′ is pure, as is (83) M n ≥ BPS
SSN Π Q ,n · i ′ ∼ = Sym ⊕ red M n ≥ ∆ SSN red ,n, ∗ Q A . Remark 6.11.
Applying Verdier duality to (82) makes for a cleaner looking statement of theresult. The Verdier dual of (82) is the isomorphism M n ≥ (cid:16) M SSN n · i ′ (Π Q ) → M n · i ′ ( Q ) (cid:17) ! Q M SSN n · i ′ (Π Q ) ∼ = Sym ⊕ M n ≥ ,m ≥ ∆ SSN red ,n, ∗ Q A ⊗ L − m − . From (81) we deduce that(84) Ψ : g SSN Π Q , d ∼ = H ( A , Q ) ∼ = Q as a cohomologically graded vector space, i.e. g SSN Π Q , d is one dimensional and concentrated incohomological degree zero. We denote by(85) α i ′ ,n = Ψ − (1)a basis element, so g SSN Π Q , d = Q · α i ′ ,n . One can see directly (e.g. without the aid of Corollary 6.9) that for m, n ∈ Z ≥ with m = n [ α i ′ ,m , α i ′ ,n ] = 0 . (86)For example this follows because the Lie bracket g SSN Π Q ,n · i ⊗ g SSN Π Q ,m · i → g SSN Π Q ,m + n · i is defined by applying H ̟ ′ red , ∗ ̟ ′ ! red to the morphism of mixed Hodge modules(87) ∆ red ,n, ∗ IC A ⊠ ⊕ red ∆ red ,m, ∗ IC A → ∆ red ,m + n, ∗ IC A . PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 35
Since m = n the morphism ⊕ red ◦ (∆ red ,n × ∆ red ,m ) : A → M m + n ( Q )is injective. It follows that the left hand side of (87) is simple, and not isomorphic to the (simple)right hand side, which has 2-dimensional support. It follows that (87) is the zero map, since it isa morphism between distinct simple objects. Proposition 6.12.
Let i ′ ∈ Q iso0 , and set d = n · i ′ as above. Up to multiplication by a scalar,there is an identity α i ′ ,n = [ Λ ( d ) (1 n ) ] ∈ H A SSN Π Q , d . Proof.
Let Λ ( d ) ◦ (1 n ) ⊂ Λ ( d ) (1 n ) be the complement to the intersection with the union of compo-nents Λ ( d ) π for π = (1 n ). Denote by j : Λ ( d ) (1 n ) / GL d → M d ( Q ) j ◦ : Λ ( d ) ◦ (1 n ) / GL d → M d ( Q )the inclusions. Define G = j ∗ j ! Q M d ( Q ) ⊗ L χ e Q ( d , d ) / G ◦ = j ◦∗ j ◦ , ! Q M d ( Q ) ⊗ L χ e Q ( d , d ) / ∼ = j ◦∗ Q Λ ( d ) ◦ (1 n ) where the isomorphism is due to the fact that Λ ( d ) ◦ (1 n ) is smooth and has codimension − χ e Q ( d , d ) / M d ( Q ). Then since j is closed, and Λ ( d ) ◦ ( n ) is open in Λ ( d ) ( n ) , we have a diagram(88) JH red , ∗ G ◦ JH red , ∗ G ξ o o q / / JH red , ∗ DT SSN Π Q , d Applying H , ξ is an isomorphism, and [ Λ ( d ) (1 n ) ] is defined to be H ( q )( H ( ξ )) − (1). Applying τ con , ≤ , the truncation functor induced by the non-perverse t structure, the diagram (88) becomes∆ SSN red ,n, ∗ Q A ∆ SSN red ,n, ∗ Q A ∼ = o o / / τ con , ≤ JH red , ∗ DT SSN Π Q , d . The element α i ′ ,n is likewise obtained by applying H to a homomorphism q ′ : ∆ SSN red ,n, ∗ Q A → JH red , ∗ DT SSN Π Q , d . The domain is a mixed Hodge module, shifted by cohomological degree 1, so that q ′ factors throughthe morphism τ ≤ JH red , ∗ DT SSN Π Q , d → JH red , ∗ DT SSN Π Q , d . By (82) there is an isomorphism τ ≤ JH red , ∗ DT SSN Π Q , d ∼ = ∆ SSN red ,n, ∗ Q A and so we deduce that dim (cid:16) Hom (cid:16) ∆ SSN red ,n, ∗ Q A , JH red , ∗ DT SSN Π Q , d (cid:17)(cid:17) = 1, and the proposition follows. (cid:3) Hyperbolic vertices.
Suppose that i ′ ∈ I hyp , i.e. i ′ supports l edge-loops with l ≥
2. Let n ∈ Z > , and set d = n · i ′ . The variety M d (Π Q ) is an irreducible variety of dimension 2+2( l − n by [9, Thm.1.3]. We set C u i ′ ,n := IC M d (Π Q ) ∈ MHM ( M d ( Q )) C u SSN i ′ ,n := ̟ ′ red , ∗ ̟ ′ ! red IC M d (Π Q ) ∈ D b ( MHM ( M d ( Q ))) cu SSN i ′ ,n := H (cid:0) M d ( Q ) , C u SSN i ′ ,n (cid:1) . The following will be proved as a special case of the results in § Proposition 6.13.
Set B n = τ ≤ RA Π Q , d ∈ MHM ( M d (Π Q )) . There is an inclusion C u i ′ ,n ֒ →B n and C u i ′ ,n is primitive, i.e. the induced morphism (89) C u i ′ ,n → B n / X n ′ + n ′′ = nn ′ =0 = n ′′ Image (cid:16) B n ′ ⊠ ⊕ B n ′′ ⋆ −→ B n (cid:17) is injective, where ⋆ is the CoHA multiplication of § By Theorem 4.7, (89) is a morphism of semisimple objects, as well as being injective, and so ithas a left inverse. Applying H ̟ ′ red , ∗ ̟ ′ ! red we deduce that there is an injective morphism(90) c SSN i ′ ,n ֒ → g SSN Π Q ,n and moreover that the induced morphism(91) c SSN i ′ ,n → g SSN Π Q ,n / X n ′ + n ′′ = nn ′ =0 = n ′′ Image (cid:18) g SSN Π Q ,n ′ ⊗ g SSN Π Q ,n ′′ [ • , • ] −−−→ g SSN Π Q ,n (cid:19) is injective. Taking the zeroth cohomologically graded piece, we deduce from Corollary 6.9 that(92) dim( H C u SSN i ′ ,n ) ≤ . The inclusion j : M simp d ( Q op ) → M d ( Q op ) is an open embedding, so there is a morphism(93) C u SSN i ′ ,n → ̟ ′ red , ∗ j ∗ j ! ̟ ′ ! red IC M d (Π Q ) ∼ = ̟ ′ red , ∗ j ∗ IC M simp d ( Q op ) ⊗ L − χ Q ( d , d ) − . The Tate twist is given by the difference in dimensions of the smooth schemes M simp d (Π Q ) and M simp d ( Q op ). Since dim( M simp d ( Q op )) = − χ Q ( d , d ) − IC M simp d ( Q op ) = Q M simp d ( Q op ) ⊗ L χ Q ( d , d ) and thus (93) induces the morphismΨ : C u SSN i ′ ,n → ̟ ′ red , ∗ j ∗ Q M simp d ( Q op ) . Proposition 6.14.
The morphism H Ψ : H C u SSN i ′ ,n → H ( M simp d ( Q op ) , Q ) ∼ = Q is an isomor-phism.Proof. By (92) it is sufficient to show that H Ψ is not the zero morphism. Let h : M simp d ( Q ) ֒ →M d ( Q ) be the inclusion. The morphism Ψ factors through the morphism(94) ̟ ′ red , ∗ ̟ ′ ! red RA Π Q → ̟ ′ red , ∗ ̟ ′ ! red h ∗ h ∗ RA Π Q Applying H to (94) yields the morphism H A SSN Π Q → H ( M simp d ( Q op ) , Q )which is not the zero morphism, since [ Λ ( d ) ( n ) ] does not lie in the kernel. Now let F ֒ → B n be the inclusion of any summand that is not isomorphic to C u SSN i ′ ,n . Then F is supported on M d ( Q ) \ M simp d ( Q ), so that h ∗ h ∗ F is zero, and the morphism H F → H ( M simp d ( Q op ) , Q )is zero. It follows that H Ψ is not the zero morphism. (cid:3)
Corollary 6.15.
The images of the inclusions ξ n : H (cid:0) cu SSN i ′ ,n (cid:1) ֒ → H (cid:0) g Π Q ,n · i ′ (cid:1) generate L n ≥ g Π Q ,n · i ′ . PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 37
Proof.
The morphism H Ψ factors through ξ n , and so ξ n is injective. The result then followsfrom the fact that L n ≥ g Π Q ,n · i ′ has one simple imaginary root for each n , and injectivity of(91). (cid:3) We define(95) α i ′ ,n = ( H Ψ) − (1) . If we express α i ′ ,n in terms of Bozec’s basis, we have shown that the coefficient of [ Λ ( d ) ( n ) ] is 1.The question of what all of the other coefficients are (in particular, whether they are nonzero)seems to be quite difficult without an explicit description of RA SSN Π Q , d like Proposition 6.10 in thehyperbolic case. On the other hand § F from(78) does not identify n + Q and H ( g SSN Π Q ). Remark 6.16.
We have shown that the zeroth cohomologically graded pieces of cu SSN i ′ ,n for i ′ ∈ Q and n ∈ Z ≥ provide a complete set of generators for H ( g SSN Π Q ) . This provides evidence forConjecture 7.7 below. BPS sheaves and cuspidal cohomology
Generators of g S ,G, ζ Π Q , θ . In § g G, ζ Π Q , θ to a Lie algebra object in thecategory MHM G ( M ζ -ss θ ( Q )). In this section we will use this lift to produce generators for g S ,G, ζ Π Q , θ .Fix a dimension vector d . Let(96) U ⊂ M G, ζ -ss ◦ d ( e Q )be the subscheme parameterising those modules for which the underlying Q -module is ζ -stable ,and let U = ( JH ◦ ) − ( U ) ⊂ M G, ζ -ss ◦ d ( e Q )be the open substack parameterising such modules. Note that U ⊂ M G, ζ -st d ( e Q ), so the morphism U → U is a B C ∗ -torsor. Define V = U ∩ M G, ζ -ss ◦ d (Jac( e Q, f W )) V =( JH ◦ ) − ( V ) . Since P i ω i ∈ Jac( e Q, f W ) is central, it acts via scalar multiplication on any module representedby a point in V . Arguing as in the proof of Proposition 6.3 we deduce that(97) V = (cid:16) M ζ -st d (Π Q ) × A (cid:17) /G ⊂ M G, ζ -ss ◦ d ( e Q ) . The variety M ζ -st d (Π Q ) is smooth, and so both V and V are smooth stacks. We denote by V the closure of V in M G, ζ -ss ◦ ( e Q ). Similarly, arguing as in the proof of (63) we deduce that φ mon T r( f W ) f IC U ∼ = f IC V , and thus(98) ( U ֒ → M G, ζ -ss d ( e Q )) ∗ JH G ∗ φ mon T r( f W ) f IC M G, ζ -ss d ( e Q ) ∼ = f IC V ⊗ H (B C ∗ , Q ) vir . By Corollary 4.9 the object JH G ∗ φ mon T r( f W ) f IC M G, ζ -ss d ( e Q ) is pure, and so in particular its first cohomol-ogy is a semisimple monodromic mixed Hodge module. From (98) and the inclusion L / ֒ → H (B C ∗ , Q ) vir we deduce that there is a canonical morphism(99) Γ : f IC V ⊗ L / ֒ → JH G ∗ φ mon T r( f W ) f IC M G, ζ -ss d ( e Q ) for which there is a left inverse α , by purity of the target. By (97) we have V = (cid:16) M ζ -ss d (Π Q ) × A (cid:17) /G, and so r ′∗ f IC V ⊗ L / ∼ = f IC M G, ζ -ss d (Π Q ) . Recall that the right hand side of (96) parameterises those C e Q -modules ρ for which the underlying C Q -moduleof ρ is ζ -semistable. Set C u S ,G, ζ Π Q , d := ( ̟ ′ red , ∗ ̟ ′ ! red f IC M G, ζ -ss d (Π Q ) if M ζ -st d (Π Q ) = ∅ cu S ,G, ζ Π Q , d := H C u S ,G, ζ Π Q , d . Then there is an isomorphism cu S ,G, ζ Π Q , d ∼ = H (cid:16) M G, ζ -ss d ( e Q ) , ̟ ′∗ ̟ ′ ! f IC V ⊗ L / (cid:17) . Applying H ̟ ′∗ ̟ ′ ! to (99) we obtain a morphism β : cu S ,G, ζ Π Q , d ֒ → g e S ,G, ζ e Q, f W , d = P ≤ H A e S ,G, ζ e Q, f W , d which is an injection, since it has a left inverse (e.g. H ̟ ∗ ̟ ! α ).We can now prove (a generalisation of) Theorem B. Theorem 7.1.
Let ζ ∈ Q Q be a stability condition, and let θ ∈ Q be a slope. For each d ∈ Λ ζθ there is a canonical decomposition g S ,G, ζ Π Q , d ∼ = cu S ,G, ζ Π Q , d ⊕ l for some mixed Hodge structure l , and for d ′ , d ′′ ∈ Λ ζθ such that d ′ + d ′′ = d , the morphism g S ,G, ζ Π Q , d ′ ⊗ g S ,G, ζ Π Q , d ′′ [ · , · ] −−→ g S ,G, ζ Π Q , d factors through the inclusion of l .Proof. We assume that M ζ -st d (Π Q ) = ∅ , as otherwise cu S ,G, ζ Π Q , d = 0 and the statement is trivial.Recall that by (29) there is an isomorphism r ′∗ JH ◦∗ φ mon T r( f W ) f IC M G, ζ -ss ◦ d ( e Q ) ∼ = JH G red , ∗ ι ∗ ι ! Q M G, ζ -ss d ( Q ) ⊗ L χ e Q ( d , d ) / = RA G, ζ Π Q , d and by Theorem 4.7 these are pure complexes of monodromic mixed Hodge modules, so there isa decomposition(100) RA G, ζ Π Q , d ∼ = M r ∈ R n F r [ n ]where each F r is a simple mixed Hodge module, and each R n is some indexing set.The stack M G, ζ -st d (Π Q ) is smooth, of codimension d · d − M G, ζ -st d ( Q ), and so there isan isomorphism( M G, ζ -st d (Π Q ) ֒ → M G, ζ -ss d (Π Q )) ∗ ι ∗ ι ! Q M G, ζ -ss d ( Q ) ∼ = Q M G, ζ -st d (Π Q ) ⊗ L d · d − and thus an isomorphism( M G, ζ -st d (Π Q ) ֒ → M G, ζ -ss d (Π Q )) ∗ RA G, ζ Π Q , d ∼ = ( M G, ζ -st d (Π Q ) → M G, ζ -st d (Π Q )) ∗ Q M G, ζ -st d (Π Q ) ⊗ L χ e Q ( d , d ) / d · d − . Noting that M G, ζ -st d (Π Q ) is open inside M G, ζ -ss d (Π Q ), of dimension χ Q ( d , d ) − χ e Q ( d , d ) / d · d −
1, we deduce that in the decomposition (100), in cohomological degree zero there is exactlyone copy of the simple object f IC M G, ζ -ss d (Π Q ) , and furthermore the morphism r ′∗ Γ : f IC M G, ζ -ss d (Π Q ) → RA G, ζ Π Q , d is the inclusion of this object. Writing H G, (cid:16) RA G, ζ Π Q , d (cid:17) = f IC M G, ζ -ss d (Π Q ) ⊕ GG = M r ∈ R ′ F r for R ′ ⊂ R , we claim that for all d ′ , d ′′ ∈ Λ ζθ with d ′ = 0 = d ′′ and d ′ + d ′′ = d the multiplication(101) H G, (cid:16) RA G, ζ Π Q , d ′ (cid:17) ⊠ ⊕ G H G, (cid:16) RA G, ζ Π Q , d ′′ (cid:17) → H G, (cid:16) RA G, ζ Π Q , d (cid:17) PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 39 factors through the inclusion of G . This follows for support reasons: by our assumptions on d ′ , d ′′ the supports of the semisimple object on the left hand side of (101) are all in the boundary M G, ζ -ss d (Π Q ) \ M G, ζ -st d (Π Q ). Applying H ̟ ′ red , ∗ ̟ ′ ! red , there is a decomposition L ≤ H A S ,G,, ζ Π Q , d ∼ = cu S ,G, ζ Π Q , d ⊕ H ̟ ′ red , ∗ ̟ ′ ! red G and the multiplication L ≤ H A S ,G,, ζ Π Q , d ′ ⊗ C L ≤ H A S ,G,, ζ Π Q , d ′′ → L ≤ H A S ,G,, ζ Π Q , d factors through the inclusion of H ̟ ′ red , ∗ ̟ ′ ! red G = l , so that the commutator Lie bracket also factorsthrough the inclusion of l . (cid:3) The BPS Lie algebra g Π Q for Q affine. Let Q be a quiver for which the underlying graphis an affine Dynkin diagram of extended ADE type. We can use the fact that g G Π Q lifts to a Liealgebra object BPS G Π Q in MHM G ( M (Π Q )) to calculate it completely.Set d = | Q |−
1. We denote by q d : M ζ -ss d (Π Q ) → M d (Π Q ) the affinization map. Let δ ∈ N Q be the unique primitive imaginary simple root of the quiver Q . Let H ⊂ SL ( C ) be the Kleiniangroup corresponding to the underlying (finite) Dynkin diagram of Q (obtained by removing asingle vertex) via the McKay correspondence. Then (see [29, 8]) for a generic stability condition ζ ∈ Q Q there is a commutative diagram X p (cid:15) (cid:15) ∼ = / / M ζ -ss δ (Π Q ) q δ (cid:15) (cid:15) Y ∼ = / / M δ (Π Q )where p is the minimal resolution of the singularity Y = A /H . Moreover by [24] there is a derivedequivalence Ψ : D b (Coh( X )) → D b (Π Q -mod)restricting to an equivalence between complexes of modules with nilpotent cohomology sheavesand complexes of coherent sheaves with set-theoretic support on the exceptional locus of p . For d ∈ N Q we denote by 0 d ∈ M d (Π Q )the point corresponding to the unique semisimple nilpotent module of dimension vector d .Via the explicit description of the representations of KQ for Q an affine quiver, we have thefollowing identities(102) a Q, d ( t ) = d is a positive real root of g Q t + d if d ∈ Z ≥ · δ Proposition 7.2.
There are isomorphisms (103)
BPS Π Q , d ∼ = Q d if d is a positive real root of g Q ∆ n, ∗ q δ , ∗ IC M ζ -ss δ (Π Q ) if d = n · δ otherwise,where ∆ n : M δ (Π Q ) → M n · δ (Π Q ) is the embedding of the small diagonal. We sketch the proof — the complete description of affine preprojective CoHAs will appear ina forthcoming paper with Sven Meinhardt. By Proposition 4.4 there is an isomorphism
BPS Π Q , d ∼ = q d , ∗ BPS ζ Π Q , d . On the other hand, any complex of compactly supported coherent sheaves F on X that is notentirely supported on the exceptional locus admits a direct sum decomposition F ′ ⊕ F ′′ where F ′′ is supported at a single point, so that Ψ( F ) admits a direct summand N with dimensionvector a multiple of δ . It follows that all points of M ζ -ss d (Π Q ) correspond to nilpotent modulesif d is not a multiple of δ , and so q d , ∗ BPS ζ Π Q , d is supported at the origin. Since by Theorem 4.7 BPS Π Q , d is pure, and supported at a single point, it is determined by its hypercohomology g Π Q , d .This hypercohomology pure, of Tate type, with dimension given by the Kac polynomial, by themain result of [12]. This deals with the first and last cases of (103).For the second case, we consider the commutative diagram C oh n ( X ) h ∼ = / / g (cid:15) (cid:15) M ζ -ss n · δ (Π Q ) JH red (cid:15) (cid:15) Sym n ( X ) l ∼ = / / M ζ -ss n · δ (Π Q ) . By Theorem 4.7, g ∗ h ∗ ι ! Q M ζ -ss n δ ( Q ) is pure, and we claim that it contains a single copy of ∆ X,n, ∗ IC X .Since X is simply connected, we can cover X by charts U i isomorphic to A and check the claim oneach of the open subvarieties U i , at which point the claim follows by (79). Finally, the BPS sheafis supported on the small diagonal by the support lemma of [12], so ∆ X,n, ∗ IC X ∼ = l ∗ BPS ζ Π Q ,n δ .Then the second case follows by Proposition 4.4.Note that there is an isomorphism(104) q δ , ∗ IC M ζ -ss δ (Π Q ) ∼ = IC M δ (Π Q ) ⊕ Q ⊕ d δ since there are d copies of P in the exceptional fibre of q δ . We deduce from (102) and (38) that H − g Π Q ,n · δ ∼ = Q is obtained by applying H ∆ n, ∗ to the first summand of (104). Proposition 7.3.
There is an isomorphism of Lie algebras (105) g Π Q ∼ = n − Q ′ ⊕ s Q [ s ] where Q ′ is the real subquiver of Q (i.e. it is equal to Q unless Q is the Jordan quiver, in whichcase it is empty) and s Q [ s ] is given the trivial Lie bracket. The monomial s n lives in N Q -degree n · δ , and in cohomological degree − .Proof. By (102) and (38), the graded dimensions of the two sides of (105) match. Furthermore,by Theorem 6.6 there is an isomorphism of Lie algebras between the zeroth cohomology of theRHS and LHS of (105).So it is sufficient to prove that H − ( g Π Q ) is central, which amounts to showing that[ s n , r ] = 0 ∈ g Π Q , β for r ∈ H ( g Π Q , α ) where α = n ′ · δ and β = n · δ + α . On the other hand, the morphism Q ⊗ Q · s n ⊗· r −−−−→ g Π Q ,n δ ⊗ g Π Q , α [ · , · ] −−→ g Π Q , β is obtained by applying H to the morphism of mixed Hodge modules∆ n, ∗ IC M δ (Π Q ) ⊠ ⊕ red Q α → ∆ ( n + n ′ ) , ∗ IC M δ (Π Q ) ⊕ Q ⊕ d β which is a morphism between semisimple mixed Hodge modules with differing supports, and isthus zero. (cid:3) A deformed example.
In this subsection we give a curious example, which will not be usedlater in the paper. It is an example of how deforming the potential can modify the BPS Liealgebra.Let Q be the oriented ˜ A d quiver, i.e. it contains d + 1 vertices, along with an oriented cycleconnecting them all. Let W = a d +1 a d · · · a be this cycle. We will consider the quiver withpotential ( ˜ Q, f W + W ). The potential f W + W is quasihomogeneous, for example we can give thearrows a s weight 1, the arrows a ∗ s weight d , and the arrows ω i weight zero, so that f W + W hasweight d + 1.As in § BPS Π Q ,W by applying φ mon T r( W ) to BPS Π Q . For d not a multipleof the imaginary simple root, BPS Π Q , d is supported at 0 d and so it follows that BPS Π Q ,W , d ∼ = φ mon T r( W ) BPS Π Q , d ∼ = BPS Π Q , d . PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 41
In particular, it follows that there is an injective map l from the Lie subalgebra of BPS Π Q generatedby L i ∈ Q BPS Π Q , i , and the only dimension vectors for which this morphism can fail to be anisomorphism are dimension vectors e = ( n, . . . , n ) for some n .Let e be such a dimension vector. Propositions 4.4 and 7.2 together yield BPS Π Q ,W , e ∼ = ∆ n, ∗ q (1 ,..., , ∗ φ mon g IC X where X is the minimal resolution of the singular surface defined by xy = z d , and g = y is thefunction induced on it by T r( W ). The reduced vanishing locus X = g − (0) is given by theexceptional chain of d copies of P , along with a line intersecting one of them transversally. Inparticular, the cohomology of X is pure. The preimage X := g − (1) is isomorphic to a copy of A . Via the long exact sequence → H i ( X, φ mon g Q X ) → H i ( X , Q ) → H i ( X , Q ) → we deduce that there is an isomorphism H ( X, φ mon g Q X ) ∼ = H ( X, Q )[ − reduced cohomology of X . It follows by counting dimensionsthat the injective map H ( l e ) is surjective, although l e is not, since crit( g ) is not contained in theexceptional locus. We deduce that(106) g Π Q ,W ∼ = n − Q i.e. the BPS Lie algebra for the deformed potential is isomorphic to negative half of the usualKac–Moody Lie algebra for Q .It is an interesting question whether for more general quivers there is a quasihomogeneousdeformation f W + W of the standard cubic potential so that (106) holds. A related question is:does the nonzero degree cohomology of the BPS Lie algebra g Π Q ,W vanish for generic deformations W ? In other words, is the BPS Lie algebra for a generic deformed 3-Calabi–Yau completion [26]of the preprojective algebra Π Q always n − Q ?7.3. The spherical Borcherds algebra.
In this section we construct a natural Lie algebrahomomorphism Φ : g e , S Π Q → g S Π Q from the positive half of a Borcherds algebra, extending theinclusion of the Kac–Moody Lie algebra from § S = SSN the zerothcohomologically graded piece of this morphism is the inclusion of the Borcherds–Bozec algebra.The existence of the morphism Φ serves as further evidence towards Conjecture 7.7.We introduce a little notation, in order to make the presentation fairly uniform. Given a tensorcategory C we denote by C N Q the category of N Q -graded objects in C . Given F ∈ C we denoteby Lie ( F ) the free Lie algebra generated by F . I.e. we pick a symmetric monoidal embeddingVect ֒ → C , and embed C ֒ → C N Q as the category of objects concentrated in degree zero, and thusconsider L ie as an operad in C N Q , and take the free algebra over it generated by F . We denoteby Bor + ( F ) the quotient of Lie ( F ) by the Lie ideal generated by the images of the morphisms( F ⊗ d ′ ) − ( d ′ , d ′′ ) Q ⊗ F d ′′ [ · , [ · ... [ · , · ] ... ] −−−−−−−→ F (107)over all pairs of dimension vectors d ′ , d ′′ satisfying either of the conditions ( d ′ , d ′′ ) Q = 0 or d ′ = 1 i for i ∈ Q . Example 7.4.
Consider the vector space V ∈ Vect N Q which has basis e i for i ∈ I ∞ , where e ( i ′ ,n ) is given degree n · i ′ . Then n + Q = Bor + ( V ) . For i ∈ Q and n ≥ i,n : M G i ( Q ) → M Gn · i ( Q ) the embedding of the smalldiagonal. Proposition 7.5.
Let S be a Serre subcategory of C Q -mod . Set P r G Π Q , sph := M i ∈ Q real0 f IC M G i ( Q ) ⊕ M i ∈ Q iso0 n ≥ ∆ i,n, ∗ f IC M G i ( Q ) ⊕ M i ∈ Q hyp0 n ≥ C u G Π Q ,n · i pr S ,G Π Q , sph := H ̟ ′ red , ∗ ̟ ′ ! red P r G Π Q , sph . There are morphisms of Lie algebra objects J : Bor + (cid:16) P r G Π Q , sph (cid:17) →BPS G Π Q L S : Bor + (cid:16) pr S ,G Π Q , sph (cid:17) → g S ,G Π Q . extending embeddings of P r G Π Q , sph and pr S ,G Π Q , sph , respectively, where in the second morphism, pr S ,G Π Q , sph is considered as an object of C -mod .Proof.
The morphism L S is obtained as H ̟ ′ red , ∗ ̟ ′ ! red J , so we concentrate on J .Firstly, note that for i ∈ Q BPS G Π Q , i = f IC M G i ( Q ) so that the first summand of P r G Π Q , sph naturally embeds inside BPS G Π Q . Secondly, as in Proposition6.10 there is an embedding (unique up to scalar) of ∆ i,n, ∗ f IC M G i ( Q ) inside BPS G Π Q ,n · i for eachisotropic i . Thirdly, for i hyperbolic the morphism (99) provides an embedding C u G Π Q ,n · i ⊂BPS G Π Q . We claim that these embeddings induce the morphism J .To prove the claim, we need to check the relation (107). Note that if i and j are both real, thisfollows immediately from Proposition 4.5. Otherwise, we need something a little more subtle, i.e.the decomposition theorem.Let i ∈ I real , let ( j, n ) ∈ I im , and set e = 1 − (( i, , ( j, n )). Set M i = M G i ( Q ) , M j,n = ( ∆ j,n ( M j ) if j ∈ Q iso0 M Gn · j ( Q ) if j ∈ Q hyp0 Then we wish to show that the morphism J ′ : f IC M i ⊠ ⊕ G · · · ⊠ ⊕ G f IC M i | {z } e times ⊠ ⊕ G f IC M j,n → BPS G Π Q given by the iterated Lie bracket (as in (107)) is the zero morphism. For this, we note that themorphism h : M i × B G · · · × B G M i | {z } e times × B G M j,n → M e · i + d (Π Q )is injective, and so since f IC M i and f IC M j,n are simple, the domain of J ′ is a simple object. Wedenote the domain of J ′ by R . Since by Theorem 4.7 the target of J ′ is semisimple we deducethat J ′ is nonzero only if there is a direct sum decomposition BPS G Π Q ∼ = R ⊕ G and J ′ fits into a commutative diagram R ι R " " ❋❋❋❋❋❋❋❋❋ J ′ / / BPS G Π Q R ⊕ G ∼ = O O where ι R is the canonical inclusion.For a contradiction, we assume that this is indeed so. Now we apply H g ∗ g ! , where g : M SSN ,G ( Q ) → M G ( Q )is the inclusion of the strictly semi-nilpotent locus. By (84) in the case of isotropic j , and Propo-sition (6.14) in the hyperbolic case, there is an isomorphism H g ∗ g ! f IC M j,n ∼ = Q and so, since H ( f IC M i ) ∼ = Q we deduce that H g ∗ g ! R ∼ = Q , PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 43 so that H g ∗ g ! ι R = ι Q = 0. On the other hand, H g ∗ g ! J ′ : Q → g SSN Π Q is the morphism taking 1 ∈ Q to [ α i , · ] e ( α j,n ) = 0, with α i and α j,n defined as in (68), (85), (95).By Corollary 6.9 we have [ α i , · ] e ( α j,n ) = 0and so ξ = 0 after all.Now assume that both i and j are imaginary, and (1 i , j ) Q = 0. Fix m ≥
1. Then, similarly toabove, we wish to show that the morphism f IC M i,m ⊠ ⊕ G f IC M j,n → BPS G Π Q provided by the Lie bracket in BPS G Π Q is the zero map. Again, applying H g ∗ g ! this follows fromCorollary 6.9 and injectivity of the morphism ⊕ G : M i,m × B G M j,n → M Gm · i + n · j ( Q ) . We thus have defined the morphism J . (cid:3) It is of course very natural to make the following
Conjecture 7.6.
The morphisms J and L S are injective. The results of § H A SSN Π Q , which is generated by the subspaces H A SSN Π Q ,n · i by[49, Prop.5.8], the Lie algebra g S Π Q is almost never generated by the subspaces g S Π Q ,n · i , so that L is almost never surjective. For instance, if Q has no edge loops, then the image of L lies entirely incohomological degree zero, while unless Q is of finite type, g Π Q will have pieces in strictly negativecohomological degree.7.4. The main conjecture for BPS Lie algebras.
We finish the paper with our main conjec-ture regarding the structure of BPS Lie algebras for preprojective CoHAs. Put informally, theconjecture states that we have found all of the generators of g S Π G . To state the conjecture fully,we make the following definitions.First fix a Serre subcategory S , a stability condition ζ ∈ Q Q , and θ ∈ Q . Set C = { d ∈ Λ ζθ : M ζ -st d (Π Q ) = ∅} . We set S ta G, ζ Π Q , θ := M d ∈ C f IC M G, ζ -ss d (Π Q ) st S ,G, ζ Π Q , θ := H ̟ ′ red , ∗ ̟ ′ ! red S ta G, ζ Π Q , θ . We consider st S ,G, ζ Π Q , θ as a C-module below. By § S ta G, ζ Π Q , θ ֒ → BPS G, ζ Π Q , θ . In the case in which the stability condition is trivial and θ = 0, H of this inclusion is the inclusionof all the real simple roots, as well as cuspidal cohomology. To cover isotropic generators, wedefine E = { n · d : ( d , d ) Q = 0 , d ∈ C, n ≥ } . By [9], E ∩ C = ∅ . Arguing as in § d ∈ E and n ≥ n, d , ∗ f IC M G, ζ -ss d (Π Q ) ֒ → BPS G, ζ Π Q , θ , where ∆ n, d : M G, ζ -ss d (Π Q ) ֒ → M G, ζ -ss n · d (Π Q ) is the diagonal embedding. Accordingly, we define I so G, ζ Π Q , θ := M n · d ∈ E d primitive ∆ n, d , ∗ f IC M G, ζ -ss d (Π Q ) is S ,G, ζ Π Q , θ := H ̟ ′ red , ∗ ̟ ′ ! red I so G, ζ Π Q , θ . We can now state the main conjecture:
Conjecture 7.7.
The above inclusions extend to an isomorphism of Lie algebra objects in
MHM G ( M ζ -ss θ ( Q )) Bor + (cid:16) S ta G, ζ Π Q , θ ⊕ I so G, ζ Π Q , θ (cid:17) ∼ = BPS G, ζ Π Q , θ . Applying H ̟ ′ red , ∗ ̟ ′ ! red , we obtain isomorphisms Bor + (cid:16) st S ,G, ζ Π Q , θ ⊕ is S ,G, ζ Π Q , θ (cid:17) ∼ = g S ,G, ζ Π Q , θ . Setting ζ = (0 , . . . , , θ = 0 , S = C Q -mod and G = { } this conjecture implies the Bozec–Schiffmann conjecture on the Kac polynomials for Q , as well as giving a precise interpretation forthe cuspidal cohomology. References [1] P. Achar. Equivariant mixed Hodge modules.
Lecture notes to accompany talk “Mixed Hodge modules andtheir applications”; available at author’s website .[2] A. Beilinson, J. Bernstein, and P. Deligne. Faisceaux pervers.
Ast´erisque , 100, 1983.[3] G. Bellamy and T. Schedler. Symplectic resolutions of quiver varieties. https://arxiv.org/abs/1602.00164 ,2016.[4] O. Ben-Bassat, C. Brav, V. Bussi, and D. Joyce. A “Darboux theorem” for shifted symplectic structures onderived Artin stacks, with applications.
Geometry & Topology , 19(3):1287–1359, 2015.[5] T. Bozec. Quivers with loops and generalized crystals.
Compositio Mathematica , 152(10):1999–2040, 2016.[6] T. Bozec and O. Schiffmann. Counting absolutely cuspidals for quivers.
Mathematische Zeitschrift , 292(1-2):133–149, 2019.[7] T. Bozec, O. Schiffmann, and E. Vasserot. On the number of points of nilpotent quiver varieties over finitefields. https://arxiv.org/abs/1701.01797 , 2017.[8] H. Cassens and P. Slodowy. On Kleinian singularities and quivers. In
Singularities , pages 263–288. Springer,1998.[9] W. Crawley-Boevey. Geometry of the moment map for representations of quivers.
Compositio Mathematica ,126(3):257–293, 2001.[10] B. Davison. Cohomological Hall algebras and character varieties.
International Journal of Mathematics , 27(07).[11] B. Davison. The critical CoHA of a quiver with potential.
The Quarterly Journal of Mathematics , 68(2):635–703, 2017.[12] B. Davison. The integrality conjecture and the cohomology of preprojective stacks. https://arxiv.org/abs/1602.02110v3 , 2017.[13] B. Davison. BPS cohomology for quiver CoHAs in dimensions 1,2,3. 2020. preprint.[14] B. Davison and S. Meinhardt. Cohomological Donaldson–Thomas theory of a quiver with potential and quan-tum enveloping algebras.
Inventiones Mathematicae , pages 1–95, 2020.[15] B. Davison and T. P˘adurariu. Deformed dimensional reduction. https://arxiv.org/abs/2001.03275 , 2020.[16] A. Dimca and B. Szendr˝oi. The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on C . Math.Res. Lett , 16(6):1037–1055, 2009.[17] V. Ginzburg. Calabi–Yau algebras. https://arxiv.org/abs/math/0612139 , 2006.[18] I. Grojnowski. Instantons and affine algebras I: the Hilbert scheme and vertex operators.
Math. Res. Lett ,3:275–291, 1996.[19] D. Halpern-Leistner. θ -stratifications, θ -reductive stacks, and applications. Algebraic Geometry: Salt LakeCity 2015 , 97:349, 2018.[20] T. Hausel. Kac’s conjecture from Nakajima quiver varieties.
Inventiones Mathematicae , 181(1):21–37, 2010.[21] D. Joyce. A classical model for derived critical loci.
Journal of Differential Geometry , 101(2):289–367, 2015.[22] V. Kac. Root systems, representations of quivers and invariant theory. In
Invariant theory , pages 74–108.Springer, 1983.[23] D. Kaledin. Geometry and topology of symplectic resolutions.
Algebraic geometry — Seattle 2005. Part , 2:595–628, 2009.[24] M. Kapranov and E. Vasserot. Kleinian singularities, derived categories and Hall algebras.
Math. Ann. ,316:565–576, 2000.[25] M. Kapranov and E. Vasserot. The cohomological Hall algebra of a surface and factorization cohomology. https://arxiv.org/abs/1901.07641 , 2019.[26] B. Keller and M. Van den Bergh. Deformed calabi–yau completions.
Journal f¨ur die reine und angewandteMathematik , 2011(654):125–180, 2011.[27] A. D. King. Moduli of representations of finite-dimensional algebras.
Quart. J. Math. Oxford Ser. , (2) 45, no.180, 1994.[28] M. Kontsevich and Y. Soibelman. Cohomological Hall algebra, exponential Hodge structures and motivicDonaldson–Thomas invariants.
Commun. Number Theory Phys. , 5, 2011. arXiv:1006.2706.[29] P. Kronheimer. The construction of ALE spaces as hyper-K¨ahler quotients.
J. Diff. Geom , 29(3):665–683,1989.[30] G. Lusztig. Quivers, perverse sheaves, and quantized enveloping algebras.
Journal of the American Mathemat-ical Society , 4(2):365–421, 1991.
PS LIE ALGEBRAS AND THE LESS PERVERSE FILTRATION ON THE PREPROJECTIVE COHA 45 [31] D. Maulik and A. Okounkov. Quantum groups and quantum cohomology.
Ast´erisque , (408):1–+, 2019.[32] L. Maxim, M. Saito, and J. Sch¨urmann. Symmetric products of mixed Hodge modules.
Journal deMath´ematiques Pures et Appliqu´ees , 96(5):462–483, 2011.[33] S. Meinhardt and M. Reineke. Donaldson–Thomas invariants versus intersection cohomology of quiver moduli.
Journal f¨ur die reine und angewandte Mathematik (Crelles Journal) , 2019(754):143–178, 2019.[34] A. Minets. Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces.
Selecta Mathematica , 26:1–67, 2020.[35] S. Mozgovoy. Motivic Donaldson-Thomas invariants and McKay correspondence. https://arxiv.org/abs/1107.6044 , 2011.[36] S. Mozgovoy and O. Schiffmann. Counting Higgs bundles and type A quiver bundles.
Compos. Math. , 156:744–769, 2020.[37] H. Nakajima. Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras.
Duke Mathematical Jour-nal , 76(2):365–416, 1994.[38] H. Nakajima. Quiver varieties and Kac–Moody algebras.
Duke Mathematical Journal , 91(3):515–560, 1998.[39] M. Porta and F. Sala. Two-dimensional categorified Hall algebras. https://arxiv.org/abs/1903.07253 , 2019.[40] J. Ren and Y. Soibelman. Cohomological Hall Algebras, Semicanonical Bases and Donaldson–Thomas Invari-ants for 2-dimensional Calabi–Yau Categories (with an Appendix by Ben Davison). In
Algebra, Geometry, andPhysics in the 21st Century , pages 261–293. Springer, 2017.[41] M. Saito. Thom–Sebastiani Theorem for Hodge Modules. preprint 2010 .[42] M. Saito. Modules de Hodge polarisables.
Publications of the Research Institute for Mathematical Sciences ,24(6):849–995, 1988.[43] M. Saito. Duality for vanishing cycle functors.
Publications of the Research Institute for Mathematical Sciences ,25(6):889–921, 1989.[44] M. Saito. Mixed Hodge modules and admissible variations.
CR Acad. Sci. Paris , 309(6):351–356, 1989.[45] M. Saito. Mixed Hodge modules.
Publ. Res. Inst. Math. , 26:221–333, 1990.[46] F. Sala and O. Schiffmann. Cohomological Hall algebra of Higgs sheaves on a curve.
Algebraic Geometry ,7(3):346–376, 2020.[47] O. Schiffmann. Kac polynomials and Lie algebras associated to quivers and curves. In
Proc. Int. Cong. ofMath. , pages 1411–1442, 2018.[48] O. Schiffmann and E. Vasserot. Cherednik algebras, W-algebras and the equivariant cohomology of the modulispace of instantons on A . Publications math´ematiques de l’IH ´ES , 118(1):213–342, 2013.[49] O. Schiffmann and E. Vasserot. On cohomological Hall algebras of quivers: generators.
Journal f¨ur die reineund angewandte Mathematik (Crelles Journal) , 2020(760):59–132, 2020.[50] Y. Soibelman. Remarks on cohomological hall algebras and their representations. In
Arbeitstagung Bonn 2013 ,pages 355–385. Springer, 2016.[51] Y. Toda. Gopakumar-Vafa invariants and wall-crossing. https://arxiv.org/abs/1710.01843 , 2017.[52] M. Varagnolo. Quiver varieties and Yangians.
Letters in Mathematical Physics , 53(4):273–283, 2000.[53] Y. Yang and G. Zhao. On two cohomological Hall algebras.
Proceedings of the Royal Society of EdinburghSection A: Mathematics , pages 1–27, 2016.[54] Y. Yang and G. Zhao. The cohomological Hall algebra of a preprojective algebra.
Proceedings of the LondonMathematical Society , 116(5):1029–1074, 2018.
B. Davison: School of Mathematics, University of Edinburgh, James Clerk MaxwellBuilding, Peter Guthrie Tait Road, King’s Buildings, Edinburgh EH9 3FD, United Kingdom
E-mail address: [email protected]
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