Cluster algebras and singular supports of perverse sheaves
aa r X i v : . [ m a t h . QA ] F e b CLUSTER ALGEBRAS AND SINGULAR SUPPORTSOF PERVERSE SHEAVES
HIRAKU NAKAJIMA
Abstract.
We propose an approach to Geiss-Leclerc-Schroer’sconjecture on the cluster algebra structure on the coordinate ringof a unipotent subgroup and the dual canonical base. It is basedon singular supports of perverse sheaves on the space of represen-tations of a quiver, which give the canonical base.
Introduction
In [27], the author found an approach to the theory of cluster al-gebras, based on perverse sheaves on graded quiver varieties. Thisapproach gave a link between two categorical frameworks for cluster al-gebras, the additive one via the cluster category by Buan et al. [5] andthe multiplicative one via the category of representations of a quantumaffine algebra by Hernandez-Leclerc [12]. See also the survey article[20].In [27, § new idea to attack these.A problem, discussed in this paper, is a natural generalization ofthe problem (4) in the to-do list. The author asked to find a relationbetween the work of Geiss-Leclerc-Schr¨oer [8] and [27] there. But thework [8] dealt with more general cases than those corresponding to [27].A main conjecture says that every cluster monomial in the coordinatering of a unipotent subgroup is a Lusztig’s dual canonical base element.Later the theory is generalized to the q -analog, where the canonicalbase naturally lives [9]. Therefore it is desirable to find a relationbetween the cluster algebra structure and perverse sheaves on the space Mathematics Subject Classification.
Primary 13F60; Secondary 17B37,35A27. of quiver representations, which give the canonical base of the quantumenveloping algebra.Another problem is not discussed here, but possibly related to thecurrent one via [13]. In [12], Hernandez-Leclerc conjectured that theGrothendieck ring of representations of the quantum affine algebra hasa structure of a cluster algebra so that every cluster monomial is aclass of an irreducible representation. Those irreducible representationsare simple perverse sheaves on graded quiver varieties. But what wasproved in [27] is the special case of the conjecture only for a certainsubalgebra of the Grothendieck ring. The first part of the conjecturehas been subsequently proved in [13]. But the latter part, every clustermonomial is an irreducible representation, is still open.In this paper, we propose an approach to the first problem. It is notfully developed yet. We will give a few results, which indicate that weare going in the right direction. The new idea is to use the singularsupport of a perverse sheaf, which is a lagrangian subvariety in thecotangent bundle. The latter is related to the representation theory ofthe preprojective algebra, which underlies the work [8].The paper is organized as follows. In the first section, we brieflyrecall the canonical and semicanonical bases. In the second section, wereview works of Geiss-Leclerc-Schr¨oer [8, 9], where a quantum clusteralgebra structure on a quantum unipotent subgroup is introduced. Asubcategory, denoted by C w , of the category of nilpotent representationsof the preprojective algebra plays a crucial role. In the third section, westudy how the singular support behaves under the restriction functorfor perverse sheaves. The restriction functor gives a multiplication inthe dual of the quantum enveloping algebra. Our main result is theestimate in Theorem 3.2. In the final section, we give two conjectures,which give links between the theory of [8, 9] and perverse sheaves viasingular support. Acknowledgments.
The main conjecture (Conj. 4.2) was found inthe spring of 2011, and has been mentioned to various people sincethen. The author thanks Masaki Kashiwara and Yoshihisa Saito fordiscussion on the conjecture. He also thanks the referee who points outa relation between the main conjecture and a conjecture in [7, § Preliminaries
Quantum enveloping algebra.
Let g be a symmetrizable Kac-Moody Lie algebra. We assume g is symmetric, as we use an approachto g via the Ringel-Hall algebra for a quiver. Let I be the index set LUSTER ALGEBRAS AND PERVERSE SHEAVES 3 of simple roots, P be the weight lattice, and P ∗ be its dual. Let α i denote the i th simple root.Let U q be the corresponding quantum enveloping algebra, that isa Q ( q )-algebra generated by e i , f i ( i ∈ I ), q h ( h ∈ P ∗ ) with certainrelations. Let U − q be the subalgebra generated by f i . We set wt( e i ) = α i , wt( f i ) = − α i , wt( q h ) = 0. Then U q is graded by P .The quantum enveloping algebra U q is a Hopf algebra. We have acoproduct ∆ : U q → U q ⊗ U q . It does not preserve U − q , but Lusztigintroduced its modification r : U − q → U − q ⊗ U − q such that r ( f i ) = f i ⊗ ⊗ f i and r is an algebra homomorphism with respect to themultiplication on U − q ⊗ U − q given by(1.1) ( x ⊗ y ) · ( x ⊗ y ) = q − (wt x , wt y ) x x ⊗ y y , where x i , y i are homogeneous elements. We call r the twisted coproduct .Let A = Z [ q, q − ]. Then U − q has an A -subalgebra A U − q generatedby q -divided powers f ( n ) i = f ni / [ n ]!, where [ n ] = ( q n − q − n ) / ( q − q − )and [ n ]! = [ n ][ n − · · · [1]. Then r induces A U − q → A U − q ⊗ A U − q , whichis denoted also by r .1(ii). Perverse sheaves on the space of quiver representationsand the canonical base.
Consider the Dynkin diagram G = ( I, E )for the Kac-Moody Lie algebra g , where I is the set of vertices, and E the set of edges. Note that G does not have an edge loop, i.e., an edgeconnecting a vertex to itself.Let H be the set of pairs consisting of an edge together with itsorientation. So we have H = 2 E . For h ∈ H , we denote by i( h )(resp. o( h )) the incoming (resp. outgoing) vertex of h . For h ∈ H wedenote by h the same edge as h with the reverse orientation. Chooseand fix an orientation Ω of the graph, i.e., a subset Ω ⊂ H such thatΩ ∪ Ω = H , Ω ∩ Ω = ∅ . The pair ( I, Ω) is called a quiver .Let V = ( V i ) i ∈ I be a finite dimensional I -graded vector space over C . The dimension of V is a vectordim V = (dim V i ) i ∈ I ∈ Z I ≥ . We define a vector space by E V def . = M h ∈ Ω Hom( V o( h ) , V i( h ) ) . Let G V be an algebraic group defined by G V def . = Y i GL( V i ) . HIRAKU NAKAJIMA
Its Lie algebra is the direct sum L i gl ( V i ). The group G V acts on E V by B = ( B h ) h ∈ Ω g · B = ( g i( h ) B h g − h ) ) h ∈ Ω . The space E V parametrizes isomorphism classes of representationsof the quiver with the dimension vector dim V together with a linearbase of the underlying vector space compatible with the I -grading. Theaction of the group G V is induced by the change of bases.In [22, 24] Lusztig introduced a full subcategory P V of the abeliancategory of perverse sheaves on E V . Its definition is not recalled here.See [22, §
2] or [24, Chap. 9]. Its objects are G V -equivariant.Let D ( E V ) be the bounded derived category of complexes of sheavesof C -vector spaces over E V . Let Q V be the full subcategory of D ( E V )consisting of complexes that are isomorphic to finite direct sums ofcomplexes of the form L [ d ] for L ∈ P V , d ∈ Z .Let K ( Q V ) be the Grothendieck group of Q V , that is the abeliangroup with generators ( L ) for isomorphism classes of objects L of Q V with relations ( L ) + ( L ′ ) = ( L ′′ ) whenever L ′′ is isomorphic to L ⊕ L ′ .It is a module over A = Z [ q, q − ], where q corresponds to the shift ofcomplexes in Q V . Then K ( Q V ) is a free A -module with a basis ( L )where L runs over P V .Let us consider the direct sum L V K ( Q V ) over all isomorphismclasses of finite dimensional I -graded vector spaces. Let S i be the I -graded vector space with dim S i = 1, dim S j = 0 for j = i . Thenthe corresponding space E S i is a single point. Let 1 i be the constantsheaf on E S i , viewed as an element in K ( Q S i ). Then Lusztig defineda multiplication and a twisted coproduct on L V K ( Q V ) such that the A -algebra homomorphismΦ : A U − q → M V K ( Q V )with Φ( f i ) = 1 i is an isomorphism respecting twisted coproducts [22,24]. The construction was motivated by an earlier work by Ringel [30].We here recall the definition of the twisted coproduct on L V K ( Q V ).For the definition of the multiplication, see the original papers.Let W be an I -graded subspace of V . Let T = V /W . Let E ( W )be the subspace of E V consisting of B ∈ E V which preserves W . Weconsider the diagram(1.2) E T × E W κ ←− E ( W ) ι −→ E V , where ι is the inclusion and κ is the map given by assigning to B ∈ E ( W ), its restriction to W and the induced map on T . LUSTER ALGEBRAS AND PERVERSE SHEAVES 5
Consider the functorRes def . = κ ! ι ∗ ( • )[ d ] : D ( E W ) → D ( E T × E W ) , where d is a certain explicit integer, whose definition is omitted hereas it is not relevant for the discussion in this paper.It is known that Res sends Q W to the subcategory Q T,W of D ( E T × E W ) consisting of complexes that are isomorphic to finite direct sumsof complexes of the form ( L ⊠ L ′ )[ d ] for L ∈ P T , L ′ ∈ P W , d ∈ Z .Therefore we have an induced A -linear homomorphismRes : K ( Q V ) → K ( Q T,W ) ∼ = K ( Q T ) ⊗ A K ( Q W ) . We take direct sum over V , T , W to get a homomorphism of analgebra with respect to the twisted multiplication (1.1). It correspondsto r on A U − q under Φ.Recall that K ( Q V ) has an A -basis ( L ), where L runs over P V .Taking direct sum over V , and pulling back by the isomorphism Φ, weget an A -basis of A U − q . This is Lusztig’s canonical basis. Let us denoteit by B ( ∞ ).Kashiwara gave an algebraic approach to B ( ∞ ). See [15] and ref-erences therein for detail. He first introduced an A -form L ( ∞ ) of U − q , where A = { f ∈ Q ( q ) | f is regular at q = 0 } . Then he alsodefined a basis, called the crystal base of L ( ∞ ) /q L ( ∞ ). Then heintroduced the global crystal base of U − q , which descends to the crystalbase of L ( ∞ ) /q L ( ∞ ). It turns out that the global crystal base andthe canonical base are the same. See [11].In this paper, we do not distinguish the crystal base and the globalbase, that is the canonical base. We denote both by B ( ∞ ).When we want to emphasize that a canonical base element b ∈ B ( ∞ )is a perverse sheaf, we denote it by L b .1(iii). Dual canonical base.
There exists a unique symmetric bilin-ear form ( , ) on U − q satisfying(1 ,
1) = 1 , ( f i , f j ) = δ ij , ( r ( x ) , y ⊗ z ) = ( x, yz ) for x , y , z ∈ U − q . Our normalization is different from [24, Ch. 1] and follows Kashiwara’sas in [19, 9].Under ( , ), we can identify the graded dual algebra of U − q , an algebrawith the multiplication given by r , with U − q itself.Let B up ( ∞ ) denote the dual base of B ( ∞ ) with respect to ( , ). Itis called the dual canonical base of U − q . HIRAKU NAKAJIMA
For b , b , b ∈ B ( ∞ ), let us define r b ,b b ∈ A by r ( b ) = X b ,b ∈ B ( ∞ ) r b ,b b b ⊗ b . Let b up1 , b up2 , b up3 ∈ B up ( ∞ ) be the dual elements corresponding to b , b , b respectively. Then we have b up1 b up2 = X b up3 ∈ B up ( ∞ ) r b ,b b b up3 . Thus the structure constant is given by r b ,b b .1(iv). Lusztig’s lagrangian subvarieties and crystal.
Let us in-troduce Lusztig’s lagrangian subvariety in the cotangent space of thespace of quiver representations.The dual space to E V is E ∗ V = M h ∈ Ω Hom( V o( h ) , V i( h ) ) . The group G V acts on E ∗ V in the same way as on E V .The G V -action preserves the natural pairing between E V and E ∗ V .Considering E V ⊕ E ∗ V as a symplectic manifold, we have the momentmap µ = ( µ i ) : E V ⊕ E ∗ V → L i gl ( V i ) given by µ i ( B ) = X i( h )= i ε ( h ) B h B h , where B has components B h for both h ∈ Ω and Ω, and ε ( h ) = 1 if h ∈ Ω and − V [21, 22] is defined as(1.3) Λ V def . = { B ∈ E V ⊕ E ∗ V | µ ( B ) = 0 , B is nilpotent } . This space parametrizes isomorphism classes of nilpotent represen-tation of the preprojective algebra associated with the quiver ( I, Ω),together with a linear base of the underlying vector space compatiblewith the I -grading. The action of the group G V is induced by thechange of bases. The preprojective algebra is denoted by Λ in thispaper.This is a lagrangian subvariety in E V ⊕ E ∗ V . (It was proved that Λ V is half-dimensional in E V ⊕ E ∗ V in [22, 12.3]. And the same argumentshows that it is also a lagrangian. Otherwise use [26, Th. 5.8] and takethe limit W → ∞ .)Let Irr Λ V be the set of irreducible components of Λ V . Lusztig de-fined a structure of an abstract crystal (see [16, §
3] for the definition)
LUSTER ALGEBRAS AND PERVERSE SHEAVES 7 on Irr Λ V in [21], and Kashiwara-Saito proved that it is isomorphic tothe underlying crystal of the canonical base B ( ∞ ) of U − q [16].We denote by Λ b the irreducible component of Λ V corresponding toa canonical base element b ∈ B ( ∞ ).1(v). Dual semicanonical base.
Let C (Λ V ) be the Q -vector space of Q -valued constructible functions over Λ V , which is invariant under the G V -action. Lusztig defined an operator C (Λ T ) × C (Λ W ) → C (Λ V ) for V = T ⊕ W , under which the direct sum L V C (Λ V ) is an associativealgebra (see [22, § V = S i , then Λ S i is a single point. Let 1 i be the constant func-tion on Λ S i with the value 1. Let C be the subalgebra of L V C (Λ V )generated by the elements 1 i ( i ∈ I ), and let C (Λ V ) = C ∩ C (Λ V ).Then Lusztig (see [22, Th. 12.13]) proved that C is isomorphic to theuniversal enveloping algebra U ( n ) of the lower triangular subalgebra n of g by f i i .Note that we have an embedding Λ T × Λ W → Λ V given by the directsum, where V = T ⊕ W as above. Then the restriction defines an op-erator C (Λ V ) → C (Λ T ) ⊗ C (Λ W ). Geiss-Leclerc-Schr¨oer proved thatit sends C (Λ V ) to C (Λ T ) ⊗ C (Λ W ), and gives the natural cocommu-tative coproduct on U ( n ) under the isomorphism C ∼ = U ( n ) (see [7, § Y be an irreducible component of Λ V . Then consider the func-tional ρ Y : C (Λ V ) → Q given by taking the value on a dense opensubset of Y . Then { ρ Y | Y ∈ Irr Λ V } gives a base of C (Λ V ). Thisfollows from [23, §
3] together with the result of Kashiwara-Saito men-tioned above. Under the isomorphism U ( n ) ∼ = C , the base ρ Y is calledthe dual semicanonical base of U ( n ) ∗ gr , where U ( n ) ∗ gr denote the gradeddual of U ( n ).We have a natural bijection b up ρ Λ b between the dual canonicalbase and the dual semicanonical base. However ρ Λ b is different fromthe specialization of b up at q = 1 in general. See [7, § Cluster algebras and quantum unipotent subgroups
We fix a Weyl group element w throughout this section. Let ∆ + w =∆ + ∩ w ( − ∆ + ), where ∆ + is the set of positive roots. Then n ( w ) = L α ∈ ∆ + w g − α is a Lie subalgebra of g , where g − α is the root subspacecorresponding to the root − α .2(i). Quantum unipotent subgroup.
Let us briefly recall the q -analog of the universal enveloping algebra U ( n ( w )) of n ( w ), denoted HIRAKU NAKAJIMA by U − q ( w ). See [24, Ch. 40], [19, §
4] and [9] for more detail. (It isdenoted by A q ( n ( w )) in [9].)Let T i be the braid group operator corresponding to i ∈ I , where T i = T ′′ i, in the notation in [24]. Choose a reduced expression w = s i s i · · · s i ℓ . Then it gives β p = s i s i · · · s i p − ( α i p ), and we have ∆ + w = { β p } ≤ p ≤ ℓ . We define a root vector T i T i · · · T i p − ( f i p ) . Let c = ( c , . . . , c ℓ ) ∈ Z ℓ ≥ . We multiply q -divided powers of rootvectors in the order given by β , . . . , β ℓ : L ( c ) = f ( c ) i T i ( f ( c ) i ) · · · ( T i · · · T i ℓ − )( f ( c ℓ ) i ℓ ) . Then the Q ( q )-subspace spanned by L ( c ) ( c ∈ Z ℓ ≥ ) is independentof the choice of a reduced expression of w . This subspace is U − q ( w ).Moreover L ( c ) gives a basis of U − q ( w ). It can be shown that U − q ( w ) isa subalgebra of U − q .It is known that U − q ( w ) is compatible with the dual canonical base,i.e., U − q ( w ) ∩ B up ( ∞ ) is a base of U − q ( w ). This is an interpretation ofthe main result due to Lusztig [25, Th. 1.2], based on an earlier workby Saito [31]. (See [19, Th. 4.25] for the current statement.)Let B up ( w ) def . = U − q ( w ) ∩ B up ( ∞ ). Let B ( w ) ⊂ B ( ∞ ) be the cor-responding subset in the canonical base. Then [25, Prop. 8.3] gives aparametrization of B ( w ) as follows. Let L ( ∞ ) be the A -form used inthe definition of the crystal base. Then one shows L ( c ) ∈ L ( ∞ ) andthe set { L ( c ) mod q L ( ∞ ) } is equal to B ( w ), where B ( w ) is consid-ered as a subset of L ( ∞ ) /q L ( ∞ ). Let us denote by b ( c ) ∈ B ( w ) thecanonical base element corresponding to L ( c ) mod q L ( ∞ ). Therefore b ( c ) ≡ L ( c ) mod q L ( ∞ ).The argument in the proof of [2, Th. 3.13] shows that the transitionmatrix between the base { L ( c ) } and { b ( c ) } is upper triangular withrespect to the lexicographic order on { c } .It is known that { L ( c ) } is orthogonal with respect to ( , ) (see [24,Prop. 40.2.4]). Therefore we can deduce the corresponding relationbetween b up ( c ) and L up ( c ) = L ( c ) / ( L ( c ) , L ( c )), where b up ( c ) is thedual canonical base element corresponding to b ( c ). (See [19, Th. 4.29].)2(ii). B ( w ) and Kashiwara operators. Let us give a characteriza-tion of B ( w ) in terms of Kashiwara operators on B ( ∞ ). Recall that B ( ∞ ) is an abstract crystal, and hence has maps wt : B ( ∞ ) → P , ε i : B ( ∞ ) → Z , ϕ i : B ( ∞ ) → Z ( i ∈ I ) together with Kashiwara oper-ators ˜ e i : B ( ∞ ) → B ( ∞ ) ⊔ { } , ˜ f i : B ( ∞ ) → B ( ∞ ) ⊔ { } satisfyingcertain axioms. We denote by u ∞ the element in B ( ∞ ) corresponding LUSTER ALGEBRAS AND PERVERSE SHEAVES 9 to 1 in U − q . There is also an operator ∗ : B ( ∞ ) → B ( ∞ ), which cor-responds to the anti-involution ∗ : U − q → U − q given by f i f i . There-fore we have another set of maps and operators ε ∗ i = ε i ∗ , ϕ ∗ i = ϕ i ∗ ,˜ e ∗ i = ∗ ˜ e i ∗ , ˜ f ∗ i = ∗ ˜ f i ∗ .Let w = s i s i · · · s i ℓ as above. For i = i , we have ε i ( b ) = c , ˜ e i b ( c ) = ( b ( c ′ ) if c = 0 , c = 0 , ˜ f i b ( c ) = b ( c ′′ ) , where c ′ = ( c − , c , . . . , c ℓ ), c ′′ = ( c + 1 , c , . . . , c ℓ ). In particular, B ( w ) is invariant under ˜ e i , ˜ f i .Saito [31, Cor. 3.4.8] introduced a bijectionΛ i : { b ∈ B ( ∞ ) | ε ∗ i ( b ) = 0 } → { b ∈ B ( ∞ ) | ε i ( b ) = 0 } by Λ i ( b ) = ( ˜ f ∗ i ) ϕ i ( b ) (˜ e i ) ε i ( b ) b, Λ − i ( b ) = ( ˜ f i ) ϕ ∗ i ( b ) (˜ e ∗ i ) ε ∗ i ( b ) b. This is related to the braid group operator as follows. Let us consider w = s i s i · · · s i ℓ and w ′ = s i · · · s i ℓ s i = s i ws i and correspondingPBW base elements L = f ( c ) i T i ( f ( c ) i ) · · · ( T i · · · T i ℓ − )( f ( c ℓ ) i ℓ ) ,L ′ = f ( c ′ ) i T i ( f ( c ′ ) i ) · · · ( T i · · · T i ℓ )( f ( c ′ ) i ) , for ( c , . . . , c ℓ ) ∈ Z ℓ ≥ , ( c ′ , . . . , c ′ ℓ , c ′ ) ∈ Z ℓ ≥ . If c = 0, c = c ′ , . . . , c ℓ = c ′ ℓ , 0 = c ′ , we have L = T i L ′ . Let b , b ′ be the canonical base elementscorresponding to L and L ′ respectively. Then Saito [31, Prop. 3.4.7]proved that the corresponding canonical base elements are related by b = Λ i b ′ . As a corollary of this result, we have a bijection { b ∈ B ( w ′ ) | ε ∗ i ( b ) = 0 } Λ i −→ { b ∈ B ( w ) | ε i ( b ) = 0 } . This together with the invariance of B ( w ) under ˜ e i , ˜ f i gives a char-acterization of B ( w ) inductively in the length of w , starting from B (1) = { u ∞ } .2(iii). A subcategory C w . In view of § B ( w ) in terms of Lusztig’s lagrangian subvari-eties Λ V , or the representation theory of the preprojective algebra. Itturns out to be related to the subcategory introduced by Buan-Iyama-Reiten-Scott [4], and further studied by Geiss-Leclerc-Schr¨oer [8] andBaumann-Kamnitzer-Tingley [1]. We do not recall the definition of the subcategory of the category offinite-dimensional nilpotent representations of the preprojective alge-bra, denoted by C w following [8], here. This is because there are manyequivalent definitions, and the author does not know what is the bestfor our purpose. See the above papers.Let Λ wV = { B ∈ Λ V | B ∈ C w } , where we identify B with thecorresponding representation of the preprojective algebra. This is anopen subvariety in Λ V (see [10, Lem. 7.2]). We set C w (Λ V ) = { f ∈ C (Λ V ) | f ( X ) = 0 for X ∈ C w } . Therefore C (Λ V ) /C w (Λ V ) consists of constructible functions on Λ wV .Let us consider C w (Λ V ) ⊥ = { ξ ∈ C (Λ V ) ∗ | h f, ξ i = 0 for f ∈ C w (Λ V ) } . This space is spanned by an evaluation at a point X ∈ Λ wV . As C w isan additive category (in fact, it is closed under extensions), we have C w (Λ T ) ⊥ · C w (Λ W ) ⊥ ⊂ C w (Λ V ) ⊥ , where the multiplication is given by the transpose of r , the natural co-commutative coproduct on U ( n ). This follows from the interpretationof r explained in § L C w (Λ V ) ⊥ ⊂ ( C ) ∗ gr ∼ = U ( n ) ∗ gr is a subalgebra. By [8,Th. 3.3] it is the C [ N ( w )], which is the q = 1 limit of U − q ( w ) ([19,Th. 4.44]).By [8, Th. 3.2] C [ N ( w )] is compatible with the dual semicanonicalbasis, i.e., the intersection { ρ Λ b | b ∈ B ( ∞ ) } ∩ C [ N ( w )] is a base of C [ N ( w )]. This base consists of ρ Λ b such that Λ b intersects with theopen subvariety Λ wV . The intersection Λ b ∩ Λ wV is an open dense subsetof Λ b .Finally we have { ρ Λ b | b ∈ B ( ∞ ) } ∩ C [ N ( w )] = { ρ Λ b | b ∈ B ( w ) } . This follows from [1, § B ( w ) in § Cluster algebras and C w . Geiss-Leclerc-Schr¨oer [8] have in-troduced a structure of the cluster algebra, in the sense of Fomin-Zelevinsky [6], on C [ N ( w )]. One of their main results says that dualsemicanonical base of C [ N ( w )] contains cluster monomials . We reviewtheir theory only briefly here. See the original paper for more detail.The construction is based on C w in § T is rigid if Ext ( T, T ) = 0. It is easy to see fromthe formula dim Ext ( T, T ) = 2 dim Hom Λ ( T, T ) − (dim V, dim V ) (seee.g., [8, Lem. 2.1]) that this is equivalent to say that the orbit through LUSTER ALGEBRAS AND PERVERSE SHEAVES 11 T is open in Λ V , where V is the I -graded vector space underlying T ,and ( , ) is the Cartan matrix for the graph G . We say T is C w -maximal rigid if Ext ( T ⊕ X, X ) = 0 with X ∈ C w implies X is inadd( T ), the subcategory of modules which are isomorphic to finitedirect sums of direct summands of T . In C w , there is a distinguished C w -maximal module, denoted by V i in [8], where i = ( i , . . . , i ℓ ) is areduced expression of w . It is conjectured that every C w -maximal rigidmodule T is reachable , i.e., it is obtained from V i using a sequence ofoperations, called mutations . This will be recalled below.Let T be a reachable C w -maximal rigid module, and T = T ⊕ · · ·⊕ T ℓ be the decomposition into indecomposables. We assume that T is basic ,which means T i are pairwise non-isomorphic. In this case the numberof summands ℓ is known to be equal to the length of w . If R ∈ add( T ),it is a rigid Λ-module, and hence the closure of the corresponding orbitis an irreducible component of Λ V for an appropriate choice of V . Wedenote it by ρ R ∈ C [ N ( w )] the corresponding dual semicanonical baseelements.From the identification of the coproduct r on U ( n ) ∗ gr , we see that(2.1) ρ R = ρ c T · · · ρ c ℓ T ℓ , where R = T ⊕ c ⊕ · · · ⊕ T c ℓ ℓ with c i ∈ Z ≥ . In the context of the clusteralgebra theory, ρ T i (1 ≤ i ≤ ℓ ) are called cluster variables and ρ R is a cluster monomial .Let T be a basic C w -maximal rigid module, and T k be a non-projectiveindecomposable direct summand of T . The mutation µ k ( T ) is a newbasic C w -maximal rigid module of the form ( T /T k ) ⊕ T ∗ k , where T ∗ k is an-other indecomposable module. Such a µ k ( T ) exists and is uniquely de-termined from T and T k (see [8, Prop. 2.19] and the reference therein).We have dim Ext ( T k , T ∗ k ) = dim Ext ( T ∗ k , T k ) = 1 and(2.2) ρ T k ρ T ∗ k = ρ T ′ + ρ T ′′ , where T ′ , T ′′ ∈ add( T /T k ) are given by short exact sequences0 → T k → T ′ → T ∗ k → , → T ∗ k → T ′′ → T k → q -analog of the result ex-plained above, namely they have introduced a structure of the quan-tum cluster algebra, in the sense of Berenstein-Zelevinsky [3], on thequantum unipotent subgroup U − q ( w ). (This result was conjecturedin [19].) The construction is again based on C w . Let T be a reach-able C w -maximal rigid module. For R ∈ add( T ), there is an element Y R ∈ U − q ( w ), which satisfies the q -analog of (2.1, 2.2):(2.3) Y R = q − α R Y c T · · · Y c ℓ T ℓ ,Y T ∗ k Y T k = q [ T ∗ k ,T k ] ( q − Y T ′ + Y T ′′ )for appropriate explicit α R , [ T ∗ k , T k ] ∈ Z . See [9, (10.17) and Prop. 10.5].One of the main conjectures in this theory is Conjecture 2.4.
All quantum cluster monomials Y R are contained in B up ( w ).This is closely related to their earlier open orbit conjecture [8, § Conjecture 2.5.
Suppose that an irreducible component Λ b of Λ V contains an open G V -orbit, then the corresponding dual semicanonicalbase element ρ Λ b is equal to the specialization of the correspondingcanonical base element b at q = 1.In fact, this is implied by Conj. 2.4 for reachable rigid modules.3. Singular supports under the restriction
Let SS ( L ) denote the singular support of a complex L ∈ Q V . See[17] for the definition. Lusztig proved that SS ( L ) ⊂ Λ V [22, 13.6]. Infact, we have finer estimates(3.1) Λ b ⊂ SS ( L b ) ⊂ [ b ′ ∈ B ( ∞ ) ∀ i ε i ( b ′ ) ≥ ε i ( b ) Λ b ′ . See [16, Thm. 6.2.2], but note that there is a misprint. See [16,Lem. 8.2.1] for the correct statement.These estimates give us some relation between the canonical baseand F V Irr Λ V via singular supports. We study the behavior of singularsupports under the functor Res in this section.3(i). The statement.
In order to state the result, we prepare nota-tion.Let f : Y → X be a morphism. Let T X , T Y (resp. T ∗ X , T ∗ Y ) betangent (resp. cotangent) bundles of X , Y respectively. Let f − T X = Y × X T X (resp. f − T ∗ X = Y × X T ∗ X ) be the pull-back of T X (resp. T ∗ X ) by f . We have associated morphisms T ∗ Y t f ′ ←− f − T ∗ X f π −→ T ∗ X, where t f ′ is the transpose of the differential f ′ : T Y → f − T X = Y × X T X . LUSTER ALGEBRAS AND PERVERSE SHEAVES 13
We apply this construction for the morphisms ι , κ to get morphisms T ∗ E ( W ) t ι ′ ←− ι − T ∗ E V ι π −→ T ∗ E V ,T ∗ E ( W ) t κ ′ ←− κ − T ∗ ( E T × E W ) κ π −→ T ∗ ( E T × E W ) . Theorem 3.2.
We have SS (Res( L )) ⊂ κ π ( t κ ′− ( t ι ′ ( ι − π ( SS ( L ))))) . Let us remark that the proof shows the following statement. Choosea complementary subspace of W in V , and identify V with W ⊕ T .Then we have the induced embedding T ∗ ( E T × E W ) ⊂ T ∗ E V . Thenwe have(3.3) SS (Res( L )) ⊂ T ∗ ( E T × E W ) ∩ SS ( L ) . The proof occupies the rest of this section.3(ii).
Inverse image. If ι were smooth, we would have SS ( ι ∗ L ) ⊂ t ι ′ ( ι − π ( SS ( L ))) by [17, Prop. 5.4.5]. And if κ were proper, we wouldhave SS ( κ ! ι ∗ L ) ⊂ [3] κ π ( t κ ′− ( SS ( ι ∗ L ))) by [17, Prop. 5.4.4]. Therefore the assertion fol-lows. However neither are true, so we need more refined versions ofthese estimates.In order to study the behavior of the singular support under thepull-back by a non-smooth morphism, we need several more notionsrelated to cotangent manifolds from [17, Ch. VI].We first recall the normal cone to S along M briefly. Suppose that M is a closed submanifold of a manifold X . Let T M X denote the normalbundle of M in X . Then one can define a new manifold ˜ X M , whichconnects X and T M X in the following way: there are two maps p : ˜ X M → X, t : ˜ X M → R such that p − ( X \ M ), t − ( R \ { } ) and t − (0) are isomorphic to ( X \ M ) × ( R \{ } ), X × ( R \{ } ) and T M X respectively. In our application, M is the zero section of a vector bundle X , and hence the normal bundle T M X is X itself. In this case, ˜ X M is X × R and p , t are the first andsecond projections. A general definition is in [17, § R + under t , and ˜ p the restriction of p to Ω.Let S be a subset of X . The normal cone to S along M , denoted by C M ( S ) is defined by C M ( S ) def . = T M X ∩ ˜ p − ( S ) . If M is the zero section of a vector bundle X as before, we have C M ( S )is identified with S itself under T M X ∼ = X .Now we return back to a morphism f : Y → X . We assume that f is a closed embedding for simplicity. We consider the conormal bundle T ∗ Y X . We denote the projection T ∗ Y X → Y by p . We have morphisms T ∗ ( T ∗ Y X ) t p ′ ←− p − T ∗ Y p π −→ T ∗ Y as before.We consider T ∗ Y as a submanifold of T ∗ ( T ∗ Y X ) via the composition T ∗ Y ֒ → p − T ∗ Y t p ′ −→ T ∗ ( T ∗ Y X ) . See [17, (5.5.10)].We consider T ∗ Y X as a closed submanifold of T ∗ X . So we can definethe normal cone to a subset of T ∗ X along T ∗ Y X .If f : Y → X is the embedding of the zero section to a vector bundle X , we can identify T ∗ Y X with the dual vector bundle X ∗ . Then T ∗ Y X → T ∗ X is also the embedding of the zero section to a vector bundle. Infact, we have a natural identification T ∗ X ∼ = T ∗ X ∗ , therefore T ∗ Y X → T ∗ X is identified with X ∗ → T ∗ X ∗ . Therefore C T ∗ Y X ( A ) is identifiedwith A itself under the isomorphism T T ∗ Y X ( T ∗ X ) ∼ = T ∗ X .Note also that T ∗ ( T ∗ Y X ) is identified with T ∗ X ∗ ∼ = T ∗ X . Under thisidentification, we have an isomorphism p − T ∗ Y ∼ = f − T ∗ X which givesa commutative diagram(3.4) T ∗ X f π ←−−− f − T ∗ X t f ′ −−−→ T ∗ Y ∼ = y ∼ = y (cid:13)(cid:13)(cid:13) T ∗ ( T ∗ Y X ) ←−−− t p ′ p − T ∗ Y −−−→ p π T ∗ Y. Let A be a conic subset of T ∗ X , i.e., invariant under the R + -action,the multiplication on fibers. We define f ( A ) def . = T ∗ Y ∩ C T ∗ Y X ( A ) . Here we identify T T ∗ Y X T ∗ X with T ∗ ( T ∗ Y X ). See [17, (6.2.3)]. Moreoverwe also have f ( A ) = p πt p ′− ( C T ∗ Y X ( A )) . See [17, Lem. 6.2.1].If f : Y → X is the embedding of the zero section to a vector bundle X , we have(3.5) f ( A ) = T ∗ Y ∩ A = t f ′ f − π ( A )by the commutative diagram (3.4). LUSTER ALGEBRAS AND PERVERSE SHEAVES 15
Spaces.
Let us describe relevant spaces explicitly.As E ( W ) is a linear subspace of E V , we have T ∗ E ( W ) = E ( W ) × E ( W ) ∗ and E ( W ) ∗ is identified with E ∗ V /E ( W ) ⊥ , where E ( W ) ⊥ = { B ′ ∈ E ∗ V | B ′ ( W ) = 0 , Im B ′ ⊂ W } . Taking a complementary subspace of W in V , we identify T as an I -graded subspace of V . We then have the direct sum decomposition V ∼ = W ⊕ T and the induced projection V → W . Then we have matrixnotations of B and B ′ : B = (cid:18) B T T B W T B W W (cid:19) , B ′ = (cid:18) B ′ T T B ′ T W B ′ W W (cid:19) . Similarly the space ι − ( T ∗ E V ) is nothing but E ( W ) × E ∗ V , and identifiedwith the space of linear maps B , B ′ of the forms B = (cid:18) B T T B W T B W W (cid:19) , B ′ = (cid:18) B ′ T T B ′ T W B ′ W T B ′ W W (cid:19) . The morphism t ι ′ : ι − ( T ∗ E V ) → T ∗ E ( W ) is induced by the projec-tion E ∗ V → E ( W ) ∗ . In the matrix notation, it is given by forgettingthe component B ′ W T .The morphism ι π : ι − ( T ∗ E V ) → T ∗ E V is the embedding E ( W ) × E ∗ V → E V × E ∗ V .For ( B, B ′ ) ∈ T ∗ ( E T × E W ), we have B = (cid:18) B T T B W W (cid:19) , B ′ = (cid:18) B ′ T T B ′ W W (cid:19) . For (
B, B ′ ) ∈ κ − ( T ∗ ( E T × E W )), we have B = (cid:18) B T T B W T B W W (cid:19) , B ′ = (cid:18) B ′ T T B ′ W W (cid:19) . The morphisms T ∗ E ( W ) t κ ′ ←− κ − ( T ∗ ( E T × E W )) κ π −→ T ∗ ( E T × E W )are given by taking appropriate matrix entries of B , B ′ .Since we have chosen an isomorphism V ∼ = W ⊕ T , we have theprojection p : E V → E ( W ) which gives a structure of a vector bundleso that ι is the embedding of the zero section. Therefore we have thecommutative diagram (3.4) for f = ι , and hence(3.6) ι ( A ) = T ∗ E ( W ) ∩ A = t ι ′ ι − π ( A )by (3.5). Proof.
We first study the behavior of the singular support un-der the functor ι ∗ . Let L ∈ Q V . By [17, Cor. 6.4.4] we have SS ( ι ∗ L ) ⊂ ι ( SS ( L )) . In our situation, we have ι ( SS ( L )) = t ι ′ ( ι − π ( SS ( L ))) by (3.6).Next study the functor κ ! . Note that κ : E ( W ) → E T × E W is theprojection of a vector bundle. Therefore the results in [17, § F in D ( E ( W )) is conic if H j ( F ) is locallyconstant on the orbits of the R + -action for all j . In our situation, F = i ∗ L satisfies this condition. Then we have SS ( κ ! ( i ∗ L )) ⊂ T ∗ ( E T × E W ) ∩ SS ( i ∗ L ) = κ πt κ ′− SS ( i ∗ L ) . See [17, Prop. 5.5.4] for the first inclusion and [17, (5.5.11)] for thesecond equality. Combining two estimates, we complete the proof ofTheorem 3.2. The estimate (3.3) has been given during the proof.4.
Conjectures
Quantum unipotent subgroup and singular supports.
Let w be a Weyl group element as before. Motivated by § B ′ ( w ) in B ( ∞ ) by B ′ ( w ) def . = { b ∈ B ( ∞ ) | SS ( L b ) ∩ Λ wV = ∅} , where we suppose L b ∈ P V in the equation SS ( L b ) ∩ Λ wV = ∅ . Equiv-alently b / ∈ B ′ ( w ) if and only if SS ( L b ) is contained in the closedsubvariety Λ V \ Λ wV .By (3.1), the condition Λ b ∩ Λ wV = ∅ implies b ∈ B ′ ( w ). Therefore B ( w ) ⊂ B ′ ( w ) by § b / ∈ B ′ ( w ). We have SS (Res( L b )) ∩ (Λ wT × Λ wW ) ⊂ SS ( L b ) ∩ Λ wV = ∅ by (3.3) and the fact that C w is an additive category. WritingRes( L b ) = M ( L b ⊠ L b ) [ n ] ⊕ r b ,b b ; n , we get SS ( L b ⊠ L b ) ∩ (Λ wT × Λ wW ) = ∅ if r b ,b b ; n = 0 for some n . This is because SS ( L ⊕ L ′ ) = SS ( L ) ∪ SS ( L ′ )and SS ( L [1]) = SS ( L ) (see [16, Chap. V]). In the notation in § r b ,b b = P n r b ,b b ; n q n .We have an estimate SS ( L b ⊠ L b ) ⊂ SS ( L b ) × SS ( L b ) [16, Prop. 5.4.1].However this does not imply SS ( L b ) × SS ( L b ) ∩ (Λ wT × Λ wW ) = ∅ , sowe need a finer estimate. Since L b a ( a = 1 ,
2) is a perverse sheaf, it cor-responds to a regular holonomic D -module under the Riemann-Hilbert LUSTER ALGEBRAS AND PERVERSE SHEAVES 17 correspondence (see e.g., [14, Th. 7.2.5]). Then the singular supportof L b a is the same as the characteristic variety of the corresponding D -module [17, Th. 11.3.3], [14, Th. 4.4.5]. As the characteristic varietyof the exterior product is the product of the characteristic varieties [17,(11.2.22)], we deduce SS ( L b ⊠ L b ) = SS ( L b ) × SS ( L b ). Thereforewe have ( SS ( L b ) ∩ Λ wT ) × ( SS ( L b ) ∩ Λ wW ) = ∅ . Therefore either b / ∈ B ′ ( w ) or b / ∈ B ′ ( w ). In other words, b , b ∈ B ′ ( w ) and r b ,b b = 0 implies b ∈ B ′ ( w ). Therefore L b ∈ B ′ ( w ) Q ( q ) b up isa subalgebra of U − q by § Conjecture 4.1. B ′ ( w ) = B ( w ). In other words, if b / ∈ B ( w ), then SS ( L b ) ⊂ Λ V \ Λ wV .This conjecture is also equivalent to say L b ∈ B ′ ( w ) Q ( q ) b up = U − q ( w ).4(ii). Cluster algebra and singular supports.
Recall that Geiss-Leclerc-Schr¨oer [9] have introduced the structure of a quantum clusteralgebra on U − q ( w ) and conjectured that quantum cluster monomialsare contained in B up ( w ). If this is true, we should have two formulas(2.3) for dual canonical base elements corresponding to Y R , Y T k , etc.Conversely (2.3) implies that Y R , Y T ∗ k are dual canonical base elementsby induction on the number of mutations.Let us speculate why these formulas hold in terms of the correspond-ing perverse sheaves.The proposal here is the following conjecture: Conjecture 4.2.
Let T be a reachable C w -maximal rigid module and R ∈ add( T ). Let Λ R be the closure of the orbit through R and b R thecorresponding canonical base element.If another canonical base element b ∈ B ( w ) satisfiesΛ R ⊂ SS ( L b ) , we should have b = b R .If Conj. 4.1 is true, b ∈ B ( ∞ ) with Λ b R ⊂ SS ( L b ) is contained in B ( w ). Therefore the above conjecture holds for any b ∈ B ( ∞ ).This conjecture is true for a special case when Λ R is the zero section E V of T ∗ E V and G V has an open orbit in E V . In fact, if SS ( L b ) ⊃ E V ,we have supp( L b ) = E V . Then L b is G V -equivariant and gives anirreducible G V -equivariant local system on the open orbit in E V . As thestabilizer of a point is connected from a general property from quiverrepresentations, it must be the trivial rank 1 local system. Thus L b is the constant sheaf on E V . In fact, the observation that supp( L b ) = E V implies L b = the constant sheaf was used in a crucial way to prove thecluster character formula in [27].If SS ( L b ) is irreducible for all b , Conj. 4.2 is obviously true. Thiscondition is satisfied for g of type A , but not for A [16].Let us remark a relation between the above conjecture and a conjec-ture in [7, § semicanonical base { f Y } of U ( n ) as the dual base of thedual semicanonical base { ρ Y } . In [7, § b is a linear combination P m Y f Y ( m Y ∈ Z ), wherethe summation runs over irreducible components Y of SS ( L b ). (Moreprecisely it is probably given by the characteristic cycle (see [14, 2.2.2]for the definition) of L b .) Dually, an irreducible component Y = Λ b cannot be contained in other SS ( L b ′ ) ( b ′ = b ) if b up | q =1 = ρ Λ b . This isnothing but our conjecture. Thus under the conjecture in [7, § R = T ⊕ T for brevity. From the assumption Λ R containsthe product Λ ◦ T × Λ ◦ T as an open dense subset. Here Λ ◦ T i denote theopen orbit through T i . Its closure is Λ T i . Suppose that b up appears inthe product b up T b up T . Then r b T ,b T b = 0. We have SS ( L b ) ∩ (Λ ◦ T × Λ ◦ T ) ⊃ SS (Res( L b )) ∩ (Λ ◦ T × Λ ◦ T ) ⊃ SS ( L b T ⊠ L b T ) ∩ (Λ ◦ T × Λ ◦ T )= ( SS ( L b T ) ∩ Λ ◦ T ) × ( SS ( L b T ) ∩ Λ ◦ T ) , where the first inclusion is by (3.3), the second as a shift of L b T ⊠ L b T is a direct summand of Res L b , and the third equality was observedabove. The last expression is nonempty thanks to (3.1). Therefore wehave SS ( L b ) ⊃ Λ R . Then Conj. 4.2 implies that b = b R . Therefore b up T b up T is a multiple of b up R . A refinement of this argument probablyproves that b up T b up T is equal to b up R up to a power of q .Let us turn to the second formula in (2.3). The same argument aboveimplies that SS ( L b ) ∩ (Λ ◦ T ∗ k × Λ ◦ T k ) = ∅ if r b T ∗ k ,b Tk b = 0. From what we have explained in § T ′ , Λ T ′′ , where T ′ and T ′′ are Λ-modules givenby non-trivial extensions of T ∗ k and T k . As a non-trivial extension candegenerate to the trivial one, both Λ T ′ and Λ T ′′ contain Λ ◦ T ∗ k × Λ ◦ T k . Itis also easy to check that dim Λ ◦ T ∗ k × Λ ◦ T k = dim Λ V −
1, where V is theunderlying vector space of T k ⊕ T ∗ k . LUSTER ALGEBRAS AND PERVERSE SHEAVES 19
Lemma 4.3.
If an irreducible component Y of Λ V contains Λ ◦ T ∗ k × Λ ◦ T k ,we have either Y = Λ T ′ or = Λ T ′′ .Proof. Take a sequence Z n of points of Y converging to the module T ∗ k ⊕ T k , regarded as a point of Y . We may assume Z n = T ∗ k ⊕ T k .Then we have dim Hom( Z n , T ∗ k ⊕ T k ) ≤ dim Hom( T ∗ k ⊕ T k , T ∗ k ⊕ T k ) forsufficiently large n by the upper semicontinuity of the dimension of co-homology groups. If the equality holds, we can take ξ n ∈ Hom( Z n , T ∗ k ⊕ T k ) converging to the identity of Hom( T ∗ k ⊕ T k , T ∗ k ⊕ T k ) for n → ∞ .In particular, ξ n is invertible, hence Z n ∼ = T ∗ k ⊕ T k . This contra-dicts with our assumption. Therefore we have the strict inequalitydim Hom( Z n , T ∗ k ⊕ T k ) < dim Hom( T ∗ k ⊕ T k , T ∗ k ⊕ T k ). The same argu-ment gives dim Hom( T ∗ k ⊕ T k , Z n ) < dim Hom( T ∗ k ⊕ T k , T ∗ k ⊕ T k ).Thereforedim Ext ( T ∗ k ⊕ T k , Z n )= − (dim V, dim V ) + dim Hom( T ∗ k ⊕ T k , Z n ) + dim Hom( Z n , T ∗ k ⊕ T k ) ≤ − (dim V, dim V ) + 2 dim Hom( T ∗ k ⊕ T k , T ∗ k ⊕ T k ) −
2= Ext ( T ∗ k ⊕ T k , T ∗ k ⊕ T k ) − , where we have used the formula in [8, Lem. 2.1]. The upper semicon-tinuity also shows Ext ( T /T k , Z n ) = 0, hence Z n ∈ add( T /T k ).As the inequality above must be an equality, we getdim Hom( T ∗ k ⊕ T k , Z n ) = dim Hom( Z n , T ∗ k ⊕ T k )= dim Hom( T ∗ k ⊕ T k , T ∗ k ⊕ T k ) − . Therefore dim Hom( T ∗ k , Z n ) + dim Hom( T k , Z n )= dim Hom( T ∗ k , T ∗ k ⊕ T k ) + dim Hom( T k , T ∗ k ⊕ T k ) − . Note that dim Hom( T ∗ k , Z n ) ≤ dim Hom( T ∗ k , T ∗ k ⊕ T k ) and dim Hom( T k , Z n ) ≤ dim Hom( T k , T ∗ k ⊕ T k ) by the semicontinuity. Hence the above impliesthat one of inequalities must be an equality. Suppose that the first oneis an equality. Then we havedim Hom( T ∗ k , Z n ) = dim Hom( T ∗ k , T ∗ k ⊕ T k ) , dim Hom( T k , Z n ) = dim Hom( T k , T ∗ k ⊕ T k ) − . The same argument shows that dim Hom( Z n , T ∗ k ) = dim Hom( T ∗ k ⊕ T k , T ∗ k ) or dim Hom( Z n , T k ) = dim Hom( T ∗ k ⊕ T k , T k ). The first equalityis impossible, as 0 = dim Ext ( T ∗ k , Z n ) = dim Ext ( T ∗ k , T k ⊕ T k ) = 1and the above dimension formula. Therefore we havedim Hom( Z n , T ∗ k ) = dim Hom( T ∗ k ⊕ T k , T ∗ k ) − , dim Hom( Z n , T k ) = dim Hom( T ∗ k ⊕ T k , T k ) . We take η n ∈ Hom( T ∗ k , Z n ) converging to id T ∗ k ⊕ T ∗ k , T ∗ k ⊕ T k ). In particular, η n is injective for sufficiently large n . We consideran exact sequence0 → Hom( Z n / Im η n , T k ) → Hom( Z n , T k ) → Hom(Im η n , T k ) . The next term Ext ( Z n / Im η n , T k ) vanishes, as we have dim Ext ( Z n / Im η n , T k ) ≤ dim Ext ( T k , T k ) = 0 by the upper semicontinuity. Therefore we havedim Hom( Z n / Im η n , T k ) = dim Hom( T k , T k ) . We take ζ n ∈ Hom( Z n / Im η n , T k ) converging to id T k . Then ζ n isan isomorphism for sufficiently large n . Composing the projection p : Z n → Z n / Im η n with ζ n , we have an exact sequence0 → T ∗ k η n −→ Z n ζ n ◦ p −−→ T k → . This shows that Z n ∼ = T ′′ .When dim Hom( T k , Z n ) = dim Hom( T k , T ∗ k ⊕ T k ), the same argumentshows that Z n ∼ = T ′ . (cid:3) Now Conj. 4.2 implies that b up T ∗ k b up T k is a linear combination of b up T ′ and b up T ′′ . A refinement of the argument hopefully gives the second formulain (2.3). References [1] P. Baumann, J. Kamnitzer, and P. Tingley,
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