GEOMETRIC REALIZATIONS OF LUSZTIG’S SYMMETRIES JIE XIAO AND MINGHUI ZHAO Abstract.

GEOMETRIC REALIZATIONS OF LUSZTIG’S SYMMETRIES

JIE XIAO AND MINGHUI ZHAO

Abstract. In this paper, we give geometric realizations of Lusztig’s symme- tries. We also give projective resolutions of a kind of standard modules. By using the geometric realizations and the projective resolutions, we obtain the categorification of the formulas of Lusztig’s symmetries.

Contents

1. Introduction 1

2. Quantum groups and Lusztig’s symmetries 4

2.1. Quantum groups 4

2.2. Lusztig’s symmetries 5

3. Geometric realizations 5

3.1. Geometric realization and canonical basis off 5

3.2. Geometric realizations of if andⁱf 7

3.3. Geometric realization ofTi:if →ⁱf 8

3.4. T_i:_if →ⁱf and canonical bases 9

4. Hall algebra approaches 10

4.1. Hall algebra approach tof 10

4.2. Hall algebra approaches to_if and ⁱf 10

4.3. Hall algebra approach toT_i:_if →ⁱf and the proof of Theorem 3.5 13 5. Projective resolutions of a kind of standard modules 15

5.1. KLR algebras 15

5.2. Projective resolutions 15

5.3. The proof of Theorem 5.3 17

5.4. The formulas of Lusztig’s symmetries 23

References 24

1. Introduction

LetU be the quantum group and f be the Lusztig’s algebra associated with a Cartan datum. Denote byU⁺andU⁻the positive part and the negative part ofU respectively. There are two well-defined Q(v)-algebra homomorphisms ⁺ :f →U and⁻:f →Uwith imagesU⁺ andU⁻ respectively.

Date: February 19, 2017.

2000Mathematics Subject Classification. 16G20, 17B37.

Key words and phrases. Lusztig’s symmetries, Standard modules, Projective resolutions.

This work was supported by the National Natural Science Foundation of China [grant numbers 11471177, 11526037].

1

Lusztig introduced the canonical basisBoff in [15, 17, 20]. LetQ= (I, H) be a quiver corresponding tof andVbe anI-graded vector space such that dimV=ν∈ NI. He studied the varietyEV consisting of representations of Q with dimension vector ν, and a category QV of some semisimple complexes on E_V. LetK(QV) be the Grothendieck group of QV. Considering all dimension vectors, he proved thatL

ν∈NIK(Q_V) realizes f and the set of isomorphism classes of simple objects realizes the canonical basisB.

Lusztig also introduced some symmetriesT_i onUfor alli∈I in [14, 16]. Note that Ti(U⁺) is not contained in U⁺. Hence, Lusztig introduced two subalgebras

if and ⁱf of f for any i ∈ I, where if ={x ∈ f |Ti(x⁺) ∈ U⁺} and ⁱf = {x ∈ f |T_i⁻¹(x⁺)∈U⁺}. LetT_i:_if →ⁱfbe the unique map satisfyingT_i(x⁺) =T_i(x)⁺. The algebraf has the following direct sum decompositionsf =_ifLθ_if =ⁱfLfθ_i. Denote by_iπ:f →if and ⁱπ:f →ⁱf the natural projections.

Associated to a finite dimensional hereditary algebra Λ, Ringel introduced the Hall algebra and the composition subalgebraF in [22], which gives a realization of the positive part of the quantum groupU. If we use the notations of Lusztig in [19], we have the canonical isomorphism between the composition subalgebraF and the Lusztig’s algebra f. Via the Hall algebra approach, one can apply BGP-reflection functors to quantum groups to give precise constructions of Lusztig’s symmetries ([23, 19, 26, 28, 6, 29]).

To a Lusztig’s algebraf, Khovanov, Lauda ([9]) and Rouquier ([24]) introduced a series of algebrasRν respectively. The category of finitely generated projective modules of Rν gives a categorification of f and Rν are called Khovanov-Lauda- Rouquier (KLR) algebras. Varagnolo, Vasserot ([27]) and Rouquier ([25]) realized the KLR algebraRν as the extension algebra of semisimple complexes inQV and proved that the set of indecomposable projective modules ofR_ν can categorify the canonical basisB.

In [7, 8], Kato gave the categorification of the PBW-type bases of quantum groups of finite type. He constructed some modules (which are called standard modules) of the KLR algebras R_ν and proved that there standard modules can categorify the PBW-type basis off by using the geometric realizations ofRν given by Varagnolo, Vasserot and Rouquier. He proved that the length of the projective resolution of any standard module is finite, which is the categorification of the following fact: the transition matrix between the PBW-type basis of f and the canonical basisBis triangular with diagonal entries equal to 1. This result implies that the global dimensions of the KLR algebrasRν are also finite. In [21, 4, 3, 11], Brundan, Kleshchev and McNamara proved the same result by using an algebraic method.

Leti∈I be a sink (resp. source) ofQ. Similarly to the geometric realization of f, consider a subvariety iEV (resp. ⁱEV) of EV and a categoryiQ_V (resp. ⁱQV) of some semisimple complexes on_iE_V (resp. ⁱE_V). In Section 3.2, we verify that L

ν∈NIK(_iQV) (resp. L

ν∈NIK(ⁱQV)) realizes_if (resp. ⁱf).

Leti∈Ibe a sink ofQ. LetQ⁰=σ_iQbe the quiver by reversing the directions of all arrows inQcontainingi. Hence,i is a source ofQ⁰. Consider twoI-graded vector spaces V and V⁰ such that dimV⁰ =s_i(dimV). In the case of finite type, Kato introduced an equivalence ˜ω_i : _iQ_V,Q →ⁱQ_V⁰_,Q⁰ and studied the properties of this equivalence in [7, 8]. In this paper, we generalize his construction to all cases and prove that the map induced by ˜ωirealizes the Lusztig’s symmetryTi :if →ⁱf.

For the proof of the result, we shall study the relations between the map induced by ˜ωi and the Hall algebra approach toTiin [19].

In [18], Lusztig showed that Lusztig’s symmetries and canonical bases are com- patible. Let _iB =_iπ(B), which is a Q(v)-basis of _if. Similarly, ⁱB =ⁱπ(B) is a Q(v)-basis of ⁱf. Lusztig proved that T_i : _if → ⁱf maps any element of _iB to an element ofⁱB.

For any simple perverse sheafL in Q_V,Q, the restriction _iL=j_V^∗(L) on_iE_V,Q is also a simple perverse sheaf and belongs to_iQ_V,Q, wherej_V :_iE_V,Q→E_V,Q is the canonical embedding. Let ⁱL = ˜ωi(iL)∈ⁱQV⁰,Q⁰. The simple perverse sheaf

iLcan be wrote asⁱL=j_V^∗0(L⁰), whereL⁰is a simple perverse sheaf inQ_V⁰_,Q⁰ and j_V⁰ :ⁱE_V⁰_,Q⁰ →E_V⁰_,Q⁰ is the canonical embedding. Since the map induced by ˜ω_i realizes T_i :_if →ⁱf, this result gives a geometric interpretation of Lusztig’s result in [18].

For anym≤ −a_ij, let f(i, j;m) = X

r+s=m

(−1)^rv^{−r(−aij−m+1)}θ^(r)_i θjθ_i^(s)∈f, and

f⁰(i, j;m) = X

r+s=m

(−1)^rv^{−r(−aij−m+1)}θ^(s)_i θjθ_i^(r)∈f.

In [20], Lusztig proved thatT_i(f(i, j;m)) =f⁰(i, j;m⁰), wherem⁰ =−aij−m. The following formula

Ti(Ej) = X

r+s=−aij

(−1)^rv^−rE_i^(s)EjE_i^(r)

is a special case ofTi(f(i, j;m)) =f⁰(i, j;m⁰). In this paper, our main result is the categorification of these formulas. Consider anI-graded vector spaceV such that dimV=ν =mi+j. Let DG_V(EV) be the boundedGV-equivariant derived cate- gory of complexes ofl-adic sheaves onEV. We construct a series of distinguished triangles in DG_V(EV), which represent the constant sheaf 1_iE_V in terms of some semisimple complexes Ip ∈ DG_V(EV) geometrically. Note that,1_iE_V corresponds to a standard module Km of the KLR algebra Rν and Ip correspond to projec- tive modules of Rν. This result means that we find projective resolutions of the standard modulesKm. Consider two I-graded vector spaces V andV⁰ such that dimV=mi+j and dimV⁰ =si(dimV) =m⁰i+j. Applying to the Grothendieck group, 1_iEV,Q (resp. 1iE_V0,Q0) corresponds to f(i, j;m) (resp. f⁰(i, j;m⁰)). The property of BGP-reflection functors implies ˜ωi(v^−mN1_iE_V,Q) = v^−m0N1iE_V0,Q0, thereforeTi(f(i, j;m)) =f⁰(i, j;m⁰).

In Example D of [8], Kato constructed a short exact sequence 0 //P1∗P2[2] //P2∗P1 //Q21 //0,

which coincides with the projection resolution in our main result in the case of finite type. In Theorem 4.10 of [4], Brundan, Kleshchev and McNamara constructed a shout exact sequence of standard modules

0 //v^−β·γ∆(β)◦∆(γ) //∆(γ)◦∆(β) //[pβ,γ+ 1]∆(α) //0.

In the case of finite type, the projection resolution in our main result is a special case of the shout exact sequence above whereα=α_i+α_j.

2. Quantum groups and Lusztig’s symmetries

2.1. Quantum groups. LetIbe a finite index set with|I|=nandA= (aij)i,j∈I

be a generalized Cartan matrix. Let (A,Π,Π^∨, P, P^∨) be a Cartan datum associated withA, where

(1) Π ={αi |i∈I} is the set of simple roots;

(2) Π^∨={hi |i∈I} is the set of simple coroots;

(3) P is the weight lattice;

(4) P^∨is the dual weight lattice.

In this paper, we always assume that the generalized Cartan matrixAis symmetric.

Fix an indeterminate v. For any n∈Z, set [n]v = ^v_v−v^n−v−1⁻ⁿ ∈Q(v). Let [0]v! = 1 and [n]v! = [n]v[n−1]v· · ·[1]v for anyn∈Z>0.

The quantum groupUassociated with a Cartan datum (A,Π,Π^∨, P, P^∨) is an associative algebra over Q(v) with unit element 1, generated by the elements Ei, F_i(i∈I) andK_µ(µ∈P^∨) subject to the following relations

K₀=1, K_µK_µ⁰ =K_µ+µ⁰ for allµ, µ⁰∈P^∨; K_µE_iK_−µ=v^αi(µ)E_i for alli∈I,µ∈P^∨; KµFiK_−µ=v^−αi(µ)Fi for alli∈I,µ∈P^∨;

E_iF_j−F_jE_i=δ_ijKi−K_−i

v−v⁻¹ for alli, j∈I;

1−aij

X

k=0

(−1)^kE_i^(k)EjE_i^{(1−aij−k)}= 0 for alli6=j∈I;

1−aij

X

k=0

(−1)^kF_i^(k)F_jF_i^{(1−aij−k)}= 0 for all i6=j∈I.

Here,Ki=Khi andE_i⁽ⁿ⁾=Eⁿ_i/[n]v!, F_i⁽ⁿ⁾=F_iⁿ/[n]v!.

LetU⁺ (resp. U⁻) be the subalgebra of U generated byE_i (resp. F_i) for all i∈I, andU⁰be the subalgebra ofUgenerated byK_µfor allµ∈P^∨. The quantum groupU has the following triangular decomposition

U∼=U⁻⊗U⁰⊗U⁺.

Let f be the associative algebra defined by Lusztig in [20]. The algebra f is generated byθi(i∈I) subject to the following relations

1−a_ij

X

k=0

(−1)^kθ_i^(k)θ_jθ^(1−a_i ^ij−k)= 0 for alli6=j∈I,

whereθ_i⁽ⁿ⁾=θⁿ_i/[n]_v!.

There are two well-definedQ(v)-algebra homomorphisms⁺:f →Uand⁻ :f → U satisfyingE_i =θ⁺_i andF_i =θ_i⁻ for all i∈I. The images of ⁺ and⁻ are U⁺ andU⁻ respectively.

2.2. Lusztig’s symmetries. Corresponding toi∈I, Lusztig introduced the Lusztig’s symmetryTi:U→U([14, 16, 20]). The formulas ofTi on the generators are:

Ti(Ei) =−FiKi, Ti(Fi) =−K_−iEi; T_i(E_j) = X

r+s=−aij

(−1)^rv^−rE_i^(s)E_jE_i^(r) fori6=j∈I;

(1)

Ti(Fj) = X

r+s=−a_ij

(−1)^rv^rF_i^(r)FjF_i^(s) fori6=j∈I;

(2)

Ti(Kµ) =K_µ−αi(µ)h_i.

Lusztig introduced two subalgebras_if and ⁱf of f. For any j ∈ I, i 6=j, m ∈N, define

f(i, j;m) = X

r+s=m

(−1)^rv^{−r(−aij−m+1)}θ^(r)_i θ_jθ_i^(s)∈f, and

f⁰(i, j;m) = X

r+s=m

(−1)^rv^{−r(−aij−m+1)}θ^(s)_i θjθ_i^(r)∈f.

The subalgebrasif andⁱf are generated byf(i, j;m) andf⁰(i, j;m) respectively.

Note that_if ={x∈f |T_i(x⁺)∈U⁺} andⁱf ={x∈f |T_i⁻¹(x⁺)∈U⁺} ([20]).

Hence there exists a unique T_i :_if →ⁱf such thatT_i(x⁺) =T_i(x)⁺. Lusztig also showed thatf has the following direct sum decompositionsf =_ifLθ_if =ⁱfLfθ_i. Denote by_iπ:f →_if and ⁱπ:f →ⁱf the natural projections.

Lusztig also proved the following formulas.

Proposition 2.1([20]). For any−aij ≥m∈N,T_i(f(i, j;m)) =f⁰(i, j;−aij−m).

The formulas (1) and (2) are two special cases of Proposition 2.1.

3. Geometric realizations

3.1. Geometric realization and canonical basis of f . In this subsection, we shall review the geometric realization off introduced by Lusztig ([15, 17, 20]).

A quiverQ= (I, H, s, t) consists of a vertex setI, an arrow setH, and two maps s, t:H→Isuch that an arrowρ∈H starts ats(ρ) and terminates att(ρ). From now on, assume thats(ρ)6=t(ρ) for anyρ∈H.

For anyi, j∈I, let aij =

−#{i→j} −#{j→i}, ifi6=j;

2, ifi=j.

The matrix A= (aij)_i,j∈I is a symmetric generalized Cartan matrix. Let f be the Lusztig’s algebra corresponding to A. Let p be a prime and q be a power of p.

Denote byFq the finite field withqelements andK=Fq. For a finite dimensionalI-gradedK-vector spaceV=L

i∈IVi, define EV=M

ρ∈H

Hom_K(V_s(ρ), V_t(ρ)).

The dimension vector of V is defined as dimV = P

i∈I(dim_KV_i)i ∈ NI. The algebraic groupG_V=Q

i∈IGL_K(V_i) acts on E_V naturally.

Fix a nonzero elementν ∈NI. Let Y_ν ={y= (i,a)|

k

X

l=1

a_li_l=ν}, wherei= (i₁, i₂, . . . , i_k), i_l∈I,a= (a₁, a₂, . . . , a_k), a_l∈N, and

I^ν ={i= (i1, i2, . . . , ik)|

k

X

l=1

il=ν}.

Fix a finite dimensional I-gradedK-vector spaceV such that dimV=ν. For any elementy= (i,a), a flag of typey inVis a sequence

φ= (V=V^k⊃V^k−1⊃ · · · ⊃V⁰= 0)

ofI-gradedK-vector spaces such that dimV^l/V^l−1=alil. LetFybe the variety of all flags of typeyinV. For anyx∈E_V, a flagφis calledx-stable ifx_ρ(V_s(ρ)^l )⊂V_t(ρ)^l for allland allρ∈H. Let

F˜y={(x, φ)∈EV×Fy | φisx-stable}

andπ_y: ˜F_y→E_V be the projection toE_V.

Let ¯Qlbe thel-adic field andD_GV(E_V) be the boundedG_V-equivariant derived category of complexes of l-adic sheaves on E_V. For each y ∈Y_ν, considerL_y = (πy)!(1F˜_y)[dy]∈ DG_V(EV), wheredy= dim ˜Fy. SinceFy is a smooth, irreducible, projective variety and the projection ˜F_y→F_y is a vector bundle, ˜F_y is a smooth, irreducible variety. At the same time, πy : ˜Fy →EV is a proper GV-equivariant morphism. Hence, by BBD decomposition theorem ([1, 5]), Ly is a semisimple complex. LetPVbe the set of isomorphism classes of simple perverse sheavesLon EV such thatL[r] appears as a direct summand of Li for somei∈I^ν andr∈Z. LetQV be the full subcategory ofDG_V(EV) consisting of all complexes which are isomorphic to finite direct sums of complexes in the set{L[r]| L ∈ PV, r∈Z}.

LetK(QV) be the Grothendieck group ofQV. Define v^±[L] = [L[±1](±1

2)],

where L(d) is the Tate twist of L. Then, K(QV) is a free A-module, whereA= Z[v, v⁻¹]. Define

K(Q) = M

ν∈NI

K(QV).

Forν, ν⁰, ν⁰⁰∈NI such thatν =ν⁰+ν⁰⁰ and threeI-gradedK-vector spacesV, V⁰, V⁰⁰ such that dimV = ν, dimV⁰ = ν⁰, dimV⁰⁰ = ν⁰⁰, Lusztig constructed a functor

∗:QV⁰× QV⁰⁰→ QV.

This functor induces an associativeA-bilinear multiplication

~:K(QV⁰)×K(QV⁰⁰) → K(QV)

([L⁰],[L⁰⁰]) 7→ [L⁰]~[L⁰⁰] = [L⁰~L⁰⁰], whereL⁰~L⁰⁰= (L⁰∗ L⁰⁰)[m_ν⁰_ν⁰⁰](^mν₂^0ν00) andm_ν⁰_ν⁰⁰=P

ρ∈Hν_s(ρ)⁰ ν_t(ρ)⁰⁰ −P

i∈Iν_i⁰ν_i⁰⁰. ThenK(Q) becomes an associativeA-algebra and the set{[L]| L ∈ PV}is a basis ofK(QV).

Theorem 3.1 ([17]). There is a uniqueA-algebra isomorphism λA:K(Q)→fA

such that λ_A(Ly) =θ_y for all y∈Y_ν, where θ_y =θ^(a_i ¹⁾

1 θ^(a_i ²⁾

2 · · ·θ^(a_i ^k)

k andf_A is the integral form off.

LetBν ={b_L =λ_A([L])| L ∈ PV}andB=t_ν∈NIBν. ThenBis the canonical basis off introduced by Lusztig in [15, 17].

3.2. Geometric realizations of if and ⁱf . Assume that i∈I is a sink. Let V be a finite dimensionalI-graded K-vector space such that dimV=ν. Consider a subvarietyiE_V ofEV

iE_V={x∈E_V | M

h∈H,t(h)=i

x_h: M

h∈H,t(h)=i

V_s(h)→V_i is surjective}.

LetjV:iE_V →EV be the canonical embedding. For anyy= (i,a)∈Yν, let

iF˜y ={(x, φ)∈iE_V×Fy |φisx-stable}

and_iπ_y:_iF˜_y →_iE_V be the projection to_iE_V. For any y∈Yν, let iLy = (iπ_y)!(1

iF˜_y)[dy]∈ DG_V(iEV). SinceiEV is an open subvariety ofEV, it is also a smooth variety. At the same time,iπ_y:iF˜_y →iE_Vis a projection. Hence, BBD decomposition theorem implies thatiLy is a semisimple complex. Let_iP_Vbe the set of isomorphism classes of simple perverse sheavesLon

iE_V such thatL[r] appears as a direct summand ofiL_i for somei∈I^ν andr∈Z. Let_iQV be the full subcategory ofDGV(_iE_V) consisting of all complexes which are isomorphic to finite direct sums of complexes in the set{L[r]| L ∈_iP_V, r∈Z}.

LetK(_iQ_V) be the Grothendieck group of_iQ_V and K(_iQ) = M

ν∈NI

K(_iQ_V).

Naturally, we have two functorsjV!:DG_V(iEV)→ DG_V(EV) andj_V^∗ :DG_V(EV)→ DG_V(iEV).

For anyy∈Yν, we have the following fiber product

iF˜y

˜j_V //

iπy

F˜y πy

iEV

jV //EV. By the smooth base change formula, we have

(3) j_V^∗L_y=j_V^∗(π_y)_!(1F˜_y)[d_y] = (_iπ_y)_!˜j^∗_V(1F˜_y)[d_y] = (_iπ_y)_!(1

iF˜_y)[d_y] =_iL_y. That isj^∗_V(Q_V) =_iQ_V. Hence j_V^∗ :Q_V →_iQ_V and j^∗ :K(Q)→K(_iQ) can be defined.

Consider the following diagram

(4) 0 //θ_if_A ⁱ //f_A ^iπA //

λ⁰_A

if_A //

0

K(Q) ^j

∗ //K(_iQ) //0,

whereλ⁰_Ais the inverse ofλA. Sincej^∗◦λ⁰_A◦i= 0, there exists a mapiλ⁰_A:if_A→ K(iQ) such that the above diagram (4) commutes.

Proposition 3.2. The map iλ⁰_A:if_A→K(iQ)is an isomorphism ofA-algebras.

The proof of Proposition 3.2 will be given in Section 4.2.

Assume thati∈Iis a source. We can give a geometric realization ofⁱf similarly.

Consider a subvarietyⁱE_V ofE_V

iEV={x∈EV | M

h∈H,s(h)=i

xh:Vi→ M

h∈H,s(h)=i

V_t(h) is injective}.

LetjV:ⁱEV →EV be the canonical embedding. SinceⁱEV is an open subvariety of EV, it is also a smooth variety. The definitions of ⁱQV, K(ⁱQV) and K(ⁱQ) are similar to those of iQ_V, K(iQV) andK(iQ) respectively. We can also define j_V^∗ :QV→ⁱQV, j^∗:K(Q)→K(ⁱQ) andⁱλ⁰_A:ⁱf_A→K(ⁱQ).

Similarly to Proposition 3.2, we have the following proposition.

Proposition 3.3. The map ⁱλ⁰_A:ⁱf_A→K(ⁱQ)is an isomorphism ofA-algebras.

3.3. Geometric realization of T_i : _if → ⁱf . Assume that i is a sink of Q = (I, H, s, t). Soi is a source ofQ⁰ =σ_iQ= (I, H⁰, s, t), whereσ_iQis the quiver by reversing the directions of all arrows inQcontainingi. For anyν, ν⁰∈NIsuch that ν⁰ =siν =ν−ν(hi)i and I-graded K-vector spaces V, V⁰ such that dimV= ν, dimV⁰=ν⁰, consider the following correspondence ([19, 8])

(5) iEV,Q ZVV⁰

oo α ^β //ⁱEV⁰,Q⁰ ,

where

(1) Z_VV⁰ is the subset inE_V,Q×E_V⁰_,Q⁰ consisting of all (x, y) satisfying the following conditions

(a) for anyh∈H such thatt(h)6=iandh∈H⁰,x_h=y_h; (b) the following sequence is exact

0 // V_i⁰

L

h∈H0,s(h)=iy_h

//L

h∈H,t(h)=iV_s(h)

L

h∈H,t(h)=ix_h

//Vi //0.

(2) α(x, y) =xandβ(x, y) =y.

From now on,iEV,Qis denoted by iEV andⁱEV⁰,Q⁰ is denoted byⁱEV⁰. Let GVV⁰ =GL(Vi)×GL(V_i⁰)×Y

j6=i

GL(Vj)∼=GL(Vi)×GL(V_i⁰)×Y

j6=i

GL(V_j⁰), which acts onZVV⁰ naturally.

By (5), we have

(6) DG_V(iEV) ^α

∗ //DG_VV0(ZVV⁰) ^β DG_V0(ⁱEV⁰)

oo ∗ .

Sinceαandβ are principal bundles with fibersAut(V_i⁰) andAut(V_i) respectively, α^∗ and β^∗ are equivalences of categories by Section 2.2.5 in [2]. Hence, for any

L ∈ DG_V(iEV) there exists a uniqueL⁰ ∈ DG_V0(ⁱEV⁰) such that α^∗(L) = β^∗(L⁰).

Define

˜

ωi:DG_V(iEV) → DG_V0(ⁱEV⁰) L 7→ L⁰[−s(V)](−s(V)

2 ),

wheres(V) = dim GL(V_i)−dim GL(V_i⁰).Sinceα^∗ andβ^∗ are equivalences of cate- gories, ˜ω_i is also an equivalence of categories.

Proposition 3.4. It holds that ω˜_i(_iQ_V) =ⁱQV⁰.

The proof of Proposition 3.4 will be given in Section 4.3.

Hence, we can define ˜ω_i:_iQ_V →ⁱQV⁰ and ˜ω_i :K(_iQ)→K(ⁱQ). We have the following theorem.

Theorem 3.5. We have the following commutative diagram

if_A ^Ti //

iλ⁰_A

ifA

iλ⁰_A

K(iQ) ^ω˜i //K(ⁱQ).

The proof of Theorem 3.5 will be given in Section 4.3.

3.4. T_i : _if → ⁱf and canonical bases. In [18], Lusztig showed that Lusztig’s symmetries and canonical bases are compatible. In this section, we shall give a geometric interpretation of this result by using the geometric realization ofT_i.

LetBbe the canonical basis off. Sinceθ_ifis the kernel of_iπ:f →_if andB∩θ_if is aQ(v)-basis ofθif,iB=iπ(B) is aQ(v)-basis ofif. Similarly,ⁱB=ⁱπ(B) is a Q(v)-basis ofⁱf.

Lusztig proved the following theorem.

Theorem 3.6 ([18]). Lusztig’s symmetryTi:if →ⁱf maps any element of iBto an element ofⁱB. Thus, there exists a unique bijectionκi:B−B∩θif →B−B∩fθi

such that Ti(iπ(b)) =ⁱπ(κi(b)).

Leti be a sink of a quiver Q. So i is a source ofQ⁰ =σiQ. By Theorem 3.1, Proposition 3.3, the formula (3) and the commutative diagram (4), we have (7) _iB=t_ν∈NI{b_L=_iλ_A([L])| L ∈_iP_V, dimV=ν}.

Similarly, we have

(8) ⁱB=tν⁰∈NI{b_L =ⁱλ_A([L])| L ∈ⁱPV⁰, dimV⁰=ν⁰}.

Fix anyν, ν⁰∈NI such thatν⁰ =siν andI-gradedK-vector spacesV,V⁰ such that dimV=ν, dimV⁰=ν⁰.

In (6), the functorsα^∗ andβ^∗ are equivalences of categories. Hence the functor

˜

ωi:iQ_V→ⁱQV⁰

maps any simple perverse sheaf in iQ_V to a simple perverse sheaf in ⁱQV⁰. That is, ˜ωi(iP_V) =ⁱPV⁰. So the map

˜

ω_i:K(_iQ)→K(ⁱQ)

satisfies

˜

ω_i({[L]| L ∈_iP_V}) ={[L]| L ∈ⁱP_V⁰}.

By Theorem 3.5, (7) and (8), it holds that Ti(iB) = ⁱB and we get a geometric interpretation of Theorem 3.6.

4. Hall algebra approaches

4.1. Hall algebra approach to f . In this subsection, we shall review the Hall algebra approach tof ([22, 19, 12, 13]).

LetQ= (I, H, s, t) be a quiver. In Section 3.1,EV andGV are defined for any I-gradedK-vector spaceV. LetFⁿbe the Frobenius morphism. The setsE^F_Vⁿand G^F_Vⁿ consist of theFⁿ-fixed points inE_V andG_V respectively.

Lusztig definedFⁿ_V as the set of allG^F_Vⁿ-invariant ¯Ql-functions onE_V^Fn and we can give a multiplication onFⁿ =L

ν∈NIFⁿ_V to obtain the Hall algebra. For any i∈I, letVi be theI-gradedK-vector space with dimension vectoriandfi be the constant function onE_V^Fn

i with value 1. Denote byFⁿthe composition subalgebra of Fⁿ generated by f_i and F_Vⁿ =Fⁿ_V∩ Fⁿ. LetF =L

ν∈NIF_V be the generic form ofFⁿ andF_A be the integral form ofF ([19]).

Theorem 4.1 ([22, 19]). There exists an isomorphism ofA-algebras

$_A:f_A→ FA

such that $_A(θ_i) =f_i.

For any L ∈ DG_V(EV), there is a function χⁿ_L : E_V^Fn → Q¯l (Section I.2.12 in [10]). Hence, we have the following map

χⁿ:DG_V(EV) → Fⁿ_V L 7→ χⁿ_L.

The restriction of this map on the subcategoryQV is also denoted by χⁿ :QV→ Fⁿ_V.

Lusztig proved thatχⁿ(QV)⊂ F_Vⁿ in [19]. Hence, we can define χⁿ :QV → F_Vⁿ, which inducesχ:QV→ FV naturally. Hence, we get a mapχ_A:K(Q)→ F_A.

Lusztig proved the following proposition.

Proposition 4.2 ([19]). χ_A:K(Q)→ F_A is an isomorphism ofA-algebras such that χ_A([Li]) =fi and the following diagram is commutative

K(Q) ^λA //

χA

fA.

$A

||F_A

4.2. Hall algebra approaches toif andⁱf . Letibe a sink ofQ. In Section 3.2,

iE_VandjV:iE_V→EV are defined for anyI-gradedK-vector spaceV. Similarly,

iE^F_Vⁿis defined as the set ofFn-fixed points iniE_Vand we havejV:iE^F_Vⁿ →E_V^Fn. Lusztig also defined iFⁿ_V as the set of all G^F_Vⁿ-invariant ¯Ql-functions oniE^F_Vⁿ. Similarly to the case in Section 4.1, the Hall algebra is denoted by_iFⁿ =L

ν∈NI iFⁿ_V, the composition subalgebra is denoted by_iFⁿ=L

ν∈NI iFⁿ_V and the generic form is denoted by_iF :=L

ν∈NI iF_V.

Naturally, we have two mapsj_V^∗ :FV→iF_VandjV! :iF_V→ FV. Considering all dimension vectors, we havej!:iF → F andj^∗:F →iF.

Proposition 4.3 ([19]). We have the following commutative diagram

if //

∼= i$

f

∼= $

iπ //_if

∼= i$

iF ^j! //F ^j

∗ //_iF, where_i$ is the isomorphism induced by $.

Next, we shall prove Proposition 3.2.

For any L ∈ DGV(_iE_V), there is also a function χⁿ_L : _iE^F_Vⁿ →Q¯l. Hence, we have the following map

iχⁿ:DG_V(iE_V) → iFⁿ_V L 7→ χⁿ_L.

The restriction of this map on the subcategory_iQ_V is also denoted by

iχⁿ:_iQ_V→iFⁿ_V. Proposition 4.4. It holds that iχⁿ(iQ_V)⊂iFⁿ_V.

Proof. By the properties of χ and _iχ (Theorem III.12.1(5) in [10]), we have the following commutative diagram

(9) QV

j_V^∗

//

χⁿ

iQ_V

iχⁿ

F_V^{n j}

∗ V //_iFⁿ_V.

By the commutative diagram (9), j_V^∗(F_Vⁿ) ⊂ iFⁿ_V and j^∗_V(QV) =iQ_V, we have

iχⁿ(iQ_V)⊂iFⁿ_V.

Hence, we can define iχⁿ : iQ_V → iFⁿ_V, which inducesiχ : iQ_V → iF_V and

iχ_A:K(iQ)→iF_A.

The commutative diagram (9) implies the following proposition.

Proposition 4.5. We have the following commutative diagram K(Q) ^j

∗ //

χA

K(iQ)

iχ_A

FA

j^∗ //_iF_A.

Proof of Proposition 3.2. First, we shall prove the following commutative diagram

if_A ^i$A //

iλ⁰_A

iF_A.

K(_iQ)

iχA

;;

Consider the following diagram fA

//

if_A

K(Q)

//K(iQ)

FA //_iF_A.

Since three squares and the triangle in the left are commutative, the triangle in the right is also commutative.

Proposition 4.3 implies that_i$_A:_if_A→iF_Ais isomorphic. Hence_iλ⁰_A:_if_A→ K(_iQ) is injective. The commutative diagram (4) in the definition ofiλ⁰_A implies that_iλ⁰_A:_if_A→K(_iQ) is surjection. Hence,_iλ⁰_A:_if_A→K(_iQ) is isomorphic.

In the proof, we get the following proposition.

Proposition 4.6. We have the following commutative diagram K(iQ) ^iλA //

iχA

if_A,

i$A

{{

iF_A

where all maps are isomorphisms of A-algebras andiλ_A is the inverse of _iλ⁰_A. Assume thatiis a source ofQ. The notations and results in this case are com- pletely similar to the case thati is a sink. We can defineⁱFⁿ_V,ⁱFⁿ =L

ν∈NI iFⁿ_V andⁱF =L

ν∈NI

iFV. We also have two mapsj_V^∗ :FV →ⁱFV andjV! :ⁱFV → FV. Considering all dimension vectors, we havej!:ⁱF → F andj^∗:F →ⁱF.

Proposition 4.7 ([19]). We have the following commutative diagram

if //

∼= ⁱ$

f

∼= $

iπ //ⁱf

∼= ⁱ$

iF ^j! //F ^j

∗ //ⁱF,

whereⁱ$ is the isomorphism induced by $.

We can also defineⁱχ:ⁱQV→ⁱFV andⁱχA:K(ⁱQ)→ⁱFA. Proposition 4.8. We have the following commutative diagram

K(Q) ^j

∗ //

χA

K(ⁱQ)

iχ_A

F_A ^j

∗ //ⁱF_A.

Proposition 4.9. We have the following commutative diagram

K(ⁱQ)

iλA //

iχA

if_A,

i$A

{{iFA

where all maps are isomorphisms of A-algebras andⁱλA is the inverse of ⁱλ⁰_A. 4.3. Hall algebra approach to Ti :if →ⁱf and the proof of Theorem 3.5.

Letibe a sink of a quiverQ= (I, H, s, t). Soiis a source ofQ⁰=σ_iQ= (I, H⁰, s, t).

For anyνandν⁰∈NIsuch thatν⁰=s_iν, and twoI-gradedK-vector spacesVand V⁰such that dimV=νand dimV⁰ =ν⁰, the following correspondence is considered in Section 3.3

iEV,Q ZVV⁰

oo α ^β //ⁱEV⁰,Q⁰ .

Similarly,Z_VV^Fn0 is defined as theFn-fixed points set inZVV⁰ and we have

iE_V,Q^Fn Z_VV^Fn0

oo α ^β //ⁱE_V^Fn0,Q⁰ .

Note thatαandβare principal bundles with fibersAut(V_i⁰) andAut(Vi) respective- ly. Hence, for anyf ∈iFⁿ_V, there exists a uniqueg∈ⁱFⁿ_V0such thatα^∗(f) =β^∗(g).

Define

ωi:iFⁿ_V → ⁱFⁿ_V0

f 7→ (pⁿ)^−s(V)2 g.

Lusztig proved that ωi(iFⁿ_V) ⊂ ⁱFⁿ_V0. Hence, we have ωi : iFⁿ_V → ⁱFⁿ_V0 and ωi:iF_V→ⁱFV⁰. Considering all dimension vectors, we haveωi:iF →ⁱF.

Lusztig proved the following theorem.

Theorem 4.10([19]). We have the following commutative diagram

if ^Ti //

i$

if

i$

iF ^ωi //ⁱF.

Proof of Proposition 3.4. By the properties ofiχⁿ andⁱχⁿ (Theorem III.12.1(4,5) in [10]), we have the following commutative diagram

DG_V(iEV) ^ω˜i //

iχⁿ

DG_V0(ⁱEV⁰)

iχⁿ

iFⁿ_V ^ω

n

i //ⁱFⁿ_V0.

Hence, we have

D_GV(_iE_V) ^ω˜i //

Q

n∈Z≥1iχⁿ

D_GV0(ⁱE_V⁰)

Q

n∈Z≥1 iχⁿ

Q

n∈Z≥1iFⁿ_V

Q

n∈Z≥1ωⁿ_i

//Qi

n∈Z≥1Fⁿ_V0. By Proposition 4.4,_iχⁿ(_iQ_V)⊂iFⁿ_V. Hence, we have

iQ_V ^ω˜i //

iχ

DG_V0(ⁱEV⁰)

Q

n∈Z≥1 iχⁿ

iF_V ^ωi //Q

n∈Z≥1

iFⁿ_V0. Hence,

( Y

n∈Z≥1

iχⁿ)◦ω˜_i(_iQ_V)⊂ω_i◦iχ(_iQ_V).

Sinceω_i(_iF_V)⊂ⁱF_V⁰,

( Y

n∈Z≥1

iχⁿ)◦ω˜_i(_iQ_V)⊂ⁱF_V⁰.

For any two semisimple complexesLandL⁰ inD_GV0(ⁱE_V⁰) such that ( Y

n∈Z≥1

iχⁿ)(L) = ( Y

n∈Z≥1

iχⁿ)(L⁰), Lis isomorphic toL⁰ by Theorem III.12.1(3) in [10]. Since (Q

n∈Z≥1

iχⁿ)(ⁱQ_V⁰) =

iFV⁰ and the objects in ˜ωi(iQ_V) are semisimple, ˜ωi(iQ_V)⊂ⁱQV⁰.

Proposition 4.11. We have the following commutative diagram

K(_iQ) ^ω˜i //

iχ

K(ⁱQ)

iχ

iF_A ^ωi //ⁱF_A.

Proof. By the properties of _iχⁿ andⁱχⁿ, we have the following commutative dia- gram

iQ_V ^ω˜i //

iχⁿ

iQV⁰

iχⁿ

iFⁿ_V ^ω

n

i //ⁱFⁿ_V0.

Hence, we get the commutative diagram in this proposition.

At last, Theorem 4.10 and Proposition 4.11 imply Theorem 3.5.