GEOMETRIC REALIZATIONS OF LUSZTIG’S SYMMETRIES OF SYMMETRIZABLE QUANTUM GROUPS

GEOMETRIC REALIZATIONS OF LUSZTIG’S SYMMETRIES OF SYMMETRIZABLE QUANTUM GROUPS

MINGHUI ZHAO

Abstract. LetUbe the quantum group andf be the Lusztig’s algebra as- sociated with a symmetrizable generalized Cartan matrix. The algebraf can be viewed as the positive part ofU. Lusztig introduced some symmetriesTi

onUfor alli∈I. SinceTi(f) is not contained inf, Lusztig considered two subalgebrasif andⁱf off for anyi∈I, whereif ={x∈f |Ti(x)∈f}and

if = {x ∈ f |T_i⁻¹(x)∈ f}. The restriction ofTi on if is also denoted by Ti :if →ⁱf. The geometric realization off and its canonical basis are in- troduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig’s symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations ofif,ⁱfandTi:if→ⁱf by using the language

’quiver with automorphism’ introduced by Lusztig.

Contents

1. Introduction 2

1.1. 2

1.2. 2

1.3. 3

2. Quantum groups and Lusztig’s symmetries 3

2.1. Quantum groups 3

2.2. Lusztig’s symmetries 4

3. Geometric realization off 5

3.1. Quivers with automorphisms 5

3.2. Geometric realization of Lusztig’s algebra ˆf corresponding to Q 5

3.3. Geometric realization off 7

4. Geometric realizations of subalgebrasif and ⁱf 10

4.1. The algebra_iˆf 10

4.2. Geometric realization ofiˆf 11

4.3. Geometric realization ofif 15

4.4. Geometric realization ofⁱf 20

5. Geometric realization ofTi:if →ⁱf 21

5.1. Geometric realization 21

Date: February 19, 2017.

2000Mathematics Subject Classification. 16G20, 17B37.

Key words and phrases. Quantum groups, Lusztig’s symmetries, Geometric realizations, Quiv- ers with automorphism.

This work was supported by the National Natural Science Foundation of China [grant numbers 11526037,11501032].

1

5.2. The proof of Proposition 5.3 25

References 27

1. Introduction

1.1. LetUbe the quantum group andf be the Lusztig’s algebra associated with a symmetrizable generalized Cartan matrix. There are two well-definedQ(v)-algebra embeddings ⁺ : f →U and ⁻ :f → U with imagesU⁺ and U⁻, where U⁺ and U⁻ are the positive part and the negative part ofUrespectively.

When the generalized Cartan matrix is symmetric, Lusztig introduced the geo- metric realization of f and the canonical basis of it in [9, 11]. In [14], Lusztig generalized the geometric realization tof associated with a symmetrizable general- ized Cartan matrix.

Let ˜Q= (Q, a) be a quiver with automorphism corresponding to f, whereQ= (I, H). LetV be anI-graded vector space with an isomorphisma:V→ Vsuch that dimV=ν ∈NI^a. Consider the varietyEV consisting of representations ofQ with dimension vectorνand a categoryQVof some semisimple complexes ([1, 2, 6]) onEV.

The isomorphism a : V → V induces a functor a^∗ : QV → QV. Lusztig defined a new category ˜QV consisting of objects (L, φ), where L is an object in QV andφ: a^∗L → Lis an isomorphism. Lusztig considered a submodule kν of K( ˜QV), whose definition is given by Lusztig and similar to that of a Grothendieck group ([14]). Considering all dimension vectors, he proved that k=L

ν∈NIkν is isomorphic tof.

Lusztig also introduced some symmetriesTi onU for alli∈I in [8, 10]. Since Ti(U⁺) is not contained inU⁺, Lusztig introduced two subalgebras if andⁱf off for anyi∈I, whereif ={x∈f |Ti(x⁺)∈U⁺}andⁱf ={x∈f |T_i⁻¹(x⁺)∈U⁺}.

Let Ti : if → ⁱf be the unique map satisfying Ti(x⁺) = Ti(x)⁺. For any i ∈ I,

if and ⁱf are the subalgebras of f generated by f(i, j;m) and f⁰(i, j;m) for all i 6= j ∈ I and −aij ≥ m ∈ N respectively. The definitions of f(i, j;m) and f⁰(i, j;m) will be given in Section 2.2. At the same time, Lusztig pointed that

if = {x ∈ f |ir(x) = 0} and ⁱf = {x ∈ f | ri(x) = 0}. The definition of ir will be given in Section 4.1 and the definition ofri is similar to that ofir. These descriptions ofif andⁱfare closely relevant to the geometric interpretation of them.

Associated to a finite dimensional hereditary algebra, Ringel introduced the Hall algebra and its composition subalgebra in [15], which gives a realization of U⁺. Via the Hall algebra approach, one can apply BGP-reflection functors to quantum groups to give precise constructions of Lusztig’s symmetries ([16, 13, 17, 18, 3, 19]).

1.2. Assume that the Cartan matrix is symmetric and letQ= (I, H) be a quiver corresponding to f. Let i ∈ I be a sink (resp. source) of Q. Similarly to the geometric realization of f, consider a subvariety iEV (resp. ⁱEV) of EV and a category iQ_V (resp. ⁱQV) of some semisimple complexes oniEV (resp. ⁱEV). In [20], it was showed that⊕ν∈NIK(iQ_V) (resp. ⊕ν∈NIK(ⁱQV)) realizesif (resp. ⁱf).

Leti∈I be a sink ofQandQ⁰ =σiQbe 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 →ⁱQV⁰,Q⁰ and studied the properties

of this equivalence in [5], building on the technical tools he established in [4]. In [20], his construction was generalized to all symmetric cases. It was proved that the map induced by ˜ωi realizes the Lusztig’s symmetry Ti : if →ⁱf by using the relations between ˜ω_i and the Hall algebra approach toT_i in [13].

In [12], Lusztig showed that Lusztig’s symmetries and canonical bases are com- patible. The main result in [20] gives a geometric interpretation of Lusztig’s result in [12].

1.3. In this paper, we shall generalize the construction in [20] and give geometric realizations of Lusztig’s symmetries of symmetrizable quantum groups.

Let ˜Q = (Q, a) be a quiver with automorphism. Fix i ∈ I = I^a and assume that iis a sink (resp. source) for anyi∈i. Similarly to the category ˜QV, we can define iQ˜_V (resp. ⁱQ˜V). Consider a submodule ik_ν (resp. ⁱkν) of K(iQ˜_V) (resp.

K(ⁱQ˜V)). We verify that⊕ν∈NI ik_ν (resp. ⊕ν∈NIikν) realizesif (resp. ⁱf) by using the result in [20] and the relation between ˜QV andQV.

Leti∈I=I^a such thatiis a sink, for anyi∈i. LetQ⁰=σiQbe the quiver by reversing the directions of all arrows in Qcontainingi∈i. So for anyi∈i,iis a source ofQ⁰.

Consider two I-graded vector spaces V and V⁰ with isomorphismsa :V →V anda:V⁰→V⁰ such that dimV⁰ =s_i(dimV). In this paper, it is proved that the equivalence ˜ω_i :_iQ_V,Q →ⁱQ_V⁰_,Q⁰ is compatible witha^∗. Hence we get a functor

˜

ω_i : _iQ˜_V,Q →ⁱQ˜V⁰,Q⁰ and a map ˜ω_i :_ik→ ⁱk. We also prove ˜ω_i : _ik→ⁱk is an isomorphism of algebras.

Assume that dimV=mγ_i+γ_j, whereγ_i=P

i∈iiandγ_j=P

i∈ji. We construc- t a series of distinguished triangles in DG_V,Q(EV,Q), which represent the constant sheaf1_iEV,Qin terms of some semisimple complexesI_p∈ DGV,Q(E_V,Q) geometrical- ly. Applying to the Grothendieck group,1_iEV,Qcorresponds to f(i, j;m). Assume that dimV⁰ =s_i(dimV) =m⁰γ_i+γ_j. Applying to the Grothendieck group,1iE_V0,Q0

corresponds tof⁰(i, j;m⁰) similarly. The properties of BGP-reflection functors im- ply ˜ω_i(v^−mN1_iEV,Q) =v^−m0N1iE_V0,Q0.Since_if (resp. ⁱf) is generated byf(i, j;m) (resp. f⁰(i, j;m)), we have the following commutative diagram

ik ^ω˜i //

ik

if_A ^Ti //ⁱf_A.

That is, ˜ωi gives a geometric realization of Lusztig’s symmetryTi for anyi∈I.

2. Quantum groups and Lusztig’s symmetries

2.1. Quantum groups. Fix a finite index setI with |I|=n. Let A= (aij)_i,j∈I be a symmetrizable generalized Cartan matrix and D = diag(εi |i ∈ I) be a diagonal matrix such that DA is symmetric. 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.

Leth=Q⊗_ZP^∨ and there exist a symmetric bilinear form (−,−) onh^∗such that (αi, αj) =εiaij for anyi, j∈I andλ(hi) = 2_(α^(αi,λ)

i,α_i) for anyλ∈h^∗andi∈I.

Fix an indeterminatev. Letvi=v^εi. For anyn∈Z, set [n]v_i= v_iⁿ−v_i⁻ⁿ

vi−v_i⁻¹ ∈Q(v).

Let [0]_vi! = 1 and [n]_vi! = [n]_vi[n−1]_vi· · ·[1]_vi for anyn∈Z>0.

Let U be the quantum group corresponding to (A,Π,Π^∨, P, P^∨) generated by the elementsE_i, F_i(i∈ I) and K_µ(µ ∈P^∨). LetU⁺ (resp. U⁻) be the positive (resp. negative) part ofUgenerated byE_i (resp. F_i) for alli∈I, and U⁰ be the Cartan part ofU generated byK_µ for allµ∈P^∨. The quantum groupU has the following triangular decomposition

U∼=U⁻⊗U⁰⊗U⁺.

Let f be the associative algebra defined by Lusztig in [14]. The algebra f is generated by θi(i∈I) subject to the quantum Serre relations. Let A=Z[v, v⁻¹] and f_A be the integral form off. There are two well-defined Q(v)-algebra homo- morphisms⁺:f →Uand⁻:f →UsatisfyingEi=θ⁺_i and Fi=θ_i⁻ for alli∈I.

The images of⁺ and⁻ areU⁺ andU⁻ respectively.

2.2. Lusztig’s symmetries. Corresponding toi∈I, Lusztig introduced the Lusztig’s symmetryT_i:U→U([8, 10, 14]). The formulas ofT_i on the generators are:

T_i(E_i) =−F_iK˜_i, T_i(F_i) =−K˜_−iE_i; Ti(Ej) = X

r+s=−a_ij

(−1)^rv^−r_i E_i^(s)EjE_i^(r) for anyi6=j∈I;

Ti(Fj) = X

r+s=−aij

(−1)^rv_i^rF_i^(r)FjF_i^(s) for anyi6=j∈I;

Ti(Kµ) =K_µ−αi(µ)h_i for anyµ∈P^∨,

whereE_i⁽ⁿ⁾=Eⁿ_i/[n]v_i!,F_i⁽ⁿ⁾=F_iⁿ/[n]v_i! and ˜K_±i=K_±εih_i.

Let if ={x∈ f |Ti(x⁺) ∈ U⁺} and ⁱf = {x ∈ f |T_i⁻¹(x⁺) ∈ U⁺}. Lusztig symmetryTi induces a unique mapTi:if →ⁱf such thatTi(x⁺) =Ti(x)⁺.

For anyi6=j∈I andm∈N, let f(i, j;m) = X

r+s=m

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

f⁰(i, j;m) = X

r+s=m

(−1)^rv_i^{−r(−aij−m+1)}θ^(s)_i θjθ_i^(r)∈f, whereθ_i⁽ⁿ⁾=θⁿ_i/[n]v_i!.

Proposition 2.1 ( Proposition 38.1.6 in [14]). For anyi∈I,

(1)_if (resp. ⁱf) is the subalgebra of f generated by f(i, j;m)(resp. f⁰(i, j;m)) for alli6=j ∈I and−aij ≥m∈N;

(2)T_i:_if →ⁱf is an isomorphism of algebras and T_i(f(i, j;m)) =f⁰(i, j;−aij−m) for alli6=j∈I and−aij ≥m∈N.

Lusztig also showed thatf has the following direct sum decompositions f =if ⊕θif =ⁱf⊕fθi.

Denote by_iπ:f →if and ⁱπ:f →ⁱf the natural projections.

3. Geometric realization of f

In this section, we shall review the geometric realization off given by Lusztig in [9, 11, 14, 13, 7].

3.1. Quivers with automorphisms. LetQ= (I, H, s, t) be a quiver, whereIis the set of vertices,H is the set of arrows, ands, t:H →Iare two maps such that an arrow ρ∈H starts at s(ρ) and terminates at t(ρ). From now on, assume that s(ρ)6=t(ρ) for anyρ∈H.

An admissible automorphism a of Q consists of permutations a : I → I and a:H →H satisfying the following conditions:

(1) for anyh∈H,s(a(h)) =a(s(h)) andt(a(h)) =a(t(h));

(2) there are no arrows between two vertices in the samea-orbit.

From now on, ˜Q = (Q, a) is called a quiver with automorphism. Assume that aⁿ= id for a given positive integer n.

LetI=I^a be the set of a-orbits inI. For anyi, j∈I, let a_ij=

−|{i→j| i∈i,j∈j}| − |{j→i|i∈i,j∈j}|, ifi6=j;

2|i|, ifi=j.

The matrixA= (aij)i,j∈I is a symmetrizable generalized Cartan matrix.

Proposition 3.1 ( Proposition 14.1.2 in [14]). For any symmetrizable general- ized Cartan matrix A, there exists a quiver with automorphism Q, such that the˜ generalized Cartan matrix correspongding toQ˜ isA.

3.2. Geometric realization of Lusztig’s algebraˆf corresponding to Q. Let pbe a prime andq=p^e. Denote byFq the finite field withqelements andK=Fq. LetQ= (I, H, s, t) be a quiver. Consider the categoryC⁰, whose objects are finite dimensional I-graded K-vector spaces V = L

i∈IVi, and morphisms are graded linear maps. For any ν ∈ NI, let C_ν⁰ be the subcategory of C⁰ consisting of the objectsV=L

i∈IVisuch that the dimension vector dimV=P

i∈I(dim_KVi)i=ν.

For anyV∈ C⁰, define

E_V=M

ρ∈H

Hom_K(V_s(ρ), V_t(ρ)).

The algebraic groupGV=Q

i∈IGL_K(Vi) acts onEV naturally.

For any ν = νii ∈ NI, ν is called discrete if there is no h ∈ H such that {s(h), t(h)} ∈ {i∈I|νi6= 0}. Fix a nonzero elementν∈NI. Let

Yν={y= (ν¹, ν², . . . , ν^k)|ν^l∈NIis discrete and

k

X

l=1

ν^l=ν}.

FixV∈ C_ν⁰. For any elementy∈Y_ν, a flag of type yinV is a sequence φ= (V=V^k⊃V^k−1⊃ · · · ⊃V⁰= 0),

where V^l ∈ C⁰ such that dimV^l/V^l−1 =ν^l. Let Fy be the variety of all flags of typeyin V. For anyx∈EV, a flagφis called x-stable ifxρ(V_s(ρ)^l )⊂V_t(ρ)^l for all l and allρ∈H. Let

F˜y={(x, φ)∈EV×Fy | φisx-stable}

andπy: ˜Fy→EV be the projection toEV.

Let ¯Qlbe thel-adic field andDG_V(EV) be the boundedGV-equivariant derived category of complexes ofl-adic sheaves onEV. For each y∈Yν,

Ly =πy!1F˜y[dy](dy

2 )∈ DG_V(EV)

is a semisimple complex, wheredy= dim ˜Fy, [−] is the shift functor and (−) is the Tate twist. LetPVbe the set of isomorphism classes of simple perverse sheavesLon EVsuch thatL[r](^r₂) appears as a direct summand ofLyfor somey∈Yνandr∈Z. LetQV be the full subcategory ofDGV(E_V) consisting of all complexes which are isomorphic to finite direct sums of complexes in the set{L[r](^r₂)| L ∈ PV, r∈Z}.

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

2)].

Then,K(QV) is a freeA-module. Define K(Q) = M

ν∈NI

K(QV).

For anyν, ν⁰, ν⁰⁰ ∈NIsuch thatν =ν⁰+ν⁰⁰, fix V∈ C_ν⁰, V⁰ ∈ C⁰_ν0, V⁰⁰ ∈ C_ν⁰00. Consider the following diagram

EV⁰×EV⁰⁰oo ^p1 E^{0 p2} //E^{00 p3} //EV, where

(1) E⁰⁰={(x,W)}, where x∈EV andW∈ C⁰_ν is anx-stable subspace ofV;

(2) E⁰ ={(x,W, R⁰⁰, R⁰)}, where (x,W)∈E⁰⁰, R⁰⁰:V⁰⁰'W andR⁰ :V⁰ ' V/W;

(3) p1(x,W, R⁰⁰, R⁰) = (x⁰, x⁰⁰), where x⁰ and x⁰⁰ are induced through the fol- lowing commutative diagrams

V⁰_s(ρ) ^x

0 ρ //

R⁰_s(ρ)

V⁰_t(ρ)

R⁰_t(ρ)

(V/W)s(ρ)

xρ //(V/W)t(ρ)

and

V_s(ρ)⁰⁰

x⁰⁰_ρ

//

R⁰⁰_s(ρ)

V⁰⁰_t(ρ)

R_t(ρ)⁰⁰

W_s(ρ) ^xρ //W_t(ρ); (4) p₂(x,W, R⁰⁰, R⁰) = (x,W);

(5) p₃(x,W) =x.

For any two complexesL⁰ ∈ DG_V0(EV⁰) andL⁰⁰∈ DG_V00(EV⁰⁰),L =L⁰∗ L⁰⁰ is defined as follows.

LetL1=L⁰⊗ L⁰⁰ andL2=p^∗₁L1. Sincep1 is smooth with connected fibres and p₂ is a G_V⁰ ×G_V⁰⁰-principal bundle, there exists a complex L3 on E⁰ such that p^∗₂(L3) =L2. The complexLis defined as (p₃)_!L3.

Lemma 3.2 ( Lemma 3.2 in [11], Lemma 9.2.3 in [14]). For any L⁰ ∈ QV⁰ and L⁰⁰∈ QV⁰⁰,L⁰∗ L⁰⁰∈ QV.

Hence, we get 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)[mν⁰ν⁰⁰](^mν₂^0ν00) andmν⁰ν⁰⁰=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).

LetAbe the generalized Cartan matrix corresponding toQand ˆfbe the Lusztig’s algebra corresponding toA. For any

y= (a₁i₁, a₂i₂, . . . , a_ki_k)∈Y_ν, letθy=θ^(a_i ¹⁾

1 θ_i^(a2)

2 · · ·θ^(a_i ^k)

k .

Theorem 3.3( Theorem 10.17 in [11], Theorem 13.2.11 in [14]). There is a unique A-algebra isomorphism

λˆA:K(Q)→ˆfA

such that ˆλ_A([Ly]) =θ_y for all y= (a₁i₁, a₂i₂, . . . , a_ki_k)∈Y_ν. Let ˆBν ={[L]| L ∈ PV} and ˆB=F

ν∈NIBˆν, which is anA-basis ofK(Q) and is called the canonical basis by Lusztig.

3.3. Geometric realization of f .

3.3.1. Let ˜Q= (Q, a) be a quiver with automorphism, whereQ= (I, H, s, t). Let C˜be the category ofV=L

i∈IVi∈ C⁰ with a linear mapa:V→Vsatisfying the following conditions:

(1) for anyi∈I,a(Vi) =V_a(i);

(2) for anyi∈Iandk∈Nsuch thata^k(i) =i, a^k|_Vi = id_Vi.

The morphisms in ˜Care the graded linear mapsf = (f_i)_i∈Isuch that the following diagram commutes

Vi a //

f_i

V_a(i)

f_a(i)

V_i ^a //V_a(i).

LetNI^a ={ν ∈NI|νi=ν_a(i)}. There is a bijection betweenNI andNI^a sending i to γi = P

i∈ii. From now on, NI^a is identified with NI. For any ν ∈NI^a, let C˜ν be the subcategory of ˜C consisting of the objectsV = L

i∈IV_i such that the dimension vector dimV=ν.

For any (V, a)∈C,˜ EV andGV are defined in Section 3.2. Leta:GV →GV

be the automorphism defined by

a(g)(v) =a(g(a⁻¹(v)))

for anyg∈GV andv∈V. Denote bya:EV→EV the automorphism such that the following diagram commutes for anyh∈H

V_s(h) ^xh //

a

V_t(h)

a

V_a(s(h))^a(x)a(h)//V_a(t(h)).

Sincea(gx) =a(g)a(x), we have a functora^∗:DGV(EV)→ DGV(EV).

Lemma 3.4 ([14]). It holds that a^∗(QV) =QV anda^∗(PV) =PV.

Lusztig introduced the following categories ˜QV and ˜PV in Section 11.1.2 of [14]. The objects in ˜QV are pairs (L, φ), where L ∈ QV and φ : a^∗L → L is an isomorphism such that

L=a^∗nL →a^∗(n−1)L →. . .→a^∗L → L

is the identity map of L. A morphism in HomQ˜V((L, φ),(L⁰, φ⁰)) is a morphism f ∈Hom_QV(L,L⁰) such that the following diagram commutes

a^∗L ^φ //

a^∗f

L

f

a^∗L^{0 φ}

0 //L⁰.

LetObe the subring of ¯Q^l consisting of allZ-linear combinations ofn-th roots of 1. In Section 11.1.5 of [14], Lusztig introduced two O-modules K( ˜Q_V) and K( ˜P_V), whose definitions are similar to that of a Grothendieck group. In this paper, K( ˜Q_V) and K( ˜P_V) are called the ”Grothendieck groups” of ˜Q_V and ˜P_V respectively. LetO⁰ =O[v, v⁻¹]. Sincea^∗ commutes with the shift functor and the Tate twist, we can define

v^±[L, φ] = [L[±1](±1

2), φ[±1](±1 2)].

Then K( ˜Q_V) has a natural O⁰-module structure. Note that K( ˜Q_V) = O⁰ ⊗_O K( ˜P_V). Define

K( ˜Q) = M

ν∈NI

K( ˜QV).

Forν, ν⁰, ν⁰⁰∈NI=NI^a such thatν =ν⁰+ν⁰⁰, fixV∈C˜ν,V⁰∈C˜ν⁰,V⁰⁰∈C˜ν⁰⁰. Lusztig proved the following lemma in Section 12.1.5 of [14].

Lemma 3.5 ([14]). The induction functor

∗:QV⁰× QV⁰⁰→ QV

satisfies that the following diagram commutes QV⁰× QV⁰⁰

∗ //

a^∗×a^∗

QV a^∗

QV⁰× QV⁰⁰

∗ //QV.

For any (L⁰, φ⁰)∈Q˜V⁰ and (L⁰⁰, φ⁰⁰)∈Q˜V⁰⁰, Lemma 3.5 implies that a^∗(L⁰∗ L⁰⁰) =a^∗L⁰∗a^∗L⁰⁰.

Hence there is a functor

∗: ˜QV⁰ ×Q˜V⁰⁰ → Q˜V

((L⁰, φ⁰),(L⁰⁰, φ⁰⁰)) 7→ (L⁰∗ L⁰⁰, φ), where

φ=φ⁰∗φ⁰⁰:L⁰∗ L⁰⁰→a^∗L⁰∗a^∗L⁰⁰=a^∗(L⁰∗ L⁰⁰).

This functor induces an associativeO⁰-bilinear multiplication

~:K( ˜QV⁰)×K( ˜QV⁰⁰) → K( ˜QV) ([L⁰, φ⁰], [L⁰⁰, φ⁰⁰]) 7→ [L, φ],

where (L, φ) = (L⁰, φ⁰)~(L⁰⁰, φ⁰⁰) = ((L⁰, φ⁰)∗(L⁰⁰, φ⁰⁰))[m_ν⁰_ν⁰⁰](^mν₂^0ν00). ThenK( ˜Q) becomes an associativeO⁰-algebra.

3.3.2. Fix a nonzero elementν∈NI^a. Let

Y_ν^a={y= (ν¹, ν², . . . , ν^k)∈Yν |ν^l∈NI^a}.

FixV∈C˜ν. For any elementy∈Y_ν^a, the automorphisma:Fy →Fy is defined as a(φ) = (V=a(V^k)⊃a(V^k−1)⊃ · · · ⊃a(V⁰) = 0)

for any

φ= (V=V^k⊃V^k−1⊃ · · · ⊃V⁰= 0)∈Fy.

There also exists an automorphisma: ˜F_y→F˜_y, defined asa((x, φ)) = (a(x), a(φ)) for any (x, φ)∈F˜y.

The automorphisma : ˜F_y →F˜_y induces a natural isomorphism a^∗1F˜y

∼=1F˜y. Hence, there exists an isomorphism

φ0:a^∗Ly=a^∗πy!1F˜_y[dy](d_y

2 ) =πy!a^∗1F˜_y[dy](d_y

2 )∼=πy!1F˜_y[dy](d_y

2 ) =Ly. That is, (Ly, φ0) is an object in ˜QV.

Letkν be theA-submodule of K( ˜QV) spanned by (Ly, φ0) for ally∈Y_ν^a. Let k=L

ν∈NIk_ν. Lusztig proved thatk is also a subalgebra ofK( ˜Q) ( Section 13.2 in [14]).

Let i∈ I and γi =P

i∈ii. Define1i = [1,id]∈ K( ˜QV), where V ∈ C˜γ_i. Let Abe the generalized Cartan matrix corresponding to (Q, a) andf be the Lusztig’s algebra corresponding toA.

Theorem 3.6( Theorem 13.2.11 in [14]). There is a uniqueA-algebra isomorphism λ_A:k→f_A

such that λ_A(1_i) =θ_i for alli∈I.

On the canonical basis off, Lusztig gave the following theorem.

Theorem 3.7 ( Proposition 12.5.2 and 12.6.3 in [14]). (1) For anyL ∈ P_V such that a^∗L ∼=L, there exists an isomorphism φ:a^∗L ∼=L such that(L, φ)∈P˜_V and (D(L), D(φ)⁻¹) is isomorphic to (L, φ) as objects of P˜_V, where D is the Verdier duality. Moreover, φis unique, ifnis odd, and unique up to multiplication by±1, if nis even.

(2)k_ν is generated by [L, φ]in (1) as anA-submodule of K( ˜Q_V).

Ifnis odd, letBν be the subset ofkν consisting of all elements [L, φ] in Theorem 3.7. If n is even, let Bν be the subset of kν consisting of all elements ±[L, φ] in Theorem 3.7. The set Bν is called a signed basis ofkν by Lusztig. Lusztig gave a non-geometric way to choose a subsetBν ofBν such thatBν =Bν∪ −Bν andBν

is an A-basis ofkν ( Section 14.4.2 in [14]). The set B =tν∈NIBν is called the canonical basis ofk.

At last, let us recall the relation between ˆf and f. Define ˜δ : k → K(Q) by δ([L, φ]) = [L] for any [L, φ]˜ ∈ B. It is clear that this is an injection and induces an embeddingδ:f →ˆf. Note thatδ(B) = ˆB^a.

4. Geometric realizations of subalgebras ⁱf andⁱf

4.1. The algebra iˆf . LetQ= (I, H, s, t) be a quiver and ˆf be the corresponding Lusztig’s algebra. Let _iˆf be the subalgebra of ˆf generated by f(i,j;m) for all i6=j∈Iand integerm. Letibe a subset ofIand define

iˆf =\

i∈i iˆf, which is also a subalgebra of ˆf.

For anyi∈I, there exists a unique linear mapir: ˆf →ˆf such that ir(1) = 0,

ir(θ_j) = δ_ij for all j ∈ I and _ir(xy) = _ir(x)y+v^(ν,αi)x_ir(y) for all homogeneous x∈ˆf_ν and y. Denote by (−,−) the non-degenerate symmetric bilinear form on ˆf introduced by Lusztig.

Proposition 4.1 ( Proposition 38.1.6 in [14]). It holds thatiˆf ={x∈ˆf |ir(x) = 0}.

As a corollary of Proposition 4.1, we have

Corollary 4.2. It holds that _iˆf ={x∈ˆf |ir(x) = 0 for any i∈i}.

Proposition 4.3. The algebraˆf has the following decomposition

ˆf =iˆf⊕X

i∈i

θiˆf.

Proof. In the algebra ˆf, (θiy, x) = (θi, θi)(y,ir(x)). Hence the decomposition ˆfν =

iˆf_ν⊕θiˆfν−i is an orthogonal decomposition andθiˆfν−i=iˆf^⊥_ν.

For the proof of this proposition, it is sufficient to show that iˆf^⊥_ν = P

i∈iiˆf^⊥_ν. That is (T

i∈iiˆf_ν)^⊥ =P

i∈iiˆf^⊥_ν. It is clear that (T

i∈iiˆf_ν)^⊥ ⊃ P

i∈iiˆf^⊥_ν. On the

other hand, x∈(P

i∈iiˆf^⊥_ν)^⊥ impliesx∈T

i∈iiˆf_ν. Hence T

i∈iiˆf_ν ⊃(P

i∈iiˆf^⊥_ν)^⊥. That is (T

i∈iiˆf_ν)^⊥⊂P

i∈iiˆf^⊥_ν.

Denoted byiπ: ˆf →iˆf the canonical projection.

4.2. Geometric realization of iˆf .

4.2.1. Letibe a subset ofIsatisfying the following conditions: (1) for anyi∈i,i is a sink; (2) for anyi,j∈i, there are no arrows between them.

For anyV∈ C_ν⁰, consider a subvariety_iE_V ofE_V

iE_V ={x∈EV | M

h∈H,t(h)=i

xh: M

h∈H,t(h)=i

Vs(h)→Vi is surjective for anyi∈i}.

Denote by_ij_V:_iE_V →E_V the canonical embedding.

For anyy∈Y_ν, let

iF˜y ={(x, φ)∈iE_V×Fy |φisx-stable}

andiπ_y:iF˜_y →iE_V be the projection toiE_V. For anyy∈Yν, iLy =iπ_y!1

iF˜_y[dy](^d₂^y)∈ DG_V(iEV) is a semisimple complex.

Let _iP_V be the set of isomorphism classes of simple perverse sheaves L on _iE_V such thatL[r](^r₂) appears as a direct summand ofiLy for somey∈Yν andr∈Z. LetiQ_Vbe the full subcategory ofDG_V(iEV) consisting of all complexes which are isomorphic to finite direct sums of complexes in the set{L[r](^r₂)| L ∈iP_V, r∈Z}.

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

2)].

Then,K(_iQV) is a freeA-module. Define K(iQ) = M

ν∈NI

K(iQ_V).

The canonical embeddingij_V :iE_V→EV induces a functor

ij^∗_V:DG_V(EV)→ DG_V(_iEV).

Lemma 4.4. It holds that ij^∗_V(QV) =iQ_V.

Proof. For anyy∈Yν, we have the following fiber product

iF˜y

i˜j_V //

iπ_y

F˜y π_y

iEV

ij_V //EV. Hence we have

ij^∗_VLy = ij^∗_Vπy!1F˜_y[dy](dy

2 )

= _iπ_y!i˜j^∗_V1F˜_y[d_y](d_y 2 )

= _iπ_y!1

iF˜_y[d_y](dy

2 ) =_iL_y.

That isij^∗_V(QV) =iQ_V.

The restriction of ij^∗_V : DG_V(EV) → DG_V(iEV) on QV is also denoted by

ij^∗_V:QV→iQ_V. Considering all dimension vectors, we haveij^∗:K(Q)→K(iQ).

Proposition 4.5. There exists an isomorphism of vector spacesiλˆ_A:K(iQ)→iˆf_A such that the following diagram commutes

K(Q) ^ij

∗ //

ˆλA

K(iQ)

iλˆ_A

ˆf_A ^iπA //_iˆf_A. Proof. Consider the following set of surjections:

{iπ_A: ˆf_A→iˆf_A |i∈i}.

It is clear that the push out of this set isiπ_A: ˆf_A→iˆf_A. For anyi∈i, there exists a canonical open embedding

ij_V:_iE_V →E_V. SinceiE_V =T

i∈iiE_V, we have the following commutative diagram

iE_V //

iE_V

ij_V

jE_V ^jjV //E_V

for anyi,j∈i. Hence we have the following commutative diagram K(QV)

jj^∗_V

ij^∗_V

//K(_iQV)

K(_jQV) //K(_iQV).

For any simple perverse sheafLsuch that ij^∗_VL= 0, we have supp(L)⊂E_V−_iE_V=E_V−\

i∈i

iE_V=[

i∈i

(E_V−_iE_V).

Hence supp(L)⊂EV−iE_Vfor somei∈i(Section 9.3.4 in [14]). Soij^∗_VL= 0. By the definition of push out, the push out of the following set

{ij^∗:K(Q)→K(iQ)|i∈i}

is_ij^∗:K(Q)→K(_iQ).

In [20], it was proved that the following diagram commutes K(Q) ^ij

∗ //

λˆA

K(iQ)

iˆλ_A

ˆf_A ^iπA //_iˆf_A

for anyi∈i. Hence there exists an isomorphismiˆλ_A:K(iQ)→iˆf_Asuch that the following diagram commutes

K(Q) ^ij

∗ //

ˆλA

K(_iQ)

iλˆ_A

ˆf_A ^iπA //_iˆf_A.

4.2.2. For anyν, ν⁰, ν⁰⁰∈NIsuch thatν =ν⁰+ν⁰⁰, fixV∈ C_ν⁰,V⁰∈ C_ν⁰0,V⁰⁰∈ C_ν⁰00. Consider the following diagram

(1) _iE_V0×iE_V00 ij_V0×_ij_V00

iE⁰

j₁

p₁

oo ^p2 //_iE⁰⁰

j₂

p₃ //_iE_V

ij_V

EV⁰×EV⁰⁰ E^0p1oo ^p2 //E^{00 p3} //EV,

where

(1) iE⁰ =p⁻¹₁ (iE_V0×iE_V00);

(2) iE⁰⁰=p2(iE⁰);

(3) the restrictions ofp1,p2 andp3 are also denoted by p1, p2 and p3 respec- tively.

For any two complexesL⁰ ∈ DG_V0(iE_V0) andL⁰⁰∈ DG_V00(iE_V00),L=L⁰∗ L⁰⁰ is defined as follows.

LetL1=L⁰⊗ L⁰⁰ andL2=p^∗₁L1. Sincep1 is smooth with connected fibres and p2 is a GV⁰ ×GV⁰⁰-principal bundle, there exists a complexL3 on iE⁰ such that p^∗₂(L3) =L2. Lis defined as (p3)!L3. Note that,Lis not a semisimple complex in general.

The canonical embeddingij_V :iE_V→EV also induces a functor

ij_V!:DG_V(iEV)→ DG_V(EV).

Lemma 4.6. It holds that _ij_V!(L⁰∗ L⁰⁰) =_ij_V0!(L⁰)∗ij_V00!(L⁰⁰).

Proof. Let ˆL⁰=_ij_V0!L⁰ and ˆL⁰⁰=_ij_V00!L⁰⁰. Since

iE_V0×_iE_V00 ij_V0×ij_V00

iE⁰

j1

p₁

oo

E_V⁰×E_V⁰⁰oo ^p1 E⁰

is a fiber product, we have ˆL₂:=p^∗₁( ˆL⁰⊗Lˆ⁰⁰) =j_1!p^∗₁(L⁰⊗ L⁰⁰) =j_1!L₂.

There exists a complex ˆL3onE⁰such thatp^∗₂( ˆL3) = ˆL2. Sincep^∗₂are equivalences of categories, ˆL3=j2!L3.

At last, ij_V0!(L⁰)∗ij_V00!(L⁰⁰) = ˆL⁰ ∗Lˆ⁰⁰ = p3!Lˆ3 = p3!j2!L3 = ij_V!p3!L3 =

ij_V!(L⁰∗ L⁰⁰).

LetK(DG_V(iEV)) be the Grothendieck group ofDG_V(iEV) andK(DG_V(EV)) be the Grothendieck group of DG_V(EV). Since ij_V : iE_V → EV is an open embedding, the functor

ij_V! :D_GV(_iE_V)→ D_GV(E_V) induces a map

ij_V!:K(DG_V(iEV))→K(DG_V(EV)).

Lemma 4.7. It holds that ij_V!(K(iQV))∈K(QV).

Proof. Consider the following diagram K(iQ) ^ij! //L

νK(DG_V(EV)) ^ij

∗ //L

νK(DG_V(iEV))

K(Q) ^ij

∗ //

OO

K(iQ)

OO

iˆf_A

iˆλ⁻¹_A

OO //ˆf_A

λˆ⁻¹_A

OO

iπA //_iˆf_A.

iλˆ⁻¹_A

OO

Since the compositions K(iQ) ^ij! //L

νK(DG_V(EV)) ^ij

∗ //L

νK(DG_V(iEV)) and

iˆf_A //ˆfA ⁱ πA //_iˆf_A are identifies, the following diagram commutes (2) K(iQ) ^ij! //L

νK(DG_V(EV)) ^ij

∗ //L

νK(DG_V(iEV))

K(iQ)

OO

iˆf_A

iˆλ⁻¹_A

OO //ˆfA ⁱ

πA //_iˆf_A.

iλˆ⁻¹_A

OO

For any homogeneous x ∈iˆf_A, choose L ∈ iQ_V such that [L] =iˆλ⁻¹_A (x). Let L1=ij_!L. It is clear that supp(L1)∈iEV.

The subalgebra_iˆf of ˆf is generated byf(i,j;m) for alli∈i,j6∈iandm≤ −aij. Let ν^(m) = mi+j ∈ NI. Fix an object V^(m) ∈ C⁰ such that dimV^(m) = ν^(m). Denote by1_iE

V(m) ∈ DG_V(m)(_iE_V(m)) the constant sheaf on_iE_V(m). Define E^(m)=j_V(m)!(v^−mN1_iE

V(m))∈ DG

V(m)(E_V(m)).

In [20], it was proved that [E^(m)] = ˆλ⁻¹_A (f(i,j;m)). Since supp(E^(m)) ∈ _iE_V(m), there existsL2such that [L2] = ˆλ⁻¹_A (x) and supp(L2)∈iE_V.