Singular Rouquier complexes

LetF be a complex of singular Soergel bimodules. Following the notation of [EW14] we indicate the homological degree on the left, that is:

F := [. . .→ⁱ⁻¹F →ⁱF →ⁱ⁺¹F →. . .].

We denote by{−}the homological shift, so thatⁱ(F{1}) =ⁱ⁺¹F.

LetK^b(SBim^I)be the bounded homotopy category of complexes ofI-singular Soergel bimodules.

We define^pK^≥0 :=^pK^b(SBim^I)^≥0to be the full subcategory ofK^b(SBim^I)with objects complexes inK^b(SBim^I)which are isomorphic to a complexF which satisfiesτ≤−i−1iF = 0 for alli∈Z.

Similarly, we define ^pK^≤0 := ^pK^b(SBim^I)^≤0 to be the full subcategory with objects complexes inK^b(SBim^I)which are isomorphic to a complexF which satisfiesⁱF =τ≤−iiF for alli∈Z. Let^pK⁰ =^pK^≥0∩^pK^≤0.

Fors∈S let Fs denote the complex

F_s= [0→B⁰_s−→^ds R[1]→0]

where d_s(f ⊗g) = f g.² Then tensoring with F_s on the left induces an equivalence on the categoryK^b(SBim^I). The inverse is given by tensoring on the left with the complex E_s= [0→R[−1] ^d

0s

−→B⁰_s→0]. Hered⁰_s(1) =c_s= ¹₂(α_s⊗1 + 1⊗α_s).

For anyx∈W^Iwe consider the complexF_s1. . . F_sk for any reduced expressions₁. . . s_k of x. As an object in K^b(SBim) it does not depend on the chosen reduced expression [Rou06, Proposition 9.2]. Hence, also (F_s1. . . F_sk)_I does not depend on the reduced ex-pression as an object inK^b(SBim^I).

We choose F_x^I

⊕

⊆(Fs1. . . Fsk)I to be the corresponding minimal complex (see [EW14,

§6.1]), so F_x^I ∼= (Fs1. . . Fs_k)I in K^b(SBim^I) and the complex F_x^I does not contain any contractible direct summand. We callF_x^I asingular Rouquier complex.

Observe that if F_x is the Rouquier complex for x ∈ K^b(SBim), i.e. is the minimal complex forFs1. . . Fsk, then F_x^I can also be obtained as the minimal complex of Fx,I :=

(F_x)_I inK^b(SBim^I).

2We use−⁰ to indicate where the object in homological degree0is placed.

Lemma 4.4.1. Let x∈W^I and s∈S.

i) If sxw_I> xw_I then F_sB_x^I ∈^pK^≥0. ii) If sxw_I< xw_I then F_sB_x^I ∼=B^I_x[−1].

Proof. i) From Theorem 4.1.5 we have ch(BsB_x^I) = H_sH^I_x =H^I_sx+P

z∈W I

z<xs

mzH^I_z with m_z ∈Z≥0. Hence

B_sB^I_x∼=B_xs^I ⊕ M

z∈W I

z<xs

(B^I_z)^⊕mz.

Then the complex

F_sB_x^I= [0→

0

B_sB_x^I →B_x^I[1]→0]

is manifestly in^pK^≥0.

ii) We have ch(B_sB_x^I) =H_sH^I_x =H_sH_xwI = (v+v⁻¹)H^I_x. Therefore B_sB_x^I ∼=B_x^I[1]⊕ B_x^I[−1] and

FsB_x^I = [0→

0

B_x^I[1]⊕B_x^I[−1]→B_x^I[1]→0].

Tensoring withFs induces an equivalence on the categoryK^b(SBim^I), and sinceB_x^Iis indecomposable also the complexF_sB_x^I must be indecomposable. Therefore the map B_x^I[1] → B_x^I[1] cannot be 0, otherwise

0

B^I_x[1] would be a non-trivial direct summand ofFsB_x^I. Since B_x^I[1]→B_x^I[1] is non zero, it is an isomorphism and B_x^I[1]→B^I_x[1]is a contractible direct summand that we can remove from the complex. In this way we obtainF_sB_x^I∼=B_x^I[−1]∈^pK^≥0.

Lemma 4.4.2. Let F ∈^pK^≥0. ThenF_sF ∈^pK^≥0.

Proof. We denote by ω≥k the truncation of complexes, that is ω≥kF = [0→^kF →^k+1F →. . .].

We have distinguished triangles

ω≥k+1F →ω≥kF →^kF{−k}−→^[1]

Fs(ω≥k+1F)→Fs(ω≥kF)→Fs(^kF{−k})−→^[1]

SinceFs(^kF{−k})∈^pK^≥0 by Lemma 4.4.1, the statement follows by induction onkusing the analogue of [EW14, Lemma 6.1].

Corollary 4.4.3. For anyx∈W^I we have F_x^I∈^pK^≥0.

Proof. Since F_x^I ∼= Fs1Fs2. . . Fs_k(RI) ∈ K^b(SBim^I) this follows by induction on `(x) directly from Lemma 4.4.1 and Lemma 4.4.2.

4.4.1 Singular Rouquier complexes are ∆-split

We can identifyR⊗_RR^I with the ring of regular functions onh×(h/WI). The inclusion R⊗_RR^I ,→R⊗_RR corresponds to the projection map π:h×h→h×(h/WI).

For a cosetp∈W/W_I letGr^I(p)be the image Gr(p) underπ, whereGr(p)⊆h×his the twisted graph defined in §3.1. Similarly, ifC ⊆W/WI let Gr^I(C) =S

p∈CGr^I(p).

Let B^I ∈ SBim^I. For C ⊆ W/WI let ΓCB = {b ∈ B | suppb ∈ Gr^I(C)}. The functorΓ_C extends to a functorΓ_C from K^b(SBim^I) to the homotopy category of graded (R, R^I)-bimodules, which we denote by K^b(R-Mod-R^I).

Let q : W → W/WI denote the projection. For y ∈ W/WI let us denote by y− the minimal element in the coset y. The bijection W^I ∼= W/W_I induces a Bruhat order on W/W_I, i.e. we say y≤z if and only if y−≤z−. The projectionq is a strict morphism of posets:

Lemma 4.4.4. Let w≥v in W. Then q(w)≥q(v).

Proof. This follows from [Dou90, Lemma 2.2].

For any B ∈SBim and anyC ⊆W/W_I we have by [Wil11, Prop 6.1.6]

(Γ_q⁻¹_(C)B)_I = Γ_C(B_I). (4.3) Note that q⁻¹(≥ y) = {x ∈ W | x ≥ y−}. If x ∈ W^I we write Γ^I_≥x for the functor Γ_{y∈W^I_|y≥x} on SBim^I, to differentiate it from the functor Γ≥x on SBim.

We choose an enumerationy₁, y₂, y₃, . . .of W/W_I refining the Bruhat order onW/W_I. For any cosety_i ∈W/W_I we choose an enumeration y_i,1, y_i,2, y_i,3. . . of the elements in y_i refining the Bruhat order. Let

z1 =y1,1, z2 =y1,2, . . . , z|W_I|=y1,|W_I|, z|W_I|+1 =y2,1, z|W_I|+2=y2,2. . . .

By virtue of Lemma 4.4.4, z1, z2, z3. . . is also an enumeration of W which refines the Bruhat order.

We denote byΓ^I_≥m the functor Γ_{yi:i≥m} onSBim^I and by Γ≥m the functor Γ_{zi:i≥m}

onSBim. For l≥k, let

Γ^I_≥k/≥lB := (Γ^I_≥kB)/(Γ^I_≥lB).

The functorΓ^I_≥k/≥lextends to a functorΓ^I_≥k/≥l:K^b(SBim^I)→ K^b(R-Mod-R^I). Similarly, we define the functorsΓ_≥k/≥l,Γ^I_≥x/≥y,Γ_≥x/≥y. They also extend to functors between the respective homotopy categories.

Fix y = y_m ∈ W/W_I and x ∈W^I. We have (y_m)− = z_k for some k and (y_m+1)− = z_k+|WI|. Then by the hin-und-her Lemma for singular Soergel bimodules [Wil11, Lemma 6.3.2] we have

Γ^I_≥y/>y(F_x^I)∼= Γ^I_≥y/>y(Fx,I)∼= Γ^I_≥m/≥m+1(Fx,I)∼= (Γ≥k/≥k+|W_I|Fx)I ∈ K^b(R-Mod-R^I) Assume x 6∈ ym. Then x = zj with j < k or j ≥ k+|W_I|. For any i such that 1≤i≤ |W_I|we have a distinguished triangle in K^b(R-Mod-R)

Γ≥k/≥k+i−1Fx →Γ≥k/≥k+iFx →Γ≥k+i−1/≥k+iFx

−→[1]

and the last term is0by [LW14, Prop 3.7]. It follows by induction thatΓ≥k/≥k+|W_I|Fx∼= 0, hence

Γ^I_≥y/>y(F_x^I)∼= Γ^I_≥y/>y(F_x,I)∼= (Γ≥k/≥k+|WI|Fx)_I ∼= 0∈ K^b(R-Mod-R^I).

Assume nowx=y−, so thatx=zk. Let Rx,I := (Rx)I. Since Γ_≥k/≥k+1F_x=R_x[−`(x)]

the same argument as above showsΓ_{≥k/≥k+|WI|}F_x∼=R_x[−`(x)], hence Γ<sup>I</sup><sub>≥x/>x</sub>(F<sub>x</sub><sup>I</sup>)∼= Γ<sup>I</sup><sub>≥x/>x</sub>(F<sub>x,I</sub>)∼= (Γ≥k/≥k+|W<sub>I</sub>|F<sub>x</sub>)<sub>I</sub> ∼=R<sub>x,I</sub>[−`(x)].

We obtain the singular version of [LW14, Prop 3.7]:

Lemma 4.4.5. Let x, y∈W^I. Then Γ^I_≥y,>y(F_x^I) =

(0 if y6=x, Rx,I[−`(x)] if y=x.

Remark 4.4.6. If we view F_s as a complex of graded left R-modules it splits, i.e. we have Fs ∼= R[−1] in K^b(R-mod). Similarly (Fs1Fs2. . . Fsk)I ∼= R[−k]in K^b(R-mod). For x∈ W^I, letF_x^I =R⊗_RF_x^I. It is a complex of graded real vector spaces. It follows that we have:

Hⁱ(F_x^I) = (

R[−`(x)] if i= 0, 0 if i6= 0.

4.4.2 Singular Rouquier complexes are linear

For us it is important to understand how singular Rouquier complexes look. The idea is to use the first differential in a singular Rouquier complex as a replacement for “weak Lefschetz” in the inductive proof of hard Lefschetz. More precisely, the first differential will have the role of the mapφin [EW14, Lemma 2.3]. For this, we first have to show that the first differential is a map of degree1between perverse singular Soergel bimodules.

Lemma 4.4.7. Let x ∈W^I. For i >0 if ⁱF_x^I contains a direct summand isomorphic to B_z^I[j], thenⁱ⁻¹F_x^I contains a direct summand isomorphic toB_z^I0[j⁰]with z⁰ > z andj⁰< j.

Proof. The proof is the same of [EW14, Lemma 6.11]. From Theorem 4.1.5 (and the definition of the map ch, cf. [Wil11, §6.3]) we have that for anyy, z ∈W^I the bimodule Γ^I_≥z/>z(B_y^I) is generated in degree< `(z) if y > z andΓ<sup>I</sup><sub>≥z/>z</sub>(B<sub>z</sub><sup>I</sup>)∼=R<sub>z,I</sub>[−`(z)].

The image ofB^I_z[j]inⁱ⁺¹F_x^I is contained inτ<−j(ⁱ⁺¹F_x^I) because of (4.1): in fact any non-zero homomorphism in degree0is an isomorphism and thus yields a contractible direct summand.

Applying Γ^I_≥z/>z to F_x^I the direct summand B_z^I[j] induces a summand R_z,I[j−`(z)].

This cannot be a direct summand inΓ^I_≥z/>z(τ<−ji+1F_x^I) and cannot survive in the coho-mology of the complex because of Lemma 4.4.5. ThusR_z,I[j−`(z)]must be the image of a direct summandR<sub>z,I</sub>[j−`(z)]inΓ_≥z/>z(τ>−j(ⁱ⁻¹F_x)).

This implies that there is a direct summandB_y^I[k]inⁱ⁻¹F_x withy > z andk < j.

Theorem 4.4.8. Let x∈W^I and F_x^I be a singular Rouquier complex. Then:

i) ⁰F_x^I=B_x^I.

ii) For i≥1, ⁱF_x^I =L

(B_z^I[i])^⊕mz,i with z < x, z∈W^I andm_z,i∈Z≥0. In particular,F_x^I ∈^pK⁰.

Proof. We apply the previous lemma. The same proof shows that since ⁻¹(F_x^I) = 0, the only direct summands occurring in⁰(F_x^I) isB_x^I. By induction we have ⁱF_x^I =τ≤−iF_x^I for anyi >0. Now ii) follows since we already know F_x^I ∈^pK^≥0 from Corollary 4.4.3.

Remark 4.4.9. We can define the character of a complexF ∈ K^b(SBim^I) as ch(F) =X

i∈Z

(−1)ⁱch(ⁱF).

Ifx∈W^I andx=s1s2. . . sk is a reduced expression we have ch(F_x^I) = ch((F_s1F_s2. . . F_sk)_I) =H_xH_w

I =:H^I_x.

An immediate consequence of Theorem 4.1.5 is that there is a non trivial morphism of degreei between B_x^I and B_y^I for x, y ∈W^I if and only ifi and `(x)−`(y) have the same parity. Because of Theorem 4.4.8 we can write

H^I_x =X

i≥0

(−1)ⁱch(ⁱF_x) =X

y≤x

g_y,xH^I_y

with gx,x(v) = 1 and gy,x(v) = P

i>0my,i(−v)ⁱ. The polynomials gy,x are the parabolic inverse Kazhdan-Lusztig polynomials. We obtain that for any y ≤ x the polynomial (−1)^`(y)−`(x)g_y,x has non-negative coefficients.

4.4.3 Singular Rouquier complexes are Hodge-Riemann

The complex F_x^I is a direct summand of (Fs1. . . Fsm)_I for a reduced expression x = s₁. . . s_m. Hence, for anyj,^jF_x^I is a direct summand of ^j(F_s1. . . F_sm)_I, that is

jF_x^I ⊆^⊕ M

x⁰∈π(x,j)

BS(x⁰)I[j]

whereπ(x, j)is the set of all subexpressions of xobtained by omittingjsimple reflections.

Fixλ= (λ_x⁰)_x⁰∈π(x,j) a tuple of strictly positive real numbers. We define a symmetric formh−,−i^λ on L

BS(x⁰)I by hb, b⁰i^λ = X

x⁰∈π(x,j)

λ_x⁰hb_x⁰, b⁰_x⁰i_BS(x0) for allb= (b_x⁰), b⁰ = (b⁰_x⁰)∈ M

x⁰∈π(x,j)

BS(x⁰)_I (4.4) whereh−,−i_BS(x⁰₎ is the intersection form on BS(x⁰)I =BS(x⁰) defined in §3.1.1.

We say that F_x^I satisfies the Hodge-Riemann bilinear relations if we can choose an embedding F_x^I

⊕

⊆ (Fs1. . . Fsm)I such that for all tuples λ as above, multiplication by ρ on the right on^jF_x[−j]satisfies the Hodge-Riemann bilinear relations with respect to the formh−,−i^λ.

Proposition 4.4.10. Assume HR(s, y) for all s∈ S and y ∈ W^I with y < x such that sywI> ywI. Then F_x^I satisfies the Hodge-Riemann bilinear relations.

Proof. Fix a reduced expression x =s₁. . . s_m and let s=s₁, y=s₂. . . s_m. By induction assumeF_y^I satisfies Hodge-Riemann so that we can find an appropriate embeddingF_y^I ⊆^⊕ (F_s2. . . F_sm)_I.

NowF_x^I is a direct summand of FsF_y^I, so we have an embedding

jF_x^I[−j]⊆^{⊕ j}(B_sF_y^I)[−j]⊕^j−1F_y^I[−j+ 1]⊆^⊕ M

y⁰∈π(y,j)

B_sBS(y⁰)_I⊕ M

y⁰⁰∈π(y,j−1)

BS(y⁰⁰)_I =

= M

x⁰∈π(x,j)

BS(x⁰).

We fix a tuple(µ_x⁰)_x⁰_∈π(x,j), or equivalently two tuples(λ_y⁰)_y⁰_∈π(y,j) and (σ_y⁰⁰)_y⁰⁰_∈π(y,j) of positive real numbers.

From Theorem 4.4.8, we know that ^jF_y^I[−j] = L

(B^I_z)^⊕mz with mz ∈ Z≥0. Let

jF_y^I[−j] =B^↑⊕B^↓ with B^↑ = M

szwI>zwI

(B_z^I)^⊕mz and B^↓ = M

szwI<zwI

(B_z^I)^⊕mz.

The decomposition is orthogonal with respect of the restriction of the form h−,−i^λ since Hom⁰(B^↑,DB^↓) = 0. Then also BsB^↑ and BsB^↓ are orthogonal with respect of the restriction of the form h−,−i^µ on B_s(^jF_y^I[−j]). In fact, for any b ∈ B^↑ and b⁰ ∈ B^↓ we have:

hc_idb, cidb⁰i^µ=∂s(hb, b⁰i^λ) = 0 (4.5) hc_sb, cidb⁰i^µ=hc_idb, csb⁰i^µ=hb, b⁰i^λ = 0 (4.6) hc_sb, c_sb⁰i^µ=α_shb, b⁰i^λ= 0. (4.7) The bimoduleB_sB^↑ is perverse whileB_sB^↓=B^↓[−1]⊕B^↓[1]. So we have a decompo-sition

jF_x^I[−j]⊆^⊕ BsB^↓⊕BsB^↑⊕^j−1F_y^I[−j+ 1]. (4.8) The inclusion is, by definition, an isometry. This decomposition is orthogonal with respect to the form h−,−i^µ. Moreover by (4.1) the image of ^jF_x^I[−j] → BsB^↓ is contained in B^↓[1].

Claim 4.4.11. The restriction of the Lefschetz form (−,−)^−k_ρ = h−,− ·ρ^ki^µ to (B^↓[1])^−k is zero.

Proof of the claim. Let B^I_z ⊆^⊕ B^↓ and let z = t₁. . . t_l be a reduced expression. Then z⁰ := t₂. . . t_l ∈ W^I and t₁z⁰w_I > z⁰w_I. Since B_z^I ⊆^⊕ B_t1B_z^I0, the hypothesis HR(t₁, z⁰) implies that multiplication byρ satisfies hard Lefschetz onB_z^I, hence onB^↓, i.e.

ρⁱ : (B^↓)⁻ⁱ −^∼→(B^↓)ⁱ for all i≥0.

By shifting we getρⁱ :B^↓[1]⁻ⁱ⁻¹ −→^∼ B^↓[1]ⁱ⁻¹. Let P_ρ⁻¹⁻ⁱ = Kerρⁱ⁺¹ ⊆B^↓[1]⁻ⁱ⁻¹ so that for anym≤0 we have

B^↓[1]^m= M

j≥max{^m+1₂ ,0}

P_ρ^m−2j·ρ^j.

Letx∈Pρ^m−2j,y∈P_ρ^m−2k for some j≥k≥0, then (xρ^j, yρ^k)^m_ρ =hx, yρ^j+k−mi^µ= 0 becausej+k−m≥2k−m andy∈Ker(ρ^2k−m).

So the image of ^jF_x^I[−j] in BsB^↓ does not contribute to the Lefschetz form. We can consider the projection onto the other two factors

i:^jF_x^I[−j]→BsB^↑⊕^j−1F_y^I[−j+ 1]

which is an isometry.

Furthermore, the mapiis injective, in fact^jF_x^I

⊕

⊆^j(FsF_y^I)is a split inclusion and since when we project toSBim^I/rad (SBim^I), the image is contained inB_sB^↑⊕^j−1F_y^I[−j+ 1], then alsoimust be a split injective morphism.

Using the fact that^jF_x^Iis stable under the Lefschetz operator and that Hodge-Riemann holds by induction for bothBsB^↑ and ^j−1F_y^I[−j+ 1], the thesis follows.

Im Dokument Hodge theoretic aspects of Soergel bimodules and representation theory (Seite 60-66)