Towards a functor between affine and finite Hecke categories in type - - PowerPoint PPT Presentation

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Towards a functor between affine and finite Hecke categories in type A

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov

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Braid groups

Bn ≃ π1(Confn(C), ζ)

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Braid groups

Bn ≃ π1(Confn(C), ζ) Baff

n

≃ π1(Confn(C∗), ζ)

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 2 / 19

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Braid groups

Bn ≃ π1(Confn(C), ζ) Baff

n

≃ π1(Confn(C∗), ζ) C∗ ֒ → C

• Baff

n

→ Bn

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Finite Hecke algebra

W = Sn – symmetric group. I = {(1 2), (2 3), ..., (n − 1 n)} ⊂ Sn. si = (i i + 1).

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Finite Hecke algebra

W = Sn – symmetric group. I = {(1 2), (2 3), ..., (n − 1 n)} ⊂ Sn. si = (i i + 1). H(W ) = Hn – unital algebra over Z[v, v−1].

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Finite Hecke algebra

W = Sn – symmetric group. I = {(1 2), (2 3), ..., (n − 1 n)} ⊂ Sn. si = (i i + 1). H(W ) = Hn – unital algebra over Z[v, v−1]. Generators: {ts, s ∈ I}; ti := tsi. Relations:

• 1. titi+1ti = ti+1titi+1.
• 2. titj = tjti, |i − j| > 1.
• 3. t2

i = 1 + (v−1 − v)ti.

Hn has a basis {tw, w ∈ W }, defined by tw = ts1 . . . tsk for a reduced expression w = s1 . . . sk.

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Extended affine Hecke algebra

(X∗, Φ, X∗, Φ∨) – root datum of GLn. X∗ = X∗ =: X ≃ Zn = spanZ{e1, . . . , en}, Φ∨ = Φ = {ei − ej}i=j ⊂ X. W acts on X and Φ permuting ei.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 4 / 19

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Extended affine Hecke algebra

(X∗, Φ, X∗, Φ∨) – root datum of GLn. X∗ = X∗ =: X ≃ Zn = spanZ{e1, . . . , en}, Φ∨ = Φ = {ei − ej}i=j ⊂ X. W acts on X and Φ permuting ei. ∆ = {ei − ei+1}n−1

i=1 – simple roots.

X+ = {(λ1, . . . , λn), λk ≥ λk+1 for all k} – dominant weights.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 4 / 19

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Extended affine Hecke algebra

(X∗, Φ, X∗, Φ∨) – root datum of GLn. X∗ = X∗ =: X ≃ Zn = spanZ{e1, . . . , en}, Φ∨ = Φ = {ei − ej}i=j ⊂ X. W acts on X and Φ permuting ei. ∆ = {ei − ei+1}n−1

i=1 – simple roots.

X+ = {(λ1, . . . , λn), λk ≥ λk+1 for all k} – dominant weights. ˜ Haff

n

– unital algebra over Z[v, v−1]. Generators: {ts, s ∈ I, θx, x ∈ X}; ti := tsi, θi := θei. Relations:

• 1. titi+1ti = ti+1titi+1.
• 2. titj = tjti, |i − j| > 1.
• 3. t2

i = 1 + (v−1 − v)ti.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 4 / 19

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Extended affine Hecke algebra

(X∗, Φ, X∗, Φ∨) – root datum of GLn. X∗ = X∗ =: X ≃ Zn = spanZ{e1, . . . , en}, Φ∨ = Φ = {ei − ej}i=j ⊂ X. W acts on X and Φ permuting ei. ∆ = {ei − ei+1}n−1

i=1 – simple roots.

X+ = {(λ1, . . . , λn), λk ≥ λk+1 for all k} – dominant weights. ˜ Haff

n

– unital algebra over Z[v, v−1]. Generators: {ts, s ∈ I, θx, x ∈ X}; ti := tsi, θi := θei. Relations:

• 1. titi+1ti = ti+1titi+1.
• 2. titj = tjti, |i − j| > 1.
• 3. t2

i = 1 + (v−1 − v)ti.

• 4. θxθy = θx+y.
• 5. tiθj = θjti if j = i, i + 1.
• 6. tiθiti = θi+1.
• 7. θ0 = 1.

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The homomorphism Π : ˜ Haff

n → Hn.

Definition

Π(ti) = ti, Π(θ1) = 1.

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The homomorphism Π : ˜ Haff

n → Hn.

Definition

Π(ti) = ti, Π(θ1) = 1. This defines Π uniquely. Π(θk) = tk−1tk−2 . . . t2t2

1t2...tk−2tk−1 =: JMk

are called (multiplicative) Jucys-Murphy elements.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 5 / 19

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The homomorphism Π : ˜ Haff

n → Hn.

Definition

Π(ti) = ti, Π(θ1) = 1. This defines Π uniquely. Π(θk) = tk−1tk−2 . . . t2t2

1t2...tk−2tk−1 =: JMk

are called (multiplicative) Jucys-Murphy elements. λ ∈ X W λ – W-orbit.

Theorem (Bernstein)

The center of ˜ Haff

n

is a free Z[v, v−1]-module with a basis given by elements {zλ, λ ∈ X+}, zλ := ∑

µ∈W λ

θµ.

Theorem (Dipper-James, Francis-Graham)

Set of symmetric polynomials in {JMi} is the center of Hn.

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• Categorification. Finite side

G = GLn(C), B – Borel subgroup, U ⊂ B – unipotent radical.

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• Categorification. Finite side

G = GLn(C), B – Borel subgroup, U ⊂ B – unipotent radical. B = G/B – flag variety, Y = G/U – base affine space.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 6 / 19

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• Categorification. Finite side

G = GLn(C), B – Borel subgroup, U ⊂ B – unipotent radical. B = G/B – flag variety, Y = G/U – base affine space. Y × Y is a T × T-torsor over B × B. ˆ Dfin := ˆ Db

c,G,mon(Y × Y ) – finite Hecke category.

ˆ Dbc,G,mon(Y × Y ) – completed monodromic (with unipotent monodromy) bounded G-equivariant derived category of constructible sheaves on Y × Y .

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 6 / 19

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• Categorification. Finite side

G = GLn(C), B – Borel subgroup, U ⊂ B – unipotent radical. B = G/B – flag variety, Y = G/U – base affine space. Y × Y is a T × T-torsor over B × B. ˆ Dfin := ˆ Db

c,G,mon(Y × Y ) – finite Hecke category.

ˆ Dbc,G,mon(Y × Y ) – completed monodromic (with unipotent monodromy) bounded G-equivariant derived category of constructible sheaves on Y × Y . G × Y G Y × Y

π(g,x)=g a(g,x)=(x,gx)

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• Categorification. Finite side

G = GLn(C), B – Borel subgroup, U ⊂ B – unipotent radical. B = G/B – flag variety, Y = G/U – base affine space. Y × Y is a T × T-torsor over B × B. ˆ Dfin := ˆ Db

c,G,mon(Y × Y ) – finite Hecke category.

ˆ Dbc,G,mon(Y × Y ) – completed monodromic (with unipotent monodromy) bounded G-equivariant derived category of constructible sheaves on Y × Y . G × Y G Y × Y

π(g,x)=g a(g,x)=(x,gx)

hc = Db

G(G) → Db G(Y × Y ), hc = a!π∗[dim Y ] – Harish-Chandra

functor. ˆ Db(CS) – completed derived category of character sheaves, hc( ˆ Db(CS)) ⊂ ˆ Dfin.

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• Categorification. Affine side

K = C((t)) ⊃ O = C[[t]].

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• Categorification. Affine side

K = C((t)) ⊃ O = C[[t]]. I ⊂ GLn(O) – Iwahori subgroup, I 0 – its pro-unipotent radical. Flaff = GLn(K)/I – affine flag variety, Fl

aff = GLn(K)/I 0.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 7 / 19

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• Categorification. Affine side

K = C((t)) ⊃ O = C[[t]]. I ⊂ GLn(O) – Iwahori subgroup, I 0 – its pro-unipotent radical. Flaff = GLn(K)/I – affine flag variety, Fl

aff = GLn(K)/I 0.

ˆ Daff := ˆ Db

c,I 0,mon(

Fl

aff) – affine Hecke category.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 7 / 19

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• Categorification. Affine side

K = C((t)) ⊃ O = C[[t]]. I ⊂ GLn(O) – Iwahori subgroup, I 0 – its pro-unipotent radical. Flaff = GLn(K)/I – affine flag variety, Fl

aff = GLn(K)/I 0.

ˆ Daff := ˆ Db

c,I 0,mon(

Fl

aff) – affine Hecke category.

Gr = G(K)/G(O) – affine Grassmannian. PG(O)(Gr) – category of G(O)-equivariant perverse sheaves on Gr.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 7 / 19

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• Categorification. Affine side

K = C((t)) ⊃ O = C[[t]]. I ⊂ GLn(O) – Iwahori subgroup, I 0 – its pro-unipotent radical. Flaff = GLn(K)/I – affine flag variety, Fl

aff = GLn(K)/I 0.

ˆ Daff := ˆ Db

c,I 0,mon(

Fl

aff) – affine Hecke category.

Gr = G(K)/G(O) – affine Grassmannian. PG(O)(Gr) – category of G(O)-equivariant perverse sheaves on Gr. Ψ : PG(O)(Gr) → ˆ Daff – Gaitsgory’s nearby cycles construction.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 7 / 19

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• Categorification. Affine side

K = C((t)) ⊃ O = C[[t]]. I ⊂ GLn(O) – Iwahori subgroup, I 0 – its pro-unipotent radical. Flaff = GLn(K)/I – affine flag variety, Fl

aff = GLn(K)/I 0.

ˆ Daff := ˆ Db

c,I 0,mon(

Fl

aff) – affine Hecke category.

Gr = G(K)/G(O) – affine Grassmannian. PG(O)(Gr) – category of G(O)-equivariant perverse sheaves on Gr. Ψ : PG(O)(Gr) → ˆ Daff – Gaitsgory’s nearby cycles construction.

Theorem (Lusztig, Ginzburg, Mirkovi´ c, Vilonen)

PG(O)(Gr) ≃ Rep(GLn).

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 7 / 19

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• Categorification. Affine side

K = C((t)) ⊃ O = C[[t]]. I ⊂ GLn(O) – Iwahori subgroup, I 0 – its pro-unipotent radical. Flaff = GLn(K)/I – affine flag variety, Fl

aff = GLn(K)/I 0.

ˆ Daff := ˆ Db

c,I 0,mon(

Fl

aff) – affine Hecke category.

Gr = G(K)/G(O) – affine Grassmannian. PG(O)(Gr) – category of G(O)-equivariant perverse sheaves on Gr. Ψ : PG(O)(Gr) → ˆ Daff – Gaitsgory’s nearby cycles construction.

Theorem (Lusztig, Ginzburg, Mirkovi´ c, Vilonen)

PG(O)(Gr) ≃ Rep(GLn). Vλ ∈ Rep(GLn) ∑

µ d(λ, µ)zµ ∈ ˜

Haff

n , d(λ, µ) – weight multiplicities.

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Coherent affine Hecke category

g – Lie algebra of G = GLn.

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Coherent affine Hecke category

g – Lie algebra of G = GLn. ˜ g = {(x, B) ∈ g × B : x ∈ Lie(B)} – Grothendieck-Springer variety. St = ˜ g ×g ˜ g – Steinberg variety. ˆ St – completion at the nilpotent cone N ⊂ g.

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Coherent affine Hecke category

g – Lie algebra of G = GLn. ˜ g = {(x, B) ∈ g × B : x ∈ Lie(B)} – Grothendieck-Springer variety. St = ˜ g ×g ˜ g – Steinberg variety. ˆ St – completion at the nilpotent cone N ⊂ g. CohG( ˆ St) – category of G-equivariant coherent sheaves.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 8 / 19

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Coherent affine Hecke category

g – Lie algebra of G = GLn. ˜ g = {(x, B) ∈ g × B : x ∈ Lie(B)} – Grothendieck-Springer variety. St = ˜ g ×g ˜ g – Steinberg variety. ˆ St – completion at the nilpotent cone N ⊂ g. CohG( ˆ St) – category of G-equivariant coherent sheaves.

Theorem (Bezrukavnikov)

There is a (monoidal) equivalence B : ˆ Daff ≃ Db(CohG( ˆ St)).

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Coherent affine Hecke category

g – Lie algebra of G = GLn. ˜ g = {(x, B) ∈ g × B : x ∈ Lie(B)} – Grothendieck-Springer variety. St = ˜ g ×g ˜ g – Steinberg variety. ˆ St – completion at the nilpotent cone N ⊂ g. CohG( ˆ St) – category of G-equivariant coherent sheaves.

Theorem (Bezrukavnikov)

There is a (monoidal) equivalence B : ˆ Daff ≃ Db(CohG( ˆ St)). ∆ : ˜ g → St – diagonal embedding, πB : ˜ g → B – projection.

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Coherent affine Hecke category

g – Lie algebra of G = GLn. ˜ g = {(x, B) ∈ g × B : x ∈ Lie(B)} – Grothendieck-Springer variety. St = ˜ g ×g ˜ g – Steinberg variety. ˆ St – completion at the nilpotent cone N ⊂ g. CohG( ˆ St) – category of G-equivariant coherent sheaves.

Theorem (Bezrukavnikov)

There is a (monoidal) equivalence B : ˆ Daff ≃ Db(CohG( ˆ St)). ∆ : ˜ g → St – diagonal embedding, πB : ˜ g → B – projection. λ ∈ X+, Vλ ∈ Rep(GLn), µ ∈ X Vλ ⊗ O˜

g, π∗ BOB(µ) ∈ CohG(g).

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Coherent affine Hecke category

g – Lie algebra of G = GLn. ˜ g = {(x, B) ∈ g × B : x ∈ Lie(B)} – Grothendieck-Springer variety. St = ˜ g ×g ˜ g – Steinberg variety. ˆ St – completion at the nilpotent cone N ⊂ g. CohG( ˆ St) – category of G-equivariant coherent sheaves.

Theorem (Bezrukavnikov)

There is a (monoidal) equivalence B : ˆ Daff ≃ Db(CohG( ˆ St)). ∆ : ˜ g → St – diagonal embedding, πB : ˜ g → B – projection. λ ∈ X+, Vλ ∈ Rep(GLn), µ ∈ X Vλ ⊗ O˜

g, π∗ BOB(µ) ∈ CohG(g).

∆∗(Vλ ⊗ O˜

g) sλ ∈ ˜

Haff

n

∆∗π∗

BOB(µ) θµ ∈ ˜

Haff

n .

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Categorification

Z(˜ Haff

n )

˜ Haff

n

Z(Hn) Hn

Π|Z Π

(1)

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Categorification

Z(˜ Haff

n )

˜ Haff

n

Z(Hn) Hn

Π|Z Π

(1) Rep(GLn) Db(CohG( ˆ St)) ˆ Db(CS) ˆ Dfin

B◦Ψ◦Sat hc

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Categorification

Z(˜ Haff

n )

˜ Haff

n

Z(Hn) Hn

Π|Z Π

(1) Rep(GLn) Db(CohG( ˆ St)) ˆ Db(CS) ˆ Dfin

B◦Ψ◦Sat hc

Conjecture

There are monoidal functors replacing the dashed arrows and categorifying (1).

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Main result

Rep(GLn) Db(CohG( ˆ St)) ˆ Db(CS) ˆ Dfin

B◦Ψ◦Sat hc

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Main result

Rep(GLn) Db(CohG( ˆ St)) ˆ Db(CS) ˆ Dfin

B◦Ψ◦Sat hc

Theorem

There are monoidal functors ̟Z, ̟, making the following diagram commutative and categorifying a part of diagram (1). Rep(GLn) DG

perf( ˆ

St) ˆ Db(CS) ˆ Dfin

(B◦Ψ◦Sat)′ ̟Z ̟ hc′

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Functors from Rep(GLn)

C – C-linear, symmetric monoidal pseudo-abelian category. X ∈ C, k ∈ Z+ Sk-action on X k. Partition λ of k SλX ∈ C – Schur functors.

Lemma

Let X ∈ Ob(C) be such that ∧nX is invertible and ∧n+1X = 0. Then there is a monoidal functor Rep(GLn) → C sending the standard n-dimensional representation of Rep(GLn) to X.

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Braided structure on Db

G(G)

G – any algebraic group. µ : G × G → G – multiplication map.

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Braided structure on Db

G(G)

G – any algebraic group. µ : G × G → G – multiplication map. A ∗ B = µ!(A ⊠ B) – monoidal structure on Db(G) (convolution).

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Braided structure on Db

G(G)

G – any algebraic group. µ : G × G → G – multiplication map. A ∗ B = µ!(A ⊠ B) – monoidal structure on Db(G) (convolution). ξ : G × G → G × G, (g, h) → (g, g−1hg) τ : G × G → G × G, (g, h) → (h, g). µ ◦ ξ = µ ◦ τ.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 12 / 19

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Braided structure on Db

G(G)

G – any algebraic group. µ : G × G → G – multiplication map. A ∗ B = µ!(A ⊠ B) – monoidal structure on Db(G) (convolution). ξ : G × G → G × G, (g, h) → (g, g−1hg) τ : G × G → G × G, (g, h) → (h, g). µ ◦ ξ = µ ◦ τ. A, B ∈ Db

G(G) A ⊠ B ≃ ξ! (A ⊠ B) ,

A ∗ B = µ!(A ⊠ B) ≃ µ!ξ! (A ⊠ B) = µ!τ! (A ⊠ B) ≃ B ∗ A – braided structure, βA,B : A ∗ B → B ∗ A.

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Braided structure on Db

G(G)

G – any algebraic group. µ : G × G → G – multiplication map. A ∗ B = µ!(A ⊠ B) – monoidal structure on Db(G) (convolution). ξ : G × G → G × G, (g, h) → (g, g−1hg) τ : G × G → G × G, (g, h) → (h, g). µ ◦ ξ = µ ◦ τ. A, B ∈ Db

G(G) A ⊠ B ≃ ξ! (A ⊠ B) ,

A ∗ B = µ!(A ⊠ B) ≃ µ!ξ! (A ⊠ B) = µ!τ! (A ⊠ B) ≃ B ∗ A – braided structure, βA,B : A ∗ B → B ∗ A.

Lemma

If centralizers of all points in G are connected, then Db

G(G) is symmetric:

βA,B ◦ βB,A = IdB∗A

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Braided structure on Db

G(G)

G – any algebraic group. µ : G × G → G – multiplication map. A ∗ B = µ!(A ⊠ B) – monoidal structure on Db(G) (convolution). ξ : G × G → G × G, (g, h) → (g, g−1hg) τ : G × G → G × G, (g, h) → (h, g). µ ◦ ξ = µ ◦ τ. A, B ∈ Db

G(G) A ⊠ B ≃ ξ! (A ⊠ B) ,

A ∗ B = µ!(A ⊠ B) ≃ µ!ξ! (A ⊠ B) = µ!τ! (A ⊠ B) ≃ B ∗ A – braided structure, βA,B : A ∗ B → B ∗ A.

Lemma

If centralizers of all points in G are connected, then Db

G(G) is symmetric:

βA,B ◦ βB,A = IdB∗A

Corollary

For G = GLn, Db

G(G) is a symmetric monoidal category.

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Pro-units

Example

G = Gm.

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Pro-units

Example

G = Gm. Local systems on Gm with unipotent monodromy ↔ finite-dimensional C[[t]]-modules, tN = 0, N >> 0.

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Pro-units

Example

G = Gm. Local systems on Gm with unipotent monodromy ↔ finite-dimensional C[[t]]-modules, tN = 0, N >> 0. · ∗ · ↔ · ⊗C[[t]] · There is no unit in the category of finite-dimensional unipotent local systems.

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Pro-units

Example

G = Gm. Local systems on Gm with unipotent monodromy ↔ finite-dimensional C[[t]]-modules, tN = 0, N >> 0. · ∗ · ↔ · ⊗C[[t]] · There is no unit in the category of finite-dimensional unipotent local systems. C[[t]] = lim

← C[[t]]/tN ˆ

δ – pro-unit in the monodromic category.

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Pro-units

Example

G = Gm. Local systems on Gm with unipotent monodromy ↔ finite-dimensional C[[t]]-modules, tN = 0, N >> 0. · ∗ · ↔ · ⊗C[[t]] · There is no unit in the category of finite-dimensional unipotent local systems. C[[t]] = lim

← C[[t]]/tN ˆ

δ – pro-unit in the monodromic category. G = GLn. ˆ ∆e ∈ ˆ Dfin – monoidal pro-unit. ˆ ∆e = hc(ˆ δ), ˆ δ ∈ ˆ Db(CS) – monoidal pro-unit.

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Sheaves on the unipotent cone

P ⊂ G – parabolic subgroup, UP – its unipotent radical.

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Sheaves on the unipotent cone

P ⊂ G – parabolic subgroup, UP – its unipotent radical. T ∗G/P ≃ ˜ NP = {(g, xP) ∈ G × G/P : g ∈ xUPx−1}. π : NP → G – projection. SprP := π∗C

NP[2 dim UP] – parabolic

Springer sheaf.

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Sheaves on the unipotent cone

P ⊂ G – parabolic subgroup, UP – its unipotent radical. T ∗G/P ≃ ˜ NP = {(g, xP) ∈ G × G/P : g ∈ xUPx−1}. π : NP → G – projection. SprP := π∗C

NP[2 dim UP] – parabolic

Springer sheaf. Nu ⊂ GLn – unipotent variety. GLn-orbits in Nu ↔ partitions λ of n, according to the Jordan decomposition.

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Sheaves on the unipotent cone

P ⊂ G – parabolic subgroup, UP – its unipotent radical. T ∗G/P ≃ ˜ NP = {(g, xP) ∈ G × G/P : g ∈ xUPx−1}. π : NP → G – projection. SprP := π∗C

NP[2 dim UP] – parabolic

Springer sheaf. Nu ⊂ GLn – unipotent variety. GLn-orbits in Nu ↔ partitions λ of n, according to the Jordan decomposition. ICλ – intersection cohomology sheaf of the orbit numbered by a partition λ.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 14 / 19

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Sheaves on the unipotent cone

P ⊂ G – parabolic subgroup, UP – its unipotent radical. T ∗G/P ≃ ˜ NP = {(g, xP) ∈ G × G/P : g ∈ xUPx−1}. π : NP → G – projection. SprP := π∗C

NP[2 dim UP] – parabolic

Springer sheaf. Nu ⊂ GLn – unipotent variety. GLn-orbits in Nu ↔ partitions λ of n, according to the Jordan decomposition. ICλ – intersection cohomology sheaf of the orbit numbered by a partition λ. P0 ⊂ GLn = GL(V ) – maximal parabolic subgroup fixing a line l ∈ P(V ). λk – “hook” partition (k, 1, . . . , 1), λn = (n).

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 14 / 19

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Sheaves on the unipotent cone

P ⊂ G – parabolic subgroup, UP – its unipotent radical. T ∗G/P ≃ ˜ NP = {(g, xP) ∈ G × G/P : g ∈ xUPx−1}. π : NP → G – projection. SprP := π∗C

NP[2 dim UP] – parabolic

Springer sheaf. Nu ⊂ GLn – unipotent variety. GLn-orbits in Nu ↔ partitions λ of n, according to the Jordan decomposition. ICλ – intersection cohomology sheaf of the orbit numbered by a partition λ. P0 ⊂ GLn = GL(V ) – maximal parabolic subgroup fixing a line l ∈ P(V ). λk – “hook” partition (k, 1, . . . , 1), λn = (n). SprP0 ≃ ICλ1 ⊕ ICλ2.

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Exterior powers of the parabolic Springer sheaf

Theorem (Bezrukavnikov, T.)

• 1. ∧kSprP0 ≃ (ICλk ⊕ ICλk+1), for 0 < k < n.
• 2. ∧nSprP0 ≃ ICλn
• 3. ICλn ∗ ˆ

δ is invertible under convolution.

• 4. ∧n+1SprP0 = 0.

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Exterior powers of the parabolic Springer sheaf

Theorem (Bezrukavnikov, T.)

• 1. ∧kSprP0 ≃ (ICλk ⊕ ICλk+1), for 0 < k < n.
• 2. ∧nSprP0 ≃ ICλn
• 3. ICλn ∗ ˆ

δ is invertible under convolution.

• 4. ∧n+1SprP0 = 0.

Corollary

There is a monoidal functor ̟Z : Rep(GLn) → ˆ Db(CS), sending a standard n-dimensional representation V to SprP0 ∗ ˆ δ, and a monoidal functor hc ◦ ̟Z : Rep(G) → ˆ Dfin, sending representations of GLn to central objects.

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Lie algebra version

g = Lie(GLn). FT : Db

Gm(g) → Db Gm(g) – Fourier transform.

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Lie algebra version

g = Lie(GLn). FT : Db

Gm(g) → Db Gm(g) – Fourier transform.

FT(A ∗ B) ≃ FT(A) ⊗ FT(B).

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Lie algebra version

g = Lie(GLn). FT : Db

Gm(g) → Db Gm(g) – Fourier transform.

FT(A ∗ B) ≃ FT(A) ⊗ FT(B). p := Lie(P), uP := Lie(UP) ˜ N g

P = {(g, xP) ∈ g × G/P : g ∈ xuPx−1},

˜ gP = {(g, xP) ∈ g × G/P : g ∈ xpx−1}, ˜ gP, ˜ N g

P – orthogonal subbundles in g × G/P.

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Lie algebra version

g = Lie(GLn). FT : Db

Gm(g) → Db Gm(g) – Fourier transform.

FT(A ∗ B) ≃ FT(A) ⊗ FT(B). p := Lie(P), uP := Lie(UP) ˜ N g

P = {(g, xP) ∈ g × G/P : g ∈ xuPx−1},

˜ gP = {(g, xP) ∈ g × G/P : g ∈ xpx−1}, ˜ gP, ˜ N g

P – orthogonal subbundles in g × G/P.

sprP = πg∗C

N g

P [2 dim UP], gsprP = πg∗C˜

gP[dim g].

FT(sprP) = gsprP.

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Lie algebra version

g = Lie(GLn). FT : Db

Gm(g) → Db Gm(g) – Fourier transform.

FT(A ∗ B) ≃ FT(A) ⊗ FT(B). p := Lie(P), uP := Lie(UP) ˜ N g

P = {(g, xP) ∈ g × G/P : g ∈ xuPx−1},

˜ gP = {(g, xP) ∈ g × G/P : g ∈ xpx−1}, ˜ gP, ˜ N g

P – orthogonal subbundles in g × G/P.

sprP = πg∗C

N g

P [2 dim UP], gsprP = πg∗C˜

gP[dim g].

FT(sprP) = gsprP. (gsprP0)g = H∗({lines in P(V ) preserved by g})[dim g].

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 16 / 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lie algebra version

g = Lie(GLn). FT : Db

Gm(g) → Db Gm(g) – Fourier transform.

FT(A ∗ B) ≃ FT(A) ⊗ FT(B). p := Lie(P), uP := Lie(UP) ˜ N g

P = {(g, xP) ∈ g × G/P : g ∈ xuPx−1},

˜ gP = {(g, xP) ∈ g × G/P : g ∈ xpx−1}, ˜ gP, ˜ N g

P – orthogonal subbundles in g × G/P.

sprP = πg∗C

N g

P [2 dim UP], gsprP = πg∗C˜

gP[dim g].

FT(sprP) = gsprP. (gsprP0)g = H∗({lines in P(V ) preserved by g})[dim g]. rk(gsprP0)x ≤ n ⇒ ∧n+1

⊗

(gsprP0) = 0.

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Lie algebra version

g = Lie(GLn). FT : Db

Gm(g) → Db Gm(g) – Fourier transform.

FT(A ∗ B) ≃ FT(A) ⊗ FT(B). p := Lie(P), uP := Lie(UP) ˜ N g

P = {(g, xP) ∈ g × G/P : g ∈ xuPx−1},

˜ gP = {(g, xP) ∈ g × G/P : g ∈ xpx−1}, ˜ gP, ˜ N g

P – orthogonal subbundles in g × G/P.

sprP = πg∗C

N g

P [2 dim UP], gsprP = πg∗C˜

gP[dim g].

FT(sprP) = gsprP. (gsprP0)g = H∗({lines in P(V ) preserved by g})[dim g]. rk(gsprP0)x ≤ n ⇒ ∧n+1

⊗

(gsprP0) = 0.

Corollary

∧n+1

∗

(sprP0) = 0.

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Additional data on Ψ

Recall Ψ ◦ Sat : Rep(GLn) → PG(O)(Gr) → ˆ Daff – Gaitsgory’s nearby cycles construction.

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Additional data on Ψ

Recall Ψ ◦ Sat : Rep(GLn) → PG(O)(Gr) → ˆ Daff – Gaitsgory’s nearby cycles construction. V ∈ Rep(G) mV ∈ End(Ψ ◦ Sat(V )) – monodromy endomorphism.

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Additional data on Ψ

Recall Ψ ◦ Sat : Rep(GLn) → PG(O)(Gr) → ˆ Daff – Gaitsgory’s nearby cycles construction. V ∈ Rep(G) mV ∈ End(Ψ ◦ Sat(V )) – monodromy endomorphism. Recall B : ˆ Daff ≃ Db(CohG( ˆ St)) – Bezrukavnikov’s equivalence of two geometric realizations of the affine Hecke category.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 17 / 19

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Additional data on Ψ

Recall Ψ ◦ Sat : Rep(GLn) → PG(O)(Gr) → ˆ Daff – Gaitsgory’s nearby cycles construction. V ∈ Rep(G) mV ∈ End(Ψ ◦ Sat(V )) – monodromy endomorphism. Recall B : ˆ Daff ≃ Db(CohG( ˆ St)) – Bezrukavnikov’s equivalence of two geometric realizations of the affine Hecke category. ∆∗(V ⊗ O˜

g) corresponds to Ψ ◦ Sat(V ).

∆∗(V ⊗ O˜

g) has a filtration by ∆∗π∗ BOB(λ) (weight filtration).

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Additional data on Ψ

Recall Ψ ◦ Sat : Rep(GLn) → PG(O)(Gr) → ˆ Daff – Gaitsgory’s nearby cycles construction. V ∈ Rep(G) mV ∈ End(Ψ ◦ Sat(V )) – monodromy endomorphism. Recall B : ˆ Daff ≃ Db(CohG( ˆ St)) – Bezrukavnikov’s equivalence of two geometric realizations of the affine Hecke category. ∆∗(V ⊗ O˜

g) corresponds to Ψ ◦ Sat(V ).

∆∗(V ⊗ O˜

g) has a filtration by ∆∗π∗ BOB(λ) (weight filtration).

∆∗π∗

BOB(λ) ∈ Db(CohG( ˆ

St)) correspond to Wakimoto sheaves Θλ ∈ ˆ Daff, categorifying θλ ∈ ˜ Haff

n .

Ψ ◦ Sat(V ) has a filtration by Θλ.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 17 / 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Additional data on Ψ

Recall Ψ ◦ Sat : Rep(GLn) → PG(O)(Gr) → ˆ Daff – Gaitsgory’s nearby cycles construction. V ∈ Rep(G) mV ∈ End(Ψ ◦ Sat(V )) – monodromy endomorphism. Recall B : ˆ Daff ≃ Db(CohG( ˆ St)) – Bezrukavnikov’s equivalence of two geometric realizations of the affine Hecke category. ∆∗(V ⊗ O˜

g) corresponds to Ψ ◦ Sat(V ).

∆∗(V ⊗ O˜

g) has a filtration by ∆∗π∗ BOB(λ) (weight filtration).

∆∗π∗

BOB(λ) ∈ Db(CohG( ˆ

St)) correspond to Wakimoto sheaves Θλ ∈ ˆ Daff, categorifying θλ ∈ ˜ Haff

n .

Ψ ◦ Sat(V ) has a filtration by Θλ. Central sheaves + monodromy endomorphism + filtration + compatibilities + “homological boundedness” functor Db(CohG( ˆ St)) → ˆ Daff.

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Additional data on Ψ

Recall Ψ ◦ Sat : Rep(GLn) → PG(O)(Gr) → ˆ Daff – Gaitsgory’s nearby cycles construction. V ∈ Rep(G) mV ∈ End(Ψ ◦ Sat(V )) – monodromy endomorphism. Recall B : ˆ Daff ≃ Db(CohG( ˆ St)) – Bezrukavnikov’s equivalence of two geometric realizations of the affine Hecke category. ∆∗(V ⊗ O˜

g) corresponds to Ψ ◦ Sat(V ).

∆∗(V ⊗ O˜

g) has a filtration by ∆∗π∗ BOB(λ) (weight filtration).

∆∗π∗

BOB(λ) ∈ Db(CohG( ˆ

St)) correspond to Wakimoto sheaves Θλ ∈ ˆ Daff, categorifying θλ ∈ ˜ Haff

n .

Ψ ◦ Sat(V ) has a filtration by Θλ. Central sheaves + monodromy endomorphism + filtration + compatibilities functor Db

perf( ˆ

St) → ˆ Daff.

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Extension to the functor from DG

perf: filtration

Recall a basis tw ∈ Hn, w ∈ W . ˆ ∆w ∈ ˆ Dfin categorify tw.

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Extension to the functor from DG

perf: filtration

Recall a basis tw ∈ Hn, w ∈ W . ˆ ∆w ∈ ˆ Dfin categorify tw. J ⊂ I W ′ ⊂ W P ⊂ G – parabolic subgroup. W J – set of minimal length representatives of W /W ′.

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Extension to the functor from DG

perf: filtration

Recall a basis tw ∈ Hn, w ∈ W . ˆ ∆w ∈ ˆ Dfin categorify tw. J ⊂ I W ′ ⊂ W P ⊂ G – parabolic subgroup. W J – set of minimal length representatives of W /W ′.

Lemma (Grojnowski, Lusztig)

hc(SprP ∗ ˆ δ) has a filtration by ˆ ∆w ∗ ˆ ∆w−1, w ∈ W J.

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Extension to the functor from DG

perf: filtration

Recall a basis tw ∈ Hn, w ∈ W . ˆ ∆w ∈ ˆ Dfin categorify tw. J ⊂ I W ′ ⊂ W P ⊂ G – parabolic subgroup. W J – set of minimal length representatives of W /W ′.

Lemma (Grojnowski, Lusztig)

hc(SprP ∗ ˆ δ) has a filtration by ˆ ∆w ∗ ˆ ∆w−1, w ∈ W J. J0 = I\{s1}. W J0 = {1, s1, s2s1, . . . , sn−1 . . . s1}.

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Extension to the functor from DG

perf: filtration

Recall a basis tw ∈ Hn, w ∈ W . ˆ ∆w ∈ ˆ Dfin categorify tw. J ⊂ I W ′ ⊂ W P ⊂ G – parabolic subgroup. W J – set of minimal length representatives of W /W ′.

Lemma (Grojnowski, Lusztig)

hc(SprP ∗ ˆ δ) has a filtration by ˆ ∆w ∗ ˆ ∆w−1, w ∈ W J. J0 = I\{s1}. W J0 = {1, s1, s2s1, . . . , sn−1 . . . s1}. hc(SprP0) has a filtration by { ˆ ∆e, ˆ ∆2

s1, ˆ

∆s2s1 ∗ ˆ ∆s1s2, . . . , ˆ ∆sn−1...s1 ∗ ˆ ∆s1...sn−1}.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 18 / 19

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Extension to the functor from DG

perf: filtration

Recall a basis tw ∈ Hn, w ∈ W . ˆ ∆w ∈ ˆ Dfin categorify tw. J ⊂ I W ′ ⊂ W P ⊂ G – parabolic subgroup. W J – set of minimal length representatives of W /W ′.

Lemma (Grojnowski, Lusztig)

hc(SprP ∗ ˆ δ) has a filtration by ˆ ∆w ∗ ˆ ∆w−1, w ∈ W J. J0 = I\{s1}. W J0 = {1, s1, s2s1, . . . , sn−1 . . . s1}. hc(SprP0) has a filtration by { ˆ ∆e, ˆ ∆2

s1, ˆ

∆s2s1 ∗ ˆ ∆s1s2, . . . , ˆ ∆sn−1...s1 ∗ ˆ ∆s1...sn−1}. Compare: V – standard representation of G = GL(V ), Π(sV ) = Π(∑ θi) = ∑ JMi = te + t2

s1 + · · · + tsn−1...s1ts1...sn−1.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 18 / 19

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Extension to the functor from DG

perf: monodromy

L := Gm × GLn−1 – Levi subgroup, corresponding to P0.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 19 / 19

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Extension to the functor from DG

perf: monodromy

L := Gm × GLn−1 – Levi subgroup, corresponding to P0. SprP0 ∗ ˆ δ = IndG

L ˆ

δL, IndG

L – parabolic induction functor.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 19 / 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Extension to the functor from DG

perf: monodromy

L := Gm × GLn−1 – Levi subgroup, corresponding to P0. SprP0 ∗ ˆ δ = IndG

L ˆ

δL, IndG

L – parabolic induction functor.

Monodromy along Gm ⊂ Gm × GLn−1 is an endomorphism of ˆ δL. Induces an endomorphism of SprP0 ∗ ˆ δ, which corresponds to mV under ̟.

Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 19 / 19