Exotic t-structures for two-block Springer fibres Vinoth Nandakumar - - PowerPoint PPT Presentation

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Exotic t-structures for two-block Springer fibres

Vinoth Nandakumar

Massachusetts Institute of Technology

July 20, 2012

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 1 / 15

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Outline

1

Two-block Springer fibres

2

Affine tangles

3

The exotic t-structure

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 2 / 15

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Two-block Springer fibres

Two-block Springer fibres: the definitions

Fix m ∈ Z≥0, and let n ∈ Z≥0. Recall the following definitions: Definition Let G = SLm+2n(C), g = slm+2n(C), B ⊂ G be the Borel subgroup of upper triangular matrices; and G/B the flag variety: Bn = G/B = {0 ⊂ V1 ⊂ · · · ⊂ Vm+2n−1 ⊂ Vm+2n = Cm+2n | dim Vi = i}

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 3 / 15

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Two-block Springer fibres

Two-block Springer fibres: the definitions

Fix m ∈ Z≥0, and let n ∈ Z≥0. Recall the following definitions: Definition Let G = SLm+2n(C), g = slm+2n(C), B ⊂ G be the Borel subgroup of upper triangular matrices; and G/B the flag variety: Bn = G/B = {0 ⊂ V1 ⊂ · · · ⊂ Vm+2n−1 ⊂ Vm+2n = Cm+2n | dim Vi = i} Definition Let Nn = {x ∈ g | x nilpotent} be the nilpotent cone, Nn it’s Springer resolution (with the natural map πn : Nn → Nn):

• Nn := T ∗Bn = {(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n−1 ⊂ Vm+2n), x | x(Vi) ⊆ Vi−1}

Let zn ∈ Nn be the standard nilpotent with Jordan type (m + n, n), and let Bm+n,n = π−1

n (zn) be the corresponding Springer fiber.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 3 / 15

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Two-block Springer fibres

Two-block Springer fibres: transverse slices

Definition Define the Mirkovic-Vybornov transverse slice as follows: Sn = {zn +

• 1≤i≤m+n

aiem+n,i +

• 1≤j≤n

bjem+n,m+n+j +

• 1≤j≤n

cjem+2n,j +

• 1≤j≤m+2n

djem+2n,m+n+j} Let Un = π−1

n (Sn ∩ Nn) denote the resolution of the variety Sn ∩ Nn.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 4 / 15

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Two-block Springer fibres

Two-block Springer fibres: transverse slices

Definition Define the Mirkovic-Vybornov transverse slice as follows: Sn = {zn +

• 1≤i≤m+n

aiem+n,i +

• 1≤j≤n

bjem+n,m+n+j +

• 1≤j≤n

cjem+2n,j +

• 1≤j≤m+2n

djem+2n,m+n+j} Let Un = π−1

n (Sn ∩ Nn) denote the resolution of the variety Sn ∩ Nn.

Note that: Bm+n,n = {(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n−1 ⊂ Vm+2n) | znVi ⊆ Vi−1} Un = {(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n−1 ⊂ Vm+2n), x | x ∈ Sn, xVi ⊆ Vi−1} Dn := Db(CohBm+n,n(Un)), the bounded derived category of coherent sheaves on Un supported on Bm+n,n, will be our primary object of interest.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 4 / 15

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Two-block Springer fibres

Two-block Springer fibres: some identities

Define the partial flag variety Pk,n (and T ∗Pk,n) as follows: Pk,n = {(0 ⊂ V1 ⊂ · · · ⊂ Vk ⊂ · · · ⊂ Vm+2n = Cm+2n)} T ∗Pk,n ={(0 ⊂ V1 ⊂ · · · ⊂ Vk ⊂ · · · ⊂ Vm+2n = Cm+2n), x | x ∈ glm+2n, xVk+1 ⊂ Vk−1, xVi ⊂ Vi−1 for i = k, k + 1}

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 5 / 15

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Two-block Springer fibres

Two-block Springer fibres: some identities

Define the partial flag variety Pk,n (and T ∗Pk,n) as follows: Pk,n = {(0 ⊂ V1 ⊂ · · · ⊂ Vk ⊂ · · · ⊂ Vm+2n = Cm+2n)} T ∗Pk,n ={(0 ⊂ V1 ⊂ · · · ⊂ Vk ⊂ · · · ⊂ Vm+2n = Cm+2n), x | x ∈ glm+2n, xVk+1 ⊂ Vk−1, xVi ⊂ Vi−1 for i = k, k + 1} Proposition We have: Sn+1 ×glm+2n+2 T ∗Pk,n+1 ≃ Un.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 5 / 15

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Two-block Springer fibres

Two-block Springer fibres: some identities

Define the partial flag variety Pk,n (and T ∗Pk,n) as follows: Pk,n = {(0 ⊂ V1 ⊂ · · · ⊂ Vk ⊂ · · · ⊂ Vm+2n = Cm+2n)} T ∗Pk,n ={(0 ⊂ V1 ⊂ · · · ⊂ Vk ⊂ · · · ⊂ Vm+2n = Cm+2n), x | x ∈ glm+2n, xVk+1 ⊂ Vk−1, xVi ⊂ Vi−1 for i = k, k + 1} Proposition We have: Sn+1 ×glm+2n+2 T ∗Pk,n+1 ≃ Un. Proof (Sketch): Below, xVk+1 = Vk−1; ∃ a canonical isomorphism φx : xVm+2n+2 ≃ Vm+2n inducing φ(x) ∈ End(Cm+2n). Sn+1 ×glm+2n+2 T ∗Pk,n+1 ={(0 ⊂ V1 ⊂ · · · ⊂ Vk ⊂ · · · ⊂ Vm+2n+2) | x ∈ Sn+1, xVk+1 ⊆ Vk−1, xVi ⊂ Vi−1} Map this to {(0 ⊂ V1 ⊂ · · · Vk−1 ⊂ xVk+2 ⊂ · · · xVm+2n+2), φ(x)} ∈ Un.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 5 / 15

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Two-block Springer fibres

Two-block Springer fibres: some identities

Let Vm,n = C{ei, fi}1≤i≤m+2n; define z by zei = ei−1, zfi = fi−1. Let Wm,n = C{ek, fl}1≤k≤m+n,1≤l≤n, and P : Vm,n → Wm,n be the natural projection. Ym+2n = {(L1 ⊂ · · · ⊂ Lm+2n ⊂ Vm,n)| dimLi = i, zLi ⊂ Li−1}

• Um+2n = {(L1 ⊂ · · · ⊂ Lm+2n) ∈ Ym+2n|P(Lm+2n) = Wm,n}

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 6 / 15

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Two-block Springer fibres

Two-block Springer fibres: some identities

Let Vm,n = C{ei, fi}1≤i≤m+2n; define z by zei = ei−1, zfi = fi−1. Let Wm,n = C{ek, fl}1≤k≤m+n,1≤l≤n, and P : Vm,n → Wm,n be the natural projection. Ym+2n = {(L1 ⊂ · · · ⊂ Lm+2n ⊂ Vm,n)| dimLi = i, zLi ⊂ Li−1}

• Um+2n = {(L1 ⊂ · · · ⊂ Lm+2n) ∈ Ym+2n|P(Lm+2n) = Wm,n}

The categories Dn := Db(Coh(Ym+2n)) have been studied by Cautis and

• Kamnitzer. The below fact allows us to apply their results to study Dn.

Proposition There is a closed embedding Un → Um+2n (hence Un is locally closed in Ym+2n).

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Two-block Springer fibres

Two-block Springer fibres: some identities

Let Vm,n = C{ei, fi}1≤i≤m+2n; define z by zei = ei−1, zfi = fi−1. Let Wm,n = C{ek, fl}1≤k≤m+n,1≤l≤n, and P : Vm,n → Wm,n be the natural projection. Ym+2n = {(L1 ⊂ · · · ⊂ Lm+2n ⊂ Vm,n)| dimLi = i, zLi ⊂ Li−1}

• Um+2n = {(L1 ⊂ · · · ⊂ Lm+2n) ∈ Ym+2n|P(Lm+2n) = Wm,n}

The categories Dn := Db(Coh(Ym+2n)) have been studied by Cautis and

• Kamnitzer. The below fact allows us to apply their results to study Dn.

Proposition There is a closed embedding Un → Um+2n (hence Un is locally closed in Ym+2n). Proof (Sketch): Let S′

n = {zn + 1≤i≤m+2n(uiem+n,i + viem+2n,i)}. It

suffices to show that Um+2n ≃ S′

n ×glm+2n T ∗Bn, since

Un = Sn ×glm+2n T ∗Bn is closed in S′

n ×glm+2n T ∗Bn.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 6 / 15

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Affine tangles

Affine tangles: definitions

Definition If p ≡ q (mod 2), a (p, q) affine tangle is an embedding of p+q

2

arcs and a finite number of circles into the region {(x, y) ∈ C × R|1 ≤ |x| ≤ 2}, such that the end-points of the arcs are given {(1, 0), (ζp, 0), · · · , (ζp−1

p

, 0), (2, 0), (2ζq, 0), · · · , (2ζq−1

q

, 0)}; where ζk = e

2πi k . Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 7 / 15

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Affine tangles

Affine tangles: definitions

Definition If p ≡ q (mod 2), a (p, q) affine tangle is an embedding of p+q

2

arcs and a finite number of circles into the region {(x, y) ∈ C × R|1 ≤ |x| ≤ 2}, such that the end-points of the arcs are given {(1, 0), (ζp, 0), · · · , (ζp−1

p

, 0), (2, 0), (2ζq, 0), · · · , (2ζq−1

q

, 0)}; where ζk = e

2πi k .

Definition Let gi

n denote the (n − 2, n) tangle with an arc connecting (2ζi n, 0) to

(2ζi+1

n

, 0). Let f i

n denote the (n, n − 2) tangle with an arc connecting

(ζi

n, 0) and (ζi+1 n

, 0). Let ti

n(1) denote the (n, n) tangle with a strand connecting (ζi n, 0) to

(2ζi+1

n

, 0) passes beneath a strand connecting (ζi+1

n

, 0) to (2ζi

n, 0).

Let rn denote the (n, n) tangle connecting (ζj

n, 0) to (2ζj+1 n

, 0) for each 1 ≤ j ≤ n. Also let r′

n := r−1 n , ti n(2) := ti n(1)−1.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 7 / 15

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Affine tangles

Affine tangles: generators and relations

Proposition Any affine tangle is a composition of the above tangles, and any relation between those generators is a composition of the following relations: f i

n ◦ gi+1 n

= f i+1

n

• gi

n = id

ti

n(1) ◦ ti+1 n

(1) ◦ ti

n(1) =

ti+1

n

(1) ◦ ti

n(1) ◦ ti+1 n

(1) f i+k−2

n

• f i

n+2 = f i n ◦ f i+k n+2

f i

n ◦ ti+k n

(q) = ti+k−2

n−2

(q) ◦ f i

n, f i+k n

• ti

n(q) = ti n−2(q) ◦ f i+k n

ti

n(1) ◦ gi+1 n

= ti+1

n

(2) ◦ gi

n, ti n(2) ◦ gi+1 n

= ti+1

n

(1) ◦ gi

n

r′

n−2 ◦ f i n ◦ rn = f i+1 n

, f n−1

n

• r2

n =

f 1

n , r′ n ◦ ti n(q) ◦ rn = ti+1 n

(q) f i

n ◦ ti±1 n

(2) ◦ gi

n =

f i

n ◦ t′i±1 n

(1) ◦ gi

n = id

gi+k

n+2 ◦ gi n = gi n+2 ◦ gi+k−2 n

gi+k−2

n

• f i

n = f i n+2 ◦ gi+k n+2, gi n ◦

f i+k−2

n

= f i+k

n+2 ◦ gi n+2

gi

n ◦ ti+k−2 n−2

(q) = ti+k

n

(q) ◦ gi

n, gi+k n

• ti

n−2(q) = ti n(q) ◦ gi+k n

ti

n(p) ◦ ti+k n

(q) = ti+k

n

(q) ◦ ti

n(p)

r′

n ◦ gi n ◦ rn−2 =

gi+1

n

, r′2

n ◦ gn−1 n

= g1

n

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 8 / 15

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Affine tangles

Affine tangles: generators and relations

It will be more convenient to use a slightly different set of relations. Definition Let si

n denote the (n, n)-tangle with a strand connecting (ζj, 0) to (2ζj, 0) for

each j, and a strand connecting (ζi, 0) to (2ζi, 0) passing clockwise around the circle, beneath all the other strands.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 9 / 15

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Affine tangles

Affine tangles: generators and relations

It will be more convenient to use a slightly different set of relations. Definition Let si

n denote the (n, n)-tangle with a strand connecting (ζj, 0) to (2ζj, 0) for

each j, and a strand connecting (ζi, 0) to (2ζi, 0) passing clockwise around the circle, beneath all the other strands. Proposition The following relations are equivalent to the two relations mentioned involv- ing the generators rn. sn

n ◦ gi n = gi n ◦ sn−2 n−2, sn−2 n−2 ◦ f i n = f i n ◦ sn n, sn n ◦ ti n(p) = ti n(p) ◦ sn n

f n−1

n

• sn

n ◦ tn−1 n

(2) ◦ sn

n ◦ tn−1 n

(2) = f n−1

n

sn

n ◦ tn−1 n

(2) ◦ sn

n ◦ tn−1 n

(2) ◦ gn−1

n

= gn−1

n

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 9 / 15

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Affine tangles

Affine tangles: generators and relations

It will be more convenient to use a slightly different set of relations. Definition Let si

n denote the (n, n)-tangle with a strand connecting (ζj, 0) to (2ζj, 0) for

each j, and a strand connecting (ζi, 0) to (2ζi, 0) passing clockwise around the circle, beneath all the other strands. Proposition The following relations are equivalent to the two relations mentioned involv- ing the generators rn. sn

n ◦ gi n = gi n ◦ sn−2 n−2, sn−2 n−2 ◦ f i n = f i n ◦ sn n, sn n ◦ ti n(p) = ti n(p) ◦ sn n

f n−1

n

• sn

n ◦ tn−1 n

(2) ◦ sn

n ◦ tn−1 n

(2) = f n−1

n

sn

n ◦ tn−1 n

(2) ◦ sn

n ◦ tn−1 n

(2) ◦ gn−1

n

= gn−1

n

Proof (Sketch): Use the identity rn = sn

n ◦ tn−1 n

(2) ◦ · · · ◦ t1

n(2) to

interchange the generators rn and sn

n.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 9 / 15

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Affine tangles

Affine tangles: functors

We closely follow the method used by Anno in the m = 0 case below. Goal: For each (m + 2p, m + 2q) affine tangle α, we would like to construct a functor Ψ(α) : Dp → Dq that is compatible with composition (i.e. given an (m + 2q, m + 2r) tangle β, Ψ(β ◦ α) = Ψ(β) ◦ Ψ(α).)

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 10 / 15

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Affine tangles

Affine tangles: functors

We closely follow the method used by Anno in the m = 0 case below. Goal: For each (m + 2p, m + 2q) affine tangle α, we would like to construct a functor Ψ(α) : Dp → Dq that is compatible with composition (i.e. given an (m + 2q, m + 2r) tangle β, Ψ(β ◦ α) = Ψ(β) ◦ Ψ(α).) Proposition (Cautis-Kamnitzer) Given a linear (m + 2p, m + 2q) tangle α, one can construct a functor

• Ψ(α) :

Dp → Dq that is compatible with composition.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 10 / 15

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Affine tangles

Affine tangles: functors

We closely follow the method used by Anno in the m = 0 case below. Goal: For each (m + 2p, m + 2q) affine tangle α, we would like to construct a functor Ψ(α) : Dp → Dq that is compatible with composition (i.e. given an (m + 2q, m + 2r) tangle β, Ψ(β ◦ α) = Ψ(β) ◦ Ψ(α).) Proposition (Cautis-Kamnitzer) Given a linear (m + 2p, m + 2q) tangle α, one can construct a functor

• Ψ(α) :

Dp → Dq that is compatible with composition. Lemma Given a linear (m + 2p, m + 2q) tangle α, one can construct a functor Ψ(α) : Dp → Dq by “restricting” the functors Ψ(α), that is compatible with composition.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 10 / 15

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Affine tangles

Affine tangles: functors

Now we describe the functors Ψ(f i

n), Ψ(gi n) and Ψ(ti n(p)) mentioned above;

since f i

n, gi n, ti n(p) generate all linear tangles, this is sufficient to describe

Ψ(α) for arbitrary α. Below all functors are derived.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 11 / 15

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Affine tangles

Affine tangles: functors

Now we describe the functors Ψ(f i

n), Ψ(gi n) and Ψ(ti n(p)) mentioned above;

since f i

n, gi n, ti n(p) generate all linear tangles, this is sufficient to describe

Ψ(α) for arbitrary α. Below all functors are derived. Xn,k := Sn×glm+2nT ∗Pk,n ×Pk,n Bn = {(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n), x | x ∈ Sn, xVk+1 ⊂ Vk−1, xVi ⊂ Vi−1∀ i} Consider the P1-bundle πn,k : Xn,k → Sn ×glm+2n T ∗Pk,n ≃ Un−1, and the divisor in,k : Xn,k → Un. Let Vk be the vector bundle on Un corresponding to Vk; define Ek = Vk/Vk−1. Let G k

m+2n(F) = in,k∗(π∗ n,kF ⊗ Ek) for

F ∈ Dn−1, and F k

m+2n(G) = πn,k∗(i∗ n,kG ⊗E−1 k+1) for G ∈ Dn. Then Ψ(f i n) =

F i

n, Ψ(gi n) = G i n.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 11 / 15

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Affine tangles

Affine tangles: functors

Now we describe the functors Ψ(f i

n), Ψ(gi n) and Ψ(ti n(p)) mentioned above;

since f i

n, gi n, ti n(p) generate all linear tangles, this is sufficient to describe

Ψ(α) for arbitrary α. Below all functors are derived. Xn,k := Sn×glm+2nT ∗Pk,n ×Pk,n Bn = {(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n), x | x ∈ Sn, xVk+1 ⊂ Vk−1, xVi ⊂ Vi−1∀ i} Consider the P1-bundle πn,k : Xn,k → Sn ×glm+2n T ∗Pk,n ≃ Un−1, and the divisor in,k : Xn,k → Un. Let Vk be the vector bundle on Un corresponding to Vk; define Ek = Vk/Vk−1. Let G k

m+2n(F) = in,k∗(π∗ n,kF ⊗ Ek) for

F ∈ Dn−1, and F k

m+2n(G) = πn,k∗(i∗ n,kG ⊗E−1 k+1) for G ∈ Dn. Then Ψ(f i n) =

F i

n, Ψ(gi n) = G i n.

Lemma G i

m+2n has right adjoint Fm+2n[−1], and left adjoint F i m+2n[1].

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Affine tangles

Affine tangles: functors

For G ∈ Dn, define T i

n(1)G to be the cone of G i m+2nF i m+2n[−1]G → G, and

T i

n(2)G to be the cone of G → G i m+2nF i m+2n[1]G. Let Ψ(ti n(1)) = T i n(1) and

Ψ(ti

n(2)) = T i n(2). Now let Sn(F) = F ⊗ E−1 m+2n, and let Ψ(sm+2n m+2n) := Sn.

To check that this assignment satisfies the compatibility under compositions, we check the relations involving the generator sm+2n

m+2n below.

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Affine tangles

Affine tangles: functors

For G ∈ Dn, define T i

n(1)G to be the cone of G i m+2nF i m+2n[−1]G → G, and

T i

n(2)G to be the cone of G → G i m+2nF i m+2n[1]G. Let Ψ(ti n(1)) = T i n(1) and

Ψ(ti

n(2)) = T i n(2). Now let Sn(F) = F ⊗ E−1 m+2n, and let Ψ(sm+2n m+2n) := Sn.

To check that this assignment satisfies the compatibility under compositions, we check the relations involving the generator sm+2n

m+2n below.

Proposition The following identities hold, where 1 ≤ i ≤ m + 2n − 2, 1 ≤ p ≤ 2: Sn−1 ◦ F i

m+2n ≃ F i m+2n ◦ Sn, Sn ◦ G i m+2n ≃ G i m+2n ◦ Sn−1

Sn ◦ T i

m+2n(p) ≃ T i m+2n(p) ◦ Sn

F m+2n−1

m+2n

• Sn ◦ T m+2n−1

m+2n

(2) ◦ Sn ◦ T m+2n−1

m+2n

(2) ≃ F m+2n−1

m+2n

Sn ◦ T m+2n−1

m+2n

(2) ◦ Sn ◦ T m+2n−1

m+2n

(2) ◦ G m+2n−1

m+2n

≃ G m+2n−1

m+2n

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 12 / 15

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Affine tangles

Affine tangles: functors

For G ∈ Dn, define T i

n(1)G to be the cone of G i m+2nF i m+2n[−1]G → G, and

T i

n(2)G to be the cone of G → G i m+2nF i m+2n[1]G. Let Ψ(ti n(1)) = T i n(1) and

Ψ(ti

n(2)) = T i n(2). Now let Sn(F) = F ⊗ E−1 m+2n, and let Ψ(sm+2n m+2n) := Sn.

To check that this assignment satisfies the compatibility under compositions, we check the relations involving the generator sm+2n

m+2n below.

Proposition The following identities hold, where 1 ≤ i ≤ m + 2n − 2, 1 ≤ p ≤ 2: Sn−1 ◦ F i

m+2n ≃ F i m+2n ◦ Sn, Sn ◦ G i m+2n ≃ G i m+2n ◦ Sn−1

Sn ◦ T i

m+2n(p) ≃ T i m+2n(p) ◦ Sn

F m+2n−1

m+2n

• Sn ◦ T m+2n−1

m+2n

(2) ◦ Sn ◦ T m+2n−1

m+2n

(2) ≃ F m+2n−1

m+2n

Sn ◦ T m+2n−1

m+2n

(2) ◦ Sn ◦ T m+2n−1

m+2n

(2) ◦ G m+2n−1

m+2n

≃ G m+2n−1

m+2n

This gives us functors Ψ(α) : Dp → Dq for each (m + 2p, m + 2q) tangle α, compatible under composition.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 12 / 15

SLIDE 28

The exotic t-structure

Exotic t-structure: Braid group action and definition

Let Baff denotes the braid group attached to the affine Weyl group Waff = W ⋉Λ, where W is the Weyl group for G, Λ is the weight lattice and Q the root lattice. Let B+

aff be the semigroup generated by the lifts of the simple

reflections sα in the Coxeter group W ⋉ Q ⊂ Waff .

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SLIDE 29

The exotic t-structure

Exotic t-structure: Braid group action and definition

Let Baff denotes the braid group attached to the affine Weyl group Waff = W ⋉Λ, where W is the Weyl group for G, Λ is the weight lattice and Q the root lattice. Let B+

aff be the semigroup generated by the lifts of the simple

reflections sα in the Coxeter group W ⋉ Q ⊂ Waff . Lemma Baff can be identified with the group of all bijective (m + 2n, m + 2n) affine tangles (i.e. where each strand must connect a point in the inner circle with a point in the outer circle), quotiented out by the generator rn.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 13 / 15

SLIDE 30

The exotic t-structure

Exotic t-structure: Braid group action and definition

Let Baff denotes the braid group attached to the affine Weyl group Waff = W ⋉Λ, where W is the Weyl group for G, Λ is the weight lattice and Q the root lattice. Let B+

aff be the semigroup generated by the lifts of the simple

reflections sα in the Coxeter group W ⋉ Q ⊂ Waff . Lemma Baff can be identified with the group of all bijective (m + 2n, m + 2n) affine tangles (i.e. where each strand must connect a point in the inner circle with a point in the outer circle), quotiented out by the generator rn. Thus for each b ∈ Baff , we have a functor Ψ(b) : Dn → Dn. Definition The exotic t-structure on Dn is given by: D≥0

n

= {F | RΓ(Ψ(b−1)F) ∈ D≥0(Vect) ∀ b ∈ B+

aff }

D≤0

n

= {F | RΓ(Ψ(b)F) ∈ D≤0(Vect) ∀ b ∈ B+

aff }

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 13 / 15

SLIDE 31

The exotic t-structure

Exotic t-structure: Exactness of G i

m+2n

Proposition The functor G i

m+2n : Dn−1 → Dn is exact with respect to the exotic t-

structures on the two categories (i.e. it maps the heart to the heart).

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 14 / 15

SLIDE 32

The exotic t-structure

Exotic t-structure: Exactness of G i

m+2n

Proposition The functor G i

m+2n : Dn−1 → Dn is exact with respect to the exotic t-

structures on the two categories (i.e. it maps the heart to the heart). Proof (Sketch): To show that G i

m+2n is right t-exact (i.e. it maps D≤0 n−1 to

D≤0

n ) is equivalent to showing (G i m+2n)L = F i m+2n[−1] is left t-exact. We

must show that for each b′ ∈ B′

aff +, G ∈ D≥0 n , RΓ(Ψ(b−1)F i m+2n[−1]G) ∈

D≥0(Vect). One can construct ηi(b) ∈ B+

aff such that b−1 ◦ f i m+2n =

f i

m+2n ◦ ηi(b)−1; now we need to show that RΓ(F i m+2n[−1]Ψ(ηi(b)−1)G) ∈

D≥0(Vect). Since Ψ(ηi(b)−1)G ∈ D≥0

n , it remains to show that we have

RΓ(F i

m+2nG[−1]) ∈ D≥0(Vect).

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 14 / 15

SLIDE 33

The exotic t-structure

Exotic t-structure: Exactness of G i

m+2n

Proposition The functor G i

m+2n : Dn−1 → Dn is exact with respect to the exotic t-

structures on the two categories (i.e. it maps the heart to the heart). Proof (Sketch): To show that G i

m+2n is right t-exact (i.e. it maps D≤0 n−1 to

D≤0

n ) is equivalent to showing (G i m+2n)L = F i m+2n[−1] is left t-exact. We

must show that for each b′ ∈ B′

aff +, G ∈ D≥0 n , RΓ(Ψ(b−1)F i m+2n[−1]G) ∈

D≥0(Vect). One can construct ηi(b) ∈ B+

aff such that b−1 ◦ f i m+2n =

f i

m+2n ◦ ηi(b)−1; now we need to show that RΓ(F i m+2n[−1]Ψ(ηi(b)−1)G) ∈

D≥0(Vect). Since Ψ(ηi(b)−1)G ∈ D≥0

n , it remains to show that we have

RΓ(F i

m+2nG[−1]) ∈ D≥0(Vect).

Proposition The functor G i

m+2n : Dn−1 → Dn maps irreducible objects in the heart of

the exotic t-structure on Dn−1 to irreducible objects in the heart of the exotic t-structure on Dn.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 14 / 15

SLIDE 34

The exotic t-structure

Exotic t-structure: Irreducible objects in the heart

Let a (m, m + 2n) crossingless matching to be affine (m, m + 2n)-tangle whose vertical projection has no crossings, and where the m inner points are not labelled. Given a crossingless (m, m + 2n) matching α, we obtain a functor Ψ(α) : D0 → Dn. Let Ψα = Ψ(α)C, where C ∈ D0 ≃ Db(Vect). Any crossingless matching is a product gin

m+2n ◦· · ·◦gi2 m+4 ◦gi1 m+2; thus since

G i

n maps irreducibles to irreducibles, Ψα is an irreducible object in the heart

• f the exotic t-structure on Dn. The following result generalizes work of

Anno, in the m = 0 case.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 15 / 15

SLIDE 35

The exotic t-structure

Exotic t-structure: Irreducible objects in the heart

Let a (m, m + 2n) crossingless matching to be affine (m, m + 2n)-tangle whose vertical projection has no crossings, and where the m inner points are not labelled. Given a crossingless (m, m + 2n) matching α, we obtain a functor Ψ(α) : D0 → Dn. Let Ψα = Ψ(α)C, where C ∈ D0 ≃ Db(Vect). Any crossingless matching is a product gin

m+2n ◦· · ·◦gi2 m+4 ◦gi1 m+2; thus since

G i

n maps irreducibles to irreducibles, Ψα is an irreducible object in the heart

• f the exotic t-structure on Dn. The following result generalizes work of

Anno, in the m = 0 case. Theorem The irreducible objects in the heart of the exotic t-structure on Dn are precisely given by Ψα, as α ranges across (m, m+2n) crossingless matchings.

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 15 / 15

SLIDE 36

The exotic t-structure

Exotic t-structure: Irreducible objects in the heart

Let a (m, m + 2n) crossingless matching to be affine (m, m + 2n)-tangle whose vertical projection has no crossings, and where the m inner points are not labelled. Given a crossingless (m, m + 2n) matching α, we obtain a functor Ψ(α) : D0 → Dn. Let Ψα = Ψ(α)C, where C ∈ D0 ≃ Db(Vect). Any crossingless matching is a product gin

m+2n ◦· · ·◦gi2 m+4 ◦gi1 m+2; thus since

G i

n maps irreducibles to irreducibles, Ψα is an irreducible object in the heart

• f the exotic t-structure on Dn. The following result generalizes work of

Anno, in the m = 0 case. Theorem The irreducible objects in the heart of the exotic t-structure on Dn are precisely given by Ψα, as α ranges across (m, m+2n) crossingless matchings. Proof (Sketch): One shows that there are m+2n

n

• crossingless matchings,

and m+2n

n

• irreducible objects in the heart of the exotic t-structure, D0

n

(using the fact that K 0(D0

n) = K 0(Bm+n,n)).

Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 15 / 15