EXOTIC t-STRUCTURES FOR TWO-BLOCK SPRINGER FIBRES
RINA ANNO, VINOTH NANDAKUMAR
Abstract. We study the exotic t-structure on Dn , the derived category of coherent sheaves on two-block
Springer fibre (i.e. for a nilpotent matrix of type (m + n, n) in type A). The exotic t-structure has been
defined by Bezrukavnikov and Mirkovic to study representations of Lie algebras in positive characteristic.
Using work of Cautis and Kamnitzer, we construct functors indexed by affine tangles, between categories
of coherent sheaves on different two-block Springer fibres (i.e. for different values of n). After checking
some exactness properties of these functors, we describe the irreducible objects in the heart of the exotic
t-structure on Dn and enumerate them by crossingless (m, m + 2n) matchings. We also compute the Ext’s
between the irreducible objects. This is based on work of Anno, where the m = 0 case was handled.
1. Introduction
Let G be a semi-simple Lie group, with Lie algebra g, flag variety B and nilpotent cone N . It is wellknown that there is a natural map π : T ∗ B → N which is a resolution of singularities (known as the
Springer resolution). Given e ∈ N , let Be = π −1 (e); these varieties are known as Springer fibers, and
are of special interest in representation theory. For instance, in type A, Springer showed that the top
cohomology of a Springer fiber can be equipped with a representation of the Weyl group, and further
realizes an irreducible representation.
This special case when G = SL(m + 2n), and the nilpotent e has Jordan type (m + n, n), is easier
to understand, and has been studied extensively. In [9], Stroppel and Webster study the geometry
and combinatorics of these “two-block Springer” fibers and investigate connections with Khovanov’s arc
algebras. In [6], Russell studies the topology of these varieties, and studies a certain basis in the Springer
representation.
In [3], Bezrukavnikov and Mirkovic introduce “exotic t-structures” on derived categories of coherent
sheaves on Springer theoretic varieties, in order to study the modular representation theory of g. These
exotic t-structures are defined a certain action of the affine braid group Baf f on these categories, which
was defined by Bezrukavnikov and Riche (see [5]).
More precisely, let k be an algebraically closed field of characteristic p with p > h (here h is the Coxeter
number). Let λ ∈ hk be integral and regular; and let e ∈ N (k) be a nilpotent. Let Modfe g,λ (Uk ) be the
category of modules with generalized central character (λ, e). In Theorem 5.3.1 from [4] (see also Section
1.6.2 from [3]), states that there is an equivalence:
Db (CohBe,k (e
gk )) ' Db (Modfe g,λ (Uk ))
Further, it is proven that the tautological t-structure on the derived category of modules, corresponds to
the exotic t-structure on the derived category of coherent sheaves.
Here we will study exotic t-structures for the case of two-block Springer fibers in type A (ie. for a
nilpotent of Jordan type (m + n, n)), and give a description of the irreducible objects in the heart of the
t-structure. Below we describe the contents of this paper in more detail.
1
2
RINA ANNO, VINOTH NANDAKUMAR
1.1. Two-block Springer fibers. In Section 1, we recall the definition and some properties of two-block
Springer fibers, and define the categories that we will be studying. Let m ≥ 0 be fixed; and let n ∈ Z≥0
vary. Consider the Lie algebra g = slm+2n , and denote the nilpotent cone of slm+2n (the variety consisting
of nilpotent matrices of size m + 2n) by Nn . Denote by zn the standard nilpotent of type (m + n, n):


1


···




1




 0 0 ··· 0 0 0 ··· 0 
zn = 

1






···



1 
0 0 ··· 0 0 0 ··· 0
Let Bn be the flag variety for GLm+2n . The Springer resolution is T ∗ Bn :
Bn = {(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n = Cm+2n ) | dim Vi = i}
T ∗ Bn = {(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n = Cm+2n , x) | x ∈ slm+2n , xVi ⊂ Vi−1 }
The natural map πn : T ∗ Bn → Nn (i.e. by forgetting the flag) is a resolution of singularities. The
two-block Springer fiber is the variety
Bzn = πn−1 (zn ) = {(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n ) ∈ Bn | zn Vi ⊆ Vi−1 }
The Mirkovic-Vybornov transverse slices Sn ⊂ g is a variant of the Slodowy slice. The following variety
is of interest, since it is a resolution of Sn ∩ N .
Un = πn−1 (Sn ) ⊂ T ∗ Bn
= {(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n = Cm+2n , x) | x ∈ Sn , xVi ⊂ Vi−1 }
Let Dn = Db (CohBzn (Un )) be the bounded derived category of coherent sheaves on Un , which are supported on Bzn . These are the categories that we will be studying.
1.2. Affine tangles. In Section 2, we recall the definition, and some properties, of affine tangles.
Definition. Let p, q be positive integers of the same parity. 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
2πi
of the arcs are (1, 0), (ζp , 0), · · · , (ζpp−1 , 0), (2, 0), (2ζq , 0), · · · , (2ζqq−1 , 0) in some order (where ζk = e k ).
Definition. Let ATan be the category with objects {k} for k ∈ Z≥0 , and the morphisms between p and
q consist of all affine (p, q) tangles (up to isotopy).
The above definition is consistent, since a (p, q) affine tangle α, and a (q, r) affine tangle β, then β ◦ α is
a (p, r) affine tangle.
We recall the well-known presentation of this category using generators and relations. The generators
consist of “cups”, gni , which are (n − 2, n) tangles; “caps”, fni , which are (n, n − 2) tangles; and “twists”,
tin (1), tin (2), which are (n, n) tangles. The relations include the 3 Reidemeister moves.
EXOTIC t-STRUCTURES FOR TWO-BLOCK SPRINGER FIBRES
3
1.3. Functors associated to affine tangles.
Definition. Let ATanm be the full subcategory of ATan, containing the objects {m + 2n} for n ∈ Z≥0 .
A “weak representation” of the category ATanm is an assignment of a triangulated category Cn for each
n ∈ Z≥0 , and a functor Ψ(α) : Dp → Dq for each affine (m + 2p, m + 2q) tangle α, such that the relations
between tangles hold for these functors: i.e. if β is an (m+2q, m+2r)-tangle, then there is an isomorphism
Ψ(β) ◦ Ψ(α) ' Ψ(β ◦ α).
The main result of this section is a construction of a weak representation of ATanm using the categories
Dn above. To do this, we mimic the strategy used by Cautis and Kamnitzer in [7], where they construct
a weak representation of a LTan (a subcategory of ATan where morphisms consist of linear tangles),
using slightly larger categories.
1.4. The exotic t-structure on Dn . In Section 5, we recall the definition of exotic t-structures (introduced by Bezrukavnikov and Mirkovic in [3]), and describe how they are related to the action of affine
tangles constructed above.
Let Baf f be the affine braid group. As a special case of the construction in Section 1 of [3] (see also
Bezrukavnikov-Riche, [5]), we have an action of Baf f on Dn (ie. for every b ∈ Baf f , there exists a functor
Ψ(b) : Dn → Dn , and an isomorphism Ψ(b1 b2 ) ' Ψ(b1 ) ◦ Ψ(b2 ) for b1 , b2 ∈ Baf f ). It turns out that Baf f
can be identified as a subgroup of the monoid of (m + 2n, m + 2n)-tangles; and under this identification,
the action of Baf f coincides with the action constructed above.
Let B+
af f ⊂ Baf f be the semigroup generated by the lifts of the simple reflections s̃α in the Coxeter group
Cox
Waf f . Bezrukavnikov-Mirkovic’s construction in [3] specializes to give an exotic t-structure on Dn , which
is defined as follows:
Dn≥0 = {F | RΓ(Ψ(b−1 )F) ∈ D≥0 (Vect) ∀ b ∈ B+
af f }
Dn≤0 = {F | RΓ(Ψ(b)F) ∈ D≤0 (Vect) ∀ b ∈ B+
af f }
We also prove that the “cup” functors Ψ(gni ) are exact with the exotic t-structures, and send irreducible
objects to irreducible objects.
1.5. Irreducible objects in the heart of the exotic t-strucutre on Dn . In Section 6, we give a
description of the irreducible objects in the exotic t-structure on Dn , and compute the Ext’s between
irreducibles.
Let Cross(n) be the set of affine (m, m+2n) tangles, where the m inner points are not labelled, the m+2n
outer points are labelled, and whose vertical projections to C do not have crossings. We have a functor
Ψ(α) : D0 → Dn ; let Ψα = Ψ(α)(C) (here C ∈ Db (Vect) ' D0 ). We show that that {Ψα | α ∈ Cross(n)}
constitute the irreducible objects in Dn0 .
We also prove that for β ∈ Cross(n), Ext• (Ψα , Ψβ ) is given by the below formula. Here Λ denotes a
complex in Db (Vect) concentrated in degrees 1 and −1; and α̌ is the (m + 2n, m) affine tangle obtained
by “inverting” α. An a (m, m) affine tangle γ with no crossings is said to be “good” if it has no cups or
caps, and ω(γ) denote the number of circles present.
4
RINA ANNO, VINOTH NANDAKUMAR
Ext• (Ψα , Ψβ ) =
(
Λ⊗ω(α̌◦β) [−n]
if α̌ ◦ β is good
0
otherwise
1.6. Acknowledgements. I would like to thank my advisor Roman Bezrukavnikov, for suggesting this
problem to me, and for numerous helpful discussions and insights. I am grateful to thank Rina Anno,
whose paper this was inspired by; and to Joel Kamnitzer, Catherina Stroppel and Ben Webster for many
helpful discussions. I would also like to thank Anthony Henderson for the proof of Lemma 4.18, and the
University of Sydney, where part of this work was completed.
2. Two-block Springer fibres
2.1. Transverse slices for two-block nilpotents. Fix m ≥ 0. For n ∈ Z≥0 , let zn be the standard
nilpotent of Jordan type (m + n, n). Let Sn ⊂ slm+2n denote the Mirkovic-Vybornov transverse slice to
the nilpotent zn (see section 3.3.1 in [8]):
X
X
Sn = {zn +
ai em+n,i +
bi em+2n,i }
1≤i≤m+2n
i∈{1,··· ,n,m+n+1,··· ,m+2n}
Definition 2.1. Denote by Nn the nilpotent cone for slm+2n . Let Bn denote the complete flag variety
for GLm+2n (C), and for 0 < k < m + 2n define the varieties Pk,n as follows:
Pk,n = {(0 ⊂ V1 ⊂ · · · ⊂ Vbk ⊂ · · · ⊂ Vm+2n = Cm+2n )}.
Then the varieties T ∗ Bn , T ∗ Pk,n can be described as follows:
T ∗ Bn = {(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n = Cm+2n , x) | x ∈ slm+2n , xVi ⊂ Vi−1 };
T ∗ Pk,n = {(0 ⊂ V1 ⊂ · · · ⊂ Vbk ⊂ · · · ⊂ Vm+2n = Cm+2n ), x |
x ∈ slm+2n , xVk+1 ⊂ Vk−1 , xVi ⊂ Vi−1 for i 6= k, k + 1}.
Pick a basis e1 , . . . , em+n+1 , f1 , . . . , fn+1 of Cm+2n+2 so that zn+1 ei = ei−1 , zn+1 fj = fj−1 (where we set
e0 = f0 = 0).
Lemma 2.2. For any x ∈ Sn+1 such that dim(Ker x) = 2, we have Ker x = Ce1 ⊕ Cf1 , and there is a
natural isomorphism φx : xVm+2n+2 ' Cm+2n .
Proof. By the construction in [8, section 3.3.1] we can assume that xei = ei−1 + ai em+n+1 + ci fm+1 if
i ≤ m + 1, xei = ei−1 + ai em+n+1 if i > m + 1, and xfj = fj−1 + bj em+n+1 + dj fm+1 . Then we have:
!
!
X
X
X
X
X
x
λi ei +
νj f j =
λi+1 ei +
ai λ i +
bj νj em+n+1
1≤i≤m+n+1
1≤j≤m+1
1≤i≤m+n
1≤i≤m+n+1
1≤j≤m+1
!
+
X
1≤j≤m
νj+1 fj +
X
1≤i≤m+1
ai λi +
X
1≤j≤m+1
d j νj
fm+1
EXOTIC t-STRUCTURES FOR TWO-BLOCK SPRINGER FIBRES
5
P
P
So xv = x
= 0 implies that λi = νj = 0 for i, j > 1, i.e. that
1≤i≤m+n+1 λi ei +
1≤j≤m+1 νj fj
v ∈ Ce1 ⊕ Cf1 . If xv = 0 it follows that a1 = b1 = c1 = d1 = 0. So:
!
X
X
X
xVm+2n+2 =
λi e i +
ai+1 λi +
bj+1 νj em+n+1 +
1≤i≤m+n
1≤i≤m+n
1≤j≤n
!
X
1≤j≤n
µj f j +
X
ci+1 λi +
1≤i≤m
X
dj+1 νj
fn+1
1≤j≤n
L
L
L
L
Let γm,n : Cm+2n+2 = ( 1≤i≤m+n Cei ⊕ 1≤j≤n Cfj )⊕(Cem+n+1 ⊕Cfn+1 ) → ( 1≤i≤m+n Cei ⊕ 1≤j≤n Cfj )
L
L
denote the natural projection map. Now φx := γm,n |xVm+2n+2 : xVm+2n+2 → 1≤i≤m+n Cei ⊕ 1≤j≤n Cfj
is an isomorphism.
Proposition 2.3. For every 0 < k < m + 2n + 2 we have Sn+1 ×slm+2n+2 T ∗ Pk,n+1 ' Sn ×slm+2n T ∗ Bn .
Proof. By definition:
Sn+1 ×slm+2n+2 T ∗ Pk,n+1 ={(0 ⊂ V1 ⊂ · · · ⊂ Vbk ⊂ · · · ⊂ Vm+2n+2 ) |
x ∈ Sn+1 , xVk+1 ⊂ Vk−1 , xVi ⊂ Vi−1 for i 6= k, k + 1}
Sn ×slm+2n T ∗ Bn = {(0 ⊂ W1 ⊂ · · · ⊂ Wm+2n = Cm+2n )|y ∈ Sn , yWi ⊂ Wi−1 }
Since x ∈ Sn+1 , the Jordan type of x is a two-block partition, and dim(Ker(x)) ≤ 2; but xVk+1 ⊂ Vk−1 so
we must have xVk+1 = Vk−1 . Consider the flag (0 ⊂ V1 ⊂ · · · ⊂ Vk−1 = xVk+1 ⊂ xVk+2 ⊂ · · · ⊂ xVm+2n+2 ).
∼
Recall the isomorphism φx : xVm+2n+2 −
→ Cm+2n from Lemma 2.2 and denote by Φ(x) ∈ End(Cm+2n ) the
endomorphism induced on Cm+2n by the action of x on xVm+2n+2 . Construct a map α : Sn+1 ×slm+2n+2
T ∗ Pk,n+1 → T ∗ Bn as follows:
α(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n , x) =
= ((0 ⊂ φx (V1 ) ⊂ · · · ⊂ φx (Vk−1 ) = φx (xVk+1 ) ⊂ φx (xVk+2 ) ⊂ · · · ⊂ Cm+2n ), Φ(x))
We claim that α gives the required isomorphism Sn+1 ×slm+2n+2 T ∗ Pk,n+1 ' Sn ×slm+2n T ∗ Bn . First we
check that Φ(x) ∈ Sn . From the argument in Lemma 2.2, Φ(x)ei = ei−1 + ai+1 em+n + ci+1 fn if i ≤ n,
Φ(x)ei = ei−1 + ai+1 em+n if i > n, and Φ(x)fj = fj−1 + cj+1 em+n + dj+1 fn . Thus Φ gives a bijection
between {x ∈ Sn+1 ∩ Nn+1 | dim(Ker x) = 2} and Sn ∩ Nn . It follows that α has image Sn ×slm+2n T ∗ Bn
and that α is an isomorphism onto its image, as required.
Definition 2.4. Under the Springer resolution map πn : T ∗ Bn → Nn , let Bzn = πn−1 (zn ). Let Un =
Sn ×slm+2n T ∗ Bn ; define Dn = Db (CohBzn (Un )) to be the bounded derived category of coherent sheaves on
Un supported on Bzn .
Next we will prove an extension of Proposition 2.4 in [7] to the case of two-block nilpotents. Consider a
2(m+2n)-dimensional vector space Vm,n with basis e1 , . . . , em+2n , f1 , . . . , fm+2n and a nilpotent z such that
zei = ei−1 , zfi = fi−1 . Let Wm,n ⊂ Vm,n denote the vector subspace with basis e1 , . . . , em+n , f1 , . . . , fn , so
that z|Wm,n has Jordan type (m + n, n); we will identify Wm,n with Vm+2n . Let P : Vm,n → Wm,n denote
6
RINA ANNO, VINOTH NANDAKUMAR
the projection defined by P ei = ei if i ≤ m + n, P ei = 0 if i > m + n; P fi = fi if i ≤ n, P fi = fi if i > n.
Following Section 2 of [7], define:
Ym+2n = {(L1 ⊂ · · · ⊂ Lm+2n ⊂ Vm,n )| dimLi = i, zLi ⊂ Li−1 };
Um+2n = {(L1 ⊂ · · · ⊂ Lm+2n ) ∈ Ym+2n |P (Lm+2n ) = Wm,n }.
Definition 2.5.


1

..
.



1


a1 a2 · · · am+n b1 b2 · · · bn
Sn0 = {

1


...



1
c1 c2 · · · cm+n d1 d2 · · · dn






}.






Note that we have Sn ⊂ Sn0 , and Un ⊂ Sn0 ×slm+2n T ∗ Bn . The following lemma (generalizing Proposition
2.4 in [7]) gives a map ιn : Un → Ym+2n , identifying Un with a locally closed subvariety of Ym+2n .
Lemma 2.6. There is an isomorphism Um+2n ' Sn0 ×slm+2n T ∗ Bn .
Proof. Given (L1 ⊂ · · · ⊂ Lm+2n ) ∈ Um+2n , since P : Lm+2n → Wm,n is an isomorphism, we have a
nilpotent endomorphism x = P zP −1 ∈ End(Vm+2n ) (here we identify Wm,n and Vm+2n ). If P −1 ei = ei +v 0 ,
where v 0 lies in the span of em+n+1 , · · · , em+2n , fn+1 , · · · , fm+2n , then zP −1 ei = ei−1 + v 00 where v 00 is in the
span of em+n , · · · , em+2n−1 , fn , · · · , fm+2n−1 . Hence P zP −1 ei = xei ∈ span(ei−1 , em+n , fn ), and similarly
xfi ∈ span(fi−1 , em+n , fn ); so x ∈ Sn0 . Thus we have a map α : Um+2n → Sn0 ×slm+2n T ∗ Bn given by
α(L1 , · · · , Lm+2n ) = (P zP −1 , (P (L1 ), P (L2 ), · · · , P (Lm+2n ))).
For the converse direction, from the below Lemma 2.7 we know that given x ∈ Sn0 ∩ Nn there exists a
unique z-stable subspace Lm+2n ⊂ Vm,n such that P Lm+2n = Wm,n and P zP −1 = x; call this subspace
Lm+2n = Θ(x). We have an isomorphism P : Θ(x) ' Wm,n . Thus given an element ((0 ⊂ V1 ⊂ · · · ⊂
Vm+2n ), x) ∈ Sn0 ×slm+2n T ∗ Bn , let β(x) = (0 ⊂ P −1 V1 ⊂ P −1 V2 ⊂ · · · ⊂ Θx ). It is clear that α and β are
inverse to one another.
Lemma 2.7. Given x ∈ Sn0 ∩ Nn , there exists a unique subspace Lm+2n ⊂ Vm,n , with P Lm+2n = Wm,n ,
such that zLm+2n ⊂ Lm+2n and P zP −1 = x.
Proof. Since P Lm+2n = Wm,n , to specify the subspace Lm+2n it suffices to specify
X (k)
X
(l)
ẽi := P −1 (ei ) = ei +
ai em+n+k +
ci fn+l
1≤k≤n
f˜j := P −1 (fj ) = fj +
X
1≤k≤n
1≤l≤m+n
(k)
bj em+n+k
+
X
(l)
dj fn+l
1≤l≤m+n
Suppose for 1 ≤ i ≤ m + n, 1 ≤ j ≤ n, xei = ei−1 + ai em+n + ci fn , xfj = fj−1 + bj em+n + dj fn ; then
(1)
(1)
(1)
(1)
the identity P zP −1 = x is equivalent to ai = ai , ci = ci , bj = bj and dj = dj . The statement
EXOTIC t-STRUCTURES FOR TWO-BLOCK SPRINGER FIBRES
7
zLm+2n ⊂ Lm+2n , i.e. zẽi , z f˜j ∈ Lm+2n , is equivalent to saying that:
zẽi = ẽi−1 + ai ẽm+n + ci f˜n
z f˜j = f˜j−1 + bj ẽm+n + dj f˜n
Expanding the above two equations:
X (k)
X
X (k)
(l)
ei−1 +
ai em+n+k−1 +
ci fn+l−1 = ei−1 +
ai−1 em+n+k +
1≤k≤n
1≤l≤m+n
1≤k≤n
X
(l)
ci−1 fn+l +
1≤l≤m+n
!
+ ai
X
em+n +
(k)
am+n em+n+k +
1≤k≤n
fj−1 +
X
X
!
(l)
cm+n fn+l + ci
fn +
1≤l≤m+n
X
(k)
bj em+n+k−1 +
1≤k≤n
X
b(k)
n em+n+k +
1≤k≤n
X
(l)
dj fn+l−1 = fj−1 +
1≤l≤m+n
X
d(l)
n fn+l
(k)
bj−1 em+n+k +
1≤k≤n
X
(l)
dj−1 fn+l +
1≤l≤m+n
!
+ bj
em+n +
X
(k)
am+n em+n+k
1≤k≤n
+
X
(l)
cm+n fn+l
;
1≤l≤m+n
!
+ dj
fn +
1≤l≤m+n
X
X
b(k)
n em+n+k +
1≤k≤n
d(l)
n fn+l
.
1≤l≤m+n
Extracting coefficients of em+n+k and fn+l in the above two equations gives:
(k+1)
= ai−1 + ai am+n + ci b(k)
n ,
(l+1)
= ci + ai cm+n + ci d(l)
n ,
ai
ci
(k)
(l)
(k)
(k+1)
bj
(l)
(l+1)
dj
(k)
(k)
= bj−1 + bj am+n + dj b(k)
n
(l)
(l)
= dj−1 + bj cm+n + dj d(l)
n
(k)
Consider the matrix coefficients (xk )p,q for 1 ≤ p, q ≤ m + 2n. It follows by induction that we have ai =
(l)
(l)
(k)
(xk )m+n,i , bj = (xk )m+n,m+n+j , ci = (xl )m+2n,i , dj = (xl )m+2n,m+n+j . Indeed, the case where k = l = 1
P
is clear; and the induction step follows from expanding the equation (xr+1 )uv = 1≤w≤m+2n (xr )uw (x)wv
for u = m + n and u = m + 2n.
(k)
(k)
(l)
(l)
(n+1)
(n+1)
= 0
= bj
Using the above recursive definition of ai , bj , ci , dj , it remains to prove that ai
(m+n+1)
(m+n+1)
n+1
m+n+1
= 0. Thus we must show that (x )m+n,p = (x
)m+2n,p = 0 given
= dj
and ci
P
r+1
r
1 ≤ p ≤ m + 2n. Using the equation (x )uv = 1≤w≤m+2n (x)uw (x )wv , we compute that:
(xn+1 )m+n,p = (xn+2 )m+n−1,p = · · · = (xm+2n )1,p = 0
(xm+n+1 )m+2n,p = (xm+n+2 )m+2n−1,p = · · · = (xm+2n )m+n+1,p = 0
This completes the proof of the existence and uniqueness of a z-stable subspace Lm+2n ⊂ Vm,n with
P Lm+2n = Wm,n and P zP −1 = x.
3. Tangles
3.1. Affine tangles.
arcs and a finite numDefinition 3.1. If p ≡ q (mod 2), a (p, q) affine tangle is an embedding of p+q
2
ber of circles into the region {(x, y) ∈ C × R|1 ≤ |x| ≤ 2}, such that the end-points of the arcs are
2πi
(1, 0), (ζp , 0), · · · , (ζpp−1 , 0), (2, 0), (2ζq , 0), · · · , (2ζqq−1 , 0) in some order; here ζk = e k .
Remark 3.2. Given a (p, q) affine tangle α, and a (q, r) affine tangle β, we can compose them using
scaling and concatenation. This composition is associative up to isotopy. The composition β ◦ α is a
(p, r) affine tangle.
8
RINA ANNO, VINOTH NANDAKUMAR
Definition 3.3. Given 1 ≤ i ≤ n, define the following affine tangles:
• Let gni denote the (n − 2, n) tangle with an arc connecting (2ζni , 0) to (2ζni+1 , 0). Let other strands
k
k
connect (ζn−2
, 0) to (2ζnk , 0) for 1 ≤ k < i and (ζn−2
, 0) to (2ζnk+2 , 0) for i + 1 < k ≤ n − 2.
• Let fni denote the (n, n − 2) tangle with an arc connecting (ζni , 0) and (ζni+1 , 0). Let other strands
k−2
k
connect (ζnk , 0) to (2ζn−2
, 0) for 1 ≤ k < i and (ζnk , 0) to (2ζn−2
, 0) for i + 1 < k ≤ n − 2.
i
i
• Let tn (1) (respectively, tn (2)) denote the (n, n) tangle in which a strand connecting (ζni , 0) to
(2ζni+1 , 0) passes beneath (respectively, above) a strand connecting (ζni+1 , 0) to (2ζni , 0). Let other
strands connect (ζnk , 0) to (2ζnk , 0) for k 6= i, i + 1.
• Let rn denote the (n, n) tangle connecting (ζnj , 0) to (2ζnj+1 , 0) for each 1 ≤ j ≤ n, and let rn0
denote the (n, n) tangle connecting (ζnj , 0) to (2ζnj−1 , 0) for each 1 ≤ j ≤ n.
Definition 3.4. Define a linear tangle to be an affine tangle that is isotopic to a product of the generators
gni , fni , tin (1) and tin (2) for i 6= n.
Remark 3.5. Linear tangles can be moved away from the half-line e−i R≥0 where is a small positive
number. If we cut the annulus 1 ≤ |z| ≤ 2 by that line and apply the logarithm map, linear tangles turn
into the usual tangles that live between two parallel lines.
Lemma 3.6. Any affine tangle is isotopic to a composition of the above generators.
Proof. For a curve in C, define its affine critical point as a point where this curve is tangent to a circle
with center at 0. We can adjust a tangle within its isotopy class so that its projection onto C has a finite
number of transversal crossings and affine critical points. We can also assume that no two of these points
lie on the same circle with center at 0. Cut the projection of the tangle by circles with center at 0 into
annuli so that each annulus contains only one crossing or affine critical point. We can further adjust the
tangle so that we have a tangle inside each annulus, and by construction these tangles have to be gni , fni ,
or tin (p), possibly composed with a power of rn .
Definition 3.7. Let ATan (resp. Tan) denote the category with objects k for k ∈ Z≥0 , and the set of
morphisms between p and q consist of all affine (resp. linear) (p, q) tangles.
In the category ATan we record the following relations between the above generators; here let 1 ≤ i ≤
n − 1, 1 ≤ p, q ≤ 2, k ≥ 2:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(Reidemeister 0) fni ◦ gni+1 = fni+1 ◦ gni = id
i
i
i±1
i
(Reidemeister 1) fni ◦ ti±1
n (2) ◦ gn = fn ◦ tn (1) ◦ gn = id
i
i
i
i
(Reidemeister 2) tn (1) ◦ tn (2) = tn (2) ◦ tn (1) = id
i
i+1
i
i+1
(Reidemeister 3) tin (1) ◦ ti+1
n (1) ◦ tn (1) = tn (1) ◦ tn (1) ◦ tn (1).
i+k
i
◦ gni = gn+2
◦ gni+k−2
(Cup-cup isotopy) gn+2
i+k
i
(Cap-cap isotopy) fni+k−2 ◦ fn+2
= fni ◦ fn+2
i+k
i+k
i
i
(Cup-cap isotopy) gni+k−2 ◦ fni = fn+2
◦ gn+2
, gni ◦ fni+k−2 = fn+2
◦ gn+2
i+k−2
(Cup-crossing isotopy) gni ◦ tn−2 (q) = tni+k (q) ◦ gni , gni+k ◦ tin−2 (q) = tin (q) ◦ gni+k
i+k−2
i
i+k
(Cap-crossing isotopy) fni ◦ ti+k
◦ tin (q) = tin−2 (q) ◦ fni+k
n (q) = tn−2 (q) ◦ fn , fn
i+k
i
(Crossing-crossing isotopy) tin (p) ◦ ti+k
n (q) = tn (q) ◦ tn (p)
i i
i+1
i
(Pitchfork move) tin (1) ◦ gni+1 = ti+1
= ti+1
n (2) ◦ gn , tn (2) ◦ gn
n (1) ◦ gn .
(Rotation) rn ◦ rn0 = rn0 ◦ rn = id
0
(Cap rotation) rn−2
◦ fni ◦ rn = fni+1 , fnn−1 ◦ rn2 = fn1
EXOTIC t-STRUCTURES FOR TWO-BLOCK SPRINGER FIBRES
9
(14) (Cup rotation) rn0 ◦ gni ◦ rn−2 = gni+1 , rn02 ◦ gnn−1 = gn1
02
n−1
2
1
(15) (Crossing rotation) rn0 ◦ tin (q) ◦ rn = ti+1
n (q), rn ◦ tn (q) ◦ rn = tn (q).
By Lemma 4.1 from [7], any relation between linear tangles can be expressed as a composition of the
relations (1)-(11) above. We can generalize that to affine tangles:
Proposition 3.8. Any relation between affine tangles can be expressed as a composition of the relations
(1)-(15) above.
Proof. First, let us reduce any relation to a composition of relations (1)-(11) involving gni , fni , tin (p) for
1 ≤ i ≤ n (for the definition of gnn , fnn , tnn (p) see the proof of Lemma 3.6). Then, we can express the
relations (1)-(11) involving gnn , fnn , tnn (p) using relations (1)-(15), by a direct computation.
Let us call an isotopy linear if it fixes a segment of the form [(ζ, 0), (2ζ, 0)] for some ζ. Note that a linear
isotopy is a composition of elementary isotopies (1)-(11) (possibly involving gnn , fnn , tnn (p)) since the points
where the tangle intersects [(ζ, 0), (2ζ, 0)] stay fixed. Now, if two affine tangles are isotopic, then they
are also isotopic through a composition of two linear isotopies, which completes the proof.
For our purposes, it will be more convenient to replace the relations (13)-(15) by the equivalent set of
defining relations below.
Definition 3.9. Let sin 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.
Lemma 3.10. The following relations are equivalent to the relations (13)-(15) above.
n−2
n
i
i
n n
i
i
• snn ◦ gni = gni ◦ sn−2
n−2 , sn−2 ◦ fn = fn ◦ sn , sn ◦ tn (p) = tn (p) ◦ sn
n−1
n−1
n
n−1
n
n−1
• fn ◦ sn ◦ tn (2) ◦ sn ◦ tn (2) = fn
n
n−1
n−1
• snn ◦ tn−1
= gnn−1
n (2) ◦ sn ◦ tn (2) ◦ gn
Proof. It is straightforward to check that the above relations are satisfied. It suffices to now use the
1
identity rn = snn ◦ tn−1
n (2) ◦ · · · ◦ tn (2), and Proposition 3.8, to show that the above three relations imply
equations (13)-(15) above (given relations (1)-(11)):
0
n−3
n−2 −1
1
rn−2
◦ fni ◦ rn = t1n−2 (1) ◦ · · · ◦ tn−2
(1) ◦ (sn−2
) ◦ fni ◦ snn ◦ tn−1
n (2) ◦ · · · ◦ tn (2)
n−3
n−2 −1
n−2
1
= t1n−2 (1) ◦ · · · ◦ tn−2
(1) ◦ (sn−2
) ◦ sn−2
◦ fni ◦ tn−1
n (2) ◦ · · · ◦ tn (2)
n−3
= t1n−2 (1) ◦ · · · ◦ tn−2
(1) ◦ fni ◦ tnn−1 (2) ◦ · · · ◦ t1n (2) = fni+1
n−2
n−3
(2) ◦ · · · t1n−2 (2)
◦ tn−2
rn0 ◦ gni ◦ rn−2 = t1n (1) ◦ · · · ◦ tnn−1 (1) ◦ (snn )−1 ◦ gni ◦ sn−2
1
= t1n (1) ◦ · · · ◦ tnn−1 (1) ◦ (snn )−1 ◦ snn ◦ gni ◦ tn−3
n−2 (2) ◦ · · · tn−2 (2)
1
i+1
= t1n (1) ◦ · · · ◦ tnn−1 (1) ◦ gni ◦ tn−3
n−2 (2) ◦ · · · tn−2 (2) = gn
rn0 ◦ tin (q) ◦ rn = t1n (1) ◦ · · · ◦ tnn−1 (1) ◦ (snn )−1 ◦ tin (q) ◦ snn ◦ tnn−1 (2) ◦ · · · ◦ t1n (2)
= t1n (1) ◦ · · · ◦ tnn−1 (1) ◦ tin (q) ◦ tnn−1 (2) ◦ · · · ◦ t1n (2) = ti+1
n (q)
10
RINA ANNO, VINOTH NANDAKUMAR
1
n−1
n
1
fnn−1 ◦ rn2 = fnn−1 ◦ snn ◦ tn−1
n (2) ◦ · · · ◦ tn (2) ◦ sn ◦ tn (2) ◦ · · · ◦ tn (2)
n
n−1
n−1
n−2
1
= fnn−1 ◦ snn ◦ tn−1
n (2) ◦ sn ◦ tn (2) ◦ tn (1) ◦ tn (2) ◦ · · · ◦ tn (2)
1
◦ tn−1
n (2) ◦ · · · ◦ tn (2)
1
1
n−1
1
n−2
= fnn−1 ◦ tn−1
n (1) ◦ tn (2) ◦ · · · ◦ tn (2) ◦ tn (2) ◦ · · · ◦ tn (2) = fn
n −1
n −1
rn02 ◦ gnn−1 = t1n (1) ◦ · · · ◦ tn−1
◦ t1n (1) ◦ · · · ◦ tn−1
◦ gnn−1
n (1) ◦ (sn )
n (1) ◦ (sn )
1
n−2
n−1
= t1n (1) ◦ · · · ◦ tn−1
n (1) ◦ tn (1) ◦ · · · ◦ tn (1) ◦ tn (2)
n −1
n −1
◦ gnn−1
◦ tn−1
◦ tn−1
n (1) ◦ (sn )
n (1) ◦ (sn )
1
n−2
n−1
n−1
= t1n (1) ◦ · · · ◦ tn−1
= gn1
n (1) ◦ tn (1) ◦ · · · ◦ tn (1) ◦ tn (2) ◦ gn
2
1
n−1
n
n−1
n −1 2
n−1
1
2
rn02 ◦ tn−1
n (q) ◦ rn = (tn (1) ◦ · · · ◦ tn (1) ◦ (sn ) ) ◦ tn (q) ◦ (sn ◦ tn (2) ◦ · · · ◦ tn (2))
2
n−1
n−1
1
2
1
= (t1n (1) ◦ · · · ◦ tn−1
n (1)) ◦ tn (q) ◦ (tn (2) ◦ · · · ◦ tn (2)) = tn (q)
3.2. Framed tangles. All preceding constructions may be carried out for framed tangles. Define the
generators ĝni (resp. fˆni , resp. t̂in (l), resp. r̂ni ) as tangles gni (resp. fni , resp. tin (l), resp. rni ) with blackboard
framing. Introduce new generators ŵni (1) and ŵni (2), which correspond to positive and negative twists of
framing of the ith strand of an (n, n) identity tangle.
Definition 3.11. Define a framed linear tangle to be a framed affine tangle that isotopic to a product
of the generators ĝni , fˆni , t̂in (1), t̂in (2) for i 6= n, and ŵni (1), ŵni (2).
Definition 3.12. Consider the category AFTan (resp. FTan), with objects k for k ∈ Z≥0 , and the set
of morphisms between p and q consist of all framed affine (resp. framed linear) (p, q) tangles.
The relations for framed tangles are transformed as follows:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
fˆni ◦ ĝni+1 = id = fˆni+1 ◦ ĝni
i
i
(Reidemeister 1) fˆni ◦ t̂i±1
n (l) ◦ ĝn = ŵn (l)
t̂in (2) ◦ t̂in (1) = id = t̂in (1) ◦ t̂in (2)
i
i
i+1
i
t̂in (2) ◦ t̂i+1
n (2) ◦ t̂n (2) = id = t̂n (1) ◦ t̂n (1) ◦ t̂n (1)
i+k
i
◦ ĝni = ĝn+2
◦ ĝni+k−2
ĝn+2
i+k
i
fˆni+k−2 ◦ fˆn+2
= fˆni ◦ fˆn+2
i+k
i
ĝni+k−2 ◦ fˆni = fˆn+2
◦ ĝn+2
i+k
i
ĝni ◦ fˆni+k−2 = fˆn+2
◦ ĝn+2
i+k−2
i
ĝni ◦ t̂n−2
(l) = t̂i+k
ĝni+k ◦ t̂in−2 (l) = t̂in (l) ◦ ĝni+k
n (l) ◦ ĝn ,
i+k−2
ˆi
fˆni ◦ t̂i+k
fˆni+k ◦ t̂in (l) = t̂in−2 (l) ◦ fˆni+k
n (l) = t̂n−2 (l) ◦ fn ,
i+k
i
t̂in (l) ◦ t̂i+k
n (m) = t̂n (m) ◦ t̂n (l)
i
i+1
i+1
i
i
t̂n (1) ◦ ĝn = t̂n (2) ◦ ĝn , t̂in (2) ◦ ĝni+1 = t̂i+1
n (1) ◦ ĝn
r̂n ◦ r̂n0 = id = r̂n0 ◦ r̂n
0
r̂n−2
◦ fˆni ◦ r̂n = fˆni+1 , i = 1, . . . , n − 2; fˆnn−1 ◦ (r̂n )2 = fˆn1
r̂n0 ◦ ĝni ◦ r̂n−2 = ĝni+1 , i = 1, . . . , n − 2; (r̂n0 )2 ◦ ĝnn−1 = ĝn1
2
1
r̂n0 ◦ t̂in (l) ◦ r̂n = t̂i+1
(r̂n0 )2 ◦ t̂n−1
n (l);
n (l) ◦ (r̂n ) = t̂n (l)
We have the following additional relations for twists:
EXOTIC t-STRUCTURES FOR TWO-BLOCK SPRINGER FIBRES
(17)
(18)
(19)
(20)
(21)
(22)
11
ŵni (1) ◦ ŵni (2) = id, ŵni (l) ◦ ŵnj (k) = ŵnj (k) ◦ ŵni (l), i 6= j
ŵni (k) ◦ ĝni = ŵni+1 (k) ◦ ĝni , ŵni (k) ◦ ĝnj = ĝnj ◦ ŵni+1±1 (k), i 6= j, j + 1
fˆni ◦ ŵni (k) = fˆni ◦ ŵni+1 (k), ŵni (k) ◦ fˆnj = fˆnj ◦ ŵni−1±1 (k), i 6= j, j + 1
ŵni (k) ◦ t̂in = ŵni+1 (k) ◦ t̂in , ŵni (k) ◦ t̂jn = t̂jn ◦ ŵni (k), i 6= j, j + 1
t̂in ◦ ŵni (k) = fˆni ◦ ŵni+1 (k), ŵni (k) ◦ fˆnj = t̂jn ◦ ŵni (k), i 6= j, j + 1
ŵni (k) ◦ r̂n = r̂n ◦ ŵni−1 (k), ŵni (k) ◦ r̂n0 = r̂n0 ◦ ŵni+1 (k)
Note how the Reidemeister 1 move (2) is the only relation between the non-twist generators that differs
from the relations in ATan.
Proposition 3.13. Any isotopy of affine framed tangles is equivalent to a composition of elementary
isotopies (1)-(22).
Proof. There is a forgetful functor from the 2-category of framed tangles and their isotopies to the 2category of non-framed tangles and their isotopies, which forgets the framing. Thus, for every isotopy
there is a composition of relations (1)-(15) (in ATan) which differs only in framing, and that can be ruled
out by the commutation laws (17)-(22) (in AFTan) of twists with all other generators.
Lemma 3.10 still holds in this context, after replacing snn by it’s “framed” version ŝnn .
4. Functors associated to affine tangles
Definition 4.1. Recall that AFTan (resp Tan, FTan) has objects {k} for k ∈ Z≥0 , and the set of
morphisms between {p} and {q} consists of all framed affine (resp. framed linear) (p, q) tangles. Define
the category AFTanm (resp. Tanm , FTanm ) to be the full subcategory of AFTan (resp. Tan, FTan)
with objects {m + 2k} for k ∈ Z≥0 .
Definition 4.2. A “weak representation” of the category AFTanm is an assignment of a triangulated
category Ck for each k ∈ Z≥0 , and a functor Ψ(α) : Dp → Dq for each framed affine (m+2p, m+2q)-tangle,
so that the relations between tangles hold for these functors: i.e. if β is an (m + 2q, m + 2r) tangle, then
there is an isomorphism Ψ(β) ◦ Ψ(α) ' Ψ(β ◦ α).
Similarly one can define the notion of a “weak representation” of the categories Tanm , FTanm . Following
Section 4 of [1] (in which the m = 0 case was treated), the goal of this section is to construct a weak
representation of AFTanm using the categories Dk .
First we recall the weak representation of Tanm constructed by Cautis and Kamnitzer (using slightly
different categories) in Sections 4 and 5 of [7]. Then we will adapt their construction to our setting.
4.1. Cautis and Kamnitzer’s representation of the tangle calculus.
en = Db (Coh(Ym+2n )). In section 4 of [7], Cautis and Kamnitzer construct a weak representation
Let D
en . In fact, Cautis and Kamnitzer construct a weak representation of
of Tanm using the categories D
the full category Tan (which gives a weak representation of the subcategory FTanm ). Also, Cautis and
Kamnitzer deal with the C∗ -equivariant derived categories; but we will omit this C∗ -equivariance as we
do not need it.
Recall the definitions of Fourier-Mukai transforms (see Section 3 of [7]). Here all pullbacks, pushforwards,
Homs and tensor products of sheaves will denote the corresponding derived functors.
12
RINA ANNO, VINOTH NANDAKUMAR
Definition 4.3. Let X, Y be two complex algebraic varieties, and let π1 : X × Y → X, π2 : X × Y → Y
denote the two projections. Given a “Fourier-Mukai” kernel T ∈ Db (Coh(X × Y )), define the FourierMukai transform ΨT : Db (Coh(X)) → Db (Coh(Y )) by ΨT (F) = π2∗ (π1∗ F ⊗ T ).
i
i
⊂ Ym+2n × Ym+2n as follows:
⊂ Ym+2n , Zm+2n
Definition 4.4. Define the subvarieties Xm+2n
i
• Xm+2n
= {(L1 ⊂ L2 ⊂ · · · ⊂ Lm+2n )| Li+1 = z −1 (Li−1 )}.
i
• Zm+2n
= {(L, L0 ) ∈ Ym+2n × Ym+2n | Lj = L0j ∀ j 6= i}
i
Let j : Xm+2n
→ Ym+2n be the embedding of the divisor. Note that there is a natural surjection
i
→ Ym+2n−2 defined by p(L1 , L2 , · · · , Lm+2n ) = (L1 , · · · , Li−1 , zLi+2 , · · · , zLm+2n ); thus we may
p : Xm+2n
i
ei be the tautological vector bundle on Ym+2n
also view Xm+2n
as a subvariety of Ym+2n−2 × Ym+2n . Let V
ei /V
ei−1 denote the quotient line bundle.
corresponding to the vector space Li ; let Eei = V
Definition 4.5. Define the following Fourier-Mukai kernels:
i
i
= OXm+2n
⊗ π2∗ Eei ∈ Db (Coh(Ym+2n−2 × Ym+2n )),
Gem+2n
−1
i
i
Fem+2n
= OXm+2n
⊗ π1∗ Eei+1
∈ Db (Coh(Ym+2n × Ym+2n−2 ))
i
i
Tem+2n
(1) = OZm+2n
∈ Db (Coh(Ym+2n × Ym+2n ))
−1
i
i
Tem+2n
(2) = OZm+2n
⊗ π1∗ Eei+1
⊗ π2∗ Eei ∈ Db (Coh(Ym+2n × Ym+2n ))
Definition 4.6. Define the functors
i
eim+2n = Ψ(g
e m+2n
G
) = ΨGei
i
i
en−1 → D
en , Fem+2n
e m+2n
:D
= Ψ(f
) = ΨFei
en → D
en−1
:D
i
en → D
en , Tem+2n
e im+2n (2)) = Ψ e i
:D
(2) = Ψ(t
T
en → D
en
:D
m+2n
i
e im+2n (1)) = Ψ e i
Tem+2n
(1) = Ψ(t
T
m+2n (1)
m+2n
m+2n (2)
eim+2n : D
en−1 → D
en admits the following alternate description: G
eim+2n (F) =
It is easy to see that G
e
e
en−1 . Similarly, the functor Fei
j∗ (p∗ F ⊗ Eei ) for F ∈ D
m+2n : Dn → Dn−1 admits the following description:
−1
i
e
ei
Fem+2n
(G) = p∗ (j ∗ G ⊗ Eg
i+1 ) for G ∈ Dn . The following calculation of the left and right adjoints to Gm+2n ,
(2), from [7] will be of use to us.
(1), Tei
and an alternative description of the functors Tei
m+2n
m+2n
R
L
ei
ei
ei
ei
Lemma 4.7. We have (G
m+2n ) = Fm+2n [−1] and (Gm+2n ) = Fm+2n [1]. Also, for F ∈ Dn , there are
R
L
ei
ei
ei
ei
ei
ei
distinguished triangles G
m+2n (Gm+2n ) F → F → Tm+2n (1)F and Tm+2n (2)F → F → Gm+2n (Gm+2n ) .
Proof. This follows from Lemma 4.4, and Theorem 4.6 in [7].
Recall that any linear tangle can be expressed as a composition of the above generators, and that any
relation between linear tangles can be expressed via the relations (1) - (11) in Section 2.2. Hence defining
functors Ψ(α) for each (m + 2p, m + 2q)-tangle α, which are compatible under composition, is equivalent
to defining functors for each of the generators, satisfying the relations (1) - (11) (up to isomorphism).
i
i
e m+2n
e m+2n
e i
e i
Proposition 4.8. The functors Ψ(f
), Ψ(g
), Ψ(t
m+2n (1)), Ψ(tm+2n (2)) satisfy the relations (1) −
(11). Thus, given a linear (m + 2p, m + 2q) tangle, α, written as a product of generators, we can define
e
Ψ(α)
by composition (and up to isomorphism, the result does not depend on the choice of decomposition
en .
as a product of generators). This gives a weak representation of FTanm using the categories D
Proof. See Theorem 4.2 in [7].
EXOTIC t-STRUCTURES FOR TWO-BLOCK SPRINGER FIBRES
13
4.2. Constructing functors Ψ(α) : Dp → Dq indexed by linear tangles.
Remark 4.9. Our goal was to define a functor Ψ(α) : Dp → Dq for each (m + 2p, m + 2q)-tangle α,
which are compatible under composition. Using the results of Cautis and Kamnitzer in the last section,
e
ep → D
eq , that are compatible under composition. Recall also that we have
we have a functor Ψ(α)
:D
en for each n. Thus one may hope to
a map in : Un → Ym+2n , and hence also a map Rin∗ : Dn → D
e
ep → D
eq to a functor Ψ(α) : Dp → Dq - i.e. to construct a functor Ψ(α)
“descend” the functor Ψ(α)
:D
e
such that Riq∗ ◦ Ψ(α) = Ψ(α)
◦ Rip∗ . While it is possible to construct such functors, it is not clear why
we have an isomorphism Ψ(β ◦ α) ' Ψ(β) ◦ Ψ(α); this does not follow directly from the isomorphism
e ◦ α) ' Ψ(β)
e
e
Ψ(β
◦ Ψ(α).
So instead we will imitate the construction used by Cautis and Kamnitzer, and
imitate their proof to deduce that Ψ(β ◦ α) ' Ψ(β) ◦ Ψ(α).
Define the variety Xn,i := Sn ×slm+2n T ∗ Pi,n ×Pi,n Bn :
Xn,i = Sn ×slm+2n T ∗ Pi,n ×Pi,n Bn ={(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n ), x |
x ∈ Sn , xVi+1 ⊂ Vi−1 , xVj ⊂ Vj−1 ∀ j}
We have a P1 -bundle πn,i : Xn,i → Sn ×slm+2n T ∗ Pi,n ' Sn−1 ×slm+2n−2 T ∗ Bn−1 = Un−1 , and the embedding
of the divisor jn,i : Xn,i → Sn ×slm+2n T ∗ Bn = Un . Thus we can view Xn,i can be viewed as a subvariety
of Un−1 × Un . Let Vk denote the tautological vector bundle on Sn ×slm+2n T ∗ Bn corresponding to Vk ; and
define the line bundle Ek = Vk /Vk−1 .
Definition 4.10. Define the following Fourier-Mukai kernels:
i
Gm+2n
= OXn,i ⊗ π2∗ Ei ∈ Db (Coh(Un−1 × Un )),
−1
i
Fm+2n
= OXn,i ⊗ π1∗ Ei+1
∈ Db (Coh(Un × Un−1 ))
Definition 4.11. Define the functors:
i
i
Gim+2n = Ψ(gm+2n
) = ΨGm+2n
: Dn−1 → Dn
i
i
i
Fm+2n
= Ψ(fm+2n
) = ΨFm+2n
: Dn → Dn−1
Remark 4.12. A priori, the functor Gim+2n maps Db (Coh(Un−1 )) to Db (Coh(Un )). However, it is easy to
see that Gim+2n maps the subcategory Dn−1 = Db (CohBzn−1 (Un−1 )) ⊂ Db (Coh(Un−1 )) to the subcategory
i
Dn = Db (CohBzn (Un )) ⊂ Db (Coh(Un−1 )); similarly Fm+2n
maps Dn to Dn−1 .
It is easy to see that Gim+2n : Dn−1 → Dn admits the follow alternate description: Gim+2n (F) =
∗
i
jn,i∗ (πn,i
F ⊗ Ek ) for F ∈ Dn−1 . Similarly, the functor Fm+2n
: Dn → Dn−1 can be expressed as follows:
−1
i
∗
Fm+2n (G) = πn,i∗ (jn,i G ⊗ Ek+1 ) for G ∈ Dn . We will define the functors Ψ(tim+2n (1)) and Ψ(tim+2n (2)) in
the following section, by proving an analogue of Lemma 4.7 above.
In the following section we state a result of Anno, which will help us prove that the functors Ψ(α) (where
α is a generator) satisfy the relations (1) - (11) from Section 2.2 (up to isomorphism).
4.3. Tangle representations via spherical functors. Following Anno ([2]), we recall the definition
of spherical, and strongly spherical, functors.
Definition 4.13. Suppose we have two triangulated categories C and D, and a functor S : C → D, with
a left adjoint L : D → C and a right adjoint R : D → C. Then S is a spherical functor if:
14
RINA ANNO, VINOTH NANDAKUMAR
• The cone of the natural map id → RS is an exact auto-equivalence of C. Denote this functor by
FS .
• The natural map R → FS L induced by R → RSL is an isomorphism of functors.
In this case, define the twist functor TS (1), and the inverse twist functor TS (2) by the following exact
triangles. Here SR → id and id → SL denote the natural maps.
SR → id → TS (1);
TS (2) → id → SL
Definition 4.14. A spherical functor S : C → D is said to be “strongly spherical” if FS = [−2].
See Section 2 and 3 of [2] for some properties, and examples of spherical functors. The reason that we
will need to deal with them is the following result, which constructs functors indexed by tangles between
categories under certain hypotheses.
Theorem 4.15. Suppose we have a triangulated category Cm+2k for each k ∈ Z≥0 ; and for each k ≥ 1,
i
1 ≤ i < m + 2k, a strongly spherical functor Sm+2k
: Cm+2k−2 → Cm+2k . Let Lim+2k be it’s left adjoint;
i
i
i
Rm+2k
be it’s right adjoint; Tm+2k
(1) it’s twist, and Tm+2k
(2) it’s inverse twist. If the following conditions
hold:
i
(1) Sm+2k
Li±1
m+2k [−1] ' id
i+l
i+l−2
i
i
(2) Sm+2k+2 Sm+2k
' Sm+2k+2
Sm+2k
for l ≥ 2.
i+l
i+l−2
i+l
i
i
i
i
◦ Li+l−2
(3) Sm+2k ◦ Lm+2k ' Lm+2k+2 ◦ Sm+2k+2 , Sm+2k
m+2k ' Lm+2k+2 ◦ Sm+2k+2 for l ≥ 2.
then assign:
i
i
i
i
• Ψ(gm+2k
) ' Sm+2k
, Ψ(fm+2k
) = Lim+2k [−1] ' Rm+2k
[1]
i
i
i
i
• Ψ(tm+2k (1)) = Tm+2k (1), Ψ(tm+2k (2)) = Tm+2k (2)
i
i
• Ψ(wm+2k
(1)) = [−1], Ψ(wm+2k
(−1)) = [1]
These functors will give a weak representation of FTanm .
Proof. See the proof of Claim 2 from Section 4 in [2]. The set-up in Claim 2 is slightly different; there
one has a category Ck for each k ≥ 0 (which is essentially the m = 0 case). However, the proof can be
easily adapted to this setting.
i
4.4. Checking the tangle relations. We will apply the above Theorem with Cm+2k = Dk , and Sm+2k
=
i
i
Gm+2k . So we will need to prove that Gm+2n : Dn−1 → Dn are spherical functors, and check the 3 relations
from Theorem 4.15. To do this, we will imitate the techniques that Cautis and Kamnitzer use to prove
Theorem 4.8.
Lemma 4.16. Recall that we have the inclusion of the divisor jn,i : Xn,i → Un , as well as the P1 -bundle
πn,i : Xn,i → Un−1 .
−1
(1) We have OUn (Xn,i ) ' Ei+1
⊗ Ei
−1
−1
−1 −1
∗
∗
(2) We have ωXn,i ⊗ jn,i ωUn ' jn,i
(Ei+1
⊗ Ei ) ' ωXn,i ⊗ πn,i
ωUn−1
i
R
i
i
L
i
(3) We have (Gm+2n ) = Fm+2n [−1], and (Gm+2n ) = Fm+2n [1].
Proof. For the first two statements, see the argument used in parts (i) and (ii) of Lemma 4.3 in [7]. The
third statement follows from the first two using the argument in Lemma 4.4 in [7].
EXOTIC t-STRUCTURES FOR TWO-BLOCK SPRINGER FIBRES
15
i
i
(2) = Ψ(tim+2n (2)) via the
(1) = Ψ(tim+2n (1)) and Tm+2n
Definition 4.17. Define the functors Tm+2n
distinguished triangles:
i
Gim+2n (Gim+2n )R → id → Tm+2n
(1),
i
Tm+2n
(2) → id → Gim+2n (Gim+2n )L
For future use, we will need the following Lemma (which is an adaptation of Lemma 5.1 of [7] to our
setting).
Lemma 4.18. For i 6= j, the varieties Xn,i and Xn,j intersect transversely inside Un .
Proof. We will view Un (and Xn,i , Xn,j ) as a subvariety of G ×B n, and compute tangent spaces to Xn,i
and Xn,j at points in Xn,i ∩ Xn,j to show transversality.
Given (g, x) ∈ G ×B n; first we will calculate the tangent space T(g,x) (G ×B n). Given X1 ∈ g, X2 ∈ n, a
curve through (g, x) in G × n with tangent direction (g · X1 , X2 ) is (g · exp(X1 ), x + X2 ). Infinitesmally,
(g · exp(X1 ), x + X2 ) = (g, x) in G ×B n provided that X1 ∈ b (ie. exp(X1 ) ∈ B), and
exp(X1 )(x + X2 )exp(−X1 ) ≈ x
Discarding non-linear powers of , the latter translates to x + (X2 + [X1 , x]) = x, ie. X2 = −[X1 , x].
Thus the kernel of the map g ⊕ n = T(g,x) (G × n) T(g,x) (G ×B n) is the subspace {(X, −[X, x])|X ∈ b},
so:
g⊕n
T(g,x) (G ×B n) '
{(X, −[X, x]) | X ∈ b}
B
Suppose (g, x) ∈ G × n lies in Un ; or equivalently, that x̃ := gxg −1 ∈ Sn . Now given (X1 , X2 ) ∈
T(g,x) (G ×B n), we have that (X1 , X2 ) ∈ T(g,x) (Un ) when the curve (g · exp(X1 ), x + X2 ) lies in Un .
precisely when g · exp(X1 )(x + X2 )exp(−X1 ) · g −1 ∈ Sn (infinitesmally). Discarding non-linear powers
of , this is equivalent to saying that
g · (x + (X2 + [X1 , x]) · g −1 ∈ Sn
Since gxg −1 ∈ Sn , this is equivalent to X2 + [X1 , x] ∈ g −1 · Cn · g (recall that Sn = zn + Cn where Cn is a
vector subspace). Thus:
{(X1 , X2 ) ∈ g ⊕ n | X2 + [X1 , x] ∈ g −1 Cn g}
{(X, −[X, x]) | X ∈ b}
{(X, Y ) ∈ g ⊕ Cn | [X, x̃] + Y ∈ g · n}
'
g·b⊕0
T(g,x) (Un ) '
For the last isomorphism, use the substitution X = −gX1 g −1 , Y = g(X2 + [X1 , x])g −1 . Recall from the
discussion in Section 1.4 of [8] that the map π : g ⊕ Cn → g, π(X, Y ) = [X, x̃] + Y is surjective. Hence:
dim(Tg,x (Un )) = dim(n) + dim(Cn ) − dim(b)
In particular, this shows that Un is smooth. Now suppose that (g, x) ∈ Xn,i ∩ Xn,j . It is clear that
Xn,i = Un ∩ (G ×B ni ), where ni ⊂ n is the nilradical of the minimal parabolic corresponding to i. The
above argument is valid after replacing n with ni , and we obtain:
T(g,x) (Xn,i ) '
{(X, Y ) ∈ g ⊕ Cn | [X, x̃] + Y ∈ g · ni }
g·b⊕0
T(g,x) (Xn,j ) '
{(X, Y ) ∈ g ⊕ Cn | [X, x̃] + Y ∈ g · nj }
g·b⊕0
16
RINA ANNO, VINOTH NANDAKUMAR
Using the surjectivity of π, it is clear that T(g,x) (Xn,i ) and T(g,x) (Xn,j ) are distinct co-dimension 1 subspaces
in T(g,x) (Un ). Hence T(g,x) (Xn,i ) + T(g,x) (Xn,j ) = T(g,x) (Un ), and Xn,i and Xn,j intersect transversely in
Un .
Corollary 4.19. The following intersections are transverse:
−1
−1
(1) π12
(Xn,i ) ∩ π23
(Xn,j ) inside Un−1 × Un × Un−1 for i 6= j.
−1
−1
(2) π12 (Xn,i ) ∩ π23 (Xn+1,j ) inside Un−1 × Un × Un+1 .
Proof. Both statements follow using Lemma 5.3 from [7]; for the first, we also need Lemma 4.18.
Now we may check the first relation from [2] (which corresponds to first of the 11 relations in Section
2.2).
i+1
i
i
Proposition 4.20. Fm+2n
◦ Gi+1
m+2n ' id ' Fm+2n ◦ Gm+2n
Proof. This follows from the argument used in Proposition 5.6 of [7] (here one has to use the first statement
from Corollary 4.19).
i
Proposition 4.21. We have Fm+2n
◦ Gim+2n ' [−1] ⊕ [1].
Proof. This is an analogue of Corollary 5.10 from [7], and can be proved using the same arguments (see
the proofs of Proposition 5.8 and Theorem 5.9 in [7]).
Corollary 4.22. The functor Gim+2n : Dn−1 → Dn is a strongly spherical functor.
Proof. The first condition from Definition 4.13 states that we should have a triangle id → (Gim+2n )R Gim+2n →
i
[−2], ie. a triangle id → Fm+2n
Gim+2n [−1] → [−2]. This follows from Proposition 4.20.
The second condition states that the natural map (Gim+2n )R → (Gim+2n )L [−2] is an isomorphism; this
i
i
[1].
follows from Lemma 4.16, since (Gim+2n )R = Fm+2n
[−1], (Gim+2n )L = Fm+2n
Proposition 4.23. The following relations hold:
i+l
(1) Gm+2k+2
◦ Gim+2k ' Gim+2k+2 Gi+l−2
m+2k for l ≥ 2.
i+l−2
i+l
i+l−2
i+l
i
i
Gim+2k+2 for l ≥ 2
(2) Gm+2k ◦ Fm+2k ' Fm+2k+2 Gm+2k+2 , Gim+2k ◦ Fm+2k
' Fm+2k+2
Proof. This follows from the argument used in Proposition 5.16 of [7] (here one has to use the second
statement from Corollary 4.19).
Now we have verified the conditions of Theorem 4.15, so we deduce that:
Theorem 4.24. The assignments
i
i
i
Ψ(gm+2n
) = Gim+2n , Ψ(fm+2n
) = Fm+2n
i
i
Ψ(tim+2n (1)) = Tm+2n
(1), Ψ(tim+2n (2)) = Tm+2n
(2)
i
i
Ψ(wm+2n
(1)) = [−1], Ψ(wm+2n
(−1)) = [1]
give rise to a weak representation of FTanm using the categories Dk .
EXOTIC t-STRUCTURES FOR TWO-BLOCK SPRINGER FIBRES
17
4.5. Functors Ψ(α) : Dp → Dq indexed by affine tangles. At this point, we have constructed a
functor Ψ(α) : Dp → Dq for each framed linear (m + 2p, m + 2q)-tangle α. To extend this construction
to framed affine tangles, it suffices to construct a functor Ψ(sm+2n
m+2n ) : Dn → Dn satisfying the relations in
−1
m+2n
Lemma 3.10. Define Sn (F) = F ⊗ Em+2n , and let Ψ(sm+2n ) := Sn . The relations that we must check are
the following:
Proposition 4.25. The following identities hold, where 1 ≤ i ≤ m + 2n − 2, 1 ≤ p ≤ 2:
•
•
•
•
•
i
i
◦ Sn
' Fm+2n
Sn−1 ◦ Fm+2n
i
i
Sn ◦ Gm+2n ' Gm+2n ◦ Sn−1
i
i
(p) ◦ Sn
(p) ' Tm+2n
Sn ◦ Tm+2n
m+2n−1
m+2n−1
m+2n−1
m+2n−1
Fm+2n ◦ Sn ◦ Tm+2n (2) ◦ Sn ◦ Tm+2n
(2) ' Fm+2n
m+2n−1
m+2n−1
m+2n−1
Sn ◦ Tm+2n (2) ◦ Sn ◦ Tm+2n (2) ◦ Gm+2n ' Gm+2n−1
m+2n
Proof. The first three identities are clear. To prove the fourth identity, we calculate as follows (the
m+2n−1
last identity is proved similarly). Let Q ∈ Dn ; since we have Tm+2n
(2)Q ' {Q → Gm+2n−1
◦
m+2n
m+2n−1,L
m+2n−1
m+2n−1
Gm+2n
(Q)} = {Q → Gm+2n ◦ Fm+2n (Q)[1]} (using Lemma 4.7), we compute:
m+2n−1
−1
m+2n−1
−1
−1
Gm+2n−1
◦ Fm+2n
(Q ⊗ Em+2n
) = Gm+2n
◦ (πn,m+2n−1∗ (i∗n,m+2n−1 (Q ⊗ Em+2n
) ⊗ Em+2n
))
m+2n
∗
−2
= in,m+2n−1∗ [Em+2n−1 ⊗ πn,m+2n−1
πn,m+2n−1∗ (i∗n,m+2n−1 Q ⊗ Em+2n
)]
m+2n−1
−1
m+2n−1
−1
Tm+2n
(2) ◦ Sn (Q) = {Q ⊗ Em+2n
→ Gm+2n−1
◦ Fm+2n
(Q ⊗ Em+2n
)[1]}
m+2n
m+2n−1
m+2n−1
−3
Fm+2n
◦ Sm+2n ◦ Tm+2n
(2) ◦ Sm+2n (Q) = {πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 Q) →
−2
∗
−2
πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 in,m+2n−1∗ (Em+2n−1 ⊗ πn,m+2n−1
πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 Q)))[1]}
Note that we have an exact triangle OUn (−Xn,m+2n−1 )[−1] ⊗ F → i∗n,m+2n−1 in,m+2n−1∗ F → F, and that
−1
⊗ Em+2n . This gives us the following exact triangle, where we abbreviate
OUn (−Xn,m+2n−1 ) = Em+2n−1
−2
R = πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 Q):
−1
∗
−2
∗
πn,m+2n−1∗ (Em+2n
⊗ πn,m+2n−1
R) → πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 in,m+2n−1∗ (Em+2n−1 ⊗ πn,m+2n−1
R))
−2
∗
→ πn,m+2n−1∗ (Em+2n
⊗ Em+2n−1 ⊗ πn,m+2n−1
R)
−1
∗
We claim that πn,m+2n−1∗ (Em+2n
⊗ πn,m+2n−1
R) = 0. This implies that the second and third terms in the
above triangle are equal; so continuing the above computation (and using the identity πn,m+2n−1∗ (A ⊗
∗
πn,m+2n−1
B) = B ⊗ πn,m+2n−1∗ A):
m+2n−1
m+2n−1
−3
Fm+2n
◦ Sm+2n ◦ Tm+2n
(2) ◦ Sm+2n (Q) = {πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 Q) →
−2
∗
−2
⊗ Em+2n−1 ⊗ πn,m+2n−1
πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 Q))[1]}
πn,m+2n−1∗ (Em+2n
−2
−2
−3
⊗ i∗n,m+2n−1 Q)[1]}
⊗ i∗n,m+2n−1 Q) → πn,m+2n−1∗ (Em+2n
⊗ Em+2n−1 ) ⊗ πn,m+2n−1∗ (Em+2n
= {πn,m+2n−1∗ (Em+2n
Here note that since πn,m+2n−1 is proper, πn,m+2n−1∗ commutes with the functors of Grothendieck-Serre
duality on the categories Db (Coh(Xn,m+2n−1 )) and Db (Coh(Un−1 )). We have ωXn,m+2n−1 = Em+2n−1 ⊗
−1
Em+2n
, and ωUn−1 = OUn−1 since Un−1 is symplectic. Thus (noting dim(Xn,m+2n−1 ) = dim(Un−1 ) + 1):
−2
πn,m+2n−1∗ (Em+2n
⊗ Em+2n−1 )[1] = πn,m+2n−1∗ (RHom(Em+2n , ωXn,m+2n−1 ))[1]
= RHom(πn,m+2n−1∗ (Em+2n ), ωUn−1 ) = πn,m+2n−1∗ (Em+2n )∨
Consider the locally free sheaf E on Un−1 = Sn ×slm+2n T ∗ Pk,n corresponding to V /Im(x) ' Ker(x); it
is clear that the associated P1 bundle is Xn,m+2n−1 (i.e. P(E) ' Xn,m+2n−1 ). Under this identification,
18
RINA ANNO, VINOTH NANDAKUMAR
the vector bundle Em+2n on Un−1 corresponds to O(1) on P(E). Since πn,m+2n−1∗ (O(1)) = E, we obtain
πn,m+2n−1∗ (Em+2n ) = Ker(x). Returning to the above computation:
m+2n−1
m+2n−1
Fm+2n
◦ Sm+2n ◦ Tm+2n
(2) ◦ Sm+2n (Q) =
−3
−2
{πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 Q) → πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 Q) ⊗ Ker(x)∨ }
−1
−1
∗
Note that from the exact sequence 0 → Em+2n
→ πn,m+2n−1
Ker(x)∨ → Em+2n−1
→ 0, we have an exact
−3
−2
−1
−2
∗
∗
∨
∗
triangle Em+2n ⊗in,m+2n−1 Q → Em+2n ⊗πn,m+2n−1 (Ker(x)) ⊗in,m+2n−1 Q → Em+2n−1 ⊗Em+2n
⊗i∗n,m+2n−1 Q.
−3
Taking the push-forward under πn,m+2n−1 gives us the exact triangle πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 Q) →
−2
−1
−2
πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 Q) ⊗ (Ker(x))∨ → πn,m+2n−1∗ (Em+2n−1
⊗ Em+2n
⊗ i∗n,m+2n−1 Q). Thus:
−3
−2
{πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 Q) → πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 Q) ⊗ Ker(x)∨ }
−1
−2
−1
m+2n−1
= πn,m+2n−1∗ (Em+2n−1
⊗ Em+2n
⊗ i∗n,m+2n−1 Q) = πn,m+2n−1∗ (Em+2n
⊗ i∗n,m+2n−1 Q) = Fm+2n
Q
−1
−1
In the second equality above we have used the fact that the line bundle Em+2n−1
⊗ Em+2n
is trivial when
m+2n−1
m+2n−1 m+2n−1
restricted to the divisor Xn,m+2n−1 . Using the fact that Fm+2n
= Fm+2n Tm+2n [1] (see [2]), we
m+2n−1
m+2n−1
m+2n−1
m+2n−1
conclude that Fm+2n ◦ Sn ◦ Tm+2n (2) ◦ Sn ' Fm+2n ◦ Tm+2n (1), as desired.
To summarize:
Theorem 4.26. The assignments
i
i
i
Ψ(gm+2n
) = Gim+2n , Ψ(fm+2n
) = Fm+2n
i
i
Ψ(tim+2n (1)) = Tm+2n
(1), Ψ(tim+2n (2)) = Tm+2n
(2)
i
i
Ψ(wm+2n
(1)) = [−1], Ψ(wm+2n
(−1)) = [1]
Ψ(sm+2n
m+2n ) = Sn
give rise to a weak representation of AFTanm using the categories Dk .
5. The exotic t-structure on Dn
First we recall that the construction of the exotic t-structure on Dn from [3] is given by the following.
Let Baf f denotes the braid group attached to the affine Weyl group Waf f = W n Λ, where W is the Weyl
group of g = slm+2n , and Λ is the weight lattice. Let BCox
af f ⊂ Baf f denote the braid group attached to the
Cox
Cox
Waf f = W n Q where Q is the root lattice. Denote by B+
af f ⊂ Baf f denote the semigroup generated by
Cox
the lifts of the simple reflections s̃α in the Coxeter group Waf
f .
Using Bezrukavnikov and Mirkovic’s construction (see Sections 1.1.1 and 1.3.2 of [3]), there exists a weak
action of the affine braid group Baf f on Dn (i.e. for every b ∈ Baf f , there exists a functor Ψ(b) : Dn → Dn ,
such that Ψ(b1 b2 ) ' Ψ(b1 ) ◦ Ψ(b2 )). This action is related to that from the previous section using the
following result, which is easy to check.
Lemma 5.1. Baf f can be identified with the group of all bijective (m + 2n, m + 2n) affine tangles (i.e.
where each strand connects a point in the inner circle with a point in the outer circle). Under this
identification, the action of Baf f on Dn coincides with the action coming from Theorem 4.26.
Following Bezrukavnikov and Mirkovic (see section 1.5 of [3]), the exotic t-structure on Dn is defined as
follows:
EXOTIC t-STRUCTURES FOR TWO-BLOCK SPRINGER FIBRES
19
Dn≥0 = {F | RΓ(Ψ(b−1 )F) ∈ D≥0 (Vect) ∀ b ∈ B+
af f }
Dn≤0 = {F | RΓ(Ψ(b)F) ∈ D≤0 (Vect) ∀ b ∈ B+
af f }
Proposition 5.2. The functor Gim+2n : Dn−1 → Dn is t-exact with respect to the exotic t-structures on
the two categories.
Proof. We must prove that Gim+2n is right t-exact and left t-exact; or equivalently that (Gim+2n )L =
i
i
[−1] is left t-exact (see Lemma 4.7). Let F ∈ Dn≤0 . First,
[1] is right t-exact and (Gim+2n )R = Fm+2n
Fm+2n
≤0
i
i
to prove that Fm+2n
[1] is right t-exact, we must show that Fm+2n
[1]F ∈ Dn−1
:
i
RΓ(Ψ(b)Fm+2n
F[1]) ∈ D≤0 (Vect) ∀ b ∈ B0+
af f
i
⇔ RΓ(Fm+2n
Ψ(i (b))F[1]) ∈ D≤0 (Vect) ∀ b ∈ B0+
af f
i
i
Ψ(i (b)) from Lemma 5.4. Since Ψ(i (b))F ∈ Dn≤0 as i (b) ∈ B+
= Fm+2n
Above, we have Ψ(b)Fm+2n
af f ,
i
≤0
i
L
i
it suffices to show that RΓ(Fm+2n F[1]) ∈ D (Vect). Since (Gm+2n ) = Fm+2n [1], this follows from the
≤0
.
statement in Lemma 5.3 that RΓ(Gim+2n F 0 ) ∈ D≤0 (Vect) for F 0 ∈ Dn−1
≥0
i
i
Similarly we can show that Fm+2n
[−1] is left t-exact, i.e. that given G ∈ Dn≥0 , we have Fm+2n
[−1]G ∈ Dn−1
:
i
RΓ(Ψ(b−1 )Fm+2n
G[−1]) ∈ D≥0 (Vect) ∀ b ∈ B0+
af f
i
⇔ RΓ(Fm+2n
Ψ(ηi (b)−1 )G[−1]) ∈ D≥0 (Vect) ∀ b ∈ B0+
af f
i
i
Ψ(ηi (b)−1 ) from Lemma 5.4. Since Ψ(ηi (b)−1 )G ∈ Dn≥0 as ηi (b) ∈
= Fm+2n
Above, we have Ψ(b−1 )Fm+2n
i
≥0
i
R
i
B+
af f , it suffices to show that RΓ(Fm+2n G[−1]) ∈ D (Vect). Since (Gm+2n ) = Fm+2n [−1], this follows
≥0
from the statement in Lemma 5.3 that RΓ(Gim+2n G 0 ) ∈ D≥0 (Vect) for G 0 ∈ Dn−1
.
≤0
≥0
Lemma 5.3. Given F 0 ∈ Dn−1
, G 0 ∈ Dn−1
, we have RΓ(Gim+2n F 0 ) ∈ D≤0 (Vect) and RΓ(Gim+2n G 0 ) ∈
D≥0 (Vect).
Proof. From the above, we have that the functors Gim+2n are conjugate by the t-exact, invertible functor
Rn ; thus it suffices to prove that G1m+2n is left exact, and that Gm+2n−1
is right exact.
m+2n
R
≥0
First we show that Gm+2n−1
is right exact, or equivalently that (Gm+2n−1
m+2n
m+2n ) is left exact. Let F ∈ Dn .
m+2n−1
Em+2n = πn,m+2n−1∗ (i∗n,m+2n−1 OUn ) = OUn−1 :
Since Gm+2n−1L
(Em+2n [−1]) = Fm+2n
m+2n
R
m+2n−1 L
m+2n−1 R
RΓ((Gm+2n−1
m+2n ) F) = RHom((Gm+2n ) Em+2n [−1], (Gm+2n ) F)
m+2n−1 R
= RHom(Em+2n [−1], Gm+2n−1
m+2n (Gm+2n ) F)
−1
m+2n−1 R
= RΓ(Em+2n
[1] ⊗ Gm+2n−1
m+2n (Gm+2n ) F)
m+2n−1
m+2n−1 R
m+2n−1 L
−1
−1
⊗ Gm+2n−1
Thus it suffices to prove that Em+2n
m+2n (Gm+2n ) F = Em+2n ⊗ Gm+2n (Gm+2n ) F[2] ∈
m+2n−1
m+2n−1
m+2n−1
Dn≥−1 . But we have a distinguished triangle Tm+2n (2)F → F → Gm+2n (Gm+2n )L F; and hence
−1
m+2n−1
−1
m+2n−1 L
a distinguished triangle Em+2n
⊗ Tm+2n
(2)F → F → Em+2n
⊗ Gm+2n−1
m+2n (Gm+2n ) F. The functor
−1
Em+2n
⊗ · corresponds to the braid sm+2n
m+2n which leaves the first m + 2n − 1 vertices in place, and winds
the last vertex counter-clockwise around the circle underneath the other strands. Using the identity
m+2n
m+2n−1
−1
−1
m+2n−1
1
rm+2n
= sm+2n
(2)
m+2n ◦ tm+2n (2) ◦ · · · tm+2n (2), we see that the functors Em+2n ⊗ · and Em+2n ⊗ Tm+2n
−1
m+2n−1
−1
are left exact since they correspond to negative braids. Thus Em+2n ⊗ Tm+2n (2)F, Em+2n ⊗ F ∈ Dn≥0 ,
20
RINA ANNO, VINOTH NANDAKUMAR
−1
m+2n−1 L
≥−1
, as
using the long exact sequence of cohomology we obtain that Em+2n
⊗ Gm+2n−1
m+2n (Gm+2n ) F ∈ Dn
required.
Using similar techniques one shows that G1m+2n is left exact, or equivalently that (G1m+2n )L is right exact.
1
Let G ∈ Dn≤0 . First we claim that (G1m+2n )L E1 = OUn−1 . So Fm+2n
E1 = πn,1∗ (i∗n,1 E1 ⊗ E2−1 ). Since the map
πn,1 : Xn,1 → Un−1 is a P1 fibre bundle, we have πn,1∗ ωXn,1 [dim Xn,1 ] = ωUn−1 [dim Un−1 ]. Since Un−1 is
a symplectic variety, ωUn−1 = OUn−1 ; so πn,1∗ ωXn,1 = OUn−1 [−1]. We will show that ωXn,1 = i∗n,1 E1 ⊗ E2−1 ;
1
it will then follow that (G1m+2n )L E1 = Fm+2n
E1 [1] = πn,1∗ ωXn,1 [1] = OUn−1 , as claimed. Under the
embedding of the divisor in,1 : Xn,1 ,→ Un , the adjunction formula gives ωXn,1 = i∗n,1 (ωUn ⊗ OUn (Xn,1 )) =
i∗n,1 OUn (Xn,1 ).
It then follows that:
RΓ((G1m+2n )L G) = RHom((G1m+2n )L E1 , (G1m+2n )L G)
= RHom(E1 , G1m+2n (G1m+2n )L G)
m+2n−1 L
= RΓ(E1−1 ⊗ G1m+2n (Gm+2n
) G)
The functor E1−1 ⊗· is right exact as it corresponds to the positive braid s1m+2n that leaves the last m+2n−1
vertices in place, and winds the first vertex counter-clockwise around the circle underneath the other
m+2n−1 L
L
1
1
(2)F → F → G1m+2n (Gm+2n−1
strands. Using the exact triangle Tm+2n
m+2n ) F, we deduce Gm+2n (Gm+2n )
1
L
≤0
is right exact since Tm+2n
(2) is right exact. Thus E1−1 ⊗ G1m+2n (Gm+2n−1
m+2n ) G ∈ Dn , as required.
Lemma 5.4. Given b ∈ B0+
af f considered as a bijective (m+2n−2, m+2n−2)-tangle, there exists bijective
−1
i
i
i
=
◦ fm+2n
(m + 2n − 2, m + 2n − 2) tangles i (b), ηi (b) ∈ B+
af f such that b ◦ fm+2n = fm+2n ◦ i (b), b
−1
i
fm+2n ◦ ηi (b) .
Proof. Using the cap-crossing isotopy relation (9), we may define i (tjm+2n−2 ) = ηi (tjm+2n−2 ) = tj+2
m+2n if
j
j
j
i
i
j ≥ i, and i (tm+2n−2 ) = ηi (tm+2n−2 ) = tm+2n if j ≤ i − 2. Also define i (tm+2n−2 ) = ηi (tm+2n−2 ) to be
the tangle with a strand connecting (ζk , 0) to (2ζk , 0) for k 6= i − 1, i, and a strand connecting (ζi−1 , 0)
to (2ζi+2 , 0) that passes beneath a strand connecting (ζi+2 , 0) to (2ζi−1 , 0). It is straightforward to check
that with this definition, i (tjm+2n−2 ) = ηi (tjm+2n−2 ) ∈ B+
af f .
ik
i1
i2
Given b ∈ B0+
af f , choose a decomposition b = tm+2n−2 (1) ◦ tm+2n−2 (1) ◦ · · · ◦ tm+2n−2 (1); clearly i (b) =
1
k
k
1
i (tim+2n−2
(1)) ◦ · · · ◦ i (tim+2n−2
(1)) and ηi (b) = ηi (tim+2n−2
(1)) ◦ · · · ◦ ηi (tim+2n−2
(1)) satisfy the required
condition.
Definition 5.5. Let Dn0 denote the heart of the exotic t-structure on Dn .
0
Proposition 5.6. The functor Gin sends irreducible objects in Dn−1
to irreducible objects in Dn0 .
Proof. This follows from Proposition 5.2 and the Theorem in Section 4.2 of [3].
5.1. Irreducible objects in the heart of the exotic t-structure on Dn .
Definition 5.7. Let an affine crossingless (m, m + 2n) matching be a framed affine (m, m + 2n)-tangle
whose vertical projection to C has no crossings. Let an unlabelled affine crossingless matching (m, m+2n)matching be an affine crossingless matching where the m inner points are not labelled. Let Cross(n) be
the set of all unlabelled affine crossingless matchings.
EXOTIC t-STRUCTURES FOR TWO-BLOCK SPRINGER FIBRES
21
Following Section 6 of [1], we will describe the irreducible objects in the heart of the exotic t-structure
on Dn using the functors constructed in the previous section.
Lemma 5.8. We have |Cross(n)| = m+2n
.
n
Proof. It suffices to construct a bijection between unlabelled affine (m, m + 2n) crossingless matchings
and assignments of m + n plus signs and n minus to m + 2n labelled points on a circle. Given such an
m+2n−1
assignment of pluses and minuses to the points (2, 0), (2ζm+2n , 0), · · · , (2ζm+2n
, 0), for each minus, move
anti-clockwise around the circle and connect the minus to the first plus such that the number of pluses
and minuses between these two points is equal. After connecting the m remaining pluses on the outer
circle to the m unlabelled points on the inner circle without crossings, we have our desired unlabelled
affine (m, m + 2n) crossingless matching.
Lemma 5.9. Let α
e be any affine (m, m + 2n)-tangle, from which we obtain α by forgetting the labelling
on the inner circle. Then the functor Ψ(e
α) depends only on α.
Proof. It is easy to check Ψ(r̂n ) acts as the identity on D0 ; the result follows.
Thus given α ∈ Cross(n), we obtain a functor Ψ(α) : D0 → Dn . Let Ψα = Ψ(α)v denote the image of
the 1-dimensional vector space v in D0 ' Db (Vect) under the map Ψ(α).
Proposition 5.10. The irreducible objects
in the heart of the exotic t-structure on Dn are precisely
m+2n
given by Ψα , as α ranges across the
unlabelled affine (m, m + 2n) crossingless matchings.
n
Proof. It is well known that every affine crossingless (m, m + 2n) matching can be expressed as a product
i2
i1
in
n
2
1
gm+2n
◦ · · · ◦ gm+4
◦ gm+2
. Thus Ψα = Gim+2n
◦ · · · ◦ Gim+4
◦ Gim+2
v; by Proposition 5.6, Ψα is an irreducible
object in the heart of the exotic t-structure. It will follow from the arguments in the next section that these
irreducible objects are distinct. Since K 0 (Dn≥0 ∩ Dn≤0 ) = K 0 (Dn ) = K 0 (Coh(Bzn )), and K 0 (Coh(Bzn )) has
rank m+2n
, these constitute all the irreducible objects in the heart of the exotic t-structure on Dn . n
5.2. The Ext algebra. The goal of this section is to compute the algebra Ext• (
M
Ψα ). Let m > 0
α∈Cross(n)
(the m = 0 is handled in [1]).Note the following description of the right adjoint functor to Ψ(γ), where γ
is an affine (m + 2p, m + 2q) tangle. Denote by γ̆ the affine (m + 2q, m + 2p) tangle obtained by inverting
γ.
Lemma 5.11. The right adjoint to Ψ(γ) is Ψ(γ̆)[p − q].
k
k
Proof. From Section 2.3, the right adjoint to Ψ(γ) is Ψ(γ̆)[p − q], when γ = gm+2n
, fm+2n
, tkm+2n (1) or
tkm+2n (2). The result follows, since any tangle is the composition of these tangles, and if γ = γ1 ◦ γ2 ,
γ̆ = γ˘2 ◦ γ˘1 .
Now we will compute Ext• (Ψα , Ψβ ) as a vector space, where α, β ∈ Cross(n).
Ext• (Ψα , Ψβ ) = Ext• (Ψ(α)v, Ψ(β)v)
= Ext• (v, Ψ(ᾰ ◦ β)[−n]v)
= Ψ(ᾰ ◦ β)[−n]v
Let Λ ∈ Db (Vect) be a complex concentrated in degrees 1 and −1, with dimension 1 in those degrees.
22
RINA ANNO, VINOTH NANDAKUMAR
j
i
◦ gm+2
)v =
Lemma 5.12. We have: Ψ(fm+2



Λ if i = j
v if |i − j| = 1


0 if |i − j| > 1
j
Proof. First suppose 1 ≤ i, j ≤ m + 1. From Section 2.3, Ψ(gm+2
)v = i1,j∗ (Ej ⊗ OX1,j ) = i1,j∗ (Ej ), since
∗
π1,j maps X1,j to the point S0 ×slm T B0 .
(1)
j
−1
−1
i
i
◦ gm+2
)v = fm+2
Ψ(fm+2
(i1,j∗ (Ej )) = π1,i∗ [i∗1,i (i1,j∗ (Ej ) ⊗ Ei+1
)] = π1,i∗ [i∗1,i (i1,j∗ (Ej )) ⊗ Ei+1
]
Since the below conditions defining X1,i force x = z1 (the standard nilpotent of type (m + 1, 1) acting on
the Jordan basis {e1 , · · · , em+1 , f1 }), Vk = he1 , · · · , ek i for 1 ≤ k ≤ i − 1, and Vk = he1 , · · · , ek−1 , f1 i for
k ≥ i + 1, we have:
X1,i = {(0 ⊂ V1 ⊂ · · · ⊂ Vi−1 ⊂ Vi ⊂ Vi+1 ⊂ · · · ⊂ Vm+2 ), x|
x ∈ S1 , xVi+1 ⊂ Vi−1 , xVk ⊂ Vk−1 } = P(Vi+1 /Vi−1 )
Thus the intersection X1,i ∩ X1,i+1 consists of the single point {(0 ⊂ V1 ⊂ · · · ⊂ Vm+2 ), z1 }, where
Vk = he1 , · · · , ek i for 1 ≤ k ≤ i, and Vk = he1 , · · · , ek−1 , f1 i for k ≥ i + 1; while X1,i ∩ X1,j = ∅ if
|i − j| > 1.
So in equation (1), if i = j since we have an exact triangle F ⊗ OU1 (−X1,i )[1] → i∗1,i i1,i∗ F → F for F ∈
Coh(X1,i ); and that OU1 (−X1,i ) = Ei−1 ⊗ Ei+1 . Thus we have an exact triangles Ei+1 [1] → i∗1,i i1,i∗ Ei → Ei ,
−1
−1
−1
and hence OU1 [1] → (i∗1,i i1,i∗ Ei ) ⊗ Ei+1
→ Ei ⊗ Ei+1
. Noting that Ei ⊗ Ei+1
is O(−2), and using the long
exact sequence of cohomology:
−1
−1
· · · → RΓi (OU1 [1]) → RΓi ((i∗1,i i1,i∗ Ei ) ⊗ Ei+1
) → RΓi (Ei ⊗ Ei+1
) → RΓi+1 (OU1 [1]) → · · ·
−1
) = 0 if i 6= 1, 0, −1; computing the terms in the above we have an exact
Thus RΓi ((i∗1,i i1,i∗ Ei ) ⊗ Ei+1
sequence
−1
0 → C → RΓ−1 ((i∗1,i i1,i∗ Ei ) ⊗ Ei+1
)→0→0→
−1
−1
RΓ0 ((i∗1,i i1,i∗ Ei ) ⊗ Ei+1
) → 0 → 0 → RΓ1 ((i∗1,i i1,i∗ Ei ) ⊗ Ei+1
)→C→0
i
i
· gm+2
)v = Λ.
Thus Ψ(fm+2
j
i
If |i−j| > 1, since i∗1,i i1,j∗ F = 0 for any F ∈ Coh(X1,i ) (as X1,i ∩X1,j = 0), we obtain Ψ(fm+2
◦gm+2
)v = 0.
If |i − j| = 1, denote by X1,i,j the point X1,i ∩ X1,j ; denote i01,i : X1,i,j ,→ X1,j , i01,j : X1,i,j ,→ X1,i . Then
j
−1
0
0
i
i∗1,i (i1,j∗ Ej ) = i01,j∗ (i0∗
1,i Ej ) = i1,j∗ OX1,i,j , so Ψ(fm+2 ◦ gm+2 )v = π1,i∗ (i1,j∗ OX1,i,j ⊗ Ei+1 ) = v.
Definition 5.13. Call an m-link an unlabelled affine crossingless (m, m)-tangle, where both the m inner
points, and the m outer points are unlabelled. If each of the m points in the inner circle are joined to m
points in the outer circle, say that the m-link is “good” , and otherwise (i.e. if there are cups and caps)
say that the m-link is “bad”. If an m-link γ is good, denote by ω(γ) the number of loops contained in γ.
Note that any loop cannot enclose the origin if m > 0; and that a good m-link γ is essentially determined
by ωγ . Given unlabelled affine crossingless (m, m + 2n) matchings α, β, α̌ ◦ β will be an m-link; further,
any m-link γ corresponds to a functor Ψ(γ) : D0 → D0 . It follows from Lemma 5.12 that:
Proposition 5.14. If the m-link γ is bad, then Ψ(γ) is zero. If the m-link γ is good, then Ψ(γ)
corresponds to multiplication by (Λ)⊗ω1 (γ) .
EXOTIC t-STRUCTURES FOR TWO-BLOCK SPRINGER FIBRES
23
i
i
i
i
i
i
) corresponds to
◦ gm+2
) = 1; and Ψ(fm+2
◦ gm+2
is good, with ω(fm+2
◦ gm+2
Proof. The m-link fm+2
j
j
i
i
) = 0; and
multiplication by Λ. If |i − j| = 1, the m-link fm+2 ◦ gm+2 is good, with ω(fm+2 ◦ gm+2
j
i
i
i
Ψ(fm+2 ◦ gm+2 ) is the identity functor D0 → D0 . If |i − j| > 1, the m-link fm+2 ◦ gm+2 is bad, and
j
i
i
i
)v = 0 for any w ∈ D0 . Call an m-link “basic” if it is of the form fm+2
◦ gm+2
Ψ(fm+2
◦ gm+2
; then
any m-link γ can be written as a composition of basic m-links. The conclusion then follows from our
knowledge of Ψ(γ) for basic m-links γ, and the fact that m-link γ is bad iff each expression of γ in terms
of basic m-links contains at least one bad basic m-link.
i
) = Λ[−1] is isomorphic to C[x]/(x2 ), where x has degree 2.
Note that the Z-graded algebra Ext• (Ψgm+2
Indeed, this is the only possible algebra structure on Λ[−1] which respects its grading. Thus:
Theorem 5.15. We have an isomorphism of vector spaces:
(
Λ⊗ω(α̌◦β) [−n] if α̌ ◦ β is good
Ext• (Ψα , Ψβ ) =
0
otherwise
6. Further directions
6.1. Multiplication in the Ext algebra. From the results in the last section, we have a description
of the algebra
M
Ext• (
Ψα )
α∈Cross(n)
as a graded vector space. It would be interesting to compute the multiplication in this algebra.
More specifically, for α, β, γ ∈ Cross(n), we would need to describe the map:
Ext• (Ψα , Ψβ ) × Ext• (Ψβ , Ψγ ) → Ext• (Ψα , Ψγ )
Using Theorem 5.15, for all three terms to be non-zero, we require α̌ ◦ β, and β̌ ◦ γ (and hence α̌ ◦ γ) to
be good. Thus we would like to compute the resulting map:
Λω(α̌◦β) [−n] × Λω(β̌◦γ) [−n] → Λω(α̌◦γ) [−n]
To do this, one could attempt to use the method in the proof of Theorem 4 in [1] (which handles the
m = 0 case).
6.2. Decategorification. Denote by V the 2-dimensional representation of sl2 . Recall that Cautis and
Kamnitzer prove in section 6 (see Theorem 6.2 and Section 6.4) of [7] that:
K 0 (Coh(Ym+2n )) ' V ⊗m+2n
It is easy to see that:
K 0 (Dn0 ) = K 0 (Dn ) = K 0 (CohBzn (Un ))
Under the natural embedding Un → Ym+2n (constructed in Section 2.1), we have a natural map
K 0 (Dn0 ) → K 0 (Coh(Ym+2n )) ' V ⊗m+2n
By counting dimensions, it is natural to expect the following.
Conjecture 6.1. Under this map, K 0 (Dn0 ) is identified with the (m + n, n) weight space in V ⊗m+2n .
24
RINA ANNO, VINOTH NANDAKUMAR
Now the images of the irreducible objects Ψ(α) in Dn0 will form a basis in the (m + n, n) weight space
in V ⊗m+2n . Using Theorem 6.5 from [7], it should be possible to compute the image of these irreducible
objects explicitly.
References
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