Categorifications of the extended affine Hecke algebra and the affine q-Schur algebra S(n,r) for 2

We categorify the extended affine Hecke algebra and the affine quantum Schur algebra S(n,r) for 2<r<n, using Elias-Khovanov and Khovanov-Lauda type diagrams. We also define the affine analogue of the Elias-Khovanov and the Khovanov-Lauda 2-representations of these categorifications into an extension of the 2-category of affine (singular) Soergel bimodules.

The two authors were supported by the FCT -Fundação para a Ciência e a Tecnologia, through project number PTDC/MAT/101503/2008, New Geometry and Topology.

Introduction
Khovanov and Lauda [KL10], and Rouquier [Rou08] following a slightly different approach, defined a graded additive 2-category U (g) with "nice properties" for any root datum. The 2-morphisms are defined by string diagrams with regions labeled by g-weights. They are generated by a finite set of elementary diagrams which obey a finite set of relations. The split Grothendieck group of the Karoubi envelope of U (g) is isomorphic to the idempotented version of the corresponding quantum groupU(g). In Crane and Frenkel's [CF94] terminology, we say that U (g) categorifiesU(g).
Khovanov and Lauda only proved this categorification theorem for g = sl n . A key ingredient of that proof was a 2-representation of U ( sl n ) on a 2category build out of the cohomology rings of partial flag varieties. The equivariant cohomology rings of these varieties, which also give rise to a 2-representation, are equivalent to the singular Soergel bimodules of type A, introduced and studied by Williamson in his PhD thesis in 2008 and published in [Wil11]. The general categorification theorem was proved by Webster [Web10], by geometric techniques well beyond our understanding.
In [MSV13], Mackaay, Stošić and Vaz defined a quotient of U (sl n ), denoted S(n, r), and proved that it categorifies the quantum Schur algebra S(n, r), for any r ∈ Z >0 . If n ≥ r, then S(n, r) contains a full sub-2-category which categorifies the Hecke algebra H A r−1 . This sub-2-category is equivalent to the 2-category of (ordinary) Soergel bimodules of type A r−1 , as was proved in [MSV13] using Elias and Khovanov's diagrammatic presentation of the Soergel 2-category [EK10].
In the same paper, Mackaay, Stošić and Vaz also showed that Khovanov and Lauda's 2-representation of U (sl n ) on the singular Soergel bimodules descends to S(n, r). Its restriction to the aforementioned sub-2-category of S(n, r) is exactly Elias and Khovanov's 2-equivalence of their diagrammatic 2-category and the 2-category of (ordinary) Soergel bimodules.
Naturally the question arises whether the results in [MSV13] extend to affine type A. In this paper, we show that this is indeed the case for 3 ≤ r < n: • As Libedinsky explained in [Lib08a], one can define Soergel bimodules using the geometric representation of the affine Weyl group W A r−1 or Soergel's extension of that representation [Soe07]. The geometric representation is not faithful, whereas Soergel's representation is reflection faithful. Both representations give rise to categories of Soergel bimodules which categorify the affine Hecke algebra H A r−1 , as shown in [Här99,Lib08b,Lib08a,Soe07]. For more information on this topic, see also [EW13]. However, the extension of the geometric representation to the extended affine Weyl group W A r−1 is "too degenerate" and cannot be The case n < r cannot be dealt with at present, because even the decategorified story has not been worked out (see Problem 2.4.5 in [DDF12]).
Remark 0.2. There is a technical detail, which will be fully explained in Section 3 but should be mentioned here already. Just as in [MSV13], we actually define S(n, r) [y] as a quotient of U ( gl n ) [y] , which is a 2-category obtained from U ( sl n ) [y] by switching to degenerate gl n -weights of level zero for the labels of the regions in the string diagrams. We conjecture that U ( gl n ) [y] categorifies the level-zeroU( gl n ), but do not need that fact for the rest of this paper.
The results in this paper have several points of interest. The categories EBim A r−1 and DEBim A r−1 contain the new objects B ρ ± and ±, respectively, and the corresponding morphisms. As our results show, these objects and morphisms also show up naturally in S(n, r) [y] as 1 and 2-morphisms.
The y-deformation U ( sl n ) [y] and its Schur quotient S(n, r) [y] are new. As the results in this paper show, they both show up naturally when considering Soergel bimodules over Q[y, x 1 , . . . , x r ], which is the ring of polynomial functions on Soergel's reflection faithful representation of the affine Weyl group.
Furthermore, there are interesting (possible) links with other categorifications of the (extended) affine Hecke algebra and the quantum affine Schur algebra. Lusztig [Lus93,Lus99] and Ginzburg and Vasserot [GV93] gave a categorification of S(n, r) using perverse sheaves, extending Grojnowski and Lusztig's approach to the categorification of S(n, r). It would be interesting to find the precise relation with the categorification presented here and in our follow-up paper [MT13].
In this paper we also define an extended version of the affine singular Soergel bimodules. Williamson introduced and studied the 2-category of singular Soergel bimodules for any Coxeter group in his PhD thesis in 2008, the results of which were published in [Wil11], and proved that it categorifies a certain "new" algebra, which he called the Schur algebroid. In finite type A the Schur algebroid is isomorphic to the quantum Schur algebra. Williamson's affine type A Schur algebroid should also be closely related to the affine quantum Schur algebra. Whatever the precise relation turns out to be, the 2-representation of S(n, r) [y] on the extended affine singular Soergel bimodules establishes an interesting relation between Khovanov and Lauda's work and Williamson's.
Another point of interest is related to the possibility of categorifying the so called Kirillov-Reshetikhin modules ofU( sl n ). These level zero modules can be defined for any affine quantum group and have been intensively studied (see [CP94,DDF12,DF12,Kas02] for more information and references).
In affine type A (and only in that type), they are special examples of evaluation modules V λ,a , where λ is a dominant weight and a ∈ C * . If λ is an n-part partition of r, then V λ,a descends to a representation of S(n, r). More precisely, V λ,a is defined by pulling back (the technical term is inflating) the action of S(n, r) on the irrep V λ via the so called evaluation map ev a : S(n, r) → S(n, r).
If λ = (m i ), i.e. m times the i-th fundamental gl n -weight, then it is known that V λ,q i−m+2 is isomorphic to a Kirillov-Reshetikhin module and has a canonical basis. It seems likely that one can categorify the evaluation map ev q i−m+2 : S(n, mi) → S(n, mi) and therefore V λ,q i−m+2 , but such a categorification is beyond the scope of this paper. This paper is organized as follows: • In Section 1, we recall the definition of and some basic results on affine roots and weights, the (extended) affine Weyl group and the (extended) affine Hecke algebra. • In Section 2, we define EBim A r−1 and recall Härterich's categorification result. We also use Elias-Khovanov type diagrams in order to define DEBim A r−1 and show that it gives a diagrammatic presentation of EBim A r−1 , using Elias and Williamson's results in [EW13].
• In Section 3, we first recall the definition of S(n, r) and its relation to the extended affine Hecke algebra. After that, we give the definitions of U ( gl n ) [y] , U ( sl n ) [y] and S(n, r) [y] . • In Section 4, we give the 2-functor Σ n,r : DEBim * A r−1 → S * (n, r) [y] and prove that it is well-defined.
• In Section 5, we define the affine 2-representation The fact that it is well-defined follows essentially from the welldefinedness of the analogous 2-functor for finite type A and a "conjugation trick", which we will explain. • In Section 6, we use the results in the previous sections to prove that S(n, r) [y] categorifies S(n, r).
1. The affine setting 1.1. Affine roots of level zero. We use the well-known realization of sl r as the central extension of the loop algebra of sl r together with a derivation (see for example [PS86,Kac90,Fuc95]), i.e. the underlying vector space is isomorphic to sl r = L(sl r ) ⊕ Q c ⊕ Q d . In order to express its root system, consider the Cartan subalgebra The simple roots are α i = (ᾱ i , 0, 0) i = 1, . . . , r − 1 and α r = −θ, 0, 1 = δ −θ whereᾱ i = ε i − ε i+1 , for i = 1, . . . , r − 1 are the simple roots of sl r ,θ = α 1 + · · · +ᾱ r−1 = ε 1 − ε r is the highest root and δ is the dual element of d.
The elements ε i for i = 1, . . . , r are the canonical basis vectors in Z r . Weights are triples of the form whereκ is an sl r -weight and k and m are integers. The integer k is called the level of κ. The inner product between two weights κ = (κ, k, m) and κ ′ = (κ ′ , k ′ , m ′ ) is given by where κ,κ ′ is the usual inner product of sl r -weights. In particular, we have α i , α i = 2 for all i = 1, . . . , r and α r , α 1 = α i , α i+1 = −1 for all i = 1, . . . , r − 1. The simple coroots are α ∨ i = α i for all i = 1, . . . , r. In the following sections, we will also use gl r -weights κ = (κ, k, m), whereκ denotes a non-affine gl r -weight.
Remark 1.1. In this paper, we will only consider sl r and gl r -weights of level zero.
1.3. The extended affine Weyl group. See [Lus89], [DG07] or [DDF12] for more details about the extended affine Weyl group. For example in [DG07], there is a definition of this group different from the following, it is described as a subgroup of permutations of Z.
Let us now consider the translations t ε i along the simple gl r -roots ε i , i = 1, . . . , r. Their action on a level zero gl r -weight κ is given by (1.5) The extended affine Weyl group W A r−1 is defined as the semidirect product of the finite Weyl group W A r−1 and the abelian group t ε 1 , . . . , t εr of translations along the coroot lattice of gl r . It contains the affine Weyl group W A r−1 as a normal subgroup.
The group W A r−1 is generated by σ 1 , . . . , σ r−1 and t ε 1 , . . . , t εr , which satisfy the following relations: . . , r − 1 and j = 1, . . . , r. (1.6) Hence the set of generators is not minimal, e.g. one can obtain any t ε j for j = 2, . . . , r by conjugating t ε 1 by certain reflections. There is another presentation of W A r−1 , which is important for this paper. It involves the following specific element ρ = t ε 1 σ 1 . . . σ r−1 , which acts on a level zero gl r -weight κ = (κ, 0, m) by (1.7) The action of its inverse ρ −1 = σ −1 r−1 . . . σ −1 1 t −ε 1 is given by ρ −1 (κ, 0, m) = ((κ 2 , . . . , κ r , κ 1 ) , 0, m + κ 1 ) . (1.8) One then sees that W A r−1 is generated by subject to the relations where the indices have to be understood modulo r, as before. We say that i and j are distant if j ≡ i ± 1 mod r. Using this set of generators, any element w ∈ W A r−1 can be written in the following way where k ∈ Z is unique and σ i 1 · · · σ i l is a reduced expression of the element w ′ ∈ W A r−1 . Note that the conventions here are opposite to the ones chosen by Doty and Green [DG07].
1.4. The extended affine braid group and Hecke algebra. One can form the extended affine braid group B A r−1 associated to W A r−1 . It admits the same presentation as W A r−1 except that one omits the involutivity relations of the generators σ i for i = 1, . . . , r.
One can also define the extended affine Hecke algebra H A r−1 , which is the quotient of the Q(q)-group algebra of B A r−1 by the relations . . , r with q being a formal parameter. For more details about this algebra, see [Opd03,Opd04,DO08,OS09,DDF12].
A Q(q)-basis of H A r−1 is given by the set T σ i l with w, k, w ′ and σ i j as in (1.13). See [Gre99] and [DG07].
The above shows that H A r−1 is generated by as an algebra. An alternative set of algebraic generators of H A r−1 is given by where the b i := C ′ σ i = q −1 (1 + T σ i ) are the Kazhdan-Lusztig generators. The relations satisfied by these generators are the following: (1.18) In [GH07], Grojnowski and Haiman show that H A r−1 has the following Kazhdan-Lusztig basis with the usual positive integrality property.

A categorification of the extended affine Hecke algebra
Notation 2.1. Let C be a Q-linear Z-graded additive category (resp. 2category) with translation (see Section 5.1 in [Lau10] for the technical definitions).
In all examples in this paper, the vector space of morphisms (resp. 2-morphisms) of any fixed degree is finite-dimensional. The Karoubi envelope of C is denoted by KarC. By C * we denote the category (resp. 2-category) with the same objects (resp. same objects and 1-morphisms) as C, but whose hom-spaces (resp. 2-hom-spaces) are defined by A degree preserving functor F : C → D between two such categories C and D lifts to a functor between the enriched categories C * → D * and to a functor between the Karoubi envelopes KarC → KarD, which are both also denoted F .
Remark 2.2. The action above naturally extends the action of the (nonextended) affine Weyl group W A r−1 . We are working here with Soergel's original reflection faithful realization of the affine type A Coxeter system W A r−1 considered in [Soe07] and [Här99]. Indeed, when generalized to the extended affine Weyl group W A r−1 , this representation remains faithful which explains why we choose to use this precise realization to achieve a categorification of the extended affine Hecke algebra H A r−1 in the present paper. Let us set X i = x i+1 − x i for i = 1, . . . , r − 1 and X r = x 1 − x r − y.

Extended Soergel bimodules.
For any i = 1, . . . , r, we define the Rbimodule We also define the twisted R-bimodule B ρ (resp. B ρ −1 ), which coincides with R as a left R-module but is twisted by ρ (resp. by ρ −1 ) as a right R-module, i.e. any a ∈ R acts on B ρ on the right by multiplication by ρ(a) (resp. ρ −1 (a)).
Arkhipov introduced twisted bimodules associated to simple reflections and the twisted functors obtained by tensoring with them. These were used by several people in their work on category O; for some history and references see [Maz12]. Here we (only) consider twisted bimodules associated to powers of ρ.
We introduce a grading on R, R σ i , B i and B ρ ±1 by setting for all k = 1, . . . , r. Curly brackets will indicate a shift of the grading: if M = i∈Z M i is a Z-graded bimodule and p an integer, then the Z-graded Form now the category monoidally generated by the graded R-bimodules defined above. Then allow direct sums and grading shifts of these objects and consider only morphisms which are degree-preserving morphisms of R-bimodules, and denote this category EBim A r−1 . Its Karoubi envelope KarEBim A r−1 is a Q-linear graded additive monoidal category with translation, which we call the category of extended Soergel bimodules of type A r−1 .
As mentioned in Remark 2.2, we are precisely using Soergel's original realization, so the bimodules B i considered here are the ones constructed by Härterich [Här99] and Soergel [Soe07]. Therefore Soergel's category KarBim A r−1 of affine type A is equivalent to the full subcategory of KarEBim A r−1 generated by the B i , for i = 1, . . . , r. Let us recall Härterich's categorification result [Här99]: For each w ∈ W A r−1 , there exists a unique indecomposable bimodule B w in KarBim A r−1 . Conversely, any indecomposable bimodule in this category is isomorphic to B w {t}, for a certain w ∈ W A r−1 and a certain grading shift t ∈ Z.
Under the isomorphism above the Kazhdan-Lusztig basis element C ′ w is mapped to [B w {−1}], for any w ∈ W A r−1 .
In the rest of the paper, we keep the convention that subscripts are considered to be modulo r, e.g. the bimodule B r+1 is by definition equal to B 1 . We will also use the notation B ⊗k ρ for any k ∈ Z, where this bimodule is defined to be the tensor product of |k| copies of Lemma 2.4. For any i = 1, . . . , r, there exists an R-bimodule isomorphism Applying these isomorphisms r times gives an isomorphism for any i = 1, . . . , r. Proof.
• Isomorphism (2.2) First note that there exist natural isomorphisms of R-bimodules: Define the isomorphism of R-bimodules ψ : This isomorphism is well-defined, because ρ defines an isomorphism between R σ i and R σ i+1 .
• Isomorphism (2.3) Note that B ⊗r ρ ∼ = B ρ r and that ρ r leaves the ring R σ i invariant.
Under this isomorphism, the indecomposables in KarEBim A r−1 correspond exactly to the Kazhdan-Lusztig basis of H A r−1 . In particular, We define the homomorphism of algebras The homomorphism is well-defined, as follows from the following isomorphisms in EBim A r−1 : for i, j = 1, . . . , r.
Note that any tensor product of B ρ ±1 's and B i 's can be rewritten in the following way B ⊗k ρ ⊗ R B i 1 ⊗ R . . . ⊗ R B i l , by sliding all the B ρ ±1 's to the left using the isomophism (2.7).
Let us now look at the indecomposables of the category KarEBim A r−1 . First observe that if the bimodule M is indecomposable in KarEBim A r−1 then, for any k ∈ Z, the tensor product B ⊗k ρ ⊗ R M is indecomposable as well. Indeed assume that which contradicts the fact that M is supposed to be indecomposable.
So let M be an indecomposable of KarEBim A r−1 . It is a direct summand of some tensor product B ⊗k The latter tensor product belongs to the subcategory KarBim A r−1 of KarEBim A r−1 .
Thus B ⊗−k ρ ⊗ R M is of the form B w for some w ∈ W A r−1 . We can conclude that the indecomposables of the category KarEBim A r−1 are all of the form B ⊗k ρ ⊗ R B w for k ∈ Z and w ∈ W A r−1 . Their Grothendieck classes correspond bijectively to the elements of which is precisely the Kazhdan-Lusztig basis of H A r−1 in (1.19).
Remark 2.6. In the category EBim A r−1 , the B i are self-adjoint and B ρ and B −1 ρ form a biadjoint pair. Therefore, there exist isomorphisms The latter hom-space is equal to zero except when k = l, in which case it is isomorphic to the corresponding hom-space in Bim A r−1 .
Indeed any morphism is completely determined by the image p of 1. Since f is a morphism of Rbimodules, we have pa = ap for any a ∈ R. In particular, for a = r i=1 x i we have Since r i=1 x i is invariant under all the reflections σ j for j = 1, . . . , r, we also have This implies that p, and therefore the morphism f , has to be zero unless k = l.

Representation of B
be the homotopy category of bounded complexes in KarEBim A r−1 . Rouquier [Rou06] obtained a representation of B A r−1 in K(KarBim A r−1 ). We generalize his result in the extended affine setting.
Remark 2.7. In what follows, we use by matter of convenience the notation X i = x i+1 − x i , for i = 1, . . . , r − 1, and X r = x 1 − x r − y, as defined in Remark 2.2.
To each braid generator σ i ∈ B A r−1 we assign the cochain complex F (σ i ) of graded R-bimodules where B i sits in cohomological degree 0, and the R-bimodule morphism rb i maps 1 to 1 2 (X i ⊗ 1 + 1 ⊗ X i ), for i = 1, . . . , r.
where B i {−2} sits in cohomological degree 0 and the R-bimodule morphism br i is just the multiplication.
To ρ (resp. ρ −1 ) we assign the cochain complex of graded R-bimodules To the unit element 1 ∈ B A r−1 we assign the complex of graded Rbimodules where R sits in cohomological degree 0; the complex F (1) is a unit for the tensor product of complexes. Finally, to any extended affine braid word, we assign the tensor product over R of the complexes associated to the generators appearing in the expression of the word.
Proposition 2.8. The above defines a categorical representation of B A r−1 in K(KarEBim A r−1 ).
2.2. The diagrammatic version. The category of Soergel bimodules of finite type A is described via planar diagrams by Elias and Khovanov in [EK10]. They associate planar diagrams to certain generating bimodule maps and give a complete set of relations on them. Elias and Williamson [EW13] worked out the generalization of the diagrammatic approach to Soergel bimodules for any Coxeter group which does not contain a standard parabolic subgroup isomorphic to H 3 . Our aim is to define and study an extension of Elias and Williamson's diagrammatic category for extended affine type A [EW13]. In affine type A Elias and Williamson's diagrammatic category is a straightforward generalization of Elias and Khovanov's original category, which was defined for finite type A. In addition to Elias and Williamson's diagrams, we also have to introduce a new type of strand. These new strands are oriented and their endpoints are labeled + or − depending on orientations. The Karoubi envelope of the diagrammatic category DEBim A r−1 obtained in this way is equivalent to the category of extended Soergel bimodules KarEBim A r−1 of affine type A, as we will show.
2.2.1. Definition of DEBim A r−1 . First start with the category whose objects are graded finite sequences of integers belonging to {1, . . . , r} and the symbols + and −. Graphically we represent these sequences by sequences of colored points (read from left to right) of the x-axis of the real plane R 2 . The morphisms are then equivalence classes of Q-linear combinations of graded planar diagrams in R × [0, 1] (read from bottom to top) and composition is defined by vertically glueing the diagrams and rescaling the vertical coordinate. These morphisms are defined by generators and relations listed below. This category possesses a monoidal structure given by stacking sequences and diagrams next to each other.
Let DEBim A r−1 be the category containing all direct sums and grading shifts of these objects and let its morphisms be the degree-preserving diagrams. The diagrammatic extended Soergel category is by definition its Karoubi envelope KarDEBim A r−1 .
In the diagrams, the strands whose endpoints are + or −-signs are oriented and the other strands are non-oriented. The non-oriented strands can be colored with integers belonging to {1, . . . , r}. Two colors i and j are called adjacent (resp. distant) if i ≡ j ± 1 mod r (resp. i ≡ j ± 1 mod r). By convention, no label means that the equation holds for any color i ∈ {1, . . . , r}.
The morphisms of DEBim A r−1 are built out of the following generating diagrams. The non-oriented diagrams are the affine analogues of Elias and Khovanov's diagrams, the ones involving oriented strands are new.
• Generators involving only one color: It is useful to define the cap and cup as follows • Generators involving oriented strands and adjacent colored strands. The mixed 4-valent vertex of degree 0: • Relations involving boxes:  For the morphisms one only needs to specify F on the generators. A sequence of vertical strands is mapped to the identity of the corresponding bimodule and box i (resp. box y ) is mapped to multiplication by x i (resp. y). Furthermore, we define (recall that Proposition 2.10. The functor F is well-defined, degree preserving and essentially surjective. Proof. The fact that the functor F is well-defined and degree preserving amounts to a straightforward verification that it preserves the relations (2.13)-(2.58) and the degrees of the morphisms. For the relations involving only non-oriented strands, this is completely analogous to Elias and Khovanov's case. For the relations involving oriented strands, the calculations are new but easy. Furthermore, in view of the definitions of the objects of DEBim * , the functor F is clearly essentially surjective.
Proposition 2.11. The functor F is full.
Proof. Libedinsky has proved in [Lib08b] that all the morphisms of the category Bim * A r−1 are generated by the following ones: • br i , rb i , pr i and inj i for all i = 1, . . . , r • f i,j for all i, j = 1, . . . , r with i = j. In view of Remark 2.6, this implies that all the morphisms of the category EBim * A r−1 are generated by the ones listed by Libedinsky and copied above, together with fl +,i and fl i+1,+ , giving and +,− r, −,+ r, r −,+ and r +,− , giving Thus the functor F is full, since all the morphisms generating EBim * Proof. Only the faithfulness of F remains to be proved.
For a given object X in DEBim * , let k X denote the total sum of plus and minus signs in X. Given two objects X, Y in DEBim * where X ′ does not have any signs and is obtained from X by applying the commutation isomorphisms (±, i) ∼ = (i ± 1, ±) and the isomorphisms (±, ∓) ∼ = ∅.
Let D be a diagram representing a morphism from X to Y , such that Let us illustrate this by the example in Figure 1. Since the faithfulness of F follows from the faithfulness of Elias and Williamson's analogous functor for non-extended affine type A, which they proved in Theorem 6.28 of [EW13].
As always in this paper, we use the convention that the indices appearing in the relations are considered modulo n.
3.2. The (extended) affine algebras. In the following definitions we do not need to consider the derivation, which we used above.
Definition 3.1. The extended quantum general linear algebra U q ( gl n ) is the associative unital Q(q)-algebra generated by R ±1 , K ±1 i and E ±i , for i = 1, . . . , n, subject to the relations Definition 3.2. The affine quantum general linear algebra U q ( gl n ) ⊆ U q ( gl n ) is the unital Q(q)-subalgebra generated by E ±i and K ±1 i , for i = 1, . . . , n.
The affine quantum special linear algebra U q ( sl n ) ⊆ U q ( gl n ) is the unital Q(q)-subalgebra generated by E ±i and K i K −1 i+1 , for i = 1, . . . , n. We will also need the bialgebra structure on U q ( gl n ).
As a matter of fact, U q ( gl n ) is even a Hopf algebra, but we do not need the antipode in this paper. Note that ∆ and ε can be restricted to U q ( gl n ) and U q ( sl n ), which are bialgebras too.
At level 0 and forgetting the derivation, we can work with the U q (sl n )weight lattice, when considering U q ( sl n )-weight representations. Omitting the extra entry corresponding to the derivation makes this weight-lattice degenerate, because α 1 + α 2 + · · · + α n = 0, but that does not matter in this section. Similarly, we can work with the U q (gl n )-weight lattice, when considering U q ( gl n ) and U q ( gl n )-weight representations.
V ∼ = λ V λ and K i acts as multiplication by q λ i on V λ . Then V is also a U q ( sl n )weight representation with weights λ = (λ 1 , . . . , λ n−1 ) ∈ Z n−1 such that λ j = λ j − λ j+1 for j = 1, . . . , n − 1. Conversely, there is not a unique choice of U q ( gl n )-action on a given U q ( sl n )-weight representation with weights µ = (µ 1 , . . . , µ n−1 ). We first have to fix the action of K 1 · · · K n . In terms of weights, this corresponds to the observation that, for any given r ∈ Z the equations determine λ = (λ 1 , . . . , λ n ) uniquely, if there exists a solution to (3.13) and (3.14) at all. We therefore define the map ϕ n,r : if (3.13) and (3.14) have a solution, and put ϕ n,r (µ) = * otherwise. As far as weight representations are concerned, we can restrict our attention to the Beilinson-Lusztig-MacPherson idempotented version of these quantum groups, denoted U ( gl n ),U( gl n ) andU( sl n ) respectively. For each λ ∈ Z n adjoin an idempotent 1 λ to U q ( gl n ) and add the relations Definition 3.4. The idempotented extended affine quantum general linear algebra is defined by Of course one definesU( gl n ) ⊂ U ( gl n ) as the idempotented subalgebra generated by 1 λ and E ±i 1 λ , for i = 1, . . . , n and λ ∈ Z n .
Similarly for U q ( sl n ), adjoin an idempotent 1 λ for each λ ∈ Z n−1 and add the relations Definition 3.5. The idempotented quantum special linear algebra is defined byU Any weight-representation of U q ( gl n ), U q ( gl n ) or U q ( sl n ) is also a representation of U ( gl n ),U( gl n ) orU( sl n ), respectively. This is not true for non-weight representations, of which there are many. There are also other differences of course, e.g. U ( gl n ),U( gl n ) andU( sl n ) are not unital, because they have infinitely many idempotents. For that same reason, they are not bialgebras, although their action on tensor products of weight representations is well-defined.
3.3. The affine q-Schur algebra. Let us first recall Green's [Gre99,DG07] tensor space and the action of U q ( gl n ) on it. We will also recall some basic results about this action and add some of our own. Wherever we omit a proof in this section, the corresponding result was taken from [DG07]. When we give a proof, it is because the corresponding result cannot be found in the literature and we had to prove it ourselves, e.g. the inner product on tensor space is probably known to experts, but there seems to be no written reference. Let V be the Q(q)-vector space freely generated by {e t | t ∈ Z}.
Definition 3.6. The following defines an action of U q ( gl n ) on V Note that V is clearly a weight representation of U q ( gl n ), with e t having weight ε i , for i ≡ t mod n. Therefore V is also a representation of U ( gl n ). From now on, let r ∈ N >0 be arbitrary but fixed. As usual, one extends the above action to V ⊗r , using the coproduct in U q ( gl n ). Again, this is a weight representation and therefore a representation of U ( gl n ), which we call Green's tensor space.
We also define a Q(q)-bilinear form on V by e s , e t = δ st , which extends Proof. We can write v uniquely as t∈T v t e t , such that T is a finite subset of Z r and for any t = (t 1 , . . . , t r ) ∈ T we have v t ∈ Q(q) and e t = e t 1 ⊗ · · · ⊗ e tr .

Thus we get
equal to 1. Choose q 0 ∈ Q such that f t (q 0 ) = 0, g t (q 0 ) = 0 for all t ∈ T (such a number exists, because T is finite and each polynomial has only finitely many roots).
This implies that There is a right action of H A r−1 on V ⊗r which commutes with the left action of U q ( gl n ). Its precise definition, which can be found in [Gre99,DG07], is not relevant here.
Definition 3.8. The affine q-Schur algebra S(n, r) is by definition the centralizing algebra By affine Schur-Weyl duality, the image of ψ n,r : U q ( gl n ) → End(V ⊗r ) is always isomorphic to S(n, r). If n > r, we can even restrict to U q ( sl n ) ⊂ U q ( gl n ), i.e. ψ n,r (U q ( sl n )) ∼ = S(n, r). For n = r, this is no longer true.
The proof of the following lemma is a straightforward check, which we leave to the reader.
Lemma 3.11. For any X ∈ U q ( gl n ) and any v, w ∈ V ⊗r , we have Proof. By Lemma 3.10, it suffices to check the above for r = 1 and v = e i and w = e j , for any i, j ∈ Z. This is straightforward and left to the reader.
Note that ρ can also be defined onU( gl n ), such that ρ(1 λ ) = 1 λ for any λ ∈ Z r , and that it descends to S(n, r).

3.4.
A presentation of S(n, r) for n > r. In this subsection, let n > r.
Recall that in this case is surjective. This gives rise to the following presentation of S(n, r). The proof can be found in [DG07].
We will use signed sequences i = (µ 1 i 1 , . . . , µ m i m ), with m ∈ N, µ j ∈ {±1} and i j ∈ {1, . . . , n}. The set of signed sequences we denote SSeq. For a signed sequence i = (µ 1 i 1 , . . . , µ m i m ) we define We write E i for the product E µ 1 i 1 . . . E µmim . For any λ ∈ Z n and i ∈ SSeq, we have The surjection ψ n,r :U( sl n ) → S(n, r) can also be given explicitly in terms of the generators in Theorem 3.12. For any λ ∈ Z n−1 , we have where ϕ n,r : Z n−1 → Λ(n, r) ∪ { * } is the map defined in 3.2. By convention, we put 1 * = 0.
Therefore, we see that The other case follows automatically, because We can also give an explicit formula for the well-known embedding (see [DG07]) of H A r−1 into S(n, r). Let 1 r = 1 (1 r ) . We define the following map σ n,r : It is easy to check that σ n,r is well-defined. It turns out that σ n,r is actually an isomorphism, which induces the affine q-Schur functor between the categories of finite-dimensional modules of the extended affine Hecke algebra and of the affine q-Schur algebra. This functor is an equivalence (see Theorem 4.1.3 in [DDF12], for example).
The following result will be needed in Section 6.
Proposition 3.14. Let n > r. Suppose that A is a Q(q)-algebra and Proof. We first prove that f is an embedding when restricted to 1 r S(n, r)1 λ and 1 λ S(n, r)1 r , for any λ ∈ Λ(n, r). Suppose that this is not true in the first case, then there exists a non-zero element 1 r X1 λ ∈ 1 r S(n, r)1 λ in the kernel of f . By Lemma 3.13, we have 1 r Xρ(X)1 r = 0. However, we have which leads to a contradiction, because by hypothesis f is an embedding when restricted to 1 r S(n, r)1 r . The second case can be proved similarly. Now, for any λ, µ ∈ Λ(n, r), let 1 λ X1 µ ∈ S(n, r) be an arbitrary non-zero element. By Schur-Weyl duality, we have where the isomorphism is induced by left composition. Therefore, there exists an element 1 µ Y 1 r such that By the above, we have This shows that f is an embedding when restricted to 1 λ S(n, r)1 µ . Since λ and µ were arbitrary, this shows that f is an isomorphism.
Let us end this section giving an embedding between affine q-Schur algebras which we will use in Section 5.
Proposition 3.15. The Q(q)-linear algebra homomorphism ι n : S(n, r) → S(n + 1, r) defined by for any 1 ≤ i ≤ n − 1 and λ ∈ Λ(n, r), is an embedding and gives an isomorphism of algebras and S(n, r) [y] . In this section we define three 2-categories, U ( sl n ) [y] , U ( gl n ) [y] and S(n, r) [y] , using a graphical calculus analogous to Khovanov and Lauda's in [KL10]. [KL10], but the 2-HOM-spaces are tensored with Q[y], where y is a formal variable of degree two, and some of the KL-relations are ydeformed, as shown below. In order to define U ( gl n ) [y] , change the weights in the definition of U ( sl n ) [y] into (degenerate) level-zero gl n -weights (i.e. gl nweights). The 2-category S(n, r) [y] is then defined as a quotient of U ( gl n ) [y] . This is precisely the affine analogue of what was done in [MSV13].
Remark 3.16. We use the sign conventions from [MSV13] in the relations on 2-morphisms, which differ from Khovanov and Lauda's sign conventions. For more details on this change of convention, see below.
Remark 3.17. We do not prove that the 2-category U ( gl n ) [y] provides a categorification of U q ( gl n ), although we conjecture that it does. We will prove that S(n, r) [y] categorifies S(n, r).
In order to avoid giving several long definitions which are very similar, , but using degenerate level-zero sl n -weights. The 2-category S(n, r) [y] is defined as a quotient of U ( gl n ) [y] . We will show that S(n, r) [y] can also be obtained as a quotient of U ( sl n ) [y] .
To be more precise, we first define the Q[y]-linear graded 2-category with translation U ( gl n ) * [y] , whose 2-morphisms are Q[y]-linear combinations of homogeneous 2-morphisms of various degrees. The Q-linear 2-category U ( gl n ) [y] is then obtained by restricting to the degree-zero 2-morphisms.
The hom-category U ( gl n ) * [y] (λ, λ ′ ) between two objects λ, λ ′ is an additive Q[y]-linear graded category with translation defined by: for any t ∈ Z and signed sequence i ∈ SSeq such that λ ′ = λ + i Λ and λ, λ ′ ∈ Z n . The morphisms of U ( gl n ) * [y] (λ, λ ′ ) are presented by generators and relations. Multiplication by y is indicated graphically by y in the diagrams below.
-vector spaces given by linear combinations of diagrams of homogeneous degrees, modulo certain relations, built from composites of: [y] ; in particular for any i ∈ {1, . . . , n}, t ∈ Z and λ ∈ Z n , the identity 2morphisms 1 E i 1 λ {t} and 1 E −i 1 λ {t} are represented graphically by More generally, for a signed sequence i = (µ 1 i 1 , µ 2 i 2 , . . . µ m i m ), the identity 1 E i 1 λ {t} 2-morphism is represented as where the strand labeled i k is oriented up if µ k = + and oriented down if µ k = −. We will often place labels on the side of a strand and omit the labels at the top and bottom. The signed sequence can be recovered from the labels and the orientations on the strands. We might also forget the object on the left of the diagram which can be recovered from the object on the right and the signed sequence corresponding to the diagram.
ii) For any λ ∈ Z n the 2-morphisms Notation: where the degrees are given by the symmetric Z-valued bilinear form on C{1, . . . , n} • relations: ⋆ Biadjointness and cyclicity: i) 1 λ+i Λ E i 1 λ and 1 λ E −i 1 λ+i Λ are biadjoint, up to grading shifts: ii) As well for dotted lines: iii) All 2-morphisms are cyclic with respect to the above biadjoint structure. This is ensured by the relations (3.33)-(3.35), and the relations y y y y y y y y ii) A dotted bubble of degree zero equals ±1: (3.40) iii) For the following relations we employ the convention that all summations are increasing, so that a summation of the form m f =0 is zero if m < 0.
iv) Fake bubbles: Notice that for some values of λ the dotted bubbles appearing above have negative labels. A composite of with itself a negative number of times does not make sense. These dotted bubbles with negative labels, called fake bubbles, are formal symbols inductively defined by the equation and the additional condition Although the labels are negative for fake bubbles, one can check that the overall degree of each fake bubble is still positive, so that these fake bubbles do not violate the positivity of dotted bubble axiom. The above equation, called the infinite Grassmannian relation, remains valid even in high degree when most of the bubbles involved are not fake bubbles.
⋆ NilHecke relations: Note that this sign takes into account the standard orientation of the Dynkin diagram.   As already remarked, U ( sl n ) [y] is defined similarly. Only some signs in the relations involving right cups and caps have to be changed, so that all relations really depend on sl n -weights and not on gl n -weights. We use the convention which is the affine analogue of the signed-version in [KL10]. For more information on this change of signs, see (3.67).
Definition 3.21. Let U ( sl n ) and U ( gl n ) denote the Q-linear 2-categories obtained by modding out U ( sl n ) [y] and U ( gl n ) [y] by the ideal generated by y.
Note that U ( sl n ) is isomorphic to the original KL 2-category and U ( gl n ) isomorphic to its gl n -analogue.
We do not know if U ( sl n ) [y] is isomorphic to U ( sl n ) ⊗ Q Q[y], but it does not seem so.
3.5.2. Further relations in U ( gl n ) [y] . The other U ( gl n ) [y] -relations expressed below follow from the relations in Definition 3.18 and are going to be used in the sequel. ⋆ Bubble slides: (3.66) where the first sum is over all f 1 , f 2 , f 3 , f 4 ≥ 0 with f 1 + f 2 + f 3 + f 4 = λ i and the second sum is over all g 1 , g 2 , g 3 , g 4 ≥ 0 with g 1 +g 2 +g 3 +g 4 = λ i −2. Note that the first summation is zero if λ i < 0 and the second is zero when Reidemeister 3 like relations for all other orientations are determined from (3.50), (3.51), and the above relations using duality.
3.5.3. The 2-category S(n, r) [y] . As explained in Section 3.4, the q-Schur algebra S(n, r) can be seen as a quotient ofU( gl n ) by the ideal generated by all idempotents corresponding to the weights that do not belong to Λ(n, r).
It is then natural to define the 2-category S(n, r) [y] as a quotient of U ( gl n ) [y] as follows.
Definition 3.22. The 2-category S(n, r) [y] is the quotient of U ( gl n ) [y] by the ideal generated by all 2-morphisms containing a region with a label not in Λ(n, r).
We remark that we only put real bubbles, whose interior has a label outside Λ(n, r), equal to zero. To see what happens to a fake bubble, one first has to write it in terms of real bubbles with the opposite orientation using the infinite Grassmannian relation (3.44).
As in [MSV13], we define S(n, r) [y] as a quotient of U ( gl n ) [y] , rather than U ( sl n ) [y] . Therefore, just as in [MSV13] (see the introduction of Sections 3 and 4.3 in that paper), we have to show that there exists a full and essentially surjective 2-functor Ψ n,r : U ( sl n ) [y] → S(n, r) [y] , which categorifies the surjective homomorphism ψ n,r :U( sl n ) → S(n, r) defined in (3.28).
On objects Ψ n,r maps µ to λ := ϕ n,r (µ), which was defined in Section 3.2. On 1 and 2-morphisms Ψ n,r is defined to be the identity except for the left cups and caps, on which it is given by (3.67) Note that here we are simply extending the 2-functor used in [MSV13]. Just for completeness, we now state the following result without proof. is well-defined, full and essentially surjective.
Just as for U ( sl n ) [y] and U ( gl n ) [y] , we can put y = 0.
Definition 3.24. Let S(n, r) be the quotient of S(n, r) [y] by the ideal generated by y.
Of course there also exists a full and essentially surjective 2-functor Ψ n,r : U ( sl n ) → S(n, r), which is defined and denoted just as above.

A functor from DEBim *
A r−1 to S(n, r) *

[y]
In this section, we define a functor Σ n,r : DEBim * which categorifies the embedding σ n,r : H A r−1 → 1 r S(n, r)1 r .
This functor is the affine analogue of Σ n,d in Section 6.5 in [MSV13]. For diagrams with only unoriented i-colored strands for i = 1, . . . , r − 1 the definitions in that paper and in this one coincide, for diagrams with unoriented r-colored strands or oriented strands the definition here is new.
In Section 6, we will prove that Σ n,r : DEBim * is faithful. We conjecture that Σ n,r is also full and, therefore, that the two categories DEBim A r−1 and S(n, r) [y] ((1 r ), (1 r )) are equivalent. The latter equivalence would the affine analogue of the one proved in Proposition 6.9 in [MSV13] for finite type A.
Proof. We check that Σ n,r preserves all relations in DEBim * A r−1 . First of all, note that all "finite type A" relations, i.e. the relations between diagrams without r-colored or oriented strands, are preserved by precisely the same arguments as in [MSV13].
Let us also remark that, except in the checks of the box relations, we will use neither Relation (3.48) nor the bubble slides Relations (3.57)-(3.64) for {i, j} = {1, n}.
Let us now go through the list of the remaining relations and explain why they are preserved: • The Isotopy Relations (2.13)-(2.21) are straightforward.
• Relations (2.22)-(2.34) with at least one r-colored strand follow from the same relations without any r-colored strand together with Relations (2.35)-(2.45). So it suffices to prove the latter relations.
Indeed, for any of these relations with an r-colored strand, one can add an oriented bubble (or if necessary two nested ones) to the diagram on the left-hand side of the equation. By Relation (2.35) the oriented bubbles are equal to one and we will show that that relation is preserved by Σ n,r below. Using the appropriate relations involving colored and oriented strands (which are checked below), slide that bubble across the diagram until it encloses that part of the diagram which differs from the diagram on the right-hand side of the equation. It is always possible to choose the orientation of the bubble so that the diagram in its interior does not have any r-colored strands. Then apply the relevant relation for colors different from r to the part of the diagram contained inside the bubble. Finally, slide the bubble aside again. This works since all these operations are proved to be preserved by Σ n,r below.
• Relation (2.35) follows directly from the fact that in S(n, r) * [y] ((1 r ), (1 r )) all dotted bubbles of degree zero are equal to ±1. Indeed we can apply successively Relations (3.40) to the nested bubbles in the image of and .
• For Relations (2.36) and (2.37) use repeatedly Relations (3.43) and (3.42). We only give the details for Relation (2.36), the other relation being completely analogous.
Apply successively Relation (3.43) to each pair of the form i y y i λ .
For i = r + 1, . . . , n, we have λ = (0, 1, . . . , 1, 0, . . . , 0, 1, 0, . . . , 0), where the entries which are equal to one are the 2nd until the rth and the i + 1st (mod n). For i = 1, . . . , r, we have λ = (1, . . . , 1, 0, 1, . . . , 1, 0, . . . , 0) where the entries which are equal to zero are the r + 2nd until the nth and the ith. For these λ, Relation (3.43) becomes y y The first term on the right-hand side of this relation is equal to zero because the label appearing in interior region contains a negative entry. The bubble appearing in the second term on the right-hand side is equal to −1. Thus, for all i, we are left with y y Taking all strands together, we get the following nested cups and caps: which is equal to Σ n,r . • Relation (2.38) For j < i and i, j = r, this relation follows from the fact that, using repeatedly Relations (3.47), (3.48) for distant i, j, (3.50) and (3.65), one can reduce both Σ n,r and Σ n,r to the following diagram Similarly, for j > i and i, j = r, Relation (2.38) follows from the fact that, using repeatedly Relations (3.47), (3.48) for distant i, j, (3.50) and (3.65), one can reduce both Σ n,r and Σ n,r to the following diagram (1 r ) One can prove the remaining cases, in which one of the integers i or j is equal to r − 1 or r, in exactly the same way. Just use repeatedly Relations (3.47), (3.48) for i · j = 0, (3.50) and (3.65).
First when the colors (i, i + 1) differ from (r, 1) and (r − 1, r). We only give the details for Relation (2.39), because the proof of the other relation is very similar.
The image under Σ n,r of is equal to After removing the bubble, we can slide the i-colored strand over the r − i − 1 rightmost strands and slide the i + 1-colored strand over the n − r + i − 1 leftmost strands, using Relation (3.48). We get . . . , 1, 0, 1, . . . , 1, 0, . . . , 0) where the first 0 is in ith position and the last 1 in r + 1st position. Apply Relation (4.2), which has only one non-zero term as before. We get · · · · · · y y · · · n r+1 1 Then the strand with the rightmost endpoints can be slid all the way to the right, first using Relation (3.47) and then Relation (3.43), which again is simply equal to y y because λ = (1 r ). Finally, we end up with a diagram which is indeed equal to Σ n,r . We can prove Relations (2.39) and (2.40) with colors (r, 1) and (r − 1, r) in almost the same way. We can slide bubbles and strands using Relations depending on the value of λ. The difference here is that we have to iterate some of the steps that we used above, e.g. because we get nested bubbles or various pairs of strands to which we can apply Relations (3.42) and (3.43).
The image under Σ n,r of is equal to We can slide the i + 1-colored strand over the n − r + i − 1 leftmost strands using Relation (3.48), so that we end up with the following picture in the central part of the diagram: Then, in the full diagram, the rightmost i-colored strand above can be slid over the r − i − 1 rightmost parallel strands using Relation (3.47), so that we end up with We apply Relation (3.41) to the curl. For λ = (1 r ), this relation becomes Since the bubble is equal to −1, the diagram in (4.4) becomes equal to which is indeed the image under Σ n,r of . To prove Relations (2.41) and (2.42) with colors (r, 1) and (r − 1, r), we use exactly the same ingredients as above: sliding strands using Relations (3.47) and (3.48), removing curls using which is equivalent. Let us start with colors (i, i + 1) different from (r, 1) and (r − 1, r).
The image under Σ n,r of The image under Σ n,r of is We can apply to the first of these two diagrams the same arguments as in the beginning of the proof of Relation (2.39), in order to simplify it to In order to prove that this diagram is equal to the second one, it only remains to check that when λ = (1 r ), the left hand side of (4.6) is equal to So proving (4.6) is equivalent to proving that: Let us apply twice Relation (4.3) to the right hand side. This gives us Then we apply Relation (3.43), with λ = (1, . . . , 1, 0, 2, 1, . . . , 1, 0, . . . , 0) where 2 is in i + 1st position and the last 1 in rth position, which gives us Here the first and last term are equal to zero, due to Relation (3.45). To the second term we can apply Relation (3.46), so that we get Here again the first term on the right-hand side is killed by Relation (3.45), so Relation (4.8) is satisfied and Σ n,r and Σ n,r are equal. Proving Relation (4.5) for colors (r − 1, r) is not much more complicated and is left to the reader.
In order to prove these relations we bend the left end of the oriented strand downward and the right end upward in the diagrams.
Let us start with the case when there are no r-colored strands. We prove Relation (2.44) for the case in which the bottom strands are colored (i − 1, i, i − 1). Relation (2.45) for the case in which the bottom strands are colored (i, i − 1, i) can be proved similarly.
By applying Relations (3.47), (3.48), (3.50) and (3.65) to the diagrams Σ n,r and Σ n,r , we can slide the entangled parts of the strands colored i − 1, i and i + 1 to the middle of the diagrams. In this way, we can turn Σ n,r and Σ n,r into the following two diagrams: the diagram Σ n,r becomes (1 r ) and Σ n,r becomes (1 r ).
We have to prove that these two diagrams are equivalent.
First consider the diagram above which is equivalent to Σ n,r . Apply Relation (3.66) to the triangle in the center formed by three i-colored strands. Since λ = (1, . . . , 1, 0, 1, 2, 0, 1, . . . , 1, 0, . . . , 0), where 2 is in the i + 1st position and the last 1 in the r + 1st position, that relation is equal Then apply Relation (3.50) to the two triangles formed by strands colored (i−1, i, i+1). Both triangles are slightly to the right of the center and one is higher and the other is lower than the center. Sliding the i-colored strands to the left using this relation creates four bigons, two between strands colored i and i + 1 and the other two between strands colored i − 1 and i + 1. The first two bigons can be solved using Relation (3.47), the other two using Relation (3.48). Finally, we apply Relation (3.50) to the top and bottom central triangles of the diagram. This proves that Σ n,r is equivalent to (1 r ) Now apply Relation (3.51) to the triangle in the central right part of the diagram. This gives us two terms. The second term is killed because it contains a bigon between two i-colored strands with the same orientation, which is zero by Relation (3.45).
In the remaining term we can slide the vertical i-colored strand to the left using Relation (3.66), as we did in (4.10). This leaves us with a bigon between two i-colored strands with opposite orientations. Use Relation (3.43) in order to remove this bigon. Note that this relation is equal to y y . . . , 1, 0, 2, 1, 0, 1, . . . , 1, 0, . . . , 0) with 2 in the ith position and the last 1 in the r + 1st position.
This tells us that Σ n,r is equivalent to (1 r ) (4.11) Let us now prove that Σ n,r is equivalent to this same diagram. First apply Relation (3.66) to the triangle formed by the three i − 1-colored strands just right of the center. Since λ = (1, . . . , 1, 2, 0, 0, 1, . . . , 1, 0 Then apply Relation (3.50) to the two central triangles formed by strands colored (i − 1, i, i + 1). Sliding the i + 1-colored strands to the left using this relation creates two bigons between strands colored i − 1 and i + 1 respectively. These bigons can be removed using Relation (3.48).
In this way, we have proved that Σ n,r is equivalent to (1 r ) Next, apply Relation (3.51) to the central left part of the diagram. Locally we end up with a sum of two terms: The second term is killed because it contains an i − 1-colored curl which is zero by Relation (3.45). Note that λ = (1, . . . , 1, 2, 0, 0, 1, . . . , 1, 0, . . . , 0) where 2 is in the ith position and the last 1 is in the r + 1st position, sō λ i−1 = −1. The first term contains a bigon between two i−1-colored strands with opposite orientations. Remove this bigon using Relation (3.42), which in this case equals Now apply Relation (3.50) to the top central and bottom central parts of the diagram which we have obtained so far. Apply Relation (3.65) to the top right and bottom right parts of the diagram. In this way we get two more bigons between strands colored i − 1 and i with opposite orientations, which we remove using Relation (3.47). This finishes our proof that Σ n,r is equivalent to the same diagram in (4.11).
Let us now consider Relation (2.44) when one of the strands has color r (Relation (2.45) can be dealt with in a similar way). Three cases have to be considered: when the bottom strands are colored (r − 2, r − 1, r − 2), (r, 1, r) or (r − 1, r, r − 1).
Using the same sort of reasoning as above, one can show that Σ n,r and Σ n,r both reduce to the same diagram. We omit the precise computations, but just give the diagram to which both images can be reduced.
• Box relations. Just as for Relations (2.22)-(2.34), some of the box relations with i = r or j = r follow from the same box relations for i, j = r together with some other box relations. Taking into account the observations in Remark 2.9 too, we see that it suffices to prove Relations (2.47) and (2.55) here.
Let us start with Relation (2.55). It is sufficient to prove this relation for i = r − 1, because we can write · · · · · · n r+1 1 r = · · · · · · n r+1 1 r − r−1 By Relations (2.41) and (2.42), we have o o n r · · · · · · n r+1 1 r = · · · · · · n r+1 1 r r−1 . Therefore, it suffices to prove On the one hand, we observe that r • −1 · · · · · · n r+1 1 r = · · · · · · n r+1 1 r • − · · · r · · · n r+1 1 r since the bubble can be slid through the first n − r − 1 left strands and then bubble slide Relation (3.53) can be applied to the bubble and the strand colored r + 1. On the other hand, we have o o n r · · · · · · n r+1 1 r = · · · r · · · n r+1 1 r which is obtained by using repeatedly Relation (3.42), which for the relevant labels λ reduces to So it only remains to prove that · · · · · · n r+1 1 r Notice that · · · · · · n r+1 1 r by the infinite Grassmannian relation. We can apply Relation (3.42) to the bubble and the strand colored r. The first term that appears is killed, because the weight inside the bigon has a negative entry. The bubble appearing in the second term satisfies . . . , 1, 0, 1, 0, . . . , 0) with the last one in r + 1st position. Thus we get · · · · · · n r+1 1 r r • −1 = · · · · · · n r+1 1 r • Finally, we have to verify that · · · · · · n r+1 1 r • = · · · · · · n r+1 1 r • (4.12) which is true: just slide the left dotted strand over all the strands colored 1, . . . , r −1, using the first case of Relation (3.48), and then apply the second case of Relation (3.48). Observe that the term with the bigon is killed, because the weight inside the bigon has a negative entry. Note that this argument is not valid if r = n − 1. Indeed in this case, r + 1 and 1 are adjacent colors, hence the left dotted strand cannot be simply slid over the strand colored 1. In order to prove this remaining case together with Relation (2.47), let us remark that the right hand side of Equation (4.12) can also be expressed as follows: · · · · · · n r+1 1 r • = · · · · · · n r+1 1 r This expression is obtained using repeatedly kink resolutions and bubble slides. Indeed the kink on the left hand side of Relation (3.41) for i = r is equal to zero here since the label inside the kink possesses a negative entry. Hence one can express the dotted r strand as a non-dotted strand times a bubble colored r on the left. This bubble can then be slid through the r − 1 strand using Relation (3.56), creating two terms: one is a dotted r −1 strand while the other is a non-dotted r − 1 strand times a bubble colored r on the left. This bubble can be slid all the way to the left using (3.56) making appear an extra term which is the dotted r + 1 strand on the right hand side of (4.13). One applies this same trick successively to the dotted strands for colors r − 1 to 1. The only exception is that, in the end, to slide the bubble colored 1 through the strand colored n, we have to use the deformed Relation (3.64), which brings out the y term in (4.13). Finally the dotted n strand that thus appears can also be expressed as a non-dotted strand times a n-colored bubble on the left using Relation (3.41). Therefore, by (4.13), it suffices to prove (1 r ) + y = 0 (4.14) in order to show (4.12). In Section 5 we define a 2-representation of S(n, r) * [y] . Its restriction to END(1 r ) gives an algebra homomorphism which on the elements of degree two is determined by Note that F ′ maps the l.h.s. of (4.14) to zero. We are going to show that this implies (4.14) by showing that F ′ is an isomorphism. From the definition it is clear that F ′ is surjective. Injectivity follows if we can prove that END(1 r ) is generated by for i = 1, . . . , r − 1, because that implies that END(1 r ) is isomorphic to a quotient of Q[y, x 1 , . . . , x r ] by the surjectivity of F ′ , which means that the two algebras have to be isomorphic. In order to prove that END(1 r ) is indeed generated by the 2-morphisms in (4.15), first note that END(1 r ) is generated by all counter-clockwise bubbles of arbitrary degree and y . It therefore suffices to prove that any counter-clockwise bubble is in the span of the 2-morphisms in (4.15).
We first prove this fact for counter-clockwise i-bubbles with 1 ≤ i ≤ r − 1. The result follows from the recursive formula for t ≥ 0. The equation in (4.16) can be obtained by unnesting the l.h.s. of (4.17) using bubble slides from the inside to the outside. Equation (4.17) holds, because the region in the center has label λ with λ r+1 = −1.
For i = r the argument is simpler, because (1 r ) = 0 for any t ≥ 2. This holds because the inner region has label λ with λ r+1 = −1.
Similarly, for i = n we have for any t ≥ 2. By the infinite Grassmannian relation, this implies that for any t ≥ 2. For r + 1 ≤ i ≤ n − 1 there is nothing to prove. In that case, the counterclockwise i-bubbles of positive degree are all zero, because their interior is labeled by λ with λ i+1 = −1. This finishes the proof that END(1 r ) ∼ = Q[y, x 1 , . . . , x r ], which implies (4.14). Now let us consider Relation (2.47). The image under Σ n,r of This expression can be obtained using repeatedly bubble slide Relation (3.54). Observe that, at each step but the first one, only one term survives, the second being systematically zero since it includes a real bubble whose inside is a label with a negative entry.
If one replaces, in this expression, the bubble colored n using Equation (4.14), one recognizes precisely the image under Σ n,r of 1 − r − y .
We have checked that Σ n,r preserves all the relations of DEBim * A r−1 , so this ends the proof.

[y]
In this section we define a 2-category ESBim A r−1 and a 2-functor The 2-category ESBim A r−1 is an extension of the category of singular Soergel bimodules in affine type A considered by Williamson in [Wil11] (see also [MSV11,MSV13] be the parabolic subgroup which is contained in the finite Weyl group. let R i 1 ···i k ⊆ R denote the subring of S i 1 × · · · × S i k -invariant polynomials. Using these rings of partially symmetric polynomials, we can construct bimodules by induction and restriction. Induction is defined as follows: suppose i j splits into i 0 j plus i 1 j , then we define the induction functor by The subscript of the tensor product means that it is taken over R i 1 ···i k . The restriction functor is defined by Definition 5.1. Let ESBim A r−1 be the 2-category whose objects are the rings R i 1 ···i k , for all partitions (i 1 , . . . , i k ) of r.
For any two partitions (i 1 , . . . , i k ) and (j 1 , . . . , j l ) of r, the 1-morphisms between R i 1 ···i k and R j 1 ···j l are by definition the direct sums and shifts of tensor products of R i 1 ···i k -R j 1 ···j l -bimodules obtained by induction and restriction and by tensoring with the twisted bimodules R i k i 1 ···i k−1 ,ρ and R i 2 ···i k i 1 ,ρ −1 , which we will define below.
The 2-morphisms are the degree-preserving bimodule maps.
The twisted bimodules are defined like the bimodules B ρ ±1 of section 2.1.2: R i k i 1 ···i k−1 ,ρ (resp. R i 2 ···i k i 1 ,ρ −1 ) is equal to R i k i 1 ···i k−1 (resp. R i 2 ···i k i 1 ) as a left R i k i 1 ···i k−1 -module (resp. as a left R i 2 ···i k i 1 -module) whereas the action on the right is twisted. The right action of any a ∈ R i 1 ···i k on R i k i 1 ···i k−1 ,ρ and R i 2 ···i k i 1 ,ρ −1 is given by multiplication by ρ i k (a) and ρ −i 1 (a) respectively.
Recall that the action of ρ ±1 was defined in section 2.1.1. The functors defined by tensoring with twisted bimodules are denote by Lemma 5.2. The map ρ i k gives an isomorphism between R i 1 ···i k and R i k i 1 ···i k−1 , while ρ −i 1 gives an isomorphism between R i 1 ···i k and R i 2 ···i k i 1 . This implies that the twisted bimodules R i k i 1 ···i k−1 ,ρ and R i 2 ···i k i 1 ,ρ −1 are well-defined.
Proof. We only prove the lemma for ρ i k . The proof for ρ −i 1 is similar and is left to the reader.
It is clear that ρ i k is a bijection. What remains to be shown is that the The ring R i 1 ···i k is generated by the following elementary symmetric polynomials: x i 1 +···+i j−1 +q 1 . . . x i 1 +···+i j−1 +qp j for all j = 1, . . . , k and p j = 1, . . . , i j .
For j = k and p k = 1, . . . , i k , we get This product is equal to This shows that ρ i k sends the ring R i 1 ···i k isomorphically to the ring The proof of the following lemma is straightforward and is left to the reader.
Lemma 5.3. We have the following isomorphisms of bimodules relating twisting, induction and restriction: There exist analogous isomorphisms for the negative twists.
Lemma 5.4. The category EBim A r−1 is a full subcategory of ESBim A r−1 .
Proof. For i = 1, . . . , r − 1, the full embedding of The fact that the isomorphism in EBim A r−1 between B r and B ρ ⊗ R B r−1 ⊗ R B ρ −1 is unique ensures that EBim A r−1 is a full subcategory of ESBim A r−1 .

5.2.
The 2-representation. We will mostly refer the reader to [KL10,KL11,MSV13] for the definition of since F ′ is a straigthforward generalization of the equivariant Khovanov-Lauda 2-representations discussed in those papers.
Remark 5.5. Khovanov and Lauda used the equivariant cohomology rings of the varieties of partial flags in C r for the definition of their equivariant 2-representations. These cohomology rings are isomorphic to the finite type A singular Soergel bimodules which were used in [MSV13]. We do not know if the 2-representation in this paper, which we define using the extended affine singular Soergel bimodules, can be defined in terms of equivariant cohomology rings of the varieties of cyclic partial flags (or periodic lattices) in C[ε, ε −1 ] r defined in [Lus99] and [GV93]. 5.2.1. Definition of F ′ . Note that for y = 0 the restriction of F ′ •Ψ n,r to U (sl n ) is simply equal to Γ G r , where Ψ n,r : U ( sl n ) * [y] → S(n, r) *

[y]
was defined just before Proposition 3.23. We will define the 2-functor F ′ on all objects and 1-morphisms of S(n, r) * [y] , give explicitly the images of the 2-morphisms for which the color n appears and explain how they are related to Khovanov and Lauda's 2-representation Γ G r . Here we are using their notation k i = λ 1 + · · · + λ i , for i = 1, . . . , n, with the convention that k 0 = 0.
Remark 5.6. Let us explain where the shifts in the image of the new 1morphisms come from. We will denote by A n (λ) the shift appearing in F ′ (E n 1 λ ) and by B n (λ) the one appearing in F ′ (E −n 1 λ ). To understand their origin, let us go back to the decategorified level (see Section 3.4), where the embedding of S(n, r) into S(n + 1, r) sends E n 1 λ to E n E n+1 1 (λ,0) and E −n 1 λ to E −(n+1) E −n 1 (λ,0) . We want this embedding to have a categorical analogue. Although we will not work out the details of the corresponding functor in this paper, a necessary condition for the existence of such a functor is that the shifts satisfy the following recurrence relations: where λ ′ = (λ 1 , · · · , λ n−1 , λ n −1, 1). This determines the value of A n (λ) and B n (λ) up to a constant which does not depend on n. To fix these constants, we use the fact that we want the triangle of lemma 6.1 to be commutative and the 2-functor F ′ to be degree preserving. Note that F(−) has a shift equal to zero, so F ′ (E r . . . E 1 E r+1 . . . E n 1 r ) = F ′ Σ n,r (−) should have a shift equal to zero too. In this way we obtain a constraint on A n ((1 r )) and we deduce that the aforementioned overall constant is equal to −(r + k 1 ). Similarly F(+) has a shift equal to zero, so F ′ (E −n . . . E −r−1 E −1 . . . E −r 1 r ) = F ′ Σ n,r (+) has to have a zero-shift too. This gives us a constraint on B n ((1 r ) +ᾱ n ) and allows us to deduce that the aforementioned overall constant is equal to zero. The reader can verify that this choice of overall constants also fits with the −1-shift of F ′ (E −n . . . E −r E r . . . E n 1 r ) = F ′ Σ n,r (r).
•. To define F ′ on 2-morphisms, we have to give the bimodule maps which correspond to the generating 2-morphisms of S(n, r) * When the color n occurs in a generating 2-morphism, the 2-functor F ′ is as follows. Here ε α and η α denote the elementary and the complete symmetric polynomials respectively.
Remark 5.7. It is natural to wonder how the previous images relate to Khovanov and Lauda's 2-representation Γ G r . Indeed, take a generating 2morphism with n-strands. By Lemma 5.3, the images of its source and target 1-morphisms are isomorphic, up to a same shift, to conjugates of the images of 1-morphisms which do not contain factors E ±n . Here conjugation means conjugation by certain invertible twisted bimodules. One can thus ask if the image of the 2-morphism can be obtained by conjugating a 2morphism which does not contain strands of color n. Before answering this question, let us do an example to make things more concrete (we omit the shifts here). Consider the 2-morphism Res λn+1 λn,1 R ρ −1 Ind 1,λ 1 −1 λ 1 (R λ 1 ,...,λn ) .
Of course one could have used a similar conjugation trick turning n into n − 1, i.e. writing everything as conjugates R ρ λ 1 −1 ⊗ − ⊗ R ρ −λ 1 and using the bimodule map corresponding to where λ ′′ = (λ 2 , . . . , λ n , λ 1 ). In this case one obtains again the image under F ′ of our original 2-morphism. In fact, the image of any generating 2-morphism containing n-strands can be obtained by either one of the conjugation tricks, i.e. turning n into 1 or n − 1, except for the dotted n-identities. The images of these two 2-morphisms (up and downward) can be obtained by the conjugation trick which turns n into n − 1, but not by the one which turns n into 1. Indeed, if one writes F ′ (E ±n 1 λ ) as R ρ −(λn±1) ⊗ F ′ (E ±1 1 λ ′ ) ⊗ R ρ λn , one obtains 1 ⊗ 1 → (x r + y) ⊗ 1 = 1 ⊗ x 1 (resp. x 1 ⊗ 1 = 1 ⊗ (x r + y)) which differs from the image under F ′ of the oriented upward (resp. downward) dotted n-identity.
As for the non-dotted generating 2-morphisms of color n, both conjugation tricks give the same bimodule maps which we have used in the definition of F ′ . Whenever only one expression was given in our definition of F ′ , it is because the expressions obtained from both conjugation tricks were obviously equal. Whenever two expressions are given, let us prove that they are equal.
This follows from the following lemma.
The proof for the image of the left n-cup is similar, using x r ⊗ 1 ⊗ 1 ⊗ 1 = 1 ⊗ (x 1 − y) ⊗ 1 ⊗ 1 and the lemma above with −y instead of y.
The result for the cups implies the result for the caps, because of the biadjointness relations (3.33) and (3.34). If two maps corresponding to a cap both satisfy the biadjointness relations w.r.t. one fixed map associated to the corresponding cup, then the two maps have to be equal.
A similar argument proves the result for downward oriented n-crossings and the remaining cases of the crossings colored (1, n) and (n, n − 1) are easy computations.
Just to summarize, all generating 2-morphisms containing n-strands can be obtained by the conjugation trick which turns n into n − 1, while only the ones whose n-strands have no dots can also be obtained by the conjugation trick which turns n into 1.
Proof. All relations between 2-morphisms which do not have n-colored strands are satisfied by the results in Section 6 in [KL10]. Since no relation in S(n, r) [y] involves all colors at the same time, the proof that F ′ preserves a given relation can always be reduced to the fact that F ′ preserves the same relation with colors belonging to {1, . . . , n − 1} by using the conjugation trick which turns n into n − 1, except in the case of relations (3.48) and (3.49) for {i, j} = {1, n}. The proof that F ′ preserves these relations is straightforward and is left to the reader.
As a matter of fact, any twisted singular bimodule is isomorphic to the image under F ′ of a certain product of categorified divided powers (see [KLMS12] and [MSV13] for more details on divided powers and extended graphical calculus). Indeed, let n > r and let λ ∈ Λ(n, r) be arbitrary. At least one entry of λ is equal to zero, let us assume it is λ i . Then

The Grothendieck group of S(n, r) [y]
The following Lemma is the affine analogue of Lemma 6.6 in [MSV13]. is an embedding.
Proof. We already know that K Q(q) 0 (F) is injective, by Theorem 2.12 and the fact that EBim A r−1 is a full sub-2-category of ESBim A r−1 . The result now follows from the commutativity of the diagram in Lemma 6.1.
Theorem 2.12 and Lemma 6.1 also imply that Σ n,r is faithful. We do not know if it is full, as in the finite type A case (Proposition 6.9 in [MSV13]), but we conjecture that to be true. in Theorem 1.1 in [KL10]. The same arguments which proved Lemma 7.7 in [MSV13] can thus be used to prove that γ : S(n, r) → K Q(q) 0 (Kar S(n, r)) is surjective. By (6.1) this implies that the analogous homomorphism γ : S(n, r) → K Q(q) 0 (Kar S(n, r) [y] ) is surjective.
The rest of the proof follows from Lemma 3.14 and Corollary 6.2 with A = K Q(q) 0 (Kar S(n, r) [y] ).