AN 𝔰𝔩_{2}-CATEGORIFICATION OF TENSOR PRODUCTS OF SIMPLE REPRESENTATIONS OF 𝔰𝔩_{2}

RICHARD WILLIAMSON

Abstract. In [12], Zheng gave a geometric categorification of tensor products of simple U_{q}(𝔰𝔩_{2})-modules. We extend his work to a 2-categorical setting, in line with the higher representation theory programme of Rouquier.

1. Introduction

For any parabolic subgroup P ⊂ GL_{n}(ℂ), let D_{P} (Gr(i)) denote the bounded derived category of P -smooth constructible \overline{ℚ_{l}}-sheaves on Gr(i). Let V denote a tensor product of simple representations of 𝔰𝔩_{2}. We show that there is an integer n ≥ 0 and a parabolic P ⊂ GL_{n}(ℂ) such that ⨁_{i=0}^{n} D_{P} (Gr(i)), equipped with certain natural endofunctors E and F, is a triangulated categorification of V. This is based on work of Zheng ([12]).

Realising E and E^{2} as Fourier-Mukai transforms, we explain how to define 2-morphisms X ∈ End^{•}(E) and T ∈ End^{•}(E^{2}) using Chern classes of canonical vector bundles. This extends our construction to an 𝔰𝔩_{2}-categorification, in the sense of Chuang and Rouquier ([6]).

Keeping track of the Tate twist, we pass to a categorification of tensor products of simple representations of U_{q}(𝔰𝔩_{2}). Via Koszul duality, as in [12], we obtain an abelian categorification of these representations.

Naturally, our results should generalise to the case of highest weight integrable representations of arbitrary quantum groups. Using a notion of micro-local perverse sheaves on quiver varieties, Zheng has shown ([11]) how to generalise the weak categorification.

Let D_{P ×ℂ^{∗}}^{b}(T^{∗}(Gr(i))-coh) denote the bounded derived category of (P × ℂ^{∗})-equivariant coherent sheaves on the cotangent bundle. Via Saito’s mixed Hodge modules, ⨁_{i=0}^{n} D_{P} (Gr(i)) can be replaced by ⨁_{i=0}^{n} D_{P ×ℂ^{∗}}^{b}(T^{∗}(Gr(i))-coh) in the constructions above. In this picture, generalising to an arbitrary quantum group should be carried out by replacing T^{∗}(Gr(i)) by other quiver varieties.

The definition of X and T, and the proof that they satisfy the Hecke relations, was explained to me by Raphaël Rouquier. I thank him very much for his generosity, and for the many things I have learnt from him.

Contents

1. Introduction  1
2. Triangulated 𝔰𝔩_{2}-categorification of tensor products  2
2.1. Abelian 𝔰𝔩_{2}-categorification  2
2.2. Triangulated 𝔰𝔩_{2}-categorification  4
2.3. Weak categorification  5
2.4. 2-morphisms X and T  8

Date: May 14, 2012.

Supported by the EPSRC.

2.5. Action of ℤ[q, q^{−1}]  12
3. Koszul duality  14
3.1. Koszul duality  14
3.2. An abelian 𝔰𝔩_{2}-categorification of simple representations of 𝔰𝔩_{2}  15
References  15

2. Triangulated 𝔰𝔩_{2}-categorification of tensor products

2.1. Abelian 𝔰𝔩_{2}-categorification. Let H_{n}(q), for q ≠ 0, 1, denote the affine Hecke algebra of type Ã_{n−1} over a field k. Let H_{n}(0) (resp. H_{n}(1)) denote the nil (resp. degenerate) affine Hecke algebra of type Ã_{n−1}. In particular, H_{n}(1) is not the specialisation of the affine Hecke algebra to q = 1.

We shall mainly need H_{n}(0), which has generators T_{1}, …, T_{n−1}, X_{1}, …, X_{n}, subject to the following relations:

T_{i}^{2} = 0
T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1}
T_{i}T j = T_{j}T_{i} if |i − j| > 1
X_{i}X_{j} = X_{j}X_{i}
T_{i}X_{j} = X_{j}T_{i} if |i − j| > 1
T_{i}X_{i} = X_{i+1}T_{i} − 1
T_{i}X_{i+1} = X_{i}T_{i} + 1.

In [6], Chuang and Rouquier introduced the notion of an abelian 𝔰𝔩_{2}-categorification.

Definition 2.1. A weak abelian 𝔰𝔩_{2}-categorification is the data of

• a k-linear abelian category A, with finite dimensional complexified Grothendieck group K = ℂ ⊗ K_{0}(A)

• an adjoint pair of exact endofunctors (E, F) such that

• E and F induce an action of 𝔰𝔩_{2} on K

• F is isomorphic to a left adjoint of E.

Example 2.2. Let Gr(i) denote the Grassmannian variety of i-dimensional subspaces of ℂ^{n}, and let Gr(i, i + 1) denote the partial flag variety {V, W ⊂ ℂ^{n} | V ⊂ W, dim(V) = i, dim(W) = i + 1}. We have the following diagram, where p and q are the canonical projections.

\xymatrix{ & Gr(i, i + 1) \ar[dl]_{p} \ar[dr]^{q} \\ Gr(i) &  & Gr(i + 1)}

The (singular) cohomology algebra H^{•}(Gr(i)) has a unique simple module up to isomorphism, induced by the projection

\xymatrix{H^{•}(Gr(i)) \ar@{->>}[r] & H^{0}(Gr(i)) \ar[r]^{≃} & ℂ} .

The morphism q induces an inclusion of algebras H^{•}(Gr(i + 1)) ↪ H^{•}(Gr(i, i + 1)). The morphism p induces an inclusion of algebras H^{•}(Gr(i)) ↪ H^{•}(Gr(i, i + 1)).

Regarding H^{•}(Gr(i, i + 1)) as an H^{•}(Gr(i + 1))-H^{•}(Gr(i))-bimodule, define

E_{i}: H^{•}(Gr(i))-mod → H^{•}(Gr(i + 1))-mod

by H^{•}(Gr(i, i + 1)) ⊗_{H^{•}(Gr(i))} −. Regarding H^{•}(Gr(i, i + 1)) as an H^{•}(Gr(i))-H^{•}(Gr(i + 1))- bimodule, define

F_{i}: H^{•}(Gr(i + 1))-mod → H^{•}(Gr(i))-mod

by H^{•}(Gr(i, i + 1)) ⊗_{H^{•}(Gr(i+1))} −. Then E = ⨁_{i=0}^{n} E_{i} and F = ⨁_{i=0}^{n} F_{i} define endofunctors of A(n) = ⨁_{i=0}^{n} H^{•}(Gr(i))-mod.

It is classical (see 3.4 in [7], for example) that, as an H^{•}(Gr(i))-module,

H^{•}(Gr(i, i + 1)) ≅ ⨁_{j=0}^{n−i−1} H^{•}(Gr(i)).

As an H^{•}(Gr(i))-module,

H^{•}(Gr(i − 1, i)) ≅ ⨁_{j=0}^{i−1} H^{•}(Gr(i)).

It follows that

EF (H^{•}(Gr(i))) = i(n − i + 1)H^{•}(Gr(i))

and that

F E(H^{•}(Gr(i)) = (n − i)(i + 1)H^{•}(Gr(i)).

Let e and f denote the endomorphisms of K_{0}(A(n)) induced by E and F. Then

(ef − fe)([H^{•}(Gr(i))]) = (2i − n) [H^{•}(Gr(i))] .

Since ℂ ⊗ K_{0}(A(n)) = ⨁_{i=0}^{n} ℂ [H^{•}(Gr(i))], we have shown that ef − fe acts on K_{0}(A(n)_{λ}) by λ, where A(n)_{λ} = H^{•}(Gr(\frac{λ+n}{2}))-mod.

The endofunctors (E, F) are an adjoint pair, with F isomorphic to a left adjoint of E. We will see this later. Alternatively, one can prove it algebraically, as in Proposition 3.5 of [7].

Thus we have a weak 𝔰𝔩_{2}-categorification of the simple representation of 𝔰𝔩_{2} of dimension n. In this example, we merged the arguments of 5.3 in [6] and 6.2 in [7]. In the latter paper, the weak 𝔰𝔩_{2}-categorification is modified to a weak categorification of the simple representation of U_{q}(𝔰𝔩_{2}) of dimension n, using graded versions of the functors and categories above. We will pass to U_{q}(𝔰𝔩_{2}) slightly differently later.

Definition 2.3. An abelian 𝔰𝔩_{2}-categorification is the data of • a weak 𝔰𝔩_{2}-categorification A

• natural transformations X ∈ End(E), T ∈ End^{2}(E) such that

• X − a is locally nilpotent for some a ∈ k

• T_{i} → 1_{E^{n−i−1}}T 1_{E^{i−1}} and X_{i} → 1_{E^{n−i}}X1_{E^{i}} define a morphism H_{n}(q) → End(E^{n}) for all n and a fixed q.

Remark 2.4. The last property is key, allowing abelian 𝔰𝔩_{2}-categorifications to be controlled.

Example 2.5. Via Koszul duality and a special case of our main result, we will see (3.2) that the weak categorification 2.2 can be extended to an 𝔰𝔩_{2}-categorification.

Subquotients of the affine Hecke algebra can be used to give an algebraic abelian 𝔰𝔩_{2}- categorification of the simple representation of 𝔰𝔩_{2} of dimension n (see 5.3 in [6]). These ‘minimal’ categorifications play a central role in the theory of abelian higher representations of 𝔰𝔩_{2} (see 5.24 in [6]) .

Remark 2.6. The paper [6] of Chuang and Rouquier illustrates that abelian 𝔰𝔩_{2}-categorifications yield derived equivalences of importance in representation theory. There are other motivations for higher representation theory beyond classical representation theory, as we remark briefly in 2.12.

2.2. Triangulated 𝔰𝔩_{2}-categorification. We now define triangulated 𝔰𝔩_{2}-categorifications, after Rouquier.

Definition 2.7. Let V denote a finite dimensional representation of 𝔰𝔩_{2}. A weak triangulated 𝔰𝔩_{2}-categorification of V is the data of

• a triangulated category A

• an adjoint pair (E, F) of triangulated endofunctors of A such that

• F is isomorphic to a left adjoint of E

• V = ℂ ⊗ K_{0}(A).

Definition 2.8. A triangulated 𝔰𝔩_{2}-categorification of V is the data of • a weak triangulated 𝔰𝔩_{2}-categorification A

• natural transformations X ∈ End^{•}(E) and T ∈ End^{•}(E^{2}) such that

• the following diagram in End^{•}(E^{3}) commutes

\xymatrix{ & EEE \ar[dl]_{1_{E}T} \ar[dr]^{T 1_{E}} \\ EEE \ar[d]^{T 1_{E}} &  & EEE \ar[d]^{1_{E}T} \\ EEE \ar[dr]_{1_{E}T} &  & EEE \ar[dl]^{T 1_{E}} \\  & EEE}

• T^{2} = 0

• T (X1_{E}) − (1_{E}X)T = 1 = (X1_{E})T − T (1_{E}X)

• X is nilpotent.

Remark 2.9. This is the case q = 0, so that we obtain a morphism H_{n}(0) → End^{•}(E^{n}). We shall not need the other two cases.

Remark 2.10. One should also ensure that T admits a weight decomposition compatible with E and F. We explain what holds for abelian categorifications.

Suppose that A equipped with endofunctors E and F is an abelian 𝔰𝔩_{2}-categorification of a representation V of 𝔰𝔩_{2}. If V_{λ} is a weight space of V, let A_{λ} denote the full subcategory of A of objects whose class belongs to V_{λ} in ℂ ⊗ K_{0}(A). It is proved in 5.5 of [6] that A = ⨁_{λ} A_{λ}, so that the class of an indecomposable object of A is a weight vector.

Furthermore, E and F are compatible with the weight decomposition of A. Indeed, it is proved in 5.27 of [6] that if λ ≥ 0, then

EF Id_{A_{−λ}} ⨁ Id_{A_{λ}}^{⊕λ} ≅ F EId_{A_{−λ}}

and

EF Id_{A_{λ}} ≅ F EId_{A_{λ}} ⨁ Id_{A_{λ}}^{⊕λ}.

In the triangulated case, one (probably) cannot deduce these facts from the axioms, so a stronger condition is needed. We avoid the question of what such a condition should be.

Example 2.11. Let A be an abelian 𝔰𝔩_{2}-categorification of V, with endofunctors E and F and natural transformations X and T. The functors E and F pass to endofunctors E^{•} and F^{•} on the derived category D(A) of A. Similarly, X and T pass to natural transformations X^{•} ∈ End^{•}(E^{•}) and T^{•} ∈ End^{•}((E^{2})^{•}), giving D(A) the structure of a triangulated 𝔰𝔩_{2}- categorification of V.

Remark 2.12. Following a suggestion of Crane and Frenkel, Rouquier has conjectured that, after passing from triangulated categories to dg-categories, higher representations (of which 𝔰𝔩_{2}-categorifications are a special case) should give rise to a 4-dimensional TQFT. The decategorification of the TQFT should recover the 3-dimensional TQFT of Reshetikhin-Turaev.

Rouquier has also suggested that higher representation theory should allow moduli space constructions to be bypassed. This would give an algebraic approach to Donaldson-Thomas and Gromov-Witten invariants.

2.3. Weak categorification. Let G = GL_{n}(ℂ), and fix a Borel subgroup B ⊂ G. Fix a prime number l, and let \overline{ℚ_{l}} denote the algebraic closure of the field of l-adic numbers. Given a complex algebraic variety X (with its étale topology) equipped with an action of B, let D(X) denote the bounded derived category of B-smooth constructible \overline{ℚ_{l}}-sheaves on X. Thus D(X) is the full subcategory of the bounded derived category of constructible \overline{ℚ_{l}}-sheaves (see 2.2.18 in [3]) consisting of complexes whose cohomology sheaves are locally constant on B-orbits.

Let \xymatrix{X \ar[r]^{f} & Y} be a morphism of B-schemes of finite type over ℂ. The usual induced functors between the bounded derived categories of constructible \overline{ℚ_{l}}-sheaves on X and Y restrict to functors f_{∗}, f_{!} : D(X) → D(Y), and f^{∗}, f^{!} : D(Y) → D(X).

Fix a positive integer n. Let Gr(i) and Gr(i, i + 1) be as in §2.2. We have the following diagram, where p and q are the canonical projections.

\xymatrix{ & Gr(i, i + 1) \ar[dl]_{p} \ar[dr]^{q} \\ Gr(i) &  & Gr(i + 1)}

We shall use the fact that p and q are proper, so that p_{∗} = p_{!} and q_{∗} = q_{!}, without further mention. Let E_{i} = q_{!}p^{∗}: D(Gr(i)) → D(Gr(i + 1)) and F_{i} = p_{!}q^{∗}: D(Gr(i + 1)) → D(Gr(i)). Let T = ⨁_{i=0}^{n} D(Gr(i)), and define E = ⨁_{i=0}^{n} E_{i}: T → T , F = ⨁_{i=0}^{n} F_{i}: T → T .

The proof of the following proposition is borrowed from 3.3.4 in [12]. We denote the constant \overline{ℚ_{l}}-sheaf on X, regarded as an object of D(X) concentrated in degree zero, by \overline{ℚ_{l}} or (\overline{ℚ_{l}})_{X}.

Proposition 2.13. There is an isomorphism of functors

F_{i}E_{i} ⊕ ⨁_{n−i≤j<i} Id[−2j](−j) ≅ E_{i}F_{i} ⊕ ⨁_{i≤j<n−i} Id[−2j](−j),

where [−] denotes the shift functor of T , and (−) denotes the Tate twist.

Proof. Let

X = {V_{1}, V_{2} ∈ Gr(i) | dim(V_{1} + V_{2}) ≤ i + 1}

and

Y = {V_{1}, V_{2}, V_{3} | V_{1}, V_{2} ∈ Gr(i), V_{3} ∈ Gr(i + 1), V_{1} ⊂ V_{3}, V_{2} ⊂ V_{3}}.

We have the following commutative diagram.

\xymatrix{X \ar[rr]^{v} \ar[dd]_{u} &  & Gr(i) \\  & Y \ar[ul]^{r} \ar[r]^{t} \ar[d]_{s} & Gr(i, i + 1) \ar[u]_{p} \ar[d]^{q} \\ Gr(i) & Gr(i, i + 1) \ar[l]^{p} \ar[r]_{q} & Gr(i + 1)}

Here p and q are as above, and

s(V_{1}, V_{2}, V_{3}) = (V_{1}, V_{3})
t(V_{1}, V_{2}, V_{3}) = (V_{2}, V_{3})
r(V_{1}, V_{2}, V_{3}) = (V_{1}, V_{2})
u(V_{1}, V_{2}) = V_{1}
v(V_{1}, V_{2}) = V_{2}.

Note that Y is the fibred product Gr(i, i + 1) ×_{Gr(i+1)} Gr(i, i + 1). Hence, by proper base change, (see XII, 5.1 in [1]) q^{∗}q_{!} ≃ t_{!}s^{∗}. Thus F_{i}E_{i} = p_{!}q^{∗}q_{!}p^{∗} ≃ p_{!}t_{!}s^{∗}p^{∗} = v_{!}r_{!}r^{∗}u^{∗}.

By sheafified Poincaré duality (see XVIII 3.2.5 in [1], and II 7.5 in [9]), r_{!}r^{∗}(−) ≃ r_{!}\overline{ℚ_{l}} ⊗ −. Hence F_{i}E_{i} ≃ v_{!}(r_{!}\overline{ℚ_{l}} ⊗ u^{∗}(−)).

Let i: Δ ↪ X denote the inclusion of the diagonal Δ in X. Note that

Id_{Gr(i)} ≃ (vi)_{!}(ui)^{∗} ≃ v_{!}i_{!}i^{∗}u^{∗}.

We deduce from sheafified Poincaré duality that

Id_{Gr(i)}(−) ≃ v_{!}(i_{!}\overline{ℚ_{l}} ⊗ u^{∗}(−)).

Since r is an isomorphism above X \ Δ, we have the following commutative diagram.

\xymatrix{X \ Δ \ar[r]^{id} \ar[d]_{r^{−1}}^{∼} & X \ Δ \ar@{^{(}->}[dd] \\ r^{−1}(X \ Δ) \ar@{^{(}->}[d] \\ Y \ar[r]^{r} & X}

By proper base change, the restriction (r_{!}\overline{ℚ_{l}})_{|X\Δ} is isomorphic to (\overline{ℚ_{l}})_{X\Δ}.

Over Δ, r is a ℙ^{n−i−1}-bundle. Applying proper base change to the diagram

\xymatrix{r^{−1}(Δ) \ar@{^{(}->}[d] \ar[r]^{r} & Δ \ar@{^{(}->}[d] \\ Y \ar[r]^{r} & X}

we see that the restriction (r_{!}\overline{ℚ_{l}})_{|Δ} is isomorphic to r_{!}((\overline{ℚ_{l}})_{r^{−1}(Δ)}). Hence (see Lemma 5.4.12 of [3]),

(r_{!}\overline{ℚ_{l}})_{|Δ} ≃ ⨁_{j=0}^{n−i−1}(\overline{ℚ_{l}})_{Δ}[−2j](−j).

Let

Y′ = {V_{1}, V_{2}, V_{3} | V_{1} ∈ Gr(i − 1), V_{2}, V_{3} ∈ Gr(i), V_{1} ⊂ V_{2}, V_{1} ⊂ V_{3}}.

Note also that

X = {V_{1}, V_{2} ∈ Gr(i) | dim(V ∩ V′) ≥ i − 1}.

We have the following canonical commutative diagram.

\xymatrix{X \ar[rr]^{v} \ar[dd]_{u} &  & Gr(i) \\  & Y′ \ar[ul]^{r′} \ar[r]^{t} \ar[d]_{s} & Gr(i − 1, i) \ar[u]_{q} \ar[d]^{p} \\ Gr(i) & Gr(i − 1, i) \ar[l]^{q} \ar[r]_{p} & Gr(i − 1)}

Here u and v are as in the commutative diagram at the start of the proof, p and q are the canonical projections, and

s(V_{1}, V_{2}, V_{3}) = (V_{1}, V_{2})
t(V_{1}, V_{2}, V_{3}) = (V_{1}, V_{3})
r′(V_{1}, V_{2}, V_{3}) = (V_{2}, V_{3}).

As above, we find that

E_{i}F_{i}(−) ≃ v_{!}(r′_{!}\overline{ℚ_{l}} ⊗ u^{∗}(−))
(r′_{!}\overline{ℚ_{l}})_{|X\Δ} ≃ (\overline{ℚ_{l}})_{X\Δ}
(r′_{!}\overline{ℚ_{l}})_{|Δ} ≃ ⨁_{j=0}^{i−1}(\overline{ℚ_{l}})_{Δ}[−2j](−j).

We have shown that

(r_{!}\overline{ℚ_{l}})_{|Δ} ⊕ ⨁_{n−i≤j<i}(\overline{ℚ_{l}})_{|Δ} [−2j](−j) ≃ (r′_{!}\overline{ℚ_{l}})_{|Δ} ⊕ ⨁_{i≤j<n−i}(\overline{ℚ_{l}})_{|Δ} [−2j](−j).

This isomorphism, the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber (6.2.5 in [3]), and the fact that

(r_{!}\overline{ℚ_{l}})_{|X\Δ} ≃ (r′_{!}\overline{ℚ_{l}})_{|X\Δ} ≃ (\overline{ℚ_{l}})_{X\Δ},

imply that

r_{!}\overline{ℚ_{l}} ⊕ ⨁_{n−i≤j<i} i_{!}\overline{ℚ_{l}}[−2j](−j) ≃ r′_{!}\overline{ℚ_{l}} ⊕ ⨁_{i≤j<n−i} i_{!}\overline{ℚ_{l}}[−2j](−j).

The result follows by comparing this isomorphism with the realisations of E_{i}F_{i}, F_{i}E_{i}, and Id_{Gr(i)} above. □

Corollary 2.14. The functors E and F induce an action of 𝔰𝔩_{2} on ℂ ⊗ K_{0}(A).

The functors E and F are adjoint to one another in the following sense.

Proposition 2.15. Up to a shift and a twist, (E, F) and (F, E) are adjoint pairs of functors.

Proof. Note that p: Gr(i, i+1) → Gr(i) is a ℙ^{n−i−1}-fibre bundle, and q: Gr(i, i+1) → Gr(i+1) is a ℙ^{i}-fibre bundle. Hence p^{!} ≃ p^{∗}[2(n − i − 1)](n − i − 1) and q^{!} ≃ q^{∗}[2i](i) (see, for example, II.8.1 in [9]). The result follows from the adjointness of (p^{∗}, p_{∗}), (p_{!}, p^{!}), (q^{∗}, q_{∗}), and (q_{!}, q^{!}). □

Using the weight space decomposition of ℂ ⊗ K_{0}(T ), we now determine the action of 𝔰𝔩_{2} on ℂ ⊗ K_{0}(T ) induced by E and F. A different approach was taken in [12].

Let P(Gr(i)) denote the category of B-smooth perverse sheaves on Gr(i), and let D^{b}(P(Gr(i))) denote the bounded derived category of P(Gr(i)) with its standard t-structure. There exists (see [2]) a canonical t-exact triangulated functor D^{b}(P(Gr(i))) → D(Gr(i)), which is the identity on P(Gr(i)). The existence follows from the existence of a filtered counterpart to D(Gr(i)), via the formalism of filtered triangulated categories.

Proposition 2.16. The canonical functor D^{b}(P(Gr(i))) → D(Gr(i)) is an equivalence of categories.

Proof. This is 1.3 in [2]. □

Proposition 2.17. The category D^{b}(P(Gr(i))) is generated as a triangulated category by the projective objects in P(Gr(i)).

Proof. Indeed, P(Gr(i)) has enough projectives (3.3.1 in [4]), and has finite global dimension (3.2.2 in [4]). □

Corollary 2.18. As a ℂ-vector space, ℂ ⊗ K_{0}(T ) has dimension 2^{n+1}.

Proof. Let T′ = ⨁_{i=0}^{n} D^{b}(P(Gr(i))). By 2.16, ℂ ⊗ K_{0}(T ) ≃ ℂ ⊗ K_{0}(T′). It follows from 2.17 that a basis of ℂ⊗K_{0}(T′) is given by the classes of indecomposable projective perverse sheaves in T′.

Indecomposable projective perverse sheaves in T′ are in bijection with simple perverse sheaves in T′, which are in bijection with orbits of B on ⨁_{i=0}^{n} Gr(i). It is classical that there are \binom{n}{i} orbits of B on Gr(i), and the result follows. □

Proposition 2.19. Let L denote the standard representation of 𝔰𝔩_{2}. As a representation of 𝔰𝔩_{2}, ℂ ⊗ K_{0}(T ) ≃ L^{⊗n}.

Proof. By 2.18, ℂ ⊗ K_{0}(T ) has the correct dimension. Let h = ef − fe, where e and f are the endomorphisms of ℂ ⊗ K_{0}(T ) induced by E and F. By 2.13, and the fact that ℂ ⊗ K_{0}(Gr(i)) has dimension \binom{n}{i}, the eigenvalues of h on ℂ ⊗ K_{0}(T ) are correct. □

We have proved the following result.

Corollary 2.20. The endofunctors E and F give T the structure of a weak 𝔰𝔩_{2}-categorification of L^{⊗n}.

Given a parabolic subgroup P of GL_{n}(ℂ) for some n, let D_{P} (Gr(i)) denote the category of P - smooth constructible \overline{ℚ_{l}}-sheaves. Let V be a tensor product of arbitrary simple representations of 𝔰𝔩_{2}.

Proposition 2.21. There is an integer n ≥ 0, a parabolic subgroup P ⊂ GL_{n}(ℂ), and a pair of endofunctors (E_{P} , F_{P}) of T_{P} = ⨁_{i=0}^{n} D_{P} (Gr(i)) giving T_{P} the structure of a weak 𝔰𝔩_{2}-categorification of V.

Proof. Exactly as in the case V = L^{⊗n}, P = B ⊂ GL_{n}(ℂ) above. The results 2.18 and 2.19 must be modified, but we omit this. Given V, the interested reader will have no difficulty finding the corresponding parabolic and checking the details. □

2.4. 2-morphisms X and T. We now explain how to extend the weak 𝔰𝔩_{2}-categorification T of L^{⊗n} to a Chuang-Rouquier 𝔰𝔩_{2}-categorification. In order to define X ∈ End^{•}(E), we realise E_{i} as a Fourier-Mukai transform for every i.

The following diagram commutes, where p, q, p′ and q′ are the canonical projections, and j is the canonical map (V_{1} ⊂ V_{2}) → (V_{1}, V_{2}).

\xymatrix{ & Gr(i, i + 1) \ar[d]^{j} \ar[dl]_{p} \ar[dr]^{q} \\ Gr(i) & Gr(i) × Gr(i + 1) \ar[l]^{p′} \ar[r]_{q′} & Gr(i + 1)}

Proposition 2.22. There is an isomorphism of functors E_{i} ≃ q′_{∗}(j_{∗}\overline{ℚ_{l}} ⊗ p^{′∗}(−)).

Proof. Straightforward. □

Let ℒ denote the tautological line bundle on Gr(i, i+1). The fibre above the point V_{i} ⊂ V_{i+1} is V_{i+1}/V_{i}. The first Chern class c_{1}(ℒ) ∈ H^{2}(Gr(i, i + 1), \overline{ℚ_{l}}(1)) of ℒ can be viewed as a morphism, belonging to Hom_{D(Gr(i,i+1))}(\overline{ℚ_{l}}, \overline{ℚ_{l}}[2](1)). By functoriality, c_{1}(ℒ) determines a morphism in Hom_{D(Gr(i)×Gr(i+1))}(j_{∗}\overline{ℚ_{l}}, j_{∗}\overline{ℚ_{l}}[2](1)) and hence, by the proposition, determines an endomorphism of E_{i}. Assembling these endomorphisms, we obtain an endomorphism X of E.

In order to define T ∈ End^{•}(E^{2}), we realise E_{i+1}E_{i} as a Fourier-Mukai transform for every i. We have the following commutative diagram, where p, q, p′ and q′ are the canonical projections, and j is the canonical map (V_{1} ⊂ V_{2} ⊂ V_{3}) → (V_{1}, V_{3}).

\xymatrix{ & Gr(i, i + 1, i + 2) \ar[d]^{j} \ar[dl]_{p} \ar[dr]^{q} \\ Gr(i) & Gr(i) × Gr(i + 2) \ar[l]^{p′} \ar[r]_{q′} & Gr(i + 2)}

Proposition 2.23. There is an isomorphism of functors E_{i+1}E_{i} ≃ q′_{∗}(j_{∗}\overline{ℚ_{l}} ⊗ p^{′∗}(−)).

Proof. We have the following commutative diagram, where the μ_{i}, ψ_{i} and j_{i} are the canonical maps, and j is the same as in the diagram above.

\xymatrix{ &  & Gr(i, i + 1, i + 2) \ar[d]^{j} \ar[dl]_{μ_{1}} \ar[dr]^{μ_{2}} \\  & Gr(i, i + 1) × Gr(i + 2) \ar[dl]^{ϕ_{1}} \ar[dr]^{ψ_{1}} & Gr(i) × Gr(i + 2) & Gr(i) × Gr(i + 1, i + 2) \ar[dl]_{ψ_{2}} \ar[dr]_{ϕ_{2}} \\ Gr(i, i + 1) \ar[dr]^{j_{1}} &  & Gr(i) × Gr(i + 1) × Gr(i + 2) \ar[dl]^{φ_{1}} \ar[u]^{φ_{2}} \ar[dr]_{φ_{3}} &  & Gr(i + 1, i + 2) \ar[dl]_{j_{2}} \\  & Gr(i) × Gr(i + 1) &  & Gr(i + 1) × Gr(i + 2)}

It follows from 2.22 (see 12.2.2 in [10]) that there is an isomorphism of functors E_{i+1}E_{i} ≃ q′_{∗}(K ⊗ p^{′∗}(−)), where K = φ_{2}_{∗}(φ_{1}^{∗}j_{1}_{∗}\overline{ℚ_{l}} ⊗ φ_{3}^{∗}j_{2}_{∗}\overline{ℚ_{l}}). By proper base change with respect to the diamonds on the lower left and lower right of the diagram, K ≃ φ_{2}_{∗}(ψ_{1}_{∗}\overline{ℚ_{l}} ⊗ ψ_{2}_{∗}\overline{ℚ_{l}}).

Thus K ≃ φ_{2}_{∗}ψ_{1}_{∗}(ψ_{1}^{∗}ψ_{2}_{∗}\overline{ℚ_{l}}). By proper base change with respect to the upper diamond (ignoring the morphisms inside), K ≃ φ_{2}_{∗}ψ_{1}_{∗}(μ_{1}_{∗}\overline{ℚ_{l}}). The result follows from the commutativity of the upper diamond. □

Remark 2.24. A different characterisation of E_{i+1}E_{i} is given in 3.3.3 of [12]. We will see it later.

The map j factors through the canonical map π: Gr(i, i + 1, i + 2) → Gr(i, i + 2) given by (V_{1} ⊂ V_{2} ⊂ V_{3}) → (V_{1} ⊂ V_{3}). Let R^{k}π_{∗} denote the k^{th} higher direct image of π, and regard R^{2}π_{∗}(\overline{ℚ_{l}}) as a complex concentrated in degree zero. Since R^{k}π_{∗} vanishes for k > 2, there is a canonical morphism π_{∗}(\overline{ℚ_{l}}[2]) → R^{2}π_{∗}(\overline{ℚ_{l}}) in D(Gr(i, i + 2)).

Let η: R^{2}π_{∗}(\overline{ℚ_{l}}(1)) → \overline{ℚ_{l}} denote the trace morphism, which is an isomorphism of \overline{ℚ_{l}}-sheaves (see XVIII 2.9 in [1]). By composition, we obtain a canonical morphism t′: π_{!}(\overline{ℚ_{l}}[2](1)) → R^{2}π_{!}(\overline{ℚ_{l}}) → \overline{ℚ_{l}}.

Moreover, t′ extends to a natural transformation π_{!}π^{!}(K) → K for any K ∈ D(Gr(i, i + 2)), via the following commutative diagram (cf. II.8 in [9]).

\xymatrix{π_{!}π^{!}K \ar[d]_{∼} \ar@{-->}[r] & K \\ π_{!}(\overline{ℚ_{l}}[2](1) ⊗ π^{∗}(K)) \ar[r]_{∼} & π_{!}(\overline{ℚ_{l}}[2](1)) ⊗ K \ar[u]_{t′⊗id_{K}}}

Composing with the adjunction morphism K → π_{∗}π^{∗}(K), we get a natural transformation T′: π_{!}π^{!}(K) → π_{∗}π^{∗}(K). Let t denote the morphism obtained by taking K = \overline{ℚ_{l}}[−2](−1). By 2.23, t induces an endomorphism of E_{i+1}E_{i} for every i. Assembling these endomorphisms, we obtain an endomorphism T of E^{2}.

Remark 2.25. The definitions of X and T were outlined to the author by Rouquier.

Let ℰ denote the canonical rank two vector bundle on Gr(i, i + 2) whose fibre above (V_{i} ⊂ V_{i+2}) is V_{i+2}/V_{i}. The ℙ^{1}-bundle π is the projectivisation of ℰ, and thus gives rise to a tautological line bundle O_{π}(−1) on Gr(i, i + 1, i + 2). Indeed, O_{π}(−1) is a subbundle of the pull-back bundle π^{∗}ℰ, whose fibre above (V_{i} ⊂ V_{i+1} ⊂ V_{i+2}) is V_{i+1}/V_{i}. The quotient bundle π^{∗}ℰ /O_{π}(−1) is the line bundle on Gr(i, i + 1, i + 2) corresponding to the twisting sheaf O_{π}(1). The fibre of π^{∗}ℰ /O_{π}(−1) above (V_{i} ⊂ V_{i+1} ⊂ V_{i+2}) is V_{i+2}/V_{i+1}.

By 2.23, the first Chern classes c_{1}(O_{π}(−1)) and c_{1}(π^{∗}ℰ /O_{π}(−1)) induce endomorphisms of E^{2}, which we denote by x and y respectively.

Proposition 2.26. In End^{•}(E^{2}), we have 1_{E}X = x and X1_{E} = y.

Proof. The second relation can be seen by inspecting the proof of 2.23. Indeed, X1_{E} is determined by the morphism φ_{2}_{∗}(φ_{1}^{∗}j_{1}_{∗}\overline{ℚ_{l}} ⊗ φ_{3}^{∗}j_{2}_{∗}c_{1}(ℒ)), where ℒ is the tautological line bundle on Gr(i + 1, i + 2). By proper base change, this morphism identifies with φ_{2}_{∗}(ψ_{1}_{∗}\overline{ℚ_{l}} ⊗ ψ_{2}_{∗}c_{1}(ϕ_{2}^{∗}ℒ)), and hence with φ_{2}_{∗}ψ_{1}_{∗}(ψ_{1}^{∗}ψ_{2}_{∗}c_{1}(ϕ_{2}^{∗}ℒ)). By proper base change once more, this morphism identifies with j_{∗}c_{1}(μ_{2}^{∗}ϕ_{2}^{∗}ℒ). The pull-back bundle (ϕ_{2}μ_{2})^{∗}ℒ is exactly π^{∗}ℰ /O_{π}(−1), as required.

The endofunctor 1_{E}X is determined by the morphism φ_{2}_{∗}(φ_{1}^{∗}j_{1}_{∗}c_{1}(ℒ) ⊗ φ_{3}^{∗}j_{2}_{∗}\overline{ℚ_{l}}), where ℒ is the tautological line bundle on Gr(i, i + 1). As above, this morphism identifies with φ_{2}_{∗}(ψ_{1}_{∗}c_{1}(ϕ_{1}^{∗}ℒ) ⊗ ψ_{2}_{∗}\overline{ℚ_{l}}), and hence with φ_{2}_{∗}ψ_{2}_{∗}(ψ_{2}^{∗}ψ_{1}_{∗}c_{1}(ϕ_{1}^{∗}ℒ)). By proper base change, this morphism identifies with j_{∗}c_{1}(μ_{1}^{∗}ϕ_{1}^{∗}ℒ). The pull-back bundle (ϕ_{1}μ_{1})^{∗}ℒ is exactly O_{π}(−1), as required. □

We now show that T, x and y satisfy the defining relations 2.1 of the affine nilHecke algebra H_{2}(0). The proof was outlined to the author by Rouquier.

Proposition 2.27. In End^{•}(E^{2}), we have

T^{2} = 0, yT − T x = 1, T (x + y) = (x + y)T.

Proof. By the naturality of T′, the composition

\xymatrix{π_{∗}π^{∗}\overline{ℚ_{l}} \ar[r]^{∼} & π_{∗}\overline{ℚ_{l}} \ar[r]^{t} & π_{∗}\overline{ℚ_{l}}[−2](−1) \ar[r]^{∼} & π_{∗}π^{∗}\overline{ℚ_{l}}[−2](−1)}

fits into the following commutative diagram, for any α ∈ End_{D(Gr(i,i+2))}^{2}(\overline{ℚ_{l}}).

\xymatrix{π_{∗}π^{∗}\overline{ℚ_{l}} \ar[d]_{π_{∗}π^{∗}α} \ar[r]^{∼} & π_{∗}\overline{ℚ_{l}} \ar[r]^{t} & π_{∗}\overline{ℚ_{l}}[−2](−1) \ar[r]^{∼} & π_{∗}π^{∗}\overline{ℚ_{l}}[−2](−1) \ar[d]^{π_{∗}π^{∗}α[−2]} \\ π_{∗}π^{∗}\overline{ℚ_{l}}[2] \ar[r]^{∼} & π_{∗}\overline{ℚ_{l}}[2] \ar[r]^{t[2]} & π_{∗}\overline{ℚ_{l}}(−1) \ar[r]^{∼} & π_{∗}π^{∗}\overline{ℚ_{l}}(−1)}

We deduce that T commutes with x + y, since

c_{1}(O_{π}(−1)) + c_{1}(π^{∗}ℰ /O_{π}(−1)) = π^{∗}c_{1}(ℰ).

Furthermore, t[−2](−1) ∘ t factors through a morphism \overline{ℚ_{l}}[−2](−1) → \overline{ℚ_{l}}[−4](−2). This is the zero morphism, since shifting \overline{ℚ_{l}} by the dimension of Gr(i, i + 2) is a simple perverse sheaf on Gr(i, i + 2). Thus T^{2} = 0.

Let α = π_{∗}c_{1}(O_{π}(−1)) and β = π_{∗}c_{1}(O_{π}(1)). We claim that the following composition is the identity in End_{D(Gr(i,i+2))}^{•}(\overline{ℚ_{l}}).

\xymatrix{\overline{ℚ_{l}} \ar[r]^{adj} & π_{∗}\overline{ℚ_{l}} \ar[r]^{β} & π_{∗}\overline{ℚ_{l}}[2](1) \ar[r]^{t} & \overline{ℚ_{l}}}

Given a point z ∈ Gr(i, i + 2), let π′: π^{−1}(z) ≃ ℙ^{1} → {z} denote the fibre map. Taking the fibre of the above composition at z, and applying proper base change, we obtain a morphism of the following form.

\xymatrix{\overline{ℚ_{l}} \ar[r]^{adj} & π′_{∗}\overline{ℚ_{l}} \ar[r]^{c_{1}(O_{ℙ^{1}}(1))} & π′_{∗}\overline{ℚ_{l}}[2](1) \ar[r] & \overline{ℚ_{l}}}

Via the natural isomorphism between π′ and the global sections functor Γ(ℙ^{1}, −), the above morphism identifies with the following morphism, where τ denotes the trace morphism on cohomology.

\xymatrix{\overline{ℚ_{l}} \ar[r]^{c_{1}(O_{ℙ^{1}}(1))} & H^{2}(ℙ^{1}, \overline{ℚ_{l}}) \ar[r]^{τ} & \overline{ℚ_{l}}}

This is the identity, since the trace of the class of c_{1}(O_{ℙ^{1}}) in H^{2}(ℙ^{1}, \overline{ℚ_{l}}) is 1 (see Cycle, 2.1.5 in [5]).

The following composition is also the identity morphism, since −c_{1}(O_{π}(−1)) = c_{1}(O_{π}(1)).

\xymatrix{\overline{ℚ_{l}} \ar[r]^{adj} & π_{∗}\overline{ℚ_{l}} \ar[r]^{−α} & π_{∗}\overline{ℚ_{l}}[2](1) \ar[r]^{t} & \overline{ℚ_{l}}}

The composition \xymatrix{\overline{ℚ_{l}} \ar[r]^{adj} & π_{∗}\overline{ℚ_{l}} \ar[r]^{t} & \overline{ℚ_{l}}[−2](−1)} is zero, since it factors through a morphism \overline{ℚ_{l}} → \overline{ℚ_{l}}[−2](−1).

It follows that the following composition is equal to the adjunction morphism \overline{ℚ_{l}} → π_{∗}\overline{ℚ_{l}}.

\xymatrix{\overline{ℚ_{l}} \ar[r]^{adj} & π_{∗}\overline{ℚ_{l}} \ar[r]^{βt−t[2](1)α} & π_{∗}\overline{ℚ_{l}}}

The following diagram commutes, since c_{2}(π^{∗}ℰ) = c_{1}(O_{π}(−1))c_{1}(O_{π}(1)).

\xymatrix{ & \overline{ℚ_{l}}[4](2) \ar[dr]^{adj} \\ \overline{ℚ_{l}} \ar[ur]^{c_{2}(ℰ)} \ar[r]^{adj} & π_{∗}\overline{ℚ_{l}} \ar[r]^{αβ} & π_{∗}\overline{ℚ_{l}}[4](2) \ar[r]^{t[2](1)} & \overline{ℚ_{l}}[2](1)}

In particular, the row in the above diagram factors through a morphism \overline{ℚ_{l}}[4](2) → \overline{ℚ_{l}}[2](1). It is therefore zero, and the composition

\xymatrix{\overline{ℚ_{l}} \ar[r]^{adj} & π_{∗}\overline{ℚ_{l}} \ar[r]^{β} & π_{∗}\overline{ℚ_{l}}[2](1) \ar[r]^{βt−t[2](1)α} & π_{∗}\overline{ℚ_{l}}[2](1)}

is equal to the composition

\xymatrix{\overline{ℚ_{l}} \ar[r]^{adj} & π_{∗}\overline{ℚ_{l}} \ar[r]^{β} & π_{∗}\overline{ℚ_{l}}[2](1)} .

This proves that yT − T x = 1. □

Corollary 2.28. In End^{•}(E^{2}), we have T y − xT = 1 .

Proposition 2.29. In End^{•}(E^{3}), the following diagram commutes.

\xymatrix{ & EEE \ar[dl]_{1_{E}T} \ar[dr]^{T 1_{E}} \\ EEE \ar[d]^{T 1_{E}} &  & EEE \ar[d]^{1_{E}T} \\ EEE \ar[dr]_{1_{E}T} &  & EEE \ar[dl]^{T 1_{E}} \\  & EEE}

Proof. Follows from the compatibility of the trace morphism with base change and composition. We omit the details. □

Combining 2.20, 2.27, 2.28, and 2.29, we have the following result.

Proposition 2.30. The endofunctors E and F, and the endomorphisms X and T, give T the structure of an 𝔰𝔩_{2}-categorification of L^{⊗n}.

Let V be a tensor product of arbitrary simple representations of 𝔰𝔩_{2}.

Proposition 2.31. There is an integer n ≥ 0, a parabolic subgroup P ⊂ GL_{n}(ℂ), a pair of endofunctors (E_{P} , F_{P}) of T_{P} = ⨁_{i=0}^{n} D_{P} (Gr(i)), and a pair of endomorphisms X_{P} ∈ End^{•}(E_{P}), T_{P} ∈ End^{•}(E_{P}^{2} ) giving T_{P} the structure of an 𝔰𝔩_{2}-categorification of V.

Proof. The integer n ≥ 0, the parabolic subgroup P ⊂ GL_{n}(ℂ), and the endofunctors (E_{P} , F_{P}) are given by 2.21. The 2-morphisms X_{P} and T_{P} are obtained exactly as in the case V = L^{⊗n} and P = B ⊂ GL_{n}(ℂ) above. □

2.5. Action of ℤ[q, q^{−1}]. We explain how to pass to an 𝔰𝔩_{2}-categorification of the quantum group U_{q}(𝔰𝔩_{2}), where q ∈ ℂ is neither zero nor a root of unity. A slightly different approach via shifting E and F is taken in [7] and [12].

Fix a parabolic subgroup P ⊂ GL_{n}(ℂ), and let T_{P} (q) = ⨁_{i=0}^{n} D_{P} (Gr(i)). Choosing an isomorphism τ: \overline{ℚ_{l}} → ℂ, fix an element q^{1/2} ∈ \overline{ℚ_{l}}. This allows us to define a half-integral Tate twist (−)(\frac{n}{2}) on D_{P} (Gr(i)). For even n, this is the usual Tate twist.

We have the following diagram, where p and r are the canonical projections.

\xymatrix{ & Gr(i, i + 1) \ar[dl]_{p} \ar[dr]^{r} \\ Gr(i) &  & Gr(i + 1)}

Let E_{i} = r_{!}p^{∗}(\frac{n−i−1}{2}): D_{P} (Gr(i)) → D_{P} (Gr(i + 1)), F_{i} = p_{!}r^{∗}(\frac{i}{2}): D_{P} (Gr(i + 1)) → D_{P} (Gr(i)), and G_{i} = (\frac{2i−n}{2}): D_{P} (Gr(i)) → D_{P} (Gr(i)). Define E, F, G: T_{P} (q) → T_{P} (q) by E = ⨁_{i=0}^{n} E_{i}, F = ⨁_{i=0}^{n} F_{i}, and G = ⨁_{i=0}^{n} G_{i}.

There is an action of ℤ[q, q^{−1}] on K_{0}(T_{P} (q)) given by q · [K] = [K(−\frac{1}{2})]. We prove that E, F, and G induce an action of U_{q}(𝔰𝔩_{2}) on ℚ(q) ⊗_{ℤ[q,q^{−1}]} K_{0}(T_{P} (q)).

Proposition 2.32. We have [GG^{−1}] = [G^{−1}G] = 1, [GEG^{−1}] = q^{−2} [E] , [GF G^{−1}] = q^{2} [F], and [EF] − [F E] = \frac{[G]−[G^{−1}]}{q−q^{−1}}.

Proof. The last relation follows from 2.13, which implies the following isomorphism of functors.

F_{i}E_{i} ⊕ ⨁_{n−i≤j<i} Id[−2j](−j)(\frac{n−1}{2}) ≃ E_{i}F_{i} ⊕ ⨁_{i≤j<n−i} Id[−2j](−j)(\frac{n−1}{2}).

The other relations are easily checked. □

Let V denote a tensor product of simple representations of U_{q}(𝔰𝔩_{2}).

Proposition 2.33. There is an integer n ≥ 0 and a parabolic subgroup P ⊂ GL_{n}(ℂ), together with endofunctors E, F and G of T_{P} (q), such that the induced action of U_{q}(𝔰𝔩_{2}) on ℚ(q)⊗_{ℤ[q,q^{−1}]} K_{0}(T_{P} (q)) is isomorphic to the action of U_{q}(𝔰𝔩_{2}) on V.

Proof. After 2.32, this is a question of combinatorics (cf. 2.16 - 2.21). The Clebsch-Gordon decomposition of V into a direct sum of simple representations (see 1.4.4 in [8]) gives a means to calculate the dimensions of the weight spaces of V. □

Remark 2.34. The proposition can be proved geometrically. This is the approach taken in [12].

Exactly as in §2.4, there are 2-morphisms X ∈ End(E) and T ∈ End(E^{2}) satisfying the relations 2.27 and 2.29, giving rise to a morphism H_{n}(0) → End(E^{n}) for every n. Furthermore, (G, G^{−1}), (G^{−1}, G), (E, GF), and (F, G^{−1}E) are adjoint pairs of functors, up to a shift (see 2.15).

Remark 2.35. We have, in essence, constructed a U_{q}(𝔰𝔩_{2})-categorification of V, in the sense of the higher representation theory programme of Chuang and Rouquier. However, we refrain from using this terminology. Properly justifying it would take us too far afield.

Fix a parabolic subgroup P ⊂ GL_{n}(ℂ). We show that the divided powers (see 1.2 in [8]) of [E] and [F] are induced by endofunctors of T_{P} (q). We have the following diagram, where p and r are the canonical projections.

\xymatrix{ & Gr(i, i + s) \ar[dl]_{p} \ar[dr]^{r} \\ Gr(i) &  & Gr(i + s)}

Let E_{i}^{(s)} = r_{!}p^{∗}(\frac{s(n−i−s)}{2}): Gr(i) → Gr(i + s), and F_{i}^{(s)} = p_{!}r^{∗}(\frac{is}{2}): Gr(i + s) → Gr(i). Define E^{(s)}, F^{(s)} : T_{P} (q) → T_{P} (q) by E^{(s)} = ⨁_{i=0}^{n} E_{i}^{(s)}, F^{(s)} = ⨁_{i=0}^{n} F_{i}^{(s)}. The proof of the following proposition is borrowed from 3.3.3 in [12].

Proposition 2.36. We have isomorphisms of functors E^{(s−1)}E ≃ ⨁_{j=0}^{s−1} E^{(s)}(\frac{s−1−2j}{2} ) and F^{(s−1)}F ≃ ⨁_{j=0}^{s−1} F^{(s)}(\frac{s−1−2j}{2} ).

Proof. Let

Y = {V_{1} ⊂ V_{2} ⊂ V_{3} | V_{1} ∈ Gr(i), V_{2} ∈ Gr(i + 1), V_{3} ∈ Gr(i + s), V_{1} ⊂ V_{3}, V_{2} ⊂ V_{3}}.

We have the following commutative diagram. The maps are the canonical projections.

\xymatrix{Gr(i, i + s) \ar[rr]^{x} \ar[dd]_{w} &  & Gr(i + s) \\  & Y \ar[ul]^{t} \ar[r]^{v} \ar[d]_{u} & Gr(i + 1, i + s) \ar[u]_{r′} \ar[d]^{p′} \\ Gr(i) & Gr(i, i + 1) \ar[l]^{p} \ar[r]_{r} & Gr(i + 1)}

We have that E_{i+1}^{(s−1)}E_{i} = r′_{!}p^{′∗}r_{!}p^{∗}(\frac{s(n+1−i−s)−1}{2}). By proper base change, r′_{!}p^{′∗}r_{!}p^{∗} ≃ r′_{!}v_{!}u^{∗}p^{∗}, and hence E_{i+1}^{(s−1)}E_{i} ≃ x_{!}t_{!}t^{∗}w^{∗}(\frac{s(n+1−i−s)−1}{2}). Since t is a ℙ^{s−1}-bundle, t_{!}\overline{ℚ_{l}} ≃ ⨁_{j=0}^{s−1} \overline{ℚ_{l}}[−2j](−j). Thus t_{!}t^{∗} ≃ t_{!}\overline{ℚ_{l}} ⊗ − ≃ ⨁_{j=0}^{s−1}[−2j](−j), and

E_{i+1}^{(s−1)}E_{i} ≃ ⨁_{j=0}^{s−1} x_{!}w^{∗}[−2j](−j)(\frac{s(n+1−i−s)−1}{2}) ≃ ⨁_{j=0}^{s−1} E_{i}^{(s)}(\frac{s−1−2j}{2} ).

The second isomorphism is proved similarly. □

Corollary 2.37. We have [E^{(s)}] = \frac{[E^{s}]}{[s]_{q}!} and [F^{(s)}] = \frac{[F^{s}]}{[s]_{q}!}.

3. Koszul duality

3.1. Koszul duality. Given a tensor product V of simple representations of U_{q}(𝔰𝔩_{2}), let T = T_{P} (q) denote the corresponding triangulated categorification of 2.33. In the spirit of Soergel, we outline how to pass to an abelian categorification via Koszul duality, as in 3.6 of [12].

Let L_{I} denote the direct sum of the simple perverse sheaves in T . Let L_{I} denote the full subcategory of T consisting of the semisimple perverse sheaves in T and their shifts and Tate twists. Let A = End_{T}^{•} (L_{I}), regarded as an algebra via composition. Then Ext_{T}^{•} (L_{I}, −) defines a fully faithful functor from L_{I} to the category A of finitely generated graded left modules over A.

By the decomposition theorem of [3], the endofunctors E, F of T preserve L_{I}, as does G. Let x, z ∈ A, and y ∈ Ext_{T}^{•} (L_{I}, E(L_{I})). The action x · y · z = xyE(z) gives Ext_{T}^{•} (L_{I}, E(L_{I})) the structure of a graded A-bimodule, and Ext_{T}^{•} (L_{I}, E(L_{I}))⊗_{A} − defines an exact endofunctor E_{a} of A. In the same way, F and G give rise to exact endofunctors F_{a} and G_{a} of A.

After 2.32, E_{a}, F_{a}, and G_{a} induce an action of U_{q}(𝔰𝔩_{2}) on K_{0}(A). The endomorphisms X ∈ End^{•}(E) and T ∈ End^{•}(E^{2}) induce endomorphisms X_{a} ∈ End(E_{a}) and T_{a} ∈ End((E_{a})^{2}) satisfying the relations 2.27 and 2.29. This gives A the structure of an abelian categorification. As in 2.5, A essentially has the structure of a U_{q}(𝔰𝔩_{2})-categorification, in the sense of the higher representation theory programme of Chuang and Rouquier.

The indecomposable projective objects of A are the modules Ext_{T}^{•} (L_{I}, K), where K is a simple perverse sheaf in T . Thus there is an action of ℤ[q, q^{−1}] on K_{0}(A), defined by q · [Ext_{T}^{•} (L_{I}, K)] = [Ext_{T}^{•} (L_{I}, K(−\frac{1}{2})]. Let K_{0}(L_{I}) denote the Grothendieck group of L_{I} as an additive category. There is also an action of ℤ[q, q^{−1}] on K_{0}(L_{I}), defined by q · [K] = [K(−\frac{1}{2})].

The functor Ext_{T}^{•} (L_{I}, −) induces an isomorphism ℚ_{q} ⊗_{ℤ[q,q^{−1}]} K_{0}(L_{I}) ≃ ℚ_{q} ⊗_{ℤ[q,q^{−1}]} K_{0}(A) of U_{q}(𝔰𝔩_{2})-modules. The U_{q}(𝔰𝔩_{2})-modules ℚ(q) ⊗_{ℤ[q,q^{−1}]} K_{0}(T ) and ℚ_{q} ⊗_{ℤ[q,q^{−1}]} K_{0}(L_{I}) are also isomorphic, so the decategorification of A is V.

Remark 3.1. There is a canonical basis of V consisting of isomorphism classes of the indecomposable projectives in A. In 3.5.9 of [11], Zheng identifies this basis with Lusztig’s canonical basis.

3.2. An abelian 𝔰𝔩_{2}-categorification of simple representations of 𝔰𝔩_{2}. Let V denote the simple representation of 𝔰𝔩_{2} of dimension n + 1. In this case, Proposition 2.31 takes the following form.

Proposition 3.2. There are endofunctors E and F and endomorphisms X ∈ End^{•}(E), T ∈ End^{•}(E^{2}) giving T = ⨁_{i=0}^{n} D_{GL_{n}(ℂ)}(Gr(i)) the structure of an 𝔰𝔩_{2}-categorification of V.

Via Koszul duality (as in 3.1, ignoring Tate twists), T gives rise to an abelian 𝔰𝔩_{2}-categorification A of V. Indeed, A is the category of finitely generated graded left modules over the algebra A = ⨁_{i=0}^{n} End_{D(Gr(i))}^{•}(\overline{ℚ_{l}}) ≃ ⨁_{i=0}^{n} H^{•}(Gr(i)).

Let E_{a} and F_{a} denote the structural endofunctors of A. Regarding H^{•}(Gr(i, i + 1)) as an H^{•}(Gr(i))-H^{•}(Gr(i + 1))-bimodule, we have an isomorphism of functors

E_{a} ≃ ⨁_{i=0}^{n} H^{•}(Gr(i, i + 1)) ⊗_{A} −.

Regarding H^{•}(Gr(i, i+1)) as an H^{•}(Gr(i+1))-H^{•}(Gr(i))-bimodule, we have an isomorphism of functors F_{a} ≃ ⨁_{i=0}^{n} H^{•}(Gr(i, i + 1)) ⊗_{A} −.

This agrees with the weak 𝔰𝔩_{2}-categorification of V given in 2.2. The structural endomorphisms X_{a} ∈ End^{•}(E_{a}) and T_{a} ∈ End^{•}((E_{a})^{2}) enhance 2.2 to an 𝔰𝔩_{2}-categorification of V.

References

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University of Oxford

 E-mail address: williamson@maths.ox.ac.uk