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Introduction to Quantum Q-Langlands Correspondence

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Introduction to Quantum Q-Langlands Correspondence Introduction to Quantum q-Langlands Correspondence Ming Zhang∗ May 16{19, 2017 Abstract Quantum q-Langlands Correspondence was introduced by Aganagic, Frenkel and Okounkov in their recent work [AFO17]. Roughly speaking, it is a correspondence between q-deformed conformal blocks of the quantum affine algebra and the deformed W-algebra. In this mini-course, I will define these objects and discuss some results in quantum K-theory of the Nakajima quiver varieties which is the key ingredient in the proof of the correspondence (in the simply-laced case). It will be helpful to know the basic concepts in the theory of Lie algebras (roots, weights, etc.). Contents 1 Introduction 2 1.1 The geometric Langlands correspondence.............................2 1.2 A representation-theoretic statement and its generalization in [AFO17].............2 2 Lie algebras 2 2.1 Root systems.............................................3 2.2 The Langlands dual.........................................4 3 Affine Kac{Moody algebras4 3.1 Via generators and relations.....................................5 3.2 Via extensions............................................6 4 Representations of affine Kac{Moody algebras7 4.1 The extended algebra........................................7 4.2 Representations of finite-dimensional Lie algebras.........................8 4.2.1 Weights............................................8 4.2.2 Verma modules........................................8 4.3 Representations of affine Kac{Moody algebras...........................9 4.3.1 The category of representations.............................9 4.3.2 Verma modulesO........................................ 10 4.3.3 The evaluation representation................................ 11 4.3.4 Level.............................................. 11 ∗Notes were taken by Takumi Murayama, who is responsible for any and all errors. Please e-mail [email protected] with any more corrections. Compiled on May 21, 2017. 1 1 Introduction We will follow Aganagic, Frenkel, and Okounkov's paper [AFO17], which came out in January of this year. Throughout, we work over the complex numbers C, although some things still work over an algebraically closed field F of characteristic zero. 1.1 The geometric Langlands correspondence The geometric Langlands correspondence is the function field analogue of the (number-theoretic) Langlands correspondence, conjectured by Beilinson and Drinfield in the 1980s [BD]. The statement is as follows: Conjecture 1.1 (Geometric Langlands correspondence [BD]). Let G be a complex reductive group, and let Σ be a projective algebraic curve (or a Riemann surface). Let LG denote the Langlands dual of G. Then, there is an equivalence of categories ∼ -Mod LocL (Σ) D-Mod Bun (Σ) ; (1) O G −! G L where -Mod LocLG(Σ) is the category of quasi-coherent sheaves on the G-moduli stack of local systems O on Σ, and D-Mod BunG(Σ) is the category of D-modules on the moduli stack of G-principal bundles on Σ. Example 1.2. Under this (proposed) correspondence, skyscraper sheaves correspond to Hecke eigensheaves. In [BD], Beilinson and Drinfeld show that there are subcategories on both sides of (1) that correspond to each other. On the other hand, the general statement of the conjecture is false: it fails in the case where G = SL(2) and Σ = P1 [Laf09]. However, there is a refined conjecture due to [AG15] involving -categories that is still expected to hold. 1 1.2 A representation-theoretic statement and its generalization in [AFO17] A key step in the proof of the correspondence in [BD] is reducing to the following representation-theoretic statement: Key Step 1.3 [FF92]. There is an isomorphism between the center of the affine Kac{Moody algebra Lcg at L L _ the critical level k = h and the classical -algebra 1(g). − W W The goal of [AFO17] is to generalize this statement to consider deformations away from the critical level L for cg, which corresponds to a quantum deformation of 1(g). More precisely, W Lcg is replaced by the quantum affine algebra U (Lcg); • ~ β(g) is replaced by the deformed -algebra q;t(g). The•W simply-laced case (i.e., the ADE case)W of this generalizationW is proved in [AFO17]. 2 Lie algebras We start with some basic material on Lie algebras. Definition 2.1. A Lie algebra is a vector space g over an algebraically closed field F of characteristic zero, together with a bilinear operator [ ; ] satisfying the following: [ax + by; z] = a[x; z] + b[y; z]− and− [z; ax + by] = a[z; x] + b[z; y], where a; b F and x; y; z g; • [x; x] = 0 for x g; 2 2 • (Jacobi identity)2 [x; [y; z]] + [y; [z; x]] + [z; [x; y]] = 0. • The building blocks for any Lie algebra are the simple Lie algebras: Definition 2.2. Let g be a Lie algebra. An ideal of g is a subspace i g such that [i; g] i. The Lie algebra g is simple if it has no non-trivial ideals, and is not abelian. ⊂ ⊂ There is a very famous classification of simple Lie algebras over F : they are classified by Dynkin diagrams. 2 Theorem 2.3. Finite-dimensional simple Lie algebras fall into four families An;Bn;Cn;Dn with five excep- tions E6;E7;E8;F4;G2, corresponding to the Dynkin diagrams below: E6 A n ··· E7 B n ··· E8 C n ··· F4 D n ··· G2 The simply-laced case corresponds to the Lie algebras with Dynkin diagrams without double-edges, i.e., the ADE case. These diagrams encode the data of a root system: nodes correspond to roots, and edges correspond to the angles between those roots. 2.1 Root systems We now define root systems and describe how they give rise to their corresponding Dynkin diagram. Definition 2.4. Fix a finite-dimensional R-vector space V with the standard Euclidean metric ( ; ). A root system Φ is a finite collection of nonzero vectors called roots, such that − − (1) The roots span V ; (2) If x Φ and kx Φ for k R, then k = 1; (3) For any2 two roots2 x; y Φ,2 the pairing ± 2 2(y; x) y; z := h i (x; x) is integer-valued; (4) For any two roots x; y Φ, 2 2(y; x) σ (y) = y x Φ: x − (x; x) 2 Example 2.5. Let g be a Lie algebra, and let h denote the Cartan subalgebra (this correpsonds to the maximal torus under the correspondence between subgroups and subalgebras). Then, h is a nilpotent subalgebra (i.e., [h; [h;:::; [h; h]]] = 0 for some finite number of iterations of the Lie bracket) such that if y g satisfies [x; y] h for all x h, then y h. Given x g, we can consider the adjoint action 2 2 2 2 2 adx(y) = [x; y] for y g. Consider the adjoint action h g. Then, g decomposes into eigenspaces for this action: 2 M g = h g ; ⊕ α α2h∗ where gα = x g [t; x] = α(t)x for all t h : 2 2 ∗ ∗ This collection α h gα = 0 gives a root system in h . 2 6 3 Definition 2.6. The positive roots Φ+ is a set of roots such that if α Φ, then either α Φ+ or α Φ+; • if α;2 β Φ+ and α + β 2 Φ, then−α +2 β Φ+. A choice• of Weyl2 chamber gives2 a set of positive2 roots. The subset ∆ Φ+ of simple roots consists of the positive roots that cannot be written as a sum of two other positive roots.⊂ Positive roots are non-negative combinations of simple roots. If α; β ∆, then the angle 2 θα,β is defined by requiring (α; β) cos(θ ) = : α,β α β j jj j Definition 2.7. Suppose that a root system Φ is irreducible, that is, it cannot be written as a union Φ Φ 1 [ 2 where (α; β) = 0 for all α Φ1; β Φ2. Let ∆ be a set of simple roots of Φ. Then, the Dynkin diagram for Φ has nodes corresponding to2 the simple2 roots ∆, and edges given by the angle between two roots: If θ = 2π=3, then α and β are connected by an undirected simple edge; • α,β If θα,β = 3π=4, then α and β are connected by a directed double edge; • If θ = 5π=6, then α and β are connected by a directed triple edge; • α,β If θα,β = π=2, then α and β are not connected by an edge. The• direction is given by pointing to the root with smaller Euclidean norm. It is a fact that a given irreducible root system has only two possible lengths. 2.2 The Langlands dual We now define the Langlands dual groups. Definition 2.8. Let (Φ;V ) be a root system. If α Φ, we define the coroot 2 1 α_ = α: (α; α) This gives rise to another root system (Φ_;V ), called the dual root or coroot system. ∗ _ Now let G be a complex connected reductive group. This gives the lattice data (X ; ∆;X∗; ∆ ), where X∗ the lattice of characters of G (homomorphisms G C∗); • ∆ the set of simple roots; ! • ∗ X∗ the lattice of one-parameter subgroups (homomorphisms C G); • ∆_ the set of simple coroots. ! • Definition 2.9. The Langlands dual LG of G is another complex connected reductive group, with lattice _ ∗ data (X∗; ∆ ;X ; ∆). ∗ _ Remark 2.10. The lattice data (X ; ∆;X∗; ∆ ) is more data than just the root system: GL(n) and SL(n) are both of type An, but the lattice for GL(n) is larger. Example 2.11. Under this dualizing procedure, the families An;Dn;En are sent to themselves, while Cn and Dn get switched. For example, GL(n) GL(n) A A ! n ! n SO(2n) SO(2n) D D ! n ! n SL(n) PGL(n) A A ! n ! n SO(2n + 1) Sp(n) B C ! n ! n 3 Affine Kac{Moody algebras We now discuss the basic definitions of affine Kac{Moody algebras, following [Her06].
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