Representation theory, geometric Langands duaity and categorification Joe Kamnitzer October 21, 2014 Abstract The representation theory of reductive groups, such as the group GL n of invertibe compex matrices, is an important topic, with appications to number theory, agebraic geometry, mathematica physics, and quantum topoogy. One way to study this representation theory is through the geometric Satae correspondence (aso nown as geometric Langands duaity). This correspondence reates the geometry of spaces caed affine Grassmannians with the representation theory of reductive groups. This correspondence was originay deveoped from the viewpoint of the geometric Langands program, but it has many other interesting appications. For exampe, this correspondence can be used to construct not homoogy theories in the framewor of categorification. In these ectures, we wi begin by expaining the representation theory of GL n, beginning with cassification of irreducibe representations. We wi aso give a presentation of the category of representations; such a presentation is nown as sein or spider theory. We wi aso discuss quantum GL n and see how it can be used to define not invariants. We wi then define the affine Grassmannian for GL n and expain the geometric Satae correspondence from the perspective of sein theory. We wi concude by expaining how these ideas, aong with the theory of derived categories of coherent sheaves, can be used to construct not homoogy theory. 1 Representation theory of GL n We wi begin by reviewing the representation theory of the group GL n of invertibe compex matrices. Good references are [FH] and [GW]. 1.1 Preiminaries and exampes Definition 1.1. An representation of GL n is a pair (V, ρ) where V is a finitedimensiona compex vector space V and ρ : GL n GL(V ) is a group homomorphism which is aso a morphism of agebraic varieties. Saying that ρ is a morphism of agebraic varieties means the foowing. Given any v V and α V, we can define a map ρ v,α : GL n C, ρ v,α (g) = α(ρ(g)(v)). 1
ρ is said to be a morphism of agebraic varieties if, for a v, α, ρ v,α is a rationa function in the entries of g. (Equivaenty, if we choose a basis for V, we get a matrix [ρ(g)] and we require that the entries of this matrix be rationa functions in the entries of g.) We wi often drop ρ from the notation and we wi write ρ(g)(v) as g v. Exampe 1.2. The simpest representation of GL n is the standard representation which is the action of GL n on C n (so V = C n and the map GL n GL(V ) is the identity). Exampe 1.3. More generay, we can aso consider actions of GL n on vector spaces buit out of C n. The first exampes are the actions of GL n on the th symmetric powers, Sym C n, and th exterior powers, C n. The action of GL n on Sym C n is given by and simiary for C n. g v 1 v = (g v 1 ) (g v ) Exercise 1.4. Identify Sym C n with the vector space of homogeneous poynomias of degree in n variabes. Consider the case n = 2. Describe the action of an invertibe matrix g = [ ] ab cd on a homogeneous poynomia p(x, y). Using this description, write the matrix for the action of g on Sym n C 2. Exampe 1.5. As a specia case of the previous exampe, we may consider V = n C n. This is a 1-dimensiona vector space. Let e 1,..., e n be the standard basis for C n. Then e 1 e n is a basis for n C n. With respect to this basis, the action of GL n becomes a group homomorphism GL n GL 1 = C. This homomorphism is the map taing a matrix to its determinant. Hence we wi refer to n C n as the determinant representation. The representations C n, for = 1,..., n 1 are usuay caed the fundamenta representations of GL n. For the purposes of this paper, we wi incude the determinant representation n C n as a fundamenta representation as we. Exampe 1.6. Tae n = 1 and tae V = C. Consider the map ρ : C C given by ρ(z) = z (compex conjugation). This map is a group homomorphism, but it is not agebraic. So this is not a representation that we wi study. Remar 1.7. There is a cose connection between the representation theory of GL n and the smooth representation theory of the subgroup U(n) of unitary matrices. If we have a representation (V, ρ) of GL n, we can restrict it to U(n) and we get a continuous representation of U(n). On the other hand, if we have a continuous representation of U(n) on a finite-dimensiona compex vector space V, then it extends uniquey to a representation of GL n on the the same vector space. Simiary, there is a cose connection with integra weight representations of the Lie agebra g n. Definition 1.8. If V, W are representations of GL n, then their direct sum V W is naturay a representation of GL n, with g (v, w) := (g v, g w). Simiary, their tensor product V W is aso naturay a representation, with g v w = g v g w. 2
1.2 Representations of tori Our goa now is to cassify a representations of GL n. Let us begin with the case of GL 1 = C. A one dimensiona representation of C is given by a (agebraic) group homomorphism C C. Proposition 1.9. Every agebraic group homomorphism C C is given by z z n for some integer n. Proof. Suppose that ρ : C C is an agebraic group homomorphism. Then since ρ(z) is a rationa function and since it is we-defined for any z C, we see that ρ(z) C[z, z 1 ]. The condition that ρ(z) is a group homomorphism forces ρ(z) = z n for some n Z. More generay, et (V, ρ) be a representation of C. Definition 1.10. A vector v V is said to have weight n if ρ(z)v = z n v for a z C. We write V n = {v V : ρ(z)(v) = z n v, for a z C } for the subspace of vectors of weight n. Theorem 1.11. Let V a representation of C. Then we have a direct sum decomposition V = n Z V n. We wi not give a proof of this resut. One way to prove it is to appea to U(1) C and to construct a U(1)-invariant inner product on V. Another approach is to use that morphisms of agebraic groups tae semisimpe eements to semisimpe eements. In this theorem, we have written V as a direct sum of eigenspaces for the eements of C. This theorem captures two important facts about the representation theory of C. First, a representations are semisimpe; meaning that they can be written as a direct sum of irreducibe subrepresentations. Second, a irreducibe representations of C are 1-dimensiona (this hods since C is abeian). More generay, et us consider a group T = (C ) n. Such a group is caed a torus. As before, we begin by studying the one dimensiona representations of this group. These are a of the form z = (z 1,..., z n ) µ(z) := z µ 1 1 zµn n for some sequence of integers µ = (µ 1,..., µ n ) Z n. Definition 1.12. A homomorphism µ : T C as above is caed a weight of T. The set of a weights P = Z n is caed the weight attice of T. Now et V be an arbitrary representation of T. For any weight µ, we can consider the subspace of vectors of weight µ V µ := {v V : z v = µ(z)v} The foowing resut generaizes Theorem 1.11. Proposition 1.13. We have a direct sum decomposition V = µ Z nv µ. 3
1.3 Weight spaces of representations of GL n Within GL n, there is a arge torus T = (C ) n consisting of the invertibe diagona matrices. We wi study representations of GL n by restricting them to representations of this maxima torus. Let (V, ρ) be a representation of GL n. Then it is aso a representation of the maxima torus T and so we get a weight decomposition V = µ V µ. We can record some numerica information about the representation by recording the dimensions of these weight spaces. Definition 1.14. The forma expression χ V := µ Z n dim(v µ )µ Z[P ]. is caed the character of the representation V. Note that if g (C ) n is any diagona eement of GL n,, then we have an equaity χ V (g) = µ Z n dim(v µ )µ(g) = tr(ρ(g)) Thus the character χ V determines the trace of the diagonaizabe eements of GL n acting on V. Since the diagonaizabe eements of GL n are dense in GL n, the character χ V determines the trace of a eements of GL n acting on V. So there is reay a ot of information in χ V. Here is the first fundamenta resut about the representation theory of GL n. Theorem 1.15. (i) Consider the action of the symmetric group S n on Z n given by w(µ 1,..., µ n ) = (µ w(1),..., µ w(n) ). For any representation V, we have dim V µ = dim V wµ for any w S n. (ii) If V, W are two representations and χ V = χ W, then V = W. Thus the character competey determines the representation. We wi not prove this resut. (One approach to (ii) is to restrict to U(n) and use resuts about the character of compact groups.) Exercise 1.16. Prove (i). Hint: consider the incusion S n GL n which taes permutations to permutation matrices. Exampe 1.17. Consider the representation Sym 3 C 2. It has a basis given by Note that if z = [ z 1 0 0 z 2 ], then e 3 1, e 2 1e 2, e 1 e 2 2, e 3 2. z e a 1e b 2 = (z 1 e 1 ) a (z 2 e 2 ) b = z a 1z b 2e a 1e b 2 and hence e a 1 eb 2 has weight (a, b). Hence the non-zero weight spaces of Sym3 C 2 are (3, 0), (2, 1), (1, 2), (0, 3) and each weight space has dimension 1. 4
Exampe 1.18. More generay consider the representation Sym C n. It has a basis given by monomias e a 1 1... ean n of tota degree. Such a monomia has weight (a 1,..., a n ). Hence µ is a weight if and ony if (µ 1 + + µ n ) =. and a weight spaces have dimension 1. Exercise 1.19. Find the character of the representation C n. Definition 1.20. Let R(GL n ) denote the representation ring of GL n. This is the abeian group generated by isomorphisms casses [V ] of representations of GL n, moduo the reation [V W ] = [V ] + [W ]. The mutipication in this ring is given by [V ][W ] := [V W ]. The map χ : R(GL n ) Z[P ] given by [V ] χ V is easiy seen to be a ring homomorphism (the mutipication on Z[P ] is defined using addition in P ). Theorem 1.15 shows that the χ is injective and that its image ands in the subspace Z[P ] Sn of invariants for the symmetric group. 1.4 Irreducibe representations of GL n Definition 1.21. A subrepresentation W V of a representation of GL n is a subspace of W V which is invariant under the action of GL n. A representation V of GL n is caed irreducibe if it has no subresentations (other than 0, V ). Let V, W be representations of GL n. A morphism of representations A : V W is a inear map such that A(g v) = g A(v) for a g GL n, v V. The set of a morphisms from V to W is denoted Hom GLn (V, W ). The foowing basic resut is caed Schur s Lemma. Its proof is straightforward. Theorem 1.22. Suppose that V, W are irreducibe representations. Then Hom GLn (V, W ) is 1-dimensiona if V = W and is 0-dimensiona otherwise. We aso have the foowing semisimpicity resut. Theorem 1.23. Every representation V can be written as a direct sum of irreducibe subresentations. There are a number of proofs of this theorem. U(n)-invariant inner product on V. One approach is to construct a Exampe 1.24. Consider the representation of GL n on C n C n. We have a direct sum decomposition into subrepresentations C n C n = 2 C n Sym 2 C n. Later we wi see that 2 C n and Sym 2 C n are irreducibe representations. Now we woud ie to describe the irreducibe representations of GL n. Before we do this, we wi need a few definitions. 5
Definition 1.25. The positive roots of GL n are the eements of Z n of the form where the 1 occurs before the 1. We say λ µ if we can write β = (0,..., 0, 1, 0..., 0, 1, 0,..., 0) λ µ = β β β for some non-negative integers β, where the sum varies over a positive roots. A weight µ = (µ 1,..., µ n ) is caed a dominant weight if µ 1 µ n or equivaenty if µ, β 0 for a positive roots β. We write P + for the set of dominant weights. A representation V is said to have highest weight λ, if V λ 0 and whenever V µ 0, then µ λ. Note that there is a unique dominant weight in every orbit of S n on the weight attice P = Z n. Aso note that because the set of weights of a representation is invariant under S n, if λ is the highest weight of a representation, then λ must be dominant. Exampe 1.26. Consider Sym 3 C 2. As cacuated in exampe 1.17, the weights are (3, 0), (2, 1), (1, 2), (0, 3). Hence this representation has highest weight (3, 0). Here is the fina main theorem concerning the representation theory of GL n. Theorem 1.27. For each λ P +, there exists a unique representation V (λ) which is irreducibe and of highest weight λ. Proof. Let us prove the existence. Since λ is dominant, we can write λ = n =1 m ω, with m 0 for < n. Consider the tensor product W = n ( C n ) m. =1 We can see that W is of highest weight λ, as the sum of the highest weights of each factor is λ. We can write W = W 1 W p for some irreducibe subrepresentations W 1,..., W p. At east one of these must have a non-zero weight space for λ and this one wi be an irreducibe representation of highest weight λ. Exercise 1.28. Use the injectivity of the character map R(GL n ) Z[P ] Sn the uniqueness of V (λ). to prove Exampe 1.29. Tae λ = (, 0,..., 0). Then V (λ) = Sym C n. 6
Exercise 1.30. Prove that C n is irreducibe and is of highest weight ω = (1,..., 1, 0,..., 0) (where there are 1s). Concude that C n = V (ω ). Exampe 1.31. Consider s 3 the vector space of 3 3 trace 0 matrices. We have an action of GL 3 on this space by conjugation. This is the irreducibe representation V (1, 0, 1). In particuar 0 0 1 0 0 0 0 0 0 has weight (1, 0, 1). The weights of this representation are (1, 0, 1), (0, 1, 1), (1, 1, 0), (0, 0, 0), (0, 1, 1), ( 1, 1, 0), ( 1, 0, 1). A of these weight spaces are 1-dimensiona with the exception of the (0, 0, 0) weight space which consists of the diagona matrices and hence is 2-dimensiona. 1.5 Determinant representation and poynomia representations The determinant representation described above is the irreducibe representation of highest weight (1,..., 1). Its dua representation has highest weight ( 1,..., 1). In this representation, we have ρ(g) = det(g) 1. Tensoring with these representations moves us around the set of dominant weights in the sense that V (1,..., 1) V (λ) = V (λ 1 + 1,..., λ n + 1) V ( 1,..., 1) V (λ) = V (λ 1 1,..., λ n 1). Let P ++ be the set of dominant weights λ such that λ n 0 for a i.. We say that an irreducibe representation V (λ) is poynomia, if λ P ++. More generay, a representation of GL n is caed poynomia if it is the direct sum of poynomia irreducibe representations. Starting with poynomia irreducibe representations, we can get to a irreducibe representations by repeatedy tensoring with V ( 1,..., 1). For this reason it is enough to consider just these poynomia irreducibe representations which we wi do in ater sections. 2 GL n spider/sein theory We wi now give a diagrammatic description of the category of GL n representations. 7
2.1 The representation category From Theorems 1.22, 1.23, 1.27, the category Rep(GL n ) of representations of GL n is pretty easy to describe. Every representation V can be decomposed uniquey as V = λ V (λ) M(λ) where M(λ) is a vector space with trivia GL n action, caed the mutpicity space. If V = λ V (λ) M (λ) is another representation, then we have Hom GLn (V, V ) = λ Hom(M(λ), M (λ)) (1) However, we can get a more interesting combinatoria structure if we wor with a restricted set of representations. Definition 2.1. Rep f (GL n ) denotes the category whose objects are isomorphic to tensor products of the fundamenta representations C n of GL n, for = 1,..., n and whose morphisms are morphisms of representations. Of course, one can describe morphisms between these tensor product representation by decomposing into irreducibes and then appying (1). However, we wi pursue a different approach here. We wi try to describe morphisms in this category as being buit from the natura maps C n C n + C n and + C n C n C n which we depict diagrammaticay as foows (where we read from the bottom up) + and. (2) + It is reativey easy to show that these are indeed generators, i.e. that every GL n -inear map between tensor products of fundamenta representations can be written as tensor products and compositions of these maps. We define a diagrammatic category, the free spider category FSp(GL n ), whose objects are sequences = ( 1,..., m ) (with j {1,..., n}) and whose morphisms are inear combinations of webs, trivaent graphs made up by gueing together the pieces in (2). The edges in these webs are oriented and abeed by {1,..., n}. Theorem 2.2. We have a fu and dominant functor FSp(GL n ) Rep(GL n ). We wi now describe the erne of this functor. This is a probem which was first posed by Kuperberg [Ku], who found the reations when n = 3. 8
2.2 Definition of the spider category The spider category Sp(GL n ) is the quotient of FSp(GL n ) by the foowing reations: + + ( ) + = + (3) + ( ) n = (4) + + m + + m + + m m = m (5) s r s r s + s + r + s ( ) s r + s + r r + s r + s = r (6) s+r s r s +s r +s = t s+r ( ) + r s +r t t s t r t +s r r+t (7) together with the mirror refections of these. It is fairy easy to chec that a these reations hod in Rep f (GL n ) and thus the functor FSp(GL n ) Rep f (GL n ) descends to a functor Γ n : Sp(GL n ) Rep f (GL n ). The foowing resut was proved with Cautis and Morrison [CKM]. Theorem 2.3. The functor Γ n : Sp(GL n ) Rep f (GL n ) is an equivaence of categories. The idea behind the proof is to use the theory of sew Howe duaity. More precisey, we consider the representation K C n C m which carries commuting actions of GL n and GL m. 9
2.3 Quantum groups The group GL n and its representations admit q-defomations. More precisey, there exists a Hopf agebra U q (g n ) over the ring C[q, q 1 ]. It has representations V q (λ) which are free modues over C[q, q 1 ] which speciaize to V (λ) when q = 1. Everything written above about representations of GL n carries over to U q (g n ). In particuar, there are U q (g n ) representations q Cn q which are free C[q, q 1 ] modues of dimension ( n ). The category Repf (U q (g n )) can be defined in a simiar way, as can F Sp(U q (g n )) and Sp(U q (g n )) except that in a the reations (3)-(7) we repace ( ) n with [ ] n which is defined as q [ ] n [n] q [1] q =, where [] q = q 1 + q 3 + q 1 [] q [1] q [n ] q [1] q q The anaog of Theorem 2.3 hods in this context. tensoring with C(q).) (More precisey, it hod after 2.4 Braiding There is one interesting new feature about U q (g n ) representations, caed braiding. Suppose that V, W are representations of GL n. Then of course we may form V W or W V and the map σ V,W : V W W V v w w v is easiy seen to be an isomorphism between these two representations. Now, if V, W are U q (g n ) representations, there is a natura way to give V W the structure of a U q (g n ) representation. However, it turns out that the map σ V,W is not a map of U q (g n ) representations. However, there does exist a natura isomorphism β V,W : V W W V, which is caed the braiding. When we set q = 1, then β becomes σ. Exampe 2.4. Consider n = 2 and V = W = C[q, q 1 ] 2. Then with respect to the standard basis of C[q, q 1 ] 2 C[q, q 1 ] 2, the braiding β C[q,q 1 ] 2,C[q,q 1 ] 2 is given by the matrix 1 0 0 0 0 0 q 0 0 q 1 q 2 0 0 0 0 1 In terms of the diagrammatics, there is a nice expression for the braiding of fundamenta representations. Proposition 2.5. The braiding β q C n q, qc n q is given by the foowing sum of webs 10
a ( 1) + q n a,b 0 b a= ( q) b b b +b Now tae V to be any representation. If consider V n, then the braidings β V,V of neighbouring pairs in this tensor product can be used to generate an action of the braid group. Definition 2.6. The braid group B n is defined topoogicay as π 1 (C n /S n ), the fundamenta group of the configuration space of n points on C = R 2. It has generators s 1,..., s n 1 and reations s i s i+1 s i = s i+1 s i s i+1, s i s j = s j s i, if i j 2. The foowing important resut is due to Drinfed. Theorem 2.7. There is an action of B n on V n where s i acts by I V (i 1) β V,V I V (n i 1). 2.5 Knot invariants The representation theory of U q (g n ) can be used to define not invariants. These were deveoped by Reshetihin-Turaev foowing earier wor by Jones, Witten, and others. Reca that a not is an embedding of finitey many circes into R 3. We wi consider nots up to isotopy. Given a not, we can consider projections of the not to R 2 to produce a not diagram, a panar graph with vertices mared by over- or undercrossings. There is a we-nown combinatoria theory of Reidemeister moves, which determine when two not diagrams come from isotopic nots. Now suppose that we have not K whose components are cooured with representations of U q (g n ). Then we can compose together the braidings which given by the crossings of a not diagram of K, as we as coevauation C[q, q 1 ] V V and evauation V V C[q, q 1 ] maps given by the cups and caps of the diagram. Reshetihin- Turaev [RT] showed that the resuting poynomia P (K) C[q, q 1 ] ony depends on the isotopy cass of K. When n = 2 and we use the standard representation C 2 q, then P (K) is caed the Jones poynomia and was earier discovered by Jones [J]. If K is abeed by fundamenta representations, then we can compute the not poynomia P (K) by repacing a crossings with webs using Proposition 2.5 and then evauating the resuting webs using reations (3)-(7). The specia case where a strands are abeed by C n q was investigated by Muraami-Ohtsui-Yamada [MOY]. 11
3 Geometric Satae correspondence We wi now give an exposition of the geometric Satae correspondence which is due to Lusztig [L], Ginzburg [G], and Mirovic-Vionen [MV]. 3.1 The varieties Gr λ Consider the (infinite-dimensiona) C-vector space C[z] C n. So if C n has basis e 1,..., e n, then C[z] C n has a basis {z e i } where = 0, 1,... and i = 1,..., n. Define a inear operator w : C[z] C n C[z] C n { z z 1 e i, if 1 e i 0, if = 0 Definition 3.1. Let Gr = {L C[z] C n : wl L} be the set of a w-invariant finite-dimensiona subspaces of C[z] C. This is caed the positive part of the affine Grassmannian. Let L Gr. We can consider the restriction of w to L. This wi be a nipotent operator and hence it wi have a Jordan type. Note that since dim er(w) = n, it foows that dim er(w L ) n which impies that w L has at most n Jordan bocs. Hence the Jordan type of w L can be written as (λ 1,..., λ n ) with λ 1 λ n 0 and thus can be regarded as an eement of P ++. We define Gr λ := {L C[z] C n : wl L, and w L has Jordan type λ}. It is a ocay cosed subset in Gr and we may tae its cosure to obtain Gr λ. Proposition 3.2. Gr λ is a projective variety of dimension (n 1)λ 1 + (n 3)λ 2 + + ( 1 n)λ n. It is easy to see exacty which subspaces we pic up in the cosure in the cosure the Jordan type of w L becomes more specia, i.e. more ie the zero matrix. This can be expressed as foow. Lemma 3.3. Gr λ = Gr µ µ P ++ :µ λ Exampe 3.4. Suppose that λ = ω = (1,..., 1, 0,..., 0). Having Jordan type ω means that w L is 0 and that dim L =. Hence we see that Gr λ is isomorphic to the Grassmannian G(, n) of dimensiona subspaces of er(w) = C n. Note that in this case, Gr λ = Gr λ. Exercise 3.5. Prove that Gr (2,0) is isomorphic to the singuar projective variety in P 3 defined by the equation x 2 + yz = 0 (in other words, the projective cosure of the 2-dimensiona quadratic cone defined by the same equation). 12
It is not immediatey obvious that these varieties Gr λ are non-empty. Let us demonstrate this now. For each µ N n, we define a subspace L µ = span C (z e i : < µ i ) C[z] C n. Exercise 3.6. L λ Gr λ. More generay L wλ Gr λ for a w S n. Now suppose = ( 1,..., m ) is a sequence with j {1,..., n}. We define a fag-ie variety Gr := {0 =L 0 L 1 L m C[z] C n : wl i L i 1, dim L i = dim L i 1 + i } Exercise 3.7. Consider the map Gr ( 1,..., m) Gr ( 1,..., m 1 ). Prove that this map is a fibre bunde with fibre G( m, n). The geometric Satae correspondence reates the representation V (λ) to the variety Gr λ and the representation 1 m to the variety Gr. To expain this more precisey, we wi need some genera facts about sheaves. 3.2 Sheaf theory Let X be a reasonabe topoogica space. Definition 3.8. A sheaf F on X is a choice of a vector space F(U) for every open set U X aong with restriction maps F(U) F(V ) for every incusion U V. These restriction maps must compose propery. Finay, we require that whenever we have a decomposition U = i U i and sections s i F(U i ) which agree (after restriction) in F(U i U j ) for a i, j, then there exists a unique s F(U) which restricts to a of the s i. The simpest exampe of a sheaf is the constant sheaf C X, with C X (U) = C for every connected open set U. If F is a sheaf, then one can define the cohomoogy of X with coefficients in F, denoted H (X, F) using Cech cohomoogy. When F = C X, then H (X, C X ) is the same as the singuar or simpicia cohomoogy of X. There is another specia sheaf on X, caed the intersection cohomoogy sheaf, denote IC X. The cohomoogy IH (X) := H (X, IC X ) is caed the intersection homoogy of X and it satisfies Poincaré duaity for any reasonabe space X. When X is a manifod, then IC X = CX. If p : X Y is a continuous map, then we can define the push-forward sheaf 1 p F by p F(U) = H (p 1 (U), F) 1 Actuay, p F is not a sheaf, but rather a compex of sheaves (more precisey, an object in the derived category of sheaves on X). For the purposes of these ectures, the above approximate definition wi be sufficient. Simiary, when we spea about morphisms of sheaves, we reay mean morphisms in the derived category. 13
Now suppose that we have two maps p : X Y and p : X Y with X, X manifods. The foowing resut of Chriss-Ginzburg [CG] wi be quite usefu for us, since it reates morphisms between pushforwards of constant sheaves and homoogy of fibre products. Theorem 3.9. We have an isomorphism for some m. Hom(p C X, p C X ) = H m (X Y X ) (If the maps p, p are semisma, then m equas the (rea) dimension of X Y X.) To expain the second part of the theorem, suppose that p : X Y is a third variety with a map Y. Then we can define a convoution product by the formua : H (X Y X ) H (X Y X ) H (X Y X ) c 1 c 2 = (π 13 ) (π 12(c 1 ) π 23(c 2 )), where denotes the intersection product (with support), reative to the ambient manifod X X X and p 12 : X X X X X, etc. For more detais about this construction, see [CG, Sec. 2.6.15] or [F, Sec. 19.2]. The foowing resut is aso due to Chriss-Ginzburg. Theorem 3.10. The isomorphism of Theorem 3.9 interwines the composition of Homs with the convoution product on homoogy. 3.3 Geometric Satae We are now in a position to roughy formuate the geometric Satae correspondence. Let P(Gr) denote the category of sheaves on Gr which direct sums of the sheaves IC Gr λ and et P f (Gr) denote the category of sheaves which are of the form p C Gr. The decomposition theorem shows us that P f (Gr) P(Gr). Theorem 3.11. We have an equivaence of categories Rep(GL n ) = P(Gr), taing Rep f (GL n ) = P f (Gr). On objects, this equivaence is given by V (λ) IC Gr λ, 1 C n m C n p C Gr Moreover, this equivaence is compatibe with the functors to vector spaces on both sides, so that IH (Gr λ ) = V (λ), and H (Gr ) = 1 C n m C n. Exercise 3.12. Tae λ = ω. As we have seen Gr ω = Gr ω = G(, n). Show that IH (Gr ω ) = H (G(, n)) has a basis abeed by eement subsets of {1,..., n}. Using this construct an isomorphism IH (Gr ω ) V (ω ). 14
3.4 Weight space decomposition As we now, representations of GL n have a weight space decomposition V = µ V µ. It is interesting to examine this decomposition from the perspective of perverse sheaves. We have an action of the torus T on C n and thus on C[z] C n and thus on Gr. The fixed points of this torus action on Gr are precisey the points L µ, for µ P. Let us pic a generic C C n, for exampe given by the diagona matrix with entries (t, t 2,..., t n ). Then for each µ P, we can consider the attracting set S µ := {L Gr : im t t L = L µ } This set S µ provides a ind of ce-decomposition of Gr, except the pieces S µ are not finite-dimensiona. They are caed semi-infinite ces. Nonetheess, we can use them to study the intersection homoogy of each Gr λ. The foowing resut is due to Mirovic-Vionen [MV]. Theorem 3.13. For each λ, we have a decomposition IH (Gr λ ) = µ H top (Gr λ S µ ) Moreover, this decomposition matches the weight space decomposition of V (λ) under the isomorphism provided by geometric Satae. The vector space H top (Gr λ S µ ) has a basis given by the irreducibe components of Gr λ S µ. The irreducibe components are caed Mirovic-Vionen cyces. 3.5 Spiders and geometric Satae We have previous described Rep f (GL n ) using webs. It is natura to as how this description oos ie after passing to P f (Gr) using geometric Satae. This wi aso aow us to understand geometric Satae on the eve of morphisms. Reca that by Theorem 3.9, for any two sequences = ( 1,..., m ), = ( 1,..., m ), we have Hom Pf (Gr)(p C Gr, p C Gr ) = H top (Z(, )) where Z(, ) = Gr Gr Gr = {(L, L ) Gr Gr Gr : L m = L m Reca that we have the equivaences of categories Sp(GL n ) Rep f (GL n ) P f (Gr) and thus an isomorphism Hom Sp(GLn)(, ) Hom Pf (Gr)(p C Gr, p C Gr ) = H top (Z(, )) (8) An eement of the LHS is represented by a web with bottom endpoints and top endpoints. On the RHS, we have a basis given by the irreducibe components of Z(, ). Thus under the geometric Satae correspondence, the web wi be mapped to a inear combination of these components. In [FKK], with Fontaine and Kuperberg, 15
we showed that for each web w, there exists a configuration space X(w) in the affine Grassmannian which maps to Z(, ) and such that (under good circumstances) the image of the web under (8) is a inear combination of the components of Z(, ) with coefficients being the Euer characteristics of the generic fibres of X(w) Z(, ). Now, et us focus on the case = (, ) and = ( + ). We can see that Z((, ), ( + )) = {(L 1, L 2 ), (L 1) : L 1 = L 2 } is just isomorphic to a G(, ) bunde over G( +, n). On the other hand, Hom Sp(GLn)((, ), (+)) is just spanned by the web consisting of a singe trivaent vertex. Under (8), this trivaent vertex is mapped to [Z((, ), ( + ))]. References [C2] [CK1] [CK2] Sabin Cautis, Casp technoogy to not homoogy via the affine Grassmannian; arxiv:1207.2074 Sabin Cautis and Joe Kamnitzer, Knot homoogy via derived categories of coherent sheaves I, s 2 case, Due Math. J. 142 (2008), no. 3, 511 588. math.ag/0701194 Sabin Cautis and Joe Kamnitzer, Knot homoogy via derived categories of coherent sheaves. II. s m case, Invent. Math. 174 (2008), no. 1, 165 232, arxiv:0710.3216. [CKL1] Sabin Cautis, Joe Kamnitzer and Anthony Licata, Categorica geometric sew Howe duaity; Inventiones Math. 180 (2010), no. 1, 111 159; arxiv:0902.1795 [CKM] [CG] Sabin Cautis, Joe Kamnitzer, and Scott Morrison. Webs and quantum sew Howe duaity, Math. Annaen, 360 (2014), 351 390; arxiv:1210.6437. Nei Chriss and Victor Ginzburg, Representation theory and compex geometry, Birhäuser Boston Inc., 1997. [FKK] Bruce Fontaine, Joe Kamnitzer, and Greg Kuperberg, Buidings, spiders, and geometric Satae, Compos. Math., 149 no. 11 (2013) 1871 1912; arxiv:1103.3519. [F] Wiiam Futon, Intersection theory, 2nd ed., Springer-Verag, 1998. [FH] Wiiam Futon and Joseph Harris, Representation theory: a first course, Springer-Verag, 2004. [G] [GW] Victor Ginzburg, Perverse sheaves on a oop group and Langands duaity, arxiv:ag-geom/9511007. Roe Goodman and Noan R. Waach, Symmetry, Representations, and Invariants, Springer-Verag, 2009. 16
[H] [J] [Kh] Danie Huybrechts, Fourier-Muai Transforms in Agebraic Geometry, Oxford University Press, 2006. V. F. R. Jones, A poynomia invariant for nots via von Neumann agebras, Bu. Amer. Math. Soc. 12 (1985) 103 112. Mihai Khovanov, A categorification of the Jones poynomia, Due Math. J., 101 no. 3 (1999), 359 426; arxiv:math.qa/9908171. [Ku] Greg Kuperberg, Spiders for ran 2 Lie agebras, Comm. Math. Phys. 180 (1996), no. 1, 109 151, arxiv:q-ag/9712003. [L] [MV] [MOY] [RT] George Lusztig, Singuarities, character formuas, and a q-anaog of weight mutipicities, Anaysis and topoogy on singuar spaces, II, III (Luminy, 1981), Astérisque, vo. 101, Soc. Math. France, 1983, pp. 208 229. Ivan Mirović and Kari Vionen, Geometric Langands duaity and representations of agebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), no. 1, 95 143, arxiv:math/0401222. Hitoshi Muraami, Tomotada Ohtsui, and Shuji Yamada, Homfy poynomia via an invariant of coored pane graphs, Enseign. Math. (2), 44, 325 360, 1998. Nicoai Yu. Reshetihin and Vadimir G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1 26. 17