Introduction to affine Springer fibers and the Hitchin fibration

Introduction to affine Springer fibers and the Hitchin fibration Zhiwei Yun Lecture notes by Tony Feng Contents 1 Classical Springer Theory 2 1.1 Springer fibers................................. 2 1.2 Geometric properties of B e.......................... 5 1.3 The Springer Correspondence........................ 5 2 Affine Springer fibers 10 2.1 Affine Grassmannian............................. 10 2.2 Symmetries of X γ............................... 11 2.3 Finiteness of affine Springer fibers....................... 13 2.4 Dimension computations........................... 14 2.5 Affine Weyl Group Actions.......................... 15 3 Orbital Integrals 17 3.1 Setup..................................... 17 3.2 Orbital integrals................................ 17 3.3 Geometric interpretation........................... 20 4 The Hitchin Fibration 23 5 Conclusion 25 1

1 CLASSICAL SPRINGER THEORY 1 Classical Springer Theory Springer Theory is now a classical piece of geometric representation theory, on which much other theory is built. 1.1 Springer fibers We work over an algebraically closed field k = k. Fix a simple algebraic group G/k. Furthermore, assume that ch k is large with respect to G (more precisely, the invariants of G) - this means that the representation theory over k is like that in characteristic 0. Example 1.1. The main examples will be G = SL n and Sp 2n. Notation. As usual, we denote g = Lie(G), the Lie algebra of G. B the flag variety of G, which can be defined as the space of Borel subgroups of G. This has the structure of a smooth projective variety, and is isomorphic to G/B. N g the nilpotent cone (evidently a cone because the property of being nilpotent is preserved by scalar multiplication). Example 1.2. If we have an embedding of Lie algebras g gl n, then N consists of nilpotent matrices in g. Definition 1.3. For a Borel subgroup B G, the nilradical n B of Lie(B), is the nilpotent cone of Lie(B). Let Ñ = {(e, B) N B e n B }. Then Ñ admits an obvious projection to B, which makes it a vector bundle (in fact, the cotangent bundle) of B: Ñ T B. Exercise 1.4. Check this by checking that n B may be canonically identified with the cotangent space of B at B. Definition 1.5. The map π: Ñ N is the Springer resolution. Implicit in the definition is the statement that Ñ N is a resolution of singularities (a birational morphism from a smooth variety). Indeed, Ñ is a cotangent bundle of a smooth variety, hence is itself smooth. Definition 1.6. Pick e N. Then the fiber π 1 (e) =: B e = {B G e Lie n B } is the Springer fiber of e. Example 1.7. If G = SL n = SL(V), then the Borel subgroups of G are in bijection with flags 0 V 1 V 2... V n 1 V, with the Borel subgroup being the stabilizer of its corresponding flag. 2

1 CLASSICAL SPRINGER THEORY Let e be a nilpotent matrix. Then we can recover B e as B e = {0 V 1... V n 1 V ev i V i 1 }. If e = 0, then this is no restriction, so we find that B e = B. At the other extreme, if e has a single Jordan block 0 1 0 1 e...... 0 then B e is a point. In terms of the standard basis, the unique full flag in the Springer fiber over e is 0 v 1... v 1,..., v n 1 V. 0 0 1 Example 1.8. We study the Springer fiber B e for G = SL 3 and e = 0 0 (there are two 0 Jordan blocks). Then we claim that B e is the union of two copies of P 1 along a point: B e = P 1 pt P 1. To see why, consider a flag 0 V 1 V 2 V. Either e(v 2 ) = 0 or not. If not, then evidently we must have V 1 = v 1, since that is the (one-dimensional) image of e. Therefore, any such flag has the form 0 v 1 V 2 V, so the choice of V 2 corresponds to any plane in k 3 containing v 1, i.e. a line in the twodimensional quotient V/ v 1, which is parametrized by P 1. 3

1 CLASSICAL SPRINGER THEORY Now consider the other case, where e(v 2 ) = 0. Since ker e = v 1, v 2, such flags are of the form 0 V 1 v 1, v 2 V. In this case, the freedom is choosing a line in the two-dimensional subspace v 1, v 2, which is again parametrized by P 1. The two choices evidently intersect at the standard flag 0 v 1 v 1, v 2 V. Remark 1.9. From this example we see that the Springer fibers may be singular. Example 1.10. Let G = SL 4 and 0 1 0 1 e =. 0 0 Note that e has two Jordan blocks, each of size 2. Then B e is the union of two surfaces, isomorphic to P(O( 2) O) and P 1 P 1, over P 1. This P 1 is embedded via the canonical section of the Hirzebruch surface P(O( 2) O) and diagonally in P 1 P 1. To see why, consider a flag 0 V 1 V 2 V 3 V. Definition 1.11. Let g = {(x, B) x g, B B, x Lie B}. This is again a vector bundle over the flag variety. The map π g : g g is called the Grothendieck alteration. 4

1 CLASSICAL SPRINGER THEORY If you restrict π g to N, we almost get Ñ. The subtle difference is that in this case we only asked that x Lie(B) instead of n B, and that makes a difference that you don t see at the level of points. That is, the pre-image of N under π g is not reduced, but its reduced structure is Ñ. Exercise 1.12. Work out an example of this. 1.2 Geometric properties of B e We list some properties without proof. (See the exercises.) B e is connected (usually singular, and in fact reducible). dim B e = 1 2 (dim N dim O e), where O e the orbit of e under the adjoint action. This is obviously integral, reflecting that the codimension of O e is always even. Symmetries. We have an action of C G (e), the centralizer of e in G, on B e. But there is more. We have an action of G := G G m on N, where the G m acts by dilation. Then Stab G (e) (which contains C G(e)) acts on B e. The reason is that G acts not only on N but on the Springer resolution Ñ (with G m acting trivially on the flag variety part). A nice thing about this is that Stab G (e) surjects to G m. This gives a torus action on N, which you wouldn t necessarily have if you didn t consider the G m action. In H (B e ), where we take étale cohomology with Q l -coefficients or singular cohomology with Q-coefficients, we have H odd (B e ) = 0 (DeConcini-Lusztig-Procesi) and H (B e ) is pure (Springer) in the sense of weights or Hodge theory, respectively. Remark 1.13. If G = SL n, then B e can be stratified into cells, each isomorphic to some affine space, and then these two statements follow immediately. 1.3 The Springer Correspondence Let W be the Weyl group of G, which we can define as N(T)/T for some maximal torus T in G. Theorem 1.14 (Springer). There is a natural action of W on H (B e ). Remark 1.15. This is subtle because there is no complex-geometric action of W on B e inducing this action on cohomology. However, W does act on B in the category of real algebraic geometry, in a way that induces this action on cohomology. The idea is that instead of presented the flag variety as G/B, one uses a compact real form for G. For instance, if G = GL n then B = G/B = U n (R)/T c. Concretely, this can be thought of as orthogonal framings in C n with the standard hermitian form, C n = L 1 L 2... L n. 5

1 CLASSICAL SPRINGER THEORY It s clear how W S n acts on the set of such framings. However, this action is only realanalytic and not complex-analytic. Sketch of construction. (Lusztig) We require some facts. Fact. The map π g : g g is a small map, i.e. if then codim Y d > 2d for all d > 0. Y d = {x g dim π 1 g (x) d} g Example 1.16. If g = sl 2, then π g is generically 2:1. The fiber over 0 is the whole flag variety, P 1. All other fibers are finite. Since the point 0 has codimension 3 in g, the map is small. By another general result, this fact implies that Rπ g is a perverse sheaf. Moreover, it is IC(g ss, L[dim g]) where L = (π g ) Q g ss (in general one would take the derived pushforward, but that is unnecessary here as the pushforward is already exact, because the map is a finite cover). Over g ss, the map π g ss : g ss g ss is an étale Galois cover with Galois group W. The induced W-action on L is the obvious one. By the functoriality of IC, W acts on Rπ g Q. Taking stalks at e N, we get a W action on (Rπ g Q) e = H (B e ). Example 1.17. If e is regular, then B e is a point and the action is trivial. If e = 0 then B e = B and we get W acting on the cohomology of the full flag variety H (B). Borel s presentation of the cohomology of the flag variety is H (B) Sym(t )/ Sym(t ) W +. The action of W is induced by the obvious one of W on t. N is a union of orbits. The largest is typically called the regular orbit O reg. Then next largest orbit is called the subregular orbit, O subreg. This has codimension 2 in N. So for subregular e, we find that dim B e = 1, and in fact is always a union of P 1 s. Their configuration is interesting, and turns out to be related to the Dykin diagram of G. The rule 6

1 CLASSICAL SPRINGER THEORY is that each node of the Dynkin diagram becomes a node of the curve. More generally, in the simply laced case the Dynkin diagram is the dual graph to the the Springer fiber. Example 1.18. For G = Sp 2n, the short roots are as before, and the long roots corresponds to two P 1 s. What is the action of the Weyl group on this union of P 1 s? For e subregular and g of type A, D, E we have H 2 (B e ) t. Then the Weyl group action is the usual reflection action on H 2 (B e ) t. The analysis is more complicated in the non simply-laced cases. Example 1.19. For type G 2, C G (e) is a disconnected algebraic group. The component group is isomorphic to S 3, and the action on cohomology factors throught this component group. 7

1 CLASSICAL SPRINGER THEORY So there is an S 3 action on H 2 (B e ). In fact we we have simultaneous actions of W and S 3 on H 2 (B e ), and these actions commute (the Weyl group commutes with centralizer, of course), and that decomposes it into t and a two-dimensional piece. The action of S 3 on the second piece is its irreducible two-dimensional representation. Now let us state the Springer correspondence. This says that each irreducible representation of W can be realized in the cohomological representations constructed by Springer. Theorem 1.20 (Springer correspondence). For each irreducible representation V of W, there is a pair (e, ρ) where e N and ρ is an irreducible representation of A G := π 0 (C G (e)) such that V Hom Ag (ρ, H 2d e (B e )) (where 2d e = dim B e ) as W-representations. Furthermore, the pair (e, ρ) is unique up to G-conjugation. We can reformulate this as the existence of a natural injection Irr(W) {(e, ρ) e N, ρ Irr(A G (e))}/conjugation. A more geometric perspective on the right hand side is that it is the set of isomorphism classes of G-equivariant irreducible local systems on nilpotent orbits. Sketch of proof. We require a more refined result. Theorem 1.21 (Borho-MacPherson). If π: Ñ N is the Springer resolution, then Rπ Q[dim N] is a perverse sheaf and End(Rπ Q) Q[W]. 8

1 CLASSICAL SPRINGER THEORY Since the Weyl group acts on Rπ Q[dim N], the Borho-MacPherson Theorem implies a decomposition Rπ Q[dim N] = F χ V χ (V χ,χ) Irr(W) Note that every (V χ, χ) has a non-zero summand on the right hand side because End(Rπ Q) Q[W]. Now F χ = IC(O, E) where O N is a nilpotent orbit and E is a G-equivariant local system on O (i.e. an irreducible representation of π 0 (C G (e)). This gives the injection Irr(W) {(O, E)} predicted by the Springer correspondence. 9

2 AFFINE SPRINGER FIBERS 2 Affine Springer fibers Motivation. First off, why does anybody care about (affine) Springer fibers? By work of Lusztig, Springer fibers turn out to have applications to the representation theory of finite groups of Lie type, e.g. SL n (F p ). Analogously, affine Springer fibers are closely related to the representation theory of p-adic groups. 2.1 Affine Grassmannian We work over an algebraically closed field k = k. Let F = k((t)) and O F = k[[t]]. Then the affine Grassmannian Gr G satisfies Gr G (k) = G(F)/G(O F ). (See the lectures of Xinwen Zhu for a more thorough introduction to the affine Grassmannina.) For G = GL n, we can also interpret Gr G (k) = {O F lattices in F n }. For γ gl n (F) and Λ F n a lattice, we obtain a subgroup γλ F n (which is not necessarily a lattice, as γ could have been 0). Definition 2.1. The affine Springer fiber of γ is the ind-subscheme X γ Gr G classifying lattices that are stable under γ: X γ (k) = {Λ F n such that γλ Λ}. Example 2.2. Let G = GL n. What is the bijection Gr G (k) {O F lattices in F n }? For a coset gg(o F ), we may associate the lattice go n F where On F Fn is the standard lattice. We claim that this is surjective (i.e. all lattices arise in this way), and that all ambiguity comes from right multiplication by G(O n F ). The surjectivity is obvious. If γλ Λ, write Λ = go n F. Then the condition is reformulated as γgo n F gon F, or g 1 γg gl n (O F ). This motivates the more general definition: Definition 2.3. Let G be a reductive group over k. If γ g(f) = g k F, then we define the affine Springer fiber X γ of γ as the ind-subscheme that classifies cosets gg(o F ) such that Ad(g 1 )γ g(o F ) =: g k O F. This can be huge, e.g. for γ = 0 we get the whole affine Grassmannian. We re not really interested in studying those huge fibers. So from now on we restrict to the case γ g(f) rs= regular semisimple. For G = GL n, that means that the eigenvalues are distinct. In practice, X γ is usually non-reduced, but we want to work with its reduced subscheme. Therefore, from now on we rename X red γ = X γ. 10

2 AFFINE SPRINGER FIBERS ( ) x 0 Example 2.4. Let G = SL 2 and γ = with x k 0 x (we should assume that the characteristic is not 2 to have distinct eigenvalues). Then X γ is a discrete set of points parametrized by Z. Indeed, the lattices stable under γ of volume 1 are of the form t n O F t n O F. ( ) t 0 For γ = g the affine Springer fiber X 0 t γ is an infinite chain of P 1 : ( ) t 0 There is a Z-action on X γ corresponding to Λ 0 t 1 (since this matrix commutes with γ, it preserves ( the) property of being in the affine Springer fiber). 0 1 For γ =, the affine Springer fiber X t 0 γ is just a point (the standard lattice). ( ) 0 t For γ = t 2, the affine Springer fiber X 0 γ is a single P 1. Note that in the last two cases, we actually obtain schemes (rather than ind-schemes). The geometry of the affine Springer fiber appears to depend heavily on whether or not the matrix can be diagonalized. The third example has eigenvalues being square roots of t, and thus require a quadratic extension in order to diagonalize. The general principle is that the more diagonalizable the element γ, the more infinite the affine Springer fiber X γ. 2.2 Symmetries of X γ Let G γ be the centralizer of γ in G(F). Since we assumed that γ is regular semisimple, G γ is a torus over F (i.e. if we extend scalars to F, it becomes a product of G m s). ( ) ( ) t a Example 2.5. If γ =, then C t G (γ) is the diagonal torus a 1 with a F. 11

( If γ = t 2 AFFINE SPRINGER FIBERS ) 1, then over F = k((it 1/2 )) F the matrix γ becomes diagonalizable, so {( ) a b G γ (F) = bt a This is a one-dimensional torus, nonsplit over F. Definition 2.6. The co-character group of G γ is We always have a map } a, b F, a 2 b 2 t = 1 = ker((f ) Nm F ). X (G γ ) = Hom F alg (G m, G γ ). G m Z X (G γ ) G γ (1) which describes the inclusion of the maximal split subtorus in G γ. For any cocharacter µ: G m G γ, we can take F-points to obtain µ: F[t, t 1 ] G γ (F). Denote by t µ the image of t in G γ (F) under µ(f). Putting this together with (1) gives a map t X (G γ ) X (G γ ) F Z X (G γ ) G γ (F). Let L γ = t X (G γ ) G γ (F), which is a lattice. Since G γ (F) acts on X γ, we find that the lattice L γ does as well. Theorem 2.7 (Kazhdan-Lustzig). If γ g(f) rs, then L γ acts freely on X γ and L γ \X γ is proper. Example 2.8. Taking the quotient of the infinite chain of P 1 from Example 2.4 by the Z- action, since Z = X (G γ ) is the lattice corresponding to the diagonal torus, yields the nodal (projective) cubic: ( ) 0 1 If γ = then we have that X t 0 (G γ ) = 0, so L γ = {0}. However, the theorem is not vacuous because then it says that the affine springer fiber X γ is proper. 12

2 AFFINE SPRINGER FIBERS 2.3 Finiteness of affine Springer fibers. Theorem 2.9 (Kazhdan-Lusztig). Let γ g(f) rs and F = k((t)). Then L γ \X γ is proper over k. Let G γ G(F) be the centralizer of γ. Recall that we considered the group X (G gγ ) of all homomorphisms G m G γ (F), and a map X (G γ ) G γ (F) sending a cocharacter λ to λ(t). The image is a lattice L γ acting on X γ. Proof. We consider only the case γ t(f) rs. We choose ( a ) maximal torus T G over k, so a T(F) G(F). For example, in sl 2 such a choice is. We want to show that there exists a finite type scheme Y X γ such that X γ is covered by translations of Y under the action of L γ. Once we know this, we see that the quotient L γ \X γ admits a surjective map from Y. Then it s easy to show that L γ \X γ is actually proper. TONY: [ehhhh?] Recall the Iwasawa decomposition, which implies that G(F)/G(O) = N(F)t λ G(O)/G(O) λ X (T) where N is the unipotent radical (think of sl 2, where N = affine Springer fiber lying in the cell where λ = 0, a 1 Y := X γ (N(F)G(O)/G(O)). ( ) 1 ). Consider the part of the 1 Sinced we assumed that γ t(f) rs, we have G γ (F) = T(F) and hence L γ = {t λ λ X (T)}. Then the action of the lattice L γ is described as follows: t µ L γ takes N(F)t λ G(O)/G(O) N(F)t λ+µ G(O)/G(O). So it s clear that L γ Y will cover all of X γ. All we need to show now is that Y is of finite type. A point in Y is an element u N(F)/N(O) such that Ad(u 1 )γ g(o). Since Ad(u 1 ) is upper-triangular and γ is diagonal, this will necessarily lie in b(o). Also note that we must have γ t(o), or the affine Springer fiber will be empty. By the root decomposition, we have Ad(u 1 )γ = γ + τ α, τ α g α (F). α>0 (We know that we can sum over positive roots only because the result is in the Borel.) Let s try to digest the terms that appear here. Observe that since u N(F)/N(O), we may write u = α>0 x α (c α ) where x α : G a G corresponds to the root α and c α F. Then it is a fact that τ α = α, γ c α (where the pairing of α t and γ t is the natural one between t and t) plus something involving only those c β for roots β lower than α, i.e. such that α β is positive. This means that if we order our matrix compatibly with this 13

2 AFFINE SPRINGER FIBERS partial order, and we think about solving for c α inductively from the lower weights, then the diagonal terms are α, γ..... β, γ c β... τ β τ α... = α, γ c α........ Our constraint is that all of the entries of the matrix lie in O, which amounts to some inequalities saying that the valuation is non-negative. Now think about solving for the c α. Inductively, we can view c β as having bounded valuation for all β < α. When we solve for c α in terms of c β, the key point is that α, γ 0 for all roots α, precisely because γ is regular semisimple, so this inequality puts a non-trivial bound on c α. That forces Y to be of finite type over k. Example 2.10. For sl 2, we have ( ) 1 cα u = 1 ( ) a1 so if γ = (with regularity being equivalent to a 1 a 2 ), then a 2 2.4 Dimension computations ( ) ( ) Ad(u 1 a1 c )γ = α (a 1 a 2 ) a1 c = α γ, α. Now we want to compute the dimension of X γ. a 2 Split case. Suppose γ t(o F ), so that the fiber is non-empty. From the proof in the preceding section, one sees: dim X γ = val F α, γ. α>0 For each γ g(f) rs, consider ad(γ): g(f) g(f). This is not invertible, since the kernel is the Lie algebra g γ (F) of the centralizer of γ, which is a torus. So that means that it descends to an injection g(f)/g γ (F) g. Since we would like to have a square matrix, we consider the reduction ad γ: g(f)/g γ (F) g/g γ (F) which we know is an isomorphism. Define (γ) := det(ad γ) F. a 2 14

2 AFFINE SPRINGER FIBERS When γ t(f) n, we claim that val F α, γ = 1 2 val F (γ). α>0 Indeed, think of g(f)/g γ (F) as the sum of the all the root spaces (both positive and negative). On the root space g α, ad γ acts by α, γ by definition. So val F (γ) is equal to the sume of val F α, γ over all the negative and positive roots. This suggests that Theorem 2.11 (Bezrukavnikov). We have where c(γ) = rank G rank X (G γ ) } {{ } L γ dim X γ 1 2 val F (Y). dim X γ = 1 2 val F (γ) c(γ) Think ok c(γ) as an error term. If γ is diagonalizable, then X (G γ ) has rank equal to rank G and c(γ) = 0. If not then it s smaller, so c(γ) is larger and the affine Springer fiber is smaller. Proof. The idea is to reduce the dimension of this complicated variety to calculating the dimension of a certain commutative group. We consider Some properties of this subset: X reg γ, X reg γ X reg γ X reg γ = {gg(o) Ad(g 1 )γ mod t g(k) reg } X γ. is open (clear) and dim X reg γ X γ is dense), = dim X γ (in fact, Ngo Bau Chau later proved that G γ (F) acts transitively on X reg γ, and we have X reg γ = G γ (F)/ compact open subgroup. This quotient is a finite-dimensional commutative group over k, possibly with infinitely many components. 2.5 Affine Weyl Group Actions We can construct an analogue of the affine Springer fibers for affine flag varieties: Y γ Fl, the affine flag varietyof G. For G = GL n, Fl has an interpretation in terms of lattices, like the affine Grassmannian, but instead of a single lattice we consider a chain of lattices: Fl = {... Λ 0 Λ 1... lattices in F n dim k Λ i /Λ i 1 = 1, Λ i+n = t 1 Λ i for all i}. 15

2 AFFINE SPRINGER FIBERS The set of such chains should be naturally identified with Fl(k). Note that this admits a map to Gr(k) by sending the chain to Λ 0, which should be be the k-points of an algebraic map Fl Gr. If G = GL n, then we can define, in analogy to the affine Springer fiber, In general, we have Y γ = {Λ Fl γλ i Λ i }. Fl = LG/I = G(F)/I where I is an Iwahori subgroup of G(O), which is a subgroup reducing to the Borel modulo t. Iwahori subgroups are all conjugate. You can think of them as integral analogues of Borel subgroups. I G(O) B(k) G(k) If G is simply-connected, then we may think of Fl as the space of Iwahori subgroups in LG, completely analogous to how B is the space of Borel subgroups in G. Then we may define: mod t Y γ = {Iwahori subgroups I LG γ Lie(I)}. Definition 2.12. For general G and a fixed Iwahori I, we define Y γ = {gi Fl Ad(g 1 )γ Lie(I)}. The Y γ have similar properties to the X γ, e.g. they have lattice actions such that the quotients by these actions are proper. However, we want to move on to discuss Springer s realizations of affine Weyl group actions. Theorem 2.13 (Lustzig, Sage). Let γ g(f) rs. There is a natural action of the affine Weyl group W = X (T) W on H (Y γ ). Remark 2.14. Since things are horribly infinite, it s better to consider homology rather than cohomology. This action may be constructed along the lines of the arguments we gave, not involving perverse sheaves. However, it is not well understood how to actually compute them. 16

3 ORBITAL INTEGRALS 3 Orbital Integrals 3.1 Setup We now choose k = F q and F = k((t)). Let G/k be a reductive group. Then G(F) is a locally compact topological group, hence carries a (left) Haar measure dg, unique up to scalar. (Though it turns out that G(F) is unimodular, so the left and right Haar measures coincide.) Let S(G(F)) be the space of locally constant C-valued functions on G(F) with compact support. Example 3.1. The characteristic function 1 G(O) is locally constant and compactly supported, because G(O) is open and compact. For f S(G(F)), we may consider G(F) f (g) dg. Assume that f is right invariant under G(O), so f descends to a function f : G(F)/G(O) C. Normalize dg so that vol(g(o), dg) = 1. Then f (g) dg = 3.2 Orbital integrals G(F) g G(F)/G(O) f (g). For orbital integrals, we need a variant of the preceding discussion. For our starting point, we have a function ϕ S(g(F)) and γ g(f) rs. We want to make sense of the integral ϕ(ad(g 1 )γ). We haven t yet chosen a measure. If we naïvely use the Haar measure, then the integral ϕ(ad(g 1 γ) dg G(F) may not converge. The problem is that ϕ isn t a Schwartz function on G(F). Indeed, the the integrand is the composition which is not Schwartz. G(F) Ad( ) 1 γ g(f) ϕ C 17

3 ORBITAL INTEGRALS Example 3.2. A compactly supported function ϕ on F restricts to a compactly supported function function ϕ on F if and only if ϕ(0) = 0. However, note that the function is invariant under G γ, since it commutes with g. Therefore, the above actually factors through Now, this is a compactly supported function. Definition 3.3. We define the orbital integral O γ (ϕ) = G γ (F)\G(F) Ad( ) 1 γ g(f) ϕ C G γ (F)\G(F) ϕ(ad(g 1 )γ) dg d γ g where dg is a Haar measure on G and d γ g is a Haar measure on G γ (F). Relation to affine Springer fibers. Consider what this gives for the function ϕ = 1 g(o). Then the integrand of O γ (1 g(o) ) is 1 if Ad(g 1 )γ g(o) and 0 otherwise. In other words, it is precisely the characteristic function of the affine Springer fibers X γ, so O γ (1 g(o) ) measures the volume of X γ. Define the set X γ = X γ (k) = {gg(o) G(F)/G(O) Ad(g 1 )γ g(o)}. The g which contribute to the integral are precisely those coming from X γ. Now pick a lattice L 0 G γ (F) which is cocompact. This is the analogue of the group L γ that we introduced in the geometric setting. For example, we can take L 0 = L γ (k). (This is possibly a smaller lattice than L γ, since its k-structure can be twisted by Galois.) Roughly speaking, we have O γ (1 g(o) ) = #(L 0 \X γ ). Of course this formula can t quite be true, because the right hand side depends on the choice of L 0 while the left hand side doesn t, and the left hand side depends on a normalization of measure while the right hand side doesn t. The precise relation is as follows. Lemma 3.4. We have O γ (1 g(o) ) = vol(l 0 \G γ, d γ g) 1 #(L 0 \X γ ). Remark 3.5. We are still assuming that dg is normalized so that vol(g(o), dg) = 1. Warning: (L γ \X γ )(k) always admits a map from L γ (k)\x γ (k), but these are not equal in general. Example 3.6. We discuss an example which is on the problem sheet. Let k = F q, F = k((t)), and G = SL 2 /F. For a k \ (k ) 2, we consider the affine Springer fibers for the two elements ( ) ( ) 0 at γ =, γ 0 at 2 =. t 0 1 0 These are conjugate under GL 2 (F), but not SL 2 (F). However, they become conjugate after making a quadratic extension k = k( a), E = Fk. 18

3 ORBITAL INTEGRALS Let me give the geometric picture. The affine Springer fibers X γ and X γ are indschemes over k, such that if we pass to k, then each looks like a chain of P 1. However, the Galois actions look different. In each case Galois operates roughly as a reflection, exchanging pairs of components. However, the affine Springer fiber X γ has a special (i.e. stable) point, while X γ has a special component. In particular, on X γ the group Gal(k /k) fixes only the special point, so O γ = 1. On the other hand, on X γ the group Gal(k /k) fixes the central P 1. Therefore, it descends to some form of P 1 over k. But since there are no twisted forms of P 1 over a finite field, it must actually descend to P 1 k, so O γ = q + 1. Now what about the quotient by the lattice action? If L γ = X (G γ k k), then there should be a relation between O γ and L γ \X γ (k). In this example, Gal(k /k) Gal(E/F) σ acts on Z = L γ by 1. Concerning X γ, the quotient L γ \X γ is obtained by identifying two points on P 1. But while each of these points is not defined over k, they form Galois conjugates and hence their pair is identified over k. Therefore, the quotient gains an additional point: #(L γ \X γ )(k) = q + 2. 19

3 ORBITAL INTEGRALS It turns out that #L γ \X γ (k) has the same answer, but the picture is more complicated. Definition 3.7. Let γ g(f) rs and ϕ S(g(F)). 1. We say that γ and γ are stably conjugate, and denote γ st γ, if they are conjugate under G(F), and we define the stable conjugacy class This is a finite set. 2. We define the stable orbital integral StConj(γ) = {γ st γ}/g(f). SO γ (ϕ) = [γ ] StConj(γ) O γ (ϕ). Theorem 3.8 (Goresky-Kottwitz-MacPherson, Ngô). We have SO γ (1 g(o) ) = vol(k γ, d γ g) 1 #(L γ \X γ )(k) where K γ G γ (F) is the parahoric (in this case, unique maximal compact) subgroup. Example 3.9. In our example, G γ = ker(nm: E Nm F ) and K γ = G γ (O). 3.3 Geometric interpretation Theorem 3.10 (Grothendieck-Lefschetz Trace Formula). We have SO γ = vol(k γ, d γ g) 1 i ( 1) i Tr(Frob, H (L γ \X γ ) k) 20

3 ORBITAL INTEGRALS This is the starting point of the geometric interpretation of the Fundamental Lemma. Example 3.11. For G = SL 3 and γ = 1 1 t 4, we have X γ = {Λ F 3 γλ Λ, vol(λ) = vol(o 3 F )}. (the volume condition comes from the fact that we are considering SL 3 instead of GL 3 ). Since G γ (F) has no split torus, L 0 = 0. The conditions are that Λ is stable under O F and γ. Now we have the commutative subalgebra R := O F [γ] Mat 3 (F). In fact, R O F [y]/(y 3 t 4 ), so E := Frac(R) is a (possibly non-galois) cubic extension of F, isomorphic to k((t 1/3 )). Then we may write X γ = {R submodules Λ Frac(R) vol(λ) = vol(r)}. This is a familiar object from number theory: E is a local field (admittedly not necessariliy) and R is an order i E (but possibly not a Dedekind domain). So X γ consists of fractional ideals of R satisfying some volume condition. If R was actually a Dedekind domain, then this set would be trivial, since fractional ideals of a local Dedekind domain, i.e. discrete valuation ring, are just determined by the power of the uniformizer, which in turn determines the volume. If our R is not a Dedekind domain, then we get more. There is an extra symmetry: we have an action of k on R k[[s 3, s 4 ]] O E = k[[s]], where s = t 1/3 (this is the normalization of R). The action is simply by scaling s. The G m action on X γ induces a cell decomposition (this is a general phenomenon, going by the name of the Balynicki-Birula theorem, whenever we have G m acting on any proper variety), indexed by fixed points of G m. So what are the fixed points here? If Λ E is fixed under G m, since the eigenvalues of G m are the monomials in s, Λ must be (topologically) spanned by the monomials. This gives a bijection { Gm -fixed fractional ideals Λ for R } { non-empty subsets M Z }. stable under +3, +4, and bounded below The answer is that O γ = 1 + q + 2q 2 + q 3. This is clearly indicating a cell decomposition, with one 0-cell, one 1-cell, two 2-cells, and a 3-cell. In the preceding example everything was quite general up until the discussion of the extra symmetries. We could interpret the affine Springer fibers as a set of modules of some commutative order. Example 3.12. For G = GL n and γ g(f) rs, one can form a commutative O F -algebra R such that Frac(R) is an F-algebra of degree n, and interpret X γ as a set of fractional R-ideals. 21

3 ORBITAL INTEGRALS Geometrically, think of Spec R as a local curve C forming a degree n cover of the formal disc Spec O F. Then X γ Pic(C) (the compactification of the Picard group of C). This is because we are classifying a line bundle plus a trivialization over the punctured disk, or alternatively a line bundle plus a meromorphic section. When C is not smooth then we need to consider more: the compactification consists of torsion-free coherent sheaves of generic rank 1. Note that here, the number n doesn t play a significant role, and the description is independent of it. 22

4 THE HITCHIN FIBRATION 4 The Hitchin Fibration Because of time constraints, we ll talk only about G = GL n. Definition 4.1. Fix X an algebraic curve over k = k and L a line bundle on X. The Hitchin moduli stack is M = { } E rank n vector bundle on X (E, ϕ) ϕ: E E L. Example 4.2. If L = ω X, then M T Bun n. Construction of the Hitchin fibration. There is a map f : M A (the base is yet to be described) sending (E, ϕ) to the coefficient of the characteristic polynomial of ϕ. What does this mean? The first coefficient a 1 should be tr(ϕ, E), but since ϕ isn t an endomorphism, this isn t a function but a section a 1 H 0 (X, L). Similarly, we have a 2 = tr(ϕ,. = 2 ɛ) H 0 (X, L 2 ) a n = det ϕ H 0 (X, L n ). So the base is A = n i=1 H0 (X, L i ). This is the Hitchin fibration. M = {(E, ϕ)} A = n i=1 H0 (X, L i ). Fix a = (a 1,..., a n ) A. Then what is M a = f 1 (a)? The picture is similar to the discussion of 3.3. The fiber is called a spectral curve. We can form Y a = O X [y]/(y n a 1 y n 1 +... ± a n ). This is a flat, degree n cover of X Y a f p a n:1 X If the discriminant (a) 0 then Y a is a reduced curve. This Y a is an analogue of the local curve Spec R. Theorem 4.3 (Hitchin). There is a canonical identification M a Pic(Y a ). How does this work? That is, how do we relate a torsion-free coherent sheave of generic rank 1 on Y a to a vector bundle on X? The answer is that for F, p a is a rank n vector bundle on X. However, the fact that it comes from upstairs implies that p a F is equipped with extra structure. Think to the discussion of the example in 3.3: the pushforward will not just be an O F -module but an R-module, which is stable by γ.. 23

4 THE HITCHIN FIBRATION f Remark 4.4. For general G, we can define M G A G analogously. There is then a similar interpretation of M a using Pic, Prym, etc. Now let s try to relate M a and X γ. The whole point of the Hitchin fibration is that it helps us to understand the geometry of affine Springer fibers, and indeed leads to the solution of the Fundamental Lemma. Consider the spectral curve Y a = spectral curve Spec R x p a n:1 X Spec O x Then we have an action Pic(Y a ) on Pic(Y a ) by the tensor product. For each x X and a x g//g(o x ) there is a canonical lift of a x to g(o x ). That defines an affine Springer fiber X ax, and it admits an action of G ax. This action factors through some local analogue of the Picard group, P ax. (A large compact open subgroup of the torus acts trivially; the action factors through a finite-dimensional quotient.) Theorem 4.5 (Ngô). We have a homeomorphism of stacks [Pic(Y a )\Pic(Y a )] [P ax \X ax ]. x X This is the key point that makes the global fundamental lemma work. 24

5 CONCLUSION 5 Conclusion We conclude with a panorama of the objects that we have introduced. Springer fibers Affine Springer fibers Hitchin fibers (flag variety verson) (flag variety version) Setting k F = k((t)) X a curve over k Symmetry W (Weyl) W (affine Weyl) W Hecke graded affine graded double affine graded double affine Hecke algebra Hecke algebra Hecke algebra Applications: rep ns of G(F q ) orbital integrals on G(F) (finite Lie type) characters of G(F)-rep ns trace formula for G/k(X) 25