INTRODUCTION TO LOCAL GEOMETRIC LANGLANDS MASOUD KAMGARPOUR

1. Introduction Let F be a local field. Fix a separable closure F¯ of F . Let G be a connected split reductive group over F . The Langlands correspondence predicts a close relationship between N -dimensional ˇ Homomorphisms Gal(F¯ /F ) → G

⇐⇒

irreducible smooth representations of G(F )

According to Frenkel and Gaitsgory [FG06], [Fre07], local geometric Langlands is a correspondence between ˇ G-connections on the formal disk Spec(C((t)))

⇐⇒

Categorical representation of the loop group G((t))

Moreover, the categorical representations of G((t)) should arise in two ways: as D-modules on ˆcrit (which is a central quotients of G((t)) and as representations of the affine Kac-Moody algebra g extension of the loop algebra g((t))). The relationship between these two realization should be given by an infinite dimensional analogue of Beilinson-Bernstein localization. In this note, I will try to motivate Frenkel-Gaitsgory correspondence, starting from the classical Langlands correspondence for F = Fq ((t)). This text is based on a talk at Max Planck Institute for Mathematics during the School on Analysis on Lie groups - August-September 2011. It is meant as an introduction for non-specialist. Interested reader is urged to consult the references for precise statements.

2. Geometrizing Galois representations 2.1. From Galois group to fundamental group. What is the geometric analogue of the Galois group? According to Grothendieck (SGA 1), the answer is the ´etale fundamental group. More ¯ is a separable precisely, given a field Ω, one has a geometric object: the point Spec(Ω). If Ω closure, then by definition ¯ = Gal(Ω/Ω). ¯ π et (Spec(Ω), Spec(Ω)) 1

Now suppose Ω = k((t)), where k is an arbitrary field. The algebra k((t)) should be thought of as the algebra of functions on the formal punctured disk D× , meromorphic at center. Similarly, k[[t]] is the algebra of of functions on the punctured disk D. As the first step of in our “geometrization quest”, we have the tautological equivalence Representations of Gal(Fq ((t))/Fq ((t))) 1

'

−→

Representations of π1et (DF×q )

2.2. From representations to local system. Suppose X is a connected topological space. A local system on X is a sheaf (say of vector spaces over some fixed field) which is locally isomorphic to the constant sheaf and has finite dimensional stalks. Let us fix a base point x ∈ X. Then we have an equivalence of categories '

Representations of π1top (X, x)

−→

Local systems on X

The above equivalence holds in algebraic geometry as well; that is, if we work with the ´etale fundamental group and `-adic local systems. In particular, taking X = Dk× , where k is an arbitrary field, we have an equivalence '

Representations of π1et (Dk× )

−→

Local systems on Dk×

2.3. From positive characteristic to characteristic zero. 2.3.1. Too few Galois representations. Our ultimate goal is to formulate a geometric version of Langlands conjectures over C. We have seen that Galois representations of Fq ((t)) are the same thing as local systems on DF×q . What should we take as analogous objects over C? The naive guess is to take exactly the same object also in characteristic zero; that is, to consider representations of π1et (DC× ). The absolute Galois group C((t)) is, however, rather simple. One can 1 show that the finite Galois extensions of C((t)) are precisely the fields C((t n )) where n ranges over b In particular, every representation of this Galois group the integers. So the Galois group equals Z. × (equivalently, every local system on DC ) factors through a finite quotient Z/nZ. Such a theory is not rich enough to provide meaningful generalization of Langlands program. 2.3.2. From local systems to connections. Let X be a scheme over Z (soon we will take X to be the disk). What should the characteristic zero analogue of local systems on XFq be? According to a theorem of Deligne [Del70], we have an equivalence of categories (2.1) ' Local systems on XC Flat connections on XC with regular singularity at boundary −→ Here, the regularity condition means, roughly, that the poles of the meromorphic connection at the boundary is at worst of order 1. (This condition is vacuous if XC is compact). We saw previously that the category of local systems on XC is not big enough. So it is natural to consider the category of all flat meromorphic connections; i.e., to drop the regularity condition. In fact, we have the following analogy (2.2)

Local Systems on XFq

⇐⇒

Flat connections on XC meromorphic at the boundary

For more information on this analogy, see [Kat87]. 2.3.3. Back to the formal disk. As a special case of the analogy (2.2), we take X = D× and propose the analogy (2.3)

Representations of Gal(F¯ /F )

⇐⇒

Connections on DC× meromorphic at the center

Let us give an explicit description of the RHS. On the formal disk every connection is flat and every vector bundle is trivial. Choosing a trivialization of the vector bundle we can write the connection as ∇ = ∂t + A(t), A(t) ∈ glN (C((t))); Note that we can also write A(t) as An tn + An+1 tn+1 + · · · , where n ∈ Z and Ai ∈ glN (C). (If ˇ then instead of representations, we want to consider analogues of homomorphisms Gal(F¯ /F ) → G, 2

ˇ). Changing trivialization by an element g ∈ G(C((t))) corresponds we take Ai ’s to be elements of g to a gauge transformation ∇ = ∂t + A(t) 7→ ∂t + gA(t)g −1 + (∂t g)g −1

(2.4)

A connection ∇ has regular singularity if, after applying a gauge transformation if necessary, it can be put in the form ∂t + A−1 t−1 + A0 t0 + · · · . For more details, see [Fre07]. 3. Local geometric class field theory Langlands correspondence is a generalization of Class Field Theory (CFT). Therefore, before geometrizing the Langland correspondence, it is sensible to try to geometrize CFT. Recall that CFT provides a close relationship between the following abelian groups Characters of Gal(F¯ /F )

⇐⇒

Characters of F ×

We already saw that the geometric analogue of the LHS is rank one connections on DC× . What is the geometric analogue of a character of F × ? Note that characters are special kinds of functions, so we will first discuss what the geometric analogues of functions are. 3.1. From functions to constructible sheaves. 3.1.1. Trace of Frobenius. As we explain above, in the ´etale topology, the category of sheaves on a point Spec(Ω) is equivalent to the category of representation of the fundamental group ¯ = Gal(Ω/Ω) ¯ π1et (Spec(Ω), Spec(Ω)) (since all sheaves are locally constant). Now suppose our field is a finite field Ω = Fq . Then there is a very special element in the Galois group called the geometric Frobenius: Fr ∈ Gal(Fq /Fq ), Fr(x) = xq . This is a topological generator for the Galois group of Fq . Starting with a sheaf on Spec(Fq ) (which is just a representation of the above Galois group), we can take the trace of Frobenius and end up with a number. The catch is that this is an `-adic number. This is because the sheaves we consider in ´etale topology are sheaves of Q` -vector spaces. It will take me too far to explain why that is the case. However, the good news is that we can once and for all fix an isomorphism Q` ∼ = C, and think of the trace of Frobenius as being a complex number.1 3.1.2. Trace of Frobenius function. More generally, suppose X is a variety over Fq and suppose x is a point in X(Fq ). In other words, x is a morphism Spec(Fq ) → X. Given a sheaf F on X, the pullback x∗ F is a sheaf on Spec(Fq ). Let us denote the trace of Frobenius function by tF (x). Then we have defined a function tF on X(Fq ). The passage from sheaves to functions satisfies many natural properties; see [Lau87]. A natural question is which functions can be realized as trace of Frobenius of some sheaves which are constructed naturally? We don’t know the answer to this question; however, it seems that Most interesting functions on X(Fq ) come naturally from (perverse) sheaves on X Automorphic functions (and representations) provide many interesting functions and one expects that they also come from sheaves. This observation is at the heart of the geometric Langlands program. One may wonder why it is better to have sheaves than functions. The category of sheaves has functors that are not visible at the level of functions: for example, Verdier Duality, nearby cycles, Goresky-Machpherson IC exensions, etc. 1In fact, one usually takes trace with respect to the inverse of the geometric Frobenius. This inverse is known as the

arithmetic Frobenius. 3

3.1.3. From characteristic p to characteristic zero: from sheaves to D-modules. Let X be a scheme over Z. What should the characteristic zero analogue of the category of sheaves on XFq be? Again, looking at the category of sheaves on XC is not sufficient, but provides a clue. Kashiwara has proved the following generalization of Deligne’s Theorem (2.1): Constructible sheaves on XC

'

−→

Holonomic regular singularity D-modules on XC

This is known as the Riemann-Hibert correspondence and is only true at the level of derived categories (or, one has to work with perverse sheaves); see, for instance, [HTT08]. Motivated by the above considerations, we propose the following analogy (3.1)

Constructible sheaves on XFq

⇐⇒

D-modules on XC

The upshot of the discussions of this subsection is that we have the following analogy (3.2)

Functions on X(Fq )

⇐⇒

D-modules on XC

3.2. Geometric nature of F × . We are interested in geometrizing functions on F × , where F = Fq ((t)). According to the previous section, if we can realize F × as the set of Fq -points of a variety X, then we can consider the category of D-modules on X as the natural geometric analogue. Claim 1. Let Y be an affine variety over a field k. Define functors Yn , Y [[t]], and Y ((t)) by Yn (R) = Y (R[t]/tn ),

Y [[t]](R) = Y (R[[t]]),

Y ((t))(R) = Y (R((t))),

where R is a test k-algebra. Then Yn (resp. Y [[t]], resp. Y ((t))) is representable by a variety (resp. a provariety, resp. an ind-scheme) over k. For instance, suppose Y = A1 . Then it is easy to see that Yn = An , Y [[t]] = A∞ . For a proof of the above claim, see [Gai]. 3.3. Multiplicative local systems and connections. Recall that F = Fq ((t)). We know that F × is the set of Fq -points of the ind-scheme Gm ((t)). (The latter is known as the loop group of Gm ). The geometric analogue of functions on Gm (Fq ((t))) is the category of D-modules on Gm ((t)). Actually, in this case, it is enough to look at integrable connections. Characters are functions f on Gm (Fq ((t))) f (xy) = f (x)f (y). The geometric analogue of characters are, therefore, connections L on the ind-scheme Gm ((t)) over C, satisfying Lxy ∼ = Lx ⊗ Ly where Lx is the stalk of L at x (ditto for y and xy). Such connections are known as multiplicative connections. Changing from characteristic p to characteristic zero, we see that the complex algebraic geometric analogue of characters of Gal(F¯ /F ) are multiplicative connections on Gm ((t)). Therefore, we can guess that the geometric analogue of local class field theory should be an equivalence of categories multiplicative connections on Gm ((t)) In fact, this is a theorem; see [BBDE05]. 4

'

−→

rank 1 connections on DC×

3.3.1. Conclusion. We apply several metamorphosis to the isomorphism of local class field theory, to arrive at a guess for what geometric version of local geometric class field theory should look like. While some of the steps are meta-mathematical (eg. moving from characteristic p to characteristic 0) the final outcome is a precise theorem. This is prototype of what happens in geometrization: one starts with a classical statement and applies several metamorphosis, similar to the ones outlined above, and arrives at a geometric statement, which one can then try to prove.

4. Geometrizing representations of G Let G be a simple group over Z. A natural representation of G(Fq ) is realized on the space of functions G(Fq )/B(Fq ). What should the geometric analogue of this representation be? According to our discussion, the geometric analogue of the vector space of functions on G(Fq )/B(Fq ) is the category of D-modules on (G/B)C . The group GC acts naturally on this category via its action on (G/B)C . Thus, a meaningful geometric analogue of the notion of a representation of G(Fq ) is a category equipped with the action of GC . Moreover, there is another realization of the representation of GC on (G/B)C which is important for us. Namely, according to a theorem of Beilinson and Bernstein (cf. [HTT08]), taking global sections defines an equivalence of categories D-modules on (G/B)C

'

−→ Representations of g with the augmentation central charcter

4.1. Geometrizing representations of GC . Instead of representations of G(Fq ), in this subsection, we work with rational finite dimensional representations of the algebraic group GC = G ⊗ C. This will lead us to a twisted version of Beilinson-Bernstein isomorphism. Let λ be a weight of T (i.e., a character of T = B/[B, B]). Then λ defines a G-equivariant line bundle Lλ . A natural representation Vλ of G occurs on the cohomology of G with coefficients in Lλ . In fact, Borel-Weil-Bott theorem states that if λ is anti-dominant and regular, then this cohomology is concentrated in degree zero and the corresponding representation of G on the space of section of Lλ is an irreducible representation with lowest weight λ. What is the natural geometric analogue of this representation? Using Lλ , we can also define the ring Dλ of λ-twisted differential operators on G. Roughly speaking, Dλ is the ring of differential operators on the space of sections of Lλ . The category of Dλ -modules on G/B is then the natural geometric analogue of Vλ , at least for λ regular and anti-dominant. Beilinson-Bernstein theorem states that when λ is regular and anti-dominant, we have an equivalence of categories between Dλ -modules on (G/B)C

'

−→

Representations of g with central character χλ

where χλ is the central character of the Verma module indgb λ = U (g) ⊗U (b) λ. '

Remark 2. One can describe χλ explicity. We have the Harish-Chanda isomorphism γ : z −→ ∗ (Fun(t∗ ))W ρ where the latter is the algebra of W -invariant polynomials on t shifted by ρ (the latter is the sum of all positive simple roots, or equivalently, 12 times the sum of all positive roots). Thus, every weight λ ∈ t∗ defines a point of the center, hence a central character χλ . Remark 3. Instead of a character of the complex torus, one can probably start with a character λ : T (Fq ) → C× . Such a character then defines a one-dimensional character sheaf Lλ on TFq . So one can consider the representation of G(Fq ) on H∗ (GFq , Lλ ). I don’t know a reference which pursue this. 5

4.2. Conclusion. For every integral weight λ ∈ t∗ (that is, those characters which come from a character of the group T → Gm ), we have an irreducible representation Vλ of GC of lowest weight λ. For anti-dominant regular weights, we can geometrize this representations in two ways: as category of Dλ -modules on G/B, and as the category of representations of g with central character χλ . The equivalence between these two geometrizations is given by Beilinson-Bernstein localization. 5. Towards geometrizing representations of G(Fq ((t))) 5.1. A format for the geometric Langlands correspondence. As mentioned above, the geometric analogue of a representation of G(Fq ((t)))) is a category on which G((t)) acts. (We don’t make precise here the notion of action of a group scheme on a category). So the Langlands correspondence should take the form ˇ G-connections on D× ⇐⇒ Categories equipped with action of G((t)) Moreover, one hopes that the category Cσ , corresponding to a connection σ, can be constructed in two ways: via representations of Lie algebras and via D-modules on quotients of the loop group G((t)). Remark 4. (i) Note that the above format for local geometric Langlands is more of a suggestion than a real conjecture. Frenkel and Gaitsgory present a more precise version in [FG06] which is still short of a precise mathematical conjecture. Nevertheless, as we shall see, one can derive interesting precise mathematical conjectures from the above statement. (ii) Recall that the usual Langlands conjectures is characterized by matching of the  and Lfactors. At the moment, there is no local characterization of the correspondence σ 7→ Cσ . For instance, it appears that we have no notion of -factors for categorical representations of G((t)). (However, we do have such notion for connections; see [BBE02]). 5.2. Representations of the affine Kac-moody algebra at the critical level. Let g be a ˆκ denote the corresponding central simple Lie algebra. Given an invariant inner product κ on g, let g extension of the formal loop algebra g ⊗ C((t)), called the affine Kac-Moody algebra, ˆκ → g ⊗ C((t)) → 0, 0 → C1 → g with the two-cocycle defined by the formula x ⊗ f (t), y ⊗ g(t) 7→ −κ(x, y).Rest=0 f dg. We will only be concerned with the so called “critical case”: i.e., when κ is − 21 times the Killing form. ˆcrit which are discrete; that is, any vector is Let Rep(ˆ gcrit ) denote the category modules over g annihilated by the subalgebra g ⊗ tN C((t)) for sufficiently large N ≥ 0. Feigin and Frenkel ([Fre07]) have proved the following Spec(Center of Rep(ˆ gcrit ))

'

−→

ˇ-opers on D× g

forgetful

−→

ConnGˇ (D× )

ˇ ˇ-oper is a G-connection Ag equipped with extra data. The extra data rigidify the objects. In other words, opers have no nontrivial automorphisms, where as connections may (thus, ConnGˇ (D× ) should be understood as a stack). The above theorem means that the central character of a given ˆcrit , should be understood as a g ˇ-oper on D× . The upshot is that to V one discrete module V on g × ˇ can canonically associated a G-connection on D . Remark 5. The center of the category Rep(ˇ gκ ) is trivial for κ 6= κcrit . This is one reason we only consider the critical level. 6

5.2.1. Implication for the Langlands program. Suppose σ is a connection on DC× . Let χ be an oper whose underlying connection is σ (According to [FZ10] such an oper exists). Let Repχ (ˆ gcrit ) denote the full subcategory of representations with central character χ. Note that this category is naturally equipped with an action of G((t)). Frenkel and Gaitsgory propose to take Cσ = Repχ (ˆ gcrit ) '

The expectation is, then, if χ and χ0 have the same underlying connection, then Repχ (ˆ gcrit ) −→ ˆcrit . Repχ0 (ˆ gcrit ). This, by itself, is a nontrivial conjecture about representations of g Next, we want to discuss how to realize Cσ via D-modules on the loop group. Here, we know even less! We will consider two case: the unramified representation and the ramified principal series representations. ˇ 5.3. Unramified representations. Let us assume that the G-connection σ0 is unramified. This ˇ According to Frenkelmeans that σ0 can be trivialized (the trivializations are parameterized by G). g) where χ is an Gaitsgory proposal, the corresponding category Cσ0 should be equivalent Repχ (ˆ oper whose underlying local system is trivial. On the other hand, we also expect that Cσ0 should in some way model an unramified representations of G(F ); that is, those irreducible representations of G(F ) which have a G(O) fixed vector, where O = Fq [[t]]. One can show that all unramified representations are principal series; that is, they can be induced from a character of B(F ) (the restriction of character to B(O) is trivial). However, this description is not very convenient for doing geometry, essentially because the quotient G((t))/B((t)) is infinite dimensional. This fact is the source of many problem in geometric representation theory. In this case, the Satake isomorphism allows for an alternative realization of unramified representations which is more convenient for doing geometry. 5.3.1. Satake Isomorphism. Let H = H (G(F ), G(O)) denote the space of G(O)-biinvariant functions on G(F ). Satake Isomorphism states that we have a canonical isomorphism (5.1)

' ˇ H = C[Tˇ/W ] −→ K0 (Rep(G)).

ˇ we denote by FV the corresponding element of H . For an element [V ] ∈ K0 (Rep(G)), Now H acts on the space of G(O)-vectors of an unramified irreducible representation. It is known that (isomorphism class of) the unramified representation is completely determined by the associated character of H . In other words, we have bijections unramified rep. ⇐⇒ Characters of H

⇐⇒ point of Tˇ/W

ˇ ⇐⇒ semisimple conj. classes in G

For a conjugacy class γ, we let πγ denote the corresponding unramified representation. 5.3.2. Realization of πγ in Fun(G(F )/G(O)). Define a homomorphism πγ → Fun(G(F )/G(O)) by u ∈ πγ 7→ fu ,

fu (x) :=< u, gvγ >

where < ., . > is an invariant bilinear form on πγ , and vγ is the spherical vector of πγ . The functions fu are right G(O) invariant and satisfy the condition (5.2)

f ? FV = Tr(γ, V )f.

(Here, following [FG06], we omit some powers of q.) Let Fun(G(F )/G(O))γ denote the space of right G(O)-invariant locally constant functions on G(F ) satisfying the above equation. Then one can show that for generic γ the map πγ 7→ Fun(G(F )/G(O))γ is an isomorphism. 7

5.3.3. Geometric Satake equivalence. The geometric analogue of Fun(G(F )/G(O)) is the category of D-modules on the affine Grassmaian Gr = G((t))/G[[t]]. It is known that this is a union of projective schemes. Therefore, we can do geometry on it. The geomeric analogue of left G(O) invariant function on G(F )/G(O) is the category of G[[t]]-equivariant D-modules on Gr. The geometric Satake Isomorphism [Gin99], [MV07] states that we have a canonical equivalence of categories (5.3)

DG[[t]] (Gr)

'

ˇ Rep(G)

−→

This is the geometric analogue of (5.1). Moreover, according to a theorem of Gaitsgory [Gai01], ˇ = DG[[t]] (Gr) on D(Gr). This is the geometric convolution of D-modules defines an action of Rep(G) analogue of the fact that H acts on Fun(G(F )/G(O)). ˇ on 5.3.4. Geometrization of unramified representations. Notice that we have two actions of Rep(G) D(Gr): one is given by taking tensor product with the vector space underlying the representation, the other is by convolution (via the geometric Satake). Let DG[[t]] (Gr)Hecke on which these two actions are identified. More precisely, the objects of this category should be the data (F, {αV }), where F ∈ D(Gr), and αV are isomorphisms '

ˇ V ∈ Rep(G).

αV : F ? FV −→ V ⊗ F,

According to Frenkel-Gatisgory, the above isomorphisms are the natural geometric analogue of (5.2). (We cannot motivate this observation here, since it involves global considerations and previous Beilinson-Drinfeld work). Therefore, DG[[t]] (Gr)Hecke is a geometric analogue of an unramified representation. 5.3.5. Conclusion. We now have two categories which are suppose to be associated to σ0 via the local geometric Langlands correspondence. The conjecture is, of course, that they are equivalent by infinite-dimensional version of Beilinson-Bernstein localization theorem. Actually, to be more precise, we should not consider plain D-modules on Gr but rather, critically twisted D-modules. After this modification, the desired Beilinson-Bernstein localization should look like (5.4)

Hecke Dcrit G[[t]] (Gr)

⇐⇒

Repχ (ˆ gcrit )

where χ is an oper whose underlying connection is trivial; see [FG09] for the progress on this conjecture. 5.4. Ramified principal series representations. 5.4.1. Families of principal series representations. Unramified representations are obtained by parabolic induction from characters of B(F ) whose restriction to B(O) is trivial. Let us fix a maximal split torus T ⊂ B. Then, we expect that for every character µ ¯ : T (O) → C× , there should be a family of principal series representation and a “Satake Isomorphism”.2 We refer the reader to [KS11, §1.4] for the precise conjectures in this regard. Henceforth, we assume that µ ¯ is regular; that is, its stabilizer in W is trivial. In this case, the corresponding family has been defined by Howe, Bushnell-Kutzko, and Roche. Namely, we prove that there exists a compact open subgroup J such that µ ¯ extends to a character µ : J → C× . The space W of (J, µ)-invariant functions on G(F ) is isomorphic (as a G(F )-module) to the family of G(F ) principal series representations indB(O) µ ¯. Moreover, we have a Satake-type isomorphism (5.5)

H = H (G(F ), J, µ)

' −→ K0 (Rep(Tˇ))

2This is true only if G = GL . For more general G, we have to restrict ourself to “parabolic characters” of T (O); N

that is, those characters whose stabilizer in W is the Weyl group of a parabolic in G. 8

5.4.2. Geometrization of principal series. The analogue of a regular character µ ¯ is a regular multiplicative connection M on T [[t]]. One can show that J is the group of points of a proalgebraic group J ⊂ G[[t]]. Moreover, the connection M extends to a multiplicative connection M on J. Thus, we can define Wgeom (resp. Hgeom ) as the cateogy of (J, M)-equivariant (resp. bi-equivariant) Dmodules on G((t)). These are the analogues of D(Gr) (resp. DG[[t]] (Gr)) in the geometric setting. In particular, the geometric analogues of principal series representation appear in Wgeom (similar to the fact unramified representations appear in Fun(G(F )/G(O))). In analogy with geometric Satake, we have proved [KS11] an equivalence of monoidal categories Hgeom

'

−→

Rep(Tˇ)

Moreover, we have shown that Hgeom acts on Wgeom via convolution (this is the analogue of Gaitsgory’s theorem quoted above). One can define an analogue of DG[[t]] (Gr)Hecke in this situation as well. 5.4.3. Implication for Langlands conjecture. Let us fix a maximal split torus T ⊆ G. Now suppose ˇ which, up to equivalence, factor we are interested in those Galois representations Gal(F¯ /F ) → G ¯ ˇ through a representation Gal(F /F ) → T . What does the Langlands conjecture predict about this Galois representation? Then starting from a homomorphism Gal(F¯ /F ) → Tˇ, class field theory defines a character T (F ) → C× . By parabolic induction, one obtains a principal series representation of G(F ). In generic enough situation, we expect that this representation is irreducible, and that it should correspond to the original representation of the Galois group, under the local Langlands correspondence. ˇ What is the geometric analogue of the above consideration? Let σ be a G-connection on D× ˇ Let us assume, further, that the stabilizer M ˇ equipped with a reduction to a Tˇ-connection M. under the action of the Weyl group W is trivial (this is the geometric analogue of “generic”). By Hecke . ˇ defines a multiplicative connection M on T [[t]]. Let us denote this by Hgeom class field theory M In analogy with Conjecture 5.4, we expect to have a Beilinson-Bernstein type equivalence crit,Hecke Hgeom

⇐⇒

CM ˇ

References [BBDE05] A. Beilinson, S. Bloch, P. Deligne, and H. Esnault. Periods for irregular connections on curves. http://www.uni-due.de/ mat903/preprints/helene/69-preprint-per051206.pdf, 2005. [BBE02] Alexander Beilinson, Spencer Bloch, and H´el`ene Esnault. -factors for Gauss-Manin determinants. Mosc. Math. J., 2(3):477–532, 2002. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. ´ [Del70] Pierre Deligne. Equations diff´erentielles a ` points singuliers r´eguliers. Lecture Notes in Mathematics, Vol. 163. Springer-Verlag, Berlin, 1970. [FG06] Edward Frenkel and Dennis Gaitsgory. Local geometric Langlands correspondence and affine Kac-Moody algebras. In Algebraic geometry and number theory, volume 253 of Progr. Math., pages 69–260. Birkh¨ auser Boston, Boston, MA, 2006. [FG09] Edward Frenkel and Dennis Gaitsgory. Localization of g-modules on the affine Grassmannian. Ann. of Math. (2), 170(3):1339–1381, 2009. [Fre07] Edward Frenkel. Langlands correspondence for loop groups, volume 103 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2007. [FZ10] Edward Frenkel and Xinwen Zhu. Any flat bundle on a punctured disc has an oper structure. Math. Res. Lett., 17(1):27–37, 2010. [Gai] D. Gaitsgory. Seminar notes: affine grassmanian and loop groups. http://www.math.harvard.edu/ gaitsgde/grad2009/SeminarNotes/Oct13(AffGr).pdf. [Gai01] D. Gaitsgory. Construction of central elements in the affine Hecke algebra via nearby cycles. Invent. Math., 144(2):253–280, 2001. [Gin99] V. Ginzburg. Perverse sheaves on a loop group and langlands’ duality. arXiv:alge-geom/9511007v4, 1999. 9

[HTT08]

Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki. D-modules, perverse sheaves, and representation theory, volume 236 of Progress in Mathematics. Birkh¨ auser Boston Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. [Kat87] Nicholas M. Katz. On the calculation of some differential Galois groups. Invent. Math., 87(1):13–61, 1987. [KS11] M. Kamgarpour and T. Schedler. Geometrization of principal series representations of reductive groups. http://arxiv.org/abs/1011.4529, 2011. [Lau87] G. Laumon. Transformation de Fourier, constantes d’´equations fonctionnelles et conjecture de Weil. Inst. ´ Hautes Etudes Sci. Publ. Math., (65):131–210, 1987. [MV07] I. Mirkovi´c and K. Vilonen. Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. of Math. (2), 166(1):95–143, 2007. E-mail address: [email protected] Max Planck Institute for Mathematics

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