A Rigid Irregular Connection on the Projective Line

ANNALS OF MATHEMATICS

A rigid irregular connection on the projective line

By Edward Frenkel and Benedict Gross

SECOND SERIES, VOL. 170, NO. 3 November, 2009

anmaah

Annals of Mathematics, 170 (2009), 1469–1512

A rigid irregular connection on the projective line

By EDWARD FRENKEL and BENEDICT GROSS

To Victor Kac and Nick Katz on their 65th birthdays

Abstract

In this paper we construct a connection on the trivial G-bundle on ސ1 for any r simple complex algebraic group G, which is regular outside of the points 0 and , 1 has a regular singularity at the point 0, with principal unipotent monodromy, and has an irregular singularity at the point , with slope 1=h, the reciprocal of the 1 Coxeter number of G. The connection , which admits the structure of an oper in r the sense of Beilinson and Drinfeld, appears to be the characteristic 0 counterpart of a hypothetical family of `-adic representations, which should parametrize a specific automorphic representation under the global Langlands correspondence. These `-adic representations, and their characteristic 0 counterparts, have been constructed in some cases by Deligne and Katz. Our connection is constructed uniformly for any simple algebraic group, and characterized using the formalism of opers. It provides an example of the geometric Langlands correspondence with wild ramification. We compute the de Rham cohomology of our connection with values in a representation V of G, and describe the differential Galois group of as a subgroup of G. r 1. Introduction The Langlands correspondence relates automorphic representations of a split reductive group G over the ring of adeles` of a global field F and `-adic representations of the Galois group of F with values in a (slightly modified) dual group of G (see 2). On the other hand, the trace formula gives us an effective tool to find the multiplicities of automorphic representations satisfying certain local conditions. In some cases one finds that there is a unique irreducible automorphic

The ﬁrst named author is partially supported by DARPA through the grant HR0011-09-1-0015. The second named author is supported by NSF, through the grant DMS 0901102.

1469 1470 EDWARD FRENKEL and BENEDICT GROSS representation with prescribed local behavior at finitely many places. A special case of this, analyzed in [Gro], occurs when F is the function field of the projective line ސ1 over a finite field k, and G is a simple group over k. We specify that the local factor at one rational point of ސ1 is the Steinberg representation, the local factor at another rational point is a simple supercuspidal representation constructed in [GR], and that the local representations are unramified at all other places of F . In this case the trace formula shows that there is an essentially unique automorphic representation with these properties. Hence the corresponding family of `-adic representations of the Galois group of F to the dual group G should also be unique. { An interesting open problem is to find it. Due to the compatibility of the local and global Langlands conjectures, these `- adic representations should be unramified at all points of ސ1 except for two rational points 0 and . At 0 it should be tamely ramified, and the tame inertia group 1 should map to a subgroup of G topologically generated by a principal unipotent { element. At it should be wildly ramified, but in the mildest possible way. 1 Given a representation V of the dual group G, we would obtain an `-adic { sheaf on ސ1 (of rank dim V ) satisfying the same properties. The desired lisse `-adic sheaves on އm have been constructed by P. Deligne [Del77] and N. Katz [Kat90] in the cases when G is SL ; Sp ; SO or G and V is the irreducible { n 2n 2n 1 2 representation of dimension n; 2n; 2n 1, and 7C, respectively. However, there are C no candidates for these `-adic representations known for other groups G. { In order to gain a better understanding of the general case, we consider an analogous problem in the framework of the geometric Langlands correspondence. Here we switch from the function field F of a curve defined over a finite field to an algebraic curve X over the complex field. In the geometric correspondence (see, e.g., [Fre07b]) the role of an `-adic representation of the Galois group of F is played by a flat G-bundle on X (that is, a pair consisting of a principal G-bundle { { on X and a connection , which is automatically flat since dim X 1). Hence r D we look for a flat G-bundle on ސ1 having regular singularity at a point 0 ސ1 { 2 with regular unipotent monodromy and an irregular singularity at another point ސ1 with the smallest possible slope 1=h, where h is the Coxeter number of 1 2 G (see [Del70] and Section 5 for the definition of slope). By analogy with the { characteristic p case discussed above, we expect that a flat bundle satisfying these 1 properties is unique (up to the action of the group އm of automorphisms of ސ preserving the points 0; ). 1 In this paper we construct this flat G-bundle for any simple algebraic group G. { { A key point of our construction is that this flat bundle is equipped with an oper structure. The notion of oper was introduced by A. Beilinson and V. Drinfeld [BD05] (following the earlier work [DS85]), and it plays an important role in the geometric Langlands correspondence. An oper is a flat bundle with an additional ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1471 structure; namely, a reduction of the principal G-bundle to a Borel subgroup B { { which is in some sense transverse to the connection . In our case, the principal r G-bundle on ސ1 is actually trivial, and the oper B-reduction is trivial as well. If N { { is a principal nilpotent element in the Lie algebra of a Borel subgroup opposite to B and E is a basis vector of the highest root space for B on g Lie.G/, then our { { L D { connection takes the form dt (1) d N Edt; r D C t C where t is a parameter on ސ1 with a simple zero at 0 and a simple pole at . 1 We also give in Section 5.1 a twisted analogue of this formula, associated to an automorphism of G of finite order preserving B (answering a question raised by { { P. Deligne). For any representation V of G our connection gives rise to a flat connection { on the trivial vector bundle of rank dim V on ސ1. We examine this connection more closely in the special cases analyzed by Katz in [Kat90]. In these cases Katz constructs not only the `-adic sheaves, but also their counterparts in characteristic 0, so we can compare with his results. These special cases share the remarkable property that a regular unipotent element of G has a single Jordan block in the { representation V . For this reason our oper connection can be converted into a scalar differential operator of order equal to dim V (this differential operator has the same differential Galois group as the original connection). We compute this operator in all of the above cases and find perfect agreement with the differential operators constructed by Katz [Kat90]. This strongly suggests that our connections are indeed the characteristic 0 analogues of the special `-adic representations whose existence is predicted by the Langlands correspondence and the trace formula. Another piece of evidence is the vanishing of the de Rham cohomology of the 1 intermediate extension to ސ of the Ᏸ-module on އm defined by our connection with values in the adjoint representation of G. This matches the expectation that the { corresponding cohomology of the `-adic representations also vanish, or equivalently, that the their global L-function with respect to the adjoint representation of G is { equal to 1. We give two proofs of the vanishing of this de Rham cohomology. The first uses nontrivial results about the principal Heisenberg subalgebras of the affine Kac-Moody algebras due to V. Kac [Kac78], [Kac90]. The second uses an explicit description of the differential Galois group of our connection and its inertia subgroups [Kat87]. Since the first de Rham cohomology is the space of infinitesimal deformations of our local system (preserving its formal types at the singular points 0 and ) 1 ([Kat96], [BE04], [Ari08]), its vanishing means that our local system on ސ1 is rigid. We also prove the vanishing of the de Rham cohomology for small representations 1472 EDWARD FRENKEL and BENEDICT GROSS considered in [Kat90]. This is again in agreement with the vanishing of the cohomology of the corresponding `-adic representations shown by Katz. Using our description of the differential Galois group of our connection and a formula of Deligne [Del70] for the Euler characteristic, we give a formula for the dimensions of the de Rham cohomology groups for an arbitrary representation V of G. { Finally, we describe some connections which are closely related to , and r others which are analogous to coming from subregular nilpotent elements. We r also use to give an example of the geometric Langlands correspondence with r wild ramification. The paper is organized as follows. In Section 2 we introduce the concepts and notation relevant to our discussion of automorphic representations. In Section 3 we give the formula for the multiplicity of automorphic representations from [Gro]. This formula implies the existence of a particular automorphic representation. In Section 4 we summarize what is known about the corresponding family of `-adic representations. We then switch to characteristic 0. In Section 5 we give an explicit formula for our connection for an arbitrary complex simple algebraic group, as well as its twisted version. In Section 6 we consider the special cases of representations on which a regular unipotent element has a single Jordan block. In these cases our connection can be represented by a scalar differential operator. These operators agree with those found earlier by Katz [Kat90]. We then take up the question of computation of the de Rham cohomology of our connection. After some preparatory material presented in Sections 7–9 we prove vanishing of the de Rham cohomology on the adjoint and small representations in Sections 10 and 11, respectively. We also show that the de Rham cohomology can be nontrivial for other representations using the case of SL2 as an example in Section 12. In Section 13 we determine the differential Galois group of our connection. We then use it in Section 14 to give a formula for the dimensions of the de Rham cohomology for an arbitrary finite-dimensional representation of G. { In particular, we give an alternative proof of the vanishing of de Rham cohomology for the adjoint and small representations. In Section 15 we discuss some closely related connections. Finally, in Section 16 we describe what the geometric Langlands correspondence should look like for our connection. Acknowledgments. We met and started this project while we were both visiting the Institut Mathematique´ de Jussieu in Paris. E.F. thanks Fondation Sciences Mathematiques´ de Paris for its support and the group “Algebraic Analysis” at Universite´ Paris VI for hospitality during his stay in Paris. We have had several discussions of this problem with Dennis Gaitsgory and Mark Reeder, and would like to thank them for their help. We thank Dima Arinkin and Pierre Schapira for answering our questions about Ᏸ-modules and deformations, ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1473 and Pierre Deligne for his help on the setup of the global Langlands correspondence. We owe a particular debt of gratitude to Nick Katz, who explained his beautiful results carefully, and guided us in the right direction.

2. Simple algebraic groups over global function ﬁelds

Let k be a finite field, of order q. Let G be an absolutely almost simple algebraic group over k (which we will refer to as a simple group for brevity). The group G is quasi-split over k, and we fix a maximal torus A B G contained in a Borel subgroup of G over k. Let k0 be the splitting field of G, which is the splitting field of the torus A, and put Gal.k =k/. Then is a finite cyclic group, of D 0 order 1, 2, or 3. Let Z denote the center of G, which is a finite, commutative group scheme over k. Let G denote the complex dual group of G. This comes with a pinning { T B G, as well as an action of which permutes basis vectors X ˛ of the L { { P simple negative root spaces. The principal element N X ˛ in g Lie.G/ is D L D { invariant under ([Gro97b]). Let Z denote the finite center of G, which also has L { an action of . There is an element " in Z.G/ which satisfies "2 1 and is defined as { D follows. Let 2 be the co-character of T which is the sum of positive co-roots, and { define " .2/. 1/: D Since the value of on any root is integral, " lies in Z.G/. It is also fixed by . { We have " 1 is a co-character of T: D ! { In order to avoid choosing a square root of q in the construction of Galois representations, we will use the following modification G of G, which was suggested by {1 { Deligne. Let G1 G އm=." 1/. We have homomorphisms އm G1 އm { D { 2 ! { ! with composite z z , The group acts on G1, trivially on އm. The co-character 7! { group X .T1/ contains the direct sum X .T/ X .އm/ with index 2. The advantage { { of passing to G is that we can choose a co-character {1 އm T1 W ! { fixed by which satisfies ; ˛ 1 h i D for all simple roots ˛ of G. (This is impossible to do for G when " 1.) Having { { ¤ w./ chosen , we let w./ ޚ be defined by composite map އm G1 އm, z z . 2 ! { ! 7! Then w./ is odd precisely when " 1. ¤ Let X be a smooth, geometrically connected, complete algebraic curve over k, of genus g. Let F k.X/ be the global function field of X. We fix two disjoint, D 1474 EDWARD FRENKEL and BENEDICT GROSS nonempty sets S;T of places v of F , and define the degrees X X deg.S/ deg v 1; deg.T / deg v 1: D D v S v T 2 2 A place v of F corresponds to a Gal.k=k/-orbit on the set of points X.k/. The degree deg v of v is the cardinality of the orbit. Let M MG V .1 d/ D d d 2 be the motive of the simple group G over F k.X/ ([Gro97a]). The spaces V , D d of invariant polynomials of degree d, are all rational representations of the finite, unramified quotient of Gal.F s=F /. The Artin L-function of V V , relative to D d the sets S and T , is defined by

Y s 1 Y 1 s LS;T .V; s/ det.1 Frv q V/ det.1 Frv q V /: D v j v j v S v T 62 2 deg v q deg v Here Frv Fr , where Fr is the Frobenius generator of , x x , and qv q . D 7! D This is known to be a polynomial of degree dim V .2g 2 deg S deg T/ in s C C q with integral coefficients and constant coefficient 1 ([Wei71]). We define Y LS;T .MG/ LS;T .V ; 1 d/; D d d 2 which is a nonzero integer. In the next section, we will use the integer LS;T .MG/ to study spaces of automorphic forms on G over F . We end this section with some examples. Let 2 d1; d2; : : : ; d h be the degrees of generators of the algebra of D rk.G/ D invariant polynomials of the Weyl group, where rk.G/ dim A is the rank of G D over the splitting field k0 and h is the Coxeter number. If G is split,

rk.G/ Y LS;T .MG/ S;T .1 di /; D i 1 D where S;T is the zeta-function of the curve X S relative to T . Now assume that G is not split, but that G is not of type D2n. Then each Vd has dimension 1, has order 2, and acts nontrivially on Vd if and only if d is odd. Hence Y Y LS;T .MG/ S;T .1 di / LS;T ."; 1 di /; D di even di odd where " is the nontrivial quadratic character of . ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1475

3. Automorphic representations

Let ށ be the ring of adeles` of the function field F k.X/. Then G.F / is a dis- D crete subgroup, with finite co-volume, in G.ށ/. Let L denote the discrete spectrum, which is a G.ށ/-submodule of L2.G.F / G.ށ//. Any irreducible representation n of G.ށ/ has finite multiplicity m./ in L. We will count the sum of multiplicities over irreducible representations D v with specified local behavior. Specifically, for v S T , we insist that v ˝b 62 [ be an unramified irreducible representation of G.Fv/, in the sense that the open compact subgroup G.ᏻv/ fixes a nonzero vector in v. At places v S, we insist 2 that v is the Steinberg representation of G.Fv/. Finally, at places v T , we insist 2 that v is a simple supercuspidal representation of G.Fv/, of the following type (cf. [GR]). We let v Pv p be a given affine generic character of a pro-p-Sylow W ! subgroup Pv G.ᏻv/. We recall that v is nontrivial on the simple affine root spaces of the Frattini quotient of Pv. (This is the affine analogue of a generic character of the unipotent radical of a Borel subgroup.) Extend v to a character of Z.qv/ Pv which is trivial on Z.qv/; then the compactly induced representation G.Fv/ Ind . v/ of G.Fv/ is multiplicity-free, with #Z.G/ irreducible summands. Pv Z.qv/ { We insist that v be a summand of this induced representation. The condition imposed on v only depends on the Iv-orbit of the generic character v, where Iv is the Iwahori subgroup of G.Fv/ which normalizes Pv. The quotient group Iv=Pv is isomorphic to A.qv/, where A is the torus in the Borel subgroup, and there are .qv 1/ #Z.qv/ different orbits. We note that affine generic ad characters form a principal homogeneous space for the group A .qv/ ކ , where qv the latter group maps a local parameter t to t. Using the simple trace formula, and assuming that some results of Kottwitz [Kot88] on the vanishing of the local orbital integrals of the Euler-Poincare´ function on nonelliptic classes extend from characteristic zero to characteristic p, we obtain the following formula for multiplicities [Gro]: Assume that p does not divide #Z.G/. Then { X #Z.G/ #Z.q/ m./ LS;T .MG/ { ; D Q v rk.G/v Q #Z.q /. 1/rk.G/v as above v S #Z.G/ . 1/ v T v 2 { 2 where rk.G/v is the rank of G over Fv. In the special case where X ސ1 has genus 0, S 0 and T , we D D f g D f1g have

LS;T .V ; s/ 1; for all d; s: d D Hence

LS;T .MG/ 1: D 1476 EDWARD FRENKEL and BENEDICT GROSS

Since v and qv q in this case, we find that D D X m./ 1: D Hence there is a unique automorphic representation in the discrete spectrum which is Steinberg at 0, simple supercuspidal for a fixed orbit of generic characters at , 1 and unramified elsewhere. This global representation is defined over the field of definition of , which is a subfield of ޑ.p/. We will consider its Langlands parameter in the1 next section.

4. `-adic sheaves on އm Consider the automorphic representation of the simple group G over F D k.t/, described at the end of Section 3. This representation is defined over the field ޑ.p/. Its local components v are unramified irreducible representations of G.Fv/, for all v 0; . The representation 0 is the Steinberg representation, ¤ 1 and is a simple supercuspidal representation. 1 Associated to and the choice of a co-character އm T1 G1, as described W ! { { in Section 2, we should (conjecturally) have a global Langlands parameter. This will be a continuous homomorphism s ' Gal.F =F / G1.ޑ.p/ / Ì W ! { for every finite place of ޑ .p/ dividing a rational prime ` p. Here we view ` ¤ G as a pinned, split group over ޑ. The homomorphism ' should be unramified {1 outside of 0 and , and map Frv to the -normalized Satake parameter for v 1 [Gro99], which is a semi-simple conjugacy class in G .ޑ. // Ì . If this is true, {1 p v the projection of ' to އm.ޑ.p// will be the w./-power of the `th cyclotomic character. At 0, '` should be tamely ramified. The tame inertia group should map to a ޚ`-subgroup of G.ޑ`.p//, topologically generated by a principal unipotent { 1=h element. At , '` should be wildly ramified, but trivial on the subgroup I C in 1 1 the upper numbering filtration, where h is the Coxeter number of G. If p does not divide h, the image of inertia should lie in the normalizer N.T/ of a maximal torus, L and should have the form E n . Here E T Œp is a regular subgroup isomorphic h i { to the additive group of the finite field ކp.h/ and n maps to a Coxeter element w of order h in the quotient group N.T /=T . The element n is both regular and L L semi-simple in G, and satisfies nh " Z.G/, with "2 1. { D 2 { D If

G1 Ì GL.V / W { ! is a representation over ޑ, we would obtain from ' a lisse -adic sheaf Ᏺ on ı އm over k, with rank.Ᏺ/ dim.V /. Katz has constructed and studied these lisse D ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1477 sheaves in the cases when the Coxeter element in the Weyl group has a single orbit on the set of nonzero weights for T on V (cf. Theorem 11.1 in [Kat90]). In all of L P these cases, the principal nilpotent element N X ˛ has a single Jordan block D on V . They are as follows:

G SLn Sp2n SO2n 1 G2 { W C dim V n 2n 2n 1 7 W C In all of these cases, there are no I -invariants on Ᏺ, the Swan conductor at 1 1 , and sw .Ᏺ/ 1. If j އm , ސ is the inclusion, Katz has shown that 1 1 D W ! H i .ސ1; j Ᏺ/ 0 for all i. Hence L.; V; s/ 1. D D More generally, consider the adjoint representation g. Then the sheaf Ᏺ has L rank equal to the dimension of G. In this case, we predict that { 1) I has rk.G/ Jordan blocks on ; 0 { Ᏺ 2) I has no invariants on Ᏺ, and sw .Ᏺ/ rk.G/; 1 1 D { 3) H i .ސ1; j Ᏺ/ 0 for all i; D 4) L.; g; s/ 1. L D Katz has verified this in the cases tabulated above, using the fact that the adjoint representation g of G appears in the tensor product of the representation V L { and its dual. It is an open problem to construct the -adic sheaf Ᏺ on އm over the finite field k for general groups G. We consider an analogous problem, for local systems { on އm.ރ/ ރ , in the next section. D 5. The connection

Let G be a simple algebraic group over ރ and g its Lie algebra. Fix a Borel { L subgroup B G and a torus T B. For each simple root ˛i , we denote by X ˛i { { { { a basis vector in the root subspace of g Lie.G/ corresponding to ˛i . Let E be L D { a basis vector in the root subspace of g corresponding to the maximal root . Set L rk.g/ XL N X ˛i : D i 1 D We define a connection on the trivial G-bundle on ސ1 by the formula r { dt (1) d N Edt; r D C t C where t is a parameter on ސ1 with a simple zero at 0 and a simple pole at . 1 The connection is clearly regular outside of the points t 0 and , where rdt D 1 the differential forms t and dt have no poles. We now discuss the behavior of (1) 1478 EDWARD FRENKEL and BENEDICT GROSS near the points t 0 and . It has regular singularity at t 0, with the monodromy D 1 D being a regular unipotent element of G. It also has a double pole at t , so { D 1 is irregular, and its slope there is 1=h, where h is the Coxeter number. Here we adapt the definition of slope from [Del70, Th. 1.12]: a connection on a principal G-bundle with irregular singularity at a point x on a curve X has slope a=b > 0 at { this point if the following holds. Let s be a uniformizing parameter at x, and pass to the extension given by adjoining the bth root of s: ub s. Then the connection, D written using the parameter u in the extension and a particular trivialization of the bundle on the punctured disc at x should have a pole of order a 1 at x, and its C polar part at x should not be nilpotent. To see that our connection has slope 1=h at the point , suppose first that 1 G has adjoint type. In terms of the uniformizing parameter s t 1 at , our { D 1 connection has the form ds ds d N E : s s2

Now take the covering given by uh s. Then the connection becomes D du du d hN hE : h 1 u u C If we now make a gauge transformation with g .u/ in the torus T , where is D { the co-character of T which is given by half the sum of the positive co-roots for B, { { this becomes du du (2) d h.N E/ : C u2 u The element N E is regular and semi-simple, by Kostant’s theorem [Kos63]. C Since the pole has order .a 1/ 2 with a 1, the slope is indeed 1=h. If G is C D D { not of adjoint type, then g .u/ might not be in T , but it will be after we pass to D { the covering obtained by extracting a square root of u. The resulting slope will be the same. Note that exp.=h/ is a Coxeter element in G, which normalizes the maximal { torus centralizing the regular element N E. We therefore have a close analogy C between the local behavior of our connection and the desired local parameters in r the -adic representation ' at both zero and inﬁnity.

The connection we have defined looks deceptively simple. We now describe how we used the theory of opers to find it. We recall from [BD05] that a (regular) oper on a curve U is a G-bundle with a connection and a reduction to the Borel { r subgroup B such that, with respect to any local trivialization of this B-reduction, { { ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1479 the connection has the form rk.g/ XL (3) d i X ˛i v; r D C C i 1 D where the i are nowhere vanishing one-forms on U and v is a regular one-form taking values in the Borel subalgebra b Lie.B/. Here X ˛i are nonzero vectors L D { in the root subspaces corresponding to the negative simple roots ˛i with respect to any maximal torus T B. Since the group B acts transitively on the set of such { { { tori, this definition is independent of the choice of T . { Any B-bundle on the curve U އm may be trivialized. Therefore the space { D of G-opers on އ may be described very concretely as the quotient of the space of { m connections (3), where 1 ޚ i ރŒt; t dt ރ t dt; 2 D and v bŒt; t 1dt, modulo the gauge action of BŒt; t 1. 2 L { We now impose the following conditions at the points 0 and . First we 1 insist that at t 0 the connection has a pole of order 1, with principal unipotent D 1 monodromy. The corresponding BŒt; t -gauge equivalence class contains a repre- 1 { sentative (3) with i ރt dt for all i 1; : : : ; rk.g/ and v bŒtdt. By making 2 D L 2 L dt a gauge transformation by a suitable element in T , we can make i for all i. { D t Second, we insist that the one-form v has a pole of order 2 at t , with D 1 leading term in the highest root space g of b, which is the minimal nonzero orbit for L L B on its Lie algebra. Then v.t/ Edt, with E a nonzero vector of g . These basis { D L 1 vectors are permuted simply-transitively by the group އm of automorphisms of ސ preserving 0 and (that is, rescalings of the coordinate t). Our oper connection 1 therefore takes the form dt (4) d N Edt: r D C t C This shows that the oper satisfying the above conditions is unique up to the automorphisms of ސ1 preserving 0 and . 1 We note that opers of this form (and possible additional regular singularities at other points of ސ1) have been considered in [FF07], where they were used to parametrize the spectra of quantum KdV Hamiltonians. Opers of the form (4), where E is a regular element of g (rather than nilpotent element generating the L maximal root subspace g , as discussed here), have been considered in [FFTL06], L [FFR07]. Finally, irregular connections (not necessarily in oper form) with double poles and regular semi-simple leading terms have been studied in [Boa01].

5.1. Twisted version. In this paper we focus on the case of a “constant” group scheme over އ corresponding to G. More generally, we may consider group m { 1480 EDWARD FRENKEL and BENEDICT GROSS schemes over އm twisted by automorphisms of G. The connection has analogues { r for these group schemes which we now describe. It is natural to view them as complex counterparts of the -adic representation ' for nonsplit groups. This answers a question raised by Deligne. Let be an automorphism of G of order n. It defines an automorphism of the { Lie algebra g, which we also denote by . Define a group scheme G on X އm L { D as follows. Let Xe އm be the n-sheeted cyclic cover of X with a coordinate z n D such that z t. Then G is the quotient of Xe G, viewed as a group scheme over D { { X, by the automorphism which acts by the formula .z; g/ .ze2i=n; .g//. It e 7! is endowed with a (flat) connection. Hence we have a natural notion of a G -bundle { on X with a (flat) connection. We now give an example of such an object which is a twisted analogue of the flat bundles described above. Take the trivial G -bundle on X. Then a connection on it may be described { concretely as an operator

d A.z/dz; r D C where A.z/dz is a -invariant g-valued one-form on Xe. e L Note that G and G are isomorphic if and differ by an inner automor- { { 0 0 phism of G. Hence, without loss of generality, we may, and will, assume that { is an automorphism of G preserving the pinning we have chosen (thus, has { order 1; 2 or 3). In particular, it permutes the elements X ˛i g. Then, for A.z/ 2 L as above, zA.z/ deﬁnes an element of the twisted afﬁne Kac-Moody algebra bg ; L see [Kac90]. The lower nilpotent subalgebra of this Lie algebra has generators

Xb ˛i ; i 0; : : : ; ` , where ` is the rank of the -invariant Lie subalgebra of g, D L and Xb ˛i ; i 1; : : : ; ` , are -invariant linear combinations of the elements X ˛j D of g (viewed as elements of the constant Lie subalgebra ofbg /. Explicit expressions L L for Xb ˛i in terms of g may be found in [Kac90]. L In the untwisted case the corresponding generators of the afﬁne Kac-Moody 1 algebra (which is a central extension of gŒt; t ; see 10) are Xb ˛i ; i i; : : : ; ` L D D rk.g/, and Xb ˛0 Et. Hence our connection (4) may be written as L D ` X dt d Xb ˛i : r D C t i 0 D We now adapt the same formula in the twisted case and deﬁne the following connection on the trivial G -bundle on އ : { m

` X dz (5) d Xb ˛i : r D C z i 0 D ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1481

We propose (5) as a complex counterpart of the -adic representation ' discussed above in the case when the group G is quasi-split and . D h i The notion of oper may be generalized to the twisted case as well. Namely, for any smooth complex curve X equipped with an unramiﬁed n-sheeted covering X we deﬁne the group scheme G in the same way as above. Since preserves e { a Borel subgroup B of G, the group scheme G contains a group subscheme B . { { { { The G -opers on X (relative to Xe) are then G -bundles on X with a connection { { r and a reduction to B satisfying the condition that locally the connection has the { form

` X (6) d i Xb ˛i v; r D C C i 1 D where i are nowhere vanishing one-forms and v takes values in the Lie algebra of

B . (As above, if acts by permutation of elements X ˛j of g, then the Xb ˛i are { L -invariant linear combinations of the X ˛j .) It is clear that (5) is a G -oper connection on އ . { m

6. Special cases

If V is any ﬁnite-dimensional complex representation of the group G, a { connection on the principal G-bundle gives a connection .V / on the vector r { r bundle Ᏺ associated to V . In our case, this connection is

dt (1) .V / d N.V / E.V /dt; r D C t C where N.V / and E.V / are the corresponding nilpotent endomorphisms of V . In this section, we will provide formulas for the ﬁrst order matrix differential operator .V / , for some simple representations V of G. We will be able to r td=dt { convert these matrix differential operator into scalar differential operators, because in these cases N.V / will be represented by a principal nilpotent matrix in End.V /. This will allow us to compare our connection with the scalar differential operators studied by Katz in [Kat90].

6.1. Case I. G SLn and V is the standard n-dimensional representation { D with a basis of vectors vi ; i 1; : : : ; n, on which the torus acts according to the D weights ei . Since ˛i ei ei 1; i 1; : : : ; n 1 and e1 en, we can normalize D C D D the X ˛i and E in such a way that 1482 EDWARD FRENKEL and BENEDICT GROSS 0 1 0 1 0 0 0 ::: 0 0 0 0 0 ::: 0 1 B 1 0 0 ::: 0 0 C B 0 0 0 ::: 0 0 C B C B C B C B C B 0 1 0 ::: 0 0 C B 0 0 0 ::: 0 0 C (2) N.V / B C ; E.V / B C : D B 0 0 1 ::: 0 0 C D B 0 0 0 ::: 0 0 C B C B C @::: ::: ::: ::: ::: :::A @::: ::: ::: ::: ::: :::A 0 0 0 ::: 1 0 0 0 0 ::: 0 0

Therefore the operator of the connection (1) has the form rtd=dt 0 1 0 0 0 ::: 0 t B 1 0 0 ::: 0 0 C B C d B C B 0 1 0 ::: 0 0 C (3) td=dt t B C r D dt C B 0 0 1 ::: 0 0 C B C @::: ::: ::: ::: ::: :::A 0 0 0 ::: 1 0 and hence corresponds to the scalar differential operator

(4) .td=dt/n . 1/n 1t: C C Now let V be the dual of the standard representation. Then, choosing as a basis the dual basis to the basis vn 1 i i 1;:::;n (in the reverse order), we obtain f C g D the same matrix differential operator (3). Hence the ﬂat vector bundles associated to the standard representation of SLn and its dual are isomorphic.

6.2. Case II. G Sp and V is the standard 2m-dimensional representation. { D 2m We choose the basis vi ; i 1; : : : ; 2m, in which the symplectic form is given by D the formula

vi ; vj vj ; vi ıi;2m 1 j ; i < j: h i D h i D C The weights of these vectors are e1; e2; : : : ; em; em;:::; e2; e1. Since ˛i ei ei 1; i 1; : : : ; m 1, ˛m 2em and 2e1, we can normalize N.V / D C D D D and E.V / in such a way that they are given by formulas (2). Hence is also given by formula (3) in this case. It corresponds to the rtd=dt operator (4) with n 2m. D

6.3. Case III. G SO2m 1 and V is the standard .2m 1/-dimensional { D C C representation of SO2m 1 with the basis vi ; i 1; : : : ; 2m 1, in which the inner C D C product has the form i vi ; vj . 1/ ıi;2m 2 j : h i D C The weights of these vectors are e1; e2; : : : ; em; 0; em;:::; e2; e1. Since ˛i D ei ei 1; i 1; : : : ; m 1, ˛m em and e1 e2, we can normalize the X ˛i C D D D C ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1483 and E in such a way that 0 1 0 1 0 0 0 ::: 0 0 0 0 0 ::: 1 0 B 1 0 0 ::: 0 0 C B 0 0 0 ::: 0 1 C B C B C B C B C B 0 1 0 ::: 0 0 C B 0 0 0 ::: 0 0 C (5) N.V / B C ; E.V / B C : D B 0 0 1 ::: 0 0 C D B 0 0 0 ::: 0 0 C B C B C @::: ::: ::: ::: ::: :::A @::: ::: ::: ::: ::: :::A 0 0 0 ::: 1 0 0 0 0 ::: 0 0

Therefore 0 1 0 0 0 ::: t 0 B 1 0 0 ::: 0 t C B C d B C B 0 1 0 ::: 0 0 C (6) td=dt t B C : r D dt C B 0 0 1 ::: 0 0 C B C @::: ::: ::: ::: ::: :::A 0 0 0 ::: 1 0 We can convert this first order matrix differential operator into a scalar differential operator. In order to do this, we need to find a gauge transformation by an upper-triangular matrix which brings it to a canonical form, in which we have 1’s below the diagonal and other nonzero entries occur only in the first row. This matrix is uniquely determined by this property and is given by 0 1 0 0 ::: t 1 B 0 1 0 ::: 0 C B C B 0 0 1 ::: 0 C : B C @::: ::: ::: ::: :::A 0 0 0 ::: 1 The resulting matrix operator is 0 1 0 0 0 ::: 2t t B 1 0 0 ::: 0 0 C B C d B C B 0 1 0 ::: 0 0 C t B C dt C B 0 0 1 ::: 0 0 C B C @::: ::: ::: ::: ::: ::: A 0 0 0 ::: 1 0 which corresponds to the scalar operator .td=dt/2m 1 2t 2d=dt t: C 6.4. CaseIV. G G2 and V is the 7-dimensional representation. { D The Lie algebra g2 of G2 is a subalgebra of so7. Both the nilpotent elements N and E in so7 may be simultaneously chosen to lie in this subalgebra, where 1484 EDWARD FRENKEL and BENEDICT GROSS they are equal to the corresponding elements for g2. Hence is equal to the rtd=dt operator (6) with m 3, which corresponds to the scalar differential operator D .td=dt/7 2t 2d=dt t: There is one more case when N is a regular nilpotent element in End.V /; n 2 namely, when G SL2 and V Sym .ރ / is the irreducible representation of { D D dimension n 1. We have already considered the cases n 1 and n 2 (the latter C D D corresponds to the standard representation of G SO3). These representations, { D and the cases considered above, are the only cases when an oper may be written as a scalar differential operator. The scalar differential operators we obtain agree with those constructed by Katz in [Kat90]. More precisely, to obtain his operators in the case of SO2m 1 and 1 C G2 we need to rescale t by the formula t t. 7! 2 In the above examples, the connection matrix was the same for Sp2n as it was for SL2n, and was the same for G2 as it was for SO7. The same phenomenon will occur for the pairs SO2n 1 < SO2n 2;G2 < D4, and F4 < E6, for the reason explained in Section 13. C C

7. De Rham cohomology

In the next sections of this paper, we will calculate the cohomology of the intermediate extension of our local system to ސ1, with values in a representation V of G. In particular, we will show that this cohomology vanishes for the adjoint { representation g, as well as for the small representations tabulated in Section 6. L This is further evidence that our connection is the characteristic 0 analogue of the `-adic Langlands parameter. It also implies that our connection is rigid, in the sense that it has no infinitesimal deformations preserving the formal types at 0 and , as 1 such deformations form an affine space over the first de Rham cohomology group [Kat96], [BE04], [Ari08]. We begin with some general remarks on algebraic de Rham cohomology for a principal G-bundle with connection on the affine curve U އm. Any complex { r D representation V of G then gives rise to a flat vector bundle .V / on U , where the { Ᏺ connection is .V /. r Since U is affine, with the ring of functions ރŒt; t 1, the connection .V / r gives a ރ-linear map dt (1) .V / V Œt; t 1 V Œt; t 1 : r W ! t Any G-bundle on U may be trivialized. Once we pick a trivialization of our bundle { Ᏺ, we represent the connection as d A, where A is a one-form on U with r D C values in the Lie algebra g. Let A.V / be the corresponding one-form on U with L ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1485 values in End.V /. We may write X dt A.V / An.V / D n t with An.V / End.V /. If 2 X n f .t/ vnt ; D n we find that X n dt .V /.f / wnt r D n t has coefficients wn given by the formula X wn nvn Aa.V /.v /: D C b a b n C D For our specific connection dt d N.V / Edt r D C t C we find wn nvn N.V /.vn/ E.V /.vn 1/: D C C The ordinary de Rham cohomology groups H i .U; Ᏺ.V // are defined as the cohomology of the complex (1): H 0.U; Ᏺ.V // Ker .V /; D r H 1.U; Ᏺ.V // Coker .V /: D r Thus elements of H 0.U; Ᏺ.V // are solutions of the differential equation .V /.f / P n r 0. For our particular connection, a solution f vnt corresponds to a solution D D to the system of linear equations

(2) nvn N.V /.vn/ E.V /.vn 1/ 0 C C D for all n. For example, if v is in the kernel of both N.V / and E.V /, then f v is D a constant solution, with vn 0 for all n 0 and v0 v. D ¤ D We can also study the complex (1) with functions and one-forms on various subschemes of U . For example, the kernel and cokernel of dt (3) .V / V ..t// V ..t// r W ! t 0 1 deﬁne the cohomology groups H .D0; Ᏺ.V // and H .D0; Ᏺ.V //, where D0 is the punctured disc at t 0. Likewise, the kernel and cokernel of D dt (4) .V / V ..t 1// V ..t 1// r W ! t 1486 EDWARD FRENKEL and BENEDICT GROSS

0 1 define the cohomology groups H .D ; Ᏺ.V // and H .D ; Ᏺ.V //, where D is the punctured disc at t . We will1 also identify the kernel1 and cokernel of1 D 1 dt (5) .V / V ŒŒt; t 1 V ŒŒt; t 1 r W ! t as cohomology groups with compact support in Section 9. The flat bundle Ᏺ.V / on U defines an algebraic, holonomic, left Ᏸ-module on U (which has the additional property of being coherent as an ᏻ-module). In the next section we will recall the definition of the intermediate extension jŠ Ᏺ.V / in the category of algebraic, holonomic, left Ᏸ-modules, where j U, ސ1 is the W ! inclusion. The de Rham cohomology of jŠ Ᏺ.V / may be calculated in terms of some Ext groups in this category. We will establish the following result, which is in agreement with the results of N. Katz on the `-adic cohomology with coefficients in the adjoint representation and small representations for the analogous `-adic representations (in those cases in which they have been constructed).

dt THEOREM 1. Assume that d N Edt and that V is either the adjoint r D C t C representation g of G or one of the small representations tabulated in Section 6. L { Then i 1 H .ސ ; jŠ Ᏺ.V // 0 D for all i. We will provide two proofs of this result. The ﬁrst, given in Sections 10 and 11, uses the theory of afﬁne Kac-Moody algebras and the relation between the cohomology of the intermediate extension of Ᏺ.V / and solutions of the equation .V /.f / 0 in various spaces. The second, given in Section 14, uses Deligne’s r D Euler characteristic formula and a calculation of the differential Galois group of . i 1 r The latter proof gives a formula for the dimensions of H .ސ ; jŠ Ᏺ.V // for any representation V of G. {

8. The intermediate extension and its cohomology

Here we follow [Kat90, 2.9] and [BE04] (see also [Ari08]). Let j U, ސ1 W ! be the inclusion. We consider the two functors (1) j direct image; D jŠ j D ı ı from the category of left holonomic Ᏸ-modules on U to the category of left holonomic Ᏸ-modules on ސ1. The functor j is right adjoint to the inverse image functor, and jŠ is deﬁned using the duality functors on these categories (see, e.g. [GM94, 5]). ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1487

We have

i 1 i i 1 i H .ސ ; j Ᏺ/ H .U; Ᏺ/; H .ސ ; jŠᏲ/ Hc .U; Ᏺ/: D D The cohomology groups on ސ1 are the Ext groups in the category of holonomic Ᏸ-modules. The ﬁrst equality follows from the adjointness property of j , and the second can be taken as the deﬁnition of the cohomology with compact support. Poincare´ duality gives a perfect pairing

H i .U; Ᏺ.V // H 2 i .U; Ᏺ.V // ރ; c ! where V is the representation of G that is dual to V . Thus, we have H 0.U; .V // { c Ᏺ 0 and D (2) H i .U; Ᏺ.V // H 2 i .U; Ᏺ.V // ; i 1; 2: c ' D From the adjointness property of j , we obtain a map of Ᏸ-modules on ސ1

jŠᏲ j Ᏺ ! whose kernel and cokernels are Ᏸ-modules supported on 0; . Let jŠ Ᏺ be the f 1g image of jŠᏲ in j Ᏺ. We will now show that 0 1 0 (3) H .ސ ; jŠ Ᏺ.V // H .U; Ᏺ.V //; D 1 1 1 1 (4) H .ސ ; jŠ Ᏺ.V // Im Hc .U; Ᏺ.V // H .U; Ᏺ.V // ; D ! 2 1 2 (5) H .ސ ; jŠ Ᏺ.V // Hc .U; Ᏺ.V //: D

We will also describe an exact sequence involving the cohomology groups on D0 1 1 and D which allows us to compute H .ސ ; jŠ Ᏺ.V //. 1 1 Let t˛ be a uniformizing parameter at ˛ 0; (t and t , respectively) and D 1 let

ı˛ ރ..t˛//=C ŒŒt˛ D be the left delta Ᏸ-module supported at ˛. We then have an exact sequence of Ᏸ-modules on ސ1

M 0 M 1 (6) 0 H .D˛; Ᏺ/ ı˛ jŠᏲ j Ᏺ H .D˛; Ᏺ/ ı˛ 0: ! ˛ ˝ ! ! ! ˛ ˝ !

By the deﬁnition of jŠ Ᏺ, this gives two short exact sequences M 0 0 H .D˛; Ᏺ/ ı˛ jŠᏲ jŠ Ᏺ 0; ! ˛ ˝ ! ! ! M 1 0 jŠ Ᏺ j Ᏺ H .D˛; Ᏺ/ ı˛ 0: ! ! ! ˛ ˝ ! 1488 EDWARD FRENKEL and BENEDICT GROSS

We now take long exact sequence in cohomology and use the fact that 0 1 H .ސ ; ı˛/ 0; D 1 1 H .ސ ; ı˛/ ރ: D This gives a proof of (4)–(5), and patching our two long exact sequences along 1 1 H .ސ ; jŠ Ᏺ/ gives a six-term exact sequence 0 M 0 1 (7) 0 H .U; Ᏺ/ H .D˛; Ᏺ/ Hc .U; Ᏺ/ ! ! ˛ ! ! 1 M 1 2 H .U; Ᏺ/ H .D˛; Ᏺ/ Hc .U; Ᏺ/ 0: ! ! ˛ ! ! We will compare it later with an exact sequence obtained from the snake lemma. From the exact sequence (7) we deduce the following condition for the vanishing of cohomology. i 1 PROPOSITION 2. For a ﬂat vector bundle Ᏺ on U , we have H .ސ ; jŠ Ᏺ/ 0 D for all i if and only if (1) H 0.U; Ᏺ/ H 0.U; Ᏺ / 0; D D 0 0 1 (2) dim H .D0/ dim H .D / dim Hc .U; Ᏺ/. C 1 D 9. The dual complex

We have seen that the de Rham cohomology of the flat vector bundle Ᏺ.V / on U can be calculated from the de Rham complex (1). Since compactly supported cohomology of Ᏺ.V / is dual to the (ordinary) de Rham cohomology of Ᏺ.V /, it can be calculated from the complex dual to dt (1) .V / V Œt; t 1 V Œt; t 1 : r W ! t In this section we will identify this dual complex with the complex dt (2) .V / V ŒŒt; t 1 V ŒŒt; t 1 r W ! t by using the residue pairing at t 0, which is described in detail below. 1 D 2 Hence Hc .U; Ᏺ.V // is identified with the kernel of (2) and Hc .U; Ᏺ.V // with its cokernel. Using this identification, we will compare the six-term exact sequence (7) to the one obtained from the snake lemma. We can also rewrite Proposition 2 in a form that depends only on solutions to .f / 0. r D 1 COROLLARY 3. The cohomology of the intermediate extension jŠ Ᏺ on ސ vanishes if and only if (1) Ker .V / 0 on V Œt; t 1 and Ker .V / 0 on V Œt; t 1; r D r D ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1489

1 (2) Every solutionf .t/ to .V /.f / 0 in V ŒŒt; t can be written (uniquely) r D 1 as a sum f0.t/ f .t/, with f0 and f in Ker .V / on V ..t// and V ..t //, C 1 1 r respectively. We now turn to the identification of (2) with the dual of (1). Define a bilinear pairing on f V ŒŒt; t 1 and ! V Œt; t 1 dt by 2 2 t (3) f; ! Rest 0 S.f !/; h i D D 1 dt where f ! is the product in V V ŒŒt; t t and S V V ރ is the natural ˝ 1 dt W ˝ ! P n contraction, so S.f !/ is an element of ރŒŒt; t t . Explicitly, if f vnt P m dt D and ! !mt , then D t X f; ! S.vn !m/: h i D ˝ n m 0 C D 1 This pairing identifies the direct product vector space V ŒŒt; t with the dual of 1 dt 1 the direct sum vector space V Œt; t t . A similar pairing identifies V ŒŒt; t 1 dt with the dual of the direct sum vector space V Œt; t t . To complete the proof that this pairing identifies the dual of (1) with (2), we must show that the adjoint of .V / is .V /. Write r r X m dt d Amt ; r D C m t P n with Am g. Then, for g wnt and f as before we have 2 L D X m n dt .V /.g/ dg Am.V /.wn/t C ; r D C m;n t

X m n dt .V /.f / df Am.V /.vn/t C : r D C m;n t The desired adjoint identity .V /.f /; g f; .V /.g/ 0 hr i C h r i D then follows from the two identities

Rest 0.g df f dg/ 0; D ˝ C ˝ D S.A.V /v; w/ S.v; A.V /w/ 0: C D

We end this section with a reconstruction of the six-term exact sequence (7). The maps ˛.f / .f; f / and ˇ.f; g/ f g give an exact sequence of vector D D spaces ˛ ˇ 0 V Œt; t 1 V ..t// V ..t 1// V ŒŒt; t 1 0: ! ! ˚ ! ! 1490 EDWARD FRENKEL and BENEDICT GROSS

Using .V /, we obtain a commutative diagram with exact rows r (4) ˇ 1 ˛ 1 1 0 V Œt; t V ..t// V ..t // V ŒŒt; t 0 ! ? ! ˚? ! ? ! ? . ; /? ? ry r r y ry ˛ ˇ 0 V Œt; t 1 dt V ..t// dt V ..t 1// dt V ŒŒt; t 1 dt 0: ! t ! t ˚ t ! t ! The three kernels and the three cokernels have been identiﬁed with the cohomology groups in (7). Do the morphisms in (7) come from an application of the snake lemma to (4)? We note that we have made two sign choices: in the pairing (3) we took the residue at t 0, not at t (which would have changed the sign), and D D 1 in the map ˇ we took f0 f , not f f0. We expect that with consistent choice 1 1 of signs the morphisms in (7) do indeed come from (4) via the snake lemma.

10. The vanishing of adjoint cohomology

We now turn to the proof of Theorem 1, using Corollary 3. Specifically, when V is the adjoint representation of G or a small representation we will show that P n { 1 any solution f .t/ vnt of .V /.f / 0 in V ŒŒt; t satisfies D r D (1) vn 0 for all n < 0: D Next, we will use the following lemma. P n 1 LEMMA 4. Suppose that any solution f vnt V ŒŒt; t to .V /.f / D 2 r 0 satisfies property (1) and the same property holds if we replace V by V . Then D i 1 H .ސ ; jŠ Ᏺ.V // 0 for all i. D Proof. The equation .V /.f / 0 implies that the components vn satisfy r D (2) nvn N.V /.vn/ E.V /.vn 1/ 0: C C D If (1) is satisfied, then it follows that v0 lies in the kernel of N.V / on V . This also shows that there is a unique solution X n f vnt D n 0 for any v0 in the kernel of N.V /, because N.V / is nilpotent so that the operator n Id N.V / is invertible on V for all n 0. Clearly then, if v0 0, this solution C ¤ ¤ has

(3) vn 0 for all n 0: ¤ Hence there cannot be a nonzero solution to .V /.f / 0 that has ﬁnitely many r D nonzero components for positive powers of t. We obtain that 0 0 H .U; Ᏺ.V // H .D ; Ᏺ.V // 0 D 1 D ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1491 and H 0.D ; Ᏺ.V // H 1.U; Ᏺ.V // Ker N.V /: 0 ' c ' Together we the same properties for V replaced by V , this implies that all of the criteria of Corollary 3 for the vanishing of cohomology are met. We now turn to the proof of the property (1) in the case when V g. We will D L drop V in our notation and write for .V /, etc. The vector space gŒt; t 1 is a r r L ޚ-graded Lie algebra, with the Lie bracket Œxt n; yt m Œx; yt n m: D C There is a G-invariant inner product { g g ރ W L ˝ L ! given by the Killing form (which is a unique such inner product up to scaling). We deﬁne an inner product on gŒt; t 1 by L DX n X mE X (4) xnt ; ymt .xn; ym/: D n m 0 C D 1 The ޚ-grading on gŒt; t is given by the differential operator td=dt. If we write L P n a solution f .t/ to .f / 0 in terms of its graded pieces: f vnt , then (2) r D D becomes

(5) nvn ŒN; vn ŒE; vn 1 0 C C D for all n. To ﬁnd the solutions to (5), it is convenient to switch to a different ޚ-grading of gŒt; t 1 called the principal grading. Let be again half the sum of positive L coroots for the Borel subgroup B. Then { Œ; N N; Œ; E .h 1/E: D D The operator d (6) d h t ad D dt has integer eigenvalues on gŒt; t 1, and deﬁnes the principal grading with respect L to which the element

(7) p1 N Et D C P has degree 1. If we write a solution f yn of .f / 0 in its components for D r D the principal grading, then (5) gives rise to the identities

(8) nyn Œ; yn hŒp1; yn 1 0 C C D for all n. 1492 EDWARD FRENKEL and BENEDICT GROSS

Since the eigenvalues of ad on g are the integers in the interval Œ1 h; h 1, L we see that the eigenvalues of d on gt n are the integers of the form nh e, with L C 1 h e h 1. In particular, as eigenvector ym with eigenvalue m has the form Ä Ä n ym t t ; 2 L if m nh, where t g is the unique Cartan subalgebra containing (so t is the D L L L kernel of ad on g). If n nh e with 0 < e < h, then ym has the form L D C n n 1 ym at bt D C C with a of degree e and b of degree .e h/ for ad . From this one deduces that the component of gŒt; t 1 of degree m with respect to the principal grading has L dimension equal to the rank of G, except when m is congruent to an exponent e { of G (mod h), when the dimension is the rank of G plus the multiplicity of that { { exponent. Note that the original grading operator td=dt preserves the components with respect to the principal grading. 1 V. Kac has studied the decomposition of gŒt; t under the action of ad p1, L where p1 N Et. His results are summarized in the following proposition. Set D C a Ker.ad p1/; c Im.ad p1/: D D PROPOSITION 5 ([Kac78, Prop. 3.8]). (1) The Lie algebra gŒt; t 1 has an orthogonal decomposition with respect to L the inner product (4), gŒt; t 1 a c: L D ˚ 1 (2) a is a commutative Lie subalgebra of gŒt; t . With respect to the principal L L grading, a i I ai , where I is the set of all integers equal to the exponents D 2 of g modulo the Coxeter number h, and dim ai is equal to the multiplicity of the L exponent i mod h. L (3) With respect to the principal grading, c j ޚ cj ; where dim cj rk.g/, D 2 D L and the map ad p1 cj cj 1 is an isomorphism for all j ޚ. W ! C 2 P Let f be a solution to .f / 0 and f yn its decomposition with r D D respect to the principal grading. Then the components satisfy (8). We now have the following crucial lemma.

LEMMA 6. Suppose that the yn satisfy the equations (8) and yn an for some 2 n. Then ym 0 for all m n. D Ä Proof. Applying (8), we obtain d m 1 (9) t yn yn Œ; yn cn; dt D h C h 2 where cn is the degree n homogeneous component of c Im.ad p1/. Let us show D that this is impossible to satisfy if yn an and yn 0. 2 ¤ ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1493

Consider the affine Kac-Moody algebra bg, which is the universal central L extension bg of gŒt; t 1 by one-dimensional center spanned by an element 1, L L 0 ރ1 bg gŒt; t 1 0: ! ! L ! L ! The commutation relations in bg read L n m n m ŒAt ; Bt ŒA; Bt C n.A; B/ın; m1: D C According to [Kac90, Lemma 14.4], the inverse image of a inbg is a (nondegenerate) L Heisenberg Lie subalgebra a ރ1. Hence there exists z a n such that ˚ 2 Œyn; z 0 in ރ1 bg: ¤ L Write, for n kh e (where e is an exponent of g), D C L k k 1 k k 1 yn at bt ; z a t b t ; D C C D 0 C 0 as above, where a; b; b; b are homogeneous elements of g with respect to the 0 L grading defined by ad of degrees e; e h; e; h e, respectively. Then we find that

Œyn; z .k .a; a / .k 1/ .b; b //1 0: D 0 C C 0 ¤ But d k .a; a / .k 1/ .b; b / t yn; z ; 0 C C 0 D h dt i where in the right-hand side we use the inner product defined by formula (4). This d contradicts the condition that t dt yn cn, which is orthogonal to a n. Hence 2 yn 0. D Now, (8) shows that if yn 0, then yn 1 an 1. Hence we find by induction D 2 that ym 0 for all m n. D Ä Setting n 0 in (8), we obtain D Œ; y0 Œp1; y 1 0: C D 1 Since gŒt; t 0 t Ker.ad /, this shows that L D L D Œp1; y 1 0: D Hence y 1 a 1. Lemma 7 then implies that yn 0 for all n < 0. Thus, any 2 1 D P solution f gŒŒt; t to .f / 0 has the form f n 0 yn, and in particular 2 L r D P Dn belongs to gŒŒt. Writing this solution as f vnt , we obtain that it satisfies L D property (1). Theorem 1 for the adjoint representation now follows from Lemma 4 because g g. L D L 1494 EDWARD FRENKEL and BENEDICT GROSS

11. Vanishing for small representations

Let V be one of the small representations considered in Section 6; that is, n-dimensional representation of SLn, 2n-dimensional representation of Sp , .2n 2n C 1/-dimensional representation of SO2n 1, or 7-dimensional representation of G2. C We denote the corresponding flat bundle on U by Ᏺ.V /. Since our connection for Sp2n and G2 is the same as for SL2n and SO7, respectively, it is sufficient to consider only the cases of SLn and SO2n 1. We will now prove Theorem 1 when VC is one of these representations. We will follow the argument used in the proof of Theorem 1 for the adjoint representation. Define a ޚ-grading on V Œt; t 1 compatible with the principal ޚ-grading on gŒt; t 1 L defined by the operator d. The representation V has a basis v1; : : : ; vp in which N appears as a lower Jordan block. We set

k deg vi t i 1 kh: D C All graded components are one-dimensional in the case of SLn. For SO2n 1 the components of degrees kh; k ޚ, are two-dimensional. Components of allC other 2 degrees are one-dimensional. It is easy to see that the operator td=dt preserves the graded components. We will use the operators N.V / and E.V / from Section 6. Denote again 1 N.V / E.V /t by p1. Let f V ŒŒt; t be a solution to .V /.f / 0. De- C P 2 r D composing f yr with respect to the above grading, we obtain the following D system: d (1) t yr p1 yr 1; r ޚ: dt D 2 The role of Lemma 6 is now played by the following result.

LEMMA 7. Suppose that the yr satisfy the equations (1) and p1 yr 0 for D some r. Then ym 0 for all m r. D Ä Proof. In the case of SLn we have Ker p1 0, so if p1 yr 0, then yr 0. D k k D1 D For SO2n 1, Ker p1 is spanned by the vectors v1t v2n 1t ; k ޚ. Hence C C 2 yr is one of these vectors. Equation (1) tells us that

d k k 1 k k 1 t .v1t v2n 1t / kv1t .k 1/v2n 1t dt C D C is in the image of p1. But the image of p1 in this graded component is spanned by k k 1 the vector v1t v2n 1t . Since k ޚ, it is impossible to satisfy this condition. C C 2 Therefore yr 0. D Now, if yr 0, then p1 yr 1 0, by (1). Hence we obtain by induction that D D ym 0 for all m r. D Ä ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1495

1 Now observe that td=dt annihilates the component of V Œt; t of degree 1 (in fact, all components of degrees 1; : : : ; h 1). Hence we ﬁnd from (1) with r 1 D that p1 y0 0. Therefore it follows from Lemma 6 that ym 0 for all m 0. D P n D Ä Hence any solution f fnt to .V /.f / 0 is a formal power series in t, D r D that is, fn 0 for all n < 0. Theorem 1 for small representations now follows D from Lemma 4 and the fact that V V in the case of SO2n 1, and in the case of D C SLn the ﬂat bundles associated to V and V are isomorphic as well (see Case I in Section 6).

12. The case of SL2 In the previous two sections we have shown that the de Rham cohomology of our connection vanishes in the adjoint representation as well as the small representations. This is because solutions to the equation .V /.f / 0 enjoy special r D properties which do not hold for a general representation V . The key property of g L and the small representations that we have used is the fact that the weights of the torus of a principal SL2 on V are all small (that is, the eigenvalues of ad on V have absolute value less than h). But for most other representations this is not the case and the de Rham cohomology does not vanish. As an example, we consider in this section the faithful 2k 1 irreducible representations V Sym of SL2 of even dimension 2k n 2. D D We will see that there are k n=2 solutions in the space of formal power series in 1 D 1 t and t with coefﬁcients in V , but that only the zero solution lies in V ..t //, and 1 1 only one line of solutions lies in V ..t//. Hence H .ސ ; jŠ Ᏺ.V // has dimension k 1 .n=2/ 1. D We will see how to compute the dimensions of the de Rham cohomology of 1 jŠ Ᏺ.V / on ސ , for any representation V of G, in Section 13. { In the case of SL2 our connection looks as follows: dt d F Edt; r D C t C where F X ˛1 and E are the standard generators of sl2. Let J diagŒ1; 2; : : : ; n. D D Make a change of variables t z2 and apply gauge transformation by zJ . We then D obtain dz d J 2.E F /dz: r D z C C

Let Veven (resp., Vodd) be the subspace of V on which the eigenvalues of J are even (resp., odd). The de Rham complex on U އm, D

V Œt; t 1 r V Œt; t 1dt; ! 1496 EDWARD FRENKEL and BENEDICT GROSS is identiﬁed with the subcomplex

2 2 2 2 2 2 dz 2 2 (1) VevenŒz ; z VoddŒz ; z z r VevenŒz ; z VoddŒz ; z dz ˚ ! z ˚ of the de Rham complex

(2) V Œz; z 1 r V Œz; z 1dz ! on the double cover of U . Introduce a ޚ-grading on this complex by setting deg vzk k. D Note that the operator E F is conjugate to 2 diagŒn 1; n 3; : : : ; n 1. C D C From now on we assume that n is even. Then the operator E F is invertible. P m C Suppose that y.z/ ymz is in the kernel of (2). Then we obtain the following D system of equations:

(3) .m Id J/ ym 2.E F/ ym 1 D C on its homogeneous components. Let m be the largest integer such that ym 0 D but ym 1 0. Then we should have .E F/ ym 1 0, which is impossi- ¤ 0 1 C D ble. This shows that H .ސ ; jŠ .Ᏺ.Vn/// 0. Using duality, we obtain that 2 1 D H .ސ ; jŠ .Ᏺ.Vn/// 0. D 1 1 In order to compute H .ސ ; jŠ .Ᏺ.Vn/// 0, we first compute the kernel of D (4) V ŒŒz; z 1 r V ŒŒz; z 1dz: ! Note that the operator m Id J is invertible for all m 1; : : : ; n. Let us choose any ¤ yn Vn. Then we can find ym; m > n, by using (3) and inverting m Id J , and we 2 can find ym; m < n, by using (3) and inverting E F . Thus, the kernel of (4) is C isomorphic to Vn. An element of this kernel belongs to

2 2 2 2 VevenŒŒz ; z VoddŒŒz ; z z ˚ 1 if and only if yn Veven. Thus, we obtain that H .U; Ᏺ.Vn// is isomorphic to 2 1 1 Veven. Using the exact sequence (7), we obtain that H .ސ ; jŠ .Ᏺ.Vn/// is the quotient of the above space of solutions of (3) by the subspace of those solutions for which ym 0 for m 0 or m 0. D Those are precisely the solutions for which there exists m 1; : : : ; n such that D ym 1 0, but ym 0. We claim that there are no such solutions for m n. Indeed, D ¤ ¤ denote by vi ; i 1; : : : ; n, an eigenvector of J with eigenvalue i. If ym 1 0, but D D ym 0, then ym is a multiple of vm. But 2.E F /.vm/ contains vm 1 with ¤ C C nonzero coefﬁcient if m < n. Hence 2.E F /.vm/ cannot be in the image of C .m 1/ Id J , and so (3) cannot be satisﬁed. If, on the other hand, yn vn, then C D ym 0 for all m < n. D ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1497

Thus, we obtain that there is a unique solution (up to a scalar) for which 1 1 ym 0 for m 0 or m 0. Hence dim H .ސ ; jŠ .Ᏺ.Vn/// .n=2/ 1, which D D is nonzero if n 4. 13. The differential Galois group

In this section, we determine the differential Galois group of our rigid irregular connection on އm, as well as its inertia subgroups (up to conjugacy) at t 0 r D and t . D 1 We first review some of the general theory, which is due to N. Katz [Kat87]. Fix the point t 1 on އm. Then the fiber at this point gives a fiber functor from D the category of flat complex algebraic vector bundles .Ᏺ; / on އm to the category r of finite-dimensional vector spaces. The automorphism group of this fiber functor is, by definition, the differential Galois group of އm. This is a pro-algebraic group diff over ރ, which we denote by DG.އm/; Katz calls this group 1 .އm; 1/. Since the fiber functor preserves tensor products, the fundamental theorem of [DM82, Ch. 2], gives an equivalence between the category of flat bundles on އm with the category of finite-dimensional representations of the differential Galois group DG.އm/. If

' DG.އm/ GL.V / W ! corresponds to the ﬂat bundle .Ᏺ; /, then r 0 DG.އm/ H .އm; Ᏺ/ Ker V : D r D Under this equivalence, a principal G-bundle with connection on އm deﬁnes { r a homomorphism

' DG.އm/ G r W ! { up to conjugacy. The image G is an algebraic subgroup of G, which we call the { { differential Galois group of .r r 1 At the two points t 0 and t on ސ އm, we have local inertia groups D D 1 I0 and I in DG.އm/, well-deﬁned up to conjugacy. Each inertia group I I˛ 1 D is ﬁltered by normal subgroups P x P x P I C for rational x > 0 (called the slopes). The wild inertia subgroup P is a pro-torus over ރ. As a pro-algebraic group over ރ, I=P is isomorphic to the product of އa with a pro-group A of multiplicative type, with character group ރ=ޚ. The additive part comes from local systems with regular singularity at ˛ with unipotent monodromy, and the rest from the one-dimensional local systems d a dt˛=t˛ with a 2ia solutions t near t˛ 0, with a ރ=ޚ. The tame monodromy is then e ރ . ˛ D 2 2 1498 EDWARD FRENKEL and BENEDICT GROSS

The torsion in the character group of A is ޑ=ޚ, so the component group of A is the dual group bޚ.1/ lim n, which is the proﬁnite Galois group of ރ..t˛//. D The quotient group I=P acts on the pro-torus P by conjugation. Its connected component centralizes P ; only the component group bޚ.1/ acts nontrivially. On the x x character group ރ of the quotient torus P =P C, the element a 2im acts by ax D multiplication by the root of unity e . Hence the component group acts through its ﬁnite quotient bޚ.1/=nbޚ.1/ n, where nx 0 (mod ޚ). D Á Now let dt d N Edt r D C t C be the connection introduced in Section 5. We will determine the differential Galois group G of G by studying the images ' .I / and '.I / in G, and we begin with { { 0 { a discussionr of the two homomorphismsr 1

' I0 G; r W ! { ' I G r W 1 ! { up to conjugacy. Since has a regular singular point at t 0, the restriction of ' to I0 is r D r trivial on the wild inertia subgroup P0. The resulting homomorphism is given by the analytic monodromy, which maps a 2in in Z.1/ to exp. aN / in G. In D { particular, the image ' .I0/ is an additive subgroup އa exp.zN / of G, containing r D { the principal unipotent element u exp. 2iN /. D To determine the image of the inertia group at t , we first make the D 1 assumption that is a co-character of G. Let uh s t 1. Then by our earlier { D D results in Section 5, over the extension ރ..u// of ރ..s// our connection is equivalent to du du d h.N E/ : C u2 u By a fundamental result of Kostant [Kos63], the element .N E/ is regular and C semi-simple. Hence the highest order polar term of our connection over ރ..u// is diagonalizable, in any representation V of G. It follows that the slopes of this { connection over ރ..u//, as defined by Deligne [Del70, Th. 1.12], are either 0 or 1, the former occurring at the zero eigenspaces for N E on V and the latter occurring C at the nonzero eigenspaces. Since Katz has shown [Kat87, 1 and 2.2.11.2] that the slopes over the extension ރ..u// are h times the slopes over the original completion ރ..s//, we see that the original slopes are either 0 or 1=h. In particular, I is trivial 1=h 1 on the subgroup P C. A similar argument works when is not a co-character, using the extension of degree 2h. The image S '.P / of wild inertia subgroup P is the smallest torus { WD 1 1 in G, whose Lie algebra contains the regular, semi-simple element N E. The { C ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1499 irregularity Irr .V / of a representation V of G at an irregular singular point ˛ was ˛ { defined by Deligne [Del70, p. 110], and shown by Katz [Kat87, 1, 2.3] to be the sum of the slopes. From the above analysis, we deduce that

S h Irr .V / dim V dim V {: 1 D The full image H '.I / normalizes S, and the quotient is generated by the { D 1 { element n .2/.ei=h/. The element n is regular and semi-simple in G. Further, D { " nh is a central involution in G which is equal to identity if and only if is a WD { co-character of G. The element n acts on N E by multiplication by e 2i=h. The { C irreducibility of the hth cyclotomic polynomial over ޚ shows that S has dimension { .h/ #.ޚ=hޚ/ and that the eigenvalues of n on Lie.S/ are the primitive hth D { roots of unity. Since n normalizes S, it also normalizes the centralizer T of S in G, which { {0 { { is a maximal torus (note that it is different from the maximal torus T considered in { 5). The image w of n in N.T /=T is a Coxeter class in the Weyl group, and has {0 {0 order h. The character group X .S/ is the free quotient of X .T / where w acts by { {0 primitive hth roots of unity. Thus the characters of T which restrict to the trivial {0 character of S are generated by those T އm where the w -orbit of has { W {0 ! h i size less than h. This completes the description of the local inertia groups, and we now turn to the global differential Galois group G . {r PROPOSITION 8. Let G0 G be an algebraic subgroup which contains '.I / { { 1 H and '.I0/ exp.zN /. Then G0 is reductive and contains the image of a D { D { principal SL in G. 2 { Proof. Let R.g0/ be the unipotent radical of G0, and let Z Lie.Z/ be its L { D center. We will show that Z 0. D The group H '.I / acts on R.G0/ and Z. Since S contains regular { D 1 { { elements, every root ˛ T އm restricts to a nontrivial character of S. Hence the W {0 ! { action of H S; n on g decomposes as { D h { i L rk.g/ ML Lie.T / Wi ; {0 ˚ i 1 D where the W are irreducible representations of dimension h whose restriction to S i { contains an entire w-orbit of roots. Since Z is nilpotent and Lie.T / is semi-simple, Z must be the sum of certain {0 Ph i Wi . But each Wi contains a nonzero vector v0 i 1 n .v/ ﬁxed by n (as h D D h i n " is central and acts trivially on g). Since n is regular and semi-simple, v0 is D L a semi-simple element and hence cannot be contained in Z. Therefore Z 0. D 1500 EDWARD FRENKEL and BENEDICT GROSS

Since G is reductive and contains the principal unipotent element u, it contains {0 a principal SL2 [Car85]. We note that a principal embedding SL2 G maps the ! { central element 1 SL2 to the element " H G. 2 2 { { Proposition 8 is a serious constraint on the global image G , as the reductive { subgroups of G containing a principal SL are severely limited byr a result of [SS97] { 2 which goes back to the work of Dynkin [Dyn52]. They are all simple, and their Lie algebras appear in one of the following maximal chains:

sl2 / sp2n / sl2n

sl2n 1 9 C

sl2 / so2n 1 C

% so2n 2 C sl7 =

sl2 / g2 / so7

" so8

sl2 / f4 / e6

sl2 / e7

sl2 / e8 In some of these cases, the subgroup G cannot contain H, as its Coxeter {0 { number is less than the Coxeter number of G. Looking at the minimal cases { remaining, we obtain

COROLLARY 9. If G is simple of type A2n; n 1; Cn; n 1; Bn; n 4, G2, { F4, E7, E8, then G G. {r D { Indeed, by the above list of embeddings and Proposition 8, any G0 G { { containing H and n must be equal to G. { h i { ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1501

For the remaining cases, observe that the automorphism group † of the pinning of G is known to be isomorphic to the outer automorphism group of G. This finite { { group fixes N and acts on the highest root space. If G is not of type A , it also { 2n fixes E. Hence † fixes the connection , and its differential Galois group G is r {r contained in G†. Thus Corollary 9 gives the differential Galois group in all cases. {

COROLLARY 10. If G is of type A , then G is of type C , with center the { 2n 1 { n kernel of the center of G on the second exterior powerr representation. { If G is of type D2n 1 with n 4, then G is of type Bn, with the center the { C {r kernel of the center of G on the standard representation. { If G is of type D4 or B3, then G is of type G2. { {r If G is of type E6, then G is of type F4. { {r

14. The dimension of cohomology

We now use the calculation of the differential Galois group G of our con- { nection, with its inertia subgroups at 0 and , to determine the dimensionsr of 1 1 the cohomology groups of the Ᏸ-modules j Ᏺ.V /, jŠᏲ.V / and jŠ Ᏺ.V / on ސ associated to a representation V of G. { We will assume that we are in one of the cases described in Corollary 9. Otherwise, the computation of cohomology reduces to one in a smaller group given by Corollary 9. We will also assume that V is irreducible nontrivial representation of G, so {

0 1 0 G H .ސ ; j Ᏺ.V // H .U; Ᏺ.V // V { 0; D D D 2 1 2 H .ސ ; jŠᏲ.V // Hc .U; Ᏺ.V // HomG.V ; ރ/ 0: D D { D We will use Deligne’s formula [Del70, 6.21.1] for the Euler characteristic X .H .U; Ᏺ// .Hc.U; Ᏺ// .U / rank.Ᏺ/ Irr˛.Ᏺ/: D D ˛

In our case, .U / 0 and is regular at ˛ 0. Hence D r D 1 1 1 1 dim H .ސ ; j Ᏺ/ dim H .ސ ; jŠᏲ/ Irr .Ᏺ/: D D 1 The kernel of the map H 1.U; Ᏺ/ H 1.U; Ᏺ/ is isomorphic to the direct sum c !

0 0 I0 I H .D0; Ᏺ/ H .D ; Ᏺ/ V V 1 ˚ 1 D ˚ by (7). Hence we obtain 1502 EDWARD FRENKEL and BENEDICT GROSS

PROPOSITION 11. If G G and V is an irreducible, nontrivial representa- {r D { tion of G with associated ﬂat vector bundle .V / on އ , then { Ᏺ m 0 1 2 1 H .ސ ; jŠ Ᏺ.V // H .ސ ; jŠ Ᏺ.V // 0; D D 1 1 I0 I0 (1) d.V / dim H .ސ ; jŠ Ᏺ.V // Irr .V / dim V dim V : WD D 1 If we don’t assume that G G or that V is an irreducible, nontrivial repre- {r D { sentation of G, we obtain the formulas { 0 1 2 1 G dim H .ސ ; jŠ Ᏺ.V // dim H .ސ ; jŠ Ᏺ.V // dim V {r ; D D 1 1 I0 I0 G dim H .ސ ; jŠ Ᏺ.V // Irr .V / dim V dim V 2 dim V {r : D 1 C Since S h Irr .V / dim V dim V {; 1 D as we have seen above, this allows us to compute the dimensions of the cohomology of the middle extension for any representation V of G, provided that we know the { restriction of V to the three subgroups S, H and G , and the restriction of V to a { { {r principal SL2. We will now make this more explicit. k The irreducible representations of SL2 all have the form Sym ; k 0. Hence we may write the restriction of V to the principal SL2 as M V .Symk/ m.k/ D ˚ k 0 with multiplicities m.k/ 0. We then have X (2) dim V I0 m.k/ D k 0 as '.I0/ exp.tN / ﬁxes a unique line on each irreducible factor. We note that the D parity of these k is determined by V : if m.k/ > 0 we have

k . 1/ " V : D j The irreducible complex representations of H have dimensions either h or 1. { The irreducible W of dimension h restrict to a sum of h distinct nontrivial characters of S, in a single n -orbit. The irreducible of dimension 1 are the representations { h i trivial on S, and determined by .n/, which lies in . We may therefore write { 2h

M m. i / M m.Wj / (3) V ˚ W ˚ : D i ˚ j i j

If mi > 0, then h i .n/ " V : D j ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1503

In terms of the decomposition (3), we have

I (4) dim V 1 m. 0/; D where 0 is the trivial character, 0.n/ 1. Each one-dimensional representation D of H is tame, and each irreducible h-dimensional representation has irregularity { 1 h .1=h/. Hence D X (5) Irr .V / m.Wj / 1 D j 1 # nontrivial weight spaces for S on V : D h f { g

If we know the restriction of V to a principal SL2 and to H , then formulas (1), { 1 1 (2), (4), and (5) allow us to determine the dimension d.V / of H .ސ ; jŠ Ᏺ.V //. For example, when V g is the adjoint representation, we have " 1 on V and D L DC rk.g/ ML V Sym2di 2; D i 1 D rk.g/ rk.g/ ML ML i Wj ; D ˚ i 1 j 1 D D where the di are the degrees of invariant polynomials and i 1 for all i. Hence I0 I ¤ Irr dim V rk.g/, V 1 0, and so d.V / 0. This gives the second proof of 1 D D L D D the rigidity of our connection (Theorem 1), for the groups G in Corollary 9. (When { G is a proper subgroup of G, we find that g g V 0, where the representation {r { L D Lr ˚ V 0 of G also has d.V 0/ 0.) {r D For example, the adjoint representation e6 of G E6 decomposes as a sum { D of two irreducible representations e6 f4 V for the differential Galois group D ˚ 0 G F4, with dim V 0 26. The restriction of V 0 to the principal SL2 is the sum {r D D Sym16 Sym8. The restriction of V to H is the sum W W , with 0 { 1 2 1 2 ˚ I0 ˚ ˚ I˚ the i nontrivial of order 3. Hence Irr .V 0/ dim V 0 2; dim V 0 1 0, and 1 D D D d.V 0/ 0. D n Similarly, for the nontrivial irreducible representations V Sym of G SL2, D { D we find that d.V / .n 1/=2 when n is odd, d.V / .n 2/=2 when n is congruent D D to 2 (mod 4), and d.V / .n 4/=2 when n is divisible by 4, in agreement with D the results of Section 12. In particular, d.V / > 0 whenever n > 4. For the spin representation V of dimension 2n of G Spin , we find that d.V / > 0 for all { D 2n 1 n > 7. C 1504 EDWARD FRENKEL and BENEDICT GROSS

We now investigate the difference

I0 I d.V / Irr dim V dim V 1 D 1 X X m.Wj / m.k/ m. 0/ D j k 0 further, assuming that V is irreducible and nontrivial. This argument, which was shown to us by Mark Reeder, breaks into two cases, depending on whether " 1 DC or 1 on V . If " 1 on V , we may assume (by passing to a quotient that acts faithfully DC on V ) that " 1 in G. Then nh 1 and H is a semi-direct product n Ë S. In D { D { h i { this case, the map SL2 G factors through the quotient P GL2. ! { We now count the dimension of the span of invariants for n on V . Since n h i h i is a subgroup of H which fixes a line in each W , we find that { j n X dim V m.Wj / m. 0/ h i D C Irr m. 0/: D 1 C Since n is conjugate to the element n .e2i=h/, which lies in the maximal torus 0 D A of the principal P GL2, we have X dim V n m.k/ # weights of A on Symk with a 0 (mod h) : h i D f Á g k 0 if m.k/ 0, then k is even and the weight a 0 occurs once in the irreducible ¤ D representation Symk. Since the weights a and a occur with the same multiplicity in V , we find that X dim V n m.k/ 2# weights a > 0 of A on V with a 0 (mod h) h i D C f Á g k 0 dim V I0 2# weights a > 0 of A on V with a 0 (mod h) : D C f Á g Hence, when " 1 we find that DC I0 d.V / Irr dim V m. 0/ D 1 2.# weights a > 0 of A on V with a 0 (mod h) m. 0//: D f Á g The fact that d.V / is even is consistent with the fact that when V is self-dual and 1 1 " V 1, then V is orthogonal. Hence H .ސ ; jŠ Ᏺ.V // is symplectic. j DC If the highest weight L of V satisfies ; h 1; hL i Ä then there are no weights a > 0 with a 0 (mod h). Since d.V / 0, this forces Á m. 0/ 0 and d.V / 0. This is what happens for the adjoint representation. D D ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1505

A similar analysis when " V 1 gives the formula j D d.V / # weights 2k 1 for the torus of SL2 on V D f C with k 0 (mod h) and k 0 m. 1/; Á ¤ g i=h h where 1 is the character of H which is trivial on S and maps n to e . if " n { { D lies in S, then m. 1/ 0, as all weights of S on V are nontrivial. { D { 15. Nearby connections

There are several connections closely related to the rigid irregular connection that we have studied in this paper. First, there is the connection r dt 1 d N r D C t which has regular singularities at both t 0 and t . The monodromy of 1 is D D 1 r generated by the principal unipotent element u exp.2iN / and the differential D Galois group is the additive group exp.zN / of dimension 1 in G. { A second related connection is the canonical extension of our local differential equation at infinity, defined by Katz [Kat87, 2.4] This is the connection Â 1 Ã dt 2 d N Edt: r D C h t C If we pass to the ramified extension given by uh s t 1, and make a gauge D D transformation by g .u/ (assuming that is a co-character), the connection 2 D r becomes equivalent to the connection du d h.N E/ : C u2 This is regular at the unique point lying above t 0. The differential Galois group D of 2 and its inertia group at infinity are both isomorphic to H . The inertia group r { at zero is cyclic, of order h or 2h. Finally, we have the following generalization of for the exceptional groups r G of types G , F , E , E , and E , which was suggested by the treatment of { 2 4 6 7 8 nilpotent elements and regular classes in the Weyl group in [Spr74, 9]. We thank Mark Reeder for bringing this argument to our attention. Let

' sl2 g 0W ! L be a subregular sl2 in the Lie algebra. The Dynkin labels of the semi-simple element ' .h/ are equal to 2 on all of the vertices of the diagram for G, except the vertex 0 { corresponding to the (unique) root with the highest multiplicity m in the highest root, where the label is 0. 1506 EDWARD FRENKEL and BENEDICT GROSS

Let d h m. Then the highest eigenspace gŒ2d 2 for ' .h/ on g has D L 0 L dimension equal to 1. Let E be a basis, and let N ' .f / in gŒ 2. Then 0 0 D 0 L Springer shows that the element N E is both regular and semi-simple in g. This 0 C 0 L element is normalized by the semi-simple element n ' .h.ei=d //, which has 0 D 0 order either d or 2d in G. { Let M be the maximal torus which centralizes N E . Then the image of n 0 C 0 0 in the Weyl group of M is a regular element of order d, in the sense of Springer [Spr74]. It has r 2 free orbits on the roots of g, where r is the rank of G. We C L { tabulate these numerical invariants for our groups below, as well as the characteristic polynomial F of n0 acting on the character group of M , as a product of cyclotomic polynomials F .n/.

G m d r 2 F { C G2 3 3 4 F .3/ F4 4 8 6 F .8/ E6 3 9 8 F .9/ E7 4 14 9 F .14/ F .2/ E8 6 24 10 F .24/

Deﬁne the subregular analog of as follows: r dt d N E dt: r0 D C 0 t C 0 Then has a regular singularity at t 0, with monodromy the subregular unipotent r0 D element u exp. 2iN /. It has an irregular singularity at , with slope 1=d 0 D 0 1 and local inertia group S ; n , where S is the subtorus of M on which n acts by h {0 0i {0 0 the primitive d th roots of unity. The connection is rigid, with differential Galois group given as follows: r0 G G2 F4 E6 E7 E8 { W G SL3 Spin9 E6 E7 E8 r0 W In the ﬁrst two cases, we note that a subregular SL in G is a regular SL in 2 { 2 G . Hence the connection 0 on G is obtained from the rigid connection for {r0 r { r the differential Galois group. In the three other cases, the connection is new; the r0 third gives us a connection with differential Galois group of type E6. In particular, the differential Galois group of އm has any simple exceptional group as a rigid quotient. We get two further rigid connections with differential Galois group E8 from the two nilpotent classes that Springer lists in [Spr74, Table 11]. These have slopes 1=20 and 1=15 at respectively. Since there is a misprint in that table, we give 1 ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1507 the Dynkin labeling of these nilpotent classes below.

2 2 0 2 0 2 2

2 0 2 0 2 0 2

0 In both cases, the action of the corresponding regular element in the Weyl group on the character group of the torus is by the primitive 20th and 15th roots of unity, respectively.

16. Example of the geometric Langlands correspondence with wild ramiﬁcation

The Langlands correspondence for function fields has a counterpart for complex algebraic curves, called the geometric Langlands correspondence. As in the classical setting, there is a local and a global pictures. For simplicity we will restrict ourselves to the case when G is a simple connected simply-connected algebraic group, so that G is a group of adjoint type. { The local geometric Langlands correspondence has been proposed in [FG06] (see also [Fre07a] for an exposition). According to this proposal, to each “local Lang- lands parameter” , which is a G-bundle with a connection on D Spec ރ..t//, { D there should correspond a category Ꮿ equipped with an action of the formal loop group G..t//. This correspondence should be viewed as a “categorification” of the local Langlands correspondence for the group G.F /, where F is a local non- archimedian field, F ކq..t//. This means that we expect that the Grothendieck D group of the category Ꮿ , equipped with an action of G..t//, attached to a Langlands parameter , should “look like” an irreducible smooth representation of G.F / attached to an `-adic representation of the Weil group of F with the same properties as . In particular, an object of Ꮿ should correspond to a vector in the representation . Thus, the analogue of the subspace .K;‰/ of .K; ‰/-invariant vectors in (that is, vectors that transform via a character ‰ under the action of a subgroup .K;‰/ K of G.F /) should be the category Ꮿ of .K; ‰/-equivariant objects of Ꮿ .

Denote by LocG.D/ the set of isomorphism classes of ﬂat G-bundles on D. { { In [FG06], a candidate for the category Ꮿ has been proposed for any LocG.D/. 2 { 1508 EDWARD FRENKEL and BENEDICT GROSS

Namely, consider the category bgcrit -mod of discrete modules over the afﬁne Kac- Moody algebra bg of critical level. The center of this category is isomorphic to the algebra of functions on the space OpG.D/ of G-opers on D (see [Fre07a]). { { Hence for each point Op .D / we have the category gcrit -mod of those 2 G b modules on which the center acts{ according to the character corresponding to . Consider the forgetful map p Op .D / Loc .D /. It was proved in W G ! G [FZ08] that this map is surjective. Given a{ local Langlands{ parameter Loc .D /, 2 G choose p 1./. According to the proposal of [FG06], the sought-after category{ 2 Ꮿ should be equivalent to bgcrit -mod (these categories should therefore be equiva- 1 .K;‰/ lent to each other for all p ./). In particular, Ꮿ should be equivalent .K;‰/2 to the category bgcrit -mod of .K; ‰/-equivariant objects in bgcrit -mod . We now consider the global geometric Langlands correspondence (see [FG06], [Fre07b], [Fre07a] for more details). Let X be a smooth projective curve over ރ. The Langlands parameters are now G-bundles on X with connections which { are allowed to have poles at some points x1; : : : ; xN . Let BunG;.xi / be the moduli stack of G-bundles on X with the full level structure at each point xi ; i 1; : : : ; N D (that is, a trivialization of the G-bundle on the formal disc at xi ). The global correspondence should assign to the category Aut;.xi / of Hecke eigensheaves on BunG;.xi / with eigenvalue . The compatibility between the local and global correspondence should be that [FG06] N O (1) Aut;.x / Ꮿ ; i ' xi i 1 D where xi is the restriction of to the punctured disc at xi . This equivalence should give rise to an equivalence of the equivariant categories. Let Kxi be a subgroup of G.ᏻxi /, where ᏻxi is the completed local ring at xi , and ‰xi be its character. .K ;‰ / Then the equivariant category of Aut is the category Aut xi xi of Hecke ;.xi / ;.xi / eigensheaves on BunG;.xi / with eigenvalue which are .Kxi ;‰xi /-equivariant. We should have an equivalence of categories N .Kxi ;‰xi / O .Kxi ;‰xi / (2) Aut Ꮿ : ;.xi / ' xi i 1 Now suppose that comes from anD oper which is regular on X x1;::: nf : : : ; xN . Let x be the restriction of to D . Then we have the categories g i xi .Kxi ;‰xi / g -mod and g -mod , which we expect to be equivalent to Ꮿ bcrit xi bcrit xi xi .Kx ;‰x / and Ꮿ i i , respectively. The stack Bun may be represented as a double xi G;.xi / quotient N N Y Y G.F / G.Fx /= G.ᏻx /; n i i i 1 i 1 D D ARIGIDIRREGULARCONNECTIONONTHEPROJECTIVELINE 1509 where F ރ.X/ is the ﬁeld of rational functions on X and Fx is its completion D i at xi . A loop group version of the localization functor of Beilinson-Bernstein gives rise to the functors

N O (3) gcrit -mod Aut;.x /; W b xi ! i i 1 D

N .K ;‰ / .K ;‰ / .Kx ;‰x / O xi xi xi xi (4) i i gcrit -mod Aut ; W b xi ! ;.xi / i 1 D and it is expected [FG06] that these functors give rise to the equivalences (1) and (2), respectively. This is a generalization of the construction of Beilinson and Drinfeld in the unramified case [BD]. We now apply this to our situation, which may be viewed as the simplest example of the geometric Langlands correspondence with wild ramification (i.e., connections admitting an irregular singularity). We note that wild ramification has been studied by E. Witten in the context of S-duality of supersymmetric Yang-Mills theory [Wit08]. Let be our G-oper on ސ1 with poles at the points 0 and . Let denote { 1 the corresponding flat G-bundle. We choose K to be the Iwahori subgroup I , { 0 with ‰0 the trivial character (we will therefore omit it in the formulas below), and 0 0 0 0 rank.G/ 1 K to be its radical I . Note that I =ŒI ;I .އa/ C . We choose a 1 ' nondegenerate additive character ‰ of I 0 as our ‰ . Thus, we have the global I;.I 0;‰/ 1 categories Aut;.0; / and Aut;.0; / on BunG;.0; /. According to1 Section 3, there1 is a unique automorphic1 representation of G.ށ/ corresponding to an `-adic analogue of our oper. Moreover, the only ramified local factors in this representation are situated at 0 and . The former is the Steinberg 1 representation, whose space of Iwahori invariant vectors is one-dimensional. The latter is the simple supercuspidal representation constructed in [GR]. Its space of .I 0;‰/-invariant vectors is also one-dimensional. We recall that the geometric analogue of the space of invariant vectors is the corresponding equivariant category. Hence the geometric counterpart of the multiplicity one statement of I;.I 0;‰/ Section 3 is the statement (conjecture) that the category Aut;.0; / has a unique nonzero irreducible object. (Here and below “unique” means “unique1 up to an isomorphism.”) The compatibility of the local and global correspondences gives us a way to construct this object. Namely, we have two local categories g -modI and bcrit 0 .I 0;‰/ bgcrit -mod attached to the points 0 and , respectively. The oper 0 on D0 1 1 has regular singularity and regular unipotent monodromy. Using the results of 1510 EDWARD FRENKEL and BENEDICT GROSS

[FG06], [Fre07a], one can show that the category g -modI has a unique nonzero bcrit 0 irreducible object ލ . 0/, which is constructed as follows. It is the quotient of the Verma module bgcrit ލ IndLie.I / ރ1.ރ / D ˚ over gcrit with highest weight , by the image of the maximal ideal in the center b corresponding to the central character 0. On the other hand, has irregular singularity with the slope 1=h. To 1 .I 0;‰/ construct an object of the category of bgcrit -mod , we imitate the construction of [GR] (see 3). Deﬁne the afﬁne Whittaker module1

ޗ Indbgcrit .‰/ ‰ Lie.I 0/ ރ1 D ˚ over bgcrit (here we denote by the same symbol ‰ the character of the Lie algebra 0 0 Lie.I / corresponding to the above character ‰ of the group I ). Let ޗ‰. / be 1 the quotient of ޗ‰ by the image of the maximal ideal in the center corresponding to 0 the central character . By construction, it is an .I ;‰/-equivariant bgcrit-module 1 .I 0;‰/ and hence it is indeed an object of our local category bgcrit -mod . Applying the I;.I;‰/ 1 localization functor of (4) to ލ . 0/ ޗ‰. /, we obtain an object of I;.I 0;‰/ ˝ 1 the category Aut;.0; / . It is natural to conjecture that this is the unique nonzero irreducible object of1 this category. One can show (see [FF07, Lemma 5]) that the image of the center of the completed enveloping algebra in End ޗ‰ is the algebra of functions on the space of opers which have representatives of the form ds ds ds d N E v ; s s2 C s where v bŒŒs. Since our oper belongs to this space, we obtain that the 2 1 quotient ޗ‰. / is nonzero. This provides supporting evidence for the above conjecture describing1 an example of the geometric Langlands correspondence with wild ramiﬁcation.

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(Received February 12, 2009) (Revised April 27, 2009)

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