aa r X i v : . [ m a t h . AG ] D ec BESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS DAXIN XU, XINWEN ZHU
Abstract.
We construct the Frobenius structure on a rigid connection Be ˇ G on G m for a split reductivegroup ˇ G introduced by Frenkel-Gross. These data form a ˇ G -valued overconvergent F -isocrystal Be † ˇ G on G m, F p , which is the p -adic companion of the Kloosterman ˇ G -local system Kl ˇ G constructed by Heinloth-Ngô-Yun. By studying the structure of the underlying differential equation, we calculate the monodromygroup of Be † ˇ G when ˇ G is almost simple (which recovers the calculation of monodromy group of Kl ˇ G dueto Katz and Heinloth-Ngô-Yun), and establish functoriality between different Kloosterman ˇ G -local systemsas conjectured by Heinloth-Ngô-Yun. We show that the Frobenius Newton polygons of Kl ˇ G are genericallyordinary for every ˇ G and are everywhere ordinary on | G m, F p | when ˇ G is classical or G . Contents
1. Introduction 21.1. Bessel equations and Kloosterman sums 21.2. Generalization for reductive groups 41.3. Strategy of the proof and the organization of the article 72. Review and complements on arithmetic D -modules 92.1. Overconvergent ( F -)isocrystals and their rigid cohomologies 92.2. (Co)specialization morphism for de Rham and rigid cohomologies 112.3. Six functors formalism for arithmetic D -modules 132.4. Complements on the cohomology of arithmetic D -modules 162.5. Equivariant holonomic D -modules 172.6. Intermediate extension and the weight theory 192.7. Nearby and vanishing cycles 202.8. Universal local acyclicity 212.9. Local monodromy of an overconvergent F -isocrystal 232.10. Hyperbolic localization for arithmetic D -modules 253. Geometric Satake equivalence for arithmetic D -modules 263.1. The Satake category 263.2. Fusion product 283.3. Hypercohomology functor 303.4. Semi-infinite orbits 313.5. Tannakian structure and the Langlands dual group 333.6. The full Langlands dual group 344. Bessel F -isocrystals for reductive groups 374.1. Kloosterman F -isocrystals for reductive groups 374.2. Comparison between Kl dRˇ G and Kl rigˇ G dRˇ G and Be ˇ G F -isocrystals for reductive groups 494.5. Monodromy groups 51 Date : December 20, 2019.
5. Applications 545.1. Functoriality of Bessel F -isocrystals 555.2. Hypergeometric F -isocrystals 565.3. Bessel F -isocrystals for classical groups 575.4. Frobenius slopes of Bessel F -isocrystals 62Appendix A. A 2-adic proof of Carlitz’s identity and its generalization 63References 691. Introduction
Bessel equations and Kloosterman sums.1.1.1.
The classical Bessel differential equation (of rank n ) with a parameter λ (1.1.1.1) (cid:18) x ddx (cid:19) n ( f ) − λ n x · f = 0has a unique solution which is holomorphic at 0 :(1.1.1.2) I ( S ) n − exp λ (cid:18) z + z + · · · + z n − + xz · · · z n − (cid:19) dz · · · dz n − (2 πi ) n − z · · · z n − = X r ≥ r !) n ( λ n x ) r . One may reinterpret this fact using the language of algebraic D -modules as follows. Let K be a field ofcharacteristic zero. The Bessel equation (1.1.1.1) can be converted to a connection Be n on the rank n trivialbundle on the multiplicative group G m,K (1.1.1.3) Be n : ∇ = d + . . .
00 0 1 . . . . . . λ n x . . . dxx , which we call the Bessel connection (of rank n ) . On the other hand, we consider the following diagram(1.1.1.4) G nm add ! ! ❈❈❈❈❈❈❈ mult } } ④④④④④④④④ G m A , where add (resp. mult) denotes the morphism of taking sum (resp. product) of n coordinates of G nm , anddefine the Kloosterman D -module on G m,K as(1.1.1.5) Kl dR n := R n − mult ! (add ∗ ( E λ )) , where(1.1.1.6) E λ = ( O A K , ∇ = d − λdx )is the exponential D -module on A K . With these notations, the fact that (1.1.1.2) is a solution of (1.1.1.1)reflects an isomorphism of algebraic D -modules on G m,K Be n ≃ Kl dR n . The connection Be n is regular singular at 0 with a unipotent monodromy with a single Jordan block, andis irregular at ∞ with irregularity = 1. Its differential Galois group was calculated by Katz [55]. ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 3 There is a parallel theory in positive characteristic. Let p a prime number. For every finite extension F q / F p and a ∈ F × q , the Kloosterman sum
Kl( n ; a ) in n -variables is defined by (1.1.2.1) Kl( n ; a ) = (cid:18) − √ q (cid:19) n − X z i ∈ F × q exp (cid:18) πip Tr F q / F p (cid:0) z + z + · · · + z n − + az · · · z n − (cid:1)(cid:19) . It admits a sheaf-theoretic interpretation by Deligne [37]. Namely, the analog of the exponential D -modulein positive characteristic is the Artin-Schreier sheaf AS ψ on A F p associated to a non-trivial character ψ : F p → Q ℓ ( µ p ) × . Deligne defined the Kloosterman sheaf Kl n as the following complex on G m, F p :(1.1.2.2) Kl n = R mul ! (add ∗ (AS ψ ))[ n − (cid:18) n − (cid:19) , and showed the following properties ([37] 7.4, 7.8):(i) Fix an embedding ι : Q ℓ ( µ p ) → C such that ιψ ( x ) = exp(2 πix/p ) for x ∈ F p . The Frobenius traceof Kl n at each closed point a ∈ F × q is equal to the Kloosterman sum Kl( n ; a ).(ii) The complex Kl n is concentrated in degree 0 and is an irreducible local system of rank n and ofweight 0, which implies the Weil bound of the Kloosterman sum | Kl( n ; a ) | ≤ n .(iii) The sheaf Kl n is tamely ramified at 0, and the monodromy is unipotent with a single Jordan block.(iv) The sheaf Kl n is wildly ramified at ∞ with Swan conductor Sw ∞ (Kl n ) = 1.In ([56] § 11), Katz calculated the (global) geometric and arithmetic monodromy group of Kl n as follow:(1.1.2.3) G geo (Kl n ) = G arith (Kl n ) = Sp n n even, SL n pn odd, SO n p = 2 , n odd, n = 7 ,G p = 2 , n = 7 . Surprisingly, the exceptional group G appears as the monodromy group. In 70’s [44], Dwork showed that there exists a Frobenius structure on the Bessel connection (1.1.1.3)whose Frobenius traces give the Kloosterman sum. Here a
Frobenius structure is a horizontal isomorphismbetween the Bessel connection and its pullback by the “Frobenius endomorphism” F : G m,K → G m,K over K defined by x x p . Although the Bessel connection is an algebraic connection, such a horizontal isomorphismis not algebraic but of p -adic analytic nature.To explain Dwork’s result, we need to introduce certain ring of p -adic analytic functions. We set K = Q p ( µ p ), equipped with a p -adic valuation | - | p normalised by | p | p = p − , and denote by A † the ring of p -adicanalytic functions with a convergence radius > A † = (cid:26) + ∞ X n =0 a n x n | a n ∈ K, ∃ ρ > , lim n → + ∞ | a n | p ρ n = 0 (cid:27) . This ring is called the ring of p -adic analytic functions on P overconvergent along {∞} by Berthelot [21].We take an algebraic closure K of K and fix an isomorphism ι : K → C . There exists a unique element π of K which satisfies π p − = − p and corresponds to the character exp 2 πi ( − p ) : F p → C × (cf. 2.1.5(i)). Theorem 1.1.4 (Dwork, Sperber [44, 78, 79]) . Let n be an integer prime to p and set λ = − π as above.There exists a unique ϕ ( x ) ∈ GL n ( A † ) such that The sum (1.1.2.1) is slightly different from the standard definition by a factor ( − √ q ) n − . DAXIN XU, XINWEN ZHU (i)
The matrix ϕ satisfies the differential equation: x dϕdx ϕ − + ϕ . . .
00 0 1 . . . ... ... . . . . . . ... . . . λ n x . . . ϕ − = p . . .
00 0 1 . . . ... ... . . . . . . ... . . . λ n x p . . . . That is, ϕ defines a horizontal isomorphism F ∗ (Be n ) ∼ −→ Be n . (ii) For a ∈ F × q , we have ι Tr ϕ a = Kl( n ; a ) , where ϕ a = Q deg( a ) − i =0 ϕ ( e a p i ) and e a ∈ K denotes theTeichmüller lifting of a . (iii) If { α , · · · , α n } denote the eigenvalues of ϕ a , then we have | α i | p = p n +1 − i deg( a ) after reordering α i . Overconvergent ( F -)isocrystals on a variety X over F p are p -adic analogues of ℓ -adic (Weil) local systems.Roughly speaking, they are vector bundles with an integrable connection and a Frobenius structure on somerigid analytic space associated to certain lifting of X to characteristic zero. The data (Be n , ϕ ) form anoverconvergent F -isocrystal on G m, F p (relative to K ), which we call the Bessel F -isocrystal (of rank n ) anddenote by Be † n . By (ii), Be † n is the p -adic companion of the Kloosterman sheaf Kl n in the sense of [38, 3].1.2. Generalization for reductive groups.1.2.1.
Recently, there are two generalizations of above results (corresponding to the GL n -case) for reductivegroups from different perspectives. The first one is due to Frenkel and Gross [47] from the viewpoint of theBessel equations. Namely, for each (split) reductive group ˇ G over a field K of characteristic zero, Frenkel-Gross wrote down an explicit ˇ G -connection Be ˇ G on G m , which specializes to Be n when ˇ G = GL n . We willcall Be ˇ G the Bessel connection of ˇ G in this paper. Another one, due to Heinloth, Ngô and Yun [52], isfrom the viewpoint of the Kloosterman sums. Namely, the authors explicitly constructed, for each (split)reductive group G over the rational function field F p ( t ), a Hecke eigenform of G , and defined Kl ˇ G as itsLanglands parameter, which is an ℓ -adic ˇ G -local system on G m that specializes to Kl n if ˇ G = GL n . Theauthors call Kl ˇ G the Kloosterman sheaf of ˇ G .The main subject of this article is to study the p -adic aspects of this theory and to unify the previous twoconstructions. Our main results can be summarized as follows:(i) we construct the Frobenius structure on Be ˇ G and obtain the Bessel F -isocrystal Be † ˇ G of ˇ G , which isthe p -adic companion of Kl ˇ G in appropriate sense;(ii) we calculate its geometric and arithmetic monodromy group;(iii) we show that the Frobenius Newton polygons of Be ˇ G (and therefore Kl ˇ G ) are generically ordinaryand when ˇ G is classical or G they are everywhere ordinary on | G m, F p | .It turns out that our p -adic theory also has applications to the ℓ -adic theory and the arithmetic property ofexponential sums associated to Kl ˇ G . Namely,(iv) we obtain a different (and more conceptual) calculation of the monodromy group of Kl ˇ G ((1.1.2.3)and one of the main results of [52]), based on the structure of the connection Be ˇ G ;(v) we prove a conjecture of Heinloth-Ngô-Yun on the functoriality of Kloosterman sheaves ([52] conjec-ture 7.3) and therefore obtain identities between different exponential sums.We discuss these results in more details in the sequel. Let ˇ G be a split almost simple group over a field K of characteristic zero. Fix a Borel subgroupˇ B ⊂ ˇ G , and a principal nilpotent element N in the Lie algebra of ˇ B . Let E denote a basis vector of thelowest root space in ˇ g = Lie( ˇ G ). In [47], Frenkel and Gross considered a connection on the trivial ˇ G -bundleover G m :(1.2.2.1) ∇ = d + N dxx + λ h Edx,
ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 5 where x is a coordinate of G m , λ ∈ K is a parameter and h is the Coxeter number of ˇ G . We may regard itas a tensor functor from the category of representations of ˇ G to the category of connections on the trivialbundles on G m (1.2.2.2) Be ˇ G : Rep ( ˇ G ) → Conn( G m ) , ( ρ : ˇ G → GL( V )) → d + dρ ( N dxx + λ h Edx ) . This connection is rigid and has a regular singularity at 0 and an irregular singularity at ∞ . Let G be a split almost simple group over F p ( t ) whose dual group is ˇ G . In [52], Heinloth-Ngô-Yun wrote down a cuspidal Hecke eigenform f on G , and defined the Kloosterman sheaf Kl ˇ G for ˇ G as theLanglands parameter of f . For simplicity, we assume that G is simply-connected. If we fix opposite Iwahorisubgroups I (0) op ⊂ G ( O ) and I (0) ⊂ G ( O ∞ ) at 0 , ∞ , and a non-degenerate character ϕ : I (1) /I (2) → Q ( µ p ) × , where I ( i ) denotes the i th step in the Moy-Prasad filtration of I (0), then f is the unique (up toscalar) non-zero function on G ( F p ( t )) \ G ( A ) that is, • invariant under G ( O x ) for every place x = 0 , ∞ ; • invariant under I (0) op at 0; • ( I (1) , ϕ )-equivariant at ∞ .Then Heinloth-Ngô-Yun defined Kl ˇ G : Rep ( ˇ G ) → LocSysm( G m, F p ) as a tensor functor from the categoryof representations of ˇ G (over Q ℓ ) to the category of ℓ -adic local systems on G m, F p , such that for every V ∈ Rep ( ˇ G ) and every a ∈ | G m, F p | , T V,a ( f ) = Tr(Frob a , (Kl ˇ G,V ) a ) f where T V,a is the Hecke operator associated to (
V, a ). The actual construction of Kl ˇ G uses the geometricLanglands correspondence (see 4.1.12).Our first main result is the existence of a Frobenius structure on Bessel connections for reductive groups. Theorem 1.2.4 (4.4.4, 5.4.2) . Let K = Q p ( µ p ) , K an algebraic closure of K and set λ = − π as in . (i) There exists a unique ϕ ( x ) ∈ ˇ G ( A † ) satisfying the differential equation x dϕdx ϕ − + Ad ϕ ( N + λ h xE ) = p ( N + λ h x p E ) and such that via a (fixed) isomorphism K ≃ Q ℓ , for every a ∈ F × q and V ∈ Rep ( ˇ G )Tr( ϕ a , V ) = Tr(Frob a , (Kl ˇ G,V ) a ) , where ϕ a = Q deg( a ) − i =0 ϕ ( e a p i ) and e a ∈ K denotes the Teichmüller lifting of a .In particular, the analytification of the Bessel connection Be ˇ G on G an m,K is overconvergent and underliesa tensor functor from Rep ( ˇ G ) to the category of overconvergent F -isocrystals on G m, F p : (1.2.4.1) Be † ˇ G : Rep ( ˇ G ) → F - Isoc † ( G m, F p /K ) , which can be regarded as the p -adic companion of Kl ˇ G . (ii) Let ρ ∈ X • ( T ) = X • ( ˇ T ) be the half sum of positive roots. When ˇ G is of type A n , B n , C n , D n or G , forevery a ∈ | G m, F p | , the set of p -adic order of eigenvalues of ϕ a ∈ ˇ G ( K ) (also known as the Frobenius slopesat a ) is same as that of ρ ( p deg( a ) ) ∈ ˇ G ( K ) . When ˇ G is of other exceptional type, the same assertion holdsgenerically on | G m, F p | . Remark 1.2.5. (i) For a ˇ G -valued overconvergent F -isocrystal on a smooth variety X over F p , we say itsNewton polygon is ordinary at a if the Frobenius slopes at a are given by ρ (in the above sense). We expectthat the Newton polygons of Be † ˇ G are always ordinary at each closed point of G m, F p .(ii) V. Lafforgue [63] showed that ρ is the upper bound for the p -adic valuations of Hecke eigenvalues ofHecke eigenforms (cf. 5.4.1 for a precise statement). Drinfeld and Kedlaya [42] proved an analogous resultfor the Frobenius slopes of an indecomposable convergent F -isocrystal on a smooth scheme. DAXIN XU, XINWEN ZHU
In ([47] Cor. 9,10), Frenkel and Gross calculate the differential Galoisgroup G alg of Be ˇ G over K , which we list in the following table (up to central isogeny):(1.2.6.1) ˇ G G gal A n A n A n − , C n C n B n , D n +1 ( n ≥ B n E E E E E , F F B , D , G G . If we denote by G geo the geometric monodromy group of Be † ˇ G over K , there exists a canonical homomorphism(1.2.6.2) G geo → G gal . Theorem 1.2.7 (4.5.2) . (i) If either ˇ G is not of type A n , or p > , the above morphism is an isomorphism. (ii) If p = 2 and ˇ G = SL n +1 , then G geo ≃ SO n +1 if n = 3 and G geo ≃ G if n = 3 . (iii) The arithmetic monodromy group G arith of Be † ˇ G is isomorphic to G geo . In fact, the second part of the theorem follows from the first part and theorem 1.2.8(ii) below. Bycompanion, this theorem allows us to recover Katz’s result on the monodromy group of Kl n (1.1.2.3) andHeinloth-Ngô-Yun’s result on the geometric monodromy group of Kl ˇ G [52] in a different way. For instance, the G -symmetry on Be † when p = 2 (1.1.2.3) appears naturally in our approach, compared with Katz’ originalapproach via point counting. In addition, we also avoid some difficult geometry related to quasi-minusculeand adjoint Schubert varieties, as analyzed in [52].Inspired by the rigidity properties of hypergeometric sheaves proved by Katz [57], Heinloth, Ngô and Yunconjectured certain functoriality between Kloosterman sheaves for different groups ([52] conjecture 7.3). Asan application of our p -adic theory, we prove this conjecture. Theorem 1.2.8 (5.1.4, 5.3.10) . (i) For ˇ G ′ ⊂ ˇ G appearing in the same line in the left column of the abovediagram, Kl ˇ G is isomorphic to the push-out of Kl ˇ G ′ along ˇ G ′ → ˇ G . (ii) If p = 2 , Kl SL n +1 is the push-out of Kl SO n +1 along SO n +1 → SL n +1 . The above theorem allows us to identify various exponential sums associated to Kloosterman sheavesdefined by different groups. Here are some examples (cf. corollary 5.3.11):(i) When ˇ G = SO ≃ PGL , we have the following identity for a ∈ F × q : (cid:18) X x ∈ F × q ψ (Tr F q / F p ( x + ax )) (cid:19) − q (1.2.9.1) = − √ q X x ,x ∈ F × q ψ (cid:18) Tr F q / F p ( x + x + ax x ) (cid:19) , p = 2 , G ( ψ − , ρ − ) X x x x =4 ay,x i ∈ F × q ψ (cid:18) Tr F q / F p ( x + x + x − y ) (cid:19) ρ − ( y ) , p > ψ ( − ) = exp πip ( − ), ρ denotes the quadratic character of F × q and G ( ψ − , ρ − ) the associated Gausssum. The identity is due to Carlitz [26] when p = 2 and Katz ([58] § 3) when p >
2. Our method iscompletely different from these works.
ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 7 (ii) For n ≥
2, via the inclusion SO n +1 → SO n +2 , for a ∈ F × q , we have X uv = a,u,v ∈ F × q (cid:18)(cid:0) X x ∈ F × q ψ (Tr F q / F p ( x + ux )) (cid:1) − q (cid:19)(cid:18) X x i ∈ F × q ψ (Tr F q / F p ( x + · · · + x n − + vx · · · x n − )) (cid:19) (1.2.9.2)= − √ q (cid:18) X x i ∈ F × q ψ (cid:18) Tr F q / F p ( x + x + · · · + x n + a x + x x x · · · x n ) (cid:19) − q n − (cid:19) . One can obtain other identities between different exponential sums, whose sheaf-theoretic incarnations wereobtained by Katz [58].
We have partial results about the local monodromy of Be † ˇ G (and Kl ˇ G ) at ∞ . Namely, we show thatthe nilpotent monodromy operator is trivial and the local Galois representation ϕ ∞ : I ∞ → ˇ G is a simplewild parameter in the sense of Gross-Reeder [51] § 6 (see corollary 4.5.7 (ii)). If p ∤ n , one can even showthat the local monodromy of Be † n at ∞ coincides with the Galois representations constructed in [51] § 6.2,by studying the solutions of Bessel differential equation (1.1.1.1) at ∞ . (The corresponding ℓ -adic statementfor Kl n was proved by Fu and Wan [48] thm. 1.1.)The above result, together with theorem 1.2.8(ii), implies that when p = 2 and n is an odd integer, theassociated local Galois representation for Be † SO n (and Kl SO n ) at ∞ coincides with the simple wild parameterconstructed in [51] § 6.3. For example, the image of the inertia group I ∞ in the case ˇ G = SO is isomorphicto A . Together with Be † SO , Std ≃ Be † SL , Sym (4.1.9.1), this allows us to recover André’s result on the localmonodromy group of Be † at ∞ when p = 2 ([9] sections 7, 8).1.3. Strategy of the proof and the organization of the article.1.3.1.
Now we outline the strategy to prove the above results. Part (i) of theorem 1.2.4 follows by combiningfollowing three ingredients:(i) We first mimic Heinloth-Ngô-Yun’s construction to produce a ˇ G -valued overconvergent F -isocrystalKl rigˇ G on G m, F p and a ˇ G -bundle with connection Kl dRˇ G on G m,K (section 4.1). A key step is to developthe geometric Satake equivalence for arithmetic D -modules, which we will discuss latter (1.3.5).(ii) Then we show that the overconvergent isocrystal Kl rigˇ G is isomorphic to the analytification of theˇ G -connection Kl dRˇ G (section 4.2) by comparing certain relative de Rham cohomologies and relativerigid cohomologies.(iii) We strengthen a result of the second author [89] to identify Kl dRˇ G with Be ˇ G (section 4.3). The local monodromy of Be † ˇ G at 0 is principal unipotent, which implies that its geometric monodromy G geo contains a principal SL . This puts strong restrictions on the possible Dynkin diagrams of G geo (cf.4.5.4 for a possible list). A result of Baldassarri [13] (cf. [9] 3.2), which implies that the p -adic slope of Be † ˇ G at ∞ is less or equal to the formal slope of Be ˇ G at ∞ , allows us to exclude the case G geo = PGL (or SL )in most cases. Together with certain symmetry on Be † ˇ G , this implies theorem 1.2.7(i). The analogous functoriality for Bessel connections Be ˇ G (theorem 1.2.8(i)) follows from their definition.Then we deduce the functoriality between Be † ˇ G ’s by theorems 1.2.4(i) and 1.2.7(i). For theorem 1.2.8(ii) (andtherefore theorem 1.2.7(ii)), we construct an isomorphism between the maximal slope quotients of Be † n +1 and Be † SO n +1 , Std using a refinement of Dwork’s congruences [43] in the 2-adic case. Then we conclude thatBe † n +1 ≃ Be † SO n +1 , Std by a recent theorem of Tsuzuki [82] (cf. appendix A). Since Be † ˇ G is the p -adiccompanion of Kl ˇ G , theorem 1.2.8 follows. By functoriality, we reduce theorem 1.2.4(ii) to the corresponding assertion for (Frobenius) Newtonpolygon of Be † SL n , Std and of Be † SO n +1 , Std , which are isomorphic to Be † n and a hypergeometric overconvergent DAXIN XU, XINWEN ZHU F -isocrystal [67] respectively. In these cases, the assertion follows from the results of Dwork, Sperber andWan [44, 79, 84]. As mentioned above, in order to carry through the first step of 1.3.1, we need to establish a versionof the geometric Satake equivalence for arithmetic D -modules. This is based on the recent developmentof the six functors formalism, weight theory and nearby/vanishing cycle functors for arithmetic D -modules[28, 29, 5, 6, 4]. We will review these theories in subsections 2.1–2.7.To state our result, we first introduce some notations. Let k be a finite field with q = p s elements and K a finite extension of Q q . Suppose that there exists a lift σ : K → K of the t -th Frobenius automorphism of k for some integer t . Let G be a split reductive group over k , ˇ G its Langlands dual group over K , Gr G theaffine Grassmannian of G , and L + G the positive loop group of G .Given a k -scheme X , one may consider the category Hol( X/K ) of holonomic arithmetic D -modules on X and the category Hol( X/K F ) of objects of Hol( X/K ) with a Frobenius structure, which are the analoguesof the category of ℓ -adic sheaves on X k and the category of Weil sheaves on X respectively. We denote byHol L + G (Gr G /K ) (resp. Hol L + G (Gr G /K F )) the category of L + G -equivariant objects in Hol(Gr G /K ) (resp.Hol(Gr G /K F )).The geometric Satake equivalence (for geometric coefficients) states that the category Hol L + G (Gr G /K ) isa neutral Tannakian category over K whose Tannakian group is ˇ G (3.5.1). The Tannakian structure and theFrobenius structure on Hol L + G (Gr G /K F ) allows us to define a homomorphism ι : Z → Aut( ˇ G ( K )) (3.6.2)and hence a semi-direct product ˇ G ( K ) ⋊ Z . Theorem 1.3.6. (i) (Geometric coefficients 3.5.1)
There exists a natural equivalence of monoidal categoriesbetween
Hol L + G (Gr G /K ) and Rep ( ˇ G ) . (ii) (Arithmetic coefficients 3.6.7) There exists an equivalence of monoidal categories between
Hol L + G (Gr G /K F ) and the category Rep ◦ K,σ ( ˇ G ( K ) ⋊ Z ) of certain σ -semi-linear representations of ˇ G ( K ) ⋊ Z (cf. ). Although the strategy of the proof of this theorem is same as the ℓ -adic case, we need to establishsome foundational results in the setting of arithmetic D -modules. We introduce a notion of universal localacyclicity (ULA) for arithmetic D -modules and discuss its relation with the nearby/vanishing cycle functorsintroduced by Abe-Caro and Abe [5, 4] in subsection 2.8. We also prove a version of Braden’s hyperboliclocalization theorem is this setting in subsection 2.10.Recall that there are motivic versions of the geometric Satake equivalence [90, 75]. The above theoremcan be regarded as their p -adic realization. (But as far as we know, there is no general construction of therealization functor as we need so the above theorem is not a formal consequence of loc. cit. .) On the otherhand, there is a very recent work of R. Cass [30] on the geometric Satake equivalence for perverse F p -sheaves.It would be very interesting to see whether there is a version of the geometric Satake equivalence for some Z p -coefficient sheaf theory, which after inverting p and mod p specializes to our version and Cass’ versionrespectively.We hope our article will lead further investigation of the p -adic aspect of the geometric Langlands programin the future. We briefly go over the organization of this article. Section 2 contains a review of and some comple-ments on the theory of arithmetic D -modules and overconvergent ( F -)isocrystals. In section 3, we establishthe geometric Satake equivalence for arithmetic D -modules (1.3.6). Subsections 4.1-4.4 are devoted to theproof of theorem 1.2.4(i) (cf. 1.3.1). We calculate the monodromy group of Be † ˇ G in subsection 4.5 (cf. theo-rem 1.2.7 and 1.3.2). In subsection 5.1, we prove the functoriality of Bessel F -isocrystals and of Kloostermansheaves (cf. theorem 1.2.8(i) and 1.3.3). In subsections 5.2 and 5.3, we identify the Bessel F -isocrystals forclassical groups with certain hypergeometric differential equations studied by Katz and Miyatani [57, 67].In particular, we obtain identities in 1.2.9. In the last subsection (5.4), we study the Frobenius Newtonpolygon of Be † ˇ G and prove theorem 1.2.4(ii). Appendix A is devoted to a proof of theorem 1.2.8(ii) from theperspective of p -adic differential equations. ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 9 In this article, we fix a prime number p . Let s be a positive integer and set q = p s . Let k be aperfect field of characteristic p , k an algebraic closure of k and R a complete discrete valuation ring withresidue field k . We set K = Frac( R ). We fix an algebraic closure K of K . We assume moreover that the s -th Frobenius endomorphism k ∼ −→ k, x x q lifts to an automorphism σ : R ∼ −→ R .By a k -scheme (resp. R -scheme ), we mean a separated scheme of finite type over k (resp. over R ). Acknowledgement.
We would like to thank Benedict Gross, Shun Ohkubo, Daqing Wan, Liang Xiaoand Zhiwei Yun for valuable discussions. X. Z. is partially supported by the National Science Foundationunder agreement Nos. DMS-1902239.2.
Review and complements on arithmetic D -modules Overconvergent ( F -)isocrystals and their rigid cohomologies. In this subsection, we brieflyrecall the definition of overconvergent (resp. convergent) isocrystal following [20].
Let X be a k -scheme. A frame of X is a quadruple ( Y, j, P , i ) (written as ( Y, P ) for short) consistingof an open immersion j : X → Y of k -schemes, and a closed immersion i : Y → P , where P is a separatedformal R -scheme which is smooth over Spf( R ) in a neighborhood of X . We denote by P rig the rigid analyticspace associated to P and by ] X [ P , ] Y [ P the tube of X, Y in P rig respectively ([20] § 1). A strict neighborhood of ] X [ P in ] Y [ P is an admissible open subspace V of ] Y [ P such that V ∪ ] Y \ X [ P forms an admissible covering of ] Y [ P ([20] 1.2). There exists an exact functor j † from the category Ab ( V )of abelian sheaves on V to itself ([20] 2.1.1), defined by(2.1.1.1) j † E = lim −→ U j U,V ∗ j − U,V ( E ) , where the inductive limit is taken over all strict neighborhoods j U,V : U → V of ] X [ P in V . It is knownthat j † O V is coherent as a sheaf of rings.The notion of a morphism of frames is naturally defined, and a morphism u : ( Y ′ , P ′ ) → ( Y, P ) of framesinduces a tensor functor: u ∗ : (Coherent j † O ] Y [ P -modules) → (Coherent j † O ] Y ′ [ P ′ -modules) . We denote by Conn( j † O ] Y [ P ) the category of coherent j † O ] Y [ P -modules M equipped with a K -linearmorphism ∇ : M → M ⊗ j † O ] Y [ P j † Ω Y [ P satisfying the Leibniz rule and the usual integrability condition. For n = 1 , ,
3, we denote by P n the fiber product of n -copies of P over Spf( R ). Then ( Y, P n ) is aframe of X and we have projections p i : ( Y, P ) → ( Y, P ) ( i = 1 , p ij : ( Y, P ) → ( Y, P ) (1 ≤ i < j ≤ Y, P ) → ( Y, P ).We denote by Isoc † ( X, Y /K ) the category of pairs ( M , ε ) consisting of a coherent j † O ] Y [ P -module M and an isomorphism ε : p ∗ ( M ) ∼ −→ p ∗ ( M )satisfying ∆ ∗ ( ε ) = id and p ∗ ( ε ) = p ∗ ( ε ) ◦ p ∗ ( ε ). Such a pair is called an isocrystal on X overconvergentalong Y − X (relative to K ) . This category is independent of the choice of P up to canonical equivalences([20] 2.3.1).When Y = X , we have j † O ] Y [ P = O ] X [ P . Such a pair is also called a convergent isocrystal on X/K . Thecategory Isoc † ( X, X/K ) is simply denoted by Isoc(
X/K ).When X is a compactification of X over k , the category Isoc † ( X, X/K ) is independent of the choice of X up to canonical equivalences ([20] 2.3.5) and is simply called the category of overconvergent isocrystals on X/K , denoted by Isoc † ( X/K ). There exists an exact and fully faithful functor ([20] 2.2.5, 2.2.7)Isoc † ( X, Y /K ) → Conn( j † O ] Y [ P ) , ( M , ε ) ( M , ∇ ) . When X = Y (resp. Y = X is a compactification of X ), we say that ( M , ∇ ) is convergent (resp. overconvergent ) if it is contained in the essential image of the above functor.Let M be an overconvergent isocrystal on X relative to K and M ⊗ j † O ] X [ P j † Ω • ] X [ P the associated deRham complex with respect to a frame ( X, P ). The rigid cohomology R Γ rig ( X/K, M ) is defined by(2.1.3.1) R Γ rig ( X/K, M ) = R Γ(] X [ P , M ⊗ j † O ] X [ P j † Ω • ] X [ P ) . The category Isoc † ( X/K ) is functorial with respect to pullbacks ([20] 2.3.6). The absolute s -thFrobenius morphism F X : X → X and endomorphism σ : K → K induce the Frobenius pullback functor:(2.1.4.1) F ∗ X : Isoc † ( X/K ) → Isoc † ( X/K ) . An overconvergent F -isocrystal on X/ ( K, σ ) (or simply X/K ) is an overconvergent isocrystal M togetherwith an isomorphism ϕ : F ∗ X ( M ) ∼ −→ M , called ( s -th) Frobenius structure of M .We denote by F - Isoc † ( X/K ) the category of overconvergent F -isocrystals on X/K and by Isoc †† ( X/K )the thick full subcategory of Isoc † ( X/K ) generated by those that can be endowed with an s ′ -th Frobeniusstructure for some integer s | s ′ . In the following, we explain some examples of overconvergent isocrystals.(i)
Dwork F -isocrystal. Let k = F p (i.e. s = 1), K = Q p ( µ p ), R = O K and σ = id. We choose π ∈ K such that π p − = − p and take P = b P R , Y = P k , X = A k . Then ] Y [= b P , rig R and ] X [ is the closed unit disc.If t denotes a coordinate of A , the connection on j † O ] Y [ defined by ∇ = d + πdt, is overconvergent and is called Dwork isocrystal , denoted by A π .By considering the lifting of the Frobenius of P to R given by t → t p , F ∗ A k ( A π ) is the module j † O ] Y [ equipped with the connection ∇ σ defined by ∇ σ = d + πpt p − dt. We define a Frobenius structure ϕ : F ∗ A k ( A π ) → A π by the multiplication by θ π ( x ) = exp( π ( x − x p )), whichis a section of j † O ] Y [ . This gives Dwork F -isocrystal associated to π on A k /K .There exists a unique nontrivial additive character ψ : F p → K × satisfying ψ (1) = 1 + π mod π . For each x ∈ F p , we denote by e x the Teichmüller lifting of x in Q p . Then θ ψ ( e x ) = ψ ( x ) ([18] 1.4). So theFrobenius trace function of A π is equal to ψ ◦ Tr F q / F p ( − ). We also denote A π by A ψ , as it plays a similarrole of Artin-Schreier sheaf associated to ψ in the ℓ -adic theory.(ii) Kummer F -isocrystal. Let k be a finite field with q = p s elements. Set K = Q q , R = O K and σ = id. We choose a ∈ R and take P = b P R , Y = P k and X = G m,k . If x denotes a coordinate of G m , theconnection on j † O ] Y [ defined by ∇ = d − a dxx , is overconvergent, denoted by K a . With the lifting of Frobenius as in (i), F ∗ G m ( K a ) is the module j † O ] Y [ equipped with connection ∇ σ defined by ∇ σ = d − ap dxx . The isocrystal K a has a Frobenius structure if and only if a ∈ q − Z . This Frobenius structure is given bymultiplication by t ( q − a . Then we obtain the Kummer F -isocrystal K a .Let χ be a character of k × such that χ ( x ) = e x ( q − a , where e x denotes the Teichmüller lifting of x . Wealso denote K a by K χ because the Frobenius trace function of K a is equal to χ ◦ Nm k ′ /k . ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 11 (Co)specialization morphism for de Rham and rigid cohomologies. In this subsection, we re-view the specialization and cospecialization morphisms between the de Rham and rigid cohomology following([14] § 1) and show the compatibility of these two morphisms in proposition 2.2.5.The results of this subsection will mainly be used in subsection 4.2.
In this subsection, X denotes a smooth R -scheme of pure relative dimension d and X k (resp. X K )its special (resp. generic) fiber. We use the corresponding calligraphic letter X to denote the rigid analyticspace X an K associated to X K and the corresponding gothic letter X to denote the p -adic completion of X .We denote by X rig the rigid generic fiber of X . Let ε : X → X K denote the canonical morphism of topoi.Let ( M, ∇ ) be a coherent O X K -module endowed with an integrable connection (relative to K ). Wedenote by ( M an , ∇ an ) its pullback to X along ε . Then the canonical morphism of de Rham complexes ε − ( M ⊗ O XK Ω • X K ) → M an ⊗ O X Ω •X induces a morphism from algebraic de Rham cohomology to analyticde Rham cohomology(2.2.1.1) R Γ dR ( X K , ( M, ∇ )) = R Γ( X K , M ⊗ O XK Ω • X K ) → R Γ( X , M an ⊗ O X Ω •X ) = R Γ an ( X , ( M an , ∇ an )) . We assume that there exists a smooth proper R -scheme X and an open immersion j : X → X . Let X be the p -adic completion of X . Then the two rigid spaces X rig and X = X an K are isomorphic, and X rig is thetube ] X k [ X of X k in X . In particular, X is a strict neighborhood of X rig in X rig . We denote by Conn( X K )(resp. Conn( X )) the category of coherent O X K -modules with an integrable connection.We associate to M an a j † O X rig -module M † = j † ( M an ) (2.1.1.1), endowed with the corresponding connec-tion. In this setting, we have the following diagram (2.1.3):(2.2.2.1) Conn( X K ) ( − ) an / / ( − ) † ( ( Conn( X ) j † / / Conn( j † O X rig ) | X rig / / Conn( O X rig ) F - Isoc † ( X/K ) / / Isoc †† ( X/K ) (cid:31) (cid:127) / / Isoc † ( X/K ) | X rig / / ?(cid:31) O O Isoc(
X/K ) ?(cid:31) O O where the vertical arrows are fully faithful (2.1.3). When X k \ X k is a divisor, the functor | X rig is exact andfaithful ([21] 4.3.10). In the following, we assume moreover that the connection on M † is overconvergent (see 2.1.3 for thedefinition). The rigid cohomology R Γ rig ( X k /K, M † ) (2.1.3.1) can be calculated by(2.2.3.1) R Γ rig ( X k /K, M † ) ∼ −→ R Γ( X , M † ⊗ O X Ω •X ) . The adjoint morphism id → j † (2.1.1.1) induces a canonical morphism on X (2.2.3.2) M an ⊗ O X Ω •X → M † ⊗ O X Ω •X . By composing with (2.2.1.1), we deduce a canonical morphism, denoted by ρ M and called specializationmorphism for de Rham and rigid cohomologies:(2.2.3.3) ρ M : R Γ dR ( X K , ( M, ∇ )) → R Γ rig ( X k /K, M † ) . Let R Γ ] X k [ be the (derived) functor of local sections supported in the tube ] X k [ X on X (or on X ) ([20]2.1.6). The rigid cohomology with compact supports and coefficients in M † is defined as: R Γ rig , c ( X k /K, M † ) = R Γ( X , R Γ ] X k [ ( M an ⊗ Ω •X )) . (2.2.3.4)The canonical morphism(2.2.3.5) R Γ ] X k [ ( M an ⊗ Ω •X ) → M an ⊗ Ω •X and (2.2.3.2) induce a morphism(2.2.3.6) ι rig : R Γ rig , c ( X k /K, M † ) → R Γ rig ( X k /K, M † ) . We recall the definition of de Rham cohomology with compact supports and coefficients in ( M, ∇ )and the cospecialization morphism, following ([14] 1.8 and [11] Appendix D.2).Let I be the ideal sheaf of the reduced closed subscheme X K − X K in X K . Take a coherent O X K -module M extending M . The connection ∇ extends to a connection on the pro- O X K -module ( I n M ) n ([11] D.2.12).This allows us to define the de Rham pro-complex I • M ⊗ O XK Ω • X K := ( I n M ) n ⊗ Ω • X K . The algebraic deRham cohomology with compact supports and coefficients in ( M, ∇ ), denoted by R Γ dR , c ( X K , ( M, ∇ )), isdefined as ([11] D.2.16) R Γ dR , c ( X K , ( M, ∇ )) = R Γ( X K , R lim ←− I • M ⊗ Ω • X K )(2.2.4.1) ≃ R lim ←− R Γ( X K , I • M ⊗ Ω • X K ) . Let j K denote the open immersion X K → X K . There exists a canonical isomorphism on X K :(2.2.4.2) j ∗ K ( R lim ←− ( I • M ⊗ Ω • X K )) ∼ −→ M ⊗ Ω • X K . We deduces from its adjoint R lim ←− ( I • M ⊗ Ω • X K ) → R j K ∗ ( M ⊗ Ω • X K ) a canonical morphism:(2.2.4.3) ι dR : R Γ dR , c ( X K , ( M, ∇ )) → R Γ dR ( X K , ( M, ∇ ))By the rigid GAGA, there are canonical isomorphisms R lim ←− R Γ( X K , I • M ⊗ Ω • X K ) ∼ −→ R lim ←− R Γ( X , I • M an ⊗ Ω •X ) ∼ −→ R Γ( X , R lim ←− I • M an ⊗ Ω •X ) . (2.2.4.4)We denote the right hand side by R Γ an , c ( X , ( M an , ∇ an )). Let j an be the inclusion X → X . Similarly, thereexists a canonical morphism(2.2.4.5) R lim ←− ( I • M an ⊗ Ω •X ) → R j an ∗ ( M an ⊗ Ω •X ) , which induces a morphism on analytic de Rham cohomologies(2.2.4.6) ι an : R Γ an , c ( X , ( M an , ∇ an )) → R Γ an ( X , ( M an , ∇ an )) . Since ( X , ]( X − X ) k [ X ) is an admissible covering of X , the canonical morphisms(2.2.4.7) R Γ ] X k [ ( R j an ∗ ( E )) → R j an ∗ ( R Γ ] X k [ ( E )) , R Γ ] X k [ ( E ) → R Γ ] X k [ R j an ∗ ( j an ∗ ( E ))are isomorphic for any complex of abelian sheaves E on X (resp. X ). Then (2.2.4.5) induces an isomorphism(2.2.4.8) R Γ ] X k [ ( R lim ←− ( I • M an ⊗ Ω •X )) ∼ −→ R Γ ] X k [ ( R j an ∗ ( M an ⊗ Ω •X )) . The cospecialization morphism , denoted by ρ c ,M , is defined as the composition ρ c ,M : R Γ rig , c ( X k /K, M † ) (2.2.4.7) ≃ R Γ( X , R Γ ] X k [ R j an ∗ ( M an ⊗ Ω •X ))(2.2.4.9) (2.2.4.8) ≃ R Γ( X , R Γ ] X k [ ( R lim ←− ( I • M an ⊗ Ω •X ))) → R Γ( X , R lim ←− ( I • M an ⊗ Ω •X )) (= R Γ an , c ( X , ( M an , ∇ an ))) ≃ R Γ dR , c ( X K , ( M, ∇ )) . Proposition 2.2.5.
With the above notation and assumption, the following diagram is commutative: R Γ rig , c ( X k /K, M † ) ι rig / / ρ c ,M (cid:15) (cid:15) R Γ rig ( X k /K, M † ) R Γ dR , c ( X K , ( M, ∇ )) ι dR / / R Γ dR ( X K , ( M, ∇ )) . ρ M O O ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 13 Proof.
The algebraic de Rham cohomology with compact supports is isomorphic to the analytic one (2.2.4.4).It suffices to show the following diagram is commutative(2.2.5.1) R Γ rig , c ( X k /K, M † ) ι rig / / ρ c ,M (cid:15) (cid:15) R Γ rig ( X k /K, M † ) R Γ an , c ( X , ( M an , ∇ an )) ι an / / R Γ an ( X , ( M an , ∇ an )) O O where right vertical arrow is induced by (2.2.3.2).The morphism R Γ rig , c ( X k /K, M † ) → R Γ an ( X , ( M an , ∇ an )) is induced by the composition on X : R Γ ] X k [ ( R lim ←− ( I • M an ⊗ Ω •X )) → R lim ←− ( I • M an ⊗ Ω •X ) (2.2.4.5) −−−−−→ R j an ∗ ( M an ⊗ Ω •X ) . The restriction of the above morphism to X coincides with the canonical morphism (2.2.3.5), which induces ι rig (2.2.3.6). Then the commutativity of (2.2.5.1) follows. (cid:3) Six functors formalism for arithmetic D -modules. Rigid cohomology theory is a p -adic Weilcohomology for a variety in characteristic p . Overconvergent F -isocrystals are “local systems” in the coeffi-cients theory of rigid cohomology. However, the category of overconvergent F -isocrystals is not stable undercertain cohomological operators. Inspired by the theory of algebraic D -modules, Berthelot introduced thenotion of arithmetic D -modules [21, 22]. A six functor formalism for these coefficients is recently achievedby Caro, Abe and etc.We use the notation of arithmetic D -modules [22]. For a smooth formal R -scheme X and a divisor Z ofthe special fiber of X , let O X , Q ( † Z ) (resp. D † X , Q ( † Z )) denote the sheaf of rings of functions (resp. differentialoperators) on X with singularities overconvergent along Z ([21] 4.2.4). Note that O X , Q ( † Z ) is isomorphic toSp ∗ ( j † O X rig ) for a frame ( X k , X ) of X k − Z (see 2.1.1) ([21] 4.3.2). We omit ( † Z ) if Z is empty. We denote D † X , Q ( † Z ) by D † X , Q ( Z ) (or D † X , Q ( ∞ )) for short. Let us begin by recalling basic notions of p -adic coefficients used in [3]. Let L be an extension of K in K and T = { k, R, K, L } the associated geometric base tuple ([3] 1.4.10, 2.4.14).We will also work in the arithmetic setting ( p -adic coefficients with Frobenius structure). For this purpose,we need to assume moreover that there exists an automorphism L → L extending σ : K → K that we stilldenote by σ , and that there exists a sequence of finite extensions M n of K in L satisfying σ ( M n ) ⊂ M n and ∪ n M n = L . Then we obtain an arithmetic base tuple T F = { k, R, K, L, s, σ } ([3] 1.4.10, 2.4.14). We set L = L σ =1 .Let X be a k -scheme. There exists an L -linear (resp. L -linear) triangulated category D( X/L ) (resp.D(
X/L F )) relative to the geometric base tuple T (resp. arithmetic base tuple T F ). This category is denotedby D bhol ( X/ T ) or D bhol ( X/L ) (resp. D bhol ( X/ T F ) or D bhol ( X/L F )) in ([3] 1.1.1, 2.1.16). When L = K and X isquasi-projective, there exists a classical description of D( X/K ) in terms of arithmetic D -modules introducedby Berthelot [22]: If X → P is an immersion into a smooth proper formal R -scheme P , then D( X/K ) isa full subcategory of D bcoh ( D † P , Q ) with objects satisfy certain finiteness condition called overholonomicity ,certain support condition, and can be equipped with some Frobenius structure (cf. [3] 1.1.1, [5]).The category D( X/L ) (resp. D(
X/L F )) is equipped with a t-structure, called holonomic t-structure , whoseheart is denoted by Hol( X/L ) (resp. Hol(
X/L F )), called category of holonomic modules . These categoriesare analogue to the category of perverse sheaves in the ℓ -adic cohomology theory. The category Hol( X/L ) isNoetherian and Artinian ([3] 1.2.7). We denote by H ∗ the cohomological functor for holonomic t-structure.When X = Spec( k ), there exists an equivalence of monoidal categories between D( X/L ) and the derivedcategory of bounded complexes of L -vector spaces with finite dimensional cohomology. To define Frobenius structure on objects of D bcoh ( D † P , Q ) and the category D( X/L ), we need to assume the existence of apair ( s, σ ) (as in 1.3.8). However, the category D(
X/L ) is independent of the choice of data ( s, σ ) up to equivalences ([3] 1.1.2).
The six functor formalism for D(
X/L ) (resp. D(
X/L F )) has been established recently. We refer to[5, 6] and ([3] 2.3) for details and to ([3] 1.1.3) for a summary. Here we only collect some results needed inthe sequel.Let f : X → Y be a morphism of k -schemes. For N ∈ {∅ , F } , there exist triangulated functors(2.3.2.1) f + , f ! : D( X/L N ) → D( Y /L N ) , f + , f ! : D( Y /L N ) → D( X/L N ) , such that ( f + , f + ), ( f ! , f ! ) are adjoint pairs. These functors satisfy following properties:(i) The category D( X/L N ) is a closed symmetric monoidal category, namely it is equipped with a tensorproduct functor ⊗ and the unit object L X = π + ( L ), where π : X → Spec( k ) is the structure morphismand L is the constant module in degree 0. The functor ⊗ admits a left adjoint functor H om X , called the internal Hom . The functor f + is monoidal.(ii) There exists a duality functor D X = H om X ( − , p ! L ) : D( X/L N ) ◦ → D( X/L N ) ([3] 1.1.4). Thecanonical morphism id → D X ◦ D X is an isomorphism. We set ( − ) e ⊗ ( − ) = D X ( D X ( − ) ⊗ D X ( − )).(iii) There exists a canonical morphism of functors f ! → f + , which is an isomorphism if f is proper.(iv) (Base change). Consider the following Cartesian diagram of k -schemes(2.3.2.2) X ′ g ′ / / f ′ (cid:15) (cid:15) X f (cid:15) (cid:15) Y ′ g / / Y. Then we have a canonical isomorphism g + f ! ≃ f ′ ! g ′ + . When f is proper, this isomorphism is the base changehomomorphism defined by the adjointness of ( f + , f + ).(v) (Berthelot-Kashiwara’s theorem). Let i be a closed immersion. Then i + is exact and fully faithful.The restriction of i ! to the essential image of i + is exact and is a quasi-inverse to i + ([6] 1.3.2(iii)).(vi) Let i be a closed immersion of k -schemes and j the open immersion defined its complement. Thereexists a canonical isomorphism j + ∼ −→ j ! . We have distinguished triangles ([3] 1.1.3(10), 2.2.9): j ! j + → id → i + i + → , i + i ! → id → j + j + → , where the first and second morphisms are defined by adjunctions.(vii) (Poincaré duality) We refer to ([3] 1.4.13) for the definition of Tate twist functor ( − ). Let f : X → Y be a smooth morphism of relative dimension d . Then there exists a canonical isomorphism ϕ : f + ( d )[2 d ] ∼ −→ f ! ([3] 1.5.13). Moreover, the functors f + [ d ] , f ! [ − d ] are exact.(viii) There exists a canonical equivalence of categories D( X/L N ) ≃ D b (Hol( X/L N )) ([5], [3] 2.2.26).(ix) Let X , X be two k -schemes and p i : X × X → X i the projection for i = 1 ,
2. There exists acanonical isomorphism of functors p +1 ( − ) ⊗ p +2 ( − ) ≃ p !1 ( − ) e ⊗ p !2 ( − ) (ii), denoted by − ⊠ − and called externaltensor product . This functor is exact ([6] 1.3.3). Remark 2.3.3. (i) If L is a finite extension of K , an object of Hol( X/L ) is defined as an object of Hol(
X/K )equipped with an L -structure ([3] 1.4.1). In general, Abe used the 2-inductive limit method of Deligne toconstruct D( X/L ) ([3] 2.4.14). If L ′ is an algebraic extension of L in K , we have an extension of scalarsfunctor ι L ′ /L : D( X/L ) → D( X/L ′ ), which is exact and commutes with cohomological functors.(ii) The category D( X/ T ) does not depend on the choice of the base field k under certain conditions.More precisely, if T ′ = { k ′ , R ′ , K ′ , L } is another geometric base tuple over T . Then there exists a canonicalequivalence ([3] 1.4.11): D( X ⊗ k k ′ / T ′ ) ∼ −→ D( X/ T ) , which commutes with cohomological functors.(iii) Let T F be an arithmetic base tuple. The s -th Frobenius morphism F X : X → X induces a σ -semi-linear equivalence of categories F ∗ X : D( X/L ) ∼ −→ D( X/L ) commuting with cohomological functors, called ( s -th) Frobenius pullback ([3] 1.1.3 lemma). An object of Hol( X/L F ) is an object E of Hol( X/L ) equippedwith a (n s -th) Frobenius structure ϕ : F ∗ X ( E ) ∼ −→ E (cf. [3] 1.4). ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 15 Let X be a smooth k -scheme of dimension d : π ( X ) → N . There exists a full subcategory Sm( X/L N )of Hol( X/L N )[ − d ] ⊂ D( X/L ) consisting of smooth objects ([3] 1.1.3(12) and 2.4.15). In general, we say acomplex M ∈ D( X/L N ) is smooth if H i ( M )[ − d ] belongs to Sm( X/L N ) for every i .When L = K , there exists an equivalence f Sp ∗ between Sm( X/K ) (resp. Sm(
X/K F )) and Isoc †† ( X/K )(resp. F - Isoc † ( X/K )) (2.1.4). If X admits a smooth compactification X with a smooth lifting X to Spf( R ),this equivalence is induced by the specialisation morphism Sp ∗ : X rig → X :(2.3.4.1) f Sp ∗ = Sp ∗ ( − d )[ − d ] : Isoc †† ( X/K ) (resp. F - Isoc † ( X/K )) ∼ −→ Sm(
X/K N ) ⊂ D bhol ( X/K N ) . In the following, we identify these two categories by f Sp ∗ and we use alternatively these two notations.Let f : X → Y be a morphism between smooth k -schemes. Via f Sp ∗ , we can identify the functor f + andthe pullback functor of overconvergent ( F -)isocrystals f ∗ ([3] 2.4.15). If d denotes dim( X ) − dim( Y ), forany object M of Sm( X/L N ), there exists a canonical isomorphism:(2.3.4.2) f ! ( M ) ≃ f + ( M )( d )[2 d ] . Let X be a k -scheme. There exists a constructible t-structure ( c-t-structure in short) on D( X/L ) (cf.[3] 1.3, 2.2.23). When X = Spec( k ), the constructible t-structure coincides with the holonomic one (2.3.2).If X a smooth k -scheme, any object of Sm( X/L ) is constructible.The heart of c-t-structure is denoted by Con( X ), called the category of constructible modules , and isanalogue to the category of constructible sheaves in the ℓ -adic theory. The cohomology functor of c-t-structure is denoted by c H ∗ .Let f : X → Y be a morphism between k -schemes. The functor f + is c-t-exact and f + is left c-t-exact.If i is a closed immersion, then i + is c-t-exact. If j is an open immersion, then j ! is c-t-exact ([3] 1.3.4).A constructible module M on X is zero if and only if i + x M = 0 for any closed point i x : x → X ([3]1.3.7). In the end, we present a generalization of the specialization morphism (2.2.3.3) in a relative situationusing the direct image of arithmetic D -modules.Let f : X = Spec( B ) → S = Spec( A ) be a smooth morphism of affine smooth R -schemes of relativedimension d and let ( M, ∇ ) be a coherent O X K -module endowed with an integrable connection relative to K . Consider M as a D X K -module. The direct image f dR+ ( M ) of D -modules is calculated by the relative deRham complex M ⊗ Ω • X/S . Since f is affine, the above complex is calculated by(2.3.6.1) Γ( S, f dR+ ( M )) ≃ DR B/A ( M, ∇ ) = M → M ⊗ B Ω B/A → · · · , where we denote abusively by M the global section Γ( X K , M ). We assume moreover that f admits a good compactification , i.e. f can be extended to a smoothmorphism f : X → S of smooth projective R -schemes X, S such that X k − X k , S k − S k are ample divisors.We keep the notation of 2.2.2 and assume that M † = j † ( M an ) is overconvergent as in 2.2.3. We denoteabusively the D † X , Q ( ∞ )-module Sp ∗ ( M † ) (2.3.4) by M † . The direct image of M † along f k : X k → S k iscalculated by a relative de Rham complex: f k, + ( M † ) ∼ −→ R f k, ∗ (Sp ∗ ( M † ⊗ O X Ω •X / S )) . (2.3.7.1)The above complex is a complex of overholonomic (and hence coherent) D † S , Q ( ∞ )-modules.We set A † = Γ( S , O S , Q ( † ∞ )), B † = Γ( X , O X , Q ( † ∞ )) and D † S ( ∞ ) = Γ( S , D † S , Q ( ∞ )) (1.3.8). By D † -affinity ([53] 5.3.3), the complex (2.3.7.1) is equivalent to a complex of coherent D † S ( ∞ )-modules: R Γ( S , f k, + ( M † )) ≃ R Γ( X , Sp ∗ ( M † ⊗ O X Ω •X / S )) ≃ ( M ⊗ B K B † ) ⊗ B Ω • B/A . We denote the complex in the second line by DR † B/A ( M † ). Note that this complex is A † -linear.If we set D S K = Γ( S K , D S K ), there exists a canonical D S K -linear morphism, called the (relative) special-isation morphism (2.3.7.2) DR B/A ( M, ∇ ) → DR † B/A ( M † ) . Complements on the cohomology of arithmetic D -modules.2.4.1. Let f : X → Spec( k ) be a k -scheme and F an object of D( X/L ). We set(2.4.1.1) H ∗ ( X, F ) = H ∗ f + ( F ) , H ∗ c ( X, F ) = H ∗ f ! ( F ) , and call them cohomology groups of F , compact support cohomology groups of F , respectively. Note thatthey are finite dimensional L -vector spaces. If F is an object of D( X/L F ), then above cohomology groupsare equipped with a Frobenius structure. If there is no confusion, we simply write H ∗ ( X, L ) for H ∗ ( X, L X ).We collect some properties that we will use in the following:(i) If X has dimension ≤ d , then for any M ∈ Con( X ), the compact support cohomology groups H i c ( X, M )are concentrated in degrees 0 ≤ i ≤ d ([3] 1.3.8).(ii) Suppose X admits a smooth compactification X such that X possesses a smooth lifting over R andthat X − X is a divisor. Given an object M of Isoc †† ( X/K ) (resp. F - Isoc † ( X/K )) (2.1.4), we have canonicalisomorphisms ([1] 5.9):(2.4.1.2) H ∗ rig ( X, M ) ≃ H ∗ ( X, f Sp ∗ ( M )) , H ∗ rig , c ( X, M ) ≃ H ∗ c ( X, f Sp ∗ ( M )) , as objects of Vec K (resp. F - Vec K ). Via (2.4.1.2), the canonical morphism H ∗ c ( X, f Sp ∗ ( M )) → H ∗ ( X, f Sp ∗ ( M ))induced by f ! → f + is compatible with ι rig (2.2.3.6).In particular, we have H ( A n , L ) ≃ L , H i ( A n , L ) = 0 for i = 0 and H n ( A n , L ) ≃ L , H i ( A n , L ) = 0 for i = 2 n .(iii) If X is smooth over k , then the dimension of H ( X, L X ) is equal to the number of geometricallyconnected components of X . The Frobenius acts on H ( X, L X ) as identity. Let Y be a closed subscheme of X and F an object of D( X/L ). In view of the distinguished triangle2.3.2(iv), there exists a long exact sequence of cohomology groups:(2.4.2.1) · · · ∂ −→ H i c ( X − Y, F ) → H i c ( X, F ) → H i c ( Y, F ) ∂ −→ · · · In general, suppose that there exists a finite filtration of closed subschemes { X i } i ∈ Z of X , with closedimmersions X i +1 ֒ → X i such that X i = X for i small enough and X i = ∅ for i big enough. Then we deducea spectral sequence (cf. [37] *2.5)(2.4.2.2) E ij = H i + j c ( X i − X i +1 , F ) ⇒ H i + j c ( X, F ) . Corollary 2.4.3.
Let d be the dimension of X . Then the dimension of the top degree compact supportcohomology H d c ( X, L X ) is equal to the number of geometrically irreducible components of X . The Frobeniuson H d c ( X, L X ) acts by multiplication by q d .Proof. We denote by X sm (resp. X sing ) the smooth (resp. singular) locus of X . Then the assertion followsfrom the long exact sequence (2.4.2.1) for ( X sm , X sing , X ), Poincaré duality and 2.4.1(iii). (cid:3) We show an analogue of ([16] 4.2.5) for arithmetic D -modules. Proposition 2.4.4.
Let f : X → Y be a smooth morphism of k -scheme of relative dimension d with geo-metrically connected fibers. Then the functor f + [ d ] induces a fully faithful functor Hol(
Y /L N ) → Hol(
X/L N ) for N ∈ {∅ , F } . Lemma 2.4.5.
Let M be an object of D ≤ ( X/L ) and N an object of D ≥ ( X/L ) . Then H om X ( M , N ) belongs to c D ≥ ( X/L ) (2.3.5) . ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 17 Proof.
We prove by induction on the dimension of X . The assertion is clear if dim X = 0. To prove theassertion, we can reduce to the case where M , N ∈ Hol(
X/L ). Then there exists a dense smooth opensubscheme j : U → X such that M | U , N | U are smooth. Let i : Z → X be the complement of U andconsider the triangle i + i ! H om X ( M , N ) → H om X ( M , N ) → j + j + H om X ( M , N ) → . Since i ! H om X ( M , N ) ≃ H om X ( i + M , i ! N ) ([3] 1.1.5), the first term belongs to c D ≥ ( X/L ) by inductionhypotheses. Note that H om U ( M | U , N | U ) ≃ D U ( M | U ⊗ D U ( N | U )) is a smooth module and of constructibledegree 0. Then j + j + H om X ( M , N ) belongs to c D ≥ ( X/L ) and the assertion follows. (cid:3)
Proof of proposition − /L ). Let M , N be two objects of Hol( Y /L ). Since f is smooth, we deducefrom f ! H om Y ( M , N ) ≃ H om X ( f + M , f ! N ) ([3] 1.1.5) an isomorphism f + H om Y ( M , N ) ∼ −→ H om X ( f + M , f + N ) . By applying c H f +c H ( − ) to the above isomorphism and lemma 2.4.5, we have(2.4.6.1) c H f + f + (cid:0) c H ( H om Y ( M , N )) (cid:1) ∼ −→ c H f +c H ( H om X ( f + M [ d ] , f + N [ d ])) . We claim that for any constructible module F on Y , there is a canonical isomorphism(2.4.6.2) F ∼ −→ c H f + f + F . Then, by 2.4.5, the proposition follows by applying H ( Y, − ) to the composition of (2.4.6.1) and (2.4.6.2).By smooth base change and 2.3.5, to prove (2.4.6.2), we can reduce to the case where Y is a point. Afterextending the scalar L and the base field k (2.3.3), we may assume moreover that Y = Spec( k ). In this case,the isomorphism (2.4.6.2) follows from 2.4.1(iii). (cid:3) Equivariant holonomic D -modules. In this subsection, we study the notion of equivariant holonomic D -modules over a k -scheme (or an ind-scheme). We write simply D( X ) (resp. Hol( X )) for D( X/L ) orD(
X/L F ) (resp. Hol( X/L ) or Hol(
X/L F )). Let X → S be a morphism of k -schemes, H a smooth affine group scheme over S with geometricallyconnected fibers and act : H × S X → X an action of H on X . We denote by pr : H × S X → X theprojection. We define the category Hol H ( X ) of H -equivariant holonomic modules on X as follow. An objectof Hol H ( X ) is a pair consisting of a holonomic module M on X and an isomorphism θ : act + ( M ) ∼ −→ pr +2 ( M )in D( H × S X ), satisfying:(i) e + ( θ ) = id, where e : X → H × S X is induced by the unit section of H ;(ii) a cocycle condition on H × S H × S X .A morphism between ( M , θ ) and ( M , θ ) is a morphism ϕ : M → M of Hol( X ) such that(2.5.1.1) pr +2 ( ϕ ) ◦ θ ≃ θ ◦ act + ( ϕ ) . It is clear that Hol H ( X ) is an abelian subcategory of Hol( H ).Suppose that [ X/H ] is represented by a separated scheme of finite type X over S . By smooth descent ofholonomic modules ([3] 2.1.13), the pullback functor along the canonical morphism q : X → X induces anequivalence of categories:(2.5.1.2) q + [ d H ] : Hol( X ) ∼ −→ Hol H ( X ) . Lemma 2.5.2.
The canonical functor
Hol H ( X ) → Hol( X ) is fully faithful.Proof. Given two objects ( M , θ ), ( M , θ ) of Hol H ( X ) and a morphism ϕ : M → M of Hol( X ), oneneed to show (2.5.1.1) θ ◦ act + ( ϕ ) ◦ ( θ ) − ≃ pr +2 ( ϕ ) . By proposition 2.4.4, pr +2 is fully faithful. To show the above isomorphism, it suffices to show e + ( θ − ◦ act + ( ϕ ) ◦ θ ) ≃ e + (pr +2 ( ϕ )), which follows from 2.5.1(i). (cid:3) Lemma 2.5.3.
Let H ⊂ H be a closed normal subgroup scheme over S . Suppose that H/H , H are smoothover S and that the action of H on X factors through H/H . Then, the canonical functor (2.5.3.1) Hol H/H ( X ) → Hol H ( X ) is an equivalence of categories.Proof. The essential surjectivity follows from smooth descent ([3] 2.1.13). The full faithfulness follows from2.5.2. (cid:3)
Keep the notation of 2.5.1. Let Y be a separated S -scheme of finite type and ̟ : E → Y an H -torsorover S with trivial action of H on Y . We denote by Y e × S X the quotient of E × S X by H , where H acts on E × S X diagonally.Let M be a holonomic module on Y and N an H -equivariant holonomic module on X . Assume that M ⊠ S N is a holonomic module on Y × S X (Note that it is true if the base S = Spec( k )). Then ( ̟ + M [dim H ]) ⊠ S N is holonomic on E × S X and is H -equivariant by construction. By (2.5.1.2), it descents to a holonomicmodule on Y e × S X , denoted by M e ⊠ S N and called the twisted external product of M and N . We say an fpqc sheaf X on the category of k -algebras is a (strict) ind-scheme over k if there existsan isomorphism of fpqc-sheaves X ≃ lim −→ i ∈ I X i for a filtered inductive system ( X i ) i ∈ I of k -schemes, whosetransition morphisms are closed immersion. The inductive system ( X i ) i ∈ I is called an ind-presentation of X . We have following properties:(i) If Z is a k -scheme and u : Z → X is a closed subfunctor, then there exists an index i such that u factors through Z → X i .(ii) If X ≃ lim −→ j ∈ J X ′ j is another ind-presentation, the for any i , there exists an index j such that X i is aclosed subscheme of X ′ j and vice versa.Given an ind-scheme X = lim −→ i ∈ I X i , we denote by X red = lim −→ i ∈ I X i, red the reduced ind-subscheme of X .For a transition morphism ϕ : X i → X j , the functor ϕ + : D( X i ) → D( X j ) is exact and fully faithful. Wedefine a triangulated category D( X ) as the 2-inductive limitD( X ) = lim −→ i ∈ I D( X i ) . The definition is independent of the choice of a ind-presentation of X . Since ϕ + is exact, D( X ) is alsoequipped with a t-structure, whose heart is denoted by Hol( X ). Note that Hol( X ) coincides with the fullabelian subcategory lim −→ i ∈ I Hol( X i ) of D( X ).Given a morphism f = ( f i ) i ∈ I : X = lim −→ X i → S to a k -scheme S , the cohomology functors f i, ! ’s and f i, + ’s allow us to define f ! , f + : D( X ) → D( S ). Let X = lim −→ i ∈ I X i be an ind-scheme and f : X → S a morphism to a k -scheme. Let ( H j ) j ∈ J be aprojective system of smooth affine S -group schemes with geometrically connected fibers, whose transitionmorphisms are quotient. We set H = lim ←− j ∈ J H j its projective limit, which is an affine group scheme over S . Assume that there exists an action of H on f : X → S such that it stabilizes each subfunctor f | X i andthat the H -action factors through a quotient H j i on X i → S for each i ∈ I . Then we define Hol H ( X i ) tobe Hol H ji ( X i ). By lemma 2.5.3, the category Hol H ( X i ) is independent of the choice of H j i up to canonicalequivalences. Therefore, for i ≤ j , we have a fully faithful functor Hol H ( X i ) → Hol H ( X j ). We define the category Hol H ( X ) of H -equivariant holonomic modules on X as the inductive limit:Hol H ( X ) = lim −→ i ∈ I Hol H ( X i ) . Let Y = lim −→ i ∈ I Y i an ind-scheme over S and ̟ : E → Y an H -torsor. We can define an ind-scheme Y e × S X as follows. For i, l ∈ I , we denote by E l,j i the H j i -torsor E | Y l × H H j i → Y l and by Y l e × S X i = E l,j i × S X i /H j i the twisted product (2.5.4). For a surjection H j ′ ։ H j i , there exists a canonical isomorphism ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 19 E l,j ′ × S X i /H j ′ ∼ −→ E l,j i × S X i /H j i . Then this allows us to represent the fpqc sheaf Y e × S X as an inductivelimit of Y l e × S X i .Let M be an object of Hol( Y ) supported in Y l and N an object of Hol H ( X ) supported in X i . Assumethat M ⊠ S N is a holonomic module on Y × S X . Then we can define an object M e ⊠ S N in Hol( Y l e × S X i )(2.5.4) and then in Hol( Y e × S X ). The construction is independent of the choice of i, l ∈ I .2.6. Intermediate extension and the weight theory.2.6.1.
Let u : Y → X be a locally closed immersion. Then the functor u + (resp. u ! ) is left exact (resp. rightexact) ([6] 1.3.13). For N ∈ {∅ , F } and E ∈ Hol(
Y /L N ), we consider the homomorphism θ u, E : H ( u ! E ) →H ( u + E ) and we define u !+ ( E ) to be ([6] 1.4.1)(2.6.1.1) u !+ ( E ) = Im( θ u, E : H ( u ! E ) → H ( u + E )) . This defines a functor u !+ : Hol( Y /L N ) → Hol(
X/L N ), called the intermediate extension functor . We recallthe following results and refer to ([6] §1.4) for general properties of this functor:(i) ([6] 1.4.7) Suppose E is irreducible. Then, u !+ ( E ) is the unique irreducible subobject of H ( u + E )(resp. irreducible quotient of H ( u ! E )) in Hol( X/L N ).(ii) ([6] 1.4.9) Let F be an irreducible object of Hol( X/L N ). Then there exists a locally closed immersion u : Y → X from a smooth k -scheme Y and a smooth holonomic module E on Y such that F ≃ u !+ ( E ). Corollary 2.6.2.
Let j : U → X be an open subscheme of X and i : Z → X its complement. (i) Given a holonomic module E on U , j !+ ( E ) is the unique extension F of E to Hol(
X/L N ) such that i + F ∈ D ≤− ( Z/L N ) and that i ! F ∈ D ≥ ( Z/L N ) . (ii) If X is smooth and F is a smooth holonomic module on X , then j !+ ( F | U ) ≃ F .Proof. (i) Since j ! , i + are right exact ([6] 1.3.2), H i + ( H ( j ! ( E ))) = 0. By applying i + to 0 → Ker( θ j, E ) →H ( j ! ( E )) → j !+ ( E ) →
0, we obtain i + ( j !+ ( E )) ∈ D ≤− ( Z/L ). We prove i ! F ∈ D ≥ ( Z/L ) in a dual way.Conversely, given such an extension F , we can prove that the adjunction morphism H j ! ( E ) → F (resp. F → H j + ( E )) is surjective (resp. injective) by the Berthelot-Kashiwara theorem. The assertion follows.(ii) The intermediate extension is stable under composition ([6] 1.4.5). Then we can reduce to the casewhere Z is smooth over k . In this case, assertion (ii) follows from (i) and (2.3.4.2). (cid:3) We briefly recall the theory of weights for holonomic F -complexes developed by Abe and Caro [6].In the rest of this subsection, we assume k has q = p s elements and we consider the arithmetic base tuple T F = { k, R, K, L, s, σ = id } (2.3.1). We fix an isomorphism ι : K ≃ C . We refer to ([6] 2.2.2, [3] 2.2.30) forthe notion of being ι -mixed (resp. ι -mixed of weight ≤ w , ι -mixed of weight ≥ w , ι -pure ) for M ∈ D( X/L F ).The weight behaves like the one in the ℓ -adic theory:(i) ([6] 4.1.3) The six operations preserve weights. More precisely, given a morphism f : X → Y of k -schemes, f + , f ! send ι -mixed F -complexes of weight ≥ w to those of weight ≥ w , f ! , f + send ι -mixed F -complexes of weight ≤ w to those of weight ≤ w . The dual functor D X exchanges ι -mixed F -complexesof weight ≤ w to ≥ w and ⊗ sends ι -mixed F -complexes of weight ( ≤ w, ≤ w ′ ) to ≤ w + w ′ .(ii) ([6] 4.2.4) Intermediate extension functor of an immersion preserves pure F -complexes and weights.Moreover, we have a decomposition theorem for pure holonomic F -module. Theorem 2.6.4 ([6] 4.3.1, 4.3.6) . Let X be a k -scheme. (i) An ι -pure F -holonomic module E on X is semisimple in the category Hol(
X/L ) (not in Hol(
X/L F ) ). (ii) An ι -pure F -holonomic complex F is isomorphic, in D( X/L ) to ⊕ n ∈ Z H n ( F )[ n ] . The original form of ([6] 4.3.1, 4.3.6) states the decomposition in the category of overholonomic modules(resp. complexes) over X . We remark that the same argument shows the decomposition in the categoryHol( X/L ) (resp. D(
X/L )).
Nearby and vanishing cycles.
In a recent preprint [4], Abe formulated the nearby and vanishingcycle functors for holonomic arithmetic D -modules, based on the unipotent nearby and vanishing cyclefunctors introduced by himself and Caro in [5]. We briefly recall these constructions in this subsection.We write simply D( X ) for D( X/L ) or D(
X/L F ). The construction are parallel in two cases. WhenD( X ) = D( X/L ), the Tate twist ( n ) denotes the identity functor. Let f : X → A k be a morphism of k -schemes. We denote by j : U = f − ( G m ) → X the openimmersion and by i : X = X − U → X its complement. Following Beilinson [15], Abe and Caro constructedthe unipotent nearby and vanishing cycle functors ([5], §2)(2.7.1.1) Ψ un f : Hol( U ) → Hol( X ) , Φ un f : Hol( X ) → Hol( X ) . We briefly recall the definition of Ψ un f . We denote O b P R , Q ( † { , ∞} ) simply by O G m (see 1.3.8). For n ≥ O G m -module Log n of rank n Log n = ⊕ n − i =0 O G m · log [ i ] , generated by the symbols log [ i ] . There exists a unique D † b P R , Q ( { , ∞} )-module structure on Log n defined for i ≥ g ∈ O G m by ∇ ∂ t ( g · log [ i ] ) = ∂ t ( g ) · log [ i ] + gt · log [ i − , where t is the local coordinate of G m and log [ j ] = 0 for j <
0. There exists a canonical Frobenius structureon Log n sending log [ i ] to q i log [ i ] . This defines an overconvergent F -isocrystal on G m and then a smoothobject of Hol( G m /K F ). We still denote by Log n the extension of scalars ι L/K (Log n ) in Hol( G m ).We set Log nf = f + Log n ∈ Hol( U ) and define for F ∈ Hol( U ) :(2.7.1.2) Ψ un f ( F ) = lim −→ n ≥ Ker( j ! ( F ⊗ Log nf ) → j + ( F ⊗ Log nf )) . This limit is representable in Hol( X ) by ([5] lemma 2.4).The functors Ψ un f , Φ un f are exact ([5] 2.7) and extend to triangulated categories. There exists a distin-guished triangle i + [ − → Ψ un f → Φ un f +1 −−→ . Proposition 2.7.2 ([5] 2.5) . There exist canonical isomorphisms: (2.7.2.1) ( D X ◦ Ψ un f )(1) ≃ Ψ un f ◦ D U , D X ◦ Φ un f ≃ Φ un f ◦ D X . To define the full nearby/vanishing cycle functors of a morphism over a henselian trait, one needto extend the definition of holonomic arithmetic D -modules to a larger class of schemes, which are closedunder henselization. We denote by Pro ( k ) the full subcategory of Noetherian schemes over k which can berepresentable by a projective limit of a projective system of k -schemes whose transition morphisms are affineand étale. In the rest of this subsection, we will work with schemes in the category Pro ( k ).Given a morphism of finite type X → S , if S is an object of Pro ( k ) then so is X . The category Pro ( k )is closed under henselization (resp. strict henselization) ([4] 1.3).Let X = lim ←− i ∈ I X i be an object of Pro ( k ) with a representation by k -schemes X i . For each transitionmorphism ϕ : X i → X j (which is affine and étale), we have a canonical isomorphism ϕ + ≃ ϕ ! : D( X j ) → D( X i ). We define a triangulated category D( X ) of arithmetic D -modules on X as an inductive limit D( X ) :=lim −→ i ∈ I ◦ D( X i ). Since ϕ + is exact, D( X ) is equipped with a t-structure whose heart is denoted by Hol( X ).Moreover, we can extend the definition of cohomological functors (2.3.2.1) to D( X ) (cf. [4] 1.4). Let (
S, s, η ) be a strict henselian trait in
Pro ( k ) and f : X → S a morphism of finite type. Withabove preparations, we can define the unipotent nearby and vanishing cycles functors for f (cf. [4] 1.7,1.8)(2.7.4.1) Ψ un f , Φ un f : Hol( X ) → Hol( X s ) . We adopt the definition of [4], which is different from that of [5] by a Tate twist.
ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 21 We denote by Hen( S ) the category of henselian traits over S which is generically étale over S . Given anobject h : S ′ → S of Hen( S ), we denote abusively by h the canonical morphism X S ′ → X , by h s : X s ′ ∼ −→ X s the isomorphism on the special fibers and by f ′ : X S ′ → S ′ the base change of f by h .Using (2.7.4.1), the full nearby and vanishing cycle can be defined as (cf. [4] 1.9):(2.7.4.2) Ψ f = lim −→ ( S ′ ,h ) ∈ Hen( S ) h s + ◦ Ψ un f ′ ◦ h + , Φ f = lim −→ ( S ′ ,h ) ∈ Hen( S ) h s + ◦ Φ un f ′ ◦ h + . By [4] 2.2, they are well-defined functorsΨ f , Φ f : Hol( X ) → Hol( X s ) . Universal local acyclicity.2.8.1.
Following Braverman-Gaitsgory ([25], 5.1), we propose a notion of (universal) local acyclicity forarithmetic D -modules with respect to a morphism to a smooth target.For a smooth k -scheme X , we denote by d X : π ( X ) → N the dimension of X . Let g : X → X be amorphism of k -schemes and F , F ′ two objects of D( X ). We consider the composition g ! ( g + ( F ) ⊗ g ! ( F ′ )) ≃ F ⊗ g ! ( g ! ( F ′ )) → F ⊗ F ′ and its adjunction:(2.8.1.1) g + ( F ) ⊗ g ! ( F ′ ) → g ! ( F ⊗ F ′ ) . Now let S be a smooth k -scheme and f : X → S a morphism of k -schemes. We set X = X , X = X × S , F ′ = L X and take g to be the graph of f . By Poincaré duality, we have L X ( − d S )[ − d S ] ∼ −→ g ! ( L X ).Then, we obtain a canonical morphism(2.8.1.2) g + ( F ) → g ! ( F )( d S )[2 d S ] . By taking F to be M ⊠ N , we obtain a canonical morphism (2.3.2(ix))(2.8.1.3) M ⊗ f + ( N ) → ( M e ⊗ f ! ( N ))( d S )[2 d S ] . Definition 2.8.2.
Let S be a smooth k -scheme and f : X → S a morphism of k -schemes. We say thatan object M of D( X ) is locally acyclic (LA) with respect to f , if the morphism (2.8.1.3) is an isomorphismfor any object N of D( S ). We say that M is universally locally acyclic (ULA) with respect to f , if for anymorphism of smooth k -schemes S ′ → S , the +-inverse image of M to X × S S ′ is locally acyclic with respectto X × S S ′ → S ′ . Proposition 2.8.3.
Keep the notation of and let M be an object of D( X ) . (i) Any object M of D( X ) is ULA with respect to the structure morphism X → Spec( k ) . (ii) Let g : Y → X be a smooth (resp. smooth surjective) morphism. Then g + ( M ) on Y is LA withrespect to f ◦ g if (resp. if and only if) M is LA with respect to f . (iii) If g : S → S ′ is a smooth morphism of smooth k -schemes and M is LA with respect to a morphism f : X → S , then M is LA with respect to g ◦ f . (iv) Let h : Y → S be a morphism of finite type and g : X → Y a proper S -morphism (resp. a closedimmersion). Then g + ( M ) is LA with respect to h if (resp. if and only if) M is LA with respect to f . (v) If M is LA with respect to f , then so is its dual D X ( M ) .Proof. (i) Let S be a smooth k -scheme and N an object of D( S ). We need to show that the canonicalmorphism (id X × ∆) + ( M ⊠ p +2 ( N )) → (id X × ∆) ! ( M ⊠ p +2 ( N ))( d S )[2 d S ]is an isomorphism, where ∆ : S → S × S is the diagonal map and p : S × S → S is the projection in thesecond component. Then we reduce to show that the canonical morphism N → ∆ ! ( p +2 ( N ))(2 d S )[2 d S ]is an isomorphism. After taking dual functor, the assertion follows from ([3] 1.5.14). Assertions (ii) and (iii) follow from (2.3.2(vii)) and the smooth descent for D( X ) ([3] 2.1.13).Assertion (iv) follows from the projection formula ([3] 1.1.3(9)) and the Berthelot-Kashiwara theorem.If we apply the dual functor D X to the morphism (2.8.1.3), then we obtain the morphism (2.8.1.3) D X ( M ) ⊗ f + ( D S ( N )) → ( D X ( M ) e ⊗ f ! ( D S ( N )))( d S )[2 d S ] , for the pair ( D X ( M ) , D S ( N )). Then assertion (v) follows. (cid:3) Remark 2.8.4.
Let S be a smooth k -scheme and f : X → S a morphism from an ind-scheme to S . In viewof proposition 2.8.3(iv), we can define the notion of LA (resp. ULA) with respect to f for objects of D( X ). Proposition 2.8.5.
Keep the notation of and let D be a smooth effective divisor in S , i : Z = f − ( D ) → X the closed immersion and j : U → X its complement. Let M be an object of D( X ) such thatit is LA with respect to f and that M | U is holonomic. (i) There exists canonical isomorphisms: (2.8.5.1) M ≃ j !+ ( M | U ) , i + M [ − ∼ −→ i ! M (1)[1] . In particular, M and i + M [ − are holonomic. (ii) The holonomic module i + M [ − is LA with respect to f ◦ i and f | Z : Z → D .Proof. (i) By étale descent for holonomic modules ([3] 2.1.13), we may assume that there is a smoothmorphism g : S → A such that D = g − (0). By proposition 2.8.3(iii), M is LA with respect to g ◦ f : X → A . Then we can reduce to the case f : X → A and Z = f − (0).We will show that Φ un f ( M ) = 0, i.e. the canonical morphism(2.8.5.2) i + M [ − → Ψ un f ( M )is an isomorphism.We denote by j : G m → A be the canonical morphism and abusively by f the restriction f | U : U → G m .By the projection formula, we have j ! ( M | U ⊗ f + Log n ) ∼ −→ M ⊗ j ! f + Log n ≃ M ⊗ f + j ! Log n . On the other hand, by the projection formula and the LA property of M , we have j + ( M | U ⊗ ( f + Log n )) ∼ −→ j + ( M | U e ⊗ ( f ! Log n ))( d X )[2 d X ] ∼ −→ M e ⊗ ( j + f ! Log n )( d X )[2 d X ] ≃ M e ⊗ ( f ! j + Log n )( d X )[2 d X ] ≃ M ⊗ ( f + j + Log n ) . Via the above isomorphisms, the canonical morphism j ! ( M | U ⊗ ( f + Log n )) → j + ( M | U ⊗ ( f + Log n ))coincides with the canonical morphism M ⊗ ( f + ( j ! Log n → j + Log n )) . To prove that (2.8.5.2) is an isomorphism, we can reduce to the case where f is the identity map of A and M is the constant module L A [1] on A . If we denote by N n the action induced by t∂ t on the fiber(Log n ) of Log n at 0 (called residue morphism in [6] 3.2.11), then Ker( j ! (Log n ) → j + (Log n )) is isomorphicto Ker( N n ) (cf. [5] proof of lemma 2.4). In this case, (2.8.5.2) is an isomorphism.In particular i + M [ −
1] is holonomic. By propositions 2.7.2 and 2.8.3(v), the second isomorphism of(2.8.5.1) follows from (2.8.5.2):(2.8.5.3) Ψ un f ( M ) ≃ D X Ψ un f ( D X ( M ))(1) ∼ −→ i ! M (1)[1] . Then we deduce M ≃ j !+ ( M | U ) by (2.6.2(i)).Assertion (ii) follows from the six functor formalism. We left the proof to readers. (cid:3) ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 23 Corollary 2.8.6.
If an object M of D( X ) is ULA with respect to f , then, for any strict henselian trait T and any morphism g : T → S , we have Φ un f T ( M | X T ) = 0 and Φ f T ( M | X T ) = 0 , where f T : X T → T is thebase change of f by g .Proof. By definition, it suffices to show that the unipotent vanishing cycle Φ un f T ( M | X T ) vanishes.By ([54] 8.8.2), there exists a smooth k -scheme S ′ , a smooth effective divisor D of S ′ with generic point η D and a morphism h : S ′ → S such that the strict henselization of S ′ at η D is isomorphic to T and that g is induced by h . We denote by f S ′ : X S ′ → S ′ the base change of f by h . After shrinking S ′ , we mayassume that there exists a smooth morphism π : S ′ → A k with D = π − (0).By definition (cf. [4] 1.7-1.8), we reduce to show that Φ un π ◦ f S ′ ( M | X S ′ ) = 0. But this follows from proposi-tion 2.8.3(iii) and the proof of (2.8.5.2). Then the assertion follows. (cid:3) Corollary 2.8.7.
Let X be a smooth k -scheme. If an object M of D( X ) is ULA with respect to the identitymorphism, then each constructible cohomology module c H i ( M ) is smooth (resp. each cohomology module H i ( M ) is smooth).Proof. When M is constructible, it follows from 2.8.6 and ([4] 3.8). We prove the general case by inductionon the cohomological amplitude of M . (cid:3) In 4.1, we will use the notion of holonomic modules over a stack and apply cohomological functors of a schematic morphism of algebraic stacks, that we briefly explain in the following. Let X be an algebraic stackof finite type over k . We refer to ([3] 2.1.16) for the definition of category Hol( X ) of holonomic modules on X and the category D( X ) (corresponds to the category D bhol ( X ) in loc. cit ). The dual functor D X is definedin ([3] 2.2). Let f : X → Y be a schematic morphism, Y • → Y a simplicial algebraic space presentation. Bypullback, we obtain a simplicial presentation X • → X and a Cartesian morphisms f • : X • → Y • . Then theconstructions of ([3] 2.1.10 and 2.2.14) allow us to define cohomological functors: f + : D( X ) ≃ D bhol ( X • ) ⇄ D bhol ( Y • ) ≃ D( Y ) : f ! . Given a object M of D( X ) and a morphism g : X → S to a smooth k -scheme S , we say M is ULA withrespect to g if its +-pullback to a presentation U → X is ULA with respect to U → S .Suppose S is moreover a curve. Let s be a closed point of S and S ( s ) the strict henselian at s .Since nearby/vanishing cycle functors commute with smooth pullbacks, we can extend the definition ofnearby/vanishing cycle functors for g × S S ( s ) .2.9. Local monodromy of an overconvergent F -isocrystal.2.9.1. We briefly recall the local monodromy group of p -adic differential equations over the Robba ringfollowing [10, 65].We denote by R K the Robba ring over K , by MC( R K /K ) (resp. MC( R /K )) the category of ∇ -modulesof finitely presented over R K (resp. over R = R ⊗ K K ). Each object of MC( R /K ) comes from the extensionof scalar of an object of MC( R L /L ) for some finite extension L of K . We denote by MC uni ( R /K ) the fullTannakian subcategory of MC( R /K ) consisting of unipotent objects, i.e. objects which are isomorphic tosuccessive extension of the trivial object (cf. [65] § 4).There is an equivalence between the category Vec nil K of finite dimensional K -vector space with a nilpotentendomorphism and MC uni ( R /K ), given by the functor ( V , N ) ( V ⊗ K R , ∇ N ), where the connection ∇ N is defined by ∇ N ( v ⊗
1) =
N v ⊗ dx/x ([65] 4.1). In particular, the Tannakian group of MC uni ( R /K ) over K is isomorphic to G a .We denote by MCF( R /K ) the full subcategory of MC( R /K ) consisting of objects admitting a Frobeniusstructure ([10] 3.4). The category MCF( R /K ) is a Tannakian category over K , whose Tannakian group isdenoted by G . Christol and Mebkhout introduce the notion of p -adic slope for objects of MCF( R /K ) andshow a Hasse-Arf type result [32]. This allows one to define a Hasse-Arf type filtration on MCF( R /K ) andthen a decreasing filtration of closed normal subgroups {G >λ } λ ≥ of G (cf. [10] § 1, 3.4). If we denote by I (resp. P ) the inertia (resp. wild inertia) subgroup of Gal( k (( t )) sep /k (( t ))), regarded as pro-algebraic groups,then there exist canonical isomorphisms of affine K -groups ([10] 7.1.1)(2.9.1.1) G ≃ I × G a , G > ≃ P. The local monodromy theorem says that any object of MCF( R /K ) is quasi-unipotent [10, 59, 66]. Givenan object M of MCF( R /K ), the action of P on M factors through a finite quotient. By a theorem ofMatsuda-Tsuzuki [81, 65] (cf. [10] 7.1.2), the irregularity of M , defined by p -adic slopes, is equal to the Swanconductor of the representation of I on a fiber of M . We denote by K { x } the K -algebra of analytic functions on the open unit disc | x | <
1, i.e.(2.9.2.1) K { x } = { X n ≥ a n x n ∈ K J x K ; | a n | p ρ n → n → ∞ ) ∀ ρ ∈ [0 , } . Let Ω K { x } (log) be the free K { x } -module of rank 1 with basis dx/x and consider the following canonicalderivation d : K { x } → Ω K { x } (log) , f xf ′ ( x ) dx/x . An unipotent object ( M, ∇ ) of MC( R K /K ) extendsto a log ∇ -module over K { x } . Let ( V, N ) be the object of Vec nil K associated to ( M, ∇ ). In view of 2.9.1,there exists a canonical isomorphism between Coker( N ) and the solution space Sol( M ) of ( M, ∇ ):(2.9.2.2) Coker( N ) ∼ −→ Sol( M ) = Hom K { x } (( M, ∇ ) , ( K { x } , d )) ∇ =0 . When the connection ∇ is defined by a differential operator D , then Sol( M ) is isomorphic to the solutionspace of D . Let X be a smooth curve over k , X a smooth compactification of X and x a k -point in the boundary X \ X . There exists a canonical functor defined by restriction at x :(2.9.3.1) | x : Isoc † ( X/K ) → MC( R K /K ) . We refer to [77] and ([60] § 6) for the definition of log convergent ( F -)isocrystals on Y = X ∪{ x } with a logpole at x . Let E be an object of Isoc †† ( X/K ) (resp. F - Isoc † ( X/K )). A log-extenbility criterion of Kedlaya([60] 6.3.2) says that if E | x is unipotent, then E extends to a log convergent isocrystal (resp. F -isocrystal) E log on Y with a log pole at x .The fiber E log x of E log at x is a K -vector space equipped with a nilpotent operator. If E moreover hasa Frobenius structure, then E log x is a ( ϕ, N ) -module , that is a K -vector space V equipped with a nilpotentoperator N : V → V and a σ -semilinear automorphism ϕ : V → V such that ϕ − N ϕ = qN . We can describeit in terms of the nearby cycle of E around x . Proposition 2.9.4.
Let X = G m , X = P and x = 0 . Suppose that E is unipotent at and let Ψ be thenearby cycle functor defined by id (2.7.4) . Then there exists a canonical isomorphism of K -vector spaces(resp. K -vector spaces with Frobenius structure): (2.9.4.1) E log0 ∼ −→ Ψ( E ) . Proof.
The argument of ([5] 2.4(1)) implies the assertion, that we briefly explain in the following. Since E is unipotent at x , we have Ψ un ( E ) ≃ Ψ( E ). By ([6] 3.4.19, cf. [5] 2.4(1)), we have a Frobenius equivariantisomorphism of K -vector spaces:Ψ un ( E ) ≃ lim −→ n ≥ Ker( N n : ( E log ⊗ Log n ) → ( E log ⊗ Log n ) ) , where Log n is the log convergent F -isocrystal on ( A ,
0) defeind in 2.7.1 and N n = N E log0 ⊗ id + id ⊗ N Log n is the tensor product of two nilpotent operators. Then the isomorphism (2.9.4.1) follows. (cid:3) ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 25 Hyperbolic localization for arithmetic D -modules.2.10.1. Let X be a quasi-projective k -scheme such that X ⊗ k k is connected and normal. We suppose thatthere exists an action µ : G m × X → X of the torus G m over k . Following [41], we denote by X the closedsubscheme of fixed points of X ([41] 1.3), by X + (resp. X − ) the attractor (resp. repeller) of X ([41] 1.4,1.8). We have a commutative diagram(2.10.1.1) X + π (cid:4) (cid:4) g ! ! ❇❇❇❇❇❇❇❇ X f = = ③③③③③③③③ f ′ ! ! ❉❉❉❉❉❉❉❉ XX − π ′ Y Y g ′ = = ⑤⑤⑤⑤⑤⑤⑤⑤ where f, f ′ are closed immersions and are sections of π, π ′ , respectively, the restriction of g (resp. g ′ ) to eachconnected component of X + (resp. X − ) is a locally closed immersion ([41] 1.6.8). Note that each connectedcomponent of X + is the preimage of a connected component of X under π .We define hyperbolic localization functors ( − ) !+ , ( − ) +! : D( X ) → D( X ), for F ∈ D( X ) by:(2.10.1.2) F !+ = f ! ( g + ( F )) , F +! = f ′ + ( g ′ ! ( F )) . We say an object F of D( X ) is weakly equivariant if there exists an isomorphism µ + ( F ) ≃ L [ − ⊠ F for some smooth module L on G m . Theorem 2.10.2 (Braden [24]) . (i) There exists a canonical morphism ι F : F +! → F !+ , which is anisomorphism if F is weakly equivariant. (ii) The canonical morphisms π ! → f ! , π ′ + → f ′ + induce morphisms (2.10.2.1) π ! g + F → F !+ , π ′ + g ′ ! → F +! , which are isomorphisms if F is weakly equivariant. Recall the construction of ι F . The canonical morphism i = ( f, f ′ ) : X → Z = X + × X X − is bothan open immersion and a closed one ([41] 1.9.4). We denote by h : Z → X + , h ′ : Z → X − the canonicalmorphisms.We set F + = g + ( g + ( F )) and denote by β : F → F + the adjunction morphism. By the base change,there exists a canonical isomorphism(2.10.3.1) ( F + ) +! = f ′ + g ′ ! g + g + ( F ) ≃ f ′ + h ′ + h ! g + ( F ) ≃ F !+ . Then we define the morphism ι F to be the composition of (2.10.3.1) and β +! : F +! → ( F + ) +! .By the base change, the morphism ι F is compatible with inverse image by the inclusion of a G m -equivariant open subscheme and with direct image by the inclusion of a G m -equivariant closed subscheme. To prove canonical morphisms in (2.10.2) are isomorphisms, we can extend the scalar and assumethat L is an extension of the maximal unramified extension of Q p (2.3.3).Let Y be a k -scheme and Y = Y ⊗ k k . The category D( Y /L ) is independent of the choice of base field k (2.3.3) and we denote it by D( Y /L ). Given a morphism f : Y → Z of k -schemes, it descents to a morphismof k ′ -schemes for some finite extension k ′ of k . This allows us to define the cohomological functors betweenD( Y /L ) and D(
Z/L ). To prove 2.10.2, we can replace the involved schemes by their base change to k .By a result of Sumihiro [80], we may assume moreover that X is isomorphic to an affine space over k ,equipped with a linear G m -action. Let Y be a k -scheme on which G m acts trivially and W a G m -equivariant vector bundle over Y .Suppose that there exists a decomposition of vector bundles W ≃ W + ⊕ W − such that all the weights of W + are larger than all the weights of W − . We set E = P ( W ) − P ( W + ) and B = P ( W − ), where P ( − ) denotes the associated projective bundle over Y . We denote by p : E → B the canonical morphism defined by p ([ w + , w − ]) = [0 , w − ], by i : B → E thecanonical morphism, which is a section of p and by ϕ : B → Y the projection.Based on the same argument of ([24] lemma 6), we can show a similar result. Lemma 2.10.6 ([24] lemma 6) . Let F be a weakly equivariant object of D( E ) . There exists canonicalisomorphisms (2.10.6.1) h + p + F ≃ h + i + F , h + p ! F ≃ h + i ! F . By 2.10.6, we deduce 2.10.2(ii). Using 2.10.6 and [80], we prove 2.10.2(i) in the same way as in ([24] § 4).3.
Geometric Satake equivalence for arithmetic D -modules In this section, we establish the geometric Satake equivalence for arithmetic D -modules.We assume that k is a finite field with q = p s elements and keep the assumption and notation in § 2.We work with holonomic modules (resp. complexes) over the geometric base tuple T = { k, R, K, L } and weomit /L from the notations Hol( − /L ) , D( − /L ) for simplicity.Let G denote a split reductive group over k . Let T be the abstract Cartan of G , which is defined upto a canonical isomorphism as the quotient of a Borel subgroup B by its unipotent radical. We denote by X • = X • ( T ) the weight lattice and by X • = X • ( T ) the coweight lattice. Let Φ ⊂ X • (resp. Φ ∨ ⊂ X • ) theset of roots (resp. coroots). Let Φ + ⊂ Φ be the set of positive roots and X • ( T ) + ⊂ X • ( T ) the semi-groupof dominant coweights, determined by a choice of B . (But they are independent of the choice of B .) Given λ, µ ∈ X • ( T ), we define λ ≤ µ if µ − λ is a non-negative integral linear combinations of simple corootsand λ < µ if λ ≤ µ and λ = µ . This defines a partial order on X • ( T ) (and on X • ( T ) + ). We denote by ρ ∈ X • ( T ) ⊗ Q the half sum of all positive roots.3.1. The Satake category.3.1.1.
We briefly recall affine Grassmannians following ([88] § 1, § 2). The loop group LG (resp. positiveloop group L + G ) is the fpqc sheaf on the category of k -algebras associated to the functor R G ( R (( t )))(resp. R G ( R J t K ) ). Then L + G is a subsheaf of LG and the affine Grassmannian Gr G is defined as thefpqc-quotient Gr G = LG/L + G. The sheaf Gr G is represented by an ind-projective ind-scheme over k . We write simply Gr instead of Gr G ,if there is no confusion.For any dominant coweight µ ∈ X • ( T ) + , we denote by Gr µ the corresponding ( L + G )-orbit, which issmooth quasi-projective over k of dimension 2 ρ ( µ ) ([88] 2.1.5). Let Gr ≤ µ be the reduced closure of Gr µ inGr, which is equal to ∪ λ ≤ µ Gr λ . Let j µ : Gr µ → Gr ≤ µ be the open inclusion. We have an ind-presentationGr red ≃ lim −→ µ ∈ X • ( T ) + Gr ≤ µ . Since we will work with holonomic modules, we can replace Gr by its reducedind-subscheme ([3] 1.1.3 lemma), and omit the subscript red to simplify the notation.For i ≥
0, let G i be the i -th jet group defined by the functor R G ( R [ t ] /t i +1 ). Then G i is representableby a smooth geometrically connected affine group scheme over k and we have L + G ≃ lim ←− i G i . If we considerthe left action of L + G on Gr, then the action on Gr ≤ µ factors through G i for some i . We can define thecategory of ( L + G )-equivariant holonomic modules on Gr (see 2.5.6), denoted as Sat G and called Satakecategory . It is a full subcategory of Hol(Gr) (2.5.2).
Proposition 3.1.2.
The category
Sat G is semisimple with simple objects IC µ := j µ, !+ ( L Gr µ [2 ρ ( µ )]) (2.6.1) . Lemma 3.1.3.
For µ ∈ X • ( T ) + , the category Sm(Gr µ ) (2.3.4) is semisimple with simple object L Gr µ .Proof. The ( L + G )-orbit Gr µ is geometrically connected and satisfies π ét1 (Gr µ ⊗ k k ) ≃ { } (cf. [74] proofof proposition 4.1). Every irreducible object M of Sm(Gr µ ) has a Frobenius structure with respect to thearithmetic base tuple T F = { k, R, K, L, s, id } with finite determinant ([2] 6.1). By the companion theorem ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 27 for overconvergent F -isocrystals over a smooth k -scheme ([7] 4.2) and Čebotarev density ([3] A.4), we deducethat M ≃ L Gr µ in the category Sm(Gr µ ).To show the semisimplicity, it suffices to show that H (Gr µ , L ) = 0. There exists a morphism π : Gr µ → G/P µ realizing Gr µ as an affine bundle over the partial flag variety G/P µ , where P µ is the parabolic subgroupcontaining B and associated with { α ∈ Φ , ( α, µ ) = 0 } . In view of (2.3.4.2) and the cohomology of affinespaces (2.4.1(ii)), the canonical morphism L G/P µ → π + ( L Gr µ ) is an isomorphism. Then the cohomologyH i (Gr µ , L ) is isomorphic to H i ( G/P µ , L ). Since the partial flag variety admits a stratification of affinespaces, we deduce that H i ( G/P µ , L ) = 0 if i is odd by (2.4.2.2). Then the assertion follows. (cid:3) To prove proposition 3.1.2, we need a parity result on the constructible cohomology of IC µ . Lemma 3.1.4.
The constructible module c H i (IC µ ) vanishes unless i ≡ dim(Gr µ ) (mod 2) .Proof. We follow the argument of Gaitsgory ([49] A.7, cf. [12] §4.2 for a detailed exposition) in the ℓ -adic case, whose proof is based on following ingredients: 1) the decomposition theorem; 2) the fiber of theBott-Samelson resolution of a cell in affine flag variety is paved by affine spaces.In our case, the assertion follows from the same argument using the decomposition theorem (2.6.4),the spectral sequence (2.4.2.2) and the parity of the compact support p -adic cohomology of affine spaces(2.4.1(ii)). (cid:3) Proof of proposition ℓ -adic case (cf. [49] prop. 1). By 2.6.1(i),holonomic modules IC µ are irreducible of Sat G . Let E be an irreducible object of Sat G . There exists an( L + G )-orbit Gr µ such that E | Gr µ is a smooth object. By 2.5.2 and 3.1.3, we deduce that E is isomorphic toIC µ .To prove the semisimplicity, it suffices to show that for λ, µ ∈ X • ( T ) + , we have(3.1.5.1) Ext (IC λ , IC µ ) = Hom D(Gr) (IC λ , IC µ [1]) = 0 . (i) In the case λ = µ , (3.1.5.1) follows from Ext µ ) ( L Gr µ , L Gr µ ) = H (Gr µ , L ) = 0 (see the proof oflemma 3.1.3).(ii) Then we consider the case either λ < µ or µ < λ . Since the dual functor D induces an anti-equivalence,we may assume that µ < λ . We denote by i : Gr ≤ µ → Gr ≤ λ the close immersion and we haveHom D(Gr) (IC λ , i + IC µ [1]) ≃ Hom
D(Gr ≤ µ ) ( i + IC λ , IC µ [1]) . Note that i + IC λ has cohomological degrees ≤ − L + G )-equivariant holonomic module H i ( i + IC λ | Gr µ ) is smooth and hence is constant (3.1.3). If there existed a non-zero morphism g : i + IC λ → IC µ [1], then it would induce a non-zero morphism h : H − ( i + IC λ | Gr µ ) → L Gr µ [2 ρ ( µ )]. Since i + is c-t-exact,it contradicts to 3.1.4. The equality (3.1.5.1) in this case follows.(iii) In the case λ (cid:2) µ and µ (cid:2) λ , we prove (3.1.5.1) by base change in the same way as in ([12] 4.3). (cid:3) We consider the action of L + G on LG × Gr defined by a ( g, [ h ]) = ( ga − , [ ah ]), where [ h ] denotes thecoset h · L + G of an element h ∈ LG . We denote by Gr e × Gr the twisted product LG × Gr /L + G (2.5.4).The morphism LG × Gr → Gr, defined by ( g, [ h ]) [ gh ], induces an ind-proper morphism(3.1.6.1) m : Gr e × Gr → Gr , called the convolution morphism . The morphism m is ( L + G )-equivariant with respect to the left ( L + G )-actions.Given two objects A , A of Sat G , we denote by A e ⊠ A their external twisted product on Gr e × Gr (see2.5.4 and 2.5.6), and define the convolution product by(3.1.6.2) A ⋆ A = m + ( A e ⊠ A ) . Similarly, there exists an n -fold twisted product Gr e × · · · e × Gr and a convolution morphism m : Gr e × · · · e × Gr → Gr ([88] 1.2.15). This allows us to define the n -fold convolution product A ⋆ · · · ⋆ A n . We will show that A ⋆ A is an object of Sat G and that ⋆ defines a symmetric monoidal structureon Sat G . To do it, we will interpret the convolution product as the specialization of a fusion product onBeilinson-Drinfeld Grassmannians in the next subsection.3.2. Fusion product.3.2.1.
Let X be a smooth, geometrically connected curve over k , n an integer ≥ X n the n -folded selfproduct of X over k . We briefly recall the definition of Beilinson-Drinfeld Grassmannians ([88] § 3).For any k -algebra R and any point x = ( x i ) ni =1 ∈ X n ( R ), we set Γ x = ∪ ni =1 Γ x i the closed subscheme of X R defined by the union of graphs Γ x i ֒ → X R of x i : Spec( R ) → X . The Beilinson-Drinfeld Grassman-nian Gr G,X n (associated to G over X n ) is the functor which associates to every k -algebra R the groupoidGr G,X n ( R ) of triples ( x, E , β ) { x ∈ X n ( R ) , E a G -torsor on X R , β : E | X R − Γ x ∼ −→ E := G × ( X R − Γ x ) a trivialisation } . The above functor is represented by an ind-projective ind-scheme over X n ([88] 3.1.3). We denote by q n : Gr G,X n → X n the canonical morphism. If there is no confusion, we will write simply Gr X n instead ofGr G,X n . Note that the fiber of Gr X at a closed point x of X is isomorphic to the affine Grassmannian.We refer to ([88] 3.1) the definition of global loop groups ( L + G ) X n and ( LG ) X n over X n . The sheaf( L + G ) X n is represented by a projective limit of smooth affine group scheme over X n . There exists a canonicalisomorphism of fpqc-sheaves ( LG ) X n / ( L + G ) X n ∼ −→ Gr G,X n . We consider the left action of ( L + G ) X n onGr G,X n over X n and denote by Hol ( L + G ) Xn (Gr X n ) the category of ( L + G ) X n -equivariant holonomic moduleson Gr X n (2.5.6). In the following, we take the curve X to be the affine line A k . Then there exists an isomorphismGr X ≃ Gr × X . Given a holonomic module A on Gr, the holonomic module A X = A ⊠ L X [1] is ULA withrespect to q : Gr X → X (2.8.4). If A is moreover ( L + G )-equivariant, then A X is ( L + G ) X -equivariant.By proposition 2.4.4, we obtain a fully faithful functor(3.2.2.1) ι : D(Gr) → D(Gr X ) , A 7→ A X . We denote the essential image of Sat G via ι by Sat X , which is a full subcategory of Hol ( L + G ) X (Gr X ). To define the fusion product on Sat X , we will use the factorization structure of Beilinson-DrinfeldGrassmannians.The diagonal immersion ∆ : X → X a morphism Gr X → Gr X sending ( x, E , β ) to (∆( x ) , E , β ), whichis compatible with ( LG ) X -actions. Moreover, it induces a canonical isomorphism(3.2.3.1) ∆ : Gr X ∼ −→ Gr X × X , ∆ X. Let U be complement of ∆ : X → X . Then there exists a canonical isomorphism, called the factorizationisomorphism ([88] 3.2.1(iii))(3.2.3.2) c : Gr X × X U ∼ −→ (Gr X × Gr X ) × X U. The involution σ : X → X , ( x, y ) ( y, x ), induces an involution ∆( σ ) : Gr X → Gr X . If we considerthe S -action on Gr X × Gr X by the permutation of two factors, then c is moreover S -equivariant. The convolution morphism (3.1.6.1) also admits a globalization. The convolution Grassmannian Gr X e × Gr X is the ind-scheme representing R (cid:26) ( x, ( E i , β i ) i =1 , ) (cid:12)(cid:12)(cid:12)(cid:12) x = ( x , x ) ∈ X ( R ), E , E are G -torsors on X R β i : E i | X R − Γ xi ∼ −→ E i − | X R − Γ xi where E = E is trivial (cid:27) . There exists a convolution morphism(3.2.4.1) m : Gr X e × Gr X → Gr X , ( x, ( E i , β i ) i =1 , ) ( x, E , β ◦ β ) . By definition, the restriction of m on U is an isomorphism. ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 29 We can view Gr X e × Gr X as a twisted product (2.5.4) in the following way. There exists a ( L + G ) X -torsor E → Gr X × X classifying R E ( R ) = (cid:26) ( x , β , E ) ∈ Gr X ( R ); x ∈ X ( R ); η : E ∼ −→ E | b Γ x (cid:27) , where η is a trivialisation of E on the formal completion b Γ x of X R along Γ x . Using the torsor E , we canidentify Gr X e × Gr X with the twisted product (Gr X × X ) e × X Gr X (2.5.6). In summary, we have the followingdiagram over X (3.2.4.2) Gr X × Gr X = (Gr X × X ) × X Gr X ← E × X Gr X → Gr X e × Gr X m −→ Gr X . Let A , A be two objects of Sat X . Note that ( A ⊠ L X ) ⊠ X A ≃ A ⊠ A is holonomic. We denote by A e ⊠ A the twisted product of A ⊠ L X and A on Gr X e × Gr X (2.5.6). Proposition 3.2.5. (i)
There exists a canonical isomorphism of holonomic modules on Gr X : (3.2.5.1) m + ( A e ⊠ A ) ≃ j !+ ( A ⊠ A | U ) . The left hand side, denoted by A (cid:24) A , is ULA with respect to q : Gr X → X and is ( L + G ) X -equivariant. (ii) There exists a canonical isomorphism of holonomic modules on Gr X : ∆ + [ − A (cid:24) A ) ∼ −→ ∆ ! [1]( A (cid:24) A ) . We denote one of the above module by A ⊛ A and call it fusion product of A , A . This holonomic moduleis ULA with respect to q : Gr X → X .Proof. (i) The holonomic module A ⊠ A on Gr X × Gr X is the inverse image of a holonomic moduleon Gr × Gr and hence is ULA with respect to the projection Gr X × Gr X → X . Recall that A e ⊠ A isconstructed by descent along a quotient by a smooth group scheme over X (2.5.6, 3.2.4.2). Hence it is ULAwith respect to the projection to X by proposition 2.8.3(iii). Since m is ind-proper, then m + ( A e ⊠ A ) isULA with respect to q : Gr X → X .Since m | U is an isomorphism, under the isomorphism (3.2.3.2) we have A e ⊠ A | U = A ⊠ A | U , which is holonomic. Then we deduce the isomorphism (3.2.5.1) from proposition 2.8.5(i). The morphism m is ( L + G ) X -equivariant. By proper base change, we deduce that m + ( A e ⊠ A ) is ( L + G ) X -equivariant.Assertion (ii) follows from proposition 2.8.5. (cid:3) Corollary 3.2.6.
Let A , A be two objects of Sat G . (i) There exists a canonical isomorphism on Gr X (3.1.6.2)(3.2.6.1) ( A ⋆ A ) X ≃ A ,X ⊛ A ,X . (ii) The convolution product A ⋆ A is still holonomic and belongs to Sat G . (iii) The category
Sat G (resp. Sat X ) equipped with the bifunctor ⋆ (resp. ⊛ ) and the unit object IC (resp. IC ,X ) forms a monoidal category.Proof. (i) There exists a canonical isomorphism(Gr e × Gr) × X ≃ (Gr X e × Gr X ) × X , ∆ X, compatible with projections to Gr X . Via the above isomorphism, we have ( A e ⊠ A ) X ≃ ∆ + [ − A ,X e ⊠ A ,X ).Then the isomorphism of (3.2.6.1) follows.(ii) Taking a k -point i x : x → X and applying the functor i + x [ −
1] to (3.2.6.1), we deduce that A ⋆ A isholonomic by propositions 2.8.5 and 3.2.5.(iii) It suffices to show the assertion for Sat G . By identifying ( A ⋆ A ) ⋆ A and A ⋆ ( A ⋆ A ) with A ⋆ A ⋆ A , we obtain the associative constraint. We can verify the pentagon axiom and the unit axiomby n -fold convolution product. (cid:3) Hypercohomology functor.3.3.1.
We define the hypercohomology functor H ∗ byH ∗ : Sat G → Vec L , A 7→ M n ∈ Z H n (Gr , A ) . (3.3.1.1)Since Sat G is semisimple (3.1.2), H ∗ is exact and faithful.Let A be an object of Sat G and π : Gr → Spec( k ) the structure morphism. By the Künneth formula ([3]1.1.7), there exists a canonical isomorphism(3.3.1.2) q + ( A X )[ − ≃ π + ( A ) ⊠ L X . Lemma 3.3.2.
Given two objects A , A of Sat X , there exists a canonical isomorphism (3.3.2.1) q + ( A ⊛ A )[ − ≃ ( q + ( A )[ − ⊗ ( q + ( A )[ − . Proof.
It suffices to construct a canonical isomorphism(3.3.2.2) q ( A (cid:24) A ) ≃ q + ( A ) ⊠ q + ( A ) . By (3.2.5.1) and the Künneth formula ([3] 1.1.7), such an isomorphism exists on U = X − ∆( X ).Let τ : X → X be the morphism sending ( x, y ) to x − y . Both sides of (3.3.2.2) are ULA with respectto τ by propositions 2.8.3 and 3.2.5. By proposition 2.8.5, we deduce a canonical isomorphism on X ∆ ! (cid:0) q ( A (cid:24) A ) (cid:1) ≃ ∆ ! (cid:0) q + ( A ) ⊠ q + ( A ) (cid:1) . Then the isomorphism (3.3.2.2) follows from the distinguished triangle ∆ + ∆ ! → id → j + j + → . (cid:3) By (3.2.6.1), (3.3.1.2) and lemma 3.3.2, we deduce that:
Corollary 3.3.3.
The functor H ∗ is monoidal. Remark 3.3.4.
Let A , A be two objects of Sat G , both equipped with a Frobenius structure with respect tothe arithmetic tuple T F = { k, R, K, L, s, id L } . The proof of corollary 3.3.3 applies to arithmetic D -moduleswith Frobenius structures. Then we deduce that the following isomorphism is compatible with Frobeniusstructure H ∗ ( A ⋆ A ) ≃ H ∗ ( A ) ⊗ H ∗ ( A ) . In the following, we will construct a commutativity constraint on (Sat G , ⋆ ) which makes the functorH ∗ into a tensor functor.The permutation σ : { , } → { , } induces an involution ∆( σ ) : Gr X → Gr X over the involution σ : X → X , ( x, y ) ( y, x ) (3.2.3). Let A , A be two objects of Sat G . We deduce from (3.2.3.2) and(3.2.5.1) a canonical isomorphism(3.3.5.1) ∆( σ ) + ( A ,X (cid:24) A ,X ) ∼ −→ A ,X (cid:24) A ,X . We denote by c gr the isomorphism which fits into the following diagram σ + ( q ( A (cid:24) A )) ∼ / / ≀ (cid:15) (cid:15) q (∆( σ ) + ( A (cid:24) A )) ∼ / / q ( A (cid:24) A ) ≀ (cid:15) (cid:15) σ + ( q + ( A ) ⊠ q + ( A )) c gr / / q + ( A ) ⊠ q + ( A )The cohomology H n ( i +( x,x ) ( c gr )) of the fiber of c gr at ( x, x ) ∈ X is the composition M i + j = n H i (Gr , A ) ⊗ H j (Gr , A ) ≃ H n (Gr × Gr , A ⊠ A ) ≃ H n (Gr × Gr , A ⊠ A ) ≃ M i + j = n H j (Gr , A ) ⊗ H i (Gr , A ) , where the first and third isomorphisms are given by the Künneth formula and the second one is induced bythe symmetry of external tensor products. It sends s ⊗ t to ( − ij t ⊗ s for s ∈ H i (Gr , A ) and t ∈ H j (Gr , A ). ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 31 Taking the fiber of (3.3.5.1) at ( x, x ), we obtain a canonical isomorphism(3.3.5.2) c ′A , A : A ∗ A ≃ A ∗ A , which fits into a commutative diagram(3.3.5.3) H ∗ ( A ∗ A ) c ′A , A / / ≀ (cid:15) (cid:15) H ∗ ( A ∗ A ) ≀ (cid:15) (cid:15) H ∗ ( A ) ⊗ H ∗ ( A ) c gr / / H ∗ ( A ) ⊗ H ∗ ( A ) . We regard H ∗ as a functor from Sat G to the category Vec gr L of Z -graded vector spaces over L by consideringthe Z -grading on cohomology degree (3.3.1.1). The above diagram means H ∗ is compatible with the constraint c ′A , A on Sat G and the supercommutativity constraint c gr on Vec gr L . In 3.4.7, we will modify the constraint c ′ and make it compatible with the usual constraint on Vec L .3.4. Semi-infinite orbits.
In this subsection, we study the p -adic cohomology of objects of Sat G on semi-infinite orbits of Gr G following Mirković and Vilonen [68]. Let B op be the opposite Borel subgroup. The inclusion B, B op → G and projections B, B op → T induce morphisms(3.4.1.1) Gr T π ←− Gr B i −→ Gr G , Gr T π ′ ←− Gr B op i ′ −→ Gr G . Via i , each connected component of (Gr B ) red is locally closed in Gr G . To simplify the notation, we willomit the subscript red in the following. The affine Grassmannian Gr T is discrete, whose k -points are givenby L λ = t λ T ( k J t K ) /T ( k J t K ) ∈ Gr T ( k ) , λ ∈ X • ( T ). For λ ∈ X • ( T ), we define ind-subschemes S λ and T λ ofGr G to be(3.4.1.2) S λ = i ( π − ( L λ )) , T λ = i ′ ( π ′− ( L λ )) . For i ∈ Z , we set cohomology functors H i c ( S λ , − ) and H iT λ (Gr G , − ) to beH i c ( S λ , − ) = H i (( π ! i + ( − )) λ ) , H iT λ (Gr G , − ) = H i ( π ′ + i ′ ! ( − )) λ ) . Proposition 3.4.2 ([68] 3.1, 3.2, [88] 5.3.6) . (i) The union S ≤ λ = ∪ λ ′ ≤ λ S λ ′ is closed in Gr G and S λ isopen and dense in S ≤ λ . (ii) For µ ∈ X • ( T ) + , the intersection Gr G,µ ∩ S λ (resp. Gr G,µ ∩ T λ ) is non-empty if and only if L λ ∈ Gr G, ≤ µ (equivalently there exists w ∈ W such that wλ ≤ µ ). In the non-empty case, Gr G,µ ∩ S λ (resp. Gr G,µ ∩ T λ ) has pure dimension ρ ( λ + µ ) . Proposition 3.4.3. (i)
For any object A of Sat G , there exists a functorial isomorphism (3.4.3.1) H i c ( S λ , A ) ≃ H iT λ (Gr G , A ) . Both sides vanish if i = 2 ρ ( λ ) . (ii) For µ ∈ X • ( T ) + , the dimension of H ρ ( λ )c ( S λ , IC µ ) is equal to the number of geometrically irreduciblecomponents of S λ ∩ Gr G,µ . If we work with the arithmetic base T F = { k, R, K, L, s, id L } , the Frobenius actson H ρ ( λ )c ( S λ , IC µ ) by multiplication by q ρ ( λ + µ ) . (iii) For any integer i , there exists a functorial isomorphism (3.4.3.2) H i (Gr G , A ) ≃ M λ ∈ X • ( T ) H i c ( S λ , A ) . The proposition can be proved in the same way as in ([68] 3.5, 3.6) by Braden’s theorem (2.10.2).
Proposition 3.4.4.
Given two objects A , A of Sat G , there exists a canonical isomorphism (3.4.4.1) H ρ ( λ )c ( S λ , A ∗ A ) ≃ M λ + λ = λ H ρ ( λ )c ( S λ , A ) ⊗ H ρ ( λ )c ( S λ , A ) . To prove the above proposition, we need to extend semi-infinite orbits S λ , T λ to Beilinson-DrinfeldGrassmannians Gr G,X n . For simplicity, we only consider the case where X = A and n = 1 , n = 1, we have ind-representation Gr G,X ≃ lim −→ Gr G, ≤ µ,X , where Gr G, ≤ µ,X ≃ Gr G, ≤ µ × X isnormal for µ ∈ X • ( T ) + . When n = 2, for µ, ν ∈ X • ( T ) + , we denote by Gr G, ≤ ( µ,ν ) ,X the closure ofGr G, ≤ µ,X × Gr G, ≤ ν,X | U in Gr G,X . These closed subschemes form an ind-representation of Gr X . Thescheme Gr G, ≤ ( µ,ν ) ,X is flat over X and satisfies Gr G, ≤ ( µ,ν ) ,X × X , ∆ X ≃ Gr G, ≤ µ + ν,X ([86] 1.2, [88] 3.1.14).The composition Gr G, ≤ ( µ,ν ) ,X → X → X with X → X, ( x, y ) x − y , is flat with reduced fiber at 0and is normal on X − { } . Then we deduce that Gr G, ≤ ( µ,ν ) ,X is normal (cf. [72] 9.2).We consider the action of G m on Gr G,X n induced by 2 ˇ ρ , which is compatible with the action of G m onGr G on each fiber of x ∈ | X n | (3.4.2(i)). Then Gr T,X n is the ind-subscheme of fixed points. For λ ∈ X • ( T ),we set C λ ( X ) = Gr T, ≤ λ,X − Gr T,<λ,X . Its fiber at x = ( x, x ) ∈ ∆( X ) ⊂ X is isomorphic to { L λ } andits fiber at x = ( x, y ) ∈ X − ∆( X ) is isomorphic to Q λ + λ = λ { L λ } × { L λ } . Connected components ofGr T,X are parametrized by { C λ ( X ) } λ ∈ X • ( T ) .We denote by S λ ( X n ) (resp. T λ ( X n )) the connected component of Gr + G,X n (resp. Gr − G,X n ) correspondingto C λ ( X ). (See 2.10.1 for the notation). The fiber of S λ ( X ) (resp. T λ ( X )) at x = ( x, x ) ∈ ∆( X ) ⊂ X isisomorphic to S λ (resp. T λ ) and its fiber at x = ( x, y ) ∈ X − ∆( X ) is isomorphic to Q λ + λ = λ S λ × S λ (resp. Q λ + λ = λ T λ × T λ ). Proof of proposition A , A be two objects of Sat G , A ,X , A ,X their extensions to Gr G,X (3.2.2.1) and B = A ,X (cid:24) A ,X . Consider the following diagram of ind-schemes:(3.4.6.1) S λ ( X ) j / / i λ S ≤ λ ( X ) i λ / / Gr G,X q / / X For i ∈ Z , we define the constructible module L iλ ( A , A ) on X to be(3.4.6.2) L iλ ( A , A ) = c H i ( q ( i λ, ! ( i + λ B ))) ≃ c H i ( q ( i ′ λ, + ( i ′ ! λ B ))) , where the second isomorphism follows from Braden’s theorem (2.10.2). By the base change, the stalk of L iλ ( A , A ) at a k -point ( x , x ) of X is isomorphic to(3.4.6.3) L iλ ( A , A ) ( x ,x ) ≃ (cid:26) H i c ( S λ , A ⋆ A ) if x = x , L λ + λ = λ H i c ( S λ × S λ , A ⊠ A ) if x = x . By 3.4.3, L iλ ( A , A ) vanishes unless i = 2 ρ ( λ ) and we deduce from the Künneth formula that(3.4.6.4) H ρ ( λ )c ( S λ × S λ , A ⊠ A ) ≃ H ρ ( λ )c ( S λ , A ) ⊗ H ρ ( λ )c ( S λ , A ) . The adjunction morphisms id → i λ, + i + λ and j ! j + → id (3.4.6.1) induce canonical morphisms(3.4.6.5) c H ρ ( λ ) ( q ( A ,X (cid:24) A ,X )) ։ c H ρ ( λ ) (( q ◦ i λ ) + i + λ B ) ∼ ←− L ρ ( λ ) ( A , A ) , where the first arrow is an epimorphism and the second arrow is an isomorphism in view of the calculationof their fibers (3.4.3).By Braden’s theorem and a dual argument for T λ ( X ), we obtain a section of (3.4.6.5): L ρ ( λ ) ( A , A ) → c H ρ ( λ ) ( q ( A ,X (cid:24) A ,X )) . In view of proposition 3.4.3, we deduce a decomposition(3.4.6.6) c H i ( q ( A ,X (cid:24) A ,X ) ≃ M ρ ( λ )= i L iλ ( A , A ) . ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 33 The left hand side is a constant module with value H i (Gr , A ⋆ A ) by (3.3.1.2), (3.3.2.2). Then each summand L iλ ( A , A ) is also a constant module. Hence fibers of L iλ ( A , A ) (3.4.6.3), (3.4.6.4) are isomorphic. Theproposition follows. (cid:3) We modify the constraints c ′A , A (3.3.5.2) by a sign as follows (see [68] after Remark 6.2) and makeit compatible with the usual constraint c Vec on Vec L defined by s ⊗ t t ⊗ s .The morphism p : X • ( T ) → Z / Z , µ ( − ρ ( µ ) defines a Z / Z -grading on simple objects of Sat G . Bypropositions 3.4.2 and 3.4.3, we have(3.4.7.1) H i (Gr , IC µ ) = 0 , if i = 2 ρ ( µ ) (mod 2) . Given two simple objects A , A of Sat G , we define a new constraint c A , A to be(3.4.7.2) c A , A = ( − p ( A ) p ( A ) c ′A , A . Since Sat G is semisimple, the definition of c A , A extends to any pair ( A , A ) of objects of Sat G . By (3.3.5.3)and (3.4.7.1), the following diagram is commutative(3.4.7.3) H ∗ ( A ∗ A ) c A , A / / ≀ (cid:15) (cid:15) H ∗ ( A ∗ A ) ≀ (cid:15) (cid:15) H ∗ ( A ) ⊗ H ∗ ( A ) c Vec / / H ∗ ( A ) ⊗ H ∗ ( A ) , where the isomorphism c Vec is the usual commutativity constraint on vector spaces, i.e c Vec ( v ⊗ w ) = w ⊗ v . Proposition 3.4.8 ([88] 5.2.9) . The monoidal category
Sat G equipped with the constraints c forms a sym-metric monoidal category. The functor H ∗ (3.3.1.1) is a tensor functor.Proof. We need to verify c A , A ◦ c A , A = id and the hexagon axiom. Since the functor H ∗ is faithful, itsuffices to prove these assertions after applying H ∗ . By (3.4.7.3) and the fact that c = id, we deducethat c A , A ◦ c A , A = id. We verify the hexagon axiom in a similar way. The second assertion follows fromcorollary 3.3.3 and (3.4.7.3). (cid:3) Tannakian structure and the Langlands dual group.Theorem 3.5.1.
The symmetric monoidal category (Sat G , IC , ∗ , c ) (3.4.8) , equipped the hypercohomologyfunctor H ∗ (3.3.1.1) forms a neutral Tannakian category over L . We prove it in the same way as in ([88] 5.2.9) using proposition 3.4.3(ii).
Proposition 3.5.2.
The Tannakian group e G = Aut ⊗ H ∗ of the Tannakian category Sat G is a connectedreductive group scheme over L .Proof. For µ , µ ∈ X • ( T ) + , IC µ ⋆ IC µ is defined by direct image through the birational morphismGr ≤ µ e × Gr ≤ µ → Gr ≤ µ + µ . Hence it is supported on Gr ≤ µ + µ and is isomorphic to L Gr µ µ [2 ρ ( µ + µ )] on Gr µ + µ . Then bydecomposition theorem (2.6.4), IC µ + µ is a direct summand of IC µ ⋆ IC µ . Hence the semisimple categorySat G is generated by { IC µ i } i ∈ I with a finite set of generators of X • ( T ) + . Then e G is algebraic by ([40]2.20). There is no tensor subcategory which contains only direct sums of finite collection of IC’s. Then e G isconnected ([40] 2.22). Finally, since Sat G is semisimple, e G is reductive ([40] 2.23). (cid:3) Theorem 3.5.3.
The reductive group e G is the Langlands dual group of G over L . More precisely, the rootdatum of e G with respect to a maximal torus e T is dual to that of ( G, T ) . Since Gr T, red is a discrete set of points indexed by X • ( T ) (3.4.1), we have: Lemma 3.5.4.
In the case G = T is a torus, theorem holds. We denote by CT : Sat G → Sat T the functor(3.5.5.1) A 7→ (H ∗ c ( S λ , A )) λ ∈ X • ( T ) , and by H ∗ G (resp. H ∗ T ) the fiber functor of the Tannakian category Sat G (resp. Sat T ). By 3.4.3, there existsa canonical isomorphism of functors:(3.5.5.2) H ∗ G ≃ H ∗ T ◦ CT : Sat G → Vec L . By 3.4.4, CT is a tensor functor and therefore induces a homomorphism:(3.5.5.3) ˇ T ≃ e T → e G. By ([40] 2.21(b)), ˇ T is a closed sub-torus in e G .Using 3.4.3, we prove the following in the same way as in the ℓ -adic case (cf. [88] 5.3.17, [12] §9.1). Lemma 3.5.6.
The torus ˇ T is a maximal torus of e G . Proof of theorem e B ⊂ e G containing ˇ T such that 2 ρ ∈ X • ( T ) is adominant coweight for the choice positive roots of e G with respect to e B . Then we can show that the setof dominant weights X • ( ˇ T ) + with respect to e B is equal to the set of dominant coweights X • ( T ) + of T byproposition 3.4.3(ii) (cf. [88] 5.3, and [12] 9.5 for more details). In particular, e B is uniquely determined.We denote by e Q + the semi-subgroup of X • ( ˇ T ) generated by positive roots of e G . A weight λ belongs to e Q + if and only if there exists a highest weight representation V µ (= H ∗ (IC µ )) such that µ − λ is also a weightof V µ . By proposition 3.4.3, this is equivalent to L µ − λ ∈ Gr ≤ µ , and equivalent to λ being a sum of positivecoroots of G . Therefore the semigroup ( Q ∨ ) + ⊂ X • ( T ) = X • ( ˇ T ) generated by positive coroots of G coincideswith the semigroup e Q + . Then, the set of simple coroots of G coincide with the set of simple roots of e G .The theorem follows from the fact that a root datum is uniquely determined by the semigroup ( X • ) + ofdominant weights and the set ∆ of simple roots. (cid:3) The full Langlands dual group.
For our applications of the geometric Satake equivalence for arith-metic D -modules, it is important to consider the Frobenius structure on the Satake category. In this subsec-tion, we study the full Langlands dual group constructed by the Satake category equipped with Frobeniusstructures. We suppose that the geometric base tuple { k, R, K, L } is underlying to an arithmetic base tuple { k, R, K, L, t, σ } , where t is an integer (which may be different from the degree s of k over F p ) and σ is anautomorphism of L and extends a lifting of t -th Frobenius automorphism on k to K (2.3.1).The Frobenius pullback functor F ∗ Gr : Hol(Gr /L ) ∼ −→ Hol(Gr /L ) (2.3.3) induces a σ -semi-linear equiva-lence of tensor categories F ∗ Gr : Sat G ∼ −→ Sat G . We denote by F - Sat G the category of pairs ( X, ϕ ) consistingof an object X of Sat G and a Frobenius structure ϕ : F ∗ Gr X ∼ −→ X . Morphisms are morphisms of Sat G compatible with ϕ (cf. [3] 1.4.6). We will show that F - Sat G is a Tannakian category. We first study some general constructions in the Tannakian formalism following [76].For n ∈ Z , we denote abusively by σ n the equivalence of categories ( − ) ⊗ L,σ n L : Vec L ∼ −→ Vec L .Let ( C , ω ) be a neutralized Tannakian over L . We suppose that, for each n ∈ Z , there exists a σ n -semi-linear equivalence of tensor categories τ n : C → C and an isomorphism of tensor functors α n : ω ◦ τ n ∼ −→ σ n ◦ ω . For any pair n, m ∈ Z , we suppose moreoverthat there exists an isomorphism of tensor functors ε : τ m ◦ τ n ≃ τ m + n such that(id ◦ α n ) ◦ ( α m ◦ id) = α m + n ◦ ω ( ε ) : ω ◦ τ m ◦ τ n ≃ σ m + n ◦ ω. Since ω is faithful, such an isomorphism ε is unique.Let H be the Tannakian group of ( C , ω ). The above structure defines a homomorphism(3.6.2.1) ι : Z → Aut( H ( L )) , ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 35 by letting ι ( n ) send h : ω → ω to ω α − n −−→ σ − n ◦ ω ◦ τ n h ◦ id −−−→ σ − n ◦ ω ◦ τ n α n −−→ ω. We define the category C Z of Z -equivariant objects in C as follows. An object ( X, { c n } n ∈ Z ) consists ofan object X of C and isomorphisms c n : τ n ( X ) ∼ −→ X satisfying cocycle conditions c n + m = c n ◦ τ n ( c m ). Amorphism between ( X, { c n } n ∈ Z ) and ( X ′ , { c ′ n } n ∈ Z ) is a morphism of C compatible with c n , c ′ n . Let Γ be an abstract group and ϕ : Γ → Z a homomorphism. We say an action of Γ on an L -vectorspace V is σ -semi-linear (with respect to ϕ ) if it is additive and satisfies γ ( av ) = σ ϕ ( γ ) ( a ) γ ( v ) for γ ∈ Γ , a ∈ L and v ∈ V . We denote by Rep
L,σ (Γ) the category of σ -semi-linear representations of Γ on finite dimensional L -vector spaces.We denote by H ( L ) ⋊ Z the semi-direct product of H ( L ) and Z via ι (3.6.2.1). The short exact sequence1 → H ( L ) → H ( L ) ⋊ Z → Z → Rep
L,σ ( H ( L ) ⋊ Z ). Proposition 3.6.4.
Let H be a split reductive group over L , Rep L ( H ) the category of algebraic representa-tions of H and Rep L ( H ( L )) the category of finite dimensional representations of the abstract group H ( L ) .Then the following canonical functor is fully faithful: Rep L ( H ) → Rep L ( H ( L )) , ρ ρ ( L ) . Proposition 3.6.5.
Keep the assumption and notation as above. (i)
The category C Z is a Tannakian category over L = L σ =1 neutralized by ω over L ([40] § 3) . (ii) Suppose that the Tannakian group H of ( C , ω ) is a split reductive group over L . Then ω induces anequivalence of tensor categories (3.6.5.1) C Z ∼ −→ Rep ◦ L,σ ( H ( L ) ⋊ Z ) , where Rep ◦ L,σ ( H ( L ) ⋊ Z ) is the full subcategory of Rep
L,σ ( H ( L ) ⋊ Z ) (3.6.3) consisting of representationswhose restriction to H ( L ) is algebraic.Proof. (i) We define a monoidal structure on C Z by letting( X, { c n } ) ⊗ ( X ′ , { c ′ n } ) = ( X ′′ , { c ′′ n } ) , where X ′′ = X ⊗ X ′ and c ′′ n is the composition τ n ( X ′′ ) ≃ τ n ( X ) ⊗ τ n ( X ′ ) c n ⊗ c ′ n −−−−→ X ⊗ X ′ . This defines a structure of symmetric monoidal category on C Z .We apply ([39] 2.5) to show that ( C Γ , ⊗ ) is rigid. Given an object ( X, { c n } ) of C Z , we denote by X ∨ bethe dual of X in C and then we have τ n ( X ∨ ) ≃ τ n ( X ) ∨ . For each n , we have an isomorphism c ∨ n : X ∨ ∼ −→ ( τ n ( X )) ∨ ≃ τ n ( X ∨ ) . Then we define ( X ∨ , { ( c ∨ n ) − } ) to be the dual of ( X, { c n } ) in C Z . In view of ([40] 1.6.5), the evaluation andcoevaluation morphisms of X and of τ n ( X ) are compatible via τ n . Then we obtain the evaluation and thecoevaluation morphisms of ( X, { c n } ) in C Z satisfying the axiom of ([39] 2.1.2). Hence C Z is a rigid abeliantensor category.Since τ n is σ n -semi-linear, we have End(id C Z ) ≃ L . The forgetful tensor functor C Z → C is exact andfaithful. Hence the fiber functor ω of C defines a fiber functor ω : C Γ → Vec L ([40] 3.1). Then the assertionfollows from ([39] 1.10-1.13, see also [40] footnote 12).(ii) It suffices to construct an equivalence of tensor categories(3.6.5.2) Ψ : Rep L ( H ) Z ∼ −→ Rep ◦ L,σ ( H ( L ) ⋊ Z ) . Let ((
V, ρ ) , { c n } ) be an object of Rep L ( H ) Z . Then we define a representation ( V, e ρ ) of Rep ◦ L,σ ( H ( L ) ⋊ Z ),for any element ( h, n ) ∈ H ( L ) ⋊ Z , by letting e ρ ( h, n ) to be the composition(3.6.5.3) σ n ( ω ( V, ρ )) α − n −−→ ω ( τ n ( V, ρ )) h ◦ id −−−→ ω ( τ n ( V, ρ )) c n −→ ω ( V, ρ ) . Using the cocycle condition, one checks that the above formula defines a representation. Then we obtain thefunctor Ψ (3.6.5.2). By 3.6.4, the canonical morphismHom H ( ρ, ρ ′ ) → Hom H ( L ) ( ρ ( L ) , ρ ′ ( L ))is bijective. In view of (3.6.5.3), we deduce that Ψ is fully faithful. We leave the verification of the essentialsurjectivity to readers. (cid:3) The Frobenius pullback functor F ∗ Gr = F +Gr /k ◦ σ ∗ : Sat G ∼ −→ Sat G satisfies H ∗ ◦ F ∗ Gr ≃ σ ◦ H ∗ . We takefor every integer n the tensor equivalence τ n on Sat G to be | n | -th composition of F ∗ Gr (or a quasi-inverse of F ∗ Gr if n <
0) (3.6.2). These functors satisfy the assumption of 3.6.2. With the notation of 3.6.2, F - Sat G isequivalent to the category Sat Z G . In this case, we obtain the following result by 3.6.5. Theorem 3.6.7. (i)
The category F - Sat G is a Tannakian category over L , neutralized by the fiber functor H ∗ over L . If t = s and σ = id L , then F - Sat G is a neutral Tannakian category. (ii) There exists a canonical equivalence of tensor categories (3.6.7.1) F - Sat G ∼ −→ Rep ◦ L,σ ( ˇ G ( L ) ⋊ Z ) , compatible with fiber functors. We work with the arithmetic tuple T F = { k, R, K, L, s, id L } and we suppose there exists a square-root p / of p in L . This allows to define half Tate twist functor ( n ) for n ∈ Z by sending each object M ∈ D( X/L F ), equipped with the Frobenius structure Φ, to ( M , p − sn/ · Φ).For µ ∈ X • ( T ), we denote by IC Weil µ = j µ, !+ ( L Gr µ )[2 ρ ( µ )]( ρ ( µ )) the holonomic module in F - Sat G withweight 0, and by S the full subcategory of F - Sat G consisting of direct sums of IC Weil µ ’s.The category S is closed under the convolution on F - Sat G , i.e. IC Weil λ ⋆ IC Weil µ is isomorphic to a directsum of IC Weil ν . Indeed, by proposition 3.4.3(ii), the Frobenius acts on the total cohomology H ∗ (IC Weil µ ) by adiagonalizable automorphism with eigenvalues q n/ , n ∈ Z . Since H ∗ is compatible with Frobenius structure(3.3.4), so is the Frobenius action on H ∗ (IC Weil λ ⋆ IC Weil µ ). We have a decomposition IC λ ⋆ IC µ ≃ ⊕ IC ν . Thenthe claim follows from the fact that the the action of Frobenius on cohomology determines the isomorphismclass of an object of F - Sat G whose underlying holonomic module is isomorphic to a direct sum of IC ν ’s.The canonical functor F - Sat G → Sat G induces an equivalence of tensor categories S ∼ −→ Sat G . Inparticular, we obtain equivalences of tensor categories(3.6.8.1) Sat : Rep L ( ˇ G ) ≃ Sat G ≃ S . In the end, we briefly review the action of outer automorphism group of G on the Satake categorySat G (resp. S ).Let ( C , ω ) be a Tannakian category over L and H the associated Tannakian group. We denote byAut ⊗ ( C , ω ) the set of isomorphism classes of pairs ( τ, α ) of a tensor equivalence τ : C ∼ −→ C and an isomorphismof functors α : ω ∼ −→ ω ◦ τ . This set has a natural group structure. A similar construction as in 3.6.2 definesa canonical morphism Aut ⊗ ( C , ω ) → Aut( H ), which is an isomorphism ([52] lemma B.1). We apply this tothe Satake category S (or Sat G ) equipped with the fiber functor H ∗ . The action of Aut( G ) on Gr G inducesan action on ( S , H ∗ ), and therefore an action of Aut( G ) on ˇ G , i.e. a homomorphism ι : Aut( G ) → Aut( ˇ G ). Lemma 3.6.10.
There is a natural pinning ( ˇ B, ˇ T , N ) of ˇ G such that that map ι factors as Aut( G ) ։ Out( G ) ∼ −→ Aut † ( ˇ G, ˇ B, ˇ T , N ) ⊂ Aut( ˇ G ) . The lemma can be shown in the same way as in ([52] lemma B.2 or [76] lemma A.6). In particular, for σ ∈ Aut( G ) and V ∈ Rep ( ˇ G ), we have σ ∗ Sat( V ) ≃ Sat( ι ( σ ) V ). ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 37 Bessel F -isocrystals for reductive groups In this section, we construct Bessel F -isocrystals for reductive groups and calculate their monodromygroups. We use notations from 1.3.8, with k being a finite field of q = p s elements. We assume moreoverthat there exists an element π ∈ K satisfying π p − = − p and a square root of p in K . We fix an arithmeticbase tuple { k = F q , R, K, L, s, id L } (2.3.1) and an isomorphism K ≃ C (in order to talk about weight).We fix { , ∞} ⊂ P (over some base that we will specify in each subsection), and set X = P − { , ∞} .Although X ≃ G m , it is more convenient to regard X as an algebraic curve equipped with a simply transitiveaction of G m .Throughout this section, let G be a split reductive group (over some base). We fix a Borel subgroup B ⊂ G and a maximal torus T ⊂ B . Let U ⊂ B be the unipotent radical of B , and U op ⊂ B op the oppositeBorel and its unipotent radical. Let T ad ⊂ B ad ⊂ G ad denote the quotients of T ⊂ B ⊂ G by the center Z ( G ) of G . We denote by ( ˇ G, ˇ B, ˇ T ) the Langlands dual group of G over L , constructed by the geometricSatake equivalence (3.5).4.1. Kloosterman F -isocrystals for reductive groups. In this subsection, we follow the method ofHeinloth-Ngô-Yun [52] to produce overconvergent F -isocrystals on X by applying the geometric Langlandscorrespondence.We work with schemes over k . We will consider with both geometric coefficients and arithmetic coefficients,but for simplicity, we omit L N from the notation Hol( − /L N ) , D( − /L N ) and L from Rep L ( − ). Let G = G × P . For a coordinate x on P , so y = x − is a local coordinate around ∞ , we denote by I (0) = { g ∈ G ( k J y K ) | g (0) ∈ B } the Iwahori subgroup ,I (1) = { g ∈ G ( k J y K ) | g (0) ∈ U } the unipotent radical of I (0) ,Z ( G )(1) = { g ∈ Z ( G )( k J y K ) | g (0) ≡ y } ,I (2) = Z ( G )(1)[ I (1) , I (1)] ,I ( i ) op ⊂ G ( k J x K ) the analogous groups obtained by opposite Borel subgroup . If G is semisimple, I (2) = [ I (1) , I (1)]. On the other hand, if G is a torus, then I (2) = I (1). (So our definitionof I (2) is slightly different from [52] 1.2 when G is not semisimple, but for G = GL n coincides with the onein [52] 3.1.) These groups are independent of the choice of x .By abuse of notations, we use the same notations for the corresponding (ind)-group schemes over k . Then(4.1.1.1) I (1) /I (2) ≃ M α affine simple U α where U α ( k ) ⊂ G ( k J s K ) is the root subgroup corresponding to α . We also writeΩ = N G ( k (( x ))) ( I (0) op ) /I (0) op , which is regarded as a discrete group over k .We denote by G ( m, n ) the group scheme over P such that ([52] 1.2) G ( m, n ) | X = G × X, G ( m, n )( O ) = I ( m ) op ⊂ G ( O ) , G ( m, n )( O ∞ ) = I ( n ) ⊂ G ( O ∞ ) . We denote by Bun G ( m,n ) the moduli stack of G ( m, n )-bundles on P . Let Bun G ( m,n ) denote its connectedcomponent containing the trivial G ( m, n )-bundle ⋆ : Spec( k ) → Bun G ( m,n ) . For each γ ∈ Ω, there is acanonical isomorphism Hk γ : Bun G (0 ,n ) ≃ Bun G (0 ,n ) given by the Hecke modification of G (0 , n )-bundles at0 ∈ P corresponding to γ ([52] Corollary 1.2). This induces a canonical bijection between Ω and the setof connected components of Bun G (0 ,n ) (and therefore all Bun G ( m,n ) ). Let Bun γ G ( m,n ) denote the connectedcomponent corresponding to γ under the bijection. For γ ∈ Ω, let i γ = Hk γ ( ⋆ ) : Spec( k ) → Bun γ G (0 ,n ) .There is also the action of I (1) /I (2) on Bun G (0 , by modifying G (0 , ∞ . Let(4.1.1.2) j : Ω × I (1) /I (2) → Bun G (0 , , be the open immersion of the big cell, defined by applying the action of I (1) /I (2) × Ω to the trivial G (0 , j γ : I (1) /I (2) → Bun γ G (0 , denote its restriction to the component corre-sponding to γ . The stack of Hecke modifications of G ( m, n )-torsors (over X ) is:Hecke X G ( m,n ) ( S ) := (cid:26) ( E , E , x, β ) (cid:12)(cid:12)(cid:12)(cid:12) E i ∈ Bun G ( m,n ) ( S ), x : S → Xβ : E | X S − Γ x ∼ −→ E | X S − Γ x (cid:27) . There exist natural morphisms(4.1.2.1) Hecke X G ( m,n )pr x x ♣♣♣♣♣♣♣♣♣♣♣ pr ' ' PPPPPPPPPPPP q / / X Bun G ( m,n ) Bun G ( m,n ) × X, where pr (resp. pr , resp. q ) sends ( E , E , x, β ) to E (resp. ( E , x ), resp. x ).Following [52], we denote by GR the Beilinson-Drinfeld Grassmannian of G ( m, n ) with modifications on X . Note that GR ≃ Gr G,X ≃ Gr G × X and therefore is independent of ( m, n ). There exists a smooth atlas ̟ : U → Bun G ( m,n ) such that U × Bun G ( m,n ) , pr Hecke X G ( m,n ) ≃ U × GR , (4.1.2.2) ( U × G m ) × (Bun G ( m,n ) × X ) , pr Hecke X G ( m,n ) ≃ U × GR . (4.1.2.3)For V ∈ Rep ( ˇ G ), we associate a holonomic module Sat( V ) on Gr G by the geometric Satake equivalence(3.6.8.1). We denote abusively by IC V the holonomic module on Hecke X G ( m,n ) defined by smooth descent of K U × X ⊠ Sat( V ) on U × X × Gr G (supported in a subscheme U × X × Gr G,V ). Then IC V is supported in asubstack Hecke X G ( m,n ) ,V of Hecke X G ( m,n ) .The geometric Hecke operators is defined as a functorHk : Rep ( ˇ G ) × D(Bun G ( m,n ) ) → D(Bun G ( m,n ) × X ) , (4.1.2.4) ( V, M ) Hk V ( M ) := pr , ! (cid:0) pr +1 ,V ( M ) ⊗ IC V (cid:1) . Here pr ,V : Hecke X G ( m,n ) ,V → Bun G ( m,n ) and pr | Hecke X G ( m,n ) ,V : Hecke X G ( m,n ) ,V → Bun G ( m,n ) × X areschematic (4.1.2.2, 4.1.2.3), which allows us to apply cohomological functors of pr ,V , pr (2.8.8).We call a tensor functor E : Rep ( ˇ G ) → Sm(
X/L ) (resp. Sm(
X/L F ))ˇ G -valued overconvergent isocrystal (resp. F -isocrystal) E on X . We denote by E V its value on V ∈ Rep ( ˇ G ). A Hecke eigen-module with eigenvalue E is a holonomic module M on Bun G ( m,n ) together with isomorphismsHk V ( M ) ∼ −→ M ⊠ E V , V ∈ Rep ( ˇ G ) , which are compatible with tensor structure on Rep ( ˇ G ) and composition of Hecke operator. We refer to [17,5.4.2] for the precise definition and detailed discussions. We take a non-trivial additive character ψ : F p → K × and denote by π ∈ K the associated elementsatisfying π p − = − p (2.1.5). Let A ψ be the Dwork F -isocrystal on A (2.1.5).We fix a generic linear function φ of I (1) /I (2), that is, a homomorphism φ : I (1) /I (2) → A of algebraicgroup over k whose restriction to each U α is an isomorphism(4.1.3.1) φ α := φ | U α : U α ≃ A . ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 39 Let A ψφ = φ + ( A ψ ). (Note that our notation is slightly abusive as this sheaf depends only on the character ψ ◦ tr k/ F p ◦ φ of I (1) /I (2) as a p -group). We denote by Hol(Bun G (0 , ) I (1) /I (2) , A ψφ the category of holonomicmodules on Bun G (0 , which are ( I (1) /I (2) , A ψφ )-equivariant.By repeating the argument of ([52] 2.3), we obtain a parallel result for holonomic modules. Lemma 4.1.4 ([52] 2.3) . (i) The canonical morphism j γ, ! ( A ψφ ) ∼ −→ j γ, + ( A ψφ ) is an isomorphism. (ii) The functor
Hol( X ) → Hol(Bun γ G (0 , × X ) I (1) /I (2) , A ψφ M j γ, ! ( A ψφ ) ⊠ M is an equivalence of categories, with a quasi-inverse given by N ( i γ × id X ) + ( N ) ≃ ( i γ × id X ) ! ( N ) . We denote by A ψφ the object of Hol(Bun G (0 , ) I (1) , A ψφ defined by ( j γ, ! ( A ψφ )[dim Bun G (0 , ]) γ ∈ Ω . Theorem 4.1.5. (i)
For ( m, n ) = (0 , , the holonomic module A ψφ (4.1.4) is a Hecke eigen-module withHecke eigenvalue a ˇ G -valued overconvergent F -isocrystal (4.1.5.1) Kl rigˇ G ( ψφ ) : Rep ( ˇ G ) → Sm(
X/L F ) . (ii) For every representation V of ˇ G , Kl rigˇ G,V ( ψφ ) is pure of weight zero. If ψ (resp. ψ and φ ) is clear from the context, we simply write Kl rigˇ G ( ψφ ) by Kl rigˇ G ( φ ) (resp. Kl rigˇ G ). In theremainder of this section, we prove the above theorem by repeating the strategy in the ℓ -adic case, following[52]. The first step is to show holonomicity. Lemma 4.1.6.
For every V ∈ Rep ( ˇ G ) , the complex Hk V ( A ψφ )[1] is holonomic.Proof. For γ ∈ Ω, we denote by j γ : I (1) /I (2) → Bun G (0 , the open immersion and by j ′ γ : pr − ,V ( I (1) /I (2)) → Hecke X G (0 , the base change of j γ to Hecke stack. Then the restriction of pr to pr − ,V ( I (1) /I (2))pr : pr − ,V ( I (1) /I (2)) → Bun G (0 , × X is affine ([52] remark 4.2). We claim that the canonical morphism(4.1.6.1) j ′ γ, ! (pr +1 ,V ( A ψφ ) ⊗ IC V ) ∼ −→ j ′ γ, + (pr +1 ,V ( A ψφ ) ⊗ IC V )is an isomorphism. Indeed, since both j γ, ! and j γ, + commute with smooth base change, it suffices to show theisomorphism after taking inverse image to U × GR (4.1.2.2). In this case, the morphism pr ,V corresponds tothe projection U × GR → U . Then the isomorphism (4.1.6.1) follows from j γ, ! ( A ψφ ) ∼ −→ j γ, + ( A ψφ ) (4.1.4).Then we deduce that j ′ γ, ! (pr +1 ,V ( A ψφ ) ⊗ IC V )[1] is holonomic and we haveHk V ( A ψφ ) | Bun γ G (0 , × X ≃ (pr ◦ j ′ γ ) ! (pr +1 ,V ( A ψφ ) ⊗ IC V )(4.1.6.2) ≃ (pr ◦ j ′ γ ) + (pr +1 ,V ( A ψφ ) ⊗ IC V ) . Since (pr ◦ j ′ γ ) is affine, (pr ◦ j ′ γ ) + (resp. (pr ◦ j ′ γ ) ! ) is right (resp. left) exact ([6] 1.3.13). Then the assertionfollows. (cid:3) Proof of I (1) /I (2) on Bun G (0 , extends to an action on the diagram (4.1.2.1).For each γ ∈ Ω, Hk V ( A ψφ ) | Bun γ G (0 , × X is ( I (1) /I (2) , A ψφ )-equivariant. By 4.1.4, for each γ ∈ Ω, we haveHk V ( A ψφ ) | Bun γ G (0 , × X ≃ A γψφ ⊠ E γV , where E γV [1] is a holonomic module on X . By the same argument as in ([52] 4.2), we show that E γV iscanonically isomorphic to E V . So we will drop the index γ in the following. Since IC V is ULA with respect to the projection GR ≃ Gr G,X → X (3.2.2), we have Φ(IC V ) = 0 (2.8.6).Since taking vanishing cycle functor commutes with smooth pull-back and proper push-forward ([4] 2.6), wededuce that A ψφ ⊠ Φ( E V ) ≃ Φ( A ψφ ⊠ E V ) ≃ pr , ! (Φ(pr +1 ,V ( A ψφ ) ⊗ IC V )) ≃ pr , ! (pr +1 ,V ( A ψφ ) ⊗ Φ(IC V )) = 0 . By 2.8.7, E V is smooth. Then the assertion follows. (cid:3) Proof of ⋆ ∈ Bun G (0 , the base point corresponding to the trivial bundle G (0 , ⋆ × X can be written as(4.1.8.1) GR p z z tttttttttt p ! ! ❇❇❇❇❇❇❇❇ Bun G (0 , X. We denote by GR V ⊂ GR ≃ Gr × X the support of Sat( V ) ⊠ L X , by GR ◦ the inverse image of the big cell j ( I (1) /I (2) × Ω) by p , and by GR ◦ V = GR V ∩ GR ◦ . Consider the following diagram:(4.1.8.2) GR ◦ Vp ◦ ,V w w ♦♦♦♦♦♦♦♦♦♦♦♦ (cid:31) (cid:127) j ′ / / p ◦ (cid:21) (cid:21) GR Vp ,V z z tttttttttt p ! ! ❈❈❈❈❈❈❈❈❈ A I (1) /I (2) × Ω φ o o (cid:31) (cid:127) j / / Bun G (0 , X. By the base change and (4.1.6.2), we have(4.1.8.3) E V ≃ p ◦ , ! ( p ◦ , +1 ,V ( A ψφ ) ⊗ IC V | GR ◦ ) . By cleanness (4.1.6.1), E V can be calculated by either + or ! pushforward. More precisely, the followingcanonical morphism is an isomorphism(4.1.8.4) p ◦ , ! ( p ◦ , +1 ,V ( A ψφ ) ⊗ IC V | GR ◦ ) ∼ −→ p ◦ , + ( p ◦ , +1 ,V ( A ψφ ) ⊗ IC V | GR ◦ ) . In particular, the overconvergent F -isocrystal E V is pure of weight zero. Theorem 4.1.5(ii) follows. (cid:3) There is the following “trivial” functoriality between Kloosterman F -isocrystals. We fix ψ . Let G ′ → G be a homomorphism of reductive groups induces the same adjoint quotient G ′ ad ≃ G ad . Thenit induces an isomorphism I ′ (1) /I ′ (2) ≃ I (1) /I (2), and therefore we can abusively use the notation φ todenote the “same” linear functions on these spaces under the identification. On the other hand, it inducesa homomorphism of dual groups ˇ G → ˇ G ′ and therefore a tensor functor Res : Rep ( ˇ G ′ ) → Rep ( ˇ G ) byrestrictions. Then Kl rigˇ G ′ is the push-out of Kl rigˇ G along ˇ G → ˇ G ′ . Concretely, this means that there is acanonical isomorphism of tensor functors (we omit both ψ and φ from the notations)Kl rigˇ G ′ ≃ Kl rigˇ G ◦ Res :
Rep ( ˇ G ′ ) → Sm(
X/L F )This allows use to reduce certain questions of Kl rigˇ G to the case when ˇ G is simply-connected. We also obtainthe following exceptional isomorphisms (due to coincidences of Dynkin diagrams in low rank cases)Kl rigSL , Sym ≃ Kl rigSO , Std , (4.1.9.1) Kl rigSp , ker( ∧ → ) ≃ Kl rigSO , Std , (4.1.9.2) Kl rigSO , Std ≃ Kl rigSL × SL , Std ⊠ Std , (4.1.9.3) Kl rigSO , Std ≃ Kl rigSL , ∧ , (4.1.9.4) ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 41 where denotes the trivial representation, Std the standard representation, Sym • and ∧ • the symmetricpowers and wedge powers of the standard representation. There is a natural action of G m on X ⊂ P . On the other hand, the group of automorphismsAut( G, B, T ) acts on G ( m, n ). It follows that G m × Aut(
G, B, T ) acts on (4.1.2.1), and therefore on(4.1.8.1). It also acts on I (1) /I (2) × Ω as group automorphisms such that the open embedding (4.1.1.2) is G m × Aut(
G, B, T )-equivariant. Recall that the natural action of Aut( G ) on the Satake category inducesa homomorphism ι : Aut( G ) → Aut( ˇ G, ˇ B, ˇ T , N ) (3.6.10). Given δ = ( a, σ ) ∈ ( G m × Aut(
G, B, T ))( k ) and V ∈ Rep ( ˇ G ), then there is a canonical isomorphism(4.1.10.1) Kl rigˇ G,V ( ψ ( φ ◦ δ )) ≃ a + Kl rigˇ G,ι ( σ − ) V ( ψφ ) , given by the composition p , ! ( p +1 ,V ( j ! ( φ ◦ δ ) + A ψ ) ⊗ IC V ) ≃ p , ! ( δ + p +1 ,V ( j ! φ + A ψ ) ⊗ IC V ) ≃ a + p , ! ( p +1 ,V ( j ! φ + A ψ ) ⊗ ( δ − ) + IC V ) ≃ a + p , ! ( p +1 ,V ( j ! φ + A ψ ) ⊗ (IC ι ( σ − ) V )) . In particular, given t ∈ T ad ( k ) ⊂ Aut(
G, B, T ), the element δ = (1 , t ) induces an isomorphism(4.1.10.2) Kl rigˇ G ( ψ ( φ ◦ δ )) ≃ Kl rigˇ G ( ψφ ) . That is, Kl rigˇ G ( ψφ ) depends only on the T ad -orbit of φ . On the other hand, let a be an element of G m ( k ), ψ a the additive character defined by ψ a ( − ) = ψ ( a − ), t a ∈ T ad the unique element such that α ( t a ) = a for everysimple root α of G and h the Coxeter number of G . By applying δ = ( a h , t a ) in (4.1.10.1), we deduce that(4.1.10.3) Kl ˇ G ( ψ a φ ) ≃ Kl ˇ G ( ψ ( φ ◦ δ )) ≃ ( a h ) + Kl ˇ G ( ψφ ) . In addition, given a generic linear function φ of I (1) /I (2), the collection { φ α } from (4.1.3.1) for those α being simple roots of G , provide a pinning of ( G, B, T ), and therefore induces a splitting Out( G ) → Aut(
G, B, T ). If G is almost simple, not of type A n , then every element σ ∈ Out( G ) fixes the remaining φ α .If G is of type A n , the unique non-trivial element σ ∈ Out( G ) send the remaining φ α to − φ α . Therefore,if either ˇ G is almost simple not of type A n , or if p = 2, then for every σ ∈ Out( G ), we have φ ◦ (1 , σ ) = φ and a canonical isomorphism(4.1.10.4) Kl rigˇ G,V ( ψφ ) ≃ Kl rigˇ G,ι ( σ − ) V ( ψφ ) , compatible with the tensor structures. On the other hand, if G is almost simple of A n and if p >
2, thenthe element δ = ( − , σ ) induces a canonical isomorphism (4.1.10.1)(4.1.10.5) Kl rigˇ G,V ( ψφ ) ≃ ( − + Kl rigˇ G,V ∨ ( ψφ ) , where V ∨ denotes the dual representation of V , compatible with the tensor structures. There is a variant with multiplicative characters, which slightly generalizes A ψφ . Note that T ≃ I (0) op /I (1) op . Let e T = N G ( k (( x ))) ( I (0) op ) /I (1) op , which fits into the exact sequence 1 → T → e T → Ω →
1. The group e T acts on Bun G (1 , by modifying G (1 , j : e T × I (1) /I (2) → Bun G (1 , . We choose a splitting s : T ⋊ Ω ∼ −→ e T . For γ ∈ Ω, j sends T × γ × I (1) /I (2) to the connected componentBun γ G (1 , and we denote it by j γ : T × γ × I (1) /I (2) → Bun γ G (1 , . A character χ : T ( k ) → K × defines a rank one overconvergent F -isocrystal K χ on the torus T (cf. 2.1.5(ii)).If χ γ : T ( k ) → K × denotes the character defined by χ γ ( t ) = χ (Ad γ ( t )), then lemma 4.1.4 also holds for( T × I (1) /I (2) , K χ γ ⊠ A ϕ )-equivariant holonomic modules on Bun γ G (1 , .We denote by A ψφ,χ,s the holonomic module on Bun G (1 , defined by ( j γ, ! ( K χ γ ⊠ A ψφ )[dim Bun G (1 , ]) γ ∈ Ω .By replacing (0 ,
2) by (1 ,
2) in theorem 4.1.5 and repeating the arguments, we obtain a ˇ G -valued overcon-vergent F -isocrystal Kl rigˇ G ( ψφ, χ, s ) : Rep ( ˇ G ) → Sm(
X/L F ) , such that for every representation V of ˇ G , Kl rigˇ G,V ( ψφ, χ ) is pure of weight zero. Note that by (4.1.8.3),Kl rigˇ G ( ψφ ) = Kl rigˇ G ( ψφ, , s ) for the trivial character and does not depend on the choice of the splitting s . Let ℓ be a prime different from p . We take an isomorphism ι : K ≃ Q ℓ . Using the ℓ -adic Artin-Schreier sheaf AS ψ on A k associated to ψ , and the Kummer local system on G m,k associated to χ , Heinloth,Ngô and Yun construct a ℓ -adic ˇ G local system(4.1.12.1) Kl ét ,ℓ ˇ G ( ψφ, χ, s ) : Rep ( ˇ G ) → LocSysm( X ) . By the trace formula ([45], [6] 4.3.9) and Gabber-Fujiwara’s ℓ -independence ([6] 4.3.11), the Frobenius tracesof Kl ét ,ℓ ˇ G,V ( ψφ, χ, s ) and of Kl rigˇ G,V ( ψφ, χ, s ) at each closed point of X k coincide via ι .When χ is the trivial character, we omit χ and s from the notation. There is a variant of Heinloth-Ngô-Yun’s construction using algebraic D -modules instead of ℓ -adicsheaves to produce a ˇ G -connection on X K in zero characteristic ([52] 2.6). Note that all the geometricobjects used in the above construction can be also defined over K . We choose a generic linear function φ : I (1) /I (2) → A over K and a linear function χ : Lie( T ) → K . Via an isomorphism T K ≃ G rm,K , ⊠ ri =1 (cid:0) K h x, x − , ∂ x i / ( x∂ x − χ ( i )) (cid:1) defines an algebraic D -module on T K , which is independent of the choice of trivialisation that we denote by K χ . We replace the Artin-Schreier sheaf AS ψ on A k by the exponential D -module (4.1.13.1) E λ = K h x, ∂ x i / ( ∂ x − λ ) , λ ∈ K, on A K . We choose a splitting s : T ⋊ Ω ≃ e T over K . Then we obtain a tensor functor(4.1.13.2) Kl dRˇ G ( λφ, χ, s ) : Rep ( ˇ G ) → Conn( X K ) , where the target denotes the category of vector bundles with connection on X K . Here we identify homo-morphisms φ : I (1) /I (2) → A of algebraic group over K with Hom K (Lie I (1) /I (2) , K ) via differentiation,so λφ is regarded as a linear function on Lie( I (1) /I (2)).When χ = 0, we omit χ and s from the notation and Kl dRˇ G ( λφ ) is constructed in the same way as Kl rigˇ G ( φ )by E λ .4.2. Comparison between Kl dRˇ G and Kl rigˇ G . In this subsection, we work with schemes over R and we keepthe notation of 4.1. We say a linear function φ : I (1) /I (2) → A over R is generic , if it is generic modulothe maximal ideal of R . We take such a function φ and we denote abusively its base change to k (resp. K )by φ . Let χ : T ( k ) → K × be a character. There exists a homomorphism χ : T → G m such that χ ( e x ) = χ ( x )for x ∈ T ( k ) and some lifting e x of x in T ( K ). We denote abusively χ : Lie( T ) → K the differential of χ . Wechoose a splitting s : T × Ω ≃ e T over R and we denote abusively its base change to k (resp. K ) by s .We denote the sheaf O b P , Q ( † { , ∞} ) (2.3) by O X for short. The following theorem is our main result ofthis subsection. Theorem 4.2.1.
We set L = K . For every representation V of ˇ G , there exists a canonical isomorphism of O X -modules with connection (2.2.2)(4.2.1.1) ι V : (Kl dRˇ G,V ( − πφ, χ, s )) † ∼ −→ Kl rigˇ G,V ( φ, χ, s ) , ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 43 compatible with tensor structures. In the following, we will present the proof in the case where χ is trivial for simplicity and the general casefollows from the same argument. We will omit − πφ, χ, φ, χ, s from the notation. We first consider the case where V is associated to a minuscule coweight λ . In this case, Gr λ isisomorphic to a partial flag variety and is smooth and projective, and IC V is isomorphic to K Gr λ [dim Gr λ ]supported on GR V ≃ Gr λ × X . We show the above theorem by comparing the relative twisted de Rhamcohomologies and the relative twisted rigid cohomologies along the morphism p ◦ : GR ◦ V → X in (4.1.8.2). To do it, we first show that the associated de Rham and rigid cohomologies at each fiber of X are isomorphic.We regard (4.1.8.2) as a diagram of schemes over Spec( R ). We denote M := p +1 ( E − π )[dim Gr λ ], which isa line bundle with connection on GR ◦ V,K . With the notation of 2.2.2, the bundle with connection M † on(GR ◦ V,K ) an is overconvergent and underlies to the arithmetic D -module p +1 ( A ψ )[dim Gr λ ] on GR ◦ V,k , denotedby M . Lemma 4.2.3.
Let s be a point of X ( k ) . We choose a lifting in X ( R ) and still denote it by s . Thespecialisation morphism (2.2.3.3) on the fiber GR ◦ V,s of GR ◦ V above s (4.2.3.1) H ∗ dR ((GR ◦ V,s ) K , M s ) → H ∗ rig ((GR ◦ V,s ) k , M s ) is an isomorphism. Moreover, these cohomology groups vanish except for the middle degree .Proof. We set Y = GR ◦ V,s and we write M (resp. M ) instead of M s (resp. M s ). Since Y admits a smoothcompactification Gr λ whose boundary is a divisor, we can calculate above cohomology groups by direct imageof corresponding algebraic (resp. arithmetic) D -modules (2.4.1). Note that Kl dRˇ G,V (resp. Kl rigˇ
G,V ) is a bundlewith connection (resp. overconvergent F -isocrystal) of rank dim V . By the base change, cohomology groupsin (4.2.3.1) vanish except for the middle degree and have dimension dim V in the middle degree. By (4.1.8.4),the canonical morphism ι rig : H ∗ rig , c ( Y k , M ) → H ∗ rig ( Y k , M ) is an isomorphism. In view of proposition 2.2.5,we deduce that the specialisation morphism (4.2.3.1) is surjective. Then the assertion follows. (cid:3) Proof of theorem in the minuscule case . Now we use the relative specialization morphism(2.3.7.2) to compare (Kl dRˇ
G,V ) † and Kl rigˇ G,V . Let Gr P → P be the Beilinson–Drinfeld Grassmannian of G over P and ̟ : Gr λ, P → P the closed subscheme associated to λ . Note that ̟ is a locally trivial fibration over P with smooth projective fibers Gr λ and defines a good compactification of p ◦ (2.3.7).We take again the notation of 2.3.7 for the smooth R -morphism p ◦ . We set A = Γ( X, O X ), A K = A [ p ], A = b A [ p ] the ring of analytic functions on b X rig and A † = Γ( P k , O G m ) the ring of analytic functions on b P overconvergent along { , ∞} . We have inclusions A K ⊂ A † ⊂ A . If D X K denotes the ring of algebraicdifferential operators on X K , there exists a canonical D X K -linear specialization morphism (2.3.7.2)(4.2.4.1) Γ( X K , Kl dRˇ G,V ) → Γ( X k , Kl rigˇ G,V ) , where the left (resp. right) hand side is coherent over A K (resp. A † ). The above morphism induces ahorizontal A † -linear morphism ι V : Γ( X K , Kl dRˇ G,V ) ⊗ A K A † → Γ( X k , Kl rigˇ G,V ) , which gives rise to the morphism (4.2.1.1). Recall that the homomorphism A † → A is faithfully flat ([21]4.3.10). To prove ι V is an isomorphism, it suffices to show that the induced horizontal A -linear morphism:(4.2.4.2) ι V ⊗ A † A : Γ( X K , Kl dRˇ G,V ) ⊗ A K A → Γ( X k , Kl rigˇ G,V ) ⊗ A † A
04 DAXIN XU, XINWEN ZHU is an isomorphism. Let b A → R be a continuous homomorphism and s : A → R the associated R -point of G m . By (2.3.4.2) and the base change, the fiber ι ⊗ A † K coincides with the morphism (4.2.3.1) associatedto the point s ∈ X ( R ) and is an isomorphism (4.2.3). Since both sides of (4.2.4.2) are coherent A -modules,the morphism ι V ⊗ A † A is an isomorphism and the assertion follows. (cid:3) Next, we consider the case where V is associated to the quasi-minuscule coweight λ . In this case,Gr ≤ λ contains a smooth open subscheme Gr λ whose complement is isomorphic to Spec( R ), and admitsa desingularisation f Gr ≤ λ (cf. [69] § 7). We take an isomorphism GR V ≃ X × Gr ≤ λ and set GR ◦◦ V =GR ◦ V ∩ ( X × Gr λ ) to be the smooth locus of GR ◦ V (4.1.8). We denote by j : GR ◦◦ V → GR ◦ V the openimmersion and by(4.2.5.1) τ = p ◦ ◦ j : GR ◦◦ V → X the canonical morphism, which admits a good compactification f Gr ≤ λ × P → P in the sense of 2.3.7.We denote by M the line bundle with connection p +1 ( E − π )[dim Gr λ ] | GR ◦◦ V,K and by M the smooth arith-metic D -module p +1 ( A ψ )[dim Gr λ ] | GR ◦◦ V,k . The holonomic module IC V is constant on GR ◦◦ V . Then we deducethat j !+ ( M ) ≃ p +1 ( E − π ) ⊗ IC V | GR ◦ V,K , j !+ ( M ) ≃ p +1 ( A ψ ) ⊗ IC V | GR ◦ V,k . Note that j !+ ( M )[1] , j !+ ( M )[1] are holonomic. Lemma 4.2.6. (i)
The complex τ k, + ( M )[1] (resp. τ K, + ( M )[1] ) is holonomic. (ii) Let s be a point of X ( k ) . We choose a lifting in X ( R ) and still denote it by s . If we denote by M s (resp. M s ) the + -pullback of M (resp. M ) along the fiber at s , then the specialisation morphism (2.2.3.3)(4.2.6.1) H ∗ dR ((GR ◦◦ V,s ) K , M s ) → H ∗ rig ((GR ◦◦ V,s ) k , M s ) induces an isomorphism (4.2.6.2) H ((GR ◦ V,s ) K , j !+ ( M s )) ∼ −→ H ((GR ◦ V,s ) k , j !+ ( M s )) . Proof. (i) Let i : Z → GR ◦ V be the complement of GR ◦◦ V in GR ◦ V , which is isomorphic to X . Consider thedistinguished triangle on GR ◦≤ λ,k j !+ ( M )[1] → j + ( M )[1] → C → . By 2.6.2, C ≃ i ! ( j !+ ( M ))[2] has degree ≥ Z . Applying p ◦ , + to the above triangle, weobtain p ◦ , + ( j !+ ( M ))[1] → τ + ( M )[1] → p ◦ , + ( C ) → , where the first term is holonomic (cf. 4.1.6), and the second term has cohomological degrees ≤ τ is affine and the last term has cohomological degrees ≥ p ◦ | Z is the identity. Then we deduce thateach term in the above triangle is holonomic.(ii) We set Y = GR ◦ V,s , U = GR ◦◦ V,s and we write simply M (resp. M ) instead of M s (resp. M s ). By apply-ing the argument of (i), we deduce that the canonical morphism of cohomology groups H ( Y k , j !+ ( M )) → H ( U k , M ) is injective. By a dual argument, we deduce that the canonical morphism H , c ( U k , M ) → H ( Y k , j !+ ( M )) is surjective. In summary, we have a sequence:(4.2.6.3) H , c ( U k , M ) ։ H ( Y k , j !+ ( M )) ∼ −→ H ( Y k , j !+ ( M )) ֒ → H ( U k , M ) , where the middle isomorphism is due to the cleanness (4.1.8.4) and the composition is the canonical morphism ι rig . ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 45 We construct an analogue sequence of (4.2.6.3) for de Rham cohomology of M on U K . These two sequencesfit into a commutative diagram (2.2.5)H , c ( U K , M ) / / / / H , c ( Y K , j !+ ( M )) ∼ / / H ( Y K , j !+ ( M )) (cid:31) (cid:127) / / H ( U k , M ) ρ M (cid:15) (cid:15) H , c ( U k , M ) / / / / ρ M, c O O H ( Y k , j !+ ( M )) ∼ / / H ( Y k , j !+ ( M )) (cid:31) (cid:127) / / H ( U k , M )Let E be the image of H , c ( U k , M ) → H ( Y K , j !+ ( M )). Then the specialisation morphism ρ M sends E surjectively to the subspace H ( Y k , j !+ ( M )). Since dim E ≤ dim H ( Y K , j !+ ( M )) = dim H ( Y k , j !+ ( M )),we deduce that E = H ( Y K , j !+ ( M )) and that ρ M induces an isomorphism (4.2.6.2). (cid:3) Proof of theorem in the quasi-minuscule case . By 4.2.6(i), we have a diagram of D X K -modules(4.2.7.1) Γ( X K , Kl dRˇ G,V ) / / Γ( X K , τ K, + ( M )) (cid:15) (cid:15) Γ( X k , Kl rigˇ G,V ) / / Γ( X k , τ k, + ( M ))where the vertical arrow is the relative specialization morphism (2.3.7.2). Let U be an open dense subschemeof X k such that τ k, + ( M ) | U is smooth, U the corresponding formal open subscheme of b X and Z = P k \ U .We denote the sheaf of rings O b P , Q ( † Z ) by O U for short. By 4.2.6 and the same argument of 4.2.4, the abovediagram induces an injective morphism of O U -modules with connection (Kl dRˇ G,V ) † ⊗ O X O U → τ + ( M ) ⊗ O X O U and then induces an isomorphism of O U -modules with connection:(4.2.7.2) (Kl dRˇ G,V ) † ⊗ O X O U ∼ −→ Kl rigˇ G,V ⊗ O X O U . In particular, the left hand side is overconvergent along Z . Since the convergency of an O b X rig -module withconnection can be checked by restricting to a dense open subscheme of X k ([70] 2.16), the O b X rig -module withconnection (Kl dRˇ G,V ) † | b X rig is convergent. Then we deduce that the O X -module with connection (Kl dRˇ G,V ) † isoverconvergent along { , ∞} . The restriction functor Isoc † ( X k /K ) → Isoc † ( U/K ) is fully faithful (cf. [60]6.3.2). Then the isomorphism (4.2.7.2) gives rise to an isomorphism (4.2.1.1) and the assertion follows. (cid:3)
In the end, we show the general case of theorem 4.2.1. Let V , · · · , V n be minuscule and quasi-minuscule representations of ˇ G . Then we have a decomposition of representations(4.2.8.1) V ⊗ V ⊗ · · · ⊗ V n ≃ M W ∈ Rep ( ˇ G ) m W W, where m W denotes the multiplicity of W . Each representation W of Rep ( ˇ G ) appears as a summand of theabove decomposition for some minuscule and quasi-minuscule representations V , · · · , V n .Then we obtain the associated decomposition of bundles with connection on X K and overconvergent F -isocrystals on X K respectively: n O i =1 Kl dRˇ G,V i ≃ M W ∈ Rep ( ˇ G ) m W Kl dRˇ G,W , (4.2.8.2) n O i =1 Kl rigˇ G,V i ≃ M W ∈ Rep ( ˇ G ) m W Kl rigˇ G,W . (4.2.8.3) Theorem 4.2.1 in the minuscule and quasi-minuscule cases provides an isomorphism of overconvergentisocrystals(4.2.8.4) ( ⊗ ni =1 Kl dRˇ G,V i ) † ∼ −→ ⊗ ni =1 Kl rigˇ G,V i . By ([20] 2.2.7(iii)), the connection on left hand side, restricted on each component (Kl dRˇ
G,W ) † , is overconver-gent. We denote abusively the associated overconvergent isocrystal on X k by (Kl dRˇ G,W ) † .The isomorphism (4.2.8.4) induces a commutative diagram(4.2.8.5) End Rep ( ˇ G ) ( N ni =1 V i ) Kl dRˇ G t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Kl rigˇ G * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ End
Conn( X K ) ( N ni =1 Kl dRˇ G,V i ) / / End
Sm( X k /K ) ( N ni =1 Kl rigˇ G,V i )Indeed, choose a k -point s of X k and a lift e s to X ( K ). The isomorphism (4.2.8.4) induces an isomorphismbetween fibers (Kl dRˇ G,V i ) e s and (Kl rigˇ G,V i ) s . The composition of the functor Kl dRˇ G (resp. Kl rigˇ G ) with the fiberfunctor at e s (resp. s ) is the forgetful functor Rep ( ˇ G ) → Vec K . Since fiber functors are faithful, we deducethe commutativity of (4.2.8.5) by considering their fibers.If e denotes the idempotent of End Rep ( ˇ G ) ( ⊗ ni =1 V i ) corresponding to a summand W , then its image vialeft (resp. right) vertical arrow is the idempotent corresponding to Kl dRˇ G,W (resp. Kl rigˇ
G,W ) (4.2.8.2, 4.2.8.3).By (4.2.8.4) and (4.2.8.5), we deduce a canonical isomorphism of overconvergent isocrystals on X k ι W : (Kl dRˇ G,W ) † ∼ −→ Kl rigˇ G,W . One verifies that the above isomorphism is independent of the choice of idempotent e and then of the choice ofminuscule representations { V i } ni =1 . Isomorphisms ι W are compatible with tensor structures due to (4.2.8.4).Now theorem 4.2.1 follows. (cid:3) Comparison between Kl dRˇ G and Be ˇ G . In this subsection, we recall the Bessel connection Be ˇ G ( ˇ ξ ) on X constructed by Frenkel and Gross [47] of ˇ G and identify it with Kl dRˇ G ( φ ) (4.1.13).We work with schemes over K . Let (ˇ g , ˇ b , ˇ t ) denote the Lie algebras of ( ˇ G, ˇ B, ˇ T ) over K . Let A K denote the ring of algebraic functions of X . There exists a grading on the affine Lie algebraˇ g aff := ˇ g ⊗ A K , which on ˇ g -part is given by Ad ρ ( G m ), and on A K -part is given by the ˇ h -multiple of thegrading induced by the natural action of G m on X . Here as before ρ ∈ X • ( T ) ⊗ Q is the half sum of positiveroots of G (and therefore is a cocharacter of ˇ G ad ), and ˇ h is the Coxeter number of ˇ G .Let ˇ g aff (1) ⊂ ˇ g aff be the subspace of degree 1. Thenˇ g aff (1) = M ˇ α affine simple ˇ g aff , ˇ α , where ˇ g aff , ˇ α is the root subspace corresponding to the affine simple root ˇ α of ˇ g aff . Let ˇ ξ ∈ ˇ g aff (1) be a generic element, by which we mean each of its ˇ α -component ˇ ξ ˇ α = 0. In [47], Frenkel and Gross defined a ˇ g -valuedconnection on the trivial ˇ G -bundle on X by the following formula:(4.3.1.1) Be ˇ G ( ˇ ξ ) = d + ˇ ξ dxx . Here x is a coordinate x : X ∪ { } ≃ A . Note that dxx itself is independent of the choice of the coordinate x , and is a generator of the module of log differentials on X ∪ { } with logarithmic pole at 0.We may write N = P ˇ α ˇ ξ ˇ α , where the sum is taken over simple roots of ˇ g (instead of ˇ g aff ). This is aprincipal nilpotent element of ˇ g . The remaining affine root subspaces are of the form x ˇ g − ˇ θ i , where x is a ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 47 coordinate as above and ˇ θ i is the highest root of the simple factor ˇ g i of ˇ g . So we may write the sum of theremaining affine root vectors as xE for some E ∈ P ˇ g − ˇ θ i . Then the connection can be written as(4.3.1.2) Be ˇ G ( ˇ ξ ) = d + ( N + xE ) dxx , which is the form as used in [47]. In particular, this connection is regular singular with a principal unipotentmonodromy at 0. On the other hand, it has an irregular singularity at ∞ , with maximal formal slope 1 / ˇ h ([47] §5).We regard Be ˇ G ( ˇ ξ ) as a tensor functor from the category of representations of ˇ G to the category of bundleswith connection on X :(4.3.1.3) Be ˇ G ( ˇ ξ ) : Rep ( ˇ G ) → Conn( X ) . We will identify Kl dRˇ G ( λφ ) and Be ˇ G ( ˇ ξ ) as ˇ G -bundles with integrable connections on X . For thispurpose, we need to discuss how these connections depend on parameters. We identify the dual space g ∗ aff of g aff := g ⊗ A K with g ∗ ⊗ A K via the canonical residue pairing( g ⊗ A K ) ⊗ ( g ∗ ⊗ ω X ) → K, ( ξ ⊗ f, ˇ ξ ⊗ g ) = ( ξ, ˇ ξ )Res x = ∞ f g dxx . Recall that λφ is a linear function Lie( I (1) /I (2)) → K . We identify Hom K (Lie I (1) /I (2) , K ) with g ∗ aff (1) = M α affine simple g ∗ α . where g ∗ α ⊂ g ∗ aff is the dual of the root subspace corresponding to α .By (4.1.10.2) (applied to the D -module setting), Kl dRˇ G ( λφ ) depends only on the T ad -orbit of this functional.In addition, T ad -orbits of generic linear functions on Lie( I (1) /I (2)) are parameterized by the GIT quotient g ∗ aff (1) //T ad .On the other hand, the group G m × Aut( ˇ G, ˇ B, ˇ T ) acts on ˇ g aff preserving the grading. For ˇ δ = ( a, ˇ σ ), asimple gauge transform implies that the analogue of (4.1.10.1) holds, namely(4.3.2.1) Be ˇ G,V (ˇ δ ( ˇ ξ )) ≃ a + Be ˇ G, ˇ σV ( ˇ ξ ) . It follows that the analogue of (4.1.10.2) and of (4.1.10.3) also hold for Bessel connections. In particular,Be ˇ G ( ˇ ξ ) only depends on the ˇ T ad -orbit of ˇ ξ . Again, ˇ T ad -orbits of generic ˇ ξ are parameterized by the GITquotient ˇ g aff (1) // ˇ T ad .Here is the main theorem of this subsection. Theorem 4.3.3.
There exists a canonical isomorphism of affine schemes (4.3.3.1) g ∗ aff (1) //T ∼ −→ ˇ g aff (1) // ˇ T , such that if the T ad -orbit through λφ and the ˇ T ad -orbit through ˇ ξ match under this isomorphism, then Kl dRˇ G ( λφ ) ≃ Be ˇ G ( ˇ ξ ) as ˇ G -bundles with connection on X . If ˇ G is of adjoint type, a weaker version of this theorem was the main result of [89]. We first explain the isomorphism (4.3.3.1). Let ω X denote the canonical bundle on X and by abuseof notation, we sometimes also use it to denote the space of its global sections. Via the open embedding j γ : I (1) /I (2) ֒ → Bun γ G (0 , , we identify I (1) /I (2) × g ∗ aff (1) with T ∗ Bun γ G (0 , | j γ ( I (1) /I (2)) . The Hitchin map(e.g. see [17] Sect. 2, and [89]) h cl : T ∗ Bun γ G (0 , → Hitch( X ) := Γ( X, c ∗ × G m ω X ) Here Γ( X, c ∗ × G m ω X ) denotes abusively the affine space associated to the K -vector space Γ( X, c ∗ × G m ω X ). induces a closed embedding h cl : g ∗ aff (1) //T ֒ → Hitch( X ), where c ∗ := g ∗ //G is the GIT quotient of g ∗ by theadjoint action of G , equipped with a G m -action induced by the natural G m -action on g ∗ . (For an explicitdescription of the image of the map when g is simple, see the discussions before [89] lemma 18).On the other hand, there exists a canonical morphismˇ g aff (1) dxx ⊂ ˇ g ⊗ ω X → Γ( X, ˇ c × G m ω X )where ˇ c := ˇ g // ˇ G , which also induces a closed embedding ˇ g aff (1) // ˇ T → Γ( X, ˇ c × G m ω X ). The identifica-tion (Lie T ) ∗ = Lie ˇ T induces a canonical isomorphism c ∗ ∼ −→ ˇ c . One checks easily that there is a uniqueisomorphism g ∗ aff (1) //T ∼ −→ ˇ g aff (1) // ˇ T that fits into the following commutative diagram g ∗ aff (1) //T ∼ / / (cid:127) _ (cid:15) (cid:15) ˇ g aff (1) // ˇ T (cid:127) _ (cid:15) (cid:15) Γ( X, c ∗ × G m ω X ) ∼ / / Γ( X, ˇ c × G m ω X )where the bottom isomorphism is induced by c ∗ ∼ −→ ˇ c .In the case G and ˇ G are almost simple, unveiling the definition, we see that λφ and ˇ ξ match to each otherif the following holds: Let r be the rank of G and ˇ G . Recall that the ring of invariant polynomials on g ∗ (resp. ˇ g ) has a generator P r (resp. ˇ P r ), homogeneous of degree h = ˇ h . We choose them to match each otheras functions on c ∗ ≃ ˇ c . Then λφ matches ˇ ξ if and only if(4.3.4.1) λ h P r ( φ ) = P r ( λφ ) = ˇ P r ( ˇ ξ ) . This condition is independent of the choice of P r and ˇ P r (as soon as they match to each other).For concrete computations, it is convenient to fix a coordinate x ∈ A ⊂ P , and a pinning N = P ˇ α ∈ ˇ∆ ˇ ξ ˇ α of ( ˇ G, ˇ B, ˇ T ). Then we may rewrite (4.3.3.1) as an isomorphism(4.3.4.2) g ∗ aff (1) //T ≃ ˇ g aff (1) // ˇ T ≃ N + x X i ˇ g − ˇ θ i ≃ x X i ˇ g − ˇ θ i . We prove theorem 4.3.3 by quantizing (4.3.3.1) and applying the Galois-to-automorphic direction ofgeometric Langlands correspondence. For this, we need to review the notion of ˇ g -opers ([17] §3). By descent,it suffices to prove the theorem after base change from K to K . So we assume that all the geometric objectsbelow are defined over K , and omit the subscript. Let ˇ G ad denote the adjoint group of ˇ G .Let Y be a smooth curve over K . Let Op ˇ g ( Y ) denote the moduli spaces of ˇ G ad -opers on Y ([17] 3.1.11).By ([17] 3.1.11, 3.4.3), Op ˇ g ( Y ) is an ind-affine scheme. There is a natural free and transitive action of the(ind)-vector space Γ( Y, ˇ c × G m ω Y ) on Op ˇ g ( Y ) ([17] 3.1.9). This induces a natural filtration on the ring ofregular functions FunOp ˇ g ( Y ), whose associated graded is the ring of regular functions FunΓ( Y, ˇ c × G m ω Y ).Back to our case Y = X . We consider the subscheme of Op ˇ g := Op ˇ g ( P ) (0 ,̟ (0)) , ( ∞ , / ˇ h ) ⊂ Op ˇ g ( X ), whichis the moduli of ˇ G ad -opers on X which are • regular singular with principal unipotent monodromy at 0; • possibly irregular of maximal formal slope ≤ / ˇ h at ∞ .See the discussions before ([89] lemma 20) (where slightly different notations were used). In this case, theaction of Γ( X, ˇ c × G m ω X ) on Op ˇ g ( X ) induces a free and transitive action of x P i ˇ g − ˇ θ i ≃ ˇ g aff (1) // ˇ T (4.3.4.2)on Op ˇ g . In particular, FunOp ˇ g has a natural filtration whose associated graded is (Funˇ g aff (1)) ˇ T .On the other hand, the space Op ˇ g has a distinguished point, corresponding to the ˇ G ad -oper that is tame atboth 0 and ∞ . Therefore, we obtain a canonical isomorphism x P i ˇ g − ˇ θ i ∈ ˇ g aff (1) // ˇ T ≃ Op ˇ g ( X ). Explicitly,this isomorphism sends xE ∈ x P i ˇ g − ˇ θ i to the connection d + ( N + xE ) dxx on the trivial ˇ G -bundle which has ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 49 a natural oper form. Now the quantization of (4.3.3.1) gives a canonical isomorphism of filtered algebras([89] lemma 21)(4.3.5.1) U (Lie I (1) /I (2)) T ≃ FunOp ˇ g , whose associated graded gives back to (4.3.3.1). Here U ( V ) is the universal enveloping algebra of V =Lie I (1) /I (2), equipped with the usual filtration. As V is abelian, it is also canonically isomorphic to(Fun V ∗ ) T . Putting all the above isomorphisms together, we obtain the following commutative diagram(Fun g ∗ aff (1)) T ∼ / / ∼ (cid:15) (cid:15) (Funˇ g aff (1)) ˇ T ∼ (cid:15) (cid:15) U (Lie I (1) /I (2)) T ∼ / / FunOp ˇ g Together with the main result of [89], we obtain the proof of theorem 4.3.3 in the case when ˇ G = ˇ G ad . Next, we explain how to extend it to allow G to be a general semisimple group.One approach is to generalize the work of [17] to allow certain level structures, as what [89] did for simply-connected groups. In this approach, one must deal with the subtle question of the construction of “squareroot” of the canonical bundle on the moduli of G -bundles.In our special case, we have another short and direct approach, using the isomorphism Kl dRˇ G ad ( λφ ) ≃ Be ˇ G ad ( ˇ ξ ) just established.First, we claim that up to isomorphism, there exists a unique de Rham ˇ G -local system on X , whichinduces Be ˇ G ad ( ˇ ξ ), and has unipotent monodromy at 0. Indeed, any two such de Rham ˇ G -local systems differby a de Rham ˇ Z -local system on X ∪ { } ≃ A (i.e. one is obtained from the other by twisting a de Rhamˇ Z -local system). As ˇ Z is a finite group, the wild part of the differential Galois group at ∞ of this localsystem must be trivial, and therefore this local system itself is trivial.Now since both Kl dRˇ G ( λφ ) and Be ˇ G ( ˇ ξ ) have the property as in the claim (to see that Kl dRˇ G ( λφ ) has unipotentmonodromy at 0, one uses the same argument as [52] theorem 1 (2)), they must be isomorphic.4.4. Bessel F -isocrystals for reductive groups. In this subsection, we construct Bessel F -isocrystalsfor reductive groups, by putting the above ingredients together. We keep the notation of 4.2. We take a non-trivial additive character ψ : F p → K × and a generic linear function φ : I (1) /I (2) → A over R (4.2). We set λ = − π ∈ K corresponding to ψ (as in 2.1.5). Let ˇ ξ ∈ ˇ g aff (1) match − πφ under theisomorphism (4.3.3.1).We write Be ˇ G ( ˇ ξ ) more explicitly as follows. Choose a coordinate x of X ∪ { } over R , and a pinning N = P ˇ α ∈ ˇ∆ ˇ ξ ˇ α of ( ˇ G, ˇ B, ˇ T ). By (4.3.4.2), there is a unique element E = E φ ∈ P i ˇ g − ˇ θ i such that(4.4.1.1) Kl dRˇ G (1 · φ ) ≃ d + ( N + xE ) dxx , By (4.3.4.1), we deduce that(4.4.1.2) Kl dRˇ G ( − πφ ) ≃ d + ( N + ( − π ) h xE ) dxx = Be ˇ G ( ˇ ξ ) . Now we can define the object appearing in the title of the paper. Let Be † ˇ G ( ˇ ξ ) denote the compositionof Be ˇ G ( ˇ ξ ) : Rep ( ˇ G ) → Conn( X K ) with the ( − ) † -functor from (2.2.2.1). By theorem 4.2.1, a choice ofisomorphism (4.4.1.2) endows Be † ˇ G ( ˇ ξ ) with a Frobenius structure, i.e. a lifting of Be † ˇ G ( ˇ ξ ) as a functor Rep ( ˇ G ) → F - Isoc † ( X k /K ), or alternatively, an isomorphism of tensor functors ϕ : F ∗ X k ◦ Be † ˇ G ( ˇ ξ ) ∼ −→ Be † ˇ G ( ˇ ξ ) : Rep ( ˇ G ) → Isoc † ( X k /K ) , where F ∗ X k : Isoc † ( X k /K ) → Isoc † ( X k /K ) denotes the s -th Frobenius pullback functor (2.1.4.1). Fromthe calculation of the differential Galois group of Be ˇ G in [47] coro. 9, coro. 10 (see (1.2.6.1)) that theautomorphism group of Be ˇ G is Z G ( K ). Therefore, the Frobenius structure on Be † ˇ G ( ˇ ξ ) is independent ofthe choice of the isomorphism Be ˇ G ( ˇ ξ ) ≃ Kl dRˇ G ( λφ ). We use (Be † ˇ G ( ˇ ξ ) , ϕ ) (or simply Be † ˇ G ( ˇ ξ ) if there is noconfusion) to denote the ˇ G -valued overconvergent F -isocrystal(4.4.1.3) (Be † ˇ G ( ˇ ξ ) , ϕ ) : Rep ( ˇ G ) → F - Isoc † ( X k /K ) , which we call the Bessel F -isocrystal of ˇ G . For each representation ρ : ˇ G → GL( V ), the restriction of Be † ˇ G,V ( ˇ ξ ) at 0 defines an object Be † ˇ G,V ( ˇ ξ ) | of MCF( R K /K ) (2.9.1), which is solvable at 1 ([61] 12.6.1). By (4.3.1.1), the p -adic exponents of Be † ˇ G,V ( ˇ ξ ) | are 0. Then it is equivalent to the connection d + dρ ( N ) over the Robba ring by ([61] 13.7.1). Hence,Be † ˇ G,V ( ˇ ξ ) | satisfies the Robba condition (i.e. it has zero p -adic slope) and is unipotent.We denote by F - Isoc log , uni (cid:0) ( A k , /K (cid:1) the category of log convergent F -isocrystals on A k with a logpole at 0 relative to K and nilpotent residue, and are overconvergent along ∞ (2.9.3). By ([60] 6.3.2), thiscategory is equivalent to the full subcategory of F - Isoc † ( X k /K ) consisting of objects which are unipotentat 0. Then the ˇ G -valued overconvergent F -isocrystal (Be † ˇ G ( ˇ ξ ) , ϕ ) (4.4.1.3) factors through:(4.4.2.1) (Be † ˇ G ( ˇ ξ ) , ϕ ) : Rep ( ˇ G ) → Isoc log , uni (cid:0) ( A k , /K (cid:1) . Here is a more concrete description of the Frobenius structure on Be † ˇ G ( ˇ ξ ). Note that its underlyingbundles of Be † ˇ G,V ( ˇ ξ ) are free O b P , Q ( † {∞} )-modules. If we set A † = Γ( P k , O b P , Q ( † {∞} )), by the Tannakianformalism, the Frobenius structure on Be † ˇ G ( ˇ ξ ) is equivalent to an element ϕ ∈ ˇ G ( A † ) satisfying(4.4.3.1) x dϕdx ϕ − + Ad ϕ ( N + ( − π ) h xE ) = q ( N + ( − π ) h x q E ) . Given a point a ∈ | A k | and e a : A † → K its Teichmüller lifting, we denote by ϕ a = Q deg( a ) − i =0 ϕ ( e a q i ). When a = 0, the Frobenius trace of (Be † ˇ G ( ˇ ξ ) , ϕ ) at a can be calculated by the trace of ϕ a . Now we rephrase theabove discussions as follows, which is the first main result of our article. Theorem 4.4.4.
There is a unique element ϕ ∈ ˇ G ( A † ) satisfying the differential equation (4.4.3.1) suchthat via a (fixed) isomorphism K ≃ Q ℓ , for every a ∈ | X | and V ∈ Rep ( ˇ G )(4.4.4.1) Tr( ϕ a , V ) = Tr(Frob a , Kl ét ,ℓ ˇ G,V, ¯ a ( ψφ )) . When a = 0, we can describe ϕ more precisely. Proposition 4.4.5.
Let ρ be the sum of positive coroots in X • ( ˇ T ) . Then ϕ = 2 ρ ( √ q ) in the semisimpleconjugacy classes Conj ss ( ˇ G ( K )) of ˇ G ( K ) .Proof. The Frobenius endomorphism ϕ at 0 satisfies ϕ − N ϕ = qN (4.3.1). Since N is a principal nilpotentelement and Ad ρ ( q ) N = q − N , we deduce that ϕ = ερ ( q ) in Conj ss ( ˇ G ( K )) for some element ε in the center Z ˇ G ( K ).To show ε = id, it suffices to investigate Frobenius eigenvalues of Ψ(Be † ˇ G,V ) (2.9.4) for V ∈ Rep ( ˇ G ),which is same as those of Ψ(Kl ét ,ℓ ˇ G,V ) by 4.1.13 and Gabber-Fujiwara’s ℓ -independence ([3] 4.3.11). By a resultof Görtz and Haines [50], the i th graded piece of the weight filtration of Ψ(Kl ét ,ℓ ˇ G,V ) has the same dimensionas the dimension of H i (Gr G , IC V ) and is equipped with a Frobenius action by × q i (cf. [52] 4.3). Then wededuce that ε = id. (cid:3) ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 51 Monodromy groups.4.5.1.
In this subsection, we keep the notation of 4.4 and we take L to be K . We drop φψ from the notation.We denote by h Be † ˇ G i (resp. h Be † ˇ G , ϕ i , resp. h Be ˇ G i ) the full subcategory of Sm( X k /K ) (resp. Sm( X k /K F ),resp. Conn( X K )) whose objects are all the sub-quotients of objects Be † ˇ G,V (resp. (Be † ˇ G,V , ϕ ), resp. Be ˇ G,V )for V ∈ Rep ( ˇ G ). Then h Be † ˇ G i (resp. h Be † ˇ G , ϕ i , resp. h Be ˇ G i ) forms a Tannakian category over K and wedenote by G geo (resp. G arith , resp. G alg ) the associated Tannakian group (with respect to a fiber functor ω ,but is independent of the choice of the fiber functor up to isomorphism [39]). The tensor functors on theleft side of the following diagrams induce closed immersions of algebraic groups on the right side(4.5.1.1) h Be † ˇ G , ϕ i { { ✈✈✈✈✈✈✈✈✈ G arith ❅❅❅❅❅❅❅❅❅ h Be † ˇ G i Rep ( ˇ G ) y y ssssssssss d d ❏❏❏❏❏❏❏❏❏ G geo " " ❊❊❊❊❊❊❊❊❊ = = ③③③③③③③③③③ ˇ G h Be ˇ G i d d ■■■■■■■■■ G alg > > ⑤⑤⑤⑤⑤⑤⑤⑤⑤ . In ([47] Cor. 9 and Cor. 10), Frenkel and Gross showed that the differential Galois group G alg of theˇ G -connection Be ˇ G : Rep ( ˇ G ) → Conn( X K ) is a connected closed subgroup of ˇ G and explicitly calculated itwhen ˇ G is almost simple. The result can be found in (1.2.6.1). The main theorem of this subsection is asfollows. Theorem 4.5.2.
Let G be a split almost simple group over R and ˇ G its Langlands dual group over K . Wedenote by Σ the outer automorphism group of ˇ G and by Out(ˇ g ) the outer automorphism group of ˇ g . (i) If ˇ G is not of type A n or char( k ) > , then G geo → G alg is an isomorphism. In particular, • G geo ∼ −→ ˇ G Σ , ◦ , if ˇ G is not type A n ( n ≥ ) or B or D n ( n ≥ ) with Σ = Out(ˇ g ) . • G geo = ˇ G , if ˇ G is of type A n , • G geo ∼ −→ G , if ˇ G is of type B or of type D . • G geo ∼ −→ Spin n − if ˇ G is of type D n with Σ ≃ { } ( n ≥ ). (ii) If ˇ G = SL n +1 and char( k ) = 2 , then G geo (Be † SL n +1 ) = G geo (Be † SO n +1 ) . In particular, • G geo ∼ −→ SO n +1 , if n = 3 , • G geo ∼ −→ G , if n = 3 .In particular, G geo = G alg in this case. (iii) The map G geo → G arith is always an isomorphism.Proof. We first study the local monodromy at 0 and ∞ .In view of 4.4.2, the restriction functor at 0 (2.9.3.1) induces Rep ( ˇ G ) → h Be † ˇ G i | −→ MC uni ( R /K ) ∼ −→ Vec nil K , sending each representation ρ : ˇ G → GL( V ) to ( V, dρ ( N )) ∈ Vec nil K . Then, it induces closed immersions ofTannakian groups(4.5.2.1) G a → G geo → ˇ G, whose composition sends 1 ∈ K ≃ Lie( G a ) to N ∈ ˇ g . Lemma 4.5.3.
The restriction functor | ∞ : h Be † ˇ G i → MCF( R /K ) to ∞ ∈ P k induces homomorphisms I ∞ × G a → G geo which is non-trivial on P ∞ . Proof.
If the image P ∞ in ˇ G geo were trivial, by the Grothendieck–Ogg–Shafarevich formula, Kl ét ,ℓ ˇ G wouldalso be tame at 0 , ∞ . Then the associated ℓ -adic representation π ( X k ) → ˇ G would factor through the tamequotient π tame1 ( X k ), which is isomorphic to I tame ∞ as X ≃ G m . Since Kl ét ,ℓ ˇ G,V is pure of weight zero for every V ∈ Rep ( ˇ G ), the geometric monodromy group of Kl ˇ G would be semisimple and then finite. This contradictsto fact that Kl ét ,ℓ ˇ G has a principal unipotent monodromy at 0 ([52] Thm. 1). (cid:3) Since every overconvergent F -isocrystal Be † ˇ G,V is pure of weight 0 and is therefore geometricallysemi-simple ([6] 4.3.1), the neutral component G ◦ geo is semi-simple [34]. Therefore, (4.5.2.1) implies that itcontains a principal unipotent element and hence its projection to the adjoint group ˇ G ad of ˇ G contains aprincipal PGL . Then it is almost simple and its Lie algebra appears in one of the following chains: sl / / sp n / / sl n sl n +1 sl / / so n +1 : : tttttttttt % % ❏❏❏❏❏❏❏❏❏❏ so n +2 sl sl / / g / / so = = ④④④④④④④④ ! ! ❉❉❉❉❉❉❉❉ so sl / / f / / e sl / / e sl / / e Lemma 4.5.5. If ˇ G is not of type A , and not of type A when p = 2 , the image G geo → ˇ G ad cannot becontained in a principal PGL of ˇ G ad .Proof. The image of the wild inertia group P ∞ (resp. I ∞ ) in PGL is a finite p -group (resp. a solvablegroup). In view of the all possible finite groups contained in PGL , there are two possibilities:(a) the image of P ∞ is contained in G m ⊂ PGL ;(b) p = 2 and the image of I ∞ (resp. P ∞ ) is isomorphic to the alternative group A (resp. the group Z / Z × Z / Z ).To prove the lemma, we follow a similar argument of ([52] 6.8), but with the quasi-minuscule representationreplaced by the adjoint representation Ad. In any case, by a result of Baldassarri [13] (cf. [9] 3.2), the maximal p -adic slope of Be † ˇ G, Ad is less or equal to the maximal formal slope 1 / ˇ h of Be ˇ G, Ad (4.3.1). Let r be the rankof ˇ G and h the Coxeter number of ˇ G . Then we deduce that(4.5.5.1) Irr ∞ (Be † ˇ G, Ad ) ≤ rank Adˇ h = ˇ h + 1ˇ h r < r + 1 , and hence Irr ∞ (Be † ˇ G, Ad ) ≤ r . ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 53 On the other hand, we have a decomposition Ad ≃ ⊕ ri =1 S ℓ i as representations of principal PGL , where { ℓ + 1 , · · · , ℓ r + 1 } is the set of exponents of ˇ g .Case (a). Since Irr ∞ (Be † ˇ G ) = 0, the image of P ∞ in PGL contains µ p and the image of I ∞ is contained in N ( G m ). By a similar argument of ([52] 6.8), we deduce Irr ∞ ( S ℓ ) ≥ ℓ − ⌊ ℓ/p ⌋ ≥
1. Under our assumption,max i { ℓ i , p } >
2, so there is least one i such that ℓ i − ⌊ ℓ i /p ⌋ >
1. Then Irr ∞ (Be † ˇ G, Ad ) > r . Contradiction!Case (b). Recall that there are four irreducible representations of A : id, two non-trivial one dimensionalrepresentation V ′ , V ′′ , the standard representation V . Via the inclusion A → PGL , we have S ≃ V , S ≃ V ′ ⊕ V ′′ ⊕ V , S ≃ id ⊕ V ⊕ , S ≃ id ⊕ V ′ ⊕ V ′′ ⊕ V ⊕ ,S ≃ V ′ ⊕ V ′′ ⊕ V ⊕ , S ≃ id ⊕ ⊕ V ′ ⊕ V ′′ ⊕ V ⊕ , S ≃ id ⊕ V ′ ⊕ V ′′ ⊕ V ⊕ . In particular, we have Irr ∞ ( S ℓ ) ≥ ℓ = 3 , , , ,
7. In general, I ∞ acts non-trivially on S n and wehave Irr ∞ ( S ℓ ) ≥
1. Then we deduce that Irr ∞ (Ad) ≥ r ( G ) + 1. Contradiction! (cid:3) Now we prove theorem 4.5.2. By the “trivial” functoriality (4.1.9), it is enough to prove the theoremwhen ˇ G is simply-connected, so that ˇ G Σ is connected.(a) The case where ˇ G is not of type A n . In view of lemma 4.5.5, and the calculation of G alg (1.2.6.1),we deduce that G ◦ geo → G geo → G alg are isomorphisms. Using 4.1.10, we see G arith ⊂ ˇ G Σ . This implies that G arith = G geo unless ˇ G is of type B . In this last case, if ˇ G = Spin , and G arith ⊂ G × Z ( ˇ G ). Taking intoaccount of the Frobenius at 0 (4.4.5), we see that G arith = G geo .(b) The case where ˇ G is of type A n and p >
2. It suffices to exclude that G geo is contained in SO n +1 .Suppose it is true by contrast. Let σ be the generator of Σ and ˇ δ = ( − , σ ) in G m × Aut(
G, B, T ). Thenwe deduce isomorphisms of overconvergent isocrystals on X k Be † SL n +1 , Std ( ˇ ξ ) ≃ ( − + Be † SL n +1 , Std ∨ ( ˇ ξ ) ≃ ( − + Be † SL n +1 , Std ( ˇ ξ ) , where the first isomorphism follows from (4.1.10.5), and the second one is due to Std ∨ ≃ Std as represen-tations of SO n +1 . Since char k >
2, this isomorphism provides a “descent datum” so that Be † SL n +1 , Std ( ˇ ξ )descends to G m /µ . It follows that its Swan conductor at ∞ is at least two, if non-zero. On the other hand,using lemma 4.5.3 and the result of Baldassarri [13] (cf. [9] 3.2) again, the Swan conductor of Be † SL n +1 , Std ( ˇ ξ )at ∞ is 1, contradiction!(c) The case where ˇ G is of type A n and p = 2. In appendix (A.1), we will identify Be † SO n +1 , Std withBe † SL n +1 , Std . Then we reduce to the case (a). (cid:3)
We end this section by some corollaries of our calculation of the monodromy groups.
Corollary 4.5.6.
Assume that ˇ G is almost simple. The monodromy groups G ℓ geo , G ℓ arith of the Kl ét ,ℓ ˇ G ( ψφ ) over Q ℓ (4.1.12.1) are calculated as in theorem . Note that this gives a different proof of the main result of [52] theorem 3 (where some explicit small p are excluded). Our method avoids analyzing some difficult geometry related to quasi-minuscule and adjointSchubert varieties. Proof.
The monodromy group G ℓ arith (resp. G arith ) can be calculated by that of Kl ét ,ℓ ˇ G,V (resp. Be † ˇ G,V ) for afaithful representation V of ˇ G . The semisimplification of Kl ét ,ℓ ˇ G,V and Be † ˇ G,V are semi-simple and have sameFrobenius traces. Then by ([36] 4.1.1, 4.3.2), there exists a surjective morphism G ℓ arith ։ G arith . Since theyare both closed subgroups of ˇ G , they must be isomorphic to each other and the assertion follows. (cid:3) Corollary 4.5.7.
Assume that ˇ G is almost simple. Let Ad be the adjoint representation of ˇ G . (i) We have H i ( P , j !+ (Be † ˇ G, Ad )) = 0 for all i . (ii) We have
Irr ∞ (Be † ˇ G, Ad ) = r ( ˇ G ) , the rank of ˇ G . In addition, Ad I ∞ = 0 , and the nilpotent monodromyoperator N ∞ = 0 (2.9.1) . Therefore, the local Galois representation I ∞ → ˇ G is a simple wild parameter inthe sense of Gross-Reeder ( [51] § 6).Proof. The corresponding assertions for the algebraic connection Be ˇ G, Ad are proved in ([47] §14). Set E =Be † ˇ G, Ad , which is self dual. We have H ( X, E ) = Ad G geo = 0 and H ( X, E ) = 0 by D † -affinity. We obtainH i c ( X, E ) = 0 for i = 0 , ( X, E ) = Irr ∞ ( E ) ≤ r ( ˇ G ) . Let j : X → P be the inclusion. We have a distinguished triangle j ! ( E ) → j !+ ( E ) → H i +0 j + ( E ) ⊕ H i + ∞ j + ( E ) → , which induces a long exact sequence:0 → H ( P , j !+ ( E )) → H i +0 j + ( E ) ⊕ H i + ∞ j + ( E ) d −→ H ( X, E ) → (4.5.7.1) H ( P , j !+ ( E )) → → H ( X, E ) = 0 → H ( P , j !+ ( E )) → . By the Poincaré duality, we conclude that H i ( P , j !+ ( E )) = 0 for i = 0 , x ∈ { , ∞} , the restriction of E at x gives rise to an action of the inertia group I x on Ad and acommuting nilpotent monodromy operator N x : Ad → Ad (2.9.1). Then H i + x ( j + ( E )) is calculated byAd I x , N x := Ker( N x : Ad I x → Ad I x ) . The Bessel isocrystal is unipotent at 0 with N = [ − , N ] (4.4.2). We have Ad I , N = Ad N , which hasdimension r ( ˇ G ). Then the morphism d in (4.5.7.1) is both injective and surjective. We deduce thatAd I ∞ , N ∞ = 0 , H ( P , j !+ ( E )) = 0 . Since N ∞ is still a nilpotent operator on Ad I ∞ , we conclude assertions (i) and (ii). (cid:3) Remark 4.5.8. (i) By corollary 4.5.6 and the same arguments, we recover [52] prop. 5.3 on the analogousstatements for Kl ˇ G (and remove the restriction of the characteristic of k in loc. cit. ).(ii) It follows from [51] prop. 5.6 that when p does not divide the order ♯W of Weyl group, the only non-zero break of Be † ˇ G, Ad (and Kl ˇ G ) at ∞ is 1 / ˇ h . Indeed, the local Galois representation I ∞ → ˇ G is describedexplicitly in [51] prop. 5.6 and § 6.2.(iii) It is expected that the description in (ii) of the local monodromy of Be † ˇ G (and Kl ˇ G ) at ∞ should holdwhen ( p, h ) = 1. When ˇ G = GL n , this is indeed the case. For Kl n , this was proved by Fu and Wan ([48]theorem 1.1). For Be † n , the can be shown by studying the solutions of Bessel differential equation (1.1.1.1)at ∞ . We omit details and refer to ([71] 6.7) for a treatment in the case when n = 2.(iv) Using theorem 4.5.2 (ii), which will be proved in the appendix A.1, we see that when p = 2 and n is an odd integer, the associated local Galois representation of Be † SO n at ∞ coincides with the simple wildparameter constructed by Gross-Reeder in [51] § 6.3. In particular, the image of the inertia group I ∞ in thecase ˇ G = SO is isomorphic to A . Together with Be † SO , Std ≃ Be † SL , Sym (4.1.9.1), this allows us to recoverAndré’s result on the local monodromy group of Be † at ∞ in the case p = 2 ([9] § 7, 8).5. Applications
In this section, we give some applications of our study of Bessel F -isocrystals for reductive groups. ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 55 Functoriality of Bessel F -isocrystals. We may ask all possible Frobenius structure on Be † ˇ G ( ˇ ξ ) (notnecessarily the one from 4.4.1), i.e. all possible isomorphisms of tensor functors ϕ : F ∗ X ◦ Be † ˇ G ∼ −→ Be † ˇ G . Lemma 5.1.1.
The Frobenius structure on Be † ˇ G ( ˇ ξ ) is unique up to an element in the center Z ˇ G ( K ) of ˇ G .Proof. Given two Frobenius structures ϕ , ϕ , u := ϕ ◦ ϕ − is an isomorphism of tensor functors Be † ˇ G ( ˇ ξ ) ∼ −→ Be † ˇ G ( ˇ ξ ). If ω denotes a fiber functor of h Be † ˇ G ( ˇ ξ ) i , then ω ◦ u is an element in ˇ G ( K ) commuting with G geo ( K )by the Tannakian formalism. Then the assertion follows from Z ˇ G ( G geo ) = Z ˇ G . (cid:3) Let
G, G ′ be two split, almost simple groups over R whose Langlands dual groups ˇ G ′ ⊂ ˇ G over K appear in the same line in the left column of the (1.2.6.1). Up to conjugation, we can assume that theinclusion ˇ G ′ ⊂ ˇ G preserves the pinning. Then it induces a natural inclusion ˇ g ′ aff (1) ⊂ ˇ g aff (1). Let φ ′ bea generic linear function of G ′ over R (4.4.1) and ˇ ξ the generic element in ˇ g ′ aff (1) corresponding to − πφ ′ (4.4.1). Note that ˇ ξ is also a generic element in ˇ g aff (1). Proposition 5.1.3. (i)
There exists a generic linear function φ of G over R such that − πφ matches ˇ ξ ∈ ˇ g aff (1) under the isomorphism (4.3.3.1) . (ii) Let (Be † ˇ G ′ ( ˇ ξ ) , ϕ ′ ) (resp. (Be † ˇ G ( ˇ ξ ) , ϕ ) ) be the Bessel F -isocrystal of ˇ G ′ (resp. ˇ G ) constructed by φ ′ (resp. φ ) in . Then (Be † ˇ G ( ˇ ξ ) , ϕ ) is the push-out of (Be † ˇ G ′ ( ˇ ξ ) , ϕ ) .Proof. (i) Let φ be the generic linear function of G over K such that − πφ corresponds to ˇ ξ under theisomorphism (4.3.3.1). We will show that φ is naturally integral.By construction, Be ˇ G ( ˇ ξ ) is the push-out of Be ˇ G ′ ( ˇ ξ ). In particular, for V ∈ Rep ( ˇ G ), the connection(Be ˇ G,V ( ˇ ξ )) † has a Frobenius structure and is overconvergent. Let χ be a generic linear function of G over R and ˇ η ∈ ˇ g aff (1) the corresponding generic element. Then there exists an element c ∈ K × such that we canrewrite two Bessel connections for the adjoint representation of ˇ G as follow (4.3.1.2):(5.1.3.1) Be ˇ G, Ad (ˇ η ) = d + ( N + xE ) dxx , Be ˇ G, Ad ( ˇ ξ ) = d + ( N + cxE ) dxx . Via (4.3.3.1), it suffices to show that c ∈ R × .Both the above two connections admit Frobenius structures and decompose in the categories Conn( X K ),Sm( X k /K ) and Sm( X k /K F ) in the same way (according to the decomposition of Ad in Rep ( ˇ G ′ ) by theorem4.5.2). Let V be a non-trivial irreducible component of Ad in Rep ( ˇ G ′ ) and V (ˇ η ) , V ( ˇ ξ ) the correspondingoverconvergent F -isocrystal. Since V (ˇ η ) | is unipotent, if { e i } denotes a basis of V , there exists a solution u : e i f i ( x ) ∈ Sol( V (ˇ η ) | ) (2.9.2.2)whose convergence domain is the open unit disc of radius 1. Then u c : e i f i ( cx ) belongs to Sol( V ( ˇ ξ ) | )and has the same convergent radius. If c is not a p -adic unit, then V (ˇ η ) (or V ( ˇ ξ )) admits the trivialoverconvergent isocrystal on X k as a quotient, which contracts to their irreducibility. The assertion follows.(ii) By (i), the ˇ G -valued overconvergent isocrystal Be † ˇ G ( ˇ ξ ) is the push-out of Be † ˇ G ′ ( ˇ ξ ). It remains toidentify two Frobenius structures on ˇ G -valued overconvergent isocrystals Be † ˇ G ( ˇ ξ ) ≃ Be † ˇ G ′ ( ˇ ξ ) × ˇ G ′ ˇ G , whichare different by an element ε in the center Z ˇ G ( K ) by (5.1.1). Taking account of the extension of Frobeniusstructures to 0 (4.4.5), we deduce that ε = id and the assertion follows. (cid:3) Now we can prove the following conjecture of Heinloth-Ngô-Yun ([52] conjecture 7.3).
Theorem 5.1.4.
We keep the notation of and fix a non-trivial additive character ψ . Assume that ˇ G ′ ⊂ ˇ G over Q ℓ appear in the same line in the left column of the (1.2.6.1) . For every generic linear function φ ′ of G ′ over k , there is a generic linear function φ of G over k such that Kl ét ,ℓ ˇ G ( ψφ ) is isomorphic to thepush-out of Kl ét ,ℓ ˇ G ′ ( ψφ ′ ) along ˇ G ′ ⊂ ˇ G as ℓ -adic ˇ G -local systems on X k . Proof.
By the “trivial” functoriality (4.1.9), we may assume that ˇ G is simply connected. We lift φ tobe a generic linear function of G ′ over R and take φ ′ as in 5.1.3. We need to show that Kl ét ,ℓ ˇ G ( ψφ ) ≃ Kl ét ,ℓ ˇ G ′ ( ψφ ′ ) × ˇ G ′ ˇ G as ˇ G -local systems. It follows from theorem 4.4.4 and proposition 5.1.3 that for everyrepresentation V ∈ Rep ( ˇ G ), regarded as a representation of ˇ G ′ , and every a ∈ | X k | , we haveTr(Frob a | Kl ét ,ℓ ˇ G,V, ¯ a ) = Tr(Frob a | Kl ét ,ℓ ˇ G ′ ,V, ¯ a ) . Note that if Σ is the group of pinned automorphisms of ˇ G , then the closed embedding ˇ G Σ → ˇ G inducesa surjective homomorphism of K-rings K ( Rep ( ˇ G )) ⊗ Q ℓ → K ( Rep ( ˇ G Σ )) ⊗ Q ℓ . Then the homomorphism K ( Rep ( ˇ G )) ⊗ Q ℓ → K ( Rep ( G geo )) ⊗ Q ℓ is also surjective. It follows that if we replace V by any represen-tation W of G geo ( ⊂ ˇ G ′ ⊂ ˇ G ), the above equality holds. This implies that the Frobenius conjugacy classesof Kl ét ,ℓ ˇ G and of Kl ét ,ℓ ˇ G ′ have the same image in G geo //G geo . Now, for a faithful representation W of G geo ,two representations Kl ét ,ℓ ˇ G , Kl ét ,ℓ ˇ G ′ : π ( X k , x ) → G geo ( Q ℓ ) are conjugated in GL( W ) by an element g . Thiselement g induces an automorphism of G geo . It fixes every Frobenius conjugacy class and therefore fixes G geo //G geo . Then g must be inner. That is these two representations are conjugate in G geo and the assertionfollows. (cid:3) Hypergeometric F -isocrystals. To describe Bessel F -isocrystals for classical groups, we need toreview some basic facts about the hypergeometric F -isocrystals. In ([57] 5.3.1), Katz interpreted hypergeometric D -modules on G m as the multiplicative convolutionof hypergeometric D -modules of rank one. Besides the hypergeometric D -modules, Katz also studied ℓ -adictheory of hypergeometric sheaves using multiplicative convolution. The Frobenius traces of these sheavesare called hypergeometric functions (over finite fields) which generalize Kloosterman sums (1.1.2.1).Let ψ be a non-trivial additive character on F p , n an integer ≥ ρ = ( ρ , · · · , ρ m ) a sequence ofmultiplicative characters on k × . The hypergeometric function H ψ ( n, ρ ) is defined for any finite extension k ′ /k and a ∈ k ′× by(5.2.1.1) H ψ ( n, ρ )( a ) = X ψ (cid:18) Tr k ′ / F p ( n X i =1 x i − m X j =1 y j ) (cid:19) · m Y j =1 ρ − j (Nm k ′ /k ( y j )) , where the sum take over ( x , · · · , x n , y , · · · , y m ) ∈ ( k ′× ) m + n satisfying Q ni =1 x i = a Q mj =1 y j .Recently, Miyatani studied the p -adic counterpart of this theory [67]. Using the multiplicative convolu-tion of arithmetic D -modules, he constructed the Frobenius structure on hypergeometric D -modules whoseFrobenius traces are hypergeometric sums. In the following, we briefly recall his results in some special cases.Let M , N be two objects of D( G m,k /L N ) and µ : G m × G m → G m the multiplication morphism. Recallthat the (multiplicative) convolution ⋆ is defined by(5.2.1.2) M ⋆ N = µ ! ( M ⊠ N ) . Let n > m be two non-negative integers, π ∈ K associated to ψ (2.1.5) and β = ( β , · · · , β m ) a sequenceof elements of q − Z − Z . We denote by Hyp π ( n, β ) the p -adic hypergeometric differential operator on G m (5.2.1.3) Hyp π ( n, β ) = δ n − ( − n + mp π n − m x m Y i =1 ( δ − β j ) , where x is a coordinate of G m and δ = x ddx . We denote by H yp π ( n, β ) the D † b P , Q ( { , ∞} )-module(5.2.1.4) H yp π ( n, β ) = D † b P , Q ( { , ∞} ) / ( D † b P , Q ( { , ∞} )Hyp π ( n, β )) . It corresponds to the hypergeometric function associated to ψ , n trivial characters χ ’s and m characters ρ ’s defined in ([57]8.2.7) ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 57 Theorem 5.2.2 (Miyatani [67]) . We fix an isomorphism Q p ≃ C . (i) The arithmetic D -module H yp π ( n, β ) underlies to a pure overconvergent F -isocrystal on G m,k of rank n and weight n + m − . (ii) The Frobenius structure on the overconvergent isocrystal H yp π ( n, β ) is unique (up to a scalar). (iii) The Frobenius trace of H yp π ( n, β ) on G m,k is equal to the hypergeometric function ( − n + m − H ψ ( n, ρ )(5.2.1.1) , where ρ i is defined for ξ ∈ k × and e ξ the Teichmüller lifting of ξ , by ρ i ( ξ ) = e ξ ( q − β i . Assertions (i) and (iii) are stated in ([67] Main theorem). One can apply the method of ([67] 4.2.1) toshow that H yp π ( n, β ) is irreducible in the category D( G m,k /K ) and hence assertion (ii).The arithmetic D -module H yp π ( n, β ) depends only on ψ and ρ (the class of β modulo Z ) that we alsodenote by H yp ψ ( n, ρ ). Let F be the hypergeometric ℓ -adic sheaf on G m,k associatedto ψ , n trivial multiplicative characters and non-trivial multiplicative characters ρ = ( ρ , · · · , ρ m ). The space F I of I -invariants is one-dimensional and Frob k acts on it as the monomial in Gauss sums ([58] 2.6.1) α = ( − m m Y j =1 G ( ψ − , ρ − j ) , where G ( ψ − , ρ − j ) denotes the Gauss sum associated to ψ − and ρ − j .On the other hand, note that the action of I is maximal unipotent. Any lifting F in the decompositiongroup D at 0 of the Frobenius automorphism has eigenvalues set { α, qα, · · · , q n α } (cf. [56] 7.0.7). Aftertwisting a geometrically constant lisse rank one sheaf (resp. overconvergent F -isocrystal), we denote by f F (resp. f H yp ψ ( n, ρ )) the normalised hypergeometric sheaf (resp. F -isocrystal) whose the Frobenius eigen-values at 0 is { q − ( n − / , · · · ., q ( n − / } . Its Frobenius trace function, called the normalised hypergeometricsum e H ψ ( n, ρ ) is defined for a ∈ F × q by(5.2.3.1) e H ψ ( n, ρ )( a ) = 1( −√ q ) n − Q mj =1 G ( ψ − , ρ − ) H ψ ( n, ρ )( a ) . When m = 0, we have f F = Kl ét n (1.1.2.2) and f H yp ψ ( n, ∅ ) = Be † n (1.1.4).5.3. Bessel F -isocrystals for classical groups.5.3.1. The Kloosterman sheaf and the Bessel F -isocrystal for ( G = GL n , ˇ G = GL n ) have been extensivelystudied. As usual, let E ij denote the n × n -matrix with the ( i, j )-entry 1 and all other entries 0. We choosethe standard Borel B of the upper triangular matrices and the standard torus T of the diagonal matrices.We choose a coordinate x of A . Then there is a canonical isomorphism G na ≃ I (1) /I (2) , ( a , . . . , a n ) n − X i =1 a i E i,i +1 + a n x − E n, . We choose φ : G na → G a to be the addition map. Under the isomorphism (4.3.3.1) and (4.3.4.1), φ correspondsto ˇ ξ = N + Edx (4.4.1.1) with(5.3.1.1) N = . . .
00 0 1 . . . . . .
10 0 0 . . . , E = . . .
00 0 0 . . . . . .
01 0 0 . . . . On the other hand, by 4.1.9 and ([52] §3), we have(5.3.1.2) Kl étSL n , Std ≃ Kl étGL n , Std ≃ Kl ét n . Indeed, diagram (4.1.8.2) reduces to diagram (1.1.1.4) in this case. Therefore, the Kloosterman connectionis isomorphic to the classical Bessel connection (1.1.1, 1.1.4)(5.3.1.3) Kl dRSL n , Std ( λφ ) ≃ Be n , Kl rigSL n , Std ( φ ) ≃ Be † n . Recall that the connection Be n corresponds to the Bessel differential equation (1.1.1.1). Consider G = SO n +1 , ˇ G = Sp n = { A ∈ SL n | AJA T = J } , where J is the anti-diagonal matrix with J ij = ( − i δ i, n +1 − j . Then matrices ( N, E ) as in (5.3.1.1) are in ˇ g and Be ˇ G ( ˇ ξ ) is given by the same formula as GL n case. Then we deduce an isomorphism of overconvergent F -isocrystals Be † Sp n , Std ( ˇ ξ ) ≃ Be † n by (5.1.3). Consider G = SO n ˇ G = SO n = { A ∈ SL n , AJA T = J } , where J is the anti-diagonal matrix with J ij = ( − max { i,j } δ i, n +1 − j . There exists a canonical isomorphism G n +1 a ≃ I (1) /I (2) , ( a , · · · , a n +1 ) n − X i =1 ( E i,i +1 + E n − i, n − i +1 )+( E n − ,n +1 + E n,n +2 )+ x − ( E , n − + E , n ) . Then we take φ : G n +1 a → G a to be the addition map. When n ≥
3, under the isomorphism (4.3.3.1) and(4.3.4.1), λφ corresponds to ˇ ξ = N + λ n − Ex (4.4.1.1) with (5.3.3.1) N = . . . . . . . . . . . . ...... ... 0 1 1 0 . . . . . . . . . . . . , E = . . . . . . . . . . . . ...... ... 0 0 0 0 . . . . . . . . . . . . . The corresponding Bessel connection is written as(5.3.3.2) Be SO n , Std ( ˇ ξ ) = d + ( N + λ n − Ex ) dxx . If e , · · · , e n denote a basis for the above connection matrix, the restriction of the above connection to thesubbundle generated by e n − e n +1 is trivial. The other horizontal subbundle, generated by e n + e n +1 andother basis vectors, is isomorphic to the Bessel connection Be SO n − , Std ( ˇ ξ ) discussed below (5.3.6.5). In [64], T. Lam and N. Templier identified the diagram (4.1.8.2) with the Laudau-Ginzburg modelfor quadrics [73] and used it to calculate the associated Kloosterman D -modules. We briefly recall thisconstruction following ([73] § 3). Let Q n − = G/P be the (2 n − p : · · · : p n − : p ′ n − : p n : · · · : p n − ) be the Plücker coordinates of Q n − satisfying(5.3.4.1) p n − p ′ n − − p n − p n + · · · + ( − n − p p n − = 0 . Consider the open subscheme(5.3.4.2) Q ◦ n − = Q n − − D, ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 59 with the complement D = D + D + · · · + D n − + D ′ n − , where D i is defined by(5.3.4.3) D := { p = 0 } D ℓ := nP ℓk =0 ( − k p ℓ − k p n − − ℓ + k = 0 o for 1 ≤ ℓ ≤ n − D n − := { p n − = 0 } D n − := { p n − = 0 } D ′ n − := (cid:8) p ′ n − = 0 (cid:9) The divisor D is anti-canonical in Q n − . For simplicity, we set δ ℓ = ℓ X k =0 ( − k p ℓ − k p n − − ℓ + k , for 0 ≤ ℓ ≤ n − . If x denotes a coordinate of G m , we define a regular function W : Q ◦ n − × G m → A to be(5.3.4.4) W ( p i : p ′ n − ; x ) = p p + n − X ℓ =1 p ℓ +1 p n − − ℓ δ ℓ + p n p n − + p n p ′ n − + x p p n − . The Kloosterman overconvergent F -isocrystal and connection are calculated by(5.3.4.5) Kl rigSO n , Std ( φ ) ≃ pr , ! ( W ∗ ( A ψ ))[2( n − n − , Kl dRSO n , Std ( λφ ) ≃ pr , ! ( W ∗ ( E λ ))[2( n − . We deduce that the Frobenius trace Kl SO n , Std of Kl rigSO n , Std ( φ ) is defined for a ∈ F × q by(5.3.4.6)Kl SO n , Std ( a ) = 1 q n − X ( p i ,p ′ n − ) ∈ Q ◦ n − ( F q ) ψ (cid:18) Tr F q / F p (cid:18) p p + n − X ℓ =1 p ℓ +1 p n − − ℓ δ ℓ + p n p n − + p n p ′ n − + a p p n − (cid:19)(cid:19) . Proposition 5.3.5. (i)
When n = 2 , we have (5.3.5.1) Kl SO , Std ( a ) = Kl(2; a ) . (ii) When n ≥ , we can simplify above sum as (5.3.5.2) Kl SO n , Std ( a ) = 1 q n − (cid:18) X x i ∈ F × q ψ (cid:18) Tr F q / F p ( x + x + · · · + x n − + a x + x x x · · · x n − ) (cid:19) + ( q − q n − (cid:19) . Proof.
Assertion (i) is easy to prove and is left to readers. It also follows from (4.1.9.3).(ii) The equality follows from subdividing the sum (5.3.4.6) in the following parts:(a) Case p n , p n +1 , · · · , p n − = 0: we replace p i , p ′ n − by x i , y i ∈ F × q as follows: p k = k = 0 x . . . x k − ( x k + y k ) if 1 ≤ k ≤ n − x . . . x n − x n − if k = n − x . . . x n − x n − y n − if k = nx . . . x n − x n − y n − y n − . . . y n − − k otherwise p ′ n − = x · · · x n − y n − . Then the sum (5.3.4.6) becomes the toric exponential sum in (5.3.5.2).(b) Case p n = 0 and p n − − ℓ = 0 for some ℓ ∈ { , · · · , n − } : we assume ℓ is maximal. By divid-ing p ′ n − , we consider the affine coordinates p , · · · , p n − and we replace p n − by the equation (5.3.4.1).Since p n , · · · , p n − − ℓ − = 0, p ℓ +1 can be taken in F q regardless of the condition δ ℓ = 0. Then we have P p ℓ +1 ∈ F q ψ ( p ℓ +1 p n − − ℓ δ ℓ ) = 0 and that the sum (5.3.4.6) equals to zero in this case.(c) Case p n = p n +1 = · · · = p n − = 0: it is easy to show that the sum (5.3.4.6) equals to q − q , which isthe constant part of (5.3.5.2). (cid:3) Consider G = Sp n , ˇ G = SO n +1 = { A ∈ SL n +1 | AJA T = J } , where J is the anti-diagonal matrix with J ij = ( − i δ i, n +2 − j . There exists a canonical isomorphism G n +1 a ≃ I (1) /I (2) , ( a , · · · , a n +1 ) n − X i =1 ( E i,i +1 + E n − i, n − i +1 ) + E n − ,n + x − E n, . Then we take φ : G n +1 a → G a to be the addition map. Under the isomorphism (4.3.3.1) and (4.3.4.1), λφ corresponds to ˇ ξ = N + λ n Ex (4.4.1.1) with N as in (5.3.1.1), which belongs to ˇ g , and(5.3.6.1) E = . . . . . . . . . . . . . . .
00 2 0 . . . ∈ ˇ g . Then we can write the Bessel connection as(5.3.6.2) Kl dRSO n +1 , Std ( λφ ) ≃ Be SO n +1 , Std ( ˇ ξ ) = d + ( N + λ n Ex ) dxx . After taking a gauge transformation by the matrix . . . . . .
00 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . λ n x . . . . we obtain the scalar differential equation associated to Be SO n +1 , Std ( ˇ ξ ):(5.3.6.3) ( x ddx ) n +1 − λ n x (4 x ddx + 2) = 0 . When n ≥
2, we can rewrite ˇ ξ as(5.3.6.4) ˇ ξ = . . . √ √ . . .
10 0 . . . + λ n . . . . . . . . .
01 0 . . . . . .
00 1 0 . . . x, where √ K and appears in positions ( n, n + 1) and ( n + 1 , n + 2). Via the naturalinclusion so n +1 → so n +2 the above element ˇ ξ ∈ ( so n +1 ) aff (1) corresponds to ˇ ξ ∈ ( so n +2 ) aff (1) definedin (5.3.3.1). The standard (2 n + 2)-dimensional representation of so n +2 decomposes as a direct sum of thetrivial representation and the standard (2 n + 1)-dimensional representation of so n +1 as representations of so n +1 . Then we obtain decompositions of Bessel connections and Bessel F -isocrystals by proposition 5.1.3(5.3.6.5) Be SO n +2 , Std ( ˇ ξ ) ≃ Be SO n +1 , Std ( ˇ ξ ) ⊕ ( O G m,K , d ) , Be † SO n +2 , Std ( ˇ ξ ) ≃ Be † SO n +1 , Std ( ˇ ξ ) ⊕ ( O G m , d ) . In the remaining ot this subsection, we omit ˇ ξ from the notation. ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 61 Remark 5.3.7.
The fact that matrix E in (5.3.6.1) takes value 2 in its non-zero entries is delicate. On theone hand, it comes from the calculation of invariant polynomials. On the other hand, it ensures the existenceof a Frobenius structure on the differential equation (5.3.6.3) with parameter λ = − π . For instance, for everyprime number p , the convergence domain of the unique solution of (5.3.6.3) ( λ = − π ) at 0 : F ( x ) = X r ≥ (2 r − r !) n +1 (2 π n ) r x r , is the open unit disc of radius 1. In particular, F ( x ) belongs to K { x } (2.9.2.1) and it justifies (2.9.2.2). Proposition 5.3.8. (i)
When p > , there exists an isomorphism of overconvergent F -isocrystals (5.2.3)(5.3.8.1) Be † SO n +1 , Std ≃ [ x x ] + f H yp ψ (2 n + 1 , ρ ) , where ρ denotes the quadratic character of k × . (ii) When p = 2 , there exists an isomorphism of overconvergent F -isocrystals (5.3.8.2) Be † SO n +1 , Std ≃ Be † n +1 . Proof. (i) If we rescale x by x x , the differential equation (5.3.6.3) turns to the hypergeometric differentialequation Hyp ψ (2 n + 1; ρ ) associated to ρ (5.2.1). Frobenius structures on two sides of (5.3.8.1) are ofweight zero and have Frobenius eigenvalues { q − n , · · · , q − , , q, · · · , q n } at 0 (4.4.5, 5.2.3.1). Then these twoFrobenius structures coincide by theorem 5.2.2(ii) and the isomorphism (5.3.8.1) follows.(ii) We will prove the assertion in Appendix A. (cid:3) It follows that there exists an isomorphism of overconvergent F -isocrystals (5.2.1.2)(5.3.9.1) Be † SO n +1 , Std ≃ Be † SO , Std ⋆ Be † n − . by the convolution interpretation of hypergeometric overconvergent F -isocrystals ([67] Main theorem (ii)and 3.3.3). Corollary 5.3.10.
Suppose p = 2 . The SL n +1 -valued overconvergent F -isocrystals Be † SL n +1 is the push-outof Be † SO n +1 along SO n +1 → SL n +1 . It follows from 5.3.8(ii).
Corollary 5.3.11. (i)
The Frobenius trace function Kl SO n +1 , Std of Be † SO n +1 , Std is equal to Kl SO n +1 , Std ( a ) = X x,y ∈ k × ,xy = a Kl SO , Std ( x ) Kl(2 n − y )(5.3.11.1) = (cid:26) Kl(2 n + 1; a ) , p = 2 , e H ψ (2 n + 1; ρ )(4 a ) , p > . (5.3.11.2)(ii) We have an identity of exponential sums (5.3.5.2)(5.3.11.3) Kl SO n +2 , Std ( a ) − SO n +1 , Std ( a ) . Proof. (i) The first equality follows from (5.3.9.1). The second one follows from 5.3.8(i-ii).(ii) It follows from proposition 5.3.5 and (5.3.6.5). (cid:3)
In particular, by (4.1.9.1) and corollary 5.3.11(i), we obtain (1.2.9.1). Using the triviality functoriality4.1.9 and the exceptional isomorphism for groups of low ranks (4.1.9.1)-(4.1.9.4), one can similarly obtainother identities between exponential sums, whose sheaf-theoretic incarnations were obtained by Katz [58].
Frobenius slopes of Bessel F -isocrystals.5.4.1. We first recall the definition of the
Newton polygon of a conjugacy class in ˇ G ( K ). Let X • ( ˇ T ) + be theset of dominant coweights of ˇ G and X • ( ˇ T ) + R the positive Weyl chamber, equipped with the following partialorder ≤ : µ ≤ λ if λ − µ can be written as a linear combination of positive coroots of ˇ G with coefficients in R + . We identify ( X • ( ˇ T ) ⊗ Z R ) /W and X • ( ˇ T ) + R . Recall that ρ denotes the half sum of positive roots of Gρ = 12 X α ∈ Φ + α ∈ X • ( T ) = X • ( ˇ T ) . Let v : K → Q ∪ {∞} be the p -adic order, normalised by v ( q ) = 1. It induces a homomorphism of groups v : ˇ T ( K ) → X • ( ˇ T ) ⊗ Z R . By identifying ˇ T ( K ) /W and the set of semisimple conjugacy classes Conj ss ( ˇ G ( K ))in ˇ G ( K ), we deduce a homomorphism:(5.4.1.1) NP : Conj ss ( ˇ G ( K )) = ˇ T ( K ) /W → ( X • ( ˇ T ) ⊗ Z R ) /W = X • ( ˇ T ) + R . In the case where ˇ G = GL n , NP is equivalent to the classical p -adic Newton polygon. Indeed, we have X • ( ˇ T ) + R = { ( λ , · · · , λ n ) ∈ R n , λ ≤ · · · ≤ λ n } , and we can associate to ( λ , · · · , λ n ) a convex polygon with vertices ( i, λ + · · · + λ i ) for i ∈ { , · · · , n } . For λ = ( λ , · · · , λ n ) , µ = ( µ , · · · , µ n ) in X • ( ˇ T ) + R , µ ≤ λ if and only if the polygon associated to µ lies abovethat of λ with the same endpoint. Theorem 5.4.2.
Let x ∈ | A k | be a closed point and ϕ x ∈ ˇ G ( K ) the Frobenius automorphism of (Be † ˇ G , ϕ ) at x (4.4.3) . Let v be the p -adic order normalised by v ( q deg( x ) ) = 1 and NP defined as above. (i) Except for finitely many closed points of | A k | , we have NP( ϕ x ) = ρ . (ii) Suppose that ˇ G is of type A n , B n , C n , D n or G , then we have NP( ϕ x ) = ρ for every x ∈ | A k | .Proof. (i) In ([63] 2.1), V. Lafforgue shows that the Newton polygon (5.4.1.1) of the Hecke eigenvalue of acuspidal function is ≤ ρ . In particular, we deduce that NP( ϕ x ) ≤ ρ for all x ∈ | G m,k | . By 4.4.5, we haveNP( ϕ ) = NP( ρ ( q )) = ρ . That is the Newton polygon achieves the upper bound ρ at 0. We take a finite setof tensor generators { V , · · · , V n } of Rep ( ˇ G ). Then the assertion follows by applying Grothendieck-Katz’theorem (cf. [33] 1.6) to log convergent F -isocrystals Be † ˇ G,V i .(ii) (a) The case where ˇ G is of type A n , C n . By functoriality (5.1.3), we reduce to study the Frobeniusslope of Bessel F -isocrystal Be † n of rank n (1.1.4). After the work of Dwork, Sperber and Wan [44, 79, 84],the Frobenius slope set of Be † n (normalised to be weight 0) at each closed point x ∈ | A k | is equal to {− n − , − n − , · · · , n − } . Then the assertion follows.(b) The case where ˇ G is of type B n , D n , G . By functoriality (5.1.3), we reduce to show that the Frobeniusslope set of Be † SO n +1 , Std at each closed point is equal to {− n, − n +1 , · · · , n } . If p = 2, it follows from 5.3.8(ii)and the case (a). If p >
2, in view of 5.2.3 and 5.3.8(i), it follows from the following lemma. (cid:3)
Lemma 5.4.3.
The Frobenius slope set of H yp ψ (2 n + 1; ρ ) (5.2.2) at each closed point is equal to (cid:26) , , · · · , n + 12 (cid:27) . Proof.
We deduce this fact from Wan’s results on Frobenius slope of certain toric exponential sums [84, 85].For a ∈ F × q and a divisor d of p −
1, consider the following Laurent polynomial in F q [ x ± , · · · , x ± n +1 ] f d ( x , · · · , x n +1 ) = x + · · · + x n − x d n +1 + ax d n +1 x x · · · x n . ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 63 For m ≥
1, we denote by S m ( f d ) the exponential sum associated a Laurent polynomial: S m ( f d ) = X x i ∈ F × qm ψ (cid:18) Tr F qm / F p f d ( x , · · · , x n +1 ) (cid:19) . Then we have an identity(5.4.3.1) S m ( f ) = S m ( f ) + X x ··· x n +1 = ayx i ∈ F × qm ψ (cid:18) Tr F qm / F p ( x + · · · + x n +1 − y ) (cid:19) · ρ − (cid:0) Nm F qm/ F q ( y ) (cid:1) , where the last term is the Frobenius trace of H yp (2 n + 1; ρ ).The L-function associated to these exponential sums is a rational function:L( f d , T ) = exp (cid:18) X m ≥ S m ( f d ) T m m (cid:19) , We denote by ∆( f d ) the convex closure in R n +1 generated by the origin and lattices defined by exponentsappeared in f d : { (0 , · · · , , (1 , · · · , , · · · , (0 , · · · , , , (0 , · · · , , d ) , ( − , · · · , − , d ) } . The polyhedron ∆( f d ) is (2 n + 1)-dimensional and has volume d n ! . The Laurent polynomials f d is non-degenerate (cf. [85] Def. 1.1). After Adolphson-Sperber [8], the L-function L( f d , T ) is a polynomial of degree d (2 n + 1).We denote by NP( f d ) the (Frobenius) Newton polygon associated to L-functions L( f d , T ) (cf. [85] 1.1)and by HP( f d ) the Hodge polygon defined in term of the polyhedron ∆( f d ) (cf. [85] 1.2). The (multi-)setof slopes of HP( f d ) is(5.4.3.2) (cid:26) , d , d , · · · , n + d − d (cid:27) . The Newton polygon lies above the Hodge polygon [8]. A Laurent polynomial is called ordinary if thesetwo polygons coincide. Let δ be a co-dimension 1 face of ∆ which does not contain the origin and f δd therestriction of f d to δ which is also non-degenerate. The Laurent polynomial f δd is diagonal in the sense of([85] § 2). If V , · · · , V n +1 denote the vertex of δ written as column vectors, the set S ( δ ) of solutions of( V , · · · , V n +1 ) r ... r n +1 ≡ , r i rational, 0 ≤ r i < , forms an abelian group of order d (cf. [85] 2.1). Since d is a divisor of p −
1, we deduce that for each δ , f δd isordinary by ([85] Cor. 2.6). By Wan’s criterion for the ordinariness [84] (cf. [85] Thm. 3.1), f d is ordinary.In view of (5.4.3.1) and the slope sets of HP( f ) , HP( f ) (5.4.3.2), the assertion follows. (cid:3) Appendix A. A -adic proof of Carlitz’s identity and its generalization As mentioned in introduction, Carlitz [26] proved the following identity between Kloosterman sums:Kl(3; a ) = Kl(2; a ) − , ∀ a ∈ F × s . In this appendix, we reprove and generalize this identity by establishing an isomorphism between two Bessel F -isocrystals Be † n +1 and Be † SO n +1 , Std . The following is a restatement of proposition 5.3.8(ii).
Proposition A.1.
There exists an isomorphism between following two overcovergent F -isocrystals on G m, F (5.3.6.3) : (A.1.1) Be † n +1 : ( x ddx ) n +1 + 2 n +1 x = 0 , Be † SO n +1 , Std : ( x ddx ) n +1 − n +1 x (2 x ddx + 1) = 0 . Our strategy is first to show that their maximal slope quotient convergent F -isocrystals are isomorphic.Then we conclude the proposition by a dual version of a minimal slope conjecture (proposed by Kedlaya [62]and recently proved by Tsuzuki [82]) that we briefly recall in the following. A.2.
We keep the notation of section 5. Let X be a smooth k -scheme. Let M † be an overconvergent F -isocrystal on X/K . We denote the associated convergent F -isocrystal on X/K by M . When the (Frobenius)Newton polygons of M are constant on X , M admits a slope filtration , that is an increasing filtration(A.2.1) 0 = M ( M ( · · · ( M r − ( M r = M of convergent F -isocrystals on X/K such that • M i / M i − is isoclinic of slope s i and • s < s < · · · < s r .By Grothendieck’s specialization theorem, for any convergent F -isocrystal M on X/K , there exists anopen dense subscheme U of X such that the Newton polygons of M are constant.We remark that for a log convergent F -isocrystal with constant Newton polygons over a smooth k -schemewith normal crossing divisor, such a slope filtration (of sub log convergent F -isocrystals) also exists.In a recent preprint, Tsuzuki showed a dual version of Kedlaya’s minimal slope conjecture ([62] 5.14): Theorem A.3 ([82] theorem 1.3) . Let X be a smooth connected curve over k . Let M † , N † be two irreducibleoverconvergent F -isocrystals such that the corresponding convergent F -isocrystals M , N admit the slopefiltrations { M i } , { N i } respectively. We renumber the slope filtration by (A.3.1) M = M ) M ) M ) · · · ) M r − ) M r = 0 with slopes s > s > · · · > s r − . Suppose there exists an isomorphism h : N / N ∼ −→ M / M of convergent F -isocrystals between the maximal slope quotients. Then there exists a unique isomorphism g † : N † ∼ −→ M † of overconvergent F -isocrystals, which is compatible with h as morphisms of convergent F -isocrystals. A.4.
Following Dwork’s strategy ([43] § 1-3), we study the maximal slope quotients of Be † n +1 and ofBe † SO n +1 , Std in terms of their unique solutions at 0.In the following, we assume k = F p . We first recall Dwork’s congruences and show a refinement of hisresult in the 2-adic case. Consider for every i ≥
0, a map B ( i ) ( − ) : Z ≥ → K × and the following congruencerelation for 0 ≤ a < p and n, m, s ∈ Z ≥ :(a) B ( i ) (0) is a p -adic unit for all i ≥ B ( i ) ( a + np ) B ( i +1) ( n ) ∈ R for all i ≥ B ( i ) ( a + np + mp s +1 ) B ( i +1) ( n + mp s ) ≡ B ( i ) ( a + np ) B ( i +1) ( n ) mod p s +1 for all i ≥ p = 2, B ( i ) ( a + n m s +1 ) B ( i +1) ( n + m s ) ≡ u ( i, s, m ) B ( i ) ( a + n B ( i +1) ( n ) mod 2 s +1 for all i ≥
0, where u ( i, s, m ) =1 if s = 1 and u ( i, , m ) = 1 or − i and m .If conditions (a-c) (or (a,b,c’)) are satisfied, then B ( i ) ( n ) ∈ R for all i, n ≥
0. We set F ( i ) ( x ) = P ∞ j =0 B ( i ) ( j ) x j ∈ K J x K ,F ( i ) m,s ( x ) = P ( m +1) p s − j = mp s B ( i ) ( j ) x j ∈ K [ x ] , s ≥ . We write F ( i )0 ,s by F ( i ) s for simplicity. Theorem A.5. (i) [[43] theorem 2]
If conditions (a-c) are satisfied, then (A.5.1) F (0) ( x ) F (1) m,s ( x p ) ≡ F (0) m,s +1 ( x ) F (1) ( x p ) mod p s +1 B ( s +1) ( m ) J x K . ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 65 (i’) If conditions (a,b,c’) are satisfied (in particular p = 2 ), then (A.5.2) F (0) ( x ) F (1) m,s ( x ) ≡ F (0) m,s +1 ( x ) F (1) ( x ) mod 2 s B ( s +1) ( m ) J x K . (ii) [[43] theorem 3] Under the assumption of (i) or (i’) and suppose moreover that (d) B ( i ) (0) = 1 for i ≥ . (e) B ( i + r ) = B ( i ) for all i ≥ and some fixed r ≥ .Let U be the open subscheme of A k defined by (A.5.3) F ( i )1 ( x ) = 0 , for i = 0 , , · · · , r − . Then the limit (A.5.4) f ( x ) = lim s →∞ F (0) s +1 ( x ) /F (1) s ( x p ) defines a global function on the formal open subscheme U of b A R associated to U , which takes p -adic unitvalue at each rigid point of U rig . We prove assertion (i’) in the end (A.13). We briefly explain Dwork’s result (ii) in the language of formalschemes. The condition (A.5.3) implies that F ( i ) s = 0 on U (cf. [43] 3.4). For s ≥
1, the congruences (A.5.1)and (A.5.2) imply that F (0) s +1 ( x ) /F (1) s ( x p ) = F (0) s ( x ) /F (1) s − ( x p ) ∈ Γ( U , O U /p s − O U ) . This allows us to use (A.5.4) to define a global function f of O U . A.6.
Let F ( x ) = P j ≥ B ( j ) x j be a formal power series in R J x K . We say F satisfies Dwork’s congruences ifby setting B ( i ) ( j ) = B ( j ) for every i ≥
0, conditions of theorem A.5(ii) are satisfied.We take such a function F and then we obtain a function f ∈ Γ( U , O U ) coinciding with F ( x ) /F ( x p )in K { x } (2.9.2.1) (i.e. the open unit disc). Moreover, by ([43] lemma 3.4(ii)), there exists a function η ∈ Γ( U , O U ) coinciding with F ′ ( x ) /F ( x ) in K { x } defined by η ( x ) ≡ F ′ s +1 ( x ) /F s +1 ( x ) mod p s . The functions f ( x ) and η ( x ) satisfy a differential equation: f ′ ( x ) f ( x ) + px p − η ( x p ) = η ( x ) . Note that f (0) = F (0) /F (0) = 1. Then we deduce that the following corollary. Corollary A.7.
The connection d − η on the trivial bundle O U rig and the function f form a unit-rootconvergent F -isocrystal E F on U/K , whose Frobenius eigenvalue at is . A.8.
Let M † be an overconvergent F -isocrystal on G m,k over K of rank r whose underlying bundle is trivialand the connection is defined by a differential equation:(A.8.1) P ( δ ) = δ r + p r δ r − + · · · + p = 0 , where δ = x ddx , p i ∈ Γ( b A R , O b A R )[ p ]. We assume moreover that M † is unipotent at with a maximalunipotent local monodromy . Then M † extends to a log convergent F -isocrystal M log on ( A ,
0) and itsFrobenius slopes at 0 are s < s = s + 1 < · · · < s r = s + r − . Note that M † is indecomposable in F - Isoc † ( G m,k /K ) and so is M in F - Isoc( G m,k /K ). Then by Drinfeld-Kedlaya’s theorem on the generic Frobenius slopes [42], we deduce property (i):(i) The generic Frobenius slopes (mult-)set is { s , · · · , s r } with s i = s + i − (ii) In view of (2.9.2.2), the differential equation D = 0 admits a unique solution at 0: F ( x ) = X n ≥ A ( n ) x n ∈ K { x } , with A (0) = 1 . Proposition A.9.
Suppose the function F ( x ) satisfies Dwork’s congruences (A.6) and let E F be the asso-ciated unit-root convergent F -isocrystal on U ⊂ A k . Then (i) There exists an epimorphism of log convergent isocrystals M log → E F on ( U, . (ii) As log convergent isocrystals, E F coincides with the maximal slope quotient M log / M log , of M log (A.3.1) .Proof. (i) We set A = Γ( U , O U )[ p ]. We claim that there exists a decomposition of differential operators:(A.9.1) P ( δ ) = Q ( δ )( δ − xη ) , Q ( δ ) = δ r − + q r − δ r − + · · · + q , q i ∈ A. Indeed, by the Euclidean algorithm ([61] 5.5.2), there exists r ∈ A such that P = Q ( δ − xη )+ r . By evaluatingthe above identity at F (in the ring K { x } containing A ), we obtain P ( δ )( F ) = 0 = Q ( δ )( δ − xη )( F ) + rF = rF. Then we deduce r = 0 and (A.9.1) follows.Let e , · · · , e r be a basis of M such that ∇ δ ( e i ) = e i +1 , ≤ i ≤ r − ∇ δ ( e r ) = − ( p r e r + · · · + p e ).We consider a free O U rig -module with a log connection N with a basis f , · · · , f r − and the connectiondefined by ∇ δ ( f i ) = f i +1 , ∇ δ ( f r − ) = − ( q r − f r − + · · · + q f ). By (A.9.1), the morphism f e − xηe induces a horizontal monomorphism N → M log whose cokernel is isomorphic to E F .(ii) Note that Pic( U rig ) ≃ Pic( U ) ([83] 3.7.4) is trivial. Then the rank one convergent isocrystal M log / M log , can be represented as a connection d − λ on the trivial bundle O U rig .Since M log has a maximal unipotent at 0, the rank one quotient of the restriction M log | of M log at theopen unit disc around 0 is unique (2.9.1). In particular, d − λ kills the unique solution F of P ( δ ) = 0. Byanalytic continuation, we have λ = η and the assertion follows. (cid:3) Remark A.10.
The unique solution F ( x ) belongs to the ring K J x K = R J x K ⊗ R K of bounded functions onopen unit disc, which is a subring of K { x } . Assertion (ii) can be viewed as an example of Dwork-Chiarellotto-Tsuzuki conjecture on the comparison between the log-growth filtration (of solutions) and Frobenius slopefiltration [31]. This conjecture was recently proved by Ohkubo [71]. A.11.
Proof of proposition
A.1 . We set k = F and apply the above discussions to overconvergent F -isocrystals M † = Be † n +1 and N † = Be † SO n +1 , Std on G m, F /K (A.1.1). Their unique solutions at 0 are: F ( x ) = X r ≥ ( − (2 n +1) r ( r !) n +1 x r , G ( x ) = X r ≥ (2 n +1) r (2 r − r !) n +1 x r . In the following lemma, we show that F and G satisfy Dwork’s congruences and that the associatedmaximal slope quotients E F and E G (A.9) are isomorphic. Then proposition A.1 follows from theorem A.3and the following lemma. (cid:3) Lemma A.12. (i)
The functions F ( x ) and G ( x ) satisfy Dwork’s congruences (A.6) and define unit-rootconvergent F -isocrystals E F and E G on A k respectively. (ii) The function F ( x ) /G ( x ) extends to a global function of b A R and induces an isomorphism E G ∼ −→ E F .Proof. (i) Conditions (a,b,d,e) are easy to verified. The coefficients of F ( x ) (resp. G ( x )) satisfy condition(c’) (resp. (c)), that is( − (2 n +1)( a + ℓ m s +1 ) / (( a + ℓ m s +1 )!) n +1 ( − (2 n +1)( ℓ + m s ) / (( ℓ + m s )!) n +1 ≡ u ( s, m ) ( − (2 n +1)( a + ℓ / (( a + ℓ n +1 ( − (2 n +1) ℓ / ( ℓ !) n +1 mod 2 s +1 , ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 67 where u (1 , m ) = ( − m and u ( s, m ) = 1 if s = 1, and(2( a + ℓ m s +1 ) − (2 n +1)( a + ℓ m s +1 ) / (( a + ℓ m s +1 )!) n +1 (2( ℓ + m s ) − (2 n +1)( ℓ + m s ) / (( ℓ + m s )!) n +1 ≡ (2( a + ℓ − (2 n +1)( a + ℓ / (( a + ℓ n +1 (2 ℓ − (2 n +1) ℓ / ( ℓ !) n +1 mod 2 s +1 . Since F ( x ) ≡ G ( x ) ≡ F -isocrystals E F , E G are defined over A k .(ii) We set B (0) ( r ) = ( − (2 n +1) r ( r !) n +1 and B (1) ( r ) = (2 n +1) r (2 r − r !) n +1 and B ( i +2) = B ( i ) . Then these sequencessatisfy conditions (a,b,c’,d,e). For condition (c’), the constants u ( i, , m ) are given by u (0 , , m ) = 1 , u (1 , , m ) = ( − m , u ( i + 2 , , m ) = u ( i, , m ) . Since F ( x ) ≡ G ( x ) ≡ F ( x ) /G ( x ) extends to a global function of O b A R by theorem A.5 and so is F ( x ) /G ( x ). Then the assertion follows. (cid:3) A.13.
Proof of theorem
A.5(i’). We prove assertion (i’) by modifying the argument of ([43] theorem 2).Note that condition (c’) implies the following congruence relation:(A.13.1) B ( i ) ( a + n m s +1 ) B ( i +1) ( n + m s ) ≡ B ( i ) ( a + n B ( i +1) ( n ) mod 2 s . When n <
0, we set B ( i ) ( n ) = 0. We set A = B (0) , B = B (1) and for a ∈ { , } , j, N ∈ Z , we set U a ( j, N ) = A ( a + 2( N − j )) B ( j ) − B ( N − j ) A ( a + 2 j ) ,H a ( m, s, N ) = ( m +1)2 s − X j = m s U a ( j, N ) . Then the assertion is equivalent to(A.13.2) H a ( m, s, N ) ≡ s B ( s +1) ( m ) , for s ≥ , m ≥ , N ≥ . By condition (b), we have A ( a + 2 m ) /B ( m ) ∈ R and hence U a ( m, N ) ≡ B ( m ) . Then equation (A.13.2) for s = 0 follows from the fact that H a ( m, , N ) = U a ( m, N ).We now prove by induction on s . We write the induction hypothesis α s : H a ( m, u, N ) ≡ u B ( u +1) ( m ) , for u ∈ [0 , s ) , m, N ≥ . We may assume α s for fixed s ≥
1. The main step is to show for 0 ≤ t ≤ s that β t,s : v ( s, t, m ) H a ( m, s, N + m s ) ≡ s − t − X j =0 B ( t +1) ( j + m s − t ) H a ( j, t, N ) /B ( t +1) ( j ) mod 2 s B ( s +1) ( m ) . where v ( s, t, m ) = 1 or − s, t, m .We list some elementary facts (cf. [43] 2.5-2.7) P Tm =0 H a ( m, s, N ) = 0 if ( T + 1)2 s > N (A.13.3) H a ( m, s, N ) = H a (2 m, s − , N ) + H a (1 + 2 m, s − , N ) if s ≥ B ( t ) ( i + m s ) ≡ B ( s + t ) ( m ) if 0 ≤ i ≤ s − , s, t ≥ . (A.13.5)We first prove β ,s . We have H a ( m, s, N + m s ) = P s − j =0 U a ( j + m s , N + m s ) ,U a ( j + m s , N + m s ) = A ( a + 2( N − j )) B ( j + m s ) − B ( N − j ) A (cid:0) a + 2 j + m s +1 (cid:1) . (A.13.6) By (A.13.1), we have A (cid:0) a + 2 j + m s +1 (cid:1) = A ( a + 2 j ) B ( j + m s ) /B ( j ) + X j s B ( j + m s ) , where X j ∈ R . Then the right hand side of (A.13.6) is B ( j + m s ) (cid:18) U a ( j, N ) /B ( j ) − s X j B ( N − j ) (cid:19) . Since U a ( j, N ) = H a ( j, , N ), we obtain H a ( m, s, N + m s ) = s − X j =0 B ( j + m s ) H a ( j, , N ) /B ( j ) − s s − X j =0 X j B ( j + m s ) B ( N − j ) . Since X j B ( N − j ) ∈ R , it follows from (A.13.5) ( B = B (1) ) that the second sum is congruent to zero modulo2 s B ( s +1) ( m ). This proves β ,s with v ( s, , m ) = 1.With s fixed, s ≥ t fixed, 0 ≤ t ≤ s −
1, we show that β t,s together with α s imply β t +1 ,s . To do thiswe put j = µ + 2 i in the right side of β t,s and write it in the form X µ =0 2 s − t − X i =0 B ( t +1) (cid:0) µ + 2 i + m s − i (cid:1) H a ( µ + 2 i, t, N ) /B ( t +1) ( µ + 2 i ) . By condition (c’), we have, B ( t +1) (cid:0) µ + 2 i + m s − t (cid:1) = u ( t + 1 , s − t − , m ) (cid:16) B ( t +1) ( µ + 2 i ) B ( t +2) (cid:0) i + m s − t − (cid:1) /B ( t +2) ( i ) (cid:17) + X i,µ s − t B ( t +2) (cid:0) i + m s − t − (cid:1) , where X i,µ ∈ R . Thus the general term in the above double sum is u ( t + 1 , s − t − , m ) (cid:18) B ( t +2) ( i + m s − t − ) H a ( µ + 2 i, t, N ) /B ( t +2) ( i ) (cid:19) + Y i,µ , where the error term: Y i,µ = X i,µ s − t B ( t +2) (cid:0) i + m s − t − (cid:1) H a ( µ + 2 i, t, N ) /B ( t +1) ( µ + 2 i ) . For this error term, since t < s , we can apply α s to conclude that Y i,µ ≡ B ( t +2) ( i + m s − t − )2 s . Then we can use (A.13.5) to conclude that Y i,µ ≡ s B ( s +1) ( m ) . After modulo 2 s B ( s +1) ( m ), the right side of β t,s is equal to u ( t + 1 , s − t − , m ) X µ =0 2 s − t − − X i =0 B ( t +2) (cid:0) i + m s − t − (cid:1) H a ( µ + 2 i, t, N ) /B ( t +2) ( i ) . By reversing the order of summation and using (A.13.4), the above sum is the same as u ( t + 1 , s − t − , m ) s − t − − X i =0 B ( t +2) (cid:0) i + m s − t − (cid:1) H a ( i, t + 1 , N ) /B ( t +2) ( i ) , which proves β t +1 ,s . In particular, we obtain β s,s , which states(A.13.7) v ( s, s, m ) H a ( m, s, N + m s ) ≡ B ( s +1) ( m ) H a (0 , s, N ) /B ( s +1) (0) mod 2 s B ( s +1) ( m ) . We now consider the statement (with s fixed before) γ N : H a (0 , s, N ) ≡ s . ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 69 We know that γ N is true for N <
0. Let N ′ (if it exists) be the minimal value of N for which γ N ′ fails. For m ≥
1, since B ( s +1) (0) is a unit, we have by (A.13.7) H a ( m, s, N ′ ) ≡ v ( s, s, m ) B ( s +1) ( m ) H a (0 , s, N ′ − m s ) /B ( s +1) (0) ≡ s . Applying this to (A.13.3), we obtain that H a (0 , s, N ′ ) ≡ s . Thus γ N is valid for all N and equation (A.13.7) implies α s +1 . This proves assertion (i’). (cid:3) References [1] T. Abe,
Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic D -modules ,Rend. Sem. Math. Univ. Padova 131, p.89–149 (2014).[2] T. Abe, Langlands program for p -adic coefficients and the petits camarades conjecture , J. Reine Angew. Math. 734 (2018),59–69.[3] T. Abe, Langlands correspondence for isocrystals and existence of crystalline companion for curves , J. Amer. Math. Soc.31 (2018), 921–1057.[4] T. Abe,
Around the nearby cycle functor for arithmetic D -modules , preprint, https://arxiv.org/abs/1805.00153 (2018).[5] T. Abe, D. Caro, On Beilinson’s equivalence for p -adic cohomology , Sel. Math. New Ser. (2018) 24: 591–608.[6] T. Abe, D. Caro, Theory of weights in p -adic cohomology , American Journal of Mathematics, Vol. 140, Number 4, pp.879–975.[7] T. Abe and H. Esnault, A Lefschetz theorem for overconvergent isocrystals with Frobenius structure , Ann. Sci. Éc. Norm.Supér. (to appear), http://arxiv.org/abs/1607.07112.[8] A. Adolphson and S. Sperber,
Exponential sums and Newton polyhedra: Cohomology and estimates , Ann. Math., 130(1989), pp. 367–406.[9] Y. André,
Représentations galoisiennes et opérateurs de Bessel p -adiques , Ann. Inst. Fourier (Grenoble) 52 (2002) no. 3,pp. 779–808.[10] Y. André, Filtrations de type Hasse-Arf et monodromie p -adique , Invent. Math. 148 (2002), 285–317.[11] Y. André, F. Baldassarri, De Rham cohomology of differential modules on algebraic varieties . Progress in Mathematics,189. Birkhäuser Verlag, Basel, 2001. viii+214 pp.[12] P. Baumann and S. Riche,
Notes on the geometric Satake equivalence . In: Relative Aspects in Representation Theory,Langlands Functoriality and Automorphic Forms (CIRM Jean-Morlet Chair, Spring 2016), pp. 1–134, Lecture Notes inMath., vol. 2221, Springer, Cham, 2018[13] F. Baldassarri,
Differential modules and singular points of p -adic differential equations , Advances in Math, Tome 44(1982), pp. 155-179[14] F. Baldassarri, P. Berthelot, On Dwork cohomology for singular hypersurfaces , In Geometric Aspects of Dwork Theory(Ed. A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz, F. Loeser), vol. I, 177–244, De Gruyter (2004).[15] A. Beilinson,
How to glue perverse sheaves , K-theory, arithmetic and geometry (Moscow, 1984), Lecture Notes in Math.,vol. 1289, Springer, Berlin, 1987, pp. 42–51[16] A. Beilinson, J. Bernstein, P. Deligne, O. Gabber,
Faisceaux pervers , Astérisque 100 (1982).[17] A. Beilinson and V. Drinfeld,
Quantization of Hitchin’s integrable system and Hecke eigensheaves
Cohomologie rigide et théorie de Dwork: le cas des sommes exponentielles , in P -adic cohomology. AstérisqueNo. 119-120 (1984), 3, 17–49.[19] P. Berthelot, Géométrie rigide et cohomologie des variétés algébriques de caractéristique p , Mem. Soc. Math. France no.23 (1986), 7–32.[20] P. Berthelot, Cohomologie rigide et cohomologie ridige à supports propres, première partie , preprint (1996).[21] P. Berthelot, D -modules arithmétiques. I. Opérateurs différentiels de niveau fini , Ann. Sci. École Norm. Sup. (4) 29(1996), no. 2, 18–272.[22] P. Berthelot, Introduction à la théorie arithmétique des D -modules , In Cohomologies p -adiques et applications arithmé-tiques II (Ed. P. Berthelot, J.-M. Fontaine, L. Illusie, K. Kato, M. Rapoport), Astérique 279 (2002), 1–80.[23] P. Berthelot, A. Ogus, Notes on Crystalline Cohomology (MN-21). Princeton University Press, (2015).[24] T. Braden,
Hyperbolic localization of Intersection Cohomology , Transformation Groups 8 (2003), no. 3, 209–216.[25] A. Braverman, D. Gaitsgory,
Geometric Eisenstein series , Invent. math. 150 (2002), 287–384.[26] L. Carlitz,
A note on exponential sums , Pacific J. Math. 30 (1969), 35-37.[27] D. Caro,
Fonctions L associées aux D -modules arithmétiques. Cas des courbes , Compositio Math.,142 (2006), 169–206.[28] D. Caro, D -modules arithmétiques surholonomes. Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 1, 141–192.[29] D. Caro et N. Tsuzuki
Overholonomicity of overconvergent F -isocrystals over smooth varieties , Annals of Math. 176(2012), 747-813. [30] R. Cass, Perverse F p sheaves on the affine Grassmannian , preprint, https://arxiv.org/abs/1910.03377 (2019).[31] B. Chiarellotto and N. Tsuzuki, Logarithmic growth and Frobenius filtrations for solutions of p -adic differential equations ,J. Inst. Math. Jussieu 8 (2009), no. 3, 465–505.[32] G. Christol, Z. Mebkhout, Sur le théorème de l’indice des équations différentielles p-adiques III , Ann. of Maths 151(2000), 385–457[33] R. Crew,
Specialization of crystalline cohomology . Duke Math. J., Volume 53, Number 3 (1986), 749–757.[34] R. Crew, F -isocrystals and their monodromy groups . Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 4, 429–464.[35] R. Crew, Kloosterman sums and monodromy of a p -adic hypergeometric equation . Compositio Math. 91 (1994), no. 1,1–36.[36] M. D’Addezio, The monodromy groups of lisse sheaves and overconvergent F-isocrystals , preprint,https://arxiv.org/abs/1711.06669 (2017).[37] P. Deligne,
Applications de la formule des traces aux sommes trigonométriques , In: Cohomologie Étale (SGA 4 ), LectureNotes in Math. 569 (1977).[38] P. Deligne, La conjecture de Weil. II . Inst. Hautes Études Sci. Publ. Math. No. 52 (1980), 137–252.[39] P. Deligne,
Catégories tannakiennes . In “The Grothendieck Festschrift”. Modern Birkhäuser Classics. Birkhäuser Boston,2007.[40] P. Deligne and J. S. Milne,
Tannakian categories , Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math-ematics, vol. 900, Springer-Verlag, Berlin-New York, (1982), 101–228.[41] V. Drinfeld, D. Gaitsgory,
On a theorem of Braden . Transform. Groups 19 (2014), no. 2, 313–358.[42] V. Drinfeld, K. Kedlaya,
Slopes of indecomposable F-isocrystals.
Pure Appl. Math. Q. 13 (2017), no. 1, 131–192.[43] B. Dwork, P -adic cycles . Publ. Math. Inst. Hautes Études Sci. 37 (1969), 27–115.[44] B. Dwork, Bessel functions as p-adic functions of the argument , Duke Math. J. Volume 41, Number 4 (1974), 711–738.[45] J.-Y. Étesse, B. Le Stum,
Fonctions L associèes aux F-isocristaux surconvergent I. Interprétation cohomologique , Math.Ann. 296 (1993), 557–576.[46] G. Faltings,
Algebraic loop groups and moduli spaces of bundles , J. Eur. Math. Soc. (JEMS) 5 (2003), no. 1, 4168.[47] E. Frenkel and B. Gross,
A rigid irregular connection on the projective line , Ann. of Math. 170 (2009), 1469–1512.[48] L. Fu and D. Wan,
L-functions for symmetric products of Kloosterman sums . J. Reine Angew. Math. 589, 79–103 (2005).[49] D. Gaitsgory,
Construction of central elements in the affine Hecke algebra via nearby cycles , Invent. Math. 144 (2001),253–280.[50] U. Görtz and T. J. Haines,
The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties ,J. Reine Angew. Math. 609 (2007), 161–213.[51] B. Gross and M. Reeder,
Arithmetic invariants of discrete Langlands parameters , Duke Math. J. 154 (2010), 431–508.[52] J. Heinloth, B-C Ngô and Z. Yun,
Kloosterman sheaves for reductive groups . Ann. of Math. (2) 177 (2013), no. 1, 241–310.[53] C. Huyghe, D † ( ∞ ) -affinité des schémas projectifs . Ann. Inst. Fourier (Grenoble) 48 (1998), no. 4, 913–956.[54] A. Grothendieck, J. Dieudonné, Éléments de Géométrie Algébrique, IV Étude locale des schémas et des morphismes deschémas . Publ. Math. Inst. Hautes Études Sci. 20 (1964), 24 (1965), 28 (1966), 32 (1967).[55] N. Katz,
On the calculation of some differential Galois groups , Invent. Math. 87 (1987), 13–61.[56] N. Katz,
Gauss Sums, Kloosterman Sums, and Monodromy Groups , Ann. Math. Studies 116, Princeton Univ. Press,Princeton, NJ, 1988.[57] N. Katz,
Exponential Sums and Differential Equations , Ann. Math. Studies 124, Princeton Univ. Press, Princeton, NJ,1990.[58] N. Katz,
From Clausen to Carlitz: low-dimensional spin groups and identities among character sums , Mosc. Math. J. 9(2009), no. 1, 57-89.[59] K. Kedlaya, A p -adic local monodromy theorem , Ann. of Math. Vol. 160, No. 1 (2004), pp. 93–184.[60] K. Kedlaya, Semistable reduction for overconvergent F-isocrystals. I. Unipotence and logarithmic extensions . Compos.Math. 143 (2007), no. 5, 1164–1212.[61] K. Kedlaya, p-adic differential equations , Cambridge Studies in Advanced Mathematics, 125. Cambridge University Press,Cambridge, 2010. xviii+380 pp.[62] K. Kedlaya,
Notes on isocrystals . preprint, https://arxiv.org/abs/1606.01321 (2016).[63] V. Lafforgue,
Estimées pour les valuations p -adiques des valeurs propres des opérateurs de Hecke , Bull. Soc. Math. France139 (2011), no. 4, 455–477.[64] T. Lam and N. Templier, The mirror conjecture for minuscule flag varieties , preprint, https://arxiv.org/abs/1705.00758(2017).[65] S. Matsuda,
Katz Correspondence for Quasi-Unipotent Overconvergent Isocrystals , Compositio Math. 134(1), 1-34.[66] Z. Mebkhout,
Analogue p -adique du théorème de Turrittin et le théorème de la monodromie p -adique , Invent. Math.148(2) (2002) 319–351.[67] K. Miyatani, P-adic generalized hypergeometric functions from the viewpoint of arithmetic D-modules , Amer. J. Math.(to appear).
ESSEL F -ISOCRYSTALS FOR REDUCTIVE GROUPS 71 [68] I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings ,Ann. of Math. 166 (2007), 95–143.[69] B. C. Ngô and P. Polo,
Résolutions de Demazure affines et formule de Casselman-Shalika géométrique , J. AlgebraicGeom. 10 (2001), no. 3, 515–547.[70] A. Ogus,
F-isocrystals and de Rham cohomology. II. Convergent isocrystals , Duke Math. J. 51 (1984), no. 4, 765–850.[71] S. Ohkubo,
Logarithmic growth filtrations for ( ϕ, ∇ ) -modules over the bounded Robba ring . preprint,https://arxiv.org/abs/1809.04065 (2018).[72] G. Pappas and X. Zhu, Local models of Shimura varieties and a conjecture of Kottwitz , Invent. Math.194 (2013), no. 1,147–254.[73] C. Pech, K. Rietsch, and L. Williams,
On Landau-Ginzburg models for quadrics and flat sections of Dubrovin connections ,Adv. Math. 300 (2016), 275–319.[74] T. Richarz,
A new approach to the geometric Satake equivalence , Doc. Math. 19 (2014), 209–246.[75] T. Richarz, J. Scholbach,
The motivic Satake equivalence , preprint, https://arxiv.org/abs/1909.08322 (2019).[76] T. Richarz and X. Zhu,
Appendix to [87].[77] A. Shiho,
Crystalline fundamental groups. II. Log convergent cohomology and rigid cohomology . J. Math. Sci. Univ. Tokyo9 (2002), no. 1, 1–163.[78] S. Sperber, p -adic hypergeometric functions and their cohomology , Duke Math. J. 44 (1977), no. 3, 535–589.[79] S. Sperber, Congruence properties of the hyper-Kloosterman sum . Compos. Math. 40 (1980), no. 1, 3–33.[80] H. Sumihiro,
Equivariant completion , J. Math. Kyoto Univ. 14 (1974), 1–28.[81] N. Tsuzuki,
The local index and the Swan conductor , Compos. Math. 111 (1998), 245–288.[82] N. Tsuzuki,
Minimal slope conjecture of F -isocrystals , preprint, https://arxiv.org/abs/1910.03871 (2019).[83] J. Fresnel, M. van der Put, Rigid analytic geometry and its applications . Progress in Mathematics, 218. Birkhäuser Boston,Inc., Boston, MA, 2004. xii+296 pp.[84] D. Wan,
Newton polygons of zeta functions and L-functions , Ann. Math. 137 (1993), 247–293.[85] D. Wan,
Variation of p-adic Newton polygons for L-functions of exponential sums , Asian J. Math. Vol. 8 (2004), No. 3,427–472.[86] X. Zhu,
Affine Demazure modules and T -fixed point subschemes in the affine Grassmannian , Adv. Math. 221 (2009), no.2, 570–600.[87] X. Zhu, The geometric Satake correspondence for ramified groups , Ann. Sci. Éc. Norm. Supér. 48 (2015), 409–451.[88] X. Zhu,
An introduction to affine Grassmannians and the geometric Satake equivalence , IAS/Park City MathematicsSeries, 24. (2017) 59–154.[89] X. Zhu,
Frenkel-Gross’ irregular connection and Heinloth-Ngô-Yun’s are the same . Selecta Math. (N.S.) 23 (2017), no. 1,245–274.[90] X. Zhu,
Geometric Satake, categorical traces, and arithmetic of Shimura varieties . Current Developments in Mathematics,(2016), 145–206.
Daxin Xu, Xinwen Zhu, Department of Mathematics, California Institute of Technology, Pasadena, CA 91125.
E-mail address : [email protected] ,,