Unipotent Nearby Cycles and the Cohomology of Shtukas 3

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Our main theorem is a result of this type for parahoric level structure. Theorem 1.1. Let π be a cuspidal automorphic representation with finite order central character for the split group GpAK q where K is the function field of the smooth connected curve C. Suppose π contains a nonzero vector v P πQ fixed under a group 1 (1.1) Q “ Q Ă GpK q“ GpA q ź x ź x K xPC xPC Q p C I1 I K K where p is a standard parahoric subgroup for fixed P . Let Kp Ă Kp Ă Weilp p{ pq Ă WeilpK{Kq be the wild inertia, inertia, and local Weil group at the place p. (1) (Special case of Theorem 6.4) Let ρ be the WeilpK{Kq representation corresponding to π ρ I1 γ I I1 under V. Lafforgue’s construction. Then p Kp q “ 1 and for P Kp { Kp a topological generator, ρpγqĂ G is unipotent. (2) (Theorem 6.5) Keepingp the above assumptions, if Qp is a standard parahoric containing ` a fixed Iwahori I, and if wP is the longest element of the Weyl group of the Levi Qp{Qp ` where Qp denotes the pro-unipotent radical of Qp, and uP is the unipotent orbit closure corresponding to the two-sided cell containing wP , then ρpγqP uP .

In the “spherical” case when Qp is GpOpq for Op the ring of integers in Kp, it is known that

ρpIKp q“ 1. Recently, C. Xue gave a proof of this result [21] by proving a stronger result: smoothness of cohomology sheaves of shtukas with arbitrarily many legs. Let C be a smooth connected curve over Fq and ChtG,N,I,W be the moduli stack of shtukas for the group G with I legs and I-tuple of representations W P ReppGI q, with level structures given by a divisor N Ă C. Xue’s result I ˚ I implies that the action of WeilpK, Kq on H pChtG,N,I,W , Qℓqq factors through WeilpCzN, ηq . An important part of Xue’s technique uses the fusion structure of the perverse sheaves on ChtG,I together with Zorro’s lemma to construct an inverse to the natural specialization map for the sheaves ˚ j j (1.2) sp : HI,W |s Ñ HI,W |η. Such an inverse map is known automatically if one has the existence of a compactification of ChtG,N,I,W such that the singularities of the compactification are no worse than the singularities of ChtG,N,I,W . Constructing such a compactification seems to be a challenging problem, and a construction which satisfies the desired properties concerning singularities only seems to be available ˚ ˚ in the case G “ GL2, I “ 2, W “ Std b Std [5] and G “ GLn, N “ ∅, I “ 2, W “ Std b Std [11]. A key intermediate result is an analogue of Xue’s smoothness result for the case of parahoric level structure, using unipotent nearby cycles instead of the specialization map. At parahoric level structure, we prove that the local Weil group actions factor through tame inertia, and tame inertia acts unipotently. Analogous to Xue’s result, we use the fusion structure and Zorro’s lemma in a systematic way to construct an inverse to the natural map (the notation will be explained later)

(1.3) can : Rp!ΨJ FIYJ,W bV Ñ ΨJ Rp!FIYJ,V bW . Once again, a standard fact about nearby cycles is that this canonical map is an isomorphism if p is proper, and we use similar techniques to those of Xue in order to avoid the need to construct compactiﬁcations. UNIPOTENT NEARBY CYCLES AND THE COHOMOLOGY OF SHTUKAS 3

To conclude useful results about the cohomology of moduli spaces of shtukas, we need integral models, at least at the parahoric level. Such integral models and their local models are available in large generality by combining the results of [16] and [1]. For this reason, for the purpose of the paper, we do not assume that G is a reductive group, but only that it is a smooth group scheme over a smooth projective curve C over a finite field Fq, and that it has reductive generic fiber. The moduli stack of shtukas ChtG will automatically incorporate the level structures, constructed in general via dilatation. An overview of this work will occupy section 2. In section 3 we are interested in the analogue of Xue’s constancy results of étale sheaves on the product of the generic fiber of a curve. Such a result is key for proving smoothness results and uses the Eichler-Shimura relation in a central way. Indeed, most of the section is dedicated to writing down a general context in which the Eichler-Shimura relation holds. In place of the fusion structure of the affine Grassmannian, we use a fusion property of nearby cycles, first identified by Gaitsgory in the proof that the nearby cycles functor is a central functor from the spherical Hecke category to the Iwahori Hecke category [8]. In section 4, we explain Gaitsgory’s construction and subsequent generalizations to prove the desired fusion property on nearby cycles over the moduli stack of shtukas. This is used in section 5 to prove cases where nearby cycles commutes with the pushforward Rp!. Finally, in section 6, we draw conclusions about the Langlands correspondence and prove our main result. The result of two-sided cells uses the work of Lusztig and Bezrukavnikov to draw conclusions about nearby cycles on the Beilinson-Drinfeld Grassmannian at parahoric level, the local model to draw analogous conclusions about nearby cycles on the stacks of shtukas, and the framed Langlands parameters of Lafforgue and Zhu to draw conclusions about the Langlands correspondence and the image of the tame generator.

Acknowledgements. I would like to thank my advisor, Zhiwei Yun, for introducing me to shtukas and for many discussions and suggestions about this document. I would like to thank Cong Xue for encouraging me to pursue questions of nearby cycles at the parahoric level and for making many helpful comments on this paper.

2. Moduli spaces of parahoric shtukas We introduce some standard notation for shtukas, first developing our maps in the context of a smooth affine group scheme over a curve, then specializing to Néron blow-ups of reductive groups to show how these maps allow us to produce perverse sheaves on our stacks of shtukas [16]. We fix a base field Fq and a smooth connected projective curve C over that base field. For G be a smooth affine group scheme over a curve, we can form the stack BunG of G-torsors. We also define the Hecke stack and the Beilinson-Drinfeld Grassmannian. Definition 2.1. Let I be a fixed finite set and G be a smooth affine group scheme over C. For a scheme S over Fq, the ind-stack HeckeG,I pSq classifies the data of

(1) Two bundles E1 and E2 in BunGpSq, I (2) A collection of points pxiqiPI P C pSq, E E (3) An isomorphism τ : 1|Cz Γx » 2|Cz Γx away from the graphs of xi. Ťi i Ťi i Definition 2.2. The global (Beilinson-Drinfeld) affine Grassmannian is the fiber of the map HeckeG,I Ñ BunG sending the tuple ppxiqiPI , E1, E2, τq to E2 over the trivial bundle.

Moreover, we can consider the moduli stack of global G-shtukas, which we denote ChtG,I . 4 ANDREW SALMON

Deﬁnition 2.3. The moduli stack of global G-shtukas, ChtG,I , is deﬁned by the pullback square:

ChtG,I HeckeG,I (2.1)

BunG BunG ˆ BunG Let us explain what the arrows are. The lower arrow is the graph of Frobenius pullback that on ˚ S-points sends a bundle E on C ˆ S to pE, p1 ˆ FrobSq Eq as a pair of G-bundles on C ˆ S. The vertical arrow is the forgetful map that on S-points sends a tuple ppxiqiPI , E1, E2, τq to pE1, E2q. ˚ τ We will adopt the convention of denoting p1 ˆ FrobSq E as E. Let us now be more specific about the sorts of smooth affine group schemes that we are talking about. We consider groups G over our curve C which have generic fiber a quasi-split reductive group GK . Following Lafforgue [12, Section 12], there is an open U Ă C such that GK extends to a smooth affine group scheme over C that is a reductive group scheme over U with parahoric reduction at CzU. Let R be the complement CzU. Next, we can modify the group by dilatation. Let N be an effective divisor, not necessarily reduced, and let H Ă G|N be a closed, smooth group subscheme of the restriction of G to the divisor N. The construction of [16] produces a smooth group scheme G over C by dilatation such that ChtG is an integral model of shtukas for the quasi- split group with level structure according to H along the divisor N. Let us fix some divisor N and let U be UzN so that the complement of U is N :“ N Y R. Let P be the subset of N where the dilatatedp group G has parahoric reduction.p p p After dilatation, we have produced a smooth affine group scheme G such that GpAq is the adeles of our quasi-split group, which contains an open subgroup GpOq corresponding to the level structures H along N. Let Z be the center of G. To make our moduli spaces of shtukas finite type, we need to quotient by the action of a discrete group Ξ, which is a fixed lattice in ZpAq such that Ξ X ZpOqZpF q“t1u. This acts on ChtG such that the quotient ChtG {Ξ has finitely many components. Moreover, this action can be made compatible with Harder-Narasimhan truncation, ďµ and after truncation, the stacks ChtG,I,W {Ξ are of finite type. We now want to construct sheaves over these finite type stacks. We can consider a global I version of the positive loop group that we write GI,8, living over C . There is an important map ǫG : ChtG,I Ñ rGI,8z GrG,I s. This follows as a straightforward generalization of [19, Definition 1.1.13]. Definition 2.4. For a divisor D, we can view the divisor as a subscheme (with non-reduced structure around points with multiplicity) and pull back the trivial G-bundle to D. We let GI,d classify tuples ppxiqiPI ,gq where xi P CpSq gives a tuple of points and g gives an automorphism of the trivial bundle on the graph of dxi. Let G “ lim G ř. We have a map I,8 ÐÝd I,d (2.2) ǫG : ChtG,I ÑrGI,8z GrG,I s defined by starting with the isomorphism E τ E (2.3) τ : |Cz Γx Ñ |Cz Γx , Ťi i Ťi i τ restricting it around formal neighborhoods of pxiqiPI , and forgetting the relationship of E and E. Since every G-bundle restricted to formal discs is trivial, rGI,8z GrG,I s classifies tuples ppxiqiPI , E, F, τq where E and F are GI,8-torsors around the points pxiq and τ is an isomorphism along the punctured formal disks around pxiq. UNIPOTENT NEARBY CYCLES AND THE COHOMOLOGY OF SHTUKAS 5

Given a lattice Ξ in ZpF qzZpAq as above, ǫG is Ξ-equivariant and factors through Ξ ad (2.4) ǫG : ChtG,I {Ξ ÑrGI,8z GrG,I s. I We view ChtG,I {Ξ and all of its substacks as living over X with respect to a structure map p. We note that the map p often fails to be proper. I Definition 2.5. Let p : ChtG,I {Ξ Ñ X send the tuple ppxiqiPI , E, τq to pxiqiPI . Geometric Satake for ramified groups gives a functor L I (2.5) Repp G q Ñ PervGI,8 pGrG,I |U I q, p and we denote the sheaves produced by SI,W . We can pull back these equivariant sheaves along Ξ the map ǫG to sheaves on ChtG,I |U I . The details of the pullback and compatibility with Ξ are explained in more detail in [19, Definition 2.4.7]. We denote these sheaves by FI,W and we denote the Zariski closures of the supports of these sheaves as ChtG,Ξ,I,W . Similarly, we denote the Zariski closures of the supports of the sheaves SI,W by GrG,I,W . We now recall the local models theorem, which is [1, Theorem 3.2.1]. We state for the stacks ChtG,Ξ,I,W and GrG,I,W , as the stratification by W form a bound in the sense of [1, Definition 3.1.3].

Theorem 2.6. For every geometric point y of ChtG,I,W there is an etale neighborhood Uy such that

(2.6) ChtG,Ξ,I,W Uy GrG,I,W is a diagram with both arrows ´etale. Proof. The statement with Ξ follows by checking that the closures of strata by W form a bound in the sense of [1, Theorem 3.2.1], which is true because the sheaves produced by geometric Satake are equivariant and their supports are stable under the action of GI,8. So it suﬃces to check that the left and right arrow can be made Ξ-equivariant. But the left arrow is a cover coming from adding level structures, and such covers are Ξ-equivariant, and the right arrow is a version of ǫG.

The sheaves SI,W and FI,W have well-known fusion properties. In [12], V. Laﬀorgue constructed the automorphic to Galois direction of the global Langlands correspondence by using the fusion and partial Frobenius structure in the moduli stack of shtukas. Recently, C. Xue proved that the I sheaves Rp!FI,W form an ind-local system, and not just an ind-constructible sheaf, on U . Our main result extends these ideas to describe how the local Galois groups act at points p P CzU when G has parahoric reduction at p. To do this, we will use nearby cycles along the base curves in various directions i P I. Our notational convention will be to suppress the fact that Ψi “ RΨi is a derived functor and also suppress similar notation for tensor products b“bL.

3. Excursion operators on nearby cycles Let FinS be the category of finite sets with set maps as morphisms. We will restate some of the general theory in terms of objects that exist over arbitrarily many legs at once. By sending I to CI , we get a contravariant functor C‚ from sets to schemes. For a group G instead of C, we get a contravariant functor G‚ from sets to algebraic groups that makes ReppGp‚q into a category cofibered over FinS. This categoryp has objects representations pairs pI, W q wherep I is a finite set and W is a direct sum of I-tuples of representations of G in finite dimensional E-vector spaces. Morphisms in this category are generated by two types ofp arrows. I-tuples of homomorphisms of 6 ANDREW SALMON

G-representations, W Ñ V , give arrows pI, W q Ñ pI, V q. A morphism of finite sets ζ : I Ñ J gives ap fusion map pI, W q Ñ pJ, W ζ q where W ζ gives tensor products over the finite fibers of the map ζ where the empty tensor product yields the trivial representation 1. The functor that forgets the representation data and just remembers the finite set shows that ReppG‚q is a cofibered category over FinS and the fusion maps are precisely the cocartesian arrows. p b ‚ Let us describe another category cofibered over FinS, which we call DcpX q for a variety X over Fq. Objects in this category are pairs pI, Lq of a finite set I and an object in the bounded b I derived category of constructible sheaves L in DcpX q. Morphisms in this category are generated b I ˚ J I by morphisms in DcpX q for all I and fusion morphisms pI, Lq Ñ pJ, ∆ζ Lq where ∆ζ : X Ñ X is induced by ζ : I Ñ J. The fusion morphisms are the cocartesian arrows in the cofibered category b ‚ DcpX q. Geometric Satake produces a full subcategory of PervG8,‚ pGrG,‚ |CzN q that is equivalent to ‚ x ReppG q as a category cofibered over FinS. The essential image consists of the sheaves SI,W I ďµ,Ξ rangingp over finite sets I and W P ReppG q. Pulling back under ǫG produces perverse sheaves ďµ FI,W on ChtG,I |pXzNqI {Ξ as above, compatiblep under specialization to diagonals and thus a full x ďµ subcategory of perverse sheaves on ChtG,‚ |pCzNq‚ {Ξ cofibered over FinS generated by the sheaves FI,W . Taking the proper pushfoward Rp!FI,Wxgives a cocartesian functor

ďµ ‚ b ‚ H‚ : ReppG q Ñ DcppCzNq q p p between categories coﬁbered over FinS. This description includes both the functoriality of H in the representation W and also the fusion morphisms, which, by the fact that Hďµ is cocartesian, must be isomorphisms Hďµ ˚Hďµ J,W ζ – ∆ζ I,W . We also note that in addition to this structure described above we have commuting partial ďµ ďµ`κ Frobenius maps HI,W Ñ HI,W . It is to such a context that we want to abstract the formation of excursion operators, the Eichler-Shimura relation [12, Section 7], and the smoothness results of [21, Section 1]. Ultimately, we will apply our results to sheaves of the form

Rp!Ψj1 ¨ ¨ ¨ Ψjn FIYJ,W bV for J “tj1,...,jnu. Let µ take values in a commutative monoid S with order such that 0 is minimal and it is a directed poset. In the cases we consider, µ takes values in dominant coroots and corresponds to the Harder-Narasimhan ﬁltration.

Deﬁnition 3.1. With the above conventions on µ, an S-ﬁltered collection of functors C1 Ñ C2 is a functor from the poset of S to the category of functors from C1 to C2.

b ‚ In the cases we consider, C2 is DcpX q, S is the monoid of dominant coroots of our reductive group G, and we will suppress the dependence on dominant coroots. For such ﬁltered collections ďµ ďµ of functors G , lim G will a functor from C1 to IndpC2q. Since the functors we consider are ÝÑµPS a only ind-constructible sheaves, we need a tool to get control over their essential image so we can treat them as if they were constructible. Such control will be given by the Eichler-Shimura relation.

ďµ ‚ b ‚ Definition 3.2. Let G be a filtered collection of cocartesian functors from ReppG q Ñ DcpX q cofibered over FinS, and let v be a closed point of X. We say that Gďµ is filteredp with respect to UNIPOTENT NEARBY CYCLES AND THE COHOMOLOGY OF SHTUKAS 7 partial Frobenius maps F‚ if for κ sufficiently large only depending on the isomorphism class of Wi P ReppGq we have maps of sheaves À Âi p ˚ Gďµ Gďµ`κ (3.1) FI0 : FrobI0 I,W Ñ I,W that are natural transformations that commute with each other for i P I and such that FI0 together with the canonical map

ďµ`κ ďµ`|I0|κ (3.2) GI,W Ñ GI,W is F . In particular, they should be compatible with the fusion isomorphisms. We denote iPI0 tiu ś ´1 FI0 “ Ftiu. If ζ : I Ñ J, and J0 Ă J such that I0 “ ζ pJ0q, we want to demand that śiPI0 ∆˚F ˚ ˚ Gďµ ζ I0 ˚Gďµ`κ ∆ζ FrI0 I,W ∆ζ I,W

˚ (3.3) Fr fusζ fusζ J0 can˝F ˚ Gďµ J0 Gďµ`κ. FrJ0 J,W ζ J,W ζ

Finally, we demand that in the case |I|“ 1, that FI factors as the composition of the action of Frobenius ˚ ďµ ďµ (3.4) Frob GI,W Ñ GI,W and the canonical map ďµ ďµ`κ (3.5) GI,W Ñ GI,W . Now suppose the base X is a dense open subset of a smooth projective curve C. We note that passing to the limit lim Gďµ and taking the stalk at a geometric generic point, our assumptions ÝÑn show that the maps Fi give rise to an action of

F WeilpηI , ηI q I which is an extension of the kernel of GalpFI {FI q Ñ Z by Z [12, Remarque 8.18]. By Drinfeld’s lemma, modules finitely generated under the action ofp some algebra commuting with partial Frobe- nius factor through Weilpη, ηqI . We would like to show that this factorization occurs over the whole lim Gďµ. This is the main result of the section, and the proof consists of applying the proof of ÝÑ [21, Proposition 1.4.3]. We note that everything in the proof goes through as long as we have an analogue of the Eichler-Shimura relation. The majority of the remaining section will be devoted to establishing this result. For this to be a useful definition, we will need an example of such a collection other than the ďµ proper pushforward HI,W used by Lafforgue. Proposition 3.3. As always, let G be a smooth group scheme whose generic fiber is reductive, ďµ and let p P C be a closed point of parahoric reduction. Consider the sheaves FIYK,W bV on ďµ ChtG,Ξ,IYK,W bV |pCzNqI as constructed above. Let K “ tk1,...,knu and consider iterated nearby ďxµ cycles Ψk1 ¨ ¨ ¨ Ψkn FIYK,W bV , taking the nearby cycle Ψki with respect to the coordinate ki P K along pCzNqIYK with special point a geometric point above p. ‚ b ‚ Considerp the functors ReppG q Ñ DcppCzNq q defined by p p ďµ Rp!Ψk1 ¨ ¨ ¨ Ψkn FIYK,W bV . 8 ANDREW SALMON

These functors are cocartesian functors between categories coﬁbered over FinS and are ﬁltered with respect to partial Frobenius maps Ftiu for i P I.

Proof. The functoriality with respect to maps pI, W q Ñ pI, V q for W Ñ V a morphism in the I ďµ categories ReppG q living above the identity 1I follows immediately from the functoriality of F , nearby cycles, andp proper pushforward. So to prove that our functor exists and is cocartesian, it suﬃces to check the fusion property, namely that for ζ : I Ñ J, the cocartesian morphisms pI, W q Ñ pJ, W ζ q are sent to cocartesian morphisms

˚ ďµ ďµ ∆ζ Rp!Ψk1 ¨ ¨ ¨ Ψkn FIYK,W bV – Rp!Ψk1 ¨ ¨ ¨ Ψkn FJYK,W ζ bV .

Note that we always have a canonical map going in one direction

˚ ďµ ˚ ďµ ďµ ∆ζ Rp!Ψk1 ¨ ¨ ¨ Ψkn FIYK,W bV Ñ Rp!Ψk1 ¨ ¨ ¨ Ψkn ∆ζ FIYK,W bV – Rp!Ψk1 ¨ ¨ ¨ Ψkn FJYK,W ζ bV . by proper base change. By the local models theorem, it suﬃces to construct the corresponding isomorphisms on the aﬃne Grassmannian,

˚ S S ζ (3.6) ∆ζ Ψk1 ¨ ¨ ¨ Ψkn IYK,W bV – Ψk1 ¨ ¨ ¨ Ψkn JYK,W bV .

This, in turn, follows by smooth pullback, restricting to the locus xk ‰ xi for all k P K, i P I, which reduces the isomorphism to a more classical fusion property on the aﬃne Grassmannian

˚ S b S S ζ b S ∆ζ I,W Ψk1 ¨ ¨ ¨ Ψkn K,V – J,W Ψk1 ¨ ¨ ¨ Ψkn K,V as sheaves on GrG,J ˆFlp. Now we want to show that

ďµ Rp!Ψk1 ¨ ¨ ¨ Ψkn FIYK,W bV are ﬁltered by partial Frobenius. This results from the fact that partial Frobenius maps exist in our setting, and they are local homeomorphisms. Therefore they commute with nearby cycles. So we get an isomorphism

˚ pI1,...,In,Kq ďµ`κ ďµ (3.7) Fr Ψk ¨ ¨ ¨ Ψk F – Ψk ¨ ¨ ¨ Ψk F . ´ I1 ¯ ´ 1 n IYK,JbW ¯ 1 n IYK,JbW

Now, using the above isomorphism, the same cohomological correspondence as in [12, Section 4.3] shows that the sheaves in our setting are ﬁltered by partial Frobenius.

The application of the above proposition will be to show that these sheaves satisfy an analogue of the Eichler-Shimura relations, which in turn will be used to prove a smoothness property their cohomology sheaves are smooth via a Drinfeld lemma argument.

Definition 3.4 (Generalized S-excursion operators). Let Gďµ be a collection of cocartesian functors ‚ b ‚ from ReppG q Ñ DcpX q cofibered over FinS and filtered with respect to partial Frobenius F‚. Let v be a closedp point of X. We define the S-excursion operator as the composition of creation, F on UNIPOTENT NEARBY CYCLES AND THE COHOMOLOGY OF SHTUKAS 9 a created leg, and annihilation.

ďµ ďµ G G I I,W – IYt0u,W b1|X ˆv ďµ G I Ñ IYt0u,W bpV bV ˚q|X ˆv ďµ G I – IYt1,2u,W bV bV ˚ |X ˆvˆv ďµ`κ G I ÑF1 IYt1,2u,W bV bV ˚ |X ˆvˆv ďµ`κ G I – IYt0u,W bpV bV ˚q|X ˆv ďµ`κ G I Ñ IYt0u,W b1|X ˆv ďµ`κ – GI,W where ÑF1 denotes the action of the operator F1 along the leg 1. We thus write SV,v for this operator, suppressing the dependence on µ, I, and W .

Theorem 3.5 (Eichler-Shimura). Let Gďµ be a collection of cocartesian functors from ReppG‚q Ñ b ‚ DcpX q coﬁbered over FinS and ﬁltered with respect to partial Frobenius maps F‚. Then, p

dim V i degpvqi (3.8) p´1q F ˝ S dim V ´i “ 0 ÿ 0 ^ V,v i“0

ďµ ďµ`κ G I G I as a map from IYt0u,W bV |X ˆv Ñ IYt0u,W bV |X ˆv. This proof is essentially contained in [12, Section 7] and the proof we give here involves essentially no new ideas. For the reader’s convenience, we review the argument almost verbatim from [12]. As a warm-up, we review Lafforgue’s point of view on the Cayley-Hamilton theorem. Before proceeding to the proof, we first make some definitions. In what follows, let J be a finite set and V a finite-dimensional vector space over E. For a finite set I, let V bI be the tensor power V b¨¨¨b V . If T P EndpV q and U P EndpV bt0uYJ q. ˚ ˚ We denote by δV the creation map 1 Ñ V b V and evV the annihilation V b V Ñ 1. We define CJ pT,Uq as the composition

1bδbJ VV V b pV b V ˚qbJ

U 1 V bt0uYJ b pV ˚qbJ b V bt0uYJ b pV ˚qbJ (3.9)

1 T bJ 1 V b V bJ b pV ˚qbJ b b V b V bJ b pV ˚qbJ

1bevbJ V b pV b V ˚qbJ V V.

At the end we are left with CJ pT,Uq as an endomorphism of V . CJ is linear in the argument U. We apply this in the speciﬁc case that the endomorphism U is the antisymmetrizer. Recall that if SI denotes the group of permutations of I, and spσq is the sign of a permutation σ P SI , then 10 ANDREW SALMON the antisymmetrizer AI is 1 A “ spσqσ. I |I|! ÿ σPSI

bI The antisymmetrizer naturally lives in the group algebra QrSI s which acts naturally on V for V a vector space. We also refer to the antisymmetrizer AI as the image of this element under the map to EndpV bI q. Laﬀorgue shows that

|J| 1 C pT, A q“ p´1qiTrp^níV qT i J t0uYJ n ` 1 ÿ i“0 which can be thought of as a generalization of the Cayley-Hamilton theorem, as the antisymmetrizer is the endomorphism 0 if |J|“ dim V . The Cayley-Hamilton theorem is a special case of the Eichler-Shimura relation for the functor ‚ b ‚ ReppN q Ñ DcpX q defined by the rule that it sends a tuple of vector spaces pt1,...,nu, pW1,...,Wnqq n to the constant sheaf over X which is an external tensor product W1 b ¨ ¨ ¨ b Wn. Here, partial Frobenius acts trivially as the sheaves are constant, and linear maps T can be represented by the endomorphism 1 P N in each coordinate. Under these identifications, the excursion operator becomes related to the trace. Moreover, the method of proof of the Cayley-Hamilton theorem above will generalize to the general case. The key subtleties are that we use the fusion isomorphisms to connect each of the lines in Equation 3.9. In what follows that the fact that Gďµ is cofibered over FinS implies that we have fusion isomorphisms

˚Gďµ Gďµ (3.10) fusζ : ∆ζ I,W Ñ J,W ζ with a natural compatibility with F‚. ďµ 7 5 The functoriality of G induces creation maps CV and annihilation maps CV as

7 Gďµ Gďµ (3.11) CV : IYt0u,W bW 1 Ñ IYt0u,W bpW 1ˆV ˆV ˚q

5 Gďµ Gďµ (3.12) CV : IYt0u,W bpW 1ˆV ˆV ˚q Ñ IYt0u,W bW 1 of sheaves over XIYt0u. For a morphism u : V Ñ W in ReppGI q, we denote p ďµ ďµ ďµ (3.13) G puq : GI,V Ñ GI,W the morphism induced by functoriality. We have now deﬁned the basic ingredients we need for our proof.

ďµ I Proof. Let us define the generalization of CJ to G . We fix I be a finite set and W P ReppG q. We also fix an elements 0, 1, 2 R I Y J and V P ReppGq. For a map U : V bt0uYJ Ñ V bt0uYJp of ďµ G-representations, let us write CJ pG ,Uq for the compositionp of the four maps below. The maps belowp are maps of sheaves on XI ˆ ∆pvq where ∆pvq is the diagonal inclusion of the closed point UNIPOTENT NEARBY CYCLES AND THE COHOMOLOGY OF SHTUKAS 11 v in the remaining copies of X.

C 7 Gďµ V bJ Gďµ IYt0u,W bV IYt0,1,2u,W bV bV bJ bpV ˚qbJ

Gďµ b b Gďµ p1 U 1q Gďµ IYt0uYt2u,W bV bt0uYJ bpV ˚qbJ IYt0uYt2u,W bV bt0uYJ bpV ˚qbJ (3.14) F degpvq Gďµ śjPJ tju Gďµ`κ IYt0uYJYt2u,W bV bt0uYJ bpV ˚qbJ IYt0uYJYt2u,W bV bt0uYJ bpV ˚qbJ

C 5 Gďµ`κ V bJ Gďµ`κ IYt0,1,2u,W bV bV bJ bpV ˚qbJ IYt0u,W bV .

All the lines are connected by fusion isomorphisms, and κ is chosen suﬃciently large so that the map jPJ Ftju is well deﬁned. ś ďµ Now consider the identity CJ pG , p|J|` 1qAJ q“ 0 for J “t1,..., dim V u. We will express the left hand side as the left hand side in the Eichler-Shimura relation 3.8. For σ P St0uYJ , we denote ℓpσ, 0q the length of the longest cycle containing 0. We rewrite our equation as

dim V 1 C ¨Gďµ, spσqσ˛ “ 0. ÿ J |J|! ÿ i“0 ˝ ℓpσ,0q“i`1 ‚ We note that by symmetry and counting possible permutations,

1 ¨ 1 ˛ C ¨Gďµ, spσqσ˛ “ C Gďµ, spσqσ J dim V ! ÿ J pdim V ´ iq! ÿ ˝ ℓpσ,0q“i`1 ‚ ˚ σPSris ‹ ˝ t0uYJ ‚

ris where St0uYJ denotes the set of permutations such that σpjq “ j ` 1 for 0 ď j ă i and σpiq “ 0. The terms σ in the sum now all factor as σ “ p0 ...iqτ and the ith term is now

¨ ďµ i 1 ˛ i ďµ CJ G , p´1q p0 ...iq spτqτ “ p´1q CJ G , p0 ...iqA . pdim V ´ iq! ÿ ` ti`1,...,nu˘ ˝ τPSti`1,...,nu ‚ bti`1,...,nu dim V í Ati`1,...,nu acts by an idempotent on V projecting onto the subspace ^ V . The ďµ cycle p0 ...iq and Ati`1,...,nu are disjoint and so the operator CJ pG , p0 ...iqAti`1,...,nuq splits as ďµ the composition of Ct1,...,iupG , p0 ...iqq and S^dim V íV,v. So it suffices to show that Gďµ degpvqi Ct1,...,iup , p0 ...iqq “ Ft0u,v . This will follow by induction on i, where the base case i “ 0 is trivial. We can write the left hand operation as creation in pairs p1, 2q¨¨¨p2n ´ 1, 2nq, application of partial Frobenius to legs 0, 2,..., 2n´2, then annihilation in pairs p0, 1q¨¨¨p2n´2, 2n´1q. The creationof the pair p2n´1, 2nq commutes with all the partial Frobenius, but by Zorro’s lemma, the composition of that creation with annihilation at p2n ´ 2, 2n ´ 1q is the identity. We are left with an extra application of partial 12 ANDREW SALMON

Frobenius to the leg 2n ´ 2 which identiﬁes with the remaining leg 2n after applying Zorro’s lemma. This shows the induction step.

ďµ ‚ b ‚ Theorem 3.6. Let G be a collection of functors from ReppG q Ñ DcpX q cofibered over FinS and filtered with respect to partial Frobenius maps F‚. Suppose alsop that X is a Zariski dense open subset of C , where C is a proper curve over a finite field. Let η be a geometric generic point of Fq ďµ X. Then lim G | I is constant as an ind-constructible sheaf. ÝÑµ I,W η Proof. Now that we have the Eichler-Shimura relation, the proof of [21, Proposition 1.4.3] goes through verbatim. Since specialization maps are isomorphisms, we are done.

4. Nearby cycles over a power of a curve We begin with some general theory, following Beilinson and Gaitsgory [2] [8]. Let us consider a general setup. Let I “t1,...,nu and f : Y Ñ XI a stack of finite type, and let s P X be a special u u fiber. Let G be a sheaf on Y . We can consider iterated unipotent nearby cycles Ψ1 ˝¨¨¨˝ ΨnG . We can also restrict G to the preimage f ´1p∆pXqq of the diagonal ∆pXqĂ XI and perform unipotent u u nearby cycles Ψ∆pG |∆pXqr1ńsq there, which we will abuse notation and denote simply Ψ∆G when the base XI is clear. The explanation for the shift is to preserve perversity. Consider first the case when our base is 1-dimensional, that is n “ 1. In Beilinson’s construction of nearby cycles, we have local systems Lm on Xztsu such that monodromy acts according to an m ˆ m nilpotent Jordan block. We have natural inclusions of local systems Lm Ñ Lm1 when m ď m1. Let i be the inclusion of f ´1psq and j the inclusion of the complement f ´1pXztsuq. One of Beilinson’s results is that unipotent nearby cycles can be written as u i˚Rj f ˚ . (4.1) Ψ F – ÝÑlim ˚ pF b Lmqr´1s m To compare the iterated unipotent nearby cycles with the diagonal unipotent nearby cycles, we introduce a functor ΥpG q that maps naturally to both. In some favorable cases we consider, u u u ´1 ΥpG q will be isomorphic to both Ψ∆pG q and Ψ1 ¨ ¨ ¨ Ψn. Let i be the inclusion of f p∆psqq “ f ´1pps,...,sqq and j the inclusion of the complement f ´1ppXztsuqI q. We define our functor on objects as (4.2) Υp q– lim i˚Rj p b f ˚p b ¨ ¨ ¨ b qqrńs. G ÝÑ ˚ G Lm1 Lmn m1,...,mn The following proposition contains all the properties that we will use about the functor Υ. Proposition 4.1. We abbreviate all mentions of unipotent nearby cycles Ψu as Ψ. (1) There are natural maps

(4.3) Ψ1 ¨ ¨ ¨ ΨnG ΥG Ψ∆G |f ´1p∆pXqqr1 ´ ns.

(2) Let π : Y Ñ Y 1 be of ﬁnite type over XI . There is a natural transformation of functors Rπ!Υ Ñ ΥRπ! such that the following diagram commutes

Rπ!Ψ1 ¨ ¨ ¨ Ψn Rπ!Υ Rπ!Ψ∆ (4.4)

Ψn ¨ ¨ ¨ ΨnRπ! ΥRπ! Ψ∆Rπ!. Moreover, the vertical arrows are isomorphisms if π is proper. UNIPOTENT NEARBY CYCLES AND THE COHOMOLOGY OF SHTUKAS 13

(3) Let g : Y Ñ Y 1 be of ﬁnite type over XI . There is a natural transformation of functors g˚Υ Ñ Υg˚ such that the following diagram commutes ˚ ˚ ˚ g Ψ1 ¨ ¨ ¨ Ψn g Υ g Ψ∆ (4.5)

˚ ˚ ˚ Ψ1 ¨ ¨ ¨ Ψng Υg Ψ∆g . Moreover, the vertical arrows are isomorphisms if g is smooth. (4) (K¨unneth formula) Suppose that n “ 2 and Y – Y1 ˆ Y2 and the map to X ˆ X comes from maps Yi Ñ X for i “ 1, 2, and suppose that G – G1 b G2. Then the maps Υ Ñ Ψ ˝ Ψ and Υ Ñ Ψ∆ are isomorphisms, and the composition coincides with the isomorphism

(4.6) Ψ1G1 b Ψ2G2 – Ψ∆pG1 bX G2q observed by Gabber (see Illusie [9, p. 4.3]).

Proof. The construction of the maps is a straightforward generalization of [8]. We consider ∆X : I X Ñ X the diagonal inclusion and j∆ : Xztsu Ñ X where we imagine X as living in the diagonal I I of X . Let f∆ be the base change of the structure map f along the diagonal ∆pXq Ă X . The canonical base change map ˚ ˚ ∆X Rj˚ Ñ Rj∆pXq,˚∆Xztsu gives a map from ˚ ˚ ∆X Rj˚ pG b f pLm1 b ¨ ¨ ¨ b Lmn qqrńs to ˚ Rj G | r1 ´ nsb f pLm b¨¨¨b Lm q r´1s. ∆pXq,˚ ` ∆pXztsuq ∆ 1 n ˘ Note that we have a map Ln b Lm Ñ Lmaxpn,mq, which gives a map ˚ ˚ Rj G | b f pLm b¨¨¨b Lm q Ñ Rj G | b f pL q . ∆pXq,˚ ` ∆pXztsuq ∆ 1 n ˘ ∆pXq,˚ ` ∆pXq maxpm1,...,mnq ˘ After shifting, composing the two maps, pulling back along the inclusion of is,∆ : tsu Ñ ∆pXq, and passing to the limit over m1,...,mn, the left hand side becomes ΥpG q while the right hand side becomes Ψ∆pG q. k´1 n´k`1 k n´k Let ik be the inclusion X ˆ s Ñ X ˆ s along the kth coordinate, and let jk be the inclusion pXztsuqk Ñ Xk which we distinguish from jn : pXztsuqn Ñ pXztsuqn´1 ˆ X. We note first that by unwinding the definition of b, we have ˚ ˚ ˚ f pLm1 b ¨ ¨ ¨ b Lmn q– f pLm1 b ¨ ¨ ¨ b Lmn´1 b L1qb f pL1 ¨ ¨ ¨ b L1 b Lmn´1 q. Next, we consider the following diagram i pXztsuqn´1 ˆtsun pXztsuqn´1 ˆ X

(4.7) jn´1ˆtsu jn´1ˆX i Xn´1 ˆtsu n Xn to produce a map n ˚ ˚ (4.8) Rpjn´1 ˆ Xq˚ Ñ Rpjn´1 ˆ Xq˚pinq˚pi q – pinq˚Rpjn´1 ˆtxuq˚in ˚ ˚ which by adjunction gives us inRpjn´1 ˆ Xq˚ Ñ Rpjn´1 ˆtxuqin. Using these maps, and writing n j as the composition of pjn´1 ˆ Xq˝ j we can now produce a map from ˚ ˚ b b inRj˚ pG b f pLm1 ¨ ¨ ¨ Lmn qq 14 ANDREW SALMON to

˚ n ˚ ˚ Rpjn 1 ˆtsuq i Rj pG b f pL1 b ¨ ¨ ¨ b L1 b Lm qq b f Lm b ¨ ¨ ¨ b Lm b L1 . ´ ˚ ` n ˚ n ` 1 n´1 ˘˘ By pulling back, applying appropriate shifts, and passing to the limit, this gives a map

ΥnpG q Ñ Υn´1ΨnpG q which inductively gives the map we want. The second and third parts of the proposition follow by proper and smooth base change. The base change theorems can be used to give maps Rπ!Rj˚ Ñ Rj˚Rπ! by adjunction, which is an isomorphism if π is a proper map. Since the other maps in the diagram arise naturally from base change theorems, the diagrams will commute. Finally, in the product case, we can write

˚ ˚ ˚ ˚ i∆Rj˚pG1 b G2 b f pLm1 b Lm2 qq – Rj∆,˚pG1 b f1 Lm1 b G2 b f2 Lm2 q and the result follows by observing that the right hand side is also isomorphic to

˚ ˚ Rj˚pG1 b f1 Lm1 qb Rj˚pG2 b f2 Lm2 q where Rj˚ is here understood as the inclusion Xztsu ãÑ X.

We claim we also have the following map where Ψ∆ denotes specialization from ∆pδq to ∆psq over pCzNqI .

Lemma 4.2. Consider the following cases: (1) Nearby cycles is taken with respect to a geometric point s over a closed point p such that G has parahoric reduction at p. (2) Nearby cycles Ψ is unipotent nearby cycles and taken with respect to a geometric point s over a point p. G is split on the generic ﬁber of the completion and has reduction according to the unipotent radical of the Iwahori.

For I “t1,...,nu and W – W1 b ¨ ¨ ¨ b Wn, we have an isomorphism natural in W and V .

(4.9) Ψ∆Ft∆uYJ,p WiqbV – Ψ1 ... ΨnFIYJ,W bV . ÂiPI Both sides of the equation are sheaves over p´1ps ˆ pCzNqJ q for which suitable Schubert cells in pFl q ˆ Gr | J are local models. p Fq Fq G,J pCzNq p x Proof. It suffices to check this theorem on sufficiently large open subsets, and the subsets given by Harder-Narasimhan truncation suffice for this purpose. The local model theorem for shtukas, recalled above, says that for µ a fixed Harder-Narasimhan truncation of G-bundles (and therefore of G-shtukas) we can form the following diagram over pCzNqIYJ :

u1 ďµ GrG,I,W u U ChtG,Ξ,I,W UNIPOTENT NEARBY CYCLES AND THE COHOMOLOGY OF SHTUKAS 15 where the arrow on the right is an ´etale cover. Since ´etale maps are a fortiori smooth, we can consider 1˚ 1˚ u Ψ∆Ft∆uYJ,p WiqbV » Ψ∆u Ft∆uYJ,p WiqbV ÂiPI ÂiPI ˚S » Ψ∆u t∆uYJ,p WiqbV ÂiPI ˚ S » u Ψ∆ t∆uYJ,p WiqbV ÂiPI 1˚ 1˚ u Ψ1 ... ΨnFIYJ,W bV » Ψ1 ... Ψnu FIYJ,W bV ˚ » Ψ1 ... Ψnu SIYJ,W bV ˚ » u Ψ1 ... ΨnSIYJ,W bV We now show the following lemma: Lemma 4.3. Under the assumptions of the above lemma, the correspondence S S S Ψ∆ t∆uYJ,p WiqbV Υ IYJ,W bV Ψ1 ¨ ¨ ¨ Ψn IYJ,W bV ÂiPI induces an isomorphism natural in W and V , S S (4.10) Ψ∆ t∆uYJ,p WiqbV – Ψ1 ... Ψn IYJ,W bV . ÂiPI Both sides of the equation are sheaves over the preimage of s ˆ pCzpN Y P qqJ in the Beilinson- Drinfeld Grassmannian GrG,t1uYJ . Since the correspondence

Ψ∆Ft∆uYJ,p WiqbV ΥFIYJ,W bV Ψ1 ¨ ¨ ¨ ΨnFIYJ,W bV Âi pulls back to an isomorphism by the lemma, it is an isomorphism to begin with. Proof of Lemma 4.3. This statement is etale local on the curve, so we can reduce to the case where G has parahoric reduction only at a point p P C. We denote the complement by C˝. ˝ t∆uYJ Consider the left hand side. We consider the locus of the base pC q where x∆ ‰ xj for all j P J. The inclusion of this locus is smooth, and smooth pullback for nearby cycles tells us that we can restrict to this locus to compute our nearby cycles sheaf over GrG,t∆uYJ |tpuˆpC˝qJ . Over this locus, the Beilinson-Drinfeld Grassmannian splits as it is factorizable, and the Satake sheaf splits b as an external product St∆u, Wi SJ,V . Taking nearby cycles now gives the central sheaf ÂiPI b (4.11) Z Wi SJ,V , ÂiPI following the work of Gaitsgory [7] and Zhu [24]. We will prove the right hand side is equal to 4.11 by induction on |I|. The case |I|“ 1 is done above, and there is nothing to prove. Now consider the case |I|“ 2. The right hand side is

Ψ1Ψ2St1,2uYJ,W1bW2bV .

By restricting to the locus where x1 ‰ xj and x2 ‰ xj for all j P J (which we note contains tpu2 ˆ pC˝qJ ), the Satake sheaf splits as an external tensor product and the right hand side is now b Ψ1Ψ2St1,2u,W1bW2 SJ,V . The case |I|“ 2 is proved for the Iwahori case by Gaitsgory [8] by using a small resolution of the 2-dimensional Beilinson-Drinfeld Grassmannian which is etale locally a product of 1-dimensional Beilinson-Drinfeld Grassmannians. On the product we can apply the K¨unneth formula, and the small resolution allows us to transfer this result to the Grassmannian itself. 16 ANDREW SALMON

For the unipotent radical of the Iwahori, the relevant result is due to Bezrukavnikov [4, Propo- sition 13]. We note that the central sheaves in that paper compute the case of unipotent nearby cycles at this level and in this setting Bezrukavnikov explains how to get around the failure of properness for the convolution product map. Now the inductive step follows from the case |I|“ 2, as we want to show

b b Ψ0pZ Wi St0uYJ,UbV q– Z WibU SJ,V ÂiPI ÂiPI and we can rewrite the left hand side as

Ψ0Ψ∆pSt∆,0uYJ, WibUbV q. ÂiPI

Remark 4.4. As noted by Cong Xue, the use of the local models theorem is not necessary, as one can also use the smoothness property of ǫΞ instead, which has the advantage of being more canonical.

Let ΨJ be the iterated nearby cycles Ψj1 ¨ ¨ ¨ Ψjk for J “ tj1,...,jku. Let J1, J2, and J3 be identical, disjoint copies of J. We use the following lemma, which generalizes Lemma 4.2.

Lemma 4.5. For pJ1, V1q and pJ2, V2q, let pJ, V q be deﬁned by having a cocartesian map pJ1 Y J2, V1 b V2q Ñ pJ, V q for J1 Y J2 Ñ J the map identifying both J1 and J2 with J.

(4.12) ΨJ FIYJ,W bV – ΨJ1 ΨJ2 FIYJ1YJ2,W bV1bV2 .

Both sides of the equation are sheaves over p´1ps ˆ pCzNqJ q – pFl q ˆ Gr | . p Fq Fq G,J pCzNqJ p x Proof. We can naturally identify both sides to the sheaf

Ψ∆FIYt∆u,W bp V1,ibV2,iq. ÀÂ The lemma can also be proved directly, following the proof of Lemma 4.2.

The above proof seems to destroy information about the potential Weil group action, except on the diagonal, but we do not need it for that purpose: only to construct an inverse to the canonical map on nearby cycles, which is automatically Galois equivariant.

Corollary 4.6. Let pJ1, V1q, pJ2, V2q, and pJ, V q be as in the above lemma. The isomorphisms in Lemma 4.2 and 4.5 are compatible with the natural !-pushforward map. In particular,

Rp!ΨJ FIYJ,W bV ΨJ Rp!FIYJ,W bV

(4.13) » »

Rp!ΨJ1 ΨJ2 FIYJ1YJ2,W bV1bV2 ΨJ1 ΨJ2 Rp!FIYJ1YJ2,W bV1bV2

Proof. This follows immediately from Proposition 4.1 (2), where we note that the vertical arrows in the diagram are compositions of the corresponding arrows in that diagram by the proof of Lemma 4.2. UNIPOTENT NEARBY CYCLES AND THE COHOMOLOGY OF SHTUKAS 17

5. Nearby cycles commute with pushforward via factorization structure Let S be a trait with generic point η and special point s. When we deal with multiple copies of S, let Ψi “ Ψi,δÑs denote the vanishing cycle functor along the ith copy of S as before. In what follows, we often suppress mention of the group scheme G, which itself contains the information of the ramiﬁcation and level structure along N. We also suppress the lattice Ξ in ChtG,Ξ,IYJ,W bV “: ChtIYJ,W bV . We pick p P P Ă C and pickp a geometric point s above p. Using this, we reduce to the local situation by taking a Henselian trait S giving the specialization from a geometric generic point δ of C to s. We have the following maps which are a priori deﬁned from the geometric Satake functor and the coalescence of legs.

Ft1,2u,W b1 Ñ Ft1,2u,W bpW ˚bW q.

Ft2,3u,pW bW ˚qbW Ñ Ft2,3u,1bW . The fact that the composition of these maps restricted to the diagonal is the identity is known as “Zorro’s lemma.”

Lemma 5.1. The sheaf FIYt1u,W b1 on ChtIYt1u,W b1 |pXzpNYP qqIYt1u is identiﬁed with the sheaf FI,W |pXzpNYP qqI b QℓXzpNYP qr´1s under the identiﬁcation

(5.1) ChtIYt1u,W b1 – ChtI,W ˆpXzNq

Proof. The identification of moduli spaces of shtukas is by sending ppx, x1q, E, pφI , φ1qq to the pair of px, E, φI q and x1, noting that φ1 must be the identity. After restricting the locus, the identification of sheaves is by noting that this splitting is compatible with the corresponding splitting on the Beilinson-Drinfeld Grassmannian. In the following lemma, for L a local system over pηqI with partial Frobenius structure, that is, I a representation of Weilpη, ηq , we denote the nearby cycles functor ΨI to be the iterated nearby cycles along all coordinates. It is a tautology that this nearby cycles does essentially nothing: it gives back L as a representation of WeilI . Lemma 5.2. Let I and J be finite sets, and let I Ñ J be a map. Let L be a smooth ind-constructible sheaf over pηqI with partial Frobenius structure, that is a representation of Weilpη, ηqI . Let s be a geometric point above a closed point p P C and let η be a geometric generic point over η. Let ΨI denote the iterated nearby cycles along each coordinate of I for the Henselian trait with a geometric J I generic point η specializing to s. Let ∆ζ : X Ñ X be the natural map, and let ΨJ be the same iterated nearby cycles. Then, we have a natural identification ˚ (5.2) ΨJ ∆ζ L – ΨI L which is equivariant with respect to the map Weilpη, ηqJ Ñ Weilpη, ηqI . Proof. Both sides are vector spaces with Weil group actions. The underlying topoi of sheaves smooth on pηqI , resp. J, are of the form BG for G a suitable power of a Weil group, and the J I pullback ∆ζ is the restriction functor for the group homomorphism Weilpη, ηq Ñ Weilpη, ηq . We will give the result for general I and J. This will be an exact analogue of the result of [21, Proposition 4.1.4, Proposition 3.2.1] but using nearby cycles instead of specialization maps. In fact, this result can be seen to imply the smoothness result of loc. cit. over very special points by showing that vanishing cycles vanish on the sheaves over pXzNqI . p 18 ANDREW SALMON

Theorem 5.3. Consider the following cases: (1) Nearby cycles is taken with respect to a geometric point s over a closed point p such that G has parahoric reduction at p. (2) Nearby cycles Ψ is unipotent nearby cycles and taken with respect to a geometric point s over a point p. G is split on the generic ﬁber of the completion and has reduction according to the unipotent radical of the Iwahori. Then the natural map

(5.3) can: Rp!ΨJ FIYJ,W bV Ñ ΨJ Rp!FIYJ,W bV is an isomorphism.

Proof. We begin by constructing an inverse map. Let J1,J2,J3 be three identical, disjoint copies of J.

ΨJ Rp!FIYJ,W bV – ΨJ1 Rp!ΨJ2 FIYJ1YJ2,W bV b1

Ñ ΨJ1 Rp!ΨJ2 Ft1,2u,W bV bpV ˚bV q

˚ – ΨJ1 Rp!ΨJ2 ΨJ3 FIYJ1YJ2YJ3,W bV bV bV

˚ Ñ ΨJ1 ΨJ2 Rp!ΨJ3 FIYJ1YJ2YJ3,W bV bV bV

˚ – ΨJ2 Rp!ΨJ3 FIYJ2YJ3,W bpV bV qbV

Ñ ΨJ2 Rp!ΨJ3 FIYJ2YJ3,W b1bV

– Rp!ΨJ FIYJ,V bW We now justify that this gives a well-defined map. (1) The first arrow is an isomorphism because of Lemma 5.1. I (2) The second arrow follows by functoriality of ReppG q Ñ PervpChtG,I q. (3) The third arrow exists and is an isomorphism by Lemma 4.5. (4) The fourth arrow is the canonical map Rf!Ψ Ñ ΨRf!. (5) The fifth arrow is an isomorphism by 5.2 and Lemma 3.3. (6) The sixth arrow follows by functoriality. (7) The seventh arrow is an isomorphism because of Lemma 5.1. We now want to show that the map constructed is a left inverse of can, and in particular can is injective. This can be seen because the squares commute in Figure 5.1. In Figure 5.1, the bottom row is an isomorphism because the sheaves are constant in the J2 direction when the corresponding tuple of representations is trivial. The left composition is the identity when specialized to the diagonal by Zorro’s lemma. The top arrow is the map can. The composed vertical line on the right, together with applications of Lemma 5.1, is the map we constructed above as the left inverse of can. To show that this map is a left inverse, it suffices to check that all the squares in Figure 5.1 commute. (1) Square p1q expresses the fact that can is a natural transformation, as the map on sheaves comes from the functoriality of F applied to the map 1 Ñ V ˚ b V . (2) Square p2q again expresses that can is a natural transformation, this time using the isomorphism in Lemma 4.5. (3) Square p3q is an instance of Corollary 4.13. (4) The bottom square p4q follows from naturality of can, with the map on sheaves coming from F applied to the map V b V ˚ Ñ 1. (5) The middle triangle composes two instances of can. UNIPOTENT NEARBY CYCLES AND THE COHOMOLOGY OF SHTUKAS 19

Rp!ΨJ1 ΨJ2 FIYJ1YJ2,W bV b1 ΨJ1 Rp!ΨJ2 FIYJ1YJ2,W bV b1

p1q

˚ ˚ Rp!ΨJ1 ΨJ2 FIYJ1YJ2,W bV bpV bV q ΨJ1 Rp!ΨJ2 FIYJ1YJ2,W bV bpV bV q

p2q

˚ ˚ Rp!ΨJ1 ΨJ2 ΨJ3 FIYJ1YJ2YJ3,W bV bV bV ΨJ1 Rp!ΨJ2 ΨJ3 FIYJ1YJ2YJ3,W bV bV bV

˚ p3q ΨJ1 ΨJ2 Rp!ΨJ3 FIYJ1YJ2YJ3,W bV bV bV

˚ ˚ Rp!ΨJ2 ΨJ3 FIYJ2YJ3,W bpV bV qbV ΨJ2 Rp!ΨJ3 FIYJ2YJ3,W bpV bV qbV

p4q

Rp!ΨJ2 ΨJ3 FIYJ2YJ3,W b1bV ΨJ2 Rp!ΨJ3 FIYJ2YJ3,W b1bV

Figure 5.1.

ΨJ1 Rp!ΨJ2 FIYJ1YJ2,W bV b1 ΨJ1 ΨJ2 Rp!FIYJ1YJ2,W bV b1

˚ ˚ ΨJ1 Rp!ΨJ2 FIYJ1YJ2,W bV bpV bV q ΨJ1 ΨJ2 Rp!FIYJ1YJ2,W bV bpV bV q

˚ ˚ ΨJ1 Rp!ΨJ2 ΨJ3 FIYJ1YJ2YJ3,W bV bV bV ΨJ1 ΨJ2 ΨJ3 Rp!FIYJ1YJ2YJ3,W bV bV bV

˚ ΨJ1 ΨJ2 Rp!ΨJ3 FIYJ1YJ2YJ3,W bV bV bV

˚ ˚ ΨJ2 Rp!ΨJ3 FIYJ2YJ3,W bpV bV qbV ΨJ2 ΨJ3 Rp!FIYJ2YJ3,W bpV bV qbV

ΨJ2 Rp!ΨJ3 FIYJ2YJ3,W b1bV ΨJ2 ΨJ3 Rp!FIYJ2YJ3,W b1bV

Figure 5.2. 20 ANDREW SALMON

Now we want to show that it is a right inverse. This can be seen because the squares commute in Figure 5.2. The justiﬁcations for isomorphism of the top row and commutativity are the same as those in the previous paragraph, but in reverse order.

6. Consequences about the Galois action and Langlands parameters Now that the key result about nearby cycles is available, we can use results about central sheaves from local geometric Langlands to draw a number of conclusions about the action of local Galois groups on cohomology. This in turn will allow us to draw conclusions about the image of tame ramification in the Galois representation that Lafforgue associates to any automorphic form. Corollary 6.1. Suppose G has split parahoric reduction at a place v. On the cohomology of shtukas, wild inertia acts trivially, and tame inertia acts unipotently. Proof. By the result of Gaitsgory, generalized by Zhu [24], nearby cycles on the affine Grassmannian are unipotent. By the local models theorem, the action of the wild inertia on stalks of nearby cycles of Ft1u,W is trivial and hence it acts trivially on the sheaf itself. After pushing forward along Rp!, the wild inertia will still act trivially because the transformation Rp!Ψ Ñ ΨRp! is Galois- equivariant, arising from proper base change theorem, and is an isomorphism in the case we are interested in. Corollary 6.2. Suppose G is ramified at v and after base change along a Galois extension C1 Ñ C, 1 1 1 for a point v lying over v, G splits over Kv1 and has parahoric reduction at v . Then wild inertia 1 IK1 acts trivially on the cohomology of shtukas. Moreover, if it acquires very special parahoric v1 1 reduction, inertia of Kv1 acts trivially. Proof. This is the same as the above, except applying the work of Zhu [25] and later Richarz [17]. We note that the formation of the Beilinson-Drinfeld Grassmannian commutes with base change [24, Lemma 3.2], and we get trivial action of inertia on nearby cycles after base change. Let K1{K be a finite Galois extension of the function field such that the generic fiber of G splits, and let C1 be the normalization of the curve C in K1. Let P 1 be the image of parahoric points P in C1. Let N 1 be the “deep” level structures NzP . 1 For a point p P P , we consider the completion of K at p, Kp, and the extension Kp1 for any point 1 1 1 1 1 p over p in P . Consider the local Weil groups WeilpKp1 , Kp q with its inertia subgroup IK p1 , wild 1 t 1 1 1 1 1 inertia subgroup IK , and the tame inertia quotient IK 1 :“ IK p1 {IK 1 . p1 p p 1 Let Weilt,P pCzN 1, ηq be the quotient of WeilpCzN, ηq by the normal subgroup generated by all wild inertia at all places in P 1, that is, so that we havep

1 t,P 1 1 p1PP 1 IK1 WeilpCzN, ηq Weil pCzN , ηq 1. p1 ś p The above theorems imply the following corollary.

I t,P 1 1 I Corollary 6.3. The action of WeilpCzN, ηq on HI,W factors through Weil pXzN , ηq , and tame inertia subgroups along P 1 act unipotently.p We now conclude with some consequences about the global Langlands correspondence. Suppose f is a cuspidal automorphic form associated to parahoric level structure at a point p P P Ă C, and 1 let ρ : Weilt,P pCzN 1q Ñ LG be the associated Langlands parameter. We note that this Langlands UNIPOTENT NEARBY CYCLES AND THE COHOMOLOGY OF SHTUKAS 21

t,P 1 1 1 t parameter restricts to a representation Weil pC zN q Ñ G and the tame inertia I 1 is a subset K p1 of the latter. We would like to draw some conclusions aboutp the image of tame generator of the local Galois group. Let us briefly review some necessary facts about Langlands parameters that we will use. Consider cuspidal automorphic forms for G with finite order central character corresponding to the lattice Ξ which is a fixed vector under an open compact subgroup corresponding to the level structure we have constructed via dilatation can be viewed as a function on BunGpFqq{Ξ. The corresponding Langlands parameters of cuspidal automorphic forms are characters of the algebra of excursion cusp operators on Ht0u,1. Any such character ν will give a semisimple Galois representation ρ. The Galois representation is compatible with the character of the excursion algebra in the sense that if I I is a finite set, f P OpGzG {Gq is a function and pγiqiPI is a tuple of Galois elements, then the value of the character atp thep correspondingp excursion operator SI,f,γ is equal to the value of f at the corresponding tuple of GI under the Langalnds parameter. That is, [13, Proposition 5.7] p

νpSI,f,pγiqiPI q“ fppρpγiqqiPI q.

For unipotence of the tame generator, we only need the case I “t1, 2u and pγ1,γ2q “ pγ, 1q for γ the tame generator. Theorem 6.4. Let ρ be a Langlands parameter associated to a cuspidal automorphic form with 1 1 t parahoric reduction at P , with K and P as above. The image of the tame generator γ P I 1 Ă K p1 t,P 1 L Weil pCzN, ηq Ñ GpQℓq is unipotent. Proof. We can justify unipotence on the level of Galois representations up to conjugation and we only need to use the fact that nearby cycles acts unipotently. The image of the generator of tame ramification is unipotent if and only if its image in G{{G is the identity. On the other hand, the unipotence of the action of γ on cohomology impliesp thatp the excursion operator St1,2u,V,pγ,1q acts unipotently for every V . So for any character of the excursion algebra, St1,2u,V,pγ,1q ´ 1 is sent to 0. Running over all matrix coefficients up to conjugation, the image of γ is 1 P G{{G. p p At parahoric levels, we would like to know which unipotent conjugacy class contains the image of the tame generator. This computation of monodromy is analogue for shtukas of the computation of monodromy for automorphic sheaves in [22, Section 4], see also [23, Proposition 4.2.4]. For the formulation of our next theorem, we introduce some notation. For notational simplicity, we will assume that the generic fiber of G splits. Fix a pinning of G and a fixed Iwahori I which reduces modulo the uniformizer to a standard Borel B over the base field. A standard parahoric is a parahoric P containing I. For such a parahoric, we can form P `, the pro-unipotent radical of ` P , and the corresponding Levi MP “ P {P , which is a reductive group. Let wP be the longest element of the Weyl group of MP . For example, for the Iwahori I, wI is the identity. We can view all the wP for different P as elements of the affine Weyl group W˜ . Moreover, these are each contained in a unique two-sided cell, which we denote cP . In fact, they are Duflo involutions in the two-sided cells. We now recall Lusztig’s order-preserving bijection [15] ttwo-sided cells in W˜ u»tunipotent conjugacy classes in Gu. p Let cP be the two-sided cell, and let uP the unipotent orbit corresponding to the two-sided cell cP in the Langlands dual group. Let uP be the closure of the unipotent orbit uP in G. In general this is the closure of the unipotent orbit containing a dense open subset of the unipotentp radical of 22 ANDREW SALMON the parabolic P Ă G. We note that the two-sided cell containing the identity is the largest with respect to thep orderp on the left hand side, and the orbit closure uI is the unipotent cone N of the Langlands dual group. We recall the main result of Bezrukavnikov [3, Theorems 1 and 2] in a format that is easy for us to use. For any w P W˜ , let Lw be the corresponding simple object in the Iwahori-Hecke category PervpIzLG{Iq. To a two-sided cell c, Bezrukavnikov associates a tensor category Ac with tensor structure ‚ given by truncated convolution. For a Duflo involution d P c, Ld is idempotent under truncated convolution in Ac. Let Ad be the category generated by subquotients of Ld ‚ ZpV q where Z : ReppGq Ñ PervpIzLG{Iq is Gaitsgory’s central functor into the Iwahori-Hecke category. This gives us ap tensor functor ReppGq Ñ Ad which is in fact a central functor. This central functor inherits an automorphism fromp monodromy, and viewing this as a fiber functor and remembering only the monodromy automorphism of the fiber functor gives an element of G. The result of Bezrukavnikov now implies that the conjugacy class of G punder Tannakian for- malism is a unipotent conjugacy class corresponding to the two-sided cellpc containing the Duflo involution d. Specializing to the case d “ wP , we have a smooth map of the Iwahori affine flag variety to the parahoric affine flag variety which intertwines the central sheaf functors and their monodromy. That is, pulling back along this map sends central sheaves ΨpSV q for the parahoric to truncated convolutions LwP ‚ZpV q where ZpV q is Gaitsgory’s central sheaf functor for the Iwahori, since LwP is just the constant sheaf in PervpIzP {Iq for parahoric P containing I.

t Theorem 6.5. In the setting of the above theorem, the image of the tame generator γ P I 1 Ă K p1 t,P 1 L Weil pCzN, ηq Ñ GpQℓq lives in the unipotent orbit closure uP inside the Langlands dual group. Proof. To prove the claim on two-sided cells, we will freely use the framed excursion algebra and related ideas from the paper of Lafforgue and Zhu [14]. Consider the affine space of moduli space 1 of representations of Weilt,P pCzN, ηq, over which (6.1) Hcusp “ Hcusp b V t0u,Reg à t0u,V V PReppGq p is a module. Choosing a generator γ of tame ramification at P 1 chooses a map of this affine space to G. This map is a restriction from all excursion operators to those involving just integer powers of cusp thep tame generator γ, including 1. We want to argue that the support of Ht0u,Reg on the excursion algebra is contained in the preimage of the unipotent orbit closure uP . The Langlands parameters in V. Lafforgue’s construction are the generalized eigenspaces given by the excursion algebra, and any character corresponds to a representation ρ which must send the tame generator γ to an element of uP . Let J be the ideal that defines the unipotent orbit closure uP . It is generated by relations involving matrix coefficients that all elements in uP satisfy. Let R be such a relation. We first show that R generates a relation that the monodromy of nearby cycles over GrG satisfies. Then we use the local model to transport this relation to a relation satisfied by nearby cycles on ChtG. Then, these relations will produce an ideal that exactly cuts out uP in G over which the framed moduli of Langlands parameters lives. p For any fixed f a regular function on G, by the algebraic Peter-Weyl theorem, f can be written as a matrix coefficient g ÞÑ xv˚, gvy for somep representation V (generally not irreducible) and vectors v˚ P V ˚, v P V . For V a G-representation, let V denote the underlying vector space with trivial G p p UNIPOTENT NEARBY CYCLES AND THE COHOMOLOGY OF SHTUKAS 23 action. Following Lafforgue and Zhu we have a map of G representations (6.2) θ : Reg bV Ñ RegpbV defined by θpf b vqpgq“ fpgqg ¨ v. For NP P uP fixed, consider the map F : Reg bV Ñ Reg bV of vector spaces (in fact it is a map in ReppHwP q in Bezrukavnikov’s notation where wP is considered as a Duflo involution) defined by ´1 (6.3) F pf b vqpgq“ fpgqg NP g ¨ v.

We note that F implicitly depends on our choice of NP . This deﬁnes an endomorphism of ΨSReg bV – ΨSReg b V considered as a map in the truncated convolution category. Moreover, if f P J represented by matrix coeﬃcients pairing maps x :1 Ñ V with ξ : V Ñ 1, we know that the composition

x F ξ (6.4) ΨSt0u,Reg ΨSt0u,Reg bV ΨSt0u,Reg bV ΨSt0u,Reg is 0. We note that there is a functor between equivariant sheaves on the parahoric affine flag variety to equivariant sheaves on the Iwahori affine flag variety sending ΨSt0u,W to the object LwP ‚ ZpV q. This functor restricts to a central functor from central sheaves on the parahoric affine flag variety to the truncated convolution category AwP . We also use F to denote the map

LwP ‚ ZpRegq‚ ZpV q Ñ LwP ‚ ZpRegq‚ ZpV q under the equivalence of categories given by [3, Theorem 1].

Lemma 6.6. For suitable NP P uP , we have a commutative square in the truncated convolution category AwP

F LwP ‚ ZpRegq‚ ZpV q LwP ‚ ZpRegq‚ ZpV q

(6.5) θ θ

1‚MV LwP ‚ ZpRegq‚ ZpV q LwP ‚ ZpRegq‚ ZpV q where the vertical arrows are isomorphisms and the bottom row is the unipotent monodromy endomorphism coming from unipotent nearby cycles. Proof. ´1 θ ˝ F pf b vqpgq“ θpf b g NP gvqpgq“ fpgqNP g ¨ v.

Under the identiﬁcation AwP – ReppHwP q, the fact that the right hand side identiﬁes with the composition of 1 ˚ MV ˝ θ follows by Theorem 1 of [3]. We return to the proof of Theorem 6.5. Using the above lemma, we build a diagram

x F ξ ΨSt0u,Reg ΨSt0u,Reg bV ΨSt0u,Reg bV St0u,Reg

(6.6) θ θ

˚ ˚ ∆ Ψ1Ψ0St0,1u,Reg bV ∆ Ψ1Ψ0St0,1u,Reg bV where the vertical arrows come from fusion of nearby cycles, the ﬁrst and last arrow come from realizing the regular function f as a matrix coeﬃcient, and the lower horizontal arrow is the action of monodromy on the leg 1. The composition of the top row is 0. 24 ANDREW SALMON

Such a relation is clearly preserved under pulling back under a smooth map, and we also know that pulling back etale sheaves under a surjective map is a surjective functor. Thus, we conclude that the composition

x ξ ΨFt0u,Reg ΨFt0u,Reg bV ΨFt0u,Reg bV Ft0u,Reg (6.7)

˚ ˚ ∆ Ψ1Ψ0Ft0,1u,Reg bV ∆ Ψ1Ψ0Ft0,1u,Reg bV is 0. After pushing forward by p! and using Theorem 5.3, the composition becomes an element Ff,γ in the framed excursion algebra, using notation in [14]. Thus, we get relations in the framed excursion algebra Ff,γ “ 0 for every element f of the ideal J , which shows that the image of the tame generator is contained in the orbit closure uP .

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