Equivariant localization, parity sheaves, and cyclic base change functoriality
aa r X i v : . [ m a t h . N T ] N ov EQUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLICBASE CHANGE FUNCTORIALITY
TONY FENG
Abstract.
Lafforgue and Genestier-Lafforgue have constructed the global and (semisim-plified) local Langlands correspondences for arbitrary reductive groups over functionfields. We apply equivariant localization arguments, inspired by work of Treumann-Venkatesh, to moduli spaces of shtukas, in order to prove properties of these correspon-dences regarding functoriality for cyclic base change.Globally, we establish the existence of functorial transfers of mod p automorphic formsthrough p -cyclic base change. Locally, we prove that Tate cohomology realizes cyclicbase change functoriality in the mod p Genestier-Lafforgue correspondence, verifying afunction field version of a conjecture of Treumann-Venkatesh.The proofs draw upon new tools from representation theory, including parity sheavesand Smith-Treumann theory. In particular, we use these to establish a categorificationof the base change homomorphism for Hecke algebras, in a joint appendix with GusLonergan.
Contents
1. Introduction 12. Generalities on Smith theory 73. Parity sheaves and the base change functor 124. Functoriality and the excursion algebra 205. Cyclic base change in the global setting 256. Cyclic base change in the local setting 40Appendix A. The base change functor realizes Langlands functorialityby Tony Feng and Gus Lonergan 47References 481.
Introduction
Global results.
Let G be a reductive group over a global function field F , of char-acteristic = p . Let k be an algebraic closure of F p . We regard the Langlands dual group L G over k . Vincent Lafforgue has constructed in [Laf18a, §13] a global “mod p ” Langlandscorrespondence irreducible cuspidalautomorphic representationsof G over k → (cid:26) Langlands parameters
Gal( F s /F ) → L G ( k ) (cid:27) / ∼ . For split groups G , Lafforgue’s correspondence has been generalized beyond the case of cuspforms by work of Cong Xue [Xue20, Xuea]. Langlands’ principle of functoriality predicts that given a map of L -groups φ : L H → L G and an automorphic form f for H , there should be a transfer f φ to G . In this paper we areconcerned with a specific type of functoriality: base change functoriality, arising from thecase where H is a reductive group over F , and G = Res E/F ( H E ) for a cyclic p -extension E/F . The relevant map φ : L H → L G is the diagonal embedding on the dual groups. Weemphasize that it is crucial for our results that the degree of the extension coincides withthe characteristic of our automorphic functions. Theorem 1.1 (Existence of global base change) . Assume p is an odd good prime for b G .Let φ : L H → L G be as above. Let ρ be a Langlands parameter arising from an automorphicform on H by Lafforgue(-Xue)’s correspondence. Then φ ◦ ρ arises from an automorphicform on G by Lafforgue(-Xue)’s correspondence. Remark 1.2.
The base change of a cuspidal automorphic representation may no longer becuspidal, so the theorem really requires Xue’s generalization of Lafforgue’s correspondence.Also because of this, the notion of a Langlands parameter “arising from an automorphicform” is a bit subtle, and is explained in §5.2.4 (it is the analog of footnote 2 below for theexcursion algebra instead of the Hecke algebra).Our proof is inspired by work of Treumann-Venkatesh [TV16]. The analog of [TV16]in the function field context would guarantee that for every Hecke eigensystem “appearingin” the space of automorphic forms for H , the transferred eigensystem “appears in” thespace of automorphic forms for G . For general groups, our theorem is more refined in viewof the failure of Multiplicity One. Indeed, Lafforgue’s correspondence can assign differentLanglands parameters to Hecke eigenfunctions with the same unramified eigensystem; infact, it can even assign different parameters to different automorphic forms generating iso-morphic automorphic representations, with examples occurring already for SL n when n ≥ [Bla94, Lap99]. The reason for this is the failure of local conjugacy to imply global conju-gacy; see [Laf18a, §0.7] for more discussion of this phenomenon. Our theorem guarantees atransfer with the correct Langlands parameter, which is a subtler property than cannot ingeneral be detected by Hecke operators; the proof thus requires more work. Remark 1.3.
In fact, the statement of the theorem is conjecturally true with characteristiczero coefficients. This is already established for G = GL n in which the full global Langlandscorrespondence is already known, using the trace formula. For general groups it does notseem like trace formula methods can prove the characteristic zero analog of Theorem 1.1,because of the issues mentioned in the previous paragraph.Moreover, our method can be used to prove analogous base change results for somecohomology classes in the moduli of shtukas, which do not necessarily lift to characteristiczero. At present our results towards this are somewhat messy, so we postpone a precisestatement. Our results are conditional on the extension of Xue’s results to non-split G , which are announced in[Xueb, Slide 30] but have not yet appeared in writing at this time. Explicitly, this means that we require p > if b G is of type A, B, C or D ; p > if b G is of type G , F , E , E ; and p > if b G is of type E . Here say that a Hecke eigensystem “appears in” the space of automorphic forms for H if, regarding thespace of automorphic forms for H as a module over the Hecke algebra for H , the corresponding maximalideal is in the support of this module. We are not necessarily saying that there is actually a function withthat eigensystem. QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 3
Still, for some groups such as GL n , our theorem gives no more information than a transferof Hecke eigenvalues, since two semisimple representations into GL n with the same charac-teristic polynomials are automatically isomorphic by the Brauer-Nesbitt Theorem. However,even in this case our method has the advantage that it also gives information about the local Langlands correspondence, which we explain next. This allows us to prove a conjecture ofTreumann-Venkatesh on the behavior of base change functoriality in the local Langlandscorrespondence, which is one of the main motivations for this paper.1.2.
Local results.
Genestier-Lafforgue have constructed a semi-simplified form of the Lo-cal Langlands correspondence over function fields [GL]. More precisely, let F v be a functionfield of characteristic = p and W v the Weil group of F v . For any reductive group G over F v ,[GL] constructs a map (cid:26) irreducible admissiblerepresentations of G ( F v ) over k (cid:27) / ∼−→ (cid:26) Langlands parameters W v → L G ( k ) (cid:27) / ∼ . Now let H be a reductive group over F v and G = Res E v /F v ( H E v ) , where E v /F v is a cyclic p -extension, and take φ : L H → L G as above. Let σ be a generator of Gal( E v /F v ) ; it acts on G and its induced action on G ( F v ) = H ( E v ) is the Galois action. If the isomorphism class ofa k -representation Π of G ( F v ) is preserved by σ , then it should come from base change. Forany irreducible admissible representation Π of G ( F v ) whose isomorphism class is fixed by σ , there is a unique σ -action on Π compatible with the G ( F v ) -action (Lemma 6.1). Hencewe can form the Tate cohomology groups T (Π) , T (Π) with respect to the σ -action, whichretain actions of H ( F v ) = G ( F v ) σ , and are conjecturally admissible H ( F v ) -representations.We prove: Theorem 1.4 (Tate cohomology realizes local functoriality) . Assume p is an odd good primefor b G . Let Π be as above and Π ( p ) := Π ⊗ k, Frob k the Frobenius twist of Π . Let π be any irre-ducible admissible subquotient of T ∗ (Π) as an H ( F v ) -representation and ρ π : Weil( F v /F v ) → L H ( k ) be the corresponding Langlands parameter constructed by Genestier-Lafforgue. Then φ ◦ ρ π is the Langlands parameter attached to Π ( p ) by Genestier-Lafforgue. This verifies, for the Genestier-Lafforgue local Langlands correspondence, a conjectureof Treumann-Venkatesh that “Tate cohomology realizes functoriality”; see §6.1 for morediscussion of this.
Remark 1.5.
Over local fields of characteristic zero, forthcoming work of Fargues-Scholzewill construct a semisimplified local Langlands correspondence for all reductive groups.Moreover, their construction seems likely to be compatible with our methods, so we areoptimistic that our arguments will generalize to prove the analog of Theorem 1.4 withrespect to the Fargues-Scholze correspondence.1.3.
Elements of the proof.
In this subsection we hint at the ingredients in the proofs ofTheorem 1.1 and Theorem 1.4.1.3.1.
The excursion algebra.
In order to convey the substance of the argument, we need toexplain a bit more about the correspondences of Lafforgue and Genestier-Lafforgue. Theyare based on the notion of the excursion algebra . We summarize this very briefly below; amore complete discussion appears in §4.To abstract the situation a bit, given a group Γ and a reductive group L G over analgebraically closed field k , Lafforgue introduces the excursion algebra Exc(Γ , L G ) (which iscommutative) whose key property is that (see §4.3): TONY FENG
There is a canonical bijection between homomorphisms
Exc(Γ , L G ) → k andsemi-simple Langlands parameters Γ → L G ( k ) .So, if Exc(Γ , L G ) acts on a vector space, then to each (generalized) eigenvector v of thisaction we get a maximal ideal m v ⊂ Exc(Γ , L G ) , and therefore a “Langlands parameter” ρ v : Γ → L G ( k ) which is well-defined modulo b G -conjugacy. For Γ = Gal(
F /F ) , Lafforgueconstructs an action of Exc(Γ , L G ) on the space of cuspidal automorphic functions for G ,thus defining a global Langlands correspondence by this mechanism.For Γ = Weil( F v /F v ) , Genestier-Lafforgue construct an action of Exc(Γ , L G ) on anyirreducible admissible representation of G ( F v ) . Since the action is G ( F v ) -equivariant, theirreducibility forces it to factor through a character of Exc(Γ , L G ) , which gives the localLanglands correspondence of [GL]. Remark 1.6 (The excursion algebra as functions on the representation stack) . The follow-ing perspective, due to Drinfeld and explained in [Laf18b], offers a more conceptual wayto picture the situation. There is a “representation stack”
Rep(Γ , L G ) which parametrizes L G -valued parameters of Γ , meaning homomorphisms Γ → L G modulo the action of b G -conjugation. If k had characteristic zero then Exc(Γ , L G ) would be the ring of functions onthe representation stack Rep(Γ , L G ) [Zhu, Remark 2.1.20]. When k has positive character-istic (which is the situation in this paper) we speculate that the same is true up to issues ofderivedness and reducedness; in any case the interpretation of k -points remains valid.The excursion algebra has an explicit presentation with generators S I,f, ( γ i ) indexed by: I a finite set, f ∈ O ( b G \ ( L G ) I / b G ) , ( γ i ) i ∈ I ∈ Γ I . If we imagine S { ,...,n } ,f, ( γ ,...,γ n ) asa function on the representation stack, its value on a representation ρ : Γ → L G ( k ) is f (( ρ ( γ γ n ) , ρ ( γ γ n ) , . . . , ρ ( γ n − γ n ) , ρ ( γ n ))) .1.3.2. Equivariant localization.
We now explain the strategy of our proof. It will be in-structive to compare it to work of Treumann-Venkatesh [TV16], so we begin by recallingtheir setup. Momentarily assuming that F is a characteristic number field, let Y G , Y H be locally symmetric spaces associated to G, H , with compatible level structures. Then
Gal(
E/F ) acts on Y G through its action on G , and for good choices of level structures Y H is a connected component of Y Gal(
E/F ) G . Treumann-Venkatesh show that for any Heckeeigensystem { h v,V χ ( h v,V ) } occurring in the action of the Hecke algebra for H actingon H ∗ ( Y H ; k ) , a certain transferred eigensystem { h w,W χ ( h w,φ ∗ W ) } occurs in the Heckealgebra for G acting on H ∗ ( Y G ; k ) .Now suppose that F is a global function field, where Lafforgue(-Xue) has constructed anaction of the excursion algebra on the space of compactly supported automorphic forms forany reductive G . We show that for any eigensystem { S I,f, ( γ i ) i ∈ I χ ( S I,f, ( γ i ) i ∈ I ) } occurringin the action of Exc(Gal( F s /F ) , L H ) on the space of compactly supported automorphicfunctions for H , a transferred eigensystem occurs in the action of Exc(Gal( F s /F ) , L G ) onthe space of compactly supported automorphic functions for G . This gives control over theSatake parameters because Hecke operators at unramified places are among the excursionoperators, but it also gives a lot of additional information. In particular, if one believes inlocal-global compatibility then taking all the γ i to be in the Weil group at a particular place v should give information about the semi-simplified local Langlands correspondence at v ,and this is indeed the source of our traction on the local functoriality. See §4.1.5 for the precise definition of this.
QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 5
The method of Treumann-Venkatesh is based on relating H ∗ ( Y G ; k ) and H ∗ ( Y H ; k ) using equivariant localization theorems for a space with Z /p Z -action, which fall under the headingof Smith theory. (We note that the core idea first occurs in [Clo14], in the context ofquaternion algebras over Q , wherein the topological aspect becomes trivial.) In general,this can be phrased as an isomorphism of Tate cohomology , which is the composition ofTate’s construction with the usual cohomology, and it says: T ∗ ( X ; k ) ∼ = T ∗ ( X Z /p Z ; k ) . In the setting of arithmetic manifolds, Treumann-Venkatesh show that these equivariantlocalization isomorphisms are “sufficiently Hecke-equivariant” to establish a transfer of Heckeeigensystems. We show that in the function field situation, the equivariant localizationtheorems are similarly “sufficiently equivariant” for the excursion operators.The proof of this equivariance is very different from that of Treumann-Venkatesh, becausethe excursion action arises in a much less direct manner than the Hecke action (which is thereason for the name “excursion algebra”). Lafforgue’s construction of the excursion actionworks by chasing cohomology classes through a plethora of auxiliary cohomology groups,of moduli spaces of shtukas with coefficients in perverse sheaves indexed by
Rep k (( L G ) I ) (ultimately coming from the Geometric Satake equivalence). The upshot is that we needto prove compatible equivariant localization theorems for “enough” of these cohomologygroups. This resembles the situation of Treumann-Venkatesh, except that we must comparecohomologies not only with constant coefficients, but with coefficients in various perversesheaves.The difficulty here is that the theory of perverse sheaves (and consequently the GeometricSatake equivalence) does not interface well with restriction to subvarieties. Because of this,it is very unclear how to even relate the coefficient sheaves whose cohomologies should becompared. The one exception is the constant sheaves on the trivial Schubert strata, whichin our context can be thought of as corresponding to the trivial representation of ( L G ) I ;this case is the function-field analog of [TV16].1.3.3. Smith theory for sheaves.
Our solution to the difficulty raised above hinges on apurely representation-theoretic problem. The Geometric Satake equivalence asserts that Rep k ( L G ) is equivalent to P G ( O ) (Gr G ; k ) . Therefore, given a map φ : L H → L G over k thereis a corresponding functor Res( φ ) : P G ( O ) (Gr G ; k ) → P H ( O ) (Gr H ; k ) . To utilize it, we needa “geometric” description of the functor Res( φ ) (e.g., which does not pass through the aboveequivalence).We solve this problem in the context of p -cyclic base change functoriality, giving a cate-gorification of the Brauer homomorphism of Treumann-Venkatesh. Since it would take muchmore setup to say anything substantial about the content, let us just touch on some of thenovel ingredients. For one, we invoke the theory of parity sheaves introduced in [JMW14].The reason they come up is that we want to employ “sheaf-theoretic Smith-Treumann the-ory” [Tre19, LL, RW]. This necessitates passing through certain “exotic” categories, whichcan be interpreted as categories of sheaves on the affine Grassmannian with coefficients in E ∞ -ring spectra. These are morally derived categories but they have no t-structure; becauseof this, they interact poorly with the theory of perverse sheaves. However, it turns out thatthese exotic categories have enough structure to support a well-behaved theory of paritysheaves. For this equivalence, one has to be careful with how the L -group is defined. See §4.1 for a precisediscussion. TONY FENG
Remark 1.7 (Analogy to the twisted trace formula) . For automorphic forms in character-istic , cyclic base change is established for some groups by comparison of the trace formulafor H with the twisted trace formula for G . The idea of the twisted trace formula is that“twisting” an operator by the automorphism σ picks out the contribution from the σ -fixedsummands, which should come from base change.Our argument can, to some extent, be viewed as a categorification of such a comparison. Itwas modeled on certain trace computations carried out in a very special situation in [Fen20].Here, instead of relating traces of (Hecke and Frobenius) operators acting on vector spacesof automorphic forms, as one would do in the classical theory, we relate certain cohomologygroups of shtukas which can (at least morally) be viewed as traces of (Hecke and Frobenius)operators acting on categories of automorphic sheaves by the formalism of [Gai]. The analogof the twisted trace is Tate cohomology, which functions to “pick out” the contribution from σ -fixed isomorphism classes (but also forces us to work modulo p ).1.4. Further questions. (1) Some version of our story should go through the generality of any group G with Z /p Z -action, as was treated in [TV16]. Our arguments mostly work at this level ofgenerality; the most serious problem is that the additional examples are nearly allin bad characteristics, and this screws up the representation-theoretic input aboutparity sheaves – in particular, parity sheaves need no longer be perverse in badcharacteristic. A notable exception is a type of automorphic induction studied in[Clo17], which we hope to address in future work.(2) Xinwen Zhu has formulated a conjectural description of the cohomology of shtukasin terms of coherent sheaves on the moduli stack of Langlands parameters [Zhu].Is it possible to view our results in terms of his picture, perhaps as some kind of( K -theoretic) equivariant localization on this stack of Langlands parameters?1.5. Organization of the paper.
The structure of this paper is as follows. • In §2, we review the basic framework of sheaf-theoretic Smith theory from [Tre19].We introduce the notion of Tate categories, the Smith functor
Psm and its proper-ties, Tate cohomology, and explain the relation to classical equivariant localizationtheorems for Z /p Z -actions. • In §3, we recall the fundamentals of parity sheaves due to Juteau-Mautner-Williamson,and the analogous notion of “Tate-parity sheaves” due to Leslie-Lonergan. We ex-plain how to combine these with the functor
Psm to construct a base change func-tor for parity objects in the Satake category. In terms of the analogy between ourmethod and the twisted trace formula (Remark 1.7), this functor plays the categori-fied role of the base change homomorphism for Hecke algebras. • In §4, we define excursion algebras and recall their relation to Langlands parameters.We explain functoriality from the perspective of excursion algebras. • In §5, we prove Theorem 1.1. First we recall background on moduli spaces ofshtukas and Lafforgue’s global Langlands correspondence in terms of actions ofthe excursion algebra on the cohomology of shtukas. Then we establish certainequivariant localization isomorphisms for the Tate cohomology of shtukas in thesetting of p -cyclic base change, which gives relations between excursion operators inthe context of functoriality. • In §6 we recall the conjectures of Treumann-Venkatesh, and the relevant aspects ofthe Genestier-Lafforgue correspondence. Then we use the results established earlierto prove Theorem 1.4.
QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 7
Notation. • (Coefficients) We let k be an algebraic closure of F p (considered with the discretetopology).In general we will consider geometric objects over fields of characteristic = p , andétale sheaves over p -adically complete coefficients. • ( σ -actions) Throughout the paper, σ denotes a generator of a group isomorphic to Z /p Z . When we say that a widget has a “ σ -action”, what we mean is that thewidget has an action of a cyclic group of order p with chosen generator σ .Let N := 1 + σ + . . . + σ p − ∈ Z [ σ ] . We will also denote by N the inducedoperation on any Z [ σ ] -module. If A is a ring or module, then A σ denotes the σ -invariants in A . • (Reductive groups) For us, reductive groups are connected by definition. The Lang-lands dual group b G is considered as a split reductive group over k . For our conven-tions on the L -group, see §4.1.For any group, denotes the trivialization representation (with the group madeclear by context). • (Equivariant derived categories) If a (pro-)algebraic group Σ acts on X , then wedenote by D Σ ( X ) or D ( X ) Σ the Σ -equivariant derived category of constructiblesheaves with coefficients in k .1.7. Acknowledgments.
We thank Gus Lonergan, David Treumann, Geordie Williamson,Zhiwei Yun, and Xinwen Zhu for helpful conversations related to this work. We thankLaurent Clozel and Michael Harris for comments on a draft. During the writing of this paper,the author was supported by an NSF Postdoctoral Fellowship under grant No. 1902927, aswell as the Friends of the Institute for Advanced Study.2.
Generalities on Smith theory
We shall require some general formalism from [Tre19], which we recall here. While [Tre19]operates in the setting of complex algebraic varieties in the analytic topology, most of theresults generalize in a well-known way to ℓ -adic sheaves on algebraic stacks, as will beformulated here. Much of what we will say is also covered in more detail in [RW, §2,3],which also works in the context of ℓ -adic sheaves.2.1. The Tate category.
Let Λ be a p -adic coefficient ring; we will be interested in thecases where Λ = k or W ( k ) . We will denote by Λ[ σ ] the group ring of h σ i with coefficientsin Λ . Our geometric objects will be over a field of characteristic = p and we will consider Λ -adic sheaves.For an algebraic stack Y with a σ -action, there is an equivariant (constructible) derivedcategory D bσ ( X ; Λ) . If σ acts trivially on Y , then we have an equivalence of derived categories D bσ ( Y ; Λ) ∼ = D b ( Y ; Λ[ σ ]) . (2.1)We will also be interested in the (full) subcategory Perf( Y ; Λ[ σ ]) ⊂ D b ( Y σ ; Λ[ σ ]) consistingof complexes whose stalks at all points of Y are perfect. This is to be contrasted with the operation Nm , which will mean Nm( a ) = a ∗ σ ( a ) ∗ . . . ∗ σ p − ( a ) inthe context where there is a monoidal operation ∗ . For us, this includes by definition the conditions of being locally finite type and separated.
TONY FENG
Definition 2.1.
The
Tate category of Y (with respect to Λ ) is the Verdier quotient category D b ( Y ; Λ[ σ ]) / Perf( Y ; Λ[ σ ]) .According to [Tre19, Remark 4.1], the category D ( Y ; Λ[ σ ]) / Perf( Y ; Λ[ σ ]) can be regardedas a derived category of sheaves for a certain E ∞ -ring spectrum T Λ . So we will denote thecorresponding Tate categories by Shv( Y ; T Λ ) . For our purposes T Λ can be thought of as justa notational device.We denote the tautological projection map from D b ( Y ; Λ[ σ ]) to Shv( Y ; T Λ ) by T : D b ( Y ; Λ[ σ ]) → Shv( Y ; T Λ ) . Example 2.2 ([Tre19, Proposition 4.2]) . The Tate category over a point (meaning thespectrum of a separably closed field) is D b (Λ[ σ ]) / Perf(Λ[ σ ]) . In this category the shift-by-2functor is isomorphic to the identity functor, as one sees by considering the nullhomotopiccomplex → V → V ⊗ Λ[ σ ] − σ −−−→ V ⊗ Λ[ σ ] → V → whose middle two terms project to in the Tate category.2.2. The Smith operation.
Let X be a stack with an action of Z /p Z ∼ = h σ i . The σ -fixedpoints of X are defined by the cartesian square X σ XX X × X i σ × Id∆
Note that the map i : X σ → X may not necessarily be a closed embedding when X is nota scheme.Given a σ -equivariant complex F ∈ D bσ ( X ; Λ) , we can restrict it (via i ∗ ) to X σ to get anobject of D bσ ( X σ , Λ) , putting ourselves in the situation of the previous subsection. Definition 2.3 ([Tre19, Definition 4.2]) . We define the
Smith operation
Psm := T ◦ i ∗ : D σ ( X ; Λ) → Shv( X σ ; T Λ ) to be the composition of i ∗ : D σ ( X ; Λ) → D σ ( X σ ; Λ) (2.1) ∼ = D ( X σ ; Λ[ σ ]) with the projection T to the Tate category. Lemma 2.4 ([Tre19, Theorem 4.1]) . Let i : X σ ֒ → X . The cone on i ! → i ∗ belongs to Perf( X σ ; Λ[ σ ]) .Proof. The point is that the stalks of the cone are cohomology of neighborhoods on which σ acts freely, which implies that they are perfect complexes. See [RW, Lemma 3.5]. (cid:3) Six-functor formalism.
The Tate category enjoys a robust 6-functor formalism. Wewill just recall what we need; see [Tre19, §4.3] for a more complete discussion. Functorsbetween derived categories, e.g. f ! , f ∗ , f ! , f ∗ , will always denote the derived functors.Let f : X → S be a σ -equivariant morphism. • (Pullback) As f ∗ preserves stalks, it preserves perfect complexes, and so descendsto the Tate category to give f ∗ : Shv( S ; T Λ ) → Shv( X ; T Λ ) . • (Pushforward) If S has the trivial σ -action, then proper pushforward preserves per-fect complexes by [Tre19, Proposition 4.3], so it descends to an operation on theTate category f ! : Shv( X ; T Λ ) → Shv( S ; T Λ ) . QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 9 • Verdier duality descends to the Tate category, hence so do the operations f ! and (if S has the trivial σ -action) f ∗ .We now list some properties which could be remembered under the slogan , “The Smithoperation commutes with all operations”.2.3.1. Compatibility with pullback. If f : X → S is a σ -equivariant map, then the diagramsbelow commute: D bσ ( X ; Λ) D bσ ( S ; Λ)Shv( X σ ; T Λ ) Shv( S σ ; T Λ ) Psm Psm f ∗ f ∗ D bσ ( X ; Λ) D bσ ( S ; Λ)Shv( X σ ; T Λ ) Shv( S σ ; T Λ ) Psm Psm f ! f ! The proof for the first square is formal; from the second it follows immediately from thefirst plus Lemma 2.4.2.3.2.
Compatibility with pushforward.
Let f : X → S be a σ -equivariant map where S hasthe trivial σ -action. The following diagrams commute: D bσ ( X ; Λ) D bσ ( S ; Λ)Shv( X σ ; T Λ ) Shv( S ; T Λ ) Psm f ∗ Psm f ∗ D bσ ( X ; Λ) D bσ ( S ; Λ)Shv( X σ ; T Λ ) Shv( S ; T Λ ) Psm f ! Psm f ! (Note that we have used S σ = S , since the σ -action on S was trivial by assumption.)2.4. Tate cohomology.
Given a bounded-below complex of Λ[ σ ] -modules C • , we defineits Tate cohomology as in [LL, §3.3]. Because of the importance of this notion for us, wewill spell out some of the details.The exact sequence → Λ → Λ[ σ ] − σ −−−→ Λ[ σ ] → Λ → induces a morphism in the derived category of Λ[ σ ] -modules, Λ → Λ[2] ∈ D b (Λ[ σ ]) . (2.2) We copied this slogan from Geordie Williamson.
Consider the double complex below, where N denotes multiplication by σ + . . . + σ p − (cf. §1.6) ... ... ... ... . . . C C . . . C n . . . . . . C C . . . C n . . . . . . C C . . . C n . . . . . . C C . . . C n . . . Row − dN dN N Nd − σ d − σ − σ − σN d dN N Nd − σ d − σ − σ − σ (2.3)We define H n ( ǫ ! C • ) to be the n th cohomology group of the totalization of this doublecomplex. We define T i ( C • ) to be lim −→ n H i +2 n ( ǫ ! C • ) , where the transition maps are inducedby (2.2).If C • is bounded, the double complex (2.3) is eventually periodic, and T i ( C • ) can becomputed as the i th cohomology group of the totalization of the double complex Tate( C • ) below: Tate( C • ) := ... ... ... ... . . . C C . . . C n . . . . . . C C . . . C n . . . . . . C C . . . C n . . . . . . C C . . . C n . . . Row − ... ... ... ... dN dN N Nd − σ d − σ − σ − σN d dN N Nd − σ d − σ − σ − σN N N N (2.4)The formation of Tate cohomology descends to the derived category, so we can view Tatecohomology as a collection of functors T i : D b (Λ[ σ ]) → Λ − Mod . The functors T i are evidently -periodic, i.e. T i ∼ = T i +2 . Since Tate cohomology of perfect Λ[ σ ] -complexes vanishes, this construction further factors through the Tate category. QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 11
Remark 2.5.
There is also a more abstract description of Tate cohomology in terms of“Homs in the Tate category”: [LL, Proposition 4.6] implies that for C • ∈ D (Λ[ σ ]) , we have T i ( C • ) ∼ = Hom Shv(pt; T Λ ) ( T Λ , T C • [ i ]) . Lemma 2.6.
Suppose C • ∈ D b (Λ[ σ ]) is inflated from D b (Λ) , i.e. σ acts trivially on C • .Then T ∗ C • ∼ = H ∗ ( C • ) ⊗ T ∗ (Λ) , where Λ is equipped with the trivial σ -action in the formationof T ∗ (Λ) .Proof. In this case (2.4) decomposes as the tensor product of C • and the Tate doublecomplex for Λ ; the result then follows from the Künneth theorem. (cid:3) The long exact sequence for Tate cohomology.
Given a distinguished triangle F ′ →F → F ′′ ∈ D b (Λ[ σ ]) , we have a long exact sequence . . . T − F ′′ → T F ′ → T F → T F ′′ → T F ′ → T F → T F ′′ → T F ′ → . . . Tate cohomology of a space.
Suppose X is a space with a σ -action, and F is a σ -equivariant sheaf on X , then (picking injective resolutions) we can form the cohomology of X with coefficients in F , as a complex C • ( X ; F ) ∈ D + (Λ[ σ ]) . Then T i C • ( X ; F ) is “theTate cohomology of X with coefficients in F ”, which we will abbreviate T i ( X ; F ) . Remark 2.7.
In all our later applications we will take care to only form Tate cohomologyof F when C • ( X ; F ) is bounded.2.4.3. Tate cohomology sheaves.
Given
F ∈
Shv( Y ; T Λ ) , we have by an analogous construc-tion to (2.4) Tate cohomology sheaves T i F on Y , which are étale sheaves of T (Λ) -modules.2.4.4. The Tate cohomology spectral sequence. If C • is bounded, then the double complex(2.4) induces a spectral sequence E ij = H j ( C • ) = ⇒ T i + j ( C • ) . The second page is E ij = T i ( H j ( C • )) . Hence we find that the Tate cohomology of C • hasa filtration whose graded pieces are subquotients of the ordinary cohomology H j ( C • ) .2.5. Equivariant localization.
We will explain how the six-functor formalism capturesequivariant localization theorems. For f : X → S a σ -equivariant map where S has thetrivial σ -action, consider the commutative diagram D bσ ( X ; Λ) D bσ ( S ; Λ)Shv( X σ ; T Λ ) Shv( S ; T Λ ) Psm f ! Psm f ! from §2.3.2. This says that for a sheaf F ∈ D bσ ( X ; Λ) , we have T ( f ! F ) ∼ = ( f | X σ ) ! Psm( F ) ∈ Shv( S ; T Λ ) . In particular, taking S = pt , and then applying Tate cohomology, we obtain T i ( X ; F ) ∼ = T i ( X σ ; Psm( F )) . (2.5)This is one formulation of classical equivariant localization theorems for Z /p Z -actions, e.g.[Qui71, Theorem 4.2]. Parity sheaves and the base change functor
We begin by indicating where this section is headed.The Geometric Satake equivalence P L + G (Gr G ; k ) ∼ = Rep k ( b G ) provides the link between G and its Langlands dual group. In the situation of functoriality, we have a map b H → b G and we would ideally like to describe the induced restriction operation Rep k ( b G ) → Rep k ( b H ) on the other side of the Geometric Satake equivalence, as a geometric operation on perversesheaves.In the context of base change it is even the case that there is an embedding Gr H ֒ → Gr G ,and when seeking to describe functoriality it is natural to look to the Smith operation.(One motivation is that the papers [Tre19, TV16] verify that the function-theoretic Smithoperation is indeed related to functoriality for Hecke algebras.) However, the Smith oper-ation lands in a Tate category, and in Example 2.2 we saw that in the Tate category, theshift-by- functor is isomorphic to the identity functor. This makes it seem unlikely thatone can capture the notion of “perverse sheaf” in the Tate category.Juteau-Mautner-Williamson invented the theory of parity sheaves , which could be seenas a variant of perverse sheaves that seems to behave better in the setting of modularcoefficients. Parity sheaves are cut out in the derived category by constraints on the parityof cohomological degrees, and can therefore make sense in a context where cohomologicaldegrees are only defined modulo . The notion of Tate-parity sheaves was introduced in[LL] as an analog of parity sheaves for the Tate category, and found to enjoy analogousproperties.After briefly reviewing the notions of parity and Tate-parity sheaves in §3.1 and §3.2, wewill establish that the Smith operation respects the parity property, at least under certainconditions satisfied in our application of interest. Using “coefficient lifting” properties ofparity sheaves, this will allow us to ultimately define a functor BC from parity sheaves on Gr G to parity sheaves on Gr H , which realizes base change functoriality on the geometricside.3.1. Parity sheaves.
We begin with a quick review of the theory of parity sheaves. Wewill take coefficients in a ring Λ , which in our applications of interest will be either k or O := W ( k ) .Let Y be a stratified variety over a field of characteristic = p , with stratification S = { Y λ } .For the theory to work, we need to assume that the (induced) stratification on Y is JMW,meaning: • for any two local systems L , L ′ on a stratum Y λ , we have Ext i ( L , L ′ ) is free over Λ for all i , and vanishes when i is odd.This holds for Kac-Moody flag varieties over separably closed fields, and in particular foraffine flag varieties over separably closed fields [JMW14, §4.1].Fix a pariversity † : S → Z / Z . In this paper we will always take the dimension pariver-sity † ( λ ) := dim Y λ mod 2 , so we will sometimes omit the pariversity from the discussion.Recall that [JMW14] define even complexes (with respect to the pariversity † ) to be those F ∈ D bS ( Y ; Λ) such that for all i λ : Y λ ֒ → X , for λ ∈ S , i ∗ λ F and i ! λ F have cohomol-ogy sheaves supported in degrees congruent to † ( λ ) modulo , and odd complexes analo-gously. They define parity complexes to be direct sums of even and odd complexes. The fullsubcategory of ( S -constructible) Tate-parity complexes (with coefficients in Λ ) is denoted Parity S ( Y ; Λ) . QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 13
Theorem 3.1 ([JMW14, Theorem 2.12]) . Let F be an indecomposable parity complex.Then: • F has irreducible support, which is therefore of the form Y λ for some λ ∈ Λ , • i ∗ λ F is a shifted local system L [ m ] , and • Any indecomposable parity complex supported on Y λ and extending L [ m ] is isomor-phic to F . A parity sheaf (with respect to † ) is an indecomposable parity complex (with respectto † ) with Y λ the dense stratum in its support and extending L [dim Y λ ] . Given L [dim Y λ ] ,it is not clear in general that a parity sheaf extending it extends exists. If it does exist,then Theorem 3.1 guarantees its uniqueness, and we denote it by E ( λ, L ) . The existence isguaranteed for Gr G with the usual stratification by L + G -orbits; E ( λ, L ) is moreover L + G -equivariant if p is not a torsion prime for G [JMW16, Theorem 1.4]. If E ( λ, L ) exists for all λ and L , we will say that “all parity sheaves exist”.3.2. Tate-parity sheaves.
As we have seen, the cohomological grading in the Tate categoryis only well-defined modulo , so it does not seem to make sense to talk about perversesheaves in the Tate category. However, elements of the Tate category have Tate cohomologysheaves (§2.4.3), which are indexed by Z / Z , so it could make sense to talk about an analogof parity sheaves in the Tate category. As [LL] observed, for this to work we must takecoefficients in the integral version of the Tate category, meaning Λ = O = W ( k ) , becausethen Ext ∗ Shv( T O ) ( T ( O ) , T ( O )) = M i ∈ Z k [2 i ] (3.1)is supported in even degrees. This is necessary for the assumption of non-vanishing oddExts in the definition of the JMW stratification.For a stratification S on Y , we define Shv S ( Y ; T O ) ⊂ Shv( Y ; T O ) to be the full subcategorygenerated by objects in D bS ( Y ; O [ σ ]) . Letting Perf S ( Y ; O [ σ ]) ⊂ Perf( Y ; O [ σ ]) be the fullthick subcategory of S -constructible objects, we have by [LL, Corollary 4.7] that D bS ( Y ; O [ σ ]) / Perf S ( Y ; O [ σ ]) ∼ −→ Shv S ( Y ; T O ) . Definition 3.2 ([LL, Definition 5.3]) . Let
F ∈
Shv S ( Y ; T O ) . Fix a pariversity † : S → Z / Z .Let ? ∈ {∗ , ! } .(1) We say F is ? - Tate-even (with respect to † ) if for each λ ∈ S , we have T † ( λ )+1 ( i ? λ F ) = 0 . (2) We say F is ? - Tate-odd (with respect to † ) if F [1] is ?-Tate-even.(3) We say F is Tate-even (resp.
Tate-odd ) if F is both ∗ -Tate even (resp. odd) and ! -Tate even (resp. odd).(4) We say F is Tate-parity complex (with respect to † ), if it is isomorphic within Shv S ( Y ; T O ) to the direct sum of a Tate-even complex and a Tate-odd complex. The full subcategory of ( S -equivariant) Tate-parity complexes (with coefficients in T O ) isdenoted Parity S ( Y ; T O ) .Parallel to Theorem 3.1, we have the following result in this context: Proposition 3.3 ([LL, Theorem 4.13]) . Let F be an indecomposable Tate-parity complex.(1) The support of F is of the form Y λ for a unique stratum Y λ . This is to be distinguished from the (upcoming) notion of
Tate-parity sheaf , which is more restrictive. (2) Suppose G and F are two indecomposable Tate-parity complexes such that supp( G ) =supp( F ) . Letting j λ : Y λ ֒ → Y be the inclusion of the unique stratum open in thissupport, if j ∗ λ G ∼ = j ∗ λ F then G ∼ = F .Proof. The same argument as in [JMW14, Theorem 2.12] works. (cid:3)
We define ǫ ∗ : D bc ( Y ; O ) → D bc ( Y ; O [ σ ]) for the inflation through the augmentation O [ σ ] ։ O . Recall that T : D bc ( Y ; O [ σ ]) → Shv( Y ; T O ) denotes projection to the Tate category. Weare interested in Tate complexes that come from the composite functor T ǫ ∗ : D bS ( Y ; O ) → D bS ( Y ; O [ σ ]) → Shv S ( Y ; T O ) . Definition 3.4. A Tate-parity sheaf
F ∈
Shv S ( Y ; T O ) is an indecomposable Tate-paritycomplex with the property that its restriction to the unique stratum Y λ which is dense inits support is of the form T ǫ ∗ L [dim Y λ ] for an indecomposable Λ -free local system L on Y λ .If such an F exists then it is unique, and we denote it by E T ( λ, L ) .If E T ( λ, L ) exists for all λ ∈ S and all L , we will say that “all Tate-parity sheaves exist”(for Y, S ).3.3.
Modular reduction.
We now explain that the functor T has good properties that onewould expect from “base change of coefficients” functors for categories of sheaves in classicalrings. We will suppression mention of the pariversity † . Proposition 3.5 ([LL, Proposition 5.16, Theorem 5.17]) . (1) If F ∈ D bS ( X ; O ) is even/odd, then T ǫ ∗ F ∈ Sh S ( X ; T O ) is Tate-even/odd.(2) If the parity sheaf E = E ( λ, L ) exists and satisfies Hom D b ( Y ; O ) ( E , E [ n ]) = 0 for all n < (this holds for example if E is perverse ) then E T ( λ, L ) exists and we have T ǫ ∗ E ( λ, L ) ∼ = E T ( λ, L ) . Proof.
We reproduce the proof because it highlights the importance of using O -coefficientsinstead of k -coefficients. The operation T ǫ ∗ is compatible with formation of i ∗ λ or i ! λ . Henceto prove (1) we reduce to examining T i ǫ ∗ L for a local system L of free O -modules, with thetrivial σ -action. This reduces to the fact that the Tate cohomology of O is supported ineven degrees, which is (3.1).For (2), we just need to check that T ǫ ∗ E ( λ, L ) is indecomposable. Since Parity S ( Y ; T O ) is Krull-Remak-Schmidt by [LL, Proposition 5.8], the endomorphism ring of T ǫ ∗ E ( λ, L ) islocal. According to [LL, §4.6], for F , G ∈ D b ( Y ; O ) we have Hom
Shv( Y ; T Λ ) ( T F , T G ) ∼ = M i ∈ Z Hom D b ( Y ; k ) ( F F , F G [2 i ]) . (3.2)We apply this to F = G = ǫ ∗ E ( λ, L ) . Since E ( λ, L ) is indecomposable the subalgebra in(3.2) indexed by i = 0 is local, and the assumption that the summands of (3.2) indexed bynegative i vanish. This implies the desired locality of the graded algebra (3.2). (cid:3) Remark 3.6.
The Proposition (and its proof) are analogous to the following results ofparity sheaves [JMW14, §2.5]. Let F denote the base change functor F = k L ⊗ O ( − ) : D S ( Y ; O ) → D S ( Y ; k ) . The functor F enjoys following properties. In fact this is an equivalence by [MR18, Lemma 6.6], which we thank Simon Riche for pointing out tous.
QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 15 (1)
E ∈ D bS ( X ; O ) is a parity sheaf if and only if F ( F ) ∈ D bS ( X ; k ) is a parity sheaf.(2) If E ( λ, L ) exists, then E ( λ, F L ) exists and we have F E ( λ, L ) ∼ = E ( λ, F L ) . What we have seen can be summarized by the slogan:If all parity sheaves exist and have vanishing negative self-Exts, then allTate-parity sheaves exist and T ◦ ǫ ∗ induces a bijection between paritysheaves and Tate-parity sheaves.3.4. The lifting functor.
We will now define a functor lifting Tate-parity sheaves to paritysheaves. In fact the preceding slogan already tells us what to do about objects, so we justneed to specify what happens on morphisms.
Definition 3.7. A normalized (Tate-)parity complex is a direct sum of Tate-parity sheaves with no shifts . Hence, under our assumptions, their restrictions to the dense open stratumin their support are isomorphic to L [dim Y λ ] (resp. T ǫ ∗ L [dim Y λ ] ). We denote the full sub-categories of such by Parity S ( Y ; O ) ⊂ Parity S ( Y ; O ) and Parity S ( Y ; T O ) ⊂ Parity S ( Y ; T O ) ,and called them the categories of normalized (Tate)-parity sheaves .Under the assumption that all parity sheaves exist and have vanishing negative self-Exts,we then have a lifting functor [LL, Theorem 5.19] L : Parity S ( Y ; T O ) → Parity S ( Y ; k ) sending E T ( λ, L ) to E ( λ, L ⊗ O k ) on objects, and on morphisms inducing projection tothe summand indexed by i = 0 under identification (3.2). It can be thought of as an“intermediate” reduction between O and k in the sense that the following diagram commutes: Parity S ( Y ; O ) Parity S ( Y ; T O )Parity S ( Y ; k ) T ǫ ∗ F L Parity sheaves on the affine Grassmannian and tilting modules.
We now con-sider the preceding theory in the context of the affine Grassmannian Gr G over a separablyclosed field F , with the stratification by L + G -orbits. Since this is a special case of a Kac-Moody flag variety, the stratification is JMW by [JMW14, §4.1].If p is a good prime for b G , [MR18, Corollary 1.6] implies that all parity sheaves exist,and that all normalized parity sheaves are perverse. Therefore, the category of normalizedparity sheaves corresponds under the Geometric Satake equivalence to some subcategoryof Rep k ( b G ) , and it is natural to ask what this is. The answer is given in terms of tiltingmodules for b G (recall that these are the objects of Rep k ( b G ) having both a filtration bystandard objects, and a filtration by costandard objects). Let Tilt k ( b G ) ⊂ Rep k ( b G ) denotethe full subcategory of tilting modules. Theorem 3.8 ([MR18, Corollary 1.6], generalizing [JMW16, Theorem 1.8]) . If p is goodfor G , then the Geometric Satake equivalence restricts to an equivalence Parity L + G (Gr G ; k ) ∼ = Tilt k ( b G ) . We need a few facts about the representation theory of tilting modules. For our arithmeticapplications, the key point is that there are “enough” tilting modules to generate the derivedcategory of
Rep k ( b G ) , as articulated by the Theorem below. Theorem 3.9 ([BBM04]) . The subcategory
Tilt k ( b G ) generates the bounded derived categoryof Rep k ( b G ) . More precisely, the natural projection from the bounded homotopy category K b (Tilt k ( b G )) to D b (Rep k ( b G )) is an equivalence.Proof. This follows from general highest weight theory. A convenient reference is [Ric,Proposition 7.17]. (cid:3)
Base change functoriality for the Satake category.
We now consider a specificgeometric situation relevant to Langlands functoriality for p -cyclic base change. Let F be afield of characteristic = p . We will consider reductive groups, and their affine Grassmanni-ans, over F .3.6.1. The base change setup.
We now specialize the situation a bit further: H is anyreductive group over a separably closed field F and G = H p . We let σ act on G by cyclicrotation, sending the i th factor to the ( i + 1) st (mod p ) factor. Then it is clear that thestratification on Gr G by L + G -orbits induces by restriction the stratification on Gr H by L + H -orbits.Evidently the “diagonal” embedding H ֒ → G realizes H as the fixed points of G underthe automorphism σ . This map H ֒ → G also induces a diagonal map Gr H → Gr G . Lemma 3.10.
The diagonal map induces an isomorphism Gr H ∼ = Gr σG as subfunctors of Gr G .Proof. We have Gr G ∼ = (Gr H ) p , with σ acting by cyclic rotation of the factors, from whichthe claim is clear. (cid:3) Henceforth we assume that p is odd and good for b G , so that the results of §3.5 apply.The restriction functor along the diagonal embedding b H k ֒ → b G k induces a restrictionfunctor Res BC : Tilt k ( b G ) → Tilt k ( b H ) . We aim to give a “geometric” description of the corre-sponding functor under the Geometric Satake equivalence, Parity(Gr G ; k ) → Parity(Gr H ; k ) ,in terms of Smith theory. (Of course, one could give an “ad hoc” description using that G = H p . The point is to define a functor that does not make reference to this, which willthen descend well to the situation where G = Res E / F ( H ) where E / F is a non-trivial fieldextension.) Definition 3.11.
Given
F ∈ P L + G (Gr G ; k ) , we define Nm( F ) := F ∗ σ F ∗ . . . ∗ σ p − F ∈ P L + G ⋊ σ (Gr G ; k ) , Strictly speaking, the cited references employ the trivial pariversity instead of the dimension pariversity.Since dimensions of Schubert strata in Gr G have constant parity on connected components, the trivialpariversity and dimension pariversity lead to the same notion of parity complexes in this case, so the onlydifference is in the notion of “normalization”. We follow [LL] in the use of the dimension pariversity so thatperverse sheaves are † -even. QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 17 equipped with the σ -equivariant structure coming from the commutativity constraint for (P L + G (Gr G ; k ) , ∗ ) : σ Nm( F ) = σ F ∗ . . . ∗ σ p − F ∗ F ∼ −→ F ∗ σ F ∗ . . . ∗ σ p − F = Nm( F ) . (3.3)There is a realization functor P L + G ⋊ σ (Gr G ; k ) → D L + G ⋊ σ (Gr G ; k ) due to Beilinson, whichwe will use to view Nm( F ) ∈ D L + G ⋊ σ (Gr G ; k ) (so that we may apply the Smith functor,for example). Equipping a general object of D L + G (Gr G ; k ) with a σ -equivariant structure ismuch more involved than just specifying isomorphisms (3.3) (satisfying cocycle conditions),so we emphasize that we construct Nm( F ) first as a σ -equivariant perverse sheaf, and thenapply the realization functor to get a σ -equivariant object of D L + G (Gr G ; k ) . Remark 3.12.
In our applications we will assume that p is large enough so that all paritysheaves are perverse. The properties of being L + G -constructible and L + G -equivariant areequivalent for perverse sheaves on Gr G . Therefore, we will not need to worry about anyextra complications coming from the equivariance. For Tate categories, Shv ( L + G ) (Gr G ; T Λ ) means by definition the category of L + G -stratified sheaves. Lemma 3.13.
Let i : Gr H ∼ = Gr σG ֒ → Gr G . For F ∈ D bL + G (Gr G ; O ) , regard Nm( F ) ∈ D bL + G ⋊ σ (Gr G ; O ) as in Definition 3.11 above.(i) The stalks of i ∗ Nm( F ) have Jordan-Hölder constituents being either trivial or free O [ σ ] -modules.(ii) The costalks of i ! Nm( F ) have Jordan-Hölder constituents being either trivial or free O [ σ ] -modules.Proof. By filtering F into its Jordan-Hölder constituents, we may assume that F itself issimple. Any simple L + G ≈ ( L + H ) p -equivariant sheaf F on a stratum Gr λG is of the form F ≈ F ⊠ . . . ⊠ F p , since the stratum is a product of homogeneous spaces for (a finite typequotient of) L + H . Then Nm( F ) ≈ ( F ∗ F ∗ . . . ∗ F p ) ⊠ ( F ∗ . . . ∗ F p ∗ F ) ⊠ . . . ⊠ ( F p ∗ F ∗ . . . ∗ F p − ) , with σ acting by rotating the tensor factors, and the σ -equivariant structure coming fromthe commutativity constraint.Write F ′ := F ∗F ∗ . . . ∗F p ∈ P L + H (Gr H ; O ) . Since i may be identified with the diagonalembedding Gr H ֒ → Gr pH , we have i ∗ (Nm F ) ≈ ( F ′ ) ⊗ p , with σ -equivariant structure givenby cyclic rotation of the tensor factors. In particular, the stalk of i ∗ (Nm F ) at x ∈ Gr H isthe tensor-induction of the stalk of F ′ x from O to O [ σ ] .Hence it suffices to prove that any such tensor induction has Jordan-Hölder constituentsbeing either trivial or free. This is verified by explicit inspection: choosing a basis for F ′ x ,the induced basis of ( F ′ x ) ⊗ p is grouped into either trivial or free orbits under the σ -action.The argument for (ii) is completely analogous (we could also apply Verdier duality to(i)). (cid:3) Smith theory for parity sheaves.
We return momentarily to the general setup forSmith theory: X has a σ -action and Y = X σ . Proposition 3.14 (Variant of [LL, Theorem 6.3]) . Suppose
E ∈ D bS,σ ( X ; O ) is a paritycomplex satisfying the condition: (*) all ∗ and ! -stalks of cohomology sheaves of E at fixed points x ∈ X have O [ σ ] -moduleJordan-Hölder constituents being trivial or free.Then Psm( E ) ∈ D S ( Y ; T O ) is Tate-parity. Proof.
This theorem is closely related to Theorem 6.3 of [LL], except [LL, Theorem 6.3]imposes the stronger condition that the σ -action on all stalks is trivial. This is satisfiedin their application (to the loop-rotation action), but not in ours, so we need to re-do theargument in the requisite generality.Let Y = X σ and i : Y → X , i Xλ : X λ ֒ → X , i Yλ : Y λ ֒ → Y , i λ : Y λ ֒ → X λ . Withoutloss of generality suppose E is an even complex on X . We are given that ( i Xλ ) ? E has O -free cohomology sheaves supported in degrees congruent to † X ( λ ) mod , where ? ∈ {∗ , ! } ;we want to show that ( i Yλ ) ? Psm( E ) has Tate-cohomology sheaves supported in degreescongruent to † Y ( λ ) mod . Unraveling the definitions, we have ( i Yλ ) ∗ Psm( E ) = ( i Yλ ) ∗ T i ∗ E∼ = T ( i Yλ ) ∗ i ∗ E∼ = T ( i λ ) ∗ ( i Xλ ) ∗ E . Similarly, using Lemma 2.4 we have ( i Yλ ) ! Psm( E ) ∼ = T ( i λ ) ! ( i Xλ ) ! E . (3.4)By hypothesis, ( i Xλ ) ∗ E has its cohomology sheaves supported in degrees congruent to † X ( λ )(mod 2) . Moreover, by assumption (*), all the stalks and costalks have Jordan-Hölder con-stituents being even shifts of either trivial or free O [ σ ] -modules. So the stalks of ( i λ ) ∗ ( i Xλ ) ∗ E are supported in degrees congruent to † X ( λ ) (mod 2) , and we must verify that their Tatecohomology groups are also supported in degrees of a single parity.For trivial O [ σ ] -modules the odd Tate cohomology groups vanish by (3.1), while for free O [ σ ] -modules all the Tate cohomology groups vanish. Hence for any O [ σ ] whose Jordan-Hölder constituents are all trivial or free, all odd Tate cohomology groups vanish by thelong exact sequence for Tate cohomology (§2.4.1). This shows that the Tate cohomologysheaves of ( i λ ) ∗ ( i Xλ ) ∗ E are supported in degrees congruent to † X ( λ ) (mod 2) .A completely analogous argument, using (3.4) instead, shows that ( i λ ) ! ( i Xλ ) ! E also hasTate cohomology sheaves supported in degrees congruent to † X ( λ ) (mod 2) . (cid:3) For an O -linear abelian category C , with all Hom-spaces being free O -modules, we denoteby C ⊗ O k the k -linear category obtained by tensoring all Hom-spaces with k over O . Lemma 3.15.
Suppose that all the strata X λ are simply connected and all parity sheaves E ( λ, L ) exist. Then we have that Parity S,σ ( X ; O ) ⊗ O k ∼ −→ Parity S,σ ( X ; k ) . Proof.
To see that the functor is well-defined, we note: • The Hom-spaces of
Parity S,σ ( X ; O ) are all free O -modules by [JMW14, Remark2.7], so that the domain is well-defined. • The functor lands in parity sheaves since the modular reduction of a O -parity sheafis a k -parity sheaf by Remark 3.6.It is essentially surjective because every k -parity sheaf lifts to a O -parity sheaf under ourassumption that all parity sheaves exist and all strata are simply connected (which impliesthat all k -local systems on strata lift to O , since they are trivial). The fact that the functoris fully faithful again follows from [JMW14, Remark 2.7]. (cid:3) QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 19
The base change functor.
We return now to the base change setup of §3.6.1. Let
F ∈
Parity L + G (Gr G ; O ) . Then F ∈ P L + G (Gr G ; O ) is perverse since p is good for b G (this isa part of Theorem 3.8), and Nm( F ) ∈ Parity L + G ⋊ σ (Gr G ; O ) is a parity sheaf by [JMW16,Theorem 1.5]. Furthermore, the σ -equivariant structure on Nm( F ) satisfies the assumption(*) of Proposition 3.14 by Lemma 3.13. Hence we may apply Proposition 3.14 to deducethat Psm(Nm( F )) ∈ Parity ( L + H ) (Gr H ; T O ) is Tate-parity.We claim that moreover Psm(Nm( F )) ∈ Parity L + H ) (Gr H ; T O ) , i.e. is normalized as longas p > . Indeed, suppose Gr λ is the unique orbit dense in the support of Nm( F ) . Then Gr λH = (Gr λG ) σ , and their dimensions are congruent modulo (since [2] ∼ = Id in the Tatecategory). To verify this latter claim, writing λ = ( λ , . . . , λ p ) for λ i ∈ X ∗ ( H ) + , we have ( Gr λG ∩ Gr H = Gr λ H λ = ( λ , . . . , λ ) , Gr λG ∩ Gr H = ∅ otherwise.By [Zhu17, Proposition 2.1.5] we have dim Gr λG = h ρ G , λ i . So we just have to verify that h ρ G , ( λ , . . . , λ ) i ≡ h ρ H , λ i (mod 2) . Indeed, ρ G = ( ρ H , . . . , ρ H ) , so h ρ G , ( λ , . . . , λ ) i = p h ρ H , λ i , and p is odd. Thanks to the claim of the preceding paragraph, we can apply the lifting functor L to Psm(Nm( F )) . At this point we have constructed the diagram Parity L + G (Gr G ; O ) Parity L + H ) (Gr H ; T O )Parity L + G (Gr G , k ) Parity L + H (Gr H ; k ) . Psm ◦ Nm F L By Lemma 3.15, the composite functor factors uniquely through a functor
Parity L + G (Gr G , k ) → Parity L + H (Gr H ; k ) . Definition 3.16.
We define BC ( p ) : Parity L + G (Gr G ; k ) → Parity L + H (Gr H ; k ) to be the functor unique filling in the commutative diagram Parity L + G (Gr G ; O ) Parity L + H ) (Gr H ; T O )Parity L + G (Gr G ; k ) Parity L + H (Gr H ; k ) . Psm ◦ Nm F L BC ( p ) One more step is required to obtain the desired base change functor. On a k -linearadditive category there is an auto-equivalence Frob p of the underlying category, which isthe identity on objects and the Frobenius automorphism ( − ) ⊗ k, Frob p k on morphisms. Wedefine BC := Frob − p ◦ BC ( p ) : Parity L + G (Gr G ; k ) → Parity L + H (Gr H ; k ) . Remark 3.17 (Galois equivariance) . If H base changed from some subfield F ⊂ F , then Aut( F / F ) acts on H F , G F and therefore also on Gr H F , Gr G F . It will be important for uslater that BC is equivariant with respect to this action. This is because the constituentfunctors Nm , i ∗ , T , L , and F all have this property, and F is essentially surjective and full. The use of p being odd is rather superficial here. We could adjust the definition of normalized complexesin the case p = 2 , but ultimately this only extends the final results in type A since is a bad prime in allother types. Remark 3.18.
The construction of BC was motivated by a similar functor “ LL ” appearingin [LL, §6.2], which gives a partial geometric description of the Frobenius contraction functor.Another motivation was the “normalized Brauer homomorphism” of [TV16, §4.3], which ourconstruction categorifies. Theorem 3.19.
Let
Res BC : Rep k ( b G ) → Rep k ( b H ) be restriction along the diagonal embed-ding. We also denote by Res BC the same functor restricted to the subcategories of tiltingmodules. The following diagram commutes:
Parity (Gr G ; k ) Parity (Gr H ; k )Tilt k ( b G ) Tilt k ( b H ) ∼ BC ∼ Res BC Proof sketch.
The argument is given in Appendix A. For now let us just explain the keytrick (which we learned from the proof of [LL, Theorem 7.3]): since
Psm commutes withhyperbolic localization by §2.3, and the restriction functor to a maximal torus
Rep( b H ) → Rep( T b H ) is faithful and injective on tilting objects, one can reduce to the case where H isa torus . In this case the functor can be computed explicitly, since the affine Grassmannianof a torus is simply a discrete set. (cid:3) Functoriality and the excursion algebra
In this section we formalize the abstract excursion algebra
Exc(Γ , L G ) , a device used todecomposable a space into pieces indexed by Langlands parameters. This notion appearsimplicitly in [Laf18a], but there it is the image of the abstract excursion algebra in acertain endomorphism group which is emphasized.Since we work with non-split groups, we first clarify in §4.1 our conventions regarding L -groups. This is a bit subtle, as one finds (at least) two natural versions of the L -groupin the literature: the “algebraic L -group” L G alg , following Langlands, and the “geometric L -group” L G geom , derived from the Geometric Satake equivalence. The difference betweenthem is parallel to the difference between L -algebraicity and C -algebraicity emphasized in[BG14].We emphasize that the unadorned notation L G denotes the algebraic L -group, to be con-sistent with [Laf18a], although the geometric L -group is really what appears more naturallyin our arguments.We introduce two explicit presentations for the excursion algebra in §4.2 and §4.4. Thefirst presentation is more natural for making the connection to Langlands parameters, whichwe recall in 4.3. The second presentation is more amenable to constructing actions of theexcursion algebra, which makes it more convenient for our purposes, and it is the only onethat will be used in the sequel.Finally in §4.5 we explain how functoriality is interpreted in terms of excursion algebras.4.1. Conventions on L -groups and Langlands parameters. For a reductive group G over a field F with separable closure F s , we regard its Langlands dual group b G as asplit reductive group over k . The L -group is a certain semi-direct product L G = b G ⋊ Note that it is not obvious that
Res BC preserves the tilting property, but this follows from the non-trivial theorem (building on work of many authors – see the discussion around [JMW16, Theorem 1.2]) thattensor products of tilting modules are tilting. This image is denoted B in [Laf18a]. QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 21
Gal( F s / F ) . Actually, in the case where F is a local field we shall instead work with the“Weil form” b G ⋊ Weil( F s / F ) . (This is just for consistency with [GL]; because we considermod p representations, in our case it would make no difference to work with the Galoisform.)4.1.1. Algebraic L -group. In fact there are at least two conventions for the definition of the L -group. The one which is more traditionally used in the literature is what we shall callthe algebraic L -group , denoted L G alg , defined as in [TV16, §2.5]. The root datum Ψ( G ) of G F s determines a pinning for b G , which in turns gives a splitting Out( b G ) → Aut( b G ) andan identification Aut(Ψ( G )) ∼ = Out( b G ) . The Gal( F s / F ) -action on Ψ( G ) transports to anaction act alg of Gal( F s / F ) on b G , and we define L G alg to be the semidirect product L G alg := b G ⋊ act alg Gal( F s / F ) . Since the action act alg factors through a finite quotient, we may regard L G alg as a pro-algebraic group over k .4.1.2. Geometric L -group. We now make a different construction of the L -group, using theTannakian theory, following [Zhu15, Appendix A] and [Zhu17, §5.5]. We begin with theGeometric Satake equivalence, P L + G F s (Gr G, F s ; k ) ∼ = Rep k ( b G ) . The Galois group
Gal( F s / F ) acts on Gr G, F s , inducing an action on the neutralized Tan-nakian category (P L + G F s (Gr G, F s ; k ) , H ∗ ( − ) | {z } fiber functor ) . By [Zhu15, Lemma A.1] this in turn in-duces an action act geom of Gal( F s / F ) on b G k . We define L G geom := b G k ⋊ act geom Gal( F s / F ) . In the case at hand we shall see that act geom also factors through a finite quotient of
Gal( F s / F ) , so we may also regard L G geom as a pro-algebraic group.4.1.3. Relation between the two L -groups. The relation between these two actions is as fol-lows. We let ρ be the half sum of positive coroots of G ∨ , and we denote by ρ : G m → G ∨ ad thecorresponding cocharacter. With cyc p : Gal( F s / F ) → F × p denoting the mod p cyclotomiccharacter, let χ denote the composite Gal( F s / F ) cyc p −−−→ F × p ֒ → k × ρ −→ b G ad ( k ) . This induces a homomorphism Ad χ : Gal( F s / F ) → Aut( b G ) . Proposition 4.1 ([Zhu15, Proposition 1.6]) . We have act geom = act alg ◦ Ad χ . Given a choice of lift e χ : Gal( F s / F ) → b G ( k ) of χ , which could for example come from asquare root of the mod p cyclotomic character, we get an isomorphism L G alg ∼ −→ L G geom by ( g, γ ) ( g e χ ( γ − ) , γ ) . (4.1) The cited reference operates over Q p instead of k . However, the stated result follows by reducing thestatement over W ( k ) modulo p . Alternatively, we can apply the same proof as in [Zhu15, Proposition 1.6];the appearance of the cyclotomic character is based on the fact that the first Chern class of a line bundlelies in H (Gr G, F s ; k (1)) Gal( F s / F ) . By [Zhu17, Remark 5.5.8], we can always choose a square root of the cyclotomic characterwhen char( F ) > . However, in general it can happen that L G alg and L G geom are notisomorphic; for an example see [Zhu17, Example 5.5.9].At different points we will want to consider both versions of L -groups. If we write L G without a superscript, then by default we mean the algebraic L -group L G alg .4.1.4. Representation categories.
For any Galois extension F ′ / F such that G F ′ is split, theanalogous construction to §4.1.1 gives a “finite form” algebraic L -group b G ⋊ act alg Gal( F ′ / F ) .We define the category of ( k -linear) algebraic representations of L G alg to be Rep k ( L G alg ) := lim −→ F ′ Rep k ( b G ⋊ act alg Gal( F ′ / F )) . Let
Rep k ( L G geom ) := Rep k ( b G ) Gal( F s / F ) , geom denote the category of continuously Gal( F s / F ) -equivariant objects in Rep k ( b G ) with respect to the geometric action. The Geometric Satakeequivalence induces by descent an equivalence P L + G (Gr G ; k ) ∼ = Rep k ( b G ) Gal( F s / F ) , geom (4.2)where the action of Gal( F s / F ) on Rep k ( b G ) on the right side is via act geom , and on the lefthand side, Gr G is considered over F . By definition, on the right side we take are taking ob-jects on which Gal( F s / F ) acts continuously with its Krull topology. Since k is algebraic over F p , in this case Rep k ( b G ) Gal( F s / F ) , geom can be identified with lim −→ F ′ / F Rep k ( b G ) Gal( F ′ / F ) , geom where the limit runs over finite Galois extensions F ′ / F over which the geometric actionfactors.An isomorphism (4.1) gives an embedding Rep k ( L G alg ) ֒ → Rep k ( b G ) Gal( F s / F ) , geom , whichas just remarked is an equivalence for our choice of k . See [Zhu15, Proposition A.10] for adescription of the essential image in general.4.1.5. Langlands parameters.
Definition 4.2.
Let Γ be a group and Γ be a quotient of Γ acting on b G . A Langlandsparameter from Γ into b G ( k ) ⋊ Γ is a b G ( k ) -conjugacy class of continuous homomorphisms ρ : Γ → b G ( k ) ⋊ Γ , which has the property that the composite map Γ → b G ⋊ Γ → Γ is thegiven quotient Γ ։ Γ .Equivalently, we may view ρ as an element of the continuous cohomology group H (Γ , b G ( k )) ,where the action of Γ on b G ( k ) is the given one (via Γ → Γ ) in the semi-direct product.We will consider Langlands parameters with b G ( k ) ⋊ Γ being either L b G alg ( k ) or L b G geom ( k ) ,and Γ being either Gal( F s /F ) for a global field F or Weil( F v /F v ) for a local field F v .Note that the algebraic Γ -action on b G ( k ) factors through a finite quotient Γ ։ Gal( F ′ / F ) .It is clear that Langlands parameters into L G alg ( k ) are in bijection (under restriction) withLanglands parameters into b G ( k ) ⋊ Gal( F ′ / F ) for any such F ′ .We say that a representation ρ : Γ → L G alg ( k ) is semisimple if the Zariski-closure ofthe image of ρ in b G ( k ) ⋊ Gal( F ′ / F ) , for any finite extension F ′ / F over which the Γ -actionfactors, has reductive component group. Also called “completely reducible” in [BHKT19].
QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 23
Presentation of the excursion algebra.
Let Γ be a group, which is either Gal( F s /F ) for a global field F or Weil( F s /F ) for a local field F . Let G be a reductive group over F and L G alg the algebraic L -group as defined in §4.1.1.We will define the excursion algebra Exc(Γ , L G alg ) to be the commutative algebra over k presented by explicit generators and relations given below. (The topology on Γ will not berelevant for the definition of Exc(Γ , L G alg ) .) For a more conceptual perspective see [Zhu,§2], wherein the excursion algebra is denoted k [ R Γ , L G alg // b G ] .4.2.1. Generators.
We define O ( L G alg k ) := lim −→ O ( b G k ⋊ Gal( F ′ /F )) where the limit runs overfinite extensions F ′ /F over which the Γ -action on b G k factors.Generators of Exc(Γ , L G alg ) will be denoted S I,f, ( γ i ) i ∈ I , where the indexing set ( I, f, ( γ i ) i ∈ I ) consists of:(i) I is a finite (possibly empty) set,(ii) f ∈ O ( b G k \ ( L G alg k ) I / b G k ) := O (( L G alg k ) I ) b G k × b G k , where the quotient is for the actionsof b G k by diagonal left and right translation, respectively, and(iii) γ i ∈ Γ for each i ∈ I .4.2.2. Relations.
Next we describe the relations. (Compare [Laf18a, §10].)(i) S ∅ ,f, ∗ = f (1 G ) .(ii) The map f S I,f, ( γ i ) i ∈ I is a k -algebra homomorphism in f , i.e. S I,f + f ′ , ( γ i ) i ∈ I = S I,f, ( γ i ) i ∈ I + S I,f ′ , ( γ i ) i ∈ I ,S I,ff ′ , ( γ i ) i ∈ I = S I,f, ( γ i ) i ∈ I · S I,f ′ , ( γ i ) i ∈ I , and S I,λf, ( γ i ) i ∈ I = λS I,f, ( γ i ) i ∈ I for all λ ∈ k. (iii) For all maps of finite sets ζ : I → J , all f ∈ O ( b G k \ ( L G alg k ) I / b G k ) , and all ( γ j ) j ∈ J ∈ Γ J ,we have S J,f ζ , ( γ j ) j ∈ J = S I,f, ( γ ζ ( i ) ) i ∈ I where f ζ ∈ O ( b G k \ ( L G alg k ) J / b G k ) is defined by f ζ (( g j ) j ∈ J ) := f (( g ζ ( i ) ) i ∈ I ) .(iv) For all f ∈ O ( b G k \ ( L G alg k ) I / b G k ) and ( γ i ) i ∈ I , ( γ ′ i ) i ∈ I , ( γ ′′ i ) i ∈ I ∈ Γ I , we have S I ⊔ I ⊔ I, e f, ( γ i ) i ∈ I × ( γ ′ i ) i ∈ I × ( γ ′′ i ) i ∈ I = S I,f, ( γ i ( γ ′ i ) − γ ′′ i ) i ∈ I , where e f ∈ O ( b G k \ ( L G alg k ) I ⊔ I ⊔ I / b G k ) is defined by e f (( g i ) i ∈ I × ( g ′ i ) i ∈ I × ( g ′′ i ) i ∈ I ) = f (( g i ( g ′ i ) − g ′′ i ) i ∈ I ) . (v) If f is inflated from a function on Γ I , then S I,f, ( γ i ) i ∈ I equals the scalar f (( γ i ) i ∈ I ) . Moregenerally, if J is a subset of I and f is inflated from a function on ( b G k \ ( L G alg k ) J / b G k ) × Γ I \ J , then we have S I,f, ( γ i ) i ∈ I = S J, ˇ f, ( γ j ) j ∈ J where ˇ f (( g j ) j ∈ J ) := f (( g j ) j ∈ J , ( γ i ) i ∈ I \ J ) . (Compare [Laf18a, p. 164].) Definition 4.3.
The excursion algebra
Exc(Γ , L G alg ) is the k -algebra with generators andrelations specified as above. Constructing Galois representations.
The following result of Lafforgue (general-ized to modular coefficients by Böckle-Harris-Khare-Thorne) explains how to obtain Lang-lands parameters from characters of
Exc(Γ , L G alg ) . Proposition 4.4 ([BHKT19, Theorem 4.5], [Laf18a, §13]) . For any character ν : Exc(Γ , L G alg ) → k , there is a semisimple representation ρ ν : Γ → L G alg ( k ) , unique up to conjugation by b G ( k ) ,which is characterized by the following condition:For all n ∈ N , f ∈ O ( b G k \ ( L G alg k ) n +1 / b G k ) , and ( γ , . . . , γ n ) ∈ Γ n +1 , we have ν ( S { ,...,n } ,f, ( γ ,γ ,...,γ n ) ) = f (( ρ ( γ γ n ) , ρ ( γ γ n ) , . . . , ρ ( γ n − γ n ) , ρ ( γ n ))) . (4.3)4.4. Another presentation for the excursion algebra.
We will now describe a secondpresentation of
Exc(Γ , L G alg ) , following [Laf18a, Lemma 0.31], which is more useful forconstructing actions of Exc(Γ , L G alg ) in practice.4.4.1. Generators.
We take a set of generators indexed by tuples of data of the form ( I, W, x, ξ, ( γ i ) i ∈ I ) , where:(i) I is a finite set,(ii) W ∈ Rep k (( L G alg ) I ) (cf. §4.1.4),(iii) x ∈ W is a vector invariant under the diagonal b G k -action,(iv) ξ ∈ W ∗ is a functional invariant under the diagonal b G k -action,(v) γ i ∈ Γ for each i .The corresponding generator of Exc(Γ , L G alg ) will be denoted by S I, ⊠ i ∈ I V i ,x,ξ, ( γ i ) i ∈ I ∈ Exc(Γ , L G alg ) .4.4.2. Relations.
Next we describe the relations.(i) S ∅ ,x,ξ, ∗ = h x, ξ i .(ii) For any morphism of ( L G alg k ) I -representations u : W → W ′ and functional ξ ′ ∈ ( W ′ ) ∗ invariant under the diagonal b G k -action, we have S I,W,x, t u ( ξ ′ ) , ( γ i ) i ∈ I = S I,W ′ ,u ( x ) ,ξ ′ , ( γ i ) i ∈ I , (4.4)where t u : ( W ′ ) ∗ → W ∗ denotes the dual to u .(iii) For two tuples ( I , W , x , ξ , ( γ i ) i ∈ I ) and ( I , W , x , ξ , ( γ i ) i ∈ I ) as in §4.4.1, wehave S I ⊔ I ,W ⊠ W ,x ⊠ x ,ξ ⊠ ξ , ( γ i ) i ∈ I × ( γ i ) i ∈ I = S I ,W ,x ,ξ , ( γ i ) i ∈ I ◦ S I ,W ,x ,ξ , ( γ i ) i ∈ I . (4.5)Letting ∆ : → ⊕ be the diagonal inclusion, and ∇ : ⊕ → the addition map,we also have S I ⊔ I ,W ⊕ W , ( x ⊕ x ) ◦ ∆ , ∇◦ ( ξ ⊕ ξ ) , ( γ i ) i ∈ I × ( γ i ) i ∈ I = S I ,W ,x ,ξ , ( γ i ) i ∈ I + S I ,W ,x ,ξ , ( γ i ) i ∈ I . (4.6)Furthermore, the assignment ( I, ⊠ i ∈ I V i , x, ξ, ( γ i ) i ∈ I ) S I, ⊠ i ∈ I V i ,x,ξ, ( γ i ) i ∈ I ∈ Exc(Γ , L G alg ) is k -linear in x and ξ .(iv) Let ζ : I → J be a map of finite sets. Suppose W ∈ Rep(( L G ) I ) , x : → W | ∆( b G ) , ξ : W | ∆( b G ) → , and ( γ j ) j ∈ J ∈ Γ J . Letting W ζ be the restriction of W under thefunctor Rep(( L G ) I ) → Rep(( L G ) J ) induced by ζ , we have S J,W ζ ,x,ξ, ( γ j ) j ∈ J = S I,W,x,ξ, ( γ ζ ( i ) ) i ∈ I . (4.7) QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 25 (v) Letting δ W : → W ⊗ W ∗ and ev W : W ∗ ⊗ W → be the natural counit and unit, wehave S I,W,x,ξ, ( γ i ( γ ′ i ) − γ ′′ i ) i ∈ I = S I ⊔ I ⊔ I,W ⊠ W ∗ ⊠ W,δ W ⊠ x,ξ ⊠ ev W , ( γ i ) i ∈ I × ( γ ′ i ) i ∈ I × ( γ ′′ i ) i ∈ I . (4.8)(vi) If W is inflated from a representation of ( L G alg ) J × Γ I \ J , then we have S I,W,x,ξ, ( γ i ) i ∈ I = S J,W | ( LG alg) J , ((1 j ) j ∈ J , ( γ i ) i ∈ I \ J ) · x,ξ, ( γ j ) j ∈ J . Relation between the presentations.
The two presentations in §4.2 and §4.4 are relatedas follows. The generator S I, ⊠ i ∈ I V i ,x,ξ, ( γ i ) i ∈ I corresponds to S I,f x,ξ , ( γ i ) i ∈ I where f x,ξ is thefunction on ( L G k ) I given by ( g i ) i ∈ I
7→ h ξ, ( g i ) i ∈ I · x i . The assumptions on ξ and x imply that f x,ξ is invariant under the left and right diagonal b G k -actions. The relations in §4.4.2 implythat S I,W,x,ξ, ( γ i ) i ∈ I depends only on f x,ξ (and not on the choice of x, ξ ) [Laf18a, Lemme10.6].4.5. Functoriality for excursion algebras.
A homomorphism of L -groups φ : L H alg → L G alg is admissible if it lies over the identity map on Γ , i.e. the diagram below commutes. L H alg L G alg Γ Γ φ Id Lemma 4.5.
Let φ : L H alg → L G alg be an admissible homomorphism. Then there is ahomomorphism φ ∗ : Exc(Γ , L G alg ) → Exc(Γ , L H alg ) which on k -points sends a parameter ρ ∈ H (Γ , b H ( k )) to φ ◦ ρ ∈ H (Γ , b G ( k )) .Proof. The map φ induces Res φ : Rep k ( L G alg ) → Rep k ( L H alg ) . At the level of generators,the map φ ∗ sends S V,x,ξ, { γ } i ∈ I S Res φ ( V ) , Res φ ( x ) , Res φ ( ξ ) , { γ i } i ∈ I . We verify by inspection that this map sends relations to relations. To see that this indeedinduces composition with φ at the level of Langlands parameters, use (4.3). (cid:3) Definition 4.6 (Base change) . In the base change situation, where H is a reductive groupover F and G = Res E/F ( H E ) , the relevant morphism of L -groups φ BC : L H alg → L G alg is defined by the formula ( h, γ ) (∆( h ) , γ ) . In fact this same formula also defines thecorrresponding map of geometric L -groups φ geom BC : L H geom → L G geom , so φ geom BC and φ BC arecompatible with (4.1) if we use the same choice of square root of the cyclotomic characterin the latter to define isomorphisms L H alg ≈ L H geom and L G alg ≈ L G geom . We denote φ ∗ BC : Exc(Γ , L G alg ) → Exc(Γ , L H alg ) the induced map of excursion algebras.5. Cyclic base change in the global setting
In this section we will prove Theorem 1.1. This will require knowledge of how Lafforgue’sparametrization works, which we summarize in §5.2. It is based on interpreting the spaceof automorphic functions as the cohomology of moduli spaces of shtukas, and constructingan action of the excursion algebra on it using geometry. We briefly recall the definitions ofthe relevant geometric objects in §5.1.The main work occurs in §5.3, where we use a variant of Lafforgue’s ideas to constructand analyze an action of the “ σ -equivariant excursion algebra” on the Tate cohomology of moduli spaces of shtukas. In the base change situation, equivariant localization mediatesbetween the Tate cohomology of shtukas for G and for H , allowing us to relate certainexcursion operators for the two groups. This is then used in §5.4 to establish the existenceof base change for mod p automorphic forms; it will also be the crucial input for our localresults in the next section.5.1. Moduli of shtukas.
We will use the theory of moduli stacks of shtukas, due to Drinfeldand generalized by Varshavsky. Here we very briefly recall the relevant definitions in orderto set notation. More comprehensive references include [Var04] and [Laf18a].5.1.1.
Shtukas.
Fix a smooth projective curve X over a finite field F ℓ of characteristic = p .For an affine group scheme G → X and a finite set I , the stack Sht
G,I represents thefollowing moduli functor on F ℓ -schemes S : Sht
G,I : S ( x i ) i ∈ I ∈ X I ( S ) E = fppf G -torsor over X × Sϕ : E| X × S − S i ∈ I Γ xi ∼ −→ τ E| X × S − S i ∈ I Γ xi , where τ is the Frobenius Frob ℓ on the S factor in X × S , and τ E is the pullback of E underthe map × τ : X × S → X × S .Geometrically, Sht
G,I has a Schubert stratification whose strata are Deligne-Mumfordstacks locally of finite type. We regard it as an ind-(locally finite type) Deligne-Mumfordstack.5.1.2.
Hecke stack.
The Hecke stack Hk G,I classifies Hk G,I : S ( x i ) i ∈ I ∈ X I ( S ) E , E ′ = fppf G -torsors over X × Sϕ : E| X × S − S Γ xi ∼ −→ E ′ | X × S − S Γ xi . The Geometric Satake equivalence provides a functor
Rep k (( L G ) I ) → D (Hk G,I ; k ) , whichwe normalize as in [Laf18a, Theorem 0.9].5.1.3. Satake sheaves.
There is a map
Sht
G,I → Hk G,I sending ( { x i } i ∈ I , E , ϕ ) to ( { x i } i ∈ I , E , τ E , ϕ ) .Composing with the ∗ -pullback through Sht
G,I → Hk G,I induces a functor
Sat geom : Rep k ( b G I ) Gal( F s /F ) , geom → D b (Sht G,I ; k ) . Finally, we may identify
Rep k (( L G alg ) I ) ∼ −→ Rep k ( b G I ) Gal( F s /F ) , geom as in §4.1.4, giving afunctor (cf. [Laf18a, Theorem 0.11]) Sat : Rep k (( L G alg ) I ) → D b (Sht G,I ; k ) . The Schubert stratification is defined by the support of the sheaves in the image of
Sat ,with the closure relations corresponding to the Bruhat order. (In particular,
Sat lands inthe derived category of sheaves constructible with respect to the Schubert stratification on
Sht
G,I .)5.1.4. There is a map π I : Sht G,I → X I projecting a tuple ( { x i } i ∈ I , E , ϕ i ) to { x i } i ∈ I .5.1.5. Level structures.
For D ⊂ X a finite-length subscheme, there are level covers Sht
G,D,I → Sht
G,I | ( X − D ) I which parametrize the additional datum of a τ -equivariant trivialization of E over S × D . Note that by definition, the “legs” { x i } i ∈ I ∈ ( X − D )( S ) I avoid D . QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 27
Iterated shtukas.
Let I , . . . , I r be a partition of I . We define Sht ( I ,...,I r ) G,D,I (sometimescalled a moduli stack of iterated shtukas ) to be the stack
Sht ( I ,...,I r ) G,D,I : S ( x i ) i ∈ I ∈ X I ( S ) E , . . . , E r = fppf G -torsors over X × Sϕ j : E j − | X × S − S i ∈ Ij Γ xi ∼ −→ E j | X × S − S i ∈ Ij Γ xi j = 1 , . . . , rϕ : E r ∼ −→ τ E trivialization over D × S . There is a map ν : Sht ( I ,...,I r ) G,D,I → Sht
G,D,I . A key property of this morphism is that it isstratified small (with respect to the Schubert stratification), which is a consequence of thesame property of the convolution morphism for Beilinson-Drinfeld Grassmannians.5.1.7.
Partial Frobenius.
There is a partial Frobenius F I : Sht ( I ,...,I r ) G,D,I → Sht ( I ,...,I r ,I ) G,D,I sending x i ( τ x i i ∈ I x i otherwise ( E , . . . , E r ) ( E , . . . , E r , τ E )( ϕ , . . . , ϕ r ) ( ϕ , . . . , ϕ r , τ ϕ ) . It lies over the partial Frobenius
Frob I on X I (applying Frob ℓ to the coordinates indexedby i ∈ I ), so that the diagram below is commutative (and cartesian up to radiciel maps): Sht ( I ,...,I r ) G,D,I
Sht ( I ,...,I r ,I ) G,D,I X I X IF I ν ν Frob I (5.1)5.1.8. Base change setup.
We now consider the following “base-change setup”. Let F be thefunction field of X and H F a reductive group over F . We choose a parahoric extension of H F to a smooth affine group scheme H over X .Let E/F be a cyclic extension of F having degree p , so E corresponds to the functionfield of a smooth projective curve X ′ . Define G := Res X ′ /X ( H X ′ ) , which is an affine groupscheme over X with generic fiber G F ∼ = Res E/F ( H E ) . The group scheme G → X comeswith an induced action of h σ i = Aut( X ′ /X ) .5.2. Review of V. Lafforgue’s global Langlands correspondence.
Write
Γ = Gal( F s /F ) .In [Laf18a, §13], Lafforgue constructs an action of Exc(Γ , L G alg ) on the space of cusp formsfor G with coefficients in k . This has been improved by Cong Xue, who extended the actionto all compactly supported functions [Xuea, §7]. We summarize the construction of the excursion action, as we shall make use of someof its internal aspects, and we also need to explain why it can be used to construct someexcursion actions on Tate cohomology. The cited paper is written for split G , but the argument can be generalized, as will appear in forth-coming work of Xue (announced in [Xueb]). Constructing actions of the excursion algebra.
We will explain an abstract setup thatgives rise to actions of the excursion algebra.
Definition 5.1.
Let A be a (not necessarily commutative) ring. A family of functors H I : Rep k (( L G ) I ) → Mod A (Γ I ) , where I runs over (possibly empty) finite sets, is admissible if it satisfies the two conditions below.(1) (Compatibility with fusion) For all ζ : I → J , there is a natural isomorphism χ ζ between the functors H I ◦ Res ζ and Res ζ ◦ H J in the diagram: Rep k (( L G ) I ) Mod A (Γ I )Rep k (( L G ) J ) Mod A (Γ J ) H I χ ζ Res ζ Res ζ H J (5.2)(2) (Compatibility with composition) For I ′ ζ ′ −→ I ζ −→ J , we have χ ζ ◦ ζ ′ = χ ζ ◦ χ ζ ′ . Construction 5.2.
Let denote the trivial representation of L G . Given an admissiblefamily of functors H I : Rep k (( L G ) I ) → Mod A (Γ I ) , we get an A -linear action of Exc(Γ , L G ) on H { } ( ) as follows.For a tuple ( I, W, x, ξ, ( γ i ) i ∈ I ) we define an endomorphism, which gives the image of S I,W,x,ξ, ( γ i ) i ∈ I in End A ( H { } ( )) , by the following composition: H { } ( ) H { } ( W ζ ) H I ( W ) H I ( W ) H { } ( W ζ ) H { } ( ) . H { } ( x ) ∼ χ ζ ( γ i ) i ∈ I ∼ χ − ζ H { } ( ξ ) From the assumptions of admissibility it is straightforward to check the relations in §4.4.2.
Remark 5.3.
Note that it follows from admissibility that the A -module underlying H I ( ) for any I is identified with H ∅ ( ) by χ ∅→{ } . Proposition 4.4 then attaches a Galois repre-sentation to each generalized eigenvector for the Exc(Γ , L G ) -action on H ∅ ( ) . (Of course,such an eigenvector is not guaranteed to exist in general.)5.2.2. Excursion action on the cohomology of shtukas.
Let H G be the Hecke algebra actingon Sht
G,D ; it is the tensor product of local Hecke algebras with the level structure dictatedby D . For any finite set I , we have a map Rπ I : Sht G,D,I → ( X − D ) I remembering the points of the curve indexed by I (which avoid D by definition). Let η I denote the generic point of X I and η I the spectum of an algebraic closure, viewed as ageometric generic point of X I . When I is a singleton, we will just abbreviate these by η and η .We will define a family of functors indexed by finite sets I : H I : Rep k (( L G alg ) I ) → Mod H G (Γ I ) (5.3)sending V ∈ Rep k (( L G alg ) I ) to R π I ! (Sht G,D,I | η I ; Sat( V )) . (5.4)Here and throughout, we use the perverse t-structure in formation of R π I ! . Note that apriori H I ( V ) has an action of π ( η I , η I ) , which maps to Γ I but neither injectively norsurjectively. The map is non-canonical: it depends on a choice of specialization as in [Laf18a, Remark 8.18].
QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 29 π ( η I , η I ) extends canonically to an action of Γ I . Assume I isnon-empty, since otherwise there is nothing to prove. The Satake functor of §5.1.3 admitsa generalization Sat ( I ,...,I r ) : Rep k (( L G ) I ) → D b (Sht ( I ,...,I r ) G,D,I ; k ) , such that the map ν : Sht ( I ,...,I r ) G,D,I → Sht
G,D,I has the property that Rν ! Sat ( I ,...,I r ) ( V ) ∼ = Sat( V ) . Furthermore, there are natural iso-morphisms F ∗ I Sat ( I ,...,I r ) ( V ) ∼ = Sat ( I ,...,I r ,I ) ( V ) , where F I is the partial Frobenius from§5.1.7.Write I = { , . . . , n } . Thanks to the above properties and (5.1), the partial Frobeniusmaps on Sht ( { } ,..., { n } ) G,D,I then induce maps
Frob ∗{ } H I ( V ) ∼ −→ H I ( V ) . That equips H I ( V ) with the action of the larger group FWeil( η I , η I ) that we now recall,summarizing [Laf18a, Remarque 8.18]. Let F I denote the function field of X I , so η I =Spec F I , and F I an algebraic closure, so we may take η I = Spec F I . Write ( F I ) perf forthe perfect closure of F I , and Frob { i } for the “partial Frobenius” automorphism of ( F I ) perf induced by Frob q on the i th factor. We define FWeil( η I , η I ) := { γ ∈ Aut F q ( F I ) : ∃ ( n i ) i ∈ I ∈ Z I such that γ | ( F I ) perf = Y i ∈ I (Frob { i } ) n i } . Writing π geom1 ( η I , η I ) := ker( π ( η I , η I ) deg −−→ b Z ) , this fits into an extension → π geom1 ( η I , η I ) → FWeil( η I , η I ) → Z I → . Fixing a specialization morphism η I ∆( η { } ) induces a surjection FWeil( η I , η I ) ։ Weil( η, η ) I . A form of Drinfeld’s Lemma [Xuea, Lemma 7.4.2] is used to show that the action of
FWeil( η I , η I ) on H I ( V ) factors through Weil( F s /F ) I ; continuity considerations then implythat the action extends uniquely to one of Γ I . Example 5.4.
Let us unravel H { } ( ) = R π { } ! (Sht G,D, { } | η { } ; Sat( )) . (5.5)By Remark 5.3 the underlying Hecke module of H { } ( ) is isomorphic to H ∅ ( ) . Accordingto [Laf18a, Remarque 12.2], this is the space of compactly supported k -valued functions onthe discrete groupoid Bun
G,D ( F ℓ ) = a α ∈ ker ( F,G ) G α ( F ) \ G α ( A F ) / Y v K v ! , (5.6)where G α is the pure inner form of G corresponding to α , K v = G ( O v ) for v / ∈ D , and K v = ker( G ( O v ) → G D ) .The family of functors H I is admissible; this is an immediate consequence of the fact that Sat is already compatible compatible with composition and fusion. Hence Construction 5.2applies to define an action of
Exc(Γ , L G ) on C ∞ c (Bun G,D ( F ℓ ); k ) . Elements of the image of Exc(Γ , L G ) in End( C ∞ c (Bun G,D ( F ℓ ); k )) are called “excursion operators”. Xue’s generalization.
The subspace C ∞ cusp (Bun G,D ( F ℓ ); k ) ⊂ C ∞ c (Bun G,D ( F ℓ ); k ) ofcusp forms is finite-dimensional and stable under the Exc(Γ , L G ) -action, and therefore de-composes into a direct sum of generalized eigenspaces under the action of Exc(Γ , L G ) . UsingProposition 4.4, this decomposition corresponds to a parametrization by Langlands param-eters.We cannot find a larger finite-dimensional subspace of C ∞ c (Bun G,D ( F ℓ ); k ) stable un-der Exc(Γ , L G ) . However, we can find finite-dimensional quotient spaces on which the Exc(Γ , L G ) -action descends.For example, quotients of the following form arise in [Xue20, Theorem 3.6.7]. Since Exc(Γ , L G ) acts Hecke-equivariantly on C ∞ c (Bun G,D ( F ℓ ); k ) , and the latter is a finite H G,u -module for u / ∈ D , any finite-codimension ideal I ⊂ H G,u for such u gives a (possiblyzero) finite-dimensional quotient space C ∞ c (Bun G,D ( F ℓ ); k ) ⊗ H G,u ( H G,u / I ) which carriesa Exc(Γ , L G ) -action.We will consider any Langlands parameter arising via Proposition 4.4 from the Exc(Γ , L G ) -action on any finite-dimensional Exc(Γ , L G ) -equivariant quotient of C ∞ c (Bun G,D ( F ℓ ); k ) to“arise from an automorphic form” for the purpose of Theorem 1.1 .By the finiteness of C ∞ c (Bun G,D ( F ℓ ); k ) over Exc(Γ , L G ) , we can state this equivalentlyas: a Langlands parameter ρ “arises from an automorphic form” if the corresponding maximalideal m ρ ⊂ Exc(Γ , L G ) is in the support of C ∞ c (Bun G,D ( F ℓ ); k ) as an Exc(Γ , L G ) -module.5.3. Excursion action on the Tate cohomology of shtukas.
For a category C with σ -action, we let C σ -eq denote the category of σ -equivariant objects in C . This comes equippedwith a forgetful functor to C .5.3.1. Tate cohomology of shtukas. If σ acts on G , it induces an action V σ V on Rep( L G ) .Given V ∈ Rep k (( L G alg ) I ) σ -eq , we can form Rπ I ! (Sht G | η I ; Sat( V )) as above. The σ -equivariant structure on V equips this with a σ -equivariant structure; more formally, because Sat and π I are σ -equivariant, Rπ I ! (Sht G , Sat( − )) lifts to a functor Rep k (( L G alg ) I ) σ -eq → D ( X I ; k ) σ -eq . Hence we can form T j ( Rπ I ! )(Sht G,D,I | η I ; Sat( V )) , the Tate cohomology(§2.4) of ( Rπ I ! )(Sht G | η I ; Sat( V )) ; we shall always do this with respect to the perverse t-structure. To ease notation, we will abbreviate T j (Sht G,D,I ; V ) := T j ( Rπ I ! )(Sht G,D,I | η I ; Sat( V )) . (5.7)Let us explain in what category we regard (5.7). Since ( Rπ I ! )(Sht G,D,I | η I ; Sat( V )) hascommuting actions of FWeil( η I , η I ) and the Hecke algebra H G (the former commutingwith the σ -action), its Tate cohomology has commuting actions of FWeil( η I , η I ) and of T ( H G ) , where Tate cohomology is formed with respect to the σ -action. We regard (5.7)as a T ( H G )[FWeil( η I , η I )] -module, a priori. (Later we will see that the FWeil( η I , η I ) -action factors uniquely through a π ( η, η ) I -action, and it will be natural to regard (5.7) asa T ( H G )[ π ( η, η ) I ] -module.) Remark 5.5 (Automorphisms of shtukas) . For any G -torsor E on X and any point v ∈ X ,we have a restriction map Aut( E ) ev v −−→ Aut( E| v ) ∼ = G. The kernel of ev v is unipotent, since Aut( E ) embeds into the group of automorphisms of E restricted to a formal disk around v , which is G ( O v ) , and the kernel of the evaluation map G ( O v ) → G is pro-unipotent. By [Xuea] for split groups, and its forthcoming generalization for non-split groups.
QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 31
Hence as soon as D is non-empty, the support of Sat( V ) in Sht
G,D,I is locally finitetype with stabilizers being finite (char( F ℓ ) = p ) -groups, which therefore have trivial groupcohomology with coefficients in k . Therefore, ( Rπ I ! )(Sht G,D,I | η I ; Sat( V )) lies in the bounded derived category D b ( η I ; k ) , so we may apply the results on Tate cohomology of boundedcomplexes from §2.4. We will always assume that D is non-empty so that this holds. Lemma 5.6.
The diagonal map H → G induces an isomorphism Sht ( I ,...,I r ) H,D,I ∼ = (Sht ( I ,...,I r ) G,D,I ) σ as subfunctors of Sht ( I ,...,I r ) G,D,I .Proof.
For notational convenience we just treat the case of non-iterated shtukas,
Sht
G,D,I ;the general case is essentially the same but with cumbersome extra notation.The main point that the “diagonal” map”
Bun
H,D ∼ −→ Bun σG,D is already an isomorphism.Indeed, there is an equivalence of categories between
Res X ′ /X ( H ) -bundles on X S and H -bundles on X ′ S , which we denote E 7→ E ′ . Then straightforward definition-chasing showsthat the datum of an isomorphism of E with its σ -twist, exhibiting E as a σ -fixed point of Bun
G,D , translates to a descent datum for E ′ to descend to an H -bundle on X S . This iscompatible with level structures: a level structure on E ′ descends to E if and only if it is σ -equivariant.More generally, notate the S -points of Sht
G,D,I as (the groupoid) { ( { x i } i ∈ I , E , ϕ ) } . Thesubfunctor (Sht G,D,I ) σ parametrizes the groupoid of such data where E is equipped witha σ -equivariant structure, and ϕ : E| X × S \ S Γ xi ∼ −→ E| X × S \ S Γ xi is σ -equivariant. By thepreceding paragraph, E is induced by an H -bundle on X S . It is similarly straightforwardto check that the σ -equivariance of ϕ is equivalent to it being induced by a map of H -bundles. (cid:3) From Lemma 2.6 we deduce the following simple but important identity:
Lemma 5.7.
Suppose σ acts trivially on Sht H and F . Then T ∗ ( Rπ I ! )(Sht H,D,I | η I ; F ) ∼ = R ∗ π I ! (Sht H,D,I | η I ; F ) ⊗ T ∗ ( k ) . σ -equivariant excursion algebra. Definition 5.8.
We define the
Exc(Γ , L G ) σ -eq to be the algebra on generators S V,x,ξ, ( γ i ) i ∈ I where • V ∈ Rep k (( L G ) I ) σ -eq , • x : → V | ∆( b G ) and ξ : V | ∆( b G ) → are σ -equivariant morphisms of b G -representations,and • ( γ i ) i ∈ I ⊂ Γ I ,with the following relations.(i) S ∅ ,x,ξ, ∗ = h x, ξ i .(ii) For any σ -equivariant morphism of σ -equivariant ( L G ) I -representations u : W → W ′ and functional ξ ′ ∈ ( W ′ ) ∗ invariant under the diagonal b G ⋊ σ -action, we have S I,W,x, t u ( ξ ′ ) , ( γ i ) i ∈ I = S I,W ′ ,u ( x ) ,ξ ′ , ( γ i ) i ∈ I , (5.8)where t u : ( W ′ ) ∗ → W ∗ denotes the dual to u .(iii) For two tuples ( I , W , x , ξ , ( γ i ) i ∈ I ) and ( I , W , x , ξ , ( γ i ) i ∈ I ) as above, we have S I ⊔ I ,W ⊠ W ,x ⊠ x ,ξ ⊠ ξ , ( γ i ) i ∈ I × ( γ i ) i ∈ I = S I ,W ,x ,ξ , ( γ i ) i ∈ I ◦ S I ,W ,x ,ξ , ( γ i ) i ∈ I . (5.9) Letting ∆ : → ⊕ be the diagonal inclusion, and ∇ : ⊕ → the addition map,we also have S I ⊔ I ,W ⊕ W , ( x ⊕ x ) ◦ ∆ , ∇◦ ( ξ ⊕ ξ ) , ( γ i ) i ∈ I × ( γ i ) i ∈ I = S I ,W ,x ,ξ , ( γ i ) i ∈ I + S I ,W ,x ,ξ , ( γ i ) i ∈ I . (5.10)Furthermore, the assignment ( I, ⊠ i ∈ I V i , x, ξ, ( γ i ) i ∈ I ) S I, ⊠ i ∈ I V i ,x,ξ, ( γ i ) i ∈ I ∈ Exc(Γ , L G alg ) σ -eq is k -linear in x and ξ .(iv) Let ζ : I → J be a map of finite sets. Suppose W ∈ Rep(( L G ) I ) σ -eq , x : → W | ∆( b G ) , ξ : W | ∆( b G ) → , and ( γ j ) j ∈ J ∈ Γ J . Letting W ζ be the restriction of W under thefunctor Rep(( L G ) I ) σ -eq → Rep(( L G ) J ) σ -eq induced by ζ , we have S J,W ζ ,x,ξ, ( γ j ) j ∈ J = S I,W,x,ξ, ( γ ζ ( i ) ) i ∈ I . (5.11)(v) Letting δ W : → W ⊗ W ∗ and ev W : W ∗ ⊗ W → be the natural counit and unit, wehave S I,W,x,ξ, ( γ i ( γ ′ i ) − γ ′′ i ) i ∈ I = S I ⊔ I ⊔ I,W ⊠ W ∗ ⊠ W,δ W ⊠ x,ξ ⊠ ev W , ( γ i ) i ∈ I × ( γ ′ i ) i ∈ I × ( γ ′′ i ) i ∈ I . (5.12)(vi) If W is inflated from a representation of ( L G alg ) J × Γ I \ J , then we have S I,W,x,ξ, ( γ i ) i ∈ I = S J,W | ( LG alg) J , ((1 j ) j ∈ J , ( γ i ) i ∈ I \ J ) · x,ξ, ( γ j ) j ∈ J . In short,
Exc(Γ , L G ) σ -eq has the same type of generators and relations as in §4.4, but alldata must be σ -equivariant. Remark 5.9 ( σ -action on the excursion algebra) . Since σ acts on G , it acts on Exc(Γ , L G alg ) by transport of structure. Concretely, we have σ · S V,x,ξ, ( γ i ) i ∈ I = S σ ( V ) ,σ ( x ) ,σ ( ξ ) , ( γ i ) i ∈ I . (5.13)There is an obvious map Exc(Γ , L G ) σ -eq → Exc(Γ , L G ) sending S V,x,ξ, ( γ i ) i ∈ I ∈ Exc(Γ , L G ) σ -eq to the element with the same name in Exc(Γ , L G ) .It seems natural to ask if this map is injective and identifies Exc(Γ , L G ) σ -eq with the σ -invariants on Exc(Γ , L G ) σ ⊂ Exc(Γ , L G ) . We believe this is true at least in characteristic . Lemma 5.10.
Recall the Tate cohomology spectral sequence §2.4.4, E ij = R j π I ! (Sht G,D,I | η ; Sat( V )) = ⇒ T i + j (Sht G,D,I ; V ) . (i) There is an Exc(Γ , L G ) σ -eq -action on the Tate cohomology spectral sequence E rij = ⇒ T ∗ (Sht G,D,I ; V ) , such that the diagrams E rij ker( d rij ) ker( d rij ) / Im ( d ri − r,j + r − ) = E r +1 ij are all Exc(Γ , L G ) σ -eq -equivariant. The Exc(Γ , L G ) σ -eq -action on every term for r ≥ factors through the map Exc(Γ , L G ) σ -eq → Exc(Γ , L G ) from Remark 5.9. QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 33 (ii) There is an
Exc(Γ , L G ) σ -eq -action on T j (Sht G,D,I ; V ) , which preserves the (increas-ing) filtration F • T j (Sht G,D,I ; V ) induced by the Tate cohomology spectral sequence2.4.4, so that the diagrams T j (Sht G,D,I ; V ) F i ( T j (Sht G,D,I ; V )) F i ( T j (Sht G,D,I ; V )) /F i − ( T j (Sht G,D,I ; V )) = E ij ∞ are all Exc(Γ , L G ) σ -eq -equivariant, with the action on E ij ∞ being the same as in part(i).Proof. For part (i), the existence of the action is formal from the fact that the
Exc(Γ , L G ) σ -eq -action on Rπ I ! (Sht G,D,I ; Sat( V )) commutes with σ , and the definition of the Tate doublecomplex (2.4). The factorization of the action through Exc(Γ , L G ) σ -eq → Exc(Γ , L G ) fol-lows from the fact that E ij = H jc (Sht G,D,I | η I ; Sat( V )) , on which the action factors through Exc(Γ , L G ) by Lafforgue-(Xue)’s construction.For part (ii), we begin by constructing the action. We will define a family of functors T jI : Rep(( L G ) I ) σ -eq → Rep T H G (Γ I ) which is compatible with composition and fusion.From this, the action of Exc(Γ , L G ) σ -eq is defined as in Construction 5.2. We set T jI ( V ) := T j (Sht G,D,I ; V ) regarded a priori as a T ( H G )[FWeil( η I , η I )] -module. The compatibility with fusion andcomposition follow formally from these same properties of the functor Sat . The extension ofthe natural π ( η I , η I ) -action to an FWeil( η I , η I ) -action using partial Frobenius is the sameas in §5.2. The only issue is to check that the FWeil( η I , η I ) -action on T ∗ (Sht G ; V ) factorsthrough π ( η, η ) I .This will follow from Drinfeld’s Lemma in the form [Xuea, Lemma 7.4.2] as soon as weestablish that T j (Sht G , V ) is a finite module over some A -algebra such that the A -actioncommutes with the action of FWeil( η I , η I ) . We take A = T ( H G,u ) for some u where G is hyperspecial. By the generalization of [Xuea, Theorem 0.0.3] to non-split groups (toappear in [Xueb]), we know that R j π I ! (Sht G,D,I | η I ; Sat( V )) is a finite H G,u -module. Bythe Artin-Tate Lemma, H G,u is a finite H σG,u -algebra, so R j π I ! (Sht G,D,I | η I ; Sat( V )) is alsofinite over H σG,u . As H σG,u is Noetherian, the subquotient T i ( R j π I ! (Sht G,D,I | η I ; Sat( V ))) isalso a finite H σG,u -module, and therefore a finite T ( H G,u ) -module (since the H σG,u -actionfactors through T ( H G,u ) ). Finally, each E ab ∞ is a further T ( H G,u ) -equivariant subquotientof such a module, therefore also a finite T ( H G,u ) -module. As these are the subquotientsin a finite filtration of T j (Sht G ; V ) , the latter is also a finite T ( H G,u ) -module.Since the formation of the Tate double complex (2.4) is functorial with respect to thesheaf, the filtration is functorial is as well. Therefore we have natural transformations F r T jI ( V ) → T jI ( V ) , compatible with fusion and composition. This implies the desired equiv-ariance of excursion operators. Concretely, the action of S V,x,ξ, ( γ i ) i ∈ I on T j (Sht G,D,I ; ) and F r T j (Sht G,D,I ; ) are given by the two rows in the diagram T j (Sht G,D,I ; ) T j (Sht G,D,I ; V ) T j (Sht G,D,I ; V ) T j (Sht G,D,I ; ) F r T j (Sht G,D,I ; ) F r T j (Sht G,D,I ; V ) F r T j (Sht G,D,I ; V ) F r T j (Sht G,D,I ; ) x ( γ i ) i ∈ I ξx ( γ i ) i ∈ I ξ and the commutativity of the outer rectangle is exactly the desired equivariance. (cid:3) Equivariant localization for excursion operators.
We define
Nm : Rep k (( L G ) I ) → Rep k (( L G ) I ) σ -eq to be the functor taking a representation V to V ⊗ k σ V ⊗ k . . . ⊗ k σ p − V ,with the σ -equivariant structure σ Nm( V ) = σ V ⊗ k σ V ⊗ k . . . ⊗ k σ p − V ⊗ k V ∼ −→ V ⊗ k σ V ⊗ k . . . ⊗ k σ p − V = Nm( V ) given by the commutativity constraint for tensor products. It corresponds under GeometricSatake to Definition 3.11. Given h : V → V ′ ∈ Rep k (( L G ) I ) , we set Nm( h ) := h ⊗ σ h ⊗ . . . ⊗ σ p − h : Nm( V ) → Nm( V ′ ) . Note that Nm is not an additive functor, nor is it even k -linear. We linearize it by defining Nm ( p − ) := Frob − p ◦ Nm , where (as in §3.6.3) Frob − p is the identity on objects and onmorphisms it is ( − ) ⊗ k, Frob − p k . Then Nm ( p − ) : Rep k (( L G ) I ) → Rep k (( L G ) I ) σ -eq is k -linear, although still not additive.For V ∈ Rep k (( L G ) I ) , we denote by N · V the σ -equivariant representation V ⊕ σ V ⊕ . . . ⊕ σ p − V , with σ -equivariant structure σ ( N · V ) = σ V ⊕ σ V ⊕ . . . ⊕ σ p − V ⊕ V ∼ −→ V ⊕ σ V ⊕ . . . ⊕ σ p − V = ( N · V ) given by the commutativity constraint for direct sums. For h : V → V ′ ∈ Rep k (( L G ) I ) , wedenote N · h : N · V → N · V ′ the σ -equivariant map h ⊕ σ h ⊕ . . . ⊕ σ p − h . Let ∆ p : → ⊕ p denote the diagonal map and ∇ p : ⊕ p → denote the sum map.Our goal in this subsection is to prove the theorem below. Theorem 5.11. (i) The action of S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I on T ∗ (Sht G ; ) isidentified with the action of S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I on T ∗ (Sht H ; ) .(ii) The action of S N · V, ( N · x ) ◦ ∆ p , ∇ p ◦ ( N · ξ ) , ( γ i ) i ∈ I on T ∗ (Sht G ; ) is . We first establish a key technical proposition giving an equivariant localization theoremfor shtukas.
Proposition 5.12.
Let V ∈ Rep k (( L G ) I ) . Then we have a natural isomorphism of functors T ∗ (Sht G,D,I ; Nm ( p − ) ( V )) ∼ = T ∗ (Sht H,D,I ; Res BC ( V )) : Rep(( L G ) I ) → Mod k ( π ( η, η ) I ) (5.14) which is compatible with fusion and composition. Remark 5.13.
Note that for this proposition, we forget the T ( H G ) -action on T ∗ (Sht G,D,I ; − ) .In fact, the proposition can be enhanced to give a compatible family of natural isomor-phisms including the Hecke-module structure, where T ∗ (Sht H,D,I ; Res BC ( V )) is regarded as a T ( H G ) -module via the “Brauer homomorphism” (to be defined later in §6.3) Br : T ( H G ) →H H . However, this is unnecessary for us and would lengthen the already lengthy argument,so we omit it. QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 35
Proof.
Since the
FWeil( η I , η I ) -actions on T ∗ (Sht G ; Nm ( p − ) ( V )) and on T ∗ (Sht H ; Res BC ( V )) are determined by their respective π ( η I , η I ) -actions plus partial Frobenius morphisms,we can and will focus on these two equivariance structures separately, starting with the π ( η I , η I ) -actions.The basic idea is that our geometric description of V Res BC ( V ) in Theorem 3.19 impliesthe statement in the case where V is a tilting module, after passing to a base extension, byequivariant localization. We will then deduce the full statement using descent and the factthat there are “enough” tilting modules by Theorem 3.9.Now we begin the argument. Consider the commutative diagram Rep k ( b G I ) P ( L + G ) Fs ((Gr G ) F s ; k ) ⊗ I D b (Rep k ( b G I )) D b (Hk G,I | η I ; k ) K b (Tilt k ( b G I )) D b (Sht G,D,I | η I ; k ) Mod k ∼ geom. Satake ∼ Theorem 3.9 T j (5.15)All the geometric objects appearing in the second column of (5.15), as well as the mapsbetween them inducing the functors depicted there, are defined over η I . Therefore, thereis a π ( η I , η I ) -action on all the categories involved, with the action on Rep k ( b G I ) factoringthrough the map π ( η I , η I ) → Γ I , and the Γ -action on Rep k ( b G ) coming from the GeometricSatake equivalence plus descent for sheaves on Gr G (i.e. the “geometric action” of §4.1.2).Furthermore, all the functors in (5.15) are π ( η I , η I ) -equivariant, hence we may considerthe π ( η I , η I ) -equivariantization of (5.15), obtaining the diagram below. Rep k (( L G geom ) I ) P L + G (Gr G ; k ) ⊗ I Rep k ( b G I ) Γ I , geom (P ( L + G ) Fs ((Gr G ) F s ; k ) ⊗ I ) Γ I D b (Rep k ( b G I )) π ( η I ,η I ) , geom D b (Hk G,I | η I ; k ) π ( η I ,η I ) K b (Tilt k ( b G I )) π ( η I ,η I ) , geom D b (Sht G,D,I | η I ; k ) π ( η I ,η I ) Mod k ( π ( η I , η I )) ∼∼ descent = ⇒ ∼∼∼ Theorem 3.9 T j (5.16)We emphasize here that D b ( − ) π ( η I ,η I ) denotes the equivariant derived category for theaction of π ( η I , η I ) .The functor Res BC : Rep k ( b G I ) → Rep k ( b H I ) extends to the derived category, and thenlifts to the π ( η I , η I ) -equivariant derived category and intertwines diagram (5.16) compati-bly with the analogous one for b H . The resulting composition of functors Rep k ( b G I ) π ( η I ,η I ) , geom Res BC −−−→ Rep k ( b H I ) π ( η I ,η I ) , geom (5.16) for H −−−−−−−−→ Mod k ( π ( η I , η I )) is the rightmost functor of (5.14). Let T j : D b (Rep k ( b G I )) π ( η I ,η I ) , geom → Mod k ( π ( η I , η I )) be the composite functor D b (Rep k ( b G )) π ( η I ,η I ) , geom Res BC −−−→ D b (Rep k ( b H )) π ( η I ,η I ) , geom (5.16) for H −−−−−−−−→ Mod k ( π ( η I , η I )) so that the rightmost functor of (5.14) is the pullback of T j to Rep k ( b G I ) π ( η I ,η I ) , geom .Then T j is the π ( η I , η I ) -equivariantization of the functor ( T j ) de − eq : D b (Rep k ( b G )) → Mod k given by the composition of functors D b (Rep k ( b G )) Res BC −−−→ D b (Rep k ( b H )) (5.15) for H −−−−−−−−→ Mod k . We claim that the π ( η I , η I ) -equivariant functor V ∈ Tilt k ( b G I ) T j (Sht G,D,I | η I ; Nm ( p − ) ( V )) ∈ Mod k extends (necessarily uniquely) to a π ( η I , η I ) -equivariant functor ( T j ) de − eq : K b (Tilt k ( b G I )) → Mod k . Note that this is not obvious because V Nm ( p − ) ( V ) is not even additive, and so Nm ( p − ) itself certainly does not extend to a functor out of the homotopy category. Nevertheless,we will see that the composite functor is well-behaved (in particular, composing with Tatecohomology restores the additivity). Indeed, we have T j (Sht G,D,I ; Nm ( p − ) ( V )) := T j (Sht G,D,I ; Sat(Nm ( p − ) ( V ))) Lemma 5.6 and §2.5 = ⇒ ∼ = T j (Sht H,D,I ; Frob − p ◦ Psm(Nm(Sat( V )))) Theorem 3.19 = ⇒ ∼ = T j (Sht H,D,I ; Sat(Res BC ( V ))) . (5.17)(Above, Frob − p is the automorphism of the k -linear category of sheaves on Sht
H,D,I obtainedby applying ( − ) ⊗ k, Frob − p k to spaces of morphisms.) Moreover, these isomorphisms arenatural in V , and in particular π ( η I , η I ) -equivariant. Hence we have presented the functorin question as a composition of two functors, Sat ◦ Res BC and T j (Sht H,D,I , − ) , which bothextend π ( η I , η I ) -equivariantly to the homotopy categories of their domains.The upshot is that ( T j ) de − eq is π ( η I , η I ) -equivariant, and the preceding computationshowed that there is a natural (in particular π ( η I , η I ) -equivariant) isomorphism ( T j ) de − eq ∼ =( T j ) de − eq as functors K b (Tilt( b G )) → Mod k . By Theorem 3.9 we may equivalently view ( T j ) de − eq and ( T j ) de − eq as functors on D b (Rep k ( b G I )) , and so we have a natural isomor-phism ( T j ) de − eq ∼ = ( T j ) de − eq as functors D b (Rep k ( b G I )) → Mod k . Then their π ( η I , η I ) -equivariantizations are naturally isomorphic functors D b (Rep k ( b G I )) π ( η I ,η I ) → Mod k ( π ( η I , η I )) . We draw attention to a subtlety in the computation below which is suppressed by the notation. Weare using that there is a natural isomorphism between the two functors D (Hk G,I ; k ) → D (Sht G ; k ) → D (Sht H ; k ) and D (Hk G,I ; k ) → D (Hk H,I ; k ) → D (Sht H ; k ) , coming from the commutative diagram Sht H Hk H,I
Sht G Hk G,I in order to identify
Psm(Nm ( p − ) (Sat( V ))) on Sht
H,D,I with the pullback of the complex with the samename in D (Hk H,I ; k ) . QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 37
Finally, the pullbacks of these functors to
Rep k (( L G geom ) I ) ∼ = Rep k ( b G I ) Γ I , geom are natu-rally isomorphic, and these two pullbacks are exactly the two sides of (5.14) after using(4.1) to identify Rep( L G ) ∼ = Rep( L G geom ) and Rep( L H ) ∼ = Rep( L H geom ) , which can bedone compatibly as discussed in Definition 4.6.Finally, we check the compatibility with partial Frobenius. We want to show that thediagram F ∗{ } T j (Sht G,D,I ; Nm ( p − ) ( V )) T j (Sht G,D,I ; Nm ( p − ) ( V )) F ∗{ } T j (Sht H,D,I ; Res BC ( V )) T j (Sht H,D,I ; Res BC ( V )) ∼∼ ∼∼ (5.18)commutes, where the vertical isomorphisms (as k -modules) have just been established. ByLemma 5.6, the σ -fixed points of F { } : Sht ( { } ,..., { n } ) G,D,I → Sht ( { } ,..., { n } , { } ) G,D,I are identified with F { } : Sht ( { } ,..., { n } ) H,D,I → Sht ( { } ,..., { n } , { } ) H,D,I . This implies that the isomorphisms (5.17) are compatible with the maps F ∗{ } . This estab-lishes the de-equivariantized version of the desired compatibility with coefficients in tiltingmodules; the π ( η I , η I ) -equivariant version then follows by re-running the same argumentas for the first part. (cid:3) Proof of Theorem 5.11. (i) Proposition 5.12 gives a chain of compatible identifications T ∗ (Sht G,D,I ; ) T ∗ (Sht G,D,I ; Nm ( p − ) ( V )) T ∗ (Sht G,D,I ; Nm ( p − ) ( V )) T ∗ (Sht G,D,I ; ) T ∗ (Sht H,D,I ; ) T ∗ (Sht H,D,I ; Res BC ( V )) T ∗ (Sht H,D,I ; Res BC ( V )) T ∗ (Sht H,D,I ; ) Nm ( p − ( x ) ∼ ∼ ( γ i ) i ∈ I ∼ Nm ( p − ( ξ ) ∼ x ( γ i ) i ∈ I ξ The operator S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I on T ∗ (Sht G ; ) is obtained by tracingalong the upper row, while the operator S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I on T ∗ (Sht H ; ) is obtained bytracing along the lower row. Hence they coincide under the vertical identifications.(ii) By Lemma 5.6 and §2.5 we have a chain of compatible identifications T ∗ (Sht G,D,I ; ) T ∗ (Sht G,D,I ; N · V ) T ∗ (Sht G,D,I ; N · V ) T ∗ (Sht G,D,I ; ) T ∗ (Sht H,D,I ; ) T ∗ (Sht H,D,I ; Psm( N · V )) T ∗ (Sht H,D,I ; Psm( N · V )) T ∗ (Sht H,D,I ; ) ( N · x ) ◦ ∆ p ∼ ∼ ( γ i ) i ∈ I ∼ ∇ p ◦ ( N · ξ ) ∼ ( N · x ) ◦ ∆ p ( γ i ) i ∈ I ∇ p ◦ ( N · ξ ) The operator S N · V, ( N · x ) ◦ ∆ p , ∇ p ◦ ( N · ξ ) , ( γ i ) i ∈ I on T ∗ (Sht G,D,I ; ) is obtained by tracing alongthe upper row. But the stalks and costalks of N · Sat( V ) | Gr H are all induced O [ σ ] -modules,so in particular they are perfect. Hence Psm( N · V ) is equivalent to in the Tate category,so T ∗ (Sht H,D,I ; Psm( N · V )) = 0 . Therefore the endomorphism in question factors throughthe zero map, hence is itself zero. (cid:3) Applications to base change for automorphic forms.
In §5.2 we described Laf-forgue’s action of
Exc(Γ , L G ) on H ∅ ( ) . By (5.6), we have H ∅ ( ) = M α ∈ ker ( F,G ) C ∞ c ( G α ( F ) \ G α ( A F ) / Y v K v ; k ) . Here ker ( F, G ) := ker( H ( F, G ) → Q v H ( F v , G )) is the isomorphism class of the genericfiber of the G -torsor. More generally, this defines a decomposition Sht
G,D,I = a α ∈ ker ( F,G ) (Sht G,D,I ) α (5.19)according to the isomorphism class of the generic fiber of E . The construction outlinedin §5.2 preserves the decomposition (5.19), and so gives an action of Exc(Γ , L G ) on each H c (Sht G,D, ∅ ; ) α := C ∞ c ( G α ( F ) \ G α ( A F ) / Q v K v ; k ) .In the base change situation, the “diagonal embedding” map φ : H → G induces a map φ ∗ : ker ( F, H ) → ker ( F, G ) .Theorem 1.1 is evidently implied by the theorem below, whose proof occupies this sub-section. Theorem 5.14.
Let [ ρ ] ∈ H (Γ F , b H ( k )) be a Langlands parameter appearing in the actionof Exc(Γ , L H ) on H c (Sht H,D,I ; Sat( )) α in the sense of §5.2.4. Then the image of [ ρ ] in H (Γ F , b G ( k )) appears in the action of Exc(Γ , L G ) on H c (Sht G,D,I ; Sat( )) φ ( α ) in the senseof §5.2.4. Definition 5.15.
For an algebra A in characteristic p with σ -action, we denote by N · A the subset consisting of elements of the form (1 + σ + . . . + σ p − ) a for a ∈ A . One easilychecks that N · A is an ideal in A σ .We denote by Nm : A → A σ the set map sending a a · σ ( a ) · . . . · σ p − ( a ) . It ismultiplicative but not additive. It is an exercise to verify that the composition of Nm withthe quotient A σ ։ A σ /N · A is an algebra homomorphism. Lemma 5.16.
Let A be a commutative ring over F p . Let A ′ ⊂ A σ be a subring containing Nm( A ) and N · A . (Since N · A is an ideal in A σ , it is also an ideal in any such A ′ .) Anycharacter χ : A ′ → k factoring through A ′ /N · A extends uniquely to a character e χ : A → k ,which is given by e χ ( a ) = χ (Nm( a )) /p . (5.20) Proof.
The same proof as that of [TV16, §3.4] works, but since our situation is a little moregeneral we reproduce it. One easily checks that the given formula (5.20) works (it is a ringhomomorphism since k is in characteristic p , and it clearly extends χ ).Next we check that it is the unique extension. Note that σ acts on characters of A bypre-composition; we denote this action by e χ σ · e χ . Clearly (5.20) is the unique σ -fixedextension, so we will show that any extension e χ ′ must be σ -fixed. Indeed, since any extension e χ ′ is trivial on N · A by the assumption that χ factors through A ′ /N A , we have p − X i =0 σ i · e χ ′ = 0 . By linear independence of characters [Sta20, Tag 0CKK] we must have σ i · e χ ′ = e χ ′ for all i ,i.e. e χ ′ is σ -fixed. (cid:3) Lemma 5.17.
Inside
Exc(Γ , L G ) we have Nm( S V,x,ξ, ( γ i ) i ∈ I ) = S Nm( V ) , Nm( x ) , Nm( ξ ) , ( γ i ) i ∈ I and N · S V,x,ξ, ( γ i ) i ∈ I = S N · V, ( N · x ) ◦ ∆ p , ∇ p ◦ ( N · ξ ) , ( γ i ) i ∈ I . QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 39
Proof.
The first equality follows from repeated application of the relations (4.7), (4.5) andthe explicit description of the σ -action in (5.13). The second equality follows from repeatedapplication of relations (4.7), (4.6) and the explicit description of the σ -action in (5.13). (cid:3) Proof of Theorem 5.14.
The Langlands parameter [ ρ ] ∈ H (Γ , b H ( k )) corresponds to a char-acter χ ρ : Exc(Γ , L H ) → k under Proposition 4.4. The assumption that χ ρ appears inthe action of Exc(Γ , L H ) on H c (Sht H,D, ∅ ; ) α implies that there is a vector v ρ in a finite-dimensional quotient of H c (Sht H,D,I ; ) α on which S ∈ Exc(Γ , L H ) acts as multiplicationby χ ρ ( S ) ∈ k . Since the Exc(Γ , L H ) -action on H c (Sht H,D, ∅ ; ) α is defined over F p , the im-age of S ⊗ under Exc(Γ , L H ) ⊗ k, Frob p k ∼ −→ Exc(Γ , L H ) acts on the image – call it v ( p ) ρ – of v ρ ⊗ under H c (Sht H,D, ∅ ; ) α ⊗ k, Frob p k ∼ −→ H c (Sht H,D, ∅ ; ) α as multiplication by χ ρ ( S ) p .The decomposition (5.19) induces a compatible direct sum decomposition of Tate coho-mology and the Tate spectral sequence, and we denote by a subscript α ∈ ker ( F, G ) thesummand indexed by α . By Lemma 5.7 this eigenvector v ( p ) ρ maps to a non-zero v ( p ) ρ insome Exc(Γ , L H ) -equivariant finite-dimensional quotient of ( E ij ∞ ) α , and the latter is itself asubquotient of T ∗ (Sht H,D,I ; ) α . By Lemma 5.10, v ( p ) ρ is also an eigenvector for Exc(Γ , L H ) with the same eigensystem as v ( p ) ρ , namely χ pρ .By Theorem 5.11 and Lemma 5.10(ii), Exc(Γ , L G ) σ -eq acts on v ρ with eigensystem S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I · v ρ = χ ρ ( S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I ) v ρ , and (using Lemma 5.17) N · S acts by for any S ∈ Exc(Γ , L G ) .Note that S Nm( V ) , Nm( x ) , Nm( ξ ) , ( γ i ) i ∈ I is the image of S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I under the map Exc(Γ , L G ) ⊗ k, Frob k ∼ −→ Exc(Γ , L G ) . Let Exc(Γ , L G ) ′ ⊂ Exc(Γ , L G ) be thesubalgebra generated by N · Exc(Γ , L G ) and all elements of the form Nm( S V,x,ξ, ( γ i ) i ∈ I ) = S Nm( V ) , Nm( x ) , Nm( ξ ) , ( γ i ) i ∈ I (the equality by Lemma 5.17). Then the preceding discussionshows that v ( p ) ρ is an eigenvector for Exc(Γ , L G ) ′ with eigensystem χ ′ ρ : Exc(Γ , L G ) ′ → k given by S Nm( V ) , Nm( x ) , Nm( ξ ) , ( γ i ) i ∈ I χ ρ ( S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I ) p (5.21) N · S This defines a certain maximal ideal m ρ of Exc(Γ , L G ) ′ . The existence of v ( p ) ρ implies that m ρ appears in the support of some ( E ij ∞ ) α as an Exc(Γ , L G ) ′ -module. By Lemma 5.10(i), ( E ij ∞ ) α is an Exc(Γ , L G ) ′ -module subquotient of H c (Sht G,D,I ; ) φ ( α ) , so m ρ is also in thesupport of H ∗ c (Sht G,D, ∅ ; ) φ ( α ) as an Exc(Γ , L G ) ′ -module.Furthermore, we claim that H c (Sht G,D, ∅ ; ) is a finite module over Exc(Γ , L G ) ′ . In-deed, by [Xuea, Theorem 0.0.3] (and its generalization to non-split groups to appear in[Xueb]), H c (Sht G,D, ∅ ; ) is a finite module over H G,u for any u ∈ X \ D . According tothe “ S = T ” Theorem [Laf18a, equation (12.16)], the action of the Hecke operator h V,u at u ∈ X indexed by V ∈ Rep( L G ) agrees with the action of the particular excursion operator S { , } ,V, unit , counit , { F u , } where F u is any lift of the Frobenius at u to π ( η, η ) . We choose u sothat G is reductive and hyperspecial at u , and so that the extension E/F is split at u . In thiscase H G,u ∼ = H ⊗ pH,u , and the subalgebra H ′ G,u ⊂ H
G,u generated by all elements of the form h u, Nm( V ) and h u,N · V coincides with H σG,u . So by the Artin-Tate Lemma, H c (Sht G,D, ∅ ; ) is also a finite H ′ G,u -module. Since the endomorphisms in the image of H ′ G,u are containedin the endomorphisms in the image of
Exc(Γ , L G ) ′ by the “ S = T ” Theorem, we concludethat H c (Sht G,D, ∅ ; ) is also a finite Exc(Γ , L G ) ′ -module, as desired. Now, we have established that m ρ is in the support of H c (Sht G,D, ∅ ; ) φ ( α ) as an Exc(Γ , L G ) ′ -module. The claim implies that the localization ( H c (Sht G,D, ∅ ; ) φ ( α ) ) m ρ is a finite Exc(Γ , L G ) ′ m ρ -module, so then ( H c (Sht G,D, ∅ ; ) φ ( α ) ) / m ρ is finite-dimensional and (by Nakayama’s Lemma)non-zero over k . Since the Exc(Γ , L G ) -action obviously commutes with the Exc(Γ , L G ) ′ -action, it descends to this finite-dimensional k -vector space ( H c (Sht G,D, ∅ ; ) φ ( α ) ) / m ρ andtherefore has an eigenvector. Then Lemma 5.16 plus Lemma 5.17 show that the only possibleeigensystem for this eigenvector is S V,x,ξ, ( γ i ) i ∈ I = χ ′ ρ ( S Nm( V ) , Nm( x ) , Nm( ξ ) , ( γ i ) i ∈ I ) /p = χ ρ ( S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I ) . This is precisely the composition χ ◦ φ ∗ BC , as desired. (cid:3) Cyclic base change in the local setting
In this section we will prove Theorem 1.4. We begin by formulating a precise version ofthe Treumann-Venkatesh conjecture in §6.1. Any formulation depends on a “local Langlandscorrespondence mod p ”; we use the Genestier-Lafforgue correspondence [GL]. This is ouronly option at the generality of an arbitrary irreducible admissible mod p representation ofan arbitrary reductive group, but for GL n there are more refined correspondences for non-supercuspidal representations [Vig01, EH14, KM20], and it would be interesting to knowwhat happens in those contexts as well.We review the relevant aspects of the Genestier-Lafforgue correspondence in §6.2. In§6.3 we recall the Brauer homomorphism introduced in [TV16]. Finally, in §6.4 we combinethese with earlier global results to conclude the proof of Theorem 1.4.6.1.
Conjectures on local base change functoriality.
We recall a conjecture of Treumann-Venkatesh that “Tate cohomology realizes base change functoriality” in the mod p LocalLanglands correspondence. We shall prove a form of this conjecture, formulated preciselybelow, for cyclic base change in the function field setting.Let F v be a local function field with ring of integers O v and characteristic = p . Let G v be a reductive group over F v with a σ -action. Let Π be a smooth irreducible representationof G v with coefficients in k . Let Π σ be the representation of G v obtained by composing Π with σ : G v → G v . We say that Π is σ -fixed if Π ≈ Π σ as G v -representations. Lemma 6.1 ([TV16, Proposition 6.1]) . If Π is σ -fixed, then the G v -action on Π extendsuniquely to an action of G v ⋊ h σ i . Let H v = G σv . Using Lemma 6.1 we can form the Tate cohomology groups T (Π) and T (Π) with respect to the σ -action, which are then representations of H v . Treumann-Venkatesh conjecture that they are in fact admissible representations of H v , but we do notprove or use this. Definition 6.2 (Linkage) . An irreducible admissible representation π of H v is linked withan irreducible admissible representation Π of G v ( F v ) if π ( p ) appears in T (Π) or T (Π) ,where π ( p ) is the Frobenius twist π ( p ) := π ⊗ k, Frob k. Conjecture 6.3 ([TV16, Conjecture 6.3]) . Linkage is compatible with functorial transfer:if π is linked to Π , then π transfers to Π under the Local Langlands correspondence. QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 41
Example 6.4.
The need for the Frobenius twist can be seen in a simple example. Suppose G = H p and σ acts by cyclic permutation. Then G σ is the diagonal copy of H . In this casea representation π of H v should transfer to π ⊠ p of G v . And indeed, T ( π ⊠ p ) = ker(1 − σ | π ⊠ p ) N · π ⊠ p ∼ = π ( p ) . Let W v = Weil( F v /F v ) . To give Conjecture 6.3 a precise meaning, we need a precisemap (cid:26) irreducible admissiblerepresentations of G v over k (cid:27) / ∼−→ (cid:26) Langlands parameters W v → L G ( k ) (cid:27) / ∼ as a candidate for “the Local Langlands correspondence”. In the function field setting,Genestier-Lafforgue have constructed such a map, which is expected to be the semi-simplificationof the local Langlands correspondence. To any irreducible admissible representation Π of G v , it assigns a semi-simple local Langlands parameter, i.e. a b G ( k ) -conjugacy class of contin-uous homomorphisms ρ Π : W v → L G ( k ) which is continuous and semi-simple [GL, Theorem8.1]. Equivalently, we may view ρ Π ∈ H ( W v , b G ( k )) .Now, if π is a representation of H v and Π is a representation of G v , we say that π transfers to Π under the Genestier-Lafforgue correspondence if the image of ρ π under H ( W v , b H ( k )) → H ( W v , b G ( k )) coincides with ρ Π .Base change supplies many examples of the situation in Conjecture 6.3. (Most of theother examples are particular to p = 2 , , ; the relevant maps of L -groups are “exceptional”homomorphisms that do not lift to characteristic .) Let H be a reductive group over F v , E v /F v a cyclic extension of degree p , and G = Res E v /F v ( H E v ) . Then G has a σ -action, H = G σ , and the induced σ -action on G ( F v ) = H ( E v ) has H ( F v ) as its fixed pointsubgroup. We call this the “base change situation”. We prove Conjecture 6.3 in the basechange situation, with respect to the Genestier-Lafforgue correspondence. Theorem 6.5.
Let H v , G v be as in the base change situation. Let Π be a σ -fixed repre-sentation of G v . Let π ( p ) be an irreducible admissible representation of H v appearing asa subquotient of T (Π) or T (Π) . Then π transfers to Π under the Genestier-Lafforguecorrespondence. Remark 6.6.
A special case of the conjecture in the base change situation is proved in[Ron16], for depth-zero supercuspidal representations of GL n compactly induced from cusp-idal Deligne-Lusztig representations. Despite the very explicit nature of the Local LanglandsCorrespondence for such representations, the proof involves rather hefty calculations.Furthermore, the unramified and tamely ramified base change are handled completelydifferently in [Ron16], whereas our proof will be completely uniform in the field extension,the reductive group, and the irreducible representation.6.2. Review of Genestier-Lafforgue’s local Langlands correspondence.
We brieflysummarize the aspects of the Genestier-Lafforgue correspondence that we will need.6.2.1.
The Bernstein center.
We begin by recalling the formalism of the Bernstein center[Ber84]. Let K v ⊂ G v be an open compact subgroup. The Hecke algebra of G with respectto K v is H ( G, K v ) := C ∞ c ( K v \ G ( F v ) /K v ; k ) . This forms an algebra under convolution, where we use (for all K v ) the left Haar measureon G v for which G ( O v ) has volume . We let z ( G, K v ) := Z ( H ( G, K v )) be the center of H ( G, K v ) . The Bernstein center of G is z ( G ) := lim ←− K v z ( G, K v ) where the transition maps to z ( G, K v ) are given by convolution with K v , the unit of H ( G, K v ) , viewed as an idempotent in the full Hecke algebra of compactly supported smoothfunctions on G v .The Bernstein center of G v is isomorphic to the endomorphisms of the identity functorof the category of smooth k -representations of G v . Explicitly, smoothness implies that Π = lim −→ K v Π K v , and z ( G, K v ) acts on Π K v as an H ( G, K v ) -module; this assembles intoaction of z ( G ) on Π . In particular, any irreducible admissible representation Π of G v inducesa character of z ( G ) .6.2.2. Action of the excursion algebra.
The main result of [GL] is the construction of ahomomorphism Z G : Exc( W v , L G ) → z ( G ) . (6.1)Extend G to a parahoric group scheme over O v . For a positive integer m , let K mv :=ker( G ( O v ) → G ( O v /t mv )) be the “ m th congruence subgroup”. We write U mv := K mv ∩ H ( O v ) for the m th congruence subgroup of H . We write Z G,m : Exc( W v , L G ) → z ( G, K mv ) for the composition of Z G with the tautological projection to H ( G, K mv ) , and similarly Z H,m : Exc( W v , L H ) → z ( H, U mv ) .We will shortly give a characterization of (6.1). First let us indicate how (6.1) definesthe correspondence Π ρ Π . An irreducible admissible Π induces a character of z ( G ) , asdiscussed above. Composing with Z G then gives a character of Exc( W v , L G ) , which byProposition 4.4 gives a semisimple Langlands parameter ρ Π ∈ H ( W v , b G ( k )) . Remark 6.7.
In fact the homomorphism (6.1) is defined over F p (with the obvious F p -structures on both sides). This implies the following relation with the Frobenius twist, whichwill be needed later: if χ Π is the character giving the action of Exc( W v , L G ) on an irreducible G v -representation Π , then the character χ Π ( p ) giving the action of Exc( W v , L G ) ⊗ k, Frob k ∼ −→ Exc( W v , L G ) on Π ( p ) := Π ⊗ k, Frob p k satisfies χ Π ( p ) ( S ⊗ | {z } ∈ Exc( W v , L G ) ⊗ k, Frob p k ) = χ Π ( S ) p for all S ∈ Exc( W v , L G ) .6.2.3. Local-global compatibility.
Choose a smooth projective curve X over F ℓ and a point v ∈ X so that X v = Spec O v , such that G extends to a reductive group over the genericpoint of X . Then choose a further extension of G to a parahoric group scheme over all of X .The map (6.1) is characterized by local-global compatibility with the global excursionaction. The idea is that for ( γ i ) i ∈ I ⊂ W Iv , the action of the the excursion operator S I,f, ( γ i ) i ∈ I on H c (Sht G,D, ∅ ; ) is local at v , i.e. it acts through a Hecke operator for G v . Moreover,it commutes with other Hecke operators because all excursion operators commute with allHecke operators, hence it must actually be in the center of the relevant Hecke algebra. Thisidea is affirmed by the Proposition below. Proposition 6.8 (Genestier-Lafforgue Prop 1.3) . For ( γ i ) i ∈ I ⊂ W Iv , the operator S I,f, ( γ i ) i ∈ I acts on H c (Sht G,D, ∅ ; ) as convolution by Z G,m ( S ,I,f, ( γ i ) i ∈ I ) ∈ z ( G v , K mv ) . Remark 6.9.
By [GL, Lemme 1.4], for large enough D v the action of Z G,m ( S ,I,f, ( γ i ) i ∈ I ) on H c (Sht G,D, ∅ ; ) is faithful. Therefore, Proposition 6.8 certainly characterizes the map (6.1). QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 43
What is not clear is that the resulting Z G,m ( S ,I,f, ( γ i ) i ∈ I ) is independent of choices (of theglobal curve, or the integral model of the affine group scheme). In [GL] this is establishedby giving a purely local construction of (6.1) in terms of “restricted shtukas”, but for ourpurposes it will be enough to accept Proposition 6.8 as a black box.6.3. The Brauer homomorphism.
We introduce the notion of the Brauer homomorphismfrom [TV16], whose utility for our purpose is to capture the relationship between Π and itsTate cohomology from the perspective of Hecke algebras.Let K v ⊂ G v be an open compact subgroup, and let U v = K σv ⊂ H v . We say that K v ⊂ G v is a plain subgroup if ( G v /K v ) σ = H v /U v .We can view H ( G, K v ) as the ring of G v -invariant (for the diagonal action) functions on ( G v /K v ) × ( G v /K v ) under convolution. We claim that if K v ⊂ G v is a plain subgroup, thenthe restriction map H ( G, K v ) σ = Fun G v (( G v /K v ) × ( G v /K v ) , k ) σ (6.2) restrict −−−−→ Fun H v (( H v /U v ) × ( H v /U v ) , k ) = H ( H v , U v ) is an algebra homomorphism . It is called the Brauer homomorphism and denoted
Br : H ( G, K v ) σ → H ( H, U v ) . Proof of claim.
What we must verify is that for x, z ∈ H v /U v , and f, g ∈ H ( G v , K v ) σ , wehave X y ∈ G v /K v f ( x, y ) g ( y, z ) = X y ∈ H v /U v f ( x, y ) g ( y, z ) . (6.3)Since f and g are σ -invariant, we have f ( x, y ) = f ( σx, σy ) = f ( x, σy ) and g ( y, z ) = g ( σy, σz ) = g ( σy, z ) . If y / ∈ H v /U v , then the plain-ness assumption implies that y is not fixed by σ . Thereforethe contribution from the orbit of σ on y to (6.3) is divisible by p , hence is . (cid:3) Lemma 6.10. If K v ⊂ ker( G ( O v ) → G ( O v / m v )) , then K v is plain.Proof. By the long exact sequence for group cohomology, the plain-ness is equivalent to themap on non-abelian cohomology H ( h σ i ; K v ) → H ( h σ i ; G v ) having trivial fiber over thetrivial class. But the assumption implies that K v has a filtration, e.g. the restriction of thelower central series on ker( G ( O v ) → G ( O v / m v )) , with subquotients being abelian char( F ℓ ) -groups, so that they are acyclic for H ( h σ i , − ) as σ has order p . Therefore H ( h σ i , K v ) vanishes for such K v as in the statement of the Lemma. (cid:3) Lemma 6.11 (Relation to the Brauer homomorphism) . Assume K v ⊂ G v is plain. Suppose Π is a σ -fixed representation of G v . Then the map of Tate cohomology groups T ∗ (Π K v ) → T ∗ (Π) lands in the U v -invariants, and for any h ∈ H ( G v , K v ) σ we have the commutativediagram below. T ∗ (Π K v ) T ∗ (Π) U v T ∗ (Π K v ) T ∗ (Π) U v T h Br( h ) (Here T h is the element of T ( H ( G v , K v )) represented by h .)Proof. A direct computation similar to the proof of the claim; see [TV16, §6.2]. (cid:3)
Tate cohomology realizes base change functoriality.
Let E v /F v be a cyclic ex-tension of order p . Let H be a reductive group over F v and G = Res E v /F v ( H E v ) . We shallprove: Theorem 6.12.
Let Π be an irreducible admissible representation of G ( F v ) and let χ Π ( p ) : Exc( W v , L G ) → k the associated character of Π ( p ) . Form T ∗ (Π) as above, viewed as a smooth H ( F v ) -representation.Then for any irreducible character χ : Exc( W v , L H ) → k appearing in the action on T ∗ (Π) via Z H , the composite character Exc( W v , L G ) φ ∗ BC −−→ Exc( W v , L H ) χ −→ k agrees with χ Π ( p ) . It is clear that Theorem 6.12 implies Theorem 6.5. The rest of the section is devotedtowards proving Theorem 6.12.6.4.1. The maps
Exc( W v , L G ) Z G,m −−−→ z ( G, K mv ) → End H G ( H c (Sht G,D, ∅ ; )) induce upon applying Tate cohomology, T Exc( W v , L G ) T Z G,m −−−−−→ T z ( G, K mv ) → End T H G ( T ( H c (Sht G,D, ∅ ; ))) . For each m we choose D v large enough and non-empty so that the map z ( G, K mv ) → End H G ( H c (Sht G,D, ∅ ; )) is injective, and Remark 5.5 applies. (Of course, we do not claimthat D v can be so chosen independently of m .)6.4.2. Theorem 5.11 implies that the action of S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I on T (Sht G,mv + D v , ∅ ; ) = (cid:18) the action of S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I on T (Sht H,mv + D v , ∅ ; ) (cid:19) . The latter action factors through the action of S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I on H c (Sht H,mv + D v , ∅ ; ) ,as Lemma 5.7 implies that T (Sht H,mv + D v , ∅ ; ) ∼ = H c (Sht H,mv + D v , ∅ ; ) .6.4.3. For any set S , we let k [ S ] denote the k -vector space of k -valued functions on S .Now suppose e S is a set with an action of G v ⋊ h σ i , such that K v ⊂ G v acts freely. Then H ( G v , K v ) acts on k [ S := e S/K v ] since we may view H ( G v , K v ) = Hom G v ( k [ G v /K v ] , k [ G v /K v ]) and k [ S ] = Hom G v ( k [ G v /K v ] , k [ e S ]) . This induces an action of T ( H ( G v , K v )) on T ( k [ S ]) ∼ = k [ S σ ] , and then by inflation an action of H ( G v , K v ) σ on k [ S σ ] .By the same mechanism, there is an induced action of H ( H v , U v ) on k [ e S σ /K σv ] = k [ e S σ /U v ] . Lemma 6.13.
Assume K v ⊂ G v is a plain subgroup. Then k [ e S σ /U v ] is a H ( G v , K v ) σ -directsummand of k [ S σ ] , and for all h ∈ H ( G v , K v ) σ we have (cid:16) the action of h on k [ e S σ /U v ] (cid:17) = (cid:16) the action of Br( h ) ∈ H ( H v , U v ) on k [ e S σ /U v ] (cid:17) . Proof.
See [TV16, equation (4.2.2)]. (cid:3)
QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 45
From §6.4.1 we have the diagram T Exc( W v , L G ) T z ( G, K mv ) End T H G ( T (Sht G,mv + D v , ∅ ; ))Exc( W v , L H ) z ( H, U mv ) End H H ( T (Sht H,mv + D v , ∅ ; )) Z G,m Z H,m (6.4)
Corollary 6.14.
For all m ≥ , the action of z ∈ T z ( G, K mv ) on T (Sht G,mv + D v , ∅ ; ) in (6.4) agrees with the action of Br( z ) on T (Sht H,mv + D v , ∅ ; ) in (6.4) under the identification T (Sht G,mv + D v , ∅ ; ) ∼ = T (Sht H,mv + D v , ∅ ; ) from §2.5.Proof. Each
Sht
G,mv + D v , ∅ is a discrete groupoid with finite stabilizers. As a special caseof Remark 5.5, for all positive m the automorphisms of Sht
G,mv + D v , ∅ are finite unipotentgroups, which therefore have no cohomology. Hence we may apply the preceding discus-sion with S := [Sht G,mv + D v , ∅ ] the set of isomorphism classes in Sht
G,mv + D v , ∅ , and e S :=[Sht G, ∞ v + D v , ∅ ] = lim ←− j ≥ [Sht G, ( m + j ) v + D v , ∅ ] . Then k [ S ] is identified with the cochains on [Sht G,mv + D v , ∅ ] , and Lemma 5.6 plus §2.5 identify k [ e S σ /K σ ] with the cochains on [Sht H,mv + D v , ∅ ] .The assertions for compactly supported cochains then follows by duality. (cid:3) Corollary 6.15.
For all m ≥ , for all { V, x, ξ, ( γ i ) i ∈ I } as in §4.4, the Brauer homo-morphism sends the element Z G,m ( S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I ) ∈ z ( G, K mv ) ⊂H ( G v , K mv ) to the element Z H,m ( S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I ) ∈ z ( H, U mv ) ⊂ H ( H v , U mv ) .Proof. The discussion of §6.4.2 shows that the image of S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I | {z } ∈ T Exc( W v , L G ) in End H H ( T (Sht H,mv + D v , ∅ ; )) via (6.4) = the image of S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I ∈ Exc( W v , L H ) in End H H ( T (Sht H,mv + D v , ∅ ; )) via (6.4) . (6.5)On the other hand, the discussion of §6.4.3 shows that the left hand side of (6.5) agrees withthe image of Br( Z G,m ( S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I )) via (6.4), for all m ≥ . Weconclude by using injectivity of z ( H, U mv ) ֒ → End H H ( T ∗ (Sht H,mv + D v , ∅ ; )) in (6.4). (cid:3) Conclusion of the proof of Theorem 6.12.
Let Π be a representation of G v . Then z ( G ) acts G ( F v ) -equivariantly on Π , inducing an H ( F v ) -equivariant action of z ( G ) σ on T ∗ (Π) . Inparticular, as Z G maps the image of Exc( W v , L G ) σ -eq → Exc( W v , L G ) (cf. Remark 5.9)into z ( G ) σ , we get an H ( F v ) -equivariant action of Exc( W v , L G ) σ -eq on T ∗ (Π) .By Lemma 6.10, K mv is plain as soon as m ≥ , so in particular the Brauer homomor-phism is defined on H ( G v , K mv ) as soon as m ≥ . Taking the (filtered) colimit over m inLemma 6.11, we find that for all S ∈ Exc( W v , L G ) σ -eq , we have (cid:18) the action on T ∗ (Π) of Z G ( S ) (cid:19) = (cid:18) the action on T ∗ (Π) of Br( Z G ( S )) (cid:19) . In other words, the diagram below commutes:
Image( Z G | Exc( W v , L G ) σ -eq ) z ( G ) End G v (Π)Image( Z H ) z ( H ) End H v ( T ∗ Π) Br (6.6)On the other hand, taking the inverse limit over m in Corollary 6.15 yields that Br( Z G ( S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I )) = Z H ( S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I ) . (6.7)Combining (6.6) and (6.7) shows that (cid:18) the action on T ∗ (Π) of Z G ( S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I ) (cid:19) = (cid:18) the action on T ∗ (Π) of Z H ( S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I ) (cid:19) . (6.8)From now on, assume Π is an irreducible admissible representation of G ( F v ) . Then End G ( F v ) (Π) ∼ = k (by Schur’s Lemma applied to the Hecke action on the invariants of Π forevery compact open subgroup of G v ). The Langlands parameter attached to Π correspondsunder Proposition 4.4 to the character χ Π : Exc( W v , L G ) Z G −−→ z ( G ) → End G v (Π) ∼ = k. This induces T χ Π : T Exc( W v , L G ) T Z G −−−−→ T z ( G ) → T End G v (Π) ∼ = k. Let ι denote the natural map T End G v (Π) → End H v ( T ∗ Π) . We also consider the homo-morphism χ T Π : Exc( W v , L H ) Z H −−→ z H → End H v ( T Π) . We have just seen in (6.8) that ι ◦ T χ Π ( S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I ) = χ T Π ( S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I ) . (6.9)Note that the fact that the right hand side of (6.9) lies in k is already non-obvious. Inparticular, (6.9) implies that for any irreducible subquotient π of T Π , we have χ π ( S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I ) = χ T Π ( S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I )= ( T χ Π )( S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I )= χ Π ( S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I ) . (6.10)Using Remark 6.7, the same reasoning as in the proof of Theorem 5.14 shows that χ Π ( p ) ( S Nm( V ) , Nm( x ) , Nm( ξ ) , ( γ i ) i ∈ I ) = χ Π ( S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I ) p . (6.11)By Lemma 5.16 and Lemma 5.17, we have χ Π ( p ) ( S V,x,ξ, ( γ i ) i ∈ I ) = χ Π ( p ) ( S Nm( V ) , Nm( x ) , Nm( ξ ) , ( γ i ) i ∈ I ) /p [ (6.11) = ⇒ ] = χ Π ( S Nm ( p − ( V ) , Nm ( p − ( x ) , Nm ( p − ( ξ ) , ( γ i ) i ∈ I )[ (6.10) = ⇒ ] = χ π ( S Res BC ( V ) ,x,ξ, ( γ i ) i ∈ I )= χ π ◦ φ ∗ BC ( S V,x,ξ, ( γ i ) i ∈ I ) . This shows that χ Π ( p ) = χ π ◦ φ ∗ BC for any irreducible subquotient π of T ∗ (Π) , which completesthe proof. (cid:3) QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 47
Appendix A. The base change functor realizes Langlands functorialityby Tony Feng and Gus Lonergan
In this section we prove Theorem 3.19. We keep the setup of §3.6.1: H is any reductivegroup over a separably closed field F of characteristic = p , and G = H p . We let σ act on G by cyclic rotation, sending the i th factor to the ( i + 1) st (mod p ) factor.A.1. Proof of linearity.
We first prove that BC is additive, i.e. we exhibit a naturalisomorphism BC ( F ⊕ F ′ ) ∼ = BC ( F ) ⊕ BC ( F ′ ) . We have Nm(
F ⊕ F ′ ) = ( F ⊕ F ′ ) ∗ ( σ F ⊕ σ F ′ ) ∗ . . . ∗ ( σ p − F ⊕ σ p − F ′ )= Nm( F ) ⊕ Nm( F ′ ) ⊕ ( direct sum of free σ -orbits ) . Therefore, the restrictions of
Nm(
F ⊕ F ′ ) and Nm( F ) ⊕ Nm( F ′ ) to X σ differ by a perfectcomplex of O [ σ ] -modules, and hence project to isomorphic objects in the Tate category Shv( X σ ; T O ) . This shows that Psm ◦ Nm is additive. We conclude by using that the modularreduction functor F and the lifting functor L are both additive. (cid:3) A.2.
Reduction to the case of a torus.
Let T H be a maximal torus of H . Recallthat the restriction functor Rep( b H ) → Rep( T b H ) is intertwined under the Geometric Satakeequivalence with the hyperbolic localization functor [BR18, §5.3].Since ∗ / ! -restriction and ∗ / ! -pushforward all commute with Psm by §2.3, the hyperboliclocalization functor commutes with
Psm . As the restriction functor
Rep( b H ) → Rep( T b H ) is faithful and injective on tilting objects (i.e. “tilting modules are determined by theircharacters”) by [Don93, p. 46], it suffices to prove Theorem 3.19 in the special case where H is a torus .A.3. Proof in the case of a torus.
Finally, we examine the case when H is a torus. Sincethe theorem is compatible with products, we can even reduce to the case H = G m . For H = G m the underlying reduced scheme of Gr H is a disjoint union of points labeled by theintegers.The irreducible algebraic representations of b H are indexed by n ∈ Z , with V n correspond-ing to the constant sheaf supported on the component Gr nH labeled by n . The irreduciblealgebraic representations of b G are then labeled by p -tuples of integers ( n , . . . , n p ) ∈ Z p . Bythe linearity of BC established in §A.1 and the complete reducibility of algebraic represen-tations of tori, we may assume that F is irreducible, say F = F ( n , . . . , n p ) is the constantsheaf supported on Gr ( n ,...,n p ) G .The σ -equivariant sheaf Nm( F ) is then the constant sheaf k supported on the component Gr ( n + ... + n p ,...,n + ... + n p ) G . Its restriction to the diagonal copy of Gr H is the constant sheafwith value k supported on Gr n + ... + n p H . This is already an indecomposable k -parity sheaf,which tautologically lifts its own image in the Tate category. Hence we have shown that k Gr n ... + npH = BC ( p ) ( V n ,...,n p ) . And indeed, this is precisely the sheaf which corresponds under geometric Satake to
Res BC ( V n ⊠ V n ⊠ . . . V n p ) ∼ = V n + n + ... + n p . This confirms the commutativity of the diagram Parity (Gr G ; k ) Parity (Gr H ; k )Tilt k ( b G ) Tilt k ( b H ) ∼ BC ∼ Res BC at the level of objects. Our final step is to verify the commutativity on morphisms. Since(as H is a torus) the categories involved are all semi-simple, the commutativity at the levelof morphisms reduces to examining a scalar endomorphism of the simple object F above,which corresponds to the simple representation V n ,...,n p . The restriction functor Res BC is k -linear, so what we have to check is that BC sends multiplication by λ on F to multiplicationby λ on BC ( F ) . Now, multiplication by λ on F is sent under Nm to multiplication by λ p on Nm( F ) , which restricts to multiplication by λ p on BC ( p ) ( F ) . Then the inverse Frobeniustwist Frob − p sends it to multiplication by λ , so BC := Frob − p ◦ BC ( p ) behaves as desired. (cid:3) References [BBM04] A. Beilinson, R. Bezrukavnikov, and I. Mirković,
Tilting exercises , Mosc. Math. J. (2004),no. 3, 547–557, 782. MR 2119139[Ber84] J. N. Bernstein, Le “centre” de Bernstein , Representations of reductive groups over a local field,Travaux en Cours, Hermann, Paris, 1984, Edited by P. Deligne, pp. 1–32. MR 771671[BG14] Kevin Buzzard and Toby Gee,
The conjectural connections between automorphic representa-tions and Galois representations , Automorphic forms and Galois representations. Vol. 1, LondonMath. Soc. Lecture Note Ser., vol. 414, Cambridge Univ. Press, Cambridge, 2014, pp. 135–187.MR 3444225[BHKT19] Gebhard Böckle, Michael Harris, Chandrashekhar Khare, and Jack A. Thorne, ˆ G -local systemson smooth projective curves are potentially automorphic , Acta Math. (2019), no. 1, 1–111.MR 4018263[Bla94] Don Blasius, On multiplicities for
SL( n ) , Israel J. Math. (1994), no. 1-3, 237–251.MR 1303497[BR18] Pierre Baumann and Simon Riche, Notes on the geometric Satake equivalence , Relative aspects inrepresentation theory, Langlands functoriality and automorphic forms, Lecture Notes in Math.,vol. 2221, Springer, Cham, 2018, pp. 1–134. MR 3839695[Clo14] Laurent Clozel,
Formes modulaires sur la Z p -extension cyclotomique de Q , Pacific J. Math. (2014), no. 2, 259–274. MR 3227435[Clo17] , Sur l’induction automorphe pour des p -extensions radicielles et quelques autres opéra-tions fonctorielles (mod p ) , Doc. Math. (2017), 1149–1180. MR 3690271[Don93] Stephen Donkin, On tilting modules for algebraic groups , Math. Z. (1993), no. 1, 39–60.MR 1200163[EH14] Matthew Emerton and David Helm,
The local Langlands correspondence for GL n in families ,Ann. Sci. Éc. Norm. Supér. (4) (2014), no. 4, 655–722. MR 3250061[Fen20] Tony Feng, Nearby cycles of parahoric shtukas, and a fundamental lemma for base change ,Selecta Math. (N.S.) (2020), no. 2, Paper No. 21, 59. MR 4073972[Gai] Dennis Gaitsgory, From geometric to function-theoretic langlands (or how to invent shtukas) ,Available at https://arxiv.org/pdf/1606.09608.pdf .[GL] Alain Genestier and Vincent Lafforgue,
Chtoucas restreints pour les groupes réductifs etparamétrisation de Langlands locale , Available at https://arxiv.org/abs/1709.00978 .[JMW14] Daniel Juteau, Carl Mautner, and Geordie Williamson,
Parity sheaves , J. Amer. Math. Soc. (2014), no. 4, 1169–1212. MR 3230821[JMW16] , Parity sheaves and tilting modules , Ann. Sci. Éc. Norm. Supér. (4) (2016), no. 2,257–275. MR 3481350[KM20] Robert Kurinczuk and Nadir Matringe, A characterization of the relation between two ℓ -modularcorrespondences , C. R. Math. Acad. Sci. Paris (2020), no. 2, 201–210. MR 4118176[Laf18a] Vincent Lafforgue, Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale ,J. Amer. Math. Soc. (2018), no. 3, 719–891. MR 3787407[Laf18b] , Shtukas for reductive groups and Langlands correspondence for function fields , Pro-ceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenarylectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 635–668. MR 3966741[Lap99] Erez M. Lapid,
Some results on multiplicities for
SL( n ) , Israel J. Math. (1999), 157–186.MR 1714998[LL] Spencer Leslie and Gus Lonergan, Parity sheaves and Smith Theory , Available at https://arxiv.org/abs/1708.08174 . QUIVARIANT LOCALIZATION, PARITY SHEAVES, AND CYCLIC BASE CHANGE 49 [MR18] Carl Mautner and Simon Riche,
Exotic tilting sheaves, parity sheaves on affine Grassmannians,and the Mirković-Vilonen conjecture , J. Eur. Math. Soc. (JEMS) (2018), no. 9, 2259–2332.MR 3836847[Qui71] Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II , Ann. of Math. (2) (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694[Ric] Simon Riche, Geometric representation theory in positive characteristic , Habilitation thesis,available at https://tel.archives-ouvertes.fr/tel-01431526/document .[Ron16] Niccolò Ronchetti,
Local base change via Tate cohomology , Represent. Theory (2016), 263–294. MR 3551160[RW] Simon Riche and Geordie Williamson, Smith-Treumann theory and the linkage principle , Avail-able at https://arxiv.org/pdf/2003.08522.pdf .[Sta20] The Stacks Project Authors,
Stacks Project , https://stacks.math.columbia.edu , 2020.[Tre19] David Treumann, Smith theory and geometric Hecke algebras , Math. Ann. (2019), no. 1-2,595–628. MR 4000251[TV16] David Treumann and Akshay Venkatesh,
Functoriality, Smith theory, and the Brauer homo-morphism , Ann. of Math. (2) (2016), no. 1, 177–228. MR 3432583[Var04] Yakov Varshavsky,
Moduli spaces of principal F -bundles , Selecta Math. (N.S.) (2004), no. 1,131–166. MR 2061225[Vig01] Marie-France Vignéras, Correspondance de Langlands semi-simple pour
GL( n, F ) modulo = p ,Invent. Math. (2001), no. 1, 177–223. MR 1821157[Xuea] Cong Xue, Cohomology with integral coefficients of stacks of shtukas , Available at https://arxiv.org/pdf/2001.05805.pdf .[Xueb] ,
To be announced , In preparation.[Xue20] ,
Finiteness of cohomology groups of stacks of shtukas as modules over Hecke algebras,and applications , Épijournal de Géométrie Algébrique (2020), no. 6, 1–42.[Zhu] Xinwen Zhu, Coherent sheaves on the stack of langlands parameters , Preprint available at https://arxiv.org/abs/2008.02998 .[Zhu15] ,