Connected components of the moduli space of L-parameters
Sean Cotner



Recently, in order to formulate a categorical version of the local Langlands correspondence, several authors have constructed moduli spaces of [1/p]-valued L-parameters for p-adic groups. The connected components of these spaces over various [1/p]-algebras R are conjecturally related to blocks in categories of R-representations of p-adic groups. Dat-Helm-Kurinczuk-Moss described the components when R is an algebraically closed field and gave a conjectural description when R = [1/p]. In this paper, we prove a strong form of this conjecture applicable to any integral domain R over [1/p].


High-fidelity and Fault-tolerant Teleportation of a Logical Qubit using Transversal Gates and Lattice Surgery on a Trapped-ion Quantum Computer
    D. Hayes
    April 29, 2024
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§ INTRODUCTION


Let p be a prime number, let F be a nonarchimedean local field of residue characteristic p, and let G be a connected reductive group over F. Since its inception, one of the primary goals in the Langlands program has been to understand the smooth -representations of G(F). Recently, in order to study congruences of smooth representations, there has been considerable interest in the category _R(G(F)) of R-representations, where R is any commutative ring, e.g., R = _ℓ; see <cit.> for a recent and extensive guide to the literature. We will only consider the case that p is invertible in R, and to avoid rationality issues we will assume further that R is a [1/p]-algebra. 

When studying any category of representations, a fundamental issue is to understand the blocks, i.e., the indecomposable direct factors. If R =, the blocks of _R(G(F)) were determined by Bernstein <cit.> in terms of (super)cuspidal support. If R = _ℓ, ℓ≠ p, it is expected that a similar block decomposition holds, and there has been a good deal of recent work in this direction, for instance <cit.>, <cit.>, <cit.>. When R = W(_ℓ) for ℓ≠ p and G = _n, Helm established a similar block decomposition in <cit.>, <cit.>, <cit.>. Understanding these blocks for inner forms of _n has led to new results on the Jacquet–Langlands correspondence in <cit.>.

This paper concerns the “dual side" of this question. To describe it, let W_F be the Weil group of F, which acts on the Langlands dual Ĝ over [1/p]. Recently, <cit.> and <cit.> have constructed a space Z^1(W_F^0, Ĝ) consisting (roughly) of 1-cocycles W_F →Ĝ over [1/p], or, equivalently, L-homomorphisms W_F →Ĝ⋊ W_F.[Because of the incompatibility of the ℓ-adic and ℓ'-adic topologies for ℓ≠ℓ', it is not totally clear how to make this definition functorially. In the approach of <cit.>, which we recall in Section <ref>, this is achieved by replacing W_F by a dense subgroup W_F^0 with “discretized inertia".] Recent categorical versions of the local Langlands correspondence as in <cit.>, <cit.>, <cit.>, and <cit.> suggest that there should be a fully faithful embedding of _R(G(F)) into ([Z^1(W_F^0, Ĝ)/Ĝ]_R). The latter category breaks up into a product indexed by the components of Z^1(W_F^0, Ĝ)_R, so one expects that each block of _R(G(F)) can be associated to such a component.

Focusing on a single prime, i.e., taking R = _ℓ or _ℓ for ℓ≠ p, the connected components of Z^1(W_F^0, Ĝ)_R were determined in <cit.> in terms of restriction to prime-to-ℓ inertia. On the other end of the spectrum, instead of considering a single prime ℓ≠ p, one can consider all primes other than p at the same time by taking R = [1/p]. Representations of finite p-groups do not deform away from characteristic p, so restriction to the wild inertia subgroup P_F is a natural discrete invariant of Z^1(W_F^0, Ĝ)_[1/p]. In fact, <cit.> exhibits a decomposition

    Z^1(W_F^0, Ĝ)_[1/p] = ∐_ϕ∈Φ
    α∈Σ(ϕ)Z^1(W_F^0, Ĝ)_ϕ, α,

where membership of a cocycle  in Z^1(W_F^0, Ĝ)_ϕ, α is determined by the restriction ϕ of  to P_F and a map α from W_F to the component group of a twisted centralizer of Lϕ in . We will recall the precise definitions in a self-contained way in Section <ref>.

<cit.>, <cit.>
    Each [1/p]-scheme Z^1(W_F^0, Ĝ)_ϕ, α is connected.



    Conjecture <ref> is true.


Conjecture <ref> was previously proved in <cit.> when Z(Ĝ) is smooth, which corresponds to the condition that G has fundamental group (in the sense of Borovoi) with no torsion of order prime to p. This handles the case that G is semisimple and simply connected, as well as some other groups like G = _n or _2n, but it says very little if, for instance, G is of adjoint type. Indeed, when G = _n for an integer n which is divisible by at least two distinct primes, this condition does not hold for any p.

Roughly, Conjecture <ref> describes the way in which components of Z^1(W_F^0, Ĝ)_ conspire to connect modulo the primes ℓ≠ p. In fact, it can equivalently be phrased as the statement that if two -valued cocycles , ψ have the same restriction to P_F and the same invariant α as above, then there is a sequence of primes ℓ_1, …, ℓ_n ≠ p and a sequence = _0, _1, …, _n = ψ such that the components containing _i and _i+1 in Z^1(W_F^0, Ĝ)_ meet modulo ℓ_i+1 for all 0 ≤ i ≤ n-1. Under this interpretation, we note the similarity to <cit.>. 

As stated above, the cases R = [1/p] and R = _ℓ are in a sense at opposite extremes. We will interpolate between them by considering an arbitrary integral domain R over [1/p]. For simplicity, let us assume for now that R = [D^-1] for some multiplicatively closed subset D ⊂. Reasoning as in the case R = [1/p] suggests that there should be a discrete invariant of Z^1(W_F^0, Ĝ)_[D^-1] coming from restriction to the maximal “pro-D" subgroup of the inertia subgroup I_F. Because of topological issues alluded to above, this does not literally make sense, but nonetheless we construct in Section <ref> a decomposition

    Z^1(W_F^0, Ĝ)_[D^-1] = ∐_ϕ_D ∈Φ_D 
    α_D ∈Σ(ϕ_D)Z^1(W_F^0, Ĝ)_ϕ_D, α_D

where, informally, Φ_D is a set of cocycles from the maximal pro-D subgroup of I_F to Ĝ and Σ(ϕ_D) comes from the failure of the centralizer of Lϕ_D in Ĝ to be connected. The following theorem, which is slightly generalized in Theorem <ref>, is our main result.


    Every summand in (<ref>) is connected.


In <cit.>, one finds detailed conjectures on the blocks of _[1/p](G(F)) suggested by Conjecture <ref>. In particular, the components of Z^1(W_F^0, Ĝ)_[1/p] are expected to be in one-to-one correspondence with the so-called “stable blocks" of _[1/p](G(F)), and there are precise expectations about the decomposition of each stable block into blocks. These conjectures have been partially confirmed for the subcategory of [1/p]-representations of depth 0 in <cit.>, following the works <cit.>, <cit.> which prove similar results for _ℓ-representations of depth 0. It seems reasonable to expect that the conjectures of <cit.> can be extended to _[D^-1](G(F)) for any D. However, there are subtleties even when R = _ℓ, as noted in <cit.>. Thus we leave precise conjectures to the reader.

Our proof of Theorem <ref> is similar in several respects to the proof sketched in <cit.> in the case R = [1/p] and the action of I_F preserves a pinning, but the details are somewhat different. In fact, to remain as self-contained as possible (and since they do not significantly simplify the argument), we do not rely on any previously-established connectedness results. Let us outline the proof in the case that R = [1/p] and the action of W_F is inner, assumptions which allow us to avoid some technical complications while retaining the key features of the problem.

    
  * (Section <ref>) Reduce to proving connectedness of Z^1(W_F^0/P_F, Ĝ)_[1/p] when the W_F-action on Ĝ is tame and preserves a Borel pair. (Since R = [1/p] in this outline, this reduction was already made in <cit.>.)
    
  * (Section <ref>) Analyze the behavior of Z^1(W_F^0/P_F, Ĝ)_[1/p] under certain homomorphisms Ĝ→Ĥ to reduce to the case that Ĝ is semisimple and simply connected.
    
  * (Section <ref>) Define a map ΣZ^1(W_F^0/P_F, Ĝ) →Ĝ//Ĝ which, roughly, records the eigenvalues of a generator of tame inertia. Using “purity" of Z^1(W_F^0/P_F, Ĝ), reduce to the claims that (Σ)_[1/p] is connected, and for every field k the fibers of Σ_k are connected.
    
  * (Section <ref>) Using the theory of complete reducibility and theorems of Steinberg, show that Σ^-1(x) is connected for every field k and every x ∈ (Σ)(k).
    
  * (Section <ref>, Appendix <ref>) Prove that (Σ)_[1/p] is connected.


Step (5) is the most interesting one. In Section <ref>, we write down connected closed subschemes C_w of (Σ)_[1/p] indexed by elements w of the Weyl group W_0 such that (Σ)_[1/p] = ⋃_w ∈ W_0 C_w. This shows π_0((Σ)_[1/p]) = W_0/∼, where ∼ is the equivalence relation generated by the relations w ∼ w' if C_w ∩ C_w'≠∅. If there is a field k of characteristic ≠ p such that the action map W_F →Ĝ^ad_k lifts to a map W_F →Ĝ_k, then we exhibit a single element which lies in every C_w(k), thus proving (5). In fact, this assumption is satisfied whenever the center Z(Ĝ) is of order divisible by at most one prime ≠ p, which holds in all simple types other than A. In type A, this argument can fail: Example <ref> shows that when Ĝ = _6 and p = 7, there are two elements w, w' ∈ W_0 such that C_w ∩ C_w' = ∅. This leads to more work in type A, which is handled in Appendix <ref>. 

To aid the reader, steps (4) and (5) are described in the special case Ĝ = _2 in Section <ref>. Section <ref> contains recollections from the literature, as well as a few basic lemmas and a proof of the twisted Chevalley–Steinberg isomorphism over a general base scheme.



 §.§ Notation


Throughout this paper, let F be a nonarchimedean local field with residue field of order q, a power of the prime p. Let W_F be the Weil group of F, let K be a number field, let D be a saturated multiplicatively closed subset of , and let Ĝ be a split reductive _K[D^-1]-group scheme (in the sense of <cit.>, i.e., with connected fibers) equipped with an action of W_F which factors through a finite quotient W and preserves a Borel pair (B̂, T̂). The definitions and notation surrounding W_F^0 and Z^1(W_F^0, Ĝ) are given in Section <ref> and retained past that section. Throughout, if R is a ring, let D_R denote the set of integers which are invertible in R. If  is a prime ideal of R, let k() denote the residue field of R at . If H is a reductive R-group scheme, then Z(H) denotes the center of H. If σ is an automorphism of H, then we will use σh to denote the image of an element h ∈ H(R) under σ.



 §.§ Acknowledgements


I thank Robert Cass, Stephen DeBacker, and Alex Hazeltine for conversations which helped to clarify the ideas of this paper. I also thank the attendees of the Harish-Chandra seminar at the University of Michigan for allowing me to give a series of talks on this work and for asking many pertinent questions.













§ AN EXAMPLE


We begin by sketching the proof of Theorem <ref> in the case Ĝ = _2, along the lines that we will use to prove the general case. This section is not logically necessary for the rest of the paper; we include it only in the hope that it will help orient the reader for the general argument. There are several features of _2 which simplify the argument in this case, among which are that the group of outer automorphisms is trivial, the order of the center is only divisible by one prime, and the exponential map is defined on nilpotent matrices in arbitrary characteristic. 

For completeness, we briefly recall the construction of Z^1(W_F^0, Ĝ) from <cit.>. Naive approaches run into thorny topological issues, and a major insight is that it is useful to introduce the subgroup W_F^0 ⊂ W_F with discretized inertia. Explicitly, there is a short exact sequence

    1 → P_F → W_F → W_F/P_F → 1

where P_F is the subgroup of wild inertia. Let Fr and s be lifts of Frobenius and a topological generator of tame inertia, respectively, and let (W_F/P_F)^0 be the discrete subgroup of W_F/P_F generated by Fr and s. We define W_F^0 to be the preimage of (W_F/P_F)^0 in W_F, and we topologize W_F^0 by declaring that P_F is open with its usual profinite topology.

We retain the notation of Section <ref>. Let Z^1(W_F^0, Ĝ) be the moduli space of 1-cocycles W_F^0 →Ĝ(R) with open kernel, where R ranges over _K[1/p]-algebras. By <cit.>, this is a locally finite type syntomic _K[1/p]-scheme which, when restricted to _ℓ⊗_K-algebras for any ℓ≠ p, parameterizes ℓ-adically continuous L-parameters from W_F.

By the reduction to tame parameters of <cit.>, it is fine to assume that the action of W_F on _2 is tame and only consider Z^1(W_F^0/P_F, _2). 
Because the group of outer automorphisms of _2 is trivial, Fr and s act by (g) and (t) for some g, t ∈T̂(_K[1/p]). These are required to satisfy gtg^-1 = t^q z for some central element z ∈ Z(_2)(_K[1/p]). Moreover, if q is even then we may and do pass from t to -1 · t if necessary to assume z = 1. 

Let ΣZ^1(W_F^0/P_F, _2) →^1 be the map given by sending a cocycle  to ((s)t). The strategy of the proof of Theorem <ref> is to show that Σ has connected fibers and that (Σ)_[1/p] is connected, and then use simple point-set topology to show that Z^1(W_F^0/P_F, _2) is connected. We will punt the point-set topology arguments (which are precisely the same for _2 as for all other groups) to the coming sections and only handle properties of Σ in this section.


    Σ has geometrically connected fibers.



    Let k be an algebraically closed field, and let x ∈ (Σ)(k) be a point. By definition, there is some 1-cocycle W_F^0/P_F →Ĝ(k) such that (s)t has eigenvalues α, α^-1. Let (s)t = t'(1+N) be the Jordan decomposition of (s)t, where N is nilpotent, and define a map _-_k^1 →Σ^-1(x) functorially via _c(Fr) = (Fr) and _c(s) = t'(1 + cN)t^-1 for c ∈_k^1. The fact that _c is a cocycle for each c can be checked directly using the defining relation Fr s Fr^-1 = s^q. Note that _1 = and that _0(s)t is semisimple.

    Define ΣZ^1(W_F^0/P_F, _2) →_2 by Σ() = (s)t, and note that ^-1(x) contains a unique semisimple conjugacy class C: if x = α + α^-1 for some α∈ k^×, then this is the conjugacy class of (α, α^-1). By the previous paragraph, it suffices to show that Σ^-1(C) is connected, and indeed it is enough to show that Σ^-1((α, α^-1)) is connected. However, using the defining relation for a cocycle, one checks that Σ^-1((α, α^-1)) is isomorphic to the centralizer of (α^q, α^-q), which is either isomorphic to  or _2 and in either case is connected.



    (Σ)_[1/p] is connected.



    If k is a field and  is a cocycle W_F^0/P_F →_2(k), then the relation (Fr)(s)(Fr)^-1 = (s)^q means (Fr)g((s)t)z^-1 = ((s)t)^q. This restricts the possible eigenvalues of (s)t considerably: to see this, let {α, β} be the set of eigenvalues of (s)t. If z = 1, then {α, β} = {α^q, β^q}, and hence α, β∈μ_q - 1(k) ∪μ_q + 1(k). If z = -1, then {α, β} = {-α^q, -β^q}, so α, β∈μ_2(q-1)(k) ∪μ_2(q+1)(k). Moreover, if k is not of characteristic 2, then α, β∉μ_q-1(k) ∪μ_q+1(k). These are, however, the only restrictions; for any α as above, it is straightforward to show that there is an L-parameter W_F^0/P_F →_2(k) such that (s) = (α, α^-1). 

    If z = 1, then by the above paragraph, Σ factors (set-theoretically, at least) through a surjective map to the schematic image Y_1 of the map μ_q-1∪μ_q+1→^1 given by α↦α + α^-1. The scheme (Y_1)_[1/p] is connected: indeed, since μ_q-1 and μ_q+1 meet at 1, it suffices to note that any multiplicative type [1/p]-group scheme which is finite of order prime to p is connected. Next, suppose z = -1, so q is odd. If n ≥ 1, let μ'_2n be the nontrivial translate of the subgroup scheme μ_n in μ_2n over [1/p]. Again, the previous paragraph shows that Σ factors (set-theoretically) through a surjective map to the schematic image Y_-1 of the map μ'_2(q-1)∪μ'_2(q+1)→^1 given by α↦α + α^-1. Over  or _ℓ for ℓ≠ 2, the schemes μ'_2(q-1) and μ'_2(q+1) do not meet, but they do meet over _2 at the element 1. Moreover, μ'_2(q-1) and μ'_2(q+1) are both connected. Thus we conclude as before that Y_-1 is connected.


























§ PRELIMINARIES




 §.§ Pure schemes


We will use the notion of purity for a scheme morphism X → S from <cit.>. The precise definition is not important in this paper, but we summarize some useful properties in the following lemma.


    Let R be a ring, and let X be a flat finitely presented R-scheme.
    
        
  * If X is R-pure and R' is an R-algebra, then X_R' is R'-pure.
        
  * If R is noetherian, then X is R-pure if and only if, for every R-algebra A which is a DVR, the scheme X_A is A-pure.
        
  * If R is a field, then X is R-pure.
        
  * If R is a complete DVR with uniformizer π and X is affine, then X is R-pure if and only if the coordinate ring R[X] is π-adically separated.
        
  * If R is a DVR with fraction field K and residue field k and X is R-pure, then the Zariski closure of every irreducible component of X_K meets X_k.
    



    (1) is proved in <cit.>. (3) and (5) follow immediately from the definition in <cit.>. For (4), note that R[X] is countably generated, so by <cit.> it is a direct sum of rank 1 R-modules. In particular, R[X] is π-adically separated if and only if it is R-free, so the result follows from <cit.>. Finally, (2) is proved in <cit.>.


The main point of purity for our purposes is that moduli spaces of L-parameters are pure.


    Let K be a number field, and let H be a split reductive group scheme over _K[1/p] equipped with a finite action of W_F. The scheme Z^1(W_F^0, H) is flat and _K[1/p]-pure.



    By Lemma <ref>(1), (2), and (3), we need only show that Z^1(W_F^0, H)__ℓ⊗_K is _ℓ⊗_K-pure for every prime number ℓ≠ p. Every connected component of Z^1(W_F^0, H)__ℓ⊗_K is flat, finite type, and affine by <cit.>, and the same reference shows that the coordinate ring of each component is ℓ-adically separated. (Strictly speaking, <cit.> only applies as written when K =, but the proof never uses this assumption.) Thus Lemma <ref>(4) shows the result.


To make reductions later, we need two elementary results about purity which are mainly topological in nature.


    Let R be an integral domain, let ∈ R, and let X and Y be finitely presented R-schemes such that Y is flat and R-pure. Let f X → Y be an R-morphism, and suppose
    
        
  * f_k() X_k()→ Y_k() is surjective,
        
  * X is connected.
    
    Then Y is connected.



    By standard limit arguments, we may and do assume R is noetherian. Since Y is flat, each component of Y has nonempty (R)-fiber. By purity and Lemma <ref>(5), it follows that each component has nonempty k()-fiber. Thus it suffices to show that any two closed points y_1, y_2 of Y_k() are contained in the same component of Y. By (1), there exist points x_1, x_2 of X_k() lifting y_1, y_2. By (2), there is a sequence x_1 = s_1, η_1, s_2, …, η_n, s_n = x_2 in X such that s_i+1 specializes η_i and η_i+1 for all i. The sequence y_1 = f(s_1), f(η_1), …, f(s_n) = y_2 provides a chain in Y which shows that y_1 and y_2 lie in the same component.











    Let R be an integral domain, and let f X → Y be a surjective morphism of finitely presented R-schemes such that X is R-pure, Y is R-finite, and f^-1(y) is connected for all y ∈ Y. Then the map π_0(f)π_0(X) →π_0(Y) is bijective.


The main point making this lemma nontrival is that f is not necessarily flat; the conclusion does not hold if, for example, Y is not assumed finite or X is not assumed pure.


    It is enough to consider the case of noetherian R. First, π_0(f) is clearly surjective. To see injectivity, let C be a connected component of Y. If K = (R), let C_K = {x_1, …, x_n}, and note that {x_i}→ R is surjective by finiteness. Each {x_i} is connected, and since C is connected, we have C = ⋃_i=1^n {x_i}. Thus it suffices to show that f^-1({x_i}) is connected for all i.By finiteness of Y, the fiber f^-1(x_i) is a connected component of X_K for all i. If X_i denotes the Zariski closure of f^-1(x_i) in X, then X_i is connected for all i. We have
    
    f^-1({x_i}) = X_i ∪⋃_∈ R f^-1({x_i}_k()).

    By hypothesis, each f^-1({x_i}_k()) is connected, and by Lemma <ref>(5), the fiber (X_i)_k() is nonempty. Thus (X_i)_k() meets f^-1({x_i}_k()), and it follows that f^-1({x_i}) is connected.









 §.§ Complete reducibility


Recall <cit.> that if H is a reductive group over a field k and λ→ H is a cocharacter, then there are subgroups P_H(λ), Z_H(λ) ⊂ H defined by

    P_H(λ) = {h ∈ H lim_t → 0λ(t)hλ(t)^-1 exists} and 
    Z_H(λ) = {h ∈ H (h)λ = λ}.

If H is connected, then P_H(λ) is a parabolic subgroup of H and Z_H(λ) is a Levi of P_H(λ).

If Γ is a group, then a homomorphism fΓ→ H(k) is completely reducible if for every cocharacter λ→ H such that f factors through P_H(λ), there is some cocharacter μ→ H such that P_H(λ) = P_H(μ) and f factors through Z_H(μ). The following three results are well-known.

<cit.>
    If Γ is finitely generated, then fΓ→ H(k) is completely reducible if and only if the H-orbit of f in _k-gp(Γ, H) is closed.



    A surjection from a finitely generated free group to Γ induces an H-equivariant closed embedding _k-gp(Γ, H) → H^n, so the result follows from <cit.>.


<cit.>
    If fΓ→ H(k) is completely reducible and N ⊂Γ is a normal subgroup, then f|_N is also completely reducible.


We recall that an automorphism σ of H^0 is quasi-semisimple if it preserves a Borel pair of H^0. 


    If f→ H(k) is a homomorphism, then f is completely reducible if and only if the conjugation action of f(1) on H^0 is quasi-semisimple.



    This follows from <cit.>.










 §.§ A twisted Chevalley–Steinberg isomorphism


Let R be a ring, and let H be a reductive R-group scheme. Let σ be an R-automorphism of H which preserves a Borel pair (B, T). There is an action of H on itself defined by h ·_σ h'  h · h' ·σh^-1, which we call σ-twisted conjugacy. Denote the resulting GIT quotient by H//_σ H, and let χ_σ H → H//_σ H be the natural map. Equivalently, one can view H//_σ H as the quotient (H ⋉σ)// H, where H acts by conjugation, and one can consider χ_σ as the natural map H ⋉σ→ (H ⋉σ)// H; this is the perspective we will take in Section <ref>. 

Let W_0 be the subgroup of σ-fixed points of the Weyl group of T in H, and let A = T/(1 - σ)T. Let W_0 act on A as follows: if w_0 ∈ W_0 and n_0 is a lift of w_0 in N_H(T) (valued in some fppf R-algebra), then define w_0 · t = n_0 t (σn_0)^-1. Restricting χ_σ to T induces a map j A/W_0 → H//_σ H.

In <cit.> and <cit.>, it is shown that j is an isomorphism if R is a field; in <cit.>, this is shown if σ is the identity; in (the proof of) <cit.>, this is shown provided that R is a Dedekind domain. The aim of this section is to explain a formal way to extend these results to general R, akin to the arguments of <cit.>.


    If H is a split reductive -group scheme and σ is a pinning-preserving automorphism of H, then the natural map j A/W_0 → H//_σ H is an isomorphism.



    The argument of <cit.> can be read verbatim with the tuple (H, B, T, σ, W_0) in place of (C_Ĝ(ϕ)^0, B_ϕ, T_ϕ, β, (Ω_ϕ^∘)^β), and  in place of the base ring _K_e[1/pN_Ĝ] (with notation as in loc. cit.).


The following lemma is a straightforward generalization of (the proof of) <cit.>, and presumably well-known, but we give an argument for want of a reference.


    Let R be a noetherian ring, let H be a flat affine R-group scheme, and let X be a flat affine R-scheme with an action of H. The formation of the GIT quotient X//H  R[X]^H commutes with arbitrary base change if and only if, for every prime ideal ∈ R, the base change map X//H ⊗_R k() → X_k()//H_k() is an isomorphism.



    Suppose X =  C, and let α C → R[H] ⊗_R C be the homomorphism corresponding to the action map, so C^H = (α - 1). For each ∈ R, observe the commutative diagram
    
    C^H ⊗_R k() [r] [d]
                   C ⊗_R k() [r] [d]
                   (α - 1) ⊗_R k() [r] [d]
                   0 
    
            (C ⊗_R k())^H_k()[r]
                   C ⊗_R k() [r]
                   (α_k() - 1) [r]
                   0

    which shows, by our hypothesis and the five lemma, that the map (α - 1) ⊗_R k() →(α_k() - 1) is an isomorphism. Moreover, if i(α - 1) → R[H] ⊗_R C is the natural inclusion, then there is a commutative diagram
    
    (α - 1) ⊗_R k() [r] [d]
                   R[H] ⊗_R C ⊗_R k() [r] [d]
                   (i) ⊗_R k() [r] [d]
                   0 
    (α_k() - 1) [r]
                   (k())[H] ⊗_k() C_k()[r]
                   (i_k()) [r]
                   0

    which shows, again by the five lemma, that (i) ⊗_R k() →(i_k()) is an isomorphism. It follows that i ⊗_R k() is injective, and hence _1^R((i), k()) = 0. Thus (i) is R-flat, so the natural map C^H ⊗_R R' → (C ⊗_R R')^H_R' is an isomorphism for all R', as desired.



    Let X be a scheme, let H be a reductive X-group scheme admitting a Borel pair (B, T), and let σ be an X-automorphism of H preserving (B, T). The natural map j A/W_0 → H//_σ H is an isomorphism. The formation of each of A/W_0 and H//_σ H commutes with arbitrary base change.



    Since the formation of GIT quotients commutes with flat base change, we may pass to fppf covers of X at will. In particular, by <cit.>, we may assume that X =  R is affine and G is split, and by <cit.> we may write σ = (t) ∘σ_1, where σ_1 preserves the pinning and t ∈ T(R). The map H → H given by h ↦ ht is an isomorphism which intertwines σ-twisted conjugacy and σ_1-twisted conjugacy and thus induces an isomorphism H//_σ H → H//_σ_1 H. There is a similar isomorphism for A/W_0, and thus we may and do pass from σ to σ_1 to assume that σ preserves a pinning.

    By <cit.>, there is a tuple (, , , σ) consisting of a split reductive -group scheme , a Borel subgroup scheme ⊂, a maximal torus ⊂, and a pinning-preserving automorphism σ of  such that (, , , σ) ⊗ R ≅ (H, B, T, σ). Let  denote the group scheme of σ-coinvariants of . For every ring R', there is a commutative diagram
    
    []^, σ⊗ R' [r] [d]
                   []^W_0⊗ R' [d] 
    
            R'[_R']^_R', σ_R'[r]
                   R'[_R']^W_0.

    By Proposition <ref>, the top and right arrows are isomorphisms, so to prove the theorem it is enough to show that the bottom arrow is an isomorphism for all R'. By Lemma <ref>, we need only check this when R' = /p for some prime number p. This case is <cit.>.










§ REDUCTION TO TAME PARAMETERS


In this section, we will describe the decomposition (<ref>) promised in the introduction, and we will show that each summand in this decomposition is equivalent to a scheme of “tame" cocycles. We retain the notation of the introduction, letting R be a [1/p]-algebra and letting D be the set of integers which are invertible in R. Let D' be the multiplicatively closed subset of  generated by the prime numbers not in D. For the basic definitions associated with Z^1(W_F^0, Ĝ), see Section <ref>.



 §.§ Motivation


Our construction may initially appear strange, so we begin with two special cases, both of which appear in <cit.>. Consider first the case D = {± p^n n ≥ 0}, which will give rise to the decomposition (<ref>). There is a restriction map r_PZ^1(W_F^0, Ĝ) →Z^1(P_F, Ĝ), and it is shown in <cit.> that this is a discrete invariant: in other words, there is a decomposition

    Z^1(W_F^0, Ĝ)_[1/p] = ∐_ϕ∈ΦĜ· r_P^-1(ϕ)

for some set of sections Φ⊂Z^1(P_F, Ĝ)([1/p]). Given ϕ∈Φ, there is a closed subgroup scheme

    Z_LG(ϕ) = {(g, w) ∈ (g, w)Lϕ(w^-1pw)(g, w)^-1 = Lϕ(p)  ∀ p ∈ P_F},

and if  extends ϕ then  factors through Z_LG(ϕ). This gives rise to an induced homomorphism α_ W_F^0 →π_0(Z_LG(ϕ)). Again α_ is a discrete invariant of r_P^-1(ϕ): if Σ(ϕ) is the set of such α_, then there is a decomposition

    Z^1(W_F^0, Ĝ)_[1/p] = ∐_ϕ∈Φ
    α∈Σ(ϕ)Ĝ· r_P^-1(ϕ)_α,

where r_P^-1(ϕ)_α is the clopen subscheme of r_P^-1(ϕ) consisting of those cocycles  such that α_ = α. This is (<ref>). The content of “tame reduction" in this case is that each r_P^-1(ϕ)_α is isomorphic to a scheme of tame cocycles Z^1(W_F^0/P_F, Z_(ϕ)^0)_[1/p], where Z_(ϕ)^0 is equipped with a finite W_F-action preserving a Borel pair.[Crucially, even if the W_F-action on Ĝ is defined over [1/p] and preserves a pinning, we do not know whether we can ensure the same for the W_F-action on Z_(ϕ)^0. This justifies working in the generality we do.] This reduces one to understanding spaces of tame cocycles.

Next consider the other extreme case D = {nℓ∤ n}, where ℓ is a fixed prime, and let us assume for simplicity that R = _ℓ; the case R = _ℓ is similar. Now <cit.> shows that if A is an _ℓ-algebra, then Z^1(W_F^0, Ĝ)__ℓ(A) parameterizes continuous 1-cocycles W_F →Ĝ(A), where Ĝ(A) is equipped with the discrete topology. Thus there is a natural restriction map r^ℓZ^1(W_F^0, Ĝ)__ℓ→Z^1_cts(I_F^ℓ, Ĝ)__ℓ, where I_F^ℓ denotes the maximal closed subgroup of I_F of pro-order prime to ℓ. Using a similar argument to the above, one shows that there is a decomposition

    Z^1(W_F^0, Ĝ)__ℓ = ∐_ϕ^ℓ∈Φ^ℓ
    α^ℓ∈Σ(ϕ^ℓ)Ĝ__ℓ· (r^ℓ)^-1(ϕ^ℓ)_α^ℓ,

where Φ^ℓ is a set of continuous cocycles I_F^ℓ→Ĝ(_ℓ) and Σ(ϕ^ℓ) is a set of component group maps W_F^0 →π_0(Z_(ϕ^ℓ). Here the content of “tame reduction" is that for each nonempty (r^ℓ)^-1(ϕ^ℓ)_α^ℓ, there is an isomorphism (r^ℓ)^-1(ϕ^ℓ)_α^ℓ≅Z^1(W_F/I_F^ℓ, Z_(ϕ^ℓ)^0)__ℓ. Again, this reduces one to studying spaces of cocycles which one might call “ℓ'-tame". 

For more general R, to obtain a similar decomposition, we will consider a reduction map r_D which, informally, records the restriction of a cocycle  to the maximal pro-D subgroup of I_F. To see how to formulate this precisely, let us reframe the previous case: for simplicity, assume that W_F acts trivially on Ĝ and consider a continuous homomorphism f W_F/P_F →Ĝ(_ℓ). If f(s) = tu is the Jordan decomposition in Ĝ(_ℓ), then t is of some finite order n = ℓ^c d, where ℓ∤ d. The restriction of f to I_F/P_F is determined by c and t^ℓ^c, so there is a restriction map

    (W_F, Ĝ(_ℓ)) →_c, d(⟨ s^ℓ^c⟩/⟨ s^ℓ^c d⟩, Ĝ(_ℓ))

given by sending f to the map s^ℓ^c m↦ t^ℓ^c m, under which the image of f determines f|_I_F^ℓ.

Because each quotient ⟨ s^ℓ^c⟩/⟨ s^ℓ^c d⟩ is finite, the above discussion suggests an algebraic way to formulate a restriction map r^ℓ, which can be defined without reference to the profinite group W_F. In this form, the generalization to r_D is simple, although a modicum of care is needed since the Jordan decomposition need not exist over general rings. After defining r_D, our version of tame reduction reduces one to studying spaces of “D-tame" cocycles.



 §.§ Constructions


We now begin by defining r_D. Let _Ĝ be the unipotent scheme of Ĝ, as defined in <cit.> to be the schematic closure of the unipotent variety of Ĝ_. We note that this definition is not completely standard when Ĝ is not simply connected, but it is determined by the conditions that _Ĝ is -flat and _Ĝ(k) is the set of unipotent elements of Ĝ(k) for every field k. We warn the reader that the fibers of _Ĝ over  are not generally reduced.


    If N ∈ D, then the Nth power map [N] (_Ĝ)_[D^-1]→ (_Ĝ)_[D^-1] is a Ĝ-equivariant isomorphism.



    Equivariance is clear. By the fibral isomorphism criterion <cit.>, it suffices to show that for every field k over [D^-1], the map [N]_k is an isomorphism. If k = 0, then this is clear from the existence of the exponential map. If k = p > 0, then there is some n such that [p^n]_k kills (_Ĝ)_k: to see this, we can reduce to Ĝ = _n, in which case _Ĝ is reduced and we reduce to the fact that __n(k) is killed by p^n. If M is such that NM ≡ 1 p^n, then [NM]_k is the identity on _Ĝ and thus [N]_k is an isomorphism, as desired.



    Let P be an open subgroup of P_F which is normal in W_F^0. There is an integer n = n_P such that s^n acts trivially on P_F/P and for every [1/p]-algebra R and every cocycle W_F^0 →Ĝ(R), the power (s)^n lies in _Ĝ(R).



    By <cit.>, there is some n such that for every field k over [1/p] and every W_F^0 →Ĝ(k), the power (s)^n lies in _Ĝ(k). Thus, since Z^1(W_F^0, Ĝ)_ is reduced <cit.>, the map π_nZ^1(W_F^0, Ĝ)_[1/p]→Ĝ given by ↦(s)^n factors through _Ĝ generically. By Lemma <ref>, the source is flat and thus π_n factors through _Ĝ. Since P_F/P is finite, we may pass to a multiple of n to assume s^n acts trivially on P_F/P.


If P is an open subgroup of P_F which is normal in W_F^0 and n ≥ 1 is such that s^n acts trivially on P_F/P, let I_F, n, P^0 denote the subgroup of I_F^0 generated by s^n and P. Let I_F, n^0 = I_F, n, P_F^0. If n is as in Lemma <ref>, write n = ab for a ∈ D and b ∈ D'. Define a [D^-1]-morphism 

    r_n, P, DZ^1(W_F^0/P, Ĝ)_[D^-1]→Z^1(I_F, b^0/I_F, n, P^0, Ĝ)_[D^-1]

as follows: if R is a [D^-1]-algebra and W_F^0/P →Ĝ(R) is a cocycle, note that (s)^ab is unipotent, so by Lemma <ref> there is a unique element u ∈_Ĝ(R) whose ath power is (s)^n. Since [a] is a Ĝ-equivariant automorphism of _Ĝ, it follows that u commutes with (p) for all p ∈ P_F and with (s)^b. Thus we may define _P, D = r_n, P, D() via _n, P, D|_P_F = |_P_F and _P, D(s^b) = u^-1·(s)^b.

Since I_F, b^0/I_F, n, P^0 is a finite group of order in D for any n, <cit.> shows that there is a set Φ_P, n⊂Z^1(I_F, b^0/I_F, n, P^0, Ĝ([D^-1])) such that

    Z^1(I_F, b^0/I_F, n, P^0, Ĝ)_[D^-1] = ∐_ϕ_n ∈Φ_P, nĜ_[D^-1]·ϕ_n,

where Ĝ_[D^-1]·ϕ_n denotes the orbit of ϕ_n, isomorphic to Ĝ_[D^-1]/Z_Ĝ_[D^-1](ϕ_n). In particular, if P' ⊂ P and n | n' = a'b', where n' = n_P' as in Lemma <ref>, then the natural map Z^1(I_F, b^0/I_F, n, P^0, Ĝ)_[D^-1]→Z^1(I_F, b'^0/I_F, n', P'^0, Ĝ)_[D^-1] is a clopen embedding, and the maps r_n, P, D and r_n', P', D are compatible in the obvious sense. Therefore we may define

    r_D _n, P r_n, P, DZ^1(W_F^0, Ĝ)_[D^-1]→ X_D _n, PZ^1(I_F, b^0/I_F, n, P^0, Ĝ)_[D^-1]

because Z^1(W_F^0, Ĝ) = _P Z^1(W_F^0/P, Ĝ).

If R is a [D^-1]-algebra and ϕ_D ∈ X_D(R), we define

    Z__R(ϕ_D) = {(g, w) ∈_R (g, w)·Lϕ_D · (g, w)^-1 = Lϕ_D ∘(w)}

and Z_Ĝ_R(ϕ_D) = Z__R(ϕ_D) ∩Ĝ_R. A key observation is that, if W_F^0 →Ĝ(R) is a cocycle such that (I_F^0) is finite, then  factors through Z__R(r_D()).


    There is a set Φ_D ⊂ X_D([D^-1]) such that
    
    X_D = ∐_ϕ_D ∈Φ_DĜ·ϕ_D

    For each ϕ_D ∈Φ_D, the centralizer Z_Ĝ_[D^-1](ϕ_D) is a smooth affine [D^-1]-group scheme with split reductive identity component and constant component group.



    The points not already observed above follow from <cit.> as in <cit.>; the key point is that, if ϕ_D is represented by a cocycle ϕ_n I_F, b^0/I_F, n, P^0 →Ĝ([D^-1]), then Z_(ϕ_D) = Z_(ϕ_n).


By Theorem <ref>, for every ϕ_D ∈Φ_D there is an exact sequence

    1 →π_0(Z_Ĝ(ϕ_D)) →π_0(ϕ_D) π_0(Z_(ϕ_D)) → W.

If R is a [D^-1]-algebra and W_F^0 →Ĝ(R) restricts to ϕ_D, then there is an induced homomorphism α_ W_F^0 →π_0(ϕ_D), and we see that the image of α_ in _n, P(I_F, b^0/I_F,n,P, π_0(ϕ_D)) agrees with the element induced by ϕ_D. Moreover, the composition W_F^0 →π_0(ϕ_D) → W is the natural projection map. Let Σ(ϕ_D) be the set of homomorphisms α_D satisfying these two conditions. By Theorem <ref>, there is a disjoint union decomposition

    Z^1(W_F^0, Ĝ)_[D^-1] = ∐_ϕ_D ∈Φ_D 
    α_D ∈Σ(ϕ_D)Ĝ×^Z_Ĝ(ϕ_D) r_D^-1(ϕ_D)_α_D,

where by definition r_D^-1(ϕ_D)_α_D is the closed subscheme of Z^1(W_F^0, Ĝ)_[D^-1] consisting of those cocycles  with r_D() = ϕ_D and α_ = α_D. We call the pair (ϕ_D, α_D) admissible if Z^1(W_F^0, Ĝ)_ϕ_D, α_DĜ×^Z_Ĝ(ϕ_D) r_D^-1(ϕ_D)_α_D is nonempty. Note that, in view of this decomposition and <cit.>, each Z^1(W_F^0, Ĝ)_ϕ_D, α_D is syntomic.

We can now state the main theorem of this paper, which was stated imprecisely and in a slightly weaker form in Theorem <ref>.


    Let ϕ_D ∈Φ_D and α_D ∈Σ(ϕ_D) be admissible, and let R be an integral domain over [1/p] with D_R = D. Then Z^1(W_F^0, Ĝ)_ϕ_D, α_D⊗_[D^-1] R is connected.


Note that the statement of Theorem <ref> is general enough that it applies to R = _ℓ and R = _ℓ for any prime ℓ≠ p, so it recovers <cit.> as a special case. The first step which allows us to analyze r_D^-1(ϕ_D)_α_D is a “reduction to tame parameters" as in <cit.>, the main input for which is provided by the following theorem.


    If ϕ_D ∈Φ_D and α_D ∈Σ(ϕ_D) are admissible, then there is a 1-cocycle W_F^0 →Ĝ([D^-1]) with r_D() = ϕ_D and α_ = α_D such that (W_F^0) is finite and _ preserves a Borel pair of Z_Ĝ(ϕ_D)^0.



    Let (B, T) be a Borel pair of Z_Ĝ(ϕ_D)^0 which is defined over [D^-1]. Let  be the normalizer of (B, T) in Z_Ĝ(_D), and note that ^0 = T and  has component group π_0(ϕ_D). Thus α_D gives rise to a natural action of W_F on T. Let P ⊂ P_F be open and normal in W_F^0 such that ϕ_D|_P is trivial, and let n = ab be an integer as in Lemma <ref>, where a ∈ D and b ∈ D'. Thus ϕ_D is realized as a 1-cocycle I_F, b^0/I_F, n, P^0 →Ĝ([D^-1]). We may form the scheme Z^1_α_D(W_F^0/I_F, b^0, T) of 1-cocycles, which is a diagonalizable group scheme by the same argument as <cit.>. Similarly, let Σ(W_F^0, )_ϕ_D, α_D be the scheme of those ∈Z^1(W_F^0, Ĝ)_ϕ_D, α_D such that (W_F^0) ⊂. It is straightforward to check that Σ(W_F^0, )_ϕ_D, α_D is a pseudo-torsor for Z^1_α_D(W_F^0/I_F, b^0, T) in the sense of <cit.>, so by the discussion following <cit.> it suffices to show that there is a faithfully flat [D^-1]-algebra R such that Σ(W_F^0, )_ϕ_D, α_D admits an R-point with finite image.

    If Z^1(W_F^0, Ĝ)_ϕ_D, α_D is nonempty, then by flatness it contains a -point . The last paragraph of the section Lifting this point to characteristic 0 in the proof of <cit.> shows that there exists such a  with finite image, and by passing to a conjugate we may and do assume _ preserves (B, T). Using the fact that = _N [1/N], we may and do assume that  factors through Ĝ([D^-1, 1/N]) for some N. Let K be a number field such that  factors through Ĝ(_K[D^-1, 1/N]), and let λ be a prime of _K not dividing any element of D; it is enough to show now that, after possibly extending K, the L-homomorphism  can be conjugated to factor through Ĝ(_K_λ). Since  has finite (hence bounded) image, this follows from <cit.>.


In general, if R is a [D^-1]-algebra, let Z^1(W_F^0/P_F, Ĝ)_R^r_D = 1 denote the clopen subscheme of Z^1(W_F^0/P_F, Ĝ)_R consisting of those  satisfying r_D() = 1. For given admissible ϕ_D and α_D, choose W_F^0 →Ĝ([D^-1]) as in Theorem <ref>. The map ρ↦ρ· gives an isomorphism

    Z^1_(W_F^0/P_F, Z_Ĝ(ϕ_D)^0)_R^r_D = 1 r_D^-1(ϕ_D)_α_D⊗_[D^-1] R.

Note that the action of W_F on Z_Ĝ(ϕ_D)^0 induced by  is D-tame in the following sense.


    The action of W_F on Ĝ is D-tame if it is trivial on I_F, b^0 for some b ∈ D'.


Thus Theorem <ref> reduces to the following a priori weaker statement.


    Suppose that the action of W_F on Ĝ is D-tame and preserves a Borel pair. If R is an integral domain over [1/p] and D_R = D, then Z^1(W_F^0/P_F, Ĝ)^r_D = 1_R is connected.
































§ PROOF OF THEOREM <REF>


For the rest of this paper, we keep the notation of Section <ref> and assume that the action of W_F on Ĝ is D-pure in the sense of Definition <ref>. Moreover, let R be an integral domain over [1/p] such that D_R = D, and let D' be as in Section <ref>. As we have shown in the previous section, to prove Theorem <ref> (or the slightly stronger Theorem <ref>), it is enough to prove Theorem <ref>.





 §.§ Reduction to the simply connected case


The goal of this section is to reduce Theorem <ref> to the case that Ĝ is semisimple and simply connected. In fact, for completeness we prove slightly more than is necessary. We begin by establishing Theorem <ref> in the case that Ĝ is a torus; the case R = [1/p] is established in <cit.> by a completely similar argument.


    Let T̂ be a torus over R equipped with a finite D-tame action of W_F^0/P_F. The scheme Z^1(W_F^0/P_F, T̂)^r_D = 1_R is connected.



    Let n be an integer as in Lemma <ref>, where P = P_F, and let b be the largest factor of n lying in D'. Thus Z^1(W_F^0/P_F, T̂)^r_D = 1_R is the kernel of the map T̂_R ×T̂_R →T̂_R ×T̂_R given by
    
    (Φ_0, Σ_0) ↦ (Φ_0 ·FrΣ_0·s^qΦ_0^-1·(∏_i=0^q-1s^iΣ_0)^-1, ∏_i=0^b-1s^iΣ_0).

    By <cit.>, this is a diagonalizable R-group scheme, so it is enough to show that its character group has torsion order in D'. If the action of s is trivial, then this is clear because Z^1(W_F^0/P_F, T̂)^r_D = 1_R is a closed R-subgroup scheme of T̂×T̂[b] which surjects onto the first factor of T̂. If instead the map L_s t ↦ t (st)^-1 is an isogeny, then the map m_b t ↦∏_i=0^b-1s^it is trivial. Indeed, since the action is D-tame, s^b acts trivially on T̂, so the homomorphism m_bT̂→T̂ factors through the finite (L_s), hence is trivial. Thus the map Z^1(W_F^0/P_F, T̂)^r_D = 1_R →T̂_R given by ↦(s) is an isogeny with kernel (L_s^q), which is of order in D' by <cit.>.

    In general, the isogeny T̂→T̂_s ×T̂/T̂^s is of degree in D' by <cit.>. Thus the map
    
    Z^1(W_F^0/P_F, T̂)^r_D = 1_R →Z^1(W_F^0/P_F, T̂_s)^r_D = 1_R ×Z^1(W_F^0/P_F, T̂/T̂^s)^r_D = 1_R

    has finite kernel and cokernel of order in D' and the lemma follows.



    Let ℓ be prime, and let f H → H' be an isogeny of connected reductive _ℓ-groups which is equivariant with respect to finite actions of a finitely generated group Γ. If f is of ℓ-power degree, then the pushforward f_*Z^1(Γ, H) →Z^1(Γ, H') is a homeomorphism.



    Since Γ is finitely generated, its actions on H and H' are both defined over some finite field _ℓ^n. Note that f is finite and connected, so, after possibly increasing n, there is an isogeny h H' → H such that g ∘ f is equal to the relative Frobenius F H → H^(ℓ^n)≅ H. Note that g is automatically Γ-equivariant because f and F both are. Moreover, the pushforward F_* is equal to the relative Frobenius of Z^1(Γ, H): one can see this by embedding Z^1(Γ, H) into H^n via a surjection from a free group to Γ. Thus F_* is a homeomorphism, and a symmetric argument with g in place of f shows that f_* is a homeomorphism.





    Let fĜ' →Ĝ be a homomorphism of reductive _K[D^-1]-group schemes which is equivariant with respect to finite D-tame actions of W_F/P_F. If f induces an isogeny of derived groups, then f_*Z^1(W_F^0/P_F, Ĝ')^r_D = 1_R →Z^1(W_F^0/P_F, Ĝ)^r_D = 1_R is surjective on π_0.



    First suppose that f is an isogeny. Let n be the degree of f, and factor n = ∏_i=1^m ℓ_i^k_i, where ℓ_i ≠ℓ_j for i ≠ j. There is a unique factorization
    
    f = ( Ĝ' = Ĝ_m Ĝ_m-1⋯Ĝ_0 = Ĝ),

    where f_i is of degree ℓ_i^k_i. Note that the action of W_F on Ĝ extends uniquely to actions on Ĝ_i for all i by canonicity, and each f_i is W_F-equivariant. For each 1 ≤ i ≤ m, there is a pushforward (f_i)_*Z^1(W_F^0/P_F, Ĝ_i)^r_D = 1→Z^1(W_F^0/P_F, Ĝ_i - 1)^r_D = 1. If ℓ_i ∈ D', the map ((f_i)_*)__K/ℓ_i is a homeomorphism by Lemma <ref>, and hence (f_i)_* is surjective on π_0 by Lemma <ref>.Next suppose ℓ_i ∈ D. We claim that if k is an algebraically closed field over R, then ((f_i)_*)_k is surjective. We first reduce to the case that k is of positive characteristic. Note that by standard limit arguments, there is a saturated multiplicatively closed subset D_1 ⊂ D such that D_1 contains only finitely many primes, the action of W_F on Ĝ is defined over _K[D_1^-1], and Z^1(W_F^0/P_F, Ĝ)^r_D_1 = 1_R = Z^1(W_F^0/P_F, Ĝ)^r_D = 1_R. If k = 0, then to show that ((f_i)_*)_k is surjective it follows from a standard constructibility argument that it is enough to show that ((f_i)_*)__K/ℓ' is surjective for all ℓ' ∉D_1. So suppose that k is of characteristic ℓ'.By Lemma <ref> and the definitions, any cocycle _i-1∈Z^1(W_F^0/P_F, Ĝ_i - 1)^r_D = 1(k) satisfies (s)^b = 1 for some b ∈ D'. If N_b is the smallest normal subgroup of W_F/P_F containing s^b, then  induces a cocycle W_F/N_b →Ĝ_i-1(k) and the obstruction to lifting _i-1 to Z^1(W_F^0/P_F, Ĝ_i)^r_D = 1(k) lies in H^2(W_F/N_b, (f_i)(k)). By the Hochschild–Serre spectral sequence, there is an exact sequence
    
    H^1(W_F/I_F, H^1(I_F/N_b, (f_i)(k))) →H^2(W_F/N_b, (f_i)(k)) →H^2(I_F/N_b, (f_i)(k))^W_F/I_F

    Since I_F/N_b is finite of order coprime to the order of (f_i)(k), we see that the outer terms of this exact sequence vanish and hence the middle term vanishes as well. So we see that ((f_i)_*)_k is surjective, and by the above it follows that (f_i)_* is surjective. We have shown each (f_i)_* is surjective on π_0 and thus f_* is surjective on π_0.

    Now let Z (resp. Z') be the maximal central torus of Ĝ (resp. Ĝ'), and let (Ĝ) (resp. (Ĝ')) be its derived group. The multiplication morphism m Z ×(Ĝ) →Ĝ (resp. m' Z' ×(Ĝ') →Ĝ') is a central isogeny by <cit.>. There is a commutative diagram
    
    Z' ×(Ĝ') [r, "f"] [d, "m'"]
                   Z ×(Ĝ) [d, "m"] 
    Ĝ' [r, "f"]
                   Ĝ

    and we have shown above that m'_* and m_* are both surjective on π_0. Moreover, Z^1(W_F^0/P_F, Z) and Z^1(W_F^0/P_F, Z') are both connected by Lemma <ref>, and f(Ĝ') →(Ĝ) is an isogeny, so f_* is also surjective on π_0. Thus f_* is surjective on π_0, as desired.
















 §.§ A map to the adjoint quotient


As in Section <ref>, denote the GIT quotient (Ĝ⋊ s)//Ĝ by Ĝ//_s Ĝ, and let χ_sĜ⋊ s →Ĝ//_s Ĝ be the natural map. Let W_0 be the subgroup of s-fixed points of the Weyl group of T̂ in Ĝ, and let A = T̂/(1 - s)T̂. There is a natural map T̂→Ĝ//_s Ĝ given by t ↦χ_s(t ⋊ s), and Theorem <ref> shows that the induced map A/W_0 →Ĝ//_s Ĝ is an isomorphism.

Define an _K[D^-1]-morphism ΣZ^1(W_F^0/P_F, Ĝ) →Ĝ⋊ s by ↦L(s). Define Σ = χ_s ∘Σ.

There are maps Ĝ⋊ s →Ĝ⋊ s^q given by the qth power map and Fr, and these induce Fr^-1[q]Ĝ//_s Ĝ→Ĝ//_s Ĝ. Let B_Ĝ = (Ĝ//_s Ĝ)^Fr^-1[q]. Moreover, let B_Ĝ[D'] denote the closed subscheme of (B_Ĝ)__K[D^-1] consisting of those sections x satisfying [b]x = χ_s(1) for some b ∈ D'. Define Σ_D' and Σ_D' to be the restrictions of Σ and Σ to Z^1(W_F^0/P_F, Ĝ)^r_D = 1, and note that Σ_D' factors through B_Ĝ[D']. In fact, Z^1(W_F^0/P_F, Ĝ)^r_D = 1 = Σ^-1(B_Ĝ[D']).

For each w ∈ W_0, let A_w be the subscheme of A consisting of those a such that (Fr^-1a)^q = na(sn)^-1, where n is a lift of w in N_Ĝ(T̂). Let A_w[D'] denote the closed subscheme of A_w consisting of those a such that χ_s(a^b) = χ_s(1) for some b ∈ D'.


    Each (A_w[D'])_R is finite and connected, and the natural map ⋃_w ∈ W_0 A_w[D']_R → B_Ĝ[D']_R is surjective.



    Note that A_w[D'] is a torsor for the multiplicative type _K[D^-1]-group scheme A_w'[D'] consisting of those a ∈ A such that a^b = 1 for some b ∈ D' and (Fr^-1a)^q = wa. Since each A_w' is a multiplicative type group scheme, to show that A_w is finite and connected it suffices to show that A_w'() contains no elements of order in D. But this is clear.By Theorem <ref>, the natural map (A/W_0)^Fr^-1[q]→ B_Ĝ is an isomorphism. Moreover, the map ⋃_w ∈ W_0 A_w[D'] → (A/W_0)^Fr^-1[q][D'] is clearly surjective.


As we will see in Section <ref>, the map of Lemma <ref> is the key to showing that (Σ_D')_R is connected and thereby (using the results of this section) to proving Theorem <ref>.


    For every algebraically closed field k of characteristic ≠ p, every w ∈ W_0, and every a ∈ A_w[D'](k), there is some ∈Z^1(W_F^0/P_F, Ĝ(k)) such that Σ() = χ_s(a).



    Note that A_Fr^-1w(k) is, equivalently, the set of a ∈ A(k) such that a^q = wFra· n (s^qn)^-1. Lifting a arbitrarily to t_0 ∈T̂(k), we find that there is some t' ∈T̂(k) such that
    
    ∏_i=0^q-1s^it_0 = wFrt_0· n(s^qn)^-1· t' (st')^-1.

    Now (1 - s)T̂ = (1 - s^q)T̂: indeed, clearly (1 - s^q)T̂⊂ (1 - s)T̂, so these are equal for dimension reasons because (1 - s^q)T̂ = (Fr - Frs)Fr^-1T̂. Moreover, (1 - s)T̂ is clearly W_0-stable. In particular, there is some t”∈T̂(k) such that w(t”(s^qt”)^-1) = t'(st')^-1. We thus define ∈Z^1(W_F^0/P_F, Ĝ(k)) by (Fr) = nt” and (s) = t_0.



    The map Σ factors through a surjective map to B_Ĝ. Moreover, the schematic image Σ is finite flat. Completely similar claims hold for Σ_D' and Σ_D' with B_Ĝ[D'] in place of B_Ĝ.



    Since Z^1(W_F^0/P_F, Ĝ) is flat over the Dedekind domain _K[1/p], it is clear that Σ is also flat. The fact that Σ factors through B_Ĝ follows directly from the fact that
    
    (Fr)(s)(Fr)^-1 = (s)^q

    for ∈Z^1(W_F^0/P_F, Ĝ). Moreover, the set-theoretic image of Σ is equal to B_Ĝ by Lemma <ref> and Lemma <ref>. Finiteness of B_Ĝ, and hence Σ, thus follows from Lemma <ref>. The surjectivity of Σ follows from Lemma <ref> and Lemma <ref>. The claims for Σ_D' are immediate consequences.



    Suppose that for every field k over _K[D^-1] and every x ∈ (Σ_D')(k), the preimage Σ^-1(x) is connected. Then the map π_0(Z^1(W_F^0/P_F, Ĝ)^r_D = 1_R) →π_0((Σ_D')_R) is bijective.



    This follows from Lemma <ref>, Lemma <ref>, and Lemma <ref>.







    If Ĝ is semisimple and simply connected and the action of W_F on Ĝ is unramified, then <cit.> and <cit.> combine with fppf descent to show that B_Ĝ is flat with étale generic fiber, so in this case we have Σ = B_Ĝ. In fact, <cit.> suggests that this equality holds even for ramified actions. Since we do not need this refined result, we have not attempted to prove it. For Ĝ which are not simply connected, it is not clear that B_Ĝ is flat.









 §.§ Deformation to quasi-semisimple inertia


In this section, we use the results of Section <ref> to show that every L-homomorphism W_F^0/P_F →(k) can be deformed to a cocycle such that (s) has quasi-semisimple action on Ĝ, and use this to deduce a connectedness claim.


    Let k be a field, let x ∈ (Ĝ//_s Ĝ)(k) be a point, and let C ⊂χ_s^-1(x) be the unique closed Ĝ-orbit. The inclusion Σ^-1(C) →Σ^-1(x) induces a surjection on π_0.



    We may and do assume k = k. Let ∈Σ^-1(x)(k). By Theorem <ref> and the Hilbert-Mumford criterion, there is a cocharacter λ→Ĝ such that Lψlim_t → 0(λ(t)) ∘ exists and is completely reducible. By Theorem <ref>, the restriction Lψ|_I_F^0/P_F is completely reducible, so Lemma <ref> shows that Lψ(s) ∈ C(k), so we have produced a morphism f^1 →Σ^-1(x) with f(1) = and f(0) ∈Σ^-1(C)(k).



    Let k be a field over _K[D^-1], and let x ∈ (Σ_D')(k). If Ĝ is semisimple and |π_1(Ĝ)| ∈ D, then Σ^-1(x) is connected.



    By Lemma <ref>, it is enough to show that X = Σ^-1(C) is connected. Note that Σ restricts to a surjective k-morphism X → C. There is a natural map [q] C →Ĝ⋊ s^q given by the qth power, whose scheme-theoretic image we call C^[q]. Being a surjective morphism of homogeneous spaces for Ĝ, the map C → C^[q] is flat. Let H denote the fiber product
    
    H [r] [d]
                   Ĝ× C^[q][d, "(g, t) ↦ (g t g^-1,  t)"] 
    
            C [r, "Δ∘[q]"]
                   C^[q]× C^[q]

    so H is a “universal twisted centralizer" over C, and the diagram shows that H → C is flat. Moreover, there is an action map α H ×_C Σ^-1(C) →Σ^-1(C) given by α(h, )(Fr) = h(Fr) and α(h, )(s) = (s), which makes Σ|_X into an H-torsor over C. By <cit.>, since |π_1(Ĝ)| ∈ D and each t ∈ C(k) is quasi-semisimple of (finite) order in D', it follows that H has connected fibers over C. Since C is a single Ĝ-orbit, it is in particular connected, and thus X is connected.








 §.§ Connectedness of the scheme of eigenvalues


In this section, we complete the proof of Theorem <ref> by showing that (Σ_D')_R is connected. We retain the notation of Section <ref>.

By Lemma <ref> and Lemma <ref>, we have a surjection |⋃_w ∈ W_0 A_w[D']_R| → |(Σ_D')_R|, where each A_w[D']_R is finite and connected. The image of each A_w[D']_R in (A/W_0)_R is closed, so we see π_0((Σ_D')_R) = W_0/∼, where ∼ is the equivalence relation generated by the relation w ∼ w' whenever the images of A_w[D']_R and A_w'[D']_R in (A/W_0)_R intersect. In this section, we will show that w ∼ w' for all w, w' ∈ W_0, and thus (Σ_D')_R is connected. In fact, we will see that A_w[D']_R ∩ A_w'[D']_R ≠∅ for all w, w' ∈ W_0 whenever Z(Ĝ) is non-smooth at at most one prime in D'. However, this can fail for simple groups in type A by the following example, so this case is worked out separately in Appendix <ref>.


    Let Ĝ = _6, let q = 7, and let α be a primitive 36th root of unity. Let Fr act trivially on Ĝ, and let s act as (t), where t = (α, α, α, α, α, α^-5). If w = (1  2  3  4  5  6), we claim that A_1 ∩ A_w = ∅. Indeed, if k is a field then any element (γ_1, …, γ_6) of A'_1(k) ∩ A'_w(k) is required to satisfy γ_i^6 = α^-6 for all i and also γ_i = γ_j for all i, j. This is not possible for any field k because there is no 6th root of α^-6 in μ_6(k) (as 6 is not a power of k).


Write s = (t) ∘ s_0 and Fr = (t') ∘ F_0, where t, t' ∈T̂(_K[D^-1]) and s_0 and F_0 preserve a pinning (B̂, T̂, {X_α}); this is possible by <cit.>. The relation Fr s Fr^-1 = s^q implies (Frt· t' ·(s^qt')^-1) = (∏_i=0^q-1s^it), so Frt· t' ·(s^qt')^-1 = (∏_i=0^q-1s^it) · z for some central element z. In particular, [Frt] · [t^-q] = [z], where we use [x] to denote the image of x in A. Since the action is D-tame, we have t^b ∈ Z(Ĝ)(_K[D^-1]) for some b ∈ D'. Writing t^b = z_0 z_1 as a product of central elements, where z_0 has order in D and z_1 has order in D', we may pass to a translate of t by a bth root of z_0^-1 to assume that [z] is of finite order in D'.


    With notation as above, we have A_w = [t]^-1·{a ∈ A(Fr^-1a)^q(wa)^-1 = [Fr^-1z^-1]}.



    Let A'_w = {a ∈ A(Fr^-1a)^q(wa)^-1 = [Fr^-1z^-1]}. We need only check the equality A_w = [t]^-1· A'_w of generically étale flat closed subschemes of A on -points. By <cit.>, we may choose some n ∈ N_Ĝ(T̂)() lifting w which is fixed by s_0. We have then n(sn)^-1 = ntn^-1t^-1 = wt· t^-1, so if a ∈ A_w() then
    
    Fr^-1([t]a)^q = [t] ·[Fr^-1z^-1] ·wa· [wt] · [t^-1].

    This shows [t]a ∈ A'_w(). Moreover, the calculation can be run in reverse to show that if a' ∈ A'_w() then [t]^-1a' ∈ A_w().


Lemma <ref> shows in particular that ∼ only depends on D, z, and th e action of W_F on A.


    If Ĝ is semisimple and simply connected, then (Σ_D')_R is connected.



    By <cit.>, there is a unique direct product decomposition Ĝ≅∏_i ∈ IĜ_i for semisimple _K[D^-1]-group schemes Ĝ_i with absolutely simple fibers. The action of W_F on Ĝ permutes the Ĝ_i, so W_F acts on I and we may write I = ∐_j=1^k I_j where each I_j is a single W_F-orbit. If Ĝ_I_j∏_i ∈ I_jĜ_i, then we have Z^1(W_F^0/P_F, Ĝ)^r_D = 1≅∏_j=1^k Z^1(W_F^0/P_F, Ĝ_I_j)^r_D = 1. Since each Z^1(W_F^0/P_F, Ĝ_I_j)^r_D = 1 admits a section over _K[D^-1] (namely, the trivial cocycle), we may thus reduce from Ĝ to Ĝ_I_j to assume that W_F permutes the simple factors of Ĝ transitively. In particular, the simple factors of Ĝ are pairwise isomorphic.Supppose that the center of Ĝ is smooth over _K[(D ∪{ℓ})^-1] for some prime ℓ∈ D', so [z] is of ℓ-power order. If  is a maximal ideal of R containing ℓ, then Lemma <ref> shows t^-1∈ A_w[D'](R/) for all w ∈ W_0, so W_0/∼ is a singleton. On the other hand, if |Z(Ĝ)| is divisible by at least two primes in D', then the simple factors of Ĝ are all isomorphic to _n for some n, and thus Lemma <ref> shows that π_0((Σ_D')_R) ≅ W_0/∼ is a singleton, as desired.



    By Lemma <ref>, we may and do pass from Ĝ to the universal cover of the derived group (Ĝ) to assume that Ĝ is semisimple and simply connected. By Lemma <ref> and Lemma <ref>, the natural map π_0(Z^1(W_F^0/P_F, Ĝ)_R) →π_0((Σ)_R) is bijective. By Proposition <ref>, the scheme (Σ_D')_R is connected, so we are done.



    The proof of Proposition <ref> (and thus Theorem <ref>) can be made uniform for Ĝ such that Z(Ĝ)__K[(D ∪{ℓ})^-1] is smooth for some ℓ∉D, and hence (by splitting into simple factors) if Ĝ has no simple factors of type A. In the reduction from Theorem <ref> to Theorem <ref>, one replaces Ĝ by the identity component Ĝ' of the group of fixed points of the action of a finite group of order in D on Ĝ, and it may happen that Ĝ' has simple factors of type A. However, if Z(Ĝ)__K[(D ∪{ℓ})^-1] is smooth for some ℓ∉D, then <cit.> or <cit.> can be used to show that Z(Ĝ')__K[(D ∪{ℓ})^-1] is also smooth, so a trick using z-extensions and an analogue of <cit.> over R (akin to the argument in Section <ref>) would allow one to prove Conjecture <ref> uniformly away from type A.










§ TYPE A


In this section, we retain the notation of Section <ref>, especially <ref>. We will assume that Ĝ≅_n^m and that W_F^0/P_F permutes the simple factors of Ĝ transitively. Further, T̂ is the diagonal torus and B̂ is the upper-triangular Borel. Our aim is to show that w ∼ w' for all w, w' ∈ W_0, which will involve constructing a sequence w = w_0, w_1, …, w_N = w' such that A_w_i∩ A_w_i+1≠∅ for all i. We begin with the first nontrivial example, illustrating part of our algorithm for constructing w_i.


    We retain the notation and definitions of Example <ref>. Although A_w ∩ A_1 = ∅, we will show that w ∼ 1. Let w' = (1   2   3)(4   5   6), and let β = α^6. Because a 6th root of unity in _2 is automatically a 3rd root of unity, a short calculation using Lemma <ref> shows that the element t^-1(1, β, β^2, β^3, β^4, β^5) lies in (A_w ∩ A_w')(_2). (In fact, A_w and A_w' only intersect modulo 2.) If w” = (1   2  3), then in fact (A_w'∩ A_w”)() ≠∅: this follows from a general argument using Lemma <ref> (in the case a = 1 and σ_w = w for all w), but one can explicitly compute that t^-1(α^11, α^-1, α^23, α, α, α) lies in this intersection. Moreover, 1 ∈ (A_w”∩ A_1)(). This gives a chain which proves w ∼ 1 in this case.


Example <ref> gives the model for our argument below: break apart n-cycles by specializing modulo a well-chosen prime, then cut off individual cycles by finding intersections over . In order to deal with actions of W_F^0/P_F which are more complicated than the one in Example <ref>, the general argument requires more notation, which we now begin to introduce.

Let I_1, …, I_a be the s-orbits on {1, …, m}. Since W_F^0/P_F permutes the simple factors of Ĝ transitively, it follows that Fr permutes the I_i transitively. By reordering the I_i, we may and do assume that Fr sends each I_i+1 to I_i, where we write I_a + 1 = I_1. We first note that we may reduce to the case a = 1.


    Let F_a denote the unramified extension of F of degree a. The natural restriction map rZ^1(W_F^0, Ĝ) →Z^1(W_F_a^0, Ĝ_I_1) is smooth with geometrically connected fibers.



    The proof of <cit.> can be read directly with only very minor adaptations to show that r is isomorphic to the projection map of Z^1(W_F_a^0, Ĝ_I_1) ×Ĝ_I_1^a-1 onto its first factor, where Ĝ_I_1 is the product of the factors of Ĝ with index in I_1.


Note that s^m induces an automorphism of each factor of Ĝ; we denote its image in the group of outer automorphisms of _n by (s), and note that it is independent of the choice of factor. The relation Fr s Fr^-1 = s^q implies that (s) = (s)^q, so (s) is trivial if q is even. If S denotes the diagonal torus of _n, then (s) induces an automorphism of S by identifying S as the group of coinvariants of S^m under the permutation action induced by s. We have A = T̂/(1 - s)T̂ = S/(1 - (s))S. Similarly, Fr induces an automorphism of S, and we denote its image in the group of outer automorphisms by (Fr). When the context makes it unambiguous, we will also use (s) and (Fr) to denote their images under the canonical isomorphism (_n) ≅{± 1}.

The action of Fr on S is given by (β_1, …, β_n) ↦(β_Fr(1)^(Fr), …, β_Fr(n)^(Fr)), where we use Fr here and below to (abusively) denote the involution of {1, …, n} induced by (Fr). Similarly, we will use s to denote the permutation induced by (s). If t = (γ_j)_j=1^n ∈ S(A) for a ring A, and σ∈ S_n is a permutation, then by definition σt (γ_σ^-1(j))_j=1^n.

Keeping the notation preceding Lemma <ref>, we have s = (t) ∘ s_0 and Fr = (t') ∘ F_0, where t, t' ∈T̂(_K[D^-1]) and s_0 and F_0 preserve a pinning (B̂, T̂, {X_α}), and we have

    Frt· t' ·(s^qt')^-1 = (∏_i=0^q-1s^it) · z

for a central element z. Note that the map Z(Ĝ)^s → Z(Ĝ)^s given by t ↦Frt· t^-q has kernel of order dividing q^a - (Fr), as one can see by a straightforward calculation. Thus, after possibly extending K, we may and do alter t by an element of Z(Ĝ)^s(_K[D^-1]) to assume that every prime dividing the order of z divides q^a - (Fr). Similarly, since the action of W_F is D-tame, we may and do assume that z has order in D'.

If w ∈ W_0, we define S_w to be the closed subscheme of S consisting of those sections t such that (Fr^-1t)^q ·(wt)^-1 = [Fr^-1z^-1]. By Lemma <ref>, if S_w[b] ∩ S_w'[b] ≠∅ for some b ∈ D', then w ∼ w'. In fact, it is enough to show S_w ∩ S_w' admits a torsion point: suppose k is a field over R and t ∈ S_w(k) ∩ S_w'(k) is of order n = ab for a ∈ D, b ∈ D'. Let c ∈ D be such that ac ≡ 1 n_0, and note

    (Fr^-1t^ac)^q ·(wt^ac)^-1 = [Fr^-1z^-1]^ac = [Fr^-1z^-1],

so t^ac∈ S_w[b](k) ∩ S_w'[b](k). For convenience, if w ∈ W_0 we will write σ_w Frw ∈ W_0. 

Having set up the necessary notation, we are now ready to begin proving that w ∼ w' for all w, w' ∈ W_0. By symmetry, we need only show that w ∼Fr for all w ∈ W_0.


    Let w, w' ∈ W_0 be such that σ_w commutes with σ_w'. If S^σ_w^-1σ_w' is a torus, then S_w[b] ∩ S_w'[b] ≠∅ for some b ∈ D'.



    Define a map f S → S by f(t) = t^q ·(σ_wt)^-1. Let k be an algebraically closed field over R and note ( f)(k) has no p-torsion, so f is an isogeny. If A is an R-algebra and t ∈ S(A) is fixed by σ_w^-1σ_w', then we have
    
    σ_w^-1σ_w'f(t) = σ_w^-1σ_w'(t^q ·(σ_wt)^-1) = f(t)

    since σ_w commutes with σ_w'^-1σ_w = w^-1w'. Thus f induces an isogeny of tori S^σ_w^-1σ_w'→ S^σ_w^-1σ_w'. In particular, since Z(_n) ⊂ S^σ_w^-1σ_w', it follows that there exists a torsion element t ∈ S^σ_w^-1σ_w'(k) such that t ∈ S_w(k). But wt = w't, so also t ∈ S_w'(k), and thus w ∼ w'.


Lemma <ref> is not enough to prove w ∼ w' for all w, w' ∈ W_0, but it is enough to show this for “most" w and w', as we now explain. We will say σ∈ S_n is an s-cycle if either σ is a cycle or there is a cycle c ∈ S_n such that c and (s)c are disjoint and σ = c ·(s)c. If σ∈ S_n is fixed by (s), then there is a unique decomposition σ = c_1 ⋯ c_r of σ into disjoint s-cycles c_1, …, c_r. If σ is an s-cycle, then we define its length to be the number of i such that σ(i) ≠ i.


    If σ_w is not a single s-cycle of length n, then S_w[b] ∩ S_Fr[b] ≠∅ for some b ∈ D'.



    First, let σ_w = c_1 ⋯ c_r be the decomposition of w into disjoint s-cycles of length > 1, so either r ≥ 2 or r=1 and c_1 is of length < n. Note w ∼Frc_1 by Lemma <ref>, since S^σ is a torus for any permutation σ in S_n which has a fixed point. Thus we may assume that r = 1. In this case, another application of Lemma <ref> shows that w ∼Fr, as desired.


Thus we must show that if w ∈ W_0 is a single s-cycle of length n, then there is some w' ∈ W_0 which is not a single s-cycle of length n such that w ∼ w'. We begin with an elementary lemma for which we do not know a reference.


    Let n_0, e, q ≥ 1 be integers such that every prime number dividing n_0 divides q - 1.
    
        
  * ∑_i = 0^n_0 e - 1 q^i ≡ 0 n_0,
        
  * ∑_i = 0^n_0 e - 1 (n_0 e - 1 - i) q^i ≡ 0 n'_0, where n_0' = n_0 for n_0 odd and n_0' = n_0/2 for n_0 even.
    



    We may and do assume e = 1 and n_0 = ℓ^a for some prime number ℓ and some a ≥ 1. Write q = ℓ^c d + 1, where ℓ∤ d. For (1), note that
    
    ∑_i=0^ℓ^a - 1 q^i = (ℓ^c d + 1)^ℓ^a - 1/ℓ^c d.

    It is elementary to check that if x ≡ y ℓ^c, then x^ℓ^a≡ y^ℓ^aℓ^a+c, so the displayed equation implies (1). Note that the sum in (2) is equal to the value at X = ℓ^c d of 
    
    d/dx1 - (1 + X)^ℓ^a/X = -ℓ^a X (1 + X)^ℓ^a - 1 + (1 + X)^ℓ^a - 1/X^2 = ∑_i=2^ℓ^a(ℓ^a  i - ℓ^a ℓ^a - 1  i - 1) X^i - 2.

    Modulo ℓ^a, this sum is equal to q^ℓ^a - (1 + ℓ^a+c d)/q^2. Standard estimates using Legendre's formula show that q^ℓ^a≡ 1 + ℓ^a+c d ℓ^a+2c if ℓ≠ 2, and q^ℓ^a≡ 1 + ℓ^a+c d ℓ^a + 2c - 1 if ℓ = 2. This immediately yields (2).


An element of S is identified with a tuple (γ_j) with 1 ≤ j ≤ n such that ∏_j=1^n γ_j = 1. Note in particular that [Fr^-1z^-1] is identified with an nth root of unity α. Note that a tuple (γ_j) lies in S_w if and only if

    γ_σ_w^-1(j) = γ_j^(Fr)qα^-1

for all j.


    Suppose w is an s-cycle of length n. There is some w' ∈ W_0 which is a product of disjoint s-cycles of length < n such that S_w[b] ∩ S_w'[b] ≠∅ for some b ∈ D'.



    If n is divisible by at most one prime in D', then this follows from the argument of Proposition <ref>, so we will assume that n is divisible by at least two primes in D'. Let n' equal n if w is a single cycle of length n, and n/2 if w is a product of two disjoint s-conjugate cycles whose lengths add to n (so in particular n is even and 2 ∈ D'). Let ℓ be a prime number in D' dividing n', and write n' = ℓ^c d for some d such that ℓ∤ d. Define w' ∈ W_0 as follows: σ_w'σ_w^i(1) is equal to σ_w^i+1(1) if d ∤ i+1, and σ_w^i+1 - d(1) if d | i + 1. If n' = n, then this determines w'; otherwise, we also require that σ_w'σ_w^i(n) is equal to σ_w^i+1(n) if d ∤ i+1, and σ_w^i+1 - d(n) if d | i+1.If n' = n, then we find
    
    σ_w' = (1  σ_w(1) ⋯σ_w^d-1(1))(σ_w^d(1) ⋯ ) ⋯ (σ_w^(ℓ^c - 1)d(1) ⋯σ_w^n - 1(1)).

    If n' = n/2, then we have
    
    σ_w' = (1  σ_w(1) ⋯σ_w^d-1(1))⋯   (σ_w^(ℓ^c - 1)d(1) ⋯σ_w^n' - 1(1)) ·
    ·   (n ⋯σ_w^d - 1(n)) ⋯ (σ_w^(ℓ^c - 1)d(n) ⋯σ_w^n'-1(n)).

    We will show that, if one of the conditions
    
        
  * n' = n is odd,
        
  * n' = n and 2 ∈ D,
        
  * n' = n and ℓ = 2 ∈ D',
        
  * n' = n/2 is even and ℓ = 2 ∈ D',
        
  * n' = n/2 is odd and ℓ∉D ∪{2},
    
    holds, then w' ∈ W_0 and (S_w ∩ S_w')(_ℓ) ≠∅, and thus w ∼ w'. This is enough to prove the lemma: if n' = n/2, then 2 ∈ D' and this follows from (4) and (5) because n is divisible by at least two primes in D' by assumption. If n' = n is odd, then this follows from (1). If n' = n is even, then this follows from (2) and (3) except possibly if ℓ = 2 ∈ D' and n/2 is odd, in which case we can pass from w to w' to reduce to the already-established case n' = n/2. 

    We begin by showing w' ∈ W_0. If (s) = 1, then this is trivial, so suppose (s) = -1. In this case, it follows that n is even, since any permutation of {1, …, 2m+1} fixed by the flip fixes m+1. Moreover, 2 ∈ D'. We have
    
    (s)σ_w'σ_w^i(1) =
           σ_w^i+1(n)  if  d ∤ i+1, 
       σ_w^i+1 - d(n)  if  d | i+1.

    On the other hand, if n' = n then (s)(1) = n = σ_w^n/2(1) and thus
    
    σ_w'σ_w^i((s)(1)) =
           σ_w^i + 1(n)  if  d ∤ i+1+n/2, 
       σ_w^i + 1 - d(n)  if  d | i+1+n/2.

    If ℓ = 2, then d |n/2, so we find (s)σ_w' = σ_w'(s), i.e., σ_w'∈ W_0, and thus w' ∈ W_0 since Fr∈ W_0. Thus for the rest of this proof, we will assume ℓ = 2 whenever (s) = -1 and n' = n. If instead n' = n/2, then the definitions immediately imply (s) σ_w' = σ_w'(s) without restriction on ℓ.

    To show w ∼ w', suppose first that n' = n. If either n is odd or 2 ∈ D, then we choose ℓ∈ D' arbitrarily; if n is even and 2 ∈ D', then we take ℓ = 2. By the congruences of Lemma <ref>, there is an element t = (γ_j) of S_w(_ℓ) determined by γ_σ_w^-j(1) = ∏_i=0^j-1α^-((Fr)q)^i for all j. The key observation is now that γ_σ_w^j(1) = γ_σ_w^j + d(1) in _ℓ for all j: indeed, unraveling, we must show that ∏_i=0^d-1α^(Fr)q^i = 1 in _ℓ. This follows from Lemma <ref>(1), using the facts that α is a (d, n_0)th root of unity in _ℓ and that (Fr)q ≡ 1 ℓ' for every prime ℓ' dividing (d, n_0). This shows t ∈ S_w'(_ℓ). Since t is torsion, we find w ∼ w'. 

    Next, suppose n' = n/2. In this case, n is even and 2 ∈ D'. If either ℓ = 2, or n' is odd and ℓ∉D ∪{2}, then we let t = (γ_j) be the element of S_w(_ℓ) determined by γ_σ_w^-j(1) = γ_σ_w^-j(n) = ∏_i=0^j-1α^-((Fr)q)^i for all j ≥ 0. Again, the congruences of Lemma <ref> imply that this is well-defined, and the same argument as before implies t ∈ S_w'(_ℓ), allowing us to conclude.



    Suppose Ĝ≅_n^m, where W_F permutes the simple factors transitively. Then w ∼ w' for all w, w' ∈ W_0.



    It follows from Lemma <ref> and Lemma <ref> that w ∼Fr for all w ∈ W_0, as desired.





























halpha-abbrv





