On the Definition of Transfer Factors*

R.P. Langlands and D. Shelstad

To Friedrich Hirzebruch on his sixtieth birthday

CONTENTS

0. Introduction (1.) Preliminaries

(1.1) An Example (1.2) Notation (1.3) Point Correspondences (1.4) Transfer Factors

(2.) Key Lemmas (2.1) General Remarks (2.2) a-Data (2.3) An Application (2.4) An Explicit Splitting (2.5) χ-Data (2.6) A Second Application

(3.) Definitions (3.1) Notation (3.2) Term Δ_{I} (3.3) Term Δ_{II} (3.4) Term Δ_{III_{1}}, or Δ_{1} (3.5) Term Δ_{III_{2}}, or Δ_{2} (3.6) Term Δ_{IV} (3.7) The Factor Δ

4. Some Properties of Δ (4.1) Invariance (4.2) The Local Hypothesis (4.3) Extension to all G-Regular Elements (4.4) Passage to Central Extensions

5. Regular Unipotent Analysis (5.1) Regular Unipotent Elements (Review) (5.2) Stars and the Variety X (5.3) Regular Unipotent Elements and X (5.4) Orbital Integrals as Fiber Integrals (5.5) A Limit Formula

6. Global Consequences

* First appeared in Math. Ann., Bd 278 (1987), 219-271 (6.1) Outline (6.2) Notation (6.3) Adelic Images and Transfer Factors (6.4) Product Formulas

0. Introduction This paper is concerned with a specific question in harmonic analysis on reductive groups over a local

field. Central to it is the notion of an endoscopic group ([Sh; L_{2}], and for a perhaps definitive treatment [K-S]). This notion is still exotic and difficult to grasp, partly because its origins lie beyond the periphery of harmonic analysis, in the L-group (which first arose in the theory of Eisenstein series) and in the study of Shimura varieties, and partly because it still has not achieved in sufficient generality its original purpose, the analysis of the internal structure of L-packets of irreducible representations.

Roughly speaking, L-packets occur in the classification of the irreducible representations of the group G(F) of F -valued points on a reductive group over the local field F because there are two types of conjugacy within G(F), that realized by elements of G(F) and that realized by elements of G(F̄). Only the first appears when the harmonic analysis is treated from a strictly analytic viewpoint, but the second intervenes when the harmonic analysis is applied to problems in number theory, especially to the study of L-functions, and leads to a coarser classification of irreducible representations than equivalence. The coarse classes are called L-packets, and they are to be analyzed individually with the help of endoscopic groups.

An endoscopic group H is not a subgroup of G, but we can associate to a conjugacy class in H(F) several conjugacy classes in G(F), and the harmonic analysis on G(F) is related to that on H(F) by means of the transfer of orbital integrals. This refers to pairs of functions, one f on G(F) and one f^{H} on H(F) whose orbital integrals on associated conjugacy classes are related by transfer factors [see (1.4)].

The definition of transfer factors that not only allow one to attach to each f at least one f^{H} but also behave well with respect to functoriality has not been easy, and if it were not that they had been proved to exist over the real field [Sh], it would have been difficult to maintain confidence in the possibility of transfer or in the usefulness of endoscopy.

The contribution of this paper is not to prove the existence of the transfer, that is to attach to f at least one f^{H}, but simply to define the transfer factor, disentangling the conditions imposed or suggested by the harmonic analysis, by Galois cohomology, by the trace formula, and by the constructions over R to arrive at an explicit definition that clearly must be the transfer factor if it exists at all and that even over the real field is an improvement over the construction of [Sh] which was not sufficiently explicit. We flatter ourselves that this definition is an advance and that it is not merely our lack of skill that has made it so hard to come by, but the difficulty of the subject.

In Sect. 1 we are more explicit about the transfer of orbital integrals and transfer factors, recalling in particular the first example that was studied, the group SL(2). Their definition appears in Sect. 3, where it will be seen that the transfer factor is the product of five terms, two, Δ_{II} and Δ_{IV}, serving to meet basic requirements of harmonic analysis. A third Δ_{I} incorporates the basic idea of endoscopy and transfer, weighted sums of orbital integrals, often referred to as κ-orbital integrals. The two others, Δ_{1} and Δ_{2}, are cohomological in nature and are there to compensate the arbitrary elements that had to be used in the definition of Δ_{II}. It should be observed that Δ_{I} does not appear in [Sh] where it had to be replaced by an existence argument.

The definitions of Δ_{1} and Δ_{2} are quite elaborate, involving a number of general arguments and constructions that we have preferred to place in a separate Sect. 2 which also prepares for the product formula of Sect. 6.

Although we do not discuss the existence of the transfer, thus of f^{H} for a given f, the result of Sect. 5 is evidence that it will be available, and is in addition the source of the factors Δ_{I} and Δ_{II}. The limit formula of (5.5) shows that the dominant term near the identity of the combination of orbital integrals of f appearing in the transfer can be made equal to the dominant term of the stable orbital integral of an f^{H}. This is clearly a necessary condition for the existence of f^{H}, and is what guarantees that the choice of the correction factor Δ_{I} is correct. The factor Δ_{2} does not affect the asymptotic behavior, and is dictated by experience with the real field.

The properties of the transfer factor that in addition to (5.5) in all probability characterize it are the Local Hypothesis, which relates the transfer factor on an arbitrary group to that on a quasi-split inner twisting, the Global Hypothesis, which is a product formula, and the transfer factor over archimedean fields, already introduced in [Sh]. We prove the first two here, in Sects. 4 and 6, reserving the proof that the transfer factor of this paper coincides over R with that of [Sh] for a later paper.

It is a pleasure to dedicate this paper to Friedrich Hirzebruch, for one of us first realized the significance of L-packets during a long stay in Bonn many years ago under the auspices of the SFB, when he was able to study Shimura’s papers on automorphic forms. At the same time the other was beginning the study of character identities for real groups, and L-packets and character identities together led to endoscopy.

I. Preliminaries (1.1) An Example Suppose that F is a local field of characteristic zero and G is SL_{2}. Take H to be a one-dimensional torus split over the quadratic extension E of F and anisotropic over F. Then H is an endoscopic group for G. To γ_{H} ∈ H(E) we attach the conjugacy class in G(E) consisting of semisimple elements with eigenvalues γ_{H}^{±1}. Assume that γ_{H} ⊧ ±1 lies in H(F). Then the associated class meets SL_{2}(F) in a stable conjugacy class of regular semisimple elements. Call γ_{H} an image of any γ in this stable class.

For f ∈ C_{c}^{∞}(G(F)) form the integral Φ(γ, f) of f along the conjugacy class of regular semisiimple γ in G(F) with respect to the G-invariant measure prescribed by a choice of invariant forms of highest degree over F on G and H. Then transfer between G and H requires a function Δ, a transfer factor, such that

f^{H} : γ_{H} → ∑_{γ} Δ(γ_{H}, γ)Φ(γ, f)

extends smoothly to γ_{H} = ±1. Here ∑_{γ} indicates summation over representatives for the regular semisimple conjugacy classes in G(F); Δ(γ_{H}, γ) is to be zero unless γ_{H} is an image of γ, so that the sum contains at most two non-zero terms.

We modify slightly the prescription for Δ in [L-L]. For reasons of functoriality Δ will depend not on H alone but on a set (H, H, s, ζ) of endoscopic data: H = Ĥ ⋊ W is the L-group of H, s lies in the conjugacy class of Ĝ = P GL(2, C) determined by the elements of GL_{2}(C) with eigenvalues ± z, z ∈ C, and ξ is an embedding of H in Ĝ ⋊ W = ^{L}G that carries Ĥ into Cent (s, Ĝ). Here we may take for W the Weil group of E/F. Equivalent, or Ĝ-conjugate, data will yield the same factor Δ.

We first define a factor Δ_{0} which depends in addition on the choice of an F -splitting of G. The quotients Δ_{0}(γ_{H}, γ)/Δ_{0}(\overline{γ}_{H}, \overline{γ}) will, however, be canonical. To prescribe Δ we fix some (\overline{γ}_{H}, \overline{γ}) with \overline{γ}_{H} an image of \overline{γ}, specify Δ(\overline{γ}_{H}, \overline{γ}) arbitrarily, and then set

(1.1.1)  Δ(γ_{H}, γ) = Δ(\overline{γ}_{H}, \overline{γ})\frac{Δ_{0}(γ_{H}, γ)}{Δ_{0}(\overline{γ}_{H}, \overline{γ})}

if γ_{H} is an image of γ. Thus Δ is canonical, up to the constant Δ(\overline{γ}_{H}, \overline{γ}).

The factor Δ_{0} will be a product of several terms. Only the first depends on the choice of F -splitting. Here we will describe it for the standard splitting (B, T, X) : B is the upper triangular subgroup, T the diagonal subgroup and X = [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ]. For the general case and for the fact that the relative factor is independent of the choice of splitting we refer to (3.2).

Other choices are needed to define the terms in Δ_{0}: an admissible embedding of H in G, and a-data and χ-data for the image of H under that embedding. Let Γ = Gal(E/F). An embedding of the torus H in G is admissible if it is dual to a composition Ĥ → T → T̂, where T is some maximal torus in Ĝ containing s, T → T̂ is the isomorphism attached to (B, T ) and some pair (B, T) chosen so that T is defined over F and Ĥ → T̂ is a Γ-isomorphism. See (1.2) and (1.3). The a-data for T consist of elements a_{α}, a_{−α} of E^{×} such that \overline{a}_{α} = −a_{α} = a_{−α}, where ±α are the roots of T and the bar denotes conjugation in E.

The χ-data are characters χ_{α}, χ_{−α} on E^{×} which extend the quadratic character on F^{×} attached to E and satisfy χ_{−α} = χ_{α}^{−1}.

Here then are the terms in Δ_{0}.

(I)  ⟨λ(T), s_{T} ⟩,

where λ(T) is the class in H^{1}(T) = H^{1}(Γ, T (E)) of the cocycle

σ → h [ \begin{matrix}0 & a_{α} \\ −a_{α}^{−1} & 0 \end{matrix} ] σ(h^{−1})

for σ nontrivial in Γ. The element h is given by h^{−1}T h = T and h^{−1}Bh = B, with B some Borel subgroup containing T. Then α is the root of T in B. The element s_{T} is the unique nontrivial Γ-invariant element in T̂; it is also the image of s under T → T̂. Finally, the pairing is the Tate-Nakayama pairing.

(II)  χ_{α} (\frac{α(γ_{T}) − 1}{a_{α}}) ,

where γ_{T} is the image of γ_{H} under H → T, and α is either root of T in G.

(III_{1})  ⟨inv(γ_{H}, γ), s_{T} ⟩^{−1},

where inv(γ_{H}, γ) is the class in H^{1}(T) of the cocycle σ → gσ(g)^{−1} and g is given by g^{−1}γg = γ_{T}.

(III_{2})  ⟨a, γ_{T} ⟩,

where a is an element of H^{1}(W, T̂) which, roughly speaking, measures the difference between the Ĝ-conjugacy class of the embedding ξ : ^{L}H → ^{L}G and the class of embeddings canonically attached to the χ-data for T [see (2.6)]. The pairing is the usual one between H^{1}(W, T̂) and T (F) [B, Sect. 9].

(IV)  |(α(γ_{T}) − 1)(α(γ_{T})^{−1} − 1)|_{F}^{1/2}.

The image T of H in G may be replaced only by g^{−1}T g, where g ∈ 𝔄(T) = {g ∈ G(E) : gσ(g^{−1}) ∈ T }. If (T, {a_{α}}, {χ_{α}}) is replaced by a triple conjugate under 𝔄(T) in the obvious sense then we see that only the terms (I) and (III_{1}) are affected, but then clearly the effects cancel. Thus it remains to consider T fixed and the a-data, χ-data changed. Only (I) and (II) involve a-data. If a_{α} is replaced by a′_{α} = a_{α}b_{α} where b_{α} ∈ F^{×} then λ(T) is multiplied by the class b of the cocycle σ → b_{α}^{α^{∨}}. Note that b_{α}^{α^{∨}} lies in E^{×} ⊗ X_{∗}(T) = T (E). To show that the product of the terms (I) and (II) is unaffected by replacing a_{α} with a′_{α} we have only to check that

⟨b, s_{T} ⟩ = χ_{α}(b_{α}).

This is clear since b is trivial if and only if b lies in NmE^{×}.

On the other hand, only the terms (II) and (III_{2}) depend on χ-data. Suppose that χ_{α} is replaced by χ′_{α} = χ_{α}ζ_{α}. Then ζ_{α} must be trivial on F^{×}. We use the fact that the norm map δ → δ\overline{δ} from T (E) to T (F) is surjective to write γ_{T} as δ_{T} \overline{δ}_{T}. Then α(γ_{T}) = α(δ_{T})/\overline{α(δ_{T})} and

ζ_{α} (\frac{α(δ_{T}) − 1}{a_{α}}) = ζ_{α} (\frac{α(δ_{T}) − \overline{α(δ_{T})}}{a_{α}}) · ζ_{α}(α(δ_{T}))

or ζ_{α}(α(δ_{T})) since a_{α}^{−1}(α(δ_{T}) − \overline{α(δ_{T})}) lies in F^{×}. Thus we have to show that when χ_{α} is replaced by χ_{α}ζ_{α} the class a appearing in the term (III_{2}) is multiplied by a class a(ζ_{α}) such that

⟨a(ζ_{α}), γ_{T} ⟩ = ζ_{α}(α(δ_{T}))^{−1}.

We will see later (3.5) that in fact a is multiplied by the class of a cocycle which is given on E^{×} by χ → ζ_{α}(χ)^{−α}. That the class pairs with γ_{T} in the desired way is the Base Change Identity in this simple case, which follows from the remarks in [B, Sect. 9].

Lemma [L-L]. For f ∈ C_{c}^{∞}(G(F)) the function

f^{H} : γ_{H} → ∑_{γ} Δ(γ_{H}, γ)Φ(γ, f)

extends smoothly to H(F).

(1.2) Notation Throughout F will be a local or a global field of characteristic zero; \overline{F} will be an algebraic closure of F, and Γ and W_{F} the Galois group and Weil group of \overline{F} /F. Let G be a connected reductive group over F. Then G^{∗} will denote a quasi-split inner form of G and ^{L}G the L-group of G. More precisely, we fix:

(i) (G^{∗}, ψ) with G^{∗} quasi-split over F and ψ : G → G^{∗} an inner twist, and (ii) (Ĝ, ϱ, η_{G}) with Ĝ connected, reductive and defined over C, ϱ an L-action of Γ on Ĝ, and η_{G} : Ψ(G)^{∨} →

Ψ(Ĝ) a Γ-bijection.

Here Ψ(−) denotes canonical based root data (see [K_{2}]).

We have then for each pair (B, T) in G and (B, T ) in Ĝ a canonical isomorphism T̂ → T , where by a pair we mean a Borel subgroup and a maximal torus contained in it.

As L-group data, that is, the data of (ii), for G^{∗} we take (Ĝ, ϱ, η_{G^{∗}}) where η_{G^{∗}} is given by

Ψ(G^{∗})^{∨} ⟶^{ψ} Ψ(G)^{∨} ⟶^{η_{G}} Ψ(Ĝ).

Finally, ^{L}G will be the semidirect product Ĝ ⋊ W_{F} with W_{F} acting by W_{F} → Γ ⟶^{ϱ} Aut Ĝ.

We shall specify endoscopic data in a way useful for extension to the twisted case [K-S]. First note that if 1 → Ĝ → G → W_{F} → 1 is a split extension then G is not necessarily an L-group. Nevertheless we attach to G an L-action ϱ_{G} on Ĝ as follows. Let spl = (B, T , {X_{α}}) be a splitting of Ĝ. We shall require that spl is preserved by the ϱ appearing in (ii), that is, it is a Γ-splitting (or F -splitting). If c : W_{F} → G splits the extension then for w ∈ W_{F} we multiply Int c(w) acting on Ĝ by an element of Int Ĝ to obtain ϱ_{G}(w) preserving spl. Then ϱ_{G} is an L-action which is independent of the choice for c. Clearly if ϱ_{G} coincides with ϱ for some choice of spl then it coincides for all choices.

Let (H, H, s, ξ) be endoscopic data for G. By this we will mean:

(i) H is a quasi-split group over F. Its L-group data will be denoted (Ĥ, ϱ_{H}, η_{H}), and ^{L}H = Ĥ ⋊ W_{F}.

(ii) H is a split extension of W_{F} by Ĥ and ϱ_{H} coincides with ϱ_{H}.

(iii) s is a semisimple element of Ĝ.

(iv) ξ : H → ^{L}G is an L-homomorphism, that is, a homomorphism of extensions of W_{F}, such that (a) Int s ∘ ξ = a ⊗ ξ,

where a is a locally trivial 1-cocycle of W_{F} in the center Z(Ĝ) of Ĝ if F is global, or a trivial 1-cocycle of W_{F} in Z(Ĝ) if F is local, and

(b) ξ|Ĥ is an isomorphism of Ĥ with the connected component of the centralizer of s in Ĝ. Here a ⊗ ξ(h) = a(w(h))ξ(h), h ∈ H, with w(h) denoting the image of h under H → W_{F}.

Data (H′, H′, s′, ξ′) are equivalent to (H, H, s, ξ) if there exist an F -isomorphism α : H → H′, an L-isomorphism β : H′ → H and an element g of Ĝ such that:

(i) Ψ(H) ⟶^{α} Ψ(H′) and Ψ(Ĥ′) ⟶^{β} Ψ(Ĥ) are dual, (ii) Int g ∘ ξ ∘ β = ξ′ and: (iii) gsg^{−1} lies in Z(Ĝ)Z(ξ′)^{0}s′, where Z(ξ′) is the centralizer in G of the image of H′ under ξ′.

Up to equivalence and the choice of ξ, which amounts to the choice of an embedding of ^{L}H in ^{L}G in the case H is an L-group, these are the endoscopic data of [L2, p. 20]. Note that in the definitions of [L2] the group H generated by ^{L}H^{0} and the elements n(w), w ∈ W_{F}, is a split extension of W_{F} by ^{L}H^{0} (see L1, Lemma 4]).

If H is an L-group then we may assume that H = ^{L}H. This will be our assumption until (4.4) as it greatly simplifies notation. The minor modification needed for the general case, a passage to certain central extensions of H, will be dealt with in that section. (1.3) Point Correspondences The isomorphism T̂_{G} → T_{G} attached to pairs (B_{G}, T_{G}) in G and (B_{G}, T_{G}) in Ĝ transports the coroots of T_{G} in G to the roots of T_{G} in Ĝ, the B_{G}-simple coroots to the B_{G}-simple roots and the Weyl group of T, with contragredient action, to the Weyl group of T_{G}. If (B_{H}, T_{H}) is a pair in Ĥ then there is an x in Ĝ such that Intx ∘ ξ maps T_{H} to T_{G} and B_{H} into B_{G}. Finally, if (B_{H}, T_{H}) is a pair in H then we have an isomorphism T̂_{H} → T̂_{G} defined by the composition T̂_{H} → T_{H} ⟶ T_{G} ⟶ T̂_{G} and thus also an isomorphism T_{H} ⟶ T_{G}. These isomorphisms transport the coroots of T_{H} in H into a subsystem of the coroots of T_{G} in G, the Weyl group Ω_{H} of T_{H} into a subgroup of the Weyl group Ω_{G} of T_{G}, and the roots of T_{H} into a subset of the roots of T_{G}. The map

T_{H}/Ω_{H} ⟶ T_{G}/Ω_{G}

of orbits of Ω_{H} in T_{H} onto orbits of Ω_{G} in T_{G} is independent of all choices. Since these orbits classify the conjugacy classes of semisimple elements in H(\overline{F}) and G(\overline{F}), and the choice of tori is of no consequence, we have a canonical map

A_{H/G} : Cℓ_{ss}(H(\overline{F})) ⟶ Cℓ_{ss}(G(\overline{F})).

We call semisimple γ_{H} ∈ H(\overline{F}) G-regular if the image of its conjugacy class under A_{H/G} consists of regular semisimple elements, and strongly G-regular if the image consists of strongly regular elements, that is, elements whose centralizer is a torus. A strongly G-regular element is strongly regular.

The group Γ = Gal(\overline{F} /F) acts on conjugacy classes. Lemma 1.3.A. A_{H/G} is a Γ-map.

Proof. A_{H/G} = A_{G^{∗}/G} · A_{H/G^{∗}} and A_{G^{∗}/G} is the map induced by ψ. Since ψ is an inner twist A_{G^{∗}/G} is a Γ-bijection. Thus we may assume that G is quasi-split over F. Then if T_{H} is defined over F Steinberg’s Theorem [K1] allows us to choose (B_{G}, T_{G}) with both T_{G} and T_{H} ⟶ T_{G} defined over F. The lemma follows.

Suppose that T_{H} is defined over F. If (B_{G^{∗}}, T_{G^{∗}}) is chosen so that T_{G^{∗}}, and T_{H} ⟶ T_{G^{∗}} are defined over F, as in the proof of the lemma, then we call T_{H} ⟶ T_{G^{∗}} an admissible embedding of T_{H} in G^{∗}. It is uniquely determined up to 𝔄(T_{G^{∗}})-conjugacy, that is, up to composition with Int g^{−1}, where g lies in

𝔄(T_{G^{∗}}) = \{g ∈ G^{∗}(\overline{F}) : gσ(g^{−1}) ∈ T_{G^{∗}}(\overline{F}), σ ∈ Γ\}.

Note that we may take g in G_{sc}^{∗}, the simply-connected cover of the derived group of G^{∗}.

For strongly regular elements in G(F) or H(F) stable conjugacy is the same as conjugacy under G(\overline{F}) or H(\overline{F}), and we may apply Lemma 1.3.A directly to define a correspondence of points. Thus, if γ_{H} ∈ H(F) is strongly G-regular then its stable conjugacy class consists of the F -rational points in its conjugacy class in H(\overline{F}). The image of this class in H(\overline{F}) under A_{H/G} is a conjugacy class of strongly regular elements in G(\overline{F}). The class is defined over F and so its F -rational points are either nonexistent or form a single stable conjugacy class of strongly regular elements in G(F). We call strongly G-regular γ_{H} ∈ H(F) an image of γ_{G} ∈ G(F) if γ_{G} lies in the image under A_{H/G} of the conjugacy class of γ_{H} in H(\overline{F}). The twisted analogue of image is norm [K-S] which explains why we have labelled γ_{H}, and not γ_{G}, as the image.

For arbitrary G-regular semisimple γ_{H} in H(F) we set T_{H} = Cent(γ_{H}, H)^{0} and choose an admissible embedding T_{H} ⟶ T_{G^{∗}} of T_{H} in G^{∗}. If γ_{G} is regular semisimple in G(F) and T_{G} = Cent(γ, G)^{0} then we say that γ_{H} is an image of γ_{G} if there exists x ∈ G^{∗} such that Int x ∘ ψ maps γ_{G} to the image γ_{G^{∗}} of γ_{H} under T_{H} ⟶ T_{G^{∗}} and T_{G} to T_{G^{∗}}. The correspondence (γ_{H}, γ_{G}) is independent of the choice of admissible embedding T_{H} ⟶ T_{G^{∗}} and extends that for the strongly regular elements. Further we have:

(i) a G-regular semisimple element of H(F) is either the image of no element or the image of exactly one stable conjugacy class of regular semisimple elements in G(F), and

(ii) the images of a regular semisimple element in G(F) form a union, possibly empty, of stable conjugacy classes of G-regular semisimple elements in H(F). If F is local then the union is finite.

(1.4) Transfer Factors We assume here that F is local, leaving remarks on the global case for Sect. 6.

To normalize measures on conjugacy classes we fix invariant forms of highest degree: ω_{G} on G, ω_{H} on H and ω_{T} on some maximal torus T in G. Then if T is any maximal torus over F in either G or H we transport ω_{T} to an invariant form ω_{T} of highest degree on T, using an inner automorphism of G if T lies in G and an isomorphism provided by the choice of pairs otherwise. In either case ω_{T} depends only on ω_{T}.

To an invariant form ω of highest degree on G, H or T we attach a Haar measure as follows. There is λ_{σ} ∈ \overline{F}^{×} such that σ(ω) = λ_{σ}ω, σ ∈ Γ. Hilbert’s Theorem 90 allows us to write λ_{σ} as μσ(μ^{−1}), where μ ∈ \overline{F}^{×}. Then μω is defined over F and the Haar measure |μω| is well-defined. To obtain a measure independent of the choice of μ we replace this by |μ|^{−1}|μω| which will be denoted simply as |ω|.

It is simplest, and sufficient, to specify transfer factors on strongly regular elements. If f ∈ C_{c}^{∞}(G(F)) then for the integral of f along the conjugacy class of any regular semisimple element γ in G(F) we take

Φ(γ, f) = ∫_{T (F)\G(F)} f(g^{−1}γg)\frac{|ω_{G}|}{|ω_{T} |},

where T = Cent(γ, G)^{0}. For the integral of f^{H} ∈ C_{c}^{∞}(H(F)) along the stable conjugacy class of strongly G-regular γ_{H} in H(F) we take

Φ^{st}(γ_{H}, f^{H}) = ΣΦ(γ′_{H}, f^{H}),

where the summation is over representatives γ′_{H} for the conjugacy classes in the stable conjugacy class of γ_{H}. If γ_{H} is not strongly regular this must be modified [see (4.3)].

Suppose Δ is a function on

\{ \begin{matrix}strongly G-regular \\ elements in H(F) \end{matrix} \} × \{ \begin{matrix}strongly regular \\ elements in G(F) \end{matrix} \}

such that

(i) Δ(γ_{H}, γ) depends only on the conjugacy class of γ in G(F) and the stable conjugacy class of γ_{H} in H(F), and

(ii) Δ(γ_{H}, γ) = 0 unless γ_{H} is an image of γ. Then we say that f ∈ C_{c}^{∞}(G(F)) and f^{H} ∈ C_{c}^{∞}(H(F)) have Δ-matching orbital integrals if

Φ^{st}(γ_{H}, f^{H}) = ∑_{γ} Δ(γ_{H}, γ)Φ(γ, f)

for all strongly G-regular elements γ_{H} in H(F). The summation is over representatives γ for the conjugacy classes of strongly regular elements in G(F); only a finite number of terms in the sum are nonzero.

We call Δ a transfer factor if for each f ∈ C_{c}^{∞}(G(F)) there exists f^{H} ∈ C_{c}^{∞}(H(F)) with Δ-matching orbital integrals. It is best to demand that Δ(γ_{H}, γ) be nonzero if γ_{H} is an image of γ, and sometimes preferable to work with functions in the Schwartz space [Sh].

In Sect. 3 we will define a function Δ. If G is quasi-split over F then the procedure is exactly that for SL(2). In general, however, the term inv(γ_{H}, γ) appearing in (III_{1}) of (1.1) is no longer well defined as the torus T will be taken in G^{∗} rather than G. Since only quotients really matter we define instead a relative invariant following [K-S], and obtain a canonical relative factor Δ(γ_{H}, γ; \overline{γ}_{H}, \overline{γ}) in place of the quotient Δ_{0}(γ_{H}, γ)/Δ_{0}(\overline{γ}_{H}, \overline{γ}) in (1.1.1).

The next section describes, in a general setting, two constructions needed for Sect. 3.

2. Key Lemmas (2.1) General Remarks Suppose that k is a field of characteristic zero with algebraic closure \overline{k}, and that G is a connected reductive algebraic group defined and quasi-split over k. Let (B, T, {X_{α}}) be a k-splitting of G and Γ be a group acting on G by automorphisms which preserve (B, T, {X_{α}}). Then Γ acts on the Weyl group Ω = Ω(G, T) of T in G. For θ = ω ⋊ ϱ ∈ Ω ⋊ Γ we define n(θ) = n(ω) ⋊ ϱ ∈ G(\overline{k}) ⋊ Γ, where n(ω) ∈ Norm (T(\overline{k}), G(\overline{k})) is given by the following well-known construction [Spr].

Let α be a simple root of T in B. Then by definition ϱX_{α} = X_{ϱα}, ϱ ∈ Γ. Let H_{α} be the coroot for α regarded as an element of Lie T. Fix the root vector X_{−α} for −α by requiring that [X_{α}, X_{−α}] = H_{α}. Then ϱX_{−α} = X_{−ϱα}, ϱ ∈ Γ. Set

n(α) = exp X_{α} exp(−X_{−α}) exp X_{α},

so that n(α) is the image of [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] under the homomorphism SL(2) → G attached to the Lie triple {X_{α}, H_{α}, X_{−α}}. Let ω ∈ Ω, ω =| 1, be written in reduced form ω(α_{1}) … ω(α_{r}). Then we may set n(ω) = n(α_{1}) … n(α_{r}) since this latter product is independent of the choice of reduced expression for ω [Spr]. If ω = 1 set n(ω) = 1. Then Intn(ω) acts on T as ω and

n(ϱ(ω)) = ϱ(n(ω)), ϱ ∈ Γ .

Thus if θ = ω ⋊ ϱ then n(θ) = n(ω) ⋊ ϱ acts on T as θ and

n(θ_{1})nn(θ_{2}) = t(θ_{1}, θ_{2})n(θ_{1}θ_{2}), θ_{i} ∈ Ω ⋊ Γ ,

where t(θ_{1}, θ_{2}) is a 2-cocycle of Ω ⋊ Γ in T(\overline{k}).

Lemma 2.1.A.

t(θ_{1}, θ_{2}) = ∏_{α>0 ; {θ_{1}^{−1}α<0 \atop θ_{2}^{−1}θ_{1}^{−1}α>0}} (−1)^{α^{∨}}.

Here α > 0 means α is a root of T in B, and α^{∨} is the coroot for α as element of X_{∗}(T). Then (−1)^{α^{∨}} ∈ k^{×} ⊗ X_{∗}(T) ⊆ T(\overline{k}). Proof. We will verify in the next lemma that the right side is, like the left, a 2-cocycle. It is therefore enough to show that the equality is valid when (i) θ_{1} or θ_{2} lies in Γ and when (ii) θ_{1} and θ_{2} lie in Ω and further θ_{1} = ω(α) where α is simple. Case (i) is clear. For case (ii), let ω = ω(α_{1}) … ω(α_{r}) be a reduced expression for θ_{2}. Then R_{ω} = {β > 0, ωβ < 0} is the set of positive roots separating the positive Weyl chamber W_{+} from ω^{−1}W_{+} and contains r elements. There are two possibilities. Either ω^{−1}α > 0, ω(α)ω(α_{1}) … ω(α_{r}) is a reduced expression for ω(α)ω and R_{ω(α)ω} = R_{ω} ∪ {ω^{−1}α}, or ω^{−1}α < 0 and there is a reduced expression for ω with ω(α_{1}) = ω(α). In the first case both sides of the putative equality are 1; in the second both are (−1)^{α^{∨}}. The lemma is thus proved.

Let X be a free finitely generated Z-module and Σ be a group which acts on X and contains an element ϵ sending λ to −λ, λ ∈ X. Then with trivial action of k^{×}, Σ acts on k^{×} ⊗ X. Let R be a finite Σ-stable subset of X and p be a gauge on R, by which we mean that p : R → {±1} and p{−λ) = −p(λ), λ ∈ R. We abbreviate a product over those λ ∈ R for which

p(λ) = 1, p(σ^{−1}λ) = −1, p(τ^{−1}σ^{−1}λ) = 1

by ∏_{1,σ,τ}^{p}.

Lemma 2.1.B. t_{p}(σ, τ) = ∏_{1,σ,τ}^{p}(−1)^{λ}, σ, τ ∈ Σ,

is a 2-cocycle of Σ with values in k^{×} ⊗ X. Proof. The coboundary of this 2-cochain is the product over all λ with p(λ) = 1 of (−1)^{Nλ}, where N is the number of ordered triples in

\begin{matrix}\{(σ^{−1}λ, τ^{−1}σ^{−1}λ, ϱ^{−1}τ^{−1}σ^{−1}λ), & (λ, τ^{−1}σ^{−1}λ, ϱ^{−1}τ^{−1}σ^{−1}λ), \\ (λ, σ^{−1}λ, ϱ^{−1}τ^{−1}σ^{−1}λ), & (λ, σ^{−1}λ, τ^{−1}σ^{−1}λ)\} \end{matrix}

on which p takes alternating signs. Thus if p(ϱ^{−1}τ^{−1}σ^{−1}λ) = −1 there can be a contribution only from the first and last triples and then only if p(σ^{−1}λ) = −1, p(τ^{−1}σ^{−1}λ) = 1 when both triples contribute, so that N = 2. If p(ϱ^{−1}τ^{−1}σ^{−1}λ) = 1 and p(σ^{−1}λ) = 1 then either the first and second triples contribute or none does, so that N is 2 or 0, but if p(ϱ^{−1}τ^{−1}σ^{−1}λ) = 1 and p(σ^{−1}λ) = −1 then the third triple contributes and exactly one of the second and fourth, so that N is 2. The lemma is proved. Lemma 2.1.C. If q is also a gauge on R, then t_{q} is cohomologous to t_{p}. Proof. We may assume that Σ is transitive on R and that X is free on {λ : p(λ) = 1}. Then by Shapiro’s Lemma we reduce to the case R = {±λ} where the assertion is clear.

In the application of (2.3) and (2.6) we will have ϵ^{2} = 1 and Σ will be the product of {1, ϵ} and a subgroup Γ. Then if Σ acts transitively on R either R consists of exactly two Γ-orbits O and −O or Γ also acts transitively on R, in which case R is a Γ-orbit O, where O = −O. In the former case O is called asymmetric and in the latter symmetric. Lemma 2.1.D. Suppose that R = ∪ ± O, where O is asymmetric. Then the restriction of t_{p} is trivial. Proof. We may assume X is free on O. Then again by Shapiro’s Lemma we reduce to the case R = {±λ} where the assertion is clear. (2.2) a-Data First we consider splitting t_{p} in \overline{k}^{×} ⊗X, given an extension of the action of Σ on k to \overline{k} with ϵ acting trivially. A collection {a_{λ} : λ ∈ R} is a set of a-data for the action of Γ in R if:

(i) a_{λ} ∈ \overline{k}^{×} and a_{σλ} = σ(a_{λ}), σ ∈ Γ, λ ∈ R, and (ii) a_{−λ} = −a_{λ}, λ ∈ R.

Of course, a-data need not exist. If they do then we can split t_{p} on Γ in the following way.

For a product over those λ ∈ R such that p(σ^{−1}λ) = 1 and p(τ^{−1}λ) = −1 we write ∏_{σ,τ}^{p}. Set

u_{p}(σ) = ∏_{1,σ}^{p} a_{λ}^{λ}, σ ∈ Γ.

Lemma 2.2.A. t_{p}(σ, τ) = ∂u_{p}(σ, τ), σ, τ ∈ Γ.

Proof. ∂u_{p}(σ, τ) = u_{p}(σ)σ(u_{p}(τ))u_{p}(στ)^{−1} = ∏_{1,σ}^{p} a_{λ}^{λ} ∏_{σ,τ}^{p} a_{λ}^{λ} ∏_{1,στ}^{p} a_{λ}^{−λ}.

Fix λ ∈ R with p(λ) = 1. Then the contributions of the terms with exponents ±λ are as follows. (i) p(σ^{−1}λ) = 1, p(τ^{−1}σ^{−1}λ) = 1 : 1 (ii) p(σ^{−1}λ) = 1, p(τ^{−1}σ^{−1}λ) = −1 : a_{λ}^{λ} · a_{λ}^{−λ} = 1 (iii) p(σ^{−1}λ) = −1, p(τ^{−1}σ^{−1}λ) = 1 : a_{λ}^{λ} · a_{−λ}^{−λ} = a_{λ}^{λ}(−a_{λ})^{−λ} = (−1)^{λ} (iv) p(σ^{−1}λ) = −1, p(τ^{−1}σ^{−1}λ) = −1 : 1.

Thus ∂u_{p}(σ, τ) = ∏_{1,σ,τ}^{p}(−1)^{λ} = t_{p}(σ, τ).

As a corollary of this lemma, or by simply modifying the proof, we have: Lemma 2.2.B. Suppose that \{b_{λ} : λ ∈ R\} satisfies b_{λ} ∈ \overline{k}^{×},

b_{σλ} = σ(b_{λ}), σ ∈ Γ, and b_{−λ} = b_{λ}.

Then

v_{p}(σ) = ∏_{1,σ}^{p} b_{λ}^{λ}, σ ∈ Γ,

is a 1-cocycle of Γ in \overline{k}^{×} ⊗ X. By the usual argument with Shapiro’s Lemma we have: Lemma 2.2.C. (a) The class of v_{p} is independent of p, and

(b) if R = ±O where O is an asymmetric Γ-orbit then v_{p} is cohomologically trivial. Proof. (a) We may assume that R = {±λ}. Then the only gauges are ±p, where p(λ) = 1. Since b_{−λ}^{−λ} = b_{λ}^{λ}b_{λ}^{−2λ} we have that v_{−p} = v_{p}w, where w(σ) = b_{λ}^{−2λ} if σλ = −λ and w(σ) = 1 otherwise. But w(σ) = xσ(x)^{−1} where x = b_{λ}^{−λ}.

(b) Again assume R = {±λ}. Then v_{p} = 1. (2.3) An Application Our first task is to define an invariant for a pair (T, {a_{α}}), where T is a maximal torus defined over k in a connected reductive group G defined and quasi-split over k, and {a_{α}} is a set of a-data for T, i.e. for the action of Γ = Gal(\overline{k}/k) on the set R(G, T) of roots of T in G. Note that in this setting a-data are readily verified to exist.

We fix a k-splitting (B, T, {X_{α}}) of G. There will be no harm in assuming G semisimple and simplyconnected and we do so to conserve notation. Denote by σ the action of σ ∈ Γ on T, and set Γ = {σ : σ ∈ Γ}. Again Ω will be the Weyl group Ω(G, T).

Choose a Borel subgroup B of G containing T and h ∈ G such that (B, T)^{h} = (B, T). Denote by σ_{T} both the action of σ ∈ Γ on T and its transport to T by Int h^{−1}. Set

Γ_{T} = {σ_{T} : σ ∈ Γ} ⊂ Ω ⋊ Γ .

If ω_{T} (σ) is the element of Ω defined by Int(h^{−1}σ(h)) then σ_{T} = ω_{T} (σ) ⋊ σ.

The a-data for T serve also for the action on Γ on R^{∨}(G, T) and, after transport, for the action of Γ_{T} on R = R^{∨}(G, T); {a_{α}^{−1}} is also a set of a-data. By Lemmas 2.1.A and 2.2.A

n(σ_{T}) = n(ω_{T} (σ)) ⋊ σ, σ ∈ Γ,

satisfies

n(σ_{T})σ(n(τ_{T}))n((στ)_{T})^{−1} = ∂x(σ_{T} , τ_{T})^{−1}

where

x(σ_{T}) = ∏_{1,σ}^{p} a_{α}^{α^{∨}}

and p(α) = 1 if and only if α is a root of T in B. Then

σ_{T} → x(σ_{T})n(ω_{T} (σ)) ⋊ σ

is a homomorphism of Γ_{T} in Norm(T(k), G(k)) ⋊ Γ. Otherwise stated,

σ_{T} → m(σ_{T}) = x(σ_{T})n(ω_{T} (σ))

is a 1-cocycle of Γ_{T} in Norm(T(\overline{k}), G(\overline{k})).

Now hm(σ_{T})σ(h^{−1}) = h(m(σ_{T})σ(h^{−1})h)h^{−1} lies in T (\overline{k}) and is evidently a 1-cocycle of Γ in T (\overline{k}). Since h is unique up to left multiplication by an element of T (\overline{k}) this cocycle yields, for given k-splitting, a-data and B ⊃ T, a well-defined element λ(T) = λ_{\{a_{α}\}}(T) of H^{1}(Γ, T (\overline{k})) = H^{1}(T). We note some of its properties. (2.3.1) The splitting (B, T, \{X_{α}\}) may be replaced only by (B^{g}, T^{g}, \{X_{α}^{g}\}), where g ∈ G(\overline{k}) is such that gσ(g)^{−1} lies in the center Z of G, σ ∈ Γ. Then m(σ_{T}) is replaced by g^{−1}m(σ_{T})g and h by hg. Thus the cocycle defining λ(T) is replaced by

hg(g^{−1}m_{T} (σ)g)σ(hg)^{−1} = gσ(g)^{−1}hm_{T} (σ)σ(h)^{−1}.

Then λ(T) is multiplied by the class g in H^{1}(T) represented by σ → gσ(g)^{−1}. (2.3.2) Suppose that the a-data {a_{α}} are replaced by {a′_{α}}. Then a′_{α} = b_{α}a_{α}, where b_{σα} = σ(b_{α}) for all σ in the group Σ generated by Γ and ϵ, and hm(σ_{T})σ(h^{−1}) is replaced by

hv_{p}(σ_{T})h^{−1}hm(σ_{T})σ(h^{−1}),

where v_{p} is the 1-cocycle σ_{T} → ∏_{1,σ}^{p} b_{α}^{α^{∨}} of Γ_{T} in T(\overline{k}) (cf. Lemma 2.2.B). Note that hv_{p}(σ_{T})h^{−1} is the cocycle

b_{q} : σ ∏_{1,σ}^{q} b_{α}^{α^{∨}} of Γ in T (\overline{k}), q denoting the transport of the gauge p to R^{∨}(G, T) by Int h, and that the class b of b_{q} in H^{1}(T) is independent of the gauge q (Lemma 2.2.C). (2.3.3) Next we show that λ(T) is independent of B. Suppose that B is replaced by vBv^{−1}, v ∈ Norm(T, G). Set u = h^{−1}vh ∈ Norm(T, G), and suppose that μ is the element of Ω defined by u. We now have to transport σ on T to T by Int(h^{−1}v^{−1}). Suppose we obtain σ′_{T} = ω′_{T} (σ) ⋊ σ. Then since h^{−1}v^{−1}σ(vh) = h^{−1}v^{−1}h · h^{−1}σ(h) · σ(h^{−1}vh) = u^{−1}h^{−1}σ(h)σ(u) we have ω′_{T} (σ) = μ^{−1}ω_{T} (σ)σ(μ). Let Γ′_{T} = {σ′_{T} : α ∈ Γ}. The a-data {a′_{α}} obtained by transport for Γ′_{T} satisfy a′_{α} = a_{μα}, α ∈ R(G, T). We then define m(σ′_{T}) in the same way as m(σ_{T}) and consider

vhm(σ′_{T})σ(h^{−1}v^{−1}) = hum(σ′_{T})σ(u^{−1})σ(h^{−1}).

But now um(σ′_{T})σ(u)^{−1} = b(σ_{T})m(σ_{T}), where σ_{T} → b(σ_{T}) is a 1-cocycle of Γ_{T} with values in T. It remains to show: Lemma 2.3.A. The cocycle b is trivial. Proof. We have m(σ_{T}) = x(σ_{T})n(ω_{T} (σ)) and m(σ′_{T}) = x(σ′_{T})n(ω′_{T} (σ)); x(σ_{T}) is defined in terms of {a_{α}} and x(σ′_{T}) in terms of {a′_{α}}. Then

m(σ′_{T}) = x(σ′_{T})n(μ^{−1}ω_{T} (σ)σ(μ))

which equals

x(σ′_{T})t_{p}(μ^{−1}ω_{T} (σ), σ(μ))t_{p}(μ^{−1}, ω_{T} (σ))n(μ^{−1})n(ω_{T} (σ))n(σ(μ))

or

x(σ′_{T})μ^{−1}(x(σ_{T})^{−1})t_{p}(μ^{−1}ω_{T} (σ), σ(μ))t_{p}(μ^{−1}, ω_{T} (σ))n(μ^{−1})m(σ_{T})n(σ(μ)).

Now n(μ) = λu, where λ ∈ T, and then

n(σ(μ)) = σ(n(μ)) = σ(λ)σ(u).

Also we have n(μ^{−1}) = n(μ)^{−1}t_{p}(μ, μ^{−1}). Thus um(σ′_{T})σ(u)^{−1} is equal to

μ(x(σ′_{T}))x(σ_{T})^{−1}t_{p}(μ, μ^{−1})μ[t_{p}(μ^{−1}ω_{T} (σ), σ(μ))t_{0}(μ^{−1}, ω_{T} (σ))]λ^{−1}σ_{T} (λ)m(σ_{T}),

and b(σ_{T}) is equivalent to

μ(x(σ′_{T}))x(σ_{T})^{−1}t_{p}(μ, μ^{−1})μ[t_{p}(μ^{−1}ω_{T} (σ), σ(μ))t_{p}(μ^{−1}, ω_{T} (σ))].

We now omit the subscript T in notation. That b(σ) is trivial follows from: Lemma 2.3.B. (a) Let δ = ∏_{1,μ}^{p} a_{α}^{−α^{∨}}. Then μ(x(σ′))x(σ)^{−1} is equal to

δσ(δ^{−1}) ∏_{{p(α)=p(μ^{−1}α)=p(μ^{−1}σ^{−1}α)=1 \atop p(σ^{−1}α)=−1}}(−1)^{α^{∨}} ∏_{{p(α)=p(σ^{−1}α)=p(μ^{−1}σ^{−1}α)=1 \atop p(μ^{−1}α)=−1}}(−1)^{α^{∨}}

(b) t_{p}(μ, μ^{−1})μ[t_{p}(μ^{−1}ω(σ), σ(μ))t_{p}(μ^{−1}, ω(σ))] is equal to

∏_{{p(α)=p(μ^{−1}α)=p(μ^{−1}σ^{−1}α)=1 \atop p(σ^{−1}α)=−1}}(−1)^{α^{∨}} ∏_{{p(α)=p(σ^{−1}α)=p(μ^{−1}σ^{−1}α)=1 \atop p(μ^{−1}α)=−1}}(−1)^{α^{∨}}

Proof.

μ(x(σ′))x(σ)^{−1} = ∏_{{p(α)=1 \atop p(μ^{−1}σ^{−1}μα)=−1}}(a′_{α})^{μα^{∨}} ∏_{{p(α)=1 \atop p(σ^{−1}α)=−1}} a_{α}^{−α^{∨}}
·
= ∏_{{p(μ^{−1}α)=1 \atop p(μ^{−1}σ^{−1}α)=−1}} a_{α}^{α^{∨}} ∏_{{p(α)=1 \atop p(σ^{−1}α)=−1}} a_{α}^{−α^{∨}}.

The contributions to this product are as follows.

(i) p(α) = 1, p(μ^{−1}α) = p(σ^{−1}α) = −1 : a_{α}^{−α^{∨}}.

(ii) p(α) = p(μ^{−1}α) = p(μ^{−1}σ^{−1}α) = 1, p(σ^{−1}α) = −1 : a_{α}^{−α^{∨}}. (iii) p(μ^{−1}α) = 1, p(α) = p(σ^{−1}α) = p(μ^{−1}σ^{−1}α) = −1 : a_{α}^{α^{∨}}. (iv) p(σ^{−1}α) = p(μ^{−1}α) = 1, p(μ^{−1}σ^{−1}α) = −1 : a_{α}^{α^{∨}}. On the other hand,

δσ(δ)^{−1} = ∏_{{p(α)=1 \atop p(μ^{−1}α)=−1}} a_{α}^{−α^{∨}} ∏_{{p(σ^{−1}α)=1 \atop p(μ^{−1}σ^{−1}α)=−1}} a_{α}^{α^{∨}}.

The contributions to this product are the following.

(i) p(α) = 1, p(μ^{−1}α) = p(σ^{−1}α) = −1 : a_{α}^{−α^{∨}}.

(ii) p(α) = p(μ^{−1}α) = p(μ^{−1}σ^{−1}α) = −1, p(σ^{−1}α) = 1 : a_{α}^{α^{∨}} = (−1)^{α^{∨}}a_{−α}^{α^{∨}}. (iii) p(μ^{−1}α) = −1, p(α) = p(σ^{−1}α) = p(μ^{−1}σ^{−1}α) = 1 : a_{α}^{−α^{∨}} = (−1)^{α^{∨}}a_{−α}^{−α^{∨}}. (iv) p(σ^{−1}α) = p(μ^{−1}α) = 1, p(μ^{−1}σ^{−1}α) = −1 : a_{α}^{α^{∨}}. The assertion (a) follows.

For (b) we observe first that p(σ^{−1}α) = p(α) so that p(ω(σ)^{−1}α) = p(σ^{−1}α) for all roots α. Thus t_{p}(μ, μ^{−1})μ(t_{p}(μ^{−1}, ω(σ))) is equal to

∏_{{p(α)=1 \atop p(μ^{−1}α)=−1}}(−1)^{α^{∨}} ∏_{{p(α)=p(σ^{−1}μα)=1 \atop p(μα)=−1}}(−1)^{μα^{∨}} = ∏_{{p(α)=1 \atop p(μ^{−1}α)=−1}}(−1)^{α^{∨}} ∏_{{p(α)=1 \atop p(μ^{−1}α)=p(σ^{−1}α)=−1}}(−1)^{α^{∨}}.

Performing the obvious cancellations, we obtain

∏_{{p(α)=p(σ^{−1}α)=1 \atop p(μ^{−1}α)=−1}}(−1)^{α^{∨}}.

Also μ(t_{p}(μ^{−1}ω(σ), σ(μ)) is equal to

∏_{{p(α)=p(μ^{−1}σ^{−1}μα)=1 \atop p(σ^{−1}μα)=−1}}(−1)^{μα^{∨}} = ∏_{{p(μ^{−1}α)=p(μ^{−1}σ^{−1}α)=1 \atop p(σ^{−1}α)=−1}}(−1)^{α^{∨}}.

Then t_{p}(μ, μ^{−1})μ(t_{p}(μ^{−1}ω(σ), σ(μ))t_{p}(μ^{−1}ω(σ)) equals

∏_{{p(α)=p(σ^{−1}α)=1 \atop p(μ^{−1}α)=−1}}(−1)^{α^{∨}} ∏_{{p(μ^{−1}α)=p(μ^{−1}σ^{−1}α)=1 \atop p(σ^{−1}α)=−1}}(−1)^{α^{∨}}.

The nontrivial contributions to this product occur only for either

(i) p(α) = p(μ^{−1}α) = p(μ^{−1}σ^{−1}α) = 1, p(σ^{−1}α) = −1, or (ii) p(α) = p(σ^{−1}α) = p(μ^{−1}σ^{−1}α) = 1, p(μ^{−1}α) = −1. Thus (b) follows, and the lemma is proved. (2.3.4) Suppose now that Int g^{−1} maps T to T′ over F and carries a-data {a_{α}} for T to the a-data {a′_{α}} for T′. We construct λ(T) = λ_{{a_{α}}}(T) and λ(T′) = λ_{{a′_{α}}}(T′). Note that if B ⊃ T is used to define the cocycle m(σ_{T}) then B′ = gBg^{−1} yields m(σ_{T′}) = m(σ_{T}). From

hm(σ_{T})σ(h)^{−1} = g(g^{−1}hm(σ_{T′})σ(h)^{−1}σ(g))g^{−1} · gσ(g)^{−1}

we conclude that if g ∈ H^{1}(T) is the class of the cocycle σ → gσ(g)^{−1} then λ(T) is g times the image of λ(T′) under the homomorphism H^{1}(T′) → H^{1}(T) given by Int g. (2.3.5) Finally, suppose that k is a number field, v is a place of k and k_{v} is the completion of k at v. Then adata for T as torus over k, or global a-data serve as well as a-data for T over k_{v}, or local a-data. We therefore obtain both λ(T) ∈ H^{1}(Γ, T) and λ_{v}(T) ∈ H^{1}(Γ_{v}, T) attached to given global a-data and k-splitting of G. For σ ∈ Γ_{v} ⊂ Γ, m(σ_{T}) = x(σ_{T})n(ω_{T} (σ)) is the same whether given in terms of k or of k_{v}. Thus λ_{v}(T) is the image of λ(T) under H^{1}(Γ, T) → H^{1}(Γ_{v}, T).

(2.4) An Explicit Splitting We return to the setting of (2.1). Recall that if p, q are gauges on R then t_{p} and t_{q} are cohomologous 2-cocycles of Σ in k^{×} ⊗ X (Lemma 2.1.C). Here we shall construct an explicit splitting of t_{p}/t_{q}.

Let s_{p/q}(σ) = ∏_{(p)}(−1)^{λ} ∏_{(−q)}(−1)^{λ}, σ ∈ Γ, where ∏_{(p)} is the product over λ such that

p(λ) = 1, p(σ^{−1}λ) = −1, q(λ) = q(σ^{−1}λ) = 1

and ∏_{(−q)} the product over λ such that

p(λ) = p(σ^{−1}λ) = 1, q(λ) = −1, q(σ^{−1}λ) = 1.

Observe that s_{p/p} = s_{−p/p} = s_{p/−p} = 1. Lemma 2.4.A. The coboundary of s_{p/q} is t_{p}/t_{q}. Proof. For σ, τ ∈ Γ, λ ∈ R, set

S(λ) = (p(λ), p(σ^{−1}λ), p(τ^{−1}σ^{−1}λ), q(λ), q(σ^{−1}λ), q(τ^{−1}σ^{−1}λ)).

Then

s_{p/q}(σ)σ(s_{p/q}(τ))s_{p/q}(στ)^{−1} = ∏(−1)^{λ},

where the product is taken over the six sets given respectively by S(λ) equal to:

\begin{matrix}(1, −1, ±1, 1, 1, ±1); (±1, 1, −1, ±1, 1, 1); & (1, ±1, −1, 1, ±1, 1); \\ (1, 1, ±1, −1, 1, ±1); & (±1, 1, 1, ±1, −1, 1); (1, ±1, 1, −1, ±1, 1) \end{matrix}

. The contribution of ±λ to this product is as follows.

(a) If all entries in S(λ) are positive: 1.

(b) If exactly five entries in S(λ) are positive: (−1)^{λ} if S(λ) = (1, 1, 1, 1, −1, 1) or (1, −1, 1, 1, 1, 1), and 1 otherwise.

(c) If exactly four entries in S(λ) are positive: (−1)^{λ} if S(λ) = (1, −1, 1, −1, 1, 1), (1, −1, 1, 1, 1, −1), (1, 1, −1, 1, −1, 1), (−1, 1, 1, 1, −1, 1), (−1, 1, −1, 1, 1, 1), or (1, 1, 1, −1, 1, −1), and 1 otherwise.

(d) If exactly three entries in S(λ) are positive: (−1)^{λ} if ±S(λ) = (1, 1, −1, −1, 1, −1), (1, −1, −1, 1, −1, 1), (1, −1, 1, −1, −1, 1) or (−1, 1, −1, −1, 1, 1), and 1 otherwise. On the other hand, t_{p}(σ, τ)/t_{q}(σ, τ) = ∏(−1)^{λ} where the product is now over each of

{λ : S(λ) = (1, −1, 1, ±1, ±1, ±1)}

and

{λ : S(λ) = (±1, ±1, ±1, 1, −1, 1)}.

Consider the contribution to this product from λ. In cases (a) and (b) the contribution is the same as for ∂s_{p/q}. In cases (c) and (d) if ±λ contribute (−1)^{λ} to s_{p/q} we find that exactly one of λ and −λ contributes (−1)^{λ} to t_{p}/t_{q}, and conversely. Thus t_{p}/t_{q} coincides with ∂s_{p/q}, and the lemma is proved.

Corollary 2.4.B.

(i) s_{q/p} is cohomologous to s_{p/q}.

(ii) s_{p/q}s_{q/r} is cohomologous to s_{p/r}. Proof. In view of Lemma 2.4.A both s_{q/p}s_{p/q}^{−1} and s_{p/q}s_{q/r}s_{p/r}^{−1} are cocycles. We can then reduce in the usual way with Shapiro’s Lemma to the case R = {±λ}. Then q, r = ±p and the lemma is clear.

(2.5) χ-Data We consider the case k = C and Γ = Gal(L/F), where F is a local or a global field and L is a finite Galois extension of F. The group Σ = ⟨Γ, ϵ⟩ acts on C^{×} ⊗ X with trivial action on C^{×}. The cocycle t_{p} is nontrivial in general. Let W be the Weil group of L/F. Then W acts on C^{×} ⊗ X through W → Γ. The inflation of t_{p} to W does split [L1]. Here we shall construct a splitting of it using χ−data which are prescribed as follows.

Set Γ_{+λ} = {σ ∈ Γ : σλ = λ} and Γ_{±λ} = {σ ∈ Γ : σλ = ±λ}, λ ∈ R. Once λ has been fixed we delete it in notation. Then F_{+} ⊂ L will be the fixed field of Γ_{+}, and F_{±} the fixed field of Γ_{±}. Note that [F_{+} : F_{±}] is 2 or 1 according as the Γ-orbit of λ is symmetric or asymmetric. Set W_{±} = W_{L/F_{±}} and W_{+} = W_{L/F_{+}}. Then by χ-data for the action of Γ on R we mean a collection {χ_{λ} : λ ∈ R} such that the following hold.

(i) χ_{λ} is a character on C_{+λ}, where C_{+λ} is the multiplicative group of F_{+λ} or the idèle-class group of F_{+λ}, according as F is local or global. In either case we may regard χ_{λ} as a character on W_{+} = W_{+λ}.

(ii) χ_{−λ} = χ_{λ}^{−1} and

χ_{σλ} = χ_{λ} · σ^{−1}, σ ∈ Γ, λ ∈ R.

(iii) If [F_{+} : F_{±}] = 2 then χ_{λ}, as character on C_{+}, extends the quadratic character on C_{±} attached to the extension F_{+}/F_{±}. It is readily verified that χ-data exist.

To prescribe the splitting we shall at first assume that Σ acts transitively on R. In general, it is a product of the splittings for the orbits. Fix λ ∈ R and choose representatives σ_{1}, …, σ_{n} for Γ_{±}\Γ. Then ±σ_{1}^{−1}λ, …, ±σ_{n}^{−1}λ are the elements of R listed without redundancy. Define a gauge p on R by

p(λ′) = 1 if and only if λ′ = σ_{i}^{−1}λ, some 1 ≦ i ≦ n.

Choose w_{1}, …, w_{n} ∈ W such that w_{i} maps to σ_{i} under W → Γ. Then w_{1}, …, w_{n} are representatives for W_{±}\W. If w ∈ W then define u_{i}(w) ∈ W_{±} by

w_{i}w = u_{i}(w)w_{i}, i = 1, …, n.

Choose representatives v_{0} ∈ W_{+} and v_{1} for W_{+}\W_{±} in case [F_{+} : F_{±}] = 2 and an element v_{0} of W_{+} if F_{+} = F_{±}. Note that χ_{λ}(v_{1}vv_{1}^{−1}) = χ_{λ}(v)^{−1}. For u ∈ W_{±} define v_{0}(u) ∈ W_{+} by

v_{0} · u = v_{0}(u) · v_{i′},

where i′ = 0 or 1, as appropriate. For u ∈ W_{±} set

s(u) = χ_{λ}(v_{0}(u))

and for w ∈ W set

r_{p}(w) = ∏_{i=1}^{n} χ_{λ}(v_{0}(u_{i}(w)))^{λ_{i}} = ∏_{i=1}^{n} s(u_{i}(w))^{λ_{i}},

where λ_{i} = σ_{i}^{−1}λ. Then r_{p} is a 1-chain of W with values in C^{×} ⊗ X.

If q is any gauge on R set

r_{q} = s_{q/p}r_{p}.

Lemma 2.5.A. The coboundary of r_{q} is t_{q}. Proof. In view of Lemma 2.4.A we have only to consider the case q = p. Suppose that v, w ∈ W. Then

w_{i}vw = u_{i}(v)w_{i′}w = u_{i}(v)u_{i′}(w)w_{i′′}

so that u_{i}(vw) = u_{i}(v)u_{i′}(w). If v → σ under W → Γ then

σλ_{i′} = σσ_{i′}^{−1}λ = ϵ_{i}σ_{i}^{−1}λ = ϵ_{i}λ_{i},

where ϵ_{i} = +1 or −1 according as u_{i}(v) ∈ W_{+} or not. Observe that ϵ_{i} = p(σ^{−1}λ_{i}). Now

\begin{aligned}r_{p}(v)v(r_{p}(w)) & = ∏_{i=1}^{n} s(u_{i}(v))^{λ_{i}}s(u_{i′}(w))^{σλ_{i}} \\ & = ∏_{i=1}^{n} s(u_{i}(v))^{λ_{i}}s(u_{i′}(w))^{ϵ_{i}λ_{i}} \end{aligned}

and

r_{p}(vw) = ∏_{i=1}^{n} s(u_{i}(v))^{λ_{i}}s(u_{i′}(w))^{λ_{i}}.

We claim that

s(u_{i′}(w))^{ϵ_{i}} = t · s(u_{i′}(w))

where t = 1 unless ϵ_{i} = −1 and u_{i′}(w) ∉ W_{+}, in which case it is −1. But ϵ_{i} = p(σ^{−1}λ_{i}) and if w → τ under W → Γ then u_{i′}(w) ∉ W_{+} if and only if p(τ^{−1}λ′_{i}) = ϵ_{i}p(τ^{−1}σ^{−1}λ_{i}) = −1. We conclude that

r_{p}(v)v(r_{p}(w))r_{p}(vw)^{−1} = ∏_{{p(σ^{−1}λ_{i})=−1 \atop p(τ^{−1}σ^{−1}λ_{i})=1}}(−1)^{λ_{i}} = ∏_{1,σ,τ}^{p}(−1)^{λ} = t_{p}(v, w).

To prove the claim there is more convenient notation. Let u, v ∈ W_{±} and suppose u → σ, v → τ under W_{±} → Γ_{±}. Set ϵ = p(σ^{−1}λ) and δ = p(τ^{−1}σ^{−1}λ). Then the claim asserts that

s(u)s(v)^{ϵ}s(uv)^{−1} = \{ \begin{matrix}−1 & if ϵ = −1, δ = 1 \\ 1 & otherwise, \end{matrix}

where s(u) = χ_{λ}(v_{0}(u)). If u ∈ W_{+}, and in particular if W_{+} = W_{±}, then ϵ = 1 and v_{0}(u) = v_{0}uv_{0}^{−1}, v_{0}(uv) = v_{0}uv_{0}^{−1}v_{0}(v) so that χ_{λ}(v_{0}(u))χ_{λ}(v_{0}(v)) = χ_{λ}(v_{0}(uv)). If u ∉ W_{+} then v_{0}uv = v_{0}(u)v_{1}v. If also v ∈ W_{+} then v_{0}(u)v_{1}v = v_{0}(u)v_{1}vv_{1}^{−1}v_{1} and v_{0}(uv) = v_{0}(u)v_{1}vv_{1}^{−1}. Thus

χ_{λ}(v_{0}(uv)) = χ_{λ}(v_{0}(u))χ_{λ}(v_{1}vv_{1}^{−1}) = χ_{λ}(v_{0}(u))χ_{λ}(v)^{−1}.

If both u, v lie outside W_{+} then

\begin{aligned}v_{0}uv = v_{0}(u)v_{1}v & = v_{0}(u)v_{1}v_{0}^{−1}v_{0}(v)v_{1} \\ & = v_{0}(u)v_{1}v_{0}^{−1}v_{0}(v)v_{1}^{−1}v_{1}^{2}v_{0}^{−1}v_{0}. \end{aligned}

Thus

v_{0}(uv) = v_{0}(u)v_{1}v_{0}^{−1}v_{0}(v)v_{1}^{−1}v_{1}^{2}v_{0}^{−1}

and

\begin{aligned}χ_{λ}(v_{0}(uv)) & = χ_{λ}(v_{0}(u))χ_{λ}(v_{1}v_{0}(v)v_{1}^{−1})χ_{λ}(v_{1}v_{0}^{−1}v_{1}^{−1}v_{0}^{−1})χ_{λ}(v_{1}^{2}) \\ & = −χ_{λ}(v_{0}(u))χ_{λ}(v_{0}(v))^{−1} \end{aligned}

since χ_{λ}(v_{1}^{2}) = −1 and χ_{λ}(v_{1}xv_{1}^{−1}) = χ_{λ}(x)^{−1}, x ∈ W_{+}. The claim is thus verified, and Lemma 2.5.A proved.

It remains to check that the various choices made have no effect on r_{q}, up to 1-coboundaries. First we observe that

r_{q}(x) = r_{p}(x) = ∏_{i=1}^{n} χ_{λ}(Nm_{F_{+}}^{L}σ_{i}x)^{σ_{i}^{−1}λ}, x ∈ L^{×},

is independent of the choices for v_{0}, v_{1}, w_{1}, …, w_{n} and λ, and that r_{q}(xw) = r_{q}(x)r_{q}(w), x ∈ L^{×}, w ∈ W.

Consider a change in v_{0}, v_{1}. Then r_{p} is replaced by a cochain r_{p′}. To show that the cocycle r_{p}r_{p′}^{−1} is trivial we may assume X is free on {λ ∈ R : p(λ) = 1} and reduce to the case R = {±λ} by Shapiro’s Lemma. It is sufficient now to observe that r_{p} and r_{p′} coincide on W_{+}, for then r_{p}r_{p′}^{−1} defines an element of H^{1}(Gal(F_{+}/F_{±}), C^{×} ⊗ X). This group is readily seen to be trivial.

Another choice w′_{i}, …, w′_{n} for w_{1}, …, w_{n} leads to a gauge p′ and it has to be shown that the cocycle s_{p′/p}r_{p}r_{p′}^{−1} is trivial. Again we reduce to the case R = {±λ}. Since p′ = ±p, s_{p′/p} is trivial and, as above, it needs only to be shown that r_{p} and r_{p′} coincide on W_{+}. We may assume p(λ) = p′(−λ) = 1, w_{1} = v_{0} and w′_{1} = v_{1}. Then λ_{1} = λ, λ′_{1} = −λ and u_{1}(w) = v_{0}wv_{0}^{−1}, u′_{1}(w) = v_{1}wv_{1}^{−1}, w ∈ W. Thus for w ∈ W_{+} we have χ_{λ′}(u′_{1}(w))^{λ′_{1}} = χ_{λ}(v_{1}wv_{1}^{−1})^{−λ} = χ_{λ}(w)^{λ} = χ_{λ}(u_{1}(w))^{λ} and r_{p}(w) = r_{p′}(w), as desired.

Finally, replacing λ by −λ clearly has no effect. If we replace λ by λ′ = ϱλ and v ∈ W is a lifting of ϱ then Γ_{+λ′} = ϱΓ_{+λ}ϱ^{−1} and we may take σ′_{i} = ϱσ_{i}. Then λ′_{i} = λ_{i} and we set w′_{i} = vw_{i}, so that u′_{i}(w) = vu_{i}(w)v^{−1}. We also take v′_{0} = vv_{0}v^{−1}, so that v′_{0}u′_{i}(w) = vv_{0}u_{i}(w)v^{−1}. Since χ_{λ′}(vxv^{−1}) = χ_{λ}(x) the independence is clear. Corollary 2.5.B. Suppose {ζ_{λ} : λ ∈ R} satisfies:

(i) ζ_{λ} is a character on C_{+λ} and hence on W_{+λ}.

(ii) ζ_{σλ} = ζ_{λ} ∘ σ^{−1}, σ ∈ Γ, λ ∈ R and ζ_{−λ} = ζ_{λ}^{−1}, λ ∈ R. (iii) If [F_{+λ} : F_{±λ}] = 2 then ζ_{λ} is trivial on C_{±λ}. Then

c(w) = ∏_{i=1}^{n} ζ_{λ}(v_{0}(u_{i}(w)))^{λ_{i}} , w ∈ W ,

is a 1-cocycle of W with values in C^{×} ⊗ X. Its cohomology class is independent of the choices made in its construction.

If the action of Σ on R is not transitive we define r_{q} and c for each Σ-orbit, thus for each pair ±O of Γ-orbits, and then take products over all such pairs. The results are denoted again r_{q} and c. (2.6) A Second Application Suppose that G is a connected reductive group defined over F. Recall that ^{L}G = Ĝ⋊W_{F}. Suppose that T is a maximal torus over F in G. Then we shall attach to χ-data for T, that is, for the action of Γ = Gal(\overline{F} /F) on R(G, T), a canonical Ĝ-conjugacy class of admissible embeddings of ^{L}T in ^{L}G.

There will be no harm in replacing \overline{F} throughout by a finite Galois extension L ⊂ \overline{F} of F over which T splits, and we do so without change in notation. Denote by σ the action of σ ∈ Γ on Ĝ and set Γ = {σ : σ ∈ Γ}. Fix a Γ-splitting (B̂, T̂, {X_{α}^{∨}}) of Ĝ.

A homomorphism ξ : ^{L}T → ^{L}G is an admissible embedding if

(i) ξ maps T̂ to T̂ by the isomorphism attached to the pair (B̂, T̂) and the choice of a Borel subgroup B in G containing T, and

(ii) ξ(w) ∈ Ĝ × w, w ∈ W. The Ĝ-conjugacy class of ξ is {Int g ∘ ξ : g ∈ Ĝ}. It is independent of the choice of (B̂, T̂) and B.

We fix B. Then to specify ξ we have only to give a homomorphism w → ξ(w) = ξ_{0}(w) × w where ξ_{0}(w) ∈ Norm(T̂, Ĝ) and where, in addition, if w → σ under W → Γ then Int ξ(w) acts on T̂ as the transport by ξ of the action of σ ∈ Γ on T̂. We write this transport σ_{T} as ω_{T} (σ) ⋊ σ ∈ Ω × Γ, and set Γ_{T} = {σ_{T} : σ ∈ Γ}. The χ-data {χ_{α}} for T provide χ-data for the action of Γ_{T} on R = R^{∨}(Ĝ, T̂); {χ_{α}^{−1}} is also a set of χ-data.

By Lemma 2.1.A

n(w) = n(ω_{T} (σ)) × w ∈ Norm(T̂, Ĝ) × w, w ∈ W,

satisfies

n(w_{1})n(w_{2})n(w_{1}w_{2})^{−1} = t_{p}(σ_{1}, σ_{2}),

if w_{i} → σ_{i} under W → Γ, where p(α) = 1 if and only if α^{∨} is a root of T̂ in B̂. By Lemma 2.5.A the inflation of t_{p} to W is split by r_{p}^{−1}, where r_{p} is the 1-cochain of W in T̂ attached to {χ_{α}} as in (2.5). We now use p_{0} for the gauge fixed there so that r_{p} = s_{p/p_{0}}r_{p_{0}}, and note that r_{p}^{−1} is the cochain attached to {χ_{α}^{−1}}. Thus with

ξ(w) = r_{p}(w)n(ω_{T} (σ)) × w, w ∈ W

we obtain an admissible homomorphism ξ : ^{L}T → ^{L}G. It is determined uniquely up to T̂-conjugacy by the Γ-splitting, Borel subgroup B and χ-data for T.

(2.6.1) Suppose that the Γ-splitting is replaced by another, (B^{g}, T^{g}, {X_{α^{∨}}^{g}}). We may suppose that g ∈ Ĝ is Γ -invariant [K2]. Then n(w) is replaced by g^{−1}n(w)g, w ∈ W, and ξ : ^{L}T → ^{L}G by Int g^{−1} · ξ, so that the Ĝ-conjugacy class of ξ is not affected. (2.6.2) We show now that the Ĝ-conjugacy class of ξ is independent of the choice of B. Suppose that B is replaced by B′ = vBv^{−1}, where v ∈ Norm(T, G), and ξ′ is obtained in place of ξ. Let μ in Ω = Ω(Ĝ, T̂) be the element defined by the transport of Int v|_{T} to T̂ by ξ. Then the transport of σ from T to T̂ via ξ′ is σ′_{T} = ω′_{T} (σ) ⋊ σ, where ω′_{T} (σ) = μ^{−1}ω_{T} (σ)σ(μ). Lemma 2.6.A. We have

ξ′ = Int g^{−1} ∘ ξ,

where g ∈ Norm(T̂, Ĝ) acts on T̂ as μ. Proof. The two sets of data {χ_{α}} and {χ′_{α} = χ_{μα}} yield

ξ(w) = r_{p}(w)n(ω_{T} (σ)) × w, ξ′(w) = r′_{p}(w)n(ω′_{T} (σ)) × w

if w → σ under W → Γ. We delete the subscript T in our notation. Lemma 2.1.A shows that

ξ′(w) = r′_{p}(w)n(μ^{−1}ω(σ)σ(μ)) × w
= r′_{p}(w)t_{p}(μ^{−1}ω(σ), σ(μ))t_{0}(μ^{−1}, ω(σ))n(μ^{−1})n(ω(σ))n(σ(μ)) × w
= n(μ)^{−1}t_{p}(μ, μ^{−1})μ[t_{p}(μ^{−1}ω(σ), σ(μ))t_{p}(μ^{−1}, ω(σ))r′_{p}(w)]r_{p}(w)^{−1}ξ(w)n(μ).

By Lemma 2.3.B,

t_{p}(μ, μ^{−1})μ[t_{p}(μ^{−1}ω(σ), σ(μ))t_{p}(μ^{−1}, ω(σ))]

is equal to

∏_{{p(α)=p(μ^{−1}α)=p(μ^{−1}σ^{−1}α)=1 \atop p(σ^{−1}α)=−1}}(−1)^{α} ∏_{{p(α)=p(σ^{−1}α)=p(μ^{−1}σ^{−1}α)=1 \atop p(μ^{−1}α)=−1}}(−1)^{α} = s_{q/p}(σ),

where q is the gauge p ∘ μ^{−1}.

To complete the proof we observe that μ(r′_{p}(w)) = s_{q/p_{0}}(σ)r_{p_{0}}(w), so that

\begin{aligned}μ(r′_{p}(w))r_{p}(w)^{−1} & = s_{q/p_{0}}(σ)r_{p_{0}}(w)s_{p/p_{0}}(σ)^{−1}r_{p_{0}}(w)^{−1} \\ & = s_{q/p}(σ). \end{aligned}

(2.6.3) Suppose that the χ-data {χ_{α}} are replaced by {χ′_{α}}. Then χ′_{α} = ζ_{α}χ_{α}, where {ζ_{α} : α ∈ R} satisfies the conditions of Corollary 2.5.B. Let c be the cocycle defined there. Then the embedding ξ is replaced by c ⊗ ξ where

c ⊗ ξ(t × w) = c(w)ξ(t × w), t × w ∈ ^{L}T.

(2.6.4) Suppose that Int g^{−1} maps T to T′ over F and carries the χ-data {χ_{α}} for T to data {χ′_{α}} for T′. Now g defines a canonical isomorphism λ_{g} : ^{L}T′ → ^{L}T. Let ξ be the embedding of ^{L}T in ^{L}G defined (up to ξ(T̂)-conjugacy) by the choice of Borel subgroup B ⊃ T. Then ξ′ = ξ ∘ λ_{g} is the embedding of ^{L}T′ in LG defined by the Borel subgroup B′ = g^{−1}Bg of G. Thus the class of embeddings attached to (T′, {χ′_{α}}) is obtained from that attached to (T, {χ_{α}}) by composition with the canonical ^{L}T′ → ^{L}T. (2.6.5) There is a simple local-global relationship when F is a number field. Suppose that v is a place of F and F_{v} is the completion of F at v. Fix a place w of L dividing v, and set Γ_{v} = Gal(L_{w}/F_{v}), W_{v} = W_{L_{w}/F_{v}}. Then a homomorphism W_{v} → W for which

1 → L_{w}^{×} → W_{v} → Γ_{v} → 1
↓ ↓ ↓
1 → C_{L_{w}} → W → Γ → 1

is commutative attaches to ξ : T̂ ⋊ W → Ĝ ⋊ W a local embedding ξ_{v} : T̂ ⋊ W_{v} → Ĝ ⋊ W_{v}. Since ξ(C_{L_{w}}) ⊂ T̂× C_{L_{w}}, where T̂ = ξ(T̂), ξ_{v} is determined uniquely up to T̂-conjugacy by ξ.

On the other hand, the place w of L determines completions of the subfields, F_{+} and F_{±}. These completions coincide with F_{v,+} and F_{v,±}, the subfields of L_{w} defined by Γ_{v,+} = {σ ∈ Γ_{v} : σλ = λ} and Γ_{v,±} = {σ ∈ Γ_{v} : σλ = ±λ}. Then the natural embeddings F_{v,+} ↪ C_{+}, F_{v,±} ↪ C_{±} allow us to construct χ-data {χ_{α}^{(v)}} for the action of Γ_{v}, local data, from χ-data {χ_{α}} for the action of Γ, global data.

To prescribe an admissible ξ : T̂⋊W → Ĝ⋊W we have to give a Γ-splitting of Ĝ and a Borel subgroup B of G containing T as well as {χ_{α}}. The same splitting and Borel subgroup together with {χ_{α}^{(v)}} yield a local embedding ξ′_{v} : T̂⋊W_{v} → Ĝ⋊W_{v}. Because the choices made in our constructions do not matter we can arrange that ξ′_{v}(w_{v}) = ξ(w) if w_{v} → w under W_{v} → W. Thus, up to T̂-conjugacy, ξ′_{v} is the local embedding attached to ξ, and passage from global to local embeddings is consistent with passage from global to local χ-data.

3. Definitions (3.1) Notation Throughout Sect. 3 F will be local and γ_{H}, \overline{γ}_{H} will be strongly G-regular elements in H(F) which are images of the elements γ_{G}, \overline{γ}_{G} in G(F).

Let T_{H}, \overline{T}_{H}, be the centralizers of γ_{H}, \overline{γ}_{H} in H. We fix admissible embeddings T_{H} → T and \overline{T}_{H} → \overline{T} of T_{H} and \overline{T}_{H} in G^{∗}, the quasi-split form of G, and denote by γ and \overline{γ} the images of γ_{H} and \overline{γ}_{H} under these embeddings. Notation for tori and elements in G itself will always include the subscript G.

We denote by R the root system of T in G^{∗}, by R^{∨} the coroots, by Ω the Weyl group, by R_{H}^{∨} the subsystem of coroots from H, by R_{H} the subset of roots from H and by Ω_{H} the Weyl group generated by R_{H} or R_{H}^{∨}. The analogous objects for \overline{T} will be denoted \overline{R}, \overline{R}^{∨} and so on.

We further fix a-data and χ-data for the action of Γ = Gal(\overline{F} /F) on the roots of T and of \overline{T}. These may also be regarded as a-data and χ-data for the action on the coroots, and are unaffected by passage to the simply-connected covering G_{sc}^{∗} of the derived group G^{∗}. If T_{H} → T is replaced by an 𝔄(T)-conjugate [recall (1.3)] then we may use this conjugation to transport given a-data and χ-data to data for the new image of T_{H}.

To check the effect of our choice of embedding T_{H} → T, a-data {a_{α}} and χ-data {χ_{α}} we will have only to determine the effect of: (A) replacing (T_{H} → T, {a_{α}}, {χ_{α}}) by an 𝔄(T)-conjugate triple, and (B) changing {a_{α}}, {χ_{α}} with T_{H} → T fixed.

Recall that T_{H} → T is given by the choice of Borel subgroups B_{H} in H and B in G^{∗}, of pairs (B_{H}, T_{H}) in Ĥ and (B, T ) in Ĝ, and of x ∈ Ĝ such that Int x ∘ ξ maps T_{H} to T and B_{H} into B. Several choices yield the same embedding. We may, without loss of generality, fix Γ-splittings (B_{H}, T_{H}, {X^{H}}) of Ĥ and (B, T , {X}) of Ĝ and require that (B_{H}, T_{H}) and (B, T ) be the chosen pairs in Ĥ and Ĝ for both T_{H} → T and \overline{T}_{H} → \overline{T}. Up to equivalence of endoscopic data we may assume that ξ maps T_{H} to T and B_{H} into B. Since the endoscopic datum s is central in ξ(Ĥ), it lies in T and its preimage in T̂_{H} is independent of the choice of B_{H}; so its image s_{T} in T̂ depends only on the embedding T_{H} → T.

There is a canonical embedding of the center Z(Ĝ) of Ĝ in T̂. Let T̂_{ad} = T̂/Z(Ĝ). Then π_{0} = π_{0}(T̂_{ad}^{Γ}) will denote the component group of the Γ-invariants in T̂_{ad}. The image of s_{T} in T̂_{ad} is Γ-invariant and so defines an element s_{T} of π_{0}. Finally we recall from [K2] that Tate-Nakayama duality provides a pairing

⟨, ⟩ : H^{1}(Γ, T_{sc}) × π_{0} → C^{×}.

(3.2) Term Δ_{I} Definition.

Δ_{I}(γ_{H}, γ_{G}) = ⟨λ(T_{sc}), s_{T} ⟩,

where λ(T_{sc}) is computed relative to an F -splitting spl of G_{sc} [see (2.3)]. Lemma 3.2.A.

Δ_{I}(γ_{H}, γ_{G})/Δ_{I}(\overline{γ}_{H}, \overline{γ}_{G})

is independent of the choice of spl. Proof. Suppose spl is replaced by spl^{g}, where g ∈ G_{sc}^{∗}(\overline{F}) is such that gσ(g^{−1}) lies in the center Z_{sc} of G_{sc}^{∗}, σ ∈ Γ. Then λ(T_{sc}) is multiplied by the class g_{T} of σ → gσ(g^{−1}) in H^{1}(T_{sc}^{∗}), by (2.3.1). Thus we have to show that

⟨g_{T} , s_{T} ⟩ = ⟨g\overline{T} , s\overline{T} ⟩.

There will be no harm in replacing \overline{F} by a finite Galois extension over which T is split. We do so without change in notation. Then following [L2, Lemma 6.2] we identify H^{1}(T_{sc}) with H^{1}(X_{∗}(T_{sc})), H^{1}(Z_{sc}) with H^{−2}(X_{∗}(T_{ad})/X_{∗}(T_{sc})) and H^{1}(Z_{sc}) → H^{1}(T_{sc}) with

H^{−2}(X_{∗}(T_{ad})/X_{∗}(T_{sc})) → H^{−1}(X_{∗}(T_{sc})).

Suppose the class of σ → gσ(g^{−1}) in H^{1}(Z_{sc}) corresponds to that of σ → λ_{σ}, where λ_{σ} ∈ X_{∗}(T_{ad})/X_{∗}(T_{sc}). Let λ_{σ} be a coset representative for λ_{σ}. Then g_{T} corresponds to the class of Σ_{σ}(σ_{T}^{−1} −1)λ_{σ} in H^{−1}(X_{∗}(T_{sc})) and ⟨g_{T} , s_{T} ⟩ = (Σ_{σ}(σ_{T}^{−1} − 1)λ_{σ})(s_{T}) if we regard the sum as a character on T̂. Similarly, there is a formula

⟨gT̄, sT̄⟩ = (Σ_{σ}(σ−1 ; \overline{T} − 1)\overline{λ}_{σ})(s\overline{T}).

To compare the two, we may choose an element x of G^{∗} such that x^{−1}T x = \overline{T} and such that t → x^{−1}tx is the transfer to T, \overline{T} of an isomorphism T_{H} → \overline{T}_{H} inner in H. Then xσ(x^{−1}) represents an element of Ω_{H}. Under Int x^{−1}, \overline{T} is identified as T with twisted Galois action σ\overline{T} = ϱ(σ) ⋊ σ_{T}, where ϱ(σ) ∈ Ω_{H}, and s\overline{T} is identified with s_{T}. We may assume \overline{λ}_{σ} = λ_{σ}, so that

⟨g\overline{T} , s\overline{T} ⟩ = (∑_{σ}(σ_{T}^{−1}ϱ(σ)^{−1} − 1)λ_{σ}) (s_{T})

= (∑_{σ}(σ_{T}^{−1} − 1)λ_{σ}) (s_{T})

= ⟨g_{T} , s_{T} ⟩,

since ϱ(σ) ∈ Ω_{H} fixes σ_{T} (s_{T}). Lemma 3.2.B. If (T_{H} → T, {a_{α}}) is replaced by its g-conjugate, g ∈ 𝔄(T_{sc}), then Δ_{I}(γ_{H}, γ_{G}) is multiplied by

⟨g_{T} , s_{T} ⟩^{−1},

where g_{T} is the class of σ → gσ(g^{−1}) in H^{1}(T_{sc}). Proof. This follows immediately from (2.3.4).

Lemma 3.2.C. Suppose that the a-data {a_{α}} are replaced by {a′_{α}}. Set b_{α} = a′_{α}/a_{α}. Then Δ_{I}(γ_{H}, γ_{G}) is multiplied by:

∏_{α} χ_{α}(b_{α}),

where {χ_{α}} are χ-data and the product is over representatives α for the symmetric orbits of Γ in the roots of T that are outside H.

Recall that b_{α} ∈ F_{±α}^{×} and that if α belongs to a symmetric orbit then χ_{α} restricts to the quadratic character on F_{±α}^{×} attached to F_{+α}/F_{±α} (2.5). Also χ_{α}(b_{α}) = χ_{σα}(b_{σα}), σ ∈ Γ, so that the choice of orbit representative does not matter. Proof of the lemma. By (2.3.2) Δ_{I}(γ_{H}, γ) is multiplied by ⟨b, s_{T} ⟩ where b is represented by the cocycle σ → ∏_{1,σ}^{q} b_{α}^{α^{∨}}, q being some gauge on R. The choice of q does not matter (Lemma 2.2.C). If O is a Γ-orbit then the contribution from ±O to this product is also a cocycle. Suppose it represents the class b_{±O}. Then b = ∏_{±O} b_{±O}. If O is asymmetric then b_{±O} is trivial (Lemma 2.2.C). Thus it remains to show the following: Lemma 3.2.D. If O is symmetric then

(i) ⟨b_{O}, s_{T} ⟩ = 1 if O is contained in R_{H} and (ii) ⟨b_{O}, s_{T} ⟩ = χ_{α}(b_{α}) if O is outside R_{H}, where α represents O.

Proof. We extend the Shapiro lemma arguments of Sect. 2. Let X_{O} be the free abelian group on O_{+}^{∨} = {α^{∨} : α ∈ O, q(α) = 1} with the inherited action of Γ. Fix some α^{∨} ∈ O_{+}^{∨} and let X_{α} be the subgroup generated by α^{∨}. Then the stabilizer Γ_{±α} of {±α} in Γ acts on X_{α} and X_{O} = Ind_{Γ_{±α}}^{Γ}X_{α}. Let T_{O} be the torus over F with X_{∗}(T_{O}) = X_{O} and T_{α} be the torus over F_{±α} with X_{∗}(T_{α}) = X_{α}. Then T_{α} is one-dimensional, anisotropic over F_{±α} and split over F_{α}, and T_{O} = Res_{F}^{F_{±α}}T_{α}. From the natural homomorphism X_{O} → X_{∗}(T_{sc}) we obtain T_{O} → T_{sc} over F and an L-homomorphism T̂_{ad} → T̂_{O}. Let s_{O} be the image of s_{T}, or more precisely of the image of s_{T} in Ĝ_{ad}. Then

α^{∨}(s_{O}) = α^{∨}(s_{T}), α ∈ O.

We pull b_{O} back to H^{1}(T_{O}), as we may, without change in notation. If s_{O} is the image of s_{T} under π_{0}(T̂_{ad}^{Γ}) → π_{0}(T̂_{O}^{Γ}) then the functoriality of Tate-Nakayama duality yields

⟨b_{T} , s_{T} ⟩ = ⟨b_{O}, s_{O}⟩.

If O ⊂ R_{H} then α^{∨}(s_{T}) = 1, α ∈ O, and so s_{O} = 1. This proves the first assertion of the lemma.

If O is outside H we compute ⟨b_{O}, s_{O}⟩ by reduction to T_{α}. The image b_{α} of b_{O} under H^{1}(Γ, T_{O}) ≅ H^{1}(Γ_{±α}, T_{α}) is represented by

σ → \{ \begin{matrix}b^{α^{∨}}, & σα = −α \\ 1, & σα = α \end{matrix} .

Let s_{α} be the image of s_{O} under π_{0}(T̂_{O}^{Γ}) ≅ π_{0}(T̂_{α}^{Γ_{±α}}). Then by restriction of scalars for Tate-Nakayama duality we have:

⟨b_{O}, s_{O}⟩ = ⟨b_{α}, s_{α}⟩.

Since π_{0}(T̂_{α}^{Γ_{±α}}) consists of two elements and s_{O} is clearly nontrivial s_{α} must be nontrivial. Then, as observed in (1.1),

⟨b_{α}, s_{α}⟩ = χ_{α}(b_{α}),

and the lemma is proved.

(3.3) Term Δ_{II} Definition. Δ_{II}(γ_{H}, γ_{G}) = ∏ χ_{α} (\frac{α(γ)−1}{a_{α}}), where the product is over representatives α for the orbits of Γ in the roots of T that are outside H.

Since

χ_{σα} (\frac{σα(γ) − 1}{a_{σα}}) = χ_{α} ∘ σ^{−1} (\frac{σ(α(γ)) − 1}{σ(a_{α})}) = χ_{α} (\frac{α(γ) − 1}{a_{α}})

the choice of representative α does not matter. Lemma 3.3.A. If O is asymmetric then the contribution from ±O is χ_{α}(α(γ)), where α lies in either O or −O. Proof.

χ_{α} (\frac{α(γ) − 1}{a_{α}}) χ_{−α} (\frac{α(γ)^{−1} − 1}{a_{−α}}) = χ_{α} (\frac{α(γ) − 1}{a_{α}}) χ_{α} (\frac{−a_{α}}{α(γ)^{−1} − 1}) = χ_{α}(α(γ)).

Lemma 3.3.B. If (T_{H} → T, {a_{α}}, {χ_{α}}) is replaced by an 𝔄(T)-conjugate then Δ_{II}(γ_{H}, γ_{G}) is unchanged. Proof. This is immediate. Lemma 3.3.C. If the a-data {a_{α}} are replaced by {a′_{α}}, where a′_{α} = a_{α}b_{α}, then Δ_{II}(γ_{H}, γ_{G}) is multiplied by

∏_{α} χ_{α}(b_{α})^{−1},

where the product is over representatives for the symmetric orbits outside H. Proof. By definition Δ_{II}(γ_{H}, γ_{G}) is multiplied by ∏ χ_{α}(b_{α})^{−1}, where the product is over representatives for all orbits outside H. Since

χ_{α}(b_{α})χ_{−α}(b_{−α}) = χ_{α}(b_{α})χ_{α}^{−1}(b_{α}) = 1

we may ignore the asymmetric orbits.

It remains to consider the effect on Δ_{II}(γ_{H}, γ_{G}) of replacing the χ-data {χ_{α}} by {χ′_{α}}. Suppose ζ_{α} = χ′_{α}χ_{α}^{−1}. Then if α lies in a symmetric orbit ζ_{α} must be an extension to F_{α}^{×} of the trivial character on F_{±α}^{×}.

Suppose that O is symmetric and q is some gauge on O. We denote by X^{O} the free abelian group on O_{+} = {α ∈ O : q(α) = 1} with the inherited action of Γ, and let X^{α} be the submodule generated by some α ∈ O_{+}, so that X^{O} = Ind_{Γ_{±α}}^{Γ}X^{α}. Define the torus T^{O} over F by X^{∗}(T^{O}) = X^{O} and T^{α} over F_{±α} by X^{∗}(T^{α}) = X^{α}. Then T^{α} is one-dimensional, anisotropic over F_{±α} and split over F_{α}, and T^{O} = Res_{F}^{F_{±α}}T^{α}. From the natural homomorphism X^{O} → X^{∗}(T) we obtain a homomorphism T → T^{O} over F and then

T (F) → T^{O}(F) ⟶^{∼} T^{α}(F_{±α}).

Let γ^{α} be the image of γ in T^{α}(F_{±α}). Then

α(γ) = α(γ^{α}).

Since the norm map T^{α}(F_{α}) → T^{α}(F_{±α}) is surjective we may write

γ^{α} = δ^{α}\overline{δ^{α}},

where δ^{α} ∈ T^{α}(F_{α}) and the bar denotes conjugation in T^{α}(F_{α}) with respect to T^{α}(F_{±α}).

Although we could do without it here we describe the analogous construction for an asymmetric orbit O. Thus X^{±O} is the free abelian group on O and X^{α} is the subgroup generated by some α in O. Then Γ acts and X^{α} has stabilizer Γ_{+α} = Γ_{±α}. We define T^{±O} over F by X^{∗}(T^{±O}) = X^{±O}; T^{α} is the one-dimensional F_{α}-split torus with X^{∗}(T^{α}) = X^{α}, and T^{±O} = Res_{F_{α}}^{F}T^{α}. From X^{±O} → X^{∗}(T) we obtain

T (F) → T^{±O}(F) ⟶^{∼} T^{α}(F_{α}).

If γ^{α} is the image of γ then α(γ^{α}) = α(γ). Lemma 3.3.D. If the χ-data {χ_{α}} are replaced by {χ′_{α}}, where χ′_{α} = χ_{α}ζ_{α}, then Δ_{II}(γ_{H}, γ_{G}) is multiplied by

∏^{asymm} ζ_{α}(γ^{α}) ∏^{symm} ζ_{α}(δ^{α}),

where the product ∏^{asymm} is over representatives α for pairs ±O of asymmetric orbits outside H, and ∏^{symm} is over representatives for the symmetric orbits outside H.

Proof. The contribution ∏^{asymm} is clear from Lemma 3.3.A. If α lies in a symmetric orbit we have

α(γ) = α(γ^{α}) = α(δ^{α}\overline{δ^{α}}) = α(δ^{α})/\overline{α(δ^{α})}.

Hence

ζ_{α} (\frac{α(γ) − 1}{a_{α}}) = ζ_{α} (\frac{α(δ^{α}) − \overline{α(δ^{α})}}{a_{α}\overline{α(δ^{α})}})

= ζ_{α} (\frac{α(δ^{α}) − α(δ^{α})}{a_{α}}) · ζ_{α}\overline{(α(δ^{α}))}^{−1}

= ζ_{α}(α(δ^{α}))

since (α(δ^{α}) − \overline{α(δ^{α}))}/a_{α} lies in F_{±α}^{×} and ζ_{α}\overline{(α(δ^{α}))}^{−1} = ζ_{α}(α(δ^{α})). (3.4) Term Δ_{III_{1}} or Δ_{1}

The next two terms will be denoted Δ_{III_{1}} and Δ_{III_{2}}, or more briefly Δ_{1} and Δ_{2}, since they are combined in a single term in the twisted case [K-S].

We begin with the case that G is quasi-split over F, taking G as G^{∗} and the twist ψ to be the identity. Since γ_{H} is an image of γ_{G} there exists h ∈ G_{sc} such that gγ_{G}h^{−1} = γ. Set v(σ) = hσ(h)^{−1}. Because γ is strongly regular the class inv(γ_{H}, γ_{G}) of v(σ) in H^{1}(T_{sc}) is well-defined. Definition (G quasi-split). Δ_{1}(γ_{H}, γ_{G}) = ⟨inv(γ_{H}, γ_{G})s_{T} ⟩^{−1}.

In general, we work with the two pairs of elements γ_{H}, γ_{G} and \overline{γ}_{H}, \overline{γ}_{G}. There exist h, \overline{h} ∈ G_{sc}^{∗} such that

hψ(γ_{G})h^{−1} = γ and \overline{h}ψ(\overline{γ}_{G})\overline{h}^{−1} = \overline{γ}.

Set

v(σ) = hu(σ)σ(h)^{−1} and \overline{v}(σ) = \overline{h}u(σ)σ(\overline{h})^{−1},

where u(σ) ∈ G_{sc}^{∗} and ψσ(ψ)^{−1} = Int u(σ), σ ∈ Γ. Then v(σ) and \overline{v}(σ) are cochains of Γ in T_{sc} and \overline{T}_{sc}, each well defined up to coboundaries because γ and \overline{γ} are strongly regular.

Further, ∂v = ∂\overline{v} = ∂u, each coboundary taking values in the center Z_{sc} of G_{sc}^{∗}. Let U = U(T, \overline{T}) be the torus

T_{sc} × \overline{T}_{sc}/\{(z^{−1}, z) : z ∈ Z_{sc}\}.

Then σ → (v(σ)^{−1}, \overline{v}(σ)) defines an element of H^{1}(U) which is independent of the choices for u(σ), h and \overline{h}. We write this class as

inv (\frac{γ_{H}, γ_{G}}{\overline{γ}_{H}, \overline{γ}_{G}} ) .

From T_{sc} × \overline{T}_{sc} → U we get H^{1}(T_{sc}) × H^{1}(\overline{T}_{sc}) → H^{1}(U). If G is quasi-split then inv (\frac{γ_{H},γ_{G}}{\overline{γ}_{H},\overline{γ}_{G}} ) is the image of (inv(γ_{H}, γ_{G})^{−1}, inv(\overline{γ}_{H}, \overline{γ}_{G})).

Recall that T̂_{ad} is T̂/Z(Ĝ), the torus dual to the preimage T_{sc} of T in G_{sc}. We denote by T̂_{sc} the torus dual to T_{ad} = T/Z(G). Then the center Ẑ_{sc} of the simply-connected covering of the derived group of Ĝ, which is a finite group isomorphic to Z_{sc}, is canonically embedded in T̂_{sc} and T̂_{sc}. Set

Û = T̂_{sc} × ˆT̄_{sc}/{z, z) : z ∈ Z_{sc}}.

Then X^{∗}(Û) ⊂ X^{∗}(T̂_{sc}×ˆT̄_{sc}). At the same time, X^{∗}(U) ⊂ X^{∗}(T_{sc}×\overline{T}_{sc}). The Q-pairing between X^{∗}(T̂_{sc}×ˆT̄_{sc}) and X^{∗}(T_{sc} × \overline{T}_{sc}) yields a dual Z-pairing between X^{∗}(Û) and X^{∗}(U), and so Û is the torus dual to U.

To the endoscopic datum s we attach s_{U} ∈ π_{0}(Û^{Γ}) as follows. Suppose s̃ lies in the preimage under T_{sc} → T_{ad} of the projection of s onto T_{ad} = T /Z(Ĝ). From T → T̂ and T → ˆ T̄ we obtain T_{sc} → T̂_{sc} and T_{sc} → ˆ T̄_{sc}. The images s̃_{T} and s̃\overline{T} of s̃ depend only on the embeddings T_{H} → T and \overline{T}_{H} → \overline{T}. The image s_{U} of (s̃_{T} , s̃\overline{T}) in Û is independent of the choice of s̃. It is also Γ-invariant and so defines an element s_{U} of π_{0}(Û^{Γ}). Definition Δ_{1}(γ_{H}, γ_{G}; \overline{γ}_{H}, \overline{γ}_{G}) = ⟨inv (\frac{γ_{H},γ_{G}}{\overline{γ}_{H},\overline{γ}_{G}} ) , s_{U}⟩. Note that if G is quasi-split over F then

Δ_{1}(γ_{H}, γ_{G}; \overline{γ}_{H}, \overline{γ}_{G}) = ⟨inv(γ_{H}, γ_{G}), s_{T} ⟩^{−1}⟨inv(\overline{γ}_{H}, \overline{γ}_{G}), s\overline{T} ⟩.

Lemma 3.4.A. If T_{H} → T and \overline{T}_{H} → \overline{T} are replaced by their g- and \overline{g}-conjugates, where g ∈ 𝔄(T_{sc}) and \overline{g} ∈ 𝔄(\overline{T}_{sc}), then Δ_{1}(γ_{H}, γ_{G}; \overline{γ}_{H}, \overline{γ}_{G}) is multiplied by

⟨g_{T} , s_{T} ⟩⟨g\overline{T} , s\overline{T} ⟩^{−1},

where g_{T} is the class of σ → gσ(g^{−1}) in H^{1}(T_{sc}), and g\overline{T} the class of σ → \overline{g}σ(\overline{g}^{−1}) in H^{1}(\overline{T}_{sc}). Proof. v(σ) is replaced by g^{−1}v(σ)σ(g) and g(g^{−1}v(σ)σ(g))g^{−1} = v(σ) · (gσ(g)^{−1})^{−1}. Similarly for \overline{v}(σ) and the lemma follows. (3.5) Term Δ_{III_{2}} or Δ_{2} For the construction here we fix Borel subgroups B_{H} ⊃ T_{H}, B ⊃ T which yield the isomorphism T̂_{H} → T_{H} → T → T̂ dual to T_{H} → T. To the χ-data {χ_{α}} are attached admissible embeddings ξ_{T} : ^{L}T → ^{L}G extending T̂ → T and ξ_{T_{H}} : ^{L}T_{H} → ^{L}H extending T̂_{H} → T_{H}. Then

ξ · ξ_{T_{H}} = aξ_{T} ,

where a is a 1-cocycle of W_{F} in T for the transport of the action of W_{F} on T̂. We transport a to T without change in notation. Its class a in H^{1}(W_{F} , T̂) is independent of the choice of B_{H} and B by (2.6.2), and of the Γ-splittings (B_{H}, T_{H}, {X^{H}}) and (B, T , {X}) by (2.6.1). Further if (T_{H} → T, {χ_{α}}) is replaced by its g-conjugate, g ∈ 𝔄(T), then a is replaced by its image in H^{1}(W_{F} , T̂^{g}) under the map induced by Int g^{−1} [see (2.6.4)]. Definition. Δ_{2}(γ_{H}, γ_{G}) = ⟨a, γ⟩. Clearly, replacing (T_{H} → T, {a_{α}}, {χ_{α}}) by an 𝔄(T)-conjugate has no effect on Δ_{2}(γ_{H}, γ_{G}). Lemma 3.5.A. Suppose that the χ-data {χ_{α}} are replaced by {χ′_{α}}, where χ′_{α} = χ_{α}ζ_{α}. Then Δ_{2}(γ_{H}, γ_{G}) is multiplied by

∏^{asymm} ζ_{α}(γ^{α})^{−1} ∏^{symm} ζ_{α}(δ^{α})^{−1},

where the product ∏^{asymm} is over representatives α for the pairs ±O of asymmetric orbits outside H and ∏^{symm} is over representatives α for the symmetric orbits outside H.

The elements γ^{α} and δ^{α} were defined in (3.3), and will be recalled in the proof. Proof. According to (2.6.3) a is replaced by ac^{−1} where c is represented by the cocycle

c(w) = ∏_{α} ∏_{i=1}^{n} ζ_{α}(v_{0}(u_{i}(w)))^{σ_{i}^{−1}α}.

Here the product ∏_{α} is over representatives α for pairs ±O of Γ-orbits of roots of T that lie outside H. The elements v_{0}(u_{i}(w)), σ_{i}^{−1}α were defined in (2.5). Note also Corollary 2.5.B.

Suppose that O is a symmetric orbit outside H. As in (3.3) we have

T (F) → T^{O}(F) ≅ T^{α}(F_{α}).

Then γ^{O} denotes the image of γ in T^{O}(F) and γ^{α} its image in T^{α}(F_{α}); γ^{α} = δ^{α}\overline{δ^{α}}. We have also

H^{1}(W_{F_{±α}}, T̂^{α}) ≅ H^{1}(W_{F} , T̂^{O}) → H^{1}(W_{F} , T̂).

We pull c back to c_{O} in H^{1}(W_{F} , T̂^{O}) and let c_{α} be the corresponding element of H^{1}(W_{F_{±α}}, T̂^{α}). Then

⟨c, γ⟩ = ⟨c_{O}, γ^{O}⟩

by the functoriality of ⟨, ⟩ which follows from its definition [B, Sect. 9]. Moreover

⟨c_{O}, γ^{O}⟩ = ⟨c_{α}, γ^{α}⟩

because the pairing ⟨, ⟩ respects restriction of scalars (see [B]). To compute ⟨c_{α}, γ^{α}⟩ we introduce S_{α}, the group obtained from G_{m} by restriction of scalars from F_{α} to F_{±α}, and the obvious homomorphism S_{α} → T_{α}, surjective on F_{±α}-valued points. By functoriality, we may replace c_{α} by its image in H^{1}(W_{±α}, Ŝ_{α}) and conclude that

⟨c_{α}, γ^{α}⟩ = ζ_{α}(α(δ^{α})).

For asymmetric O we have

T (F) → T^{±O}(F) ≅ T^{α}(F_{α})

as in (3.3), and dual

H^{1}(W_{F_{α}}, T̂^{α}) ≅ H^{1}(W_{F} , { \atop T^{±O})} → H^{1}(W_{F} , T̂).

again it is sufficient to compute ⟨c_{α}, γ^{α}⟩. Now T^{α} is split over F_{α} and from c_{α}(w) = ζ_{α}(w)^{α}, w ∈ W_{F_{α}}, we obtain

⟨c_{α}, γ^{α}⟩ = ζ_{α}(α(γ^{α})).

This completes the proof of the lemma. (3.6) Term Δ_{IV} If γ ∈ T (F) then ∏_{α}(α(γ) − 1), where the product is over all roots of T in G^{∗}, lies in F. We set

D_{G^{∗}}(γ) = | ∏_{α}(α(γ) − 1)|^{1/2}.

Definition. Δ_{IV} (γ_{H}, γ_{G}) = D_{G^{∗}}(γ)D_{H}(γ_{H})^{−1}. Then Δ_{IV} (γ_{H}, γ_{G}) depends only on the stable conjugacy classe of γ_{H}. (3.7) The Factor Δ We now fix the pair \overline{γ}_{H}, \overline{γ}_{G} and specify Δ(\overline{γ}_{H}, \overline{γ}_{G}) arbitrarily. Then we define

Δ(γ_{H}, γ_{G}) = Δ(\overline{γ}_{H}, \overline{γ}_{G})Δ(γ_{H}, γ_{G}; \overline{γ}_{H}, \overline{γ}_{G}),

where

Δ(γ_{H}, γ_{G}; \overline{γ}_{H}, \overline{γ}_{G})

is equal to

\frac{Δ_{I}(γ_{H}, γ_{G})}{Δ(\overline{γ}_{H}, \overline{γ}_{G})} · \frac{Δ_{II}(γ_{H}, γ_{G})}{Δ_{II}(\overline{γ}_{H}, \overline{γ}_{G})} · \frac{Δ_{III_{2}}(γ_{H}, γ_{G})}{Δ_{III_{2}}(\overline{γ}_{H}, \overline{γ}_{G})} · \frac{Δ_{IV} (γ_{H}, γ_{G})}{Δ_{IV} (\overline{γ}_{H}, \overline{γ}_{G})} · Δ_{III_{1}}(γ_{H}, γ_{G}; \overline{γ}_{H}, \overline{γ}_{G}).

In the case that G is quasi-split over F we set

Δ_{0}(γ_{H}, γ_{G}) = Δ_{I}(γ_{H}, γ_{G})Δ_{II}(γ_{H}, γ_{G})Δ_{1}(γ_{H}, γ_{G})Δ_{2}(γ_{H}, γ_{G})Δ_{IV} (γ_{H}, γ_{G})

so that

Δ(γ_{H}, γ_{G}; \overline{γ}_{H}, \overline{γ}_{G}) = Δ_{0}(γ_{H}, γ_{G})/Δ_{0}(\overline{γ}_{H}, \overline{γ}_{G}).

Recall that Δ_{I}(γ_{H}, γ_{G}), but not Δ_{I}(γ_{H}, γ_{G})/Δ_{I}(\overline{γ}_{H}, \overline{γ}_{G}), depends on the choice of an F -splitting for G^{∗}. Theorem 3.7.A. Δ(γ_{H}, γ_{G}) is independent of the choice of admissible embeddings, a-data and χ-data. Proof. If T_{H} → T, \overline{T}_{H} → \overline{T} and their a-data, χ-data are replaced by 𝔄(T)-, 𝔄(\overline{T})-conjugates then only Δ_{I} and Δ_{1} are changed. By Lemmas 3.2.B and 3.4.A, Δ is unchanged. If the a-data and χ-data alone are changed then Δ_{I}, Δ_{II}, and Δ_{2} are affected. Again the effects cancel, by Lemmas 3.2.C, 3.3.C, 3.3.D, and 3.5.A.

The same lemmas show that the factor Δ_{0}(γ_{H}, γ_{G}) is independent of these choices. Finally, if no strongly regular element in G(F) has an image in H(F) we set Δ ≡ 0.

4. Some Properties of Δ (4.1) Invariance Lemma 4.1.A. Δ(γ_{H}^{1}, γ_{G}^{1}, γ_{H}^{2}, γ_{G}^{2})Δ(γ_{H}^{2}, γ_{G}^{2}; γ_{H}^{3}, γ_{G}^{3}) = Δ(γ_{H}^{1}, γ_{G}^{1}; γ_{H}^{3}, γ_{G}^{3}). Proof. It is enough to show this with Δ replaced by Δ_{1} = Δ_{III_{1}}, that is to show that

⟨inv_{(1,2)}, s_{U_{1,2}}⟩⟨inv_{(2,3)}, s_{U_{2,3}}⟩ = ⟨inv_{(1,3)}, s_{U_{1,3}}⟩,

where

inv_{(i,j)} = inv (\frac{γ_{H}^{i}, γ_{G}^{i}}{γ_{H}^{j}, γ_{G}^{j}} ) , U_{i,j} = U(T^{i}, T^{j}),

1 ≦ i < j ≦ 3, and T^{i} is the image of Cent(γ_{H}^{i}, H) under some admissible embedding in G^{∗}. Let

V = T_{sc}^{1} × T_{sc}^{2} × T_{sc}^{3}/{(z^{−1}, zw^{−1}, w) : z, w ∈ Z_{sc}}

in the notation of (3.4). There are F -homomorphisms U_{i,j} → V. Under the induced maps on cohomology the image of inv_{(1,3)} is the product of the images of inv_{(1,2)} and inv_{(2,3)}. Still following the notation of (3.4) we see that the dual of V is

V̂ = T̂_{sc}^{1} × T̂_{sc}^{2} × T̂_{sc}^{3}/{(z, z, z) : z ∈ Ẑ_{sc}}.

Let s_{V} be the image of (s̃_{T^{1}}, s̃_{T^{2}}, s̃_{T^{3}}) in π_{0}(V̂^{Γ}). Then

⟨inv_{(1,2)}, s_{U_{1,2}}⟩⟨inv_{(2,3)}, s_{U_{2,3}}⟩ = ⟨image(inv_{(1,2)})image(inv_{(2,3)}), s_{V} ⟩

which equals

⟨image(inv_{(1,3)}), s_{V} ⟩ = ⟨inv_{(1,3)}, s_{U_{1,3}}⟩

and the lemma is proved. Corollary 4.1.B.

(i) Δ(γ_{H}, γ_{G}; γ_{H}, γ_{G}) = 1 and

(ii) Δ(γ′_{H}, γ′_{G}) = Δ(γ_{H}, γ_{G})Δ(γ′_{H}, γ′_{G}; γ_{H}, γ_{G}) if γ_{H}, γ′_{H} are images of γ_{G}, γ′_{G} respectively.

Lemma 4.1.C. Δ(γ_{H}, γ_{G}) depends only on the stable conjugacy class of γ_{H} in H(F) and the conjugacy class of γ_{G} in G(F). Proof. Let γ′_{G} = g^{−1}γ_{G}g, g ∈ G(F). Then

Δ(γ_{H}, γ′_{G}) = Δ(γ_{H}, γ_{G})Δ(γ_{H}, γ′_{G}; γ_{H}, γ_{G}).

On examining the terms Δ_{I}, …, Δ_{IV} we see that

Δ(γ_{H}, γ′_{G}; γ_{H}, γ_{G}) = Δ_{1}(γ_{H}, γ′_{G}; γ_{H}, γ_{G})

and so it remains to check that Δ_{1}(γ_{H}, γ′_{G}; γ_{H}, γ_{G}) = 1. There is g_{1} ∈ G_{sc}(\overline{F}) such that g^{−1}γ_{G}g = g_{1}^{−1}γ_{G}g_{1}. Then g_{1}σ(g_{1})^{−1} is a cocycle with values in Ker(G_{sc} → G). If h ∈ G_{sc}^{∗} is such that hψ(γ_{G})h^{−1} = γ then hψ(g_{1})ψ(γ′_{G})(hψ(g_{1}))^{−1} = γ and so the cocycle defining inv (\frac{γ_{H},γ′_{G}}{γ_{H},γ_{G}} ) is of the form (ψ(g_{1}σ(g_{1})^{−1})^{−1}v(σ)^{−1}, v(σ)). Then Δ_{1}(γ_{H}, γ′_{G}; γ_{H}, γ_{G}) coincides with ⟨ψ(g_{1}σ(g_{1})^{−1})^{−1}, s_{T} ⟩ · Δ_{1}(γ_{H}, γ_{G}; γ_{H}, γ_{G}). The first term in this product is trivial since ψ(g_{1}σ(g_{1})^{−1}) ∈ Ker(G_{sc}^{∗} → G^{∗}); the second is trivial by (4.1.B).

Suppose γ′_{H} = h^{−1}γ_{H}h is stably conjugate to γ_{H}. Then an admissible embedding of Cent(γ′_{H}, H) in G^{∗} is obtained by composition of an admissible embedding of Cent(γ_{H}, H) with Int h. Term-by-term examination of Δ yields Δ(γ′_{H}, γ_{G}; γ_{H}, γ_{G}) = 1. (4.2) The Local Hypothesis We now examine the relation between factors for G and those for G^{∗}.

The endoscopic data (H, H, s, ξ) serve both G and G^{∗}. Suppose that strongly G-regular γ_{H} ∈ H(F) is an image of γ_{G} ∈ G(F). Then γ_{H} is also strongly G^{∗}-regular. By Steinberg’s Theorem [K1], γ_{H} is the image of a stable conjuacy class of elements in G^{∗}(F). Suppose that γ_{G^{∗}} belongs to this class. Then both Δ(γ_{H}, γ_{G}) and Δ(γ_{H}, γ_{G^{∗}}) are defined and nonzero. Set

Δ_{G/G^{∗}}(γ_{H}, γ_{G}, γ_{G^{∗}}) = Δ(γ_{H}, γ_{G})/Δ(γ_{H}, γ_{G^{∗}}).

It is clear from the definitions that this assumes a quite simple form, for we may use the same auxiliary data of admissible embeddings, a-data and χ-data to define all terms in the numerator and denominator. Since Δ(γ_{H}, γ_{G}) and Δ(γ_{H}, γ_{G^{∗}}) are canonical only up to constants we will investigate

Δ_{G/G^{∗}}(γ_{H}, γ_{G}, γ_{G^{∗}})/Δ_{G/G^{∗}}(γ′_{H}, γ′_{G}, γ′_{G^{∗}})

where γ′_{H} is an image of γ′_{G} and of γ′_{G^{∗}}.

First we shall define a number λ_{H}(γ_{G}, γ_{G^{∗}}, γ′_{G}, γ_{G^{∗}}). There are unique admissible embeddings Cent(γ_{H}, H) → Cent(γ_{G^{∗}}, G^{∗}) and Cent(γ′_{H}, H) → Cent(γ′_{G^{∗}}, G^{∗}) mapping γ_{H} to γ_{G^{∗}} and γ′_{H} to γ′_{G^{∗}}. We set

λ_{H}(γ_{G}, γ_{G^{∗}}; γ′_{G}, γ′_{G^{∗}}) = ⟨inv (\frac{γ_{H}, γ_{G}}{γ′_{H}, γ′_{G}} ) , s_{U}⟩

in the notation of (3.4). Note that inv (\frac{γ_{H},γ_{G}}{γ′_{H},γ′_{G}} ) is represented by the cocycle

(σ(h)u(σ)^{−1}h^{−1}, h′u(σ)σ(h′)^{−1}),

where h, h′ ∈ G_{sc}^{∗} and

hψ(γ_{G})h^{−1} = γ_{G^{∗}}, h′ψ(γ′_{G})h^{′−1} = γ′_{G^{∗}}.

Moreover u(σ) ∈ G_{sc}^{∗} is given by ψσ(ψ)^{−1} = Int u(σ). Then λ_{H} is independent of the choice of γ_{G}, …, γ′_{G^{∗}} within their conjugacy classes (see the proof of Lemma 4.1.C). Lemma 4.2.A. Δ_{G/G^{∗}}(γ_{H}, γ_{G}, γ_{G^{∗}})/Δ_{G/G^{∗}}(γ′_{H}, γ′_{G}, γ′_{G^{∗}}) = λ_{H}(γ_{G}, γ_{G^{∗}}; γ′_{G}, γ′_{G^{∗}}). Proof. We choose admissible embeddings as in the definition of λ_{H}. Since Δ_{I}, Δ_{II}, Δ_{2}, Δ_{IV} take the same values at (γ_{H}, γ_{G}) as at (γ_{H}, γ_{G^{∗}}) only Δ_{1} yields a nontrivial contribution to the left side. In view of Corollary 4.1.B this contribution is

Δ_{1}(γ_{H}, γ_{G}; γ′_{H}, γ′_{G})/Δ_{1}(γ_{H}, γ_{G^{∗}}; γ′_{H}, γ′_{G^{∗}}) = Δ_{1}(γ_{H}, γ_{G}; γ′_{H}, γ′_{G})

by our choice of embeddings. This equals

λ_{H}(γ_{G}, γ_{G^{∗}}; γ′_{G}, γ′_{H}),

and the lemma is proved.

The following is then immediate: Corollary 4.2.B.

\frac{Δ(γ_{H}, γ_{G})}{Δ(γ_{H}, γ_{G^{∗}})} = λ_{H}(γ_{G}, γ_{G^{∗}}; γ′_{G}, γ′_{G^{∗}})\frac{Δ(γ′_{H}, γ′_{G})}{Δ(γ′_{H}, γ′_{G^{∗}})}.

This asserts that the factors Δ satisfy the Local Hypothesis of [L2, Chap. VI]. To reconcile our notation with that of [L2] we note that if T_{H} = Cent(γ_{H}, H), T = Cent(γ_{G^{∗}}, G^{∗}) and T_{G} = Cent(γ_{G}, G)

then T_{H} → T_{H} → T → T

↑
T_{G}

is a diagram D, where T_{H} → T is the admissible embedding taking γ_{H} to γ_{G^{∗}} and Int h ∘ ψ : T_{G} → T. Similarly we have D′:

T′_{H} → T_{H} → T → T′
↑
T′_{G}.

The Local Hypothesis states that c(D, D′), the left side of Lemma 4.2.A, is given by an expression κ(θ(E, E′)) which we now show to coincide with λ_{H} = λ_{H}(γ_{G}, γ_{G^{∗}}, γ′_{G}, γ′_{G^{∗}}).

There is no harm in assuming G simply-connected. Let U = U(T, T′). Then we have

X_{∗}(U) ↪ X_{∗}(T_{ad}) × X_{∗}(T′_{ad}).

The elements λ ∈ X_{∗}(T_{ad}), λ′ ∈ X_{∗}(T′_{ad}) are defined on p. 84 of [L2]. On modifying λ′ as on p. 85 we may assume (6.12) of p. 85. Then θ(E, E′) = λ − λ′. On the other hand, (λ, λ′) ∈ X_{∗}(U) and defines an element of H^{−1}(X_{∗}(U)) which corresponds under Tate-Nakayama duality to the class of the cocycle

(σ(h)u(σ)^{−1}h^{−1}, h′u(σ)σ(h′)^{−1})

defining inv (\frac{γ_{H},γ_{G}}{γ′_{H},γ′_{G}} ) in H^{1}(U). Since κ is obtained from the endoscopic datum s [L2, p. 100] we conclude that κ(θ(E, E′)) coincides with λ_{H}. (4.3) Extension to All G-Regular Elements The definition of Δ_{1}(γ_{H}, γ_{G}; \overline{γ}_{H}, \overline{γ}_{G}) requires that G-regular γ_{H} be strongly regular. The notion of image, however, is well defined for an arbitrary G-regular semisimple element in H(F) [recall (1.3)] and we expect an identity

Φ^{st}(γ_{H}, f^{H}) = ∑_{γ_{G}} Δ(γ_{H}, γ_{G})Φ(γ_{G}, f^{G})

for all G-regular semisimple elements γ_{H} in H(F), where Δ(γ_{H}, γ_{G}) = 0 unless γ_{H} is an image of γ_{G}.

We shall extend Δ by continuity. Suppose γ_{H}^{0} ∈ T_{H}(F) is G-regular and is an image of γ_{G}^{0} ∈ T_{G}(F). Fix an admissible embedding T_{H} → T of T_{H} in G^{∗} and an isomorphism Int x ∘ ψ : T → T_{G} over F so that γ_{H}^{0} → γ_{G}^{0} under T_{H} → T_{G}. Suppose g_{1}, …, g_{n} are representatives for D(T_{G}) = T_{G}(\overline{F})\𝔄(T_{G})/G(F), where 𝔄(T_{G}) = \{g ∈ G(\overline{F}) : gσ(g)^{−1} ∈ T_{G}(\overline{F})\}. Then if γ_{G} ∈ T_{G}(F) is strongly regular the elements g_{i}^{−1}γ_{G}g_{i} are representatives for the conjugacy classes in the stable conjugacy class of γ_{G}. We may find a sequence {γ_{H}} of strongly G-regular elements in T_{H}(F) such that {γ_{H}} → γ_{H}^{0}. Let γ_{G} be the image of γ_{H} under T_{H} → T_{G}. Then the limit γ_{G}^{i} = g_{i}^{−1}γ_{G}^{0}g_{i} of {g_{i}^{−1}γ_{G}g_{i}} has γ_{H}^{0} as image. We define

Δ(γ_{H}^{0}, γ_{G}^{i}) = lim_{γ_{H}→γ_{H}^{0}} Δ(γ_{H}, g_{i}^{−1}γ_{G}g_{i}),

as an examination of the terms in Δ shows the right side to be well-defined. Then if f, f^{H} have Δ-matching orbital integrals we have

Φ^{st}(γ_{H}^{0}, f^{H}) = ∑_{i=1}^{n} Δ(γ_{H}^{0}, γ_{G}^{i})Φ(γ_{G}^{i}, f),

where Φ(γ_{G}^{i}, f) is as specified in (1.3) and

Φ^{st}(γ_{H}^{0}, f^{H}) = ∑_{j=1}^{m} Φ(γ_{H}^{j}, f^{H}),

with γ_{H}^{j} = h_{j}^{−1}γ_{H}^{0}h_{j} and {h_{j}} representatives for D(T_{H}). Thus on either side a conjugacy class may contribute several terms. (4.4) Passage to Central Extensions The center of G is canonically embedded as a central subgroup of H. If γ_{H} is an image of γ_{G} then zγ_{H} is an image of zγ_{G}, z ∈ Z(F). Still assuming H is ^{L}H we have: Lemma 4.4.A. There is a character λ^{G} on Z(F) such that

Δ(zγ_{H}, zγ_{G}) = λ^{G}(z)Δ(γ_{H}, γ_{G}), z ∈ Z(F),

for all γ_{H}, γ_{G}.

The proof will be included in another paper. There is one case which it is useful to treat here. Proof for z in the identity component Z^{0} of Z:

According to the definitions,

Δ(zγ_{H}, zγ_{G})Δ(γ_{H}, γ_{G})^{−1} = Δ_{2}(zγ_{H}, zγ_{G})Δ_{2}(γ_{H}, γ_{G})^{−1} = ⟨a, z⟩,

so that we have to show that the character z → ⟨a, z⟩ on Z^{0}(F) is independent of the choice of γ_{H} and of T_{H} → T, χ-data and a-data. The last three choices have no effect because they have no effect on Δ.

Given also \overline{γ}_{H} and \overline{T}_{H} → \overline{T} we form S = T × \overline{T} /Z^{0}, S_{H} = T_{H} × \overline{T}_{H}/Z^{0}, G̃ = G × G/Z^{0} and H̃ = H × H/Z^{0}, where Z^{0} is embedded in each case by z → (z, z^{−1}). Because Z^{0} is connected we have natural embeddings

LS ↪ ^{L}T ×^{L}\overline{T} , ^{L}S_{H} ↪ ^{L}T_{H} ×^{L}\overline{T}_{H}, ^{L}G̃ ↪ ^{L}G ×^{L}G, ^{L}H̃ ↪ ^{L}H ×^{L}H

and may form commutative diagrams

\begin{matrix}^{L}H̃ → ^{L}H & ×^{L}H \\ ξ̃^{∩} ↓ & ^{∩} ↓(ξ, ξ) \\ ^{L}G̃ → ^{L}G & ×^{L}G \end{matrix} \begin{matrix}^{L}S → ^{L}T & ×^{L}\overline{T} \\ ξ_{S}^{∩} ↓ & ^{∩} ↓(ξ_{T} , ξ\overline{T}) \\ ^{L}G̃ ⟶ & ^{L}G \end{matrix}

and so on. As a result we conclude that the element (a, \overline{a}) of H^{1}(W_{F} , T̂ × T̂) is the image of an element ã of H^{1}(W_{F} , Ŝ). Then

⟨a, z⟩⟨\overline{a}, z⟩^{−1} = ⟨(a, \overline{a}), (z, z^{−1})⟩ = ⟨ã, 1⟩ = 1

and the lemma is proved.

We now remove the assumption that H is an L-group. The data (H, H, s, ξ) given, orbital integrals of functions on G(F) will be matched not with those of functions on H(F) but with those of functions on H_{1}(F), where H_{1} is a central extension of H. We shall take H_{1} attached to a central extension of G as the arguments are more transparent. Thus we fix a z-extension 1 → Z_{1} → G_{1} → G → 1 of G [K1]. This means, in particular, that Z_{1} is a connected central subgroup of G_{1}, G_{1}(F) → G(F) is surjective and the derived group of G_{1} is simply-connected. The dual sequence 1 → Ĝ → Ĝ_{1} → Ẑ_{1} → 1 allows us to regard Ĝ as a subgroup of Ĝ_{1}. We may assume that ϱ_{G_{1}}(σ) and ϱ_{G}(σ) agree on Ĝ, σ ∈ Γ, so that ^{L}G embeds canonically in ^{L}G_{1}.

There is a central extension 1 → Z_{1} → H_{1} → H → 1 and embedding ξ_{1} : ^{L}H_{1} ↪ ^{L}G_{1} such that (H_{1},^{L}H_{1}, s, ξ_{1}) are endoscopic data for G_{1} (see [L1]). The parameter for a character λ on Z_{1}(F) is given by

W_{F} → ^{L}G_{1} → ^{L}Z_{1},

where the first arrow denotes the restriction of ξ_{1} to W_{F} and the second is the natural extension of Ĝ_{1} → Ẑ_{1}.

The matching is to be between orbital integrals of functions f on G(F), and thus of functions on G_{1}(F) invariant under Z_{1}(F), and orbital integrals of functions f^{H_{1}} on H_{1}(F) satisfying

f^{H_{1}}(zh) = λ(z)f^{H_{1}}(h), z ∈ Z_{1}(F), h ∈ H_{1}(F).

We need consider only elements γ_{H_{1}} whose image in H(F) is strongly G-regular. Then γ_{H_{1}} is an image of γ_{G} ∈ G(F) if it is an image of some γ_{G_{1}} in the preimage of γ_{G} in G_{1}(F). The element γ_{G_{1}} is uniquely determined. We say that f and f^{H_{1}} have Δ-matching orbital integrals if, as usual,

Φ^{st}(γ_{H_{1}}, f^{H_{1}}) = ∑_{γ_{G}} Δ(γ_{H_{1}}, γ_{G})Φ(γ_{G}, f)

for all such γ_{H_{1}}. Because of the transformation rule for f^{H_{1}} we must have

Δ(zγ_{H_{1}}, γ_{G}) = λ(z)Δ(γ_{H_{1}}, γ_{G}), z ∈ Z_{1}(F).
The factor Δ(γ_{H_{1}}, γ_{G_{1}}) has been defined. We set
Δ(γ_{H_{1}}, γ_{G}) = Δ(γ_{H_{1}}, γ_{G_{1}})

if γ_{H_{1}} is an image of γ_{G_{1}} and γ_{G} is the image of γ_{G_{1}} under G_{1}(F) → G(F), or Δ(γ_{H_{1}}, γ_{G}) = 0 if γ_{H_{1}} is not an image of γ_{G}.

Recall that

Δ(zγ_{H_{1}}, zy_{G_{1}}) = λ_{1}(z)Δ(γ_{H_{1}}, γ_{G_{1}}), z ∈ Z_{1}(F),

where λ_{1} is the character on the center of G_{1}(F) attached to a [see the beginning of the proof of (4.4.A)]. To conclude that

Δ(zγ_{H_{1}}, γ_{G}) = λ(z)Δ(γ_{H_{1}}, γ_{G})

we have only to show that λ_{1} coincides with λ on Z_{1}(F). But a is represented by the cocycle a defined by

ξ_{1} · ξ_{T_{H_{1}}} (w) = a(w)ξ_{T_{1}}(w), w ∈ W_{F} ,

where T_{H_{1}} = Cent(γ_{H_{1}}, H_{1}) and T_{H_{1}} → T_{1} is an admissible embedding. To compute λ_{1} on Z_{1}(F) we project a onto Ẑ_{1}, obtaining a_{1} : W_{F} → Ẑ_{1}. Since, by construction, ξ_{T_{H_{1}}} (w) ∈ Ĥ × w and ξ_{T_{1}}(w) ∈ Ĝ × w, w ∈ W_{F}, we have that W_{F} ⟶^{ξ_{1}} ^{L}G_{1} ⟶ ^{L}Z_{1} coincides with w → a_{1}(w) × w and then λ_{1} = λ on Z_{1}(F) by definition.

The group H_{1} is determined up to isomorphism by G_{1} and (H, H, s, ξ), but ξ_{1} may be replaced by b ⊗ ξ_{1} where b is a 1-cocycle of W_{F} in the center of Ĥ_{1}. This cocycle determines a character λ_{0} on H_{1}(F) and λ, λ_{1} are replaced by λ_{0}λ, λ_{0}λ_{1}.

Finally, we observe that it is only the equivalence class of (H, H, s, ξ) that matters for the definition of Δ. The choice of twist ψ : G → G^{∗} does affect Δ, but ψ may be replaced by Int x ∘ ψ, x ∈ G^{∗}, without effect.

5. Regular Unipotent Analysis (5.1) Regular Unipotent Elements (Review) Recall that the regular unipotent elements of G(\overline{F}) are characterized by the property that each lies in exactly one Borel subgroup of G. They form a single conjugacy class [St].

Suppose B is a Borel subgroup of G containing the maximal torus T. Denote by N the unipotent radical of B. For a simple root α of T in B let N_{α} be the 1-parameter subgroup of N attached to α and N^{α} be the subgroup of N generated by the 1-parameter subgroups for the remaining roots of T in B, so that N is the direct product N_{α} · N^{α}. Set N′ = ∩_{α} N^{α}. Given root vectors {X_{α}} we define x_{α}(u) for u in N by

u ≡ exp x_{α}(u)X_{α}(mod N^{α}).

Then u is regular if and only if x_{α}(u) is nonzero for all simple α. See [St, pp. 110-112] for this and the next paragraph.

Fix u_{0} regular in N. Then every regular element in N may be written in the form

u = t^{−1}u′u_{0}t,

where u′ ∈ N′ is uniquely determined and t ∈ T is determined modulo the center Z of G. Conversely, every element of this form is regular in N. Note that x_{α}(t^{−1}u′u_{0}t) = α(t)^{−1}x_{α}(u_{0}).

Let u be any regular unipotent element of G, B_{u} be the Borel subgroup containing u, and T_{u} be a maximal torus in B_{u}. For each simple root α of T_{u} in B_{u} define a root vector X_{α}^{u} by requiring exp X_{α}^{u} to be the projection of u onto N_{α}. Write spl(u) for the splitting (B_{u}, T_{u}, {X_{α}^{u}}) of G. Every splitting of G is obtained in this manner.

The following are equivalent:

(i) G is quasi-split over F.

(ii) G has an F -splitting. (iii) There are regular unipotent elements in G(F). For (ii) ⇒ (ii) we observe that if u ∈ G(F) then spl(u) is an F -splitting as long as T_{u} is chosen over F. Note also that then spl(u) is determined up to N(F)-conjugacy by u. For (i) ⇒ (iii), the existence of a Borel subgroup over F implies that the conjugacy class of regular unipotent elements is defined over F. This class then contains an F -rational point [K1].

From now on G will be quasi-split over F. The regular unipotent elements in G(F) form a single stable conjugacy class by definition [K1]. Suppose T and B = T N are defined over F. Then each G(F)-conjugacy class of regular unipotent elements in G(F) meets N(F). If u_{0} ∈ N(F) is regular then the regular element u = t^{−1}u′u_{0}t in N is F -rational if and only if u′ ∈ N′(F) and tσ(t^{−1}) ∈ Z(\overline{F}), σ ∈ Γ. Lemma 5.1.A. The correspondence u → spl(u) induces a bijection between the G(F)-conjugacy classes of regular unipotent elements in G(F) and the G(F)-conjugacy classes of F -splittings of G. Proof. As above, if u ∈ B(F) then spl(u) is determined up to N(F)-conjugacy, where N is the unipotent radical of B. On the other hand, if spl = (B, T, {X_{α}}) is an F -splitting then we can find u ∈ N such that spl(u) = spl. For σ ∈ Γ we must have spl(σu) = spl as well, which implies that u^{−1}σ(u) lies in N′. Because H^{1}(Γ, N′(\overline{F})) is trivial we can find u′ ∈ N′ such that uu′ ∈ N(F). Then spl = spl(uu′). It is clear now that u → spl u induces a surjective map from G(F)-conjugacy classes to G(F)-conjugacy classes. For injectively it is enough to show that if B = T N is over F and spl(u_{1}) = spl(u_{2}), where u_{1} and u_{2} lie in N(F), then u_{1} and u_{2} are N(F)-conjugate. But spl u_{1} = spl u_{2} implies that u_{1} = t^{−1}u′_{1}u_{0}t and u_{2} = t^{−1}u′_{2}u_{0}t with t ∈ T and u′_{1}, u′_{2} ∈ N′, for some fixed regular u_{0} ∈ N(F). Because u_{1}, u_{2} ∈ N(F) we have tσ(t^{−1}) ∈ Z, σ ∈ Γ, and u′_{1}, u′_{2} ∈ N′(F). Thus it is enough to show that u′_{1}u_{0} and u′_{2}u_{0} are conjugate under N(F). This is so because N′(\overline{F})u_{0} is the N(\overline{F})-conjugacy class of u_{0} [St, p. 112] and H^{1}(Γ, N(\overline{F})) = 1. The proof is then complete.

We now define a transfer factor Δ(u) for u regular unipotent in G(F). Fix an F -splitting spl = (B, T, {X_{α}}) of G. There exists h ∈ G_{sc} such that

spl(u)^{h} = spl,

where h acts in the obvious manner. Then hσ(h)^{−1} lies in the center Z_{sc} of G_{sc}, σ ∈ Γ. The class inv(u) of σ → hσ(h)^{−1} in H^{1}(Γ, Z_{sc}) is well-defined.

To pair inv(u) with the endoscopic datum s we choose any maximal torus T over F in G which contains regular elements with images in H(F). Inv(u) has an image inv_{T} (u) under H^{1}(Z_{sc}) → H^{1}(T_{sc}). As earlier, (3.1), s determines an element s_{T} of π_{0}(T̂_{ad}^{Γ}). We set

⟨inv(u), s⟩ = ⟨inv_{T} (u), s_{T} ⟩.

The argument used in the proof of Lemma 3.2.A shows that ⟨inv(u), s⟩ is independent of the choice for T. In addition if we define Δ(\overline{γ}_{H}, \overline{γ}_{G}) and Δ_{0}(\overline{γ}_{H}, \overline{γ}_{G}) as in (3.7), but use – for reasons that will appear later – the opposite splitting spl_{∞} = (B_{∞}, T, {X_{−α}}), where B_{∞} ∩ B = T and the root vectors X_{−α} are fixed as in (2.1), then

Δ(u) = \frac{Δ(\overline{γ}_{H}, \overline{γ}_{G})}{Δ_{0}(\overline{γ}_{H}, \overline{γ}_{G})}⟨inv(u), s⟩

is independent of the choice of spl. Finally, Δ(u) depends only on the G(F)-conjugacy class of u in G(F). (5.2) Stars and the Variety X This will be a review of some material from [L3]. We continue with G quasi-split over F and (B, T, {X_{α}}) an F -splitting of G.

Let T be a maximal torus over F in G. We fix some Borel subgroup B_{T} containing T and in the usual manner transport the roots, Weyl group, Weyl chambers and Galois action for T to T without change in notation. We denote by Ω the Weyl group, by 𝔚 the set of Weyl chambers and by W_{+} the chamber attached to B_{T} or B. If ω ∈ Ω we write W (ω) for the chamber ω^{−1}W_{+} and B_{T}^{ω} for the Borel subgroup w^{−1}B_{T} w, where w ∈ G represents ω.

Let S be the variety of stars attached to T [L3]. The choice of B_{T} affects S, but a different choice yields an F -isomorphic variety. Recall that the elements of S are functions from 𝔚 to B, the variety of Borel subgroups of G. A typical element will be denoted (B(W)). The F -structure is defined by

σ(B(W)) = (σ(B(σ_{T}^{−1}(W)))), σ ∈ Γ,

and G acts on the right:

(B(W))^{g} = (g^{−1}B(W)g).

If W, W′ are adjacent chambers, so that W = W (ω) and W′ = W (ω(α)ω) for some ω ∈ Ω and simple root α of T in B then by definition (B(W), B(W′)) lies in the closure of the orbit of (B_{T} , B_{T}^{ω(α)}) in B^{2}.

The standard star s_{0} is given by B(W (ω)) = B_{T}^{ω}, ω ∈ Ω. A star is regular if it lies in the G-orbit of s_{0}. The F -rational regular stars form the orbit of s_{0} under 𝔄(T) = \{g ∈ G(\overline{F}) : gσ(g^{−1}) ∈ T \}.

Let B_{∞} be opposite to B relative to T. Then S(B_{∞}) consists of all (B(W)) for which each B(W) is opposite to B_{∞}, and S(B_{∞}, B) consists of those stars for which we also have B(W_{+}) = B. If B_{∞} = TN_{∞} then the morphism

(n, (B(W))) → (B(W))^{n}

from N_{∞} × S(B_{∞}, B) to S(B_{∞}) allows us to identify these two varieties.

If W is a chamber and β a W -simple root the coordinate function z(W, β), or z(ω, α), is defined on S(B_{∞}, B). Here ω ∈ Ω and B-simple α are given by W = W (ω) and ωβ = α. The chamber W′ = W (ω(α)ω) is adjacent to W and there is a unique h ∈ N_{∞} such that

hB(W)h^{−1} = B,

and then

hB(W′)h^{−1} = exp(−zX_{−α})Bexp zX_{−α},

where z = z(W, β) = z(ω, α) lies in \overline{F}. We have, for F -rational (B(W)),

(5.2.1)  σ(z(ω, α)) = z(σωσ_{T}^{−1}, σα),  σ ∈ Γ.

If g ∈ G and s = (B(W)) is a star we write g ∈ s if g ∈ ∩_{W} B(W). Let X^{0} consist of all pairs (g, s), where g is regular semisimple, s is regular and g ∈ s, and let X be the closure of X^{0} in G × S. Both X^{0} and X are defined over F, and X is contained in {(g, s) : g ∈ s}. Thus ξ : (g, (B(W))) → (g, B(W_{+})) is a well-defined morphism from X to the Springer-Grothendieck variety M = {(g, B) : g ∈ B}. Let π_{M} : M → G be the projection onto the first factor and ϕ_{M} : M → T be defined by ϕ_{M}(g, B) = γ if h^{−1}gh ≡ γ mod N_{T} where B^{h} = B_{T} and N_{T} is the unipotent radical of B_{T}. Both π_{M} and ϕ_{M} are smooth and π_{M} is proper. Set π = π_{M} ∘ ξ and ϕ = ϕ_{M} ∘ ξ. Then both π and ϕ are defined over F, and ϕ is smooth and proper.

Set M^{0} = π_{M}^{−1}(G_{regss}). Then ξ : X^{0} → M^{0} is an isomorphism. If γ ∈ T is regular then ϕ^{−1}(γ) is the G-orbit of (γ, s_{0}) and so may be identified with the conjugacy class of γ in G. If also γ is F -rational then ϕ^{−1}(γ)(F) is identified with the stable conjugacy class of γ in G(F). (5.3) Regular Unipotent Elements and X Suppose u ∈ G is regular unipotent and contained in the Borel subgroup B_{u}. Then we define the star s_{u} by B(W) = B_{u}, W ∈ 𝔚, and set x_{u} = (u, s_{u}). Lemma 5.3.A.

(i) x_{u} lies in X.

(ii) If u ∈ G(F) then x_{u} ∈ X(F). (iii) ξ : X → M is invertible at x_{u}. Proof. A point in S(B_{∞}) may be written s^{h}, where s ∈ S(B_{∞}, B) and h ∈ N_{∞}. Suppose g ∈ s^{h}. Then we write g = h^{−1}tnh, where t ∈ T, n ∈ N. We calculate the z-coordinates of s in terms of t, n as follows.

Let s = (B(W)). If α is B-simple then z = z(W_{+}, α) is the solution to

B(ω(α)W_{+}) = exp(−zX_{−α})Bexp zX_{−α}.

On the other hand, write tn as

tn = t exp(x_{α}(n)X_{α})n′

where n′ ∈ N^{α}. The condition that

t exp(x_{α}(n)X_{α})n′ ∈ B(ω(α)W_{+})

is the requirement in SL(2) that

[ \begin{matrix}1 & 0 \\ z & 1 \end{matrix} ] [ \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ] [ \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 0 \\ −z & 1 \end{matrix} ] = [ \begin{matrix}a_{1} & x_{1} \\ y_{1} & b_{1} \end{matrix} ]

be upper triangular, where x = x_{α}(n), z = z(W_{+}, α) and α(t) = a^{2}.

If s is regular then z ≠ 0 and

xz = x_{α}(n)z(W_{+}, α) = 1 − a^{−2} = 1 − α(t)^{−1}.

This equation continues to hold on X. Observe also that a_{1} is then a^{−1} and x_{1} is a^{2}x. More generally, suppose W = W (ω) and ωβ = α. Then: Lemma 5.3.B. If n is regular then

z(W, β) = \frac{1 − β(t)^{−1}}{x(W, β)} ,

where x(W, β) is a rational function of tn which is defined and equal to x_{α}(n) at t = 1. Proof. Consider now z = z(ω(α_{1})W_{+}, ω(α_{1})α_{0}), where α_{0}, α_{1} are B-simple. Let z_{1} be the coordinate z(W_{+}, α_{1}) of s, so that

exp z_{1}X_{−α_{1}}B(ω(α_{1})W_{+}) exp(−z_{1}X_{−α_{1}}) = B

and

exp z_{1}X_{−α_{1}}B(ω(α_{1})ω(α_{0})W_{+}) exp(−z_{1}X_{−α_{1}}) = exp(−zX_{−α_{0}})Bexp zX_{−α_{0}}.

Thus z is the coordinate z(W_{+}, α_{0}) for the star

B_{1}(W) = exp z_{1}X_{−α_{1}}B(ω(α_{1})W) exp(−z_{1}X_{−α_{1}}), W ∈ 𝔚,

which is regular if s is. Replace tn by t_{1}n_{1} = exp z_{1}X_{−α_{1}}tn exp(−z_{1}X_{−α_{1}}).

Then if x_{α_{0}}(n_{1}) ≠ 0 we get

z(ω(α_{1})W_{+}, ω(α_{1})α_{0})\frac{1 − α_{0}(t_{1})^{−1}}{x_{α_{0}}(n_{1})} .

The earlier SL(2) calculation shows that t_{1} = ω(α_{1})(t) and also that if α_{0} = α_{1} then x_{α_{0}}(n_{1}) = α_{0}(t)x_{α_{0}}(n). Otherwise x_{α_{0}}(n_{1}) is a more complicated function of t and a but it is equal to x_{α_{0}}(n) at t = 1, for then z_{1} = 0. Thus the lemma is verified in the case of z(ω(α_{1})W_{+}, ω(α_{1})α_{0}). We repeat this procedure to obtain the lemma in general.

From this lemma we deduce immediately Lemma 5.3.A. For (h^{−1}tnh, s) ∈ X the star s is a rational function of t, n and h that is defined in a neighborhood of t = 1, n = u, where it takes the value s_{u}. Corollary 5.3.C. ϕ : X → T is smooth at x_{u}. (5.4) Orbital Integrals as Fiber Integrals From now on we assume F local and view ∑_{γ_{G}} Δ(γ_{H}, γ_{G})Φ(γ_{G}, f), f ∈ C_{c}^{∞}(G(F)), as a fiber integral on X(F). Fix for once and for all an admissible embedding T_{H} → T of T_{H} in G, which is quasi-split over F, along with the Borel subgroup B_{T} containing T. If γ is the image of γ_{H} under T_{H} → T then

γ → ∑ Δ(γ_{H}, γ_{G})ϕ(γ_{G}, f)

is a function on the regular elements of T (F). Its value at γ is an integral over the fiber ϕ^{−1}(γ)(F) in X(F) as follows.

The forms ω_{G}, ω_{T} on G, T and measures |ω_{G}|, |ω_{T} | have been specified in (1.4). To ω_{G} there is attached a G-invariant form ω_{M} of highest degree on M nowhere vanishing on M^{0} and hence, after transport by ξ, a G-invariant form ω_{X} of highest degree on X nowhere vanishing on X^{0} [L3, Lemma 2.8]. We embed the variety U_{reg} of regular unipotent elements in G as an open subvariety of ϕ^{−1}(1) under u → x_{u}. More generally if z ∈ Z we embed zU_{reg} as an open subvariety of ϕ^{−1}(z). Then ϕ is smooth at the points in zU_{reg} and we see easily that ω_{X} is nonvanishing around zU_{reg}. If γ ∈ T (F) is regular then the quotient of ω_{X} by ϕ^{∗}(ω_{T}) defines a G-invariant form ω_{γ} of highest degree along ϕ^{−1}(γ). For z ∈ Z we similarly obtain a G-invariant form ω_{z} of highest degree along zU_{reg}. For γ and z F -rational the measures |ω_{γ}| on ϕ^{−1}(γ)(F) and |ω_{z}| on Z_{reg}(F) are specified in the manner of (1.4). Recalling the definition of ω_{T} in (1.4) we note that |ω_{z}| is independent of T. On the other hand, by [L3, Lemma 2.12],

|ω_{γ}| = ∏_{α} |1 − α(γ)^{−1}| · |ω_{G}|/|ω_{T} |,

where the product is over roots α of T in B_{T}. We replace |ω_{γ}| by

|ω_{γ}^{∗}| = ∏_{α} |α(γ)|^{1/2}|ω_{γ}|

to obtain

|ω_{γ}^{∗}| = D_{G}(γ)|ω_{G}|/|ω_{T} |.

Then for γ_{H} strongly G-regular we may write

D_{H}(γ_{H}) ∑_{γ_{G}} Δ(γ_{H}, γ_{G})Φ(γ_{G}, f)

as

∫_{ϕ^{−1}(γ)(F)} Δ(x)f(π(x))|ω_{γ}^{∗}|,

where

Δ(x) = Δ(γ_{H}, π(x))D_{H}(γ_{H})D_{G}(γ)^{−1}.

With suitable conventions there is a similar formula in the case γ is not strongly regular, but we shall not need this fact.

Since G is quasi-split over F we have defined Δ(γ_{H}, γ_{G}) as

\frac{Δ(\overline{γ}_{H}, \overline{γ}_{G})}{Δ_{0}(\overline{γ}_{H}, \overline{γ}_{G})}Δ_{0}(γ_{H}, γ_{G}),

where \overline{γ}_{H}, \overline{γ}_{G} are fixed and

Δ_{0} = Δ_{I}Δ_{II}Δ_{1}Δ_{2}Δ_{IV} .

But Δ_{IV} (γ_{H}, γ_{G}) = D_{G}(γ)D_{H}(γ_{H})^{−1} and Δ_{I}(γ_{H}, γ_{G}), Δ_{II}(γ_{H}, γ_{G}), Δ_{2}(γ_{H}, γ_{G}) depend only on γ. Thus Δ_{1}(x) = Δ_{1}(γ_{H}, π(x)) alone varies along the fiber ϕ^{−1}(γ)(F).

Write x as (γ_{G}, s) and assume s ∈ S(B_{∞}). To specify Δ_{1}(x), we choose g ∈ G such that

(γ_{G}, s) = (γ, s_{0})^{g},

where s_{0} is the standard star. Then inv(γ_{H}, γ_{G}) is the class of the cocycle σ → gσ(g)^{−1} in H^{1}(T) and

Δ_{1}(x) = ⟨inv(γ_{H}, γ_{G})^{−1}, s_{T} ⟩

[see (3.4)]. More precisely, we should pass to G_{sc} to define inv(γ_{H}, γ_{G}). We do so without change in notation. Proposition 5.2 of [L3] describes the inverse cocycle σ(g)g^{−1} in terms of coordinates. We recall this next.

The cocycle λ(T) from (2.3) will be computed relative to a-data {a_{α}} and the splitting opposite to (B, T, {X_{α}}). Also p will denote the gauge on the roots of T in G attached to B_{T}, and ∏_{1,σ}^{p} will indicate a product over roots α such that p(α) = 1 and p(σ_{T}^{−1}α) = −1. Suppose ω(α_{1}) … ω(α_{r}) is a reduced expression for ω_{T} (σ), where σ_{T} = ω_{T} (σ) ⋊ σ. Set ω_{0} = 1 and ω_{k} = ω(α_{1}) … ω(α_{k}), 1 ≦ k ≦ r. If p(α) = 1 and p(σ_{T}^{−1}α) = −1 then α = ω_{k−1}(α_{k}), some 1 ≦ k ≦ r, and we may set

z(σ, α) = z(−ω_{k−1}W_{+}, −α).

Note that −ω_{k−1}W_{+} = ω_{k−1}ω_{−}W_{+} and

−α = ω_{k−1}ω_{−}\overline{α}_{k},

where ω_{−} ∈ Ω maps W_{+} to −W_{+} and \overline{α}_{k} = −ω_{−}α_{k}. Lemma 5.4.A. inv(γ_{H}, γ_{G})^{−1} is represented by the cocycle

σ → λ(T)^{−1} ∏_{1,σ}^{p} (\frac{−a_{α}}{z(σ, α)})^{α^{∨}} .

Proof. This formula has just to be reconciled with that of [L3, Proposition 5.2]. We have used h, where (B_{T} , T)^{h} = (B, T), to identify roots of T with roots of T. Suppose that (B_{T} , T)^{h_{1}} = (B_{∞}, T) and ω_{∞}(σ) = ω_{−}ω_{T} (σ)ω_{−}. Then

ω_{∞}(σ) = ω(\overline{α}_{1}) … ω(\overline{α}_{r})

is a reduced expression. We define n(ω_{∞}(σ)) and n(ω_{∞}(σ)^{−1}) as in (2.1), but relative to the splitting opposite to (B, T, {X_{α}}). Thus

n(ω_{∞}(σ)^{−1}) = n(−\overline{α}_{r}) … n(−\overline{α}_{1}).

By Lemma 2.1.A

n(ω_{∞}(σ)^{−1}) = n(ω_{∞}(σ))^{−1} ∏_{∞}(−1)^{α^{∨}},

where ∏_{∞} indicates a product over roots α such that both α and −ω_{∞}(σ)^{−1}α are positive for B_{∞}. Note that h_{1}(∏_{∞}(−1)^{α^{∨}})h_{1}^{−1} = ∏_{1,σ}^{p}(−1)^{α^{∨}}. Applying (5.2.1) we rewrite the formula of [L3] as

σ(h_{1})n(ω_{∞}(σ))^{−1} ∏_{∞}(−1)^{α^{∨}}h_{1}^{−1}

times

h_{1} (∏_{k=1}^{r} z(σ, ω_{k−1}(α_{k}))^{ω(\overline{α}_{1})...ω(\overline{α}_{k−1})\overline{α}_{k}^{∨}}) h_{1}^{−1}.

(To make the comparison with [L3] easier we note that h_{1} ↔ hw_{−}^{−1}, that n(w_{∞}(σ)^{−1}) ↔ w_{a_{j}}…w_{α_{1}}, and that all u_{α_{i}} are 1. We recall from (5.2.1) that

σ(z(w, α)) = z(σωσ_{T}^{−1}, σα).)

Since λ(T) : σ → h_{1}(∏_{∞} a_{α}^{α^{∨}})n(ω_{∞}(σ))σ(h_{1}^{−1}), we obtain

λ(T)^{−1} ∏_{1,σ}^{p} a_{α}^{α^{∨}} · ∏_{1,σ}^{p}(−1)^{α^{∨}} ∏_{1,σ}^{p} z(σ, α)^{−α^{∨}},

as desired, because Int h_{1} takes the root \overline{α} of T to the root −α of T. Lemma 5.4.B. If γ_{G} = n_{1}^{−1}tnn_{1} where n ∈ N is regular and n_{1} ∈ N_{∞} then z(σ, α) = \frac{1−α(γ)}{x(σ,α)}, where x(σ, α) → x_{\overline{α}_{k}}(n) as γ → 1. Proof. ϕ(γ_{G}, s) = γ implies that h^{−1}γh = t and we have only to apply Lemma 5.3.B to z(−ω_{k−1}W_{+}, −α).

The constructions and results of (5.3) and (5.4) are described for SL(2) in [L-S]. (5.5) A Limit Formula We assume that H_{1} is a central extension of H as constructed in (4.4). The character λ^{G_{1}} of Lemma 4.4.A defines characters λ on Z_{1}(F), the kernel of H_{1}(F) → H(F), and λ^{G} on Z_{1}^{G}(F), the preimage of the center Z(F) of G(F) under H_{1}(F) → H(F). Recall that Z(F) is canonically embedded in H(F). The projection of Z_{1}^{G}(F) onto Z(F) will be written as z_{1} → z, and γ_{1} will be an element of H_{1}(F) with strongly G-regular image γ_{H} in H(F). Theorem 5.5.A. lim_{γ_{1}→z_{1}} D_{H_{1}}(γ_{1}) ∑_{γ_{G}} Δ(γ_{1}, γ_{G})Φ(γ_{G}, f) is equal to

λ^{G}(z_{1}) ∑_{u} Δ(u)Φ(zu, f), f ∈ C_{c}^{∞}(G(F)),

where ∑_{u} indicates summation over representatives u for the G(F)-conjugacy classes of regular unipotent elements in G(F), and

Φ(zu, f) = ∫ f|ω_{z}|,

the integral being taken over the conjugacy class of zu, an open subset of zU_{reg}(F). Corollary 5.5.B. If f ∈ C_{c}^{∞}(G(F)) and f^{1} ∈ C_{c}^{∞}(H_{1}(F), λ) have Δ-matching orbital integrals then

∑_{u_{1}} Φ(z_{1}u_{1}, f^{1}) = λ^{G}(z_{1}) ∑_{u} Δ(u)Φ(zu, f).

Proof of the Theorem. Because Δ(z_{1}γ_{1}, zγ_{G}) = λ^{G}(z_{1})Δ(γ_{1}, γ_{G}) we reduce immediately to the case z_{1} = 1. Replacing G by G_{1} we may assume that H_{1} = H. By known properties of the asymptotic behavior of orbital integrals it suffices to consider a function f supported in a small neighborhood of a regular unipotent element in G(F).

Suppose that γ is strongly regular, and let x = (γ_{G}, s) lie in ϕ^{−1}(γ)(F), with x in the coordinate patch S(B_{∞}). We write x as (tn, s_{1})^{n_{1}}, where n_{1} ∈ B_{∞}, s_{1} ∈ S(B_{∞}, B) and t = h^{−1}γh, and assume n ∈ N is regular. Then all that has to be shown is that as γ → 1 we have Δ(x) → Δ(n). From (5.4) we have

Δ(x) = \frac{Δ(\overline{γ}_{H}, \overline{γ}_{G})}{Δ_{0}(\overline{γ}_{H}, \overline{γ}_{G})} · Δ_{1}(γ_{H}, γ_{G}) · Δ_{II}(γ_{H}, γ_{G}) · Δ_{1}(γ_{H}, γ_{G}) · Δ_{2}(γ_{H}, γ_{G}).

Both Δ_{0}(\overline{γ}_{H}, \overline{γ}_{G}) and Δ_{I}(γ_{H}, γ_{G}) are to be computed relative to the same F -splitting, that opposite to (B, T, {X_{α}}). Nothing else depends on a splitting.

The quotient Δ(\overline{γ}_{H}, \overline{γ}_{G})/Δ_{0}(\overline{γ}_{H}, \overline{γ}_{G}) is a constant which also appears in Δ(n). By definition

Δ_{I}(γ_{H}, γ_{G}) = ⟨λ(T), s_{T} ⟩.

Also

Δ_{II}(γ_{H}, γ_{G}) = ∏_{α} χ_{α} (\frac{α(γ) − 1}{a_{α}}) ,

where the product is over representatives α for the Γ-orbits of roots of T outside H. If O is an asymmetric orbit then the contribution for ±O is a character (which we could take to be trivial) evaluated at γ (Lemma 3.3.A). Thus we need take into account only the symmetric orbits. It remains to consider Δ_{1}(γ_{H}, γ_{G}), for Δ_{2}(γ_{H}, γ_{G}) is a character evaluated at γ and so has limit 1. But

Δ_{1}(γ_{H}, γ_{G}) = ⟨inv(γ_{H}, γ_{G})^{−1}, s_{T} ⟩

and inv(γ_{H}, γ_{G})^{−1} is represented by the cocycle

λ(T)^{−1} ∏_{1,σ}^{p} (\frac{−a_{λ}}{z(σ, α)})^{α^{∨}}

of Lemma 5.4.A. On cancellation with Δ_{I} we may replace inv(γ_{H}, γ_{G})^{−1} by the class of the cocycle

∏_{1,σ}^{p} (\frac{−a_{α}}{z(σ, α)})^{α^{∨}} = ∏_{1,σ}^{p} (\frac{a_{α}x(σ, α)}{α(γ) − 1})^{α^{∨}}

(see Lemma 5.4.B). But around 1, α(γ)^{1/2} is well defined and continuous. Thus we rewrite the cocycle as

∏_{1,σ}^{p} (\frac{a_{α}}{α(γ)^{1/2} − α(γ)^{−1/2}} )^{α^{∨}} ∏_{1,σ}^{p} (\frac{x(σ, α)}{α(γ)^{1/2}} )^{α^{∨}} .

Since a_{α}/(α(γ)^{1/2} − α(γ)^{−1/2}) lies in F_{±α}^{×} the first product is a cocycle (Lemma 2.2.B). The pairing of this cocycle with s_{T} yields

∏_{α} χ_{α} (\frac{a_{α}}{α(γ)^{1/2} − α(γ)^{−1/2}} ) ,

where the product is taken over representatives α for the symmetric Γ-orbits of roots outside H (Lemma 3.2.D). The product of this term with the contribution of the symmetric orbits to Δ_{II} is then:

∏_{α} χ_{α} (\frac{α(γ) − 1}{α(γ)^{1/2} − α(γ)^{−1/2}} ) = ∏_{α} χ_{α}(α(γ)^{1/2}),

and so approaches 1 as γ → 1. We conclude that

lim_{γ→1} Δ(x) = \frac{Δ(\overline{γ}_{H}, \overline{γ}_{G})}{Δ_{0}(\overline{γ}_{H}, \overline{γ}_{G})} lim_{γ→1}⟨C(γ), s_{T} ⟩,

where C(γ) is represented by the cocycle

∏_{1,σ}^{p}(x(σ, α)α(γ)^{−1/2})^{α^{∨}}.

But lim_{γ→1} x(σ, α)α(γ)^{−1/2} = x_{\overline{α}_{k}}(n) (Lemma 5.4.B). Thus it remains to show that ∏_{1,σ}^{p} x_{\overline{α}_{k}}(n)^{α^{∨}} represents the class inv_{T} (n) defined in (5.1).

We shall pass to G_{sc} as necessary, without change in notation. Choose t ∈ T such that

α(t) = x_{α}(n)

for all simple roots α. Then inv(n) is the class of σ → tσ(t)^{−1} in H^{1}(Z). Its image inv_{T} (n) in H^{1}(T) is, after transport to T, the class of this same cocycle but now for the σ_{T} -action on T.

Set t_{1} = ω_{−}(t^{−1}), and note that

α(t_{1}) = x\overline{α}(n)

for all simple α. We have also that

σ(t_{1})t_{1}^{−1} = ω_{−}(σt^{−1})ω_{−}(t) = ω_{−}(tσ(t^{−1})) = tσ(t^{−1})

since this last element is central. Thus it is enough to show that:

∏_{1,σ}^{p} x_{\overline{α}_{k}}(n)^{α^{∨}}is cohomologous to σ(t_{1})t_{1}^{−1}.

But ∏_{1,σ}^{p} x_{\overline{α}_{k}}(n)^{α^{∨}} = ∏_{k=1}^{n} x_{\overline{α}_{k}}(n)^{ω_{k−1}(α_{k}^{∨})}. For this, recall that ω_{T} (σ) = ω(α_{1}) … ω(α_{r}) is a reduced expression and ω_{0} = 1, ω_{k} = ω(α_{1}) … ω(α_{k}), 1 ≦ k ≦ r. Hence

∏_{1,σ}^{p} x_{\overline{α}_{k}}(n)^{α^{∨}} = ∏_{k=1}^{n} α_{k}(t_{1})^{ω_{k−1}(α_{k}^{∨})} = t_{1}ω(α_{1}) … ω(α_{r})(t_{1}^{−1}) = t_{1}ω_{r}(σ)(t_{1}^{−1}).

Since σt_{1} ≡ t_{1}(mod Z), this equals

σ(t_{1})ω_{T} (σ)(σt_{1}^{−1}) = σ(t_{1})σ_{T} (t_{1}^{−1}) = σ(t_{1})t_{1}^{−1} · t_{1}σ_{T} (t_{1}^{−1}),

and we are done.

6. Global consequences (6.1) Outline Here the results take a simple form. To explain them we continue the example of SL(2) from (1.1). Now F is a number field with adele ring A. The global matching concerns

(6.1.1)  ∑_{γ_{G}} Δ_{A}(γ_{H}, γ_{G})Φ_{A}(γ_{G}, f),

where γ_{H} ≠ ±1 lies in H(F) and is an adelic image of γ_{G} ∈ G(A), Φ_{A}(γ_{G}, f) is the integral of a function f on G(A) along the G(A)-conjugacy class of γ_{G}, and the sum is over representatives γ_{G} for G(A)-conjugacy classes. The factor Δ_{A} is prescribed as follows.

To say that γ_{H} is an adelic image of γ_{G} we mean, in this example, that γ_{G} is everywhere locally stably conjugate to the image γ of γ_{H} under some admissible embedding H → T of H in G defined over F. If γ_{G,v} is the component of γ_{G} at the place v then the element inv(γ_{H}, γ_{G,v}) of H^{1}(Γ_{v}, T (E_{v})) is defined as in (III_{1}) of (1.1). For almost all v it is trivial and the image inv(γ_{G}) of ∑_{v} inv(γ_{H}, γ_{G,v}) in the 2-element group H^{1}(Γ, T (E)\T (A_{E})) is independent of the choice of γ. Moreover, inv(γ_{G}) is trivial if and only if the G(A)-conjugacy class of γ_{G} meets G(F). Tate-Nakayama duality allows us to pair inv(γ_{G}) with the image s_{T} in T̂^{Γ} of the endoscopic datum s, that is, with the nontrivial element of T̂^{Γ}. Then we define

Δ_{A}(γ_{H}, γ_{G}) = ⟨inv(γ_{G}), s_{T} ⟩.

Thus Δ_{A}(γ_{H}, γ_{G}) = 1 if the G(A)-conjugacy class of γ_{G} meets G(F) and Δ_{A}(γ_{H}, γ_{G}) = −1 otherwise.

We use unnormalized Tamagawa measures to specify orbital integrals. The function f is to be of the form ∏_{v} f_{v} where f_{v} ∈ C_{c}^{∞}(G(F_{v})) for all v, and for almost all v the function f_{v} is to be the characteristic function of K_{v} = G(O_{v}) divided by the measure of K_{v}. Then Φ_{A}(γ_{G}, f) = ∏_{v} Φ(γ_{G,v}, f_{v}). The local factor Δ(γ_{H}, γ_{G,v}) was defined in (1.1). Inspection of the terms shows that

(6.1.2)  Δ(γ_{H}, γ_{G,v}) = 1 for almost all v

and

(6.1.3)  ∏_{v} Δ(γ_{H}, γ_{G,v}) = Δ_{A}(γ_{H}, γ_{G})

provided the fixed elements \overline{γ}_{H}, \overline{γ}_{G} are F -rational and at each place v the otherwise arbitrary Δ(\overline{γ}_{H}, \overline{γ}_{G}) is chosen so that (6.1.2) and (6.1.3) are satisfied.

Then

∑_{γ_{G}} Δ_{A}(γ_{H}, γ_{G})Φ_{A}(γ_{G}, f) = ∑_{γ_{G}} ∏_{v} Δ(γ_{H}, γ_{G,v})Φ(γ_{G,v}, f_{v})

= ∏_{v} ∑_{γ_{G,v}} Δ(γ_{H}, γ_{G,v})Φ(γ_{G,v}, f_{v})

by [L2, Lemma 8.3]. This equals ∏_{v} f_{v}^{H}(γ_{H}) where f_{v}^{H} is the smooth extension of

γ_{H} → ∑_{γ_{G,v}} Δ(γ_{H}, γ_{G,v})Φ(γ_{G,v}, f_{v})

to H(F_{v}) (see Lemma 1.1.A). For almost all v, f_{v}^{H} is the characteristic function of the maximal compact subgroup of H(F_{v}) divided by its measure.

We set f^{H} = ∏_{v} f_{v}^{H} to conclude that

f^{H}(γ_{H}) = ∑_{γ_{G}} Δ_{A}(γ_{H}, γ_{G})Φ_{A}(γ_{G}, f).

This is the global matching of orbital integrals for our example (see [L-L, p. 756]).

In general we shall define an adelic factor Δ_{A}, and verify (6.1.2) and the product formula (6.1.3) for the local factors of Sect. 3. Then suppose that f = ∏_{v} f_{v}, f^{H} = ∏_{v} f_{v}^{H} are as usual, and that strongly G-regular γ_{H} ∈ H(F) is an adelic image of γ_{G} ∈ G(A). Because of (6.1.2) and [L2, Lemma 8.3] we have for almost all v that

Φ^{st}(γ_{H}, f_{v}^{H}) = ∑_{γ_{G,v}} Δ(γ_{H}, γ_{G,v})Φ(γ_{G,v}, f_{v}),

each side having non-zero contribution from only one conjugacy class. Thus if f_{v} and f_{v}^{H} have Δ-matching orbital integrals for all v we conclude that

(6.1.4)  ΦA^{st}(γ_{H}, f^{H}) = ∑_{γ_{G}} Δ_{A}(γ_{H}, γ_{G})Φ_{A}(γ_{G}, f),

where the left side is, by definition, the sum of the integrals of f^{H} along the H(A)-conjugacy classes of elements everywhere locally stably conjugate to γ_{H}. With a little care this extends to elements γ_{H} which are G-regular but not strongly G-regular (recall (4.3) for the local analogue).

Finally we shall observe that the Global Hypothesis of [L2] is satisfied. Thus, assuming the Hasse Principle for G_{sc}, the factors of Sect. 3 will be correct for stabilization of the Arthur-Selberg Trace Formula (see [L2, Chap. VIII]). (6.2) Notation If v is a place of F then we fix an extension v of v to \overline{F} and for L ⊆ F denote by L_{v} the completion of L so determined. There will be no harm in working with some suitably large finite Galois extension L ⊂ \overline{F} of F. Thus Γ = Gal(L/F) and Γ_{v} = Gal(L_{v}/F_{v}). Set W = W_{L/F} and W_{v} = W_{L_{v}/F_{v}}. Then we fix W_{v} → W such that

1 → L_{v}^{×} → W_{v} → Γ_{v} → 1
↓ ↓ ↓
1 → C_{L} → W → Γ → 1

is commutative.

Global endoscopic data (H, H, s, ξ) yield data (H, H_{v}, ξ_{v}) for G as group over F_{v} such that each of

1 → Ĥ → H_{v} → W_{v} → 1
↓ ↓ ↓
1 → Ĥ → H → W → 1

and

\begin{matrix}H_{v} ⟶^{ξ_{v}} & ^{L}G_{v} \\ ↓ & ↓ \\ H ⟶^{ξ} & ^{L}G \end{matrix}

is commutative, where ^{L}G_{v} is the semidirect product Ĝ ⋊ W_{v} with W_{v} acting through W_{v} → W ⟶^{ϱ} Aut Ĝ.

Again H need not be an L-group and we introduce central extensions of H satisfying global analogues of the conditions of (4.4) (see [L1, K1]). In the following discussion we assume that H itself satisfies these conditions and identify H with ^{L}H. Passage to the general case is then routine, following (4.4) for the local case.

Let K = ∏_{v} K_{v}, K^{∗} = ∏_{v} K_{v}^{∗} be compact open subgroups of G(A_{f}), G^{∗}(A_{f}). We fix a finite set V_{0} of places such that for v ∉ V_{0} the groups G, G^{∗} are unramified at v and K_{v}, K_{v}^{∗} are hyperspecial maximal compact subgroups of G(F_{v}), G^{∗}(F_{v}). Outside V_{0}, ψ is defined over F_{v}, up to composition with an inner automorphism which does not affect transfer factors. Choose g_{v} ∈ G(\overline{F}_{v}) such that ψ_{v} = Int g_{v} ∘ ψ is defined over F_{v} and takes K_{v} to K_{v}^{∗}. Then ψ_{v} may be used to identify G(F_{v}) with G^{∗}(F_{v}).

(6.3) Adelic Images and Transfer Factors Suppose γ_{H} ∈ H(F) is G-regular and lies in the maximal torus T_{H} of H. At each place v we shall allow only admissible embeddings of T_{H} in G^{∗} which are defined over F.

These exist, by Steinberg’s Theorem (see [K1]). Then we say that γ_{H} is an adelic image of γ_{G} ∈ G(A) if, for every v, γ_{H} is an image of the component γ_{G,v} of γ_{G} in G(F_{v}). Thus if T_{H} maps to T and γ_{H} to γ ∈ T (F) under some admissible embedding over F then we require that for each v there exists x_{v} ∈ G^{∗}(L_{v}) such that Int x_{v} ∘ ψ maps the maximal torus in T containing γ_{G,v} to T over F_{v} and carries γ_{G,v} to γ.

Suppose γ_{H}, \overline{γ}_{H} are strongly G-regular elements of H(F) and are adelic images of γ_{G}, \overline{γ}_{G} ∈ G(A). For an L splitting T the element

μ_{v} = inv (\frac{γ_{H}, γ_{G,v}}{\overline{γ}_{H}, \overline{γ}_{G,v}} )

of H^{1}(Γ_{v}, U(L_{v})) was defined in (3.4). Lemma 6.3.A. μ_{v} = 1 for almost all v. Proof. Take an L splitting T. For almost all v ∉ V_{0} : (i) L is unramified at v; (ii) γ, γ_{G,v} lie in K_{v}; and (iii) for each root α of T in G^{∗}, α(γ) is a unit in the ring of integers of L_{v}. That γ_{H} is an image of γ_{G,v} means, for v ∉ V_{0}, that γ_{G,v} and γ are stably conjugate. If (i), (ii), and (iii) are satisfied γ_{G,v} and γ are conjugate under K_{v} ([L2, Lemma 8.3]). If K_{v} is the stabilizer of the hyperspecial point x of the Bruhat-Tits building of G(F_{v}) we denote by K_{sc} the stabilizer of x in G_{sc}(F_{v}) and by K_{sc,L} the stabilizer in G_{sc}(L_{v}). By [K3, (3.3.4)], γ_{G,v} and γ are conjugate under K_{sc,L}. Since H^{1}(Γ_{v}, T_{sc} ∩ K_{sc,L}) = 1 for almost all v and γ_{G,v}, γ are strongly regular we conclude that γ_{G,v}, γ are conjugate under K_{sc} for almost all v. Then the class inv(γ_{H}, γ_{G,v}) in H^{1}(Γ_{v}, T_{sc}(L_{v})) is trivial.

We argue similarly for \overline{γ}_{H}, and \overline{γ}_{G} to obtain for almost all v ∉ V_{0} that

μ_{v} = inv (\frac{γ_{H}, γ_{G,v}}{\overline{γ}_{H}, \overline{γ}_{G,v}} ) = \frac{inv(\overline{γ}_{H}, \overline{γ}_{G,v})}{inv(γ_{H}, γ_{G,v})} = 1,

and the lemma is proved.

Let μ be the image of Σ_{v}μ_{v} in H^{1}(Γ, U(L)\U(A_{L})) under

∑_{v} H^{1}(Γ_{v}, U(L_{v})) → H^{1}(Γ, U(L)\U(A_{L}))

given by 1 → U(L) → U(A_{L}) → U(L)\U(A_{L}) → 1 and the isomorphism Σ_{v}H^{1}(Γ_{v}, U(L_{v})) → H^{1}(Γ, U(A_{L})). The endoscopic datum s determines s_{U,v} ∈ π_{0}(Û^{Γ_{v}}) as in (3.4). By its definition s also determines s_{T} ∈ π_{0}(T̂_{ad}^{Γ}) and similarly s\overline{T} [see (1.2), (3.1)]. As in (3.4) we may define s_{U} ∈ π_{0}(Û^{Γ}) which depends only on the choice of embeddings T_{H} → T, \overline{T}_{H} → \overline{T}. Global Tate-Nakayama duality allows us to pair μ with s_{U} and the local-global relationship for the pairing yields:

(6.3.1)  ⟨μ, s_{U}⟩ = ∏_{v}⟨μ_{v}, s_{U,v}⟩.

There is another way to define μ. Strongly G-regular γ_{H} is an adelic image of γ_{G} ∈ G(A) if and only if there exists h ∈ G_{sc}^{∗}(A_{L}) such that

hψ(γ_{G})h^{−1} = γ

[see the proof of (6.3.A)]. We proceed as in the local case (3.4). Recall that ψσ(ψ)^{−1} = Int u(σ), u(σ) ∈ G_{sc}^{∗}(L), and then v(σ) = hu(σ)σ(h)^{−1} lies in T_{sc}, with ∂v = ∂u taking values in T_{sc}(L). Thus v(σ) defines an element μ_{T} of H^{1}(Γ, T_{sc}(L)\T_{sc}(A_{L})). By global Tate-Nakayama duality we may pair μ_{T} with s_{T}. Further, ⟨μ_{T} , s_{T} ⟩ is the independent of the choice of admissible embedding T_{H} → T over F, and clearly

(6.3.2)  ⟨μ, s_{U}⟩ = ⟨μ\overline{T} , s\overline{T} ⟩/⟨μ_{T} , s_{T} ⟩.

It will be more convenient to write ⟨μ_{T} , s_{T} ⟩ as d(γ_{H}, γ_{G}). Lemma 6.3.B.

(i) d(γ′_{H}, γ_{G}) = d(γ_{H}, γ_{G}) if γ′_{H} is stably conjugate to γ_{H} in H(F).

(ii) d(γ_{H}, γ′_{G}) = d(γ_{H}, γ_{G}) if γ′_{G} is G(A)-conjugate to γ_{G}. (iii) d(γ_{H}, γ_{G}) = d(\overline{γ}_{H}, \overline{γ}_{G}) if γ_{H}, \overline{γ}_{H} are adelic images of γ_{G}, \overline{γ}_{G} ∈ G(F). Proof. (i) is immediate. For (ii) we use (6.3.2) and then (6.3.1) to reduce to the proof of Lemma 4.1.C for the local case. For (iii), μ is trivial if γ_{G}, \overline{γ}_{G} ∈ G(F) for then we find h, \overline{h} ∈ G_{sc}^{∗}(L) such that hψ(γ_{G})h^{−1} = γ, \overline{h}ψ(\overline{γ}_{G})\overline{h}^{−1} = \overline{γ}. (6.3.2) now yields the result.

Fix strongly G-regular \overline{γ}_{H} ∈ H(F) and \overline{γ}_{G} ∈ G(F) such that \overline{γ}_{H} is an adelic image of \overline{γ}_{G}. We assume that such a pair \overline{γ}_{H}, \overline{γ}_{G} exists; otherwise all the following factors are to be zero. Definition. For all strongly G-regular γ_{H} ∈ H(F),

Δ_{A}(γ_{H}, γ_{G}) = d(\overline{γ}_{H}, \overline{γ}_{G})/d(γ_{H}, γ_{G})

if γ_{H} is an adelic image of γ_{G} ∈ G(A) and Δ_{A}(γ_{H}, γ_{G}) = 0 otherwise.

By Lemma 6.4.B, Δ_{A}(γ_{H}, γ_{G}) is independent of: (i) the pair \overline{γ}_{H}, \overline{γ}_{G}; (ii) γ_{H} within its stable conjugacy class; (iii) γ_{G} within its G(A)-conjugacy class. Further,

Δ_{A}(γ_{H}, γ_{G}) = 1

if γ_{H} is an adelic image of γ_{G} and the G(A)-conjugacy class of γ_{G} meets G(F). (6.4) Product Formulas To specify the local factors of (3.7) we use a pair \overline{γ}_{H}, \overline{γ}_{G} of F -rational elements, as for the adelic factor. At almost all places v we set Δ^{(v)}(\overline{γ}_{H}, \overline{γ}_{G}) = 1. At the remaining places Δ^{(v)}(\overline{γ}_{H}, \overline{γ}_{G}) is arbitrary except for the requirement that

∏_{v} Δ^{(v)}(\overline{γ}_{H}, \overline{γ}_{G}) = 1.

Then as in (3.7), but with the superscript (v) inserted, we set

Δ^{(v)}(γ_{H}, γ_{G,v}) = Δ^{(v)}(\overline{γ}_{H}, \overline{γ}_{G,v})Δ^{(v)}(γ_{H}, γ_{G,v}; \overline{γ}_{H}, \overline{γ}_{G})

for all strongly G-regular γ_{H} in H(F). The relative factor Δ^{(v)}(γ_{H}, γ_{G,v}; \overline{γ}_{H}, \overline{γ}_{G}) is the product of

\frac{Δ_{I}^{(v)}(γ_{H}, γ_{G,v})}{Δ_{I}^{(v)}(\overline{γ}_{H}, \overline{γ}_{G})} · \frac{Δ_{II}^{(v)}(γ_{H}, γ_{G,v})}{Δ_{II}^{(v)}(\overline{γ}_{H}, \overline{γ}_{G})} · \frac{Δ_{2}^{(v)}(γ_{H}, γ_{G,v})}{Δ_{2}^{(v)}(\overline{γ}_{H}, \overline{γ}_{G})}

and

\frac{Δ_{IV}^{(v)}(γ_{H}, γ_{G,v})}{Δ_{IV}^{(v)}(\overline{γ}_{H}, \overline{γ}_{G})} · Δ_{1}(γ_{H}, γ_{G,v}; \overline{γ}_{H}, \overline{γ}_{G}).

The various terms are defined using any admissible embeddings over F_{v}, and a-data, χ-data for G as group over F_{v}. We shall use embeddings over F and global a- and χ-data [see (2.2), (2.5)] in order to obtain product formulas for the individual terms as well as for Δ. Theorem 6.4.A. (i) For almost all v each of Δ_{I}^{(v)}(γ_{H}, γ_{G,v}), Δ_{II}^{(v)}(γ_{H}, γ_{G,v}), Δ_{2}^{(v)}(γ_{H}, γ_{G,v}) and Δ_{IV} (γ_{H}, γ_{G,v}) equals 1.

(ii) ∏_{v} Δ_{I}^{(v)}(γ_{H}, γ_{G,v}) = 1, and similarly for Δ_{II}^{(v)}, Δ_{2}^{(v)}, and Δ_{IV}^{(v)}. In the last section we showed that

Δ_{I}^{(v)}(γ_{H}, γ_{G,v}; \overline{γ}_{H}, \overline{γ}_{G}) = 1

for almost all v (Lemma 6.3.A) and that

∏_{v} Δ_{I}^{(v)}(γ_{H}, γ_{G,v}; \overline{γ}_{H}, \overline{γ}_{G}) = Δ_{A}(γ_{H}, γ_{G})

[(6.3.1) and (6.3.2)]. Corollary 6.4.B.

(i) Δ_{(v)}(γ_{H}, γ_{G,v}) = 1 for almost all v and (ii) ∏_{v} Δ^{(v)}(γ_{H}, γ_{G,v}) = Δ_{A}(γ_{H}, γ_{G}).

This product formula contains the Global Hypothesis of [L2, p. 149] because Δ_{A}(γ_{H}, γ_{G}) is the term κ(ϵ(D)) of [L2]. To see this we translate our terminology into that of diagrams, as at the end of (4.2). The formula for ϵ(D) on p. 137 of [L2] determines an element of H^{−1}(Γ, X_{∗}(U)) which under global Tate-Nakayama duality coincides with our μ of (6.3). Then κ(ϵ(D)) is ⟨μ, s_{U}⟩ which is the same as Δ_{A}(γ_{H}, γ_{G}). Proof of Theorem. (Δ_{I}) By definition,

Δ_{I}^{(v)}(γ_{H}, γ_{G,v}) = ⟨λ_{v}(T_{sc}), s_{T,v}⟩

[see (3.2)]. Recall from (2.3.5) that the global invariant λ(T_{sc}) ∈ H^{1}(Γ, T (L)) is defined and that λ_{v}(T_{sc}) is the image of λ(T_{sc}) under H^{1}(Γ, T (L)) → H^{1}(Γ_{v}, T (L_{v})). Thus for almost all v we have λ_{v}(T_{sc}) = 1 and then Δ_{I}^{(v)}(γ_{H}, γ_{G,v}) = 1. Further, the image of Σ_{v}λ_{v}(T_{sc}) in H^{1}(Γ, T_{sc}(L)\T_{sc}(A_{L})) is trivial and so

∏_{v} Δ_{(I)}^{(v)}(γ_{H}, γ_{G,v}) = ∏_{v}⟨λ_{v}, s_{T,v}⟩ = ⟨Image Σ_{v}λ_{v}, s_{T} ⟩ = 1.

(Δ_{II}) In (2.6.5) we attached local χ-data {χ_{α}^{(v)}} to global data {χ_{α}}. The character χ_{α}^{(v)} is defined on F_{v,α}, the fixed field in L_{v} of the stabilizer of α in Γ_{v}, and

Δ_{II}^{(v)}(γ_{H}, γ_{G,v}) = ∏_{α} χ_{α}^{(v)} (\frac{α(γ) − 1}{a_{α}}) ,

where the product is over representatives α for the orbits of Γ_{v} in the roots of T which lie outside H. Note that δ = \frac{α(γ)−1}{a_{α}} lies in F_{α}, the fixed field in L of the stabilizer of α in Γ. On the other hand we may write χ_{α} as ∏_{v} χ_{α,v} where χ_{α,v} is a character on (F_{α} ⊗ F_{v})^{×}. Then we claim that

Δ_{II}^{(v)}(γ_{H}, γ_{G,v}) = ∏′_{α} χ_{α,v}(δ),

where the product is now over representatives α for the orbits of Γ in the set of roots outside H. Then

Δ_{II}^{(v)}(γ_{H}, γ_{G,v}) = 1

for almost all v and

∏_{v} Δ_{II}^{(v)}(γ_{H}, γ_{G,v}) = ∏_{v} ∏′_{α} χ_{α,v}(δ) = ∏′_{α} ∏_{v} χ_{α,v}(δ) = ∏′_{α} χ_{α}(δ) = 1.

To prove the claim we fix a root α outside H and choose representatives σ_{1}, …, σ_{n} for Γ_{v}\Γ/Γ_{α}, where Γ_{α} is the stabilizer of α in Γ. Then σ_{1}α, …, σ_{n}α are representatives for the Γ_{v}-orbits in the Γ-orbit of α.

Thus the contribution to Δ_{II}^{(v)}(γ_{H}, γ_{G,v}) from these orbits is ∏_{i} χ_{σ_{i}α}^{(v)}(σ_{i}δ). But F_{α} ⊗ F_{v} = ∏_{i} F_{α,σ_{i}^{−1}v} = ∏_{i} σ_{i}^{−1}(F_{v,σ_{i}α}). From χ_{α} = χ_{σ_{i}α} · σ_{i} we conclude that if χ_{α,v} = ∏_{i} χ_{i} then χ_{i} = χ_{σ_{i}α}^{(v)} · σ_{i}, and the claim is proved.

(Δ_{2}) By definition,

Δ_{2}^{(v)}(γ_{H}, γ_{G,v}) = ⟨a_{v}, γ⟩,

where a_{v} is the element a of H^{1}(W_{v}, T̂) constructed in (3.5). Because we have global χ-data and T_{H} → T is defined over F we may similarly construct a ∈ H^{1}(W, T̂). By (2.6.5) we have that a_{v} is the image of a under H^{1}(W, T̂) → H^{1}(W_{v}, T̂). Thus Δ_{2}(γ_{H}, γ_{G,v}) = 1 for almost all v and ∏_{v} Δ_{2}^{(v)}(γ_{H}, γ_{G,v}) = ∏_{v}⟨a_{v}, γ⟩ = ⟨a, γ⟩ = 1 since γ ∈ T (F).

The assertions of the theorem are immediate for Δ_{IV}. This completes the proof.

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