Descent for Transfer Factors

R. Langlands and D. Shelstad

Dedicated to A. Grothendieck on his 60th Birthday

Introduction

In [I] we introduced the notion of transfer from a group over a local field to an associated endoscopic group, but did not prove its existence, nor do we do so in the present paper. Nonetheless we carry out what is probably an unavoidable step in any proof of existence: reduction to a local statement at the identity in the centralizer of a semisimple element, a favorite procedure of Harish Chandra that he referred to as descent.

The principal difficulty is to show that the transfer factors of [I] for the original group G are compatible with those on the connected centralizer G_{ϵ} of the semisimple element ϵ. After some preliminary explanations in Section 1, the compatibility is stated as Theorem 1.6.A. In Section 2 we show that this compatibility indeed reduces the problem of existence to a local problem at the identity on the groups G_{ϵ}, and in passing we note some other applications.

The remaining four sections are devoted to the proof of Theorem 1.6.A. The transfer factors are defined in a rather elaborate manner as the product of five factors that mix group-theoretical data with Galois cohomology. The first steps are to reduce to quasisplit groups and then to discard two of the five factors, leaving only three, one of which is defined in a simple fashion, and two of which involve group-theoretic and cohomological data. It is these two that are difficult to compare for G and G_{ϵ}. The principal tools are the two comparison lemmas of Section 3.

The contributions to the factors are labelled by orbits of the Galois group in sets of roots, and the first use made of the comparison lemmas is to deal in Section 4, and rather quickly, with all orbits except those lying outside both G_{ϵ} and the endoscopic group.

This leaves a rather concise but still far from trivial statement that is proved partly by an analysis of the structure of semisimple groups and partly by explicit cohomological calculations. The structural analysis is possible only after the critical lemma of Section 5.1 has been established. This lemma allows us to introduce an inductive component into the argument, and then to assume that both the element ϵ and the datum s defining the endoscopic group are essentially of order two, and moreover, that all roots are of the same length. This done, the burden of the rest of the proof is carried by explicit arguments with the constructs of local class-field theory. They all appear in Section 6.

We cannot hope that the groping, pedestrian style of the paper will appeal to Grothendieck, for it lacks the force and penetration that he achieved so readily, like Nietzche’s Philosoph der Zukunft, erfinderisch in Schematen, mitunter stolz auf Kategorien-Tafeln. Nonetheless, it is a great pleasure for us to express our admiration of his magnificient contributions to the mathematics of our time.

§1. Descent Principles 1.1. Notation 1.2. Images of semisimple elements 1.3. The function Φ_{f}^{H} 1.4. Descent for endoscopic data 1.5. Descent for Φ_{f}^{H} 1.6. Descent for transfer factors 1.7. Final formula

§2. Consequences 2.1. Local transfer 2.2. A characterization lemma 2.3. Reduction to local transfer 2.4. Equisingular transfer 2.6. Archimedean transfer

§3. Comparison Lemmas 3.1. Reduction to the quasisplit case 3.2. Remarks and notation 3.3. First Lemma of Comparison 3.4. Second Lemma of Comparison 3.5. An application

§4. Analysis of b, b̂ and a Reduction 4.1. Galois action 4.2. Calculation of a coboundary 4.3. Explicit form 4.4. Root types 4.5. Analysis of θ_{1}, θ_{2}

§5. Final Reductions 5.1. Introduction 5.2. Beginning of proof of the critical lemma 5.3. The term Θ_{2}^{(d)} 5.4. Construction of φ and φ_{s} 5.5. Reducing the dimension of G_{der}

§6. The Order-Two Case 6.1. Introduction 6.2. Liftings 6.3. Some coboundaries 6.4. Remaining steps 6.5. Symmetric orbits 6.6. Final calculations

References

§1. Descent Principles

1.1. Notation

We follow closely the notation of [I]. In particiular, G is a connected reductive group over a field F of characteristic zero, now assumed local.

As in [I, Sect. 1.2.] G^{∗}, ψ are quasisplit data and ^{L}G = Ĝ ⋊ W_{F} is the L-group. To conserve notation we fix an F -splitting (B, T , {X_{α}ˇ }) of Ĝ and given a class of endoscopic data choose a representative (H, H, s, ξ) with ξ : H ↪ ^{L}G as inclusion and s an element of T . It is also convenient to fix an F -splitting (B_{H}, T_{h}, {Y_{β}ˇ }) of Ĥ = Cent(s, Ĝ)^{0} and assume that B_{H} = B ∩ Ĥ, T_{n} = T .

For the moment we refer directly to [I] for the definition of the factor Δ. Measures also remain as there. If ϵ ∈ G(F) is semisimple we choose an invariant differential form of highest degree on Cent(ϵ, G)^{0} in order to fix a Haar measure on the F -rational points of this group. We require that differential forms on inner forms be obtained by transport. 1.2. Images of semisimple elements

For ϵ in G the identity component of Cent(ϵ, G) will be denoted G_{ϵ}. If ϵ in G(F) is semisimple then, following [K1], the stable conjugacy class of ϵ is

{g^{−1}ϵg : gσ(g)^{−1} ∈ G_{ϵ}, σ ∈ Γ},

where Γ = Gal(F̄/F). If Cent(ϵ, G) is connected then this coincides with the set of F -rational points in the conjugacy class of ϵ in G(F̄). In general, an F -rational ϵ′ = g^{−1}ϵg is stably conjugate to ϵ if and only if Int g : G_{ϵ′} → G_{ϵ} is an inner twist. If G is quasisplit over F then there is an ϵ′ stably conjugate to ϵ such that G_{ϵ′} is quasisplit over F [K1, Lemma 3.3]. We now generalize the notion of image from [I, 1.3] (see also [K2]). It is convenient to use the notation γ_{H}, γ, γ_{G} when γ_{H} is strongly G-regular and ϵ_{H}, ϵ, ϵ_{G} in general.

Suppose then that ϵ_{H} lies in the Cartan subgroup G_{H}(F) of H(F). Then we call ϵ_{H} a T_{H}- image of ϵ_{G} in G(F) if for some admissible embedding T_{H} → T of T_{H} in G^{∗} carrying ϵ_{H} to, say, ϵ there exists x in G^{∗}, or just as well in G_{sc}^{∗}, such that ψ_{x} = Int x ∘ ψ has the properties that ψ_{x}(ϵ_{G}) = ϵ and that both T_{G} = ψ_{x}^{−1}(T) and ψ_{x} : T_{G} → T are defined over F. In varying T_{H} we obtain all images of ϵ_{G}. Observe that [K2, Lemma 10.2] shows that all images of a given ϵ_{G} are obtained by simply fixing one image ϵ_{H}, if it exists, and then taking all T_{H}-images for some T_{H} fundamental in H_{ϵ_{H}}. 1.3. The function Φ_{f}^{H}

Recall that to define transfer factors for (G, H) we may need to pass to a central extension of H. Call a central extension H̃ of H admissible if it is attached to a z-extension of G as in [I, 4.4] (although a wider class of extensions could be used [K-S]). The sequence

1 → Z̃(F) → H̃(F) → H(F) → 1

is then exact, where Z̃ is a central torus in H̃, and combinations of orbital integrals of functions on G(F) are to be matched with those of functions on H̃(F) that transform under Z̃(F) according to a certain character λ̃ [I, 4.4].

Suppose ϵ_{H} is semisimple in H(F) and ϵ̃_{H} lies in its preimage in H̃(F). The factor Δ(γ̃_{H}, γ_{G}) has been defined for γ̃_{H} strongly G-regular in H̃(F), by which we mean that the image γ_{H} of γ̃_{H} in H(F) is strongly G-regular. We shall investigate the behavior of

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

for γ̃_{H} near ϵ̃_{H}.

First, following [I, 4.3] it is easy to see that if ϵ_{H} is G-regular, but not strongly so, then Φ_{f}^{H} extends continuously to ϵ̃_{H} and that this extension is in fact smooth at ϵ̃_{H}. Second, if ϵ_{H} is not the image of any semisimple element in G(F) then no strongly G-regular element in H_{ϵ_{H}} (F) can be the image of an element in G(F) and so Φ_{f}^{H} vanishes on the strongly G-regular elements in H̃ϵ̃_{H} (F). In particular Φ_{f}^{H} vanishes for all γ̃_{H} in a neighborhood of ϵ̃_{H} in H̃(F).

We may then assume that ϵ_{H} is an image of an element ϵ_{G} in G(F). There is an ϵ′_{H} = h^{−1}ϵ_{H}h stably conjugate to ϵ_{H} such that H_{ϵ′_{H}} is quasisplit over F. If ϵ_{H} is a T_{H}-image of ϵ_{G} then we may multiply h by an element of H_{ϵ′_{H}} to assume that the homomorphism Int h^{−1} : T_{H} → H_{ϵ′_{H}} is defined over F, that is, that h ∈ 𝔄(T_{H}). Then h acts on the preimage T̃_{H} of T_{H} in H̃ as an element of 𝔄(T̃_{H}) and [I, 4.1.] implies that

Φ_{f}^{H}(h^{−1}γ̃_{H}h) = Φ_{f}^{H}(γ̃_{H})

for all strongly G-regular γ̃_{H} in T̃_{H}(F). Thus we may replace ϵ_{H} by ϵ′_{H} and assume from now on that H_{ϵ_{H}} is quasisplit over F. Then H_{ϵ_{H}} is an endoscopic group for G_{ϵ_{G}} as we now explain in detail. We sometimes denote it by H_{ϵ}. 1.4. Descent for endoscopic data

We start then with semisimple ϵ_{H} in H(F), an image of ϵ_{G} in G(F), and H_{ϵ_{H}} quasisplit over F. Choose T_{H} such that ϵ_{H} is a T_{H}-image of ϵ_{G}. Let ϵ be the image of ϵ_{H} under some admissible embedding T_{H} → T of T_{H} in G^{∗}. An argument as in the last paragraph allows us to choose the embedding so that G_{ϵ}^{∗} is quasisplit over F. We will see that these choices are of no real importance, the essential data being ϵ_{H}, ϵ and ϵ_{G} and thus H_{ϵ_{H}} , G_{ϵ}^{∗} and G_{ϵ_{G}}.

To the endoscopic data (H, H, s, ξ) we shall attach an extension H_{ϵ} of W_{F} by Ĥ_{ϵ_{H}} and an admissible embedding ξ_{ϵ} : H_{ϵ} ↪ ^{L}G_{ϵ_{G}} such that H_{ϵ_{H}} , H_{ϵ}, s and ξ_{ϵ} yield endoscopic data for G_{ϵ_{G}}. Further choices will be made, for example to specify Ĝ_{ϵ_{G}}, but again they will not affect the isomorphism class of the endoscopic data. Moreover, all endoscopic data for G_{ϵ_{G}} will be so obtained up to isomorphism.

Suppose that B_{H} ⊃ T_{H} in H and B ⊃ T in G^{∗} are Borel subgroups for which the already chosen T_{H} → T is the attached embedding. Also suppose x ∈ G_{sc}^{∗} is such that ψ_{x}(ϵ_{G}) = ϵ, with both T_{G} = ψ_{x}^{−1}(T) and ψ_{x} : T_{G} → T defined over F. Note that G_{ϵ}^{∗} and ψ_{x} serve as quasisplit data for G_{ϵ_{G}}.

The embedding

T_{H} → T ⟵^{ψ_{x}} T_{G}

is by definition dual to a diagram

Ĥ ← T_{H} = T → T̂ ⟶^{ψ_{x}} T̂_{G}.

This diagram allows us to identify R(G, T_{G})ˇ with R(Ĝ, T ) and then R(G_{ϵ_{G}}, T_{G})ˇ with a subset of R(Ĝ, T ).

In fixing L-data (Ĝ_{ϵ}, ρ_{ϵ}) and ^{L}G_{ϵ} = Ĝ_{ϵ} ⋊ W_{F} for G_{ϵ_{G}} or G_{ϵ}^{∗} we may assume that Ĝ_{ϵ} contains T and that R(Ĝ_{ϵ}, T ) coincides with R(G_{ϵ}, T_{G})ˇ as a subset of R(Ĝ, T ). We set B_{ϵ} = B ∩ G_{ϵ}^{∗} and let B_{ϵ} be the Borel subgroup of Ĝ_{ϵ} generated by T and the B-positive roots in T in Ĝ_{ϵ}. The embedding T̂ → T of T̂ in Ĝ_{ϵ} provided by B_{ϵ} and B_{ϵ} will be identified with T → T̂ above. The isomorphism T̂_{G} ⟶^{ψ_{x}} T̂ ⟶ T embeds T̂_{G} in Ĝ_{ϵ} and extends to an admissible embedding of ^{L}T_{G} in ^{L}G_{ϵ} (see, for example, [I, 2.6]). The image, again denoted ^{L}T_{G}, is independent of the choice of extension.

The element s lies in T and thus in Ĝ_{ϵ}. Moreover Ĥ = Cent(s, Ĝ)^{0}. We may take the dual Ĥ_{ϵ_{H}} of H_{ϵ_{H}} as the subgroup Cent(s, Ĝ_{ϵ})^{0} of Ĝ_{ϵ}; Ĥ_{ϵ_{H}} is normalized by ^{L}T_{G}. We define H_{ϵ} to be the subgroup of ^{L}G generated by Ĥ_{ϵ_{H}} and ^{L}T_{G}, and ξ_{ϵ} to be the inclusion map. Observe that there is clearly a split exact sequence

1 ⟶ Ĥ_{ϵ_{H}} ⟶ H_{ϵ} ⟶ W_{F} ⟶ 1

and that (H_{ϵ_{H}} , H_{ϵ}, s, ξ_{ϵ}) is a set of endoscopic data for G_{ϵ_{G}}.

We also identify the embedding T̂_{H} ⟶ T given by B_{H} ∩ H_{ϵ_{H}} and B_{ϵ} ∩ Ĥ_{ϵ_{H}} with T̂_{H} → T above. Then T_{H} → T is an admissible embedding for both (G, H) and (G_{ϵ_{G}}, H_{ϵ_{H}}). Moreover any admissible embedding of a maximal torus of H_{ϵ_{H}} in G_{ϵ}^{∗} is admissible as an embedding of a maximal torus of H in G^{∗} and carries ϵ_{H} to ϵ.

It remains to examine the effects of our choice. Suppose first that B and B_{H} are changed but that T_{H} → T remains fixed. Then (Ĝ_{ϵ}, ρ_{ϵ}) is replaced by a pair (Ĝ′_{ϵ}, ρ′_{ϵ}), Γ-isomorphic to it under a map that carries R(Ĝ_{ϵ}, T ) to R(Ĝ′_{ϵ}, T ) and the image of ^{L}T_{G} in ^{L}G_{ϵ} to its image in ^{L}G′_{ϵ}. This isomorphism further fixes s and carries Ĥ_{ϵ_{H}} and H_{ϵ} to the new Ĥ′_{ϵ_{H}} and H′_{ϵ}. Thus we obtain isomorphic endoscopic data for G_{ϵ}.

With T fixed, the choice of T_{H} and ψ_{x} does not affect endoscopic data. Now suppose we replace T_{H} → T by T̄_{H} → T̄. Then ϵ_{H} lies in both T_{H}, T̄_{H} and ϵ in both T, T̄. We may assume that the new Borel subgroups are obtained from B_{H} and B by conjugation in Cent(ϵ_{H}, H) and Cent(ϵ, G^{∗}). Then again the new data are seen to be isomorphic. Note that if T̄_{H} → T̄ is admissible for (G_{ϵ_{G}}, H_{ϵ_{H}}) then we may use conjugations in H_{ϵ_{H}} and G_{ϵ}^{∗}, and the data are unchanged.

Finally it is straightforward to check that the choice of (H, H, s, ξ) within its equivalence class does not affect the class of (H_{ϵ_{H}} , H_{ϵ}, s, ξ_{ϵ}) among data for G_{ϵ_{G}}. Moreover from any class of data for G_{ϵ_{G}} we can recover s ∈ T , Ĥ = Cent(s, Ĝ)^{0}, H = ⟨Ĥ,^{L}T_{G}⟩ contained in ^{L}G and ϵ_{H} semisimple in H(F) such that (H_{ϵ_{H}} , H_{ϵ}, s, ξ_{ϵ}) lies in the class. 1.5. Descent for Φ_{f}^{H}

Continuing with ϵ_{h}, ϵ and ϵ_{G}, we assume that endoscopic data have been fixed once and for all by the choices of the last section. Since these choices will not be mentioned again we reserve no notation for them. In particular, T_{H} → T will be an arbitrary embedding of T_{H} in G_{ϵ}^{∗} which is admissible for (G_{ϵ}^{∗}, H_{ϵ_{H}}). To spare notation further we assume that G_{ϵ}^{∗}, ψ are quasisplit data for G_{ϵ_{G}}. If then ϵ_{H} is a T_{H}-image of ϵ_{G} there is an x ∈ (G_{ϵ}^{∗})_{sc} such that T_{G} = ψ_{x}^{−1}(T) and ψ_{x} : T_{G} → T are defined over F.

For γ̃_{H} strongly G-regular in the preimage of T_{H}(F) in H̃(F) we calculate Φ_{f}^{H}(γ̃_{H}) as

∑_{γ_{G}} Δ(γ̃_{H}, γ_{G})Φ(γ_{G}, f)

where the sum is over representatives γ_{G} for the G(F)-conjugacy classes in the stable conjugacy class of the image γ_{G}^{0} of γ̃_{H} under

T̃_{H} ⟶ T_{H} ⟶ T ⟶ T_{G}.

We write γ_{G} as w^{−1}γ_{G}^{0}w, w ∈ 𝔇(T_{G}), and now describe a choice of representatives for 𝔇(T_{G}).

Consider first the stable conjugacy class of ϵ_{G} in G(F). Since Cent(ϵ_{G}, G) may be disconnected we pass to a z-extension of G and pick ϵ̃ ∈ G̃(F) mapping to ϵ under G̃ → G. Suppose that {ϵ̃_{j} = w̃_{j}^{−1}ϵ̃w̃_{j} : 0 ≤ j ≤ n} is a set of representatives for the conjugacy classes in the stable class of ϵ̃, with w̃_{0} = 1. Let w_{j} be the image of w̃_{j} in G(F), ϵ_{j} = w_{j}^{−1}ϵ_{G}w_{j} and G_{j} = Cent(ϵ_{j}, G)^{0}.

Notice that the ϵ_{j} need not be distinct. We use (G_{ϵ}^{∗}, ψ_{j}), where ψ_{j} = ψ ∘ Int w_{j}, as quasisplit data for G_{j}. Define a subset S(T_{H}) of {0, 1, ⋯, n} by j ∈ S(T_{H}) if and only if ϵ_{j} has ϵ_{H} as T_{H}-image relative to (G_{j}, H_{ϵ_{H}}), that is, if and only if there exists h_{j} ∈ G_{j} such that Int h_{j}^{−1}w_{j}^{−1} maps T_{G} to G_{j} over F. Then fix some such h_{j} and set ŵ_{j} = w_{j}h_{j}, T_{j} = ŵ_{h}^{−1}T_{G}ŵ_{j}. Passage to G̃ shows that

{ŵ_{j}w′ : j ∈ S(T_{H}) and w′ a representative for 𝔇(T_{j}, G_{j})}

is a set of representatives for 𝔇(T_{G}, G).

Thus Φ_{f}^{H}(γ̃_{H}) is equal to

(1.5.1)  ∑_{j=0}^{n} ∑_{w′∈𝔇(T_{j},G_{j})} δ(j)Δ(γ̃_{H}, w^{′−1}γ_{G}^{j}w′, f)Φ(w^{′−1}γ_{G}^{j}w′, f)

where γ_{G}^{j} = ŵ_{j}^{−1}γ_{G}^{0}ŵ_{j}, 0 ≤ j ≤ n, and δ is the characteristic function of S(T_{H}).

From the well-known construction of Harish-Chandra [HC1, Sect. 22; HC2, Part VI] we can find f^{j} ∈ C_{c}^{∞}(G_{j}(F)), 0 ≤ j ≤ n, such that

Φ(δ, f) = Φ(δ, f^{j})

for all regular semisimple δ in some neighborhood of ϵ_{j} in G_{ϵ_{j}}(F). It remains to relate transfer factors for (G, H) to those for (G_{ϵ_{G}}, H_{ϵ_{H}}). 1.6. Descent for transfer factors

To conserve notation H will be assumed an L-group, but H_{ϵ} must of course remain arbitrary. Then H̃_{ϵ_{H}} will be an admissible central extension of H_{ϵ_{H}} (recall 1.3) and ϵ̃_{H} will be an element in the preimage of ϵ_{H} in H̃_{ϵ_{H}} (F). We suppose that ϵ_{H} is a T̄_{H}-image as well as a T_{H}-image, and take γ̃_{H}, ˜ γ̄ in H̃_{ϵ_{H}} (F) with strongly G-regular images γ_{H} ∈ T_{H}, γ̄_{H} ∈ T̄_{H} in H_{ϵ_{H}} (F). Then the factors

Δ = Δ(γ_{H}, γ_{G}; γ̄_{H}, γ̄_{G})

for (G, H) and

Δ_{ϵ} = Δ(γ̃_{H}, γ_{G}; ˜ γ̄_{H}, γ̄_{G})

for (G_{ϵ_{G}}, H_{ϵ_{H}}) are defined and non-zero whenever γ_{H}, γ̄_{H} are images with respect to (G_{ϵ_{G}}, H_{ϵ_{H}}) of γ_{G}, γ̄_{G} in G_{ϵ_{G}}(F). That will be our assumption on γ_{G}, γ̄_{G} throughout this section.

Let Θ = Δ/Δ_{ϵ}. Then Θ is naturally a product

Θ_{I}Θ_{II}Θ_{1}Θ_{2}Θ_{IV} ,

the factors corresponding to those of Δ and Δ_{ϵ} [I, Sect. 3]

Keeping T_{H} and T̄_{H} fixed we consider γ̃_{H}, ˜ γ̄_{H} near ϵ̃_{H} with γ_{G}, γ̄_{G} both near ϵ_{G}. We will see in 3.1 that Θ_{1}(γ̃_{H}, γ_{G}; ˜ γ̄_{H}, γ̄_{G}) = 1. A glance at the remaining factors convinces one that Θ extends continuously to (ϵ̃_{H}, ϵ_{G}; ϵ̃_{H}, ϵ_{G}) taking a nonzero value there. The extension is then seen to be smooth. Theorem 1.6.A.

lim Θ(γ̃_{H}, γ_{G}; ˜ γ̄_{H}, γ̄_{G}) = 1
γ̃_{H}, ˜ γ̄_{H} ⟶ ϵ̃_{H}
γ_{G}, γ̄_{G} ⟶ ϵ_{G}.

The proof will occupy Sections 3 to 6.

Suppose F is nonarchimedean. Then the theorem says that

Θ(γ̃_{H}, γ_{G}; ˜ γ̄_{H}, γ̄_{G}) = 1

for γ̃_{H}, ˜ γ̄_{H} near ϵ̃_{H} and γ_{G}, γ̄_{G} near ϵ_{G}. Thus for the absolute factors Δ(γ_{H}, γ_{G}), Δ(γ̄_{H}, γ̄_{G}) for G and Δ_{ϵ}(γ̃_{H}, γ_{G}), Δ_{ϵ}(˜γ̄_{H}, γ̄_{G}) for G_{ϵ} [I, 3.7] we have

\frac{Δ(γ_{H}, γ_{G})}{Δ_{ϵ}(γ̃_{H}, γ_{G})} = \frac{Δ(γ̄_{H}, γ̄_{G})}{Δ_{ϵ}(˜γ̄_{H}, γ̄_{G})},

and so we may write

Δ(γ_{H}, γ_{G}) = cΔ_{ϵ}(γ̃_{H}, γ_{G})

for γ̃_{H} near ϵ̃_{H} and γ_{G} near ϵ_{G}, where c is a constant. Observe that for γ̃_{H} near ϵ̃_{H} the factor Δ_{ϵ}(γ̃_{H}, γ_{G}) depends only on γ_{H} and γ_{G} (see [I, 4.4.A] and 3.5). We emphasize that the assertion of the theorem is that c is independent of the Cartan subgroup T_{H} containing γ_{H}. That this is crucial for the transfer of orbital integrals (and therefore characters) will be seen in Section 2.

In the archimedean case c is a function. This however presents no difficulties in our applications (2.4, 2.5). 1.7. Final formula

For the pair (G_{j}, H_{ϵ_{H}}) we write c_{j} in place of c. Then

(1.7.1)  Φ_{f}^{H}(γ_{H}) = ∑_{j=0}^{n} c_{j}Φ_{f^{j}}^{H_{ϵ}}(γ̃_{H})

for strongly G-regular γ̃_{H} sufficiently close to ϵ̃_{H} in H̃_{ϵ_{H}} (F). Note that the characteristic function δ(j) of (1.5.1) has disappeared because by definition Φ_{f^{j}}^{H_{ϵ}}(γ̃_{H}) = 0 for j ∉ S(T_{H}) (recall 1.3).

§2. Consequences

2.1. Local transfer

We say that (G, H) admits Δ-transfer if for each f ∈ C_{c}^{∞}(G(F)) there exists f H̃ ∈ C_{c}^{∞}(H̃(F), λ̃) notation of 1.3) such that f and f H̃ have Δ-matching orbital integrals, that is,

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

for all strongly G-regular γ̃_{H} in H̃(F). If H is the quasisplit form of G, so that Δ is a constant, we often refer to stable transfer rather than Δ-transfer. Suppose that F is nonarchimedean. Then for γ̃_{H} near 1 the factor Δ(γ̃_{H}, γ_{G}) depends only on the image γ_{H} of γ̃_{H} in H(F) (see [I 4.4] again); so we may denote it instead by Δ_{loc}(γ_{H}, γ_{G}). We say that (G, H) admits local Δ-transfer at the identity if for any f ∈ C_{c}^{∞}(G(F)) we can find f^{H} ∈ C_{c}^{∞}(H(F)) such that

(2.1.2)  Φ^{st}(γ_{H}, f^{H}) = ∑_{γ_{G}} Δ_{loc}(γ_{H}, γ_{G})Φ(γ_{G}, f)

for all strongly G-regular γ_{H} near 1 in H(F). 2.2. A characterization lemma

Throughout 2.2 and 2.3 we assume that F is nonarchimedean. Suppose that Φ^{st} is a stablyinvariant function on the regular semisimple elements of G(F) that is compactly supported modulo conjugation, viz., that vanishes along all conjugacy classes of regular semisimple elements that do not meet some fixed compact subset of G(F). Then we call Φ^{st} a local stable orbital integral if for each semisimple element ϵ in G(F) there exists f_{ϵ} ∈ C_{c}^{∞}(G(F)) such that Φ^{st}(γ) = Φ^{st}(γ, f_{ϵ}) for all regular semisimple γ near ϵ.

Lemma 2.2.A. Let G be a quasisplit group. If Φ^{st} is a local stable orbital integral on G(F) then there exists f ∈ C_{c}^{∞}(G(F)) such that

Φ^{st}(γ) = Φ^{st}(γ, f)

for all regular semisimple γ in G(F).

Proof. By a simple passage to a z-extension ([K1]) we reduce immediately to the case that the derived group of G is simply connected.

If ϵ is semisimple in G(F) we denote by Z_{ϵ} the center of G_{ϵ} and by Z′_{ϵ} the set of ϵ′ in Z_{ϵ} at which D_{G}/D_{G_{ϵ}}, does not vanish. For ϵ′ in Z′_{ϵ} we have G_{ϵ′} = G_{ϵ} and Z_{ϵ′} = Z_{ϵ}, while if ϵ′ ∈ Z_{ϵ} − Z′_{ϵ} then G_{ϵ′} ⫌ G_{ϵ}, so that dim G_{ϵ′} > dim G_{ϵ}, and Z_{ϵ′} ≠^{⊂} Z_{ϵ}. Notice that the group G_{ϵ} is Cent(Z_{ϵ}, G). Thus if g ∈ G(F̄) then g^{−1}ϵg is stably conjugate to ϵ if and only if Int g^{−1} : Z_{ϵ} ⟶ Z_{g^{−1}eg} is defined over F, that is, Z_{ϵ} is stably conjugate to Z_{g^{−1}ϵg}.

There are only finitely many stable conjugacy classes among groups Z_{ϵ}. We label representatives Z_{0}, ⋯, Z_{r} for these classes so that Z_{0} is the center of G, the group G(ℓ) = Cent(Z_{ℓ}, G) is quasisplit over F for each ℓ (using [K1, Lemma 3.3]), and so that dim G(ℓ) ≤ dim G(k) if k ≤ ℓ. Notice that if Z_{ℓ} = Z_{ϵ} then G(ℓ) = G_{ϵ} and Z′_{ℓ} = Z′_{ϵ}.

It is sufficient to show for each ℓ = 0, ⋯, r that if a local stable orbital integral Φ^{st} vanishes on the regular semisimple elements in a neighborhood of ∪_{k<ℓ} Z_{k}(F) then there exists f_{ℓ} ∈ C_{c}^{∞}(G(F)) such that

(2.2.1)  Φ^{st}(γ) = Φ^{st}(γ, f_{ℓ})

for all γ near ∪_{k≤ℓ} Z_{k}(F), for if ϵ′ is stably conjugate to ϵ in this set then any regular γ′ close to ϵ′ is stably conjugate to a γ close to ϵ because G_{ϵ} is quasisplit. Thus (2.2.1) implies that

Φ^{st}(γ′) = Φ^{st}(γ′, f_{ℓ}).

We then proceed inductively, replacing the original Φ^{st} by

γ ⟶ Φ^{st}(γ, f_{0}),

passing to Z_{1}, and so on.

Because G_{der} is simply connected, stable semisimple conjugacy classes are labelled by orbits of the Weyl group in a fixed Cartan subgroup T over F. Of course the orbits lie in T (F̄), and not all such orbits label stable conjugacy classes. Suppose we are given a Galois-invariant metric on T (F̄), one such orbit τ̄ = {t_{0}, t_{1}, ⋯, t_{s}}, and a δ > 0. Then the set of all g ∈ G(F) such that the semisimple part of g is conjugate in G(F̄) to an s such that |s − t_{i}| < δ for some i is an open subset of G(F) that is stably invariant. Multiplying a given function f by the characteristic function of such a set, we concentrate its stable orbital integrals on the set X(τ, δ) of regular γ such that γ is conjugate to s with |s − t_{i}| < δ for some δ.

Thus a stable conjugacy class that meets Z′_{ℓ}(F) will intersect Z′_{ℓ}(F) in a finite set {ϵ_{0}, ⋯, ϵ_{r}}, and mapping Z_{ℓ} into T over F̄ we send the elements ϵ = ϵ_{0}, ⋯, ϵ_{r} to t_{0}, ⋯, t_{r}. We complete this set to a full orbit τ = {t_{0}, ⋯, t_{s}}. Since G_{ϵ} is quasisplit, it is clear that if δ is sufficiently small then any stable conjugacy class in X(τ, δ) meets any given neighborhood of ϵ in G_{ϵ}. In verifying this, we may suppose that Z_{ℓ} ⊆ T so that t_{0} = ϵ.

A stable conjugacy class in X(τ, δ) is then represented by an s ∈ T (F̄) such that |s − ϵ| < δ and such that for any σ ∈ Gal(F̄/F) there is a u_{σ} in the normalizer of T in G(F) satisfying

(2.2.2)  σ(s) = u_{σ}^{−1}su_{σ}.

Since

|σ(s) − σ(ϵ)| = |σ(s) − ϵ| = |u_{σ}^{−1}su_{σ} − ϵ|,

we can choose δ sufficiently small that (2.2.2) forces u_{σ} to centralize ϵ. Consequently, by Steinberg’s Theorem, s determines a stable conjugacy class in G_{ϵ}(F). Moreover, if δ is small enough this class must meet the given neighborhood of ϵ in G_{ϵ}.

By assumption, there is an f_{ℓ} such that Φ^{st}(γ) = Φ^{st}(γ, f_{ℓ}) for γ in a neighborhood of ϵ in G_{ϵ}, and we conclude that this relation is valid on all of X(τ, δ). A simple partition-of-unity argument completes the proof. Observe that we could have dispensed with the introduction of the groups Z_{ℓ}, if we had wished to do so. 2.3. Reduction to local transfer

Theorem 2.3.A Suppose all pairs (G_{ϵ_{G}}, H_{ϵ_{H}}) have local Δ_{ϵ}-transfer at the identity. Then (G, H) has Δ-transfer.

Proof. We first observe that if G is replaced by a z-extension G̃ the hypothesis of the theorem remains valid. Further it is then sufficient to prove the theorem for G̃. Thus we may as well assume H an L-group. By Lemma 2.2.A we have then just to show that

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

defined so far for γ_{H} strongly G-regular in H(F), is a local stable orbital integral on H(F). We shall use the descent formula (1.7.1). Observe also that local Δ_{ϵ}-transfer at the identity for (G_{ϵ_{G}}, H_{ϵ_{H}}) implies local Δ_{ϵ}-transfer at ϵ_{H} for the same pair (see Lemma 3.5.A).

First, if ϵ_{H} is regular semisimple in H(F), so that H_{ϵ_{H}} is a Cartan subgroup of H, then local transfer at ϵ_{H} for (G_{ϵ_{G}}, H_{ϵ_{H}}) implies by (1.7.1) that Φ_{f}^{H} extends smoothly to ϵ_{H}. Thus we have Φ_{f}^{H} defined on all regular semisimple elements of H(F). It is further locally constant, stable, and compactly supported modulo conjugation.

For general ϵ_{H} we use (1.7.1) to obtain f′ ∈ C_{c}^{∞}(H_{ϵ_{H}} (F)) and thus f′′ ∈ C_{c}^{∞}(H(F)) such that

Φ_{f}^{H}(γ_{H}) = Φ^{st}(γ_{h}, f′) = Φ^{st}(γ_{H}, f′′)

for γ_{H} near ϵ_{H}. Thus Φ_{f}^{H} is a local stable orbital integral and the proof is complete. 2.4. Equisingular matching

We call ϵ_{H} and ϵ_{G} equisingular if H_{ϵ_{H}}, is an inner form of G_{ϵ_{G}}, that is, if ϵ_{H} is (G, H)-regular in the sense of [K2]. Assume that f, f^{H} have Δ-matching orbital integrals. For simplicity, we shall suppose that the derived group of G is simply connected so that neither H itself nor H_{ϵ_{H}} need be replaced by a central extension in the matchings. Note also that Cent(ϵ_{H}, H) is connected [K2, Lemma 3.2]. Following [K2], and by the homogeneity of germs, we may expect a stable combination of the integrals of f^{H} along the conjugacy classes in the stable class of ϵ_{H} to match with some suitable combination of the integrals of f along the classes in the stable class of ϵ_{G}. We shall show this is true and so verify some conjectures in [K2].

The first step is to define a factor Δ(ϵ_{H}, ϵ_{G}). Let ϵ_{H} be a T_{H}-image of ϵ_{G}. Then we set

Δ(ϵ_{H}, ϵ_{G}) = lim Δ(γ_{H}, γ_{G}),

where the limit is taken as γ_{H} ⟶ ϵ_{H} in T_{H}(F) and γ_{G} ⟶ ϵ_{G}, and γ_{H} is an image of γ_{G}. Suppose F nonarchimedean. If we consider also γ̄_{H}, γ̄_{G} with γ̄_{H} near ϵ_{H} in another Cartan subgroup T̄_{H}(F) of H_{ϵ_{H}} (F) then Δ_{ϵ}(γ_{H}, γ_{G}; γ̄_{H}, γ̄_{G}) ≡ 1 since H_{ϵ_{H}} is the quasisplit inner form of G_{ϵ_{G}}. Thus Theorem 1.6.A asserts that

Θ(γ_{H}, γ_{G}; γ̄_{H}, γ̄_{G}) = Δ(γ_{H}, γ_{G}; γ̄_{H}, γ̄_{G})

= Δ(γ_{H}, γ_{G})/Δ(γ̄_{H}, γ̄_{G}) = 1

if γ_{h}, γ̄_{H} near ϵ_{H} are images of γ_{G}, γ̄_{G} near ϵ. In other words, Δ(γ_{H}, γ_{G}) is a constant independent of the Cartan subgroup containing γ_{H} and Δ(ϵ_{H}, ϵ_{G}) equals this constant. If F is archimedean we still have Δ_{ϵ}(γ_{H}, γ_{G}; γ̄_{H}, γ̄_{G}) ≡ 1 but Theorem 1.6.A asserts only that

lim Δ(γ_{H}, γ_{G})/Δ(γ̄_{H}, γ̄_{G}) = 1.

We conclude nevertheless that Δ(ϵ_{H}, ϵ_{G}) is well defined, that is, independent of the choice of T_{H}.

We write O(ϵ_{G}, f) for the integral of f along the conjugacy class of ϵ_{G}, keeping in mind our convention for measures (1.1). A sign e(G) is defined in [K4]. If we sum over representatives ϵ′_{H} for the conjugacy classes in the stable conjugacy class of ϵ_{H} then

O^{st}(ϵ_{H}, f^{H}) = ∑_{ϵ′_{H}} e(H_{ϵ′_{H}})O(ϵ′_{H}, f^{H})

is a stable distribution [K3, S5].

Lemma 2.4.A Suppose ϵ_{H} is (G, H)-regular then

(2.4.1)  O^{st}(ϵ_{H}, f^{H}) = ∑_{ϵ_{G}} e(G_{ϵ_{G}})Δ(ϵ_{H}, ϵ_{G})O(ϵ_{G}, f)

where the sum is over representatives ϵ_{G} for the conjugacy classes in G(F) equisingular with ϵ_{H}.

Proof. (i) F nonarchimedean. Suppose γ_{H} is near ϵ_{H} in a fundamental Cartan subgroup T_{H}(F) of H_{ϵ_{H}} (F). Then by descent (1.7),

∑_{γ_{G}} Δ(γ_{H}, γ_{G})Φ(γ_{G}, f)

coincides with

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

and thus with

∑_{j} Δ(ϵ_{H}, ϵ_{j})Φ^{st}(γ_{j}, f^{j}).

The zero-degree term in the Shalika germ expansion of this expression is

λ ∑_{j} Δ(ϵ_{H}, ϵ_{j})e(G_{j})f^{j}(ϵ_{j})

where λ is a constant depending only on ϵ_{H} (see [K3, §3]). Apart from λ, this is the right side of (2.4.1). At the same time,

∑_{γ_{G}} Δ(γ_{H}, γ_{G})Φ(γ_{G}, f)

coincides with Φ^{st}(γ_{H}, f^{H}) which has λO^{st}(ϵ_{H}, f^{H}) as zero-degree term in its germ expansion. So (2.4.1) is proved.

(ii) F archimedean. Descent again yields

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

In place of germ expansions we use Harish-Chandra’s limit formula [HC3, Lemma 17.5]. Again T_{H} is taken to be fundamental in H_{ϵ_{H}}. We write γ_{H} ∈ T_{H}(F) as ϵ_{H} exp X and multiply each side of the equation

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

by

∏(α(ϵ_{H})e^{α(X)/2} − e^{−α(X)/2}),

the product being over all positive roots of T_{H} in H. We then apply the operator ϖ_{ϵ_{H}} = ∏′ H_{α}, the product being now over positive roots in H_{ϵ_{H}}, and take limits as X ⟶ 0. As a first step we obtain on the right side

∑_{j} Δ(ϵ_{H}, ϵ_{j}) lim ϖ_{ϵ_{H}} (∏′ e^{α(X)/2} − e^{−α(X)/2})Φ^{st}(γ_{j}, f^{j})

(see [S5] for the explicit form of Δ(γ_{H}, γ_{j}), especially the term Δ_{2}(γ_{H}, γ_{j}), and [W, p. 371] for a similar calculation).

We calculate this new limit by means of Harish-Chandra’s formula as in [S5, §2.9], using the results of §37 of [HC3] to keep track of constants. The contribution of the right side is then

∑_{j} Δ(ϵ_{H}, ϵ_{j})λe(G_{j})f^{j}(ϵ_{j}) = λ ∑_{j} Δ(ϵ_{H}, ϵ_{j})e(G_{j})O(ϵ_{j}, f),

λ being a constant depending only on H_{ϵ_{H}}. The left side contributes λO^{st}(ϵ_{H}, f^{H}), and so the lemma is proved.

Observe that from our product formula for Δ(γ_{H}, γ_{G}) [I, §6.4] we obtain a product formula for Δ(ϵ_{H}, ϵ_{G}) as conjectured in [K2, §6.10]. 2.5. Regular matching

Suppose δ_{G} is regular in G(F), that is, that Cent(δ_{G}, G) is of minimal dimension. Let δ_{G} = ϵ_{G}u_{G} be the Jordan decomposition; both ϵ_{G} and u_{G} belong to G(F). Assume ϵ_{G} has image ϵ_{H} in H(F). As usual, we take H_{ϵ_{H}} quasisplit over F and then choose u_{H} regular unipotent in H_{ϵ_{H}} (F). Thus δ_{H} = ϵ_{H}u_{H} is regular in H(F). We shall use descent and the regular unipotent matching of [I, §5.5] to match integrals over the classes of δ_{H} and δ_{G}. For simplicity of notation we assume H, H_{ϵ} are L-groups and that Cent(ϵ_{H}, H), Cent(ϵ_{G}, G) are connected. The stable conjugacy classes of δ_{H}, δ_{G} are then the F -points in their F̄-classes.

Set Φ^{st}(δ_{H}, f^{H}) = ∑ Φ(δ′_{H}, f^{H}), where the sum is over representatives δ′_{H} for conjugacy classes in the stable conjugacy class of δ_{H}. Then descent to H_{ϵ_{H}} and Theorem 5.5.A of [I] show immediately that

Φ^{st}(δ_{H}, f^{H}) = lim_{γ_{H}→ϵ_{H}} D_{ϵ_{H}} (γ_{H})Φ^{st}(γ_{H}, f^{H})

so that f^{H} → Φ^{st}(δ_{H}, f^{H}) is a stable distribution.

At the same time we set

Δ(ϵ_{H}, ϵ_{G}) = lim_{{γ_{H}→ϵ_{H} \atop γ_{G}→ϵ_{G}}} Δ(γ_{H}, γ_{G})/Δ_{ϵ}(γ_{H}, γ_{G}),

the limits taken as in 1.6, and

Δ_{ϵ}(δ_{G}) = λ(ϵ_{G})Δ(u_{G})

in the notation of [I, 5.5], with λ, Δ calculated with respect to (G_{ϵ_{G}}, H_{ϵ_{H}}). Then we define

Δ(δ_{H}, δ_{G}) = Δ(ϵ_{H}, ϵ_{G})Δ_{ϵ}(δ_{G}).

Suppose f and f^{H} have Δ-matching orbital integrals. Descent and the regular-unipotent matching also imply easily that

lim_{γ_{H}→ϵ_{H}} D_{ϵ_{H}} (γ_{H}) ∑_{γ_{G}} Δ(γ_{H}, γ_{G})Φ(γ_{G}, f) = ∑_{δ′_{G}} Δ(δ_{H}, δ′_{G})Φ(δ′_{G}, f),

and so we have the matching

Φ^{st}(δ_{H}, f^{H}) = ∑_{δ′_{G}} Δ(δ_{H}, δ′_{G})Φ(δ′_{G}, f).

Here we have dealt with both the nonarchimedean and archimedean cases. We also see that if δ_{H} = ϵ_{H}u_{H} is regular in H(F) and ϵ_{H} is not an image then

Φ^{st}(δ_{H}, f^{H}) = 0.

Finally we remark that if f ∈ C_{c}^{∞}(G(F)) is supported on the full regular set of G(F) then we can find f^{H} ∈ C_{c}^{∞}(H(F)) supported on the full regular set of H(F) with Δ-matching integrals.

2.6. Archimedean transfer

Suppose F is archimedean. If F = C we can define transfer factors for G over C or for the group G̃ = Res_{R}^{C}G over R obtained by restriction of scalars. On identifying G(C) with G̃(R) in the usual way we see that the transfer factors for a pair (G(C), H(C)) are the same as for (G̃(R), H̃(R)) (this is a special case of a fact used in 5.5). It is then sufficient to treat the case F = R. Again there will be no harm in taking H to be an L-group.

For real groups a transfer factor, that we now denote Δ^{(R)}(γ_{H}, γ_{G}), was defined in [S4] using diagrams. It includes moreover an implicitly defined sign, (see [S4, Section 3.5]).

Theorem 2.6.A There is a constant c such that

Δ(γ_{H}, γ_{G}) = cΔ^{(R)}(γ_{H}, γ_{G})

for all G-regular γ_{H} in H(R).

Proof. By continuity we can assume γ_{H} strongly G-regular, and because both Δ and Δ^{(R)} satisfy the Local Hypothesis ([I, 4.2.B], [L2, Lemma 6.17]) we may assume that G is quasisplit over R.

For an imaginary root α we may take the χ-datum χ_{α} as the character z ⟶ z/|z| on C^{×} and for the remaining roots we take χ_{α} trivial. Then inspection of the terms in each factor shows that

Δ(γ_{H}, γ_{G}) = c(T_{H})Δ^{(R)}(γ_{H}, γ_{G})

where T_{H} = Cent(γ_{H}, H). Up to a constant independent of T_{H}, c(T_{H}) is either real or purely imaginary. To show that c(T_{H}) is in fact independent of T_{H}, which is what the theorem asserts, we argue as follows.

By the matching of orbital integrals for Δ^{(R)} [S4] there is for each Schwartz function f on g(R) a Schwartz function f^{H} on H(R) (or an essentially Schwartz function if the embedding ξ : ^{L}H ↪ ^{L}G is not of unitary type [S3]) such that

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

for all strongly G-regular elements γ_{H}. We multiply both sides by D_{H}(γ_{H}). The left side has a limit as γ_{H} ⟶ 1 through any Cartan subgroup and the limit is independent of the choice of Cartan subgroups. This follows from applying the Harish-Chandra jump conditions to stable orbital integrals [S1, Section 4]. The right side then has the same property. But so also does

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

by [I, Theorem 5.5.A]. Because G is quasisplit over R we can choose f such that the limit is nonzero. Thus we get a contradiction unless c(T_{H}) is independent of T_{H}, and the theorem is proved.

We conclude now that for each Schwartz function f on G(R) there exists an (essentially) Schwartz function f^{H} on H(R) with Δ-matching orbital integrals. Then the Paley-Wiener results of [C-D] allow us to take f^{H} ∈ C_{c}^{∞}(H(R), K_{H}) if f ∈ C_{c}^{∞}(G(R), K), with K_{H}, K maximal compact subgroups of H(R), G(R) respectively and the notation indicating bi-K_{H}-finite or bi-K-finite functions.

Of course we have not used the Descent Theorem (1.6.A) in this proof of Δ-transfer. However the proof of Δ^{(R)}-transfer that we have used is based on the Harish-Chandra jump conditions and these come from descent to centralizers of semiregular elements. Many of the arguments in [S1-S4] for Δ^{(R)}-transfer are essentially special cases of results needed for 1.6.A. To prove Δ-transfer directly we may apply the results of Section 1 to verify the jump conditions of [S1] for Φ_{f}^{H}. Since we still need some of the arguments from [S2-S4] and overall the proof is not much shorter, we forgo the details.

§3. Comparison Lemmas

3.1. Reduction to quasisplit groups Recall from 1.6 that

Θ(γ_{H}, γ_{G}; γ̄_{H}, γ̄_{G}) = Δ(γ_{H}, γ_{G}; γ̄_{H}, γ̄_{G})/Δ_{ϵ}(γ_{H}, γ_{G}; γ̄_{H}, γ̄_{G}),

where γ_{H}, γ̄_{H} are images of γ_{G}, γ̄_{G} relative to (H_{ϵ_{H}} , G_{ϵ_{G}}). We consider the limit of Θ as γ_{H}, γ̄_{H} approach ϵ_{H} through fixed Cartan subgroups denoted respectively T_{H}, T̄_{H}, and γ_{G}, γ̄_{G} approach ϵ_{G}. Theorem 1.6.A, which we have to prove, states that this limit is 1.

Fix embeddings T_{H} ⟶ T, T̄_{H} ⟶ T̄ for (H_{ϵ_{H}} , G_{ϵ}^{∗}). Recall that we assume G_{ϵ}^{∗}, ψ to be quasisplit data for G_{ϵ_{G}}. Write γ, γ̄ for the images of γ_{H}, γ̄_{H}, as usual. If we factor Θ as we factor Δ, Δ_{ϵ} then only Θ_{1} = Θ_{III_{1}} depends on γ_{G}, γ̄_{G} rather than on γ, γ̄ alone. The next lemma allows us to replace G by G^{∗}, ϵ_{G} by ϵ and γ_{G}, γ̄_{G} by γ, γ̄.

Lemma 3.1.A.

Θ_{1}(γ_{H}, γ_{G}; γ̄_{H}, γ̄_{G}) = 1.

Proof. We use the notation of [I, 3.4], putting a subscript ϵ on those objects attached to G_{ϵ_{G}}. We may take u(σ) to be the image in G_{sc}^{∗} of the element u_{ϵ}(σ) in (G_{ϵ}^{∗})_{sc}, and h and h̄ to be the images of h_{ϵ} and h̄_{ϵ}. Then v(σ) and v̄(σ) are the images of v_{ϵ}(σ) and v̄_{ϵ}(σ). The cochains v_{ϵ} and v̄_{ϵ} have the same coboundary and it takes values in the inverse image W (a finite group) of Z_{sc}^{∗} in (G_{ϵ}^{∗})_{sc}. So we have a cocycle i with values in

V = T_{ϵ−sc} × T̄_{ϵ−sc}/W,

where we use a notational principle that admits several variants: the subscript ϵ − sc indicates inverse image in (G_{ϵ}^{∗})_{sc}. The classes inv(γ_{H}, γ_{G}; γ̄_{H}, γ̄_{G}) and inv_{ϵ}(γ_{H}, γ_{G}; γ̄_{H}, γ̄_{G}) are the images of the class of i under the homomorphisms induced by

(3.1.1)  \begin{aligned}& & U \\ & ↗ \\ V \\ & ↘ \\ & & U_{ϵ}. \end{aligned}

Dual to (3.1.1) we have

(3.1.2)  Û ; ↙ ; V̂ ; ↖ ; Û_{ϵ}.

To prove the lemma it is sufficient to show that the images of s_{U} and s_{U_{ϵ}} in V̂ are the same. The L-data for G_{ϵ}^{∗} (see 1.4) provide T ⟶^{∼} T̂ and T ⟶^{∼} ˆ T̄. We have two commutative

diagrams.

V  ⟶  U
↓  ↓
T_{ϵ−ad} × T̂_{ϵ−ad} ⟶ T_{ad} × T̄_{ad}

U_{ϵ}

T_{ϵ−ad} × T̄_{ϵ−ad}

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

Using them we extend (3.1.2) to a commutative diagram

Û ⟵ T̂_{sc} × ˆT̄_{sc}  ⟵^{∼}  T_{sc} × T_{sc}
↙
V̂ ↑ ↑
↖
Û_{ϵ} ⟵ T̂_{ϵ−sc} × ˆT̄_{ϵ−sc} ⟵^{∼} T_{ϵ−sc} × T_{ϵ−sc}.

We may write s̃ ∈ T_{sc} as the product of the image of s̃_{ϵ} in T_{ϵ−sc} with an element of the identity component R of the subgroup {x ∈ T_{sc} : α^{∨}(x) = 1, α^{∨} ∈ R(Ĝ_{ϵ}, T )}. Embed R diagonally in T_{sc} × T_{sc} without change in notation. Then we have just to show that the image of R in V̂ is trivial. But R is connected and V̂ ⟶ T̂_{ϵ−ad} × ˆT̄_{ϵ−ad} has finite kernel because T_{ϵ−sc} × T̄_{ϵ−sc} ⟶ V does. Thus it is enough to show that

R ⟶ V̂ ⟶ T̂_{ϵ−ad} × T̄_{ϵ−ad}

is trivial. That, however, is immediate, and the lemma is proved.

3.2. Remarks and notation

We assume from now on that G = G^{∗}, ϵ_{G} = ϵ, γ_{G} = γ and γ̄_{G} = γ̄. Then Θ(γ_{H}, γ; γ̄_{H}, γ̄) is a quotient Θ(γ_{H}, γ)/Θ(γ̄_{H}, γ̄). So also is each of Θ_{I}, Θ_{II}, Θ_{2} and Θ_{IV}. The embeddings T_{H} ⟶ T, T̄_{H} ⟶ T̄ being fixed, we may delete γ and γ̄ from the notation. We may further fix Borel subgroups B_{H} ⊃ B_{ϵ_{H}} ⊃ T_{H}, B ⊃ B_{ϵ} ⊃ T, B̄_{H} and so on, for which T_{H} ⟶ T, T̄_{H} ⟶ T̄ are the attached embeddings. There is then a canonical identification of the roots of T in G with those of T̄. It carries the B-positive roots to the B̄-positive ones, the (positive) roots of T in G_{ϵ} to the (positive) roots of T̄ in G_{ϵ}, and the roots from H to the roots from H. Thus we use the simpler notation R = R(G), R_{ϵ} = R(G_{ϵ}), R_{s} = R(H), and so on, for root systems. Also R(G) − R(G_{ϵ}) will be abbreviated as R(G/G_{ϵ}) and R(G) − (R(G_{ϵ}) ∪ R(H)) as R(G/G_{ϵ}, H).

In this notation Θ_{IV} (γ_{H}) is written as

∏_{α∈R(G/G_{ϵ},H)} |α(γ) − 1|^{1/2},

and so

lim_{γ_{H}⟶ϵ_{H}} Θ_{IV} (γ_{H}) = ∏_{α∈R(G/G_{ϵ},H)} |α(ϵ) − 1|^{1/2}

= lim_{γ̄_{H}⟶ϵ_{H}} Θ_{IV} (γ̄_{H}).