Orbital Integrals on Forms of SL(3), II

By R.P. Langlands and D. Shelstad

0. Introduction. In the paper [6] we described in a precise fashion the notion of transfer of orbital integrals from a reductive group over a local field to an endoscopic group. We did not, however, prove the existence of the transfer. This remains, indeed, an unsolved problem, although in [7] we have reduced it to a local problem at the identity.

In the present paper we solve this local problem for two special cases, the group SL(3), which is not so interesting, and the group SU(3), and then conclude that transfer exists for any group of type A_{2}. The methods are those of [4], and are based on techniques of Igusa for the study of the asymptotic behavior of integrals on p-adic manifolds. (As observed in [7], the existence of the transfer over archimedean fields is a result of earlier work by Shelstad.)

Since the problem is solved in so few cases, some doubts about the value of the method are justified. However, in the hands of Thomas Hales (1], [2]) it is revealing itself to be a powerful and suggestive technique, and the simple case treated here may serve as a useful introduction to his work. Moreover, the theory of automorphic forms on SU(3) or U(3) is well worth studying as an example and in its own right ([9]); many phenomena that are absent for GL(2) but present in general appear first with these groups.

1. Local transfer of Shalika germs. Recall from ([7], Section 2.1) that the problem of local transfer is to show that for any smooth, compactly supported function f on G(F) there is a function f^{H} of the same type on the endoscopic group H(F) such that

(1.1)  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 the identity in H(F). The sum is over a set of representatives for the conjugacy classes in the stable conjugacy class of which γ_{H} is an image.

Fix one γ_{G} with γ_{H} as an image, and let T_{G} be the centralizer of γ_{G}, T_{H} that of γ_{H}. Choose an admissible embedding T_{H} → T → T_{G} and let γ ∈ T ⊆ G^{∗} be the image of γ_{H}. Recall that E(T) is the image of H^{1}(T_{sc}) in H^{1}(T), and may of course be identified with E(T_{G}). The element s of the endoscopic data defining H can be transported to T̂ or to T̂_{G} and devines a character, customarily denoted κ, of X_{∗}(T), X_{∗}(T_{G}), or of E(T), E(T_{G}). Recall that Tate-Nakayama duality implies that E(T) is a subquotient of X_{∗}(T). If δ ∈ ξ(T_{G}), there is a g ∈ G(F̄) such that the cocycle {σ(g)g^{−1}} lies in T and belongs to the class of δ. Set

γ_{G}^{g} = g^{−1}γ_{G}g.

Orbital integrals on forms of SL(3) part II 2

Since Δ(γ_{H}, γ_{G}^{g}) and Φ(γ_{G}^{g}, f) depend only on δ and not on g, we write Δ(γ_{H}, γ_{G}^{δ}) and Φ(γ_{G}^{δ}, f). It is a consequence of Section 3.4 of [6] that

Δ_{loc}(γ_{H}, γ_{G}^{δ}) = κ(δ)Δ_{loc}(γ_{H}, γ_{G}),

so that the right side of (1.1) is

Δ_{loc}(γ_{H}, γ_{G}) ∑_{δ∈E(T_{G})} κ(δ)Φ(γ_{G}^{δ}, f).

For γ_{H} near the identity we may write γ = exp X, and, for any root α, define α(γ)^{1/2} to be exp α(X)/2. Then

{a_{α} = α(γ)^{1/2} − α(γ)^{−1/2}}

is a set of a-data to which we may apply the construction of [6, Section 2.3] to obtain a cohomology class

λ(T) = inv(T, γ)

in E(T). In terms of the classes introduced in [10] it is a product

inv(T, γ) = inv(γ)inv(T).

Observe that the choice of F -splitting implicit in Section 2.3 is made as in [10] and [6, Section 5.1]. To calculate Δ_{loc}(γ_{H}, γ) we may use the a-data just introduced. This yields

Δ_{II}(γ_{H}, γ) = 1

and

Δ_{I}(γ_{H}, γ) = κ(inv(T, γ)).

In addition, Δ_{1}(γ_{H}, γ) is 1 by definition, and Δ_{2}(γ_{H}, γ) is 1 near the identity. Thus, choosing the overall constant correctly, we have

(1.2)  Δ_{loc}(γ_{H}, γ) = κ(inv(T, γ))D_{G^{∗}/H}(γ),

where ([6], Section 3.6)

D_{G^{∗}/H}(γ) = D_{G^{∗}}(γ)D_{H}(γ_{H})^{−1}.

If, with the notation of [6, Section 4.2] and [3], we set

inv(T_{G}, γ_{G}) = inv(T, γ)θ(E, E′)

then the Local Hypothesis (Corollary 4.2.B of [6] yields

(1.3)  Δ_{loc}(γ_{H}, γ_{G}) = κ(inv(T_{G}, γ_{G}))D_{G/H}(γ_{G}).

The usual germ expansion on G(F) is

Φ(γ_{G}, f) ∼ ∑_{O} Γ_{O}(γ_{G})a_{O}(f).

The germ expansion for

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

is

Φ(γ_{H}, f) ∼ ∑_{O} Γ_{O}^{κ}(γ_{H})a_{O}(f)

with

Γ_{O}^{κ}(γ_{H}) = κ(inv(T_{G}, γ_{G}))D_{G/H}(γ_{G}) ∑_{δ} κ(δ)Γ_{O}(γ_{G}^{δ})

if γ_{H} is an image of an element of G. Otherwise Γ_{O}^{κ}(γ_{H}) = 0. It will be referred to as the κ germ-expansion, and the Γ_{O}^{κ} as κ-germs, even though the true parameter is the collection of endoscopic data, the character κ changing from Cartan subgroup to Cartan subgroup. Taking H = G^{∗}, so that all the κ are trivial, we obtain the stable germs Γ_{O}^{st}.

The existence of the local transfer for a group H and all f is clearly equivalent to the validity of the following assertion for all O.

Assertion A. The κ-germ Γ_{O}^{κ} is a linear combination of stable germs for H.

2. Some easy general cases. There are a few general cases for which Assertion A is either easy to verify directly or easy to deduce from known results.

The endoscopic group H, or the data defining it, will be said to be cuspidal if the maximal split subgroup S_{H} of the center of H is contained in the center of G (a condition whose precise meaning should be clear enough).

Lemma 2.1. Suppose H is not cuspidal. If local transfer exists for all endoscopic groups of all Levi factors of proper parabolic subgroups of G then it exists for H.

Proof. If an admissible embedding T_{H} → T → T_{G} exists then we say that T_{H} is an image of T_{G}. If no Cartan subgroup of H is an image, as can very well happen if, for example, G is anisotropic, then f^{H} = 0 will be the transfer of all f. Thus, suppose that some Cartan subgroup T_{H}^{0} is an image of, say, T^{0}.

We suppose that H is not cuspidal. Then S_{H} is transported by T_{H}^{0} → T^{0} to a split group S_{G} in G that is not central. The centralizer M of S_{G} is a Levi factor of a proper parabolic subgroup P over F. Any two split tori in G over F that are conjugate in G(F̄) are already conjugate in G(F). Thus if T_{H} is an image of some T_{G}, it is the image of a T_{G} contained in M.

If m ∈ M(F) set
D_{G/M}(m) = |det(1 − ad m)|_{𝔤/𝔪}|^{1/2}

and

δ_{G/M}(m) =| det m|_{𝔫}|^{1/2},

where 𝔫 is the Lie algebra of the unipotent radical of P. It follows readily from the definitions that H is also an endoscopic group for M. We denote transfer factors for M by the addition of the superscript M. We can easily arrange [7]

Δ(γ_{H}, γ) = Δ^{M}(γ_{H}, γ)D_{G/M}(γ).

Thus to prove the lemma, we need only verify the existence of a smooth compactly supported function f^{M} on M(F) such that

(2.1)  D_{G/M}(γ)Φ(γ, f) = Φ(γ, f^{M}).

However, it is well known that for this purpose we can take

f^{M}(m) = cδ_{G/M}(m) ∫_{K} ∫_{N(F)} f(k^{−1}mnk)dndk,

with a suitable constant c.

Lemma 2.2. Assertion A is valid for regular unipotent classes O and all H.

This follows readiy from Theorem 5.5.A of [6].

Lemma 2.3. Assertion A is valid for the class O = {1} and all H.

Proof. Abbreviate Γ_{{1}} to Γ_{1}. By results of Howe, Harish-Chandra and Rogawski, Γ_{1} vanishes on tori that are not anisotropic, and is a constant c_{G} on all anisotropic tori. Moreover c_{G} = c_{G^{∗}} for compatible choices of the measures. Thus it is clear that if H = G^{∗} then

Γ_{1}^{κ} = Γ_{1}^{st} = Γ_{1^{∗}}^{st},

where 1^{∗} denotes the identity in G^{∗}.

If H ≠ G^{∗} the lemma takes an even stronger form.

Lemma 2.4. If H ≠ G^{∗} then Γ_{1}^{κ} ≡ 0.

Proof. On tori that are not anisotropic this is a consequence of (2.1). On the other hand, for an anistropic torus T the group E(T) ≃ E(T_{G}) is a quotient of X_{∗}(T_{ad}). Since H ≠ G^{∗} the character κ is not trivial on E(T_{G}) and

∑_{δ} κ(δ)Γ_{1} = 0.

Corollary 2.5. If G is anisotropic modulo its center then local transfer exists for all H.

With these scraps of general information at our disposal, we now turn to forms of SL(3), excluding, because of the last corollary, anisotropic groups, and thus confining ourselves to quasi-split groups, so that we may take G = G^{∗}. We observe, moreover, that every endoscopic group H of G arises in a natural way from some H_{(sc)} for G_{sc}, and that it follows readily from [7] that if local transfer is possible for the pair G_{sc}, H_{(sc)} then it is possible for G, H. Thus we shall eventually obtain the following statement.

Theorem. If G is of type A_{2} then all pairs (G, H) admit Δ-transfer.

In treating forms of SL(3), we need only consider cuspidal endoscopic groups and subregular classes O. We now begin to examine the possible endoscopic groups.

3. Endoscopic groups. Since G is simply connected, we confine ourselves to endoscopic data (H, H, s, ξ) (see [6], (1.2)) for which H = ^{L}H, so that ξ is an embedding of ^{L}H in ^{L}G. The most important element among the data is then s. If we realize Ĝ in the usual way as P GL(3, C), then we may suppose that s is diagonal and that the diagonal matrices form part of a Γ-splitting of ^{L}H. There are then three possibilities:

(i) all eigenvalues of s are equal; (ii) two are equal to each other but not to the third, (iii) no two eigenvalues are equal.

In case (i), the group ^{L}H is ^{L}G and H = G^{∗} = G. In case (ii), we may suppose that it is the first and third eigenvalues that are equal. If α̂′ = x_{1} − x_{2}, α̂′′ = x_{2} − x_{3}, and α̂′′′ = x_{1} − x_{3} are the usual roots of T̂ then σα̂′′′ = α̂′′′ for all σ ∈ Γ. Thus σα̂′ = α̂′, σα̂′′ = α̂′′ or σα̂′ = α̂′′, σα̂′′ = α̂′. The second possibility can occur for some σ if and only if G is a unitary group, and then

s = [ \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ] .

If no two eigenvalues of s are equal then T̂ = Ĥ and H is a torus T.

When discussing a given Cartan subgroup with an image T_{H} it is convenient to fix a diagram:

T_{H} ⟶ T ⟵ T
↑
T_{G}

Then T̂ and T̂_{G} may be identified with T̂, the group of diagonal matrices in P GL(3, C). The group Γ_{T} acts on T̂.

Lemma 3.1. If H ≠ G^{∗} then the group Γ_{T} cannot contain all reflections in the Weyl group.

Proof. Since H ≠ G^{∗}, we have α̂(s) ≠ 1 for at least one root α. However, if σ_{α} is the reflection corresponding to α then there is another root β such that β̂ − σ_{α}β̂ = α̂. Thus β̂(s) ≠ σ_{α}β̂(s) and σ_{α} cannot lie in Γ_{T}.

Corollary 3.2. If G is an inner form of SL(3), H ≠ G^{∗}, and T is anisotropic then Γ_{T} is the cyclic subgroup of the Weyl group of order three and H is a torus.

Proof. Γ_{T} must be a proper subgroup of the Weyl group whose only fixed point in X_{∗}(T) is 0. Thus Γ_{T} is cyclic of order three. Since s is projectively invariant under Γ_{T} and not a scalar, its three eigenvalues must be of the form λ, ξλ, ξ^{2}λ where ξ^{3} = 1, ξ ≠ 1. Thus Ĥ is a torus, and so is H.

Since T̂ has been identified with T̂ by means of the fixed diagram, the group X_{∗}(T) is identified with triples of integers (x, y, z) whose sum is zero.

Corollary 3.3. If G is SU(3), T is anisotropic, and the element s has exactly two eigenvalues equal then we may suppose that Γ_{T} is one of the two groups:

(a)  (x, y, z) → ±(x, y, z);
(b)  (x, y, z) → ±(x, y, z), ±(z, y, x).

Proof. Choose the diagram so that (1, 0, −1) is a root that is 1 on s.

Lemma 3.4. If G is SU(3) and H = T is an anisotropic torus, then Γ_{T} is a group of order six containing the cyclic permutations of order three. It can also be assumed to contain

(x, y, z) → (−z, −y, −x).

Proof. The argument of Lemma 2.1 implies that Γ_{T} contains no reflections. Thus the intersection of Γ_{T} with the Weyl group is trivial or cyclic of order three. If it is trivial then Γ_{T} must be (x, y, z) → ±(x, y, z), because T is anisotropic, but then all eigenvalues of s_{T} have the same square, so that two are equal. Then H is not a torus. Thus Γ_{T} is of order six and, with a suitable choice of diagram, either contains (x, y, z) → (−z, −y, −x) or the direct product of the cyclic subgroup of order three with the group of order two generated by (x, y, z) → −(x, y, z). There is however no s_{T} invariant under the second group that is not scalar.

There is one more lemma to be noted. Its proof is immediate.

Lemma 3.5. If H is cuspidal but is neither G^{∗} nor a torus, then G is SU(3) and s is of type (ii).

4. Recapitulation. Up to conjugacy there are only finitely many Cartan subgroups of H over F. Suppose T_{H} is one of them, and 𝔱_{H} its Lie algebra. As a consequence of a theorem of Harish-Chandra, there is a neighborhood V_{H} of 0 in 𝔱_{H} such that for regular X ∈ V_{H} and t ∈ F^{∗}, |t| < 1,

Γ_{O}^{st}(exp t^{2}X) = |t|^{d−l}Γ_{O}^{st}(exp X),

if O is a unipotent conjugacy class and d = d_{O} is defined by the condition that 2d + l is the dimension of the centralizer of any element in it. The integer l is the rank of H, and 2d_{I} + l is the dimension of H.

A similar assertion is valid for the κ-germ. Thus if for a given unipotent class O in G(F) there is a collection of unipotent classes O_{1}, …, O_{n} in H(F) and complex numbers a_{1}, …, a_{n} such that for every Cartan subgroup T_{H} and every regular ray {tX | t ∈ F^{∗}} in 𝔱_{H} we have

Γ_{O}^{κ}(exp tX) = ∑_{i=1}^{n} a_{i}Γ_{O_{i}}^{st}(exp tX)

for t sufficiently small, then there is an equality of germs

Γ_{O}^{κ} = ∑_{i=1}^{n} a_{i}Γ_{O_{i}}^{st}

with

d_{O_{i}} = d_{O}

for all i. Hence the theorem of Harish-Chandra allows us in principle to apply the methods of Igusa.

For forms of SL(3) only the subregular classes O are not dealt with by the remarks of Section 2; so we suppose henceforth that O is subregular. There are two lemmas to be proven. The first is simple to state.

Lemma 4.1. If H is a torus and O is subregular then

Γ_{O}^{κ} ≡ 0.

Otherwise we may suppose that G is SU(3) and

s = [ \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ] .

Then the derived group of H is SL(2). Moreover d_{O} = 1 if O is subregular and d_{O_{i}} = 1 if and only if O_{i} = {1}.

Lemma 4.2. If G is SU(3), H is cuspidal but neither G^{∗} nor a torus, and if O is subregular then for some a ∈ C

Γ_{O}^{κ} = aΓ_{1}^{st}.

Recall that Γ_{1}^{st}, a stable germ for H, has been calculated in [5]. It is 0 on Cartan subgroups whose intersection with H_{der} is not anisotropic. If, however, T_{0} = T_{H} ∩ H_{der} is anisotropic then Γ_{T_{0}} is a group of order two:

Γ_{T_{0}} = {1, σ}.

Take Q = Q_{T_{H}} to be the form of P^{1} × P^{1} defined by the cocycle:

a_{σ} : (x, y) → (y, x).

Here x, y are inhomogeneous coordinates on P^{1} × P^{1}. Then, apart from a constant that is the same for all Cartan subgroups, Γ_{1}^{st} is the principal-value integral ([5])

(4.1)  |α(X)| ∮_{Q(F)} \frac{dxdy}{|x − y|^{2}}

times

D_{H}^{−1}(γ_{H}).

Here γ_{G} = e^{X} and α is one of the two roots of H_{der}. Thus, we prefer to study

D_{H}(γ_{H})Γ_{O}^{κ}(γ_{H}) = κ(inv(T_{G}, γ_{G}))D_{G}(γ_{G})Σκ(δ)Γ_{O}(γ_{G}^{δ}).

5. The method of Igusa. For a given T_{G} the method of Igusa [4] yields (for forms of SL(3)) an asymptotic expression for

D_{G}(γ_{G})Σκ(δ)Γ_{O}(γ_{G}^{δ}) = Σκ(δ)D_{G}(γ_{G}^{δ})Γ_{O}(γ_{G}^{δ})

along rays. One begins with

Σκ(δ)D_{G}(γ_{G}^{δ})Φ(γ_{G}^{δ}, f),

which in the notation of [4] is F^{κ}(γ, f), and applies Proposition 1.1 of [4].

The theorem of Harish-Chandra implies that the terms appearing in that proposition are 0 for r > 1 or β < 1. For r = 1, β ≧ 1 the factor F_{r}(θ, β, f) is a principal-value integral

∮_{E(θ,β)} h_{E}|ν_{E}|

over the union E(θ, β) of the varieties E in E(θ, β). Observe that the factor A_{r}(M) appearing in Lemma 1.3 of [4] has been incorporated into the function h_{E}.

The morphism π of [4] maps E onto the closure Ō_{E} of a unipotent orbit O_{E} in G. Departing from the notation of [4], we let Ê be the inverse image of O_{E} in E. The set O_{E}(F) is a stable orbit O^{st} and thus a finite union of conjugacy classes.

For a given β, the sum over θ

(5.1)  κ(inv(T, γ_{G})) ∑_{θ} θ(λ)|λ|^{β−1} ∮_{E(θ,β)} h_{E}|ν_{E}|

is equal to

(5.2)  ∑_{d_{O}=β−1} {∮ f(η)|η|} D_{H}Γ_{O}^{κ}.

It was observed to us by Thomas Hales that, when we are concerned with a particular O, we may as well suppose that f vanishes on all unipotent orbits O′ with d_{O′} ≧ d_{O} other than O itself. This has two advantages at present, when we are dealing with forms of SL(3) and a subregular O. First of all, the blow-up otherwise entailed by Lemma 4.5 of [4] is unnecessary. Secondly, if β = d_{O} + 1 and E ∈ ℭ(θ, β) then, by Lemma 4.7 and 4.8 of [4], necessarily O ⊆ O_{E}. Hence in (5.1) the integrals may be taken over

Ê(θ, β) = ∪_{E∈ℭ(θ,β)}Ê.

Moreover O_{E} is smooth and, as we shall see, Ê(θ, β) → O_{E} is flat, so that (5.1) is equal to

(5.3)  ∑_{θ} θ(λ)|λ|κ(inv(T, γ_{G})) ∫_{O_{E}(F)} f(n) {∫_{E_{n}(θ,β)} h_{n}|ν_{n}|} ,

where E_{n}(θ, β) is the inverse image of n in Ê(θ, β).

To prove Lemma 4.1 it is enough to show that

(5.4)  ∮_{E_{n}(θ,2)} h_{n}|ν_{n}| = 0

for the pertinent T and κ; to prove Lemma 4.2 it is enough to show that

(5.5)  ∑_{θ} θ(λ)|λ|κ(inv(T, γ_{G})) ∮_{E_{n}(θ,2)} h_{n}|ν_{n}| = a|α(X)| ∮_{Q(F)} \frac{dxdy}{|x − y|^{2}} ,

a being a constant that depends on the endoscopic data, and perhaps on n and θ, but not on the particular T under consideration. Moreover, γ_{G} = exp λX and α is a root of H_{der}.

The remainder of the note is devoted to the explicit description of h_{n}, ν_{n}, and the varieties E_{n}(θ, 2), based, of course, on the construction of [4], and to the calculation of the principal-value integrals. Observe that if β(E) = 2 then, in the notation of [4], E is one of E′_{1}, E′′_{1}, or E_{6}.

6. The fibres. If G is SL(3) then there is no supplementary blow-up, and the divisor E_{6} does not appear; if G is SU(3) the divisors E′_{1}, E′′_{1} have no rational points and thus may be discarded. Therefore it is convenient to treat the two cases separately, beginning with SL(3).

Associated to a subregular n over F in SL(3) are a distinguished point and a distinguished line in P^{2}. The point P is the range of n − 1 and the line l its null space. On the other hand, a Borel subgroup B of SL(3) is determined by a point q on a line m in P^{2}, and n ∈ B if and only if p ∈ m or q ∈ l. Let M_{n} be the join at p = l of the variety P_{l} of points on l with the variety P_{p} of lines through p, so that M_{n} is the union of two projective lines crossing normally at p = l, and parameterizes the Borel subgroups containing n.

There is a set-theoretical mapping of E_{n} = E_{n}(2), the inverse image of n in Ê′_{1} ∪ Ê′_{2}, to M_{n}. To a point in E_{n} there is associated a star and the Borel subgroup B = B(W_{+}) in this star lies in M_{n}. To investigate E_{n}(2) over the inverse image of a neighborhood of B, we choose B_{0} to be defined by p ∈ l and B_{∞} to be opposite to B. Then the inverse image is contained in

G × S_{1}(B_{∞}, B_{0}) × N_{∞}.

We may choose X_{α′}, X_{α′′} so that

n = exp w_{0}[X_{α′}, X_{α′′}].

Observe that w_{0} ≠ 0.

If, as in Section 4 of [4], we take x, y, V as coordinates on S_{1}(B_{∞}, B_{0}) then the coordinates z′_{i}, z′′_{i} figuring in the diagram on p. 489 of [4] can be calculated in terms of x, y, V. The results can be summarized in a similar diagram

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Thus there is a coordinate x, attached to each of six rays r. The extended Weyl group, viz, the Weyl group together with outer automorphisms, acts by replacing x_{r} by

ϵ(ω)xω_{r}^{−1}.

An element σ of the Galois group acts by replacing x_{r} by

ϵ(σ_{T})σ(xσ_{T}^{−1}r).

Here ϵ(ω) is the determinant of ω, and ϵ(σ_{T}) the determinant of σ_{T}. Thus the varieties S′(B_{∞}, B_{0}), S_{1}(B_{∞}, B_{0}), R(B_{∞}, B_{0}) are all defined over F.

On the open set Y_{1}^{11}(B_{∞}, B_{0}) of Section 4 of [4] we have

(6.2)  uU = b(λ)yV w, νU = c(λ)xV w.

If

n′ = exp uX_{α′} exp vX_{α′′} exp w[X_{α′}, X_{α′′}]

and if

n = (n′)^{n_{∞}}, n_{∞} ∈ N_{∞}

then m ∼ B′ = B_{0}^{n_{∞}} is the image in M_{n} and w = w_{0}. The coordinates u, v serve as coordinates of n_{∞} or as coordinates on M_{n}. We conclude from Equation 6.2 and Equation 4.6 of [4] that E_{n} → M_{n} is a morphism, and that it is smooth off V = 0.

The condition V = 0 is more invariantly defined as the condition that the point in S_{1} is defined by a point of type B_{3} (see Lemmas 3.13, 3.14 of [4]). Since the B_{j} are cyclically permuted by the Galois group, no such point has coordinates in F and E_{n} → M_{n} is smooth at every point of E_{n}(F). Moreover, as we shall see in the next section, κ′_{1} ≠ κ′_{2}. Consequently

(6.3)  ∮_{E_{n}(θ,2)} h_{n}|ν_{n}| = ∮_{M_{n}} {∮_{F_{m}} k_{m}|ν′_{m}|} |ν′′|.

Here ν′′ is any non-vanishing form on the base and k_{m}, ν′_{m} are determined by restriction and division.

The conditions for rationality are deduced from the diagram and the equation following (3.4) of [4]. For the F -valued points in F_{m}, the coordinate V is neither 0, 1, nor ∞ and the map that sends a point to (−1, V, 1 − V) or simply to V identifies F_{m}(F) with the P^{1} associated to elements of trace zero in the cyclic cubic extension K associated to the torus T.

Passing to SU(3) we observe first that if n ∈ G(F) is subregular then there is a unique Borel subgroup B_{0} over F containing it. If B_{∞} is defined over F and opposed to B_{0} then E_{n} = E_{n}(2) is contained in

G × S_{1}(B_{∞}, B_{0}) × N_{∞}.

The only pertinent divisor E is now E_{6}; it is obtained by a blow-up along the intersection of E′_{1} and E′′_{1}. Hence it fibres over this intersection with fibre P^{1}. The fibre E_{n} is contained in n × S_{1}(B_{∞}, B_{0}) × 1 and the projection on S_{1}(B_{∞}, B_{0}) of its intersection I_{n} with E′_{1} ∩ E′′_{1} is (see Section 3 of [4])

S′(B_{∞}, B_{0} ×_{T_{0}}U) − 0 ≃ R(B_{∞}, B_{0}).

Let ρ be the element of the Galois group such that

ρ_{T} : (x, y, z) → (−z, −y, −x)

as given by Lemma 3.4. Thus a condition for rationality is that

(6.4)  ρ(V − 1) · (V − 1) = 1,

so that V − 1 = ρ(R)R^{−1}. The map that sends a point with coordinate V to

(6.5)  R(−1, V, 1 − V)

identifies I_{n}(F) with the P^{1} defined by elements of trace zero.

In the case of Corollary 3.3 (a), the element ρ does not exist, and the condition for rationality to be imposed on V is that V ∈ F. In the case of Corollary 3.3 (b), the number R can be taken to be fixed by the element σ with

σ_{T} : (x, y, z) → −(x, y, z),

and the coordinate V lies in K_{0}, the fixed field of σ.

It is easy to see that

(6.6)  ∮_{E_{n}(θ,2)} h_{n}|ν_{n}| = ∮_{l_{n}} {∮_{F_{x}} k_{x}|ν′_{x}|} |ν′′|,

x denoting a point of I_{n}, and F_{x} the fibre over x. Observe that at a point of I_{n} where V is not 0, 1, nor ∞ we may use as coordinate z = x/y. The condition for rationality is somewhat elaborate to describe.

Think of the chambers, W_{i}, 0 ≦ i ≦ 5, as oriented in the counterclockwise direction for i even and in the clockwise direction for i odd. Every element of the extended Weyl group takes W_{0} either to a W_{i} or to a −W_{i}, the chamber W_{i} with the opposite orientation. We assign to a chamber a function φ(W_{i}) of V as follows:

\begin{aligned}φ(W_{0}) & = 1; φ(W_{1}) = \frac{−1}{V}; φ(W_{2}) = V^{−1}(1 − V)^{−1}; \\ φ(W_{3}) & = (V − 1)^{2}; \\ φ(W_{4}) & = −V (1 − V)^{−2}; φ(W_{5}) = V (1 − V)^{−1}; \\ φ(−W_{i}) & = φ(W_{i})^{−1}. \end{aligned}

The positive chamber is W_{0} = W_{+}. The condition for rationality of a point in I_{n} with coordinate V is that

(6.7)  ρ(φ(σ_{T} W))φ(ρ_{T} W) = φ(ρ_{T} σ_{T} W) ∀ρ, σ, W.

The condition for rationality of a point on the fibre over a rational base point is that

(6.8)  σ(z) = φ(σ_{T} W_{+})z^{ϵ_{σ}} ∀σ.

Here ϵ_{σ} is 1 if σ fixes E and −1 if it does not. The field E is the quadratic field over which the hermitian form giving the group is defined.

7. The forms. It is sufficient to calculate them where V is finite and not 0 nor 1. Beginning with the form ν_{n}, we make use of Lemmas 2.8 and 2.12 of [4]. We may suppose that the form ∧φ_{i} appearing in Lemma 2.8 is ω_{T} ∧ du ∧ dv ∧ dw. Then the form ω^{0} on Y^{0} used to define F^{κ}(γ, f) is the product of dλ ∧ du ∧ dv ∧ dw with the restriction of ∧ω_{j} to N_{∞}.

The form on N_{∞} need not be investigated further. To describe the first factor in other coordinates we need to calculate the jacobians:

\frac{∂(λ, u, v, w)}{∂(u, v, V, w)}; \frac{∂(λ, u, v, w)}{∂(x, y, V, w)}; \frac{∂(λ, u, v, w)}{∂(z, y, V, w)}.

This is a straightforward matter. If G = G^{∗} is so defined that we may take B^{∗} to be the group of uppertriangular matrices and T^{∗} to be the group of triangular matrices, then the diagram (2.0) of [4] identifies the tangent space to the curve C with a line

F [ \begin{matrix}α & 0 & 0 \\ 0 & β & 0 \\ 0 & 0 & γ \end{matrix} ] .

We choose α, β, γ so that λ is also a natural coordinate on the line; then b = b(0) = α − β, c = c(0) = β − γ, d = d(0) = α − γ. By assumption, none of these numbers is zero. The three determinants are the product of a factor of the form 1 + E(λ), E(0) = 0, with

−\frac{uv(α − γ)}{(α − β)(β − γ)V^{2}w}; AxyV^{2}w^{3}; Azy^{3}V^{2}w^{3}.

Here

A = −\frac{(α − β)(β − γ)(α − γ)}{((α − γ) − V (β − γ))^{4}} = −\frac{(α − β)(β − γ)(α − γ)}{U^{4}}.

Thus the form γ′_{m} that appears in (6.3) may be taken to be (see Section 1 of [4])

(7.1)  −(α − β)(β − γ)(α − γ)\frac{dV}{U^{2}} = (α − β)(α − γ)\frac{dU}{U^{2}} ,

for the form ν appearing on p. 469 of [4] is the product of this with wv^{−1} (at the divisor u = 0) or wu^{−1} (at the divisor v = 0) and the invariant form on N_{∞}.

It is convenient to choose (α_{0}, β_{0}, γ_{0}) = (α_{0}, σ(α_{0}), σ^{2}(α_{0})) of trace zero and linearly independent of (c, −d, b), and to set

U_{1} = \frac{d − cV}{γ_{0} + β_{0}V} .

Then U_{1} takes rational values at rational points, and (7.1) becomes

(7.2)  −(α − β)(β − γ)(γ − δ)(cγ_{0} + dβ_{0})\frac{dU_{1}}{U_{1}^{2}} .

On the divisor E_{6} the prescription of Section 1 of [4] leads to the form

(7.3)  (α − β)(β − γ)(γ − α)\frac{dV}{U^{2}} · \frac{dz}{z}

on the fibre, the form on the base being the product of wdw with the invariant form on N_{∞}. Thus we have the form (7.2) on I_{n} and the form dz/z on the fibre F_{x}. If the endoscopic group is a torus, we may once again use the coordinate U_{1}. At a rational point,

U_{1} ∈ E and U_{1} = U_{1}(U_{1} − 1)^{−1},

the bar denoting conjugation over F.

It is also convenient to replace z by

z_{1} = \frac{z}{V − 1},

for then the condition for rationality becomes z_{1} ∈ E, z̄_{1} = z_{1}^{−1}.

8. The functions. The functions h_{n} are essentially the functions on m_{κ} of Section 2 of [4] and are to be calculated as in Section 5 of that note; or rather as in Section 5.4 and Section 5.5 of [6]. Since we have multiplied by D_{H}(γ_{H}), the function m_{k}(e(·)), of [4] becomes the function Δ(·) of [6]. We calculate with Δ(·), obtaining thereby not the functions h_{n} of (5.4) and (5.5) but the functions

κ(inv(T, γ)h_{n},

without for the moment the factor A_{r}(M). Since we are close to the identity, Δ(·) may be taken to be

(8.1)  Δ_{I}(γ_{H}, π(·))Δ_{II}(γ_{H}, π(·))Δ_{1}(γ_{H}, π(·)).

On putting the first and third factors together as in Section 5.5 of [6] we obtain the value of s_{T} on the cocycle

(8.2)  σ → ζ_{σ} = ∏_{1,σ}^{ρ} [\frac{−a_{α}}{z(σ, α)}]^{α^{ν}} .

We choose, as in Section 1,

a_{α} = α(γ)^{1/2} − α(γ)^{−1/2},

so that the second factor of (8.1) becomes 1. (Notice that the symbols α, γ have two conflicting meanings.)

We begin with SL(3), and the endoscopic groups given by anisotropic tori. Then Γ_{T} is cyclic of order three, and we take the generator given by

σ : (α, β, γ) → (β, γ, α).

It is enough to calculate the cocycle (8.2) on this generator. We write σ = ω′′ω′, and find that the roots appearing in (8.2) are α′′ and α′′′ = α′ + α′′. The factors z(σ, α) are calculated in terms of the coordinates in (6.1). Thus

ξ_{σ} = [\frac{−a_{α′′}(V − 1)}{x}]^{α̂′′} [\frac{−a_{α′′′}}{yV} ]^{α̂′′′} .

In terms of the coordinates u, v this is

ξ_{σ} = [\frac{−a_{α′′}(V − 1)V wc(λ)}{vU}]^{α̂′′} [\frac{−a_{α′′′}wb(λ)}{uU}]^{α̂′′′} .

Since λ is close to 0, we may replace c(λ) by c(0) = c and b(λ) by b(0) = b. Moreover, on a fibre over a point of C close to the identity, we may replace a_{α′}, a_{α′′}, a_{α′′′} by λb, λc, and λd respectively, so that ζ_{σ} becomes

[\frac{−c}{b}(V − 1)u]^{α̂′′} ]\frac{−d}{c} \frac{v}{V} ]^{α̂′′′} .

Let K be the splitting field of T. The standard identification

(8.3)  F^{×}/Nm_{K/F} K^{×} ≃ H^{1}(T)

is obtained by choosing for a given k ∈ F^{∗} a triple e, f, g in K^{∗} with efg = k and then taking the cocycle whose value at σ is

(8.4)  [ \frac{e}{σ(g)} ; \frac{f}{σ(e)} ; \frac{g}{σ(f)} ] .

Of course, T is identified with the group of diagonal matrices of determinant 1. The contributions of u, v, w to ξ_{σ} may be factored out as a cocycle of this type with e = v, f = uv, g = 1, for u, v, w lie in F^{×}.

The identification (8.3) allows us to identify κ with a character of F^{×}, necessarily non-trivial. The contribution of u, ν, w to Δ(·) is then

κ(uν^{2}) = κ(u)κ(ν)^{−1}.

Thus κ′_{1} = κ^{−1}, κ′′_{1} = κ. In particular, κ′_{1} ≠ κ′′_{1}.

The remaining contribution to ζ_{σ} is more disagreeable. Fortunately, we need not calculate it, for it is independent of u, ν, w, as is the form (7.2). Thus the inner integral appearing in (6.3) becomes

∮_{F_{m}} k_{m}(ν′_{m}) = cκ^{2}(ν)

for θ = κ^{−1},

∮_{F_{m}} k_{m}(ν′_{m}) = cκ^{−2}(u)

for θ = κ. The constant c does not depend on n. Then the integral over the base becomes, apart from this constant,

|w| ∮_{P_{1}} κ^{2}(ν)\frac{|dν|}{|ν|} = |w| ∮_{P_{1}} κ^{−2}(u)\frac{|du|}{|u|} = 0.

(These simple principal values are calculated in Lemma 1.C of [5].)

We observe in passing that when we are dealing with table orbital integrals, the cocycle is no longer pertinent because κ = 1. Then the fibre integrals become essentially

∮_{P_{1}} \frac{|dU_{1}|}{|U_{1}|^{2}} = 0.

However, the formula (6.3) is no longer valid, for κ′_{1} = κ′_{2} = 1. Now the calculation must be made as in the appendix. The total contribution of the term (A.3) is 0, but the total contribution of the terms (A.2) is given by an integral of the form (A.4).

Lemma 4.1 is now proven for groups of type SL(3). We turn to SU(3) beginning with endoscopic groups that are tori. The following lemma is easy and simplifies the calculations.

Lemma 8.1. If G is SU(3) and H is an anisotropic torus, then the restriction map

H^{1}(Gal(K/F), T (K)) → H^{1}(Gal(K/E), T (K))

is injective.

The field E appearing here is the quadratic extension of F associated to G. To prove the lemma, one observes that the order of H^{−1}(Gal(K/F), X(T)) clearly divides three, and then appeals to Tate-Nakayama theory.

This lemma allows us to calculate the value of κ on the cocycle (8.2) as before, except that u and v are no longer the appropriate coordinates. They are to be replaced by z_{1}, y. Thus if σ is chosen in the same way, then

ζ_{σ} = [\frac{−a_{α′′}}{z_{1}y} ]^{α̂′′} [\frac{−a_{α′′′}}{yV} ]^{α̂′′′} .

When λ is close to 0, we may replace a_{α′′} by λc, a_{α′′′} by λd, and express λ with the help of (4.3) of [4] to obtain

(8.5)  ζ_{σ} = [\frac{−cV (V − 1)yw}{U} ]^{α̂′′} [\frac{−dz_{1}(V − 1)yw}{U} ]^{α̂′′′} .
If ρ is such that ρ_{T} : α′, α′′, α′′′ → α′′, α′, α′′′ then the rationality conditions for y are:
σ(y) = yV; ρ(y) = zy or ρ(yR^{−1}) = z_{1}yR^{−1}.

Since z̄_{1} = ρ(z_{1}) = z_{1}^{−1}, we can find r ∈ E^{×} such that ρ(r) = z_{1}r. Let y = y_{0}Rr. If K_{0} is the fixed field of ρ then y_{0} ∈ K_{0} and

σ(y_{0}) = y_{0}rRσ(rR)^{−1}V = y_{0}Rσ(R)^{−1}V.

Choose one such y_{0}. Then all others are of the form ty_{0}, t ∈ F. It is t rather than y that is the local coordinate defining E_{6}.

The contribution of t to (8.5) is of the form (8.3) with e = t^{−1}, f = t^{−2}, g = 1. Thus κ_{6} = 1 and the θ appearing in (6.6) are of order two.

The inner integral of (6.6) is taken over the projective line with coordinate z_{1} and rationality condition z̄_{1} = z_{1}^{−1}. The dependence of (8.5) on z_{1} is through z_{1} and r, and

σ → [ \begin{matrix}z_{1}r \\ & r \\ & & z_{1}^{−1}r^{−2} \end{matrix} ]

is of the form (8.4) with e = z_{1}r, f = z_{1}r^{2}, g = 1, efg = z_{1}^{2}r^{3}.

K is the extension K = K_{0}E of F, and there is a diagram

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The character κ may be regarded as a non-trivial character on the group of order three,

E^{∗}/Nm_{K/E}K^{∗},

and

z_{1} → κ(z_{1}^{2}r^{3}) = κ(z_{1})^{−1}

as a character on

{z_{1} ∈ E^{∗} | z̄_{1} = z_{1}^{−1}} = E^{1}.

Since

λ = z_{1}y^{2}V (V − 1)U^{−1}w

the procedure of Section 1 of [4] leads to an inner integral in (6.6) that is the product of a function on the base and

(8.6)  ∮_{P_{1}} θ(r^{2}z_{1})^{−1}κ(z_{1})^{−1} \frac{|dz_{1}|}{|z_{1}|}.

This is the integral of θ(r^{2}z_{1})κ(z_{1})^{−1} over the group E^{1} with respect to the Haar measure. To formulate the result of the procedure this way it has been necessary to extend θ from F^{×} to E^{×}.

Observe that θ(r^{2}z_{1}) is independent of the choice of r, and that it is a character on E^{1} of order two. Thus to show that (8.6) vanishes we need only verify that κ is not trivial on E^{1}. By Lemma 3.4 the field E is the maximal abelian subfield of the Galois extension K. Thus, by local classfield theory

Nm_{K/F} K^{×} = Nm_{E/F} E^{×}.

Moreover,

[E^{×}; Nm_{K/E}K^{×}] = 3.

Consequently

[E^{1} : E^{1} ∩ Nm_{K/E}K^{×}] = 3,

and κ is not trivial on E_{1}.

It remains to treat the case that H is cuspidal but not a torus. We may suppose that ±α′′′ are the roots that take the value 1 on s. Only anisotropic tori are to be considered. According to Corollary 3.3 there are two possibilities. Let σ now be the element such that

σ_{T} : (x, y, z) → −(x, y, z).

In case (b) let K_{0} be the fixed field of σ.

Lemma 8.2. The restriction map

H^{1}(Gal(K/F), T (K)) → H^{1}(Gal(K/K_{0}), T (K))

is injective.

Let ρ_{T} : (x, y, z) → −(z, y, x). Then under the restriction map
(8.7)  H^{−1}(Gal(K/F), X_{∗}(T)) → H^{−1}(Gal(K/K_{0}), X_{∗}(T))

the element μ = (1, −1, 0) in X_{∗}(T) is sent to

μ + ρμ = (1, 0, −1).

The group on the left of (8.7) is of order two and is generated by the class of μ^{∗}, the group on the right is of order four and the class of (1, 0, −1) is not zero.

In case (a) the field K_{0} is F and H^{1}(Gal(K/K_{0}), T (K)) consists of all
(8.8)  [ \begin{matrix}e \\ & f \\ & & g \end{matrix} ]

with e, f, g ∈ K_{0}^{×}, efg = 1, e, f, g modulo Nm_{K/K_{0}}K^{×}.

Consider the number

\frac{−bw}{V d/c}.

Since σ(b) = −b, σ(c) = −c, σ(d) = −d and σ(w) = −w, it belongs to K_{0} in case (b) and to F in case (c). Moreover in case (b), ρ(w) = −w and

ρ (\frac{−bw}{V d/c}) = cw (\frac{V}{V − 1})^{−1} (−\frac{d}{b}) = (V − 1) (\frac{−bw}{V d/c}) .

Thus in both cases we may take

R = \frac{−bw}{V d/c}.

The two rationality conditions are:

V_{1} = RV ∈ F; z̄_{1} = z_{1}^{−1}.

The third is

σ(y) = \frac{x}{1 − V} = −z_{1}y,

and in case (b)

ρ(y) = x = zy = z_{1}(V − 1)y.

Thus

−σ(yR^{−1}) = ρ(yR^{−1}) = z_{1}yR^{−1}.

Choose r ≠ 0 such that

σ(r) = ρ(r) = z_{1}r

and a ≠ 0 such that

ρ(a) = a = −σ(a).

Then

y = atrR,

and t ∈ F is the correct local coordinate at y = 0. Notice that a/d ∈ F, indeed, that a can be taken equal to d.

Since σ_{T} has changed, the formula (8.5) for the value of the cocycle (8.2) at σ has to be modified. It becomes

ζ_{σ} = [\frac{−a_{α′}}{(1 − V)y} ]^{α̂′} [\frac{−a_{α′′′}(1 − V)}{V x}]^{α̂′′′} [\frac{a_{α′′}}{y} ]^{α̂′′} .

For λ close to 0, this may be replaced by

(8.9)  [\frac{bz_{1}V yw}{U} ]^{α̂′} [\frac{−d(1 − V)yw}{U} ]^{α̂′′} [\frac{cz_{1}ywV (V − 1)}{U}]^{α̂′′} .
The characters on H^{1}(Gal(K/K_{0}), T (K)) are given by choosing two characters η_{1}, η_{2} of
K_{0}^{×}/Nm_{K/K_{0}}K^{×}

and then sending the class given by e, f, g to η_{1}(e)η_{2}(g). It is the character κ defined by s (or its extension, in case (b)) that is of concern. Since

s = [ \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ]

and the Tate-Nakayama isomorphism sends (l, m, n) to the class of (8.8) with

e = h^{l}, f = h^{m}, g = h^{n}, h ∈ K_{0}^{×}, h ∉ Nm_{K/K_{0}}K^{×},

the character κ corresponds to η_{1} = η_{2} ≠ 1 in case (a). In case (b) the injection (8.7) is such that we must have η_{1} ≠ η_{2} in order that κ not be trivial, for the class of (l, m, n) = (1, 0, −1) is in the image of (8.7).

Since the contribution of t to (8.9) is of the form (8.8) with e = t^{2}, f = 1, g = t^{−2}, the character κ_{6} = 1, and the θ that appear in (6.6) are quadratic. The contribution of z_{1} is through z_{1} and r and is given by

e = z_{1}r^{2} = rσ(r), f = 1, g = e^{−1}.

Thus the inner integral of (6.6) is, as before, essentially the integral over the group E^{1} of the character

θ(r^{2}z_{1}) = θ(rσ(r)).

If this is not to be 0, then θ must be 1 or the character η associated to the quadratic extension E/F. Suppose henceforth that this is so. Then we may replace the inner integral in (6.6) by 1.

For the integral over the base it is convenient to introduce the variable

V_{1} = −bw · \frac{V}{V − d/c} = bcw \frac{V}{U} = RV.

According to (7.3) the pertinent form to take on the base is still (7.1). In terms of V_{1} it becomes w^{−1}dV_{1}. The factor w^{−1} may be incorporated into the integral over the unipotent orbit, and therefore ignored, although it will in fact be cancelled in the next section. Moreover,

λ = xy \frac{V}{U} w = z_{1}(atrR)^{2}b^{−1}c^{−1}V_{1}(V − 1)

so that the function to be integrated is the product of

(8.10)  θ^{−1}(a^{2}R^{2}(V − 1)V_{1}b^{−1}c^{−1})

with the value of κ on the cocycle given by

(8.11)  σ → [\frac{−aRV_{1}}{c(1 − V)}]^{α̂′} [\frac{−adR(1 − V)V_{1}}{bcV} ]^{α̂′′′} [\frac{aRV_{1}}{b} ]^{α̂′′} .

When expressing (8.11) in the form (8.8) the factors R^{2} and V_{1}^{2} may be dropped, as may −a^{2} = aσ(a) or −c^{2} = cσ(c) and −b^{2}. This yields

e = \frac{d}{b}V^{−1}, f = \frac{c}{b}(V − 1), g = −\frac{c}{d} · \frac{V}{1 − V} .

Consider first the case that θ is η. Then we obtain in case (a)

(8.12)  ∮ η(V_{1})|dV_{1}|.

To treat the case (b), we observe that

η_{1}η_{2} = η ∘ Nm_{K_{0}/F} ,

and that

η_{1}(e)η_{2}(g) = η_{1}(e)η_{2}(e) = η (\frac{d^{2}}{bc} ) η(Nm_{K_{0}/F} V),

because

bρ(b) = bc, dρ(d) = d^{2}.

Moreover

Nm_{K_{0}/F} (V − 1) = (V − 1)ρ(V − 1) = 1

and

Nm_{K_{0}/F} V = (V/(V − 1)R)^{2}R^{2}(V − 1) ∈ (F^{×})^{2}R^{2}(V − 1).

Thus we again obtain (8.12). It is one of the simple principal-value integrals calculated in Lemma 1.C of [5] and equals 0. Hence we may take θ = 1.

Until now, we have been able to ignore the factor A_{r}(M) of Lemma 1.3 of [4] that is to be incorporated into h_{E}. However, to obtain the correct expression for (5.5), the integral (8.12) has to be multiplied by A_{r}(M). Since a_{6} = 2 and the principal part of (1 − t^{2})^{−1} at t = 1 is (1 − t)^{−1}/2, the procedure of [4] yields

A_{r}(M) = \frac{1}{2}.

9. Final calculations.— To put the left side of (5.5) in a form that can be compared directly with the right, we first observe that (8.12) is also 0 when η is replaced by 1. Thus we may add a constant times

\frac{1}{2} ∮ ∮ \frac{|dz_{1}|}{|z_{1}|}|dV_{1}|

to our integral representation of (5.5). This yields |λ| times

(9.1)  |bcd| ∮ ∮ \frac{|dz_{1}|}{|z_{1}|} \frac{dV}{|U|^{2}} .

The |w| that appears in the change of variables has cancelled the factor |w|^{−1} from the previous section. The integral is, however, to be taken over

(9.2)  \{(z_{1}, V) : z_{1} ∈ E^{1}, \frac{b}{c}(V − 1) ∈ Nm_{E/F} E^{×}\}

in case (a), and over

(9.3)  {(z_{1}, V) : z_{1} ∈ E^{1}, σ(V) = V, ρ(V − 1) = (V − 1)^{−1}, bcd^{−2}V ρ(V) ∈ Nm_{E/F} E^{×}}

in case (b). Thus we have implicitly extended, but in a rather simple way, the notion of principal value. Since |λd| is the factor |α(X)| occurring on the right side of (5.5) we suppress the |d| and show that what remains is a constant times

(9.4)  ∮_{Q(F)} \frac{dxdy}{|x − y|^{2}} .

The symbols x, y are now free of any earlier meaning and refer solely to coordinates on Q. Choose δ ∈ E^{×}, δ ∉ F^{×} and set

x′ = \frac{x − δ}{x − δ̄}, y′ = \frac{y − δ}{y − δ̄}.

Then the rationality conditions on s′, y′ are

(9.5)  \begin{matrix}σ(x′) = y^{′−1}, & σ(y′) = x^{′′−1} \\ ρ(x′) = x^{′−1}, & ρ(y′) = y^{′−1}. \end{matrix}

In case (a), the second condition is replaced by x′, y′ ∈ E.

Choose μ ∈ E^{1} and consider the morphism

(9.6)  z_{1} = μx′y′, W_{1} = \frac{c}{b}(V − 1) = \frac{x′}{y′} .

It is not defined at x′ = y′ = 0 or x′ = y′ = ∞. So we blow up these two points. Since they are not rational this has no effect on (9.4). The morphism is a double covering ramified along the two curves introduced by the blow-up. A simple calculation shows that the form

\frac{dx′dy′}{(x′ − y′)^{2}}

is the pull-back of

−\frac{1}{2}bc\frac{dz_{1}}{z_{1}} \frac{dV}{U^{2}} .

It is evident that the rationality conditions on x′, y′ imply that

z_{1} ∈ E, z̄_{1} = z_{1}^{−1}, σ(V − 1) = V − 1,

and that ρ(V − 1) = (V − 1)^{−1} in case (b). In case (a), V ∈ E.

In case (a)

\frac{x′}{y′} = x′x̄′.

Thus we can solve (9.6) for x′, y′ if and only if the condition (9.2) is satisfied and

z_{1} ∈ {μx′/x̄′|x′x̄′ = W_{1}}.

Hence if we let μ run over a set of representatives for the cosets of (E^{1})^{2} in E^{1}, the equations (9.5) have exactly two solutions. (Points with S_{1} = 0 or ∞ are exceptional.) We conclude that (9.1) is equal to |2|N times (9.4) if 2N = [E^{1} : (E^{1})^{2}].

The conditions on V in (9.3) are:

σ(V) = F; W_{1} = cV ρ(cV)^{−1}; cd^{−1}V ∈ Nm_{K/K_{0}}K^{×}.

If

cd^{−1}V = tσ(t), t ∈ K^{×}

and if we set

x′ = utρ(t)^{−1}, y′ = σ(x′)^{−1},

then the second set of conditions in (9.5) amounts to

ρ(u) = u^{−1},

and the equations (9.6) become

z_{1} = μuσ(u)^{−1}tσρ(t)ρ(t)^{−1}σ(t)^{−1}; σ(u) = u^{−1}.

If we again let μ run over a set of representatives for the cosets of (E^{1})^{2} in E^{1} this equation has a solution unique in μ and in u up to a sign for a given z_{1} ∈ E^{1}.

Conversely, if we can solve the equations (9.6) subject to (9.5) then μ is uniquely determined and

W_{1} = eρ(e)^{−1}σ(e)σρ(e)^{−1}.

It follows that V ∈ K_{0} and that (V − 1)ρ(V − 1) = 1, so that

W_{1} = cV ρ(cV)^{−1}

and cV e^{−1}σ(e)^{−1} is fixed by ρ. Since it is also fixed by σ, it lies in Nm_{K/K_{0}}K_{0}^{×} and so does cV. We conclude once again that (9.1) is equal to |2|N times (9.4).

Appendix I. Although it would be out of place to elaborate on the formal properties of the principal values of [4], there is one calculation that will be useful. Before describing it, we observe that in the definition of A(x) in Section 1 of [4], the coefficients c_{j} should be divided by (j − 1)!

Suppose that in a coordinate patch U defined by

|μ_{i}| ≦ q^{−m}, 0 ≦ i ≦ r,

we have:

λ = αμ_{0}^{a_{0}}μ_{1}μ_{2}; ω = W μ_{0}^{b_{0}−1}μ_{1}μ_{2} ∧ dμ_{i}; f = λκ_{0}(μ_{0})κ(μ_{1}μ_{2}).

Suppose also that b_{0} ≠ 2a_{0} or that κ^{a_{0}} ≠ κ_{0}. We take D_{r} = D_{r}(κ, 2) and calculate

(A.1)  ∮_{D_{1}∩U} h_{1}|ν_{1}|,

in the sense of Section 1 of [4], using the notation of [5] for a principal value.

It is the sum of two terms. The first is obtained by setting

\begin{aligned}ω′ & = W (μ_{0}, 0, 0, μ_{3}, ⋯)α^{−β}μ_{0}^{b_{0}−2a_{0}−1}dμ_{0} ∧ dμ_{3} ∧ ⋯ \\ f′ & = (ln_{q}|αμ_{0}^{a_{0}}|)(λκ_{0}(μ_{0})κ^{−a_{0}}(μ_{0})κ^{−1}(α)) \end{aligned}

and taking

(A.2)  (1 − \frac{1}{q}) ∫_{D_{2}∩U} f′|ω′|.

The second is obtained by setting:

\begin{aligned}λ′′ & = μ_{1}μ_{2}; \\ ω′′ & = μ_{1}μ_{2}dμ_{1} ∧ dμ_{2}; \\ f′′ & = ∫ λ|W ||α|^{−2}|μ_{0}|^{b_{0}−2a_{0}−1}κ_{0}(μ_{0})κ^{−a_{0}}(μ_{0})κ^{−1}(α) × |dμ_{0} ∧ dμ_{3} ⋯ |; \end{aligned}

and then taking

(A.3)  ∮_{D′′_{1}∩U′′} h′′_{1}|ν′′_{1}|,

the data for this integral being defined by λ′′, ω′′, f′′.

This is of course an easy calculation, for in the present circumstances the only c_{j} that is not 0 is c_{2} and c_{2} = 1. Then

A_{1}(y) = 1 − y

and the decomposition of (A.1) into the sum of (A.2) and (A.3) is given by

\begin{matrix}1 − m − M_{1} − M_{2} − a_{0}m(μ_{0}) \\ = (−m − a_{0}m(μ_{0})) + (1 − M_{1} − M_{2}). \end{matrix}

In the text, the integral to which the term (A.2) leads is

(A.4)  ∮_{P^{1}} ln|x − ξ|\frac{dx}{|x|^{2}} ,

in which ξ lies in a cubic extension of F. If, for example, the extension if ramified then, with no loss of generality, we may suppose that the order of ξ is r + n with r equal to 1/3 or 2/3. Then the integral is calculated to be

−rq^{−n−1} + \frac{q^{−n}}{q − 1} = \frac{q^{−n}}{q − 1} ((1 − r) + \frac{r}{q} ) .

In the context of Section 8, this is to be compared with Section 10 of [8].

Finally we note two more corrections to [4]. The exponent n′ should be removed from the middle term of (3.4) and “non-zero” in Lemma 3.13 (ii) should be “zero”.

Appendix II. Although not necessary for proving the existence of the transfer it is sometimes useful [9] to know the value of the constant appearing in Lemma 4.2. Of course, the constant only has a meaning after the measure defining the integral over the class O has been fixed.

If, to be explicit, we take G to be the group attached to the hermitian matrix

[ \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ]

and the quadratic extension E, then every subregular conjugacy class over F has a representative of the form

n(w) = [ \begin{matrix}1 & 0 & w \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ] , w ∈ E^{×}

with w + w̄ = 0, the bar denoting conjugation of E over F. It is easy to see that n(w) and n(aw), a ∈ F^{∗}, are conjugate over F if and only if a ∈ Nm_{E/F} E^{×}.

We take N_{∞} to be the group of lower-triangular unipotent matrices in G. Then

(n_{∞}, n(w)) → n_{∞}^{−1}n(w)n_{∞}, n_{∞} ∈ N_{∞}(F), w ∈ E^{×}, w + w̄ = 0,

yields a parameterization of a dense open subset of the manifold of F -valued points on the variety of subregular unipotent elements. If, as in Section 7, we take the measure dn_{∞} on N_{∞}(F) to be that associated to ∧ω_{j}, then

∫ f(n_{∞}^{−1}n(w)n_{∞})|w|dwdn_{∞}

defines a G-invariant integration on the manifold of subregular unipotent elements over F and thus on each G-orbit in it. We take

a_{O}(f) = ∫_{O} f(n_{∞}^{−1}n(w)n_{∞})|w||dw||dn_{∞}|.

It is best to observe that the absolute value on E is to be taken to be an extension of the normalized absolute value on F.

The form ν_{n}, n = n(w), appearing in (5.5) is then that given by (7.3). Thus, according to the discussion at the end of Section 7, the end of Section 8, and Section 9, the constant a appearing in (5.5) and therefore in Lemma 4.2 is |2|N.

Lemma. For any local field of characteristic zero, |2|N = 1.

Since

N = \frac{1}{2} · [E^{1} : (E^{1})^{2}]

the lemma is clear for a field with odd residual characteristic, for then N = 1. If the residual characteristic is even, let 2^{r} be the number of elements in the residue field, and let 2 = ω̃^{k} where ω̃ is a uniformizing parameter.

Let E^{j}, j ≧ 2, be

{1 + aω̃^{j−1}|a integral}.

Since (1 + aω̃^{j})^{2} = 1 + 2aω̃^{j} + a^{2}ω̃^{2j}, the index

[E^{j} : E^{j+1}(E^{j} ∩ (E^{1})^{2})]

is equal to 1 for j odd and different from 2k + 1 and for all j greater than 2k + 1. If j is even and j ≦ 2k it is 2^{r}, and it is 2 for j = 2k + 1. The lemma follows.

References

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2. Shalika germs on GSp(4), preprint.

3. R.P. Langlands, Les débuts d’une formule des traces stable, Publ. Math. Univ. Paris 13 (1983).

4. Orbital integrals on forms of SL(3), I. Amer. Jour. Math. 105 (1983), 465-506.

5. R.P. Langlands and D. Shelstad, On principal values on p-adic manifolds, in Lect. Notes in Math. 1041, 250-279.

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7. Descent for transfer factors, to appear.

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