New York Journal of Mathematics

New York J. Math. 4 (1998) 17–29.

An Example of Fourier Transforms of Orbital Integrals and their Endoscopic Transfer

Ulrich Everling

Abstract. For the Lie algebra 𝔰𝔩_{2} over a 𝔭-adic field, the Fourier transform of a regular orbital integral is expressed as an integral over all regular orbital integrals, with explicit coefficients. This expression, unlike the Shalika germ expansion, is not restricted to orbits of small elements. The result gives quite an elementary access to a simple example of Waldspurger’s recent theorem on endoscopic transfer of the Fourier transforms.

Contents

1. Introduction  17
2. Conjugacy Classes  18
3. Orbital Integrals  20
4. Fourier Transforms  21
5. Transfer Factors  25
6. The Transfer Formula  26
7. Remarks  28
References  28

1. Introduction

Integrals over conjugation classes in Lie groups, called orbital integrals, are fundamental objects of harmonic analysis. They occur in the “geometric side” of Selberg’s trace formula; the “spectral side” contains traces of representations. For applications to automorphic forms, it is useful to consider not just Lie groups but both real and p-adic (and also adelic) algebraic groups. Howe [11] conjectured and Clozel [7] proved, for reductive p-adic Lie groups or Lie algebras, that the invariant distributions of given compactly generated support, restricted to a space of uniformly locally constant functions, make a finite dimensional space. Harish-Chandra [10] uses this principle in his study of Fourier transforms and characters of admissible representations. Shalika germs [24] describe the orbital integrals of small regular elements as combinations of unipotent orbital integrals, with coefficients difficult to get hold of.

\begin{matrix}c \\ ◯ \end{matrix}1998 State University of New York ISSN 1076-9803/98

Received October 14, 1997.

Mathematics Subject Classification. 11F85, 22E35, 43A80.

Key words and phrases. orbital integral, Fourier transform, endoscopic transfer.

Endoscopic transfer tries to relate instable orbital integrals of a group and stable orbital integrals of so-called endoscopic groups attached to it via L-groups. It was developed by Shelstad, Langlands, Labesse, Kottwitz, and stated in final though conjectural form by [20]. To explain it roughly: Given an algebraic group G over a field F, with algebraic closure \overline{F}; elements in G(F) which are conjugate in G(\overline{F}) make a “stable orbit”, which may contain several orbits of G(F)-conjugacy (more precisely, see [14]). A stable orbital integral is the sum of integrals of the orbits in one stable orbit; certain other linear combinations of these integrals, such as their difference when the stable orbit contains two orbits, are called instable orbital integrals. Stable orbits have the advantage of being compatible when two groups are inner forms of each other. Transfer means that an instable integral of a function on G(F) may be expressed as stable integral of another function on another group, called an endoscopic group; the construction of that group is based on L-groups, defined by a duality of root systems using the structure theory of reductive algebraic groups. For G = SL_{2} the non-trivial endoscopic groups are one-dimensional algebraic tori defined by quadratic extensions E|F and a stable orbital integral in such a torus is just evaluation at an element.

Arthur developed trace formulae introducing truncation operators and weighted orbital integrals. Waldspurger [29] established a p-adic trace formula and suggested a transfer formula for the Fourier transforms of regular orbital integrals in the Lie algebra, (9) below, which would imply the transfer for the orbital integrals themselves; recently he reduced this conjecture to the fundamental lemma and proved it for many groups [30, Théorème 1.5], among them SL_{n}; in general, transfer remains a conjecture.

Let F be a p-adic field (p ≠ 2). Fourier transformation was thoroughly studied for orbital integrals on the group SL_{2}(F) by Sally and Shalika [22], and for nilpotent orbital integrals on the Lie algebra 𝔰𝔩_{ℓ}(F) by Assem [4]. Now for X ∈ 𝔰𝔩_{2}(F) semi-simple regular, we consider the orbital integral over Ad(SL_{2}(F)) · X. Its Fourier transform is invariant with respect to Ad(SL_{2}(F)) and is known to be an integral (over the space of regular orbits) of orbital integrals; we explicitly specify the coefficients (Propositions 4 and 10). This expression, unlike Shalika’s germ expansion, is not limited to orbits of small elements. In the last two sections we translate our Corollary 11 into the context of formula (9) recently proved by [30] for certain groups, of which SL_{2} is but one example; our elementary computation yields this example directly.

Our discussion, up to Corollary 11, does not presume knowledge of the works mentioned above; in the last two sections we quote from [20] and [29] respectively.

2. Conjugacy Classes

Matrices in the Lie algebra 𝔰𝔩_{2} will always be denoted

X = ({x \atop z} {y \atop −x}) or U = ({u \atop w} {v \atop −u}),

and the letters U, u, v, w, X, x, y, z, bare or decorated like X′ or X_{m}, will always mean this. The action of GL_{2} on 𝔰𝔩_{2} by Ad(g)(X) := gXg^{−1} leaves the bilinear form

⟨U, X⟩ := 2ux + vz + wy

invariant; it will be used for Fourier transforms.

Let us classify the orbits of G := SL_{2}(F) acting on 𝔤 := 𝔰𝔩_{2}(F), where F is a field in which 2 ≠ 0; integrals over these orbits will then be the basic objects to study. The group G acts in the set of regular semi-simple elements

𝔤^{reg} = {X ∈ 𝔤 det X ≠ 0}

and in each of the stable orbits

𝔤_{t} := {X ∈ 𝔤 ∖ {0} det X = −t} , t ∈ F.

Each X ∈ 𝔤 is G-conjugate to some ( \begin{matrix}0 & t/β \\ β & 0 \end{matrix} ) where t = − det X and β ∈ F^{×},

the latter unique up to a factor in

N_{t} := N_{tF^{×2}} := {λ^{2} − μ^{2}t λ, μ ∈ F } ∖ {0} ⊆ F^{×},

which is the norm group of the algebra

E_{t} := F [ϑ]/(ϑ^{2} − t);

in fact the orbits are parametrized by the bijection

{Orb(𝔤 ∖ \{0\}) := \{(t, b) t ∈ F, b ∈ F^{×}/N_{t}\} ⟶^{∼} Ad(G)\(𝔤 ∖ \{0\}) \atop (t, b) ↦ 𝔤_{t,b} := \{X ∈ 𝔤_{t} z ∈ b ∪ \{0\} ∋ −y\} ,}

and Orb(𝔤^{reg}) := {(t, b) t ≠ 0} corresponds to Ad(G)\𝔤^{reg}. Each stable orbit 𝔤_{t} is the union of all the 𝔤_{t,b} where b ∈ F^{×}/N_{t}, and

(1)  ∀g ∈ GL_{2}(F) :  Ad(g)𝔤_{t,b} = 𝔤_{t,det(g)b}.

(Each group F^{×}/N_{t} can also be considered as Galois cohomology group of the stabilizer in SL_{2} of the matrix (_{1}^{0} _{0}^{t}) ∈ 𝔤_{t}; in this guise they appear in [18, p. 728] when orbits in G are parametrized.)

From now on let F be a non-archimedean local field [6]; it comes with the valuation ring O ⊂ F, the maximal ideal 𝔭 ⊂ O, the residue field κ = O/𝔭 with q elements, the valuation exponent ν : F ↠ ℤ ∪ {+∞} and the absolute value | · | so that |π| = 1/q for any uniformizing element π ∈ 𝔭 ∖ 𝔭^{2}. We shall assume 2 ∉ 𝔭.

(Example: F = ℚ_{p}, O = ℤ_{p}, 𝔭 = pℤ_{p}, κ = 𝔽_{p}, π = p.)

The group F^{×}/F^{×2} has four elements and fits in the exact sequence

1 → O^{×}/O^{×2} ⟶ F^{×}/F^{×2} ⟶^{ν} ℤ/2ℤ → 0
↓≅
κ^{×}/κ^{×2}

and for each τ = tF^{×2} ∈ F^{×}/F^{×2} there is a canonical isomorphism

F^{×}/N_{τ} ≅ \{ \begin{matrix}\{1\} & if 1 ∈ τ, \\ ℤ/2ℤ & if τ = ν^{−1}(2ℤ) ∖ F^{×2}, \\ κ^{×}/κ^{×2} & if ν(τ) = ℤ ∖ 2ℤ. \end{matrix}

Thus for t ∈ F^{×} ∖ F^{×2}, the stable orbit 𝔤_{t} contains two orbits 𝔤_{t,b}.

3. Orbital Integrals

On each G-orbit in 𝔤, we need a G-invariant measure to define integrals. Instead of referring to the general fact that a quotient space of a unimodular locally compact group G by a unimodular closed subgroup has a G-invariant positive measure unique up to a positive constant, we shall construct such measures on each 𝔤_{t,b} so as to have an explicit formula to work with.

The additive Haar measure on F such that vol(O) = 1, as well as its restriction to subsets of F, will be denoted dt, or dx, dy, dz respectively; for example, dt/|t| is a Haar measure on the multiplicative group F^{×}. The F -rational points of a smooth algebraic variety over F form a totally disconnected topological space, on which measures can be defined by means of differential forms, as explained in [31, p. 13–14]. For any complex-valued locally constant compactly supported function (in short, C_{c}^{∞} function), integrals are finite sums. The characteristic function of an open set A will be called f_{A}.

For each t ∈ F, the variety {X ∈ 𝔰𝔩_{2} ∖ {0} x^{2} + yz = t} carries a GL_{2}-invariant algebraic 2-form

\frac{dy ∧ dz}{2x} = \frac{dx ∧ dy}{y} = \frac{dz ∧ dx}{z}

(the relation 0 = d(x^{2} + yz) = 2x dx + z dy + y dz implies that all three terms are equal and invariant). This 2-form defines a measure dX on 𝔤_{t}. The orbital integrals

I_{t,b}(f) := ∫_{𝔤_{t,b}} f(X)dX ∈ ℂ (f ∈ C_{c}^{∞}(𝔤), (t, b) ∈ Orb(𝔤 ∖ {0}))

are finite sums if t ≠ 0, or still convergent for t = 0, see the proof of Proposition 2 below. Let Λ_{0} := 𝔰𝔩_{2}(O) = {X x, y, z ∈ O} and Λ_{n} := 𝔭^{n}Λ_{0} for each n ∈ ℤ.

Proposition 1. Let m < ℓ ∈ ℤ, X_{m} ∈ Λ_{m} ∖ Λ_{m+1}, and (t, b) ∈ Orb(𝔤 ∖ {0}). If (i) t + det(X_{m}) ∈ 𝔭^{ℓ+m} and (ii) {z_{m}, −y_{m}} ⊂ b ∪ 𝔭^{m+1},

then I_{t,b}(f_{X_{m}+Λ_{ℓ}}) = q^{m−2ℓ}. Otherwise the integral is zero.

Proposition 2. Let ℓ ∈ ℤ, (t, b) ∈ Orb(𝔤 ∖ {0}). Then

I_{t,b}(f_{Λ_{ℓ}}) = \{ \begin{matrix}\frac{1}{2}q^{−ℓ} & if t = 0, ℓ ∈ ν(b) \\ \frac{1}{2}q^{−ℓ−1} & if t = 0, ℓ ∉ ν(b) \\ 0 & if 2ℓ > ν(t) \\ q^{−ℓ} + q^{−1−ℓ} & if 2ℓ ≤ ν(t), t ∈ F^{×2} \\ q^{−ℓ} − q^{−1−h} & if 2ℓ ≤ ν(t) = 2h, 0 ≠ t ∉ F^{×2}, ℓ ∈ ν(b) \\ q^{−1−ℓ} − q^{−1−h} & if 2ℓ ≤ ν(t) = 2h, 0 ≠ t ∉ F^{×2}, ℓ ∉ ν(b) \\ (q^{−ℓ}−q^{−h})\frac{q+1}{2q} & if 2ℓ < ν(t) = 2h − 1. \end{matrix}

Corollary 3. For every f ∈ C_{c}^{∞}(𝔤^{reg}) the function

Orb(𝔤^{reg}) ∋ (t, b) ↦ I_{t,b}(f)

belongs to C_{c}^{∞}(Orb(𝔤^{reg})). Explicitly, let n ∈ ℤ, X_{n} ∈ Λ_{n}∖Λ_{n+1}, t = − det X_{n} ≠ 0, j > ν(t) − n. Then f = f_{X_{n}+Λ_{j}} corresponds to the function q^{n−2j}f_{(t+𝔭^{n+j})×{b}}.

Proofs. For Proposition 1, fix a prime element π ∈ 𝔭 ∖ 𝔭^{2}, no matter which, so that πO = 𝔭, let X_{m} = π^{m}X_{0} and let X′ run through Λ_{0} so that X = X_{m} + π^{ℓ}X′ runs through X_{m} + Λ_{ℓ}. Then

π^{−ℓ−m}(t + det X) = π^{−ℓ−m}(t + det X_{m}) − ⟨X_{0}, X′⟩ + π^{ℓ−m} det X′.

If this wants to be zero, the condition (i) is necessary.

If y_{0}, z_{0} ∈ 𝔭, then x_{0} ∈ O^{×} and one can solve the equation t = − det X for x′. Because of t ∈ π^{2m}(x_{0}^{2} + 𝔭) ⊂ F^{×2} we have 𝔤_{t,b} = 𝔤_{t}. The integral is

∫_{𝔤_{t}∩(X_{m}+Λ_{ℓ})} \frac{dy dz}{|2x|} = ∫_{O×O} \frac{|π^{ℓ}|dy′ · |π^{ℓ}|dz′}{|x_{m}|} = q^{m−2ℓ}.

If y_{0} ∈ O^{×} one can solve for z′ and then −yN_{t} = −y_{m}N_{t}. If this class is the same as b, the integral is again

∫_{𝔤_{t,b}∩(X_{m}+Λ_{ℓ})} \frac{dx dy}{|y|} = q^{m−2ℓ},

otherwise it is zero. Likewise in the case z_{0} ∈ O^{×}.

The second proposition follows from the first one applied to each term of the sum

f_{Λ_{m}} − f_{Λ_{m+1}} = ∑_{0∉X∈Λ_{m}/Λ_{m+1}} f_{X},

by descending induction on ℓ if t ≠ 0, or with a geometric series if t = 0. The corollary follows immediately. □

In view of the two propositions, one recognizes the Shalika germs: Let ℓ ∈ ℤ, (t, b) ∈ Orb(𝔤^{reg}), ν(t) ≥ 2ℓ, and f ∈ C_{c}^{∞}(𝔤) a function that factors through 𝔤/Λ_{ℓ}. Then

I_{t,b}(f) = (const)\sqrt{|t|} · f(0) + ∑_{β∈b/F^{×2}} I_{0,β}(f)

where (const) depends only on tF^{×2}; the right hand side consists of contributions from the nilpotent orbits, which are the four 𝔤_{0,b} and {0}.

4. Fourier Transforms

Let Ψ : F → ℂ^{×} be an unramified additive character; then

(2)  Ψ(O) = 1, ∑_{a∈𝔭^{−1}/O} Ψ(a) = 0.

The Fourier transform of functions on 𝔤 is defined as

C_{c}^{∞}(𝔤) ˜ ⟶ C_{c}^{∞}(𝔤), f̃(U) := ∫_{𝔤} f(X)Ψ ⟨U, X⟩ dX

where dX is the Haar measure such that vol(Λ_{0}) = 1, which is self-dual in the sense

that ˜ f̃(X) = f(−X).

For each orbit, labeled (s, a) ∈ Orb(𝔤^{reg}), the distribution

C_{c}^{∞}(𝔤^{reg}) → ℂ, f ↦ I_{s,a}(f̃),

is G-invariant in the sense that I_{s,a}({ \atop f ∘ Ad(g)}) = I_{s,a}(f̃) for each g ∈ G. This invariant distribution can be expanded in terms of all the I_{t,b}:

Proposition 4. There exists a unique locally constant function

c_{G} : Orb(𝔤^{reg}) × Orb(𝔤^{reg}) → ℂ

such that

(3)  ∀f ∈ C_{c}^{∞}(𝔤^{reg}) : I_{s,a}(f̃) = ∫_{F^{×}} ∑_{b∈F^{×}/N_{t}} c_{G}(s, a; t, b)I_{t,b}(f)\frac{dt}{\sqrt{|t|}}.

Explicitly, for n ∈ ℤ and X_{n} ∈ 𝔤_{t,b} ∩ Λ_{n} ∖ Λ_{n+1}:

(4) \frac{c_{G}(s, a; t, b)}{\sqrt{|t|}} = I_{s,a}(f_{Λ_{−n}}) + ∑_{i>n} ∑_{Λ_{1−i} / ⊃U_{−i}∈Λ_{−i}/Λ_{−n}} Ψ ⟨U_{−i}, X_{n}⟩ I_{s,a}(f_{U_{−i}}),

and the sum over i is actually finite.

The proof will be finished before Corollary 8. First we show that (3) implies (4). Suppose c_{G} locally constant. Given n and X_{n}, let j ≫ ν(t) − n and c ∈ ℂ so that

∀X′ ∈ X_{n} + Λ_{j} : c_{G}(s, a; − det X′, b) = c\sqrt{|t|}.

Corollary 3 yields

(5)  c = q^{3j} · vol(t + 𝔭^{n+j}) · c · q^{n−2j}

= q^{3j} · I_{s,a}({ \atop f_{X_{n}+Λ_{j}}})

= I_{s,a}(f_{Λ_{−n}} + ∑_{n<i≤j} ∑_{Λ_{1−i} / ⊃U_{−i}∈Λ_{−i}/Λ_{−n}} Ψ ⟨U_{−i}, X_{n}⟩ f_{U_{−i}});

this does not depend on j and therefore the i ≫ ν(t) − n do not contribute to (4).

It remains to show that there exists a function with the property (3). (We might refer to [10] but the lemmas below are needed anyway for Proposition 10.) For each (t, b) ∈ Orb(𝔤^{reg}) let us fix n ∈ ℤ, y ∈ O, z ∈ O^{×} such that

(6)  X_{n} = π^{n}({0 \atop z} {y \atop 0}) ∈ 𝔤_{t,b} , ν(y) ≤ 2, and (t ∈ F^{×2} ⟹ y ∈ O^{×2}).

Let c_{G} be defined by (4) with this choice of X_{n}. The relations (2) imply that the apparently infinite sum (4) is actually finite as follows.

Lemma 5. Let i > n, s′ = π^{2i}s + 𝔭^{i−n}, h_{1} = i − n − 1, h_{2} = h_{1} − ν(y). The U_{−i} = π^{−i}U such that

(i) ν(v) = 0 < h_{1} ∧ (s′ = 0 ∨ h_{1} + ν(w) > min{ν(s′ − vw), h_{1}}) or (ii) ν(w) = 0 < h_{2} ∧ (s′ = 0 ∨ h_{2} + ν(v) > min{ν(s′ − vw), h_{1}})

contribute all in all nothing to c_{G}(s, a; t, b) in (4).

Corollary 6. The only contributions to the sum (4) come from values of i such that

(7) max\{n + 1, −\frac{ν(s)}{2}\} ≤ i ≤ i_{max} := max\{n + 1 + ν(y), −1 − n − ν(s)\}.

Proofs. Let us assemble the U_{−i} with property (i) as follows: v runs through one class mod𝔭^{h_{1}}, w remains fixed, and u varies so that the equation s′ = u^{2} + vw remains valid. The I_{s,a}(f_{U_{−i}}) are all equal and the Ψ ⟨U_{−i}, X_{n}⟩ = Ψ(π^{n−i}(vz +wy)) make a total of zero.

Likewise, we can assemble the U_{−i} such that (ii) ∧ ¬(i); this time w varies by 𝔭^{h_{2}}, and v remains fixed.

The upper bound in the corollary follows from the lemma because

i ≥ −n − ν(s) ⟹ s′ = 0 ⟹ vO + wO = O.

The two lower bounds come from (4) and from condition (i) of Proposition 1. □

By replacing X and U by λ^{−1}X and λU we see that the definition of c_{G} does not depend on the choice in (6) and that

(8)  ∀λ ∈ F^{×} :  c_{G}(λ^{−2}s, a; λ^{2}t, b) = c_{G}(s, a; t, b).

Lemma 7. The function c_{G} : Orb(𝔤^{reg}) × Orb(𝔤^{reg}) → ℂ is locally constant.

Proof. First let us fix (t, b) and a, and vary s. In view of Propositions 1 and 2, each term of (4) allows a neighbourhood where s can move. The set of terms which are not excluded by Corollary 6 is finite and depends only on ν(s) and (t, b). Therefore c_{G} is locally constant with respect to s. Using (8) we can barter a small change of t for a small change of s. Hence the lemma. □

Going back via (5), we now find that (3) is true for f = f_{X_{n}+Λ_{j}} when the X_{n} are chosen as in (6) and when j ≥ i_{max} as defined in (7). These functions f, and their transforms f ∘ Ad(g) by all g ∈ SL_{2}(F), span the whole linear space C_{c}^{∞}(𝔤^{reg}). The condition (3) being linear and SL_{2}(F)-invariant, Proposition 4 follows.

Corollary 8. ∀(s, a), (t, b) ∈ Orb(𝔤^{reg}) ∀λ ∈ F^{×} : c_{G}(s, λa; t, λb) = c_{G}(s, a; t, b).

Proof. All the ingredients of Proposition 4 being GL_{2}(F)-invariant, so is c_{G}; and we remember (1). □

Corollary 9. If sF^{×2} = tF^{×2} then c_{G}(s, a; t, b) depends on a and b only via a · b. If sF^{×2} ≠ tF^{×2} then c_{G}(s, a; t, b) does not at all depend on a or b.

Proof. This follows from Corollary 8; if sF^{×2} ≠ tF^{×2} one can choose λ so as to change a without changing b, or vice versa. □

We shall need auxiliary complex numbers depending on Ψ, namely

γ(c) := 1 if c ∈ F, ν(c) ∈ 2ℤ,
γ(c) := (1 + 2 ∑_{cF^{×2}⊃a∈𝔭^{−1}/O}Ψ(a))/\sqrt{q}  if c ∈ F, ν(c) ∉ 2ℤ.

If c ∈ F^{×} and ν(c) = −1 we have

\sqrt{q} · γ(c) = ∑_{λ∈κ} Ψ(cλ^{2}),

γ(c · (O^{×} ∖ O^{×2})) = −γ(c), γ(−c) = \overline{γ(c)} = (−1)^{\frac{q−1}{2}} · γ(c), |γ(c)| = 1.

In view of lemma 5 and Corollary 6 there remain only quite few U_{−i} contributing to (4) for the X_{n} chosen in (6). Counting thoroughly and using the relations (2), we obtain:

Proposition 10. Given (s, a), (t, b) ∈ Orb(𝔤^{reg}), let

S = ∑_{r^{2}=st}Ψ(2r), Q_{1} = \frac{1}{q}\sqrt{|st|}, Q_{2} = \frac{q + 1}{2q}\sqrt{|st|/q}.

Then c_{G}(s, a; t, b) is as follows:

t ∈ F^{×2} t ∉ F^{×2}, 2|ν(t)  2 ∤ ν(t)
s ∈ F^{×2} S  0
st ∈ 𝔭^{−1} 1 − Q_{1} (−1)^{ν(ab)} − Q_{1}  −Q_{1}

\begin{matrix}st ∉ 𝔭^{−1} \\ 2|ν(s) \\ s ∉ F^{×2} \end{matrix} 0 (−1)^{ν(ab)}S

if \frac{ν(st)}{2} ∈ ν(ab) 0 otherwise

0

st ∈ 𝔭^{−1} 1 − Q_{2} −Q_{2}

γ(−ab) − Q_{2} if st ∈ F^{×2}
−Q_{2} otherwise

\begin{matrix}st ∉ 𝔭^{−1} \\ 2 ∤ ν(s) \end{matrix} 0

γ(−ab) ∑_{{r^{2}=st \atop r∈−ab}} Ψ(2r)

if st ∈ F^{×2}

0 otherwise

For each s ∈ F^{×} let ε_{s} : F^{×}/N_{s} ↪ {±1} be the character. From Proposition 10 we extract the following corollary (actually some of the boxes of the table are not needed here; in particular those where s ∉ F^{×2} / ∋ st cancel out by Corollary 9.)

Corollary 11. Let s ∈ F^{×} and (t, b) ∈ Orb(𝔤^{reg}). Then

c_{G}^{ε}(s; t, b) := ∑_{a∈F^{×}/N_{s}} ε_{s}(a) · c_{G}(s, a; t, b) = ε_{t}(b) · γ(t) · ∑_{r^{2}=st}Ψ(2r);

this is zero if st ∉ F^{×2}.

The corollary describes the ε_{s}-instable integral

I_{s}^{ε}(f̃) := ∑_{a∈F^{×}/N_{s}} ε_{s}(a)I_{s,a}(f̃) = ∫_{F^{×}} ∑_{b∈F^{×}/N_{t}} c_{G}^{ε}(s; t, b)I_{t,b}(f)\frac{dt}{\sqrt{|t|}}

for f ∈ C_{c}^{∞}(Orb(𝔤^{reg})).

5. Transfer Factors

In order to derive (9) below from Corollary 11, we first spell out the definition of transfer factors which occur in I^{G,H}. As mentioned in the introduction, an instable integral in our case is the difference of two orbital integrals. The transfer factor will simply assign the signs +1 and −1 to the two orbital integrals; its definition in [20] is quite sophisticated because they make a canonical choice for all reductive groups and in general there occur coefficients other than ±1.

Let E|F be a quadratic extension, ϑ ∈ E^{×} an element of trace zero, and ε = χ_{E|F} the quadratic character on F^{×} so that ker χ_{E|F} = N_{E|F} E^{×}. Let

H = ker(R_{E|F} 𝔾_{m} ⟶^{N} 𝔾_{m}) the corresponding one-dimensional algebraic torus (so that H(F) ⊂ E^{×} is the norm-one group); its Lie algebra 𝔥 is naturally identified with the imaginary line ker(Tr_{E|F}) ⊂ E.

Now we borrow notations from [20, p. 222–223]. Let h = (\begin{matrix}1/2 \\ 1/(2ϑ) \end{matrix} _{1}^{−ϑ}) ∈ SL_{2}(E),

ι : H ⟶^{∼} T := h𝕋h^{−1} ⊂ B := h𝔹h^{−1} ⊂ G := SL_{2},

ι(a + bϑ) := h({a + bϑ \atop 0} {0 \atop a − bϑ})h^{−1} = ({a \atop b} {bϑ^{2} \atop a}).

Let

s := ({1 \atop 0} {0 \atop −1}) ∈ T := ({∗ \atop 0} {0 \atop ∗}) ⊂ B := ({∗ \atop 0} \begin{matrix}∗ \\ ∗ \end{matrix}) ⊂ Ĝ = PGL_{2}(ℂ)

and

ξ : H → (T · \{1, ({0 \atop 1} {1 \atop 0})\}) ⋊ W ⊂ ^{L}G

the embedding such that ξ^{−1} : T → Ĥ is dual to

H → 𝕋, a + bϑ ↦ h^{−1}ι(a + bϑ)h.

Then (H, H, s, ξ) are endoscopic data and ι is an admissible embedding. The algebraic character on T over E,

α : ({a \atop b} {bϑ^{2} \atop a}) ↦ (a + bϑ)^{2},

is the root of T in B; let the a-data consist of a_{α} := 2ϑ, a_{−α} := −2ϑ, and choose whichever χ_{α} : E^{×} → ℂ^{×} extending χ_{E|F}.

Let 0 ≠ Y = ηϑ ∈ 𝔥 and X = (_{z}^{x} _{−x}^{y}) ∈ 𝔤_{η^{2}ϑ^{2}}. Following the “Remarques” of [29, p. 90], we evaluate the formulae (I), (II), (III_{1}) and (III_{2}) of [20] with γ_{H} = exp(e^{2}Y) and γ = exp(e^{2}X) for any e ∈ F^{×} near 0. The cocycle λ(T) of (I) is trivial. In (II) we have α(γ_{T}) − 1 ≈ 2e^{2}ηϑ. For the cocycle in (III_{1}) we take

g = ({η \atop 0} {x \atop z})({a \atop b} {bϑ^{2} \atop a})

where a, b ∈ E, a^{2} − b^{2}ϑ^{2} = 1/ηz. In (III_{2}) we need not know a because γ_{T} ≈ 1. Omitting (IV), we obtain the transfer factor

Δ(Y, X) = (Δ_{I} = 1) · (Δ_{II} = ε(η)) · (Δ_{1} = ε(ηz)) · (Δ_{2} = 1) = χ_{E|F} (z).

(For a split torus H ≅ 𝔾_{m} we simply have Δ(Y, X) = 1 for all Y ∈ 𝔥, X ∈ 𝔤_{Y^{2}}.)

By identifying 𝔥 with the imaginary line of E|F (or with F if H splits) we may write Y · Z ∈ F for Y, Z ∈ 𝔥. Then

Δ(Y, X) = \{ \begin{matrix}ε_{t}(b) & if X ∈ 𝔤_{t,b} and t = Y^{2}, \\ 0 & otherwise. \end{matrix}

6. The Transfer Formula

For f_{𝔥} ∈ C_{c}^{∞}(𝔥) we define the Fourier transform f̃_{𝔥} with respect to Ψ(2Y · Z). Now we need formulae from [29]; quotations ‘…’ will refer to that work. In ‘III.1’ in the case M = G the factor v_{M}(x) is 1, so the J_{G}^{G} are the ordinary or-

bital integrals; so are the I_{G}^{G} by ‘p.74, remarque (b)’. Now I_{H}^{st}(Y, f_{h}) and I^{G,H}(Y, f) are defined by ‘VIII.7, (6) and (7)’.

Next, there are λ_{1}, λ_{2} ∈ F^{×} so that ⟨·, ·⟩_{g} = λ_{1} ⟨·, ·⟩ and ψ(c) = Ψ(λ_{2}c). For the construction of ⟨·, ·⟩_{h} in ‘VIII.6’, we can simply take 𝔊^{∗} = 𝔊 = SL_{2} and the same endoscopic data as in the previous section; then the whole procedure simplifies to

⟨ϑ, ϑ⟩_{h} = ⟨({0 \atop 1} {ϑ^{2} \atop 0}), ({0 \atop 1} {ϑ^{2} \atop 0})⟩_{g} = 2λ_{1} · ϑ · ϑ.

Consider the constant γ_{ψ}(g) of ‘VIII.5’. If λ ∈ O^{×}, we can take r = Λ_{0} in ‘VIII.1’ and then I(r) = 1. If ν(λ) = −1 we can still take r = Λ_{0} and then

I(r) = q^{−3} ∑_{x,y,z∈O/𝔭} Ψ(λ · (x^{2} + yz)) = q^{−3} ∑_{x} Ψ(λx^{2}) · ∑_{y} ∑_{z} Ψ(λyz);

by (2) the ∑_{z} vanishes for each y ≠ 0, so

I(r) = q^{−2} ∑_{x} Ψ(λx^{2})

whence γ_{ψ}(g) = γ(λ). If ν(λ) ∉ {0, −1} we can multiply λ by a square and adapt r accordingly.

The computation of γ_{ψ}(h) is similar.

Summing up, we obtain the following dictionary:

in [29]  here
I_{H}^{st}(Y, f_{h})  f_{𝔥}(Y)
I^{G,H}(Y, f)  C_{s} · I_{s}^{ε}(f), s = Y^{2}
⟨·, ·⟩_{g}  λ_{1} · ⟨·, ·⟩
⟨Y, Z⟩_{h}  2λ_{1} · Y · Z
ψ(c)  Ψ(λ_{2}c)
γ_{ψ}(g)  γ(λ), λ = λ_{1}λ_{2}
γ_{ψ}(h)  γ(λY^{2}) (for any Y ∈ 𝔥 ∖ {0})
f̂(X)  |λ|^{3/2} f̃(λX) (f ∈ C_{c}^{∞}(𝔤))
f̂_{h}(Y)  |λ|^{1/2} f̃_{𝔥}(λY) (f_{𝔥} ∈ C_{c}^{∞}(𝔥))

The constants C_{s} > 0 depend on the choice of Haar measures but C_{s} depends only on sF^{×2}; we do not need them explicitly.

For every function f ∈ C_{c}^{∞}(𝔤), the function f_{𝔥} defined by

f_{𝔥}(Y) := I^{G,H}(Y, f) (0 ≠ Y ∈ 𝔥)

extends to a function f_{𝔥} ∈ C_{c}^{∞}(𝔥); this was known to [18] and it is also clear in view of Propositions 1 and 2.

Let 0 ≠ Y ∈ 𝔥 and s = Y^{2}, and f ∈ C_{c}^{∞}(𝔤^{reg}). The self-dual Haar measure dZ on 𝔥 for f_{𝔥} ↦ f̃_{𝔥}, restricted to 𝔥 ∖ \{0\}, corresponds, via t = Z^{2}, to dt/\sqrt{|t|} on sF^{×2} ⊂ F. Then

I^{G,H}(Y, f̂) = C_{s} · ∑_{a∈F^{×}/N_{s}} ε_{s}(a) ∫_{X∼(_{a}^{0} _{0}^{s/a})} f̂(X)dX

= C_{s} · ∑_{a} ε_{s}(a) ∫_{λX∼(_{λa}^{0} _{0}^{λs/a})} |λ|^{3/2} f̃(λX)\frac{d(λX)}{|λ|}

= C_{s}\sqrt{|λ|} · ∑_{a} ε_{s}(a)I_{λ^{2}s,λa}(f̃)

= C_{s}\sqrt{|λ|} · ε_{s}(λ)I_{λ^{2}s}^{ε}(f̃);

by Corollary 11 this is

= C_{s}\sqrt{|λ|} · ε_{s}(λ) ∫_{F^{×}} ∑_{b∈F^{×}/N_{t}}ε_{t}(b)γ(t) ∑_{r^{2}=λ^{2}st} Ψ(2r)I_{t,b}(f)\frac{dt}{\sqrt{|t|}}

= C_{s}\sqrt{|λ|} · ε_{s}(λ)γ(s) ∫_{𝔥∖\{0\}} Ψ(2λY · Z)I_{Z^{2}}^{ε}(f)dZ

= ε_{s}(λ)γ(s)\sqrt{|λ|}f̃_{𝔥}(λY)

= ε_{s}(λ)γ(s)f̂_{h}(Y).

28 Ulrich Everling

In the four cases according to ν(λ), ν(s) mod 2 one verifies that

ε_{s}(λ)γ(s)γ(λ) = γ(λs),

and one recognizes the transfer formula of [29, VIII.7 Conj. 1] with his formulae ‘(6)’ and ‘(7)’ inserted:

(9)  γ_{ψ}(h)I_{H}^{st}(Y, f̂_{h}) = γ_{ψ}(g)I^{G,H}(Y, f̂).

7. Remarks

• We have been supposing f ∈ C_{c}^{∞}(𝔤^{reg}) so that the integrals are finite sums. Due to [10, Th.8], it is known that the Fourier transforms of regular orbital integrals are locally integrable distributions. This allows all f ∈ C_{c}^{∞}(𝔤).

• As long as 2 ∉ 𝔭, we allow F to have finite characteristic.

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Kath. Universität MGF, 85071 Eichstätt, Germany

ulrich.everling@ku-eichstaett.de http://mathsrv.ku-eichstaett.de/MGF/homes/everling/

This paper is available via http://nyjm.albany.edu:8000/j/1998/4-2.html.