Geometric side of a local relative trace formula

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Geometric side of a local relative trace formula

Patrick Delorme, Pascale Harinck, Sofiane Souaifi

To cite this version:

Patrick Delorme, Pascale Harinck, Sofiane Souaifi. Geometric side of a local relative trace formula.

Transactions of the American Mathematical Society, American Mathematical Society, 2019, 371 (3), pp.1815-1857. �10.1090/tran/7360�. �hal-01467988�

Geometric side of a local relative trace formula

P. Delorme

, P. Harinck, S. Souaifi

Abstract

Following a scheme suggested by B. Feigon, we investigate a local relative trace formula in the situation of a reductive p-adic group G relative to a symmetric subgroup H “ HpFq where H is split over the local field F of characteristic zero andG“GpFq is the restriction of scalars of H_IE relative to a quadratic unramified extension E of F. We adapt techniques of the proof of the local trace formula by J. Arthur in order to get a geometric expansion of the integral over HˆH of a truncated kernel associated to the regular representation ofG.

Mathematics Subject Classification 2000: 11F72, 22E50.

Keywords and phrases: p-adic reductive groups, symmetric spaces, local relative trace formula, truncated kernel, orbital integrals.

Contents

Introduction 2

1 Preliminaries 8

1.1 Reductive p-adic groups . . . . 8

1.2 The symmetric space H z G . . . 12

1.3 Weyl integration formula and orbital integrals . . . 14

2 Geometric side of the local relative trace formula 21 2.1 Truncation . . . 21

2.2 The truncated kernel . . . 24

2.3 Preliminaries to estimates . . . 27

2.4 Proof of Theorem 2.3 . . . 33

2.5 The function J

pf q . . . 41

∗The first author was supported by a grant of Agence Nationale de la Recherche with reference ANR-13-BS01-0012 FERPLAY.

A Appendix. Spherical character of a supercuspidal representation

as weighted orbital integral 46

Introduction

In this article, we investigate a local relative trace formula in the situation of p- adic groups relative to a symmetric subgroup. This work is inspired by the recent results of B. Feigon ([F]), where she investigated what she called a local relative trace formula on P GLp2q and a local Kuznetsov trace formula for Up2q.

Before we describe our setting and results, we would to explain on the toy model of finite groups the framework of the formulas of B. Feigon. We even start with the more general framework of the relative trace formula initiated by H. Jacquet ([J], see also [O] for an account of some applications of this relative trace formula).

Let G be a finite group and let H, H

, Γ be subgroups of G. We endow any finite set with the counting measure. We denote by r the right regular representation of G on L

pΓzGq and we consider the H-fixed linear form ξ on L

pΓzGq defined by

ξ “ ÿ

hPHXΓzH

δ

(0.1)

where δ

is the Dirac measure of the coset Γh, or in other words ξpψq “

ż

HXΓzH

ψpΓhqdh, ψ P L

pΓzGq.

We define similarly ξ

relative to H

.

We view ξ, ξ

as elements of L

pΓzGq and we form the coefficient c

pgq “ prpgqξ, ξ

q.

Integrating over functions on G, it defines a ”distribution” Θ on G which is right invariant by H and left invariant by H

. The relative trace formula in this context gives two expressions of Θpf q for f a function on G, the first one, called the geometric side, in terms of orbital integrals, and the second one, called the spectral side, in terms of irreducible representations of G.

First we deal with the geometric side. For this purpose we introduce suitable orbital integrals. For γ P Γ, we set rγs :“ pH

X ΓqγpH X Γq and one introduces two subgroups of H

ˆ H

pH

ˆ Hq

“ tph

, hq|h

γh

“ γu, pH

X Γ ˆ H X Γq

“ pH

ˆ Hq

X pΓ ˆ Γq.

Then, we define the orbital integral of a function f on G by I pr γ s , f q “

ż

pH¹ˆHqγzpH¹ˆHq

f p h

γh

q dh

dh.

Let f be a function on G. Since r p g q δ

“ δ

, the definition of ξ and ξ

gives

Θ p f q “ ÿ

gPG

f p g q Θ p g q “ ÿ

gPG

f p g q 1 volpΓ X Hq

1 volpΓ X H

q

ÿ

hPH

ÿ

h¹PH¹

p δ

, δ

q . Changing g in g

h and using the fact that p δ

, δ

q is equal to 1 for g P Γh

and to zero otherwise, one gets

Θpf q “ 1

volpΓ X Hq

1 volpΓ X H

q

ÿ

hPH

ÿ

h¹PH¹

ÿ

γPΓ

fph

γhq. (0.2) A simple computation of volumes leads to the geometric expression of Θ in terms of orbital integrals

Θpf q “ ÿ

rγsPH¹XΓzΓ{ΓXH

volppH

X Γ ˆ H X Γq

zpH

ˆ Hq

qI prγ s, fq. (0.3) Let us turn to the spectral side. We decompose L

p Γ z G q into isotypic components

‘

H

The restriction of ξ and ξ

to H

will be denoted ξ

and ξ

respectively.

The spectral formula for Θ is the simple equality Θ “

ÿ

πPGˆ

c

. (0.4)

Notice that it might be also interesting to decompose further the representation into irreducible representations and the restriction of ξ to each of them will be called a period.

There is a third interpretation of the distribution Θ. If f is a function on G, then the operator rpf q on L

pΓzGq is an integral operator whose kernel K

is the function on ΓzG ˆ ΓzG given by

K

px, yq “ ÿ

γPΓ

f px

γyq.

By (0.2), one gets easily the following expression of Θ p f q Θpfq “

ż

pH¹XΓzH¹qˆpHXΓzH

K

ph

, hqdh

dh. (0.5) This point of view is probably the best one. But it is important to have the repre- sentation theoretic meaning of Θ.

The toy model for the local relative trace formula of B. Feigon appears as a

particular case of the above relative trace formula. In that case, the groups G, H

and H

are products G

ˆG

, H

ˆH

and H

ˆH

respectively and Γ is the diagonal

of G

ˆ G

. Then Γ z G identifies with G

and the right representation corresponds

to the representation R of G

ˆ G

on L

pG

q given by rRpx, yqφspgq “ φpx

gyq.

Then, we have

ξpψq “ ż

H1

ψphqdh, ψ P L

pG

q.

The spectral side is more concrete. If p π

, H

q is an irreducible unitary represen- tation of G

then G

ˆG

acts on EndpH

q by an irreducible representation denoted by π. It is unitary if we use the scalar product associated to the Hilbert-Schmidt norm. Moreover L

p G

q is canonically isomorphic to the direct sum ‘

End p H

q . Let P

be the orthogonal projector onto the space of invariant vectors under H

. Then, the period map ξ

, which is a linear form on EndpH

q, is given by

ξ

pT q “ ż

H1

T rpπ

phqT qdh “ pT, P

q, T P EndpH

q.

One further decomposes ξ

by using an orthonormal basis (η

) of the space of H

-invariant vectors. We will use the identification of EndpH

q with the tensor product of H

with its conjugate complex vector space. In this identification, one has

P

“ ÿ

i

η

b η

.

We define similar notations for ξ

relative to H

. Then, for two functions f

, f

on G

, the spectral side (0.4) can be written

Θpf

b f

q “ ÿ

π1PGˆ1

ÿ

i,i¹

c

1,i,η¹

π1,i1

pf

qc

1,i,η¹

π1,i1

pf

q.

For the geometric side, we define the integral orbital of a function f on G

by Ipg, fq “

ż

pH₁¹ˆH1qgzH₁¹ˆH1

f ph

gh

qdhdh

which depends only on the double coset H

gH

. Then one gets by (0.3) the equality Θpf

b f

q “ ÿ

gPH₁¹zG1{H1

vpgqIpg, f

qIpg, f

q

where the v p g q ’s are positive constants depending on volumes. Hence the final form of the local relative trace formula is:

ÿ

gPH₁¹zG1{H1

v p g q I p g, f

q I p g, f

q “ ÿ

π1PGˆ1

ÿ

i,i¹

c

1,i,η¹_π

1,i1

p f

q c

1,i,η_π¹

1,i1

p f

q .

This formula allows to invert the orbital integrals Ipg, f

q. For this purpose, one chooses g

P G

and takes for f

the Dirac measure at g

. Then Ipg

, f

q “ 1 and the other orbital integrals of f

are zero. Hence

vpg

qIpg

, f

q “ ÿ

π1PGˆ1

ÿ

i,i¹

c

1,i,η_π¹

1,i1

pf

qc

1,i,η¹_π

1,i1

pf

q.

In order to make the formula more precise, one needs to compute the constants c

1,i,η_π¹

1,i1

pf

q.

The inversion of orbital integrals is one of our motivations to investigate a lo- cal relative trace formula in the situation of p-adic groups relative to a symmetric subgroup H and we will take H “ H

.

In this article, we consider a reductive algebraic group H defined over a non archimedean local field F of characteristic 0. We fix a quadratic unramified extension E of F and we consider the group G :“ Res

H obtained by restriction of scalars of H, where here H is considered as a group defined over E. We denote by H and G the group of F-points of H and G respectively. Then G is isomorphic to HpEq and H appears as the fixed points of G under the involution of G induced by the nontrivial element of the Galois group of E { F. We assume that H is split over F and we fix a maximal split torus A

of H. The groups G and H correspond to G

and H

“ H

respectively in our example of a local relative trace formula for finite groups.

The starting point of our study is the analogue to the expression (0.5). We consider the regular representation R of G ˆ G on L

pGq given by pRpg

, g

qψqpxq “ ψ p g

xg

q . Then for f “ f

b f

where f

and f

are two smooth compactly supported functions on G, the corresponding operator Rpfq is an integral operator on L

pGq with smooth kernel

K

px, yq “ ż

G

f

pxgqf

pgyqdg “ ż

G

f

pgqf

px

gyqdg.

As H may be not compact, even modulo the split component A

of the center of H, we have to truncate this kernel to integrate it. We multiply this kernel by a product of functions upx, T qupy, T q where up¨, T q is the characteristic function of a large compact subset in A

zH depending on a parameter T P a

“ RatpA

q b

R (Rat p A

q is the group of F-rational characters of A

) as in [Ar3] (cf. (2.7)). As H is split, we have A

“ A

. Hence the kernel K

is left invariant by the diagonal diagpA

q of A

and we can integrate the truncated kernel over diagpA

qzH ˆ H.

We set

K

pfq :“

ż

diagpAHqzpHˆHq

K

px

, x

qupx

, T qupx

, T qdpx

, x

q.

In [Ar3], J. Arthur studies the integral of K

px, xqupx, T q over A

zG to obtain its local trace formula on reductive groups.

We study the geometric expression of the distribution K

p f q and its dependence on the parameter T . Our main results (Theorem 2.3 and Corollary 2.11) assert that K

pfq is asymptotic as T approaches infinity to another distribution J

pf q of the form

J

p f q “

N

ÿ

k“0

p

p T, f q e

(0.6)

where ξ

“ 0, . . . ξ

are distinct points of the dual space ia

and each p

pT, f q is a polynomial function in T . Moreover, the constant term ˜ J pfq :“ p

p0, fq of J

pfq is well-defined and uniquely determined by K

p f q . We give an explicit expression of this constant term in terms of weighted orbital integrals.

These results are analogous to those of [Ar3] for the group case. Our proof follows closely the study by J. Arthur of the geometric side of his local trace formula which we were able to adapt under our assumptions to the case of double truncations.

In the first section, we introduce notation on groups and on symmetric spaces according to [RR]. The starting point of our study is the Weyl integration formula established in [RR], which takes into account the pH, H q-double classes of σ-regular elements of G (cf. (1.30) and (1.32)). These double classes are express in terms of σ- torus, that is torus whose elements are anti-invariant by σ. Under our assumptions, there is a bijective correspondence S Ñ S

between maximal tori of H and maximal σ-tori of G which preserves H-conjugacy classes.

Then the Weyl integration formula can be written in terms of Levi subgroups M P LpA

q of H containing A

and M -conjugacy classes of maximal anisotropic tori of M (cf. (1.33)):

ż

G

f pgqdg “ ÿ

MPLpA₀q

c

ÿ

SPTM

ÿ

xmPκS

c

ż

Sσ

|∆

px

γq|

ż

diagpAMqzHˆH

f ph

x

γlqdph, lqdγ where κ

is a finite subset of G, c

and c

are positive constants, T

is a suitable set of anisotropic tori of M and ∆

is a jacobian.

A fundamental result for our proofs concerns the orbital integral Mpfq of a compactly smooth function f on G. It is defined on σ-regular points by

M p f qp x

γ q “ | ∆

p x

γ q|

ż

diagpASqzHˆH

f p h

x

γl q d p h, l q ,

where S is a maximal torus of H, x

P κ

and γ P S

such that x

γ is σ-regular.

As in the group case using the exponential map and the property that each root of S

has multiciplity 2 in the Lie algebra of G, we prove that the orbital integral is bounded on the subset of σ-regular points of G (cf. Theorem 1.2).

In the second section, we explain the truncation process based on the notion of pH, M q-orthogonal sets and prove our main results. Using the Weyl integration formula, we can write

K

pfq “ ÿ

MPLpA0q

c

ÿ

SPT_M

ÿ

xmPκS

c

ż

Sσ

K

px

, γ, f qdγ where

K

px

, γ, f q “ |∆

px

γ q|

ż

diagpAMqzHˆH

ż

diagpAMqzHˆH

f

py

x

γy

q

ˆf

px

x

γx

qu

px

, y

, x

, y

, T qdpx

, x

qdpy

, y

q and

u

px

, y

, x

, y

, T q “ ż

AHzAM

upy

ax

, T qupy

ax

, T qda.

The function J

p f q is obtained in a similar way to K

p f q where we replace the weight function u

px

, y

, x

, y

, T q by another weight function v

px

, y

, x

, y

, T q.

The weight function v

is given by v

px

, y

, x

, y

, T q :“

ż

A_HzA_M

σ

ph

paq, Y

px

, y

, x

, y

, T qqda

where σ

p¨, Y q is the function of [Ar3] depending on a pH, M q-orthogonal set Y and Y

px

, y

, x

, y

, T q is a pH, M q- orthogonal set obtained as the ”minimum” of two p H, M q - orthogonal sets Y

p x

, y

, T q and Y

p x

, y

, T q (cf. (2.4), Lemma 2.2 and (2.11)). If Y

and Y

are two pH, M q-orthogonal positive sets then the ”minimum”

Z of Y

and Y

satisfies the property that the convex hull S

pZ q in a

za

of the points of Z is the intersection of the convex hulls S

p Y

q and S

p Y

q in a

z a

of the points of Y

and Y

respectively.

If }T } is large relative to }x

}, }y

}, i “ 1, 2 then σ

p¨, Y

px

, y

, x

, y

, T qq is just the characteristic function of S

p Y

p x

, y

, x

, y

, T qq . In that case, this function is equal to the product of σ

p¨, Y

px

, y

, T qq and σ

p¨, Y

px

, y

, T qq.

Our proofs consist to establish good estimates of |u

ppx

, y

, x

, y

, T q ´ v

p x

, y

, x

, y

, T q| when x

, y

, i “ 1, 2 satisfy f

p y

x

γy

q f

p x

x

γx

q ‰ 0 for some γ P S

and x

P κ

. Then, using that orbital integrals are bounded, we deduce our result on |K

pf q ´ J

pf q|.

This work is a first step towards a local relative trace formula. For the spectral side, we have to prove that K

pfq is asymptotic to a distribution k

pfq which is of general form (0.6) and constructed from spectral data. We hope that we can express the constant term of k

p f q in terms of regularized local period integrals introduced by B. Feigon in [F] in the same way than Jacquet-Lapid-Rogawski regularized period integrals for automorphic forms in [JLR]. We plan to explicit such a local relative trace formula for P GL p 2 q .

Acknowledgments. We thank warmly Bertrand Lemaire for his answers to our

many questions on algebraic groups. We thank Bertrand R´ emy and David Renard

for our helpful discussions. We thank also Guy Henniart for providing us a proof of

(1.5).

1 Preliminaries

1.1 Reductive p-adic groups

Let F be a non archimedean local field of characteristic 0 and odd residual charac- teristic q. Let | ¨ |

denote the normalized valuation on F.

For an algebraic variety M defined over F, we identify M with M pFq where F is an algebraic closure of F and we set M :“ M pFq.

We will use conventions like in [W2]. One considers various algebraic groups J defined over F, and sentences like

” let M be an algebraic group” will mean ” let M be the F-points of an algebraic group M defined over F” and ” let A be a split torus ” will mean ” let A be the group of F-points of a torus, A, defined and split over F .”

(1.1)

If J is an algebraic group, one denotes by RatpJq the group of its rational characters defined over F. If V is a vector space, V

will denote its dual. If V is real, V

will denote its complexification.

Let G be an algebraic reductive group defined over F. We fix a maximal split torus A

of G and we denote by M

its centralizer in G.

We denote by A

the maximal split torus of the center of G and we define a

: “ Hom

p Rat p G q , R q .

One has the canonical map h

: G Ñ a

which is defined by

e

“ |χpxq|

, x P G, χ P RatpGq. (1.2) The restriction of rational characters from G to A

induces an isomorphism

RatpGq b

R » RatpA

q b

R . (1.3) Notice that RatpA

q appears as a generating lattice in the dual space a

of a

and

a

» RatpGq b

R . (1.4)

The kernel of h

, which is denoted by G

, is the intersection of the kernels of

| χ |

for all character χ P Rat p G q of G. The groupe G

is distinguished in G and contains the derived group G

of G. Moreover, it is well-known that

the group G

is generated by the compact subgroups of G. (1.5)

G. Henniart has communicated to us an unpublished proof of this result by N. Abe,

F. Herzig, G. Henniart and M.F. Vigneras.

One denotes by a

(resp., ˜ a

) the image of G (resp., A

) by h

.

Then G{G

is isomorphic to the lattice a

. (1.6) If P is a parabolic subgroup of G with Levi subgroup M , we keep the same notation with M instead of G.

The inclusions A

Ă A

Ă M Ă G determine a surjective morphism a

Ñ a

(resp., an injective morphism, ˜ a

Ñ ˜ a

) which extends uniquely to a sur- jective linear map h

from a

to a

(resp., injective map between a

and a

).

The second map allows to identify a

with a subspace of a

and the kernel of the first one, a

, satisfies

a

“ a

‘ a

. (1.7)

For M “ M

, we set a

: “ a

and a

: “ a

0

. We fix a scalar product p¨ , ¨q on a

which is invariant under the Weyl group W pG, A

q of pG, A

q. Then a

identifies with the fixed point set of a

by W pG, A

q and a

is an invariant subspace of a

under W p G, A

q . Hence, it is the orthogonal subspace to a

in a

. The space a

might be viewed as a subspace of a

by (1.7). Moreover, by definition of the surjective map a

Ñ a

, one deduces that

if m

P M

then h

pm

q is the orthogonal projection of h

pm

q onto

a

. (1.8)

From (1.7) applied to pM, M

q instead of pG, M q, one obtains a decomposition a

“ a

‘ a

. From the W pG, A

q invariance of the scalar product, one gets:

The decomposition a

“ a

‘ a

is an orthogonal decomposition.

The space a

appears as a subspace of a

and, in the identification of a

with a

given by the scalar product, a

identifies with a

.

(1.9) The decomposition a

“ a

‘ a

is orthogonal relative to the restriction to a

of the W p G, A

q -invariant inner product on a

and the natural map h

is identified with the orthogonal projection of a

onto a

.

In particular, a

is the orthogonal projection of a

onto a

. More-

over, we have ˜ a

“ a

X ˜ a

(cf. [Ar3] (1.4)). (1.10) By a Levi subgroup of G, we mean a group M containing M

which is the Levi component of a parabolic subgroup of G. If P is a parabolic subgroup containing M

then it has a unique Levi subgroup denoted by M

which contains M

. We will denote by N

the unipotent radical of P .

For a Levi subgroup M , we write LpM q for the finite set of Levi subgroups of G

which contain M and we also let P pMq denote the finite set of parabolic subgroups

P with M

“ M.

Let K be the fixator of a special point in the apartment of A

in the Bruhat-Tits building. We have the Cartan decomposition

G “ KM

K. (1.11)

If P “ M

N

is a parabolic subgroup of G containing M

, then

G “ P K “ M

N

K. (1.12)

If x P G, we can write

x “ m

p x q n

p x q k

p x q , m

p x q P M

, n

p x q P N

, k

p x q P K. (1.13) We set

h

p x q : “ h

p m

p x qq . (1.14) The point m

pxq is defined up an element of K but h

pxq does not depend of this choice.

We introduce a norm } ¨ } on G as in ([W2] §I.1.) (called height function in ([Ar3])). Let Λ

: G Ñ GL

pFq be an algebraic embedding. For g P G, we write

Λ

pgq “ pa

q

, Λ

pg

q “ pb

q

. We set

}g} :“ sup

i,j

supp|a

|

, |b

|

q. (1.15) If Λ : G Ñ GL

pFq is another algebraic embedding then the norm } ¨ }

attached to Λ as above is equivalent to } ¨ } in the following sense: there are a positive constant C

and a positive integer d

such that

} g }

ď C

} g }

.

This allows us to use results of [W2] for estimates on norms.

The following properties of } ¨ } are immediate consequences of definition:

1 ď }x} “ }x

}, x P G, (1.16)

}xy} ď }x}}y}, x, y P G. (1.17)

In order to have estimates, we introduce the following notation. Let r be a positive integer. Let f and g be two positive functions defined over a subset W of G

.

We write f p x q ď g p x q , x P W if and only if there are a positive constant

c and a positive integer d such that f pxq ď cgpxq

for all x P W . (1.18)

We write f p x q « g p x q , x P W if f p x q ď g p x q , x P W and g p x q ď

f pxq, x P W . (1.19)

If f

, f

and f

are positive functions on G

, we clearly have

if f

pxq ď f

pxq, x P W and f

pxq ď f

pxq, x P W then f

pxq ď f

pxq, x P W , if f

pxq « f

pxq, x P W and f

pxq « f

pxq, x P W then f

pxq « f

pxq, x P W . Moreover, if f

, f

, g

and g

are positive functions on G

which take values greater or equal to 1, we obtain easily the following properties:

1. for all positive integer d, we have f

pxq « f

pxq

, x P W , 2. if f

pxq ď g

pxq, x P W and f

pxq ď g

pxq, x P W then

p f

f

qp x q ď p g

g

qp x q , x P W ,

3. if f

pxq « g

pxq, x P W and f

pxq « g

pxq, x P W then pf

f

qpxq « pg

g

qpxq, x P W .

(1.20)

Since }x} “ }xyy

} ď }xy}}y} and }xy} ď }x}}y}, we obtain

If Ω is a compact subset of G, then } x } « } xω } , x P G, ω P Ω. (1.21) Let P “ M

N

be a parabolic subgroup of G containing M

. Then, each x P G can be written x “ m

pxqn

pxqk where m

pxq P M

, n

pxq P N

and k P K . By ([W2]

Lemma II.3.1), we have

}m

pxq} ` }n

pxq} ď }x}, x P G. (1.22)

Recall that G

is the kernel of h

: G Ñ a

. Let us prove that

}xa} « }x}}a}, x P G

, a P A

. (1.23)

According to the Cartan decomposition (1.11), if g P G, we denote by m

pgq an element of M

such that there exist k, k

P K with g “ km

pgqk

. Notice that } h

p m

p g qq} does not depend on our choice of m

p g q . By (1.21), one has

}g} « }m

pgq}, g P G, (1.24)

and by ([W2]) 1.1.(6)) we have

} m

} « e

, m

P M

. (1.25)

Let x P G

and a P A

. Then m

pxq P G

X M

and m

pxaq “ m

pxqa. Thus, one

has h

p m

p x qq “ 0. We deduce from (1.8) that h

p m

p x qq belongs to a

. Since

h

pm

pxqaq “ h

pm

pxqq ` h

paq and h

paq P a

, we obtain by orthogonality that

1

2 p} h

p m

p x qq} ` } h

p a q}q ď } h

p m

p x q a q} ď } h

p m

p x qq} ` } h

p a q} . Hence (1.23) follows from (1.24) and (1.25).

We denote by C

pGq the space of smooth functions on G with compact support.

We normalize Haar measures according to [Ar3] §1. Unless otherwise stated, the Haar measure on a compact group will be normalized to have total volume 1.

Let M be a Levi subgroup of G. We fix a Haar measure on a

so that the volume of the quotient a

{ ˜ a

equals 1.

Let P “ M N

P P pM q. We denote by δ

the modular function of P given by δ

p mn q “ e

, m P M, n P N

,

where 2ρ

is the sum of roots, with multiplicity, of pP, A

q. Let ¯ P “ M N

be the the parabolic subgroup which is opposite to P . If dn is a Haar measure on N

then the number

γpP q “ ż

NP

e

dn

is finite. Moreover, the measure γpP q

dn is independent of the choice of dn and thus defines a canonical Haar measure on N

.

If dm is a Haar measure on M then there exists a unique Haar measure dg on G, independent of the choice of the parabolic subgroup P , such that

ż

G

fpgqdg “ 1 γpP qγp P ¯ q

ż

NP

ż

M

ż

NP¯

fpnm¯ nqδ

pmq

d¯ n dm dn,

for f P C

pGq. We say that dm and dg are compatible. Compatibility has the obvious transitivity property relative to Levi subgroups of M . Using the Iwasawa decomposition (1.12), these measures satisfy

ż

G

f pgqdg “ 1 γ p P q

ż

K

ż

M

ż

NP

f pmnkqdn dm dk.

1.2 The symmetric space H z G

Let E be an unramified quadratic extension of F. Thus E “ Frτs where τ

is not a square in F. We denote by σ the nontrivial element of the Galois group GalpE{Fq of E{F. The normalized valuation | ¨ |

on E satisfies |x|

“ |x|

for x P F.

If J is an algebraic group defined over F, as usual we denote by J its group of points over F. Let J ˆ

E be the group, defined over E, obtained from J by extension of scalars. We consider the group

J ˜ : “ Res

p J ˆ

E q

defined over F, obtained by restriction of scalars.

With our convention, one has ˜ J “ JpFq ˜ and ˜ J is isomorphic to JpEq.

Let H be a reductive group defined over F. In all this article, we assume that H is split over F and we set G :“ H ˜ and G :“ H. We fix a maximal split torus ˜ A

of H. Then A

is also a maximal split torus of G and we have A

“ A

.

The nontrivial element σ of GalpE{Fq induces an involution of G defined over F, which we denote by the same letter. This automorphism σ extends to an E- automorphism σ

on G ˆ

E.

We consider the canonical map ϕ defined over F from G to pH ˆ

Eq ˆ pH ˆ

Eq by ϕpg q “ pg, σpgqq.

Then, ϕ extends uniquely to an isomorphism Ψ defined over E from G ˆ

E to pH ˆ

Eq ˆ pH ˆ

Eq such that Ψpg q “ pg, σpgqq for all g P G and if Ψpgq “ pg

, g

q then Ψpσ

pgqq “ pg

, g

q.

(1.26) Now we turn to the description of the geometric structure of the symmetric space S “ HzG according to [RR] sections 2 and 3.

Let g be the Lie algebra of G and g be the Lie algebra of its F-points. We will say that g is the Lie algebra of G and the Lie algebra h of H consists of the elements of g invariant by σ. We denote by q the space of antiinvariant elements of g by σ.

Thus, one has g “ h ‘ q and g may be identified with h b

E.

As in ([RR] §2.), we say that a subspace c of q is a Cartan subspace of q if c is a maximal abelian subspace of q made of semisimple elements. As E “ Frτ s, the multiplication by τ induces an isomorphism between the set of Cartan subspaces of q and the set of Cartan subalgebras of h which preserves H-conjugacy classes.

We denote by P the connected component of 1 in the set of x in G such that σpxq “ x

. Then the map p from G to P defined by ppxq “ x

σpxq induces an isomorphim of affine varieties p : HzG Ñ P .

A torus A of G is called a σ-torus if A is a torus defined over F contained in P . Notice that such torus are called σ-split torus in [RR]. We prefer change the terminology as σ-tori are not necessarily split over F. Each σ-torus is the centralizer in P of a Cartan subspace of q, or equivalently of a Cartan subalgebra of h.

Let S be a maximal torus of H. We denote by S

the connected component of

S r X P . Then S

is a σ-torus defined over F which identifies with the antidiagonal

tps, s

q; s P Su of S ˆ S by the isomorphism (1.26). Thus, S

is a maximal σ-torus

and each maximal σ-torus arises in this way . The H-conjugacy classes of maximal

tori of H are in bijective correspondence with the H-conjugacy classes of maximal

σ-tori of G by the map S ÞÑ S

. The roots of S (resp.; S

) in h “ LiepHq (resp.;

q b

F) are the restrictions of the roots of ˜ ¯ S in g “ LiepGq.

Therefore, each root of S (resp.; S

) in g has multiplicity two. If ˜ S splits over a finite extension F

of F, we denote by ΦpS

, g

q (resp.; ΦpS

, h

q) the set of roots of S

p F

q in g b

F

(resp.; S p F

q in h b

F

).

Let ˜ s be the Lie algebra of ˜ S. Then, the differential of each root α of Φp S ˜

, g

q defines a linear form on ˜ s b

F

which we denote by the same letter.

(1.27)

Let GalpF{Fq be the Galois group of F{F. By ([RR] §3), the set of pH, S

q- double cosets in HS

X G are parametrized by the finite set I of cohomology classes in H

pGalpF{Fq, H X S

q which split in both H and S

. To each such classe m, we attach an element x

P G of the form x

“ h

a

with h

P H and a

P S

such that m

“ h

γph

q “ a

γ pa

q for all γ P GalpF{Fq.

1.1 Lemma. Let x P G such that x “ hs with h P H and s P S. Then, ˜ xSx

is a maximal torus of H and there exists h

P H such that x

“ h

x centralizes the split connected component A

of S.

Proof :

Replace S by a H-conjugate if necessary, we may assume that A :“ A

is contained in the fixed maximal split torus A

of H. Since H is split, A

is also a maximal split torus of G.

Since x “ hs P G, the torus S

:“ xSx

is equal to hSh

Ă H. Thus S

is defined over F and contained in H and we obtain the first assertion.

Let S

:“ S

pFq and let A

be the split connected component of S

. There exists h

P H such that h

A

h

Ă A

. We set x

“ h

x, thus we have A

:“ x

Ax

Ă A

. Let M “ Z

pAq and M

“ Z

pA

q “ x

M x

. Then A

and x

A

x

are maximal split tori of M

. Therefore, there exists y

P M

such that y

x

A

x

y

“ A

. As H is split, the Weyl group of A

in G coincides with the Weyl group of A

in H. Thus, there exist h

P N

pA

q and v P Z

pA

q such that z :“ y

x

“ h

v.

For a P A Ă A

, one has zaz

“ h

ah

“ y

x

ax

y

“ x

ax

since x

ax

P A

and y

P M

. One deduces that x

:“ h

h

x centralizes A.

Thus, for each maximal torus S of H, we can fix a finite set of represen- tatives κ

“ tx

u

of the pH, S

q-double cosets in HS

XG such that each element x

may be written x

“ h

a

where h

P H centralizes A

and a

P S

. Hence x

centralizes A

.

(1.28)

1.3 Weyl integration formula and orbital integrals

We first recall basic notions on the symmetric space according to ([RR], §3). An

element x in G is called σ-semisimple if the double coset HxH is Zariski closed. This

is equivalent to say that p p x q is a semisimple point of G. We say that a semisimple

element x is σ-regular if this closed double coset HxH is of maximal dimension.

This is equivalent to say that the centralizer of ppxq in q (resp.; P ) is a Cartan subspace of q (resp.; a maximal σ-torus of G).

We denote by G

the set of σ-regular elements of G.

For g P G, we denote by D

pgq the coefficient of the least power of t appearing nontrivially in detpt ` 1 ´ Adpg qq. We define the H-biinvariant function ∆

on G by

∆

p x q “ D

p p p x qq . Then by ([RR], Lemma 3.2. and Lemma 3.3), the set of g P G such that ∆

pgq ‰ 0 coincides with G

.

Let S be a maximal torus of H with Lie algebra s. Then ˜ s :“ s b

E identifies with the Lie algebra of ˜ S. For g P x

S

with x

P κ

, one has

∆

pgq “ D

pppgqq “ detp1 ´ Adpppgqqq

. (1.29) By ([RR] Theorem 3.4 (1)), the set G

is a disjoint union

G

“ ď

tSuH

ď

xmPκS

H `

px

S

q X G

˘ H,

where tSu

runs the H-conjugacy classes of maximal tori of H.

(1.30)

If x

P κ

then x

“ h

a

for some h

P H and a

P S

, hence p p x

q “ a

commutes with S and S

. Therefore for γ P S

, we have

ppx

γq “ ppx

qγ

and Hx

γS “ Hx

γ.

We have the following Weyl integration formula (cf. [RR] Theorem 3.4 (2)):

Let f be a compactly supported smooth function on G, then we have ż

G

f pyqdy “ ÿ

tSuH

ÿ

xmPκS

c

ż

Sσ

|∆

px

γq|

ż

SzH

ż

H

f phx

γlqdhd ¯ ldγ, (1.31) where the constants c

m

are explicitly given in ([RR] Theorem 3.4 (1)).

For our purposes, we need another version of this Weyl integration formula.

Let S be a maximal torus of H. We denote by A

its split connected component.

Since the quotient A

zS is compact, by our choice of measure, the integration over SzH in the Weyl formula above can be replaced by an integration over A

zH.

Moreover, it is convenient to change h into h

. As every x

P κ

commutes with A

(cf. (1.28)), one can replace the integration over pA

zHq ˆ H, by an integration over diagpA

qzpH ˆ Hq where diagpA

q is the diagonal of A

. This gives the following Weyl integration formula equivalent to (1.31):

ż

G

f pyqdy “ ÿ

tSuH

ÿ

xmPκS

c

ż

Sσ

|∆

px

γq|

ż

diagpASqzpHˆHq

f ph

x

γlqdph, lqdγ.

(1.32)