数理解析研究所講究録1052 巻 1998 年 81-98

81

An example of Adelic Zeta function associated toPrehomogeneous vector Spaces of Parabolic Type : Case
Iris MULLER$(F_{4}, \alpha_{1})$$(^{*})$

lnstitut de Recherche Math\’ematique Avanc\’ee, Universit6 Louis Pasteur and.R.S (UMR 7501), 67084 STRASBOURG Cedex, FRANCE-mail: muller@math.u-strasbg.fr$\mathrm{C}.\mathrm{N}$$\mathrm{e}$Abstract
In this work, we give an examP..l of an Adelic Zeta function, with its
functional equation and its poles, associated to Prehomogeneous vector Spacesof Parabolic Typein the spirit of the works of A.WeilandS.Rallis and G.Schiffmann ([R-S]) in the case where the fundamental invariant
is a quadratic form, using well known methods of calculus of Tamagawa
numbers.
$\mathrm{e}$$(F_{4}, \alpha_{1})$$([\backslash \eta \mathrm{E}1])$$([\mathrm{M}\mathrm{A}],[1\forall \mathrm{E}1])$

Introduction
Many adelic Zeta functions have been considered for Prehomogeneous
Vector Spaces
and many general results have been established.
These works begin with A.Weil in the case of a non degenerate quadratic form
[WE 1],.M. Mars for the case of a cubic form [MA],. Igusa in the caseof finitely many orbits and absolutely admissible representations, T.Shintani
and.Wright for the space of binary cubic forms and by A.Yukie when thevector space and the group acting have the same dimension.
K.Ying has proved the convergence of the Zeta function in almost all casesof irreducible , reduced , regular. A.Yulcie has studied cases where thegroup is a product ofusing smoothed version of Eisenstein series. And
many otherare done in this subject.
$(\mathrm{a}\mathrm{b}\mathrm{r}.

\mathrm{P}\mathrm{V})$$\mathrm{J}.\mathrm{G}$$\mathrm{J}.\mathrm{I}$$\mathrm{D}.\mathrm{J}$$\mathrm{P}\mathrm{V}$$GL_{\uparrow x}$$\mathrm{w}\mathrm{o}\mathrm{r}1’\backslash \mathrm{s}$$(^{*})$

Participation during the stay of the author in Japan supported by Grant-inAid for Scientific Research, The Ministry of Education, Science, Sports and..
Culture of Japan.

82
Here we shall give the adelic Zeta function , equation and poles for a particularand simple situation of Prehomogeneous Vector Spaces of parabolic type (ab.
).
PV of$\mathrm{P}\mathrm{T}$First we recall the form that the adelic Zeta function can take for PV of
PT for which the fundamental character has its values in the set of squareof the field, using mean function (we have always infinitely many genericorbits in this case) and we give suffisant conditions of absolute convergence
for it (prop. 1). Then we apply these results in the particular case of PV of
PT havingas Dynkin diagram (whith an exception), because they
are a particular case of a more general situation where it is possible to give ageneral description of the orbits by means of some quadratic forms
and it is possible to do the calculus in a general standing.
.
The cases considered in this paper are listed in table 1.
$(F_{4}, \alpha_{1})$$([\mathrm{b}’\mathrm{I}\mathrm{U}1])$

I Prehomogeneous vectors spaces of parabolic type$([\mathrm{R}\mathrm{U}1],[\mathrm{R}\mathrm{U}2]\mathrm{I}$The situation of PV of PT that we can consider is the following:Leta finite dimensional simple graded Lie algebra over a g..lobal
field of characteristic,is the element giving the gradation:$\mathfrak{g}=\oplus_{i\in \mathbb{Z}}9i$$0$$\mathrm{F}^{\mathrm{Z}}$$H_{0}$$\mathfrak{g}_{i}=\{x\in \mathfrak{g}|[H_{0}, x]=ix\}$$G’$is the centralizer of$H_{0}$

in the group$\mathrm{A}ut_{0}(\mathfrak{g})$of automorphisms of$\mathfrak{g}([\mathrm{B}\mathrm{O}$$2])$acts onby adjoint action andandis a geometric
.
mallv
Let
the Killing form of , then the dual PV of$1])$ .
We assume that1)
is an absolutely simple -module$2H0,$
2)can be completed in a{$G’$$\mathfrak{g}_{1}$$(G, \wedge 4d, \mathfrak{g}_{1})$$\mathfrak{g}_{-1}$$(90, \mathfrak{g}_{1}))$$B$$\mathrm{P}\mathrm{V}$$(\mathfrak{g}_{0}, \mathfrak{g}_{1})$$\mathfrak{g}$$\mathfrak{g}_{1}$$\mathfrak{g}_{1}’=$(denoted infinitesi-is$(\mathfrak{g}0, \mathfrak{g}_{-1})([\mathrm{R}\mathrm{U}$$\mathfrak{g}_{0}$$x\in \mathfrak{g}_{1}|(x,$$.)$$sl_{2^{-\mathrm{t}}}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}$(2) is equivalent to the regularity of the PV because of 1)}$\neq\emptyset$$([\mathrm{R}\mathrm{U}1]))$is a PV of PT regular , absolutely irreducible, having a relativeSoinvariant of minimal degree, denoted
and we call X the corresponding.
character$(G, \wedge 4d, \mathfrak{g}_{1})$$P$$([\mathrm{R}\mathrm{U}\mathrm{I}])$

83
Let $S,$
respectively the set of places,infinite places, finite places of thenumber field F.
To every $v\in S_{f}$ we associate as usual $\mathfrak{O}_{v}=\{x\in \mathrm{F}_{v} | |x|_{v}\leq 1\},$the set of unities,
the number of elements of the residual field. fl,arerespectively the ring of adeles of , the ideles of F.
Let a lattice in , for all in,is the closure of in,$L_{v}^{0}=\{x\in L_{v}||P_{v}(x)|v=1\},$ $I\iota^{\nearrow}v=\{g\in G_{v}|g(L_{v})=Lv\}$ and we denote
by $Gfl$ the adele group of $G,$ $K$ is a compact subgroup of
containing,$S_{\infty’ f}S$

$5\supset_{v}*$$q_{v}$$\mathrm{f}\mathrm{l}^{*}$$[^{=}$$L$$v$$\mathfrak{g}_{1}$$S_{f}$$L$$L_{v}$$\mathfrak{g}_{1,v}=\mathfrak{g}_{1}\otimes_{\mathrm{F}}\mathrm{F}_{v}$$G\mathrm{f}\mathrm{l}$$\prod_{v\epsilon S_{f}v}I\mathrm{i}’$$\mathfrak{g}_{1,\mathrm{f}\mathrm{l}}=(\mathfrak{g}fl)_{19\otimes_{\mathrm{F}}}=1$$\mathfrak{g}_{1,\mathrm{f}\mathrm{l}}’=$$S(\mathfrak{g}_{1,\mathrm{f}\mathrm{l}})${$x\in 91,\mathrm{f}\mathrm{l}|\forall v\in SP_{v}(X_{v})\neq 0$

fl, for almost all $v\in S_{f}x_{v}\in L_{v}^{0}$ }is the Schwartz space of functions on.
we can consider the following Zeta function:$\mathfrak{g}_{1,fl}$When$\chi(G)=\mathrm{F}^{*2}$

II An adelic Zeta function under some assumptions: case$1-$

The mean function : recallsa) The local caseLet$\theta_{t}$$\chi(G)=\mathrm{F}^{*2}$$t\in \mathrm{F}_{v}^{*}$$([\mathrm{R}- \mathrm{S}],[\mathrm{R}\mathrm{U}1],[\mathrm{I}\mathrm{G}3])$, $U_{t}=\{x\in \mathfrak{g}_{1,v} | P_{v}(x)=t\}$ , on

$U_{t}$

there is a gauge form

defined by $\theta_{t}(x)=(\frac{dx}{d(P_{v}(x))})_{x=t}$ which determines a measure on

denotedin$\mu_{v,t}$

and if([R-S])$\mathrm{J}/I_{f}(t)=\int_{U_{t}}fd\mu_{v},t$

we have for every$f\in S(\mathfrak{g}_{1,v})$

and$U_{t}$$\varphi$$\mathfrak{D}(P(9’1,v))$$\int_{\mathfrak{g}_{1,v}}\varphi(Pv(_{X}))f(X)d_{X=}\int_{P(9_{1,v}’}))\mathrm{n}_{/[}f(t)\varphi(tdt$(with the volume ofand the volume of
When is in , we denote (as usual) by
to the characteristic function of the lattice$L_{v}$$v$$S_{f}$$\mathfrak{O}_{v}$$\mathrm{J}l_{v}$$L_{v}$

equal to 1 if $v\in S_{f}$ )the mean function associated.

b) The global caseWe assume now that the following conditions, denoted $(H)$ are satisfied :84
HYPOTHESIS (H).

–1) A lmost everywherecontains
2) There is $C>0,$ $\alpha>1$ such that for almost allin$P_{v}(\mathfrak{g}_{1,v}’)$$\mathfrak{O}_{v}^{*}$

$v$

in

$S_{f}$

we have for all$t$$\mathfrak{O}_{v}^{*}|M_{v}(t)-1|\leq c.q_{v}-\alpha$First, for every in
we denote , as before, $U_{t}=\{x\in \mathfrak{g}_{1} | P(x)=t\}$ ,then by hypothesis (2) : (1) are factors of convergence of, with $t_{v}=t$
for all in ([WE 1]).
$t$$[^{=*}$$(d\mu_{t_{v}})$

$S$

$v$

Secondly, we consider for $t\in fl^{*}$ as usual $U_{t}=\{x\in \mathfrak{g}_{1,fl}’ | P(x)=t\}$ ,we define on
the measureproduct of the local measuresand forthe function $M_{f}(t)= \int_{U_{t}}fd\mu_{t}$ is a borelian function on([R-S]) and we have the following property$U_{t}$$\mu_{t}$$\mu_{v,t_{v}}$$f\in S(\mathfrak{g}_{1},\mathrm{f}\mathrm{l})$$(*)$$fl^{*}$

where$\mathbb{J}/I_{f(g}.()t)=|\chi(g)|-\kappa.+1Mf(\chi(g).t)$

Now we can define the adelic Zeta function : forunitary character of , trivial onlet$\mathrm{f}\mathrm{l}^{*}$$r_{\dot{v}}= \frac{\dim(\mathfrak{g}_{1})}{\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{P}}$$s\in \mathbb{C},$$f\in S(9_{1,\mathrm{f}\mathrm{l}}),$$\lambda$a$\mathrm{F}^{*}$$W_{f}( \lambda, s)=\sum_{\xi\in \mathrm{F}^{*/_{\mathrm{F}}2}*}\mathit{1}^{M}fl*)f(t\xi 2\lambda(t)|t|^{2S+2}d^{*}t\mathrm{f}\mathrm{l}$which corresponds to an integration on, with$D$
being the discriminant of and,if $v\in S_{f}$$\rho_{v}=1$
and else.
This is the adelic Zeta function introduced by A.Weil ([WE 1]), then by
S.Rallis and G.Schiffmann ([R-S]) in the case where $P$ is a quadratic form.
$(\mathrm{f}\mathrm{l}^{*})^{2}.P(9_{1});$$d^{*}t \mathrm{f}\mathrm{l}=|D|^{-\frac{1}{2}}\prod_{v\in sv}d*t$$\mathrm{F}$$d^{*}t_{v}= \rho_{v}(\frac{dt_{v}}{|t_{v}|})$$\rho_{v}=(1-\frac{1}{q_{v}})-1$

2 A result about convergence

NOTATIONS.

–

1) If
is not of residual characteristic 2 , let
a set ofrepresentatives of, withinand2) For and in
we denote by $(u, v)$ the value of the Hilbert symbol
and bythe corresponding character on3) Let $Z_{v}( \lambda, s)=\int_{L_{v}}\lambda(P_{v}(x))|P(vx)|^{s}dx$ (having volume 1)$[^{=}v$$\{1, \epsilon_{v’ v’ v}\pi\pi.\epsilon_{v}\}$$\mathrm{F}_{v}^{*}/_{\mathrm{F}_{v^{2}}^{*}}$$u$$\chi_{u}$$v$$\epsilon_{v}$$\mathfrak{O}_{v}^{*}$$| \pi_{v}|=\frac{1}{q_{v}}$$\#^{=*}/v\mathrm{F}_{v^{2}}*$$\mathrm{F}_{v}^{\mathrm{j}*}(\chi_{u}(v)=(u, v))$

$L_{v}$,

854) We choose an additive character of A such that $\tau(xy)$ put in duality flwith itself in such a manner that the discrete subgroup corresponds to itself
with the duality $\tau(xy)$ and for, let$\tau$$\mathrm{F}$$f\in S(\mathfrak{g}_{1},fl),$$\hat{f}(y)=\int_{\mathfrak{g}_{1}},.\mathrm{n}f(x)\mathcal{T}(B(Xy))dxfl$

5)$\rho_{\mathrm{F}}$$y\in g_{-1,\mathrm{f}\mathrm{l}}$.

is the residue in 1 of the Zeta function of $[^{=}and$

PROPOSITION 1.

–If we assume that the conditions $(Hl)$ are verified withcontains
almost everywherecontains
andsuch that almost everywhere$\dot{i})\forall v\in S_{f}P_{v}(\mathfrak{g}_{1}’,v)$$\dot{i}\dot{i})$$C_{\mathrm{F}}=2^{-\Gamma}|D|^{\frac{n-1}{2}}$$\mathfrak{O}_{v}^{*}$$\chi_{v}(K_{v})$$\mathfrak{O}_{v^{2}}^{*}$$\exists\alpha>1$$i_{\dot{i}\dot{i}})\exists c>0$$| \int_{L_{v}^{0}}(\pi_{v}, P_{v}(x))dX|\leq$$c.q_{v}^{-\alpha}$

andsuch that for almost all
number with strictly positive real part$\dot{i}v)\exists d>0$$\exists\beta>2$

$v$

in

and

$S_{f}$$s$complex.

$|Z_{v}(x_{\pi_{v}}, s)-Z_{v}(\chi\pi v\cdot\epsilon v’ S)|\leq d.q_{v}^{-\beta}$

Then for every character ofon , almost everywhere non ramified
we have for all1) for every complex number , with strictly positive real partisabsolutely convergent and for $alfA>0$ , let $S_{A}=\{s\in \mathbb{C}$$Re(s)>$, $|Im(S)|<A.Re(s)\}$ thenand$\mathrm{f}\mathrm{l}^{*},$$tr\dot{i}v\dot{i}a\iota$$\mathrm{F}^{*}$$f\in S(\mathfrak{g}_{1,\mathrm{R}})$$W_{f}(\lambda, s)$

$s$

$|$

$0$

$l\dot{i}m_{S}arrow 0_{S},\in SAfS7\eta_{/^{\mathit{7}}}(\lambda, S)--0\dot{i}f\lambda\neq id$$l\dot{i}m_{S}-0,s\in sASWf(\dot{i}d, S)=C\mathrm{F}\cdot\rho_{\mathrm{F}}\hat{f}(0)$2)$\forall t\in \mathrm{f}\mathrm{l}^{*}$$\sum_{\xi\in \mathrm{F}^{*}}\mathit{1}\lambda/If(t^{2}\xi$

I is absolutely convergent.

We can remark that conditions $(H)+\dot{i}i)$ are equivalent to ), ), ).
$\mathrm{i}$$\mathrm{i}\mathrm{i}$$\mathrm{i}\mathrm{i}\mathrm{i}$Proof1) We proceed as in the work of A.Weil

G.Schiffmann ([R-S], comparing
Integrals:$77_{f}^{\gamma}(\lambda, s)$

and S.Rallis andto an appropriate sum of Onowith$([\backslash \forall \mathrm{E}1]),\S 4.5)$$Z(f; \lambda J, s)=\int_{9_{1,\mathrm{f}\mathrm{l}}},f(x)\lambda’(P(X)d\prime x\mathrm{f}\mathrm{l}$$d’x_{\mathrm{R}}=|D|^{-\frac{n}{2}} \prod_{v\in^{s^{\rho dx_{v}}}}v$

We can deduce from a calculus analogous to that ofand the results in(
p. 124 and corollary p.288) that $l_{\dot{i}m_{Sarrow}}0,Re(S)>0s.Z(f;\dot{i}d, s)=$and if is as in theexists and is
non-nil.
2) We deduce 2) from 1), using Fubini-theorem and).
$([\backslash \forall \mathrm{E}1])$$[\backslash \forall \mathrm{E}3]_{\mathrm{C}},\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{y}2$$p_{\mathrm{F}}\hat{f}(0)$$\lambda$.ion$\mathrm{p}.\mathrm{r}\mathrm{o}_{\mathrm{P}^{\mathrm{o}\mathrm{S}}}\mathrm{i}\mathrm{t}$$\iota_{\dot{i}m_{Sarrow}}0,Re(S.)>0Z(f;\lambda, S)$$\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{n}\mathrm{l}\mathrm{P}^{\mathrm{t}}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{i}$So finally we can write for $Re(s)>0$ :$W_{f}( \lambda, S)=\int \mathrm{f}\mathrm{l}^{*}/\mathrm{F}^{*}$$( \sum\lambda I_{f}(t\xi 2))\lambda(t)|t|^{2_{S+2*}}dt\mathrm{f}\mathrm{l}$$\xi\in \mathrm{F}^{*}$$\square$.

86
III The case$1$$(\mathrm{F}_{4}, \alpha_{1})$The situation

We suppose from now that we are in the casewhich means thefollowing additional properties at section I.
If is any maximal toral subalgebra of
the associated root system isalso graded:, and the positive system
is chosenso that we have
. Then, as we havecan be conlpleted in a
, we can assume that-triple}andgenerate so the irreducible condition of I is expressed by saying that A isirreducible and there is only one simple root not in, it is in
ancl
is a maximal parabolic subalgebra of.
When
is of dimension not greater than one, there is a maximal setof orthogonal roots of, denoted’ such that $\sum_{1\leq i<_{7l}}h_{i}=2H_{0}$ ,whereis the -root of . The non-nil restrictions of tosubalgebra
is a root system denoted of rank , having the same properties
than : irreducibility, gradation and is also associated to a maximal parabolic(prop. 2.6.1 and coroll. 3.1.7 of [MU 1]).
Thecase” corresponds tois a commutative Liealgebra). This case is intensively studied and it is easy to study in a generalstanding the ordinary Zeta function, equation and poles (except in two cases)
but all these cases (except two) are lunown by case by case examination. This
is why I talk about the casehoping that it is of sonue interest.
$(F_{4}, \alpha_{1})$$a$$\triangle$$\mathfrak{g}0,$$\triangle_{i}=\{\lambda\in\triangle|\lambda(H_{0})=i\}$$\triangle^{+}$$\mathfrak{g}_{1}’=\{x\in \mathfrak{g}_{1}|(x, 2H0, .)$$\bigcup_{i\geq 1}\triangle_{i}\subset\triangle^{+}$$sl_{2}$$\neq\emptyset$$\mathfrak{g}_{1}$$\mathfrak{g}_{-1}$$\mathfrak{g}$$\triangle 0$$\triangle_{1}$$\oplus_{i\geq 0\mathfrak{g}_{i}}$$\mathfrak{g}([\mathrm{R}\mathrm{U}1])$$\mathfrak{g}_{2}$$\triangle_{1}$$h_{i}$$(\lambda_{i})_{1\leq i}\leq f\gamma$$\triangle$$\lambda_{i}$$\mathrm{c}\mathrm{o}$$R$$\oplus_{1\leq i\leq?}\mathrm{z}[^{=}hi$$\mathrm{t}\overline{\mathrm{h}}\mathrm{e}$$n$$\triangle$$\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{a}\mathrm{l},\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}\backslash$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{U}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\tau r\mathrm{e}$$\mathfrak{g}_{2}=\{0\}(\mathfrak{g}_{1}$$(F_{4}, \alpha_{1})$Ifis one dimensional it is easy to prove that $P(x)=B((ad(x))4(\omega_{-}), \omega_{-})$ ,$P$ is ofwhere
is a generator of(lemmedegree four and
4.1 of [MU 1] . If we assume that the roots
are strong orthogonal
and of same length, then the different types $(R, \alpha 0),$being the only simple
root in, are given by (prop. 6.6 of [MU 2]) :$\mathfrak{g}_{2}$$\omega_{-}$$\chi(C_{7})--\mathrm{I}^{=*2}$$\mathfrak{g}_{-2},$$)$$(\lambda_{i})_{1\leq\leq 4}i$$\alpha_{0}$$R_{1}$$(B_{4}, \alpha_{2}),$in the Bourbaki notation
the diagram)$(D_{4}, \alpha_{2}),$$([\mathrm{B}\mathrm{O}1])(\alpha_{1}$$(F_{4}, \alpha_{1})$is the long simple root at the end of

So we assume from now that the Dynkin diagram of$\mathrm{F}_{4}$$\mathrm{R}$is of type87
are chosen in the corresponding root space (as in [MU 3])

If
then

$(X_{\pm\lambda_{i}})_{1\leq i\leq 4}$$(**)$$\omega_{i}=[\omega_{+}, X_{-\lambda_{i}}],$$\omega_{+}$$\mathfrak{g}_{1}=\oplus_{1\leq i\leq 4}(\mathrm{F}\mathrm{j}x_{\lambda}\oplus i[=\omega i)\oplus 1\leq i<j\leq 4E_{i,j}$being a generator of$\mathfrak{g}_{2},$$E_{i,j}=\{x\in \mathfrak{g}[h_{i}, x]=[h_{j}, x]=$

, 1 $\leq k\neq i,j\leq 4$ }, is the common dimension of eachsubspace, we have $\dim(\mathfrak{g}_{1})=8+6d,$ $ki=2+3 \frac{d}{2}$ and can take thevalues 1, 2, 4, 8 (the case $d=8$ is treated in [IG 2]) (cf.Table 1).
$P$ is normalized such that $P( \sum_{1<i\leq}4)x_{\lambda_{i}}=1$ .
Leta non-empty subset of} and $H_{I}= \sum_{\{i\in I\}}h_{i}$ then theinis reductive , its semi-simple part , denoted
centralizer ofis a PV
is graded by $ad(H_{I})$ andof PT which is absolutely irreducible and commutative if $H_{I}\neq 2H_{0}$ because
for $|i|\geq 2$ and $P_{H_{I}}(x)=P(x+ \sum_{\{i\not\in I\}}X_{\lambda_{i}})$ is then a fundamental
invariant of it and its degree is , the number of elements of .
$x$,$[h_{k}, x]=0$

$d$

$d$

$E_{i,j}$$I$$\overline{\{}1,2,3,4$$\oplus_{i\not\in I}\mathrm{F}h_{i}$$\mathfrak{g}$$l[_{i}=\mathfrak{U}\cap \mathfrak{g}_{i},$$\mathrm{U}=\mathrm{U}(\oplus_{i\not\in Ii}\mathrm{F}:h)$$(\mathrm{U}_{0}, \mathfrak{U}_{1})$$\mathrm{U}_{i}=\{0\}$

$I$

$|I|$

2 Preliminary results
LEMMA 2- LOCAL RESULTS. $-R=F_{4}$1) For almost every
2) $\exists K>0$ suchfor almost every $v\in S_{f}$ and for all$v\in S_{f}Z_{v}(\chi_{\pi_{v}\epsilon}v’ S)--Z_{v}(\chi_{\pi_{v}}, s)$$y\in \mathfrak{O}_{v}^{*}$

we have$tf \iota at-\frac{3}{2}$$|_{i}\uparrow/[_{v}(y)-1|\leq K.q_{v}$Sketch of the proof
The detailed proof will appear somewhere later. We use similar methods asin [IG 2]
1) A change of variable
Let $H=h_{1}+h_{2}+h_{3},$ $E_{i}(H)=\{x\in \mathfrak{g}|[H, x]=ix\}$ .
can be decomposed in the form
every in
Using$(^{**})$$x$$\mathfrak{g}_{1}$$x= \sum_{0\leq i\leq 3}x_{i}$so$x_{0}=tx\lambda_{4},$$x3=u\omega 4,$with$x_{i}\in E_{i}(H)\cap \mathfrak{g}_{1}$$(t, u)\in[=\mathrm{F}\cross$

If $t\neq 0$ we have
$x=exp(.ad(c))(X’)$

with$x^{r}=tX_{\lambda_{4}}+x_{2}’+x_{3}’$, $C=- \frac{1}{t}[X1, x-\lambda_{4}]$88$x_{2}’=x_{\underline{)}}.- \frac{1}{2}ad(C)^{2}(tx\lambda_{4})$,$x_{3}’=X_{3}-[C, X2]+ \frac{1}{3}(ad(c))3(tx_{\lambda_{4}})$

So if is a $K$ -invariant function and if, we haveafter a suitable change of variable and with the choice of Haar measures :$f$$L_{v}’=\{x\in L_{v}|t\in \mathfrak{O}_{v}^{*}\}$$\int_{L_{v}’}f(x)d_{X=}\int\int\int_{\{*}\ell\in 0\}\cross\{x\in L\mathrm{n}vE2(H)\}\mathrm{x}\{u\in^{\mathrm{o}_{v}\}}tf(X\lambda 4+x+u\mathrm{t}v_{4})dtdXvdu$

2) Two calculus

We say that is ”good” if is finite , not of residual characteristic 2,
isof order and for every non empty subset of {1, 2, 3, 4} ,and all its
partial derivatives have their coefficients in(definition given in).
Let as before, if $|P_{H_{I}}(x_{0})|_{v}=1$ for somein, denoted
simply, then using Taylor formula we have for ”good”
$v$

$v$

$\tau_{v}$

$0$

$I$

$P_{H_{I}}$$\mathfrak{O}_{v}$$1,\mathrm{I}\mathrm{I}\mathrm{I}$$I$$L_{v}\cap \mathrm{u}(\oplus_{i\not\in I}\mathrm{F}h_{i})1$$x_{0}$$L_{v,H_{I}}$$v$$|P_{H_{I}}(_{X}0+ \overline{J\mathfrak{l}}Lv\cdot v,HI)-P_{H_{I}}(X0)|_{v}\leq\frac{1}{q_{v}}$

and

so

$|P_{H_{I}}(x_{0+\pi_{vv,H_{I}}}L)|_{v}=1$$I(H_{I}, u)= \iint_{\mathrm{o}_{v}\mathrm{X}L_{v}},H_{I}\tau_{v}(uwP_{H_{I}}(x_{0}+\tau_{\mathrm{t}_{v}}y))dwdy$$= \int_{0_{v}}\tau_{v}(uw)dw=1$

if$u\in \mathfrak{O}_{v}$

else

$0$

$I’(H_{I},$ $u \mathrm{I}=\int\int_{0_{v}\cross L_{v}},\mathcal{T}_{v}(u(1+\pi_{v}w)PH_{I}(x_{0}+\Gamma|y)\iota’)dwH_{I}dy$$=[ \oint_{L_{v,H_{I}}}\mathcal{T}_{v}(uP_{H}(Ix_{0+\pi}vy))dy]$$=\tau_{v}(uPH_{I}(X\mathrm{o}))$Every element of the formon the form
for some) soassociated to

if

$g$

else

is a square innormalizing$1+\pi_{v}\mathfrak{O}_{v}$$\chi_{H_{I}}(g)$$|u|_{v}\leq q_{v}$$[ \int_{0_{v}}\mathcal{T}_{\mathrm{t}},(u\pi w)vdw]$$0$$\mathfrak{O}_{v}^{*}$$L_{v,H_{I}}(\lambda H_{I}$so it can be written
being the character$P_{H_{I}}$$J(H_{I}, u)= \int\int_{0_{v}\cross L_{v},I}\tau_{v}(u(1+\tau\downarrow tv)PHI(x))dHtdx$$= \int_{L_{v,H_{I}}}\tau_{v}(uP_{H}(_{X}I))dx=l\backslash /\tau*(u)PH_{I}$893) Proof of 1) : we assume thatLet
see thatso we haveis ”good”
and with relation’ asinvariant; it is not difficult to prove thatis inand forwe have$v$$\mathrm{J}/\mathcal{I}_{v}^{*}(u)=\int_{L_{v}}\tau_{v}(u.P(vX))dXv$$M_{\underline{v}}^{*}\mathrm{i}\mathrm{s}\mathfrak{O}_{v}^{*2}$$\mathfrak{O}_{v}^{*2}\subset\chi(I\zeta_{v})$$(^{*})$$M_{v}^{*}$$\mathrm{J}l_{v}=\mathrm{J}/I_{v}^{*}$

we$L^{1}(\mathrm{F}_{v})$$\omega/_{\mathrm{O}_{v}^{*}}=\chi_{\pi_{v}}$$Z_{v}( \omega)=\int_{\mathrm{F}}M(t)\omega(t)dt=q^{\frac{1}{v^{2}}}c(\chi_{\pi}v)\sum\omega vv(\pi)^{k}+\infty 0vb_{k}+1$

being a Gauss sumandof the lemma is equivalent to show that $b_{2k}=0$ for all $k\geq 1$ .
First , for every $x\neq 0$ in, there aresuch that, with, so we can write$C(\chi_{\pi_{v}})$$([\mathrm{s}_{-}\mathrm{T}])$$b_{k}= \int_{0_{v}^{*}}\Lambda/I^{*}(\pi_{v}-k.u)x_{\pi_{v}}(u)du$$L_{v}$$u_{i}\in \mathfrak{O}_{v}^{*}\cup\{0\},$$kx= \sum_{i=1}^{4}u_{i}X_{\lambda_{i}}+z$$z\in\pi_{v}.L_{v}$and, so 1)’$k\in I\iota_{v}’$$u_{4}\neq 0$$\mathrm{J}l_{v}^{*}(u)=\sum\int_{\pi_{v}}Lv)\tau_{v}(uP(\overline{X}+y)d\overline{x}\in Lv/_{\pi}vLvy$$=q_{v}^{-(8+6} \sum_{\in/_{\pi_{v}}}d)\int_{L}T\overline{x}LvLvvv(uP(\overline{x}+\pi_{v}y))dy=q_{v}^{-}\sum_{v}(8+6d)\overline{x}\in L_{v}/_{\pi}LvI_{\overline{x}}(u)$with$I_{0}(u)=l|/\tau_{v}*(\pi_{v}^{4}u)$

and we obtain 1) by induction on$\int_{0_{v}}*I_{\overline{x}}(\pi_{v}^{-2}ku)x_{\pi_{v}}(u)du=0$

If$\overline{x}\neq 0$

for

$k$

if we prove that$\overline{x}\neq 0$

after a change of variable we have
, let
, as we have assumed that
we have, applying 1) :$\overline{x}=\sum_{i=1}^{4}u_{i}X_{\lambda_{i}}$$\sum_{i=1}^{3}u_{i}x_{\lambda_{i}}$$\overline{x}’=$$u_{4}\neq 0$$I_{\overline{x}}(u)= \int\int\int_{\mathrm{o}_{v}\mathrm{x}L_{v^{\cap E}}}H)\cross \mathrm{O}v(\tau_{v}(uP(\pi_{v}t+u_{4})x_{\lambda}+4(T+v^{X}\overline{X})’+\pi \mathcal{Z}v\omega 4)2()dtdxd\approx$

but we have for every

$x$

in

$E_{2}(H)\cap \mathfrak{g}_{1}$:$P(x+Sx_{\lambda})4^{+Z\omega_{4}}=sP_{H}(x)- \frac{1}{4}(_{ZS})^{2}$

with the choice of the basis of$\mathfrak{g}_{1}$. We have after a change of variable:$I_{\overline{x}}(u)=J_{\overline{x}}(u).J_{0}v\mathcal{T}_{v}(-u\pi Z^{2})v2d_{Z}$

90

with

$J_{\overline{x}}(u)= \iint_{\{\iota\in\}\cross_{\mathrm{t}\in}}\mathrm{o}_{v}xLv\mathrm{n}E2(H)\}(\mathcal{T}u(vt\pi_{v}+u_{4})P_{H}(\overline{|},xv+\overline{x}’))dtd_{X}$It remains to calculate each piece :$\int_{0_{v}}\tau_{v}(-u_{\mathit{1}}\mathrm{T}^{2}Z)dZ=\alpha(-u)v2q_{v}|u|_{v}-\frac{1}{2}$if$|u|_{v}>q_{v}^{2}$

1 (lemma 4.3 of [R-S])

else

and this quantity is constant on each set $|u|_{v}=q_{v}^{2k}([1\mathrm{G}2])$depends of the rank ofof$J_{\overline{x}}(u)$If ’$u_{i}\neq 0$$\overline{x}=\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}$$\cdot ank(\overline{x})=4$

If rank

then $J_{\overline{x}}(u)=I’(H, uu_{4})=\tau_{v}(uP(\overline{x}))$ if$|u|_{v}\leq q_{v}$

and

$0$

else.

then

$(\overline{x})=1$$J_{\overline{x}}(u)= \int\int_{\{^{\ell\in}\mathrm{i}}\supset_{v}\}\cross \mathrm{t}x\in Lv\cap E2(H)\}(_{\Gamma 1}\tau(vuvt+u4)PH(\overline{J\{}xv))dtdx$$= \int\int_{\mathrm{f}\ell\in \mathrm{O}_{v}\{(}\}\mathrm{x}x\in L\cap vE2H)\}(\tau_{v}(\pi_{v}u\pi_{v}t+3u_{4})P_{H}(x))dtdx$$= \int_{0_{v}}fu_{P_{H}}*(\tau^{3}(vuu_{4})$asis of odd degree( here 3) ,and is constant on each subset$P_{H}$

using $J(H, uu_{4})$$l1I_{PH}^{*}$

is$|u|_{v}=q_{v}^{k}$$\mathfrak{O}_{v}^{*}$-invariant so$J_{\overline{x}}(u)=l\downarrow/I_{P_{H}}^{*}(\pi_{v}^{3}u)$.

or 3 we can assume that
relatively to $ad(H’)$ , with $H’= \sum_{1\leq j\leq i-1}h_{j}$ :$x=x_{0}+x_{1}+X_{2}$ with $x_{j}\in E_{j}(H’)\cap E2(H)\mathrm{n}91$ (note thatand aswe have using Taylor formula:
If rank
decompose$(\overline{x})=2$$\overline{x}’=\sum_{1\leq j\leq-1}iu_{j}X\lambda i$$u\in \mathfrak{O}_{\iota^{\mathrm{t}}}^{*}$

, we

$x$

$E_{-1}(H’)\cap \mathfrak{g}_{1}=\{0\}$)$P_{H’}(\overline{x}’)\in \mathfrak{O}_{\mathrm{t})}^{*}$$|P_{H’}( \overline{x}’)-PH’(\overline{x}’+\tau|x_{2}\mathrm{t}^{\tau})|v\leq\frac{1}{q_{v}}$and$|P_{H’}(\overline{x}’+r\downarrow v^{X_{2}})|v=1$

so we can use a change of variable like in 1), let $H”=H-H’$$J_{\overline{x}}(_{\mathcal{U}})=.

\int\int\int_{F_{\underline{\circ}}}.,0\cross F0,2\cross \mathrm{O}vT(vu(\overline{\prime|}tv+\mathcal{U}_{4})P_{H^{;(}}\overline{x}’+_{J}\overline{\downarrow}2)v^{X}P_{H},,(_{J}\mathrm{T}_{v}x_{0}))dX2dx_{0}dt$$(_{X)}P0)dx2dX_{0}dt$$=. \oint\int\int_{F_{2},00}.\cross F,2\cross 0_{v}’ \mathrm{I}Tv(_{J}\overline{1}-iu(\iota 4u\pi t+4PH’(\overline{X}\prime v+\overline{\prime \mathfrak{l}}X_{2})vH$ ””$(x_{0}\mathrm{I})vdX_{2}dx_{0}$$= \int\int_{F_{2,0^{\cross F_{0,2}}}}.Tv(\overline{J(}-iuu_{4}P_{H’}(1^{7}4\overline{X}’+\tau_{1}x_{2})PH$(using $J(H$ ”, $uu_{4})$ )

91
with
If $i=2$ (resp. $i=3$ )is a quadratic form , so

$F_{2,0}=L_{v}\cap\iota\square (\oplus_{l=i}^{4}[=_{h_{l}})_{1})$and(resp.

$F_{0,2}=L_{v}\cap 1[(\oplus l\in\{1,\ldots,i-1,4\}\mathrm{F}^{\mathrm{j}}h_{l}\mathrm{I}_{1})$$F_{2,0}=\mathfrak{O}X_{\lambda_{1}}$For$F_{0,2}=\mathfrak{O}X_{\lambda_{3}}$) and$P_{H},$,(resp.

$P_{H’}$)$\dot{i}=2$$(x0)0,2)dSd_{X}0$$J_{\overline{x}}(u)= \int\int_{0_{v}\cross F}.\mathcal{T}_{v}(u\pi^{2}u_{4(\pi_{v^{S}}}+v)u_{1}PH$”$= \int_{F_{0}},2\tau_{v}(u\pi^{2}u_{41H}vuP,, (x_{0}))dx_{0}$if$= \gamma(uu_{1}u_{4}P_{H},, )q_{v}^{d+2}|u|-(\frac{d}{2}+1)$with prop.4.4. of [R-S], which gives that(and prop. 1.7 of [R-S])$J_{\overline{x}}$(using $J(H$ ”,$|u|_{\iota},$$>q_{v}^{2}$$\overline{(},,\mathcal{U}u1u\iota 24)$

else

)

1

is constant on the subset$|u|_{v}=q_{v}^{2k}$$[\mathrm{I}\mathrm{G}2]$

lf $i=3$ we have as precendetly $|P_{H’}(\overline{x}’+\pi_{v}x_{2})|v=1$ so$J_{\overline{x}}(u)= \int\int_{0_{v}\cross F_{2}}.,T_{v}(uu4TPv^{S}H;0(\overline{X}’+\tau_{\{}X2)v\mathrm{I}dsd_{X_{2}}$$=I(H’, uu4\tau_{1})v$

$=1$

if

else

$|u|_{v}\leq q_{v}$$0$Remark If we put together the results ,we obtain a formula for $l1I_{v}^{*}(u)-$which is analogous to that ofof [IG 2].

$q_{\iota}^{-(+},\mathit{1}\mathfrak{h}I_{v}^{*}86d)(\tau(,u)\mathrm{L}4$$\S_{\backslash }4$4) For the proof of 2), as we have for

$y$

in

$\mathfrak{O}_{v}^{*}$:$\mathit{1}\eta/I_{v}(y)-1=$$-. \frac{1}{2}(_{\mathit{1}\eta}/I_{v}^{*}(\Gamma^{-1}\downarrow v)+M_{v}^{*}(\pi^{-}v1\epsilon v))+.\frac{1}{2}x_{\pi_{v}}(y)c(\lambda\pi_{v}\mathrm{I}q(\frac{1}{v^{2}}\mathit{1}\mathfrak{h}/[_{v}^{*}(_{T_{\mathrm{t})}}-1)-^{\mathrm{n}_{/}[_{v}}*(\gamma \mathrm{I}-1\epsilon)vv)$

it is enough to prove that for in
we have.
do this with the usual methods, splitting the integral in order to makeappear irreducible polynomials and then we use the lemma 1 of [L-W], forthis we use changes of variable like in 1) and we comp.lete with the value of$y$$\mathfrak{O}_{v}^{*}$$|\mathrm{n}\tau_{\mathrm{t}}*()\tau|-1yv)|v\leq Kq_{v}^{-2}$$\backslash \lambda^{\gamma},\mathrm{e}$$\int_{\mathrm{i}\tau_{v}}- \mathcal{T}_{v}(uz\mathrm{I}^{d}2\mathcal{Z}.$$\square$This lemma oves that the. hypothesis of proposi.tion 1 are verified; we canconsiderfor
as in proposition 1.
$H$
Let
be the Kernel of . If is a subgroup of
and is an elementofis the centralizer of in U.
$\mathrm{p}.\mathrm{r}$$\mathrm{T}/V_{f}(\lambda, S)$$\lambda,$$s,$ $f$$\chi$$\mathfrak{g},$$U_{7}7$$U$$\eta$$Aut(\mathfrak{g})$$\eta$

92

PROPOSITION 3.

Orbital resufts
and $H$-orbits are the same.

–1) The singular2) Let $\xi\in[^{=*},$ $U_{\xi}=\{x\in \mathfrak{g}_{1}|P(x)--\xi\}$ then $H$ has a finite number oforbits inand ifis a set of representatives , we have $(U_{\xi})fl=\{x\in$, this union being disjoint, and every
is open in.
3) For every non singufar , the centralizer
is semi-simple
4) For any the centralizer
is unimodular$G$$U_{\xi}$$\Omega_{\xi}$$(\mathfrak{g}_{1})fl|P(x)=\xi\}=\cup.\eta\in\Omega_{\xi}Hfl.\eta$$H\mathrm{f}\mathrm{l}.\eta$$(U_{\xi})\mathrm{f}\mathrm{l}$$H_{\eta}$$\eta$$H_{\eta}$$\eta$

Proof
1) It is the theorem 4.3.2 of [MU 3].
2) It comes from the ”Hasse principle” : two elements are in the same G-orbit
if and only if they are in the same$v\in S$ (theorem 4.4.2 of-orbit for[MU 3] .
3) It is a calculus because we can assume that(lemma 2.3.2 of [MU 3]) and we use the explicit description of
given
in prop.3.1.3 of [MU 3].
4) For generic , it comes from 3) and for singular elements it is prop.3.3 of[MU 4].

all.

$G_{v}$$)$$x= \sum 1\leq i\leq 4X_{\lambda}a_{i}\prod 1\leq i’ i\leq 4a_{i}\neq$$0$$(\mathfrak{g}_{0})_{x}$$\cdot$$\eta$$\square$Lemma 2 and proposition 3 imply that the mean function verify for every$\xi\in[^{=*}$:$M_{f}(t^{2}\xi)=|t|^{2(1)}\kappa-$.

$( \sum_{\eta\in\Omega_{\xi}} ,\frac{1}{\tau(H_{\eta})}\int_{Hfl/H\eta}f(g’g\eta)dg\mathrm{f}\mathrm{l})$with $\chi(g’)=t^{2}$

andbeing the Tamagawa number ofandthe proportionality coefficient between the two $Hfl$ -invariant measure ofandrestricted to the set.
$T’(H_{\eta})=c_{\eta}\mathcal{T}(H_{\eta}),$$Hfl/(H\mathrm{f}\mathrm{l}^{)_{\eta}}$$H_{\eta}$

in

$x$

$c_{\eta}$$Hfl\cdot\eta$$\mu_{\xi}$For every$\mathfrak{g}_{1}$$gx= \sum_{1\leq i\leq 4}a_{i}X_{\lambda_{i}}$of.
For a real$\tau(H_{\eta})$there is . inandsuch that(lemma 2.3.2 of [MU 3]), we note $rk(x)$ the number
$g$

$G$

$a_{i}$$\in \mathrm{F},\dot{i}=1,$$4$$\ldots,$$a_{i}\neq 0$

$t$

,

$[t]$

is its integer part

ConvergenceWhen
we assume that $d\neq 2$ then for every Schwartz function
the $\dot{i}ntegra\iota Hfl\int/H(\sum_{\xi\in 91}f(g\xi))dgfl$ is absolutely convergent.

PROPOSITION 4.
$\triangle\neq R$$\mathfrak{g}_{1,fl}$–$\mathit{1})$

$f$

on

93
2) Measures on singular sets

Let$\eta$

in$\mathfrak{g}_{1}$of rank $j\in\{1,2,3\}$ then for every Schwartz function on$f$$\mathfrak{g}_{1,\mathrm{f}\mathrm{l}}$$T_{?7}(.f)= \int_{H\mathrm{f}\mathrm{l}/}f(Hfl)_{\eta}(cj\eta)dg$

is absolutely convergent and$T_{7} \overline,(f(g.))=|\lambda(g)|-j\frac{d}{\underline{9}}-\frac{1}{2}[\frac{j+1}{2}1T(\eta f)$for$g\in G\mathrm{f}\mathrm{l}$

.

Proof
1) There are two proofs.

Ifis an absolutely simple split algebra and we have two cases :$d=4$ (\S 7 typeof [IG 1] with representation ) or $d=8$ (\S 9 type E..of [IG 1] with representation ); we use the results of [IG 1].
If, we use the lemnle 5 of [WE 2]. For
and
, let
the elennent of the Cartan subgroup of associated to each root,
chap VIII, \S 1,
and$\triangle\neq R,$$\mathfrak{g}$$D_{6}$$\rho_{6}$$($$p_{1}$$\triangle=R$$\dot{i}=1,$$4$$t\in \mathrm{F}^{*}$$h_{\lambda_{i}}(t)$$\ldots,$$G$$\lambda_{i}([\mathrm{B}\mathrm{O}2]$$n^{\mathrm{o}_{5)}}$$A= \{h(t)=1\leq\leq\prod_{i4}h_{\lambda}(t_{i})i|t=(t_{1}, t_{2}, t_{3}\mathrm{I}\in(\mathrm{F}^{*})^{3}, t_{4}=(1\leq i\prod_{\leq 3}t_{i})-1\}$, $w_{2}(t)=t_{2}^{-1}t_{3}$ , $w_{3}(t)=(t_{2}t_{3})^{-2}$ are the simple roots
associated to $H$ (with Cartan subgroup A ; let $A_{\mathrm{C}}=\{t=(t_{1}, t_{2}, t_{3})\in$
for $i=1,2,3$ } then 1) is verified if we show that$w_{1}(t)=t_{1}^{-1}t_{2}$$\mathrm{I}$$(\mathbb{R}^{*+})^{3}|\omega_{i}(t)\leq 1$$-rd).(1,$”$)1\ldots$.

(titj)$t_{i}$(-2t1\leq i<j\leq\leq
1\leq\prod_{1i4}\sup)r.\prod\sup(1,$i\leq 4)^{-}1\cdots t_{i}^{-}t4$$\int_{A_{C}}\prod_{4\leq},\sup(1,$$t_{1}^{-\Gamma(+}t_{2}-4d+4t_{3}^{-}6d4).

\gamma().2r(d+2)1\leq i\leq 3\square \frac{dt_{i}}{t_{i}}$with$l\cdot=\dim$of$\mathrm{F}$over$\mathbb{Q}$is absolutely convergent which is an easy calculus.
2) 1) implies 2) because each
is unimodular.
For $d\neq 1$ we can do it using the theory of quasi-invariant measures ofhomogeneous spaces as in [WE 2]. Indeed, if is singular , we complete it
’
,
then
with
-triplein awith $N=exp(ad(\oplus i\geq 1Ei(h)\cap \mathfrak{g}_{0}))$ .
$H_{\eta}$$\eta$$sl_{2}$$([\mathrm{M}\mathrm{U}2])(\eta, h, \eta)$$h\in \mathfrak{g}0,$$\eta^{r}\in \mathfrak{g}_{-1}$$H_{l}7=\wedge i\backslash \tau_{\eta}.(H_{?7})_{\iota}$

94
To prove the absolute convergence of
convergence of$T_{\eta}$, it is enough to prove the absolute$\int_{P\mathrm{f}\mathrm{l}/fl}.f(y.\eta)\triangle_{P}-1(y)dy(H)\eta fl$

where $P=N.H_{h}$ is a parabolic subgroup of $H$ and with this integral wearrive atto compute , so we have toto theadelic Zeta function associated to a”commutative” PV of, and for $d\geq 2$
it is easy to prove the absolute convergence because we have an Ono-integral
with.
$1\mathrm{o}\mathrm{o}1<$$\int_{Pfl}.\eta f(z)|Ph(Z)|\beta d_{Z},\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\beta$$\mathrm{P}\mathrm{T}$$\beta>0([\mathrm{O}\mathrm{N}\mathrm{O}])$$\square$Remark: in the case $d=2(H_{\eta})_{h}$ has a nontrivial center of dimension one if$?’ank(\eta)=2$ (this also true for the ”commutative” case with $d=2$ )For $j=1,2,3$ let$T_{j}= \oint_{H}\mathrm{n}/_{H}(\sum_{k\{?7\in 91,r\cdot()7)=j\}},\cdot f(g\eta))dgfl=\sum_{1^{\backslash }\{\uparrow 7\in \mathfrak{g}1,k(t\overline{l})=j\}}\tau(H_{7}?)T\eta$then
We have the same results on.
. So we define in the same manner(recall that in the dual PV the$\forall g\in GflT_{j}(f(g.))=|\chi(\mathit{9})|-j\frac{d}{2}-\frac{1}{2}[\frac{j+1}{\underline{0}}]T_{j(f)}$$\mathfrak{g}_{-1}$$\tau_{j}*\mathrm{a}\mathrm{n}\mathrm{d}$$\forall g\in GflT_{j}^{*}(.f(g.))=|\chi(g)|^{j+\frac{1}{2}}\frac{d}{2}[\frac{j+1}{2}]Tj(f)$associated character is$\chi^{-1}$).

With the proposition 4 it is not difftcult to establish the following result
originally due to Mars in the ”commutative case ”of PV of PT corresponding
to $d=8$ ([MA])$\mathrm{c}\mathrm{o}\mathrm{R}\mathrm{o}\mathrm{L}\mathrm{L}\mathrm{A}\mathrm{R}\mathrm{Y}5$$d\in\{4,8\}$ .

For(MARS).
$g’\dot{i}nG^{\mathrm{t}}fl$

We assume thatand in
, let
–

$f$

$\triangle=F_{4}$or if$\triangle\neq F_{4}$then$S(\mathfrak{g}_{1})$$U_{f}(g’ \mathrm{I}=\int_{H/}\mathrm{f}\mathrm{l}H$$( \sum.f.(g’g\xi \mathrm{I})d_{jCfl}$$\xi\in 9_{1}’$then $\forall N>0\exists C_{N}$ such that $|U_{f}(g^{;})|\leq C_{N}.|\lambda(g’)|^{-N}$ for large $|\chi(g’)|$

All the precedent results are true on the dual PV
of by$\iota$$\chi^{-1}$$(G, \mathfrak{g}_{-1})$

after the change

95$\mathrm{c}\mathrm{o}\mathrm{R}\mathrm{o}\mathrm{L}\mathrm{L}\mathrm{A}\mathrm{R}\mathrm{Y}6$. –We assume that

For every non singular$\eta_{0}$we have$\triangle=F_{4}$or if$\triangle\neq F_{4}$$\tau’(H_{\eta}0)=\frac{1}{\underline{?}c_{\mathrm{F}}}\tau(H)$then $d\in\{4,8\}$.

introduce first the usual notation : let $F^{+}(t)=0$ if $|t|fl<1$ and else 1,$\backslash h^{\gamma}\mathrm{e}$$F^{-}(t)=1-F^{+}(t)$$W_{f}^{+}(\lambda, S)=Wf.F+\circ P(\lambda, S)$$V\mathrm{T}^{r}/-(f\lambda, S)=77_{f.\circ}/\gamma F-P(/\backslash , s)$

Proof
It is the method of [MA]. Let the set of -orbits in the non singular elementsof:a Schwartz function such that $f(x)=0$ for
and $/_{9_{1},fl}f(X)dX\neq 0$ (lemme 10 of [MA]) then$G$$\Omega$$\mathfrak{g}_{1}$$\mathfrak{g}_{1}’,$$.f$$x \in\bigcup_{\eta\Omega-\eta_{0}}\epsilon G\mathrm{f}\mathrm{f}\mathrm{i}\cdot\eta$$( \sum.f.(g0g_{X}))dgfl$$\mathcal{L}\sum_{\backslash ,\in \mathrm{F}^{*}}\mathrm{n}lf(g0\cdot)(\xi)=\frac{1}{\tau’(H_{70}1)}\int_{H\mathrm{f}\mathrm{l}}/Hx\in 9_{1}’$We apply the ordinary Poisson formula:$\backslash \sum_{x\in 9_{1}’}.;.(\mathit{9}0gx)=|x(g_{0})|-\kappa(\sum_{y\in 9-1}\hat{f}_{g0}g(y))-(\sum_{1}x\in_{91}-\mathfrak{g}\prime f_{g0}(gx))$$\backslash 1^{\gamma}\prime \mathrm{i}\mathrm{t}\mathrm{h}$the corollary 5,$77_{f}^{\gamma+}/(\lambda, .)$is analytic onso with proposition 1(
as in proposition 1) but for $s\in S_{A}$$\lim_{s-0,S\in S}.SV\mathfrak{s}_{f^{-}}^{\tau}4/(Id, s)=C_{\mathrm{F}p_{\mathrm{F}}\hat{f}}(0)$$\mathbb{C}$$S_{A}$

we have$s. \tau^{;}(H_{70}\overline,).\mathrm{V}V^{-}f(Id, s)=S.\int fl*/\mathrm{F}*f-(|t|)|t|^{2\kappa+S}(\wedge)$$[( \int_{Hfl/_{H}}.[|t|^{-}2\mathrm{K}(\sum_{-}\hat{f}_{g}0g(y))-(\sum_{\prime,1-1}f_{g0}y\in \mathfrak{g}1x\in_{9}9g(x)]dg\mathrm{f}\mathrm{l}]dt^{*}fl$

$=s$

.

.

with

$|\chi(g\mathrm{o})|=|t|^{2}$.

$\int_{\mathrm{A}^{*/_{\mathrm{F}}}}*f-(|t|)|t|2_{S}[\int_{H/H}\mathrm{f}\mathrm{l}(\sum_{y\in 9’-1}\hat{f}(g0gy))dg\mathrm{n}]d\#^{*}fl+\sum_{j=1}\frac{sT_{j}^{*}(\hat{f})\rho_{\mathrm{F}}}{2s+jd+[\frac{j+1}{2}]}3$$+ \frac{p_{\mathrm{F}}\tau(H)\hat{f}(0)}{2}-S$$[ \sum_{=j1}\frac{T_{j}(.f)p_{\mathrm{F}}}{2s+2_{\hslash}-jd-[\frac{j+1}{2}]}+\frac{p_{\mathrm{F}}f(0)}{2s+2_{h}}3.\cdot]$

96
and with the corollary 5 applied to the PV
the linuit of the two members for $s\in S_{A}arrow 0$ :Finally we have obtained : $(***)$$(G, \mathfrak{g}_{-1})$$\sum_{\xi\in \mathrm{F}^{*}}\sim^{\mathit{1}}\mathfrak{h}xf(t^{2}\epsilon)=\frac{2c_{\mathrm{F}}}{\tau(H)}|t|^{2(t\overline{\iota}}-1)$$3$

we obtain when we take$\tau(H\mathrm{I}=2\tau’(H?70)_{C_{\mathrm{F}}}-.$. $\int_{Hfl/H}(\sum_{x\in 9_{1}},f(g’g_{X}))dgfl$

with$\square$$\chi(g’)=t^{2}$

The result

THEOREM 6. –We assume that $\triangle=F_{4}$Letorthen $d\in\{4,8\}$a character of, trivuial on$\dot{i}f\triangle\neq F_{4}$a Schwartz
onand,almost everywhere non ramified then1)
analytic on2)
has an meromorphic extension given by the following formula$f$$funct_{\dot{i}\mathit{0}}n$$\mathrm{w}_{/}^{r_{f^{+}}}(\lambda, .)\dot{i}S$$\lambda$$\mathfrak{g}_{1}$$\mathrm{f}\mathrm{l}^{*}$$[^{=*}$$\mathbb{C}$$l,7_{f^{-}}^{\gamma}(\lambda, .)$$W_{f}^{\mathit{7}-}( \lambda, s)=7\eta/^{r}\hat{f}+(\lambda-1, -s-2-\frac{3}{2}d)-\delta_{\lambda^{\frac{f(0)}{s+2+\frac{3}{2}d}+}}\delta_{\lambda}.\frac{\hat{f}(0)}{s}$.

$- \frac{\delta_{\lambda}}{\tau(H)}(\frac{T_{1}(f)}{s+d+\frac{3}{2}}+\frac{T_{2}(f)}{s+\frac{d}{2}+\frac{3}{2}}+\frac{T_{3}(.f)}{s+1})$$+ \frac{\delta_{\lambda}}{\tau(H)}(\frac{T_{1}^{*}(\hat{f})}{s+\frac{d}{2}+\frac{1}{2}}+\frac{T_{2}^{*}(\hat{f})}{s+d+\frac{1}{2}}+\frac{T^{*}(\hat{f})}{s+\frac{3d}{2}+1})$

with$\delta_{\lambda}=c_{\mathrm{F}}\rho_{\mathrm{F}}\dot{i}f\lambda/(\mathrm{f}\mathrm{l}^{*})^{1}=\dot{i}d$

and

$0$

else

3) $Funct_{\dot{i}}onal$ equation: $W_{f}( \lambda, S)=\Gamma V_{\hat{f}}(\lambda^{-1}, -s-\frac{3}{2}d-2)$

Proof1) is due to formulaand corollary 5.
2) is the same as the proof of corollary 6 because of the formula3) Comes from 2.
$(^{***})$$(^{***})$$\square$

References[BO 1] N.Bourbaki , Groupes et alg\‘ede Lie , chap. 4, 5, 6, Hermann,
Paris (1968)[BO 2] N. Bourbaki , Groupes et alg\‘ebres de Lie , chap. 7 et 8, Hermann,
Paris (1975)$\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{S}$97[IG 1].Igusa, On certain Representations of semi-simple Algebraic Groups
and the Arithmetic of the correspondingts (1), Inventiones M.
. $12$(1971) , 62-94[IG 2].Igusa, Exponential sums associated with a Freudenthal quartic ,.
J.Fac.Sci.Univ. Tokyo, 24 (1977) , 231-246[IG 3].Igusa, Lectures on forms of higher degree, Tataoffundamental research,Bombay (1978)[L-W] S.Lang and A.Weil , Number of points of varieties in finite fields,.
Amer.J. of Math., 76 (1954) , 819-827[MA]
Mars , Les nombres de Tamagawa de certains groupes exceptionnels, Bull.Soc.Math.France , 94 (1966) , 97-140[MU 1] I. Muller, Syst\‘emes de racines orthogonales et orbites d’espaces
pr\’ehomog\‘e , Th\‘ese, Universit\’e de Strasbourg, 1996
[MU 2] I. Muller, , Racines orthogonales et orbites d’alg\‘ede Lie semisimple gradu\’ees, Journal of Algebra,, (1997), 41-74[MU 3] I. Muller, Classification d’orbites pour une classe d’espaces pre’homo
g\‘enes, to appear in Nagoya math. Journal[MU 4] I. Muller, On local Zeta functions associated to prehomogeneous
vector spaces of commutative parabolic type , Part 2 : Zeta functions andsingular measures, preprint[ONO] T.Ono, An integral attached to a hypersurface , Amer.J. of Math.,
90 (1968) , 1224- 1236
[R-S] S.Rallis and G.Schiffmann, Distributions invariantes par le groupeothogonal, Analyse harmonique sur les groupes de Lie, Lecture Notes in Math.
497 , Springer-Verlag, Berlin, Heidelberg, New-York (1975), 494-642[RU 1] H.Rubenthaler, Espaces pr\’ehomog\‘enes de type parabolique , Th\‘ese,
Universit\’e de Strasbourg (1982)[RU 2] H.Rubenthaler , Alg\‘ebres de Lie et Espaces pr\’ehomog\‘e , Travaux
en cours , Hermann (1992)[S-K] M.Sato and T.Kimura, A classification of irreducible Prehomogeneous
Vector Spaces and their relative invariants,Nagoya Math.J. 65 (1977),1-55.Sally and[S-T].Taibleson, Special functions on locally compact
fields, Acta. Math.116 (1966),279-308[WE 1] A.Weil, Adeles and algebraic groups, The Institute for Advanced
Study, Princeton (1961)[WE 2] A.Weil, Sur la formule de Siegel dans la th\’eorie des groupes classiques, Acta. Math. 113 (1965), 1-87
[WE 3] A.Weil, Basic number theory, Springer (1967)$\mathrm{J}.\mathrm{I}$Invaria.n$\mathrm{a}\mathrm{t}\mathrm{h}$$\mathrm{J}.\mathrm{I}$1$\mathrm{J}.\mathrm{I}$$\mathrm{I}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{t}\mathrm{e}\vee$$\mathrm{J}.\mathrm{G}.\dot{\mathrm{b}}’\mathrm{I}$$\mathrm{n}\mathrm{e}\mathrm{s}$$\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{S}$$\mathrm{v}\mathrm{o}\mathrm{l}193$$=$$\mathrm{n}\mathrm{e}\mathrm{s}$$\mathrm{P}.\mathrm{J}$$\mathrm{M}.\mathrm{H}$98
Table 1

List of PV of PT having root systemgiven by their Satake
diagram with the different values of (cf. section 1 III) and the corresponding
number in the algebraically closed case in M.Sato and T.Kimura- classification. The theorem 6 is verified for all of them except one split case with
Dynkin Diagram.
$(R, \alpha_{0})=(F_{4}, \alpha_{1})$$d$$([\mathrm{s}_{-}\mathrm{K}])$$(\Delta, \lambda_{0})=(E_{6}.\alpha_{2})$$\backslash \mathrm{t}^{r}\mathrm{e}$recall that ([S-K]).

(5) : ( $GL(6)$ A3,)(14) : ( $GL(1)\cross Sp(3).\square \cross$ A3.
(23) $(GL\langle 1)\cross Spin(12).0\cross half$ spin rep.,$\iota^{r}(20_{)})$$\dagger’(1)\cross l^{r}(14)$$\ddagger^{r}\text{ノ}(1)\mathrm{x}$\d ag er $r(32))$(29) $(GL(1)\cross E\overline, , \square \cross, l’(1)\cross\Lambda_{6}, l’(1)\cross V(56))$ (case of [IG 2])