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
2 over a -adic eld, the Fourier transform of a
regular orbital integral is expressed as an integral over all regular orbital integrals,
with explicit coecients. 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.
1. Introduction
2. Conjugacy Classes
3. Orbital Integrals
4. Fourier Transforms
5. Transfer Factors
6. The Transfer Formula
7. Remarks
References
sl
p
Contents
17
18
20
21
25
26
28
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 nite 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 coecients dicult
to get hold of.
Received October 14, 1997.
Mathematics Subject Classication. 11F85, 22E35, 43A80.
Key words and phrases. orbital integral, Fourier transform, endoscopic transfer.
17
c 1998 State University of New York
ISSN 1076-9803/98
Ulrich Everling
18
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 nal though
conjectural form by 20]. To explain it roughly: Given an algebraic group G over
a eld F , with algebraic closure F elements in G(F ) which are conjugate in G(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
dierence 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,
dened by a duality of root systems using the structure theory of reductive algebraic groups. For G = SL2 the non-trivial endoscopic groups are one-dimensional
algebraic tori dened by quadratic extensions E jF 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, Theoreme 1.5], among them SLn in general, transfer
remains a conjecture.
Let F be a p-adic eld (p 6= 2). Fourier transformation was thoroughly studied for
orbital integrals on the group SL2 (F ) by Sally and Shalika 22], and for nilpotent
orbital integrals on the Lie algebra sl` (F ) by Assem 4]. Now for X 2 sl2 (F )
semi-simple regular, we consider the orbital integral over Ad(SL2 (F )) X . Its
Fourier transform is invariant with respect to Ad(SL2 (F )) and is known to be an
integral (over the space of regular orbits) of orbital integrals we explicitly specify
the coecients (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 SL2 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 sl2 will always be denoted
X = xz ;yx or U = wu ;vu and the letters U u v w X x y z , bare or decorated like X or Xm , will always
mean this. The action of GL2 on sl2 by Ad(g)(X ) := gXg 1 leaves the bilinear
0
;
form
hU X i := 2ux + vz + wy
Fourier Transforms and Endoscopic Transfer
19
invariant it will be used for Fourier transforms.
Let us classify the orbits of G := SL2 (F ) acting on g := sl2 (F ), where F is a
eld in which 2 6= 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
greg = fX 2 g det X 6= 0g
r
and in each of the stable orbits
g t := fX 2 g f0g det X = ;tg t 2 F:
Each X 2 g is G-conjugate to some 0 t=
0 where t = ; det X and 2 F ,
the latter unique up to a factor in
:= 2 ; 2 t 2 F
which is the norm group of the algebra
Nt := NtF
2
r
f0g F Et := F #]=(#2 ; t)
r
r
in fact the orbits are parametrized by the bijection
Orb(g f0g) := f(t b) t 2 F b 2 F =Nt g ;! Ad(G)n(g f0g)
(t b) 7! g tb := fX 2 g t z 2 b f0g 3 ;yg and Orb(greg ) := f(t b) t 6= 0g corresponds to Ad(G)ngreg. Each stable orbit g t is
the union of all the g tb where b 2 F =Nt , and
Ad(g)g tb = g tdet(g)b :
8g 2 GL2 (F ) :
(1)
(Each group F =Nt can also ;be considered
as Galois cohomology group of the
0
t
stabilizer in SL2 of the matrix 1 0 2 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 eld 6] it comes with the
valuation ring O F , the maximal ideal p O, the residue eld = O=p with q
elements, the valuation exponent : F
f+1g and the absolute value j j so
that jj = 1=q for any uniformizing element 2 p p2 . We shall assume 2 2= p.
(Example: F = p , O = p, p = p p, = p , = p.)
The group F =F 2 has four elements and ts in the exact sequence
Z r
Q Z Z F
1 ! O =O
#
=
2 ;! F =F 2 ;!
ZZ
=2 ! 0
= 2
and for each = tF 2 2 F =F 2 there is a canonical isomorphism
8
if 1 2 ,
< f1g
F = N if = 1 (2 ) F 2 ,
= : =2
2
= if ( ) = 2 .
Thus for t 2 F
r
ZZ
;
ZrZ Zr
F 2 , the stable orbit g t contains two orbits g tb .
Ulrich Everling
20
3. Orbital Integrals
On each G-orbit in g, we need a G-invariant measure to dene 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 tb 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=jtj
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 dened by means of dierential forms, as explained in 31,
p. 13{14]. For any complex-valued locally constant compactly supported function
(in short, Cc function), integrals are nite sums. The characteristic function of an
open set A will be called fA . For each t 2 F , the variety X 2 sl2 f0g x2 + yz = t carries a GL2 -invariant
algebraic 2-form
dy ^ dz = dx ^ dy = dz ^ dx
2x
y
z
r
1
(the relation 0 = d(x2 + yz ) = 2x dx + z dy + y dz implies that all three terms are
equal and invariant). This 2-form denes a measure dX on g t . The orbital integrals
Itb (f ) :=
Z
f (X )dX 2
g tb
Z
C
r
(f 2 Cc (g) (t b) 2 Orb(g f0g))
1
r
r
Zr
are nite sums if t 6= 0, or still convergent for t = 0, see the proof of Proposition 2
below. Let 0 := sl2 (O) = fX x y z 2 Og and n := pn 0 for each n 2 .
Proposition 1. Let m < ` 2 , Xm 2 m m+1, and (t b) 2 Orb(g f0g). If
(i) t + det(Xm ) 2 p`+m
and
(ii) fzm ;ymg b pm+1 ,
then Itb (fXm +` ) = qm 2` . Otherwise the integral is zero.
Proposition 2. Let ` 2 , (t b) 2 Orb(g f0g). Then
8 1q `
if t = 0, ` 2 (b)
>
2
>
>
1q ` 1
>
if t = 0, ` 2= (b)
>
2
>
>
if 2` > (t)
<0
if 2` (t), t 2 F 2
Itb (f` ) = > q ` + q 1 `
`
1
h
>
q ;q
if 2` (t) = 2h, 0 6= t 2= F 2 , ` 2 (b)
>
>
1
`
1
h
>
(t) = 2h, 0 6= t 2= F 2 , ` 2= (b)
>
>
: q(q ` ;q; qh ) q+1 ifif 22`` <
(t) = 2h ; 1.
2q
;
;
Z
; ;
;
; ;
;
; ;
; ;
;
; ;
;
Corollary 3. For every f 2 Cc (greg) the function
Orb(greg ) 3 (t b) 7! Itb (f )
belongs to Cc (Orb(greg )). Explicitly, let n 2 , Xn 2 n n+1, t = ; det Xn 6= 0,
1
1
Z r
j > (t) ; n. Then f = fXn +j corresponds to the function qn 2j f(t+pn+j ) b .
;
f g
r
Fourier Transforms and Endoscopic Transfer
21
Proofs. For Proposition 1, x a prime element 2 p p2, no matter which, so
that O = p, let Xm = m X0 and let X run through 0 so that X = Xm + ` X
0
runs through Xm + ` . Then
0
` m (t + det X ) = ` m (t + det Xm ) ; hX0 X i + ` m det X :
; ;
; ;
0
;
0
If this wants to be zero, the condition (i) is necessary.
If y0 z0 2 p, then x0 2 O and one can solve the equation t = ; det X for x .
Because of t 2 2m (x20 + p) F 2 we have g tb = g t . The integral is
0
Z
dy dz = Z j` jdy j` jdz = qm 2` :
j2xj
jxm j
0
0
;
g t \(Xm +` )
OO
If y0 2 O one can solve for z and then ;yNt = ;ymNt . If this class is the
same as b, the integral is again
0
Z
dx dy = qm 2`
jy j
;
g tb \(Xm +` )
otherwise it is zero. Likewise in the case z0 2 O .
The second proposition follows from the rst one applied to each term of the
sum
X
fm ; fm+1 =
fX 02= X 2m =m+1
by descending induction on ` if t 6= 0, or with a geometric series if t = 0.
The corollary follows immediately.
Z
In view of the two propositions, one recognizes the Shalika germs: Let ` 2 ,
(t b) 2 Orb(greg ), (t) 2`, and f 2 Cc (g) a function that factors through g=`.
Then
X
p
Itb (f ) = (const) jtj f (0) +
I0 (f )
1
b=F
2
2
where (const) depends only on tF 2 the right hand side consists of contributions
from the nilpotent orbits, which are the four g0b and f0g.
C
4. Fourier Transforms
Let
(2)
:F !
be an unramied additive character then
(O) = 1
X
a p;1 =
2
(a) = 0:
O
The Fourier transform of functions on g is dened as
Z
e
Cc (g) ;! Cc (g) f (U ) := f (X ) hU X i dX
1
e
1
g
Ulrich Everling
22
where dX is the Haar measure such that vol(0 ) = 1, which is self-dual in the sense
that fe (X ) = f (;X ).
For each orbit, labeled (s a) 2 Orb(greg ), the distribution
C
^
Cc (greg ) ! f 7! Isa (fe)
1
is G-invariant in the sense that Isa (f Ad(g)) = Isa (fe) for each g 2 G. This
invariant distribution can be expanded in terms of all the Itb :
Proposition 4. There exists a unique locally constant function
cG : Orb(greg ) Orb(greg ) !
such that
(3)
8f 2 Cc
1
(greg ) :
Z
Isa (fe) =
Z
C
X
cG (s a t b)Itb (f ) pdt :
jtj
F b F = t
r
2
N
Explicitly, for n 2 and Xn 2 g tb \ n n+1 :
(4)
cG(spa t b) = I (f ) + X
sa ;n
jtj
i>n
X
1;i = U;i 2;i =;n
hU i Xn i Isa (fU;i )
;
C
and the sum over i is actually nite.
The proof will be nished before Corollary 8. First we show that (3) implies (4).
Suppose cG locally constant. Given n and Xn , let j (t) ; n and c 2 so that
p
8X 2 Xn + j : cG (s a ; det X b) = c jtj:
0
Corollary 3 yields
(5)
0
^
c = q3j vol(t + pn+j ) c qn 2j
= q3j Isa (fXn +j )
X
X
;
= Isa f;n +
;
n<i j 1;i = U;i ;i =;n
hU i Xn i fU;i ;
2
this does not depend on j and therefore the i (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) 2 Orb(greg ) let us x n 2 , y 2 O, z 2 O such that
(6)
Z
y 2 g (y) 2 and (t 2 F
tb
z0
Xn = n 0
2
=) y 2 O 2 ):
Let cG be dened by (4) with this choice of Xn . The relations (2) imply that the
apparently innite sum (4) is actually nite as follows.
Fourier Transforms and Endoscopic Transfer
23
Lemma 5. Let i > n, s = 2i s + pi n, h1 = i ; n ; 1, h2 = h1 ; (y). The
i U such that
0
;
U i=
(i) (v) = 0 < h1 ^ (s = 0 _ h1 + (w) > minf
(s ; vw) h1 g)
(ii) (w) = 0 < h2 ^ (s = 0 _ h2 + (v) > minf
(s ; vw) h1 g)
contribute all in all nothing to cG(s a t b) in (4).
;
;
0
or
0
0
0
Corollary 6. The only contributions to the sum (4) come from values of i such
that
maxfn + 1 ; (2s) g i imax := maxfn + 1 + (y) ;1 ; n ; (s)g:
(7)
Proofs. Let us assemble the U i with property (i) as follows: v runs through
one class mod ph1 , w remains xed, and u varies so that the equation s = u2 + vw
;
0
i Xn i = ( n i (vz + wy ))
remains valid. The Isa (fU;i ) are all equal and the hU
make a total of zero.
Likewise, we can assemble the U i such that (ii) ^ :(i) this time w varies by
ph2 , and v remains xed.
The upper bound in the corollary follows from the lemma because
;
;
;
i ;n ; (s) =) s = 0 =) vO + wO = O:
0
The two lower bounds come from (4) and from condition (i) of Proposition 1.
By replacing X and U by 1 X and U we see that the denition of cG does
not depend on the choice in (6) and that
;
C
cG ( 2 s a 2 t b) = cG (s a t b):
8 2 F :
(8)
;
Lemma 7. The function cG : Orb(greg ) Orb(greg) ! is locally constant.
Proof. First let us x (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 nite and depends only on (s) and (t b). Therefore
cG 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 nd that (3) is true for f = fXn +j when the Xn are
chosen as in (6) and when j imax as dened in (7). These functions f , and their
transforms f Ad(g) by all g 2 SL2 (F ), span the whole linear space Cc (greg ). The
condition (3) being linear and SL2 (F )-invariant, Proposition 4 follows.
Corollary 8. 8(s a) (t b) 2 Orb(greg) 8 2 F : cG(s a t b) = cG(s a t b):
Proof. All the ingredients of Proposition 4 being GL2 (F )-invariant, so is cG and
we remember (1).
1
Corollary 9. If sF 2 = tF 2 then cG(s a t b) depends on a and b only via a b.
If sF
2 6= tF 2
then cG (s a t b) does not at all depend on a or b.
Ulrich Everling
24
Proof. This follows from Corollary 8 if sF 2 =6 tF 2 one can choose so as to
change a without changing b, or vice versa.
Z
We shall need auxiliary complex numbers depending on , namely
Z
(c) := 1 if c 2 F (c) 2 2 X
(c) := 1 + 2
(a) =pq if c 2 F (c) 2= 2 :
cF
a p;1 =
2
2
O
If c 2 F and (c) = ;1 we have
X
p
q (c) =
(c2 )
(c (O
r
2
q;1
O 2 )) = ; (c) (;c) = (c) = (;1) 2 (c) j (c)j = 1:
In view of lemma 5 and Corollary 6 there remain only quite few U i contributing
to (4) for the Xn chosen in (6). Counting thoroughly and using the relations (2),
we obtain:
Proposition 10. Given (s a) (t b) 2 Orb(greg ), let
;
S=
X
r2 =st
p
p
(2r) Q1 = 1q jstj Q2 = q 2+q 1 jstj=q:
Then cG (s a t b) is as follows:
t 2 F 2 t 2= F 2 2j
(t)
S
1 ; Q1 (;1) (ab) ; Q1
0
(;1) (ab) S
if (2st) 2 (ab)
2
;1
;1
s2F
st 2 p
st 2= p
2j
(s)
s 2= F
2
st 2 p
1
-
;
0
otherwise
1 ; Q2
;Q2
st 2= p 1
2 (s)
;
0
-
2 (t)
;Q1
0
(;ab) ; Q2 if st 2 F
;Q2 otherwise
P
(;ab)
(2r)
2
r2 =st
r ab
0
2;
if st 2 F 2
otherwise
0
For each s 2 F let "s : F =Ns ,! f1g 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 2= F 2 3= st cancel out by Corollary 9.)
Fourier Transforms and Endoscopic Transfer
Corollary 11. Let s 2 F and (t b) 2 Orb(greg). Then
25
c"G (s t b) :=
X
a F = s
2
"s (a) cG (s a t b) = "t (b) (t) N
X
r2 =st
(2r)
this is zero if st 2= F 2 .
The corollary describes the "s -instable integral
Is" (fe) :=
X
a F = s
2
"s (a)Isa (fe) =
N
Z
X
c"G (s t b)Itb (f ) pdt
jtj
F b F = t
2
N
for f 2 Cc (Orb(greg )).
1
5. Transfer Factors
In order to derive (9) below from Corollary 11, we rst spell out the denition of
transfer factors which occur in I GH . As mentioned in the introduction, an instable
integral in our case is the dierence of two orbital integrals. The transfer factor
will simply assign the signs +1 and ;1 to the two orbital integrals its denition
in 20] is quite sophisticated because they make a canonical choice for all reductive
groups and in general there occur coecients other than 1.
Let E jF be a quadratic extension, # 2 E an element of trace zero, and
" = E F the quadratic character on F so that ker E F = NE F E . Let
H = ker(RE F m ;! m ) the corresponding one-dimensional algebraic torus (so
that H(F ) E is the norm-one group) its Lie algebra h is naturally identied
with the imaginary line ker(TrE F ) E .
;
Now we borrow notations from 20, p. 222{223]. Let h = 1=1(2=2#) 1# 2 SL2 (E ),
j
G G
j
j
N
j
T
B
j
;
: H ;! T := h h 1 B := h h 1 G := SL2 2
a
+
b#
0
a
b#
1
(a + b#) := h 0 a ; b# h = b a :
s := 10 ;01 2 T := 0 0 B := 0 Gb = PGL2 ( )
;
;
;
Let
and
o
C
: H ! (T f1 01 10 g) W L G
the embedding such that 1 : T ! Hb is dual to
H ! a + b# 7! h 1 (a + b#)h:
Then (H H s ) are endoscopic data and is an admissible embedding. The algebraic character on T over E ,
2
a
b#
: b a 7! (a + b#)2 ;
T
;
26
C
Ulrich Everling
is the root of T in B let the a-data consist of a := 2# a := ;2#, and choose
whichever : E ! extending ; E F .
Let 0 6= Y = # 2 h and X = xz yx 2 g2 #2 . Following the \Remarques"
of 29, p. 90], we evaluate the formulae (I), (II), (III1 ) and (III2 ) of 20] with
H = exp(e2 Y ) and = exp(e2 X ) for any e 2 F near 0. The cocycle (T ) of (I)
is trivial. In (II) we have (T ) ; 1 2e2#. For the cocycle in (III1 ) we take
;
j
;
2
g = 0 xz ab b#a
where a b 2 E , a2 ; b2 #2 = 1=z . In (III2 ) we need not know a because T 1.
Omitting (IV), we obtain the transfer factor
G
!(Y X ) = (!I = 1) (!II = "()) (!1 = "(z )) (!2 = 1) = E F (z ):
(For a split torus H = m we simply have !(Y X ) = 1 for all Y 2 h, X 2 gY 2 .)
By identifying h with the imaginary line of E jF (or with F if H splits) we may
write Y Z 2 F for Y Z 2 h. Then
j
2
!(Y X ) = "t (b) if X 2 g tb and t = Y ,
0
otherwise.
6. The Transfer Formula
For fh 2 Cc (h) we dene the Fourier transform feh with respect to (2Y Z ).
Now we need formulae from 29] quotations `: : :' will refer to that work.
In `III.1' in the case M = G the factor vM (x) is 1, so the JGG are the ordinary orbital integrals so are the IGG by `p.74, remarque (b)'. Now IHst (Y fh) and I GH (Y f )
are dened by `VIII.7, (6) and (7)'.
Next, there are 1 2 2 F so that h ig = 1 h i and (c) = (2 c). For the
construction of h ih in `VIII.6', we can simply take G = G = SL2 and the same
endoscopic data as in the previous section then the whole procedure simplies to
1
h# #ih =
0 #2 0 #2 = 21 # #:
1 0 1 0
g
Consider the constant (g) of `VIII.5'. If 2 O , we can take r = 0 in `VIII.1'
and then I (r) = 1. If () = ;1 we can still take r = 0 and then
I (r ) = q
by (2) the
P
z
3
;
X
xyz
2O
=p
( (x2 + yz )) = q
3X
;
x
(x2 ) XX
y
z
(yz )
vanishes for each y 6= 0, so
I (r ) = q
;
2X
x
(x2 )
whence (g) = (). If () 2= f0 ;1g we can multiply by a square and adapt
r accordingly.
Fourier Transforms and Endoscopic Transfer
27
The computation of (h) is similar.
Summing up, we obtain the following dictionary:
in 29] here
st
IH (Y fh) fh (Y )
GH
I (Y f ) Cs Is" (f ) s = Y 2
h ig
1 h i
hY Z ih
21 Y Z
(c)
(2 c)
(g) () = 1 2
(h) (Y 2 ) (for any Y 2 h f0g)
fb(X ) jj3=2 fe(X ) (f 2 Cc (g))
c
fh(Y ) jj1=2 feh (Y ) (fh 2 Cc (h))
The constants Cs > 0 depend on the choice of Haar measures but Cs depends only
on sF 2 we do not need them explicitly.
For every function f 2 Cc (g), the function fh dened by
fh (Y ) := I GH (Y f ) (0 6= Y 2 h)
extends to a function fh 2 Cc (h) this was known to 18] and it is also clear in
view of Propositions 1 and 2.
Let 0 6= Y 2 h and s = Y 2 , and f 2 Cc (greg ). The self-dual Haar measure
p dZ
on h for fh 7! feh , restricted to h f0g, corresponds, via t = Z 2 , to dt= jtj on
sF 2 F . Then
Z
X
I GH (Y fb) = Cs "s (a)
fb(X )dX
a F = s
X (a0 s=a
0 )
Z
X
d(X )
= Cs "s (a)
jj3=2 fe(X )
jj
a
X (a0 s=a
0 )
p X
= Cs jj "s (a)I
2 s
a (fe)
1
r
1
1
1
r
2
1
N
a
= Cs jj "s ()I
"2 s (fe)
p
by Corollary 11 this is
p
Z
p
F b FZ = t
= Cs jj "s ()
X
2
= Cs jj "s () (s)
hrf0g
p
= "s () (s) jjfeh (Y )
= "s () (s)c
fh(Y ):
N
"t (b) (t)
X
(2r)Itb (f ) pdt
jtj
r2 =
2 st
(2Y Z )IZ" 2 (f )dZ
28
Ulrich Everling
In the four cases according to () (s) mod 2 one veries that
"s () (s) () = (s)
and one recognizes the transfer formula of 29, VIII.7 Conj. 1] with his formulae
`(6)' and `(7)' inserted:
(9)
(h)IHst (Y fch ) = (g)I GH (Y fb):
7. Remarks
We have been supposing f 2 Cc (greg ) so that the integrals are nite 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 2 Cc (g).
As long as 2 2= p, we allow F to have nite characteristic.
1
1
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Kath. Universitat MGF, 85071 Eichstatt, 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.