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On the spectral decomposition of affine Hecke algebras
Opdam, E.M.
Publication date
2000
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Citation for published version (APA):
Opdam, E. M. (2000). On the spectral decomposition of affine Hecke algebras.
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ERIC M. OPDAM
Abstra t. An aÆne He ke algebra H ontains a large abelian
subalgebraAspannedbyLusztig'sbasiselements
,whereruns
over the root latti e. The enter Z of H is the subalgebra of
WeylgroupinvariantelementsinA. Thetra eoftheaÆneHe ke
algebra an be written as an integral of a rational n form (with
valuesinthelineardualofH )overa ertain y leinthealgebrai
torusT =spe (A). WeshallderivethePlan herelformulaofthe
aÆne He ke algebra by lo alization of this integral on a ertain
subsetofspe (Z).
Date:De ember31,2000.
1991Mathemati sSubje t Classi ation. 20C08,22D25,22E35,43A32.
Theauthor wouldlike tothank Erikvan den Ban,Gerrit He kman andKlaas
1. Introdu tion 2
2. Preliminaries and des ription of results 5
2.1. The aÆne Weyl group and its rootdatum 6
2.2. Rootlabels 7
2.3. The Iwahori-He ke algebraas a Hilbert algebra 8
2.4. The Plan herel measure 11
2.5. Des ription of the mainresults 12
3. De nition and lassi ation of residual osets 15
3.1. Propertiesof residual and tempered osets 19
4. Lo alizationof onSpe (Z) 21
4.1. Quasi residual osets 22
4.2. The ontour shiftand the lo al ontributions 24
4.3. Appli ation tothe tra e fun tional 33
4.4. The probability measure and the A-weights of t
38
5. Lo alizationof the He ke algebra 47
5.1. Lusztig's stru ture theorem and paraboli indu tion 50
5.2. The tra ial states t
54
5.3. The Plan herel de omposition of the tra e 58
6. Base hange invarian e of the residue Frobenius algebra 60
6.1. S alingof the rootlabels 60
6.2. Appli ation tothe residue Frobenius algebra 62
7. Appendix: the Kazhdan-Lusztig parameters 65
Referen es 70
1. Introdu tion
InthispaperIwilldis ussthespe tralde ompositionofaÆneHe ke
algebras. It is the natural sequel to the paper [24℄, in whi h I made a
basi study of the Eisenstein fun tionals of anaÆne He kealgebra H .
These Eisenstein fun tionals are holomorphi fun tions of a spe tral
parameter t 2 T, where T is a omplex n-dimensional algebrai torus
naturally asso iated to H . In [24℄, I derived a representation of \the"
tra e fun tional of the He ke algebra, as the integral of the
normal-izedEisenstein fun tional timestheholomorphi extension oftheHaar
measure of the ompa t form of T, against a ertain \global n- y le"
(a oset of the ompa t form of T)in T. The kernel of this integral is
a meromorphi (n;0)-form on T.
The present paper takes o from that starting point, and re nes
formula for the tra ial state of the He ke algebrauntil we rea h the
level of the spe tral de omposition formula for H , or rather for C, the
C
-algebrahulloftheregularrepresentationofH . Onthe simplerlevel
ofthespheri alortheanti-spheri alsubalgebra,asimilarapproa h an
befoundin[21℄and[12℄. Inthe aseofthespheri alalgebraoneshould
of oursealsomentionthe lassi alwork[20℄,althoughthepointofview
is di erent there, and based onanalysis on aredu tive p-adi group.
Anaturalin arnationoftheaÆneHe kealgebraisthe entralizer
al-gebraof aspe ial kind ofindu ed representations ofap-adi redu tive
group G, notably representations indu ed from a uspidal unipotent
representation of a parahori subgroup (see [23℄ and [19℄). The
on-stituents of these indu ed representations are alled unipotent
repre-sentations by Lusztig [19℄. In this ontext, the expli it de omposition
of the tra e in terms of irredu ible hara ters of the aÆne He ke
algebrashouldbenaturallyequivalent tothe disintegration of the
or-responding indu ed representation of G as a subrepresentation of the
left regular representation of G. In this way one an view the
restri -tion of the Plan herel measure of G to the olle tion of irredu ible
tempered unipotent representations as a juxtaposition of Plan herel
measures omingfromvariousIwahori-He kealgebras (seeforinstan e
[12℄). This is by itselfalready suÆ ient motivation forthis work.
In doing these omputations, I have tried to maintain a fresh and
personal approa h to the problems that arise at various stages. I did
not study indetail the enormous amountof lassi al material whi his
relevanttothe problemsathand,su hasthe worksofHarish Chandra
[10℄,Arthur[1℄,[2℄, Langlands[16℄,MoeglinandWaldspurger[22℄,Van
den Ban and S hli htkrull [4℄ and many others, dealing with general
prin iples of a similar nature, but in situations of greater omplexity,
whereexpli it answers are not even to beexpe ted. In all, I hope that
thishasleadtoavaluableapproa h,whi hmayperhapsbeusefuleven
beyond the relatively simple ontext of the aÆne He ke algebra. As
a drawba k, I may have re-invented the wheel at points, exhibiting
ignoran e and disregard for existing wisdom. I apologize beforehand
toevery spe ialist who iso ended by this style.
As a result, the paper is lengthy but ompletely elementary and
al-most self ontained. To understand this paper in detail one needs a
de ent edu ation in Fourier analysis and the theory of distributions,
and some additional basi knowledge of ommutative ring theory and
representation theory of linear algebrai groups and C
-algebras. No
referen e is made to more advan ed ma hinery from algebrai
geom-etry, non- ommutative geometry or K-theory. This represents, more
harming to see how far one an push the subje t in an elementary
ontext,still re overingmu hofthegeneralstru tureofrepresentation
theory of redu tive algebrai groups.
It is however lear to me that this \elementary approa h" is
insuf- ient to understand the omplete story of the spe tral theory of the
aÆne He ke algebra-thereby a knowledging the fa t that my
under-standing is more advan ed than my te hni al skills. In any ase, the
paperhopefully bringsout learly the distin tion between what is
ele-mentaryand well understood,what iselementary butnot understood,
andwhatisnotatallelementary. Itmayevenprovidesomehintsasto
whatone should lookfor ifone iswell preparedto applyhigh powered
equipment tothe remainingissues.
The expli it spe tral de omposition of H is ompletely redu ed in
thispapertotheanalogousproblem for ertain nitedimensional
sym-metri algebras. I alled these the \residue Frobeniusalgebras" of the
aÆne He ke algebra. They are asso iated to the so- alled residual
points of the spe trum of the Bernstein enter Z of H . The
under-standing of these residue Frobenius algebras themselves seems tobe a
problem of a di erent level. A fundamental approa h to this problem
should perhaps involve te hniques like y li ohomology or K-theory
ofC
-algebras, butthisismerespe ulation. Isimplyignoredthis
prob-lemaltogetherinthispaper, ex eptforoneimportantfa t,sayingthat
theseresidue algebras are invariantfor\base hange transformations".
This invarian e property is perhaps the most important point of this
paper.
It may be helpful to give the reader a rough outline of the story in
this paper, and an indi ation of the guiding prin iples in the various
stages.
(1). The de nition of the Eisenstein fun tional of the aÆne He ke
algebra, in [24℄. This is losely related to the study of intertwining
operators in minimal prin ipal series modules. It naturally leads to a
representation of the tra e of H as an integral of a ertain rational
kernel overa \global" y le.
(2). Thestudyofthelo usofthesingularitiesoftheabovekernel. It
leads to the notion \residual oset", analogous to the notion of
resid-ual subspa e whi h was introdu ed in [11℄. This olle tion of osets
an be lassi ed, and fromthis lassi ation weverifysome important
geometri propertiesof this olle tion.
(3). The study of the residues of the rational kernel for . This
involvesageneral(butbasi )s hemeforthe al ulationofmultivariable
residues. After symmetrization over the Weyl group, the result is a
expli it probability measure on the spe trum W 0
nT of the Bernstein
enter Z of H . The main tools in this pro ess are the positivity of ,
and the geometri properties of the olle tion of residual osets (2). I
alled this step the \lo alization of the tra e ".
(4). The lo al tra e (as was mentioned in (3)) de ned at an orbit
W 0
t T, arises as an integral of the Eisenstein kernel over a \lo al
y le"whi hisde nedinanarbitrarilysmallneighborhoodoftheorbit
W 0
t. This gives a natural extension of the lo al tra e to lo alizations
of the He ke algebra itself (lo alization as a module over the sheaf of
analyti fun tionson W 0
nT).
(5). The analyti lo alization ofthe He kealgebrahas a remarkable
stru ture dis overed by Lusztig in [17℄. This part of the paper is not
self- ontained, but draws heavily on the paper [17℄ (whi h is by the
way itself ompletely elementary). By Lusztig's wonderful stru ture
theorem we an now investigate the lo al tra es. We dis over in this
way that everything is organized in a ordan e with Harish-Chandra
paraboli indu tion. The problem of nding the Plan herel measure
forH redu es ompletelytothe study ofthe nitedimensionalresidue
Frobenius algebras de ned atthe 0-dimensional residual osets in this
way.
(6). Ifweassumethat therootlabels de ning the He ke algebraare
allof the formq n
for ertainnon-negative integers n
, we prove that
the residue Frobenius algebras are independent of q 2 R >1
. It proves
that the set of irredu ible dis rete series representations of H
asso i-ated with a entral hara ter r, all have a formal dimension whi h is
proportionalto the mass of W 0
r with respe t tothe expli it
probabil-ity measure onW 0
nT mentionedin(3), with aratio ofproportionality
whi h is independent of q.
2. Preliminaries and des ription of results
Thealgebrai ba kgroundforouranalysiswasdis ussedinthepaper
[24℄. Themainresultofthatpaperisaninversionformula(seeequation
4.1) whi h will be the starting point in this paper. The purpose of
this se tion is to de ne the aÆne He ke algebra H and to review the
relevantnotations and on epts involvedintheaboveresult. Moreover
we introdu e aC
-algebras hull Cof H , whi h will be the main obje t
of study in this paper. Finally we will give a more pre ise outline of
2.1. The aÆne Weyl group and its root datum
A redu ed root datum is a 5-tuple R = (X;Y;R 0 ;R _ 0 ;F 0 ), where
X and Y are free Abelian groups with perfe t pairing h;i over Z,
R 0
X is a redu ed integral root system, R _ 0
Y is the dual root
system of oroots of R 0 , and F 0 R 0
is a basis of fundamental roots.
Ea h element 2R 0 determines a re e tion s 2GL(X) by s (x)=x hx; _ i : (2.1) The groupW 0 inGL(X)generatedby the s
is alled theWeylgroup.
Asiswellknown,thisgroupisinfa tgeneratedbythesetS 0 onsisting of the re e tionss with 2F 0 . The set S 0
is alledthe set of simple
re e tions in W 0
.
By de nition the aÆne Weyl group W asso iated with a redu ed
root datum R is the group W = W 0
nX. This group W naturally
a ts on the set X. We an identify the set of integral aÆne linear
fun tions on X with Y Z via(y;k)(x) :=hy;xi+k. It is lear that
wf(x):=f(w 1
x) de nes an a tionof W on Y Z. The aÆne root
system isby de nitionthe subsetR=R _ 0
ZY Z. Noti e thatR
is aW-invariant set inY Z ontaining the set of oroots R _ 0
. Every
element a =( _
;k)2R de nes an aÆne re e tion s a 2 W, a tingon X by s a (x)=x a(x) : (2.2) The re e tions s a
with a 2 R generate a normal subgroup W a
=
W 0
nQ of W, where Q X denotes the root latti e Q = ZR 0
. We
an hoose abasis of fundamentalaÆne roots F by
F :=f( _ ;1)j 2S m g[f( _ ;0)j 2F 0 g; (2.3) where S m
onsists of the set of minimal oroots with respe t to the
dominan e ordering onY. It is easyto see that everyaÆne root isan
integral linear ombination of elementsfrom F with eitherall
nonneg-ative or all nonpositive oeÆ ients. The set R of aÆne roots is thus
a disjoint union of the set of positive aÆne roots R +
and the set of
negative aÆne roots R . The set S of simple re e tions in W is by
de nition the set of re e tions in W asso iated with the fundamental
aÆneroots. They onstituteasetofCoxetergeneratorsforthenormal
subgroup W a
W.
There exists an Abelian omplement to W a
in W. This is best
understoodby introdu ingthe importantlengthfun tionl onW. The
splittingR =R +
[R des ribedaboveimpliesthatR +
\s a
(R )=fag
when a2F. De ne, asusual, the length of anelement w2W by
l(w):=jR \w 1
It follows that, whena 2F, l(s a w)= l(w)+1if w 1 (a)2R + : l(w) 1 if w 1 (a)2R : (2.4)
For any w 2 W we may therefore write w = !w~ with w~ 2 W a
and
with l(!) = 0 (or equivalently, !(F) = F). This shows that the set
of elements of length0 is a subgroup of W whi h is omplementary
tothe normalsubgroup W a
. Hen e ' W=W
a
'X=Q is a nitely
generated Abelian group. There is a natural map m : X ! P whi h
restri tstothein lusionQP onQ. IfwewriteZ X
Xforitskernel,
we have =Z X = f where Z X is free and f = m(X)=Q P=Q is
nite. It is easy to see that Z X
is the subgroup of elements in X that
are entral in W. The nite group f
a ts faithfully onS by diagram
automorphisms.
2.2. Root labels
These ondingredientinthede nitionofHisafun tionqonS with
values in the group of invertible elements of a ommutative ring, su h
that q(s)=q(s 0 ) if s and s 0 are onjugate inW: (2.5)
A fun tion q on S, satisfying 2.5, an learly be extended uniquely to
a length-multipli ative fun tion on W, also denoted by q. By this we
mean that the extension satis es
q(ww 0 )=q(w)q(w 0 ) (2.6) whenever l(ww 0 )=l(w)+l(w 0 ): (2.7)
Conversely, every length multipli ative fun tion on W restri ts to a
fun tion on S that satis es 2.5. Another way to apture the same
information is by assigning labels q a
to the aÆne roots a 2 R . These
labels are uniquely determined by the rules
(i) q wa =q a 8w 2W; and (ii) q(s a )=q a+1 8a2F: (2.8)
Note that a translation t x
a ts on an aÆne root a =( _ ;k) by t x a = a _ (x). Hen e by(i), q a =q _, ex eptwhen _ 22Y,inwhi h ase q a = q ( _ ;k(mod2))
. This last ase o urs i W ontains dire t fa tors
whi hareisomorphi totheaÆneCoxetergroupwhosediagram equals
C a
Yet another manner of labeling will play an important role. It
in-volves apossibly non-redu ed rootsystem R nr
, whi h is de ned by:
R nr :=R 0 [f2 j _ 2R _ 0 \2Yg: (2.9)
Now de ne labels for the roots _ =2 inR _ nr nR _ 0 by: q _ =2 := q 1+ _ q _ :
This hoi e is natural, be auseit implies the formula
q(w)= Y 2Rnr ;+\w 1 Rnr; q _; (2.10) for allw2W 0 . Wedenote by R 1
the root system of longroots inR nr . In other words R 1 :=f 2R nr j2 62R nr g: (2.11)
2.3. The Iwahori-He ke algebra as a Hilbert algebra
Let R be a root datum, and q = (q s
) s2S
a set of root labels as in
the previoussubse tion. Weassumethroughout inthis paperthat the
labels are real numbers satisfying
q(s)>1 8s2S; (2.12)
The following theorem is well known.
Theorem 2.1. Thereexists aunique omplex asso iativealgebraH=
H (R;q) with C-basis (T w
) w2W
whi h satisfy the following relations:
(a) If l(ww 0 )=l(w)+l(w 0 ) then T w T w 0 =T ww 0 . (b) If s 2S then (T s +1)(T s q(s))=0.
ThealgebraH=H (R;q)is alledtheaÆneHe kealgebra(or
Iwahori-He ke algebra) asso iated to (R;q).
We equip the He ke algebra H with ananti-linear anti-involutive
operator de ned by T w =T w 1:
In addition, wede ne a tra efun tional onH , by means of (T w
)=
Æ w;e
. It is awell known basi fa tthat
(T w T w 0)=Æ w;w 0q(w);
implying that is positive and entral. Hen e the formula
(h 1 ;h 2 ):= (h h 2 );
de nes anHermitian inner produ t satisfying the following rules: (i) (h 1 ;h 2 )=(h 2 ;h 1 ): (ii) (h 1 h 2 ;h 3 )=(h 2 ;h 1 h 3 ): (2.13) The basis T w
isorthogonal for (;). Weput
N w :=q(w) 1=2 T w (2.14)
for the orthonormal basis of H that is obtained from the orthogonal
basis T w
by s aling. Let usdenote by (h) and (h) the leftand right
multipli ation operators on H by an element h 2 H . Let H be the
Hilbert spa e obtained from H by ompletion; in other words, H is
the Hilbert spa e with Hilbert basis N w
. Then (h) and (h) extend
uniquely to elements of B(H), the spa e of bounded operators on the
Hilbert spa eH.
The operator extends toanisometri involution onH.
Lemma 2.2. We have k(h)k = k(h)k, and for a simple re e tion
s2S, k(N s
)k=q(s) 1=2
.
Proof. For every w su h that l(sw) > l(w), (N s
) a ts on the two
dimensionalsubspa eV w ofHspannedby N w andN sw asaself-adjoint
operatorwitheigenvaluesq(s) 1=2
and q(s)
1=2
. Sin eHistheHilbert
sumofthesubspa esV w
,weseethattheoperatornormof(N s
)equals
q(s) 1=2
. The fa t that k(h)k=k(h)k follows from
k(h)k=k(h) k =k(h )k =k(h)k
TheabovelemmashowsthatHhasthestru tureofaHilbertalgebra
in the sense of Dixmier [8℄. Moreover, this Hilbert algebra is unital,
and the Hermitian produ t isde ned with respe t tothe tra e .
Wede netheoperatornormkk o
onHbykhk o
:=k(h)k=k(h)k.
The losureofHwithrespe ttokk o
is alledC. Clearly,khk o
khk,
andthuswe anidentify CwithasubsetofH. We anequipCwiththe
stru ture of a C
-algebra by the produ t 1 2 :=( 1 )( 2 )=( 2 )( 1 )
and the -operator oming from H. We extend () to a faithful left
(right)representation of C inthe Hilbert spa e H.
Anelementa 2His alledboundedif thereexists anelement (a)2
B(H) su h that for all h2H ,
(a)(h)=(h)(a): (2.15)
When a 2 H is bounded, there also exists a unique (a) su h that
for allh2H ,
(a)(h)=(h)(a): (2.16)
Itisobvious thatthe elementsofCarebounded. LetusdenotebyN
H the subspa e of bounded elements. Weequip N with the involutive
algebrastru turede nedbytheprodu tn 1 n 2 :=(n 1 )(n 2 )=(n 2 )(n 1 )
and the operator asbefore.
Proposition 2.3. 1. The subspa e (N) := f(a) j a 2 Ng
B(H) is the Neumann algebra ompletion of (H ). In other
words, (N) is the losure of (H ) in B(H) with respe t to the
strong topology (de ned by the semi-norms T ! kT(x)k with
x2 H). The analogous statements hold when we repla e by .
TheNeumannalgebras (N)and(N)aremutually entralizing.
2. Anelementa2Hisboundedexa tly whenitisthelimitin Hofa
sequen e h i
2H su h that the sequen ekh i k o isbounded. In this ase, (h i
) onverges to (a) in the strong topology (similarly,
(h i
) onverges strongly to (a)).
Proof. Allthis anbefoundin[8℄,ChapitreI,paragraphe5. Ingeneral,
(N) isa two-sided idealof the Neumann algebrahull of(H ), but in
the presen e of the unit 1 2H the two spa es oin ide. In fa t, when
A 2 B(H) and A is in the strong losure of (H ), it is simple to see
that A(1)2His bounded.
The pre-Hilbert stru ture oming from H gives N itself the stru ture
of a unital Hilbert algebra. The algebra N an and will be identi ed
withitsasso iated standardNeumann algebra(N). In this situation,
N issaid tobe a saturated Hilbert algebra (withunit element).
LetH
denotethealgebrai dualofH ,equipped withitsweak
topol-ogy. Noti e that extends to H
by the formula ( ) := (1). The
-operator an be extended to H by (h) := (h ). We have the
following hain of in lusions:
H CNHH
: (2.17)
Proposition 2.4. The restri tion of to N is entral, positive and
nite. It isthe natural tra e of the Hilbert algebra N, in the sense that
(a)=(b;b) (2.18)
for every positive a2N, and b 2N su h that a=b 2
.
Proof. Asquare rootbis inNandisHermitian. Then (b;b)=(1;a)=
Corollary 2.5. The Hilbert algebra N is nite.
2.4. The Plan herel measure
By a well known (unpublished) result of J.Bernstein (see [17℄), H
an be viewed as the produ t H 0
A or AH
0
of an abelian subalgebra
A (isomorphi to the group algebra of the latti e X), and the He ke
algebraH 0
ofthe nite Weyl groupW 0
. Both produ tde ompositions
H 0
A andAH
0
give alinearisomorphism ofH withthe tensorprodu t
H 0
A.
The relations between produ ts in H 0
A and in AH
0
are des ribed
by Lusztig's relations (see for example [24℄, Theorem 1.10), and with
the above additional des ription of the stru ture of A and H 0
these
give a omplete presentation of H .
The algebra A has a C-basis of invertible elements x
(with x2X)
su hthat x! x
isa monomorphismof X intothe group ofinvertible
elements of A. This basis is uniquely determined by the additional
property that x = N tx (see 2.14) when x 2 X + . As an important
orollary of this presentation of H , Bernstein identi ed the enter Z
of H as the spa e Z = A W
0 of W
0
-invariant elements in A (see [24℄,
Theorem 1.11).
Proposition 2.6. The He ke algebra H is nitely generated over its
enter Z. At amaximalideal m=m t
(witht 2T)of Z, the lo al rank
equals jW 0
j 2
if andonly ifthe stabilizergroup W t
W
0
isgeneratedby
re e tions.
Proof. Itis lear thatH 'H 0
A is nitelygenerated overZ =A W
0 .
When W
t
is generated by re e tions, it is easyto see that the rank of
m-adi ompletion ^ A m over ^ Z m isexa tly jW 0 j(seeProposition2.23(4) of [24℄).
Lemma 2.7. ThealgebraC isof nitetype I. Infa t everyirredu ible
representation of C has dimension at most equal to the order jW 0
j of
the niteWeylgroup. Furthermore, restri tiontoHindu esabije tion
from the set ^
C onto the set ^
H of irredu ible -representations of H .
Proof. Let 2 ^
C be a nonzero irredu ible representation in a Hilbert
spa eH. Thenforeverynonzerox2H,(H )(x)isadensesubspa eof
H. Also,the enterZ ofHa tsbys alarsinH. ByProposition2.6we
seethatthedimensionof(H )is nite,andthusthat(H )=End (H).
We an nd a simultaneous eigenve tor v with eigenvalue t 2 T say,
for (A). This indu es an epimorphism : I t
! H, where I t
denotes
of H is at most jW 0
j. Therefore C is liminal in the terminology of [9℄.
In parti ular, C has typeI.
Sin eCisseparable,liminalandunital,thespe trum ^
Hisa ompa tT 1
spa e with ountable base. Moreover it ontains an open dense
Haus-dor subset. Be ause C is of nite type I and unital, every irredu ible
representation of C has a ontinuous hara ter.
Thegeneraltheory ofthe entral disintegrationof separable,liminal
C
-algebras (see [9℄, paragraphe 8.8), equipped with a tra ial state
(i.e. isa tra esu hthat (1)=1),assertsthat thereexists aunique
Borel measure P on ^ C= ^ H su h that H' Z ^ H End (V )d P () (2.19)
and su h that
(h)= Z ^ H Tr ((h))d P (): (2.20) The measure P
is alled the Plan herel measure.
The enter ZofNwillbemapped ontothe algebraof diagonalizable
operators L 1 ( ^ H ; P
). This is an isomorphism of algebras, ontinuous
when we give Z the weak operator topology and L 1 ( ^ H ; P ) the weak
topology of the dual of L 1 ( ^ H ; P ). It is anisometry.
The general theory implies in addition that there exists a ompa t
metrizablespa eZ,witha\basi measure",anda-negligablespa e
N Z, su h that Z N is isomorphi as a Borel spa e to the
om-plementof a P
-negligablesubsetof thesupport of P
. The
isomor-phism transforms toa measure whi his equivalent to P
.
There may exist several (but nitely many, by 2.6) inequivalent
ir-redu ible -representations of H with the same entral hara ter t 2
Spe (Z)'W 0
nT,whereT isthe\algebrai torus"T =Hom(X;C
)=
Spe (A). We onsider T with its analyti topology. Let us put p z
:
! W
0
nT for the nite map whi h asso iates to ea h irredu ible
-representationits entral hara ter. Infa tthere may exist non
neg-ligable subsets V of su h that 1 <jp 1 z
(y)j(< 1) for all y 2p z
(V).
Inview of theaboveremarksthis implies inparti ularthat the losure
Z of Z inN isstri tly smallerthan Z, ingeneral.
2.5. Des ription of the main results
support of P
, and p z
denotes the proje tion of onto the spe trum
W 0
nT of Z.
2.5.1. Des ribe the set S :=p z ()W 0 nT. The set S :=p z () turns
out to be a nite union of osets of ompa t subtori of the algebrai
torus T, modulo the a tion of W 0
. The omplexi ations of these
osetsare alled\residual osets"ofT. Conversely,ea hresidual oset
Lhas aunique ompa t formwhi hisasubset ofS. Thisis alled the
\tempered form" of L, and is denoted by L temp
. The residual osets
will be des ribed and lassi ed in Se tion 3, and the proof that the
support ofthe Plan herel measure isrestri ted to entral hara tersin
this set isgiven in Se tion 4.
2.5.2. Des ribe the map p z
: ! S. The bers of p z
are nite
every-where, and p z
is ontinuous and open. Forea h tempered oset L temp , put S L :=W 0 nW 0 L temp and L :=p 1 z (S L
). Then there exists anopen
dense subset 0 L
L
su h that the restri tion of p z
to 0 L
is a trivial
nite overingofitsimage. Thedes riptionofthe bers ofp z
seemsto
be out of the s ope of the methods used in this paper. As we will see
in Theorem 5.17, the des ription of the generi bers p 1 z (p z ()) with 2 0 L
redu es,by theusualmethodofunitaryparaboli indu tion,to
the ase of the residual points(0-dimensional residual osets) of
para-boli subsystems ofroots. The beratresidual pointsr 2T onsists of
a olle tion r
of equivalen e lasses of irredu ible dis rete series
rep-resentations with entral hara ter r. The further des ription of the
spe ial bers boils down tothe study of the redu ibility of
representa-tionsthatareindu edfromdis reteseriesrepresentationsof\paraboli
subalgebras", at spe ial unitary indu tion parameters. Sin e this
phe-nomenon of redu ibility o urs only at a measure 0 subset of , this
problem is less urgentfor our purpose.
Forspe ial ases these problems were solved, usinggeometri
meth-ods. Kazhdan and Lusztig [15℄ solved the problem using equivariant
K-theory in the ase when the labels q i
are equal and X =P. In the
appendixSe tion7one an ndana ountoftheseresults,andthe
re-lation withresidual osets. Lusztig [18℄ solved moregeneral situations
using equivariant homology.
2.5.3. Des ribe the Plan herel measure P . We rst des ribe a \basi measure" b on, su hthat P
isabsolutely ontinuous relative to b
.
Aswasmentioned,S isa niteunionofthetemperedresidual osetsof
T,modulothea tionofW 0
. TakethenormalizedHaarmeasureonea h
of the tempered residual osets, take the push forward ofthis measure
ea h of the onne ted omponents of , we an use p z
tolift this toa
measure on . The measure b
is the measure thusobtained.
We remark that the tempered residual osets are never nested, and
thus there is no \embedded spe trum". This fa t is a fundamental
property of residual osets, but unfortunately our only proof of rests
on the lassi ation of residual subspa es for graded He ke algebras
[11℄. See Se tion 3.1.
We will nd expli itly a W 0
invariant, ontinuous, positive fun tion
m S
on S su h that the Plan herel measure P
on an be expressed
as follows. For ea h onne ted omponent L;Æ of L there exists a onstant d L;Æ 2R + su h that P j L;Æ =d L;Æ p z (m S ) b (2.21)
(seeSe tion5.3). The onstantsd L;Æ
dependonthe hoi eofq. Wewill
showhowever(in Se tion 6)that thed L;Æ
are invariantfor \s aling"in
the followingsense. Assume that
q(s)=q fs
8s2S (2.22)
for positive integer onstants f s
2 N, and q 2 R >1
a parameter.
Us-ing the results of Lusztig [17℄ we will show that the support q
of
the Plan herel measure of the C - algebra C q asso iated with H q := H (R;q)(withq=(q(s)) s2S asin2.22)isindependentofq2R >1 upto
homeomorphisms. The s aling invarian e means that in formula 2.21
the onstants d L;Æ
will be independent ofthe parameter q.
This invarian e isone of the most important points of our
onsider-ations, inview of the followingappli ation. LetG denotethe group of
rational points of a split adjoint simple algebrai group over a p-adi
eld F. We now denote by qthe ardinality of the residue eld of F.
The entralizer algebra of the indu ed representation from a uspidal
unipotentrepresentation ofaparahori subgroup isanaÆneHe ke
al-gebra with its labels given by 2.22 for suitable onstants f s
. By the
Plan herel formula of the aÆne He ke algebra we an ompute the
formal dimension of a dis rete series unipotent representation whi h
arises as a summand in the indu ed module. The invarian e implies
that we determine this formal dimension as a fun tion of q, up to the
omputation of the relevant onstant d L;Æ
.
I would like tomention here that the onstantsd L;Æ
have been
om-puted expli itly inthe ases where the He ke algebrais of ex eptional
split adjoint type (i.e. the onstants f s
in 2.22 are all equal to 1, and
X =P)by MarkReeder[26℄. He onje turesaninterpretation (in the
generalsplit adjoint ase) of the onstants d L;Æ
dis rete series representation. In the ex eptional ases he veri es this
onje ture usingaformula of S hneider andStuhler [27℄ forthe formal
degree of a dis rete series representation of an almost simple p-adi
group whi h ontains xed ve tors for the pro-unipotent radi al U of
amaximal ompa tsubgroup K. This formula ofS hneider and
Stuh-ler however is analternating sum of terms whi h doesnot explain the
produ t stru tureof the formal degrees. In addition, a lever
algorith-mi approa h to the S hneider-Stuhler formula is ne essary to t the
above ase-by- ase veri ations into a omputer.
3. De nition and lassi ation of residual osets
We x on e and for all a rational, positive de nite, W 0
-invariant
symmetri form on X. This de nes an isomorphism between X Z
Q
and Y
Z
Q, andthusalso arational, positive de nitesymmetri form
on Y. We extend this form to a positive de nite Hermitian from on
t C
:=Lie (T)=Y Z
C,whereT isthe omplextorusT =Hom(X;C
).
Via the exponential overing map exp : t C
! T this determines a
distan e fun tion on T.
Letqbeasetofrootlabels. If2 62R nr weweformallyputq _ =2 =1, andalwaysq 1=2 _ =2
denotesthepositiverootsquarerootofq
_ =2
. LetLbe
a oset fora subtorus T L T of T. Put R L :=f 2R 0 j (T L )=1g.
This is a paraboli subsystem of R 0
. The orresponding paraboli
subgroup of W 0 isdenoted by W L . De ne R p L :=f 2R L j (L)= q 1=2 _ =2 or (L)=q 1=2 _ =2 q _g (3.1) and R z L :=f 2R L j (L)=1g: (3.2) Wewrite R p;ess L =R p L nR z L and R z;ess L =R z L nR p L . We de ne anindex i L by i L :=jR p L j jR z L j: (3.3)
Wegivethe following re ursive de nition of the notion residual oset.
De nition 3.1. A oset L of a subtorus of T is alled residual if
ei-ther L = T, or else if there exists a residual oset M L su h that
dim(M)=dim (L)+1 and
i L i M +1: (3.4)
Noti e that the olle tion of residual osets is losed for the a tion
of the group of automorphisms of the root system preserving q (in
Proposition 3.2. If L is residual, then (i) R p;ess L spans a subspa e V L
of dimension odim(L) in the Q
ve -torspa e V =XQ. (ii) We have R L =V L \R 0
, and the rank of R L equals odim(L). (iii) Put L X := V L \X and X L := X= L X. Then T L = ft 2 T j x(t)=1 8x2 L Xg=Hom(X L ;C )=(T W L ) 0 . (iv) Put Y L := Y \QR _ L and L X := Y ? L \X. Let X L := X= L X. We identify R L
with its image in X L . Let F L be the basis of R L su h that F L R 0;+ . ThenR L :=(X L ;Y L ;R L ;R _ L ;F L ) is a root datum. (v) Put T L := Hom(X L ;C ) T (we identify T L
with its anoni al
image in T). Then T L
is the subtorus in T perpendi ular to L.
De ne K L :=T L \T L =Hom(X=( L X+ L X);C )T, a nite
subgroup of T. The interse tion L\T L is a K L - oset onsisting of residual points in T L
with respe t to the root datum R L
and
the root label q L
obtained from q by restri tion to R _ L;nr R _ nr . When r L 2 T L \L, we have L = r L T L . Su h r L is determined up to multipli ation by elements of K L .
Proof. By indu tion on odim(L) we may assume that the assertions
of (i) and (ii) hold true for M in 3.4. From the de nition we see that
R p;ess L nR p;ess M
is not the empty set. An element of R p;ess L nR p;ess M an
not be onstant onM, and hen e 62R M =V M \R 0 . Thus dim (V L )dim (V M )+1= odim(M)+1= odim (L): Sin e also V L Lie(T L ) ? ;
equality has to hold. Hen e R L V L and R L spans V L . Sin e R L is
paraboli , we on lude that R L
= V
L
\R
0
. This proves (i) and (ii).
The subgroup ft 2 T j x(t) = 1 8x 2 L Xg T is isomorphi to Hom(X L ;C
), whi h is a torus be ause X L
is free. By (ii) then, its
dimension equals dim (T L ). It ontains T L , hen e is equal to T L . It followsthatT L
isthe onne ted omponentofthegroupof xedpoints
for W L
,proving (iii). The statements(iv) and (v) are trivial.
For later referen e we introdu e the following notation. A residual
osetL determinesaparaboli subsystem R L
R
0
,and weasso iated
with this a root datum R L
. When R
0
is any root subsystem,
notne essarily paraboli ,weasso iate totwonew rootdata, namely
R :=(X;Y;; _ ;F ) with F
determined by the requirement F
R , and R :=(X ;Y ;; _
quotient of X by the sublatti e perpendi ular to , and Y
Y is
the sublatti e of elements of Y whi h are inthe R-linear span of _
.
There isan obvious onverse to 3.2:
Proposition 3.3. Let R 0
R
0
be a paraboli subsystem of roots, and
let T L
T be the subtorus su h that R 0 = R L . Let T L T be the
subtorus whose Lie algebra Lie(T L ) is spanned by R _ L . Let r 2T L be a
residual point with respe t to (R L ;q L ) as in Proposition 3.2(v). Then L:=rT L
is a residual oset for (R;q).
The re ursive nature of the de nition of residual osets makes it
feasible to give a omplete lassi ation of them. By Lemma 3.2, this
lassi ationproblemredu estothe lassi ationoftheresidualpoints.
In turn, Lusztig [17℄ indi ates how the lassi ation of residual points
redu es to the lassi ation of residual points in the sense of [11℄ for
ertain graded aÆne He ke algebras. This lassi ation is known by
the results in [11℄. Let usexplain this in detail. Following [17℄ we all
a root datum R = (X;Y;R 0 ;R _ 0 ;F 0
) primitive if one of the following
onditions is satis ed: (1) 8 2R 0 : _ 622Y. (2) There is a unique 2 F 0 with _ 2 2Y and fw( ) j w 2 W 0 g generates X.
A primitive root datum R satisfying (2) is of the type C a n , by whi h we mean that R =(Q(B n )=Z n ;P(C n )=Z n ;B n ;C n ;fe 1 e 2 ;:::;e n 1 e n ;e n g):
By [17℄ we know that every root datum is a dire t sum of primitive
summands.
Proposition 3.4. Let r 2 T be a residual point, and write r = s 2
T u
T rs
for its polar de omposition (with T u = Hom(X;S 1 ) and T rs = Hom(X;R +
)). The root system
R s;1
:=f 2R 1
j (s)=1g
has rank dim (T). The system
R s;0 :=f 2R 0 jk 2R s;1 for some k2Ng ontains both R p;ess frg and R z;ess frg
, and r is residual with respe t to the
aÆne He ke subalgebra H s
H whose root datum is given by R s := (X;Y;R s;0 ;R _ s;0 ;F s;0 ) (with F s;0 the basis of R s;0 ontained in R 0;+ ).
Proof. It is lear from the de nitions that R s;0 ontains R p;ess frg and R z;ess frg
, and hen e has maximal rank. Given a full ag of R-residual
weseethatthesetsR p L i ,R z L i are ontainedinR s;0 . Itfollowsbyreverse
indu tion on i (starting with L n
=T) that ea h element of the ag is
R s
-residual.
Lemma 3.5. Givenaresidualpointr=s , let s 0 2T u =Hom(X;S 1 )
be the element whi h oin ides with s on ea h primitive summand of
type C a n
and is trivial on the omplement of these summands. Then
s 0
has order 2.
Proof. To see this we may assume that R is of type C a n . Then R 1 is of type C n ,s =s 0 ,and R s;1 =f 2R 1 j (s 0 )=1g,beingof maximal rankinR 1 ,isof typeC k +C n k
forsome k. In parti ular,2e i
2R
s;1
for all i =1;:::;n. Moreover the index of ZR s;1 in ZR 1 is at most 2. Thus s 0 takes values in f1g on R 1
, and is trivial on elements of the
form2e i . It follows that s 0 has order 2 onX =Z n . Denotebyh 2Hom (Q;S 1 )theimageofs 0 inHom(Q;S 1 ). Chooseroot labels k =k s; 2R with 2R s;0 by the requirement (k depends on
the image of s in Hom(Q;S 1
), but we suppress this in the notation if
there is nodangerof onfusion)
e k =q h( )=2 _ q 1=2 _ +1 = ( q 1=2 _ =2 q _ if h( )=+1 q 1=2 _ =2 if h( )= 1 (3.5)
Theorem 3.6. Let r = s be a (R;q)-residual point. Then :=
log ( ) 2 t := Lie(T rs
) is a residual point in the sense of [11℄ for the
graded He ke algebra H s
=C[ W(R s;0
)℄Sym (t) with root system R s;0
androotlabelsk s :=(k s; ) 2R s;0
. Thismeansexpli itly thatthereexists
a full ag of aÆne linear subspa es f g = l n l n 1 l 0 = t
su h that the sequen e
i s;l i :=jR p s;0;i j jR z s;0;i j (3.6)
is stri tly in reasing, where
R p s;0;i =f 2R s;0 j (l i )=k s; g; (3.7) and R z s;0;i =f 2R s;0 j (l i )=0g: (3.8) Conversely, given a s 2 T u su h that R s;1 R 1
has rank equal to
rank(X), and a residual point 2t for therootsystem R s;0
with labels
(k s;
) de nedby 3.5,the pointr :=sexp is(R;q)-residual. This sets
up a 1 1 orresponden e between W 0
-orbits of (R;q)- residual points
elements of T u
su h that R s;1
has rank equal to rank (X), and 2 t
runs over the W(R s;0
)-orbits of residual points (in the sense of [11℄)
for R s;0
with the labels k s
.
Proof. Straightforward fromthe de nitions.
For onvenien e wein lude the following lemma:
Lemma 3.7. If the rank of R 0
equals the rank of X (a ne essary
ondition for existen e of residual points!), the W 0 -orbits of points s 2 T u su h that R s;1 R 1
has maximal rank orrespond 1 1 to the
Hom(P(R 1 )=X;S 1 ) ' Y=Q(R _ 1
)-orbits on the aÆne Dynkin diagram
R (1) 1
.
Proof. In the ompa t torus Hom(P(R 1 );S 1 ), the W 0 -orbits of su h
points orrespond to the verti es of the fundamental al ove for the
a tionof the aÆne Weyl groupW 0 n2iQ(R _ 1 )onY 2iR. Nowwe have torestri tto X P(R 1 ).
Withtheresultsofthissubse tionathand,the lassi ationofresidual
osets isnow redu ed tothe lassi ation ofresidual subspa es aswas
given in[11℄.
3.1. Properties of residual and tempered osets
In the derivation of the Plan herel formula of the aÆne He ke
al-gebra, some properties of residual osets will play an important role.
Unfortunately, I have no dire t proof of these properties. With the
lassi ation athand they an be he ked on a ase-by- ase basis. By
the previoussubse tion this veri ation redu es tothe ase ofresidual
subspa es for graded aÆne He ke algebras. In [11℄ ( f. Theorem 3.9,
3.10 and Remark 3.14) these matters have indeed been veri ed.
Theorem 3.8. De ne : T ! T by x(t
) =x(t) 1
. If r = s2 T is
a residual point, then r
2W(R s;0
)r.
Theorem 3.9. For ea h residual oset L2T wehave
i L
= odim(L): (3.9)
In other words, for every in lusion L M of residual osets with
dim(L)=dim(M) 1,the inequality 3.4 isa tually anequality. Note
that Theorem3.9 redu es tothe ase ofresidual pointsbyProposition
3.2 and Proposition3.3. This redu es tothe ase of residual pointsin
lassi a-Theorem3.9hasimportant onsequen es, aswewillseelater. Atthis
point we show that the de nition of residual osets an be simpli ed
asa onsequen e of Theorem 3.9. We begin with a simple lemma:
Lemma 3.10. Let V be a omplex ve tor spa e of dimension n, and
supposethatListheinterse tionlatti eofasetP oflinearhyperplanes
in V. Assume that ea h hyperplane H 2 P omes with a multipli ity
m H
2 Z, and de ne the multipli ity m L for L 2 L by m L := P m H ,
where the sum is taken over the hyperplanes H 2P su h that L H.
Assume that f0g2L and that m f0g
n. Then there exists a full ag
of subspa es V =V 0 V 1 V n =f0g su h thatm k :=m V k k.
Proof. We onstru t the sequen e indu tively, starting with V 0
.
Sup-pose we already onstru ted the ag up to V k
, with k n 2. Let
P k
L denote the set of elements of L of dimension n k 1
on-tained inV k
, and let N k
denote the ardinality of P k
. By assumption,
N k
n k2. Sin e every H 2P either ontains V k orinterse ts V k inan element of P k , we have X L2P k (m L m k )=m n m k : (3.10) Assume that 8L2P k : m L k. Then,be ausem k k and N k 2, m n kN k +(1 N k )m k kn 2; (3.11)
ontradi ting the assumption m n
n. Hen e there exists a L 2 P k
with m L
k+1, whi h we an de ne tobe V k+1
.
Corollary 3.11. For every oset L T one has i L
odim (L), and
L is residual if and only if i L
= odim(L).
Proof. De ne P to be the list of odimension 1 osets of T arising as
onne ted omponents of the following odimension 1 sets:
L + ;1 :=ft 2T j (t) =q _q 1=2 _ =2 g L + ;2 :=ft 2T j (t) = q 1=2 _ =2 g L ;1 :=ft 2T j (t) =1g L ;2 :=ft 2T j (t) = 1g (3.12) Here 2 R 0 , and q _ =2 = 1 when 2 62 R 1
. We give the omponents
of L + ;1 , L + ;1
the index +1, and we give the omponents of L ;1
, L ;1
index 1.
Suppose that L isany oset ofa subtorus T L
in T. Theni L
isequal
tothe sum of the indi es the elements of P ontaining L.
Assume that i L
odim(L)=k. By Lemma3.10 there exists a
ofelementsofP su hthati L k e =i L kand i L j j = odim(L j )(we
did not assume that L is a omponent of an interse tion of elements
in the list P, hen e e > 0 may o ur). If k(0) is the smallest index
su h that i L
k (0)
> k(0), then L k(0)
is by de nition residual, and thus
violates Theorem3.9. Hen e su hk(0) doesnot exist and we on lude
that i L
k
=k forallk. Thisproves thate=0andthat Lisresidual.
Remark 3.12. Thissolves thequestion raisedin Remark 3.11of [11℄.
Theorem 3.13. Suppose that L M are two residual osets. Write
L = r L T L = s L exp ( L )T L and M = r M T M = s M exp ( M )T M as before, with r L 2T L and r M 2T M . If L = M then L=M.
De nition 3.14. Let L be a residual oset, and write L = r L T L = s L exp ( L )T L with r L 2T L
. This isdeterminedup to multipli ation of
r L by elements of K L =T L \T L . We all L :=exp( L ) the \ enter" of L, and we all L temp :=r L T L u
the tempered ompa tform of L (both
notionsareindependentof the hoi eof r L sin eK L T L u ). The osets of the form L temp
in T will be alled \tempered osets".
Theorem3.13followsfromRemark3.14in[11℄,andshows inparti ular
thatatempered oset annot beasubsetof astri tlylargertempered
oset.
4. Lo alization of on Spe (Z)
Re all the de omposition of we derived in[24℄, Theorem 3.7:
= Z t2t0Tu E t q(w 0 )(t) dt q(w 0 ) (t 1 ) (t) : (4.1)
Let us brie y review the various ingredients of this formula. First of
all, T u
= Hom(X;S
1
), the ompa t form of the algebrai torus T =
Hom(X;C ), and t 0 2 T rs
, the real split part of T, su h that the
following inequality is satis ed ( f. [24℄ De nition 1.4 and Corollary
3.2): t 0 <Æ 1=2 : (4.2) In other words, t 0
is inthe shifted hamber de ned by
8i2f1;:::;ng : i (t 0 )<q(s i ) 1 : (4.3)
Theformdt denotes theholomorphi n-formonT whi hrestri ts to
The fun tion (t):= Q 2R1;+ with :=1 2A (4.4)
is the Weyldenominator, and (t)= (t;q) isMa donald's -fun tion.
This -fun tion is introdu ed as an element in F
A, the eld of
fra -tions of A, and will be interpreted as rationalfun tion on T ( f. [24℄,
De nition 1.13). Expli itly, we put
:= Y 2R 1;+ ; (4.5) where, for 2R 1 , we de ne by := (1+q 1=2 _ =2 )(1 q 1=2 _ q 1 2 _ =2 ) 1 2 F A: (4.6)
Remark 4.1. It is handy to write the formulas in the above form,
but stri tly speaking in orre t if =2 62 R 0
. However, we formally put
q 2
_
= 1 if =2 62 R 0
, and with this substitution the above formula
redu es to an expression ontaining only
. Here and below we use
this onvention.
The expression E t
2 H
is the holomorphi Eisenstein series for H ,
with the following de ning properties ( f. [24℄, Propositions 2.23 and
2.24):
(i) 8h2H ; the map T 3t!E t (h)is regular: (ii) 8x;y2X;h2H ; E t ( x h y )=t(x+y)E t (h): (iii) E t (1)=q(w 0 )(t): (4.7)
We want to rewrite the integral 4.1 representing the tra e fun tional
as an integral over the olle tion of tempered residual osets, by a
ontour shift. It turns out that su h a representation exists and is
unique. To nd it,weneed an intermediate step. We will rst rewrite
the integral asa sum of integralsover alarger set of tempered
\quasi-residual osets", andthenwewillshowthatifwesymmetrizetheresult
over W 0
, allthe ontributions of non-residual osets an el.
4.1. Quasi residual osets
The basi s heme to ompute residues has nothing to do with the
properties of rootsystems. It is therefore onvenient to formulate
De nition 4.2. Let! =pdt=q bearational(n;0)-formonT. Assume
that p;q are relatively prime, and are of the form
q(t)= Y m2M (x m (t) d m ); p(t)= Y m 0 2M 0 (x m 0(t) d m 0): (4.8)
where the produ ts are taken over nite sets M;M 0
. The index sets
M and M
0
ome equipped with maps m ! (x m ;d m ) 2 X C . An
!-residual oset Lis a onne ted omponentof aninterse tion of
odi-mension 1 sets of the form L m =ft jx m (t)=d m g with m 2M, su h
that the order i L of ! along L satis es i L :=jfm2MjLL m gj jfm 0 2M 0 jLL m 0gj odim (L):
The list of !-residual osets is denoted by L !
(it in ludesby de nition
the empty interse tion T).
We de ne the notions T L
; T L
as we did in the ase of the residual
subspa es of se tion 3. We note that the \perpendi ular torus" T L
exists be ausethe W 0
-invariantinner produ t onX t has values in Q. We denote by M L M the subset fm 2 M jx m (L) =d m g. We hoose anelement r L =s L L 2T L
\L forea hL sothat we an write
L=r L T L . We all L 2T rs
the enter of L,and note that this enter
is determined uniquely by L. We write L =exp L with L 2t L . The
set of enters of the !-residual osets is denoted by C ! . The tempered form of a !-residual L = r L T L is by de nition L temp := r L T L u (whi h
is independent of the hoi e of r L
), and su h a oset will be alled an
!-tempered oset.
Basi ally, the only properties of the olle tion L !
wewill need are
Proposition 4.3. (i) If 2C !
then the union
S :=[ fL2L ! j L = g L temp T u
is a regular support in the sense of [28℄ in T u
. This means that
a distribution on T u
with support in S
an be written as a sum
of derivativesof measures supported on S
.
(ii) If =exp 2T rs
, andLis!-residualwithj L
jj jbut L
6= ,
then there exists a m2M L su h that f(t)=x m (t) d m is non-vanishing on T u .
Proof. The set S
isa niteunion of smooth varieties, obviously
satis-fying the ondition of [28℄, Chapitre III, x9 for regularity. Hen e (i)is
trivial. As for(ii), rst note that the assumption implies that L
6=0,
hen ethatL6=T. Thusthe odimensionofLispositive,andM L 6=;. Clearly 62 L +t L = log (T rs \LT u ) sin e L
is the unique smallest
t L = (t L ) ? , we an nd a m 2 M L su h that x m ( ) 6= x m ( L ). This
implies the result.
4.2. The ontour shift and the lo al ontributions
The following theorem isessentially the same as Lemma3.1 of [11℄,
but be ause of itsbasi importan eI in lude the proof here. See also
[3℄for a more generalmethod.
Theorem 4.4. Let ! be as in De nition 4.2 and let t 0 2 T rs n [ (T rs \T u L m
). There exists a unique olle tion of distributions fX 2 C 1 ( T u )g 2C
! su h that the following onditionshold:
(i) The support of X satis es supp (X )S . (ii) For every a2A an (T) we have Z t2t 0 Tu a(t)!(t)= X 2C ! X (aj T u ): (4.9)
Proof. The existen e is proved by indu tion on the dimension n of T,
the ase of n = 0 being trivial. Suppose that the result is true for
tori of dimension n 1. Choose a smooth path in T rs
from t 0
to the
identity e whi h interse ts ea h oset T rs \T u L m transversally and in
at most one point t m
. We may assume the t m to be distin t unless t m =e. When we move t 0
along this urve to e we pi k up a residue
when we ross at a point t m
6= e. For simpli ity of notation we write
(x;d) instead of (x m ;d m ), L instead of L m , t insteadof t m et . Re all
that we have the de omposition L = r L T L = s L L T L with r L 2 T L .
LetD bethe unit onstantve tor eld onT whi his perpendi ularto
L. Write d L
t=hD;dti for the invariant(n 1;0)-formwhi hrestri ts
toHaarmeasure onT L u
. The residue ontribution we pi k up whenwe
ross L is then equal to Z t 0 2tsLT L u D i L 1 (((x d) i L p=q)a)(t 0 )d L t 0 :
This de omposes as a nitesum of the form
i L 1 X j=0 X k Z ts L T L u (D j (a)j L )! j;k ; where! j;k
isitselfarational(n 1;0)-formonLintheprodu tformas
inDe nition 4.2, withpoles alongthe interse tions L 0 n
=L\L n
(with
n 2M)whi hare of odimension 1inL. Asimple omputationshows
thatforeveryj;k andevery onne ted omponentH ofaninterse tion
of osets of the form L 0
L, the index i ! ;H
i ! j;k ;H (i !;H
1) j. Itfollowsthatthe unionoverallj;k ofthe! j;k
-residual osets of L is ontained in the olle tion of !-residual osets
of H.
By the indu tion hypotheses we an rewrite su h residues in the
desired form, where the role of the identity element in the oset L is
now played by r L
. At the identity e itself we have to take a boundary
value of ! towards T u
, whi h de nes a distributionon T u
. This proves
the existen e.
The uniqueness is proved asfollows. Suppose that wehave a
olle -tion fY 2C 1 ( T u )g 2C !
of distributions su h that
(i) supp (Y )S . (ii) 8a2C[ T℄: P 2C ! Y (aj Tu )=0. WeshowthatY
=0byindu tiononj =log ( )j. Choose 2C ! su h that Y 0 =0 for all 0 with j 0 j <j j. For ea hL 2L ! with j L jj j and L 6= we hoose a l 2 M L su h that x l (t) d l
does not vanish
on T u
(Proposition 4.3) and we set
(t):= Y fL:j L jj jand L 6= g (x l (t) d l ):
It is lear that for suÆ iently large N 2 N, Y
( N
a) = 0 for all a 2
C[T℄. Ontheotherhand,bythetheoryofFourierseriesofdistributions
on T u
, C[T℄j T
u
is a dense set of test fun tions on T u . Sin e N is nonvanishingon T u
,thisfun tionisaunitinthespa eoftestfun tions
in T u . Thusalso N C[ T℄j T u
isdense inthe spa e oftest fun tions. It
follows that Y
=0.
4.2.1. Approximating sequen es. There is an \analyti ally dual"
for-mulation ofthe result onresidue distributions that will be useful later
on. The idea todeal with the residue distributions inthis way was
in-spiredbytheapproa hin[13℄toprovethepositivityof ertainresidual
spheri al fun tions.
Lemma 4.5. For all N 2 N there exists a olle tion of sequen es
fa N; n g n2N ( 2C !
) in A with the following properties:
(i) For all n2N, P 2C ! a N; n =1.
(ii) For every onstant oeÆ ient di erentialoperator D of order at
mostN onT,D(a N; n )!D(1)uniformlyon S andD(a N; n )!0 on S 0 if 0 6= .
Proof. We onstru t the sequen es with indu tion on the norm j =
log ( )j. We x N and suppress it from the notation. Let 2 C !
and
assume that wehave already onstru tedsu h sequen es a
0
n
satisfying
the se ond part of the proof of Theorem 4.4. By Fourier analysis on
T u
it is lear that there exists a sequen e f n
g n2N
in C[T℄ su h that
forea h onstant oeÆ ient di erential operatorDof orderatmostN
there exists a onstant D su h that k(D( n ) D( (N+1) ))j T u k 1 < D =n
Applying Leibniz' rule to (N+1) n 1= (N+1) ( n (N+1) )
repeat-edly we see that this implies that there exists a onstant 0 D
for ea h
onstant oeÆ ient di erential operatorD, su h that
k(D( (N+1) n ) D(1))j T u k 1 < 0 D =n: Noti e that D( (N+1) n )= 0on all S 0 with j 0 j j j but 0 6= . On
theotherhand,forea h onstant oeÆ ientdi erential operatorE the
fun tionE(1 P f 0 jj 0 j<j jg a 0 k
) onverges uniformlytozeroonea hS
0
with j 0
j < j j. Again applying Leibniz' rule repeatedly we see that
there exist a k 2N (depending onn) su h that the fun tion
a n := (N+1) n (1 X f 0 jj 0 j<j jg a 0 k )
has the property that
kD(a n )j [S 0 k 1 < 0 D =n;
where the union is taken over all 0
with j 0
j<j j. It is learthat the
sequen e a n
thus onstru ted satis es (ii). We ontinue this pro ess
untilwe have onlyone enter left. Forthis last enter we an simply
put a n :=1 X 0 6= a 0 n :
This satis es the property (ii), and for es (i) tobe valid.
The use of su h olle tions of sequen es isthe following:
Proposition 4.6. In the situation of Theorem 4.4 and givenany
ol-le tion of sequen es fa n
g as onstru ted in Lemma 4.5 we an express
the residue distributions as (with a2A):
X (a)= lim n!1 Z t0Tu a n a!;
provided N (in Lemma 4.5) is hosen suÆ iently large.
Proof. Be ause we are working with distributions on ompa t spa es,
the orders ofthe distributions are nite. Take N larger than the
X
0 as a sum of derivatives of order at most N of measures supported
on S
0. The result now follows dire tly fromthe de ning properties of
the sequen e a n
.
4.2.2. Cy les of integration. Yet another useful way to express the
residue distribution is by means of integration of a! over a suitable
n- y le. We will need this representation later on.
Inthe next propositionwewill use thedistan e fun tiononT whi h
measures the distan e between 2iY-orbits in t C
. For Æ > 0 and ea h
L whi h isa onne ted omponent of aninterse tion of odimension 1
osets L m T with m2M, we denote by B L (r L ;Æ) aball inT L with
radius Æ and enter r L
, and by B L rs
(Æ) a ball with radius Æ and enter
d in T L rs
. We put M
L
M for the m 2 M su h that L L
m , and
M L
M for the m 2 M su h that L m \L has odimension 1 in L. Wewrite T m =ft jx m (t)=1g. Let U L (Æ) T L
be the open set ft 2T L rs j 8m2 M L :tB L (r L ;Æ)\ L m =;g. Note that U L (Æ 1 )U L (Æ 2 ) if Æ 1 >Æ 2
,and that the union of
theseopen setsisequaltothe omplementofunionofthe odimension
1 subsetsr 1 L (L\L m )T L with m2M L .
Proposition 4.7. Let >0 be su h that for all m 2M and L2L ! , L m \B L (r L ;)B L rs ()T L u 6= ; implies that L temp \L m 6= ;. Denote by M L;temp the set of m2M L su h that L temp \L m 6=;. There exist (i) 8L2L ! , a point L 2B L rs ()n[ m2M L;temp T m ,
(ii) a 0<Æ< su h that 8L2L ! ; L T L u U L (Æ), and (iii) 8L2L ! , a y le L B L (r L ;Æ)n[ m2M L L m , su h that 8 2C ! ;8 2C 1 ( T u ):X ( )= X fLj L = g X L ( ); (4.10) where X L is the distribution on T u de ned by 8a2A:X L (a)= Z L T L u L a!; (4.11) If M L;temp =; we may take L =e.
Proof. We begin the proof by remarking that (i), (ii) and (iii) imply
that the fun tional X L
on A indeed de nes a distribution on L T u , supported onL temp . Consider for t2U L
(Æ) the inner integral
Z t a! :=i(a;t)d L t: (4.12)
Theni(a;t)isalinear ombination ofderivativesD
a (normaltoL)of
a at r L
t with oeÆ ients in the ring of meromorphi fun tions on T L
whi h areregularoutsidethe odimension1interse tions r 1 L (L\L m ): i(a;t) = X f D a: (4.13) Hen e X L
(a) is equal to the sum of the boundary value distributions
BV
L ;f
of the meromorphi oeÆ ient fun tions,applied tothe
orre-sponding normalderivative of a, restri ted toL temp : X L (a)= X BV L ;f (D aj L temp ): (4.14) We see that X L is a distribution supported in L temp L T u , whi h onlydepends on L
and onthe omponent ofB L rs ()n[ m2M L;tempT m in whi h L lies.
Hen e,bytheuniqueness assertionofTheorem4.4,we on ludethat
itissuÆ ienttoprovethat we an hoose L
; Æ; L
insu h away that
8a2A: Z t 0 T u a! = X L2L ! X L (a): (4.15)
Inordertoprovethis itisenoughtoshowthatwe an hoose L
; Æ; L
as in (i), (ii) and (iii) for the larger olle tion ~ L !
of all the onne ted
omponents of interse tions of the L m
(with m2M), su h that
t 0 T u [ L2 ~ L ! L T L u L : (4.16)
Here means that the left hand side and the right hand side are
homologous y lesinTn[ m2M
L m
. Thedesiredresultfollowsfromthis,
sin ethefun tional X L
isequalto0unlessLis!-residual (be ausethe
innerintegral4.12isidenti allyequalto0fornon-residualinterse tions,
by an elementary argument whi h is given in detail in the proof of
Theorem 4.26).
Letk2f0;1;:::;n 1g. Denoteby ~ L !
(k)the olle tionof onne ted
omponentsofinterse tions oftheL m
(m2M)su hthat odim (L)<
k. Assumethatwealreadyhave onstru tedpoints L
; Æ; L
satisfying
(i),(ii)and(iii)forallL2 ~ L !
(k)andinaddition,forea hL2L !
with
odim(L)=k, a niteset of points L T L rs su hthat L T L u U L (Æ) and a y le L;w B L (r L ;Æ)n[ m2M L L m for ea h w 2 L , su h that t 0 T u ishomologous to [ L2 ~ L ! (k) ( L T L u L )[[ L2 ~ L ! (k+1)n ~ L ! (k) [ w2 L (wT L u L;w ): (4.17)
This equation holds for k = 0, with T
= ft 0
g, whi h is the starting
point of the indu tive onstru tion to be dis ussed below. We will
onstru t L , Æ and forL2 ~ L ! (k+1)n ~ L !
L 2 ~ L (k+2)n ~ L (k+1), with a y le w for ea h w 2 L su h that
equation 4.17 holds with k repla ed by k+1, and Æ by Æ 1
.
First ofall, noti ethat we may repla e Æ by any 0<Æ 0
<Æ in
equa-tion4.17, be ausewe anshrink the L
and L;w
withintheirhomology
lass to t in the smaller sets B L (r L ;Æ 0 )n[ m2M L L m . Choose Æ 0 small
enough su h that for ea h L 2 ~ L ! (k+1)n ~ L !
(k) there exists a point
L
2B L rs
() withthe property that L T L u U L (Æ 0 ).
The singularities of the inner integral are lo ated at odimension 1
osets in T L
of the form r 1 L
N, where N isa onne ted omponent of
L\L m for somem 2M L . Wehave r 1 L N =r 1 L r N T N T L ,and thus 1 L N T N rs T L rs
. Choose paths inside T L rs fromw2 L tothe point L .
We hoose ea h path su h that it interse ts the real osets 1 L N T N rs
transversallyandinatmostonepoint,andsu hthattheseinterse tion
points are distin t. If p is the interse tion pointwith the path from
w 2
L to
L
then p is of the form p = 1 L N w L;N;w 2 1 L N T N rs with w L;N;w 2 T N rs . Given N 2 ~ L ! (k+2)n ~ L ! (k+1) we denote by N the set of all w L;N;w
arising in this way, for all the L 2 ~ L ! (k+1)n ~ L ! (k)
su h that LN, and w2 L . Noti e that if v = w L;N;w 2 N and vs 2 r 1 N (N \L m ) for some m 2 M N and s 2 T u , we have that 1 L N v 2 1 L ( N T N rs \ N 0T N 0 rs ) where N 0 = L\L m . Sin e T N 0 6= T N
, this ontradi ts the assertion
thatthe interse tionpointsof thepathsinT L rs
andthe osets 1 L N T N rs
are distin t. We on lude inparti ular that the ompa t set N
T N u
is
ontainedin the unionof the open sets U N
(Æ 0
). We an thus hoose Æ 0
smallenoughsu hthatinfa t N T N u U N (Æ 0 ),asrequiredinequation 4.17. WriteT NL
fortheidentity omponentofthe1-dimensional
interse -tion T N
\T L
,and de ompose thetorus T L as theprodu t T N T NL . Let v =w L;N;w 2 N and put p= 1 L N
v for the orresponding
inter-se tionpointinT L rs
. Noti e thatfora odimension1 osetr 1 L N 0 T L with N 0 2 ~ L ! we have that pT NL;u \r 1 L N 0 = ( ; if 1 L N 0T N 0 6= 1 L N T N ; G L;N 0 ;w otherwise (4.18) whereG L;N 0 ;w
isa oset ofthe subgroup T NL
\T N
of the nitegroup
K N 0 =K N =T N \T N T N u , of the form G L;N 0 ;w =(T NL \T N )r 1 L r N 0v: (4.19) The osets G L;N 0 ;w
are disjoint. Let Æ(L;w) be the minimum distan e
the Land w2 L
. Choose Æ 1
>0 smallerthanthe minimum of Æ 0
and
Æ(k+1). Let be a ir le of radius Æ 1 =2with enter e in T NL . Next wemake Æ 0
suÆ iently smallsothat [ N 0G L;N 0 ;w U L (Æ 0 ). Forx ;x + suitably lose tox 0 with x <x 0 <x + we have in U L (Æ 0 ): (x )T NL;u (x + )T NL;u [[ N 0 G L;N 0 ;w ; (4.20)
where the union is over all N 0 L su h that 1 L N 0T N 0 = 1 L N T N . De ne L;N 0 ;v :=r 1 L r N 0 L;w : (4.21) Wethen have (x )T L u L;w (x + )T L u L;w [[ N 0 vT N u L;N 0 ;v : (4.22) By possibly making Æ 0
smaller we get that L;N;v B N (r N ;Æ 1 ) for all
possible hoi es N;L and w. If L m N but L m 6 L, then, sin e r 1 L r N U L (Æ 0 ) and L;w B L (r L ;Æ 0 ),we have L;N;v \L m =;. Ifon
the other hand L m L then L;N;v \L m =r 1 L r N ( L;w \L m )=;. Finally we put N;v :=[ (L;w) L;N;v ; (4.23)
wherewetaketheunionoverallpairs(L;w)withL2 ~ L ! (k+1)n ~ L ! (k)
su h that L N and w 2
L
su h that there is aninterse tion point
w L;N;w
with w L;N;w
=v. We have shown that
N;v B N (r N ;Æ 1 )n[ m2M N L m ; (4.24) asrequired in equation 4.17.
Applying equation 4.22 for all the interse tions of all the paths we
hose, we obtain equation 4.17 with k repla ed by k + 1 and Æ by
Æ 1 . We thus take L = [ w2 L L;w for L 2 ~ L ! (k+1)n ~ L ! (k), and for N 2 ~ L ! (k+2)n ~ L ! (k+1)we take N and N;v as onstru tedabove.
This pro ess ontinues untilwe have k=n 1in equation 4.17. In
the next step we pro eed in the same way but with N
= feg for all
N 2 ~ L ! (n+1)n ~ L !
(n). The pro ess nowstops, sin ealso N
=e. This
proves the desired result, with Æ equal to the Æ 1
obtained in the last
step of the indu tive onstru tion.
Remark 4.8. The homology lasses of the y les L
are not uniquely
determinedbytheabovealgorithm. Infa tthesplittingX = P fLj L = g X L
is not unique without further assumptions. Nonetheless, we will later
study a situation where the de omposition X = P fLj L = g X L is su h that ea h X L
is a regular measure supported on L temp
de-Welist someuseful propertiesof the y les L
. We x !, andsuppress
it fromthe notation.
De nition 4.9. Let L 2 L. Denote by L L
the on guration of real
osets M L := L T M rs
where M 2 L su h that M L, M 6= T. The
\dual" on guration, onsistingof the osets M L := L T M;rs T L with
M 2L su h that M L, is denoted by L L
. Given an (open) hamber
C in the omplement of L L , we all C d =f L exp (v)j (v;w)<08w 2 log ( 1 L
C)nf0gg the anti-dual one. This anti-dual one is the interior
of the losure of a union of hambers of the dual on guration L L in T L . Proposition 4.10. If t 0
ismovedinsidea hamberof L L we anleave L un hanged. Proof. Ift 0
ismoved withina hamberofL L
,thepath fromt 0
toe an
be hosen equal to the original path up to a path whi h only rosses
odimension one osets of the form L m
T u
\T rs
whi h do not ontain
= L
. Therefore this doesnot hange L . Proposition 4.11. Write L=r L T L = L s L T L
as usual, and let M 2
L L . ThenM temp L temp if andonly ife2M L . Inparti ular, L temp is
maximalinthe olle tionof !-tempered osetsifandonlyifeisregular
with respe t to the on guration L L . Proof. If M 2 L L , then L temp M temp , L = M (sin e then s L 2 ( 1 M M)\T u = s M T M u , implying that r L 2 M temp ). Now L = M , L 2T M ,e2M L .
Proposition 4.12. ( f. [11℄,Lemma 3.3.) If e isnotin the losure of
the anti-dual one of the hamber of L L
in whi h t 0
lies, we an take
L
=;.
Proof. The possible ontributions to the y le L
ome from ontour
shiftsinthe entral on gurationL L
. We anthereforerepla eLbythe
on guration L(L) of onne ted omponents of interse tions of osets
L m
with m 2M
L
, and apply the indu tive pro ess of Proposition 4.7
toL(L).
WeidentifyT rs
withtherealve torspa etviathemapt!log ( 1 L
t),
and we denote by h;i the Eu lidean inner produ t thus obtained on
T rs
. Noti e that the role of the origin isplayed by L . The sets M T M rs withM 2L(L)satisfy M T M rs 3 L
,and areequipped withthe indu ed
Eu lidean inner produ t.
Bythe assumption and Proposition4.10 we an hoose t 0
withinits
