Spectral transfer morphisms for unipotent affine Hecke algebras

66 Pages·2017·1.22 MB·English

by  Eric Opdam

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Spectral transfer morphisms for unipotent affine Hecke algebras
Opdam, E.
DOI
10.1007/s00029-016-0273-7
Publication date
2016
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Final published version
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Selecta Mathematica-New Series
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Opdam, E. (2016). Spectral transfer morphisms for unipotent affine Hecke algebras. Selecta
Mathematica-New Series, 22(4), 2143–2207. https://doi.org/10.1007/s00029-016-0273-7
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Sel.Math.NewSer.(2016)22:2143–2207 SelectaMathematica
DOI10.1007/s00029-016-0273-7 NewSeries
Spectral transfer morphisms for unipotent affine Hecke
algebras
EricOpdam1
DedicatedtoJosephBernsteinontheoccasionofhis70thbirthday,withadmiration
Publishedonline:5October2016
©TheAuthor(s)2016.ThisarticleispublishedwithopenaccessatSpringerlink.com
Abstract We classify the spectral transfer morphisms (cf. Opdam in Adv Math
286:912–957,2016)betweenaffineHeckealgebrasassociatedtotheunipotenttypes
of the various inner forms of an unramified absolutely simple algebraic group G
definedoveranon-archimedeanlocalfieldk.ThisturnsouttocharacterizeLusztig’s
classification (Lusztig in Int Math Res Not 11:517–589, 1995; in Represent Theory
6:243–289,2002)ofunipotentcharactersofGintermsofthePlancherelmeasure,up
todiagramautomorphisms.Asanapplicationoftheseresults,thespectralcorrespon-
dencesassociatedwithsuchmorphisms(Opdam2016),andsomeresultsofCiubotaru,
KatoandKato[CKK](alsoseeCiubotaruandOpdaminAuniformclassificationof
the discrete series representations of affine Hecke algebras. arXiv:1510.07274) we
prove a conjecture of Hiraga, Ichino and Ikeda [HII] on formal degrees and adjoint
gammafactorsinthespecialcaseofunipotentdiscreteseriescharactersofinnerforms
ofunramifiedsimplegroupsofadjointtypedefinedoverk.
Keywords Heckealgebra·Formaldegree·Spectraltransfermorphism·L-packet
MathematicsSubjectClassification Primary20C08;Secondary22D25,43A30
ThisresearchwassupportedbyERC-advancedGrantNo.268105.
B
EricOpdam
[email protected]
1 KortewegdeVriesInstituteforMathematics,UniversityofAmsterdam,SciencePark107,
1018TVAmsterdam,TheNetherlands
2144 E.Opdam
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2144
2 Unipotentrepresentationsofquasisimplep-adicgroups . . . . . . . . . . . . . . . . . . . . . 2146
2.1 Unramifiedreductivep-adicgroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2146
2.2 Unipotentrepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2150
2.3 UnramifiedlocalLanglandsparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 2153
2.4 UnipotentaffineHeckealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2155
3 ThespectraltransfercategoryofunipotentHeckealgebras . . . . . . . . . . . . . . . . . . . . 2159
3.1 Spectraltransfermorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2159
3.2 Maintheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2165
4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2180
4.1 Classificationofunipotentspectraltransfermorphisms . . . . . . . . . . . . . . . . . . . 2180
4.2 PartitioningofunramifiedsquareintegrableL-packetsaccordingtoBernsteincomponents. 2181
4.3 Parameterizationforclassicaltypes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2183
4.4 Parameterizationforsplitexceptionalgroups. . . . . . . . . . . . . . . . . . . . . . . . . 2186
4.5 Parameterizationfornon-splitquasisplitexceptionalgroups . . . . . . . . . . . . . . . . . 2188
4.6 Formaldegreeofunipotentdiscreteseriesrepresentations . . . . . . . . . . . . . . . . . . 2189
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2205
1 Introduction
Recallfrom[54]thatanormalizedaffineHeckealgebraHisessentiallydeterminedby
acomplextorusT andameromorphicfunctionμonT.Aspectraltransfermorphism
(see[54])φ :H (cid:2)H betweennormalizedaffineHeckealgebrasexpressesthefact
1 2
thatμ isequaltoaresidueofμ alongacertaincosetofasubtorusofT .Thisturns
1 2 2
outtobeaconvenienttooltocompareformaldegreesofdiscreteseriesrepresentations
ofdifferentaffineHeckealgebras.
Thenotionisbasedonthespecialpropertiesoftheμ-functionofanaffineHecke
algebra [52,53] which are intimitely related to its basic role in the derivation of
the Plancherel formula for affine Hecke algebras via residues [21,52,55,56]. This
approachtothecomputationofformaldegreeshasitsorigininthetheoryofspherical
functionsfor p-adicreductivegroups[47],andwasfurtherinspiredbyearlyobserva-
tionsofLusztig[35,38]andReeder[59,60]onthebehaviourofformaldegreeswithin
unipotent L-packets.
In the present paper we classify the spectral transfer morphisms (STMs in the
sequel)betweentheunipotentaffineHeckealgebrasofthevariousinnerformsofa
given absolutely simple algebraic group G of adjoint type, defined and unramified
over a non-archimedean local field k. In particular we will show, for any unipotent
type τ = (P,σ) of an inner form of G, existence and uniqueness (up to diagram
automorphisms)ofsuchSTMoftheHeckealgebraofτ totheIwahori–Heckealgebra
HIM(G)ofG.TheSTMsofthiskindturnouttocorrespondexactlytothearithmetic-
geometriccorrespondencesofLusztig[40,43].
Asanapplicationofthisclassification,usingthebasicpropertiesofSTMsdiscussed
in [54], we prove the conjecture [26, Conjecture 1.4] of Hiraga, Ichino and Ikeda
expressingtheformaldegreeofadiscreteseriesrepresentationintermsoftheadjoint
gammafactorofits(conjectural)localLanglandsparametersandanexplicitrational
constant factor, for all unipotent discrete series representations of inner forms of G
(whereweacceptLusztig’sparametersfortheunipotentdiscreteseriesrepresentations
Spectraltransfermorphisms 2145
asconjecturalLanglandsparameters).Itshouldbementionedthatitwasalreadyknown
fromReeder’swork[59,60](seealso[26])thatthisconjectureholdsfortheunipotent
discreteseriescharactersofsplitexceptionalgroupsofadjointtype,andforsomesmall
rankclassicalgroups.ItshouldbementionedthatthestabilityofLusztig’spacketsof
unipotentrepresentationswasshownbyMoeglinandWaldspurgerforoddorthogonal
groups[49]andbyMoeglinforunitarygroups[48].
ThroughoutthispaperweusethenormalizationofHaarmeasuresasin[17].Let
q = v2 denote the cardinality of the residue field of k. The formal degree of a
unipotentdiscreteseriesrepresentationthenfactorizesuniquelyasaproductofaq-
rationalnumber(whichwedefineasafractionofproductsofq-numbersoftheform
[n] := (vn−v−n) withn ≥2)andapositiverationalnumber.Ourproofofconjecture
q v−v−1
[26, Conjecture 1.4] involves the verification of the q-rational factors, which rests
on the existence of the Plancherel measure preserving correspondences for STMs
as discussed in [54], and the verification of the rational constants. The latter uses
theknowledge oftheserationalconstants from[60]forthecaseofequalparameter
exceptionalHeckealgebras,andcontinuityprinciplesdueto[11]and[56](also[13])
whichimplyroughlythatwecancomputetheserationalconstantfactorsintheformal
degrees of discrete series of non-simply laced affine Hecke algebras at any point in
theparameterspaceoftheaffineHeckealgebraonceweknowtheserationalconstants
in one regular point (in the sense of [56]) of the parameter space. In particular, for
(1)
classicalaffineHeckealgebrasoftypeC ;itwasshownin[11]thatatagenericpoint
n
intheparameterspace,therationalconstantsforallgenericfamiliesofdiscreteseries
charactersareequal.Theconstantsatspecialparametersfollowthenbyacontinuity
principleintheformaldegreedueto[56].
Analternativeapproachtotheconjecture[26,Conjecture1.4],restrictedtothecase
offormaldegreesofunipotentdiscreteseriesrepresentations,wasformulatedin[12].
A conjectural formula for the formal degrees of unipotent discrete series characters
is proposed in [12], which involves Lusztig’s non-abelian Fourier transform matrix
forfamiliesofunipotentrepresentations[36,44,45]andanotionofthe“ellipticfake
degree” of a unipotent discrete series character in the unramified minimal principal
seriesofG.Inthisapproachtheformulafortherationalconstantfactorsoftheformal
degrees appears in a very natural way from the basic properties of the non-abelian
Fouriertransform.
The notion of spectral transfer morphism is based on a certain heuristic idea on
the behavior of L-packets under ordinary parabolic induction (see 3.1.3 for a more
detailed discussion of this heuristic idea). The fact that this principle turns out to
hold for all unipotent representations is striking. Also striking is the fact that the
isomorphismclassoftheIwahori–HeckealgebraHIM(G)of G istheleastelement
in the poset of isomorphism classes of normalized affine Hecke algebras in the full
subcategoryofC (G)whoseobjectsaretheHeckealgebrasofunipotenttypes(P,σ)
es
oftheinnerformsofG,inthesenseof[54,Paragraph7.1.5].Moreover,ifHissuch
a unipotent affine Hecke algebra of an inner form of G, then the STM φ : H (cid:2)
HIM(G) (which exists by the above) is essentially unique, and such STMs exactly
matchLusztig’sarithmetic/geometriccorrespondences.Theproofofthesestatements
reduces,asexplanedinthispaper,tothesupercuspidalcase[19]incombinationwith
2146 E.Opdam
theaboveprinciplethatonecanparabolicallyinduceunipotentsupercuspidalSTMs
fromLevisubalgebrastoyieldnewSTMs.
It is quite clear that the definition of the notion of STM could be generalized to
Bernstein components [2,23–25] in greater generality than only for the unipotent
Bernstein components. It would be interesting to investigate the above mentioned
inductionprincipleingeneral.Inviewofourresults,thiscouldprovideacluehowL-
packetsarepartitionedbytheBernsteincenterbeyondLusztig’sunipotentL-packets
forsimplegroupsofadjointtype.
Inthefirstsectionofthispaperwewillreviewthetheoryofunipotentrepresenta-
tionsofG withanemphasisonitsharmonicanalyticaspects.Theresultshereareall
dueto[40,43,50,51]and[17].Thissectionservesanimportantpurposeofreviewing
therelevantfactsonunipotentrepresentationsforthispaperintheappropriatecontext
of harmonic analysis, and fixing notations. We kept the setup in this section more
generalthannecessaryfortheremainderofthepaper,sincethisdoesnotcomplicate
matterstoomuchandthismaybeusefulforlaterapplications.Inthesecondsection
wewilldescribethestructureoftheSTMsbetweenthenormalizedunipotentHecke
algebrasoftheinnerformsofG,anddiscusstheapplicationsofthisresult.1
2 Unipotentrepresentationsofquasisimplep-adicgroups
Thecategoryofunipotentrepresentationsofinnerformsofanunramifiedabsolutely
quasisimplep-adicgroupGisMoritaequivalenttothecategoryofrepresentationsofa
finitedirectsumoffinitelymanynormalizedaffineHeckealgebras(called“unipotent
Hecke algebras”) in such a way that the Morita equivalence respects the tempered
spectraandthenaturalPlancherelmeasuresonbothsides.
Therefore it is an interesting problem to classify all the STMs as defined in [54,
Definition5.1]betweentheseunipotentnormalizedaffineHeckealgebras.Itwillturn
outthatthistasktoclassifytheseSTMsessentiallyreducestothetaskoffindingall
STMs from the rank 0 unipotent affine Hecke algebras to the Iwahori–Matsumoto
Hecke algebra HIM(G(cid:3)) of the quasisplit G(cid:3) such that G is an inner form of G(cid:3). In
turnthisreducestosolvingequation[54,equation(55)]whered0denotestheformal
degree of a unipotent supercuspidal representation. The latter part of this task, the
classificationoftherank0unipotentSTMs,willbediscussedinasecondpaper(joint
withYongqiFeng[19]).Itshouldberemarkedthattheresultsofthepresentpaper,in
whichtheexistenceofcertainspectraltransfermorphismsisestablished,playsarole
intheproofoftheclassificationresultin[19].
2.1 Unramifiedreductivep-adicgroups
Letkbenon-archimedeanlocalfield.Fixaseparablealgebraicclosurekofk,andlet
K ⊂ k bethemaximalunramifiedextensionofk ink.LetK = O/P betheresidue
fieldof K,andlet pdenoteitscharacteristic.Let(cid:5) = Gal(k/k)denotetheabsolute
1 ItisapleasuretothankJosephBernstein,DanCiubotaru,MaartenSolleveld,DavidKazhdanandMark
Reederforusefuldiscussionsandcomments.
Spectraltransfermorphisms 2147
Galoisgroupofk,andletI =Gal(k/K)⊂(cid:5)betheinertiasubgroup.LetFrobbethe
geometricFrobeniuselementofGal(K/k)=(cid:5)/I (cid:5)Zˆ,i.e.,thetopologicalgenerator
whichinducestheinverseoftheautomorphismx → xq ofK.Hereq = pn denotes
thecardinalityoftheresiduefieldk:=KFrobofk.Wedenotebyvthepositivesquare
rootofq.
LetGbeaconnectedreductivealgebraicgroupdefinedoverk,andsplitover K.
We denote by G∨ be the neutral component of a Langlands dual group LG for G
(see[3]).TheconstructionofLG presupposesthechoiceofamaximaltorusSanda
BorelsubgroupBofGwhoseLevi-subgroupisS,andthechoiceofanépinglagefor
(G,B,S), inorder todefine asplittingofAut(G).Let X∗(Z(G∨))bethecharacter
groupofthecenter Z(G∨)ofG∨.Thenatural(cid:5)-actiononthisspacefactorsthrough
thequotientGal(K/k)sinceweareassumingthatGisK-split.Observethattheaction
ofFrobon X∗(Z(G∨))isindependentofthechoiceofasplittingofAut(G).
We will always denote the group G(K) of K-rational points of G by the corre-
spondingnon-boldfaceletter,i.e.,G =G(K).Kottwitz[31,Section7]hasdefineda
(cid:5)-equivariantfunctorialexactsequence
1→G →G −w→G X∗(Z(G∨))→1. (1)
1
In our situation there is a continuous equivariant action of the group (cid:5)/I on this
sequence.WedenotebyFtheactionofFrobonG andG,andbyθ theautomorphism
1 F
of X∗(Z(G∨))definedby F.Thissequencehasthepropertythattheassociatedlong
exactsequenceincontinuousnonabeliancohomologyyieldsanexactsequence
1→G F →G(k)→ X∗(Z(G∨))(cid:9)θF(cid:10) →1 (2)
1
andanisomorphism
H1(F,G)−∼→ X∗(Z(G∨))(cid:9)θ (cid:10). (3)
F
Nowassumethat G issemisimple.Inthissituationtheabovesequencessimplifyas
follows.LetSbeamaximalK-splittorusofG,andletX := X∗(S)beitscocharacter
lattice. Let Q := Xsc = X∗(Ssc) be the cocharacter lattice of the inverse image of
SinthesimplyconnectedcoverG → GofG(hence Q ⊂ X isthecorootlattice
sc
of (G,S); we warn the reader that we call the roots of G∨ “roots” and the roots of
(G,S) “coroots”. We apologize for this admittedly awkward convention). Let (cid:7) be
the finite abelian group (cid:7) = X/Q. Then we may canonically identify X∗(Z(G∨))
with(cid:7).Hence(2)becomes
1→G F →G(k)→(cid:7)θF →1 (4)
1
(see[29,30])and(3)becomes
∼
H1(F,G)−→(cid:7)/(1−θ )(cid:7). (5)
F
WeremarkthatG ⊂G ⊂G,andthatitcanbeshownthatG =G ifandonly
der 1 der 1
if pdoesnotdividetheorder|(cid:7)|of(cid:7).WewillfromnowonalwaysassumethatGis
absolutelyquasisimpleand K-split,unlessotherwisestated.
2148 E.Opdam
2.1.1 Inner k-rational structures of G The k-rational structures of G which are
inner forms of G are parameterized by H1(k,G ). By Steinberg’s Vanishing The-
ad
orem it follows that all inner k-forms of G are K-split and that H1(k,G ) =
ad
H1(Gal(K/k),G )(see[62,Section5.8]).Wewillfromnowonreservethenotation
ad
G forak-quasisplitrationalstructureinthisinnerclass.Welet F betheautomor-
phismofG (orG)correspondingtotheactionofFrob,andθ =θ .Wethendenote
ad F
thenonabeliancohomology H1(Gal(K/k),G )by H1(F,G ).
ad ad
ForGsemisimpleandnotnecessarilyofadjointtype,Voganconjecturedarefined
Langlandsparameterizationoftheirreducibletemperedunipotentrepresentationsof
pureinnerformsofG [68].
PureinnerformofG correspondbydefinitiontococyclesz ∈ Z1(F,G)[17,68].
Suchacocycleisdeterminedbytheimageu :=z(Frob)∈G.Thecorrespondinginner
k-formofGisdefinedbythefunctorialimagezad ∈ Z1(F,G )ofz.This“pure”inner
ad
formisdefinedbythetwistedFrobeniusaction F onG givenby F = Ad(u)◦ F,
u u
andisdenotedby Gu.Thecocycle z determinesaclassin[z] ∈ H1(F,G).Wesay
thattwopureinnerformsz andz ofGareequivalentiff[z ]=[z ].Thek-rational
1 2 1 2
isomorphismclassoftheinnerform Gu isdeterminedbytheimage[zad]of[z]via
thenaturalmap H1(F,G)→ H1(F,G ).Thereaderbewarnedhowever,thatview
ad
of (5) this map is neither surjective in general (this is obvious, G = SL provides
2
an example) nor injective (however, if G is k-split and semisimple, then the map is
injective).Inotherwords,notallk-rationalequivalenceclassesofinnerformsof G
canberepresentedbyapureinnerform,andifG isnotk-splitandsemisimple,then
aninnerformofG mayberepresentedbyseveralinequivalentpureinnerforms.
Itisinprinciplepossibletocomputewithourmethodstheformaldegreesofthe
elements of L-packets according to this refined form of the Langlands parameteri-
zation,oreventocheckexamplesoftheconjecture[26,Conjecture1.4]beyondthe
caseofpureinnerforms.Forlaterreferencewewillformulatemattersinthismore
generalsetupwherepossible,eventhoughwewillinpresentpaperlimitourselvesin
theapplicationstothecasewhereG isofadjointtype.
2.1.2 TheaffineWeylgroup Thereexistsamaximal K-splittorusSdefined overk
and maximally k-split [5, 5.1.10]. We fix such a maximal torus S of G, and denote
byS itsinverseimageforthecoveringG →G.RecallthatG isk-quasisplit,and
sc sc
that F defines an automorphism on the lattices X and X = Q denoted by θ. The
sc
extendedaffineWeylgroupW of(G,S)isdefinedby
W = NG(S)/SO. (6)
ThegroupW actsfaithfullyontheapartmentAasanextendedaffineCoxetergroup.
WedenotebySO =O×⊗XthemaximalboundedsubgroupofS.ThenX = S/SO,
andwedefinetheassociated F-stableapartmentA = A(G,S)ofthebuildingofG
by A(G,S) = R ⊗ X. As explained in [17, Corollary 2.4.3], [6, Section 3] the
isomorphism(5)canbemadeexplicitbyacanonicalbijection
∼
(cid:7)/(1−θ)(cid:7)−→ H1(F,G) (7)
Spectraltransfermorphisms 2149
sending [ω] ∈ (cid:7)/(1−θ)(cid:7) to the cohomology class of the cocycle z which maps
u
Frobto Fu,whereuSO = x ∈ X andx isarepresentativeofω∈ X/Q.
Let C be an F-stable alcove in A (such alcoves exist, see [67]). Let 1 → N →
G →G→1bethesimplyconnectedcoverofG,andletS betheinverseimage
sc sc
ofS.
Proposition2.1 TheimageofG →GisequaltothederivedgroupG ofG,and
sc der
wehaveG/G −∼→ H1(K,N)= K×⊗(cid:7).
der
Proof Indeed, it is clear that the image is contained in G because G is its own
der sc
derivedgroup[65].Theotherinclusionfollowsbyapplyingthelongexactsequence
in nonabelian cohomology to the central isogeny G → G and again appealing to
sc
Steinberg’sVanishingTheorem.ItfollowsthatthequotientofG bytheimageofG
sc
is the abelian group H1(K,N), whence the result. On the other hand, we have the
obviousexactsequence
1→Hom((cid:7)∗,K×)→ S → S → K×⊗(cid:7)→1 (8)
sc
which we can compare to the long exact sequence in cohomology (with respect to
I) associated to the canonical exact sequence of diagonalizable groups 1 → N →
S →S→1. (cid:15)(cid:16)
sc
WedenotebyWa the F-stablenormalsubgroupofW generatedbythereflectionsin
C
thewallsofC.Thisnormalsubgroup isindependent ofthechoice ofC andcanbe
∼
canonicallyidentifiedwithNGder(S)/SO∩Gder −→ Wa ⊂ W,theaffineWeylgroup
of(G ,S ).
sc sc
Returning to Kottwitz’s homomorphism we obtain the following result (compare
with[5,5.2.11]).
Corollary2.2 WehaveG1 =(cid:9)SO,Gder(cid:10).
Proof Let B be the Iwahori subgroup of G associated with C [5, 5.2.6]. By [57,
Appendix,Proposition3]wehaveB=Fix(C)∩G1.InparticularwehaveSO ⊂G1,
sothatwehaveGder ⊂G(cid:3)1 :=(cid:9)SO,Gder(cid:10)⊂G1Henceby(4),theequalityG(cid:3)1 =G1
isequivalenttoshowingthat G/G(cid:3) = G/G = (cid:7).Bythepreviouspropositionwe
1 1
have G/Gder = K×⊗(cid:7).Since SO/SO ∩Gder = O×⊗(cid:7)theresultfollowsfrom
K×/O× (cid:5)Z. (cid:15)(cid:16)
Let(cid:7) bethesubgroupofWwhichstabilizesC.Thissubgroupmaybeidentifiedwith
C
asubgroupofthegroupofspecialautomorphisms(inthesenseof[40,paragraph1.11])
oftheaffinediagramassociatedwiththechoiceofC.Wehaveasemidirectproduct
∼
decompositionW = Wa(cid:2)(cid:7) ,andthusacanonicalisomorphism(cid:7) −→(cid:7)forany
C C
choiceofC.
∼ ∼
Corollary2.3 Wehave NG1(S)/SO −→ Wa, NG1(B)=Band NG(B)/B−→(cid:7).
Proof ByCorollary2.2itfollowsthat NG1(S)= NGder(S).SO.Thisimpliesthefirst
assertion,sinceWaistheaffineWeylgroupof(G ,S )andG isthehomomorphic
sc sc der
2150 E.Opdam
imageofG .SinceanIwahori-subgroupofG isself-normalizing,wehavesimilarly
sc sc
NG1(B) = NGder(B).SO = (B∩Gder).SO = B, proving the second assertion. For
thethirdassertion,observethat(cid:7)C = (NG(B)∩ NG(S))/SO.Itiswellknownthat
B∩NG(S)= SO,hence(cid:7)C mapsinjectivelyintoNG(B)/B.Bythesecondassertion
thisgroupmapsinjectivelyintoG/G =(cid:7).Since(cid:7)(cid:5)(cid:7) arefinitethetwoinjective
1 C
homomorphismsareinfactisomorphisms.
Since G is unramified there exist hyperspecial points in the apartment A [67]. A
choiceofahyperspecialpointa ∈ A,inducesasemidirectproductdecomposition
0
W = W (cid:3) X,whereW denotestheisotropysubgroupofa inW.Thek-structure
0 0 0
of G definedby F isquasisplit,whichimpliesthatthereexistsahyperspecialpoint
a ∈ A(G,S) which is F-fixed. In this case we denote by θ the automorphism of
0
W (and of A) induced by F. We fix a , an F-fixed hyperspecial point, and an F-
0
stablealcoveChavinga initsclosure.Observethatthesubgroup(cid:7) dependsonthe
0 C
choiceofC,notofthehyperspecialpointa .Recallwehaveacanonicalisomorphism
0
∼
(cid:7) −→ (cid:7) = X/Q,whichwewilloftenusetoidentifythesetwogroups.Observe
C
thatθ stabilizesthesubgroupsWa,(cid:7) , X andW ofW.
C 0
2.2 Unipotentrepresentations
2.2.1 Parahoric subgroups Recall the explicit representation of pure inner forms
Gu as discussed in (7). Fix a representative u = ω˙ ∈ N (S) with ω ∈ (cid:7) ⊂ W.
G C
Then F acts on the apartment A(G,S) by means of the finite order automorphism
u
ωθ.Since F stabilizesC theIwahorisubgroupBis F stable.Recallthatthegroup
u u
(cid:7) can be canonically identified with the group N (B)/B. Since (cid:7) is abelian it
C G
is clear that the subgroup (cid:7)Fu = (cid:7)ωθ of F -invariant elements is independent of
C C u
ω∈(cid:7) .
C
Following [57, Appendix] we may define a “standard parahoric subgroup of G”
as a subgroup of the form Fix(F ) ∩ G where F ⊂ C denotes a facet of C.
P 1 P
By [57, Appendix, Proposition 3] this definition coincides with the definition in
[5]. In particular, a standard parahoric subgroup of G is a connected pro-algebraic
group. A parahoric subgroup of G is a subgroup conjugate to a standard parahoric
subgroup.
Itiswellknown(by“Lang’stheoremforconnectedproalgebraicgroups”,see[40,
1.3]) that any Fu-stable parahoric subgroup of G is GFu-conjugate to a “standard”
F -stableparahoricsubgroup,i.e.,an F -stableparahoricsubgroupcontainingB.It
u u
followsthattheGFu-conjugacyclassesof Fu-stableparahoricsubgroupsareinone-
to-onecorrespondencewiththesetof(cid:7)θ-orbitsofωθ-stablefacetsintheclosureof
C. Similarly, a parahoric subgroup P or a double coset of a parahoric subgroup is
Fu-stableiffitcontainspointsof GFu.LetPbean Fu-stableparahoricsubgroupof
G.WecallPFu aparahoricsubgroupofGFu.
WerecordtwoimportantpropertiesofF -stableparahoricsubgroupswhichfollow
u
easilyfromCorollary2.3.Firstofall,parahoricsubgroupsareself-normalizinginG ,
1
i.e.,
(N P)Fu ∩G =PFu. (9)
G 1
Spectraltransfermorphisms 2151
Secondly,foranF -stablestandardparahoricsubgroupPcorrespondingtoaωθ-stable
u
facetC ofC,wehave
P
(N P)Fu/PFu =(cid:7)P,θ (10)
G
where(cid:7)P ⊂(cid:7) isthesubgroupstabilizingC ,and(cid:7)P,θ ⊂(cid:7)Pitsfixedpointgroup
C P
fortheactionofθ (or F =ωθ,whichamountstothesamesince(cid:7) isabelian).We
u C
defineanexactsequence
1→(cid:7)P,θ →(cid:7)P,θ →(cid:7)P,θ →1 (11)
1 2
where(cid:7)P,θ isthesubgroupofelementswhichfixthesetof F -orbitsofverticesof
1 u
C notinC pointwise.
P
2.2.2 NormalizationofHaarmeasures LetG, F,and F beasinthepreviouspara-
u
graph.ThenGFu isalocallycompactgroup.Forany Fu-stableparahoricsubgroupP
ofG wedenotebyPFu thereductivequotientofPFu.Thisisthegroupofk-pointsof
aconnectedreductivegroupoverk.Inparticularthisisafinitegroup.
Following[17,Section5.1]wenormalizetheHaarmeasureofGFu uniquely,such
thatforall F -stableparahoricsubgroupsPofG onehas
u
Vol(PFu)=v−a|PFu| (12)
wherea ∈ZisequaltothedimensionofPoverK.Itiswellknownthattheright-hand
sideisaproductofpowersofvandcyclotomicpolynomialsinv.
2.2.3 Theanisotropiccase ItisusefultodiscussthecasewhereGFu isanisotropic
explicitly. It is well known that an anisotropic absolutely simple group GFu is iso-
morphictoPGL (D) := D×/k×,whereDisanunramifiedcentraldivisionalgebra
1
over k of degree m + 1, rank (m + 1)2 (see for instance [16]). We choose a uni-
formizer π of k. D contains an unramified extensionl of degree m +1 over k, and
wemaychooseauniformizer(cid:10)ofDwhichnormalizesl,suchthatconjugation by
(cid:10) restricted to l yields a generator for Gal(l : k), and such that (cid:10)m+1 = π. The
group P := G is the only F -stable parahoric subgoup in this situation, and obvi-
1 u
ouslyG = NP.By(4)wehave(cid:7) := GFu/GFu ≈ (cid:9)(cid:10)(cid:10) ≈ Gal(l : k),acyclicgroup
1 (cid:9)π(cid:10)
oforderm+1.GFu containsamaximalprounipotentsubgroupG+ (denotedby V1
in [16]) and we have GFu = C.G+, where C is generated by the anisotropic torus
TFu :=l×/k×and(cid:10).WeseethatthereductivequotientP/(P∩G+)isananisotropic
torus T of rank m over k, and that TFu can be identified with the group of roots of
unityoforderprimeto pinlmodulothesubgroupofthoserootsofunityink.Hence
Vol(GFu)=v−m|(cid:7)||TFu|=(m+1)[m+1]q (with[m+1]q theq-integerassociated
tom+1∈N[seeDefinition2.6)].
2.2.4 UnipotentrepresentationsandaffineHeckealgebras Let G beaquasisimple
linearalgebraicgroup,definedandquasisplitoverkandK-splitasabove.Recallthat
theautomorphisminducedbytheFrobenius F onthebuildingofG wasdenotedby