BRUHAT TITS THEORY FROM BERKOVICH'S POINT OF VIEW I REALIZATIONS AND COMPACTIFICATIONS OF BUILDINGS

Description

• exposé - matière potentielle : xxvi

BRUHAT-TITS THEORY FROM BERKOVICH'S POINT OF VIEW. I — REALIZATIONS AND COMPACTIFICATIONS OF BUILDINGS BERTRAND RÉMY, AMAURY THUILLIER AND ANNETTE WERNER Abstract: We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic ge- ometry over complete non-Archimedean fields. For every reductive group G over a suitable non-Archimedean field k we define a map from the Bruhat-Tits buildingB(G,k) to the Berkovich analytic space Gan asscociated with G. Composing this map with the projection of Gan to its flag varieties, we define a family of compactifi- cations ofB(G,k). This generalizes results by Berkovich in the case of split groups. Moreover, we show that the boundary strata of the compactified buildings are precisely the Bruhat-Tits buildings associated with a certain class of parabolics. We also investigate the stabilizers of boundary points and prove a mixed Bruhat decomposition theorem for them. Keywords: algebraic group, local field, Berkovich geometry, Bruhat-Tits building, compactification. Résumé: Nous étudions les immeubles de Bruhat-Tits et leurs compactifications au moyen de la géométrie analytique sur les corps complets non archimédiens au sens de Berkovich. Pour tout groupe réductif G sur un corps non archimédien convenable k, nous définissons une application de l'immeuble de Bruhat-Tits B(G,k) vers l'espace analytique de Berkovich Gan associé à G.

• strates au bord des immeubles compactifiés


• group


• among all parabolic


• géométrie analytique sur les corps complets


• over


• stabilisateurs des points au bord


• berkovich


• espace analytique de berkovich gan


• archimedean extension


• bruhat- tits building

Sujets

• Berkovich

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 23
• Langue: English

Extrait

B
RUHAT
-T
ITSTHEORYFROM
B
ERKOVICH
’
SPOINTOFVIEW
.
I—R
EALIZATIONSANDCOMPACTIFICATIONSOFBUILDINGS
B
ERTRAND
R
ÉMY
,A
MAURY
T
HUILLIERAND
A
NNETTE
W
ERNER

Abstract:
WeinvestigateBruhat-TitsbuildingsandtheircompactificationsbymeansofBerkovichanalyticge-
ometryovercompletenon-Archimedeanfields.ForeveryreductivegroupGoverasuitablenon-Archimedean
field
k
wedefineamapfromtheBruhat-Titsbuilding
B
(
G
,
k
)
totheBerkovichanalyticspaceG
an
asscociated
withG.ComposingthismapwiththeprojectionofG
an
toitsflagvarieties,wedefineafamilyofcompactifi-
cationsof
B
(
G
,
k
)
.ThisgeneralizesresultsbyBerkovichinthecaseofsplitgroups.
Moreover,weshowthattheboundarystrataofthecompactifiedbuildingsarepreciselytheBruhat-Tits
buildingsassociatedwithacertainclassofparabolics.Wealsoinvestigatethestabilizersofboundarypoints
andproveamixedBruhatdecompositiontheoremforthem.
Keywords:
algebraicgroup,localfield,Berkovichgeometry,Bruhat-Titsbuilding,compactification.
Résumé:
NousétudionslesimmeublesdeBruhat-Titsetleurscompactificationsaumoyendelagéométrie
analytiquesurlescorpscompletsnonarchimédiensausensdeBerkovich.PourtoutgrouperéductifGsurun
corpsnonarchimédienconvenable
k
,nousdéfinissonsuneapplicationdel’immeubledeBruhat-Tits
B
(
G
,
k
)
versl’espaceanalytiquedeBerkovichG
an
associéà
G
.Encomposantcetteapplicationaveclaprojection
surlesvariétésdedrapeaux,nousobtenonsunefamilledecompactificationsde
B
(
G
,
k
)
.Cecigénéralisedes
résultatsdeBerkovichsurlecasdéployé.
Enoutre,nousdémontronsquelesstratesauborddesimmeublescompactifiéssontprécisémentlesim-
meublesdeBruhat-Titsassociésàcertainesclassesdesous-groupesparaboliques.Nousétudionségalement
lesstabilisateursdespointsaubordetdémontronsunthéorèmededécompositiondeBruhatmixtepources
groupes.
Mots-clés:
groupealgébrique,corpslocal,géométriedeBerkovich,immeubledeBruhat-Tits,compactifica-
.noitAMSclassification(2000):
20E42,51E24,14L15,14G22.

2

I
NTRODUCTION

1.
Inthemid60ies,F.BruhatandJ.Titsinitiatedatheorywhichledtoadeepunderstandingof
reductivealgebraicgroupsovervaluedfields[
BT72
],[
BT84
].Themaintool(andaconcisewayto
expresstheachievements)ofthislong-standingworkisthenotionofa
building
.Generallyspeaking,
abuildingisagluingof(poly)simplicialsubcomplexes,allisomorphictoagiventilingnaturallyacted
uponbyaCoxetergroup[
AB08
].Thecopiesofthistilinginthebuildingarecalled
apartments
and
mustsatisfy,bydefinition,strongincidencepropertieswhichmakethewholespaceverysymmetric.
ThebuildingsconsideredbyF.BruhatandJ.TitsareEuclideanones,meaningthattheirapartments
areEuclideantilings(infact,tocoverthecaseofnon-discretelyvaluedfields,onehastoreplace
EuclideantilingsbyaffinespacesacteduponbyaEuclideanreflectiongroupwithanon-discrete,
finiteindex,translationsubgroup[
Tit86
]).AEuclideanbuildingcarriesanaturalnon-positively
curvedmetric,whichallowsonetoclassifyinageometricwaymaximalboundedsubgroupsinthe
rationalpointsofagivennon-Archimedeansemisimplealgebraicgroup.Thisisonlyaninstanceof
thestronganalogybetweentheRiemanniansymmetricspacesassociatedwithsemisimplerealLie
groupsandBruhat-Titsbuildings[
Tit75
].Thisanalogyisourguidelinehere.
Indeed,inthispaperweinvestigateBruhat-Titsbuildingsandtheircompactificationbymeansof
analyticgeometryovernon-Archimedeanvaluedfields,asdevelopedbyV.Berkovich—see[
Ber98
]
forasurvey.Compactificationsofsymmetricspacesisnowaveryclassicaltopic,withwell-known
applicationstogrouptheory(e.g.,groupcohomology[
BS73
])andtonumbertheory(viathestudy
ofsomerelevantmodulispacesmodeledonHermitiansymmetricspaces[
Del71
]).Fordeeper
motivationandabroaderscopeoncompactificationsofsymmetricspaces,werefertotherecent
book[
BJ06
],inwhichthecaseoflocallysymmetricvarietiesisalsocovered.Oneofourmainre-
sultsistoconstructforeachsemisimplegroupGoverasuitablenon-Archimedeanvaluedfield
k
,a
familyofcompactificationsoftheBruhat-Titsbuilding
B
(
G
,
k
)
ofGover
k
.Thisfamilyisfinite,
actuallyindexedbytheconjugacyclassesofproperparabolic
k
-subgroupsinG.Suchafamilyisof
coursetheanalogueofthefamilyofSatake[
Sat60
]orFurstenberg[
Fur63
]compactificationsofa
givenRiemanniannon-compactsymmetricspace—see[
GJT98
]forageneralexposition.
Infact,thethirdauthorhadpreviouslyassociated,witheachBruhat-Titsbuilding,afamilyof
compactificationsalsoindexedbytheconjugacyclassesofproperparabolic
k
-subgroups[
Wer07
]
andgeneralizingthe"maximal"versionconstructedbeforebyE.Landvogt[
Lan96
].TheBruhat-Tits
building
B
(
G
,
k
)
ofGover
k
isdefinedasthequotientforasuitableequivalencerelation,say
∼
,
oftheproductoftherationalpointsG
(
k
)
byanaturalmodel,say
Λ
,oftheapartment;wewillrefer
tothiskindofconstructionasa
gluingprocedure
.Thefamilyofcompactificationsof[
Wer07
]was
obtainedbysuitablycompactifying
Λ
toobtainacompactspace
Λ
andextending
∼
toanequivalence
relationonG
(
k
)
×
Λ
.Asexpected,foragivengroupGweeventuallyidentifythelatterfamilyof
compactificationswiththeoneweconstructhere,see[
RTW
].
OurcompactificationproceduremakesuseofembeddingsofBruhat-Titsbuildingsintheana-
lyticversionsofsomewell-knownhomogeneousvarieties(inthecontextofalgebraictransformation
groups),namelyflagmanifolds.TheideagoesbacktoV.BerkovichinthecasewhenGsplitsoverits
groundfield
k
[
Ber90
,§5].Oneaestheticaladvantageoftheembeddingprocedureisthatitissimilar
tothehistoricalwaystocompactifysymmetricspaces,e.g.,byseeingthemastopologicalsubspaces
ofsomeprojectivespacesofHermitianmatricesorinsidespacesofprobabilitymeasuresonaflag
manifold.Moreusefully(aswehope),thefactthatwespecificallyembedbuildingsintocompact
spacesfromBerkovich’stheorymaymakethesecompactificationsusefulforabetterunderstanding
ofnon-Archimedeanspacesrelevanttonumbertheory(inthecaseofHermitiansymmetricspaces).
Forinstance,thebuildingofGL
n
overavaluedfield
k
isthe"combinatorialskeleton"oftheDrinfel’d
half-space
Ω
n
−
1
over
k
[
BC91
],anditwouldbeinterestingtoknowwhethertheprecisecombinato-
rialdescriptionweobtainforourcompactificationsmightbeusefultodescribeothermodulispaces

3

forsuitablechoicesofgroupsandparabolicsubgroups.Oneotherquestionaboutthesecompactifi-
cationswasraisedbyV.Berkovichhimself[
Ber90
,5.5.2]anddealswiththepotentialgeneralization
ofDrinfel’dhalf-spacestonon-Archimedeansemisimplealgebraicgroupsofarbitrarytype.
2.
LetusnowturntothedefinitionoftheembeddingmapsthatallowustocompactifyBruhat-
Titsbuildings.LetGbea
k
-isotropicsemisimplealgebraicgroupdefinedoverthenon-Archimedean
valuedfield
k
andlet
B
(
G
,
k
)
denotetheEuclideanbuildingprovidedbyBruhat-Titstheory[
Tit79
].
Weprovethefollowingstatement(see2.4andProp.3.34):
assumethatthevaluedfieldkisalocal
field(i.e.,islocallycompact)and(forsimplicity)that
G
isalmostk-simple;thenforanyconjugacy
classofproperparabolick-subgroup,sayt,thereexistsacontinuous,
G
(
k
)
-equivariantmap
ϑ
t
:
B
(
G
,
k
)
→
Par
t
(
G
)
an
whichisahomeomorphismontoitsimage.
HerePar
t
(
G
)
denotestheconnected
componentoftype
t
inthepropervarietyPar
(
G
)
ofallparabolicsubgroupsinG(onwhichGacts
byconjugation)[
SGA3
,ExposéXXVI,Sect.3].Thesuperscript
an
meansthatwepassfromthe
k
-varietyPar
t
(
G
)
totheBerkovich
k
-analyticspaceassociatedwithit[
Ber90
,3.4.1-2];thespace
Par
(
G
)
an
iscompactsincePar
(
G
)
isprojective.Wedenoteby
B
t
(
G
,
k
)
theclosureoftheimageof
ϑ
t
andcallitthe
Berkovichcompactification
oftype
t
oftheBruhat-Titsbuilding
B
(
G
,
k
)
.
Roughlyspeaking,thedefinitionofthemaps
ϑ
t
takesupthefirsthalfofthispaper,soletusprovide
somefurtherinformationaboutit.Asapreliminary,werecallsomebasicbuthelpfulanalogies
between(scheme-theoretic)algebraicgeometryand
k
-analyticgeometry(inthesenseofBerkovich).
Firstly,theelementaryblocksof
k
-analytics