Representations of reductive normal algebraic monoids

0.08 MB·

by  Stephen Doty

Representations of reductive normal algebraic
monoids
StephenDoty
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7 DedicatedtoLexRennerandMohanPutcha

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T AbstractTherationalrepresentationtheoryofareductivenormalalgebraicmonoid
R (withone-dimensionalcenter)formsahighestweightcategory,inthesenseofCline,
. Parshall, and Scott. This is a fundamental fact about the representation theory of
h
reductivenormalalgebraicmonoids.We surveyhow this resultwas obtained,and
t
a treatsomenaturalexamplescomingfromclassicalgroups.
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1
Introduction
v
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LetM beanaffinealgebraicmonoidoveranalgebraicallyclosedfieldK.See[10,
4
1 13, 12] for generalsurveysand backgroundon algebraic monoids.Assuming that
. Misreductive(itsgroupGofunitsisareductivegroup)whatcanbesaidaboutthe
1
representationtheoryofMoverK?
0
4 Recallthatanyaffinealgebraicgroupissmoothandhencenormal(asavariety).
1 The normality of the algebraic group plays a significant role in its representation
:
v theory,forinstanceintheproofofChevalley’stheoremclassifyingtheirreducible
i representations.Thusitseemsreasonableintryingtoextend(rational)representa-
X
tiontheoryfromreductivegroupstoreductivemonoidstolookfirstatthecasewhen
r
a themonoidM isnormal.Furthermore,evenincaseswhereagivenreductivealge-
braicmonoidisnotnormal,onemayalwayspasstoitsnormalization,whichshould
becloselyrelatedtotheoriginalobject.
Renner[11]hasobtainedaclassificationtheoremforreductivenormalalgebraic
monoids under the additional assumptions that the center Z(M) is 1-dimensional
andthatMhasazeroelement.Renner’sclassificationtheoremdependsonanalge-
LoyolaUniversityChicago,DepartmentofMathematicsandStatistics,ChicagoIL60660USA,
e-mail:[email protected]
1
2 StephenDoty
braic monoidversionof Chevalley’sbig cell, whichholdsforany reductiveaffine
algebraicmonoid(withnoassumptionsaboutitscenterorazero).Asacorollaryof
itsconstruction,Rennerderivesaveryusefulextensionprinciple[11,(4.5)]which
isakeyingredientintheanalysis.
1 Reductive normal algebraicmonoids
Let M be a linear algebraicmonoid overan algebraicallyclosed field K. In other
words,M is a monoid(with unitelement1∈M) whichisalso anaffine algebraic
varietyoverK,suchthatthemultiplicationmapm :M×M→M isamorphismof
varieties.WeassumethatMisirreducibleasavariety.HencetheunitgroupG=M×
(thesubgroupofinvertibleelementsofM)isaconnectedlinearalgebraicgroupover
K andGisZariskidenseinM.
1.1. Associated with M is its affine coordinate algebra K[M], the ring of regular
functionsonM.ThereexistK-algebrahomomorphisms
D :K[M]→K[M]⊗ K[M], e :K[M]→K
K
called comultiplication and counit, respectively. For a given f ∈K[M], we have
e (f)= f(1);furthermore,ifD (f)=(cid:229) r f ⊗f′then f(m m )=(cid:229) r f(m )f′(m ),
i=1 i i 1 2 i=1 i 1 i 2
forallm ,m ∈M.ThemapsD ,e makeK[M]intoabialgebraoverK.Thismeans
1 2
thattheysatisfythebialgebraaxioms:
(1) (id⊗D )◦D =(D ⊗id)◦D ,
(2) (e ⊗id)◦D =id=(id ⊗e )◦D
wherej ⊗j ′denotesthemapa⊗a′7→j (a)j ′(a′).
We note that the commutative bialgebra (K[M],D ,e ) determines M, as the set
Hom (K[M],K) of K-algebra homomorphismsfrom K[M] into K. The multi-
K−alg
plication on this set is defined by j ·j ′ =(j ⊗ j ′)◦D and the identity element
isjustthecounite .OneeasilyverifiesthatthisreconstructsM fromitscoordinate
bialgebraK[M].
Moregenerally,givenanycommutativebialgebra(A,D ,e )overK,onedefineson
thesetM(A)=Hom (A,K)analgebraicmonoidstructurewithmultiplication
K−alg
m (j ,j ′)=j ·j ′=(j ⊗j ′)◦D .Thisgivesafunctor
{commutativebialgebrasoverK}→{algebraicmonoidsoverK}
which is quasi-inverse to the functor M 7→ K[M]. Thus the two categories are
antiequivalent.
1.2. SinceGisdenseinM,therestrictionmapK[M]→K[G](givenby f 7→ f )is
|G
injective,sowemayidentifyK[M]withasubbialgebraoftheHopfalgebraK[G]of
regularfunctionsonG.
Representationsofreductivenormalalgebraicmonoids 3
1.3. AssumethatM isreductive;i.e.,itsunitgroupG=M× isreductiveasanal-
gebraic group. Fix a maximal torus T in G. (Up to conjugationT is unique.) Let
X(T)=Hom(T,K×)bethecharactergroupofT;thisistheabeliangroupofmor-
phismsfromT intothemultiplicativegroupK×ofK.LetX∨(T)=Hom(K×,T)be
theabeliangroupofcocharactersinto T. LetR⊂X(T)betherootsystem forthe
pair(G,T)andR∨⊂X∨(T)thesystemofcoroots.Accordingtotheclassification
of reductivealgebraicgroups,the reductivegroupG is uniquelydeterminedup to
isomorphismbyitsrootdatum(X(T),R,X∨(T),R∨).
1.4. We now add the assumptionthat M is normalas a variety.LetD=T be the
ZariskiclosureofT in M. ThenT ⊂D is anaffine torusembedding.LetX(D)=
Hom(D,K) be the monoid of algebraic monoid homomorphisms from D into K.
The restriction c of any c ∈X(D) is an element of X(T), so restriction defines
|T
a homomorphismX(D)→X(T).Since T isdenseinD,thismapisinjective,and
thuswemayidentifyX(D)withasubmonoidofX(T).Rennerhasshownthatthe
additionaldatumX(D)isallthatisneededtodetermineMuptoisomorphism,under
theadditionalhypotheses(probablyunnecessary)thatthecenter
Z(M)={z∈M:zm=mz, forallm∈M}
is 1-dimensionaland that M has a zero element. (One can always add a zero for-
mally,sothelastrequirementisinsubstantial.)
Itturnsoutthattheset X(D)also determinesthe rationalrepresentationtheory
ofthereductivenormalalgebraicmonoidM,inasensemadepreciseinSection3.
1.5. Notethatitiseasytoconstructreductivealgebraicmonoids.Startwitharatio-
nal representationr :G→End (V) of a reductivegroupG in some vectorspace
K
V with dim V =n<¥ . Theimager (G)isa reductiveaffinealgebraicsubgroup
K
ofEnd (V)≃M (K),themonoidofalln×nmatricesunderordinarymatrixmul-
K n
tiplication. Desiring our monoid to have a center of at least dimension 1, we in-
cludethescalarsK×asscalarmatrices,definingG tobethesubgroupofEnd (V)
0 K
generated by r (G) and K×. Now we set M = G , the Zariski closure of G in
0 0
End (V)≃M (K).Thisisareductivealgebraicmonoid.
K n
Forexample,ifG=SL (K)andV isitsnaturalrepresentationthenG ≃GL (K)
n 0 n
and M=M (K). (In general,to obtaina monoidM closely related to the starting
n
groupG, oneshouldpickV to bea faithfulrepresentation.)Thereisnoguarantee
thatthisprocedurewillalwaysproduceanormalreductivemonoid,butifnotthen
onecanalwayspasstoitsnormalization.
2 Examples: symplecticandorthogonal monoids
The paper [4] considered some more substantial examples of reductive algebraic
monoids coming from other classical groups. LetV =Kn with its standard basis
{e ,...,e }.Puti′=n+1−iforanyi=1,...,n.
1 n
4 StephenDoty
2.1. The orthogonal monoid. Assume the characteristic of K is not 2. Define a
symmetricnondegeneratebilinearformh, ionV byputting
(a) hei,eji=d i,j′ forany1≤i,j≤n.
Hered isKronecker’sdeltafunction.LetJbethematrixofh, iwithrespecttothe
basis{e ,...,e }.ThentheorthogonalgroupO(V)isthegroupoflinearoperators
1 n
f ∈End (V)preservingtheform:
K
(b) O(V)={f ∈End (V):hf(v),f(v′)i=hv,v′i, allv,v′∈V}.
K
LetAbethematrixof f withrespecttothebasis{e ,...,e }.Thenwemayidentify
1 n
O(V)withthematrixgroup
T
(c) O (K)={A∈M (K):A JA=J}.
n n
Thisis containedin thelargergroupGO (K), thegroupoforthogonalsimilitudes
n
(seee.g.,[9])definedby
(d) GO (K)={A∈M (K):ATJA=cJ, somec∈K×}.
n n
NotethatGO (K)isgeneratedbyO (K)andK×.Wedefinetheorthogonalmonoid
n n
OM (K)tobe
n
(e) OM (K)=GO (K),
n n
theZariskiclosureinM (K).Thesemonoids(fornodd)werestudiedbyGrigor’ev
n
[8].In[4]thefollowingresultwasproved.
2.2Proposition. TheorthogonalmonoidOM (K)isthesetofallA∈M (K)such
n n
T T
thatA JA=cJ=AJA ,forsomec∈K.
Note that the scalar c∈K in the above is allowed to be zero, and the “extra”
T T
condition cJ =AJA is necessary. If c6=0 then it is easy to see that A JA=cJ
T
isequivalenttocJ=AJA ,butwhenc=0thisequivalencefails.Theequivalence
meansthatwecouldjustaswellhavedefinedGO (K)by
n
GO (K)={A∈M (K):ATJA=cJ=AJAT, somec∈K×}
n n
whichisperhapsmoresuggestiveforthedescriptionofOM (K)givenabove.
n
2.3. Thesymplecticmonoid.Assumethatn=dim V iseven,sayn=2m.Define
K
anantisymmetricnondegeneratebilinearformh, ionV byputting
(a) hei,eji=eid i,j′ forany1≤i,j≤n.
where e is 1 if i≤m and −1 otherwise. Let J be the matrix of h , i with respect
i
to the basis {e ,...,e }. Then the symplectic group Sp(V) is the group of linear
1 n
operators f ∈End (V)preservingthebilinearform:
K
Representationsofreductivenormalalgebraicmonoids 5
(b) Sp(V)={f ∈End (V):hf(v),f(v′)i=hv,v′i, allv,v′∈V}.
K
LetAbethematrixof f withrespecttothebasis{e ,...,e }.Thenwemayidentify
1 n
Sp(V)withthematrixgroup
T
(c) Sp (K)={A∈M (K):A JA=J}.
n n
ThisiscontainedinthelargergroupGSp (K),thegroupofsymplecticsimilitudes,
n
definedby
(d) GSp (K)={A∈M (K):ATJA=cJ, somec∈K×}.
n n
NotethatGSp (K)isgeneratedbySp (K)andK×.Asintheorthogonalcase,we
n n
couldjustaswellhavedefinedGSp (K)by
n
GSp (K)={A∈M (K):ATJA=cJ=AJAT, somec∈K×}
n n
T T
thankstotheequivalenceoftheconditionsA JA=cJandcJ=AJA incasec6=0.
WedefinethesymplecticmonoidSpM (K)tobe
n
(e) SpM (K)=GSp (K),
n n
theZariskiclosureinM (K).In[4]thefollowingwasproved.
n
2.4Proposition. ThesymplecticmonoidSpM (K)isthesetofallA∈M (K)such
n n
T T
thatA JA=cJ=AJA ,forsomec∈K.
Notethatthescalarc∈K in theaboveisallowedtobezero,andthecondition
T
cJ=AJA isnecessary,justasitwasintheorthogonalcase.
2.5. Sketch of proof. I want to briefly sketch the ideas involved in the proof of
Propositions2.2and2.4.Fulldetailsareavailablein[4].Themethodofproofworks
for any infinite field K (except that characteristic 2 is excluded in the orthogonal
case).Wecontinuetoassumethatn=2miseveninthesymplecticcase.
LetG=GO (K) or GSp (K) andlet M=OM (K) or SpM (K), respectively.
n n n n
LetT bethemaximaltorusofdiagonalelementsofG.Thenwehaveinclusions
(a) T ⊂G⊂M
andwedesiretoprovethatthelatterinclusionisactuallyanequality.Toaccomplish
this, we consider the action of G×G on M given by (g,h)·m=gmh−1. Suppose
thatwecanshowthateveryG×GorbitisoftheformGaG,forsomea∈T.Then
itfollowsthat
(b) M= GaG⊂G
a∈T
S
and this givesthe opposite inclusion that provesPropositions2.2 and 2.4. In fact,
as it turned out, the distinct a∈T in the above decompositioncan be taken to be
certainidempotentsinT.
6 StephenDoty
Thissuggeststheprogramthatwascarriedoutin[4],whichintheendleadsto
additionalstructuralinformationonM:
(i)ClassifyallidempotentsinT.
(ii)ObtainanexplicitdescriptionofT.
(iii)DeterminetheG×GorbitsinM.
Part(i)iseasy.Forpart(ii)oneexploitstheactionofT onT byleftmultiplication
and determines the orbits of that action. Part (iii) involves developing orthogonal
andsymplecticversionsofclassicalGaussianelimination.
2.6. The normality question. It is clear from the equalities in Propositions 2.2
and2.4thatOM (K)andSpM (K)bothhaveone-dimensionalcentersandcontain
n n
zero.Whatisnotclear,andnotaddressedin[4],iswhetherornottheyarenormal
asalgebraicvarieties.
This question was recently settled in [6], where it is shown that SpM (K) is
n
alwaysnormal,whileOM (K)isnormalonlyin caseniseven.Moreprecisely,it
n
isshownin[6]thatwhenn=2miseven,OM+(K)andOM−(K)arebothnormal
n n
varieties.Here
(a) OM (K)=OM+(K)∪OM−(K)
n n n
isthedecompositionintoirreduciblecomponents,whereOM+(K)isthecomponent
n
containingtheunitelement1.
3 Representation theory
FromnowonweassumethatMisanarbitraryreductivenormalalgebraicmonoid,
withunitgroupG=M×.Wewishtodescribesomeresultsof[5].Themainresult
isthatthecategoryofrationalM-modulesisahighestweightcategoryinthesense
ofCline–Parshall–Scott[1].
3.1. We work with a fixed maximaltorusT ⊂G, and setD=T. We assume that
dimZ(M)=1and0∈M.RecallthatrestrictioninducesaninjectionX(D)→X(T),
sowemayidentifyX(D)withasubmonoidofX(T).WefixaBorelsubgroupBwith
T ⊂B⊂GandletthesetR−ofnegativerootsbedefinedbythepair(B,T).Weset
R+=−(R−),thesetofpositiveroots.WehaveR=R+∪R−.Let
X(T)+={l ∈X(T):(a ∨,l )≥0, foralla ∈R+}
betheusualsetofdominantweights.Wedefine
X(D)+=X(T)+∩X(D).
3.2. By a (left) rationalM-modulewe meana linearaction M×V →V suchthat
the coefficientfunctionsM→K of the actionare allin K[M]. Thisis the same as
Representationsofreductivenormalalgebraicmonoids 7
havinga(right)K[M]-comodulestructureonV.Thismeansthatwehaveacomodule
structuremap
(a) D :V →V⊗ K[M].
V K
SinceK[M]⊂K[G]themapD inducesacorrespondingmapV →V⊗ K[G]mak-
V K
ingV into a K[G]-comodule;i.e., a rational G-module. Thus, rationalM-modules
mayalsoberegardedasrationalG-modules.AnyrationalM-moduleissemisimple
whenregardedasarationalD-module,withcorrespondingweightspacedecompo-
sition
(b) V = l ∈X(D)Vl
whereVl ={v∈V :d·v=l (d)v, allLd∈D}.
RecallthatanyrationalG-moduleV is semisimplewhenregardedasa rational
T-module,withcorrespondingweightspacedecomposition
(c) V = l ∈X(T)Vl
whereVl ={v∈V :t·v=l (t)v, allLt ∈T}.IfV isarationalM-modulethenthe
weight spaces relative to T are the same as the weight spaces relative to D. So
the weights of a rational M-module all belong to X(D). Conversely, we have the
following.
3.3Lemma. IfV isarationalG-modulesuchthat
{l ∈X(T):Vl 6=0}⊂X(D)
thenV extendsuniquelytoarationalM-module.
ThisisprovedasanapplicationofRenner’sextensionprinciple,whichisaver-
sionofChevalley’sbigcellconstructionforalgebraicmonoids.
3.4Remark. Aspecialcaseofthelemma(forthecaseM=M (K))canbefound
n
in[7].
3.5. Nextoneneedsanotionofinductionforalgebraicmonoids,i.e.,aleftadjoint
torestriction.Theusualdefinitionofinducedmoduleforalgebraicgroupsdoesnot
work for algebraic monoids. Instead, we use the following definition. LetV be a
rationalL-modulewhereLisanalgebraicsubmonoidofM.WedefineindMV by
L
indMV ={f ∈Hom(M,V): f(lm)=l·f(m), alll∈L,m∈M}.
L
Thisisviewedasa rationalM-moduleviarighttranslation.Onecancheckthatin
caseL,Marealgebraicgroupsthenthisisisomorphictotheusualinducedmodule.
ItiswellknownthattheBorelsubgroupBhasthedecompositionB=TU,where
U isits unipotentradical.Givena characterl ∈X(T)oneregardsK asa rational
T-modulevial ;thisisoftendenotedbyKl .OneextendsKl toarationalB-module
bylettingU acttrivially.Similarly,wehavethedecompositionB=DU.Ifl ∈X(D)
8 StephenDoty
thenwehaveKl asabove,andagainwemayregardthisasarationalB-moduleby
lettingU acttrivially.
NowwecanformulatetheclassificationofsimplerationalM-modules.
3.6Theorem. LetM be a reductivenormalalgebraicmonoid.Letl ∈X(D)and
letKl betherationalB-moduleasabove.Then
(a)indMKl 6=0ifandonlyifl ∈X(D)+.
B
(b)IfindMKl 6=0thenitssocleisasimplerationalM-module(denotedbyL(l )).
B
(c)ThesetofL(l )withl ∈X(D)+givesacompletesetofisomorphismclasses
ofsimplerationalM-modules.
Letl ∈X(T).Let(cid:209) (l )=indGBKl .Itiswellknownthat(cid:209) (l )6=0ifandonlyif
l ∈X(T)+.Thefollowingisakeyfact.
3.7Lemma. Ifl ∈X(D)+ thenindMBKl =(cid:209) (l )=indGBKl .
3.8. Nowweconsidertruncation.Letp ⊂X(T)+.GivenarationalG-moduleV,let
Op V betheuniquelargestrationalsubmoduleofV withthepropertythatthehighest
weightsofallitscompositionfactorsbelongtop .The(leftexact)truncationfunctor
Op wasdefinedbyDonkin[2].
RecallthatX(T)ispartiallyorderedbyl ≤m ifm −l canbewrittenasasumof
positiveroots;thisissometimescalledthedominanceorder.Asubsetp ofX(T)+is
saidtobesaturatedifitispredecessorclosedunderthedominanceorderonX(T).
Inotherwords,p issaturatedifforanym ∈p andanyl ∈X(T)+,l ≤m implies
thatl ∈p .
In order to show that the category of rational M-modules is a highest weight
category,wearegoingtotakep =X(D)+.Weneedthefollowingobservation.
3.9Lemma. Thesetp =X(D)+isasaturatedsubsetofX(T)+.
For l ∈X(T)+, let I(l ) be the injective envelope of L(l ) in the category of
rationalG-modules.Forl ∈X(D)+ letQ(l )betheinjectiveenvelopeofL(l )in
thecategoryofrationalM-modules.Thefollowingrecordstheeffectoftruncation
onvariousclassesofrationalG-modules.
3.10Theorem. Letp =X(D)+.Foranyl ∈X(T)+ wehavethefollowing:
(cid:209) (l ) ifl ∈p
(a) Op (cid:209) (l )=
(0 otherwise.
Q(l ) ifl ∈p
(b) Op I(l )=
(0 otherwise.
(c) Op K[G]=K[M].
Note that K[M] is regarded as a rational M-module via right translation. A (cid:209) -
filtrationforarationalG-moduleV isanascendingseries
0=V ⊂V ⊂···⊂V ⊂V =V
0 1 r−1 r
Representationsofreductivenormalalgebraicmonoids 9
ofrationalsubmodulessuchthatforeach j=1,...,r, thequotientV /V isiso-
j j−1
morphictosome(cid:209) (l ).WheneverV isarationalG-modulewitha(cid:209) -filtration,let
j
(V :(cid:209) (l )) bethe numberofl forwhichl =l . Thisnumberis independentof
j j
thefiltration.
The proof of the above theorem, which relies on results of [3], also shows the
followingfacts.
3.11Corollary. (a)Letl ∈p =X(D)+.ThemoduleQ(l )hasa(cid:209) -filtration.Fur-
thermore,itsatisfiesthereciprocityproperty
(Q(l ):(cid:209) (m ))=[(cid:209) (m ):L(l )]
foranym ∈X(D)+,where[V :L]standsforthemultiplicityofasimplemoduleLin
acompositionseriesofV.
(b)ThemoduleK[M]hasa(cid:209) -filtration.Moreover,(K[M]:(cid:209) (l ))=dim (cid:209) (l )
K
foreachl ∈X(D)+.
3.12. Fromtheseresultsoneobtainstheimportantfactthatthecategoryofrational
M-modulesisahighestweightcategory,inthesenseof[1].Inparticular,onealso
seesthatdim Q(l )isfinite,foranyl ∈X(D)+.(Incontrast,itiswellknownthat
K
dim I(l )isinfinite.)
K
3.13. Itisalsoshownin[5],exploitingtheassumptionthatZ(M)isone-dimensional,
thatthecategoryofrationalM-modulessplitsintoadirectsumof‘homogeneous’
subcategories each of which is controlled by a finite saturated subset of X(D)+.
Fromtheresultsof[1]itthenfollowsthatthereisafinitedimensionalquasihered-
itaryalgebraineachhomogeneousdegree,whosemodulecategoryispreciselythe
homogeneoussubcategoryin thatdegree.Details are givenin [5], where it is also
shown that the quasihereditary algebras in question are in fact generalized Schur
algebrasinthesenseofDonkin[2].
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