Singular blocks of parabolic category O and finite W-algebras

0.13 MB·English

by  Ben Webster

Singular blocks of parabolic category O and finite W-algebras
BenWebster1
DepartmentofMathematics,MassachusettsInstituteofTechnology,Cambridge,MA,02139
[email protected]
Abstract. We show that each integral infinitesimal block of parabolic category O (including
singular ones) forasemi-simple Liealgebracanberealizedasafullsubcategoryofa“thick”
9 categoryOoverafiniteW-algebraforthesameLiealgebra.
0
0 ThenilpotentusedtoconstructthisfiniteW-algebraisdeterminedbythecentralcharacterof
2 theblock,andthesubcategorytakenisthatkilledbyaparticulartwo-sidedidealdependingon
the originalparabolic. Theequivalencesinquestion areinducedbythose definedbyMilicˇic´-
c
e SoergelandLosev.
D
Wealsogiveaproofofaresultofsomeindependentinterest: thesingularblocksofparabolic
0 category O can be geometrically realized as “partial Whittaker sheaves” on partial flag vari-
1 eties.

]
T
In this note, we aim to clarify the relationship between the category O of Bernstein-Gelfand-
R
Gelfand[BGG76]andthecategoryOoverthefiniteW-algebradefinedbyBrundan,Goodwinand
.
h
Kleshchev[BGK08]. Weshowthat:
t
a
m
Theorem AnyintegralinfinitesimalblockofparaboliccategoryOforasemi-simpleLiealgebragisequiv-
[ alenttoacategory ofrepresentations over afiniteW-algebra forgcorresponding toanilpotent (depending

3 onthechoiceofcentralcharacter).
v
0 RecallthataninfinitesimalblockofacategoryofU(g)-modulesisdefinedtobethesubcategory
6
generatedbythesimplemodulesonwhichthecenterofU(g)actsbyafixedcentralcharacter.
8
1 For a more precise statement of this Theorem, see Corollary 6. The equivalences refered
. to above are obtained by restricting to subcategories equivalences defined in previous work by
9
0 Milicˇic´-Soergel[MS97]onrepresentationsofLiealgebrasandLosev[Losa]onfiniteW-algebras.
9
0 ThecategoryofW-algebramodulesinquestionhasaflavorsimilartothecategoryOfortheW-
:
v algebraalreadydefinedbyBrundan,GoodwinandKleshchev[BGK08],butitissubtlydifferent.
i Inordertoexplainthisdifference,considerthecaseofourparabolicbeingtheBorelitself,the
X
original definition of category O considered in [BGG76]. In this case, we show that a singular
r
a block of category O is equivalent to the category of modules over a W-algebra for the same Lie
algebra (with nilpotent dependingon the singular block) generatedby “Verma modules”for the
W-algebra defined by Brundan, Goodwin and Kleshchev [BGK08, §4.2], and which furthermore
havetrivialcentralcharacter. WewillrefertothiscategoryascategoryO′ forthefiniteW-algebra.
This category has two subtle differences from the category O’s defined in [BGG76] for U(g)
andin[BGK08]fortheW-algebra:
• ThecategoryO′ onlycontainsmoduleswherethecenteractssemi-simply.
• Wehave noanalogueoftheconditionthattheCartan act semi-simply, which isimposedin
[BGG76](ananalogueofthissemi-simplicityisalsoimposedin[BGK08]).
1SupportedbyanNSFPostdoctoralResearchFellowship
1
IfinsteadoftheBorel,webeginwithablockofparaboliccategoryOforsomelargerparabolic,
we must further impose the condition that these modules are killed by a particular two-sided
ideal.
The proof of the theorem above is organized into two sections. Section 1 covers preparatory
resultsoncategoryO andHarish-Chandrabimodules;it islikelythatmanyoftheresultstherein
are familiar to experts, but they do not seem to have been written down together in the neces-
sary form anywhere in the literature. Section 2 describes the relationship of these results to the
representationtheoryofthefiniteW-algebra,andcompletestheproof.
Finally, in Section 3, we also discuss briefly the relationship of these results to geometry. In
particular,wegiveaproofofaresultwhichhascirculatedasafolktheorem: thesingularintegral
infinitesimal blocks of parabolic category O can be geometrically realized as (N,f)-equivariant
(the so-called “partial Whittaker”) D-modules or perverse sheaves on G/Q. This is a classical
theoremwhenf isgeneric(goingbacktoapaperofKostant[?]),andtheQ = B casefollowsfrom
workofBezrukavnikovandYun[BY,Theorem4.4.1]butweknowofnoproofintheliteratureof
thegeneralcase.
In a forthcoming paper [BLPW], the author, Braden, Licata and Proudfoot will discuss the
relationship to geometry more fully and relate this isomorphism to the geometry of strata in a
Slodowysliceandtoourconjecturesonsymplecticdualitybetweensuchstrata.
Acknowledgments
I would like to thank Tom Braden, Tony Licata and Nick Proudfoot both for useful discussions
during the development of this paper, and extremely helpful comments on earlier versions of
this paper; Jim Humphreys and an anonymous referee for very useful comments on an earlier
version of the paper; Catharina Stroppel, Roman Bezrukavnikov and Ivan Mirkovic´ for some
valuableeducationonHarish-ChandrabimodulesandWhittakersheaves;andtheMathematische
ForschungsinstitutOberwolfachforhospitalitywhilethispaperwaswritten.
1 Lie theory
First, we fix notation. Let g ⊃ q ⊃ b = h⊕n be a semi-simple complex Lie algebra, a parabolic,
aBorel,and achosendecompositionofbas aCartan and itsnilpotentradical. LetG ⊃ Q ⊃ B =
HN be connected groups with these Lie algebras (which form of G we take is irrelevant to our
purposes). LetZ denotethecenteroftheuniversalenvelopingalgebraU(g).
Definition1 WeletO bethecategoryofrepresentationsofgwherenandZ actlocallyfinitely.
b
ThiscategoryhasanaturalinfinitesimalblockdecompositionO = LO(χ,f)wherethesumruns
b b
overcharacters χ: Z → Cand f: n → Cand O(χ,f)is thesubcategorywhereall theirreducible
b
constituents of the restriction to U(n)⊗Z are f ⊗χ. For any highest weight λ, we let χ be the
λ
characterofZ actingontheuniqueirreducibleg-moduleofhighestweightλ.
ThecategoryO containstwoverynaturalsubcategories:
b
• O isthesubcategoryofmodulesonwhichhactssemi-simply.
• O′ isthesubcategoryofmodulesonwhichZ actssemi-simply.
These have the same infinitesimal block decomposition as above. We note that O is the most
commonlystudiedofthesecategories,andL O(χ,0)inournotationispreciselythecategoryO
χ
2
originally defined by Bernstein-Gelfand-Gelfand [BGG76], and O(χ,0) its infinitesimal block for
thecentralcharacterχ.
It’s worth noting that for non-integral χ, the infinitesimal blocks O(χ,0) are not necessarily
blocks in the abstract sense; they may have further decompositions as the sum of orthogonal
subcategories. For several infinitesimal blocks of categories we consider, the abstract block de-
compositionscanactuallybequitesubtleandcomplicated.
WewillalsowishtoconsidertheparabolicversionofcategoryO. LetOq(χ,0)bethesubcate-
goryofO(χ,0) consistingofmodulesalso locally finitefor theaction ofq. One particular special
objectinOq(χ ,0)isthedominantparabolicVermamoduleMq = IndU(g) ,where isthetrivial
0 U(q)1 1
representationofq.
Fortheentiretyofthepaper,wewillalwaysusef andµtodenoteaLiealgebramapf: n → C
andaweightµofgsuchthatif∆ bethesetofsimplerootsαwhoserootspacesarenotkilledby
f
f, then α∨(µ) = −1 for all α ∈ ∆ and α∨(µ) ∈ Z\{−1} for all α ∈/ ∆ . In particular, note that
f f
f = 0andµ = 0arecompatibleinthesenseabove;thiswillbeanimportantspecialcaseforus.
Letp betheparabolicgeneratedbythestandardBorelbandthenegativerootspacesg for
f −α
α ∈ ∆ .
f
The startingpointfor us is an equivalence of categoriesΦ : O(χ ,0) → O(χ ,f)constructed
µ b µ b 0
byMilicˇic´ andSoergel[MS97, Theorem5.1]. Thisfunctormakessensefor anyintegralvalue ofµ
for an appropriate value of f; one case of special interest is the functor Φ : O(χ ,0) → O(χ ,0).
0 b 0 b 0
This functor was considered in earlier work of Soergel [?] to construct an equivalence between
O(χ ,0) and O′(χ ,0). Our main result of this section will be to understand the effect of this
0 0
functoronsubcategoriesofinteresttous.
WeletI = Ann Φ (Mq)andletO′(χ ,f)denotethesubcategoryofmoduleskilledbyI .
q U(g) 0 q 0 q
It may look peculiar to the reader that we are using Mq, a Verma module of central character χ
0
todefineanideal,andthenusingthisidealtodefineacategoryrelatedtoO(χ ,0);thiscreatesno
µ
problems, since all the target categories for Φ have central character χ . We note that I ∩Z =
∗ 0 q
kerχ , since Φ (Mq) is a quotient of a Verma module. Thus any object of O(χ ,f) killed by I is
0 0 b 0 q
automaticallyasemi-simpleZ module,andthereforeinO′(χ ,f).
0
Proposition2 Thesubalgebra qactslocallyfinitelyandhactssemi-simply onamoduleX ∈ O(χ ,0)if
b µ
andonlyifAnn(Φ (X)) ⊃ I ;thatis,wehaveaninducedequivalenceΦ : Oq(χ ,0) → O′(χ ,f).
µ q µ µ q 0
In particular, setting q = b, we have that h acts semi-simply on X ∈ O(χ ,0) if and only if Z acts
b µ
semi-simplyonΦ (X);thatis,wehaveaninducedequivalenceΦ : O(χ ,0) → O′(χ ,f).
µ µ µ 0
Most of the ideas necessary for the proof of this fact are already in the literature, which we now
collectintothe
Lemma3 For each object X ∈ Ob(χµ,0) and each projective functor P: Ob(χµ′,0) → Ob(χµ,0), we have
aninclusionAnnU(g)(Φµ(PX)) ⊃ AnnU(g)(Φµ′(X)).
IfL andL aresimple, thenthis statementhasaconverse; L appears asacomposition factor inPL
1 2 2 1
forsomeprojective functorifandonlyifAnnU(g)(ΦµL1) ⊂ AnnU(g)(Φµ′L2).
Proof: TheMilicˇic´-Soergelresultdependsonequivalencesofcategories
ϕ1 ϕ2
H −→ O(χ ,0) H −→ O(χ ,f)
χµ χ0 b µ χ0 χµ b 0
with certain blocks ofthecategoryH ofHarish-Chandra bimodules,definedby fixing thegener-
alized character of the left and right Z-actions. The equivalence Φ is given by composing these
µ
withthe“flipequivalence”F: H → H ,sowehaveΦ = ϕ Fϕ−1.
χµ χ0 χ0 χµ µ 2 1
3
Ifwewerebeingverycareful,wewouldincludeµandf inthenotation,sincethisequivalence
existsforanycompatiblechoice,butforsimplicity,weleavethemout.
Thefunctorsϕ arebothoftheform
i
ϕ (Y) = limY ⊗ Ln
i ←− U(g) i
for an inverse systemofrepresentationsLn for i = 1,2, and n ≥ 0. Thus,for any U(g)-bimodule
i
D,wehaveϕ (D⊗ Y)∼= D⊗ ϕ (Y).
i U(g) U(g) i
In particular, for any two-sided ideal I, we have ϕ (IY) ∼= Iϕ (Y). Also, for any projective
i i
functorP : Ob(χµ′,0) → Ob(χµ,0)whereweletPL denotethefunctoronU(g)-bimodulesgivenby
actingwiththeprojectivefunctorPontheleft,thenbythesameprinciple,wehavePϕ = ϕ P .
1 1 L
One important consequence of this fact is that the left annihilator of a bimodule Y coincides
withtheannihilatorofϕ (Y). Thus,ϕ−1(X)isaHarish-Chandrabimodulewhoseleftannihilator
i 1
isthatofX andwhoserightannihilatoristhatofΦ (X). Thus,wehave
µ
Ann(Φµ(PX)) = RAnn(PLϕ−11(X)) ⊃ RAnn(ϕ−11(X)) = Ann(Φµ′(X)).
Thefirstpartoftheresultfollows.
Now, we prove the second statement (the partial converse). Consider the Harish-Chandra
bimodulesϕ−1(L )andϕ−1(L ). Byreversingleftandrightin[Vog80,Theorem3.2],wehave
1 1 1 2
Ann(ΦµL1) = RAnn(ϕ−11(L1))⊂ RAnn(ϕ−11(L2)) = Ann(Φµ′L2)
ifandonlyifthereisaprojectivefunctorPsuchthatϕ−1(L )isacompositionfactorinP ϕ−1(L ).
1 2 L 1 1
Applyingtheequivalenceϕ ,thisistrueifandonlyifL isacompositionfactorofPL . 2
1 2 1
Proof of Proposition 2: If X is in Oq(χ ,0), then there is a projective functor P: Oq(χ ,0) →
µ 0
Oq(χ ,0) and a surjection PMq → X, (this is an old result; it follows, for example, from [Kho05,
µ
Proposition22])andthus,byLemma3,wehaveinclusions
Ann(Φ (Mq)) ⊂ Ann(Φ (PMq)) ⊂ Ann(Φ (X)).
0 µ µ
Assume Φ (X) is killed by I . By passing to composition factors, we may assume that X
µ q
is simple. Let J be the primitive ideals killing the composition factors L of Mq, in the order
i i
induced by a Jordan-Ho¨lder series. Then J ···J ⊂ I ⊂ Ann(Φ (X)). Since Ann(Φ (X)) is
1 m q µ µ
prime,wehaveJ ⊂ Ann(Φ (X))forsomei. Thus,bythesecondpartofLemma3,X appearsas
i µ
acompositionfactorofPL . SinceL isq-locally finite,soisPL ,andthusX.
i i i
Finally, we establish the relationship between h-semi-simplicity and Z-semi-simplicity. By
[MS97,Theorem5.3]inthecasewheren = 1,wehavethatabimoduleD ∈ H issemi-simple
χµ χ0
as a right Z-module if and only if ϕ (D) is a module on which the Cartan acts semi-simply, that
1
is, a weight module. Thus, X is a weight module if and only if ϕ−1(X) is semi-simple as a right
1
bimodule if and only if Fϕ−1(X) is semi-simple as a left Z-module if and only if Φ (X) is semi-
1 µ
simpleasaZ-module(applyingthecommutationofidealspastϕ again). 2
2
2 Finite W-algebras
Now,wewillconnecttheprevioussection,whichonlydealtwithrepresentationsofLiealgebras,
tothestudyoffiniteW-algebras.
4
ThefiniteW-algebraW isainfinite-dimensionalassociativealgebraconstructedfromthedata
e
of g and a nilpotent element e ∈ g; geometrically, it is obtained by taking a quantization of the
Slodowy slice to the orbit through e. It is most easily defined as the endomorphism ring of the
Gelfand-Graev representation Q = Ξ⊗ U(g) for a particular nilpotent subalgebra m ⊂ n
Ξ,m U(m)
depending on e, and a one-dimensional representation Ξ of m constructed using e. For more on
thegeneraltheoryofW-algebras,see[Pre02,GG02].
ThealgebraW hasacategoryofrepresentationsanalogoustoOabove,originatinginthework
e b
ofBrundan, Goodwinand Kleshchev[BGK08]. This categoryis associateda choice ofparabolic p
suchthat
• eisdistinguishedintheLevilofpand
• pcontainsafixedmaximaltorustofthecentralizerC (e) ={x ∈ g|[x,e] = 0}.
g
Thischoiceofparabolicallowsustoputanpreorderontheweightsoftbydeclaringthatλ ≤ µif
andonlyifµ−λisalinearcombinationofweightsoftactingonp.
We have a natural inclusion U(t) ֒→ W , (described in [BGK08, Theorem 3.3]) and thus can
e
analyzeanyW representationbyitsrestrictiontot.
e
Definition4 LetO(W ,p)bethecategoryofmodulesX suchthat
b e
• tactsonX withfinitedimensionalgeneralizedweightspaces.
• theweightswhichappearinX arecontainedinafiniteunionofsetsoftheform{µ|µ ≤ λ}.
As before, O(W ,p) denotesthe subcategoryon which t acts semi-simply, and O′(W ,p) denotes
e e
thesubcategoryonwhichZ ∼= Z(W )actssemi-simply.
e
Alternatively, following [Losa], we could define an analogue of O as the category of W -
b e
modules which are locally finite for the action of the non-negative weight spaces of an element
ξ ∈ C (e) acting by theadjoint action on W . This will recoverthe BGKdefinition whenpis pre-
g e
ciselythenon-negativeweightspacesfortheadjointactionofξong. InLosev’snotation,O(W ,p)
b e
is denoted O(ξ), and the category we and [BGK08] denote O(W ,p), Losev would denote Ot(ξ)
e
(leavingtheeimplicit).
There is an isomorphism Z ∼= Z(W ), typically referenced in the literature to a footnote to
e
Question 5.1 in the paper of Premet [Pre07], where Premet gives an argument he ascribes to
Ginzburg. This isomorphism allows us to identify central characters of U(g) and W . We de-
e
note the subcategory of O(W ,p) with generalized central character χ by O(χ,W ,p), and the
b e b e
subcategorywhereZ actssemi-simply(andthusbythecharacterχ)byO′(χ,W ,p).
e
Wewillprimarilyinterestedinthecasewhereafterfixingaparabolicp ⊃ bwithLevil,wetake
etobetheprincipalnilpotentinl(fortheBorelgivenbyintersectionwithb),andwewillassume
we are in this situation. In type A, all nilpotents will appear this way up to conjugacy, since we
cantakeptobeblockuppertriangularmatricesattachedtotheJordanblocksofe;inothertypes,
therearenilpotentsthatarenotofthisform.
Asbefore,letf: n → CbeafixedcharacterofthisnilpotentLiealgebra.
Proposition5([Losa]) IfeisprincipalintheLeviofp ,thereisanequivalence
f
L: O(χ ,W ,p ) → O(χ ,f)
b 0 e f b 0
inducingequivalences
O′(χ ,W ,p )∼= O′(χ ,f) O(χ ,W ,p ) ∼= O(χ ,f)
0 e f 0 0 e f 0
5
Furthermore, L(X) is killed by I if and only if X is killed by a certain two-sided ideal (I ) of the W-
q q †
algebra. Wedenotethesubcategory killedby(I ) byO (χ ,W ).
q † bq 0 e
Theideal(I ) arisesfromapairofmaps: I 7→ I† sendstwo-sidedidealsintheW-algebratotwo-
q †
sided ideals of U(g) and J 7→ J goes the opposite direction; these both are defined by Losev in
†
[Losb,§3.4]. Inaforthcomingpaper[BLPW],theauthor,Braden,LicataandProudfootwillshow
thatthisideal(I ) isprimitive,anditselfhasageometricinterpretation.
q †
Proof: The only part of the equivalences not stateddirectly in [Losa, Theorem4.1] is the equiva-
lenceofmostinterestforus,thatofO′(χ ,W ,p )∼= O′(χ ,f).
0 e f 0
Itfollowsfrom[Losa,Theorem4.1(1)]and[Losb,Theorem1.2.2(iii)]that
(∗): thecenteractssemi-simplyonX ∈ O(χ ,W ,p )ifandonlyifitactssemi-simplyonL(X).
b 0 e f
The “only if” portion of (∗) shows that L induces a fully faithful functor L : O′(χ ,W ,p ) →
0 e f
O′(χ ,f). The “if” portion of (∗) shows that the essential surjectivity of L implies the essential
0
surjectivityofthisrestriction.
Now, fix a U(g)-module X such that Ann(X) ⊃ I , and consider Ann(L−1(X)). We know by
q
[Losb,Theorem4.1(1)]thatAnn(L−1(X))† = Ann(X). ItfollowsthatAnn(L−1(X)) ⊃ Ann(X) ⊃
†
(I ) since Losev’s† maps both preserve inclusion (this follows from [Losb, Theorem1.2.2(i)] for
q †
(·) andisnotedbyLosevin[Losb,§3.4]for(·)†).
†
On the hand, if V ∈ O′(χ ,W ,p ) is killed by (I ) . Then L(V) is a U(g)-module killed by
0 e f q †
((I ) )†. By[Losb,Proposition3.4.4],((I ) )† ⊃ I andL(V)iskilledbyI . 2
q † q † q q
Torecap,ifµisaweightofgsuchthatµ+ρisdominantintegral,andeistheprincipalnilpotent
oftheLeviintheparabolicp correspondingtothestabilizerofµ+ρundertheusualactionofW
f
onweights,thencombiningPropositions2and5,wehave
Corollary6 ThereisanequivalenceOq(χ ,0) ∼= O′(χ ,f)∼= O′(χ ,W ,p ).
µ q 0 q 0 e f
3 Relation togeometry
Inordertogivethereadersomefeelingforthe“meaning”oftheidealI ,wewillbrieflyindicate
q
its relationship to geometry. This relation will be drawn out much more fully in a forthcoming
paperoftheauthor,Braden,Licata,andProudfoot.
We have the G-space G/Q, and as with any G-space, a map α: U(g) → D(G/Q) to the ring
ofglobaldifferentialoperatorsonG/QsendingaLiealgebraelementtothecorrespondingvector
field.
Proposition7 The map α induces an isomorphism D(G/Q) ∼= U(g)/I . In particular, gr(U(g)/I ) =
q q
C[T∗G/P],andI isprime.
q
Proof: Borho and Brylinski [BB82] show2 that D(G/Q) ∼= U(g)/I where I is the annihilator of a
simple parabolic Verma module Lq with highest weight 2ρ − 2ρ; this is the unique such with
Q
centralcharacterχ . Thus,weneedonlyshowthatI = I .
0 q
WecanunderstandthefunctorΦ restrictedtoO(χ ,0)∩O′(χ ,0)aslocalizing amoduletoa
0 0 0
B-equivariantD-moduleonG/B,thenusingthe“flip”functortothinkofthisasaB-equivariant
2Wethanktherefereeforpointingoutthisreference.
6
D-module on B\G, and taking sections. In particular, the parabolic Verma module Mq localizes
to the !-extension of the structure sheafO as a D-module (don’t confuse this with the use
Qw0B/B
of the symbol O for a category; that will not appear in the rest of this proof), the structure sheaf
of the dense Q-orbit, so Φ (Mq) localizes to the !-extension of O . Since the latter sheaf is
0 Bw0Q/B
smoothalongthefibersofthenaturalmapG/B → G/Q,thepushforwardofthissheaftoG/Qis
the!-extensionofO . Inparticular,Φ (Mq)isthesectionsofaD-moduleonG/Q,andthus
Bw0Q/Q 0
annihilatedbyI.
Ontheotherhand,bytheworkofIrving[Irv85,§4.3], thesocleofMqisin thesamerightcell
as Lq. Thus, the socle of Φ (Mq) is in the same left cell as Lq, and so these simple modules have
0
the same annihilator. Since any ring element annihilating Φ (Mq) annihilates its socle, we have
0
thatI ⊃ I ,andequalityofidealsfollows.
q
This isomorphism, followed by the symbol map on differential operatorsinduces an isomor-
phismgr(U(g)/I ) ∼= C[T∗G/Q]. SincetheassociatedgradedofU(g)/I istheglobalfunctionring
q q
ofanirreduciblequasi-projectivevariety,theassociatedgradedisanintegraldomain,andthusso
isU(g)/I . 2
q
Interestingly,thisestablishesa conjecturewhich hadappearedin an unpublishedmanuscript
ofBezrukavnikovandMirkovic´,andwhichhadbeenpreviouslystudiedbytheauthorandStrop-
pel.
Definition8 WesaythataD-moduleMonG/Qis(N,f)-equivariantiftheactionofnonMby
n·m = α m−f(n)m(whereα isthevectorfieldgivenbytheinfinitesimalactionofnonG/Q)
n n
integratestoan N-equivariant structureon M. Some authorsuse“strongly equivariant” for this
formofequivariance.
There is also a version of (N,f)-equivariance for perversesheaves on G/Q, though in the al-
gebraic category this can only be made sense of using perverse sheaves on the characteristic p
analogue of G/Q (for more on this approach, see [BBM04]). In this case, we let f′ be a character
f′: N/Fp → Ga/Fp suchthat∆f′ = ∆f andtwisttheequivariance conditionbyanArtin-Schreier
sheaf pulled back from G /F by f′. These are sometimes called “partial Whittaker” or “Whit-
a p
tavariant”sheaves.
Corollary9 The category Oq(χ ,0) is equivalent to the category of (N,f)-equivariant D-modules on
µ
G/Q (and thus also to the category of (N,f′)-equivariant perverse sheaves on G/Q, with all groups con-
sidered asalgebraic groupsoverF ).
p
Proof: By Beilinson-Bernstein [BB81], the sections functor D − mod → U(g)/I − mod is an
G/Q q
equivalence. If M is a (N,f)-equivariant D-module, then the action of U(n) on Γ(M) is lo-
cally finite, with C being the only representation which appears in the composition series, so
f
M ∈ O′(χ ,f). On the other hand, if M ∈ O′(χ ,f), then the f-twistedn-action via differential
q 0 q 0
operatorsonthelocalization willintegratetoan(N,f)-action. 2
Aswenotedearlier,theB = Qcaseofthiscanbederivedfromthe“paradromic/Whittavariant”
dualityof[BY].
7
References
[BB81] AlexandreBe˘ılinsonandJosephBernstein,Localisationdeg-modules,C.R.Acad.Sci.Paris
Se´r.IMath.292(1981), no.1,15–18.
[BB82] Walter Borho and Jean-Luc Brylinski, Differential operators on homogeneous spaces. I. Irre-
ducibilityoftheassociatedvarietyforannihilatorsofinducedmodules,Invent.Math.69(1982),
no.3,437–476. MRMR679767(84b:17007)
[BBM04] Roman Bezrukavnikov, Alexander Braverman, and Ivan Mirkovic, Some results about
geometric Whittaker model, Adv. Math. 186 (2004), no. 1, 143–152. MR MR2065510
(2005e:20068)
[BGG76] I. N. Bernsˇte˘ın, I. M. Gel′fand, and S. I. Gel′fand, A certain category of g-modules,
Funkcional.Anal.iPrilozˇen.10(1976), no.2,1–8.
[BGK08] JonathanBrundan,SimonM.Goodwin,andAlexanderKleshchev,Highestweighttheory
for finite W-algebras, Int. Math. Res. Not. IMRN (2008), no. 15, Art. ID rnn051, 53. MR
MR2438067 (2009f:17011)
[BLPW] TomBraden,AnthonyLicata,NicholasJ.Proudfoot,andBenWebster,Symplecticduality
forSlodowyslices,Inpreparation.
[BY] RomanBezrukavnikovandZhiweiYun,OntheKoszuldualityforaffineKac-Moodygroups,
arXiv:0812.1584.
[GG02] WeeLiang Gan and Victor Ginzburg, Quantization of Slodowy slices, Int.Math. Res.Not.
(2002), no.5,243–255.
[Irv85] Ronald S. Irving, Projective modules in the category O : self-duality, Trans. Amer. Math.
S
Soc.291(1985), no.2,701–732.
[Kho05] OleksandrKhomenko,Categories with projective functors, Proc. LondonMath.Soc.(3) 90
(2005), no.3,711–737.
[Losa] IvanLosev,Onthestructureofthecategory OforW-algebras, arXiv:0812.1584.
[Losb] ,Quantizedsymplectic actionsandW-algebras, arXiv:0707.3108.
[MS97] Dragan Milicˇic´ and Wolfgang Soergel, The composition series of modules induced from
Whittaker modules, Comment. Math. Helv. 72 (1997), no. 4, 503–520. MR MR1600134
(99e:17010)
[Pre02] Alexander Premet, Special transverse slices and their enveloping algebras, Adv. Math. 170
(2002), no.1,1–55,WithanappendixbySergeSkryabin.
[Pre07] ,Envelopingalgebras ofSlodowyslices andtheJoseph ideal, J.Eur.Math.Soc.(JEMS)
9(2007), no.3,487–543. MRMR2314105 (2008c:17006)
[Vog80] David A. Vogan, Jr., Ordering of the primitive spectrum of a semisimple Lie algebra, Math.
Ann.248(1980), no.3,195–203. MRMR575938(81k:17006)
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