Differential operators on G/U and the affine Grassmannian

70 Pages·2017·1.07 MB·English

Differential operators on G/U and the affine
Grassmannian
Victor Ginzburg, Simon Riche
To cite this version:
Victor Ginzburg, Simon Riche. Differential operators on G/U and the affine Grassmannian. Journal
de l’Institut de Mathématiques de Jussieu, 2015, 14. ￿hal-00839864v2￿
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DIFFERENTIAL OPERATORS ON G/U AND THE AFFINE GRASSMANNIAN
VICTORGINZBURGANDSIMONRICHE
ABSTRACT. Wedescribetheequivariantcohomologyofcofibersofsphericalperversesheavesonthe
affineGrassmannianofareductivealgebraicgroupintermsofthegeometryoftheLanglandsdual
group. In factwe givetwo equivalentdescriptions: one intermsofD-modulesofthe basicaffine
space,andoneintermsofintertwiningoperatorsforuniversalVermamodules. Wealsoconstruct
natural collections of isomorphisms parametrized by the Weyl group in these three contexts, and
provethattheyarecompatiblewithourisomorphisms. Asapplicationswereprovesomeresultsof
thefirstauthorandofBraverman–Finkelberg.
1. INTRODUCTION
1.1. ThegeometricSatakeequivalencerelatesperversesheaves(withcomplexcoefficientsinour
case)ontheaffineGrassmannianGrofacomplexconnectedreductivealgebraicgroupGˇ andrep-
resentationsoftheLanglandsdual(complex)reductive groupG. Theunderlyingvectorspaceof
q
therepresentationS(F)attachedtoaperversesheafF isgivenbyitstotalcohomologyH (Gr,F).
It turns out that various equivariant cohomology groups attached to F also carry information on
the representationS(F), see e.g. [G1, YZ, BF]. In this paper, if Tˇ is a maximal torus of Gˇ, we de-
scribe, in terms of G, the equivariant cohomology of cofibers of F at Tˇ-fixed point, with respect
to the action of Tˇ or of Tˇ × C×, where C× acts on Gr by loop rotation. In fact these groups can
bedescribedin twoequivalentways,eitherintermsofD-modulesonthebasicaffine spaceorin
terms of intertwining operators for universal Verma modules. We also describe the Weyl group
actiononthiscollectionofspacesinducedbytheactionofN (Tˇ)onGr.
Gˇ
1.2. To state our results more precisely, choose some Borel subgroup Bˇ ⊂ Gˇ containing Tˇ, and
let T,B be the maximal torus and the Borel subgroup of G provided by the geometric Satake
equivalence. Note that the Tannakian construction of G also provides no zero vectors in each
simplerootsubspaceofg := Lie(G). Inthispaperwestudythreefamiliesofgradedmodulesover
a polynomial algebra, attached to G or Gˇ, and endowedwith symmetries parametrized by their
commonWeylgroupW.
Let t := Lie(T) and S~ := S(t)[~], considered as a graded algebra where ~ and the vectors in t
areindegree2. (Here,S(t)isthesymmetricalgebraofthevectorspacet.) LetalsoX := X∗(T)be
thecharacterlattice. LetRep(G)bethecategoryoffinitedimensionalalgebraicG-modules.
Our first family of graded modules over S~ is of “geometric” nature. Let U be the unipotent
radical of B, and let X := G/U be the basic affine space. Consider the algebra D~(X) of (global)
asymptoticdifferentialoperatorsonX,i.e.theReesalgebraofthealgebraΓ(X,DX)ofdifferential
operators on X, endowedwith the order filtration (see §2.4 for details). This algebra is naturally
graded, and endowed with an action of T induced by right multiplication on X. We denote by
D~(X)λ theweightspaceassociatedwithλ ∈ X. Thenweset
MgVe,λom := V ⊗(λ)D~(X)λ G.
(cid:0) (cid:1)
TheworkofV.G.wassupportedinpartbytheNSFgrantDMS-1001677. TheworkofS.R.wassupportedbyANR
GrantsNo.ANR-09-JCJC-0102-01andNo.ANR-2010-BLAN-110-02.
1
(Herethetwistfunctor(λ)(·)willbedefinedin§2.4.)
Our secondfamily ofgradedS~-modulesis of“algebraic” nature. LetU~(g) be the asymptotic
enveloping algebra of g (i.e. the Rees algebra of the algebra U(g) endowed with the Poincare´-
Birkhoff-Witt filtration, see §2.1 for details). For λ ∈ X we let M(λ) be the asymptotic universal
Verma module associated with λ, a graded (S~,U~(g))-bimodule whose precise definition is re-
calledin§2.1. Thenweset
Malg := Hom M(0),V ⊗M(λ)
V,λ (S~,U~(g))
whereweconsidermorphismsinthecategoryof((cid:0)S~,U~(g))-bimodu(cid:1)les.
We will also construct a third family of graded modules, of “topological” nature, which is as-
sociated with the “Langlands dual data.” Let ˇt := Lie(Tˇ). We have canonical identifications
X ∼= X∗(Tˇ) and S~ ∼= S(ˇt∗)[~]. Consider the category PervGˇ(O)(Gr) of Gˇ(O)-equivariant perverse
sheavesontheaffineGrassmannianGrofGˇ. Thenforanyλ ∈ XandF inPerv (Gr)weset
Gˇ(O)
q
Mtop := H +λ(2ρˇ)(i! F).
F,λ Tˇ×C× λ
Here i is the inclusion of the point of Gr naturally associated with λ, C× acts on Gr by loop
λ
top
rotation, and ρˇis the half sum of positive coroots of G. Then M is in a natural way a graded
F,λ
S~-module.
1.3. Each ofthesefamilies is endowedwith a kindof“symmetry”governedby theWeylgroup
W of (G,T) or (Gˇ,Tˇ). (Note that these Weyl groups can be canonically identified.) Namely, we
haveisomorphismsofgradedS~-modules
A :M −→∼ wM or A : M −→∼ wM
V,λ,w V,λ V,wλ F,λ,w F,λ F,wλ
forallw ∈ W. (Herethetwistfunctorw(·)willbedefinedin§2.4.)
In the “geometric” case, the isomorphisms Ageom are constructed using a W-action on D~(X)
V,λ,w
givenbypartialFouriertransformsduetoGelfand–GraevandstudiedinparticularbyBezrukav-
nikov–Braverman–Positselskiiin[BBP]. Theseoperatorsdependonachoiceof(non-zero)simple
rootvectorsing,whichwechoosetobethoseprovidedbythegeometricSatakeequivalence.
In the “algebraic” setting, the isomorphisms Aalg are constructed using properties of inter-
V,λ,w
twining operators between a Verma module and a tensor product of a G-module and a Verma
module. Our constructions are “renormalized” variants of classical constructions appearing in
thedefinitionofthedynamical Weylgroup(see[TV,EV])but,as opposedtothoseconsideredin
loc. cit., our isomorphisms do not have poles. Again, the operators Aalg depend on a choice of
V,λ,w
simplerootvectorsing,whichwechooseasabove.
Inthe“topological”setting,theisomorphismsAtop areinducedbytheactionofN (Tˇ)onGr
F,λ,w Gˇ
byleftmultiplication.
Ineachsetting,thecollectionofoperatorsiscompatiblewiththeproductinW inthesensethat
y A ◦A = A or y A ◦A = A
V,yλ,x V,λ,y V,λ,xy F,yλ,x F,λ,y F,λ,xy
foranyλ,V,F as(cid:0)above(cid:1)andx,y ∈ W. (cid:0) (cid:1)
1.4. Inaddition,thesefamilies ofgradedmodulesareendowedwithmorphisms
Conv :M ⊗ (λ)M → M , Conv :M ⊗ (λ)M → M
V,V′,λ,µ V,λ S~ V′,µ V⊗V′,λ+µ F,F′,λ,µ F,λ S~ F′,µ F⋆F′,λ+µ
relatedtothemonoidalstructureonthecategoryRep(G)(denoted⊗)orPerv (Gr)(denoted⋆).
Gˇ(O)
2
In the “geometric” setting, morphisms Convgeom are induced by the product in the algebra
V,V′,λ,µ
D~(X). In the “algebraic” setting, morphisms ConvaVl,gV′,λ,µ are induced by composition of mor-
phisms of bimodules. In the “topological” setting, morphisms Convtop are defined using a
F,F′,λ,µ
standardconstructionconsideredinparticularin[ABG].
1.5. Ourmainresultmightbestatedasfollows(seeCorollary2.4.2,Theorem2.5.5,andProposi-
tion8.1.5).
Theorem. ForF inPerv (Gr)andλ ∈ Xthereexistcanonicalisomorphisms
Gˇ(O)
Mtop ∼= Mgeom ∼= Malg ,
F,λ S(F),λ S(F),λ
where S : Perv (Gr) −→∼ Rep(G) is the geometric Satake equivalence. These families of isomorphisms
Gˇ(O)
arecompatiblewithoperators AandwithmorphismsConv.
geom alg
TheproofofthistheoremisbasedonanothercrucialpropertyofthemodulesM ,M and
V,λ V,λ
top
M : they are all compatible with restriction to a Levi subgroup in the appropriate sense. This
F,λ
propertyisusedtoreducetheproofofourclaims tothecaseGandGˇ havesemisimplerankone,
inwhichcasetheycanbecheckedbyexplicitcomputation. Thisstrategyisratherclassicalinthis
context,seee.g.[BFM,BF,BrF,AHR].
1.6. Thepresentpaperiscloselyrelatedto,andmotivatedby,resultsof[ABG]and[BF]. Infact,in
afollow-uppapertheresultsofthepresentarticlewillbeusedtoobtainacommongeneralization
oftheequivalencesofcategoriesestablishedin thesepapers. Asimilar generalization canalsobe
obtained using recent results of Dodd [Do], but our approach is different and, we believe, more
explicit. We will follow the strategy of [ABG] and a key technical step in our approach is the
following algebra isomorphism, which is a ”quantum” analogue of [ABG, Theorem 8.5.2] and
whichfollowsfromthetheoremstatedin§1.5:
q
ExtTˇ×C×(RG,Wλ⋆RG) ∼= U~(g)⋉ D~(X)λ.
λ∈X+ λ∈X+
M M
 
Here, Wλ is the Wakimoto sheaf associated with λ, R is an ind-perversesheafon Gr correspond-
G
ing to the regular representation of G, and we refer to [ABG, §8] for this and other unexplained
notation.
1.7. We will also consider “classical analogues” of the above constructions, by which we mean
specializing~to0,hencereplacingS~ byS(t)orS(ˇt∗). TheclassicalanaloguesofMtop areeasyto
define: wesimplyset
q
Mtop := H +λ(2ρˇ)(i! F).
F,λ Tˇ λ
WealsohavemorphismsAtopandConvtopgivenbythesameconstructionsasforAtopandConvtop.
ThereisnointerestingclassicalanalogueofMalg. TheclassicalanaloguesofMgeom aredefined
usingthegeometryoftheGrothendieck–Springerresolutiong. Morepreciselyweset
geom G
MV,λ := V ⊗Γ(g,Oeg(λ))
e
geom
whereOeg(λ) is the G-equivariant line bundl(cid:0)e on g associated(cid:1)with λ. The operatorsConv are
e
inducedbythenaturalmorphisms
Γ(g,Oeg(λ))⊗Γ(g,Oeg(µe)) → Γ(g,Oeg(λ+µ)).
3
e e e
geom
Finally, theoperatorsA are definedusingtheW-action ontheregular part ofg. Again, these
operatorsdependonachoice ofsimplerootvectorsin g. Thisconstructionseemstobenew,and
hasinterestingconsequences(see§5.5). e
Thenweprovethefollowing(seeCorollary2.4.5,Theorem2.5.7andRemark8.1.6(2)).
Theorem. ForF inPerv (Gr)andλ ∈ X,thereexistcanonicalisomorphisms
Gˇ(O)
MtFo,pλ ∼= MgSe(Fom),λ.
Thisfamilyofisomorphisms iscompatiblewithoperators AandwithmorphismsConv.
Themodulesappearinginthetheorem(andthecorrespondingmorphisms)arerelatedtothose
appearinginthetheoremof§1.5bythefunctorC⊗C[~](−)(where~actsbyzeroonC). ForMtop
and Mtop, this easily follows from the parity vanishing ofHq(i! F), see Lemma 6.2.4. For Mgeom
λ
geom
and M , this requires a more subtle argument, see §3.5. In particular, our results establish
a relation between the automorphisms of D~(X) induced by partial Fourier transforms and the
W-actionontheregularpartofg,whichseemstobenew.
e
1.8. Asapplicationsofourconstructionswegivenewproofsoftworesults: ageometricdescrip-
tionoftheBrylinski–Kostantfiltrationduetothefirstauthor(see[G1]),andageometricconstruc-
tionofthedynamical WeylgroupduetoBraverman–Finkelberg(see[BrF]). Wealsoobservethat
someofourtechnicalpreliminaryresultshaveinterestingapplications: theyallowtogivesimpler
proofsofresultsonthestructureofthealgebra D(X) ofdifferential operatorson X(see§3.6)and
toconstructanactionofW ontheregularpartofT∗Xwhich“lifts”theactionontheregularpart
ofg,see§5.5.
OneimportanttoolinthefirstproofofthegeometricSatakeequivalencein[G1]wasspecialized
equeivariant cohomology of cofibers (see in particular [loc. cit., §3.5]), while in [MV] the authors
replaced this tool by cohomology of corestrictions to semi-infinite orbits T . Our descriptions of
q q λ
H (i! F),H (t! F)(wheret denotestheinclusionofT )andthenaturalmorphismbetweenthem
Tˇ λ Tˇ λ λ λ
(seeTheorem2.3.1)shedsomelightonthepreciserelationbetweenthesepointsofview.
1.9. Descriptionofthepaper. InSection2wedefineourmainplayers,andstateourmainresults.
geom
InSection3westudythemodulesM anddefinetheirsymmetries. InSection4westudythe
V,λ
alg
modulesM ,definetheirsymmetries,andrelatethisalgebraicfamilywiththegeometricone. In
V,λ
geom
Section5westudythemodulesM ,definetheirsymmetries,andrelatethemwiththemodules
V,λ
geom
M . In Section 6 we recall the construction of the geometric Satake equivalence and its main
V,λ
properties. In Section 7 we prove our main results. In Section 8 we give some complements
and applications of these results. Finally, the paper finishes with two appendices: Appendix A
collectscomputationsinsemi-simplerankonethatareneededinourproofs,andAppendixBisa
reminderonpartialFouriertransformsfor(asymptotic)D-modules.
1.10. Conventions. Throughout,wewill workoverthegroundfieldCofcomplexnumbersand
write⊗ = ⊗C. IfM = n∈ZMn isagradedvectorspaceandm ∈ Z,wedefinethegradedvector
spaceMhmibythefollowingrule: (Mhmi) = M . Notethath1iisa“homological”shift,i.e.it
n n−m
shifts graded vector spLaces to the right. We will always consider C[~] as a graded algebra where
~ has degree 2. If A and B are C[~]-algebras, by an (A,B)-bimodule we mean an (A⊗C[~] Bop)-
module. IfAisanalgebra,wewriteHom−A(−,−)forHomAop(−,−).
4
1.11. Acknowledgements. Wethank Pramod Achar for usefuldiscussionsat early stagesofthis
work, and Pierre Baumann. Part of this work was completed while the second author visited
UniversityofChicago.
2. STATEMENT OF THE MAIN RESULTS
2.1. Asymptotic Verma modules. Given a filtered C-algebra A = i∈Z≥0FiA, we let A~ be the
Rees algebra (sometimes referred to as “graded” or “asymptotic” version) of the filtered algebra
S
A. ItcanbedefinedasthefollowingsubalgebraofA[~]:
0 ifiisodd;
A~ := i∈Z Ai~ with Ai~ = (~i·Fi/2A ifiiseven. (2.1.1)
M
Thus,A~ isagradedC[~]-algebra, wheretheindeterminate~hasgradedegree2. (Thereasonfor
ourconventionwillbecomeclearlater.) Moreover,onehasanaturalisomorphism
A~/~·A~ ∼= grFA
(wheredegreesaredoubledontheleft-handside).
IfkisaLiealgebra,theenvelopingalgebraU(k)comesequippedwithanaturalascendingfiltra-
tion,thePoincare´-Birkhoff-Wittfiltration,suchthatgrU(k) = S(k). Thecorrespondingasymptotic
enveloping algebra U~(k) := U(k)~ has an alternative (equivalent) definition as the C[~]-algebra
generatedbyk,withrelationsxy−yx = ~[x,y]forx,y ∈ k. Hereelementsofkhavedegree2. We
willusethisdescriptionofU~(k), andstilldenotebyxtheimageofanelementx ∈ k. (Ifwewere
usingthedescription(2.1.1),thiselementshouldratherbedenoted~x.)
LetGbeaconnectedreductivegroupoverC withLiealgebrag. Wefixatriangular decompo-
sitiong = u⊕t⊕u−,sob= t⊕uisaBorelsubalgebra. LetT bethemaximaltorusandB = T ·U
the Borel subgroup corresponding to the Lie algebras t and b, respectively. We will denote by R
the set of roots of G (relative to T), by R+ the positive roots (i.e. the roots of u), and by W the
Weylgroup of(G,T). Letρ ∈ t∗, resp. ρˇ∈ t, be the half sumof positive roots, resp. coroots. We
also let X be the lattice of characters of T, and X+, resp. X−, be the sub-semigroup of dominant,
resp.antidominant,weights. Wewillfrequentlyconsiderelementsoft∗(resp.ofX)aslinearforms
on b (resp. characters of B) which are trivial on u (resp. on U). Also, as usual, when convenient
weidentifyXwithasubsetoft∗ viathedifferential.
We consider asymptotic C[~]-algebras U~(g) and U~(t). The latter algebra is a commutative
graded algebra which is clearly isomorphic to S~ := S(t)[~] where S(t), the symmetric algebra of
t, is equippedwith its natural grading. LetZ(g) be the centerof the algebra U(g). The Poincare´-
Birkhoff-Witt filtration on U(g) induces, by restriction, a filtration on Z(g). The corresponding
asymptotic algebra Z~(g) is the center of the algebra U~(g). One has a Harish-Chandra isomor-
phismZ~(g)∼= SW~ = S(t)W[~],agradedC[~]-algebraisomorphisminducedbythecomposition
Z~(g) ֒→ U~(g)։ U~(g)/(u·U~(g)+U~(g)·u−)←∼− S~ −→∼ S~
where the isomorphismon the right-hand side sendst ∈ t to t+~ρ(t). Using this isomorphism,
wemay(andwill)identifythealgebraZ~(g)withasubalgebraofS~.
For any λ ∈ t∗, let S~hhλii be the U~(b)-bimodule defined as follows. As a C[~]-module, it is
isomorphic to S~. The left U~(b)-module structure is given by the natural isomorphism S~ ∼=
U~(b)/U~(b)·n. Then in theright U~(b)-module structure, theLie ideal u ⊂ b acts by 0, and t ∈ t
actsbymultiplication byt+~λ(t).
We define a graded (U~(b), U~(g))-bimodule, a certain asymptotic version of the universal
Vermamodule,asfollows:
M(λ) = S~hhλ+ρii⊗U~(b)U~(g).
5
(WewillmainlyonlyconsiderM(λ)asan(S~,U~(g))-bimodule.) TheactionofU~(g)onthevector
vλ := 1⊗1∈ M(λ)inducesanisomorphismofrightU~(g)-modules
M(λ) ∼= U~(g)/(u·U~(g)). (2.1.2)
Underthis isomorphism, the right S~-module structure is such that the action oft ∈ t is induced
byrightmultiplicationbyt−~·(λ+ρ)(t)onU~(g).
It is immediate from definitions that there is well defined ‘adjoint’ action b : m 7→ adb(m) of
theLiealgebrabonM(λ),whichisrelatedtothebimodulestructurebytheequation
~·adb(m)= (b+~ρ(b))·m−m·b ∀b ∈b.
The adjoint action of the subalgebra t ⊂ b is semisimple. Therefore, one has a weight decompo-
sition M(λ) = M(λ) . In particular, we have M(λ) = C[~]·v , and M(λ) = 0 unless
µ∈t∗ µ −λ λ µ
µ ∈ −λ − Z R+. For λ ∈ X, the adjoint action on M(λ) can be exponentiated to an algebraic
≥0L
B-action.
Let Rep(G) be the tensor category of finite dimensional rational G-modules. For V ∈ Rep(G)
andλ ∈ X,letV denotetheT-weightspaceofV ofweightλ.
λ
The assignment ~ 7→ 1⊗~, x 7→ −x⊗~+1⊗x has a unique extension to an algebra homo-
morphismU~(g) → U(g)op ⊗U~(g). Via this homomorphism,for any rightU~(g)-module M and
V ∈ Rep(G), the vector space V ⊗M acquires the structure of a right U~(g)-module. This gives
an (U~(b),U~(g))-bimodule structure on V ⊗M(λ), where the left action of the algebra U~(b) on
V ⊗ M(λ) comes from its action on M(λ) on the left. If λ ∈ X, the differential of the diagonal
B-actiononV ⊗M(λ)andthebimodulestructurearerelatedasfollows: ifb ∈ b ⊂ U(b),~times
theactionofbisgivenbytheassignmentn7→ (b+~ρ(b))·n−n·b.
One has a natural morphism of left S~-modules pλ : M(λ) → S~, induced by the projection
U~(g) → U~(b) orthogonal to U~(g) · u−. If λ ∈ X, then pλ is also a morphism of T-modules
M(λ) → S~⊗C−λ. AnimportantrolebelowwillbeplayedbythemorphismofgradedS~-modules
κalg defined(forV inRep(G)andλ ∈X)asthefollowingcomposition:
V,λ
V ⊗M(λ) B ֒→ V ⊗M(λ) T idV⊗pλ // (V ⊗S~⊗C−λ)T = Vλ⊗S~. (2.1.3)
(cid:0) (cid:1) B (cid:0) (cid:1)
Notethat V ⊗M(λ) has a natural structureofZ~(g)-module, induced by the (right)action of
Z~(g) ⊂ U~(g) on M(λ). With this definition, κalg is also a morphism of Z~(g)-modules, where
(cid:0) (cid:1) V,λ
Z~(g) = SW~ actsonS~ viatherestrictionoftherightactionofS~ onS~hhλii.
2.2. The affine Grassmannian: equivariant cohomology of cofibers. Write G for the multi-
m
plicative group. LetGˇ betheLanglandsdualgroupofG. ThegroupGˇ comesequippedwiththe
maximal torus Tˇ ⊂ Gˇ, with opposite Borel subgroups Bˇ = Tˇ ·Uˇ and Bˇ− = Tˇ ·Uˇ−, and with a
canonical isomorphismX = Hom(G ,Tˇ), the cocharacter lattice ofTˇ. (To be completely precise,
m
one should first choose Gˇ,Bˇ,Tˇ, and then use the affine Grassmannian of Gˇ to define G,B,T by
Tannakianformalism;see§6.1fordetails.)
Let K = C((z)), resp. O = C[[z]]. Let Gr := Gˇ(K)/Gˇ(O), resp. Gr := Tˇ(K)/Tˇ(O), be the
Gˇ Tˇ
affine Grassmannian associated with the group Gˇ, resp. Tˇ. (We will consider the reduced ind-
scheme structure on these affine Grassmannians.) Thus, one has X = Gr and there is a natural
Tˇ
embeddingX = Gr ֒→ Gr . Forλ ∈ X,weletλbetheimageofλandleti : {λ} ֒→ Gr denote
Tˇ Gˇ λ Gˇ
theonepointembedding. ThegroupGˇ(K)⋊G actsonGr ontheleft,wherethefactorG acts
m Gˇ m
byrotationoftheloop.
Fortherestofthissection,wewill usesimplifiednotationGr := Gr . Thefollowingsubsetsof
Gˇ
theaffine Grassmannian will play an important role. For λ ∈ X+,welet Grλ := Gˇ(O)·λ. This is
6
a finite dimensional (Gˇ(O)⋊G )-stable locally closed subvariety of Gr. One has a stratification
m
Gr = ⊔λ∈X+ Grλ. Further, for any λ ∈ X, following Mirkovic´-Vilonen one puts Tλ := Uˇ−(K)·λ.
Welett :T ֒→ Grbetheinclusion.
λ λ
LetA := Tˇ×G ,atoralsubgroupofGˇ(K)⋊G . TheMirkovic´-Vilonen spaceT is A-stable.
m m λ
Further, the set X ⊂ Gr is known to be equal to the set of A-fixed points in Gr. Therefore, for
any object F of the equivariant derived category Db(Gr), there are well defined A-equivariant
q q A
cohomology groups H (T ,t! F), resp. H (i! F). These are graded modules over the graded
q A λ λ A λ
algebraH (pt) ∼= S(t)[~] = S~.
A
LetPerv (Gr),resp.Perv (Gr),bethecategoryofGˇ(O)-equivariant,resp.Gˇ(O)⋊G -
Gˇ(O) Gˇ(O)⋊Gm m
equivariant,perversesheavesonGr. LetalsoPerv (Gr)bethecategoryofperversesheaves
Gˇ(O)-mon
onGrwhichareconstructiblewithrespecttothestratificationbyGˇ(O)-orbits. Recallthatallthree
of these categories are semisimple, with simple objects parametrized by X+. In particular, the
forgetfulfunctors
Perv (Gr)→ Perv (Gr) → Perv (Gr)
Gˇ(O)⋊Gm Gˇ(O) Gˇ(O)-mon
areequivalencesofcategories(see[MV,AppendixA]forasimilar resultinamuchmoregeneral
situation). Let
S :Perv (Gr) −→∼ Rep(G)
Gˇ(O)
be the geometric Satake equivalence. By the remark above, any object of Perv (Gr) can be
Gˇ(O)
considerednaturallyasanobjectofDb(Gr).
A
The following lemma is a simple consequence of results of Kazhdan–Lusztig [KL] and Mirko-
vic´–Vilonen[MV],cf.also[YZ,BrF]. Itwillbeprovedin§6.2.
Lemma2.2.1. ForanyF inPerv (Gr )andλ ∈ X,onehas
Gˇ(O) Gˇ
q q
(1) ThegradedS~-moduleH (i! F),resp.thegradedS(t)-moduleH (i! F),isfree.
A λ Tˇ λ
(2) Thereisacanonicalisomorphism ofgraded S~-modules,resp.ofgradedS(t)-modules
q q
HA(Tλ,t!λF) ∼= S(F) λ⊗S~hλ(2ρˇ)i, HTˇ(Tλ,t!λF) ∼= S(F) λ⊗S(t)hλ(2ρˇ)i. (2.2.2)
Onemayfactorthe(cid:0)embe(cid:1)ddingi : {λ} ֒→ Grasacompositio(cid:0)n{λ}(cid:1)−ı→λ T −t→λ Gr. Hence,there
λ λ
isapush-forwardmorphism
q q q
(ı ) : H (i! F) = H (ı! t! F) −→ H (T ,t! F).
λ ! A λ A λ λ A λ λ
top
Letκ bethefollowingcompositemorphism
F,λ
κtFo,pλ : HAq (i!λF) (ıλ)! // HAq (Tλ,t!λF) (2.∼2.2) // S(F) λ⊗S~hλ(2ρˇ)i. (2.2.3)
(cid:0) (cid:1)
Thus,wegetadiagramofmorphismsofgradedS~-modules
S(F)⊗M(λ) Bhλ(2ρˇ)i κaS(lgF),λhλ(2ρˇ)i // S(F) ⊗S~hλ(2ρˇ)i oo κtFo,pλ Hq (i! F).
(2.1.3) λ (2.2.3) A λ
(cid:0) (cid:1) (cid:0) (cid:1)
Oneofourkeyresults(tobeprovedin§7.6)readsasfollows.
Theorem 2.2.4. For any F in Perv (Gr) and λ ∈ X, the morphisms κalg hλ(2ρˇ)i and κtop are
Gˇ(O) S(F),λ F,λ
injectiveandhavethesameimage. Thus,thereisanaturalisomorphismofgradedS~-modulesζF,λthatfits
7
intothefollowingcommutativediagram
S(F)⊗M(λ) Bhλ(2ρˇ)i ζF,λ // Hq(i! F)
(cid:127)_ ∼ A λ(cid:127)_
(cid:0) κa(cid:1)lg hλ(2ρˇ)i κtop (2.2.5)
S(F),λ F,λ
(cid:15)(cid:15) (cid:15)(cid:15)
S(F) ⊗S~hλ(2ρˇ)i S(F) ⊗S~hλ(2ρˇ)i.
λ λ
Remark 2.2.6. Wehave(cid:0)defin(cid:1)edin §2.1 an action ofZ~(g)(cid:0)on (S((cid:1)F)⊗M(λ))B. On the otherhand,
q
itisexplainedin [BF,§2.4]thatHA(i!λF)alsohasanaturalaction ofZ~(g) = SW~ coming fromthe
naturalmapGr = (Gˇ(K)⋊G )/(Gˇ(O)⋊G )→ pt/(Gˇ(O)⋊G ). Weclaimthatourisomorphism
m m m
ζF,λ isalsoZ~(g)-equivariant.
q
First, the action of S~⊗Z~(g) on HA(i!λF) factors through an action of S~⊗C[~]Z~(g) = C[t∗ ×
(t∗q/W)×A1]. Then, the action of S~⊗C[~]Z~(g) factors through the naturalqaction of the algebra
HA(λ). Finally, it is explained in [BF, §3.2] that the S~⊗C[~]Z~(g)-algebra HA(λ) is isomorphic to
thedirectimageunderthenaturalquotientmapofO ,where
Γλ
Γ := {(η ,η ,z) ∈ t∗×t∗×A1 | η = η +zλ}.
λ 1 2 2 1
TheclaimeasilyfollowsfromtheseremarksandtheS~-equivarianceofζF,λ.
Remark 2.2.7. Consider the case F = IC is the IC-sheaf associated with the Gˇ(O)-orbit Grν for
ν
some ν ∈ X, and λ = w ν (where w ∈ W is the longest element). Then Vν := S(IC ) is a
0 0 ν
simple G-module with highest weight ν, and λ is the lowest weight of Vν. In view of the right-
alg
hand isomorphism in Lemma 2.4.1 below, the image of the morphism κ is computed by
Vν,w0ν
Kashiwara in [Ka]: namely, withourconventions,combining Theorem1.7and Proposition1.8 in
loc.cit.weobtainthattheimageofκaVlgν,w0ν inVwν0ν⊗S~ ∼= S~isgeneratedbythefollowingelement:
−ν(w0α)−1
(αˇ−j~) . (2.2.8)
 
α∈R+ j=0
Y Y
 
Thetopologicalcontextiseasyinthiscase. Namelywehave
−ν(w0α)−1
Tw0ν ∩Grν = Tw0ν ∩Grν = Uˇ−(O)·(w0ν) ∼=  C−αˇ+j~
α∈R+ j=0
Y Y
top  
as A-varieties. One can easily deduce that the image of κ is also generated by (2.2.8), see
ICν,w0ν
§6.2. Hence,inthisparticularcase,Theorem2.2.4canbedirectlydeducedfromtheseremarks.
alg top
Inthecaseλ = ν,onecanalsodirectlycheckthatbothκ andκ areisomorphisms.
Vν,ν ICν,ν
2.3. Classical analogue. We will also prove an analogue of Theorem 2.2.4 where one replaces A
byTˇ. InthiscasetherepresentationtheoryofthealgebraU~(g)hastobereplacedbythegeometry
ofthealgebraicvarietyg∗.
We will identify t∗ with the subspace (g/u⊕u−)∗ ⊂ g∗. This way we obtain a canonical mor-
phism q : S(g/u) → S(t) induced by restriction of functions. For V in Rep(G) and λ ∈ X, the
alg alg
“classicalanalogue”ofthemorphismκ ,whichwewilldenotebyκ ,isthecomposition
V,λ V,λ
V ⊗S(g/u)⊗C B ֒→ V ⊗S(g/u)⊗C T −i−d−V−⊗−q⊗−→1 (V ⊗S(t)⊗C )T = V ⊗S(t).
−λ −λ −λ λ
This(cid:0)morphism is S(t)-e(cid:1)quivar(cid:0)iant, where the S(t(cid:1))-action on the left-hand side is induced by the
morphism(g/u)∗ → t∗ givenbyrestrictionoflinearmaps.
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NowweconsiderperversesheavesonGr. ForF in Perv (Gr)andλ ∈ X,wewilldenoteby
Gˇ(O)
top
κ thefollowingcompositemorphism
F,λ
κtop : Hq(i! F) (ıλ)! // Hq(T ,t! F) (2.2.2) // S(F) ⊗S(t)hλ(2ρˇ)i.
F,λ Tˇ λ Tˇ λ λ ∼ λ
(cid:0) (cid:1)
ThentheclassicalanalogueofTheorem2.2.4(tobeprovedin§7.7)readsasfollows.
Theorem 2.3.1. For any F in Perv (Gr) and λ ∈ X, the morphisms κalg hλ(2ρˇ)i and κtop are
Gˇ(O) S(F),λ F,λ
injective andhave the same image. Thus, there isa naturalisomorphism ofgraded S(t)-modules ζ that
F,λ
fitsintothefollowingcommutativediagram
S(F)⊗S(g/u)⊗C Bhλ(2ρˇ)i ζF,λ // Hq(i! F)
(cid:127)_ −λ ∼ Tˇ λ(cid:127)_
(cid:0) κaS(lgF),λ(cid:1)hλ(2ρˇ)i κtFo,pλ (2.3.2)
(cid:15)(cid:15) (cid:15)(cid:15)
S(F) ⊗S(t)hλ(2ρˇ)i S(F) ⊗S(t)hλ(2ρˇ)i.
λ λ
(cid:0) (cid:1) (cid:0) (cid:1)
2.4. Alternative descriptions: differential operators on G/U and intertwining operators for
Verma modules. An important role in our arguments will be played by two alternative descrip-
tionsoftheC[~]-modules V ⊗M(λ) B.
IfX isasmoothalgebraicvariety,wewriteD forthesheafofdifferentialoperatorsonX. The
X
sheaf D comes equipped(cid:0) with a na(cid:1)tural filtration by the order of differential operator. We let
X
D~,X bethecorrespondingsheafofasymptoticdifferentialoperators. Asforenvelopingalgebras,
thisalgebrahasanalternativedescriptionasthesheafofgradedC[~]-algebrasgenerated(locally)
by O in degree 0 and the left O -module T (the tangent sheaf) in degree 2, with relations
X X X
ξ · ξ′ − ξ′ · ξ = ~[ξ,ξ′] for ξ,ξ′ ∈ T and ξ · f − f · ξ = ~ξ(f) for ξ ∈ T and f ∈ O . As for
X X X
enveloping algebras, we will use this description of D~,X and still denote by ξ the image of an
elementξ ∈ T . (Ifwewereusingthedescriptionprovidedby(2.1.1),thiselementshouldrather
X
bedenoted~ξ.) NotethatD~,X actsonOX[~]viaξ·f = ~ξ(f)forξ ∈ TX andf ∈ OX.
WeputD(X) = Γ(X,DX), resp. D~(X) = Γ(X,D~,X), forthecorrespondingalgebra ofglobal
sections. TheorderfiltrationmakesD(X)afilteredalgebra;theassociatedReesalgebraD(X)~ is
canonicallyisomorphictoD~(X). ThereisalsoacanonicalinjectivemorphismD~(X)/~·D~(X) →
Γ(X,D~,X/~·D~,X), which is not surjective in general. Note finally that there exists a canonical
algebra isomorphismD~,X/~·D~,X ∼= (pX)∗OT∗X, where T∗X is thecotangentbundle to X and
p : T∗X → X istheprojection.
X
Consider the quasi-affine variety X := G/U. There is a natural G×T-action on X defined as
follows: g ×t : hU 7→ ghtU. The T-action on X also induces an action of T on D~(X) by algebra
automorphisms. Inparticular,thisT-actiongivesaweightdecompositionD~(X) = λ∈XD~(X)λ.
Thus,D~(X)0 = D~(X)T isthealgebraofrightT-invariant asymptoticdifferentialoperators.
L
Differentiating the T-action on X yields a morphism γ : S~ → D~(X) of graded C[~]-algebras.
Using this we will consider D~(X) as an S~-module where t ∈ t acts by right multiplication by
t−~ρ(t). Notealso thatdifferentiating theG-action onXweobtainamorphismZ~(g) → D~(X).
ThisdefinesaZ~(g)-modulestructureonD~(X)(inducedbymultiplicationontheleft).
If M is an S~-module and ϕ an algebra automorphism of S~, we denoteby ϕM the S~-module
whichcoincideswithM asaC-vectorspace,andwheres ∈ S~ actsasϕ(m)actsonM. Ifϕ,ψ are
algebraautomorphismsofS~ wehave ψ ϕM = ϕ◦ψM. Thisconstructionprovidesan autoequiv-
alenceofthecategoryofS~-modules,actingtriviallyonmorphisms. Wewilluseinparticularthis
(cid:0) (cid:1)
notation when ϕ = w ∈ W (extendedin the natural way to an automorphism of S~), and for the
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