Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs

0.23 MB·English

by  Avraham Aizenbud

HOLONOMICITY OF RELATIVE CHARACTERS AND APPLICATIONS TO
MULTIPLICITY BOUNDS FOR SPHERICAL PAIRS
AVRAHAMAIZENBUD,DMITRYGOUREVITCH,ANDANDREYMINCHENKO
toJosephBernstein-theteacher,themathematician,andtheperson.
ABSTRACT. In this paper, we prove that any relative character (a.k.a. sphericalcharacter) of
any admissible representation of a real reductive group with respect to any pair of spherical
subgroups is a holonomic distribution on the group. This implies that the restriction of the
6
1 relativecharactertoanopendensesubsetisgivenbyananalyticfunction. Theproofisbased
0 onanargumentfromalgebraicgeometryandthusimpliesalso analogousresultsinthep-adic
2 case.

r As an application, we give a shortproofof some results from [KO13; KS16] on bounded-
p
nessandfinitenessofmultiplicitiesofirreduciblerepresentationsinthespaceoffunctionsona
A
sphericalspace.

1 InordertodeducethisapplicationweproverelativeandquantitativeanalogsoftheBernstein-
2 Kashiwaratheorem,whichstatesthatthespaceofsolutionsofaholonomicsystemofdifferen-
tialequationsinthespaceofdistributionsisfinite-dimensional. Wealsodeducethat,forevery
T] algebraicgroupGdefinedoverR,thespaceofG(R)-equivariantdistributionsonthemanifold
R ofrealpointsofanyalgebraicG-manifoldX isfinite-dimensionalifGhasfinitelymanyorbits
onX.
.
h
t
a
m
[
CONTENTS

2
v 1. Introduction 2
9
1.1. Therelativecharacter 2
7
4 1.2. Boundson thedimensionofthespaceofsolutions 3
1 1.3. Applicationstorepresentationtheory 4
0
1.4. Thenon-Archimedeancase 4
.
1
1.5. Related results 5
0
5 1.6. Futureapplications 5
1 1.7. Structureofthepaper 5
:
v 1.8. Acknowledgements 5
i
X 2. ProofofTheorem B 6
r 2.1. ProofofLemma2.0.1 7
a
3. Dimensionofthespaceofsolutionsofa holonomicsystem 7
3.1. Preliminaries 7
3.2. Dimensionofthespace ofsolutionsofaholonomicsystem 10
3.3. FamiliesofD-modules 11
4. ProofofTheoremsA and E 12
4.1. Preliminaries 12
Date:April22,2016.
Keywordsandphrases. Realreductivegroup,HolonomicD-module,equivariantdistribution,Besselfunction,
sphericalpair,momentmap.2010MSClassification:17B08,22E45,22E46,22E50,14M27,46F05,32C38,53D20.
ThethreeauthorswerepartiallysupportedbytheMinervafoundationwithfundingfromtheFederalGerman
MinistryforEducationandResearch;A.A.wasalsopartiallysupportedbyISFgrant687/13,andD.G.andA.M.
byERCStGgrant637912andISFgrant756/12.AtthebeginningofthisprojectD.G.wasateammemberinthe
ERCgrant291612.
1
2 AVRAHAMAIZENBUD,DMITRYGOUREVITCH,ANDANDREYMINCHENKO
4.2. ProofofTheoremA and Proposition1.1.4 13
4.3. ProofofTheoremE 13
AppendixA. ProofofLemma3.1.1 14
References 15
1. INTRODUCTION
1.1. The relative character. Let G be a reductive group1 defined over R. In this paper,
we prove that a relative character (a.k.a. spherical character) of a smooth admissible Fre´chet
representation of moderate growth of G(R) is holonomic. The relative character is a basic
notion of relative representation theory that generalizes the notion of a character of a repre-
sentation. Let us now recall the notions of spherical pair, relative character and holonomic
distribution. For the notion of smooth admissible Fre´chet representation of moderate growth
and its relation to Harish-Chandra modules we refer the reader to [Cas89] or [Wal88, Chapter
11].
Definition 1.1.1. Let H ⊂ G be a (closed) algebraic subgroup defined over R. Let P denote a
minimalparabolicsubgroupofGdefinedoverRandB denoteaBorelsubgroupofG(possibly
notdefined overR). ThesubgroupH iscalled sphericalifithasfinitelymanyorbitsonG/B.
Wewillcall H stronglyreal sphericalifithas finitelymanyorbitsonG/P.
Itis knownthat apair(G,H)isspherical ifand onlyifH has an open orbitonG/B.
Definition 1.1.2. Let H ,H ⊂ G be spherical subgroups and let h be the Lie algebras of H .
1 2 i i
Let χ be characters of h . Let π be a smooth admissible Fre´chet representation of moderate
i i
growth of G(R), π∗ be the continuous dual of π, and πˆ ⊂ π∗ be the smooth contragredient
representationtoπ (i.e. theonlysmoothadmissibleFre´chet representationofmoderategrowth
with the same space underlying Harish-Chandra module as π∗). Let φ ∈ (π∗)h1,χ1 and
1
φ ∈ (πˆ∗)h2,χ2 beequivariantfunctionals. FixaHaarmeasureonG(R). Itgivesrisetoanaction
2
of the space of Schwartz functions S(G(R)) on π∗ and πˆ∗, and this action maps elements of
π∗ and πˆ∗ to elements ofπˆ and πˆˆ = π respectively. Forthedefinition ofthespace ofSchwartz
functionsS(G(R)) see, e.g., [Cas89; Wal88; AG08].
The relative character ξ of π, with respect to φ and φ , is the tempered distribution on
φ1,φ2 1 2
G(R) (i.e. a continuousfunctionalonS(G(R)))defined byhξ ,fi = hφ ,π(f)·φ i.
φ1,φ2 1 2
Definition1.1.3. Let X bean algebraicmanifolddefined overR. Let ξ ∈ S∗(X(R)) beatem-
pereddistribution. Thesingularsupport2SS(ξ)ofξ isthe zerolocusinT∗X ofallthesymbols
of (algebraic) differential operators that annihilate ξ. The distribution ξ is called holonomic if
dimSS(ξ) = dimX.
In thispaper weprovethefollowingtheorem.
Theorem A (See §4.2). In the situation of Definition 1.1.2, the relative character ξ is
φ1,φ2
holonomic.
WeproveTheoremA usingthefollowingwell-knownstatement.
Proposition1.1.4(See §4.2). Let g,h betheLiealgebrasofG andH ,i = 1,2. IdentifyT∗G
i i
with G×g∗ andlet
S := {(g,α) ∈ G×g∗ | α is nilpotent,hα,h i = 0, hα,Ad∗(g)(h )i = 0}.
1 2
Then SS(ξ ) ⊂ S.
φ1,φ2
1Byareductivegroupwemeanaconnectedalgebraicreductivegroup
2a.k.a.characteristicvariety
HOLONOMICITYOFRELATIVECHARACTERS 3
Note that the Bernstein inequality states that the dimension of the singular support of any
non-zero distribution is at least the dimension of the underlying manifold. Thus Theorem A
followsfrom thefollowingmorepreciseversion,whichisthecoreofthispaper.
Theorem B (See §2). WehavedimS = dimG.
LetU := g ∈ G | S ∩T∗G = {(g,0)} .NotethatU isZariskiopensinceS isconicand
(cid:8) g (cid:9)
closed. It iseasy to seethatTheorem Bimpliesthefollowingcorollary.
CorollaryC. ThesetU isa Zariskiopen densesubsetofG.
Thiscorollaryisusefulinviewofthenextproposition,whichfollowsfromProposition1.1.4
and Corollary 3.1.3below.
Proposition1.1.5. The restrictionξ | isan analyticfunction.
φ1,φ2 U(R)
Remark 1.1.6. In general, S has irreducible components that can not lie in SS(ξ ) for any
φ1,φ2
φ ,φ . Indeed, SS(ξ ) is coisotropic by [Gab81; KKS73; Mal79], and thus, by Theorem
1 2 φ1,φ2
A, Lagrangian. On the other hand, one can show that when G = GL4,R, and H1 = H2 =
GL2,R×GL2,R embeddedasblockmatricesinsideG,thevarietyS hasnon-isotropic(andthus
non-Lagrangian)components.
1.2. Bounds onthedimensionofthespaceofsolutions. Nextweapplyourresultstorepre-
sentationtheory. Forthis,weusethefollowingtheorem.
Theorem 1.2.1 (Bernstein-Kashiwara). Let X bean algebraicmanifolddefined over R. Let
{D ξ = 0}
i i=1...n
be a system of linear PDE on S∗(X(R)) with algebraic coefficients. Suppose that the joint
zerosetofthesymbolsofD inT∗X isdimX-dimensional. Then thespaceofsolutionsofthis
i
system isfinite-dimensional.
It seems that this theorem is not found in the literature in this formulation, however it has
two proofs, onedue to Kashiwara(see [Kas74; KK76] for similarstatements)and anotherdue
toBernstein (unpublished).
Inordertomakeourapplicationsinrepresentationtheorymoreprecise,weneedaneffective
version of this theorem. We prove such a version (see Theorem 3.2.2 below) following Bern-
stein’s approach, as it is more appropriate for effective bounds. We use this effective version
to derive a relative version. Namely, we show that if the system depends on a parameter in an
algebraicway,then thedimensionofthespaceofsolutionsisbounded(see §3.3below).
Thisrelativeversionallowsustodeduce thefollowingtheorem.
Theorem D(See§3.3). LetGbeanalgebraicgroupdefinedoverRandletX beanalgebraic
G-manifold with finitely many orbits. Let g be the Lie algebra of G. Let E be an algebraic
G-equivariant bundle on X. Then, for any natural number n ∈ N, there exists C ∈ N such
n
thatforeveryn-dimensionalrepresentationτ ofgwe have
dimHom (τ,S∗(X(R),E)) ≤ C ,
g n
where Hom denotes thespaceofall continuousg-equivariantmaps.
g
Remark 1.2.2. Note that the condition that G has finitely many orbits on X is equivalent to
G(C) having finitely many orbits on X(C) but not equivalent to (and not implied by) G(R)
havingfinitelymanyorbitson X(R).
4 AVRAHAMAIZENBUD,DMITRYGOUREVITCH,ANDANDREYMINCHENKO
1.3. Applications to representation theory. Using §3, we give a short proof of some results
from [KO13; KS16]. Namely,weprove:
Theorem E (See §4). Let G be a reductive group defined over R, H ⊂ G be a Zariski closed
subgroup,andh betheLiealgebraofH.
(i) If H is a strongly real spherical subgroup then, for every irreducible smooth admissible
Fre´chet representation of moderate growth π ∈ Irr(G(R)), and natural number n ∈ N
thereexists C ∈ N suchthatforeveryn-dimensionalrepresentationτ ofh we have
n
dimHom (π,τ) ≤ C .
h n
(ii) If H is a spherical subgroup and we consider only one-dimensional τ then the space is
universally bounded, i.e. there exists C ∈ N such that dim(π∗)h,χ ≤ C for any π ∈
Irr(G(R))) andanycharacterχof h.
CorollaryF. Let GbeareductivegroupdefinedoverR, H ⊂ GbeaZariskiclosedreductive
subgroup,andh betheLiealgebraofH.
(i) If the diagonal ∆H is a strongly real spherical subgroup in G × H then for every π ∈
Irr(G(R))) andτ ∈ Irr(H(R))) wehavefinitemultiplicities,i.e.
dimHom (π,τ) < ∞.
h
(ii) Ifthediagonal∆H isasphericalsubgroupinG×H thenthemultiplicitiesareuniversally
bounded, i.e.,thereexists C ∈ N suchthatfor everyπ ∈ Irr(G(R))), τ ∈ Irr(H(R))) we
have
dimHom (π,τ) ≤ C.
h
This corollary follows from Theorem E since Hom (π,τ) lies in the space of ∆h-invariant
h
functionalsonthecompletedtensorproductπ⊗τˆ ∈ Irr(G(R))×H(R)))(see[AG09a, Corol-
lary A.0.7 and Lemma A.0.8]). All symmetricbpairs satisfying the conditions of the corollary
wereclassified in [KM14].
TheinverseimplicationsforTheorem E(ii)and Corollary F(ii)areprovenin[KO13].
Theresultsonmultiplicitiesin[KO13;KS16]arestrongerthanTheoremEsincetheydonot
require H to be algebraic, and consider maps from the Harish-Chandra space of π to τ. Also,
in [KO13; KS16] Theorem E(i) and Corollary F(i) are proven in the wider generality of real
spherical subgroups.
In addition, [KO13, Theorem B] implies that if H ⊂ G is an algebraic spherical subgroup
there exists C ∈ N such that dimHom (π,τ) ≤ Cdimτ, for every π ∈ Irr(G(R)) and every
h
finite-dimensional continuous representation τ of H(R). It is easy to modify our proof of
TheoremE(ii)toshowtheboundednessofmultiplicitiesforanyπ ∈ Irr(G(R)) and anyτ ofa
fixeddimension,buttheproofthatthebounddependslinearlyonthisdimensionwouldrequire
morework.
Our methods are different from the methods of [KO13], which in turn differ from the ones
of[KS16], and theboundsgivenin thethreeworksare probablyverydifferent.
1.4. The non-Archimedean case. Theorem B and Corollary C hold over arbitrary fields of
characteristiczero. Theyareuseful alsoforp-adiclocalfields F,sincetheanalogsofProposi-
tions 1.1.4 and 1.1.5 hold in this case, see [AGS15, Theorem A and Corollary F]. Namely, we
havethefollowingtheorem.
Theorem 1.4.1 ([AGS15]). Let G be a reductive group defined over a non-Archimedean field
F of characteristic 0 and let ξ be a relative character of a smooth admissible representation
with respect to two spherical subgroups H ,H ⊂ G. Let S and U be the sets defined in
1 2
Proposition1.1.4andCorollaryC. Then
(i) The wavefrontsetof ξ liesin S(F).
HOLONOMICITYOFRELATIVECHARACTERS 5
(ii) The restrictionof ξ toU(F) is given bya locallyconstantfunction.
1.5. Relatedresults. Inthegroupcase,i.e. thecasewhenG = H×H andH = H = ∆H ⊂
1 2
H × H, Theorem A essentially becomes the well-known fact that characters of admissible
representationsare holonomicdistributions.
As we mentioned above, Theorem E was proven earlier in [KO13; KS16] using different
methods. An analog of Theorem E(i) over non-Archimedean fields is proven in [Del10] and
[SV, Theorem 5.1.5]formanyspherical pairs,includingarbitrary symmetricpairs.
ThegroupcaseofCorollaryC, Proposition1.1.5, andTheorem1.4.1(ii)is(theeasy partof)
the Harish-Chandra regularity theorem (see [HC63; HC65]). Another known special case of
theseresultsistheregularityofBessel functions,see[LM;AGK15;AG15].
1.6. Futureapplications. OurproofofTheoremE(ii)doesnotusetheCasselmanembedding
theorem (Theorem 4.1.3). This gives us hope that it can be extended to the non-Archimedean
case. Themain difficultyisthefact that ourproofheavilyrelies on thetheory ofmodulesover
the ring of differential operators, which does not act on distributions in the non-Archimedean
case. However, in view of Theorem 1.4.1 we believe that this difficulty can be overcome.
Namely,onecandeduceananalogofTheoremE(ii)formanysphericalpairsfromthefollowing
conjecture.
Conjecture1.6.1. LetGbeareductivegroupdefinedoveranon-Archimedeanfield F ofchar-
acteristic 0 and let H ,H ⊂ G be its (algebraic) spherical subgroups. Let χ be characters of
1 2 i
H (F). Fix acharacter λ oftheBernsteincenter z(G(F)).
i
Thenthespaceofdistributionswhich are:
(1) left (H (F),χ )-equivariant,
1 1
(2) right(H (F),χ )-equivariant,
2 2
(3) (z(G(F)),λ)-eigen,
isfinite-dimensional. Moreover,thisdimensionisuniformlyboundedwhenλ varies.
Note that Theorem B and Theorem 1.4.1(i) imply that the dimension of (the Zariski closure
of)thewavefrontsetofadistributionthatsatisfies(1-3)doesnotexceeddimG. Inmanyways
thewavefrontsetreplacesthesingularsupport,inabsenceofthetheoryofdifferentialoperators
(see, e.g., [Aiz13; AD15;AGS15; AGK15]). Thus,inordertoproveConjecture1.6.1, itisleft
toproveanalogsofTheorems 1.2.1and 3.2.2fortheintegralsystemofequations(1-3).
1.7. Structureofthepaper. In§2,weproveTheoremBusingatheoremofSteinberg[Ste76]
concerning theSpringerresolution.
In§3,weproveaneffectiveversionofTheorem1.2.1,andthenadaptittoalgebraicfamilies.
Wealso deriveTheoremD.
In §4, we derive Theorem E from Theorem B and §3. We do that by embedding the multi-
plicityspaceintoacertain spaceofrelativecharacters.
In Appendix A, we prove Lemma 3.1.1 which computes the pullback of the D-module of
distributionswithrespect to aclosedembedding. Weusethislemmain§3.
1.8. Acknowledgements. WethankEitanSayagandBernhardKroetzforfruitfuldiscussions.
We thank Joseph Bernstein for telling us the sketch of his proof of Theorem 1.2.1. We thank
ToshiyukiKobayashi,AlexanderShamov,and theanonymousreferee forusefulremarks.
A.A.andD.G.willalwaysbegratefultoJosephBernsteinforintroducingthemtotheamaz-
ingworldofalgebra,forsharinghisknowledge,hisapproachestoproblemsandhisphilosophy
foralready morethanhalfoftheirlives,andforbeing ashiningexampleforever.
6 AVRAHAMAIZENBUD,DMITRYGOUREVITCH,ANDANDREYMINCHENKO
2. PROOF OF THEOREM B
It is enough to prove the theorem for a reductive group G defined over an algebraically
closed field of characteristic 0. Since S includes the zero section of T∗G ∼=G × g∗, we have
dimS ≥ dimG. Thus,itisenoughtoprovethatdimS ≤ dimG. LetB denotetheflagvariety
ofG and N ⊂ g∗ denotethenilpotentcone. SinceG isreductive,wecan identify
T∗B ∼= {(B,X)∈ B ×g∗|X ∈ (LieB)⊥}.
Recall the Springer resolution µ : T∗B → N defined by µ(B,X) = X and consider the
followingdiagram.
(1) T∗B×T∗B G×N
❚❚❚❚❚❚❚❚µ❚×❚µ❚❚❚❚❚❚)))) uu❦❦❦❦❦❦❦α❦❦❦❦❦❦❦
N ×N
res
(cid:15)(cid:15)
h∗ ×h∗
1 2
Here, α is defined by α(g,X) = (X,Ad∗(g−1)X), and res is the restriction. Passing to the
fiber of0 ∈ h∗ ×h∗, weobtainthefollowingdiagram.
1 2
(2) L ×L S
1 2❙❙❙❙❙❙❙❙µ❙′❙❙❙❙❙❙)))) vv♠♠♠♠♠♠♠α♠′♠♠♠♠♠♠♠♠
N ×N
h1 h2
Here, N := N ∩h⊥ andL := {(B,X) ∈ T∗B|X ∈ h⊥}.Weneed toestimatedimS. We
hi i i i
do itusingthefollowinglemma.
Lemma 2.0.1 (See §2.1 below). Let ϕ : X → Y, i = 1,2, be morphisms of algebraic
i i
varieties. Supposethatϕ issurjective. Then thereexistsy ∈ Y such that
2
dimX ≤ dimX +dimϕ−1(y)−dimϕ−1(y).
1 2 1 2
By this lemma, applied to φ = α′ and φ = µ′, it is enough to estimate the dimensions of
1 2
L and ofthefibers ofµ′ and α′.
i
Lemma 2.0.2. WehavedimL = dimL = dimB.
1 2
Proof. Since H has finitely many orbits in B, it is enough to show that L is the union of the
i i
conormalbundlestotheorbitsofH inB. LetB ∈ B,andb = LieB,andidentifyT B ∼= g/b.
i B
Then T (H ·B) ∼= h /(b∩h ) and the conormal space at B to the H -orbit of B is identified
B i i i i
withb⊥ ∩h⊥. (cid:3)
i
Let(η,Ad∗(g)η) ∈ Im(α′). Thefiber(α′)−1(η,Ad∗(g)η)is isomorphictothestabilizerG ,
η
and the dimension of the fiber (µ′)−1(η,Ad∗(g)η) is twice the dimension of the Springer fiber
µ−1(η). Recall thefollowingtheorem ofSteinberg (conjecturedby Grothendieck):
Theorem 2.0.3 ([Ste76, Theorem4.6]).
dimG −2dimµ−1(η) = rkG.
η
HOLONOMICITYOFRELATIVECHARACTERS 7
UsingLemma2.0.1, weobtainforsome(η,ad∗(g)η):
dimS ≤ dim(L ×L )+dim(a′)−1(η,ad∗(g)η)−dim(µ′)−1(η,ad∗(g)η) =
1 2
= 2dimB +dimG −2dimµ−1(η) = 2dimB+rkG = dimG.
η
(cid:3)
2.1. Proof of Lemma 2.0.1. Recall that, for a dominantmorphismϕ : X → Y of irreducible
varieties, there exists an open dense U ⊂ Y such that dimX = dimY + dimϕ−1(y) for all
y ∈ U (see, e. g., [Mum99, Theorem 1.8.3]). Let Z be an irreducible component of X of
1
maximaldimensionandW ⊂ Y betheZariskiclosureofϕ (Z). SinceW isirreducible,there
1
existsan open denseU ⊂ W suchthat
(3) dimX =dimZ ≤ dimW +dimϕ−1(y)
1 1
for all y ∈ U. Let V ⊂ U be an open dense subset such that ϕ−1(V) intersects those and
2
only those irreducible components C ,...,C of ϕ−1(W) that map dominantly to W. Note
1 j 2
that j > 0 since ϕ is surjective. Moreover, without loss of generality, we may assume that,
2
for every 1 ≤ i ≤ j, all fibers overV of the restriction of ϕ to C are of the same dimension.
2 i
Since one of these dimensions has to be equal to dimϕ−1(V)−dimV, we have, that there is
2
an 1 ≤ i ≤ j suchthat,forall y ∈ V,
(4)
dimV = dimC −dim(ϕ | )−1(y) ≤ dimϕ−1(V)−dimϕ−1(y) ≤ dimX −dimϕ−1(y).
i 2 Ci 2 2 2 2
Thanksto dimV = dimW,takinganyy ∈ V, formulas(3)and (4)implythestatement. (cid:3)
3. DIMENSION OF THE SPACE OF SOLUTIONS OF A HOLONOMIC SYSTEM
Inthissection,weproveaneffectiveversionofTheorem1.2.1,andthenadaptittoalgebraic
families. WealsoderiveTheorem D.
3.1. Preliminaries.
3.1.1. D-modules. In this section, we will use the theory of D-modules on algebraic varieties
over an arbitrary field k of characteristic zero. We will now recall some facts and notions that
we will use. Fora good introductionto thealgebraic theory of D-modules, we refer the reader
to[Ber] and[Bor87]. Forashort overview,see[AG09a, AppendixB].
By a D-module on a smooth algebraic variety X we mean a quasi-coherent sheaf of right
modules over the sheaf D of algebras of algebraic differential operators. By a finitely gen-
X
erated D-module on a smooth algebraic variety X we mean a coherent sheaf of right modules
overthesheafD . Denotethecategory ofD -modulesbyM(D ).
X X X
Fora smoothaffine varietyV, we denoteD(V) := D (V). Notethat thecategory M(D )
V V
ofD-modulesonV isequivalenttothecategoryofD(V)-modules. Wewillthusidentifythese
categories.
The algebra D(V) is equipped with a filtration which is called the geometric filtration and
defined by the degree of differential operators. The associated graded algebra with respect to
this fitration is the algebra O(T∗V) of regular functions on the total space of the cotangent
bundle of V. This allows us to define the singular support of a finitely generated D-module
M on V in the following way. Choose a good filtration on M, i.e. a filtration such that the
associatedgradedmoduleisafinitely-generatedmoduleoverO(T∗V),and definethesingular
support SS(M) to be the supportof this module. One can show that the singularsupport does
notdepend onthechoiceofagoodfiltrationon M.
This definition easily extends to the non-affine case. A finitely generated D-module M on
X is called smooth if SS(M) is the zero section of T∗X. This is equivalentto being coherent
over O and to being coherent and locally free over O . The Bernstein inequality states that,
X X
8 AVRAHAMAIZENBUD,DMITRYGOUREVITCH,ANDANDREYMINCHENKO
for any non-zero finitely generated M, we have dimSS(M) ≥ dimX. If the equality holds
thenM is calledholonomic.
For a closed embedding i : X → Y of smooth affine algebraic varieties, define the functor
i! : M(D ) → M(D ) by i!(M) := {m ∈ M |I m = 0}, where I ⊂ O(Y) is the ideal
Y X X X
of all functions that vanish on X. It has a left adjoint functor i : M(D ) → M(D ), given
∗ X Y
bytensorproductwithi!(D ). Thefunctori isanequivalenceofcategoriesbetweenM(D )
Y ∗ X
and the category of D -modules supported in X. Both i and i! map holonomic modules to
Y ∗
holonomicones.
IfV isanaffinespacethenthealgebraD(V)hasanadditionalfiltration,calledtheBernstein
filtration. It is defined by deg(∂/∂x ) = deg(x ) = 1, where x are the coordinates in V. This
i i i
givesriseto thenotionofBernstein’s singularsupport,thatwe willdenoteSS (M) ⊂ T∗V ∼=
b
V ⊕V∗. It isknownthatdimSS(M) = dimSS (M).
b
We will also use the theory of analytic D-modules. By an analytic D-module on a smooth
complex analytic manifold X we mean a coherent sheaf of right modules over the sheaf DAn
X
of algebras of differential operators with analytic coefficients. All of the above notions and
statements, except those concerning the Bernstein filtration, have analytic counterparts. In
addition,all smoothanalyticD-modulesofthesamerank are isomorphic.
3.1.2. Distributions. We will use the theory of distributions on differentiable manifolds and
the theory of tempered distributions on real algebraic manifolds, see e.g. [Ho¨r90; AG08].
For an algebraic manifold X defined over R, we denote the space of real valued distribu-
tions on X(R) by D′(X(R),R) := (C∞(X(R),R))∗ and the space of real valued tempered
c
distributions (a.k.a. Schwartz distributions) by S∗(X(R),R) := (S(X(R),R))∗. Similarly,
denote by D′(X(R)) := D′(X(R),C) the space of complex valued distributions on X(R)
and by S∗(X(R)) := S∗(X(R),C) the space of complex valued tempered distributions on
X(R). Also, for an algebraic bundle E (complex or real) over X we denote D′(X(R),E) :=
(C∞(X(R),E))∗ andS∗(X(R),E) := (S(X(R),E))∗.
c
The algebra D(X) acts on the spaces D′(X(R),R) and S∗(X(R),R). Thus, for affine
X we can consider these spaces as D -modules and D′(X(R),R) also as an analytic D -
X X
module. Similarly,wewillregardthespacesD′(X(R))andS∗(X(R))asD -modules,where
XC
XC denotes the complexification of X. For non-affine X, define the DX-module SX∗,R by
S∗ (U) := S∗(U(R),R). It is easy to see that this sheaf is quasi-coherent. Denote by S∗ the
X,R X
D -moduleobtainedby complexificationofS∗ .
XC X,R
We define the singular support of a tempered distribution to be the singular support of the
D-moduleitgenerates. ItiswellknownthatthisdefinitionisequivalenttoDefinition1.1.3. We
say thatadistributionisholonomicifitgenerates aholonomicD-module.
Lemma 3.1.1 (See Appendix A). Let i : X → Y be a closed embedding of smooth affine
algebraicvarietiesdefined over R. Then
D′(X(R)) ∼= i!(D′(Y(R))) andS∗(X(R)) ∼= i!(S∗(Y(R))).
Lemma 3.1.2. Let M be a smooth D(An)-module of rank r. Embed the space An(Cn) of
C
analyticfunctionson Cn intoD′(Rn) usingtheLebesgue measure. Then
HomD(AnC)(M,D′(Rn)) = HomD(AnC)(M,An(Cn))and dimHomD(AnC)(M,D′(Rn)) = r.
Proof. Let MAn := M ⊗O(Cn)An(Cn) andDAn(AnC) := D(AnC)⊗O(Cn)An(Cn) betheanaly-
tizationsofM and D(An). Then
C
HomD(AnC)(M,D′(Rn)) ∼= HomDAn(AnC)(MAn,D′(Rn)).
SinceM is alsosmooth,M ∼= An(Cn)r. Thusit isleft toprovethat
An An
HomDAn(AnC)(An(Cn),D′(Rn)) = HomDAn(AnC)(An(Cn),An(Cn))
HOLONOMICITYOFRELATIVECHARACTERS 9
and the latter space is one-dimensional. This follows from the fact that a distribution with
vanishingpartialderivativesis amultipleoftheLebesguemeasure. (cid:3)
Corollary3.1.3. If adistributiongeneratesa smoothD-module,then itis analytic.
3.1.3. Liealgebraactions.
Definition 3.1.4. Let X be an algebraic manifold defined over a field k and g be a Lie algebra
overk.
(i) An action of g on X is a Lie algebra map from g to the algebra of algebraic vector fields
on X.
(ii) Assume that X is affine, fix an action of g on X and let E be an algebraic vector bundle
onX. LetM bethespaceofglobalregular(algebraic)sectionsofE. AnactionofgonE
isa linearmap T : g → End (M) such that,foranyα ∈ g, f ∈ O(X), v ∈ M, wehave
k
T(α)(fv) = (αf)v+fT(α)v.
(iii) Thedefinitionaboveextendsto non-affineX in astraightforwardway.
3.1.4. Weilrepresentation.
Definition 3.1.5. Let V be a finite-dimensional real vector space. Let ω be the standard sym-
plectic form on V ⊕ V∗. Denote by pV : V ⊕V∗ → V and pV∗ : V ⊕V∗ → V∗ the natural
projections. Definean actionofthesymplecticgroupSp(V ⊕V∗) on thealgebra D(V) by
(∂v)g := π(g)(∂v) := pV∗(g(v,0))+∂pV(g(v,0)), wg := π(g)w := pV∗(g(0,w))+∂pV(g(0,w))
where v ∈ V, w ∈ V∗, ∂ denotes the derivativein the direction of v, and elements of V∗ are
v
viewedaslinearpolynomialsandthusdifferentialoperatorsoforderzero. ForaD(V)-module
M and an element g ∈ Sp(V ⊕ V∗), we will denote by Mg the D(V)-module obtained by
twistingtheactionofD(V) byπ(g).
Since the above action of Sp(V ⊕ V∗) preserves the Bernstein filtration on D(V), the fol-
lowinglemmaholds.
Lemma 3.1.6. For a finitely generated D(V)-module M and g ∈ Sp(V ⊕ V∗) we have
SS (Mg) = gSS (M).
b b
Theorem 3.1.7([Wei64]). Thereexistsatwo-foldedcoverp : Sp(V ⊕V∗) → Sp(V ⊕V∗)(R)
f
andarepresentationΠofSp(V ⊕V∗)onthespaceS∗(V)oftempereddistributionsonV such
f
that,foranyα ∈ D(V), g ∈ Sp(V ⊕V∗), ξ ∈ S∗(V), we have
f
Π(g)(ξα) = (Π(g)ξ)αp(g).
Corollary 3.1.8. We have an isomorphism of D(V)-modules S∗(V)g ∼= S∗(V) for any g ∈
Sp(V ⊕V∗)(R).
In fact, thiscorollarycan bederiveddirectlyfrom theStone-von-Neumanntheorem.
3.1.5. Flatmorphisms.
Lemma 3.1.9. Let φ : X → Y be a proper morphism of algebraic varieties defined over a
field k and M be a coherent sheaf on X. Then there exists an open dense U ⊂ Y such that
M|φ−1(U) isflatover U.
Proof. By [EGA IV, The´ore`me II.3.I], the set V of scheme-theoretic points x ∈ X for which
M is φ-flat at x is open in X. Since φ is proper, the set Z := φ(X \V) is closed in Y. Note
that M is flat over U := X \Z, since φ−1(U) ⊂ V. Moreover, U contains the generic points
oftheirreduciblecomponentsofY. Hence, U ⊂ Y is dense. (cid:3)
10 AVRAHAMAIZENBUD,DMITRYGOUREVITCH,ANDANDREYMINCHENKO
Lemma 3.1.10 (See, e. g., [Mum74, Corollary on p. 50]). Let φ : X → Y be a proper mor-
phismof algebraicvarieties defined over a fieldk andM bea coherent sheaf on X thatisflat
over Y. Fora pointy ∈ Y,let M denotethepullbackofM toφ−1(y). Then thefunction
y
∞
y 7→ χ(My) = X(−1)idimk(y)Hi(My)
i=0
islocallyconstant.
Corollary 3.1.11. Let Y be an algebraic variety defined over a field k and M be a coherent
sheafon Y ×Pn. Then thereexistsan open denseU ⊂ Y suchthat theHilbertpolynomial3of
M doesnot dependon y aslongas y ∈ U.
y
3.2. Dimensionofthe spaceofsolutionsofa holonomicsystem.
Definition3.2.1. Let k beafield ofcharacteristiczero.
(i) Let M be a finitely generated D-module over an affine space An over k. Let Fi be a
k
good filtrationon M with respect to theBernstein filtration on thering DAn. Let p bethe
k
corresponding Hilbert polynomial of M, i.e. p(i) = dimFi for large enough i. Let d be
thedegreeofpanda betheleadingcoefficientofp. DefinetheBernsteindegreeofM to
d
be deg (M) := d!a . It is well-knownthat d and a do not depend on the choice ofgood
b d d
filtrationFi.
(ii) Let M be a finitely generated D-module over a smooth algebraic variety X defined over
k. Let X = l U be an open affine cover of X and let φ : U ֒→ Ani be closed
Si=1 i i i k
embeddings. Denote
l
deg{(Ui,φi)}(M) := Xdegb((φi)∗(M|Ui)).
i=1
Define the global degree of M by deg(M) := mindeg (M), where the minimum
{(Ui,φi)}
istaken overthesetofall possibleaffinecoversand embeddings.
In thissubsection,weprove
Theorem 3.2.2. Let X be an algebraicmanifold defined over R. Let M be a holonomic right
D -module. Then dimHom(M,S∗(X)) ≤ deg(M).
XC
Wewillneed thefollowinggeometriclemmas
Lemma3.2.3. LetV beavectorspaceoveranalgebraicallyclosedfield,L ⊂ V beasubspace
andC ⊂ V beaclosedconicalgebraicsubvarietysuchthatL∩C = {0}. Thentheprojection
p : C → V/Lisa finitemap.
Proof. By induction, it is enough to provethecase dimL = 1. Choose coordinates x ,...,x
1 n
on V such that thecoordinatesx ,...,x vanishon L. Let p bea homogeneouspolynomial
1 n−1
that vanishes on C but not on L. Write p = d g xi, where each g is a homogeneous
Pi=1 i n i
polynomial of degree d−i in x ,...,x . Then x | satisfies a monic polynomial equation
1 n−1 n C
withcoefficients inO(V/L). (cid:3)
Lemma 3.2.4. Let W be a 2n-dimensional real symplectic vector space, and C ⊂ WC be a
closed conicsubvarietyof dimensionn. Then there exists a real LagrangiansubspaceL ⊂ W
suchthatLC ∩C = {0}.
3ForthedefinitionofHilbertpolynomialsee[Har77,ChapterIII,Exercise5.2].