Distributions on homogeneous spaces and applications
53 Pages
English
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Distributions on homogeneous spaces and applications

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53 Pages
English

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In pr og re ss Distributions on homogeneous spaces and applications N. Ressayre? May 20, 2011 Abstract Contents 1 Introduction 2 2 Infinitesimal filtrations 6 2.1 The case of a vector space . . . . . . . . . . . . . . . . . . . . . . 6 2.2 The case of manifolds . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 The case of varieties . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Infinitesimal filtration and cohomology 13 3.1 Filtration of differential forms on a manifold . . . . . . . . . . . . 13 3.2 Filtration of the cohomology . . . . . . . . . . . . . . . . . . . . 15 3.3 Cohomology with complex coefficients . . . . . . . . . . . . . . . 16 3.4 The case of a smooth complex variety . . . . . . . . . . . . . . . 17 4 Isomorphism with Belkale-Kumar's product 19 4.1 Infinitesimal filtration of G/P . . . . . . . . . . . . . . . . . . . .

  • cominuscule then

  • then

  • tangent space

  • belkale- kumar product

  • intersecting schubert

  • kostant's harmonic

  • schubert varieties


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Contents

1 Introduction

on homogeneous spaces and
applications


N. Ressayre

May 20, 2011

Abstract

2 Infinitesimalfiltrations
2.1 Thecase of a vector space. . . . . . . . . . . . . . . . . . . . . .
2.2 Thecase of manifolds. . . . . . . . . . . . . . . . . . . . . . . .
2.3 Thecase of varieties. . . . . . . . . . . . . . . . . . . . . . . . .
3 Infinitesimalfiltration and cohomology
3.1 Filtrationof differential forms on a manifold .. . . . . . . . . . .
3.3 CohomologywithIcomnplex coepfficierntso. . .g. .r. . .e. .s. .s. . .
3.2 Filtrationof the cohomology. . . . . . . . . . . . . . . . . . . .
3.4 Thecase of a smooth complex variety. . . . . . . . . . . . . . .

2

6
6
12
13

13
13
15
16
17

4 Isomorphismwith Belkale-Kumar’s product19
4.1 Infinitesimalfiltration ofG/P. . . . . . . . . . . . . . . . . . . .19
4.2 Belkale-Kumar’sproduct .. . . . . . . . . . . . . . . . . . . . . .20
4.3 Gradeddimension of Schubert varieties. . . . . . . . . . . . . .21
4.4 Thestatements . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
4α˜p
4.5 Anupper bound fordim(F H(G/P,C)). . . . . . . . . . . .22
4.6 Kostant’sharmonic forms. . . . . . . . . . . . . . . . . . . . . .23
4.6.1 Therole of Kostant’s harmonic forms in this paper. . . .23
4.6.2 RestrictiontoK. . . . . . . . . . . . . .23-invariant forms
4.6.3 TheLie algebrar24. . . . . . . . . . . . . . . . . . . . . . .
˜
• ∗
4.6.4 TheΓ-filtration of∧r. . . . . . . . . . . . . . . . . . .25

Université Montpellier II - CC 51-Place Eugène Bataillon - 34095 Montpellier Cedex 5
France -ressayre@math.univ-montp2.fr

1

5

6

7

4.7
4.8

• −
4.6.5 ActionofLon∧u. . . . . . . . . . . . . . . . . . . . .
4.6.6 Afirst differential form. . . . . . . . . . . . . . . . . . .
4.6.7 AnHermitian product onr. . . . . . . . . . . . . . . . .
• ∗
4.6.8 Operatorson∧(r). . . . . . . . . . . . . . . . . . . . .
4.6.9 Kostant’stheorem .. . . . . . . . . . . . . . . . . . . . .
Proof of Theorem 4 .. . . . . . . . . . . . . . . . . . . . . . . . .
Belkale-Kumar fundamental class. . . . . . . . . . . . . . . . . .

Intersecting Schubert varieties

5.1 ProductsonH(G/P,C). . . . . . . . . . . .and Bruhat orders
5.2 LikeRichardson’s varieties .. . . . . . . . . . . . . . . . . . . . .
5.3 Conjecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Interpretationin terms of Harmonic forms. . . . . . . . . . . . .

The case of the complete flag varieties

The case of the Grassmannian
7.1 Schubertvarieties .. . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Liu’srule .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Liu’salgorithm and Littlewood-Richardson rule. . . . . . . . . .
7.4 Varietyassociated to shapes .. . . . . . . . . . . . . . . . . . . .
7.5 Coloredshapes and associated variety. . . . . . . . . . . . . . .

26
26
26
27
28
29
29

30
30
31
34
35

37

38
38
38
42
42
44

8 Infinitesimalfiltration ofG/Pand Schubert varieties44
8.1 Peterson’sapplication .. . . . . . . . . . . . . . . . . . . . . . .44
8.2 Alemma ofT49. . . . . . . . . . . . . . . . . . . . . . .-varieties .
8.3 Schubertvarieties .. . . . . . . . . . . . . . . . . . . . . . . . . .50
In progress
1 Introduction
LetGbe a complex semisimple group and letPbe a parabolic subgroup of
Gthis paper, we are interested in the Belkale-Kumar product. In⊙0on the
cohomology group of the flag varietyG/P.

Belkale-Kumar’s product.Fix a maximal torusTand a Borel subgroupB
such thatT⊂B⊂P. LetWandWPdenote respectively the Weyl groups ofG
P
andP. LetWbe the set of the minimal length representative in the cosets of
P
W/WP. Foranyw∈W, letXwbe the corresponding Schubert variety (that

is, the closure ofBwP /P) and let[Xw]∈H(G/P,C)be its cohomology class.
w
The structure coefficientscof the cup product are written as
uv
X
w
[X] [
u.[Xv] =cuvXw].(1)
P
w∈W

LetLbe the Levi subgroup ofPcontainingTgroup acts on the tan-. This
gent spaceTP /PG/PofG/Pat the base pointP /Pthis action is. Moreover,

2

multiplicity free and we have a canonical decomposition

P /P1⊕Vs,
T G/P=V⊕ ∙ ∙ ∙

(2)

P
as sum of irreducibleL-modules. Itturns out that, for anyw∈W, the
−1−1
paceT:=XT w
tangent sw P/P wof the varietyw Xwat the smooth point
P /Pdecomposes as

Tw= (V1∩Tw)⊕ ∙ ∙ ∙ ⊕(Vs∩Tw).

(3)

i
SetT∩V. Since[X
w:=Tw iw]has degree2(dim(G/P)−dim(Tw))in the graded
∗w
algebraH(G/P), ifc6= 0then
uv

that is

dim(Tu) + dim(Tv) = dim(G/P) + dim(Tw),

s s
X X
i ii
dim(T) + di
um(Tvdim() =Vi) + dim(T).
w
i=1i=1

(4)

(5)

The Belkale-Kumar product asks for equality (5) holds term by term.More
w
precisely, the structure constantsc˜of the Belkale- Kumar product [BK06],
uv
X
w
[X](6)
[Xu]⊙0[Xv] =c˜uv w
P
w∈W

can be defined as follows (see [RR11, Proposition 2.4]):

w ii i
≤sdim(T+ d) = dim(V) + dim(T),
wcif∀1≤iu) im(Tv iw
uv
c˜ =(7)
uv
The Belkale-Kumar produIct hnas provepn torbeothe mgorerreleevanst prsoduct for
0otherwise.
describing the Littlewood-Richardson cone (see [BK06, Res10a, Res10b]).

Motivations.IfG/Pis cominuscule thenTP /PG/Pis an irreducibleL-module
(that is,s= 1this case, the Belkale-Kumar product is simply the cup). In
product. Thispaper is motivated by the guess that several known results for
cominusculeG/Pcould be generalized to anyG/Pbut using the Belkale-Kumar
product. Inparticular, it might be a first step toward a positive geometric
w
uniform combinatorial rule for computing the coefficientsc˜. Indeed,we define
uv
v
a subvarietyYwhich be encoded by combinatorial datum (precisely a subset
u
v
of roots ofG). Wealso define a Belkale-Kumar fundamental class[Y]⊙and
u0
v
conjecture that[Y]s
u⊙0= [Xu]⊙0[Xv]conjecture is true for. ThisG/BifGi
simple of typeA,B, orCis also proved for the Grassmannians.. It

A geometric construction of Belkale-Kumar’s ring.The first aim of this
paper is to give a geometric construction of the Belkale-Kumar ring which does
not deal with the Schubert basis.Consider the connected centerZofLand

3

its character groupX(Z)Azad-Barry-Seitz theorem (see [ABS90]) asserts. The
that eachViin decomposition (2) is an isotipical component for the action
ofZassociated to some weight denoted byαi∈X(Z). ThegroupPacts on
T G/PThe groupbut does not stabilize decomposition (2).X(Z)is endowed
P /P
with a partial order<(see Section 4.1 for details), such that for anyα∈X(Z)
the sum
V<α:=⊕αi<αVi(8)
isP-stable. SinceV<αisP-stable, it induces aG-homogeneous subbundle

T G/Pof the tangent bundleT G/P. Weobtain a family of distributions

indexed byX(Z). Thisiffamily forms a filtration:α<βthenT G/Pis a

subbundle ofT G/P. Moreover,these distributions are globally integrable in
the sense that
<α<β<α+β
[T G/PT G/P,]⊂T G/P.(9)
This allows us to define a filtration (“à la Hodge”) of the De Rham complex and

so of the algebraH(G/P,C)indexed by the groupX(Z)×Z. Weconsider the
associated graded algebra.


Theorem 1The(X(Z)×Z)-graded algebraGrH(G/P,C)associated to the

(X(Z)×Z)-filtration is isomorphic to the Belkale-Kumar algebra(H(G/P,C),⊙0).

The first step to get Theorem 1 is to give it a precise sense defining the
orders onX(Z)andX(Z)×Zand the filtrations.The key point to get the

isomorphism is that the Schubert basis([Xw])w∈WofH(G/P,C)is adapted
P
to the filtration.Indeed each linear subspace is spanned by the Schubert classes
it contains.To get this result, we make use in an essential way, Kostant’s
Harmonic forms [Kos61].
mental class for the BelkalIn progress
A conjecture.The main motivation to get Theorem 1 is to define the
fundae-Kumar product of any irreducible subvarietyYof

G/P. Thisclass[Y]⊙0which belongs toGrH(G/P,C)is defined in Section 3.4.
P PP
andwbe the longest elements ofWandWrespectively. Ifv∈W
Letw0 0
∨P P
belon
thenv:=w0vw0gs toWand represent the Poincaré dual class of[Xv].
P
Consider the weak Bruhat order⋖onWare interested in the product. We
∗P
[Xu]⊙0[Xv]∈H(G/P,C), for givenuandvinW. Lemma20 below shows
∨ ∨
that if[Xu]⊙0[Xv]6= 0thenv⋖u. Assumethatv⋖uand consider the group
v−1P−1P
H:=u Bu∩w vBvw .(10)
u0 0
It is a closed connected subgroup ofGcontainingT; in particular, it can be
v vv
encoded by its setΦof roots.LetYdenote the closure of theH-orbit of
u uu
P /P:
v
v
Y=/P .H .P(11)
u u
Another characterization of this subvariety is given by the following
statement.

4

v
Proposition 1The varietyYis the unique irreducible component of the
inu
−1P−1
tersectionu Xu∩Xw vvcontainingP /P. Moreover,this intersection is
0
v
transverse alongY.
u
Our main conjecture can be stated as follow.


Conjecture 1Ifv⋖uthen
v∗
[Y]⊙= [Xu]⊙0[Xv]∈GrH(G/P,C).
u0
Unfortunately, I am not able to prove this conjecture in general.The first
evidence is the following weaker result.

Proposition 2Write

X
v w
[Y]⊙=d[Xw].
u0uv
P
w∈W

Then
w w
(i)d6= 0⇐⇒c˜6= 0;
uv uv
w w
(ii)d≤c˜.
uv uv

Known cases.Conjecture 1 generalizes a known one forG/B. Indeed,if
G/P=G/Bis a complete flag variety then Conjecture 1 is equivalent to the
following one.

w
Conjecture 2ForG/Band anyu, vandwinW, the structure constantc˜
uv
i ii
is equal to1if for any1≤i≤s,dim(T) + dim(Tim(= dT)and
u v) im(Vi) + dw
0otherwise.
geometric model for the BIelkalne-Kumarpprorducto. Cgonjecrturee2swassexplicitly
In particular, Conjecture 2 implies that we have a uniform combinatorial and
stated in [DR09].E. Richmond proved in [Ric09] and [Ric11] this conjecture
forG= SLorG= S.
np2nIn Section 6, we prove it forG= SO2n+1(this
proof is certainly known from some specialist but I have shortly included it for
convenience). Dimitrov-Rothhave also a proof forG= SO2nandG2but it is
not published.

We prove that Conjecture 1 is true for the Grassmannians in Section 7.The
proof is based on a combinatorial model of the Littlewood-Richardson rule due to
Liu [Liu10].Note that in this case, our conjecture can be actually thought as the
first step toward a combinatorial model for the Belkale-Kumar product, namely
Liu’s model.R. Vakil obtained a geometric model for computing the
LittlewoodRichardson coefficients in [Vak06].Here we give a geometric interpretation of
the usual Littlewood-Richardson rule.

Combinatorial evidences.Consider the very degenerate following version
Conjecture 1.

5