Convexity properties of Hamiltonian group a tions Vi tor Guillemin Reyer Sjamaar Department of Mathemati s, Massa husett...
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Convexity properties of Hamiltonian group a tions Vi tor Guillemin Reyer Sjamaar Department of Mathemati s, Massa husetts Institute of Te hnology, Cambridge, Massa husetts 02139-4307
E-mail address :
vwgmath.mit.edu
Department of Mathemati s, Cornell University, Itha a, New York 14853-4201
sjamaarmath. ornell.edu http://www.math. ornell.edu/~sjamaar
E-mail address : URL:
2000 Mathemati s Subje t Classi ation. 53D20 (14L24 53-02 53C55)
Contents Introdu tion Chapter 1. 1.1.
1 The onvexity theorem for Hamiltonian
G-spa es
Introdu tion
7 7
1.2.
How are abelian and non-abelian onvexity related?
10
1.3.
Variants and generalizations of the onvexity theorem
11
1.4.
Loop groups
14
1.5.
Kostant's theorem for loop groups
19
1.6.
The symple tomorphism group of the annulus
20
Chapter 2.
A onstru tive proof of the non-abelian onvexity theorem 23
2.1.
Introdu tion
23
2.2.
An outline of the proof of the Kirwan onvexity theorem
23
2.3.
The equivariant Darboux theorem
25
2.4.
The ross-se tion theorem
27
2.5.
The Frobenius re ipro ity theorem
28
2.6.
Convexity
29
2.7.
Rigidity of the moment one
31
Chapter 3.
Some elementary examples of the onvexity theorem
35
3.1.
Introdu tion
35
3.2.
The eigenvalues of the prin ipal minors of a Hermitian matrix
36
3.3.
Confo al quadri s
37
3.4.
Gelfand-Cetlin
41
3.5.
The eigenvalues of the sum of two Hermitian matri es
3.6.
Morse theory on a Hamiltonian
3.7.
Klya hko's theorem
54
3.8.
Restri tion to a subgroup
55
Chapter 4.
G-manifold
Kähler potentials and onvexity
43 46
61
4.1.
Introdu tion
61
4.2.
Stability
63
4.3.
Kähler metri s on oadjoint orbits
68
4.4.
Convexity theorems for
B-orbits iii
71
iv
CONTENTS
4.5.
The onvexity theorem for linear a tions
74
4.6.
The onvexity theorem for ompa t Kähler manifolds
76
Chapter 5.
Appli ations of the onvexity theorem
79
5.1.
The Delzant onje ture
79
5.2.
Kählerizability
83
Bibliography
89
Introdu tion A prototype for the type of onvexity theorem that we will be dis ussing in this monograph is a theorem about Hermitian matri es whi h was proved
Hn be the set of n n matri es whose eigenvalues are the numbers, 1 , 2 , : : : , n with 1 2 n , and n for ea h A 2 H , let a11 , a22 , : : : , ann be the diagonal entries of A. The
by Horn [35℄ in the mid 1950's: Let
assignment,
A 7 ! (a11 ; a22 ; : : : ; ann );
de nes a mapping
: Hn ! Rn
and Horn's theorem asserts that the image
of this mapping is a onvex polytope. image of
where
More expli itly it asserts that the
is the onvex hull of the ve tors
=
(1) ; (2) ; : : : ; (n) ;
ranges over the set of permutations of
f1; 2; : : : ; ng.
In the early 1970's Kostant showed that this result was a spe ial ase of a more general result having to do with oadjoint orbits of Lie groups.
G be a ompa t onne ted Lie group, T the Cartan subgroup G on g dualizes to give an a tion of G on g and the oadjoint orbits are by de nition the orbits of this a tion. Let : g ! t be the transpose of the in lusion map, t ! g. If O g is a oadjoint orbit, then by restri ting to O one gets a Spe i ally, let
of
G
and g and t their Lie algebras. The adjoint a tion of
map,
: O ! t ; and Kostant's theorem asserts that the image of
(I.1)
is a onvex
polytope. In
T T fa t, if O is the set of T - xed points on O, maps O bije tively onto an
orbit of the Weyl group
N(T )=T, and
the image of
is the onvex hull of
the points on this orbit. This result has a formulation whi h involves ideas from symple ti geometry. Namely by a theorem of Kirillov and Kostant, the oadjoint orbits,
O,
of
G
are symple ti manifolds, and the a tion of
G
on
O
preserves the
G is ompa t and onne ted, these O's are the G a ts transitively. In addition, the a tion of G on O is Hamiltonian : For 2 g the a tion of G on O asso iates
symple ti form. Moreover, if
only symple ti
G-manifolds
on whi h
1
2
INTRODUCTION
with
a ve tor eld
O
on
O
and this ve tor eld is a Hamiltonian ve tor
!O is the symple ti form on O, the interior produ t of O di l where i : O ! g is the in lusion map and l the linear fun tional on g oming from the pairing on 2 g with elements of g . In other words, in the language of Se tion 1.1 below, the a tion of G on O is a Hamiltonian a tion with moment map, i : O ! g . The restri tion of this a tion to the torus T is also a Hamiltonian a tion and its moment
eld. In fa t, if with
!O
is
mapping is the mapping (I.1), so what Kostant's theorem asserts is that the image of this moment mapping is a onvex polytope, and, more expli itly,
(O) = onv (OT ): (Here
onv A denotes
the onvex hull of a subset
A
of a real ve tor spa e.)
In the early 1980's it was shown by Atiyah [4℄ and Guillemin-Sternberg [27℄ that, in this symple ti version of Kostant's theorem, one an drop the
O is a transitive symple ti G-spa e, and, in fa t, get rid of G entirely. Their result asserts that if M is a ompa t symple ti manifold, T an n-torus and T M ! M a Hamiltonian a tion of T on M, and if : M ! t is the moment map asso iated with this a tion, then assumption that
the role of
(i) (ii)
(MT ) is a nite subset (M) = onv (MT ).
In parti ular the image
of t
and
(M) is a onvex polytope.
We will hen eforth refer
to this result as the abelian onvexity theorem (and we will sket h a proof of it in Se tion 1.1). Three years after this theorem was proved, Fran es Kirwan proved a
G be a ompa t onne ted G a ts in a Hamiltonian fashion
mu h deeper non-abelian onvexity theorem. Let Lie group whi h is not ne essarily abelian. If
on a ompa t symple ti manifold then, as above, one has a moment map
: M ! g ;
(I.2)
but in general its image is not onvex. However, a mu h more subtle on-
=G be the orbit spa e for the oadjoint a tion of G on g . If W := N(T )=T is the Weyl group of G, the a tion of W on t gives by duality an a tion of W on t , and the orbit spa es, g =G and t =W , vexity result is true: Let g
are isomorphi .
Let us x a ( losed) Weyl hamber, t+ , in t .
fundamental domain for the a tion of
W
This is a
on t , so one has identi ations
t =W = t+ and hen e from (I.2) a map g =G = + : M ! t+ :
(I.3)
The Kirwan onvexity theorem asserts that the image of this map is a onvex polytope.
INTRODUCTION
3
This theorem, whi h was proved by Kirwan [40℄ in 1984, is the main topi of this monograph.
We will sket h below several proofs of it:
in
parti ular, in Chapter 2 we will des ribe some of the ingredients that ame into Kirwan's original proof, and in se tion 1.3 we will outline a short and relatively simple proof whi h dates from the mid 90's and is due to Lerman, Meinrenken, Tolman and Woodward. Our main on ern in this monograph will be with onstru tive versions of Kirwan's theorem. One of the defe ts of this theorem is that, unlike the abelian onvexity theorem, it does not ome with an expli it des ription of the image of
in t+ .
However, in a number of on trete examples su h
expli it des riptions have been found. For example, suppose tonian
G-manifold
and
X
an orbit of
G
in
M is a Hamil-
M. Then from the equivariant MX an for the a tion of G in a
Darboux theorem, one gets a anoni al model
G-invariant
neighborhood of
X, and for this
anoni al model there is a on-
stru tive version of the onvexity theorem whi h we will des ribe in Chapter 2. Moreover, by oupling this with a Morse theory result whi h Kirwan uses in her proof we will obtain in Chapter 2 a lo ally onstru tive onvexity theorem. This theorem is due to Sjamaar [59℄. Chapter 3 is devoted to two spe ial ases of the onvexity theorem, both involving (like the theorem of Horn whi h we des ribed above) isospe tral sets of Hermitian matri es. The rst of these is the Kirwan theorem for the a tion of U(
n - 1)
n). In its onstru tive Hn onto Hn-1 whi h assigns to ea h n onto the set (n - 1) (n - 1) minor, maps H
on a generi oadjoint orbit of U(
form it asserts that the proje tion of Hermitian
nn
matrix its
[
Hn-1;
n - 1-tuples su h that the i 's intertwine the for i = 1, 2, : : : , n - 1. We will also dis uss some
where the union is over all
i 's,
i.e.
i
i i 1 +
tie-ins of this result with Gelfand-Cetlin theory and with a topi dear to the hearts of nineteenth entury geometers: the theory of onfo al quadri s. The se ond example of Kirwan's theorem we will dis uss in Chapter 3
n) on the produ t of two oadjoint
on erns the diagonal a tion of U(
orbits
n). The Hermitian matrix version of this theorem asserts that if and are n-tuples of real numbers, the set of 's satisfying
of U(
Hn Hn + H n
(I.4)
is a onvex polytope. The onstru tive form of this result turns out to be mu h harder to prove (and also mu h harder to formulate) than the result for
(n-1)
(n-1) minors: It is, in fa t, only in the last de ade that a ompletely
satisfa tory des ription of this moment polytope was obtained, largely due
4
INTRODUCTION
to the eorts of Klya hko [41℄. A set of ne essary onditions for
to satisfy
(I.4) an be obtained by mini-max and Morse theoreti al arguments (these we will des ribe in Se tion 3.5) and Klya hko's great a hievement was to
1
show that these onditions are su ient as well as ne essary.
The onstru tive Kirwan problem for produ ts of oadjoint orbits of
n) an be reformulated as follows: If O is a oadjoint orbit of U(n) U(n) and U(n) ! U(n) U(n) is the diagonal imbedding, what is the moment polytope asso iated with the a tion of U(n) on O? This formulation admits of the following generalization: if G and H are ompa t Lie groups, i : H ! G an embedding of H in G, and O a oadjoint orbit of G, what is the moment polytope asso iated with the a tion of H on O? Berenstein and Sjamaar
U(
answered this question in [9℄.
In the last se tion of Chapter 3 we will
des ribe a set of inequalities whi h, they showed, hara terize this polytope; and we will also prove the easy part of their result: the ne essity of these inequalities. In Chapter 4 we will dis uss yet another version of the Kirwan theorem.
T M ! M a Hamiltonian T on M with moment map . If this a tion preserves the omplex 1n stru ture, then the a tion of the n-torus T = (S ) on M an be extended C to a holomorphi (but non-Hamiltonian) a tion of the omplex torus T = M
Let
be a ompa t Kähler manifold and
a tion of
n
(C ) , where C
is the multipli ative group of omplex numbers. Atiyah
proved in [4℄ a lo al form of the abelian onvexity theorem for this
T Cp is the orbit C of T p is a onvex
T C through p 2 M,
T C-
a tion: If
of
the moment image of the
losure
polytope. Moreover, this polytope is the onvex
hull of the moment image of the set
T Cp \ MT : A non-abelian generalization of this fa t, due to Brion [16℄, on erns Kählerian a tions of a ompa t onne ted Lie group, manifold,
M.
G,
on a ompa t Kähler
As above su h an a tion extends to a holomorphi a tion
of the omplex Lie group,
GC ,
and Brion's result asserts that if
omposite moment mapping (I.3), the image with respe t to
+
+
is the
of the
GC -
M is onvex. A somewhat related result GC asso iated to the opposite hamber image with respe t to (Bp) of the B-orbit
orbit through an arbitrary point of
on erns the Borel subgroup
-t+ . We will prove that the through a point
1
p
of
M
B
of
interse ts the open hamber
Int t+
in a onvex set
It was pointed out by Woodward that some of these onditions are redundant. Re nements of Klya hko's results, eliminating these redundan ies, have been obtained by Belkale [8℄, Knutson-Tao [42℄, and Knutson-Tao-Woodward [43℄.
INTRODUCTION
5
and that this interse tion is ontained in the interse tion
Int t+ \
\
bB
T Cbp :
(I.5)
2
Moreover, if
M
is a proje tive variety,
(Bp) \ Int t+
is equal to the inter-
se tion (I.5) and, in parti ular, is a onvex polytope. In future work we will show that the set (I.5) is a lower semi ontinuous fun tion of
p, and from this
lo al onvexity theorem get a onstru tive version of Kirwan's theorem for Kähler manifolds whi h is rather dierent in spirit from those we previously des ribed. In the pages above we have surveyed the ontents of Chapters 24. It remains to say a few words about the material in Chapters 1 and 5. In the last two de ades the onvexity theorem has been generalized in a number of predi table ways and, in a few instan es, in some ompletely unanti ipated ways. There are now, for instan e, onvexity theorems for Hamiltonian a tions of ompa t Lie groups on non- ompa t symple ti manifolds, for Hamiltonian a tions of non- ompa t Lie groups, for Poisson a tions of Lie groups on Poisson manifolds, and for quasi-Hamiltonian a tions of Lie groups.
For non- ompa t groups these results are quite ompli ated, and
for this reason it is rather surprising that some of the simplest and most elegant generalizations of the onvexity theorm have to do with the a tion of in nite-dimensional groups, e.g. loop groups and groups of gauge transformations, on in nite-dimensional symple ti manifolds. We will attempt to give a brief a
ount of these results, without getting too bogged down in details, in Chapter 1. In Chapter 5 we will dis uss some appli ations of the onvexity theorem. One parti ularly beautiful appli ation is the Delzant theorem [22℄: Let be a ompa t Hamiltonian inequality,
T -manifold.
If
T
a ts faithfully on
dim M 2 dim T;
holds, and if this inequality is an equality the
M
M
then the
(I.6)
T -a tion
is alled a om-
pletely integrable or tori a tion. What Delzant proved is that for su h a tions
M
is determined, up to a
T -equivariant
symple tomorphism, by its
moment polytope. One of the most intriguing outstanding questions about Hamiltonian a tions of Lie groups has to do with the non-abelian analogue
M a onne ted G-manifold, U, of M with the property that, for all p U, the stabilizer groups, Gp and Gq , are onjugate in G, i.e. all U have the same orbit type. The group, Gp , (whi h is unique up
of this result. If
G
is a ompa t Lie group and
there exists a dense open subset, and
q
in
points in
to onjuga y) is alled the prin ipal isotropy group of the a tion; and if
6
INTRODUCTION
this group is dis rete, there is an analogue of the inequality (I.6), namely
dim M dim G + rank G: If this inequality is an equality the a tion of
G is alled multipli ity-free ; M is determined up to iso-
and for su h a tions Delzant onje tured that
morphism by its moment polytope and its prin ipal isotropy group. This
onje ture is still unsettled, but some partial results, whi h we will report on in Se tion 5.1, indi ate that it is very likely to be true. The other appli ation of the onvexity theorem whi h we will dis uss in Chapter 5 has to do with Kählerizability. Up until a few years ago the following seemed to be a highly plausible onje ture: Let Hamiltonian
M
be a ompa t
T -manifold for whi h the xed point set, MT , is nite.
Then
M admits a T -invariant omplex stru ture whi h is ompatible with its symple ti stru ture. What made this onje ture seem plausible is a theorem of Biaªyni ki-Birula [10, 11℄, whi h asserts that a nonsingular omplex proje tive variety equipped with a torus a tion with nitely many xed points admits a de omposition into a ne spa es. In parti ular, su h a variety is birationally equivalent to proje tive spa e. In view of this it seemed unlikely that dropping the Kähler assumption ould ompli ate this birational lassi ation. In 1995 Sue Tolman found a ounterexample whi h demolished this onje ture; and as we will des ribe in Se tion 5.2, the key ingredient in the proof of the non-Kählerizability of her example is a orollary of Atiyah's
onvexity theorem for
T C-orbits,
whi h imposes some onstraints on the
shape of the moment polytope when the a tion is a Kähler a tion. One aspe t of moment geometry whi h we have not dis ussed in this monograph, and whose absen e we regret, is Duistermaat-He kman theory. From this theory one sees that the moment polytope has a lot of additional stru ture whi h we have negle ted to mention: in parti ular it de omposes into a disjoint union of open onvex subpolytopes, alled a tion hambers, and asso iated with ea h of these a tion hambers is a polynomial:
the
Duistermaat-He kman polynomial. Fortunately there are many good expositions of this subje t available. In parti ular, we re ommend the a
ount of Duistermaat-He kman theory by Mi hèle Audin in [7℄ and, for an in nitedimensional version, Atiyah's arti le [5℄.
CHAPTER 1
The onvexity theorem for Hamiltonian
G-spa es
1.1. Introdu tion
M be a dierentiable manifold, G a ompa t onne ted Lie group : G ! Di (M) a dierentiable a tion of G on M. Let g be the Lie algebra of G. From one gets an in nitesimal a tion Let
and
Æ :
g
! ve tor
elds
7 ! M
;
M. Now suppose that ! 2 2 (M) is a symple ti two-form. The a tion, , preserves ! (i.e. is a symple ti a tion) if and only if, for every 2 g, the one-form, (M )!, is losed, and is alled a Hamiltonian a tion if this losed form is exa t. For instan e, if M is simply onne ted every of g on
symple ti a tion is Hamiltonian. If
is a Hamiltonian a tion one an asso iate to it a map,
: M ! g ;
alled the moment map ; and the topi of this monograph will be a onvexity property of this map whi h was dis overed twenty years ago by Atiyah [4℄
1 , i, (iM )! is
and by Guillemin and Sternberg [27℄. To de ne this map, x a basis,
2 , : : : , n ,
of g and let
i
be the dual basis of g . For ea h
i 1 exa t, so there exists a fun tion 2 C (M) su h that (iM )! = -d i : Now set
X i i : i P i
=
(1.2)
i d i is intrinsi ally de ned; i.e. is i i independent of the hoi e of the 's and the 's. Hen e is intrinsi ally It is lear from (1.1) that
d
(1.1)
=
de ned up to an additive onstant. There are various ways to x this onstant. For instan e, if the dimension of require that
Z
i !d
or if there is a base point
=
0
p0 2 M
i (p0 ) = 0
for
M is 2d and M is ompa t, i = 1, 2, : : : , n;
xed by for 7
one an
,
one an require that
i = 1, 2, : : : , n:
8
1. THE CONVEXITY THEOREM
In any ase one an always hoose this additive onstant so that the moment map (1.2) is equivariant, i.e. intertwines the a tion of
G
oadjoint a tion of
on g .
G
on
M with the has this
Hen eforth we will assume that
property. The rst of the onvexity theorems that we want to talk about in this monograph is the abelian onvexity theorem. xed point set of
Let us denote by
.
1.1.1. Theorem. Assume
M
G
ompa t and
(MG ) is a nite set, G (ii) (M) is the onvex hull of (M ). parti ular, = (M), is a onvex polytope:
MG
the
abelian. Then
(i)
In
the moment polytope of
M.
This result was obtained by Atiyah [4℄ and, independently, by Guillemin and Sternberg [27℄ in the spring of 1981. The proof in [27℄ was simple and elegant; however, the proof in [4℄ was even a bit more simple and elegant: Atiyah dedu es Theorem 1.1.1 from the following onne tivity result:
2 g , -1 ()
1.1.2. Theorem. For every
is onne ted.
We will not go into the proof of this, whi h involves Morse theory. However, we will indi ate how Theorem 1.1.2 implies Theorem 1.1.1. Let
G,
a odimension one losed subgroup of
H :
g
!h
be the dual of the in lusion map, h
has as its moment map
Take
2h.
is onne ted.
But
1 line, H ( ), in g -
! g.
Then the a tion,
H = H Æ :
Then by Theorem 1.1.2
1 H ( )
-1 ( ) H
is onne ted, so
is just the interse tion of
1 H ( ),
are just the a ne lines in g
Hen e every a ne line interse ts
jH,
with the a ne
slope. It is not hard to see that the set of rational points in
.
be
1 H ( )
. The on lusion is that every su h line interse ts
interval. But the lines,
H
let h be its Lie algebra and let
in an
with rational
is dense in
in an interval; in other words
is
onvex. We want next to des ribe a non-abelian version of the onvexity theorem. To state this theorem we will need to review some Lie theory. Let Cartan subgroup of
G,
let t be its Lie algebra and let
Weyl group. Note that sin e
W
T
be a
W = N(T )=T be
the
a ts on t, it also a ts on t . Also note that
T
t has a natural imbedding in g as the set (g ) . It is easy to see that every
G-orbit
in g
interse ts t
in a
W -orbit, so
g
one has
t =W: =G =
(1.3)
1.1. INTRODUCTION
Let
treg =
f 2 t j G
=
9
T g:
This set is a disjoint union of open polyhedral ones. Fix one of these and
denote its losure by t+ . Then every
W -orbit in
t interse ts t+ in a single
point, hen e by (1.3)
g i.e. the quotient spa e, g
=G = t =W = t+
=G, has the natural stru ture of a polyhedral one.
The non-abelian onvexity theorem asserts: 1.1.3. Theorem. If
M
is ompa t the image,
,
of the map
M ! g ! g =G = t+ is a onvex polytope. In [27℄ this theorem was proved for Kähler manifolds, and two years later Fran es Kirwan [40℄ proved it in general. We will des ribe a proof of it by Sjamaar [59℄ (a slight variant of Kirwan's proof ) in Chapter 2. One problem with this theorem is that, unlike the abelian onvexity theorem, it does not
ome with an expli it des ription of
.
The main topi of this monograph
will be a number of onstru tive versions of the Kirwan onvexity theorem whi h seek to remedy this defe t. For instan e Sjamaar's result is a rst step in this dire tion: it provides an expli it des ription of the Kirwan polytope
lo ally in the neighborhood of a point on its boundary. In the mid-nineties, Klya hko and others proved more global onstru tive versions of onvexity, and these we will report on in Chapter 3.
In Chapter 4 we will dis uss
a non-abelian generalization of a onvexity theorem of Atiyah's on erning orbits of omplex tori a ting on Kähler manifolds. Finally, in Chapter 5 we will review some appli ations of Atiyah's theorem to equivariant symple ti geometry. In Se tions 1.31.6 of this hapter we will report on a number of onvexity results whi h generalize Kirwan's theorem in various non- ompa t dire tions: allowing either the group, to be non- ompa t.
G,
or the manifold on whi h it a ts
In parti ular, we will dis uss a version of Kirwan's
theorem, due to Meinrenken and Woodward, for loop groups a ting on in nite-dimensional manifolds and a beautiful proof of this theorem by Alekseev, Malkin and Meinrenken in whi h this result gets re ast as a nitedimensional onvexity result.
We will also dis uss brie y some in nite-
dimensional versions of the abelian onvexity theorem: a onvexity theorem due to Atiyah and Pressley for loop groups and a onvexity theorem for the group of symple tomorphisms of the annulus due to Blo h, Flas hka and Ratiu.
10
1. THE CONVEXITY THEOREM
An important pre ursor of the abelian onvexity theorem was a onvexity theorem for oadjoint orbits whi h is due to Kostant, and we will begin by using Kostant's result to show, in the next se tion, how the abelian and non-abelian onvexity theorems are related.
1.2. How are abelian and non-abelian onvexity related?
Let
dual of the in lusion map, t by
T
=
G on M. Then jT is a Hamiltonian M. Moreover, if T : g ! t is the the G and T moment maps are related
be a Hamiltonian a tion of
a tion of the maximal torus
T Æ .
T
on
!
g,
How are the two moment polytopes related? The answer
in the positive f w j w 2 W g, i.e.
involves a beautiful onvexity theorem of Kostant: for every
hamber t+ let
be the onvex hull of the nite set,
the onvex hull of the Weyl group orbit through 1.2.1. Theorem. If
S
.
We laim:
is a onvex subset of t+ , then
S℄
[
f j 2 S g
:=
is a onvex subset of t . Proof. Let
0 = f (; ) 2 t+ t j 2 g:
It is easy to see that this is a onvex polytope. In fa t, it is just the Kirwan polytope for the a tion of
of t+
t
on t+ and t
so it is lear that if
S
G T on T G.
Let
respe tively. Then
pr1 and pr2 be the proje tions
1 S℄ = pr2 pr1 (S) \ 0 ; is onvex,
S℄
is onvex.
QED
Next we laim 1.2.2. Theorem. If
NA
=
(M) \ t+
and
A = T (M),
then
A = ℄NA : Proof. We re all that by Kirillov and Kostant every oadjoint orbit,
O,
in g
is a symple ti manifold.
Moreover, the a tion,
O ,
of
G
on
O
is Hamiltonian and the moment map asso iated to this a tion is just the in lusion map,
iO : O !
g .
Thus
0 jT
has for moment map,
Hen e by the abelian onvexity theorem, Theorem 1.1.1, we have
onv T (OT ).
T=t
However, (g )
OT
=
, so
O \ t
=
f w j w 2 W g;
T Æ iO . T (O) =
1.3. VARIANTS AND GENERALIZATIONS
being the unique point of interse tion of
O
with t+ .
11
Thus the abelian
onvexity theorem implies that
T (O) = : This result is the Kostant onvexity theorem, whi h was rst proved in [45℄. From it we get
A = 2 t 2 ; 2 (M) \ t+
=
℄NA : QED
One of the main goals of the next hapter will be to obtain a des ription of
NA
similar to Theorem 1.2.2.
1.3. Variants and generalizations of the onvexity theorem
There have been many interesting generalizations of the onvexity theorem, too many in fa t to do justi e to. In this se tion we will only attempt to des ribe a few of these. The Atiyah onvexity theorem for
n
(C ) -orbits. Let C
be the
C n 1n multipli ative group of omplex numbers, let G = (C ) and let G = (S ) C C be the maximal ompa t subgroup. A holomorphi a tion, , of G on a C
ompa t Kähler manifold, M, is Kähler if = jG preserves the Kähler C be su h an a tion and let be the -moment map. stru ture. Let
Atiyah's theorem asserts: 1.3.1. Theorem ([4℄). If
X
is the losure of a
(X) = onv (MG \ X):
GC -orbit,
then
We will dis uss a non-abelian variant of this result in Chapter 4. Non- ompa t manifolds. Let
ifold and let
be
M
be a non- ompa t symple ti man-
a Hamiltonian a tion of the
n-torus, G = T n,
on
M.
The
following theorem is due to Lerman, Meinrenken, Tolman and Woodward. 1.3.2. Theorem ([48℄). Suppose there exists a onvex subset,
g
ontaining
(M)
(M)
su h that the map,
: M ! C
is proper.
C,
of
Then
is onvex.
This theorem is a generalization of an earlier non- ompa t onvexity theorem of Prato [56℄ and both an be extra ted from the abelian onvexity theorem for ompa t manifolds. The key ingredient is the following sym-
1 : S1 M ! M be a manifold M, with moment map,
ple ti utting theorem of Eugene Lerman: Let Hamiltonian
1 : M ! R,
1 a tion of S
on a symple ti
and suppose that
1
is proper and bounded from below.
12
1. THE CONVEXITY THEOREM
1.3.3. Theorem ([47℄). If
disjoint union of the set,
1 1 1 (a)=S ,
S1
Ua
a ts freely on the level set, =
fp
2
M j 1 (p) < a g, Ma .
1 1 (a),
the
and the set,
is a ompa t symple ti manifold,
Ma is the set, 1 a, with 1 = a, ollapsed to points.
(i) As a topologi al spa e,
Remarks.
the ir le orbits on the boundary,
1 S1 does not a t freely on 1 (a), but there are no xed points on this set, then a weaker result is true: Ma is a symple ti orbifold. If is a Hamiltonian a tion of G on M whi h ommutes with 1 , there is an indu ed Hamiltonian a tion of G on Ma , whose moment map oin ides with the -moment map on the set > a.
(ii) If
(iii)
Here is a brief sket h of how Theorem 1.3.3 implies Theorem 1.3.2. For
i
=
1, 2, : : : , n,
let
Gi
be ir le subgroups of
omponent of the moment map,
hoose the
Gi 's
.
G
and let
i
be the
Gi
By the properness assumption, one an
2 R the set f p 2 M j i i , i = 1, 2, : : : , N g
so that for all
is ompa t, and if the
i 's
i
are hosen judi iously this set, with the ir le
orbits on the boundary omponent ollapsed to points, is a ompa t
G-
Hamiltonian orbifold. Lerman et al. [48℄ prove an orbifold version of the abelian onvexity theorem and, using it, on lude that the moment image of this set is onvex. They then show, by letting the that the moment image of
M
i 's
tend to in nity,
itself is onvex.
One appli ation of Theorem 1.3.2 (whi h an also be found in [48℄) is a proof of Kirwan's theorem whi h is mu h shorter, albeit less onstru tive, than the proof we will des ribe in Chapter 2: Let
G
be a non-abelian om-
T a Cartan subgroup, a Hamiltonian a tion of G on a ompa t -1 symple ti manifold, M, and the moment map. The set X = (Int t+ ) is a T -invariant symple ti submanifold of M and the T -a tion on it satis es the hypotheses of Theorem 1.3.2 with C = Int t+ . Moreover, the T -moment map, T , is just the restri tion to X of , and so (M) \ Int t+ = T (X) is pa t group,
onvex. The real version of the onvexity theorem. Let
(M; !) a ompa t symple ti manifold and isfying of
M
! = -!.
The set of xed points,
G
be an
n-torus,
: M ! M an involution satX, of is alled the real lo us
(sin e the anoni al example of an involution of this type is omplex
n
onjugation on C ).
ompatible with
Let
be a Hamiltonian a tion of
in the sense that
Æ (g) = (g-1 ) Æ
G
on
M
whi h is
1.3. VARIANTS AND GENERALIZATIONS
g 2 G.
for all
13
Duistermaat proved the following real version of the abelian
onvexity theorem. 1.3.4. Theorem ([24℄). If
(X).
is the
-moment
map, then
(M)
=
This theorem applies in parti ular to the real lo i of oadjoint orbits of non-abelian ompa t groups, e.g. real Grassmannians and real ag varieties. There is also, as O'Shea and Sjamaar [54℄ have shown, a non-abelian version of it, i.e. a real version of Kirwan's theorem. Convexity theorems in Poisson geometry. If
group, the spa e of smooth fun tions on
G
G
is a ompa t Lie
an be equipped with a natural
Poisson bra ket operation, the Lu-Weinstein bra ket (see [50℄); and an a tion, a tion
: G M ! M, of G on a symple ti if is a Poisson mapping, i.e.
manifold,
M,
is a Poisson
ff; gg = f f; gg
f and g, in C1 (M). For su h an a tion Jiang-Hua Lu [49℄ has shown there is a moment map of M into the dual Poisson-Lie group of G, and Flas hka and Ratiu have proved an analogue of the Kirwan theorem
for all fun tions,
for this map. We will not go into the details of this result, but one orollary of it is the following theorem about singular values of produ ts of
p
omplex matri es. (Re all that the singular values of a matrix, eigenvalues of
nn
A A.)
Given
1
A,
nn
are the
2 n, let M be the set of all
matri es with these as singular values.
1.3.5. Theorem ([25℄). The set
polytope.
f j M
M M g
is a onvex
We mention this result be ause it is a multipli ative version of a theorem about eigenvalues of sums of
n n Hermitian
matri es whi h will be one of
the main topi s of Chapter 3. The onvexity theorem whi h we alluded to above turns out not only to be analogous to the Kirwan theorem, but, in fa t, a onsequen e of it.
: G M ! M is a Poisform on M so that it be omes
Namely Anton Alekseev [2℄ has shown that if son a tion one an hange the symple ti
a Hamiltonian a tion. This enables one to translate many theorems about Poisson a tions into theorems about Hamiltonian a tions and, in parti ular, to show that the Flas hka-Ratiu theorem is implied by the Kirwan theorem. Non- ompa t groups. In order to formulate a onvexity theorem for
Hamiltonian a tions of a Lie group,
G,
ni e des ription of the orbit spa e, g
it is ne essary rst of all to have a
=G.
However, even for very simple
14
1. THE CONVEXITY THEOREM
non- ompa t groups this orbit spa e an be very bad. the simplest of all non-abelian groups, the is non-Hausdor.
ax + b
For instan e, for
group, this orbit spa e
(It onsists of two open orbits and a one-dimensional
ontinuum of zero-dimensional orbits.) Nevertheless a few interesting onvexity theorems have been proved for (very spe ial) Hamiltonian a tions of non- ompa t groups. See for instan e Paneitz [55℄ and Neeb [53℄. The one result we know of whi h is lose in spirit to Theorems 1.3.11.3.5 is a onvexity theorem of Alan Weinstein. This has to do with redu tive Lie groups,
G,
with the property that the maximal ompa t subgroup,
has the same rank as
G
K,
of
G
itself. For these groups the set of ellipti elements,
ge , in g satis es the ni eness riterion above in the sense that ge =G is a
onvex polyhedral one; and the following onvexity assertion is true:
M be a Hamiltonian G-manifold with (M) is ontained in an open G-invariant subset, U, of ge whose image in ge =G is onvex. In addition suppose : M ! U is proper. Then the image of (M) itself in ge =G is onvex. 1.3.6. Theorem ([64℄). Let
moment map,
.
Suppose
1.4. Loop groups
Given the fa t that there are very few onvexity results for non- ompa t Lie groups, it omes as a surprise that a number of the most interesting re ent results in this subje t have to do with groups that are not only non ompa t but not even nite-dimensional.
In this se tion we will dis uss
a version of Kirwan's onvexity theorem for Hamiltonian a tions of loop groups due to Meinrenken and Woodward. In their original paper on the Kirwan theorem for these groups, Meinrenken and Woodward onfronted head-on the Bana h manifold ompli ations that this involved.
However,
Alekseev, Malkin and Meinrenken subsequently found a simpler proof that skirted these ompli ations by identifying the moment image in question with the moment image of a quasi-Hamiltonian a tion of a ompa t group on a nite-dimensional manifold. We will formulate this result and give a
G is a ompa t simply onne ted LG is the loop group, C1 (S1 ; G). The Lie algebra of LG is the 1 1
ve tor spa e, Lg = C (S ; g). We will denote by L g the anoni al entral extension of Lg; i.e. the produ t sket h of its proof below. In what follows,
Lie group and
L g = R Lg with the bra ket operation
(t
1 ; 1 ); (t2 ; 2 )
Z =
S1
1 ; d2 ); [1 ; 2 ℄ :
(
1.4. LOOP GROUPS
(The (
Lg
15
; ) in the integrand is the Killing form.)
via the pairing
1 ; 2 ) =
(
Z
S1
We will identify
Lg
with
1 ; 2 )dt;
(
and this gives us an identi ation d L g = R Lg = R Lg:
The right side of (1.4) an also be identi ed with a tion of
LG
L g
on
(If it did,
R and
Lg.)
L g
LG
The adjoint
f0g xed; however, the oadjoint a tion does
would just be the dire t produ t of the Lie algebras,
What is true is that the oadjoint a tion of
invariant the sets of
However, the adjoint
is not the same as the oadjoint a tion.
a tion, for instan e, leaves R not.
L g.
(1.4)
fg Lg, for ea h 2 R.
on the set
LG
on
Ld g
leaves
We will all the oadjoint a tion
Lg = fg Lg = fg Lg
(1.5)
LG on Lg . For 6= 0 this is not a linear a tion. d fa t it annot be sin e fg Lg is not a linear subspa e of L g . To des ribe the level a tion of LG on Lg for 6= 0 it su es,
the level
a tion of
In
by
homothety, to onsider the level one a tion; and this turns out to have a very simple geometri des ription. Consider the trivial prin ipal
1 over S ,
G-bundle
P := G S1 ! S1 ;
G a t by left multipli ation on the rst fa tor. Let R 2 1 g (G) be the right-invariant Maurer-Cartan form on G and let be the 1 1 1 standard angle oordinate on S . Then for any f 2 C (S ; g) the one-form f 2 g 1 (P) de ned by (f )(g; ) := (R )g + Ad(g)f( )(d )
on whi h we let
g; ) 2 P is a onne tion form on P. Hen e we an identify the spa e C1 (S1 ; g) = Lg with the spa e of onne tions on P. The group, LG, an be interpreted as the group of gauge transformations of the bundle, P , and 1 hen e it a ts on the spa e of onne tions. The a tion of a loop : S ! G on P is given by right multiplying the rst fa tor, (g; ) 7! (g ( ); ), and the
orresponding a tion on a onne tion is given by pulling ba k, 7! . 1 1 If f is the onne tion orresponding to f 2 C (S ; g), then a al ulation gives f = f , where
for (
f := Ad( )f + R( -1 ) 0 :
0 : S1 ! TG is the derivative of the loop and R denotes right multipli ation in G. Comparing this formula with the des ription of the oadjoint
Here
16
1. THE CONVEXITY THEOREM
a tion of the loop group given by Pressley and Segal in [57, proposition 4.3.3℄, we ome to the following on lusion. 1.4.1. Proposition. The loop group a tion on the spa e of onne -
tions is pre isely the level one a tion of
LG
on
Lg .
Given a onne tion form f and any group element a 2 G, the urve it an be lifted to a unique horizontal
: [0; 2℄ ! S1 de ned by (t) = e
urve, ~(t); (t) , in G S1 starting at ~(0) = a and ending at
~(2) = Holf (a)a
(1.6)
Holf (a) 2 G. Let G be the spa e of onjuga y lasses in G. It is -1 easy to he k that Holf (ga) = g Holf (a)g and hen e that the onjuga y
lass, Hol f 2 G , of Holf (a) is independent of a. This onjuga y lass is the
for some
holonomy of the onne tion,
f .
See e.g. [3, se tion 8℄ for a dis ussion of
the following result. 1.4.2. Proposition. Two onne tions,
holonomy i they are gauge equivalent. Thus the level one oadjoint orbits of the holonomy map,
G . Lg =LG =
Hol : Lg
! G ,
f1 LG
de ned by
and
f2 ,
have the same
are just the level sets of
f ! 7 Holf.
In parti ular
t=W , where W is the =G =
T=W and sin e T is identi ation G =
Earlier in this hapter we pointed out that g Weyl group. From this we get an
T = ker expT , of t,
the quotient of t by the group latti e, Z
t=Wa ; G = where
Wa
is the a ne Weyl group
W n ZT .
=Wa
Finally, t
is isomorphi
to a+ , where a+ , the positive Weyl al ove, is a fundamental domain for the
Wa -a tion
on t. (Sin e t+ is a fundamental domain for the
W -a tion,
we
an assume that a+ is ontained in t+ .) Thus we have proved 1.4.3. Proposition. The quotient of
LG
Lg
by the level one a tion of
is the onvex polytope a+ .
M be an (in nite-dimensional) symple ti manifold and an a tion of LG on M whi h preserves the symple ti form. is a level one a tion if there exists a moment map : M ! Lg whi h intertwines the a tion of LG on M with the level one a tion of LG on Lg . Now let
Remark. The group,
LG,
has a entral extension
! LG ! 1; 1 ! S1 ! LG
1.4. LOOP GROUPS
L g,
whose Lie algebra is
a Hamiltonian a tion of moment map
1.
17
and one an regard a level one a tion of
LG
for whi h
S1
LG
as
a ts inee tively with onstant
The Meinrenken-Woodward theorem asserts
: LG M ! M is a level one Hamiltonian M with proper moment map : M ! Lg , the image Lg =LG = a+ is a onvex polytope.
1.4.4. Theorem ([52℄). If
a tion of
(M)=LG
LG in
on
In the remainder of this se tion we will sket h the proof of this theorem.
From the evaluation map,
2 LG 7! (0) 2 G, one gets a group
LG ! G, whose kernel is, by de nition, the group of based loops, G. Now re all that an element of Lg is a onne tion, f , on the 1 bundle, G S . Let Holf (a), a 2 G, be the holonomy transformation, (1.6), asso iated with this onne tion. Setting hol f = Hol f (e) we get a map homomorphism,
hol : Lg
f 7 ! holf ;
! G;
whi h intertwines the level one a tion of a tion of
G
on itself.
LG
Lg with the
G-invariant.
on
Hen e, in parti ular, it is
(1.7)
onjuga y Moreover,
sin e the level sets of the omposite map
Lg ! G ! G hol
are the
LG-orbits
one gets from
in
Lg ,
the level sets of
hol a bije tive
h
itself are the
G-orbits.
Thus
map
Lg = G ! G:
(1.8)
What makes this map interesting is that one an show:
G on Lg is with G as its base.
1.4.5. Proposition. The level one a tion of
Hen e, by (1.8),
Lg
is a prin ipal
G-bundle
free.
be a level one Hamiltonian a tion of LG on a manifold, M, and let : M ! Lg be its moment map. Sin e is G-equivariant and
G a ts freely on Lg , G a ts freely on M. Let Now let
X = M= G: Sin e the map (1.7) is
G-invariant proper,
G-invariant
and hen e indu es on
the omposition of this map with
is
: X ! G. Moreover, if is X is a ompa t (in parti ular, proje tion, : M ! X, makes M into
X
a map
has to be proper as well, and hen e
nite-dimensional) manifold; and the
18
1. THE CONVEXITY THEOREM
a prin ipal
G-bundle.
Thus one get a diagram of maps
/ Lg
M
(1.9)
hol
X
/G
G-prin ipal bundle maps. Furthermore, the a tion of LG on M indu es on X an a tion of G; and by (1.9) this a tion is intertwined by with the onjugation a tion of G on G. 1 2 m be a basis of g and let be the anoni al three-form Let , , : : : , on G de ned by the stru ture onstants of g with respe t to this basis, i.e. in whi h the verti al arrows are
X
=
ijk
ijk i ^ j ^ k : Lg is ontra tible, satisfying hol = -d$.
By the Ja obi identity this form is losed and hen e, sin e there exists an
LG-invariant
$,
two-form,
on
Lg
In fa t, as is shown in [3, se tion 8.1℄, one an write down an expli it manifestly
LG-invariant
formula for
$,
whi h we will not bother to des ribe
here. Now let
!
be the symple ti form on
se tion 8.2℄ that
! + $
M.
One an show [lo . it.,
is basi with respe t to the bration,
hen e that there exists a two-form,
,
on
X
!
and
su h that
= ! + $: In parti ular, sin e
,
(1.10)
is losed,
d = - d$ = - hol =
and thus
d =
:
To summarize: From the Hamiltonian a tion,
(1.11)
,
of
LG
on
M,
one obtains
G-manifold, X, a G-valued moment map, , on X satisfying (1.11). Alekseev, Malkin and Meinrenken de ne a triple, (X; ; ) with these properties to be a q-Hamiltonian G-spa e and note that from (1.9) and (1.10) one gets a one-to-one orresponden e between Hamiltonian LG-spa es with proper a ompa t nite-dimensional
:
X ! G, and a G-invariant
two-form,
moment maps and ompa t nite-dimensional q-Hamiltonian spa es. One ni e thing that this orresponden e does is translate the onvexity problem into a nite-dimensional problem. Namely, by (1.9), the same image in
and
have
G; so to prove Theorem 1.4.4 it su es to prove that the
into G is onvex; i.e. to prove a q-Hamiltonian
proje tion of the image of
version of Kirwan's theorem. This an be done by imitating, almost word
1.5. KOSTANT'S THEOREM FOR LOOP GROUPS
19
for word, the proof we des ribed in Se tion 1.3 (after Theorem 1.3.3) of the usual Kirwan theorem. Namely imbed the interior of the Weyl al ove, a+ , into
T,
and onsider its pre-image,
from (1.11) that the restri tion of the restri tion of
X
to
(X) is onvex.
X, with respe t to . It is easy to see into X is a symple ti form and that
T -moment
is its
map. Hen e, by Theorem 1.3.2,
1.5. Kostant's theorem for loop groups
O be a level G ! LG, one gets a Let
one oadjoint orbit of Hamiltonian a tion of
LG. From the in lusion, T ! T on O. If T : O ! t is the
asso iated moment map one an show that
T (O) = onv T (OT ):
(1.12)
However, this result is not very interesting sin e it turns out that both sides
of (1.12) are equal to t . There is a mu h more interesting Kostant theorem, due to Atiyah and Pressley, whi h has to do with a ertain distinguished level one oadjoint
C1 (S1 ; g) an be identi ed with the spa e of onne tions 1 1 bundle P = G S ! S . Consider in parti ular the trivial
orbit. Re all that on the trivial
onne tion. This onne tion is invariant under the one-parameter group of bundle automorphisms
F : S1 ! Aut(P); Moreover, if
2 LG
F (g; ) = (g; + ):
(1.13)
is a gauge transformation, then
1 F F
=
;
(1.14)
( ) = ( + ). Hen e the one-parameter group (1.13) is ontained in the normalizer of LG in Aut(P ). Thus, if O is the oadjoint orbit through 1 the trivial onne tion, one gets from (1.13) an a tion of S on O. Moreover, by (1.14) this a tion ommutes with the a tion of T , and one an show that 1 the produ t a tion, , of T S on O is Hamiltonian. The Atiyah-Pressley
where
theorem asserts (see Figure 1): 1.5.1. Theorem ([6℄). Let Z
T
be the group latti e of
kk be the metri on t asso iated with the Killing form.
LG-orbit t
through the trivial onne tion.
R of OT S1 is the set
T
Let
(v; )
and the moment image of
O
v 2 ZT , =
O
be the
Then the moment image in
in t and let
1 2 2 kvk
is the onvex hull of this set.
20
1. THE CONVEXITY THEOREM
v Figure 1. Atiyah-Pressley parabola for
S1
LG
lifts to an a tion of
and one an form the semi-dire t produ t
The
One an show that the a tion (1.14) of on
, LG
S1
G = SU(2)
on
G
= LG
o S1 .
moral of the Atiyah-Pressley result is that the interesting generalization of Kostant's theorem to loop groups involves this group rather than
LG itself.
What about other oadjoint orbits of this group? Assume for simpli ity that
G is both simple and simply onne ted.
Then Vi tor Ka£ and Dale Peterson
have shown in [37℄ that Kostant's theorem is true for all oadjoint orbits of
G
ex ept for a small lass of degenerate orbits (the analogues of the level
zero orbits of
LG).
Their proof involves Ka£-Moody theoreti te hniques
and is a bit too intri ate to des ribe here.
(However, the virtue of their
proof is that it works for an arbitrary Ka£-Moody group asso iated to a symmetrizable generalized Cartan matrix.) A more geometri ally oriented proof of the Ka£-Peterson result, for Ka£-Moody groups of a ne type, an be found in the paper [60℄ of Chuu-Lian Terng.
1.6. The symple tomorphism group of the annulus
In the loop group examples above the group is in nite-dimensional but its Cartan subgroup is nite-dimensional.
We will des ribe below a on-
vexity theorem of Kostant type in whi h both the group and its Cartan subgroup are in nite-dimensional.
This result is due to Blo h, Flas hka
and Ratiu [13℄. Let
A
be the annulus
f (r; ) j r0 r r1 ; 0 < 2 g
1.6. THE ANNULUS
21
G be the group of symple tomorphisms (area-preserving maps) of A. A natural andidate for the Cartan subgroup of G is the group, T , of twist maps in the plane and let
7
(r; ) The Lie algebra of
G
P=
is, modulo onstants, the Poisson algebra
and the Lie algebra of depend only on
of
P.
r,
f 2 C1 ([r0 ; r1 ℄):
! (r; + f(r));
f f f 2 C1 (A) (r0 ) = (r1 ) = 0
(1.15)
T is the subalgebra of P onsisting of fun tions whi h
i.e. the ommutative subalgebra
Pradial = C1 ([r0 ; r1 ℄)
(1.16)
Moreover, inside this algebra is a natural positive Weyl hamber:
fun tions,
f 2 Pradial, whi h
are monotone in reasing. In addition there is a
natural Killing form on the Lie algebra of
f; g 2 P 7 !
Z
A
G,
the
L2
pairing
f(r)g(r) r dr d;
and if we use this pairing to identify the Lie algebras of
G and T with
their
Pradial into P is the averaging map: Z 1 G : P ! Pradial; f7 ! f(r; ) d: 2
an de ne the Weyl group of G to be W = N(T )=T , as in the
duals, the dual of the in lusion map of
Finally one
nite-dimensional ase; and one is led by these analogies to pose the following onje ture.
2 C1 ([r0 ; r1 ℄) orbit G. Then
1.6.1. Conje ture. Let
let
O
be the oadjoint
be monotone in reasing and
G (O ) = onv f w j w 2 W g:
(1.17)
Unfortunately it is easy to see that this statement, as it stands, is false, i.e. that the right-hand side is a very small subset of the left-hand side. The problem has to do with the de nition of the Weyl group,
W. It is easy G onsisting
see that this group an be identi ed with the subgroup of all maps of the form
(r; )
7
(r) is a measure preserving (r) takes on the values, 1.
where and
! (r); (r)
to of
(1.18)
r0 ; r1 ℄,
dieomorphism of the interval, [
Hen e this group is the four element
dihedral group, and the onvex hull of the four elements on the right side of (1.17) is a negligeably small subset of the set on the left. Noti e, however, that if one repla es
serving transformations of
A
and
T
G
by the group of measure pre-
by the group of measure preserving
22
1. THE CONVEXITY THEOREM
twist maps and repla e their Lie algebras (1.15) and (1.16) by
L2 ([r1 ; r2 ℄),
L2 (A)
all the a tions and maps that we de ned above are still well-
de ned, but the Weyl group is now a mu h bigger group, sin e the
and
and
in (1.18) are no longer required to be smooth but merely measurable.
With these minor adjustments Blo h, Flas hka and Ratiu prove that Conje ture 1.6.1 is true, and also show that it has an interpretation in terms of doubly sto hasti matri es whi h makes its resemblan e to the lassi al Kostant theorem even more striking. [13, pp. 516518℄.)
(See the dis ussion of this point on
CHAPTER 2
A onstru tive proof of the non-abelian onvexity theorem 2.1. Introdu tion
M a ompa t symple ti manifold, a Hamiltonian a tion of G on M and : M ! g the moment map asso iated with . Composing with the map, g ! g =G = t+ , one gets a map of M into t+ . The non-abelian onvexity theorem says that the image of this map is a onvex polytope, the Kirwan polytope, . Let
G
be a ompa t onne ted Lie group,
One of the short omings of this theorem is that (unlike the abelian
onvexity theorem) it does not provide a on rete des ription of
.
In this
hapter we will outline a proof of Kirwan's theorem, due to Sjamaar [59℄
2 , gives one a lo al on rete des ription of in a of . To simplify our a
ount of this proof we will assume p 2 M the isotropy group of p is abelian.
whi h, for xed neighborhood that for every
2.2. An outline of the proof of the Kirwan onvexity theorem
Let
p
be a point of
M
whose moment image,
,
is in t+ . There are two
ingredients in the proof. One is the following lo al onvexity theorem. 2.2.1. Theorem. There exist a neighborhood,
U
borhood,
0
, of
of
p
in t+ and a onvex oni polytope,
(U) \ t+ = Cp \ U . Moreover, -1 (), then Cp = Cq .
U,
0
p
if
and
q
in
M,
Cp ,
a neigh-
su h that
are nearby points on the
level set,
The other ingredient is a non-abelian generalization of the Atiyah onne tivity theorem: 2.2.2. Theorem. The level set,
-1 (),
is onne ted.
Cp = Cq for any pair 0 0 Hen e \ U = Cp \ U for a 0 U , of in t+ . This shows that is lo ally
Combining these two theorems one on ludes that
1 (). of points, p and q, on the level set, -
su iently small neighborhood,
onvex in a neighborhood of every point, and to on lude the proof, one invokes the easily veri ed fa t that every losed set whi h is lo ally onvex is onvex. 23
24
2. A CONSTRUCTIVE PROOF
We will not attempt to des ribe the proof of Theorem 2.2.2, whi h involves Morse theory and topology, but will fo us instead on the proof of Theorem 2.2.1.
Involved in this proof is an equivariant Darboux theorem
of Marle [51℄ and Guillemin-Sternberg [31℄, whi h des ribes what the a tion of
G looks
like in a
G-invariant
neighborhood of
p in terms of a simple
anoni al model. Thus to prove Theorem 2.2.1 it su es to prove it for this
anoni al model. We will a tually be able to do more: If the stabilizer group of
p
is abelian we will obtain from this anoni al model a simple expli it
Cp .
des ription of
This des ription will be the ontent of Theorem 2.2.3
below. To state this theorem, let us re all a few elementary fa ts about the
G on g . Let F be a fa e of t+ . It is easy to see that the stabilizer group, G , 2 F, does not depend on ; so we an denote it by GF . Sin e the oadjoint a tion of T on g xes t , T is ontained in GF . ss We will denote by GF the semi-simple part, [GF ; GF ℄, of GF and by TF the ss interse tion of GF with T . The Lie algebra of TF is the subspa e
oadjoint a tion of
t = (span F)
F
of t. Let
WF
?
GF
be the Weyl group of
hull of the orbit of
WF through . F
(i) If
Remarks.
(2.1)
2 t+
and for
let
F
G
is the degenerate fa e (g )
be the onvex
of the hamber
t+ , then WF = W , so F = , the onvex hull of the full W -orbit through . F (ii) If 2 F, then = fg. Coming ba k to the lo al moment one, Cp , suppose that the moment image, , of p is in F. Let K = Gp be the stabilizer group of p. Then K GF and, by assumption, K is abelian; so, without loss of generality we
K is ontained in T . Let X be the a tion of K on TpM preserves TpX and
an assume that the identity omponent of orbit of
G
through
p.
The isotropy
TpX)? ; hen e there is a natural representation
its symple ti annihilator (
K
on the symple ti ve tor spa e
N := (TpX)? Let
1 , 2 , : : : , r
X r :
t
!k
in t+ and let
F
?
(2.2)
i=1
ti i ti
0
p
2 M.
Suppose that
be the fa e of t+ ontaining
C℄p
k
be
:
be the dual of the in lusion map, k
2.2.3. Theorem. Let
pX) \ TpX :
(T
be the weights of this representation and let
the one
Finally let
Æ
of
.
:=
! t.
(p)
is ontained
Then the lo al moment
2.3. THE EQUIVARIANT DARBOUX THEOREM
one,
Cp ,
is the set
25
2 t+ (F ) \ (C℄p + ) 6= ; :
A proof of this theorem will be sket hed in Se tions 2.32.4. however, we will des ribe a few appli ations of it. des ription of the interse tion of
and
First,
One is a simple lo al
F. U,
2.2.4. Theorem. There exists a neighborhood,
U \ = U \ 1 (C℄p + ).
of
in
F
su h that
-
Proof. This follows from the fa t that
F = fg
for
2 F.
Another appli ation is the following riterion for a point, vertex of
.
2.2.5. Theorem. A ne essary ondition for
F
2 F,
to be a
2 F to be a vertex of
is that t + k = t.
ker , the a ne plane, k +, is in -1 (C℄p ) whenever -1 (C℄p ). Hen e if 0 6= 2 span F \ k , the line, + t,
Proof. Sin e
QED
itself is in
k? =
?
?
< t < 1, is ontained in -1 (C℄p + ), and so by Theorem 2.2.4 the line segment, + t, for -" < t < ", is ontained in \ F if " is small enough. Thus is not a vertex (extremal point) of . This shows that a ne essary ?
ondition for to be a vertex is the ondition span F \ k = f0g, and by (2.1) -1
this ondition is equivalent to the ondition t Finally we note that if
F + k = t.
G is abelian the vertex
QED
riterion be omes onsid-
erably simpler. 2.2.6. Theorem. Suppose that
dition for
= (p)
Cp the
i 's
G
to be a vertex of =
X
i
is abelian. Then a ne essary on-
is that
ti i + ti
p 2 MG ,
0
in whi h ase
;
being the weights of the isotopy representation of
G
on
TpM.
2.3. The equivariant Darboux theorem
In this se tion we will prove Theorem 2.2.3 modulo a simplifying assumption (whi h we will show how to get rid of in Se tion 2.4). This assumption
(p) = is a xed point of the oadjoint a tion of G. In other G words, F = (g ) and GF = G. An important geometri onsequen e of this assumption is that the orbit, X = Gp, is isotropi . Hen e the spa e (2.2) is the symple ti normal spa e to X at p, is that
N = (TpX)? =TpX:
26
2. A CONSTRUCTIVE PROOF
Now let
K = Gp
and onsider the produ t
T G N:
G a ts on this manifold by a ting trivially on N and by its left a tion on T G, and K a ts on this manifold by a ting by its linear a tion on N and by its right a tion on T G. Moreover, these a tions are Hamiltonian and
ommute with ea h other. Sin e the a tion of K is learly free, we an redu e T G N by K at the zero level set of its moment map to obtain a symple ti manifold, M an ; and sin e the G-a tion and K-a tion ommute, there is a 0 residual Hamiltonian a tion of G on M an . Let 0 be the zero ve tor in Te G 00 0 00 and 0 the zero ve tor in N. The point, 0 = (0 ; 0 ), in T G N proje ts onto a point, p0 in M an , and the G-orbit through this point is isotropi . We will normalize the moment map an : M an ! g by requiring that an (p0 ) = . 2.3.1. Theorem ([51, 31℄). There exist a G-invariant neighborhood, U, of p in M and an open equivariant symple ti imbedding (U; p) ! (M an ; p0 ) whi h intertwines and an . This equivariant Darboux theorem is a spe ial ase of a mu h more general Darboux theorem: the isotropi tubular neighborhood theorem of Weinstein [63℄.
M an .
Let us now ompute the moment image of
This omputation will
make use of the following three fa ts.
O is a oadjoint orbit of G, the symple ti redu tion of T G at O is -O. (Redu tion in stages) Let G1 and G2 be Lie groups and let G = G1 G2 . Given a Hamiltonian G-manifold, M, and oadjoint orbits O1 and O2 of G1 and G2 we an redu e M with respe t to the produ t orbit O = O1 O2 . Let us denote this redu ed spa e by MO . The redu tion in stages theorem asserts that there is another way to de ne MO . Namely we an view M as a Hamiltonian G1 -manifold and redu e it with respe t to O1 . Let's denote this redu ed spa e by MO1 . This spa e has a residual a tion of G2 , so we an redu e it with respe t to O2 . Let's denote this redu ed spa e by MO1 O2 . The redu tion in stages theorem asserts that (i) If
Fa ts.
(ii)
MO = MO1
O2 :
As a orollary we get the ommutativity of redu tion in stages theorem,
MO 1
O2
=
MO2
O1 :
2.4. THE CROSS-SECTION THEOREM
(iii) Let
1 , 2 , : : : , r
be the weights of the representation of
(Re all our assumption that image of
N
is the one
C
℄
27
=
X r
i=1
K
is abelian.)
ti i ti
0
Then the
K on N.
K-moment
:
T G N and the groups, G and K. For 2 t+ let O be the oadjoint orbit of G through . If we regard T G N as a G-spa e and redu e with respe t to O we get the produ t -O N. This spa e is a K-spa e and its moment image in k is Let's apply these remarks to the spa e
-( ) + C + : ℄
( ) omes from Kostant's theorem, the C℄ from fa t (iii) above and the from our normalization, an (p0 ) = , of an .) Thus the redu tion of -O N with respe t to the zero ve tor in k is not the empty set i (The
Now regard
T G N
0 2 -( ) + C℄ + :
as a
K-manifold
and redu e with respe t to the zero
ve tor in k . The spa e we get is, by de nition,
M an .
Next regard
M an as a
G-spa e and redu e it with respe t to O . The spa e we get is not the empty set i O is in the moment image of M an . Thus by the ommutativity of
redu tion in stages theorem we have proved 2.3.2. Theorem. The moment image of
2
t+ ( )
\ (C
M an
6 ;
+ ) =
℄
in t+ is the set
:
2.4. The ross-se tion theorem
Let
F
be a fa e of t+ , let
the enter of
GF .
GF
We will denote by g
xed by the oadjoint a tion of
F
be the stabilizer group of
H.
℄
and let
F the set of elements in g
H
whi h are
We laim (see [31℄)
F is invariant under the oadjoint a tion ℄of GF. The anoni al proje tion, g ! gF maps gF bije tively onto gF and is GF -equivariant. ℄ Every G-orbit in g interse ts g in a nite number of GF -orbits. F ℄
(i) g (ii)
(iii) Now x a
G-invariant
inner produ t on g
B" () = 2
and for
2F
℄ gF - < " :
k
k
let
For the following symple ti ross-se tion theorem see [31℄. 2.4.1. Theorem. Let (i)
be
Y
is a
Y = -1 (B" ()).
GF -invariant
Then for
"
small
symple ti submanifold of
M,
28
2. A CONSTRUCTIVE PROOF
G-orbit
(ii) every
orbit, (iii) the set
U
Taking
\t
(U)
G-invariant
(Y ) \ t+ ,
,
interse ts
Y
in a single
GF -
M.
is open in
open set of Theorem 2.4.1(iii), we have
so to prove Theorem 2.2.3 it su es to prove
Theorem 2.2.3 for the Hamiltonian xes
Y
U = f gp j (g; p) 2 G Y g
to be the
=
+
whi h interse ts
GF -manifold, Y .
Therefore, sin e
GF
Theorem 2.2.3 is a onsequen e of Theorem 2.3.2.
2.5. The Frobenius re ipro ity theorem
Theorem 2.3.2 has an interesting interpretation as the symple ti analogue of a lassi al theorem in representation theory: Given a nite group,
G, a subgroup, K, of G and a representation, , of K on a nite-dimensional ve tor spa e, V , de ne IndG to be the ve tor spa e of maps, f : G ! V , satisfying the automorphi ity ondition
f(ak) = (k)f(a)
2
K. If f belongs to this spa e, the fun tion, Ug f, de ned by Ug f(a) = f(g-1 a) also belongs to this spa e; so one has a anoni al representation of G on IndG , alled the indu ed representation of G asso iated to . At issue is the question of how this representation breaks up into ir-
for all
k
redu ibles, and the Frobenius re ipro ity theorem provides the answer: an irredu ible representation,
,
of
G
o
urs as a subrepresentation of
K o
urs jK.
if and only if some irredu ible representation of resentation of
and as a subrepresentation of
The symple ti analogue of indu tion works as follows.
G be of K on
Let
a
G and a Hamiltonian a tion a symple ti manifold, X. Then IndG is the spa e obtained from the produ t
Lie group,
K
IndG
both as a subrep-
a losed subgroup of
T G X
M an , in Se tion 2.3 is just the spa e, IndG , where is the linear Hamiltonian a tion of K on N. What about the re ipro ity theorem? Let M be a symple ti manifold and a Hamiltonian a tion of a Lie group, G, on M. The quantization
ommutes with redu tion prin iple tells us that the analogue for of an by redu ing with respe t to
K.
For instan e, the spa e,
irredu ible subrepresentation is a oadjoint orbit o
urring in the moment image of
M.
Hen e the symple ti analogue of the re ipro ity theorem
should be: 2.5.1. Theorem. Let
of
G
on
O.
Then
O
O
be a oadjoint orbit of
G
O the a tion M if and only
and
o
urs in the moment image of
2.6. CONVEXITY
if the moment image of
29
O jK
and the moment image of
nontrivially.
M an ,
In parti ular, applied to orbit
O
=
G,
with
2t
+ , o
urs in the moment image of
K
identity omponent of is equal to and
:
t
( ),
!
this theorem asserts that the oadjoint
O jK
only if the moment image of
where
interse ts
is a subgroup of
C
T , so
k is the dual of the in lusion, k
if and only if
℄
M an
if and
non-trivially. However, the the moment image of
O jK
is the onvex hull of the Weyl group orbit of
just a spe ial ase of the re ipro ity theorem:
M an
interse t
( ) \ C℄
6= ;.
!
O
t. Thus Theorem 2.3.2 is is in the moment image of
2.6. Convexity
For every through
.
2 t+
let
be the onvex hull in t
S,
Given a subset,
of t
let
S
[
=
of the Weyl group orbit
f 2 t+ j \ S 6= ; g.
Our goal
in this se tion is to prove 2.6.1. Theorem. If
S
is onvex,
S[
is onvex.
We begin by re alling a few elementary fa ts about the a tion of the Weyl group,
W , on
t . If we x a
W -invariant inner
produ t on t ,
W
be omes a
O(V ) and, in fa t, a rather spe ial kind of nite subgroup: + More expli itly, let R be a system of positive + roots for (g; t) and for ea h 2 R let s be the orthogonal re e tion in the hyperplane, H = f 2 t j = 0 g. Then W is generated by these re e tions. In fa t given any w 2 W , w an be written as a produ t of + these re e tions in the following expli it way: Let H be the half-spa e, f j 0 g. Then nite subgroup of
a nite re e tion group.
t+ = and
treg = t Now x an element,
[
\
f H+ j 2 R+ g
H =
[
w(Int t+ ) w 2 W :
2 Int t
(2.4)
+ , and suppose that the line joining
interse ts, su
essively, root hyperplanes,
s k s k-1 s 1 lie
(2.3)
H 1 , H 2 , : : : , H k .
to w w and
Then
in the same onne ted omponent of treg , and hen e by
(2.4)
w = s k s k-1 s 1 : We will need below the following property of this fa torization.
(2.5)
30
2. A CONSTRUCTIVE PROOF
2.6.2. Lemma. For
Then if
2 Int t+
and
1 l k let wl 0i<jk
j X
wi - wj =
Proof.
s l s l-1 s 1
=
tr r ,
and let
w0
=
1.
tr > 0:
with
r=i+1 We an write wi - wj as a sum (wi - wi+1 ) + + (wj-1 - wj ):
However,
wr - wr+1 = wr - s r+1 wr and, by de nition, and wr lie in the same onne ted omponent of t + H r+1 . Hen e by (2.3), wr lies in H r+1 ; so wr - s r+1 wr is a positive multiple of r+1 . QED Now let
CR+
be the one spanned by all the positive roots,
CR+
:=
X
t t
0
:
R+ Corollary. - w 2 CR+ for all 2 t+ and w 2 W . P Corollary. CR+ = w W tw ( - w) tw 0 2
2.6.3. 2.6.4.
2 Int t+ .
2
for all
Coming ba k to the proof of Theorem 2.6.1, we will rst prove this theorem when
S
onsists of a single point.
fg[
2.6.5. Theorem.
= ( 0 + C
unique Weyl group onjugate of Proof. Let
and
P
w t w = 1.
2t
+ be in
Thus
=
X
w
R+ ) \ t +
fg[ .
for all
in t+ .
Then
0
X
w
=
fg
(
0
is the
P
w W tw w, where tw 0 2
tw ( - w);
and, by Corollary 2.6.3, the se ond summand is in 0
where
tw = 0 + [
2 t ,
R+ ) \ t+ :
+C
CR + .
Thus
Int t+
C be the one spanned by the polytope, , with apex, . By de nition C is the onvex hull of the rays, + t(w - ), t 0, so by Corollary 2.6.4, C = - CR+ . To prove the reverse in lusion, let
It is lear that
be in
and let
\
f wC j w 2 W g: (2.6) Now belongs to ( + CR+ ) \ t+ i is in ( - CR+ ) \ t+ = C \ t+ ; so to [ prove that ( + CR+ ) \ t+ is ontained in fg it su es to show that C \ t+ =
0
0
0
2.7. RIGIDITY OF THE MOMENT CONE
31
. Be ause of (2.6) it is enough to show that C \ t+ is
ontained in wC for every w 2 W . Let 2 C \ t+ . Then w 2 wC . -1 Moreover, - w 2 CR+ by Corollary 2.6.3, so - w 2 -wCR+ . Hen e = w + ( - w) is in wC - wCR+ = w(C - CR+ ) = wC . Thus QED C \ t+ wC .
is ontained in
Now let
S be an
arbitrary onvex subset of t . From Theorem 2.6.5 one
gets the following des ription of
S
[
:
w 2 W , let Sw
2.6.6. Theorem. For
S[
=
[
=
w-1 S \ t+ .
Then
w + CR+ ) \ t+ :
(S
wW
2
Thus to prove Theorem 2.6.1 it su es to prove 2.6.7. Proposition. Suppose
S
interval,
w2
2 Sw1 and 2 Sw2 , Sw + CR+ , for all t in the
is onvex. If
the point, (1 - t) + t, is in one of the sets,
0 < t < 1.
Proof. Without loss of generality one an assume that
=
v.
Suppose
=
id
and
and v are in S, so (1 - t) + tv is in S by onvexity. (1 - t) + tv 2 S \ wt+ , with w 2 W . Then by Corollary 2.6.3 Then
(1 - t) + t = (1 - t)w is in
w1
1 + tw-1 v + (1 - t)( - w-1 ) + t( - w-1 v)
-
Sw + CR+ .
QED 2.7. Rigidity of the moment one
Our goal in this se tion is to verify the se ond of the two assertions in
-1 (). By the
ross-se tion theorem, Theorem 2.4.1, it su es to assume that = 0 and hen e to prove this assertion for the anoni al model, M an , des ribed in Se tion 2.3, with p being the base point, p0 . In this model the identity
omponent, K0 , of K := Gp is a subgroup of T , 1 is the proje tion, g ! k , N is a symple ti ve tor spa e, K N ! N a linear symple ti a tion of K on N and M an the quotient by K of the set Theorem 2.2.1:
Cp
Cq
(g; ; v)
if
p
and
q
are nearby points on
2 G g N 1 () = N(v)
g; ; v℄ for the image in M an G-moment map on M an is the map
Let us write [ the
=
:
(2.7)
g; ; v) in (2.7).
of a point (
([g; ; v℄) = Ad (g):
Then
(2.8)
It is lear from the de nition that the lo al moment one is rigid along orbits, so it su es to prove the rigidity at points in
q = [1; ; v℄, where 1 is the identity element in G.
-1 (0)
G-
of the form,
For the points of this form
32
2. A CONSTRUCTIVE PROOF
one an prove something stronger: One an show that the one
-1 (C℄q ) t
q 2 -1 (0). Note that by (2.8) 0, i.e. q = [1; 0; v℄. To ompute C℄q
guring in Theorem 2.2.3 is independent of the point,
q,
lies on
-1 (0)
i
=
at su h a point we have to ompute the symple ti normal spa e to the orbit
Gq
at
q,
the isotropy group,
Gq ,
and the representation of
Gq
on the
symple ti normal spa e. 2.7.1. Lemma. Let
q = [1; 0; v℄ 2 M an .
(i) The symple ti normal spa e at
Gq = Kv .
(ii)
(iii) The representation of
Gq
q
is
N.
on the symple ti normal spa e is
the restri tion of the representation of Proof. The
K
on
N
to
Kv .
G-orbit through (1; 0; v) 2 G g N is f (g; 0; v) j g 2 G g,
so the normal spa e to it is
f (; w) j 1 () = d v (w) g:
(2.9)
1 is k? , the annihilator of k in g , whi h is the dual spa e to Tq(Gq) = g=k, and hen e the symple ti normal spa e an be identi ed with N by (2.9). Moreover, the stabilizer group in G of q is the set of all elements, g, in G satisfying g(1; 0; v) = (1; 0; v)k for some k 2 K, i.e. (g; 0; v) = (k; 0; kv). Thus g = k and kv = v, whi h shows that Gq = Kv . Finally, by (2.9) it is lear that the representation of Gq on the symple ti normal spa e at q is just the representation of Kv on N. QED
The kernel of
1 , 2 , : : : , r 2 k N, and let v be the
Let
be the weights of the isotropy representation of
2.7.2. Proposition. If
1 v 2 N (0),
K on proje tion of k onto kv . Then v 1 , v 2 , : : : , ℄ v r , are theweights of the isotropy representation of Kv on N. Let C k P ℄ ℄ ℄ ℄ be the one i ti i ti 0 and let Cv = v (C ). (Note that C = Cp0 , where p0 is the base point, [1; 0; 0℄.) If (1; 0; v) is in the set (2.7) and q = ℄ ℄ ℄ [1; 0; v℄ 2 M an , then Cq = Cv by Lemma 2.7.1, so to show that -1 (Cq ) is independent of q it su es to prove
then
C℄
=
℄ v 1 (Cv ).
This is of ourse just the rigidity theorem for a linear a tion of a torus. For the sake of ompleteness let us review the proof. Let k
v in k
lator of k
. Then
v
?
be the annihi-
v 1 (C℄v ) = -v 1 (v (C℄ )) = C℄ + k?v ;
so Proposition 2.7.2 is equivalent to 2.7.3. Proposition. For every
1 v 2 N (0),
v
k? is ontained in
C℄ .
2.7. RIGIDITY OF THE MOMENT CONE
T
33
℄ H+ +i i 2 k be a representation of C as an interse tion of half-spa es, H being the half-spa e, f 2 k j (i ) 0 g. i Let h be the span of the i 's and let H be the subtorus of K0 having h as ? its Lie algebra. Clearly h C℄, so the result follows from Lemma 2.7.4
Proof. Let
C℄
=
below.
QED
2.7.4. Lemma. For every
1 v 2 N (0),
v h.
k
1 , 2 , : : : , r , so that i 62 h? i 1 i l. l+2 , : : : , r . We assert that
Proof. Order the weights,
Let m be the span of
l+1 ,
?
m=h
:
(2.10)
2 h? , 6= 0, of the form, = i=1 ti i , with ti 0. However, for every i , 1 i l, there exists a j su h that i (j ) 6= 0; so ti = 0 for all i and = 0, whi h is The ontrapositive of this assertion is that there exists a
Pl
a ontradi tion.
N with Cr so that K0 be omes a subgroup of S1 S1 S1 of K0 on N the a tion
Now identify and the a tion
1 1 () v1 ; e -1 2 () v2 ; : : : ; e -1 r ()vr : (2.11) Pr moment map is N (v) = i=1 jvi j2 i . We laim that if v 2 p
(exp )v = The resulting
1 N (0),
then
1
Indeed, if
e
i
ontained in
Hj , +
p
-
v1 = v2 = = vl = 0: l, then i (j ) > 0 for some j; and
vi = 0.
Now suppose
v
r X k=1
1 ; v2 : : : ; vr )
= (v
is on the zero level set of
ontained
k 's
are
v
-1 N (0), then, by (2.12), vi in spanf i j 1 i l g = m.
=
v h.
ontains h. Con lusion: k
N .
Let
algebra, k , of the stabilizer group
spanf i j i 2 I g:
v
sin e all the
jvk j2 k (j ) = hN (v); j i = 0;
I = f i j vi 6= 0 g. Then by (2.11) the Lie of v is the annihilator in k of the spa e But if
(2.12)
we have
0 jvi j2 i (j ) and hen e
p
=
0
(2.13) for
1
i
l,
so (2.13) is
Hen e, by (2.10), its annihilator QED
CHAPTER 3
Some elementary examples of the onvexity theorem 3.1. Introdu tion
The abelian onvexity theorem omes with a built-in des ription of the moment polytope.
One of the short omings of the non-abelian onvexity
theorem is that it does not.
In on rete examples one an determine the
moment polytope, to some extent, by the kinds of lo al methods whi h we dis ussed in the last hapter.
For instan e using the vertex riterion,
Theorem 2.2.5 in the last hapter, one an often determine the verti es whi h lie in the interior of the positive Weyl hamber or on its odimension one fa es. However, more often than not, one has to supplement these methods by other methods in order to get a omplete des ription of the moment polytope. We will dis uss in this hapter some examples of the onvexity theorem for whi h these other methods are of onsiderable interest in their own right. The problems we will be talking about here appear mu h more elementary than those that were dis ussed in the rst two hapters. instan e, Se tions 3.23.4 will be on erned with the problem: If
(n + 1)
A
For is an
(n + 1) Hermitian matrix with eigenvalues, 1, 2, : : : , n 1, what k k minor for 1 k n? +
an one say about the eigenvalues of its prin ipal
Se tions 3.53.7 will be on erned with a somewhat related problem: Let
A and B be n n Hermitian matri es with eigenvalues 1 , 2 , : : : , n and 1 , 2 , : : : , n . What an one say about the eigenvalues of A + B? For both these problems a Kirwan polytope is involved, and we will see that, even though these problems are elementary, symple ti geometry and moment te hniques turn out to be very helpful in understanding the answers. The se ond problem, by the way, was solved a few years ago by Klya hko [41℄, and a beautiful a
ount of his result an be found in Fulton's arti le [26℄ (from whi h we have borrowed some of the material in Se tions 3.6 and 3.7.) A generalization of Klya hko's result, by Berenstein and Sjamaar [9℄, is dis ussed in Se tion 3.8. (Their result is a partial response to a question we raised in the last hapter: is there a global des ription of the Kirwan polytope analogous to the lo al des ription whi h we gave in the previous
hapter?) 35
36
3. ELEMENTARY EXAMPLES
3.2. The eigenvalues of the prin ipal minors of a Hermitian matrix
Let
Hn+1
be the spa e of Hermitian (
n + 1) (n + 1)
matri es and let
: Hn+1 ! Hn A = (aij ) 2 Hn+1 , its n n prin ipal minor, (aij )1i;jn . Let = (1 ; 2 ; : : : ; n+1 ) be an (n + 1)-tuple n+1 of real numbers with 1 > 2 > > n+1 and let H be the set of all Hermitian (n + 1) (n + 1) matri es with eigenvalues 1 , 2 , : : : , n+1 .
be the map whi h assigns to ea h matrix,
The question we want to dis uss below is what onditions the assumption,
A 2 Hn+1, imposes on the
eigenvalues of
(A).
In other words, what is the
set
:=
2 Rn 1 2 n ;
Hn Hn+1
?
One thing whi h is easy to see is that this set is a onvex polytope. Letting
G = U(n)
and
~ = U(n + 1) G
one has identi ations
g =g=
~g
~= =g
p
n = Hn ; n+1 = Hn+1; -1 H
p
-1 H
and, modulo these identi ations, the oadjoint orbits of
~ are the G
isospe -
n+1, and is the Kirwan polytope asso iated with the Hamiltral sets, H tonian a tion of G on H . As for the expli it determination of this polytope, we note that if
A
2
Hn+1 and the eigenvalues of (A) are 1 2 n , then by a simple mini-max argument the
1
's
satisfy the interla ing ondition,
1 2 2 n 1 n n n 1;
and we will prove below that
-
+
(3.1)
is exa tly the set of 's satisfying (3.1). The
proof is not hard, but it is interesting be ause it reveals some unexpe ted
onne tions between the spe tral theory of prin ipal minors and the theory of onfo al quadri s. Let's see why onfo al quadri s are onne ted with this
satis es (3.1) there exists an A 2 Hn+1 su h that (A) has spe trum f1 ; 2 ; : : : ; n g. Without loss of generality we
problem. We want to show that if
an assume that
1 > 1 > 2 > 2 > > n-1 > n > n > n+1 ;
(3.2)
3.3. CONFOCAL QUADRICS
sin e these
's
37
are an open dense subset of (3.1), and after onjugating
n + 1)
by an element of U(
we an assume that it has the form
0
1
B B A=B
..
0
.
1
z1
0
C C C: zn A . . .
(3.3)
n zn 2x0
z1
In fa t we an do a bit more. The a tion of the Cartan subgroup,
n) on matri es of the form above just zi 's are all real and non-negative, i.e.
U(
the
0
1
B B A=B with
xi 0
for
i 1.
3.2.1. Lemma.
0 x1
..
.
xi =(i
-
)
1
rotates the
zi 's
T n,
of
so we an assume
1
x1
0
C C C xn A . . .
(3.4)
n xn 2x0
We laim
det(A - ) = (1 - ) (n - )q(; x), n x2 X i q(; x) = 2x0 - +
Proof. For
A
i=1 - i
where
:
i n multiply the ith row of the matrix, A - , by
and subtra t it from the last row.
triangular matrix with diagonal entries,
P 2x0 - + i x2i =( - i ).
1
-
This gives one an upper
, 2
-
, : : : , n
-
and
QED
From this lemma we on lude
Hn+1 if and only if (x0 ; x1 ; : : : ; xn ) is a ommon point of interse tion of the n + 1 3.2.2. Theorem. If (3.2) holds, then the matrix (3.4) is in
onfo al quadri s
Qi
=
n X x2i n +1 x2R i=1 i - i
=
i - 2x0 :
3.3. Confo al quadri s
1 ; 2 ; : : : ; n ) be a xed n-tuple satisfying 1 > 2 > 2 R let Q be the family of onfo al quadri s
Let ( and for
x21 - 1
+
x22 - 2
+
2
+ -xn
n
=
> n
- 2x0 :
We will prove below a number of properties of these quadri s and dedu e from these properties a very strong version of the onvexity theorem that we des ribed in Se tion 3.2.
38
3. ELEMENTARY EXAMPLES
x
3.3.1. Lemma. Let
non-zero. Then
Q .
x
is on
Moreover, if these
where
0 ; x1 ; : : : ; xn ),
x1 , x2 , : : : , xn are all the interse tion of exa tly n + 1 of the quadri s interse ting quadri s are Q1 , Q2 , : : : , Qn+1 , = (x
1 > 2 > > n+1 ,
Proof. Fix
x0 , : : : , xn
f() =
x21
- 1
where
then (3.2) holds.
with
+
xi 6= 0
x22
- 2
for
i1
and onsider the fun tion
x2n
+ -
+
n
0
+ 2x - :
This fun tion has the following obvious properties:
f 0 () < 0
for
6= i ;
!+1
lim f() = -1
and
lim f() = +1
!-1
lim f() = +1;
!+i
and
!-i
lim f() = -1:
x2i > 0 for i 1 and the i are distin t, this shows that f has exa tly n+1 zeroes, 1 > 2 > > n+1 , and that these zeroes satisfy (3.2). QED Sin e
x = (x0 ; : : : ; xn ), of the Q1 , Q2 , : : : , Qn+1 , the normal ve tors to these quadri s are
3.3.2. Lemma. At ea h point of interse tion,
quadri s,
mutually perpendi ular. Proof. Let
A be the symmetri matrix (3.4).
By Theorem 3.2.2 the
i 's
are the eigenvalues of this matrix. Moreover, the eigenve tor orresponding to
i
must satisfy
0
a1
1
0
1
B .. C B B C B i B . C = B an A
1
i.e.
i aj
=
j aj + xj ,
or
aj
=
0
..
.
0
10
a1
1
C B .. C CB . C CB C; xn A an A . . .
n xn 2x0
x1
x1
xj =(i - j ).
Sin e
1
A
is symmetri its eigen-
ve tors are mutually perpendi ular, so the ve tors
vi
=
xn ; ;:::; ;1 i - 1 i - 2 i - n x1
x2
are mutually perpendi ular. Noti e, however, that the ve tor, normal ve tor to the quadri ,
Qi ,
x 1 ; x 2 ; : : : ; x n ).
at (
BAB-1
vi , is just the QED
wi = vi =jvi j be the unit normal to Qi at (x0 ; x1 ; : : : ; xn ), B be the orthogonal matrix whose olumn ve tors are the wi 's. Then
Remark. Let
and let
(3.5)
is the diagonal matrix
0
D=
1
0
..
.
0 n+1
1 A:
3.3. CONFOCAL QUADRICS
In parti ular, the tra e of
D is 1 + 2 + + n+1 = 1 + 2 + + n + 2x0 ,
X
so
2x0
n For xed 2 R
39
de ne
=
i
i -
X
i
i :
(3.6)
to be the region of all
2 Rn 1 satisfying +
ondition (3.2), i.e.
= 2 Rn+1 1 > 1 > 2 > 2 > > n-1 > n > n > n+1 : 2 Rn+1
For ea h
let
I
be the interse tion lo us
I = Q1 \ Q2 \ Qn+1 : I
A
ording to Lemma 3.3.1, if
is nonempty, then
2 .
2 . Then ea h interse tion point, x 2 I , satis es xi 6= 0 for i 1. Moreover, if K is a ompa t subset of , the S n+1
olle tion of ve tors 2K I is a ompa t subset of R . 3.3.3. Lemma. Let
Proof. Let
2 and x 2 I , and suppose xi
Then we see from (3.4) that the matrix,
i
=
i ,
A,
has
=
= i
0
1.
i
for some
as an eigenvalue, so
ontradi ting our hypothesis that the inequalities (3.2) are stri t.
Furthermore, Theorem 3.2.2 tells us we have a mapping, ing to the ve tor
x
the matrix
A
I ! Hn+1 , assign-
given in (3.4). This mapping is evidently
proper, so the se ond assertion now follows from the observation that the union
n+1 K H is ompa t.
S
2
QED
2 the interse tion with xi > 0 for i 1.
3.3.4. Theorem. For ea h
a point,
x = (x0 ; x1 ; : : : ; xn ),
Proof. De ne
property.
0
to be the olle tion of all
We must show that
0
=
and it follows from Lemma 3.3.1 that show that
0
is open and losed in
.
. 0
.
2 . For > 0 for i 1.
whi h satis es
x(ik)
k the sequen e x
ea h
is onne ted
2
k
Hen e
1 , (2) , : : : , ( )
the quadri s
1 sele t a ve tor
is open
0
and
x k 2 I(k) ( )
and
xi
0
shows that QED
x, above is the unique interse tion of these quadri s with xi > 0, for i 1. (Repla ing xi by xi we get n interse tion points. In other words, there are 2 interse tion points
We will next show that the point,
other
Qi
By Lemma 3.3.3 we may assume that
for
point
0
lying in
onverges to a limit x. By ontinuity, x 2 I i 1. But then xi > 0 by Lemma 3.3.3, sin e 2 . This 2 0 and hen e that 0 is losed in . ( )
ontains
is nonempty; hen e it su es to
If for some
Consider now a sequen e of ve tors
onverging to a limit,
I ,
2 having the desired
It is lear that
interse t, they interse t transversely by Lemma 3.3.2. in
lo us,
in all.) To prove this we will need
40
3. ELEMENTARY EXAMPLES
2 1 B 1 - 1 B
3.3.5. Lemma. For ea h
0
the matrix
.. .
C=B B
1
n+1 - 1
1
1 - n .. .
1
n+1 - n
1
1 C
.. C C .C
1
A
is non-singular.
I , ontains a point (x0 ; x1 ; : : : ; xn ) su h that xi > 0 for i 1. Multiplying the ith olumn of C by xi one obtains Proof. By Theorem 3.3.4 the lo us,
the matrix whose row ve tors are the ve tors (3.5) and sin e these ve tors are linearly independent,
C
is non-singular.
QED
2 the interse tion lo us I onsists 2n points, and in parti ular there is exa tly one interse tion x = (x0 ; x1 ; : : : ; xn ), with the property, xi > 0 for i 1.
3.3.6. Theorem. For every
of exa tly point,
0 ; x1 ; : : : ; xn ) be one interse tion point. By Lemma 3.3.3 the omponents x1 , x2 , : : : , xn are non-zero, so the points (x0 ; x1 ; : : : ; xn ) n are a set of 2 distin t interse tion points. On the other hand, x0 is om2 2 2 pletely determined by (3.6) and by Lemma 3.3.5, x1 , x2 , : : : , xn are omProof. Let
(x
pletely determined by the system of equations
n X
x2j
j=1 i - j for
=
i - 2x0
i = 1, 2, : : : , n + 1.
QED
Coming ba k to the eigenvalue problem of Se tion 3.2, we get from Theorems 3.2.2 and 3.3.6 the following result: 3.3.7. Theorem. For all ve tors
2 Rn+1
and
2 Rn
satisfying
1 > 1 > 2 > 2 > > n-1 > n > n > n+1 there exist exa tly
2n
matri es
A
2 H of the form xi > 0 for i 1.
(3.4) with the
real and a unique su h matrix with
xi 's
This result has an important impli ation for the a tion of the group,
Hn+1 . We an think of Hn+1 as a oadjoint orbit, O, of ~ = U(n + 1), and hen e, sin e G sits inside U(n + 1), as a the group, G Hamiltonian G-manifold. Let : O ! g be the moment map. As we
G
= U(n), on
pointed out in Se tion 3.2 the polytope
= f 2 Rn j 1 1 n n 1 g +
3.4. GELFAND-CETLIN
is the Kirwan polytope of
O.
3.3.7 there exists a unique matrix,
xn
positive, and
-1 ()
U(
n),
i 's
i = 1, 2, : : : , n:
are distin t the entralizer,
zi 's,
Then by Theorem
A 2 H , of the form (3.4) with x1 , x2 , : : : ,
for
is the diagonal subgroup,
by rotating the
2 Int .
onsists of all matri es of the form (3.3) with
jzij = xi Sin e the
Now suppose
41
T n,
G ,
of
(3.7)
inside the group,
G
=
and it a ts on the set of matri es (3.3)
i.e. by the a tion
z 7 ! ei1 z1 ; ei2 z2 ; : : : ; ein zn : Sin e
jzij = xi > 0 this a tion is a free a tion and by (3.7) the set, 1(), -
is a single
G -orbit.
Thus we have proved
3.3.8. Theorem. For every
and transitively on
-1 ().
2 Int
the entralizer
Hen e the redu ed spa e,
G
a ts freely
Ored () = -1 ()=G ;
onsists of a single point. This shows that the a tion of
G
on
O
is multipli ity-free. (For more
about multipli ity-free a tions see [30℄.)
3.4. Gelfand-Cetlin
Let
Ak
denote the prin ipal
kk
minor of a matrix
A
2 Hn 1 . +
The
question we want to dis uss in this se tion is what onditions the assump-
A 2 Hn+1, imposes on the eigenvalues k the map, A 7! Ak , what is the set
tion, by
k
( )
For
k = n,
:=
2 Rk 1 2
of
Ak .
k ;
In other words, denoting
Hk k Hn+1
?
we answered this question in the last se tion, and we will show
below that we an extra t from that result the answer to this question for arbitrary
k.
Let us denote the eigenvalues of
Ak
by
1k , 2k , : : : , kk , and let 1k 2k kk .
assume these are arranged in de reasing order:
us
3. ELEMENTARY EXAMPLES
ik 's
prin ipal minor are
Ak-1
22 11
ik 2 Hn 1, for whi h the eigenvalues of
Moreover, for every sequen e of ditions there exists a matrix,
Proof. Consider
nn
n-1;n-1 12
kth
1;n-1
its
n-1;n
2n
n+1
n
1n
n-1
3
2
1
satisfy the interla ing onditions
3.4.1. Theorem. The
42
A
's satisfying these interla ing on+
1k , 2k , : : : , kk .
k - 1) (k - 1)
as the prin ipal (
minor of
and apply to it the results we proved in the last se tion. The inverted pyramid of Theorem 3.4.1, with top row,
1n , 2n , : : : , nn
1 , 2 , : : : , n+1
Ak
QED along its
along the se ond row from the top and so on, is
alled a Gelfand-Cetlin diagram, and the set of all Gelfand-Cetlin diagrams with xed top row,
,
is a onvex polytope,
r, alled the Gelfand-Cetlin
polytope. As a orollary of Theorem 3.4.1 we have:
k
( )
3.4.2. Theorem. The set
r
! Rk ;
is the image of the proje tion
ij 7 ! (1k ; 2k : : : ; kk ):
( )
In parti ular it is a onvex polytope.
As in Se tion 3.2 we an think
Hn+1 as a oadjoint orbit, O , of ~ = U(n + 1). Let G = U(k) and imbed G in G ~ as the group, U(k) In+1-k. G (k) Then O is a Hamiltonian G-manifold and the polytope is its Kirwan
of it as a Kirwan polytope by regarding
polytope.
What about the polytope,
r, itself? This an also be regarded, in some
sense, as a moment polytope.
pkk ()
Namely for ea h
be the elementary symmetri fun tions in
: Hn+1 ! r
k let p1k (), p2k (), : : : , 1k , 2k , : : : , kk , and let
A 2 H onto the Gelfand-Cetlin diagram (ik ), with Ak . This map is not a smooth n+1 with map, but the pullba ks, pik , are smooth. Hen e if we identify H the orresponding oadjoint orbit, O , of U(n + 1), the fun tions, pik , 1 be ome smooth fun tions on O . Noti e that there are exa tly n(n + 1) of 2
be the map whi h maps
1k , 2k , : : : , kk
being the eigenvalues of
3.5. THE SUM OF TWO HERMITIAN MATRICES
these fun tions and that
dim O = n(n + 1).
these fun tions are independent on the set
43
Moreover, it is easy to see that
-1 (Int r ),
and hen e by the
following theorem of Thimm [61℄ they form a ompletely integrable system.
pik ,
3.4.3. Theorem. The Poisson bra kets of the fun tions,
1ik
and
1 k n,
are all identi ally zero.
This system is alled the lassi al Gelfand-Cetlin system.
for
In [28℄ a
number of its properties are dis ussed and in parti ular the following theorem is proved. 3.4.4. Theorem. The
grable system
f pik g,
i;k 's
are the a tion oordinates of the inte-
and from the angle oordinates asso iated with
these a tion oordinates one gets a Hamiltonian a tion of the torus,
T n(n+1)=2,
on the open subset,
-1 (Int r ),
of
O
with moment map,
.
3.5. The eigenvalues of the sum of two Hermitian matri es
Let H be the set of all n n Hermitian matri es, and let H be the set n n Hermitian matri es with eigenvalues = (1 ; 2 ; : : : ; n ), where 1 2 n . What an one say about the eigenvalues of A + B, where A 2 H and B 2 H ? In other words what are properties of the set
of
:=
f j H H + H g?
Let's rst of all prove that this set is a onvex polytope.
onsequen e of Kirwan's theorem. Let
G
p
This is an easy
= U(n). Then g =
-1 H =
H.
A; B) 7! tra e AB, whi h is nonG-invariant; so it gives one a G-equivariant identi ation of g with g and of H with a oadjoint orbit, O , of G. Hen e the set is just the Kirwan polytope of the Hamiltonian G-manifold, O O . To get a more on rete pi ture of , let us try to determine this set when A is a su iently generi element of H and B is a small perturbation of A. For every multi-index
The Killing form on g is the bilinear form, ( degenerate and
I = (i1 ; i2 ; : : : ; ir );
1 i1 < i2 < < ir
n;
i I i , and let us assume that has the property I 6= J for I 6= J. (In parti ular, i 6= j for i 6= j.) Let Sn be the group of permuta tions of f1; 2; : : : ; ng and for 2 Sn write = (1) ; (2) ; : : : ; (n) . We let
I
P
be the sum,
2
laim: 3.5.1. Theorem. If
ti al with the set
is large ompared with
onv f + j 2 Sn g.
,
the set
is iden-
44
3. ELEMENTARY EXAMPLES
H and H with o-adjoint orbits, O and O , in g . The assumption that is small ompared with guarantees that the Kirwan polytope of O O is ontained in the interior of t+ . So Proof. As above we an identify
by the vertex riterion of the previous hapter, Theorem 2.2.5, the verti es of the moment polytope are the images in
OT OT =
and sin e
2 Sn .
Int t+
of the xed point set
(1 ; 2 ) 1 , 2
2 Sn
;
is large ompared with this image onsists of the points, +, QED
A of C
This perturbation problem also has a slightly dierent solution. Let be in
H , B
has to satisfy
for
H
C = A + B. Then the largest eigenvalue, 1 , 1 1 + 1 and, more generally, by mini-max te hniques,
in
i + j n + 1.
i+j-1 i + j
(3.8)
In parti ular
i
i + 1:
(3.9)
r Cn . From C we get a selfadjoint linear operator (whi h we will also all C) mapping V to V , and (r) this extends to a selfadjoint operator C : r ! r whi h is given by
Now let
V = Cn
and
and let
r
be the exterior power
r X r v1 ^ v2 ^ ^ Cvi ^ ^ vr C (v1 ^ v2 ^ ^ vr ) = i=1 (r) on de omposable elements. For C in H , the eigenvalues of C are the P n sums, I = (r) (r) (r) i I i . Now apply (3.9) to A , B and C . Sin e the r I 's are all distin t and is small ompared with , I is the kth largest (r) (r) eigenvalue of A if and only if I is the kth largest eigenvalue of C . (r) Moreover, the largest eigenvalue of B is 1 + 2 + + r , so by (3.9) r X I I + i : (3.10) i=1 ( )
2
If
I
is of length
n,
this inequality is an equality
n X i=1
sin e
i
=
n X i=1
i +
n X i=1
i ;
tra e C = tra e(A + B) = tra e A + tra e B.
are equivalent to Theorem 3.5.1. su ient ondition for
(3.11)
We laim that (3.10)(3.11)
Sin e Theorem 3.5.1 is a ne essary and
to be in the set , it implies (3.10)(3.11); and the
impli ation the other way an easily be dedu ed from a onvexity theorem of Birkho for doubly sto hasti matri es. Re all that a doubly sto hasti matrix is a square matrix with non-negative entries whose row and olumn
3.5. THE SUM OF TWO HERMITIAN MATRICES
45
sums are equal to one. Let us denote the set of all doubly sto hasti matri es by
Ms2 .
nn
This set is onvex, and Birkho 's theorem asserts that it
is a onvex polytope whose verti es are the permutation matri es. Thus
onvf + j 2 Sn g = f + M j M 2 Ms2 g: Now let
0
=
- and let = (1 ; 2 ; : : : ; n ) be n-tuple, 0 , so that 1 2
oe ients of the
a re-ordering of the
n.
Then (3.10)
(3.11) redu e to
r X i=1
n X i=1
i i
=
r X i=1
n X i=1
i
for all
r,
i :
(3.10)
0
(3.11)
0
, is said to majorize the sequen e, , if (3.10) 0 (3.11) 0 hold. .) Thus, to prove the impli ation, (3.10)(3.11) =)
The sequen e, (In symbols,
Theorem 3.5.1, it su es to prove
1
3.5.2. Lemma. Let
= (1 ; 2 ; : : : ; n )
there exists a doubly sto hasti matrix,
for
= (1 ; 2 ; : : : ; n ), where if and only if Pn (bij ), su h that i = j=1 bijj and
2 n and 1 2 n . i = 1, 2, : : : , n.
Then
The proof of this fa t is ompletely elementary and an be found in [32, se tion 2.20℄. The proof we have just sket hed that (3.10)(3.11) imply Theorem 3.5.1 is due to Helmut Wielandt [65℄. Thus the perturbation version of this problem about eigenvalues of sums of Hermitian matri es is ompletely solved by (3.10)(3.11). Our solution also suggests that, for arbitrary
and , the set should be hara terized
by the mini-max inequalities (3.8) and the analogues of these inequalities for
A(r) , B(r) and C(r) .
We will dis uss in the next se tion a theorem of Klya hko
whi h says that, morally speaking, this statement is true.
However, this
statement annot, unfortunately, be redu ed to a ni e set of inequalities of Take, for instan e, r = 2. Then 1 + 2 is the largest A(2) , 1 + 2 the largest eigenvalue of B(2) and 1 + 2 the (2) eigenvalues of C , so from (3.9) we get the inequality
the form (3.10). eigenvalue of largest
1 + 2 1 + 2 + 1 + 2 : Moreover,
1 + 3
C 2 , so by (3.9) ( )
and
1 + 3
are the se ond largest eigenvalues of
1 + 3 1 + 2 + 1 + 3 :
B(2)
and
46
3. ELEMENTARY EXAMPLES
However, what about
1 + 4
and
1 + 4 ?
These an either be third largest
or fourth largest eigenvalues; so (3.10) does not imply
1 + 4 1 + 2 + 1 + 4 ;
(3.12)
but only the weaker statement: either (3.12) or
1 + 4 1 + 2 + 2 + 3 : We will show in Se tion 3.6 that the problem of ordering the eigenvalues of
A(r) , B(r) , and C(r) so as to make sense of the r for an, to a ertain extent, be solved by
mini-max analogues of (3.8) topology.
the eigenve tors orresponding to the eigenvalues
More expli itly,
r (r) and C(r) of A , B ( )
are
de omposable elements of r , and hen e, up to s alar multiples, elements n of the Grassmann manifold Gr(C ; r), and the problem of ordering these (r) (r) eigenvalues an, to a ertain extent, be solved by regarding A , B and n (r) C as Bott-Morse fun tions on Gr(C ; r) and ordering their riti al values by topology. 3.6. Morse theory on a Hamiltonian
G
Let
M
(for the moment) be a torus, and let
2d-dimensional
G-manifold
be a ompa t onne ted
a Hamiltonian a tion of G on M p 2 MG , let 1;p , 2;p , : : : , d;p be the weights of the isotropy representation of G on TpM and let P g symple ti manifold and
with nite xed point set
MG .
For ea h point,
be the omplement of the weight hyperplanes,
P = 2 g i;p () 6= 0
for all
The onne ted omponents of
P
p 2 MG
and
i = 1, 2, : : : , d :
(3.13)
are open onvex polyhedral ones, whi h
we will all the a tion hambers.
For
2 g, let
=
h; i be the -
omponent of the moment map. We will prove: 3.6.1. Lemma. The riti al points of
2 P.
Moreover, if
and the index of
2 P,
then
are isolated if and only if
is a Morse fun tion,
at a riti al point
p
rit = MG ,
is equal to twi e the number
#f i;p () j i;p () < 0 g:
hi denote the subtorus of G generated by 2 g. The d = -(M )! implies that rit = Mhi , the xed-point set of the torus hi. Now suppose 2 g - P. Then j;p () = 0 for some j and G some p 2 M . By the equivariant Darboux theorem, Theorem 2.3.1, we d
an assume that (M; p) = (C ; 0), that the symple ti form is the standard p d -1 P dzi ^ dzi , and that the a tion of G is a symple ti form on C , i Proof. Let
identity
d
linear a tion with weight spa e de omposition, C and weights,
i;p .
Then
a ts trivially on the
jth
= C
C C,
weight spa e and so
3.6. MORSE THEORY
dim M
h i
2, i.e. M
h i
47
is not isolated. Conversely, suppose
be a onne ted omponent of M
2 P.
Let
X
and let p be a G- xed point in X. (Su h a X is a Hamiltonian G-manifold in its own right.) Again using the lo al normal form at p, we see that TpX = f0g, be ause a ts nontrivially on ea h weight spa e, and hen e X = fpg. Thus we see that Mhi = MG , and in parti ular Mhi is isolated. The moment map in the P equivariant Darboux hart at p is (z) = (0)+ i;p ()jzi j2 , from whi h h i
point exists be ause
the statement on erning the Morse index follows immediately. Let
2 P.
The fa t that the riti al points of
dex implies that
M
QED
are all of even in-
is homotopy equivalent to a CW omplex with only
even-dimensional ells. Therefore
is a perfe t Morse fun tion, i.e. the
homology is torsion-free and
rank Hi (M; Z) = # p 2 MG index p = i :
In parti ular,
Hi (M; Z)
is zero for
asso iate with the set
bases of
H2i (M; Z)
and
i
odd.
Using Morse theory one an
p 2 MG index p = 2i
H2d-2i (M; Z).
We will review how these bases are
onstru ted, starting for simpli ity with the Kähler ase and nishing with a brief dis ussion of the general symple ti ase. The Kähler ase. Suppose
M admits a G-invariant
omplex stru ture
J whi h is ompatible with the symple ti form in the sense that !(; J) is a M. Let v be the gradient ve tor G the eld asso iated with , and let exp tv be its ow. For ea h p 2 M stable (resp. unstable ) manifolds of v at p are de ned to be the sets
(positive de nite) Riemannian metri on
W (p; ) = q 2 M lim (exp tv )(q) = p :
t! 1
These submanifolds are ells,
R2i W +(p; ) = where
2i
=
spa es and the a tion
index p.
In fa t,
and
TpW +(p; )
R2d-2i ; W -(p; ) = is the sum of the positive eigen-
TpW (p; ) is the sum of the negative eigenspa es of TpM for + of . Hen e W (p; ) and W (p; ) interse t transversely at p. -
The two de ompositions
M=
[
p MG 2
W +(p; ) =
[
p MG
W -(p; )
2
are the stable, resp. unstable Morse de ompositions of
M.
Extending work
of Biaªyni ki-Birula [10, 11℄, Carrell and Sommese [19, 20℄ showed that, if
48
3. ELEMENTARY EXAMPLES
2 P is rational (and thus generates a holomorphi a G-equivariant biholomorphi embedding
ir le a tion), there is
TpW (p; ) ! M with image
W (p; ).
Therefore
W (p; )
is a
G-invariant
omplex sub-
manifold. Even better, its losure is an analyti subvariety, ontaining the
ell as a Zariski open subset. In fa t, Carrell and Sommese showed the ells
an be de ned independently of the metri in the following manner. The group of biholomorphi transformations of
M
is a omplex Lie group and
extends uniquely to a holomorphi a tion For ea h q in M the map C ! M whi h 1 C sends z 2 C to (z)q extends holomorphi ally to a map CP ! M. It follows from the moment map ondition d = -(M )! that the gradient C ve tor eld v is equal to -JM . This implies (exp tv )(q) = (exp -tJ)q for all q 2 M, and hen e therefore the holomorphi a tion
C
of the omplexi ed torus
GC .
W (p; ) = q 2 M lim C (z1 )q = p :
z!0
We will denote the fundamental lasses of the subvarieties
(p; ).
By [20, theorem 1℄ the olle tion
is a basis of
is a basis of
H2i (M; Z)
+ (p; ) index p = 2i
and the olle tion
- (p; ) index p = 2i
W (p; )
by
H2(d-i) (M; Z).
What are the interse tion properties of these bases? From the fa t that in reases along its gradient traje tories we see that
=) (x) < (p) =) (y) > (q)
x 2 W +(p; )
y 2 W -(q; ) for any xed points
and
q.
In parti ular, if
x = p;
or
y=q
(p) < (q) the losures and if (p) = (q) they
W (q; ) do not interse t, interse t if and only if p = q. Thus the interse tion matrix is lower tri + angular relative to the bases (p; ). Moreover, the losures of W (p; ) and W (p; ) have a single, transverse, point of interse tion of multipli ity +1. By Poin aré duality, the interse tion matrix is invertible, so this inter se tion annot be removed by moving the losures of W (p; ) within their
of
W (p; )
p
or
+
and
-
homology lasses. To what extent do these homology lasses depend on the hoi e of Let's x an a tion hamber, in
P+ .
Then
i;p () > 0
P+ ,
and onsider two rational points
if and only if
i;p () > 0,
so from the
?
and
G-invariant
3.6. MORSE THEORY
49
TpW (p; ) we see that limt!1 (exp tv )(q) = p W (p; ) = q 2 W (p; ). Thus W (p; ) = W (p; ) and we have proved
identi ation for all
3.6.2. Theorem. To every a tion hamber
asso iate a basis
of
of
H2i (M; Z)
H2(d-i) (M; Z).
- (p) p 2 MG ; index p = 2i
Moreover, if
some (and hen e all)
2 P+ ,
p, q
2 MG
satisfy
(3.14)
(p)
(3.15)
(q)
for
then
Æ
(p) (q) = +
one an anoni ally
+ (p) p 2 MG ; index p = 2i
and a basis
P+
-
fpointg 0
if if
p = q; p 6= q;
(3.16)
where denotes the interse tion produ t in homology. If G is any ompa t onne ted Lie group, these results are appli able to M regarded as a T -spa e, where T is a Cartan subgroup of G, provided that MT is nite. In fa t, the a tion of G imposes additional symmetries, sin e T the Weyl group, W = N(T )=T, of G permutes the points of M and hen e permutes the set of weights
Thus
i;p 2 ZT p 2 MT ;
1id :
(3.17)
W preserves the subset P of t and in parti ular it a ts on the olle tion Ad(G)P of P is a dense subset of g. Regarding
of a tion hambers. The orbit
the omponents of the moment map we have the following statement. 3.6.3. Lemma. Let
, of the (i)
(ii)
= g
with
g2G
and
moment map is Morse if and only
rit
2 t. The - omponent, if 2 P. If 2 P, then
gMT and, for ea h p 2 MT , the stable and unstable manifolds of at q = gp are W (q; ) = gW (p; ); the homology lass de ned by W (q; ) is equal to (p; ). =
Proof. All assertions but part (ii) follow from Lemma 3.6.1 and the
equivarian e of the moment map. being onne ted, its a tion on
Part (ii) follows from the fa t that,
H (M; Z)
The Weyl group symmetry of
P
is trivial.
G
QED
implies the following basi observation
about the size of the a tion hambers. 3.6.4. Lemma. If the a tion of (3.17) , ontains the roots of
for the
Int t+ .
T -a tion
G.
G
on
M
is faithful, the set of weights,
Hen e there exists an a tion hamber
P+
whi h is ontained in the open positive Weyl hamber
50
3. ELEMENTARY EXAMPLES
Proof. Let
Gss
= [G; G℄ be the semi-simple part of
G
and let
G1 ,
T G2 , : : : , Gr be the simple omponents of We re all that t = (g ) ; T T hen e if p 2 M , (p) 2 t . We will show for ea h Gi there exists a p 2 M su h that (p) is not xed by Gi . Indeed, if Gi xed all the (p)'s, then by the onvexity theorem it would x all points in the image of T . However, every G-orbit in (M) interse ts this image, so this would imply that (M) is xed pointwise by Gi , and this shows that this image has to be ontained ? in the annihilator, gi , of gi in g . Thus, in parti ular, for every 2 gi , = 0 and hen e M = 0. Thus Gi a ts trivially on M, whi h ontradi ts our assumption that the G-a tion is faithful. T Now let p 2 M be a point for whi h = (p) is not xed by Gi and let O be the orbit Gi . Among the weights, i;p , i = 1, 2, : : : , d, are the weights of the isotropy representation of T on TO. However, O is the quotient of Gi by a subgroup, Hi , whi h is the entralizer of a subtorus of T ; so these weights are (up to plus or minus sign) the roots of Gi not o
urring among the roots of Hi , and these in lude the dominant long root and the dominant short root. However, every root of Gi is Wi- onjugate to one of these two roots. Hen e all the roots of Gi o
ur in the set (3.17). QED 3.6.5. Remark. If
Gss .
M
=
G,
the oadjoint orbit through
2 g , then
the set of weights (3.17) is a subset of the root system, and therefore has a unique a tion hamber
P+
whi h ontains the open hamber
Combining this fa t with Lemma 3.6.4, we see that
oadjoint orbits. (More pre isely,
onto h is non-zero for P+ = Int t+ if 2 Int t+ .)
of
P+
=
Int t+
P+
=
Int t+
M
Int t+ .
for most
exa tly when the proje tion
ea h simple omponent h of g. In parti ular,
3.6.6. Remark. It is important to keep in mind that the bases (3.14) and (3.15) are labelled by the xed points, i.e. they should be regarded as
: MT ! H (M; Z), determined by a hoi e of an a tion hamber P+ . It is easy to see that the assignment P+ 7! is W -equivariant in the sense that if we translate the hamber by a Weyl group element w the bases T remain the same, but the labels are permuted by the a tion of w on M .
fun tions
We will on lude this dis ussion by proving an elementary result about the riti al values of the fun tions,
, whi h brings us ba k to our problem
of hara terizing the eigenvalues of sums of Hermitian matri es.
P+ for the T -a tion on M. i = 1, 2, 3, let i 2 P+ and gi 2 G. Put i = Ad(gi )i and suppose -1 that 3 = 1 + 2 . Let qi be a riti al point of i and put pi = gi qi . 3.6.7. Theorem. Fix an a tion hamber
For
Then
3 (q3 ) 1 (q1 ) + 2 (q2 );
(3.18)
3.6. MORSE THEORY
provided that
51
+ (p1 ) + (p2 ) - (p3 ) 6= 0:
(3.19)
Proof. The basi observation involved in this proof is that, for any
2 Ad(G)P, the value of at a point, q 0 2 W +(q; ), q 0 6= q, is less than 00 00 its value at q and its value at q 2 W (q; ), q 6= q, is greater than its value at q. (This is a simple onsequen e of the fa t that is in reasing along integral urves of grad .) Now suppose that (3.19) holds. By Lemma + + 3.6.3(ii), this for es the losure of W (q1 ; 1 ), the losure of W (q2 ; 2 ) and the losure of W (q3 ; 3 ) to have a ommon point of interse tion, q, and at this point of interse tion
3 (q) = 1 (q) + 2 (q);
3 (q) is greater than the left-hand side of QED (3.18) and 1 (q) + 2 (q) is less than the right-hand side of (3.18).
but, by the observation above,
The symple ti ase. We will now say a few words about how the
dis ussion above an be adapted to the general ase of a Hamiltonian manifold
M
whi h does not ne essarily possess a
G-invariant
omplex stru ture (nor indeed any omplex stru ture).
G
G-
ompatible
As before, we let
be a ompa t onne ted Lie group a ting in a Hamiltonian fashion on a
ompa t symple ti manifold element
M.
We hoose a maximal torus
of a xed a tion hamber
of the Morse fun tion
P+
T
of
G,
an
and we onsider the gradient ow
with respe t to a Riemannian metri on
M.
We
wish to de ne ( o)homology lasses supported by the losures of the stable and unstable ells for this ow. We rst note that, sin e
G a ts trivially
on homology, it is not really es-
W (p; ), be G-invariant.
ase they are not G-invariant if G is nonessential that the Riemannian metri on M
sential that the stable and unstable manifolds, (Indeed, even in the Kähler abelian.) be
Therefore it is not
G-invariant,
and dropping this assumption gives us a lot more latitude
in the onstru tion of these manifolds.
In fa t, by results of Harvey and
Lawson [33, se tion 14℄ the metri an be hosen in su h a way that the gradient ow of
is Morse-Stokes, whi h means among other things that
the stable and unstable ells have nite Riemannian volume. A typi al way to produ e a Morse-Stokes metri for a Morse fun tion is to start with an arbitrary Riemannian metri whi h is tame with respe t to the Morse fun tion, i.e. Eu lidean in suitable Morse oordinates about ea h riti al point, and then perturb this metri in the omplement of a neighborhood of the
riti al set in su h a way that it be omes Morse-Smale, whi h means that the stable manifolds
W +(p; )
interse t the unstable manifolds
W -(q; )
transversely. Su h a metri is always Morse-Stokes, and in fa t this line of
52
3. ELEMENTARY EXAMPLES
argument shows that tame Morse-Stokes metri s are
C1
of all tame Riemannian metri s. (See [33, se tion 14℄.) Given a Morse-Stokes metri for the Morse fun tion grate dierential forms over
dense in the spa e
,
one an inte-
W (p; ), and this integration operation de nes
a linear fun tional on dierential forms, that is to say a de Rham ur-
i
=
dim W (p; ).
For any riti al point
(p; ) of degree
rent (form with distributional oe ients)
p
of index
i
2d - i,
where
Harvey and Lawson
[34, equation (4.3)℄ prove that
d
where the oe ients
(p; ) =
X
+
npq
q
rit index q=i-1
npq
2
are integers. Sin e our Morse fun tion,
has riti al points of even index, the urrents, de ne ohomology lasses
(q; );
+
+ (p) = [
,
only
(p; ), are losed and so
+
(p; )℄. In fa t, by [34, theorem 4.3℄
+
+ (p), where p ranges over all riti al points of index i, are a basis 2d-i (M; Z). Similarly, the urrents -(p; ) for the ohomology group H i de ne a basis (p) for the ohomology group H (M; Z). Now let g be in G and put q = gp and = g. Let W (q; ) denote the (un)stable ell for the Morse fun tion with respe t to the g-translate of the Morse-Stokes metri . Then W (q; ) = gW (p; ), so we nd, as in the Kähler ase, that the lass de ned by W (q; ) is equal to (p). Thus the lasses
the analogue of Lemma 3.6.3 holds in the general symple ti ase. The one aspe t of Theorem 3.6.2 that may fail in the general ase is the independen e of the lasses of
P+ .
(p)
from the metri and the element
(See the remark at the end of this se tion.) Even so, we have the
1 be points in P and let h0 and h1 0 and 1 . Put t = (1 - t)0 + t1 for 0 t 1. Then t 2 P, be ause P is onvex, and so t is Morse for all t. There exists a path of metri s ht starting at h0 and ending at h1 su h that ht is tame Morse-Smale for t for all t. For every T - xed point S p the union 0t1 W -(p; t ) ftg de nes an i + 1- urrent on M [0; 1℄, where i = index p. The pushforward of this urrent to M has boundary (p; 1 ) - -(p; 0 ), and hen e -(p; 1 ) is ohomologous to -(p; 0 ). See
following partial result.
Let
0
and
be tame Morse-Smale metri s for
Laudenba h's appendix to [12℄ for a dis ussion of this point. We on lude that the lasses
(p)
depend only on the a tion hamber, as long as we
on ne ourselves to the universe of tame Morse-Smale metri s. The proof of Theorem 3.6.7 is also easily adapted to the general symple ti setting. Sin e we are now working in ohomology rather than homology,
3.6. MORSE THEORY
53
the interse tion produ t (3.19) has to be repla ed by the up produ t
+ (p1 ) [ + (p2 ) [ - (p3 );
(3.20)
and unfortunately we an't ompute this by wedging the forms
(p
+
2 ; 2 ) and
(p
-
3 ; 3 ), sin e
(p
+
1 ; 1 ),
these forms have distributional oe ients.
However, by de Rham's theorem [58, hapitre IV℄ the in lusion of the omplex of smooth forms into the omplex of urrents is a homotopy equivalen e, with a homotopy inverse de ned by onvolving urrents with approximate identities, one for ea h hart in an atlas of
M.
This shows that every losed
urrent is ohomologous to a losed smooth form whose support is ontained in an arbitrarily small neighborhood of the support of the urrent. Thus,
W +(q1 ; 1 ), W +(q2 ; 2 ) and W -(q3 ; 3 ) do + + not interse t, there exist smooth losed forms ~ (q1 ; 1 ), ~ (q2 ; 2 ) and ~ -(q3 ; 3 ) whi h are ohomologous to respe tively +(q1 ; 1 ), +(q2 ; 2 ) and (q3 ; 3 ), and whose supports do not interse t. Hen e the wedge + + produ t ~ (q1 ; 1 ) ^ ~ (q2 ; 2 ) ^ ~ (q3 ; 3 ) is zero and, onsequently, so if the losures of the manifolds
is the up produ t (3.20). Conversely, if (3.20) is non-zero, the losures of
W +(q1 ; 1 ), W +(q2 ; 2 )
and
W -(q3 ; 3 )
have to interse t, and one gets as
before the inequality (3.18). Two on luding remarks. It is shown in [34, se tion 8℄ that in the Kähler
ase the gradient ow asso iated with a C -a tion on
M
is always Morse-
Stokes. This means that the Harvey-Lawson approa h to Morse theory on a Hamiltonian
G-manifold leads in the Kähler ase to the same ohomology
basis as the Carrell-Sommese approa h, provided one takes the metri to be the Kähler metri . Let us, however, point out that dierent hoi es of Morse-Stokes metri s may give rise to dierent ( o)homology bases. Suppose we equip tame Morse-Smale metri for the fun tion
.
M
with a
It follows from the Morse-
p and q are points of equal index, then W +(p; ) does not interse t W (q; ) unless p = q. If p = q the interse tion onsists + of the single point p. This implies that (p) [ (q) = Æpq [M℄, where [M℄ denotes the fundamental lass of M. In other words, the bases Smale ondition that, if
f + (p) j index p = i g
and
f - (p) j index p = i g
are dually paired under the Poin aré duality isomorphism
(Hi (M; Z))
H2d-i (M; Z) !
. Easy examples (su h as [11, example 1℄) show that the orre-
sponding statement fails to be true for the lasses
(p)
of Theorem 3.6.2.
In parti ular, the ohomology basis de ned by a generi tame metri may be distin t from that de ned by a Kähler metri .
54
3. ELEMENTARY EXAMPLES
3.7. Klya hko's theorem
G to be the group, U(n), and M to be the Grassmannian,
Let's now take
Gr(Cn; k); and see what the inequalities, Se tion 3.5 we will let
(3.18), tell us in this ase. As in
H be the spa e of n n Hermitian matri es and make
our usual identi ations
g= g =
p
-1 H =
H:
(3.21)
Under this identi ation
t=
Hdiag = diagonal
matri es with real entries
Hdiag)+ , the olle tion of diagonal matri es with diagonal entries d11 d22 dnn . We an identify Gr(Cn ; k) with the onjuga y lass of Hermitian matri es, H , where i = 1 for 1 i k and i = 0 for k + 1 i n. The identi ation is given by the orresponden e and t+ = (
V
! V ;
n and V 2 H being the Hermitian n matrix orresponding to the orthogonal proje tion of C onto V . Thus, via n (3.21), Gr(C ; k) gets identi ed with a oadjoint orbit, O , of G and via n these identi ations the moment map, : Gr(C ; k) ! g , is just the map, V 7! V . Thus if A 2 H the A omponent, A , of the moment map is the
V being
a
k-dimensional
subspa e of C
fun tion
V 7 ! tra e(AV ):
(3.22)
A = D , the diagonal matrix with entries, dii = i , where 1 > 2 > > n . The eigenve tors of this matrix are the standard basis
In parti ular, let
ve tors
v1
= (1; 0; : : : ; 0),
v2 = (0; 1; 0; : : : ; 0), : : : , vn = (0; : : : ; 0; 1)
n I = (i1 ; i2 ; : : : ; ik ), 1 i1 < i2 < < ik n, A let VI = spanf vi j i 2 I g and let be the fun tion, , A = D . By Lemma 3.6.3 and Remark 3.6.5, is a Morse fun tion and we laim of C . For every multi-index,
3.7.1. Theorem. The riti al points of
VI, i2I i .
subspa es,
I =
P
of
Gr(Cn ; k)
are the
k-dimensional
and the orresponding riti al values are
are the k-dimensional subspa es of Cn xed by the a tion of the diagonal subgroup of U(n); and hen e these are the spa es, VI. Moreover, by (3.22), (VI) = I . QED Proof. The riti al points of
For ea h
I
let
WI
be the stable and unstable ells of
at
VI
with
n respe t to a U(n)-invariant Kähler metri on the Grassmannian Gr(C ; k).
3.8. RESTRICTION TO A SUBGROUP
55
These ells are alled the Bruhat ells and the fundamental lasses of their
losures are the S hubert lasses u t,
and
I J , +
I
+
+
I
an be written as a sum
+ + I J =
J K 6= 0 i K +
-
IJ ,
The numbers,
KIJ 6= 0
of
Gr(Cn ; k).
The interse tion prod-
X K +
IJ K
K
(provided that
dim +K
=
dim(+I +J )).
are known as the Littlewood-Ri hardson oe ients,
and there are simple ombinatorial re ipes for omputing them. (See [26℄.) Now let
Ui
2 U(n), i = 1, 2, 3, and let's see what Theorem 3.6.7 tells
us about the matri es -1 1 A = U1 D U1 , B = U2 D U2 ,
and
1 C = U3 D U3 .
A (resp. B , C ) are the points, n U1 VI (resp. U2 VJ, U3 VK) of Gr(C ; k) and the orresponding riti al values A (resp. B , C ) are I (resp. J , K ). Hen e from Theorem 3.6.7 we of By Theorem 3.7.1 the riti al points of
dedu e 3.7.2. Theorem. The inequalities
K I + J
hold whenever
KIJ
6= 0.
These remarkable inequalities are due to Klya hko [41℄ and others. (See the survey paper [26℄ for omplete attributions.) Even more remarkable is the following onverse result (whi h we will not attempt to prove). 3.7.3. Theorem ([41℄). The inequalities of Theorem 3.7.2 are su -
ient as well as ne essary for
H
to be ontained in
H + H .
Agnihotri and Woodward [1℄ have proved a striking analogue of this theorem for produ ts of unitary matri es, whi h involves the quantum Littlewood-Ri hardson oe ients.
3.8. Restri tion to a subgroup
We will give in this se tion another appli ation of the Morse theoreti results of Se tion 3.6. Let
G
be a ompa t onne ted Lie group and
G. Let T and T1 be Cartan subgroups W and W1 be the Weyl groups:
losed onne ted subgroup of and
G1
with
T1 T
and let
NG (T )=T
of
a
G
NG1 (T1)=T1:
G we an regard this orbit as a Hamiltonian G1 -spa e, whose moment map O ! g1 is the restri tion to O of the anoni al proje tion, : g ! g1 . By Kirwan's theorem the interse tion of (O) with Given a oadjoint orbit,
a hamber in t
O,
and
G1
of
1 is a onvex polytope and in this se tion we will des ribe this
56
3. ELEMENTARY EXAMPLES
polytope in terms of inequalities whi h are similar to (and in fa t generalize) the Klya hko inequalities of Se tion 3.7. Our result will be formulated in terms of homology lasses on adjoint rather than oadjoint orbits.
However, it is easy to restate the results of
Se tion 3.6 in terms of adjoint orbits, sin e by hoosing a produ t (
G-invariant
j ) we obtain an G-equivariant identi ation g $ g .
identi ation the proje tion,
:
g
inner
Via this
! g1 , be omes the orthogonal proje tion
1 g1 ! g1. Choose a hamber (t1 ) in t1 for the W1-a tion, x an element, 2 Int(t1 ) , and let G be its entralizer in G. The adjoint orbits
:
g = g?
+
G and G1 an
+
G=G
be identi ed with
and
G1 =T1 via
the moment maps
gG 7 ! Ad(g)
: G=G ! g; and
gT1 7 ! Ad(g):
1 : G1 =T1 ! g1 ;
i : G1 =T1 ! G=G be the embedding indu ed by the in lusion G1 ! G. Sin e G1 is ontained in G, these moment maps satisfy i = 1 . For 2 g and 1 2 g1 we will denote by = ( j ) and 1 1 = (1 j 1 ) the and 1 omponents of and 1 . Let t+ be a hamber in t for the W -a tion. We an, and will, hoose this hamber in su h a way that its Let
1 is ontained in the hamber (t1 )+ (but we will not require to be in t+ ). In addition, for the T1-spa e G1 we will
hoose the a tion hamber (P1 )+ = (t1 )+ , and for the T -spa e G we will
interse tion with the subspa e t
hoose the unique a tion hamber
P+ ontaining t+ .
hoi es determine bases of the homology groups of
G1 )T1
the xed point sets (
=
W1,
G)T
resp. (
By Theorem 3.6.2 these
G1 and G labelled by = W. From Theorem
3.6.2 and Lemma 3.6.3 we have 3.8.1. Lemma. Let
the fun tion
g
2 Int(t1 )
+.
Then for all
is a Morse fun tion on
the osets
gwG ;
G=G
g
2 G and 2 Int t
w 2 W:
(3.23)
g1 2 G1 and 1 2 Int(t1 )+ the fun tion 1g1 1 on G1 =T1 with riti al points at the osets
Similarly, for all Morse fun tion
g1 w1 T1; Now suppose that
(g) = g1 1 .
w1 Sin e
2 W1:
i = 1 ,
i g = 1g1 1 ;
+
with riti al points at
is a
(3.24) we have (3.25)
and we will show that this identity an be exploited to give us inequalities between
g and g1 1 that are very similar to the Klya hko inequalities whi h
3.8. RESTRICTION TO A SUBGROUP
57
we derived (by essentially the same methods as those we will use below) in Se tion 3.7.
g and g1 W (1 ; w1 )
For the moment let
W (; w) and 1 1 at the riti al
and let
and
be the identity elements in
G
and
G1
be the stable and unstable manifolds of
points, (3.23) and (3.24). The losures of these
w and w1 j w1 2 W1 g are
manifolds represent homology lasses
and by Theorem 3.6.2
bases for the homology f w1 j w1 2 W1 g and f w1 H (G1 =T1; Z) of G1 =T1. Now let g and g1 be arbitrary. The stable and unstable manifolds of g and 1g1 1 at the riti al points, (3.23) and (3.24), are gW (; w) and g1 W (1 ; w1 ), and by Lemma 3.6.3 their losures also represent the homol ogy lasses w and w . We will now prove 1 -
+
the sets group
G, 2 Int t+ , g1 (g) = g1 1 . Then
3.8.2. Theorem. Suppose that
Int(t1 )+ for all
satisfy the ondition
2 Int(t1 )+
( and all
g
2
2 G1, 1 2
j w) (1 j w1 )
w2W
and
(3.26)
w1 2 W1
satisfying
i + w1 -w 6= 0:
(3.27)
gW -(; w) point, p, in the
Proof. If (3.27) holds, the losure of the unstable manifold
has to interse t the image of the in lusion map,
losure of
and sin e
g1 W (; w1 ). +
folds of the riti al
in a
However, (3.25) says that at this point,
g (p) = 1g1 1 (p);
(3.28)
g1 W +(1 ; w1 ) are the points, gwG , and g1 w1 T1,
gW -(; w)
i,
and
g (gwG ) g (p)
and
Finally, sin e the moment maps,
unstable and stable mani-
1g1 1 (g1 w1 T1) 1g1 1 (p):
and
1 ,
are equivariant,
g (gwG ) = (wG ) = ( j w);
1g1 1 (g1 w1 T1) = 1 1 (w1 T1) = (1 j w1 ); and putting (3.28)(3.30) together we get (3.26). By Theorem 3.6.2, the homology lasses,
and 1 ;
however,
G
does depend on
.
(3.29)
w
(3.30)
QED
+ w1 , do not depend on there is a -dependen e
and
Hen e
built into the ondition (3.27). (We will dis uss at the end of this se tion, however, an important lass of examples for whi h the ondition (3.27) is also independent of
.)
This means that in general the inequalities of Theorem
3.8.2 are not a su ient set of inequalities for the ondition to hold.
Indeed, let
2
Int(t1 )+
and suppose
v
2
W
(g) = g1 1
is a Weyl group
58
3. ELEMENTARY EXAMPLES
= v is also in Int(t1 )+ . Let g 2 G, 2 Int t+ , g1 2 G1 , and assume that (g) = g1 1 . Then by Theorem 3.8.2 we
element su h that
1
2 Int(t1)
+
have additional inequalities
(
j w) (1 j w1);
w and w1 satisfying j + w1 -w 6= 0. Here j denotes the embedding G1 =T1 ! G=G indu ed by the in lusion G1 ! G, and w denotes the fundamental lass in H (G=G ; Z) of the losure of the unstable de ned by
ell of the riti al point wG 2 G=G of the Morse fun tion (gG ) = ( j g). However, the adjoint orbits through and = v are obviously the same. In fa t, there is a G-equivariant dieomorphism f : G=G ! G=G sending gG to gvG . The embedding iv = f Æ j from G1 =T1 into G=G is then given by iv (gT1) = gvG . From the ommutative whi h hold for all
diagram
G=G
iv
:
O
v vvv v
vv vv
G1 =T1
HH HH HH j HHH$
f
DD DD DD DD D! =R zz z zz zz zz
G=G
we see that
f
maps the unstable ell at the xed point
unstable ell at the xed point implies
where
s
wvG 2 G=G .
wG 2 G=G to the f w = -wv . This
Therefore
v + + f (j + w1 w ) = s (f j w1 ) (f w ) = s i w1 wv ; =
1
orientation.
if
f
s
preserves the orientation and
Thus
j w1 w -
+
6=
0
if and only if
= -1 if
f
reverses the
iv + w1 -wv 6= 0.
This
establishes the following generalization of Theorem 3.8.2.
g 2 G, 2 Int t+ , g1 (g) = g1 1 . Then
3.8.3. Theorem. Suppose that
Int(t1 )+ for all
satisfy the ondition
2 Int(t1 )
v 2 Int(t1 )+
( +
j wv) (1 j w1 v)
and all
and
w
2
W, v
2
W
and
2 G1, 1 2 (3.31)
w1
2 W1
su h that
iv + w1 -wv 6= 0:
(3.32)
Theorems 3.8.2 and 3.8.3 are due to Berenstein and Sjamaar [9℄. Moreover, they show that the inequalities of Theorem 3.8.3 are su ient as well as ne essary for
(g)
=
g1 1
to hold.
In other words, the inequalities
(3.31) are a de ning set of inequalities for the Kirwan polytope
(G) \
3.8. RESTRICTION TO A SUBGROUP
1
(t )+ .
59
(Many of these inequalities turn out to be redundant.
For in-
stan e, it is shown in [9℄ that the inequalities for whi h the lass (3.32) is positive-dimensional follow from the inequalities for whi h the lass is zerodimensional. Thus the inequalities for whi h the lasses
iv + w1
and
wv have
omplementary dimension and nontrivial interse tion su e to des ribe the moment polytope.) In the spe ial ase where
n)
the diagonal opy of U( 3.7.2.
G = U(n) U(n)
and
G1
is
Theorem 3.8.3 redu es to Klya hko's Theorem
It is also easy to see that the inequalities (3.31) an be written in
a form whi h makes them look strikingly like the lo al formula, Theorem 2.2.3, for the Kirwan polytope. For simpli ity let us assume
Int(t1 )+ = t1 \ Int t+ ; in whi h ase the inequalities of Theorem 3.8.2 su e to des ribe the mo-
G = T for all 2 Int(t1 )+ ; so G=T and the homology lass - (w) and the interse tion-produ t ondition (3.27) do not depend on . Let 1 , 2 , : : : , N 2 t1 be the positive roots of the group, G1 , and let
ment polytope.
Under this assumption
the homogeneous spa e
G=G
CR+
=
is just
X N
ti i ti
0 t1 :
i=1 This one is dual to the one (t1 )+ in t1 . Now let us, by abuse of language, think of the and 1 in (3.26) as being elements of t , resp. t1 . In other
words, let's make use of the equivariant indenti ation between g and g to onvert (3.26) into a statement about a oadjoint orbit
:
t
! t1
be the (now anoni al) proje tion of t
(3.26) be ome
for all
w
and
w1
1 w1 1
2 (w 1) + CR
satisfying (3.27).
-
+
1
G.
Letting
onto t , the inequalities
CHAPTER 4
Kähler potentials and onvexity 4.1. Introdu tion
T be the n-dimensional torus, (S1 )n , and T C = (C )n its omplexi ation. If is a holomorphi a tion of T on a ompa t omplex manifold, M, C one an show that extends to a holomorphi a tion of T . A elebrated theorem of Atiyah [4℄ asserts that if M is Kähler and a Hamiltonian a tion C with moment map, T , then the moment images, T (T p), p 2 M, of the C are onvex polytopes. The purpose of this hapter is to give orbits of T Let
an a
ount of a non-abelian analogue of this result for Borel subgroups of
omplex redu tive Lie groups. with maximal torus
T
and
M
Let
G
be a ompa t onne ted Lie group
a Kähler manifold (whi h we will not for the
moment assume to be ompa t). Let
be
a Hamiltonian a tion of
G on M
: M ! g . Assume that extends to a holomorphi GC . Let t+ be a losed Weyl hamber in t , let Int t+ be its interior C generated by the root spa es whi h and let B be the Borel subgroup of G are negative with respe t to t+ . Suppose p is a point of M at whi h B a ts
with moment map,
a tion of
freely. Then we will prove that the set
(Bp) \ Int t+
is an open onvex subset of t . a tion of
G.
(Note that
(4.1)
Bp
is not invariant under the
Thus this statement is slightly dierent in nature from the
other onvexity theorems dis ussed in this monograph.) We will also prove that, if
N
is the unipotent radi al of
interse tion of
Int t+
and
\
nN
B,
the set (4.1) is ontained in the
T (T Cnp):
(4.2)
2
The main result of this hapter asserts that these two sets oin ide if
M
is
ompa t and if the ohomology lass of the Kähler form is rational. Hen e, by Atiyah's theorem, (4.1) is a onvex polytope. We also obtain the following des ription of (4.1) in terms of the
N-orbit
through
Mumford riterion of geometri invariant theory.
p,
akin to the Hilbert-
Let
L ! M
be a
G-
equivariant ample holomorphi line bundle whose Chern lass is a positive integer multiple of the ohomology lass of the Kähler form. Then the spa e 61
62
4. KÄHLER POTENTIALS AND CONVEXITY
(L) is a
of global holomorphi se tions
G-module
d M
(L) =
i=1
Let
i 2 t
I
Vi and
for all
T
T.
let
s 2 Vi g:
f1; : : : ; dg put XI = i I Xi and let CI be the interior of f i j i 2 I g. We will show that (4.2) is the interse tion
of
the onvex hull of
Int t+
a ts on
Xi = f q 2 M j s(q) = 0
For any subset
of
T
be the weight with whi h
Vi
(L) with respe t to the a tion of
be the weight-spa e de omposition of
in a natural way. Let
2
with the open polytope
\
I
f CI j Np \ XI 6= ; g:
(4.3)
We hope to dis uss elsewhere appli ations of these results to theorems of Klya hko type: onstru tive versions of the Kirwan onvexity theorem for Kähler manifolds. A few words about the organization of this hapter. Mumford's on ept of stability for orbits of
GC -a tions will play a major role in our proof of the
results above and we will re all a number of basi fa ts about stability in Se tion 4.2, following the approa h of Guillemin and Sternberg [29℄. At the end of this se tion we will also review a basi relationship between Kähler forms on omplex redu tive tori and the familiar Legendre transform of
lassi al me hani s. This relationship was employed by Burns and Guillemin [17℄ to give an alternative proof of Atiyah's theorem and motivated our treatment of the non-abelian ase.
In Se tion 4.3 we will prove a result
whi h seems to be of interest in its own right, independent of the appli ations whi h we will make of it here and whi h we were unable to nd a itation for in the literature. Given
2 t
let
F : GA ! R
p
F (ka) = ( Via the Iwasawa de omposition,
GC
=
be the fun tion
-1 log a):
GAN,
one gets an identi ation of
GA with GC =N and, from this, a omplex stru ture on GA. Now let O be the oadjoint orbit through , and ! the anoni al symple ti form on O . -1 The map G ! O de ned by k 7! Ad(k ) gives one an identi ation of O with G=G , where G is the entralizer of , and hen e a proje tion, : GA ! O . We will prove that F is a potential fun tion for ! in the C sense that ! = dd F .
In Se tion 4.4 we will prove that the set (4.1) is onvex and ontained in the interse tion of
Int t+
with (4.2). Using stability theory we will also
4.2. STABILITY
63
see that the sets (4.1) and (4.2) themselves are stable under bounded perturbations of the Kähler potential.
In Se tions 4.54.6 we will prove the
M is ompa t and the Kähler lass is rational. We will show in Se tion 4.5 that this is the ase if M is proje tive spa e and a linear a tion, and then in Se tion 4.6 ombine the stability of the sets
ontainment the other way if
(4.1)(4.2) with an equivariant version of the Kodaira imbedding theorem to dedu e the general result.
4.2. Stability
The proofs of the onvexity theorems whi h we will dis uss in Se tions
GC -
4.44.5 make heavy use of Mumford's notion of stability for orbits of a tions.
The version of stability we will need is that des ribed in [29,
se tion 2℄ and is based on an existen e theorem for potential fun tions, Theorem 4.2.2 below. Let
G be
GC its ! on GC ,
a ompa t onne ted Lie group and
omplexi ation. Consider any left whi h we allow to be degenerate.
G-invariant
losed two-form
Just as in the symple ti ase, the left
a tion is alled Hamiltonian if there exists a moment map as in (1.1)(1.2). This is always the ase if
G
is semisimple, and for general
G
we an make
the following statement.
G-invariant losed two-form ! on GC is exa t G-a tion on GC is Hamiltonian with respe t to !.
4.2.1. Lemma. A left
if and only if the left
Proof. Exa tness of
!
is learly su ient for the a tion to be Hamil-
tonian. Now assume the a tion is Hamiltonian. The ompa t real form
C a deformation retra t of G , so it su es to show that !jG is exa t. suppose
G
is a produ t of a semisimple group
H2 (G; R) =
M
i+j=2
K
and a torus
T.
G is
First
Then
Hi (K; R) Hj (T; R) = H2 (T; R)
by the Künneth theorem and the Whitehead lemmas, so it su es to show
!jT is exa t. Sin e the left T -a tion on GC is Hamiltonian, all T -orbits in GC are !-isotropi . In parti ular, T itself is isotropi , so in fa t !jT = 0. ~ of the form K T , so the lift of A general ompa t G has a nite over G ~ ! to G is exa t. By averaging over the sheets of the over one sees that ! itself is exa t.
QED
C
! is of type (1; 1). Let d be the p 1 real operator JdJ = -1 ( - ), with J : TGC ! TGC being the omplex p C C C stru ture on G . Note that dd = -d d = 2 -1 . A mu h stronger statement is true if -
64
4. KÄHLER POTENTIALS AND CONVEXITY
! be a left G-invariant exa t (1; 1)-form on GC with ! = !. Then there exists a real-valued left G-invariant fun tion f su h that ! = ddC f. 4.2.2. Theorem. Let
Proof. By Lemma 4.2.1 there exists a one-form,
Write = + with 2 01(GC ).
= ,
with
! = d.
Then
! = d = + + + ; = =0
so
be ause
p
! 2 11 (GC ).
Sin e
GC
is stri tly pseudo onvex
(see for instan e Lassalle's paper [46℄), there exists a fun tion,
. Moreover, by averaging, = -1 h . G-invariant. Now let f = h + h
we an assume that
h, su h that and h are QED
A basi identity involving the operator
dC
is
L(Jv)f = -(v)dC f; whi h holds for any fun tion ifold
M.
f
(4.4)
and any ve tor eld
v
on any omplex man-
This follows from the Cartan-Weil identity:
L(Jv)f = (Jv)df = (Jv)( + )f = ( + )f(Jv)
where we used
M is
and for
dC =
2g
G-invariant;
let
=
p
p
-1
C v;
f(v) f(v) -
= -(v)d
- ). Now suppose -1 (
M
be the ve tor eld on
then the form
!
=
ddC f
is
G a ts holomorphi ally on M indu ed by . Suppose f
G-invariant.
The identity (4.4)
enables us to write a simple formula for a moment map on the form
!
in terms of the potential
: GC ! R by
The
G-a tion
f.
2g
Namely, for
= (M )dC f = -L(JM )f:
is holomorphi , so the operators
L(M )
and
G-invarian e
Hen e, by the Cartan-Weil identity and the
M
relative to
de ne a fun tion
of
(4.5)
dC
ommute.
f,
d(M )dC f = L(M )dC f - (M )ddC f = dC L(M )f - (M )! = -(M )!; so
d
M )!.
= -(
is equivariant, so
Ad(g))M
Moreover, the identity (
is a moment map with respe t to
k; g) 7! kg
the ase of the left a tion (
of
G
on
GC ,
=
!.
g M
implies
Returning to
we obtain the following
result. 4.2.3. Theorem. Let
fun tions
!
and
: GC ! R given in
with respe t to
!
for the left
f
be as in Theorem 4.2.2.
(4.5) de ne a moment map
G-a tion
on
GC .
Then the :
GC ! g
4.2. STABILITY
GC
We shall denote the ve tor elds on
65
indu ed by the left
G-a tion
by
. The identities = -L(J[ )f an be regarded as a system of dierential C ! R up to equations for f, whi h determines the invariant fun tion f : G [
an additive onstant.
Thus the potential is determined up to a onstant
by the moment map. (In parti ular, if is uniquely determined by up to a onstant by 0
moment map
!.)
for the
!,
is semisimple, the moment map
so then the potential is uniquely determined
Conversely, the next statement says that every
G-a tion
GC
on
f 0.
for a suitable potential
G
4.2.4. Theorem. Let
!
is of the form
h
0
; i
= -L([ )f 0
be as in Theorem 4.2.2 and let
0
be any
moment map with respe t to ! for the left G-a tion on GC . Then there exists a G-invariant fun tion f : GC ! R, unique up to an additive 0
; i = C 0 dd f = !.
onstant, su h that fun tion satis es Proof. Let
is of the form
h
f and 0
=
L(J[ )f
0
0
-L(J[ )f 0 for all
2 g.
Moreover, this
be as in Theorem 4.2.3. Then the moment map
+ for some fun tional
=-
h
0
; i = -h
i
2 g .
+ ; =
Hen e for all
2g
L(J[ )f - ();
h = f 0 - f,
or equivalently, putting
L(J[ )h = -():
(4.6)
G-invariant and the ve tor elds [ and J[ span the tangent C bundle of G , the equations (4.6) determine h up to a onstant. If h is a solution, then for all 2 g Sin e
h
0
is
([ )ddC h = L([ )dC h - d([ )dC h = dC L([ )h + d() = 0;
where we used (4.4), and
(J[ )ddC h = L(J[ )dC h - d(J[ )dC h = dC L(J[ )h - d(J[ )dC h =
C () - d([ )dh = 0 - L([ )h = 0;
-d
dC . Be ause [ and J[ span TGC, ddC f 0 = !. It remains to show that the
where we used (4.6) and the de nition of this implies
ddC h = 0
and hen e
equations (4.6) have a solution. The equivarian e of
is a hara ter of g. Therefore
L(J )h [
=
0
for
and
G
with
G=[G; G℄,
we may assume
that ase the fun tion given by
for
t2G
and
a 2 exp
p
G
implies that
2 [g; g℄ and hen e any GC ; GC ℄.
solution is invariant under the ommutator subgroup [ repla ing
0
Thus, after
to be a torus. However, in
log a)
(4.7)
-1 g is easily seen to be a solution of (4.6).
QED
p
h(ta) = (
-1
66
4. KÄHLER POTENTIALS AND CONVEXITY
!
Assume now that the form
is Kähler, i.e. the Hermitian form
is positive de nite. Then the potential
f
is stri tly plurisubharmoni . By
f: GnGC !
a theorem of Lassalle [46℄ this is equivalent to the fun tion indu ed by
f
!(; J) R
being stri tly geodesi ally onvex (with respe t to the metri
on the non ompa t symmetri spa e
GnGC).
From this one sees that
f
either has no riti al points, or is proper and bounded from below. In the se ond ase the riti al set of
f
onsists of a unique
G-orbit,
along whi h
f
attains its unique minimum value. Let us all a real-valued fun tion on a topologi al spa e stable if it is proper and bounded from below. 4.2.5. Theorem. Let
!
be a left
G-invariant
GC , f a left G-invariant potential for !, and map. Then
f
exa t Kähler form on
the asso iated moment
is stable if and only if zero is in the image of (x) =
Proof. Theorem 4.2.3 shows that
is a riti al point of
f.
Hen e
0 is
0
for some
x
in the image if and only if
.
if and only if
f has
x
a riti al
point.
QED
The moment map image of a ompa t Hamiltonian
G-manifold
does
not hange if we move the symple ti form inside its ohomology lass. A limited version of that statement is true in our present non ompa t setting. 4.2.6. Corollary. Let !, f and be as in Theorem 4.2.5. Let h : GC ! R be a left G-invariant smooth fun tion. Put f 0 = f + h, ! 0 = ddC f 0 and h 0 ; i = -L(J[ )f 0 . Assume h is bounded and ! 0 is
Kähler. Then zero is in the image of 0
of
if and only if it is in the image
.
Proof. Sin e
h
is bounded,
f
is stable if and only if
f0
is stable. Thus
the result follows immediately from Theorem 4.2.5.
QED
We will see in Theorem 4.4.5 how to use Theorem 4.2.5 as a test to dete t any point in the image of the moment map. This is easy, however, if
G
is a torus, sin e then we an subtra t from
any ve tor
2 g .
This
f by subtra ting the fun tion (4.7) and p-1 g by subtra ting p-1 . f on GnGC =
hanges the orresponding potential hen e the indu ed fun tion
Thus Theorem 4.2.5 leads to the following des ription of the image of 4.2.7. Corollary. Let
that
!, f
G is a torus and let 2 g
and
. Then
.
p
be as in Theorem 4.2.5. Assume
2
C
(G ) if and only if
f-
-1
is stable. As an aside we mention an attra tive alternative des ription of the moment map
in the ase of a torus, whi h was rst used by Burns and
Guillemin [17℄ to prove Corollary 4.2.7.
Let
T
be a torus and let
TC
be
4.2. STABILITY
TC
p
-1 t and
Let us put a =
its omplexi ation.
67
A
Rn exp a; then A = f : T C ! R be the T let : T C ! t be the =
F : a ! R, let F(log a), and C
orresponding moment map for the form ! = dd f. Re all that the Legendre transform of a fun tion F : V ! R, where V is a real ve tor spa e, is the map LF : V ! V de ned by hLF (v); wi = dFv (w). Let pr : TA ! A be
and
=
TA.
Given a fun tion
invariant fun tion given by
f(ta)
=
the proje tion. We an summarize the situation in the following diagram:
T C = TA
pr
log
/A p
t
/
R
LF
1
-
F
/a
(4.8)
/ a
and we assert that this diagram ommutes.
p
-1
4.2.8. Theorem.
(ta) =
LF (log a)
for all
Proof. The left-hand side of the identity is
t = 1. for all
Also we an write
2 t.
p
-1
a = exp
(exp );
F
=
with
2 a.
-1 =
and
T -invariant, so
a 2 A. we may set
We must show that
LF ();
p
-1
(4.9)
The left-hand side of (4.9) is equal to
(exp ) = L(J[ )fexp :
Writing
p
t2T
f Æ exp,
where
exp : a ! A
is the exponential map, we see the
right-hand side of (4.9) is equal to
p p -1 ) = dfexp Æ d exp ( -1 ): p Noti ing that d exp ( -1 ) = J (exp ) we nd p dfexp Æ d exp ( -1 ) = dfexp (J ) = (J )dfexp = L(J )fexp : dF (
[
[
[
[
Therefore (4.9) holds.
QED
A
ording to Lassalle [lo . it.℄
!
only if
F
is stri tly onvex in the ordinary sense.
A standard result says
that the image of the Legendre transform of a stri tly onvex fun tionals
su h that F - is stable.
F Æ log is A, i.e. if and
is Kähler if and only if
stri tly geodesi ally onvex on the Eu lidean symmetri spa e
F is the set of
(See e.g. [17℄ or Cannas' book [18℄.)
Thus Theorem 4.2.8 provides an alternative proof of Corollary 4.2.7. An example of a potential that will be important later is that of a logarithm of an exponential sum.
2, : : : , d,
Let
and de ne
F() = log
i
d X i=1
2a
i e i () :
and
i > 0,
where
i
=
1,
(4.10)
68
4. KÄHLER POTENTIALS AND CONVEXITY
The Hessian of
F
at
is given by
X
d2 F (x; y) = e-2F
1 i<j d
for
i j eh i + j ;i h i - j ; xih i - j ; yi
x, y 2 a, so F is stri tly onvex if and only if the dieren e ve tors i - j
span the dual spa e a . This is equivalent to the polytope
P = onvf 1 ; 2 ; : : : ; d g not being ontained in an a ne hyperplane.
Assuming
F
stri tly onvex,
one an see quite easily that it is stable if and only if zero is in the interior of
P.
Sin e
F-
=
log
Pd
statement.
i=1 i e
i -
for
2 a , this implies the following
4.2.9. Lemma. The image of the map
the polytope
P.
C -1 i 2 t
1 , 2 , : : : , d ,
where
i
=-
a
!
a is the interior of
(T ) is the interior of the onvex
p
Hen e, by Theorem 4.2.8 the image hull of
LF :
. This was used in [17℄ to
give an alternative proof of Atiyah's onvexity theorem, and we will see it
omes up in the non-abelian version as well. 4.3. Kähler metri s on oadjoint orbits
Another pie e of Kähler theory we will need is a Biquard-Gaudu hon
1; 1)-formspon oadjoint orbits. Fix a maximal -1 t of gC and let A = exp a be the torus T of G. Let a be the subalgebra C C = TA is a maximal torus of GC.
orresponding subgroup of G . Then T C C and let N be the maximal Let B be a Borel subgroup of G
ontaining T C C and unipotent subgroup [B; B℄ of G . The Iwasawa de omposition for G B takes the form GC = GAN; B = TAN: C and G ,! GC indu e dieomorphisms Thus the in lusions GA ,! G type theorem for invariant (
GA =! GC =N
G=T =! GC =B:
and
Via these dieomorphisms the manifolds
GA
and
G=T
a quire omplex
stru tures, and the map
sending bundle.
: GA ! G=T ka 2 GA to the oset kT 2 G=T is Given 2 t , let F : GA ! R be the
p
F (ka) = ( O be the oadjoint torus T (indeed, is equal
Let
T
for
T C-
fun tion
-1 log a):
(4.11)
. The entralizer G ontains the regular ); hen e from the map k 2 G 7!
orbit through to
a holomorphi prin ipal
4.3. KÄHLER METRICS ON COADJOINT ORBITS
Ad(k-1 ) one gets a G-equivariant map G=T ! O .
to
G=T of the anoni al symple ti o = 1T 2 G=T is given by
form on
O .
Let
69
! be the pullba k
Its value at the identity
oset
for
, 2 g.
!o ([ ; [ ) = h; [; ℄i
The
G-a tion on G=T is Hamiltonian relative to ! with moment
map
: G=T ! g
given by the proje tion
ourse,
!
G=T ! O
(4.12)
followed by the in lusion
O ,! g .
Of
is not exa t and so has no potential, but the main result of this
GA
se tion asserts that its pullba k to
does have a potential, namely
F .
More expli itly:
!
4.3.1. Theorem. The pullba k of
to
GA
satis es
! = ddC F .
Before proving Theorem 4.3.1 we will rst prove a few elementary results
2 tC = t a let be the ve tor eld on GA indu ed by the right a tion (t; ka) ! 7 kat 1 of T C on GA. p -1 2 a. Then 4.3.2. Lemma. Let 2 t and = about
F .
For
℄
-
L(℄ )F
=
0
L(℄ )F
and
=
Proof. The rst identity follows from the right
the se ond follows from the identity
F ka exp(-t)
4.3.3. Lemma. Let
=
p
2t
T -invarian e of F , and
-1 log (a exp(-t)) =
and
dC F (℄ ) = -()
Proof. The right
T -a tion
=
p
-1
GA
F (ka) + t(): QED
and
on
():
2 a.
Then
dC F (℄ ) = 0:
is holomorphi , so
℄
=
J℄ .
Using
(4.4) and Lemma 4.3.2 we nd
dC F (℄ ) = -L(J℄ )F
Similarly,
= -L( )F = -(): ℄
dC F (℄ ) = dF (J℄ ) = -dF (℄ ) = 0.
QED
As in (4.5) de ne
= ddC F
for
G.
2 g, where
Then
[
is the ve tor eld on
1; 1)-form on GA G-a tion on GA.
is a (
map for the left
and
= ([ )dC F GA
indu ed by the left a tion of
and the map
:
GA !
g is a moment
70
4. KÄHLER POTENTIALS AND CONVEXITY
4.3.4. Proposition. The form,
basi with respe t to the bration,
F
, .
and the map,
:
GA !
g , are
T -invariant, is right T -invariant. MoreC over, by Lemma 4.3.2 dF is right A-invariant, so = -d dF is right A-invariant. Therefore is right T C-invariant. Sin e the T C-a tion on GA ℄ C C ℄ C is holomorphi , we have L( )d = d L( ) for all 2 t . By Lemma 4.3.2 Proof. Sin e
this implies
C for all 2 t .
is right
L(℄ )dC F = dC L(℄ )F
that
2 tC ,
=
d(℄ )dC F + (℄ )ddC F
2 tC
= ( ) ℄
where in the last equality we used Lemma 4.3.3. This shows
is horizontal for the T C-a tion.
for all
0
Hen e, by the Cartan-Weil identity,
0 = L(℄ )dC F
for all
=
We on lude that
L(℄ ) = L(℄ )([ )dC F by Lemma 4.3.2, so
is basi .
Likewise,
C L(℄ )F = 0
= ( )d [
is basi .
QED
Proof of Theorem 4.3.1. By Proposition 4.3.4 there exist a unique
G-invariant two-form, !, on G=T with = ! and a unique map, : G=T ! g with = . Furthermore, d = -([ )!, where, again, [ denotes the ve tor eld on G=T indu ed by 2 g. To on lude the proof we must show that ! = ! . This we will do by a moment map argument. Sin e G=T is homogeneous under the G-a tion, the moment map determines the form !; so to prove that ! = ! it su es to prove that oin ides with , the ! -moment map given in (4.12). In fa t, sin e takes the value, , at the identity oset, o = 1T 2 G=T , it su es to prove T that (o) = . Note that, sin e o is xed under T , (o) 2 (g ) = t by the G-equivarian e of . In other words, it su es to show that
losed
(o) = ()
for all
2 t.
To see this, we restri t the identity
C F
= ([ )d
to the
1 (o) = TA GA of . On TA the left and right T -a tions agree up ber -1 [ ℄ to the automorphism t 7! t of T , so = - for 2 t. Hen e = ([ )dC F = -(℄ )dC F = () -1 by Lemma 4.3.3. But on (o) we have = (o) and so (o) = () -
for all
2 t.
QED
2 t and any Borel subgroup B ontaining The form ! is degenerate if and only if is in a singular hyperplane. If is regular, the signature of ! depends on the hamber of t ontaining T C.
Theorem 4.3.1 holds for any
4.4. CONVEXITY THEOREMS FOR
.
B
-ORBITS
71
The Kähler ase will be important for us, so let us review the proof of
the following well-known result (if only as a he k on our sign onventions).
4.3.5. Lemma. Let t+
B
t be the positive hamber for
and let
Int t+ denote its interior. Then ! is positive de nite pre isely 2 Int t+ and negative de nite pre isely when 2 - Int t+ .
when
Proof. Let
R be the root system of the pair (GC ; T C) and R+
the set of
C C L positive roots, i.e. the set of roots su h that b = t R+ g p. Re all that the positive hamber t+ onsists of all 2 t su h that (2 -1 h ) 0 C C for 2 R+ , where h is the unique ve tor h 2 [g ; g- ℄ satisfying (h) = 2. C Let f e j 2 R g be a Chevalley system in g , i.e. a olle tion of root ve tors C e 2 g su h that h = -[e ; e- ℄ and the map de ned by e 7! e- and h 7! -h for h 2 tC is an automorphism of gC . By [14, se tion 9.3℄ we may 2
assume that g is the real linear span of t and the ve tors
for
2 R+ .
x = e + e- ; x ; y ℄ = 2
Note that [
m=
M
R+
p
y
=
p
-1 (e
- e- )
-1 h . De ne
Ry );
(R x
n
-
=
R+
T -invariant
C
omplement to b in g .
C:
g-
2
2
Then m is the unique
M
-
omplement to t in g, and n
Consider the linear isomorphism m
!n
is a linear -
obtained
by omposing the maps
! g=t ! gC =b ! n- ;
m
= !
where g t
2R
g
C =b
x
+ this map sends
omplex stru ture all
is the derivative of the map
2 R+
J
to
e-
and
on m is given by
y
Jy
to -
=
x .
pG=T
! GC=B.
- 1 e- .
Now let
For all
Therefore the
2t .
Then for
p
o [ ; Jy[ ) = -([y ; x ℄) = ([x ; y ℄) = (2 -1 h ): Moreover, it is easy to see that J is ! -orthogonal and that the span of the x 's and the span of the y 's are omplementary Lagrangianspin m with respe t to ! . Therefore ! is positive de nite if and only if (2 -1 h ) > (! ) (y
0 for all
positive roots
,
ase follows by repla ing
i.e. if and only if
2 Int t+ .
with - .
QED
4.4. Convexity theorems for
M; !)
Let ( on
M.
be a Kähler manifold and
moment map,
: M !
g . Identify t
B-orbits
a holomorphi a tion of
We will assume that the restri tion of
The negative de nite
to
G
GC
is Hamiltonian with
T
with the subspa e (g )
of g
and
72
4. KÄHLER POTENTIALS AND CONVEXITY
hoose a losed Weyl hamber t+ in t . Let
B
be the Borel subgroup of
GC
generated by the negative root spa es, i.e. the Borel opposite to the one
onsidered in Lemma 4.3.5. 4.4.1. Theorem. Let
p2M
be a point at whi h
(Bp) \ Int t+
is onvex and is ontained in
B
a ts freely. Then
T
C n N T (T np). 2
We will prove this assertion after establishing a number of preliminary
GC of the symple ti form ! on M via C the map G ! M de ned by g 7! (g)p. Let 2 t be regular and let O be the oadjoint orbit through . As in Se tion 4.3 we will identify O with GC =B and let o 2 O denote the identity oset. Let be the pullba k of C the symple ti form ! + ! on M O via the map G ! M O de ned
results. Let
0
be the pullba k to
by
g 7 ! (g)p; go :
G-a tions on M and O are Hamiltonian GC is Hamiltonian with respe t to both !0
(4.13)
G-a tion ! , with moment maps 0 and obtained by pulling ba k the moment maps on M and M O . By Theorems 4.2.24.2.4 there exist left G-invariant fun tions, f0 and f , C satisfying the onditions on G
The
and hen e the left
on
and
0 = ddC f0 ;
for all
2
onstants.
= ddC f ;
= ([ )dC f0 ; 0
g.
= ([ )dC f
Moreover, both these fun tions are unique up to additive
Let
~F
be the pullba k to
GC
of the fun tion,
F ,
on
GC =N
de ned by (4.11). 4.4.2. Lemma. Up to an additive onstant, Proof. Let
o
be the identity oset in
f = f0 + ~F .
GC =N.
The mapping (4.13) is
the omposite of the mapping
GC ! M GC =N;
g 7 ! (g)p; go
and the proje tion map
idM : M GC=N ! M GC =B:
Hen e we have
= 0 + ! ~ ;
= 0 + ;
(4.14)
! ~ is the pullba k of the symple ti form ! on O via the proje tion C G ! GC =B = O and is the omposition of the proje tion GC ! O
where
4.4. CONVEXITY THEOREMS FOR
O !
and the in lusion potentials for
B
-ORBITS
f
g . Thus the fun tions
and
73
f0
+
de ning the same moment map for the left
~F
are both
G-a tion
so, by Theorem 4.2.4, they dier by a onstant.
are
1 , 2 2 t stable, then f
s1 + (1 - s)2
4.4.3. Lemma. Let
If
f 1
and
f2
Proof. If
=
and
and
QED
= s1 + (1 - s)2
with
0 s 1.
is stable. then
F
=
sF1
+ (1 - s)F 2 . Therefore,
by Lemma 4.4.2,
f
=
f + ~F
= s(f + ~ F 1 ) + (1 - s)(f + ~F 2 ) =
up to additive onstants. Hen e, if below, so is
f .
f1
and
f 2
sf1 + (1 - s)f2
are proper and bounded from QED
4.4.4. Lemma. Assume that
- is a Kähler form on GC Proof. Sin e we hose
B
p
B
is a point where
for every
2 Int t+ .
a ts freely. Then
to be the negative Borel with respe t to t+ ,
2 Int t+ the form !- is Kähler. Hen e ! + !- is a Kähler form on M O . Sin e B a ts freely at p, the map (4.13) is an imbedding, and so - is Kähler. QED Lemma 4.3.5 says that for all
We an now establish a shifted version of Theorem 4.2.5. 4.4.5. Theorem. Assume
is in
(Bp)
a2A
0 (g) = (gp) for and
a ts freely at
p.
if and only if the potential fun tion
Proof. Re all that
that
B
n2N
0 all
2 Int t+ . Then f- : GC ! R is stable. Let
is the pullba k of the Kähler form
g 2 GC .
De omposing
g
=
kan
!
M, so k 2 G,
on
with
and using (4.14) we an therefore write
(kan) = 0 (kan) + - (kan) = Ad(k-1 ) ((anp) - ): By Lemma 4.4.4, - is Kähler. Hen e, by Theorem 4.2.5, f- is stable if C and only if 0 2 - (G ). Hen e f- is stable if and only if (anp) = for
-
some
a2A
and
n 2 N.
Proof of Theorem 4.4.1. That
QED
(Bp) \ Int t+
is onvex follows from
Lemma 4.4.3 and Theorem 4.4.5. It remains to show that this interse tion
T (T Cnp) for all n 2 N. Suppose 2 Int t+ is ontained in (Bp). Then f- is stri tly plurisubharmoni by Lemma 4.4.4 and it is stable by Theorem 4.4.5. Let n 2 N be arbitrary. Then the restri tion of f- to the oset T Cn GC is likewise stri tly plurisubharmoni and stable. C Sin e B a ts freely at p, T a ts freely at np. Applying Theorem 4.4.5 to the C group T instead of G, we on lude that is ontained in T (T np). QED is ontained in
Our proof leads also to a stability result for the various images.
74
4. KÄHLER POTENTIALS AND CONVEXITY
!0
is a Kähler metri on
M
of the form
be the moment map with respe t to
!0
given by
4.4.6. Theorem. Suppose
! = ! + ddC H, where H is a bounded G-invariant smooth fun tion on 0
M.
Let
0 : M ! g
( ) = 0
Then, for any
p2M
+ (J[ )dC H: B
at whi h
a ts freely,
0 (Bp) \ Int t+ T0 (T Cnp) = T (T Cnp)
and
Proof. Let
(Bp) \ Int t+ ;
n 2 N.
for all
H
be the pullba k of
via the map
2 Int t+ . Repla ing the form ! by ! has the ee t f0 + h and hen e f- by f- + h. The result now follows
Let by
h : GC ! R
=
0
of
g 7! (g)p. repla ing f0
from Corollary
4.2.6 and Theorem 4.4.5.
QED
4.5. The onvexity theorem for linear a tions
M is omplex proje tive spa e and G (Bp) \ Int t+ is an open onvex
: GC ! GL(V ) be a representation of GC
In this se tion we will show that, if a ts on
M by ollineations,
the onvex set
polytope. More expli itly, let on a
d + 1-dimensional
ve tor spa e. Then by hoosing an appropriate basis
V one an arrange that V = Cd+1 and that maps G into U(d + 1) and B into the lower triangular Borel subgroup of GL(d + 1). (As in Se tion 4.4, B of
denotes the negative Borel subgroup asso iated with the hamber t+ .) From
,
GC
d
whose restri tion to G is be the moment map asso iated with this a tion and let
omposition of with the proje tion, g ! t . Our main goal
one gets an indu ed a tion,
of
on CP
Hamiltonian. Let
T
be the
in this se tion is to prove the following partial onverse to Theorem 4.4.1. 4.5.1. Theorem. Let
Then
p
2
(Bp) \ Int t+
d
CP
=
be a point at whi h
\
nN
T (T Cnp) \ Int t+ :
B
a ts freely.
(4.15)
2
Remark. By Atiyah's theorem, Theorem 1.3.1 in Chapter 1, the right-
hand side of (4.15) is a onvex polytope. Indeed, Atiyah's theorem asserts that ea h of the sets
T (T Cnp)
is the open onvex hull of the nite set
T Cnp \ (CPd )T :
In parti ular, sin e the set
f T (T Cnp) j n 2 N g
d T is nite, the olle tion of polytopes
(CP )
is nite and hen e the interse tion is a polytope.
4.5. THE CONVEXITY THEOREM FOR LINEAR ACTIONS
!FS be the Fubini-Study form on CPd and let d+1 - f0g onto CPd . Then !FS = ddCH, where C Let
of
H(z0 ; z1 ; : : : ; zd ) =
75
be the proje tion
1 log jz0 j2 + + jzd j2 ; 4
p = (q) the pullba k of !FS to the GC -orbit through p is ddC f0 , where f0 (g) = H( (g)q). Writing g = kan, where k 2 G, a = exp , and n = exp , with 2 a and 2 n, we nd
so for
f0 (kan) = where, for
i
=
d
X 4p-1 () 1 i jz ()j2 ; e log i 4
(4.16)
i=0
0, 1, : : : , d, zi ()
is the
ith
oordinate of the point,
(n)q,
2 HomZ(ZT ; Z) is the ith weight of the representation, . (Here we C a ts on the ith adopt the onvention for weights a
ording to whi h 2 t p weight spa e by zi ! 7 2 -1 i()zi .) From the fa t that n a ts on Cd 1 and
i
+
by lower triangular matri es we obtain the following. 4.5.2. Lemma. The fun tion,
the Lie algebra, n.
7! zi (),
is a polynomial fun tion on
Proof of Theorem 4.5.1. Theorem 4.4.1 says that the left-hand side
of (4.15) is ontained in the right-hand side, so we only have to prove ontainment the other way. for all
n
2 N.
Let
2
Int t+ .
Assume that
By Theorem 4.4.5, to prove that
GC
is in
2 T (T Cnp) (Bp) we must
f- on is stable. By Lemma 4.4.2, p f- (kan) = f0 (kan) - ( -1 log a) up to a onstant. Putting a = exp and n = exp with 2 a and 2 n, and using (4.16) we nd show that the Kähler potential
f (kan) =
d
X 4p-1 () p 1 i jz ()j2 - ( -1 ) e log i 4 i=0
d
= where
i
for every
=
i - .
2a
show that
To show
f
X 4p-1 () 1 i jz ()j2 ; log e i 4
i=0
is stable it su es to show that
tlim !1 f(exp t exp t) = 1
and
2n
su h that
limt!1 l(t) = 1, where l(t) =
d X e4 i=0
+ 6= 0.
p
In other words it su es to
1 t i () jzi (t)j2 :
-
(4.17)
76
4. KÄHLER POTENTIALS AND CONVEXITY
zi (t), are polynomials in t by Lemma 4.5.2. Let I be the set 2 of indi es, i, for whi h zi (t) is not identi ally zero. Suppose that 6= 0. C By Lemma 4.2.9 the assumption 2 T (T np) implies that 0 is ontained p in the interior of the onvex hull of f i j i 2 I g, and hen e -1 i () > 0 for some i 2 I. Therefore the ith summand in (4.17) tends to 1 as t ! 1 The fun tions,
and hen e so does the whole sum. Suppose that = 0. Then the fun tion (4.17) is a positive polynomial in t; so it tends to 1 as t tends to 1, provided that the fun tions, jzi (t)j, for i 2 I, are not all ontants. If this were the ase, however, then for i 2 I the polynomials, zi (t), would have to be onstant as fun tions of t, and hen e we would have (exp t)q = q for all t, whi h violates our assumption that B a ts freely at p. QED One ni e way to hara terize the interse tion
(Bp) \ Int t+
is in terms
Np with the weight hyperplanes. Namely let S be the I, for whi h Np interse ts the plane
of the interse tions of set of multi-indi es,
[z
0 ; z1 ; : : : ; zd ℄ 2 CPd zi = 0; i 2 I :
Then by Lemma 4.2.9 and Theorem 4.5.1 an element of
Int t+
is in
(Bp)
if and only if it is ontained in the set
\
f CI j I 2 S g;
where
CI
is the interior of the onvex hull of
(4.18)
f i j i 2 I g.
4.6. The onvexity theorem for ompa t Kähler manifolds
Let
M
be a ompa t Kähler manifold for whi h the Kähler form,
!,
is rational, i.e. its ohomology lass is in the image of the anoni al map
H2 (M; Q) ! H2 (M; R).
Let
be a Hamiltonian a tion of
G
on
M
by
M is a omplex extends to an a tion of GC . As before let B be the Borel C C C L subgroup of G with Lie algebra b = t 2R+ g- . We will prove below biholomorphisms. Sin e the group of biholomorphisms of
Lie group,
the following analogue of Theorem 4.5.1. (Our proof will be similar to the proof of Atiyah's theorem given in [17℄.) 4.6.1. Theorem. Let
(Bp) \ Int t+
p2M
is equal to
T
be a point at whi h
n2N T (T Cnp) \ Int t+ .
B
a ts freely. Then
! by k!, where k is a large positive integer, we
an assume that [!℄ is the Chern lass of a very ample line bundle, L ! M. Let (L) be the spa e of global holomorphi se tions, and for p 2 M let Æp : (L) ! Lp be the delta fun tion, Æp (s) = s(p). Then Æp 6= 0, so the Proof. After repla ing
4.6. THE CONVEXITY THEOREM FOR COMPACT KÄHLER MANIFOLDS
kernel of
Æp
dual spa e
(L) and its annihilator,
is a hyperplane in
(L) . The Kodaira map is the assignment
: M ! P( (L)) = CPd ;
p ,
is a line in the
p 7 ! p ;
whi h, by the Kodaira imbedding theorem, imbeds
M in
77
d
CP . Let
G-invariant Hermitian bre metri on L whose urvature is Kähler form !. By a theorem of Kostant's [44, theorem 4.5.1℄ a
h; i be
equal to the the group of
bundle automorphisms preserving the metri is a entral extension of the symple tomorphism group the bres of
L.
Let
G
of
M
by the ir le
S1 ,
whi h a ts by spinning
^ !G !1 1 ! S1 ! G
G ! G.
be the pullba k of this extension via the homomorphism
Sin e
G
^ preserves the unique Hermitian onne tion a ts holomorphi ally on M and G ^ a ts holomorphi ally on L. Sin e the group of
ompatible with ; , G
h i
holomorphi bundle automorphisms is a omplex Lie group, this shows that the a tion of
GC
automorphisms.
on
M lifts to an a tion
The indu ed a tion of
des ends to an a tion of
GC
GC
of ^
^C G
on
L by holomorphi bundle (L)
on the spa e of se tions
d
on the proje tive spa e CP
d C Kodaira map intertwines the G -a tions on M and CP .
and plainly the The pullba k by
of !FS is not equal to !; however, lifts anoni ally to a map from L into the hyperplane bundle O(1) on CPd whi h indu es an isomorphism ^ -invariant Hermitian bre metri h; i 0 on O(1) L ! O(1). Choose a G 0 su h that h; i = h; i. This is possible be ause is an imbedding and ^ G is ompa t. Let ! 0 2 11 (CPd ) be the urvature form of this metri . 0 In general ! is not equal to !FS , but its ohomology lass is the Chern
lass of O(1) and therefore oin ides with that of !FS . Hen e there exists a G-invariant one-form, , on CPd with !0
= + with = = 0. Sin e
Write
=
!FS + d:
2 01(CPd).
Then
(4.19)
!0
-
!FS
=
+
and
H1 (CPd ; C) = H10 (CPd ) H01 (CPd ) = f0g;
p
f for some f 2 C1(CPd ). By averaging we an = -1 and using (4.19) we obtain assume that f is G-invariant. Setting h = f + f 0 C ! = !FS + dd h, and hen e
=
0
implies
! = !FS + ddC H
on
M,
4.5.1.
where
H
=
h.
The result now follows from Theorems 4.4.6 and QED
78
4. KÄHLER POTENTIALS AND CONVEXITY
4.6.2. Corollary. Let
p 2 M be a point at whi h B a ts freely.
(Bp) \ Int t+ is equal to the interse tion of Int t+ T I f CI j Np \ XI 6= ; g de ned in (4.3).
with the open polytope
Proof. This follows from (4.18) by pulling ba k by
fun tions,
zi .
Then
the oordinate QED
CHAPTER 5
Appli ations of the onvexity theorem 5.1. The Delzant onje ture
In this hapter we will dis uss some examples of problems in equivariant symple ti geometry whi h, by means of the onvexity theorem, an be onverted into problems involving onvex polytopes. In parti ular, the topi of this se tion is a theorem of Delzant whi h asserts that the lassi ation of ompletely integrable Hamiltonian
T -a tions
the lassi ation of a ertain type of onvex polytope.
is equivalent to
This theorem also
has a onje tured generalization to non-abelian groups, about whi h we will say a few words at the end of this se tion.
In Se tion 5.2 we will dis uss
an appli ation of the Atiyah onvexity theorem for lem of Kählerizability: does a Hamiltonian
T C-orbits
T -manifold admit
to the proba ompatible
T -invariant omplex stru ture? In what follows G will be a ompa t onne ted Lie group, M a ompa t symple ti manifold, : G M ! M a Hamiltonian a tion of G, : M ! g , the moment map asso iated with , T a maximal torus of G, t+ a ( losed) Weyl hamber in t , and M = (M) \ t+ the Kirwan polytope. We will begin by proving a few elementary fa ts about the derivative of at a point p, and dedu ing from these fa ts some onstraints on the dimension of M relative to the dimension and rank of G. We will then de ne, in terms of these onstraints, a lass of manifolds for whi h the stru ture of the manifold is more or less ompletely determined by its moment polytope. 5.1.1. Proposition. Let
Gp
be the stabilizer group of
its Lie algebra. Then the image of Proof. If
M )!.
-(
Hen e
dp
is an element of g, then by de nition
hdp; i = 0
G
at
p
and g
p
p d =
in g is the annihilator of g .
() M (p) = 0.
hd; i =
QED
This result implies in parti ular: 5.1.2. Corollary.
subgroup of
G.
dp
is surje tive if and only if
The result above hara terizes the range of des ription of the kernel of
dp .
79
dp .
Gp
is a dis rete
The next result is a
80
5. APPLICATIONS
2 TpM, is in the kernel of dp if and only if !p (M (p); v) = 0 for all 2 g. Proof. As above, this follows from the identity hd; i = -(M )!. 5.1.3. Proposition. A ve tor,
v
QED 5.1.4. Corollary. The kernel of
plement in
TpM
dp
is the symple ti ortho om-
of the tangent spa e to the orbit,
Gp,
at
p.
One onsequen e of Corollary 5.1.2 is that the dimension of the polytope,
M , is the same as the dimension of t if and only if Gp is dis rete for some p 2 M. Indeed, M is of dimension n = dim t if and only if the interior of M is non-empty, and this is the ase if and only if the image of ontains a non-empty open set. But by Sard's theorem this is true if and only if, for some
p 2 M, dp is surje tive and hen e, by Corollary 5.1.2, Gp is dis rete.
We will only onsider in this se tion manifolds whi h have this property: i.e. for whi h the a tion,
, is lo ally
free at some point,
that if it has this property at some point, this property generi ally ; i.e. if whi h
Gp
M
p,
and if
M
p.
It is easy to see
is onne ted, it has
is onne ted, the set of points,
p,
for
is dis rete is either empty (no point has this property) or is an
open dense subset of
M.
This result follows from a slightly stronger result,
for the proof of whi h we refer to [15℄: 5.1.5. Theorem (Prin ipal orbit type theorem). If
M
is onne ted,
G-invariant open dense subset, U, of M with the property q are in U the stabilizer groups, Gp and Gq , are onjugate subgroups of G, i.e. Gp = aGq a-1 for some a 2 G.
there exists a
p
that if as
For
and
p 2 U the
Gp , whi h is unique up to onjuga y, is alled the group of M. Our main appli ation of Proposition 5.1.3
group,
prin ipal isotropy
is the following dimension inequality: 5.1.6. Theorem. If the prin ipal isotropy group of
dim M dim G + rank G: Proof. Sin e
is equivariant it maps
Gp
M
is dis rete,
onto a oadjoint orbit,
O.
For this orbit we have the inequality
dim O dim G - rank G: Let
be the restri tion of
dimension as
G
to
Gp.
If
Gp
(5.1)
is dis rete
and hen e, by (5.1), the kernel of
d
Gp
p
has the same
is of dimension
rank G. Hen e the kernel of dp is of dimension rank G and therefore by Proposition 5.1.3, the dimension of
rank G.
M is greater than or equal to dim G + QED
5.1. THE DELZANT CONJECTURE
81
Let us investigate the impli ations of the results above in the spe ial
ase where
G = T , i.e.
where
G
is an
n-dimensional
torus. In this ase the
prin ipal orbit type theorem tells us that there exists an open dense set on whi h the isotropy group of
p is the
same for all
p.
Thus, if the a tion of
T
is faithful (whi h we will assume to be the ase from now on), the prin ipal isotropy group is the identity. Moreover, by Theorem 5.1.6
dim M 2 dim T: We will all
M a Delzant
manifold if this inequality is an equality. We will
prove that for a Delzant manifold the moment polytope has some rather spe ial distinguishing features. We re all that if nential map,
exp : t ! T is
T
dis rete subgroup, Z , of
T
T
the expo-
T . Moreover, exp is t=ZT = T . In addition the
T is just the dual latti e to ZT in t
weight latti e Z
then for every vertex,
n-torus
alled the group latti e of
surje tive, so there is an isomorphism of groups:
5.1.7. Theorem. If
is an
a group homomorphism and that its kernel is a
is the , of
.
moment polytope of a Delzant manifold,
there are exa tly n edges meeting in . 1; , 2; , : : : , n; , of the weight latti e, meeting at lie on the rays, + t i; ,
Moreover, there exists a basis, Z
T,
for
su h that the
n
edges of
i = 1, 2, : : : , n.
Proof. By the onvexity theorem is the image of a xed point, p 2 MT , and by the onstru tive version of the onvexity theorem that we proved in Chapter 2 the image of at oin ides with the image at of the moment map asso iated with the anoni al model for the a tion of T at p. Sin e p is a xed point and T is abelian the anoni al model is extremely simple: Let 1; , 2; , : : : , n; be the weights of the isotropy representation of T on TpM. Then this anoni al model is the standard representation of T on n C with weights, i; , and moment map
n X n z2C 7 !+ i; jzi j2 : i=1
Thus the image of
is, lo ally at
,
as des ribed above. Finally it is lear
from this anoni al model that the prin ipal isotropy group of where
n; .
is the latti e in t dual to the latti e generated by
Thus
= ZT ,
sin e the a tion of
A onvex polytope,
T
M
1; ,
=ZT 2; , : : : , is
is assumed to be faithful.
QED
t , with the properties des ribed in Theorem
5.1.7 is alled a simple regular polytope and hen e we an rephrase Theorem 5.1.7 as asserting
82
5. APPLICATIONS
5.1.8. Theorem. The moment polytope of a Delzant manifold is sim-
ple and regular. Delzant's theorem is a very strong onverse to this statement.
is a simple regular polytope, there exists a Delzant manifold, M, with = M . Moreover, M is unique up to isomorphism, i.e. if M1 and M2 are Delzant manifolds and M1 = M2 , there exists a T -equivariant symple tomorphism mapping M1 onto M2 . 5.1.9. Theorem ([21℄). If
In fa t, Delzant gives an expli it onstru tion of moment polytope. The onstru tion shows that
M
M
starting from its
is a tori variety in the
sense of Demazure [23, se tion 5℄ and of Kempf, Knutsen, Mumford and Saint-Donat [39℄. In parti ular, it establishes the following Kählerizability result. 5.1.10. Theorem. A Delzant manifold admits a
T -invariant omplex
stru ture ompatible with its symple ti stru ture. We will on lude this se tion by saying a few words about analogues of these results for non-abelian groups. We will ontinue to assume, as above, that the prin ipal isotropy group,
Gp ,
of
is dis rete.
, is multipli ity free if the inequality if dim M = dim G + rank G.
5.1.11. Definition. The a tion, of Theorem 5.1.6 is an equality, i.e.
For su h a tions the following onje ture is, as of now, still not proved; however, over the ourse of the last fteen years there has been mounting eviden e that it is true. 5.1.12. Conje ture (Delzant). A multipli ity free Hamiltonian
G-
a tion on a ompa t symple ti manifold is determined up to isomorphism by its moment polytope and (the onjuga y lass of ) its prin ipal isotropy group. For rank one groups this theorem was proved by Iglesias [36℄ and for rank two groups by Delzant [22℄. Moreover, for groups of arbitrary rank, Woodward [66℄ proved that this onje ture is true modulo two hypotheses: (i) The prin ipal isotropy group is the identity. (ii) The moment mapping
: M ! g
is transverse to t .
There are also a number of results known about spheri al a tions: algebrai a tions of the omplexi ed group, for whi h the a tion of
GC , on omplex proje tive varieties
G is multipli ity-free;
and this suggests the following
very plausible Kählerizability onje ture: 5.1.13. Conje ture. If
G-invariant
is a multipli ity-free a tion,
M
admits a
Kähler stru ture ompatible with its symple ti stru ture.
5.2. KÄHLERIZABILITY
83
5.2. Kählerizability
M is a ompa t Hamiltonian T -manifold and T a ts faithfully, then as we saw in the last se tion, dim M 2 dim T . Moreover, by Delzant's theorem, one has a omplete lassi ation of spa es for whi h dim M = 2 dim T . The next simplest ase is the omplexity one ase: dim M = 2(dim T + 1). If
The lassi ation of su h spa es began in 1994 with the following result of Yael Karshon:
M
5.2.1. Theorem ([38℄). Let
ple ti manifold and
a faithful Hamiltonian a tion of
isolated xed points. Then a tion of
T 2;
be a ompa t four-dimensional sym-
S1
on
M
with
an be extended to a faithful Hamiltonian
and hen e, in parti ular,
M
is Delzant.
We will be on erned in this se tion with the next higher dimension:
dim M
=
6
and
dim T
=
2.
Here one might onje ture that the analogue
of Karshon's result is true: that Hamiltonian a tion of a
M
3-torus.
M
is Delzant and
the restri tion of a
By Theorem 5.2.1 this would imply that
is Kählerizable and hen e settle a rmatively the onje ture: 5.2.2. Conje ture. Every omplexity one manifold of dimension
6
is Kählerizable. In fa t, in the early nineties the following mu h more auda ious onje ture was regarded as very plausible (mainly be ause there were no obvious
ounterexamples). 5.2.3. Conje ture. If
M
is a ompa t Hamiltonian
its xed point set is nite, then
M
T -manifold and
is Kählerizable.
In 1995 Sue Tolman su
eeded in onstru ting a ounterexample whi h demolished all of the three onje tures 5.1.13, 5.2.2 and 5.2.3. Her ounterexample is a Hamiltonian a tion of
M,
with
dim M
=
6, dim T
=
2
T
and
is the trapezoid depi ted in Figure 2.
on a ompa t symple ti manifold,
#MT
=
6,
whose moment polytope
In this gure the weight latti e Z
T
is the triangular latti e designated by the white dots. The four verti es of
and are the through and
the trapezoid are images of four of the six xed points, and images of the remaining two xed points. The lines going are the images of the one-skeleton of
M,
i.e. the set
f p 2 M j dim Tp = 1 g of one-dimensional orbits of
T.
We will des ribe below how this manifold
is onstru ted, but rst we must explain why a manifold with the moment polytope depi ted in Figure 2 annot be given a ture.
Re all from Chapters 1 and 4 that if
M
T -invariant
Kähler stru -
is a Kähler manifold the
84
5. APPLICATIONS
Figure 2. Tolman trapezoid
onvexity theorem is a onsequen e of a mu h stronger onvexity theorem
T C-orbits. More expli itly, if M is a ompa t Kähler manifold and : T M ! M a Hamiltonian a tion of T whi h preserves the omplex C C stru ture, extends to a holomorphi a tion, , of T , and Atiyah's theC orem, Theorem 1.3.1, asserts that if X is the losure of a T -orbit then the moment image, X , of X is the onvex hull of the moment image of the set, XT , of T - xed points lying on X. In fa t Atiyah proved somewhat more. Let : M ! t be the moment map. of Atiyah's for
5.2.4. Proposition. If
X \ -1 (F)
is a
T C-orbit.
F
is an open fa e of the polytope,
X ,
then
This result turns out to impose a lot of onditions on the polytope,
p 2 MT
X .
(p) and let 1 , 2 , : : : , d , be the weights of the isotropy representation of T on the tangent spa e to M at p. For every subset, I, of the set, f1; : : : ; dg, let For
let
=
CI
=
+
X
iI
si i si
0
:
2
We will all these sets the isotropy ones at
.
From Proposition 5.2.4 one
easily dedu es: 5.2.5. Proposition. For every orbit losure
vertex
X
in
M
and for every
of X there exists an isotropy one, CI , su h that U\CI = U\X U, of in t .
for some neighborhood,
Tolman proves a result in the onverse dire tion:
p 2 MT
and an isotropy one, CI , at C = (p), there exists a T -orbit su h that p is ontained in the losure, X, of this orbit and su h that U \ CI = U \ X for some neighborhood, 5.2.6. Theorem. Given a point
U,
of
in t .
5.2. KÄHLERIZABILITY
85
Figure 3. Non-existent image of orbit losure
Let us now ome ba k to Figure 2 and explain why a manifold with this as its moment polytope annot admit a
T -invariant ompatible
Kähler
stru ture: Suppose that it did admit su h a stru ture and hen e that the a tion of
T extended
lines going through
to an a tion of
and
5.2.6 there exists an orbit of
T C.
The one at
whose sides are the
are an isotropy one and hen e by Theorem
TC
whose losure,
X,
ontains the xed point
and whose moment image oin ides at with this one. It is X are the xed points whi h have as their moment images and ; so by Atiyah's theorem the moment image of X is the triangle in Figure 3. However, this is impossible sin e the one having as a vertex and edges going through the points and is not one of the three moment ones at !
mapping onto
lear from Figure 2 that the other xed points lying on
This ounterexample an be onsiderably beefed up; and, in fa t, Tolman uses the ideas involved in the onstru tion of this ounterexample in [62℄ to give a more or less omplete lassi ation of
T 2-a tions
on six-
manifolds whi h do admit equivariant Kähler stru tures and shows on the other hand that most su h a tions do not. We will now give a rough sket h of how Tolman onstru ts a Hamiltonian
T 2-manifold of dimension six with the trapezoid in Figure 2 as its moment 3 polytope. Consider the following two examples of a tions of the torus T . 5.2.7. Example. The standard a tion,
1 CP2 .
CP
The moment polytope for
, of T 3 = S1 T 2 on the produ t, is the produ t of an interval and
a triangle, i.e. the triangular prism depi ted in Figure 4(a). Let
3 1 kernel of the map from T to S de ned by ei1 ; ei2 ; ei3 7 ! ei(2 +3 -1 ) :
Restri ting
to this subtorus of
T 3 one
T2
be the
gets a Hamiltonian a tion of
T 2 on
1 CP2 , whose moment polytope is the proje tion of the prism onto R2 via the map, (x1 ; x2 ; x3 ) 7! (x2 + x1 ; x3 + x1 ). This proje tion is depi ted in
CP
Figure 4(b).
86
5. APPLICATIONS
5.2.8. Example. The standard a tion of this a tion is a
3-simplex
T3
3
on CP .
The image of
3
(i.e. a triangular pyramid). If one blows up CP
at the xed point orresponding to the apex of this pyramid one gets a Delzant spa e whi h, as a dierentiable manifold, is the onne ted sum
3 # CP3 , and its moment polytope is the trun ated pyramid in Figure 2 1 1 3 5(a). Restri ting this a tion to the subgroup, T = S S f1g, of T one 3 3 2 gets a Hamiltonian a tion of T on the manifold CP # CP whose moment polytope is just the trun ated pyramid proje ted onto the (x1 ; x2 )-plane, i.e.
CP
the polygon in Figure 5(b). To onstru t a manifold with moment polytope being the trapezoid of Figure 2, one uts Figures 4(b) and 5(b) along the dotted lines and glues the top part of Figure 4(b) to the bottom part of Figure 5(b) to get the polygon of Figure 2. Lerman's symple ti utting theorem, Theorem 1.3.3, enables one to dupli ate these utting and gluing operations upstairs on the manifolds: i.e. ut the manifold in Example 5.2.7 along the preimage of the dotted line to get a manifold whose moment polytope is the top part of Figure 4(b), and ut the manifold in Example 5.2.8 along the dotted line to get a manifold whose moment polytope is the bottom part of Figure 5(b). The result is a pair of Hamiltonian
(a)
T 2-manifolds M1 and M2 .
The preimages
(b)
Figure 4. Triangular prism and proje tion
(a)
(b)
Figure 5. Trun ated pyramid and proje tion
5.2. KÄHLERIZABILITY
87
of the dotted lines are odimension two symple ti submanifolds and
X2 of M2 .
The manifolds
and
M2
of
M1
X1 and X2 are equivariantly symple tomorphi
and they have opposite symple ti normal bundles.
M1
X1
together to get a Hamiltonian
Hen e we an glue
T 2-manifold
with the moment
polytope in Figure 2. This spa e also turns out to be a ounterexample to Conje ture 5.1.13. Namely Chris Woodward [67, 68℄ noti ed that Tolman's example has some
T 2-a tion an be extended to a Hamiltonian a tion of U(2); and, sin e dim U(2) + rank U(2) = 6, this means that it is a multipli ity-free Hamiltonian U(2)-manifold. Sin e Tolman's manifold does 2 not possess a T -invariant ompatible Kähler stru ture, a fortiori it does not possess a U(2)-invariant su h stru ture, i.e. Conje ture 5.1.13 is false.
hidden symmetries: the
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