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[Q,R]=0 and Kostant partition functions
SZENES, Andras, VERGNE, Michele
Abstract
On a polarized compact symplectic manifold endowed with an action of a compact Lie group, in analogy with geometric invariant theory, one can define the space of invariant functions of degree k. A central statement in symplectic geometry, the quantization commutes with reduction hypothesis, is equivalent to saying that the dimension of these invariant functions depends polynomially on k. This statement was proved by Meinrenken and Sjamaar under positivity conditions. In this paper, we give a new proof of this polynomiality property. The proof is based on a study of the Atiyah-Bott fixed point formula from the point of view of the theory of partition functions, and a technique for localizing positivity.
SZENES, Andras, VERGNE, Michele. [Q,R]=0 and Kostant partition functions. , 33 p.
arxiv : 1006.4149v1
Available at:
http://archive-ouverte.unige.ch/unige:11901
Disclaimer: layout of this document may differ from the published version.
1 / 1
arXiv:1006.4149v1 [math.SG] 21 Jun 2010
A. SZENES AND M. VERGNE
1. The theorem
Let M be a compact manifold endowed with an almost complex struc- ture J ∈ ΓEnd(TM). We denote by TJM the complex vector bundle of +i-eigenspaces of J acting on TM ⊗ C, and by ¯TJM the bundle of −i- eigenspaces of J. Then we have the splitting TM ⊗C= TJM⊕T¯JM. If M is a complex manifold endowed with an Hermitian metric, then TJM may be identified with the complex tangent bundle, while ¯TJM with the complex cotangent bundle of M. Informally, we will use this terminology even when M has only an almost complex structure.
To every complex vector bundle E → M over M one can associate an integer as follows (cf. (2)). Set the notationΩ•J(M,E) = Γ(Λ•( ¯TJM)∗⊗ E) for the anti-holomorphic differential forms with values in E, and consider the twisted Dolbeault-Dirac operator [5]
DE :ΩevenJ (M,E)→ΩoddJ (M,E).
This is a first-order elliptic differential operator, and its “virtual space of solutions” is well-defined as a formal difference of two finite-dimensional vector spaces:
(1) Q(M,E)=Ker(DE)⊖Coker(DE).
The dimension of this virtual vector space is defined to be the integer (2) dim Q(M,E)=dim Ker(DE)−dim Coker(DE).
This number may be computed by the Atiyah-Singer index formula:
(3) dim Q(M,E)=
Z
M
ch(E) Todd(TJM);
here ch(E) is the Chern character ofEand Todd(TJM) is the Todd class of M.
Now assume that a compact, connected Lie group G acts compatibly on the manifold M and the bundleE, and preserves the almost complex struc- ture J. Then Q(M,E) becomes a virtual representation of G, whose charac- ter we denote byχE.
To make this more explicit, we introduce the following notation for the Lie data:
• Denote by T the maximal torus of G, and
The support of FNS grants 122084 and 126817 is gratefully acknowledged.
1
• bygandtthe Lie algebras of G and T , respectively;
• we will identify t∗ with the T -invariant subspace of g∗ under the coadjoint action.
• LetΛstand for the weight lattice of T thought of as a subspace of t∗.
• We will use the notation eλfor the character T →C∗corresponding toλ∈Λ, and write tλfor the value of this character on t in T . Thus we have eλ(t) = tλ for t ∈ T , and also tλ = eihλ,Xi if X ∈ t and t=exp(X).
• Denote the set of roots of G byR; these split into a positive and a negative part:R= R+∪R−.
• Write dt for the Haar measure on T satisfyingR
T dt= 1.
Further, for X ∈g, we denote by V X the vector field V X : M →TM, V X : q 7→ d
dte−tXq|t=0 on M induced by the G-action.
Atiyah-Bott-Segal-Singer [2, 3, 4] give a fixed point formula for χE(t).
The Fourier transform FχE ofχE is then a function with finite support on Λ; its value
FχE(λ)= Z
T
e−λχEdt
is an integer, called the multiplicity of the weightλinχE. It is then possible to express FχE(λ) in terms of partition functions, the first example of such an expression being Kostant’s formula for the multiplicity of a weightλin a finite dimensional representation of a compact Lie group in terms of the number of ways a weight can be expressed as a sum of positive roots.
Our focus will be the calculation of the dimension of the G-invariant part Q(M,E)Gof Q(M,E), obtained by taking G-invariants on the right hand side of (1):
Q(M,E)G= Ker(DE)G⊖Coker(DE)G.
According to the Weyl character formula, this integer may be expressed via the multiplicities as follows
(4) dim Q(M,E)G =
Z
T
Y
α∈R−
(1−eα)χEdt.
Consider an equivariant line bundleLover M, endowed with a G-invariant Hermitian structure and an Hermitian connection ∇. Then the curvature
∇2 will be equal to −iΩ, where Ω is a closed real 2-form on M. The G- invariant connection ∇ determines a G-equivariant map, the moment map µG: M →g∗by the formula:
(5) ihµG,Xi= LX − ∇V X,
where LX is the Lie derivative acting on the sections ofL. Observe that if p ∈ M is a fixed point of the T -action, thenµG(p) is int∗ ⊂ g∗, moreover,
µG(p) is exactly the T -weight of the fiberLp. Differentiating (5), we obtain the key equality
(6) hdµG,Xi+ Ω(V X,·)=0.
The spaceµ−1G(0)/G is called the reduced space of M with respect to G.
The philosophy of quantization commutes with reduction (or [Q,R]=0 for short) is that the virtual space Q(M,E ⊗ Lk)G may be identified with the virtual space of solutions of a Dirac operator associated to a vector bundle of the formE0⊗Lk0on the reduced spaceµ−1G(0)/G. This idea was introduced in [10] in the form of a precise conjecture (cf. Theorem 6, and the discussion below). Ifµ−1G (0)/G is smooth, then, using the conjecture and applying the Atiyah-Singer formula (3) to the bundle E0 ⊗ Lk0, we can conclude that dim Q(M,E ⊗ Lk)Gdepends polynomially on k. This polynomiality thus is a key manifestation of the [Q,R]=0 principle, and it will be our main object of study in this article.
This reduction principle comes from considering the special case when M is a complex projective G-manifold, L is the ample bundle, and E is a holomorphic vector bundle on M. Then the G-action on M may be extended to a holomorphic action GC×M → M of the complexification of the com- pact Lie group G, and [Q,R]=0 follows from the fact that (cf. [16]) the orbit ofµ−1G (0) by this complexified action of GCis dense in M.
Returning to the almost complex setting, where no complexified action exists, consider the case where G =T is abelian. In this case, we will write µ : M → t∗for the moment map, omitting the index T . A special case of the quantization commutes with reduction principle then reads as follows.
Theorem 1. LetE+andE−be T -equivariant vector bundles over the almost complex manifold M. Let L be an equivariant line bundle with associated moment map µ : M → t∗. Suppose that the bundleE+is equivalent to E− overµ−1(0). Then, for k large, the multiplicitiesFχE+⊗Lk(0) andFχE−⊗Lk(0) are equal.
Following Meinrenken, we give a proof of this theorem in§5 based on the stationary phase principle applied to the integral formula of [7] forχE⊗Lk. Example 2. Let us consider the simplest example: M = P1(C),Lis the dual of the tautological bundle, endowed with an action of the group SU(2). The maximal torus T of this group corresponds to the set of diagonal matrices in SU(2). Then we have the following table:
k . . . −4 −3 −2 −1 0 1 2 3 . . .
dim Q(Lk) . . . −3 −2 −1 0 1 2 3 4 . . . dim Q(Lk)T . . . −1 0 −1 0 1 0 1 0 . . . dim Q(Lk)SU(2) . . . 0 0 −1 0 1 0 0 0 . . . Thus we see that
• dim Q(Lk)=k+1; it is thus a polynomial for all k∈Z.
• dim Q(Lk)T =
1, if 0 ≤k is even,
−1, if 0>k is even, 0, if k odd.
In particular, this is a quasi-polynomial for all k≥ 0.
• dim Q(Lk)SU(2) is, however, only polynomial for k ≥ 1. Note that dim Q(L−k)SU(2) is not polynomial for k≥ 1.
This example shows, that, in general, dim Q(M,E ⊗ Lk)G is not polyno- mial for small k. To make a stronger statement, we set E to be the trivial line bundle, and introduce a key condition onLas follows.
Definition 3. Given an almost complex manifold (M,J), we say that a line bundleLover M is positive if for an Hermitian structure, and a compatible connection∇onL, the corresponding curvature−iΩsatisfies
(7) Ωq(V,JV)> 0 for all 0, V ∈TqM at every point q∈ M.
Note that in this caseΩis a symplectic form on M.
Remark 4. The data of a positive line bundle is the same as that of a pre- quantizable symplectic manifold endowed with a Kostant line bundle ([12]);
this latter is a G-equivariant line bundle with first Chern class equal to the class of Ω/2π. Indeed, if (M,Ω) is a symplectic manifold, then we can choose a unique (up to homotopy) positive almost complex structure J, and then the Kostant property ofLis equivalent to the existence of a G-invariant connection with property (7).
Definition 5. Let Ξ be a freeZ-module. A function P : Ξ → C is quasi- polynomial if there exists a sublattice Ξ0 ⊂ Ξof finite index such that for everyλ∈Ξthe function P restricted toλ+ Ξ0coincides with the restriction of a polynomial function fromΞtoΞ0.
In particular, a function P : Z → C is quasi-polynomial if, for some nonzero d ∈Z, the function l7→ P(ld+r) is polynomial for every r ∈Z.
Now we are ready to formulate the polynomiality statement for which we give a new proof. As we explain below, this is a corollary of results of [15].
Theorem 6. Let (M,J) be a compact, connected almost complex manifold, endowed with the action of a connected compact Lie group G, and letLbe a positive G-equivariant line bundle on M. Then
• the integer function
k→dim Q(M,Lk)G is quasi-polynomial for k≥1, and
• this quasi-polynomial is identically zero if 0<µG(M).
Let us further comment on the relation between this theorem and the original Guillemin-Sternberg conjecture [10]. One may also consult [19]
and [23] for more details and references.
Let M be a symplectic manifold equipped with a Kostant line bundleL.
If 0 is a regular value ofµG, then the reduced spaceµ−1G (0)/G is a symplectic orbifold equipped with a Kostant line bundleL0. Guillemin-Sternberg for- mulated the conjecture that Q(M,L)Gmay be identified to Q(µ−1G (0)/G,L0).
Meinrenken, in his first approach to the Guillemin-Sternberg conjecture [13], determined the asymptotic behavior of dim Q(M,Lk)G for k large, under the assumption that 0 is a regular value. By a “stationary phase”
method (that we borrowed in part for our proof of Theorem 1), he showed that dim Q(M,Lk)Gis indeed equal to dim Q(µ−1G (0)/G,Lk0) for k sufficiently large, and that the equality holds for all k ≥ 0 if G is abelian. He has thus shown that the Guillemin-Sternberg conjecture for general compact con- nected Lie group G is equivalent to the fact that dim Q(M,Lk)G is quasi- polynomial in k for k ≥1.
Meinrenken-Sjamaar in [15] formulated the Guillemin-Sternberg conjec- ture for the case when 0 is not necessarily a regular value of the moment map, and, using techniques of symplectic cutting, proved this more general statement. There is also an analytic proof of this generalized Guillemin- Sternberg conjecture by Tian and Zhang [21], and another proof by Paradan [17] using transversally elliptic operators. Theorem 6 is a consequence of this statement.
In the present paper, we prove that dim Q(M,Lk)G is quasi-polynomial in k for k ≥ 1 directly, without making the assumption that 0 is a regular value of the moment map. However, we will not pursue here the task of identifying geometrically the quasi-polynomial dim Q(M,Lk)G.
Our main purpose is to show that this result may be obtained from the Atiyah-Bott fixed point formula for χLk, using Theorem 1 as the only ana- lytic input. The rest of the argument is based on combinatorial manipula- tions of Kostant partition functions and some simple geometric arguments
“localizing positivity”. The ideas underlying our paper originated in the works of Paradan [17, 18].
The paper is structured as follows: in§2 we describe the calculus of the expansions of the terms of the Atiyah-Bott fixed point formula forχE⊗Lk us- ing partition functions (Corollary 10), and then introduce a quasi-polynomial character∆µ[E,a] encoding the asymptotic behavior of this expansion. We begin§3 by Paradan’s combinatorial formula decomposing a partition func- tion in terms of convolution products of partitions functions in lower dimen- sions, then we apply this formula to our geometric setup (Proposition 35), which results in a decomposition ofχE in terms of certain characters of the type∆µ enumerated by fixed-point sets of subtori of T . This combinatorial decomposition is analogous at the level of characters to the stratification of the manifold M using the Morse function kµk2 used by Witten [24] to compute intersection numbers on reduced spaces. We finish the proof of Theorem 6 in §4, by studying the terms of this expansion. The crucial part
of the argument is a “localization of positivity” result: Proposition 44. Fi- nally, we give a quick proof of Theorem 1 in §5. A list of notations given in§6 helps the reader to navigate the paper.
2. Fixed point formula and a formal character.
2.1. The Fixed Point Formula. As in the previous section, let us start with a connected, compact, almost complex T -manifold M, and a pair (E,L), consisting of a complex equivariant vector bundle and a line bundle on M.
In this section, we embark on the study of the charactersχE andχE⊗Lk. Our starting point is the Atiyah-Bott fixed point formula [2]. To simplify our situation, we assume that the set F of T -fixed points in M is finite. Let us introduce some notation for the fixed point data.
We will use the term list for set with multiplicities. A listΦthus consists of a set{Φ}, and a multiplicity function mΦ:{Φ} →Z>0. We will write
• ψ∈Φifψ∈ {Φ};
• ifψ ∈ Φ and mΦ(ψ) > 1, then Φ− {ψ}will denote the list Φ with the multiplicity ofψdecreased by 1; if mΦ(ψ) =1, thenΦ− {ψ}will denote the listΦwithψremoved;
• for a list Φ and a set S , we will writeΦ ∩S for the list with un- derlying set{Φ} ∩S and multiplicity function coinciding with that ofΦon this set; we will writeΦ\S for the list with underlying set {Φ} \S and multiplicity function coinciding with that ofΦon this set;
• the productQ
ψ∈Φψwill stand for the productQ
ψ∈{Φ}ψmΦ(ψ).
For example, for each fixed point p ∈ F, the weights of the T -action on Epform a list, which we will denote byΨp. Letεp be the function T →C obtained by taking the trace of the T -action on the fiberEp. Thus we have εp = P
η∈Ψpeη. Similarly, we denote by Φp the list of T -weights of the complex vector space ¯TJpM.
Without loss of generality, we can make the additional simplifying as- sumption that the generic stabilizer of the T -action on M is finite; this is equivalent to the condition thatΦpspanst∗for all p∈F. With these prepa- rations we can state the Atiyah-Bott fixed point formula for our case: it is an equality between two functions defined on an open and dense subset of T .
(8) χE =X
p∈F
εp Q
φ∈Φp(1−eφ). Indeed, the right hand side is meaningful on the set
{t∈T|tφ,1∀p∈F andφ∈Φp},
while the left hand side is regular on T . Let us see a few simple examples.
Example 7. Let M = P1(C) with the action of U(1) given by t·(x : y)=(tx : t−1y), and letE=Lkbe the kth tensor power of the dual of the tautological
line bundle. There are 2 fixed points p+ = (1 : 0) and p− = (0 : 1), and we have
χE(t) = tk
(1−t−2)+ t−k (1−t2) =
k
X
j=0
tk−2 j. The graph of the functionFχE is pictured below for k=4.
−4 −3 −2 −1 0 1 2 3 4
1
Example 8. Let M be the flag variety of C3 endowed with the action of the group U(3). The subgroup T = {(t1,t2,t3); t1,t2,t3 ∈ U(1)} ⊂ U(3) of diagonal matrices serves as the maximal torus in this case, and the weight lattice has a canonical diagonal bases: Λ = Zθ1 + Zθ2 + Zθ3. Then the coordinate flag{Ce1 ⊂Ce1⊕Ce2 ⊂Ce1⊕Ce2⊕Ce3}is invariant under T , and the rest of the fixed points in MT may be obtained by applying to this flag the elements of the permutations groupΣ3in a natural manner. We will use the notation w ∈Σ37→ pw ∈ MT for this correspondence; in particular, the coordinate flag will be denoted by p123.
Consider the line bundleLinduced from the character t14t2−1t33of T . Then χLk = X
w∈Σ3
w∗ t4k1 t−k2 t3k3
(1−t2/t1)(1−t3/t2)(1−t3/t1), where, again, w∗stands for the natural action ofΣ3on the indices.
The functionFχLk(for k =1) is depicted below on the root lattice gener- ated by the simple rootsα= θ1−θ2andβ= θ2−θ3. The weightµ123of the bundleLat p123is 4α+3β.
= 3
= 2
= 1 µ123
β α
2.2. The partition function. Recall that χE is determined by its Fourier coefficientsFχE : Λ→ Z, and that this latter function has finite support in Λ. Our immediate goal is to write down (8) as an equality of two functions in the Fourier dual space{Λ→Z}.
Before we proceed, we need to introduce a few basic notions.
• We denote by R(T ) the set of finite integral linear combinations of the characters eλ, λ∈Λ, and
• by ˆR(T ) the space of formal, possibly infinite, integral linear com- binations of these characters. Thus the elements of ˆR(T ) are in one-to-one correspondence with the functions m(λ) : Λ → Z via θ := P
λ∈Λm(λ)eλ ∈ R(T ). We will writeˆ Fθ for the function m in this case. Conversely, given a function m, we will call the cor- responding series θ its character. If we extend the weights λ ∈ Λ to linear functions ontC, then we can also think of the elements of R(T ) as formal series of holomorphic exponential functions onˆ tC.
• Informally, we will callδ∈R(T ) a quasi-polynomial character if itsˆ Fourier transformFδ : Λ → Cis quasi-polynomial (cf. Definition 5).
Observe that ˆR(T ) is a module over R(T ) and the set of quasi-polynomial characters forms a linear subspace in ˆR(T ) which is stable under multipli- cation by R(T ). In addition, elements of ˆR(T ) whose Fourier transforms are supported on a fixed acute cone may be multiplied.
With these preparations, we are ready to introduce the basic building block of our constructions. Let Φbe a list of nonzero elements of Λ. We will call X ∈tpolarizing forΦifhφ,Xi, 0 for everyφ∈Φ. For nonempty Φand polarizing X, define the partitionΦ = Φ+∪Φ−ofΦinto
Φ+= {φ∈Φ| hφ,Xi> 0} and Φ−= {φ∈Φ| hφ,Xi<0}, and introduce the formal character
(9) Θ[Φ↑X]= (−1)|Φ−| Y
φ∈Φ−
e−φ× Y
φ∈Φ−
∞
X
k=0
e−kφ× Y
φ∈Φ+
∞
X
k=0
ekφ.
It is easy to verify that the products in this formula are meaningful, and hence the series Θ[Φ↑X] defines an element of ˆR(T ). We also setΘ[∅ ↑ X]=1 for any X ∈t.
We record a few basic properties ofΘ[Φ↑X].
Lemma 9. (1) Θ[Φ↑X] satisfiesQ
φ∈Φ(1−eφ)Θ[Φ↑X]=1.
(2) Considered as a series of holomorphic functions of the form eiλ with λ ∈ Λ, the series (9) converges absolutely in a neighborhood of iX ∈tC.
(3) FΘ[Φ↑X] is supported in the pointed cone generated in t∗ by the setΦ+∪(−Φ−).
Note that if two formal characters in ˆR(T ) absolutely converge, and co- incide on an open set oftC, then they also coincide as elements of ˆR(T ). In
particular, it follows from Lemma 9 that in their domain of convergence the formal charactersΘ[Φ↑X] are equal to the functionQ
φ∈Φ(1−eφ)−1 for any X.
This allows us to rewrite (8) as follows:
Corollary 10. We have the following equality in ˆR(T ):
(10) χE =X
p∈F
εpΘ[Φp↑X].
Remark 11. The function FΘ[Φ ↑ X] : Λ → Z, traditionally, has been called the partition function, since, assumingΦ = Φ+, its value atµequals the number of ways one can writeµas a nonnegative integral linear combi- nations of vectors fromΦ. In particular, the equality (10) applied to Weyl’s formula for the characters leads to Kostants’s formula for the multiplicity of a weight in an irreducible representation of a reductive Lie group.
A key fact is that the Fourier transformFΘ[Φ↑X], as a function onΛ, is piecewise quasi-polynomial. Let us explain this in more detail:
Definition 12. Given a list Φspanning t∗, we will call an element γ ∈ t∗ Φ-regular if it is not the linear combination of fewer than dim(t) elements ofΦ.
The set of Φ-regular elements form the complement of a hyperplane ar- rangement int∗, and we will use the termΦ-tope for the connected compo- nents of this set. It will be convenient to use the notationT(γ) for the tope containing theΦ-regular elementγ. Note that topes are open convex cones, which are invariant under rescaling.
Lemma 13. GivenΦand X as above, and aΦ-topeT, there exists a quasi- polynomial character δ[Φ ↑ X,T] such that FΘ[Φ ↑ X] coincides with Fδ[Φ↑X,T] onΛ∩ T.
This lemma is proved in [9] (see also [8]).
Example 14. Let t∗= Rα, Λ = Zα,Φ = [α] and let X ∈tto be the vector satisfyinghα,Xi=1. Then
Θ[Φ↑X]=
∞
X
k=0
ekα. ThenT := {tα,t > 0}is a tope andδ[Φ↑X,T]= P
k∈Zekα. The character δ[Φ ↑ X,T] is quasi-polynomial as the multiplicity Fδ[Φ ↑ X,T] is the constant function 1 onZα.
2.3. The asymptotics of the character. Now we return to our geometric setup. We continue to assume that the torus T acts on the compact al- most complex manifold M with a finite set of fixed points. Also, recall our notation µ : M → t∗ for the moment map associated to the Hermitian T - equivariant line bundleL as in §1, and the fact that for p ∈ F,µ(p) is the weight of the T -action on the fiberLp.
Definition 15. For p ∈ F, and a subset S ⊂ t∗ denote by Sp the shifted subset S −µ(p). In particular, we haveγp =γ−µ(p) forγ∈t∗.
The moment map µ gives rise to a real affine hyperplane arrangement whose complement is the open set
(11) \
p∈F
nγ ∈t∗|γpisΦp-regularo
⊂ t∗.
We will use the term alcove for the connected components of the set (11).
The alcoves are thus minimal nonempty intersections of the translated poly- hedral conesT +µ(p), where p ∈ F, andT is a tope ofΦp. Just as in the case of topes, we will use the notation a(C) for the alcove containing the connected subset C of the set (11).
Remark 16. If Lis a positive line bundle as in Definition 3, thenµ(M) is the convex hull of the pointsµ(p), where p∈F, and the set (11) is contained in the set of regular values ofµ.
Next, we define a quasi-polynomial character by formally replacing the generating function for the partition function Θ[Φp↑X] in (10) by appro- priately chosen corresponding quasi-polynomialsδ[Φp↑X,T] (cf. Lemma 13).
Definition 17. Given a T -equivariant vector bundle Eover M, and an al- covea⊂t∗, we define the formal character
(12) ∆µ[E,a]=X
p∈F
εp·δ[Φp↑X,T(ap)], whereap =a−µ(p).
Remark 18. Note that we omitted the dependence on X in the notation (cf.
Corollary 23).
This character acquires a geometric meaning for the sequence of bundles E ⊗ Lk, k=1,2, . . . (cf. Lemma 21 and Proposition 22).
The relation of the character ∆µ[E,a] to symplectic reduction may be described as follows. In the case when the moment map µ : M → t∗ is associated to a positive line bundle (and hence Ω is a nondegenerate 2- form), any element γin an alcoveais a regular value ofµ, and the torus T acts with finite stabilizers on µ−1(γ). Then the setµ−1(γ)/T is an orbifold, which does not change if γremains ina, and thus we can denote it by Ma. The bundleEdescends to an orbifold bundleEa on Ma, and each character λallows us to twistEa by the associated line bundle Lλ = µ−1(γ)×TCλover Ma. We then consider the integer dim(Q(Ma,Ea ⊗ Lλ)), which, according to the Atiyah-Singer index formula, is a polynomial function of λif Ma is smooth. In the general case, when Ma is an orbifold, the index formula for orbifolds ([1], see also [22]) implies that the functionλ→dim(Q(Ma,Ea⊗
Lλ)) is a quasi-polynomial. In this setup, the character ∆µ[E,a] appears as the generating function of this quasi-polynomial:
∆µ[E,a]= X
λ
dim Q(M,E ⊗Lλ) eλ. We will not use this interpretation in what follows.
Example 19. We return to Example 7, with µ is associated to the dual of the tautological bundle over P1(C). Then there are 3 alcoves
a1:=]− ∞,−1[; a2:=]−1,1[; a3:=]1,∞[
ForEthe trivial bundle, we have
∆µ[E,a1]=0; ∆µ[E,a2](n)= 1; ∆µ[E,a3]= 0.
Example 20. We return to Example 8, with µassociated to the line bun- dle L. The straight lines cut the plane into alcoves. The support of the multiplicity functionFχLis the highlighted hexagon.
Since∆µ[E,a] is a linear combination of quasi-polynomial characters, it is itself quasi-polynomial. The following extension of this fact holds.
Lemma 21. The function (λ,k) 7→ F∆µ[E ⊗ Lk,a](λ) is quasi-polynomial on the latticeΛ×Z.
Proof. Indeed, we have
∆µ[E ⊗ Lk,a]= X
p∈F
εpekµ(p)δ[Φp↑X,ap].
The statement now follows from the fact that for a formal character θ ∈ R(T ), andˆ λ, ν∈Λ, we haveFekµθ(λ)=Fθ(λ−kµ).
For small k, in particular for k = 0, ∆µ[E ⊗ Lk,a] does not have any direct relationship with χE⊗Lk. Sinceεp for p ∈ F is the trace of a finite- dimensional representation, its multiplicity function Fεp, p ∈ F has finite support inΛ. For large k, we have the following statement.
Proposition 22. Letbbe a compact subset of an alcovea. Then there exist a positive integer K such that for every k >K andλ∈kb∩Λ, the equality (13) F∆µ[E ⊗ Lk,a](λ)= FχE⊗Lk(λ)
holds.
Proof. Recall the notation Ψp introduced for the list of weights of Ep at p∈F. According to (10), we have
FχE⊗Lk(λ)=X
p∈F
X
η∈Ψp
FΘ[Φp↑X](λ−η−kµ(p)), while, by Lemma 21,
F∆µ[E ⊗ Lk,a](λ)=X
p∈F
X
η∈Ψp
Fδ[Φp↑X,T(ap)](λ−η−kµ(p)).
Hence, by the definition of the quasi-polynomial characterδgiven in Lemma 13, these two expressions coincide as long as for each p ∈ F and η ∈ Ψp, we have λ−η−kµ(p) ∈ T(ap). Since topes are invariant under rescaling, andap =a−µ(p), we can conclude that (13) holds if
(14) λ
k − η
k ∈a for eachη∈ ∪p∈FΨp.
As the set ∪p∈FΨp is finite, for large enough k, we will haveb−η/k ∈a for everyηfrom this set. Hence (14) holds for large enough k, uniformly in λ∈kb∩Λ. This completes the proof.
Corollary 23. The quasi-polynomial character∆µ[E,a] does not depend on the choice of the polarizing vector X.
Indeed, assume that the subsetb ⊂ ain Proposition 22 contains an open set U, and observe that the set of pairs (ξ,t) satisfying the condition t >
K, ξ ∈tbwith K as in Proposition 22 contains a translate of the open cone {(rγ,r)|r > 0, γ ∈ U} ⊂ t∗× R. According to Lemma 21, F∆µ[E,a] is quasi-polynomial onΛ×Z, and hence the right hand side of (13) completely determines the left hand side. SinceFχE⊗Lk does not depend on X, we can conclude that neither doesF∆µ[E,a].
In what follows, we will need the extension of the definition of ∆µ[E,a]
to the case when the generic stabilizer of the T action is not finite.
Definition 24. Suppose the Lie group G with Lie algebragacts on a mani- fold M. Then for a subset C ⊂ M we denote by
gC ={X∈g; V X vanishes on C}
the stabilizer subalgebra ofg, and by GC the connected subgroup of G with Lie algebragC.
In particular, TM is the connected component of the generic stabilizer of M containing the identity element, and tM ⊂ t is the Lie algebra of TM. Then for every p∈F, the weightsΦpspan the annihilatort⊥M ⊂ t∗.
Clearly, the group TM acts on each of the fibers Eq, q ∈ M, and since M is connected, this representation does not depend on q. In particular, for two fixed points p and q in F, the weights µ(p) andµ(q) of T differ by an element oft⊥M. Thus the affine-linear subspace
(15) AM = µ(p)+t⊥M
of t∗ does not depend on p ∈ F. Note that according to equation (6), the imageµ(M) is contained in AM.
Now we can repeat the definitions (11) and (12) witht∗replaced by t⊥M. More precisely, we consider the open set in AM consisting of those elements γfor whichγp ∈t⊥MisΦp-regular for any p∈F. An alcoveais a connected component of this open set. For an alcove a, we denote byT(ap) the Φp- tope int⊥M containingap. The formal character∆µ[E,a] may be defined by equation (12). The multiplicity function F∆µ[E,a] is then supported on a finite number of translates (by vectors from the set∪p∈F{Ψp}) oft⊥M∩Λ, and it is quasi-polynomial on each translate.
Denote byCλ the trivial line bundle over M endowed with the action eλ of T . As an immediate consequence of the decomposition
(16) E= M
λ∈Λ/Λ∩t⊥M
Cλ⊗(E ⊗C−λ)TM, one has the explicit formula
(17) ∆µ[E,a]= X
λ∈Λ/Λ∩t⊥M
eλ∆µ[(E ⊗C−λ)TM,a],
where the sum is understood as taken over any system of representatives of the quotient. Note that this formula expresses the formal T -character
∆µ[E,a] through quasi-polynomial characters for the torus T′ = T/TM, which acts on M with generic finite stabilizer.
We have the following simple corollary of (17):
Lemma 25. If for some λ ∈ Λ, the multiplicity F∆µ[E,a](λ) is not zero, then the restriction of λto tM is a weight of the representation of TM on a fiber ofE.
We end this section with a comment on the situation, where the affine space AM given by equation (15) is linear, i.e. passes through the origin.
This is equivalent to the condition that TM acts trivially on the fibers ofL, i.e. Lis a T/TM-line bundle.
Lemma 26. LetEbe a T -bundle, andLbe a T/TM-line bundle on M. Then k 7→ F∆µ[E ⊗ Lk,a](0) is a quasi-polynomial function of k.
Proof. Applying (17) to the bundle E ⊗ Lk, and using the condition on L, we obtain the equality
∆µ[E ⊗ Lk,a](0)= ∆µ[ETM ⊗ Lk,a](0).
SinceETM is a T/TM-equivariant vector bundle, we can replace T by T/TM. According to Lemma 21,F∆µ[ETM⊗Lk,a](λ) is quasi-polynomial in (λ,k)∈ (t⊥M∩Λ)×Z, and henceF∆µ[ETM⊗ Lk,a](0) is a quasi-polynomial function
of k.
3. Decomposition of partition functions
3.1. The decomposition formula. In this section, we prove a decomposi- tion formula for the generating function Θ[Φ↑X] of the partition function introduced in (9). This formula is due to Paradan [18], and it will serve as the combinatorial engine of our proof of Theorem 6.
Definition 27. Given a list Φ of weights in Λ ⊂ t∗, introduce the set of Φ-rational subspaces
RS(Φ)={S ⊂t∗linear; Φ∩S spans S}.
This is the set of linear subspaces oft∗spanned by some subset ofΦ:
Remark 28. 1. Note that{0} ∈ RS(Φ), andt∗∈ RS(Φ) ifΦspanst∗. 2. Comparing this definition to Definition 12, we see that all subspaces S ∈ RS(Φ), except for S = t∗, consist of nonregular elements.
Now, fix a positive definite scalar product (·,·) on t∗, and denote by γ 7→ γ∗ the induced linear bijection between t∗ and t. For each rational subspace S ∈ RS(Φ) and γ ∈ t∗, introduce the notationγS for the orthog- onal projection of γ onto S , and YS,γ for the vector (γS −γ)∗ ∈ t(see the diagram below).
0
γ
γS S YS,γ
Finally, recall from Lemma 13 that, on aΦ-topeT, the partition function FΘ[Φ↑X] coincides with a quasi-polynomialFδ[Φ↑X,T] : Λ → Z. It is thus natural to compare the two functions at all points ofΛ. As we will see, the difference may be expressed as a sum of (convolution) products of partition functions and quasi-polynomials coming from lower-dimensional systems.
Proposition 29. (Paradan) Let Φbe a list of vectors in Λ, and let X be a polarizing vector forΦ(see the definition ofΘin (9)). Assume thatγ ∈ t∗ is such that for every S ∈ RS(Φ), the projectionγS ∈S is (Φ∩S )-regular, while the orthogonal component YS,γ is polarizing forΦ\S . Then
(18) Θ[Φ↑X]= X
S∈RS(Φ)
Θ[Φ\S↑YS,γ] · δ[Φ∩S↑X,T(γS)].
Observe that if−γis in the dual cone to the cone generated byΦ+∪ −Φ−, then all the terms but the one corresponding to S = {0}vanish, and hence, in this case, the identity (18) is tautological.
Example 30. Lett∗= Rα, Λ =Zα,Φ:= [α] and set X ∈tto be the vector satisfyinghα,Xi=1. Then
Θ[Φ↑X]=
∞
X
k=0
ekα. The setRS(Φ) has two elements: S = {0}and S =t∗.
If we let γ = tαfor some t > 0, then on the right hand side of (18) we have
• δ[Φ↑X,T(γS)]= P
k∈Zekαfor S = t∗, and
• Θ[Φ↑YS,γ]=−P
k>0e−kα, for S ={0}.
Then formula (18) reads:
Θ[Φ↑X]= δ[Φ↑X,T(γS)]+ Θ[Φ↑YS,γ].
Proof of Proposition 29. Replacing γ by its orthogonal projection on the subspace generated byΦ, we may assume that V is spanned byΦ.
Observe that for each term on the right hand side of (18), the Fourier transform restricted to a tope of Φis quasi-polynomial. We show that the Fourier transforms of the two sides of (18) coincide.
We start by showing that the Fourier coefficients of the two sides coincide on the tope T(γ). The term corresponding to S = t∗ is δ[Φ ↑ X,T(γ)], whose Fourier coefficients coincide with those of Θ[Φ ↑ X] on the tope T(γ). For any S ∈ RS(Φ) different from t∗, by construction, the Fourier transform of the corresponding term Θ[Φ\S ↑YS,γ]·δ[Φ∩S ↑X,T(γS)]
is a function on Λsupported on the subset {λ; hλ,YS,γi ≥ 0}(cf. Lemma 9). Since hγ,YS,γi = −|γS −γ|2 < 0, we see that this function vanishes on a conic neighborhood of the half line R+γ. This implies that the Fourier coefficients of all these terms vanish on all of T(γ), and thus indeed, the Fourier coefficients of the two sides of (18) coincide onT(γ).
To extend the equality of Fourier coefficients to the rest of Λ, we use induction on the number of elements inΦ. IfΦis empty, then both sides are equal to 1. Now pick an elementφ∈Φ, and considerΦ′ = Φ− {φ}(cf. the beginning of§2 for our conventions). Clearly (1−eφ)·Θ[Φ↑X]= Θ[Φ′↑X].
If we restrict the Fourier transform of this equation to a topeT, we obtain (1−eφ)δ[Φ↑X,T)]= δ[Φ′↑X,T′]
ifΦ′generates V andT′is the tope ofΦcontainingT, while (1−eφ)δ[Φ↑X,T)]=0
ifΦ′does not generate V.
We multiply both sides of (18) by (1−eφ), and compare the results. On the left hand side, we end up with Θ[Φ′↑X]. For a term on the right hand side corresponding to S ∈ RS(Φ), we separate 3 cases:
1. φ<S : In this case, S ∈ RS(Φ′), Φ∩S = Φ′∩S and (1−eφ) · Θ[Φ\S↑YS,γ]= Θ[Φ′\S↑YS,γ].
Thus, after multiplication by (1−eφ), we end up with the term (19) Θ[Φ′\S↑YS,γ] · δ[Φ′∩S↑X,T(γS)].
2. φ∈S , and S ∈ RS(Φ′): In this caseΦ\S = Φ′\S while (Φ∩S )− {φ}= Φ′∩S , which implies that
(1−eφ)δ[Φ∩S↑X,T(γS)]=δ[Φ′∩S↑X,T′(γS)].
Thus we end up with the term (19) again.
3. φ∈S , and S <RS(Φ′): In this case,
(1−eφ)δ[Φ∩S↑X,T(γS)]=0.
Thus multiplying the right hand side of (18) by (1−eφ) has the effect of replacing Φby Φ′. Using the inductive assumption, we can conclude that after multiplying both sides of (18) by (1−eφ) for anyφ ∈ Φ, we obtain an identity. As Φ spanst∗, this implies that the Fourier coefficients of the difference of the two sides of (18) form a periodic function with respect to the sublattice of finite index in Λ generated byΦ. Since we also know that these coefficients vanish on T(γ), they must vanish on all ofΛ. This
completes the proof.
3.2. Paradan’s decomposition of a character. In this paragraph, we sub- stitute the decomposition (18) into formula (10), and then find a geometric interpretation of the resulting expression.
Note that while performing this substitution, we have the freedom of varying the vectorγin (18) depending on the fixed point p. We take advan- tage of this possibility, by shiftingγ byµ(p) for p∈ F, and thus replacing γ by the vectorγp= γ−µ(p).
With this choice, we obtain (20) χE = X
p∈F
X
S∈RS(Φp)
εp·Θ[Φp\S↑YS,γp]·δ[Φp∩S↑X,T(γSp)].
Now we turn to the geometric meaning of this sum. Recall that each X ∈t defines a vector field V X on M; this vector field vanishes on the fixed point set F.
Definition 31. For p ∈F and S ∈ RS(Φp), denote by C(p,S ) the connected component of the set
MS⊥ = {m∈M|V X(m)=0 for every X ∈S⊥}
which contains p. Let FPC(M) stand for the set of all connected fixed point components C(p,S ) obtained this way:
FPC(M)={C(p,S )|p∈F, S ∈ RS(Φp)}.
Observe that the set MS⊥ is also the fixed point set of the subtorus of T with Lie algebra S⊥.
Lemma 32. For p ∈ F and S ∈ RS(Φp), consider C(p,S ) ∈ FPC(M).
Then
(1) the set C(p,S ) is a smooth, T -invariant submanifold of M,
(2) the Lie algebra of the generic stabilizertC(p,S ) of C(p,S ) intis ex- actly S⊥, and
(3) for q∈C(p,S )∩F, we have S ∈ RS(Φq) and C(q,S )= C(p,S ).
Proof. Recall the basic fact that a torus action on a manifold can be lin- earized near a fixed point. It is sufficient then to verify these statements in a linear model. Let V be a vector space with a linear T -action, with a list of weights Ψ, and let S ∈ RS(Ψ). Then the set of points stabilized by the torus exp(S⊥) is the direct sum of all eigenspaces of V whose weight is in S . This easily implies all three statements.
It follows then that there is a one-to-one correspondence
{(p,S )|p∈F, S ∈ RS(Φp)} ↔ {(p,C)|C∈FPC(M), p∈C∩F}, and hence we can regroup the terms of the sum in (20) according to the fixed point component C to which it corresponds. To write down the resulting formula, we need to introduce new notation for the vectors γS and YS,γ in terms of the component C.
The manifold C inherits a T -invariant almost complex structure, and the set of weights of the fiber of the complex vector bundle ¯TJC at p ∈ C∩F isΦp∩t⊥C. Consider the T -equivariant line bundleLC on C obtained as the restriction ofLto C, and recall from (15) the definition of the affine space AC.
Using our scalar product, we decomposet∗=tC⊕t⊥C. If p and q∈C∩F, thenµ(p)−µ(q) belongs totC⊥, thus the projection (µ(p)−γ)t(C) ofµ(p)−γ to t(C) does not depend on p ∈C∩F.This is the polarizing vector in any term of the sum in (20) associated to the component C.
Using this observation, we introduce the following notations.
Definition 33. From now on, we will writeγC for the orthogonal projection ofγon the affine space AC. We introduce the notation
YC def= (µ(p)−γ)t(C) = γC−γ
for the polarizing vector int(C), omitting the dependence onγ.
µ(p) µ(q)
γ
AC γC
YC 0
Regrouping the terms of (20), and using these notations, we obtain the formula
(21) χE = X
C∈FPC(M)
X
p∈C∩F
εp·Θ[Φp\tC⊥↑YC]·δ[Φp∩t⊥C↑X,T(γC−µ(p))].
Recall thata(γC) stands for the alcove containingγC. We claim that the term
∆C := X
p∈C∩F
εp·Θ[Φp\tC⊥↑YC]·δ[Φp∩tC⊥↑X,T(γC−µ(p))]
has the form ∆µ[ ˜EC,a(γC)] for an infinite-dimensional bundle ˜EC over C, whose fiber at the fixed point p has characterεp·Θ[Φp\tC⊥↑YC].
This bundle may be constructed as follows. The bundle KC =T¯JM/T¯JC is a T -equivariant complex bundle on C, whose TC-weights are constant along C; denote the set of these TC-weights byΦC. These weights thus may be obtained by restricting Φp\t⊥C to tC for any p ∈ C ∩F. Now, we can splitΦC into the disjoint union ΦC+∪Φ−C according to the sign of the value of the weights on the polarizing vector YC ∈tC, and we obtain a direct sum decomposition of KC in KC = KC+⊕KC−, where KC+ and KC− are the subspaces generated by eigenvectors of TC with weights from ΦC+ andΦ−C, respectively.
Then, recalling the definition of the formal character Θfrom (9), we ob- tain the following statement.
Lemma 34. Define the infinite-dimensional T -equivariant virtual bundle (22) S•(KC↑YC)= (−1)rank KC−det(KC−∗)⊗
∞
M
m=0
S[m](KC−∗ ⊕KC+) over C, where S[m](V) stands for the mth symmetric tensor product of the vector space V, and det(V) for its top exterior product. The fibers of this bundle over points of C form a TC-representation with finite multiplicities, moreover, for p ∈ C ∩ F, the T -character of the fiber S•(KC ↑ YC)p is Θ[Φp\tC⊥↑YC].
Denote by EC the restriction of E to C. Then the combination of the fixed point formula with Paradan’s decomposition leads to the following statement.
Proposition 35 (Paradan). Letγbe a generic point int∗. Then we have the following equality in ˆR(T ):
(23) χE = X
C∈FPC(M)
∆µ[EC ⊗ S•(KC↑YC),a(γC)].
If C consists of a single fixed point p ∈ F, then the corresponding term is εp ·Θ[Φp↑(µ(p)−γ)∗]. It is reassuring to compare this to (10), which contains a similar Θ-term functions but oriented differently. According to Lemma 9 (1), the two terms, interpreted as generalized functions, coincide on the set{t∈T ; tφ, 1∀φ∈Φp}.
We also observe that these are the only terms which correspond to smooth functions on some open sets of T ; all the other terms correspond to gener- alized functions supported in positive-codimensional subtori of T . One can think of this formula then as a refinement of the Atiyah- Bott formula (8).
Remark 36.
(1) The alcovea(γC) is an open set in AC =µ(p)+tC⊥when p∈C.
(2) The moment mapµ: M → t∗of the line bundleµsatisfies equation (6), and this implies thatµ(C) is contained in the affine space AC. Moreover, ifLis a positive line bundle as in Definition 3, thenµ(C) is the convex hull of the finite set{µ(p); p∈C∩F}.
(3) Proposition 52 implies the vanishing of those the terms of the sum in (23) which corresponding to a fixed point component C satisfying a(γC)1 µ(C).
(4) We note that the terms of the sum (23) are not quasi-polynomial.
Rather, the term corresponding to the fixed point component C is quasi-polynomial along each element an infinite set of linear trans- lates oft⊥C ⊂ t∗.
(5) When C = M, then the term of the sum (23) reduces to∆µ[E,a(γ)], which is an actual quasi-polynomial character (cf. Lemma 21 and Proposition 22).
Example 37. In Example 7, the elements of FPC(M) are easy to list:
FPC(M) := {M,p+,p−}.
If we considerγ = 0, the corresponding decomposition ofχE reads as fol- lows:
t−4+t−2+1+t2+t4 =X
k∈Z
t2k−t6
∞
X
k=0
t2k−t−6
∞
X
k=0
t−2k.
Example 38. In Example 8 (see also Example 20), the set of fixed point components FPC(M) consists of the following elements:
• the complex 3-dimensional manifold M itself,
