Vector partition function and generalized Dahmen-Micchelli spaces

VECTOR P AR TITION FUNCTIONS AND GENERALIZED D AHMEN-MICCHELLI SP A CES C. DE CONCINI, C. PR OCESI. M. VERGNE Abstract. This is the first of a series of papers on partition functions and the index theory of transv ersally elliptic op erators. In this pap er w e only discuss algebraic and com binatorial issues related to partition functions. The applications to index theory will appear in [4]. Here we introduce a space of functions on a lattice which general- izes the space of quasi–p olynomials satisfying the difference equations asso ciated to co circuits of a sequence of vectors X . This space F ( X ) con tains the partition function P X . W e pro v e a ”localization formula” for an y f in F ( X ). In particular, this implies that the partition function P X is a quasi–p olynomial on the sets c − B ( X ) where c is a big cell and B ( X ) is the zonotop e generated by the vectors in X . 1. Introduction Recall some notions. W e take a lattice Γ in a vector space V and X := [ a 1 , . . . , a m ] a list of non zero elemen ts of Γ, spanning V as vector space. If X generates a p oin ted cone C ( X ), the p artition function P X ( γ ) counts the n um b er of w a ys in whic h a v ector γ ∈ Γ can b e written as P m i =1 k i a i with k i non negative integers. A quasi–p olynomial is a function on Γ which coincides with a p olyno- mial on eac h coset of some sublattice of finite index in Γ. A theorem [6],[10], generalizing the theory of the Ehrhart polynomials [5], sho ws that P X ( γ ) is a quasi–p olynomial on certain regions c − B ( X ), where B ( X ) := { P m i =1 t i a i , 0 ≤ t i ≤ 1 } is the zonotop e generated by X while c denotes a big c el l , that is a connected comp onen t of the complement in V of the singular ve ctors which are formed by the union of all cones C ( Y ) for all the sublists Y of X which do not span V . The complement of C ( X ) is a big cell. The other cells are inside C ( X ) and are conv ex. The quasi–p olynomials describing the partition function b elong to a re- mark able finite dimensional space introduced and describ ed b y Dahmen– Micc helli [6] and which in this pap er will b e denoted b y D M ( X ). This is the space of solutions of a system of difference equations. In order to de- scrib e it, let us call a subspace r of V r ational if r is the span of a sublist of X . W e need to recall that a c o cir cuit Y in X is a sublist of X suc h that X \ Y do es not span V and Y is minimal with this prop ert y . Th us The first tw o authors are partially supp orted by the Cofin 40 %, MIUR. 1 2 C. DE CONCINI, C. PROCESI. M. VERGNE Y is of the form Y = X \ H where H is a rational h yp erplane. Giv en a ∈ Γ, the differ enc e op er ator ∇ a is the op erator on functions defined by ∇ a ( f )( b ) := f ( b ) − f ( b − a ). F or a list Y of v ectors, w e set ∇ Y := Q a ∈ Y ∇ a . W e then define: D M ( X ) := { f | ∇ Y f = 0 , for ev ery co circuit Y in X } . It is easy to see that D M ( X ) is finite dimensional and consists of quasi– p olynomial functions (cf. [3]). In this article, w e in tro duce F ( X ) := { f | ∇ X \ r f is supp orted on r for every rational subspace r } . Clearly P X as w ell as D M ( X ) are con tained in F ( X ). The space F ( X ) is of in terest ev en when X do es not span a p ointed cone and o ccurs in studying indices of transversally elliptic op erators on a vector space. The main result of this article is a “localization formula” for an elemen t f in F ( X ). In particular, giv en a c hamber c , our localization form ula allo ws us to write explicitly the partition function P X as a sum of a quasi–p olynomial function P c X ∈ D M ( X ) and of other functions f r ∈ F ( X ) supp orted outside c − B ( X ). This allows us to give a short pro of of the quasi–p olynomialit y of P X on the regions c − B ( X ). F urthermore this decomp osition implies P aradan’s wall crossing form ulae [8] for the quasi–p olynomials P c X . Our ap- proac h is strongly inspired b y P aradan’s lo calization form ula in Hamiltonian geometry , but our methods here are elemen tary . W e wish to thank Michel Duflo and Paul-Emile P aradan for several suggestions and corrections. 2. Special functions 2.1. Basic notations. Let Γ b e a lattice and E = Z , Q , R , C . Consider the space C E [Γ] of E v alued functions on Γ. When E = Z , w e shall simply write C [Γ]. W e display suc h a function f ( γ ) also as a formal series Θ( f ) := X γ ∈ Γ f ( γ ) e γ . Of course, under suitable conv ergence conditions, the series P γ ∈ Γ f ( γ ) e γ is a function on the torus G whose c haracter group is Γ, and it is the Laplace– F ourier transform of f . In fact the functions that w e shall study are F ourier co efficien ts of some imp ortan t generalized functions on G . This fact and the sev eral implications for the index theory of transv ersally elliptic op erators will b e the sub ject of a subsequent pap er [4]. The space C E [Γ] is in an obvious wa y a mo dule o ver the group algebra E [Γ], m ultiplication b y e λ on the series Θ( f ) corresp onding to the translation op erator τ λ defined by ( τ λ f )( γ ) := f ( γ − λ ) on the function f . Th us m ultiplication by 1 − e λ corresp onds to the difference op erator ∇ λ . VECTOR P AR TITION FUNCTIONS 3 W e denote by δ 0 the function on Γ iden tically equal to 0 on Γ, except for δ 0 (0) = 1. Remark that the product of tw o formal series Θ( f 1 )Θ( f 2 ), whenev er it is defined, corresponds to c onvolution f 1 ∗ f 2 of the functions f 1 and f 2 . The function δ 0 is the unit elemen t. R emark 2.2 . Notice that, for a difference op erator ∇ a acting on a conv olu- tion, we hav e: ∇ a ( f 1 ∗ f 2 ) = ∇ a ( f 1 ) ∗ f 2 = f 1 ∗ ∇ a ( f 2 ) . Let no w X := [ a 1 , . . . , a m ] b e a list of non zero elements of Γ and let V := Γ ⊗ Z R b e the real v ector space generated b y Γ. W e assume that X generates the vector space V , but w e do not necessarily assume that X generates a p ointed cone in V . If X generates a p oin ted cone, then we can define Θ X = Y a ∈ X ∞ X k =0 e ka . W e write Θ X = X γ ∈ Γ P X ( γ ) e γ where P X ∈ C [Γ] is the partition function. “ Morally” the series Θ X is equal to Q a ∈ X 1 1 − e a , but 1 1 − e a has to b e understo o d as the geometric series expansion P ∞ k =0 e ka . R emark 2.3 . W e easily see that the partition function satisfies the difference equation ∇ X P X = δ 0 . Clearly this equation has infinitely man y solutions. The fact that P X is uniquely determined by the recursion expressed by this equation comes from the further prop erty of this solution of having supp ort in the c one C ( X ). W e shall see other functions of the same type app earing in this pap er. Definition 2.4. i) A subspace of V generated b y a subset of the ele- men ts of X will b e called rational (relative to X ) . ii) Given a rational subspace r , we denote b y C [Γ , r ] the set of elemen ts in C [Γ] which ha v e supp ort in the lattice Γ ∩ r . iii) Given a rational subspace r , we set ∇ X \ r := Q a ∈ X \ r ∇ a . With these notations, the space D M E ( X ) defined by Dahmen–Micc helli is formed by the set of functions f ∈ C E [Γ] satisfying the system of difference equations ∇ X \ r f = 0 as r v aries among all prop er rational subspaces relative to X . It is easy to see that D M E ( X ) consists of quasi–p olynomials. It follows from their theory (see also [3]) that, for each E , the space D M E ( X ) is a free E − mo dule of dimension δ ( X ), the volume of the zonotop e B ( X ). In particular D M E ( X ) = E ⊗ Z D M Z ( X ) for all E . Therefore from no w on we shall work directly o ver Z and drop the subscript E . 4 C. DE CONCINI, C. PROCESI. M. VERGNE The smallest sub–lattice of Γ for whic h eac h function of D M ( X ) is a p olynomial on its cosets is the intersection of all the sublattices of Γ gener- ated by all the bases of V that one can extract from X (the least common m ultiple). Giv en a rational subspace r , we will iden tify the space C [Γ ∩ r ] with the subspace C [Γ , r ] of C [Γ] by extending the functions with 0 outside r . 2.5. The sp ecial functions P F r X \ r . Giv en a rational subspace r , X \ r de- fines a h yp erplane arrangemen t in the space r ⊥ ⊂ U orthogonal to r . T ak e an open face F r in r ⊥ with respect to this h yp erplane arrangement. W e shall call such a face a regular face for X \ r . If F r is a regular face, then − F r is also a regular face. By definition a v ector u ∈ r ⊥ and such that h u, a i 6 = 0 for all a ∈ X \ r lies in a unique suc h regular face F r and u will b e called a regular vector for X \ r . Giv en a regular face F r for X \ r , w e divide X \ r in to tw o parts A, B , according to whether they take p ositive or negativ e v alues on our face. W e denote the cone C ( A, − B ), generated by the list [ A, − B ], b y C ( F r , X \ r ). If w e take u ∈ F r , C ( F r , X \ r ) is con tained in the closed half space of v ectors v where u is non negative. W e are going to consider the series Θ F r X \ r whic h is c haracterized by the follo wing tw o prop erties: Lemma 2.6. Ther e exists a unique element Θ F r X \ r = X γ P F r X \ r ( γ ) e γ such that i) Q a ∈ X \ r (1 − e a )Θ F r X \ r = 1 , e quivalently ∇ X \ r P F r X \ r = δ 0 . ii) P F r X \ r is supp orte d in − P b ∈ B b + C ( F r , X \ r ) . Pr o of. Set: (1) Θ F r X \ r = ( − 1) | B | e − P b ∈ B b Y a ∈ A ( ∞ X k =0 e ka ) Y b ∈ B ( ∞ X k =0 e − kb ) . It is easily seen that this element satisfies the tw o prop erties and is unique.  In particular, if r = V , w e ha v e F V = { 0 } and P { 0 } X = δ 0 . Morally , Θ F r X \ r = Q a ∈ X \ r 1 1 − e a = Q a ∈ A 1 1 − e a Q b ∈ B − e − b 1 − e − b . W e indeed need to rev erse the sign of some of the vectors in X \ r in order that the conv olution pro duct of the corresp onding geometric series makes sense. Although a function f ∈ C [Γ , r ] ma y ha ve infinite supp ort, w e easily see that the con volution P F r X \ r ∗ f is w ell defined. In fact w e claim that, given an y γ ∈ Γ, we can write γ = λ + µ with µ ∈ r ∩ Γ , and λ ∈ ( − P b ∈ B b + VECTOR P AR TITION FUNCTIONS 5 C ( A, − B )) ∩ Γ only in finitely many w a ys. T o see this, tak e u ∈ F r . Then h u | γ i = h u | λ i and λ = P a ∈ A k a a + P b ∈ B h b ( − b ) with k a ≥ 0 , h b ≥ 1. Th us the equalit y h u | γ i = P a ∈ A k a h u | a i + P b ∈ B h b h u | − b i yields that the v ector λ is in a b ounded set, intersecting the lattice Γ in a finite set. Cho ose tw o rational spaces r , t and an regular face F r for X \ r in r ⊥ . The image of F r mo dulo t ⊥ is a regular face for ( X ∩ t ) \ r . T o simplify notations, we still denote this face b y F r . W e hav e: Prop osition 2.7. i) ∇ ( X \ t ) \ r P F r X \ r = P F r ( X \ r ) ∩ t . ii) F or g ∈ C [Γ ∩ r ] : (2) ∇ X \ t ( P F r X \ r ∗ g ) = P F r ( X \ r ) ∩ t ∗ ( ∇ ( X ∩ r ) \ ( t ∩ r ) g ) . Pr o of. i) F rom Equation (1), we see that the series associated to the function ∇ ( X \ t ) \ r P F r X \ r equals Θ F r ( X \ r ) ∩ t = ( − 1) | B ∩ t | e − P b ∈ B ∩ t b Y a ∈ A ∩ t ( ∞ X k =0 e ka ) Y b ∈ B ∩ t ( ∞ X k =0 e − kb ) . ii) Let g ∈ C [Γ ∩ r ]. T ake any rational subspace t , we hav e that ∇ X \ t = ∇ ( X ∩ r ) \ ( t ∩ r ) ∇ ( X \ t ) \ r , thus ∇ X \ t ( P F r X \ r ∗ g ) = ( ∇ ( X \ t ) \ r P F r X \ r ) ∗ ( ∇ ( X ∩ r ) \ ( t ∩ r ) g ) . As ∇ ( X \ t ) \ r P F r X \ r = P F r ( X \ r ) ∩ t from part i) , we obtain F ormula (2), which is the mother of all other form ulae of this article.  In particular, for r = t , F ormula (2) implies the following. Prop osition 2.8. If f ∈ C [Γ ∩ r ] , we have f = ∇ X \ r ( P F r X \ r ∗ f ) . 3. A remarkable sp a ce 3.1. The space F ( X ) . W e let S X denote the set of all rational subspaces relativ e to X . Definition 3.2. W e define the space of interest for this article by: (3) F ( X ) := { f ∈ C [Γ] | ∇ X \ r f ∈ C [Γ , r ] , for all r ∈ S X } . One of the equations (corresp onding to r = { 0 } ) that must satisfy Θ( f ) when f ∈ F ( X ) is the relation Q a ∈ X (1 − e a )Θ( f ) = c , where c is a constan t. This equation was the motiv ation for in tro ducing the space F ( X ). Indeed the first imp ortant fact on this space is the follo wing: Lemma 3.3. i) If F is a r e gular fac e for X , then P F X lies in F ( X ) . ii) The sp ac e DM ( X ) is c ontaine d in F ( X ) . 6 C. DE CONCINI, C. PROCESI. M. VERGNE Pr o of. i) Indeed, ∇ X \ r P F X = P F X ∩ r ∈ C [Γ , r ]. ii) Is clear from the definitions.  In particular, if X generates a p oin ted cone, then the partition function P X lies in F ( X ). Example 3.4. Let us give a simple example. Let Γ = Z ω and X = [2 ω , − ω ]. Then it is easy to see that F ( X ) is a free Z mo dule of dimension 4, with corresp onding basis θ 1 = X n ∈ Z e nω , θ 2 = X n ∈ Z ne nω , θ 3 = X n ∈ Z ( n 2 + 1 − ( − 1) n 4 ) e nω , θ 4 = X n ≥ 0 ( n 2 + 1 − ( − 1) n 4 ) e nω . Here θ 1 , θ 2 , θ 3 are a Z basis of D M ( X ). In fact, there is a m uc h more precise statemen t of whic h Lemma 3.3 is a v ery sp ecial case and which will b e the ob ject of Theorem 3.8. 3.5. Some prop erties of F ( X ) . Let r b e a rational subspace and F r b e a regular face for X \ r . Prop osition 3.6. i) The map g 7→ P F r X \ r ∗ g gives an inje ction fr om F ( X ∩ r ) to F ( X ) . Mor e over (4) ∇ X \ r ( P F r X \ r ∗ g ) = g , ∀ g ∈ F ( X ∩ r ) . ii) ∇ X \ r maps F ( X ) surje ctively to F ( X ∩ r ) . iii) If g ∈ D M ( X ∩ r ) , then ∇ X \ t ( P F r X \ r ∗ g ) = 0 for any r ational subsp ac e t such that t ∩ r 6 = r . Pr o of. i) If g ∈ F ( X ∩ r ), then ∇ ( X ∩ r ) \ ( t ∩ r ) g ∈ C [Γ ∩ t ∩ r ], hence F orm ula (2) in Proposition 2.7 shows that ∇ X \ t ( P F r X \ r ∗ g ) ∈ C [Γ , t ], so that P F r X \ r ∗ g ∈ F ( X ) as desired. F orm ula (4) follows from the fact that ∇ X \ r P F r X \ r = δ 0 . ii) If f ∈ F ( X ), we hav e ∇ X \ r f ∈ F ( X ∩ r ). In fact take a rational subspace t of r , we hav e that ∇ ( X ∩ r ) \ t ∇ X \ r f = ∇ X \ t f ∈ C [Γ ∩ t ]. The fact that ∇ X \ r is surjective is a consequence of F ormula (4). iii) Similarly , if g ∈ D M ( X ∩ r ), F ormula (2) in Prop osition 2.7 implies the third assertion of our prop osition.  Prop osition 3.6 allows us to asso ciate, to a rational space r and a regular face F r for X \ r , the op erator Π r,F r X \ r : f 7→ P F r X \ r ∗ ( ∇ X \ r f ) on F ( X ). F rom F orm ula (4), it follows that the op erator Π r,F r X , on F ( X ), is a pro jector with image P F r X \ r ∗ F ( X ∩ r ) . VECTOR P AR TITION FUNCTIONS 7 3.7. The main theorem. Cho ose, for ev ery rational space r , a regular face F r for X \ r . The following theorem is the main theorem of this section. Theorem 3.8. With the pr evious choic es, we have: (5) F ( X ) = ⊕ r ∈ S X P F r X \ r ∗ D M ( X ∩ r ) . Pr o of. Denote by S ( i ) X the subset of subspaces r ∈ S X of dimension i . Con- sider ∇ X \ r as an op erator on F ( X ) with v alues in C [Γ]. Define the spaces F ( X ) i := ∩ t ∈ S ( i − 1) X k er( ∇ X \ t ) . Notice that b y definition F ( X ) { 0 } = F ( X ), that F ( X ) dim V is the space D M ( X ) and that F ( X ) i +1 ⊆ F ( X ) i . Lemma 3.9. L et r ∈ S ( i ) X . i) The image of ∇ X \ r r estricte d to F ( X ) i is c ontaine d in the sp ac e D M ( X ∩ r ) . ii) If f is in DM ( X ∩ r ) , then P F r X \ r ∗ f ∈ F ( X ) i . Pr o of. i) First we know, by the definition of F ( X ), that ∇ X \ r F ( X ) i is con tained in the space C [Γ , r ]. Let t b e a rational hyperplane of r , so that t is of dimension i − 1. By construction, w e ha v e that for every f ∈ F ( X ) i 0 = Y a ∈ X \ t ∇ a f = Y a ∈ ( X ∩ r ) \ t ∇ a ∇ X \ r f . This means that ∇ X \ r f satisfies the difference equations giv en by the cocir- cuits of X ∩ r , that is, it lies in D M ( X ∩ r ). ii) F ollo ws from the third item of Prop osition 3.6.  Consider the map µ i : F ( X ) i → ⊕ r ∈ S ( i ) X D M ( X ∩ r ) giv en b y µ i f := ⊕ r ∈ S ( i ) X ∇ X \ r f and the map P i : ⊕ r ∈ S ( i ) X D M ( X ∩ r ) → F ( X ) i giv en by P i ( ⊕ g r ) := X P F r X \ r ∗ g r . Theorem 3.10. The se quenc e 0 − → F ( X ) i +1 − → F ( X ) i µ i − → ⊕ r ∈ S ( i ) X D M ( X ∩ r ) − → 0 is exact. F urthermor e, the map P i pr ovides a splitting of this exact se quenc e: µ i P i = Id . Pr o of. By definition, F ( X ) i +1 is the kernel of µ i , thus we only need to show that µ i P i = Id. Given r ∈ S ( i ) X and g ∈ D M ( X ∩ r ), by F orm ula (4) w e ha v e ∇ X \ r ( P F r X \ r ∗ g ) = g . If instead we tak e t 6 = r another subspace of S ( i ) X , r ∩ t is a prop er subspace of t . Item iii) of Prop osition 3.6 says that for 8 C. DE CONCINI, C. PROCESI. M. VERGNE g ∈ D M ( X ∩ r ), ∇ X \ t ( P F r X \ r ∗ g ) = 0 . Th us, given a family g r ∈ D M ( X ∩ r ), the function f = P t ∈ S ( i ) X P F t X \ t ∗ g t is such that ∇ X \ r f = g r for all r ∈ S ( i ) X . This prov es our claim that µ i P i = Id.  Putting together these facts, Theorem 3.8 follo ws.  A collection F = ( F r ) of faces F r ⊂ r ⊥ regular for X \ r , indexed by the rational subspaces r ∈ S X will b e called a X –r e gular c ol le ction . Giv en a X -regular collection F , w e can write an elemen t f ∈ F ( X ) as f = X r ∈ S X f r , (Theorem 3.8) with f r ∈ P F r X \ r ∗ D M ( X ∩ r ) . This expression for f will b e called the F decom- p osition of f . In this decomp osition, w e alwa ys ha ve F V = { 0 } , P F V X = δ 0 and the comp onent f V is in D M ( X ) . The space P F r X \ r ∗ D M ( X ∩ r ) will b e referred to as the F r -c omp onent of F ( X ). F rom Lemma 3.10, it follo ws that the op erator Id − P i µ i pro jects F ( X ) i to F ( X ) i +1 with kernel ⊕ r ∈ S ( i ) X P F r X \ r ∗ D M ( X ∩ r ) (this operator dep ends of F ). Thus the ordered pro duct Π F i := (Id − P i − 1 µ i − 1 )(Id − P i − 2 µ i − 2 ) · · · (Id − P 0 µ 0 ) pro jects F ( X ) to F ( X ) i ; therefore, we ha v e Prop osition 3.11. L et F b e a X -r e gular c ol le ction and r a r ational subsp ac e of dimension i . The op er ator P F r = Π r,F r X (Id − P i − 1 µ i − 1 )(Id − P i − 2 µ i − 2 ) · · · (Id − P 0 µ 0 ) = Π r,u r X Π u i is the pr oje ctor of F ( X ) to the F r -c omp onent P F r X \ r ∗ D M ( X ∩ r ) of F ( X ) . In p articular, if dim( V ) = s , the op er ator P V := (Id − P s − 1 µ s − 1 )(Id − P s − 2 µ s − 2 ) · · · (Id − P 0 µ 0 ) is the pr oje ctor F ( X ) → DM ( X ) asso ciate d to the dir e ct sum dec omp osi- tion: F ( X ) = D M ( X ) ⊕  ⊕ r ∈ S X | r 6 = V P F r X \ r ∗ D M ( X ∩ r )  . Let F = ( F r ) be a X –regular collection. If t is a rational subspace and, for eac h r ∈ S X ∩ t , we tak e the image of F r mo dulo t ⊥ w e get a X ∩ t –regular collection. W e still denote it by F in the next prop osition. The pro of of this prop osition is skipp ed, as it is very similar to preceding pro ofs. VECTOR P AR TITION FUNCTIONS 9 Prop osition 3.12. L et t b e a r ational subsp ac e. L et f ∈ F ( X ) and f = P r ∈ S ( X ) f r b e the F dec omp osition of f and ∇ X \ t f = P t ∈ S X ∩ t g r b e the F dec omp osition of ∇ X \ t f , then • ∇ X \ s f r = 0 if r / ∈ S X ∩ t , • ∇ X \ s f r = g r if r ∈ S X ∩ t . R emark 3.13 . It follo ws from the previous theorems and the properties of D M ( X ) that, for e v ery E , w e could define a space F E ( X ) of E v alued functions as in Definition 3.2 and w e ha ve F E ( X ) = E ⊗ Z F ( X ) . 3.14. Lo calization theorem. Definition 3.15. By the w ord top e , w e mean a connected comp onen t of the complement in V of the union of the h yp erplanes generated b y subsets of X . W e in tro duce also B ( X ) := { P m i =1 t i a i , 0 ≤ t i ≤ 1 } the zonotop e gener- ated by X . B ( X ) is a compact con vex p olytope which app ears in sev eral w a ys in the theory and plays a fundamen tal role. W e will show that, for ev ery elemen t f ∈ F ( X ), the function f ( γ ) coin- cides with a quasi–p olynomial on the sets ( τ − B ( X )) ∩ Γ as τ v aries o ver all top es (we simply say f is a quasi–p olynomial on τ − B ( X )). Definition 3.16. Let τ b e a top e and r b e a prop er rational subspace. W e sa y that a regular face F r for X \ r is non–positive on τ if there exists u r ∈ F r and x 0 ∈ τ suc h that h u, x 0 i < 0. Giv en x 0 ∈ τ , it is alwa ys p ossible to choose a regular face F r ⊂ r ⊥ for X \ r such that x 0 is negativ e on some v ector u r ∈ F r , since the pro jection of x 0 on V /r is not zero. Let F = { F r } be a X –regular collection. W e shall sa y that { F } is non– p ositiv e on τ if each F r is non–p ositive on τ . Let f ∈ F ( X ) and let f = P f r b e the F decomp osition of f . R emark 3.17 . This c hoice of F has the effect of pushing the supp orts of the elemen ts f r ( r 6 = V ) aw ay from τ . See Figures 1, 2 whic h describ e the F decomp osition of the partition function P X for X := [ a, b, c ] with a := ω 1 , b := ω 2 , c := ω 1 + ω 2 in the lattice Γ := Z ω 1 ⊕ Z ω 2 . Th us the con ten t of Theorem 3.18 is v ery similar to P aradan’s lo calization theorem [9]. Our previous claim then follows from the explicit construction b elo w. Theorem 3.18 (Lo calization theorem) . L et τ b e a t op e. L et F = { F r } b e a X –r e gular c ol le ction non–p ositive on τ . The c omp onent f V of the F dec omp osition f = P r ∈ S X f r is a quasi– p olynomial function in D M ( X ) such that f = f V on τ − B ( X ) . 10 C. DE CONCINI, C. PROCESI. M. VERGNE n 1 n 2 ( n 2 + 1) ( n 1 + 1) 0 0 0 0 Figure 1. The partition function of X := [ a, b, c ] Pr o of. Let r b e a prop er rational space and f r = P F r X \ r ∗ k r where k r ∈ D M ( X ∩ r ). In the notation of Lemma 2.6, the supp ort of f r is contained in the p olyhedron r − P b ∈ B b + C ( F r , X \ r ) ⊂ r + C ( F r , X \ r ). This last p olyhedron is conv ex and, b y construction, it has a b oundary limited b y h yp erplanes which are rational with resp ect to X . Thus, either τ ⊂ r + C ( F r , X \ r ) or τ ∩ ( r + C ( F r , X \ r )) = ∅ . T ak e u r ∈ F r and x 0 ∈ τ so that u r ( x 0 ) < 0. As u r ≥ 0 on r + C ( F r , X \ r ) it follo ws that τ is not a subset of r + C ( F r , X \ r ), so that τ ∩ ( r + C ( F r , X \ r )) = ∅ . In fact, w e claim that τ − B ( X ) do es not in tersect the supp ort r − P b ∈ B b + C ( F r , X \ r ) of f r . Indeed, otherwise, w e w ould ha v e an equa- tion v − P x ∈ X t x x = s + P a ∈ A k a a + P b ∈ B h b ( − b ) with v ∈ τ , 0 ≤ t x ≤ 1, k a ≥ 0 , h b ≥ 1 , s ∈ r . This would imply that v ∈ r + C ( F r , X \ r )), a con- tradiction. Thus f coincides with the quasi–polynomial f V on τ − B ( X ).  One should remark that a quasi–p olynomial is completely determined b y the v alues that it takes on τ − B ( X ), th us f V is indep enden t on the construction. Definition 3.19. W e shall denote b y f τ the quasi–polynomial coinciding with f on τ − B ( X ). Let us remark that the op en subsets τ − B ( X ) cov er V , when τ runs o ver the top es of V (with p ossible o v erlapping). Th us the element f ∈ F ( X ) is en tirely determined by the quasi–p olynomials f τ . Example 3.20. In Figure 3, for each top e τ , the set of in tegral p oin ts in τ − B ( X ) is con tained in one of the affine closed cones limited by thic k VECTOR P AR TITION FUNCTIONS 11 n 1 n 2 ( n 2 + 1) ( n 1 + 1) x 0 + n 1 + n 2 − ( n 2 + 1) x 0 n 1 + n 2 − ( n 1 + 1) x 0 n 1 n 2 ( n 1 − n 2 ) x 0 + n 1 n 2 ( n 2 + 1) x 0 Figure 2. F decomp osition of the partition function of X := [ a, b, c ] for F non–p ositiv e on τ lines. W e are sho wing in color the con vex env elop of the integral p oin ts in τ − B ( X ) and not the larger op en set τ − B ( X ). 12 C. DE CONCINI, C. PROCESI. M. VERGNE a c b n 1 n 2 Figure 3. T ranslated top es of X . The zonotop e − B ( X ) is in black. F or collections arising from scalar products, one can give an explicit for- m ula for the decomp osition of an elemen t f ∈ F ( X ). Let us c ho ose a scalar pro duct on V , and identify V and U with resp ect to this scalar pro duct. Given a p oin t β in V and a rational subspace r in S X , we write β = p r β + p r ⊥ β with p r β ∈ r and p r ⊥ β in r ⊥ . Definition 3.21. W e sa y that β ∈ V is generic with resp ect to r if p r β is in a top e τ ( p r β ) for the sequence X ∩ r , and p r ⊥ β ∈ r ⊥ is regular for X \ r . W e clearly hav e Prop osition 3.22. The set of β which ar e not generic with r esp e ct to r is a union of finitely many hyp erplanes. By Theorem 3.18, if f ∈ F ( X ), the element ∇ X \ r f is in F ( X ∩ r ) and coincides with a quasi–p olynomial ( ∇ X \ r f ) τ ∈ DM ( X ∩ r ) on each top e τ for the system X ∩ r . Theorem 3.23. L et β ∈ V b e generic with r esp e ct to al l the r ational sub- sp ac es r . L et F β r b e the unique r e gular fac e for X \ r c ontaining p r ⊥ β . Then f = X r ∈ S X P − F β r X \ r ∗ ( ∇ X \ r f ) τ ( p r β ) . VECTOR P AR TITION FUNCTIONS 13 Pr o of. By the hypotheses made on β the collection F = { F r } is a X − regular collection. Set f = P r ∈ S X P F r X \ r ∗ q r with q r ∈ D M ( X ∩ r ). W e apply Prop osition 3.12. It follows that q r is the comp onent in D M ( X ∩ r ) in the decomp osition of ∇ X \ r f ∈ F ( X ∩ r ), with resp ect of the X ∩ r -regular collection induced by F . Set u t = − p t ⊥ β . Remark that, for t ⊂ r , w e hav e h u t , p r β i = −k u t k 2 , so that each u t is negativ e on p r β . Thus the formula follo ws from Theorem 3.18.  3.24. W all crossing formula. W e first dev elop a general form ula describ- ing how the functions f τ c hange when crossing a w all. Then w e apply this to the partition function P X and deduce that it is a quasi–p olynomial on c − B ( X ), where c is a big cell. Let H b e a rational hyperplane, and let u ∈ U b e an equation of the h yp erplane. Then the tw o op en faces in H ⊥ are the half-lines F H = R > 0 u and − F H . Lemma 3.25. If q ∈ D M ( X ∩ H ) , then w := ( P F H X \ H − P − F H X \ H ) ∗ q is an element of D M ( X ) . R emark 3.26 . In [1], a one-dimensional residue form ula is given for w allo w- ing us to compute it. Pr o of. If t ∈ S X is differen t from H , ∇ X \ t ( P F H X \ H ∗ q ) = ∇ X \ t ( P − F H X \ H ∗ q ) = 0, as follo ws from Prop osition 3.6 iii) . If r = H , then ∇ X \ H ( P F H X \ H ∗ q − P − F H X \ H ∗ q ) = q − q = 0.  Assume that τ 1 , τ 2 are tw o adjacent top es, namely τ 1 ∩ τ 2 spans a hyper- plane H . The hyperplane H is a rational subspace. Let τ 12 b e the unique top e for X ∩ H such that τ 1 ∩ τ 2 ⊂ τ 12 (see Figure 4). Example 3.27. Let C b e the cone generated b y the vectors a := ω 3 + ω 1 , b := ω 3 + ω 2 , c := ω 3 − ω 1 , d := ω 3 − ω 2 in a 3-dimensional space V := R ω 1 ⊕ R ω 2 ⊕ R ω 3 . Figure 4 represents the section of C cut by the affine h yp erplane containing a, b, c, d . W e consider X := [ a, b, c, d ]. W e show in section, on the left of the picture, t wo top es τ 1 , τ 2 adjacen t along the hyperplane H generated b y b, d and, on the right, the top e τ 12 . The list X ∩ H is [ b, d ]. The closure of the top e τ 12 is “t wice bigger ” than τ 1 ∩ τ 2 . Let f ∈ F ( X ). The function ∇ X \ H f is an elemen t of F ( H ∩ X ), thus, by Theorem 3.18, there exists a quasi–p olynomial ( ∇ X \ H f ) τ 12 on H such that ∇ X \ H f agrees with ( ∇ X \ H f ) τ 12 on τ 12 . Theorem 3.28. L et τ 1 , τ 2 , H , τ 12 b e as b efor e and f ∈ F ( X ) . L et F H b e the half line in H ⊥ p ositive on τ 1 . Then (6) f τ 1 − f τ 2 = ( P F H X \ H − P − F H X \ H ) ∗ ( ∇ X \ H f ) τ 12 . 14 C. DE CONCINI, C. PROCESI. M. VERGNE a b c d x 0 τ 1 τ 2 b d τ 12 x 0 Figure 4. Tw o adjacen t top es of X := [ a, b, c, d ] Pr o of. Let x 0 b e a p oint in the relative in terior of τ 1 ∩ τ 2 in H . Then x 0 do es not b elong to an y X -rational hyperplane different from H (see Figure 4). Therefore we can choose a regular vector u r for X \ r for every rational subspace r differen t from H, V suc h that u r is negativ e on x 0 . By con tin uity , there are p oin ts x 1 ∈ τ 1 and x 2 ∈ τ 2 sufficien tly close to x 0 and where these elemen ts u r are still negativ e. W e c ho ose u ∈ H ⊥ p ositiv e on τ 1 . Consider the sequences u 1 = ( u 1 r ) where u 1 r = u r for r 6 = H and u 1 H = − u and u 2 = ( u 2 r ) where u 2 r = u r for r 6 = H and u 2 H = u . Corresp ondingly w e get t w o X -regular collections F 1 and F 2 . F or i = 1 , 2 let f = f i V + f i H + P r 6 = H ,V f i r b e the F i decomp osition of f . W e write f 1 H = P − F H X \ H ∗ q (1) with q (1) ∈ D M ( X ∩ H ). Now the sequence u 1 tak es a negativ e v alue at the p oin t x 1 of τ 1 , thus by Theorem 3.18, the comp onen t f 1 V is equal to f τ 1 . By Prop osition 3.12, ∇ X \ H f = q (1) + X r ⊂ H ,r 6 = H ∇ X \ H f 1 r is the F 1 decomp osition of ∇ X \ H f so that, again b y Theorem 3.18, q (1) = ( ∇ X \ H f ) τ 12 . W e thus ha v e f 1 V = f τ 1 and f 1 H = P − F H X \ H ∗ ( ∇ X \ H f ) τ 12 . Similarly u 2 tak es a negative v alue at the p oint x 2 of τ 2 , so f 2 V = f τ 2 and f 2 H = P F H X \ H ∗ ( ∇ X \ H f ) τ 12 . No w from Prop osition 3.11, when dim( r ) = i , f 1 r = Π r,F 1 r X Π F 1 i f , f 2 r = Π r,F 2 r X Π F 2 i f , and, for an y i < dim V , the op erators Π F 1 i and Π F 2 i are equal. Thus f 1 r = f 2 r for r 6 = V , H . So we obtain f 1 V + f 1 H = f 2 V + f 2 H , and our form ula.  Consider no w the case where X spans a p oin ted cone. Let us in ter- pret F ormula (6) in the case in whic h f = P X . W e kno w that for a given top e τ , P X agrees with a quasi–p olynomial P τ X on τ − B ( X ). Recall that ∇ X \ H ( P X ) = P X ∩ H as we hav e seen in Lemma 3.3. It follo ws that given VECTOR P AR TITION FUNCTIONS 15 t w o adjacent topes τ 1 , τ 2 as ab o v e, ( ∇ X \ H f ) τ 12 equals ( P X ∩ H ) τ 12 (extended b y zero outside H ). So we deduce the identit y (7) P τ 1 X − P τ 2 X = ( P F H X \ H − P − F H X \ H ) ∗ P τ 12 X ∩ H . This is Paradan’s form ula ([8], Theorem 5.2). Example 3.29. Assume X = [ a, b, c ] as in Remark 3.17. W e write v ∈ V as v = v 1 ω 1 + v 2 ω 2 . Let τ 1 = { v 1 > v 2 > 0 } , τ 2 = { 0 < v 1 < v 2 } . Then one easily (see Figure 1) sees that P τ 1 X = ( n 2 + 1) , P τ 2 X = ( n 1 + 1) , P τ 12 X ∩ H = 1 . Equalit y (7) is equiv alen t to the follo wing identit y of series whic h is easily c hec ked: X n 1 ,n 2 ( n 2 − n 1 ) x n 1 1 x n 2 2 = ( − X n 1 ≥ 0 ,n 2 < 0 x n 1 1 x n 2 2 + X n 1 < 0 ,n 2 ≥ 0 x n 1 1 x n 2 1 )( X h x h 1 x h 2 ) . Recall that a big cell is a connected comp onent of the complement in V of the singular ve ctors , which are formed by the union of all cones C ( Y ) for all the sublists Y of X whic h do not span V . A big cell is usually larger than a top e. See Figure 5 whic h shows a section of a cone in dimension 3 generated by 3 indep endent vectors a, b, c . Here X = [ a, b, c, a + b + c ]. On the drawing, the v ertices a , b , c , d represents the intersection of the section with the half lines R + a , R + b , R + c , R + d . a b c d a b c d Figure 5. T op es and cells inside C ( X ) for X := [ a, b, c, d := a + b + c ] Let us now consider a big cell c . W e need Lemma 3.30. Given a big c el l c , let τ 1 , . . . , τ k b e al l the top es c ontaine d in c . Then: c − B ( X ) = ∪ k i =1 ( τ i − B ( X )) . Pr o of. Notice that ∪ k i =1 τ i is dense in c . Giv en v ∈ c − B ( X ), v + B ( X ) has non em pt y interior and thus its non empt y in tersection with the op en set c has non empt y interior. It follows that v + B ( X ) meets ∪ k i =1 τ i pro ving our claim.  No w in order to prov e the statemen t for big cells, we need to see what happ ens when we cross a wall betw e en tw o adjacen t top es. 16 C. DE CONCINI, C. PROCESI. M. VERGNE n 1 n 2 ( n 2 + 1) ( n 1 + 1) x 0 Figure 6. The function f Theorem 3.31. On c − B ( X ) , the p artition function P X agr e es with a quasi–p olynomial f c ∈ D M ( X ) . Pr o of. By Lemma 3.30, it suffices to show that given t wo adjacen t top es τ 1 , τ 2 in c , P τ 1 X = P τ 2 X . But now notice that the p ositiv e cone spanned by X ∩ H , supp ort of P X ∩ H , is formed of singular vectors and therefore it is disjoint from c b y definition of big cells. Therefore P X ∩ H v anishes on τ 12 . Thus P τ 12 X ∩ H = 0 and our claim follo ws from F orm ula (7).  This theorem w as pro ven [6] by Dahmen-Micchelli for top es, and by Szenes-V ergne [10] for cells. In many cases, the sets c − B ( X ) are the max- imal domains of quasi–p olynomialit y for P X . R emark 3.32 . If c is a big cell con tained in the cone C ( X ), the op en set c − B ( X ) contains c so that the quasi–polynomial f c coincides with P X on c . This is usually not so for f ∈ F ( X ) and a top e τ : the function f does not usually coincide with f τ on τ . Figure 6 describ es the function f := −P F r X in F ( X ) with X := [ a, b, c ] as in Remark 3.17 and u ∈ F r strictly negative on a, b . W e see for example that f is equal to 0 on the set n 1 = 0 , n 2 ≤ 0 whic h is in the closure of the top e τ 3 := { v 2 < v 1 < 0 } while the quasi–p olynomial f τ 3 = ( n 1 + 1) takes the v alue − 1 there. W e finally give a form ula for P X due to Paradan. Let us c ho ose a scalar pro duct on V . W e use the notations of Theorem 3.23. As ∇ X \ r P X = P X ∩ r , we obtain as a corollary of Theorem 3.23 VECTOR P AR TITION FUNCTIONS 17 Theorem 3.33. (Par adan). L et β ∈ V b e generic with r esp e ct to al l the r ational subsp ac es r . Then, we have P X = X r ∈ S X P − F β r X \ r ∗ ( P X ∩ r ) τ ( p r β ) . R emark 3.34 . The set of β ∈ V generic with resp ect to all the rational subspaces r decomp oses into finitely man y op en p olyhedral cones. The decomp osition dep ends only on the cone in whic h β lies. In this decomp osition, the comp onent in D M ( X ) is the quasi-polynomial whic h coincides with P X on the cell con taining β . Finally if X spans a p ointed cone and β has negative scalar pro duct with X , the decomp osition reduces to P X = P X , the comp onent for r = { 0 } . 4. A second remarkable sp a ce 4.1. A decomp osition form ula. In this section, we w an t to present the analogue for distributions, the pro ofs are essentially the same or simpler than in the previous case, so w e skip them. W e shall freely use the notations of the previous sections. Let V b e a finite dimensional v ector space, consider the space D ( V ) of distributions on V . W e denote b y δ 0 the delta distribution on V . D ( V ) is in an ob vious wa y a mo dule o ver the algebra of distributions with compact supp ort, under conv olution. Let no w X := [ a 1 , . . . , a m ] b e a list of non–zero elements of V . Definition 4.2. i) Given a rational subspace r , we denote by D ( V , r ) the set of elemen ts in D ( V ) whic h v anish on all test functions v an- ishing on r . ii) Given a vector a 6 = 0, we denote by ∂ a the directional deriv ative asso ciated to a . F or a list Y of non–zero vectors, w e denote by ∂ Y := Q a ∈ Y ∂ a . The restriction map C ∞ c ( V ) → C ∞ c ( r ) on test functions induces, by dual- it y , an iden tification b et w een the space of distributions on r and the space D ( V , r ). If X spans V , we consider the space defined b y Dahmen–Micchelli, which w e denote D ( X ), formed by the distributions f ∈ D ( V ) satisfying the system of differential equations ∂ Y f = 0, as Y v aries among all co circuits of X . It is easy to see that an elemen t of D ( X ) is a p olynomial density P ( x ) dx on V . Assume that X spans a p ointed cone. Recall that the multivariate spline T X is the temp ered distribution defined, on test functions f , by: (8) h T X | f i = Z ∞ 0 · · · Z ∞ 0 f ( m X i =1 t i a i ) dt 1 · · · dt m . 18 C. DE CONCINI, C. PROCESI. M. VERGNE If W is the span of X and if w e choose a Leb esgue measure dx on W , w e ma y interpret T X as a function on W supp orted in the cone C ( X ) by writing h T X | f i = R W f ( x ) T X ( x ) dx. If Y is a sublist of X , one has that ∂ X \ Y T X = T Y . W e next define the vector space of interest for this section: (9) G ( X ) := { f ∈ D ( V ) | ∂ X \ r f ∈ D ( V , r ) , for all r ∈ S X } . Lemma 4.3. i) If X gener ates a p ointe d c one, the multivariate spline T X lies in G ( X ) . ii) The sp ac e D ( X ) is c ontaine d in G ( X ) . Pr o of. i) ∂ X \ r T X = T X ∩ r ∈ D ( V , r ) . ii) is clear from the definition.  As for the partition functions, this lemma is a very sp ecial case of Theorem 4.5 which follows. Giv en a rational subspace r , take a regular face F r for X \ r . Divide as b efore the set X \ r into t wo parts A, B , of p ositiv e and negativ e v ectors on F r W e w an t to define an elemen t T F r X \ r ∈ D ( V ) whic h is c haracterized by the follo wing tw o prop erties: Lemma 4.4. Ther e exists a unique element T F r X \ r char acterize d by the pr op- erties ∂ X \ r T F r X \ r = δ 0 and T F r X \ r is supp orte d in C ( A, − B ) . Pr o of. Set: T F r X \ r = ( − 1) | B | T [ A, − B ] . It is easily seen that this elemen t satisfies the tw o properties. The uniqueness is also clear.  Iden tify the space of Dahmen–Micchelli D ( X ∩ r ) with a subspace of D ( V , r ). Although a distribution f ∈ D ( V , r ) may ha ve non compact sup- p ort, w e easily see that the conv olution pro duct T F r X \ r ∗ f is well defined. In fact, given any γ ∈ V , we can write γ = λ + µ with µ ∈ r , and λ ∈ C ( A, − B ) only in a b ounded p olytop e. The analog of the “mother form ula” (2) of Prop osition 2.7 is the following form ula. F or g ∈ D ( V , r ): (10) ∂ X \ t ( T F r X \ r ∗ g ) = T F r ( X \ r ) ∩ t ∗ ( ∂ ( X ∩ r ) \ ( t ∩ r ) g ) . F ollowing the same scheme of pro of as for Theorem 3.8, the follo wing theorem follows: VECTOR P AR TITION FUNCTIONS 19 Theorem 4.5. Cho ose for every r ational sp ac e r , a r e gular fac e F r for X \ r . Then: G ( X ) = ⊕ r ∈ S X T F r X \ r ∗ D ( X ∩ r ) . W e associate, to a rational space r and a regular face F r for X \ r , the op erator on G ( X ) defined by π r,F r X : f 7→ T F r X \ r ∗ ( ∂ X \ r f ) . This is well defined. Indeed ∂ X \ r f is supp orted on r so that the conv olution is well defined. W e see that π r,F r X maps G ( X ) to G ( X ) and that it is a pro jector. Giv en a X –regular collection F , w e can write an elemen t f ∈ G ( X ) as f = X r ∈ S X f r with f r ∈ T F r X \ r ∗ D ( X ∩ r ) . This expression for f will b e called the F de- comp osition of f . In this decomp osition, the comp onent f V is in D ( X ) . The space T F r X \ r ∗ D ( X ∩ r ) will b e referred to as the F r -c omp onent of G ( X ). Let F b e a X –regular collection. One can write in the same wa y as in Prop osition 3.11 the explicit pro jectors to the v arious comp onen ts. 4.6. P olynomials. Let f ∈ G ( X ) and let τ b e a top e. Prop osition 4.7 (Lo calization theorem) . L et F b e a X -r e gular c ol le ction non-p ositive on τ . L et f = P f r b e the F -dec omp osition of f . Then the c omp onent f V of this dec omp osition is a p olynomial density in D ( X ) such that f = f V on τ . Pr o of. W rite f = P r ∈ S X f r with f r = T F r X \ r ∗ k r where k r ∈ D ( X ∩ r ). The distribution f r = T F r X \ r ∗ k r is supp orted on r + C ( F r , X \ r ). As in Theorem 3.18, we know that τ ∩ ( r + C ( F r , X \ r )) = ∅ .  R emark 4.8 . Thus the distribution f is a lo cally p olynomial density on V . In particular this is a temp ered distribution. Here the distribution f coincides with the p olynomial density f V only on τ and not on the bigger op en set τ − B ( X ). This extension prop erty is replaced the regularit y property that f is of class C r − 1 , where r is the minimum of the cardinality of the co circuits of X (see [3]). W e shall denote b y f τ the polynomial density in D ( X ) coinciding with f on the top e τ . Let H b e a rational h yp erplane, let u ∈ U b e an equation of the hyper- plane and F H the half-line containing u . 20 C. DE CONCINI, C. PROCESI. M. VERGNE Lemma 4.9. If q ∈ D ( X ∩ H ) , then w := ( T F H X \ H − T − F H X \ H ) ∗ q is an element of D ( X ) . Let us use the notations τ 1 , τ 2 , H , τ 12 as in Theorem 3.28. Let f ∈ G ( X ). The distribution ∂ X \ H f is an element of G ( H ∩ X ), thus b y Prop osition 4.7, there exists a p olynomial density ( ∂ X \ H f ) τ 12 ∈ D ( X ∩ H ) on H suc h that ∂ X \ H f agrees with ( ∂ X \ H f ) τ 12 on τ 12 . Theorem 4.10. L et F H b e the half line in H ⊥ p ositive on τ 1 . Then (11) f τ 1 − f τ 2 = ( T F H X \ H − T − F H X \ H ) ∗ ( ∂ X \ H f ) τ 12 . When X spans a p oin ted cone, w e interpret F orm ula (11) for f = T X . On a given top e τ , T X agrees with a p olynomial density T τ X . Since ∂ X \ H ( T X ) = T X ∩ H w e deduce, the identit y (12) T τ 1 X − T τ 2 X = ( T F H X \ H − T − F H X \ H ) ∗ T τ 12 X ∩ H . No w the statement for big cells: Theorem 4.11. On c , the multivariate spline T X agr e es with a p olynomial density in D ( X ) . R emark 4.12 . Once Theorems 3.31 and 4.11 hav e b een pro ven, it is easy to deduce from them that the generalized Khov anskii–Pukhliko v formula relating volumes and n umber of p oin ts holds [2]. Indeed, one can pro v e it easily sufficiently far from the walls (cf. [3]). Finally , using a scalar pro duct, we give P aradan’s formula for T X . Theorem 4.13. (Par adan). L et β ∈ V b e generic with r esp e ct to al l the r ational subsp ac es r . Then, we have T X = X r ∈ S X T − F β r X \ r ∗ ( T X ∩ r ) τ ( p r β ) . References [1] Boysal A., V ergne M., Paradan’s wall crossing formula for partition functions and Kho v anski-Pukhliko v differential op erator. Preprint ArXiv (2008): 0803.2810 [2] Brion M. and V ergne M., Residue formulae, vector partition functions and lattice p oin ts in rational p olytop es, J. Amer. 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