Splines and index theorem

In this article we want to present a meeting ground between rather different mathematical topics from numerical analysis to index theory and symplectic geometry. The unifying idea is that of linear representation of tori inside which the combinatorics, analysis and geometry is developed.


Introduction
In 1968 appears the fundamental work of Atiyah and Singer on the index theorem of elliptic operators, a theorem formulated in successive steps of generality [3][4][5][6][7].
This theorem is the crowning point of a long sequence of ideas starting from the theory of algebraic functions of Riemann and passing through the theory of De Rham and Hodge on cohomology on manifolds. The index theorem generalizes the Riemann-Roch theorem of Hirzebruch, a cornerstone in modern algebraic geometry [25].
What is an index? Recall that the index i(A) of a linear operator, when defined, is i(A) := dim(ker A) − dim(ker A * ), where ker(A) is the space of solutions of the homogeneous equation Au = 0 and A * denotes the adjoint.
• The linear operators consider by Atiyah and Singer are differential (or rather pseudodifferential) operators on manifolds. • Precisely they operate not simply on functions but, as usual in global geometry, on sections of vector bundles. The condition of being elliptic (like the Laplace operator) is needed to insure that the index is well defined and a finite integer. The index theorem produces a formula for the index through cohomological data associated to the operator, the manifold and the bundles involved (Chern character and Todd class).
The need for K -theory An essential step in the index theory is given by the construction of a generalized cohomology theory, called K -theory.
Then the index theorem passes through several steps: • to an operator one associates the symbol , which is a (matrix valued) function on the cotangent bundle. • to the symbol an element of a suitable group of K -theory, the index can be computed this way.
• Alternatively from K -theory one passes to cohomology using the Chern character ch( ) of . • The final formula is proved by showing enough properties of all these steps which reduce the formula to some basic cases.
One general and useful setting is for operators on a manifold M which: (i) satisfy a symmetry with respect to a compact Lie group G (ii) are elliptic in directions transverse to the G-orbits.
In this case the values of the index are generalized functions on G. In fact in this case the kernel of A or of A * is no more finite dimensional, but it is a particularly well behaved representation ρ : G → U (H), called of trace class, from the group of symmetries G to unitary operators on a Hilbert space. This means that we can integrate the C ∞ functions of G to operators T f := G f (g)ρ(g) which have a trace and f → T r(T f ) is a distribution.
Being of trace class implies that both spaces of solutions are • a (possibly infinite) direct sum of irreducible representations of G each with finite multiplicity • the multiplicities have some moderate growth so that they can be interpreted as Fourier coefficients of generalized functions (distributions) on G.
The analytic index of the operator A is the virtual trace class representation of G, obtained as difference of the spaces of solutions of A and its adjoint A * in an appropriate Sobolev space. This is equivalently described by an integral valued function on the set of irreducible representations, the index multiplicity.
The main topological ingredient of the theory is the group K 0 G,c (T * G M) of equivariant K -theory with compact support, which is defined in a topological fashion.
If T * M is the cotangent bundle. we denote by T * G M the closed subset of T * M, union of the conormals to the G orbits. Using a Riemannian metric we may replace the cotangent bundle with the tangent bundle, and then conormal is replaced with orthogonal.
Example 1.1 G = S 1 acting by rotations on the plane R 2 , the orbits are the circles centered at the origin, we identify T * M = R 4 with pairs of vectors.
A pair of vectors (a, b) ∈ R 2 × R 2 is in T * S 1 M if either a = 0, in this case its orbit reduces to 0 and b arbitrary, or a = 0 its orbit is a circle and b is orthogonal to the orbit, that is it is proportional to a.
Assume first that M is compact. The condition of being transversally elliptic is expressed by the fact that the symbol is invertible on T * G M minus the zero section. So one repeats the analysis for the index, substituting to K -theory equivariant K -theory.
This index depends only of the class defined by in K 0 G,c (T * G M), so that in the end the index defines a R(G) module homomorphism from K 0 G,c (T * G M) to virtual trace class representations of G.
Alternatively, from equivariant K -theory one passes to equivariant cohomology with compact supports using the equivariant Chern character ch( ) of . This is an element of a suitable G-equivariant cohomology group, with compact supports, of T * G M.
In his Lecture Note describing joint work with I.M. Singer, Elliptic operators and compact groups, [1], Atiyah explains how to reduce general computations to the case in which • G is a torus, e.g. G = (S 1 ) s ,Ĝ = Z s its character group.
• The manifold M is a complex linear representation M X = ⊕ a∈X L a , • X ⊂Ĝ = Z s is a finite list of characters and L a denotes the one dimensional complex line where G acts by the character a ∈ X .
He then computes explicitly in several cases and ends his introduction saying "…for a circle (with any action) the results are also quite explicit. However for the general case we give only a reduction process and one might hope for something explicit. This probably requires the development of an appropriate algebraic machinery, involving cohomology but going beyond it." A solution to this question is given in the paper Vector partition functions and index of transversally elliptic operators, by C. De Concini, C. Procesi, M. Vergne [20].
In this paper I want to give an idea of the algebraic machinery which provides a complete solution to this question. This turns out to be a spinoff of the theory of splines, a developement of the classic Theory of Schoenberg, [31] as in the classic book [15] by de Boor, Höllig, and Riemenschneider. For most of this material the reader can consult the recent book by De Concini, and Procesi Topics in hyperplane arrangements, polytopes and box-splines [23].
The main goal is to describe explicitly the map induced by the index, from K 0 G,c (T * G M) to virtual trace class representations of G. This is described combinatorially.
Once we identify the character group of G with the free abelian group Z s we have, as values of the index, a space of integer valued functions on Z s . Our main result is for a linear representation M X associated to a list of characters X in Z s : • We identify the space of functions, image of the index, for M = M X .
• We prove that the index is an isomorphism between K 0 G,c (T * G M) and its image in the space of functions.
We need the basic notion of Partition function for a list X of vectors in Z s . If there is a linear form φ so that φ | a > 0, ∀a ∈ X we have a Z valued function P X (b) on Z s defined as: The function P X (b) is best expressed via its generating series If X is the list of characters of a linear representation M X the partition function can be viewed as the trace of the representation of S s in the symmetric algebra S(M X ).
Atiyah-Singer constructed a "pushed" ∂ operator on M X , with index the partition function.
For a general list X consider a linear function φ which is non-zero on each element of X, then X is divided into two parts A where φ is positive, and B respectively where φ is negative. We may thus consider the partition function P A,−B . Lemma 1. 2 The function P A,−B depends only on the chamber F, of the hyperplane arrangement defined by X in the dual space, in which φ lies and will be denoted P F X . Each of these partition functions can be viewed as multiplicity index of a "pushed" ∂ operator on M X by changing the complex structure.
Integer valued functions on Z s are a module (using translation) over the group algebra Z[Z s ] (also denoted by R(T ), if T is the torus with character groupT = Z s ). We can thus consider the Z[Z s ] moduleF(X ) of functions on Z s generated by all the partition functions P A,−B for all the chambers.
The main Theorem 1 states that the index induces an isomorphism between K 0 G,c (T * G M X ) andF(X ). Moreover there is a very precise description of the group F(X ).
1.2.1. The case S 1 . For s = 1, or G = S 1 we denote by t the basic character of . We write generalized functions by their generating series n∈Z a n t n .
When S 1 acts by homotheties on C k+1 we have C k+1 = M X and X = [t, t, . . . , t], k + 1 times. We start by describing Theorem 1 in this special case.
By the "pushed" ∂ operator we get the generalized function In this case we only have two chambers and the other partition function The map n → n+k k is a polynomial function on Z which, for any n positive or negative, represents the dimension of a virtual space, the alternate sum of the cohomology spaces of the sheaf O(n) on k-dimensional projective space.
One easily sees that this function gives the Fourier coefficients of a generalized function θ k (t) := ∞ n=−∞ n+k k t n , on S 1 supported at t = 1. In fact for k = 0 the generalized function θ 0 (t) := ∞ n=−∞ t n , is the Fourier expansion of the delta function δ 1 and we obtain θ k (t) by applying suitable derivatives to θ 0 (t).
One can prove that the tangential Cauchy-Riemann operator, on the unit sphere S 2k+1 of C k+1 , is a transversally elliptic operator with index θ k (t).
In fact the R(S 1 ) module generated by X is free over R(S 1 ).
The R(S 1 ) module generated by θ X is the torsion submodule.
• This submodule is the module of polynomial functions on Z of degree at most k so it is a free Z-module of rank k + 1. • It corresponds to indices of operators on C k+1 − {0}, the set where S 1 acts freely • It is the space of solutions of the difference equation

Higher dimension.
In higher dimensions, the single difference equation ∇ k f = 0 must be replaced by a system of difference equations, discovered by Dahmen-Micchelli as the natural generalization of a system of differential equations associated to splines in approximation theory. We need a basic definition of combinatorial nature. Consider a sequence X of integral vectors (i.e. weights of a torus G with character group ). Definition 1. 4 We say that a sublist Y ⊂ X is a cocircuit, if the elements in X \Y do not span V and Y is minimal with this property.
For an integral vector a ∈ (often identified with Z s ) define the translation operator τ a and the difference operator ∇ a by These operators act on functions on Z s . For a list Y of integral vectors we set ∇ Y := a∈Y ∇ a and introduce thus the system of difference equations ∇ Y f = 0 associated to cocircuits Y of X ⊂ Z s . The space of solutions is denoted by DM(X ). There is a beautiful Theory of this space which we will in part recall presently (cf. Definition 7.5).
Recall the Z[Z s ] moduleF(X ) of functions on Z s generated by all the partition functions P A,−B for all the chambers.

Theorem 2 The index map induces an isomorphism between K
For the general case we need some associated spaces of functions, generalizing DM(X ). We thus introduce the spacesF i (X ) associated to the orbit types in M X .
Any linear subspace r generated by vectors in X is called a rational subspace, we denote by S X the set of all rational subspaces. We define: Clearly DM(X ) is contained in F(X ). One can show thatF(X ) is the space of functions generated by F(X ) under translations by integral vectors.
Denote by S (i) X the subset of subspaces r ∈ S X of dimension i. Define the spaces Denote byF i (X ) the space of functions generated by F i (X ) under translations by integral vectors. Set M ≥i as the open set of points in M X with the property that the orbit has dimension ≥ i.

Theorem 3
For each s ≥ i ≥ 0, the index multiplicity map ind m gives an isomorphism between K 0 G (T * G M ≥i ) and the spaceF i (X ).
Finally one has very explicit descriptions of DM(X ) and of the spacesF i (X ) which are built from various spaces DM(X i ).
We then investigate the formulas allowing us to pass from equivariant K -theory to equivariant cohomology.
In order to understand these explicit formulas in [18] we have introduced the infinitesimal index, called for short "infdex", a map from the equivariant cohomology, with compact supports, of the zeroes of the moment map to distributions on g * .
We have proved several properties for this map which, at least in the case of the space T * G M, in principle allow us to reduce the computations to the case in which G is a torus and the manifold is a complex linear representation M X of G. Where X is a list of characters as before.
The equivariant cohomology of the open sets M X,≥i can be computed from the structure of the algebra S[g * ][( a∈X a) −1 ] as a module over the Weyl algebra studied in [17].
The equivariant cohomology with compact supports is isomorphic, as a S[g * ]-module, to a remarkable finite dimensional space D(X ) of polynomial functions on g * , where S[g * ] acts by differentiation. The space D(X ) is the continuous analogue of DM(X ) and it is defined as the space of solutions of a set of linear partial differential equations combinatorially associated to X and has been of importance in approximation theory (see for example [13,14]). In fact in a natural way D(X ) is a component of DM(X ) C and it coincides with DM(X ) C in the special case of X unimodular 7.2.
At this point the notion of infinitesimal index comes into play. One can show, Theorem 34, that the infinitesimal index gives an isomorphism between H * G,c (T * G M f in X ) and D(X ). After this one shows that, for each i, the infinitesimal index establishes an isomorphism between H * G,c (T * G M X,≥i ) and a space of splinesG i (X ), introduced in [19] and analogous to the arithmetic spacesF i (X ), and generalizing D(X ).
Then Theorem 37 is a deconvolution formula which allows us to compute the function ind m (A) in function of the distribution infdex(ch( )) by applying to it a Todd operator.
In order to develop this formula one needs results obtained by Dahmen-Micchelli in the purely combinatorial context of the semi-discrete convolution with the Box spline.
In this paper we do not give details for the proofs which can be found in the papers quoted.

Part 1. Polytopes and splines 2 Polytopes
Polytopes play a special role in the Theory. A convex polytope can be defined as the convex envelop of finitely many points or by finitely many linear inequalities. We take this second point of view and study polytopes which vary into families.

From a list of vectors
We start from a real, sometimes integer n × m matrix X . We always think of X := (a 1 , . . . , a m ) as a list of vectors in V = R n , its columns. We assume that 0 is NOT in the convex hull of its columns.
From A we make several constructions, algebraic, combinatorial, analytic etc. Many people have contributed to the Theory from various branches of mathematics.
The columns a i , b are vectors with n coordinates a j,i , b j , j = 1, . . . , n.
As in Linear Programming Theory we deduce and want to study the, convex and bounded, variable polytopes: , the cone of positive combinations of the a i . The hypothesis that 0 is NOT in the convex hull of the a i implies that C(X ) is pointed, i.e. there is a linear function φ with φ(a i ) > 0 for all i so that φ(a i ) > c > 0/ In other words φ is strictly positive on all non zero points of C(X ).
The property of being bounded is trivial for 1 The object of study are two basic functions to be the number of solutions of the system in which the coordinates x i are non negative integers. • In other words P X (x) is the number of integral points in the variable polytope Up to a multiplicative normalization constant: T X (x) is the Multivariate-spline, B X (x) the Box-spline, P X (x) is called the partition function. We are interested in computing the three functions T X (x), B X (x), P X (x) and describe their qualitative properties.
We shall discuss applications of these functions to arithmetic, numerical analysis, Lie theory and Index Theory.
An important example is when X is the list of positive roots of a root system, e.g. B 2 : The associated cone C(X ) has three big cells (cf. Sect. 2.9): In the literature of numerical analysis the Box spline associated to the root system B 2 is called the Zwart-Powell or ZP element, 2T X is 0 outside the cone and on the three cells ( Fig. 1): The computation of the box-spline has some geometric, combinatorial and algebraic flavor. It appears as a piecewise polynomial function on a compact polyhedron. From simple data we get soon a complicated picture! (Fig. 2).

The partition function
When X, b have integer entries, it is natural to think of an expression like: b = t 1 a 1 + · · · + t m a m , with t i not negative integers, as a partition of b with the vectors a i , Fig. 1 The area and the shape as b = (x, y) varies in the cone Fig. 2 The box spline 2B X , ZP element in t 1 + t 2 + · · · + t m parts, hence the name partition function for the number P X (b), thought of as a function of the vector b.
Of special interest is the case when the a i are numbers. Given positive integers h := (h 1 , . . . , h m ), the problem of counting the number of ways in which a positive integer n can be written as a linear combination n i=1 k i h i , with the k i again positive integers, is a basic question in arithmetic. Denote by P h (n) this number.
The answer depends on the class of n modulo 6 A function on a lattice which is a polynomial on each coset of some sublattice M of finite index is called a quasi-polynomial.  For the root system A 2 we may take X = 1 0 1 0 1 1 . The corresponding partition function P X is piecewise polynomial with top degree coinciding with T X (Fig. 3).

Example 2.4
In Lie theory the Kostant partition function counts in how many ways can you decompose a weight as a sum of positive roots. This is used in many computations.

The case of integers.
The number P h (n) of ways in which n is written as combination of m numbers h i is already studied by Euler and called by Sylvester a denumerant. The first results on denumerants are due to Cayley and Sylvester, who proved that such a denumerant is a polynomial in n of degree m − 1 plus a periodic polynomial of lower degree called an undulant. A different approach is also developed by Bell in (1943). One starts with Lemma 2.5 P h (n) is the coefficient of x n in the power series expansion of: In order to compute for a given n ≥ 0, the coefficient P h (n) of x n in S h (x) we can use two essentially equivalent strategies: 1. Develop S h (x) in partial fractions. 2. Compute the residue 1 2πi In both cases first we must expand in a suitable way the function S h (x). Given k let us denote by ζ k := e 2πi k , a primitive kth root of 1; we can write (2) Let μ be the least common multiple of the numbers h i , and write μ = h i k i . If ζ = e 2πi/μ , we have ζ h i = ζ k i therefore we have where the integer b is the number of k i which are divisors of .
In particular the function x −n−1 S h (x), n ≥ 0 has poles at 0 and at the μth roots of 1 (but not at ∞).
The classical method starts from the fact that there exist numbers c i for which: We then use the simple formula: to get: We have thus obtained a formula for the coefficient Let us remark now that k − 1 + n n = (n + 1)(n + 2) · · · (n + k − 1) (k − 1)! is a polynomial of degree k −1 in n, while the numbers ζ in depend only from the coset of n modulo μ. Given an 0 ≤ a < μ and restricting us to the numbers n = μk + a, we have:

Theorem 4 The function P h (mk + a) is a polynomial in the variable k of degree
The fact that P h (n) is a polynomial on every coset means that it is a quasi-polynomial or a periodic polynomial.
Second method: computation of residues Here the strategy is the following: shift the computation of the residue to the remaining poles, taking advantage of the fact that the sum of residues at all the poles of a rational function is 0.
From the theory of residues we have: where C j is a small circle around ζ − j . In order to compute the term 1 2πi we perform the change of coordinates x = w + ζ − j obtaining: Finally we have that (6) equals: Summing over j we obtain an explicit formula for P h (n), again as a quasipolynomial.
Remark that, in order to develop these formulae it suffices to compute a finite number of coefficients a j,h .
In these formulae roots of unity appear while the final partition functions are clearly integer valued. An algorithmic problem remains. We need to know how to manipulate these expressions involving roots on 1. This is an elementary but computationally very complex problem on cyclotomic polynomials. Set ζ k := e 2πi k a primitive root of 1. Define: This is a periodic integral valued function, computable through the Möbius function and the Euler φ function. One can use these functions in order to find explicit formulas for the partition function. For instance: Example 2.7 (X = 2, 3, 4, 4, 6, 9, 10, 40).
• The formula is over all the roots of 1 of order a divisor of an element in X .
This has been computed with Mathematica and it is:

Main objects associated to X
We now go back to the general case of a list X = {a 1 , . . . , a m } of vectors. We think of the elements a i as linear equations defining hyperplanes in the dual space U = V * . This is a central hyperplane arrangement. One way to study this arrangement is to study the algebra of rational functions which have poles only on this arrangement. That is we consider the algebra The algebra R X consists of those rational functions which have at the denominator a product of powers of the linear forms a i . It is clearly a module under the Weyl algebra of differential operators with polynomial coefficients and we analyze in depth the module structure.
When the coordinates of the elements a i = (a i,1 , . . . , a i,s ) are integers we take instead a multiplicative point of view and think of a i as a character s j=1 x a i, j j . This defines a subgroup of the torus (C * ) s the points in which the character is 1 or vanishes. This is call a central toric arrangement (by the exponential map this corresponds in fact to a periodic hyperplane arrangement).
One way to study this arrangement is to study the algebra of Laurent polynomials We study also this as a module over the algebra of differential operators with coefficients Laurent polynomials.

The theorems
There are several general formulas to compute the previous functions which are obtained by a mixture of techniques.
A main geometric notion that plays a role is that of big cells. Assume that X spans V .
• The singular points C sing (X ) are the points in the cone C(X ) lying in some cone C(Y ) for any sublist Y of X which does NOT span the ambient space. • The other points are called regular. • A big cell is a connected component of the set or regular points.
In other words, take all the hyperplanes H spanned by sublists of X and then the cones C(H ∩ X ). The union of all these cones forms the set of singular vectors.
It is easy to see that the big cells are convex polyhedra. Take all bases b extracted from X, for each basis consider the cone C(b) generated by b. Its boundary is made of singular points.
A standard fact of polyhedra is that . It follows that the big cell c in which p lies is the intersection of all the

Example 2.10 The positive roots of type
We want to decompose the cone C(X ) into big cells and see its singular and regular points. We do this on a transversal section (Fig. 4).

Towards the Jeffrey-Kirwan formula.
Assume X spans R n . Let m(X ) denote the minimum number of columns that one can remove from X so that the remaining columns do not span R n .
coincides with a homogeneous polynomial of degree m − n on each big cell.
In order to compute T X (x) we need to (i) Determine the decomposition of C(X ) into cells.
(ii) Compute on each big cell the homogeneous polynomial of degree m − n coinciding with T X (x).
One can find explicit polynomials p b,X (x), indexed by a combinatorial object called unbroken bases (see Sect. 3.1) and characterized by certain explicit differential equations so that, given a point x in the closure of a big cell c we have the (cf. Sect. 9.6.1) From T X one computes B X . For a given subset S of X define a S := a∈S a; the basic formula is: So T X is the fundamental object. Notice that the local pieces of B X are no more homogeneous polynomials.

Unbroken bases
We now explain some ideas relating unbroken bases with cells.
From the theory of matroids to cells.
Let c := a i 1 , . . . , a i k ∈ X, i 1 < i 2 · · · < i k , be a sublist of linearly independent elements. Definition 3. 2 We say that a i breaks c if there is an index 1 ≤ e ≤ k such that: • a i is linearly dependent on a i e , . . . , a i k .
Example 3.3 Take as X the list of positive roots for type A 3 .
We have 16 bases 10 broken and 6 unbroken, all contain necessarily α 1 : The overlapping theorem states that, by overlapping the cones generated by the unbroken bases one obtains the entire decomposition into big cells!! PROBLEM: Describe the previous pictures for root systems. For type A n the unbroken bases are known and can be indexed by certain binary graphs or by permutations of n elements. The decomposition into cells is unknown.
is supported in the compact polytope: called the zonotope: When X is the set of positive roots of a root system and ρ := 1/2 a∈X a we have: The zonotopes associated to the root system A 3 B 3 We now want to describe the faces of B(X ). Consider a sublist A of X spanning a hyperplane H 0 and such that A = H 0 ∩ X .
Take a linear equation φ H 0 for H 0 and set Let A, B be two disjoint sublists of X such that A does not span V and B(A) its associated zonotope. Let Let H X be the real hyperplane arrangement defined, in dual space, by the vectors of X thought of as linear equations.

Theorem 6 • There is an order reversing 1-1 correspondence between faces of B(X ) and faces of H X . • To a face G of the hyperplane arrangement we associate the set of points in B(X )
where any element f ∈ G takes its maximum value.
The zonotope B(X ) has a nice combinatorial structure, proved by Shephard, it can be paved by a set of parallelepipeds indexed by all the bases which one can extract from A! Example 3.5 In the next example we have 15 bases and 15 parallelograms.
This can be constructed stepwise adding a vector at a time.

Splines
where f (x) is any continuous function with compact support.
In general T X (x) is a tempered distribution, supported on the cone C(X ), only when X has maximal rank T X (x) is a function.
, and given by the formula:

Properties of box-splines.
In the case of integral vectors, we have that the translates B X (x − λ), as λ runs over the integral vectors form a partition of 1 (Fig. 5).

Approximation theory
The box spline, when the a i are integral vectors, can be effectively used in the finite element method to approximate functions. We define the Cardinal spline space as in [31].
The function a i on Z s is called a mesh function.
Define D(X ) to be the space of polynomials contained in S X .
For a function f we have the semi-discrete convolution The theory of Dahmen-Micchelli consists of several Theorems. The basic results are: • The dimension of D(X ) equals the number of bases which one can extract from X .

Algebra
How to compute T X ? or the partition function P X ? We use the Fourier-Laplace trans- dv which will change the analytic problem to one in algebra.

Basic properties.
Given p ∈ U, w ∈ V, write p, D w for the linear function p | v and the directional derivative on V . We then have: , An easy computation gives the Laplace transforms: We need to rewrite LT X = a∈X a −1 , for this we need to develop a theory of partial fractions in several variables, in this case for the algebra We do this using non commutative algebra. Set W (V ), W (U ) be the two Weyl algebras of differential operators with polynomial coefficients on V and U . Both algebras are generated by V ⊕ U with the same relations so there is an algebraic Fourier isomorphism between these two algebras, so any W (V ) module M becomes a W (U ) module, denoted byM.

D-modules in Fourier duality.
The Laplace transform gives rise, using the previous construction, to an isomorphism of two modules under the two algebras X obtained from the polynomials on U by inverting the element d X := a∈X a. The algebra R X is the coordinate ring of the open set A X complement of the union of the hyperplanes of U of equations a = 0, a ∈ X . It is a cyclic module under W (U ) generated by d −1 X . It is well known that, once we invert an element in a polynomial algebra, we get a holonomic module over the algebra of differential operators (cf. [12]).
In particular R X is holonomic, cyclic and it has a finite composition series. In order to understand this composition series we observe that for each subspace W of U we have an irreducible module N W (over W (U )) generated by the δ function of To be explicit, take coordinates x 1 , . . . , x n ∈ V so that W = {x 1 = x 2 = · · · = x k = 0}. N W is generated by an element δ W satisfying: In particular the composition factors of R X are of the form N W as W runs over the subspaces of the hyperplane arrangement, in U, given by the equations a i = 0, a i ∈ X . In order to explain this we exhibit an explicit filtration.

The filtration of R X by polar order
Definition 5. 1 We denote by R X,k the span of all the fractions f a∈X a −h a , h a ≥ 0 for which the set of vectors a, with h a > 0, spans a space of dimension ≤ k.
• R X is filtered by the W (U )-submodules R X,k .
• We have R X,s = R X .
• For all k we have that R X,k /R X,k−1 is semisimple.
• The isotypic components of R X,k /R X,k−1 are of type N W as W runs over the subspaces of the arrangement of codimension k.
• The space R X,s /R X,s−1 is a free module over In fact a more precise statement holds. Denote by R s X the linear span of all the fractions X is a complement of R X,s−1 in R X,s and it is a free module over It is important to choose a basis.

Theorem 7 A basis for R s X ≡ R X,s /R X,s−1 over S[U ] is given by the elements a∈b a −1 as b runs over the set of unbroken bases.
Denote by N B the unbroken bases extracted from X . We have in particular an expansion of Example 5.2 A 2 also called Courant element in the Theory of splines.
Example 5.3 B 2 also called ZP element in the Theory of splines.
We now need the basic inversion. Let X = {a 1 , . . . , a n } be a basis, d := | det(a 1 , . . . , a s )| and χ C(X ) the characteristic function of the positive quadrant C(X ) generated by X .
From the basic example and the properties we get The same analysis for the ZP element leads to Fig. 1.

The theory of Dahmen-Micchelli
Given a vector a denote by D a the directional derivative, a first order operator. For a list Y of vectors set D Y := a∈Y D a , a differential operator of order |Y | with constant coefficients. We now fix, as usual, a list X of vectors spanning V . The graded dimension of D(X ) is given by H X (q) = b∈B(X ) q m−n(b) . The number n(b) (called external activity in the Theory of Matroids) is the number of elements of X breaking b.

Approximation
The aim of this section is to explain some motivations, for the study of box-splines, coming from numerical analysis.

The Strang-Fix conditions
The interest of the space of polynomials D(X ) comes in approximation theory from the problem of studying the approximation of a function f (x), on R s , by the finite element method (cf. Strang and Fix [33]). Using a box-spline we construct the approximations: Or at order n ∈ N: In general, given a function M(x) with compact support (as B X ) we want to find optimal weights c i (n) in order to approximate f by f (x) → i∈Z s M(nx −i/n)c i (n) and determine a constant k ∈ N so that on any given bounded region there is a constant C, independent of n with: The maximum k is the approximation power of M(x).
Given a spline M(x) on R s , with compact support one may define the cardinal spline space to be the space S M of all (infinite) linear combinations: The approximation power of M(x) is related to two questions: The power of approximation by discrete convolution is measured by the maximum degree of the space of polynomials which reproduce under discrete convolution.
6.1.1. Superfunctions. Consider the following algorithm applied to a function g: There are functions F in the cardinal spline space such that this transformation is the identity on polynomials of degree < m(X ), these are the super-functions. For such functions the previous algorithm satisfies the requirements of the Strang-Fix approximation Theorem 9 We have, under the explicit algorithm previously constructed that, for any domain G: For every multi-index α ∈ N s with |α| ≤ m(X ) − 1, we have: Theorem 10 When p ∈ D(X ) we have that also B X * p ∈ D(X ).
This defines a linear isomorphism F of D(X ) to itself, given explicitly by the invertible differential operator a∈X • If R is any difference operator inverting B X * − on D(X ) then R B X is a superfunction. • We can then take as R a truncation of Let us give as example some cases of the function s m (x) obtained by this procedure from b m (x).

The partition function
The partition function is a quasi polynomial on each big cell, in fact on the larger neighborhood c − B(X ) of the big cell c.
We think of the partition function P . There are explicit formulas relating the partition function P X with the multivariate splines T A , A ⊂ X . One could derive the Formula of Example 2.7 in this way.
We used hyperplane arrangements in order to study multivariate splines, for partition functions we use toric arrangements. The elements of the toric arrangement are ordered by reverse inclusion, particular importance is given to the points of the arrangement, P(X ) which are the zerodimensional, i.e. points, elements of the arrangement.
A very special case is when P(X ) reduces to the point 1, this is the unimodular case. For root systems the only unimodular case is A n .

The filtration
The formulas we are aiming at are built from basic functions. These are analogues, in the discrete case, of the Laplace transforms of the characteristic functions of the cones generated by bases extracted from X . Given a basis b of V = ⊗ R, contained in , we introduce the set Notice that this is a set of coset representatives of ∩ b / b . Given a character φ : → C * which is 1 on b set ξ φ is a tempered distribution supported on the cone C(b) and its Laplace transform has an analytic meaning and we get: .
We filter S X by polar order that is S X,k is formed by all fractions which have at the denominator a product a∈X (1 − e a ) h a so that the a with h a > 0 span a subspace of dimension ≤ k.
A connected component of the toric arrangement is of the form p P for some subtorus P. For each connected component p P we have an irreducible module generated by the delta function.
Concretely, if p P has codimension k we construct a copy N p P,b in S X,k /S X,k−1 as generated by the class of an element | det(b)| e(φ) as b is a basis for the span of the sublist X p P of the elements in X which are 1 on p P.
Theorem 11 S X,k /S X,k−1 , decomposes as direct sum of the modules N p P,b as p P runs over all components of the arrangement of codimension k and b over the unbroken bases on p P. For each p P, F p P is the isotypic component of type N p P .
Take the space of polar parts S P X := S X /S X,s−1 , its isotypic components are F φ indexed by points of the arrangement.
Let us consider the element v X , in S P X , class of the function a∈X (1 − e −a ) −1 . Decompose it uniquely as a sum of elements v X φ in F φ , each one of these elements is uniquely expressed as a sum The main formula that one can effectively use for computing the partition function P X :

Theorem 12 Let be a big cell, B(X ) the box associated to X . For every x ∈ − B(X ),
Finally we compute the contributions by residues. The isotypic components appearing in grade k correspond to the connected components of the toric arrangement of codimension k.
For the top part S X,n /S X,n−1 d we have a sum over the points of the arrangement P(X ).
The isotypic component associated to a point e φ decomposes as direct sum of irreducibles indexed by the unbroken bases in X φ := {a ∈ X | e a | φ = 1}.

Local structure of P X .
The previous formula shows in particular, that the partition function is on each cell a quasi polynomial, as for the case of the multivariate spline: The quasi polynomials appearing in the formula for P X satisfy special difference equations.
Denote by C[ ] the abelian group of Z-valued functions on .
Given X a list in spanning the ambient space we define as Y runs over all cocircuits of X .

Second Theorem of Dahmen-Micchelli. The dimension of DM(X ) is the Volume δ(X ) of the box B(X ):
This follows from a more precise statement.

The restriction of D M(X ) to Z valued functions on δ(r | X ) is an isomorphism.
In other words the values on δ(r | X ) of a function in DM(X ) determine uniquely this function and in particular DM(X ) is a free abelian group of rank δ(X ). In this case we only have one equation The space of solutions has as integral basis the elements Take a regular point close to 0, 0 < a < 1 then Formula (11) has a strict connection with the paving of the box.

Example 7.7 Let us take
See that δ(X ) = 1 + 1 + 1 + 1 + 1 + 2 = 7 is the number of points in which the box B(X ), shifted generically a little, intersects the lattice! 7.8 Combinatorics is a lattice in a real vector space V and X a list of vectors in . In coordinates V = R s , = Z s . Assume that X generates a pointed cone C(X ). We have already seen in Sect. 2.9 the notions of singular, regular points and big cells. We now extend these notions as follows: When we translate we have: We draw the zonotope and the cut-locus If X is a basis of the lattice we have that the chambers are open parallelepipeds.

Difference theorem
One of the main results of the Theory is the fact that a partition function P X is a quasi-polynomial on the regions − B(X ), for each big cell . The fact that P X is a quasi-polynomial not just on the big cells but in fact in the larger regions − B(X ) may be considered as a discrete analogue of the differentiability properties of the multivariate spline T X . In this section we discuss some properties of the quasi-polynomials describing P X .
• P X on − B(X ) coincides with an element of DM(X ).
• The element is uniquely determined by its initial values, that is the values it takes on δ(u | X ) where u ∈ is close to zero. • In this case δ(u | X )∩C(X ) = {0} so that, in order to coincide with P X on − B(X ) we must have that f (0) = 1 while f (a) = 0 for all non-zero elements of δ(u | X ). • These are the defining initial conditions. An idea of a possible proof will be given in Sect. 9.20.
Let us start to explain why the system of difference equations gives a system of recursion which allow to determine the values of a function f ∈ DM(X ) by the values that it takes in a set δ(u | X ). To begin notice that

Thus the recursive formula for a function
easily implies that a function satisfying ∇ 2 ∇ 3 f (x) = 0 is determined by the values that it takes on the interval [−4, 0].
A function f satisfying ∇ Y f = for all cocircuits Y extracted from X is determined by the values on a set δ(c 0 | X ) where c 0 is a given chamber.
• First = ∪ c δ(c | X ) as c runs over all chambers. • Next we want to prove that if f vanishes on a given δ(c | X ) for a chamber c then it also vanishes on δ(g | X ) for a chamber g which is adjacent to c. • Then we go by recursion.
The fact that the values of a function f ∈ DM(X ) are determined by the values it takes on a single set δ(c | X ) comes from a wall crossing formula when we pass from δ(c | X ) to δ(g | X ) for a chamber g which is adjacent to c.

Connection between D M(X ) and D(X ).
There is a formal machinery which allows us to interpret, locally around a point, difference equations as restriction to the lattice of differential equations, we call it the logarithm isomorphism. We have this for any module over the periodic Weyl algebra C[ ∂ ∂ x i , e x i ] as soon as for algebraic reasons (nilpotency) we can deduce from the action of e x i also an action of x i .
The main point is that the ideal J X generated by the elements a∈Y (1−e a ) as Y runs over the cocircuits defines (as a scheme) the set P(X ) of points of the arrangement. Thus Where C[ ]/J X ( p) is a local algebra supported at p. This implies that the dual of C[ ]/J X which is the complexified form decomposes as direct sum of contributions at the various points in P(X ). Choose representativesP(X ) for P(X ): where D φ is the space of polynomials f such that e φ | v f ∈ DM C (X ).
This explains why DM(X ) is formed by quasi-polynomials.
• if v ∈ V, the difference operator ∇ v and the differential operator • T v is invertible on the space of polynomials and commutes with all the ∇ a .
• Hence ∇ Y = AD Y where A is invertible on polynomials.
• The equations ∇ Y f = 0 are equivalent to D Y f = 0 for polynomials.
• Thus we obtain that e φ | v f ∈ DM C (X ) if and only if f ∈ D(X p ).
• In particular the contribution of the point 1 to DM C (X ) is the space of polynomials D(X ). • In the unimodular case DM C (X ) = D(X ).
• In general we also have the contribution of the other points, that will be denoted by

Supports.
We want now to explain the nature of DM C (X ) as Fourier coefficients of distributions on the compact torus T, supported at the points of the arrangement. For this we consider the space D(X ) (or D(X p )) as a space of polynomial differential operators on U and hence also on T with constant coefficients.
The Haar measure on T of total mass 1. This allows us to identify generalized functions on T and distributions on T .
Call DM(X ) the space of distributions on T of which DM C (X ) gives the Fourier coefficients.
We deduce

Proposition 8.4 DM(X ) is the direct sum of the spaces of distributions D(X p )δ p as p ∈ P(X ).
In particular we see that DM(X ) is supported at the finite set P(X ).

Wonderful models
There is an approach to compute the partition function based on residues (as for numbers in Sect. 2.4.1), also in higher dimension. In order to define multidimensional residues we need divisors with normal crossings. That is we consider functions which, in some coordinates x i , have poles only on the hyperplanes x i = 0, i = 1, . . . , s. The residue is the coefficient of i x −1 i . The construction of these divisors with normal crossings follows [16] and in our case starts with a Definition 9.2 Given a subset A ⊂ X := {x 1 , . . . , x s } the list A := X ∩ A will be called the completion of A. In particular A is called complete if A = A.
The space of vectors φ ∈ U such that a|φ = 0 for every a ∈ A will be denoted by A ⊥ . Notice that clearly A equals to the list of vectors a ∈ X which vanish on A ⊥ . From this we see that we get a bijection between the complete subsets of X and subspaces of the arrangement defined by X .
A central notion in what follows is given by Definition 9.3 Given a complete set A ⊂ X, a decomposition is a decomposition A = A 1 ∪ A 2 in non empty sets, such that: Clearly the two sets A 1 , A 2 are necessarily complete. We shall say that: a complete set A is irreducible if it does not have a non trivial decomposition.

Theorem 14 Every set A can be decomposed as A = A 1 ∪ A 2 ∪ · · · ∪ A k with the A i irreducible and:
This decomposition is unique up to order.

Example 9.4 An interesting example is that of the configuration space of s-ples of point in a line (or the root system A s−1 ). In this case
In this case, irreducible sets are in bijection with subsets of {1, . . . , s} with least 2 elements. If S is such a subset the corresponding irreducible is I S ={z j −z j |{i, j} ⊂ S}.
Given a complete set C, the irreducible decomposition of C corresponds to a family of disjoint subsets S 1 , . . . , S k of {1, . . . , s} each with at least 2 elements. Definition 9.5 A family S of irreducibles A i is called nested if, given elements A i 1 , . . . , A i h ∈ S mutually incomparable we have that C := A 1 ∪ A 2 ∪ · · · ∪ A i is complete and C := A 1 ∪ A 2 ∪ · · · ∪ A i is its decomposition into irreducibles.

Consider the hyperplane arrangement H X and the open set
complement of the union of the given hyperplanes.
Let us denote by I the family of irreducible subsets in X . We construct a minimal smooth variety Z X containing A X as an open set with complement a normal crossings divisor, plus a proper map π : Z X → U extending the identity of A X .
For any irreducible subset A ∈ I take the vector space V /A ⊥ and the projective space P(V /A ⊥ ).
Notice that, since A ⊥ ∩ A X = ∅ we have a natural projection π A : A X → P(V /A ⊥ ). If we denote by j : A X → U the inclusion we get a map The model There is a very efficient approach to computations by residue at points at infinity in the wonderful compactification of the associated hyperplane arrangement.
Points at infinity correspond to maximal nested sets. Around each such point one can consider a s-dimensional torus and its class in homology A basis of the homology or of the corresponding residues corresponds to the tori around special points indexed by unbroken bases. 9.6.1. The non linear coordinates. We now apply the previous Theory to the multivariate spline or the partition function associated to a list X . One can find explicit polynomials p b,X (x) (given in Formula (16)), indexed by the points at infinity associated to the maximal nested set generated by unbroken bases so that, given a point x in the closure of a big cell c we have Jeffrey-Kirwan residue formula [26].
There is a parallel theory for the partition function, as a result we can compute a set of polynomials q b,φ (−x) (given in Formula (17)) indexed by pairs, a character φ of finite order and a unbroken basis in The analogue of the Jeffrey-Kirwan formula is:

Theorem 15 Given a point x in the closure of a big cell c we have a Residue formula for partition function
Finally one can deduce the partition function from some combinatorics and multivariate splines (with parameters): Construct new coordinates z A , A ∈ S using the monomial expressions:

Theorem 16 For the points x in the interior of big cells c we have
The residue at the point 0 for these coordinates is denoted by res b .
Partition function. (17) The proof is easy applying formal properties of the residue. This gives a simple algorithm to compute the polynomials q b,X .

The abelian group F(X ).
We denote by C[ , r ] the subgroup in C[ ] consisting of the elements supported in ∩ r . The following abelian group will play a key role in what follows Notice that if f ∈ F(X ) then f must in particular satisfy the relation corresponding to the space r = {0}) that is with c ∈ Z. In particular F(X ) ⊂ R (the rational elements) and L r (F(X )) ⊂ C( ) is the one dimensional abelian group spanned by Example 9.9 Let us give a simple example. Let = Z, and X = [2, −1]. Then it is easy to see that F(X ) has as integral basis Here θ 1 , θ 2 , θ 3 is an integral basis of DM(X ).

Proposition 9.10 F(X ) is a free abelian group whose rank equals the number of integral points in the Zonotope B(X ).
Recall that, in Lemma 1.2 we have defined the partition functions P F X associated to the faces of the hyperplane arrangement given by X .
The first important fact on this abelian group is the following: (i) If F is a regular face for X, then P F X lies in F(X ). (ii) The abelian group DM(X ) is contained in F(X ).
Each P F X is a partition function. In particular, if X generates a pointed cone, then the partition function P X equals P F X for the face F which is positive on X .

Some properties of F(X ).
Let r be a rational subspace and F r be a regular face for X \r . Proposition 9.11 (i) The map g → P F r X \r * g gives an injection from F(X ∩ r ) to F(X ). Moreover (ii) ∇ X \r maps F(X ) surjectively to F(X ∩ r ).
(iii) If g ∈ DM(X ∩ r ), then ∇ X \t (P F r X \r * g) = 0 for any rational subspace t such that t ∩ r = r .

Recall that S (i)
X the set of rational subspaces of dimension i. Define the abelian groups Choose, for every rational space r , a regular face F r for X \r .

Theorem 18
With the previous choices, we have: Definition 9.12 A collection F = {F r } of faces F r ⊂ r ⊥ regular for X \r , indexed by the rational subspaces r ∈ S X will be called a X -regular collection.
Given a X -regular collection F, we can write, using Theorem 18, an element f ∈ F(X ) as This expression for f will be called the F decomposition of f . In this decomposition, we always have F V = {0}, P F V X \V = δ 0 and the component f V is in DM(X ).

Localization theorem
There is a more restricted notion than that of big cell, it is the notion of tope by this we mean a connected component of the complement of the union of all proper rational subspaces generated by subsets of X .
In this section we are going to discuss the fact that every element f ∈ F(X ) coincides with a quasi-polynomial on the sets (τ − B(X )) ∩ as τ varies over all topes (we simply say f is a quasi-polynomial on τ − B(X )). Definition 9.14 Let τ be a tope and r be a proper rational subspace. We say 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 such that u r , x 0 < 0. Given x 0 ∈ τ, it is always possible to choose a regular face F r ⊂ r ⊥ for X \r such that x 0 is negative on some vector u r ∈ F r , since the projection of x 0 on V /r is not zero. Let f ∈ F(X ) and let f = f r be the F decomposition of f . The following result is quite similar to Paradan's localization theorem [29].  Figure 7 describes the F decomposition relative to the tope containing x 0 . Notice how the choice of F has the effect of pushing the supports of the elements f r (r = V ) away from τ .
A quasi-polynomial is completely determined by the values that it takes on (τ − B(X )) ∩ . Thus f V is independent on the construction so: Definition 9. 17 We shall denote by f τ the quasi-polynomial coinciding with f on The open subsets τ − B(X ) cover V, when τ runs over the topes of V (with possible overlapping). Thus the element f ∈ F(X ) is entirely determined by the quasi-polynomials f τ .
If f ∈ F(X ), the element ∇ X \r f is in F(X ∩ r ) and coincides with a quasi-polynomial (∇ X \r f ) τ ∈ DM(X ∩ r ) on each tope τ for the system X ∩ r .

Theorem 20
Let β ∈ V be generic with respect to all the rational subspaces r. Let F β r be the unique regular face for X \r containing p r ⊥ β.  Remark 9.18 In [11], a one-dimensional residue formula is given for w allowing us to compute it.
On the left of the picture we show the intersection of C with the two topes τ 1 , τ 2 adjacent along the hyperplane H generated by b, d and, on the right, that with the tope τ 12 . The list X ∩ H is [b, d]. The closure of the tope τ 12 is "twice bigger " than τ 1 ∩ τ 2 .
Let f ∈ F(X ). The function ∇ X \H f is an element of F(H ∩ X ), thus there exists a quasi-polynomial (∇ X \H f ) τ 12 on H such that ∇ X \H f agrees with (∇ X \H f ) τ 12 on τ 12 .

Theorem 21
Let τ 1 , τ 2 , H, τ 12 be as before and f ∈ F(X ). Let F H be the half line in H ⊥ positive on τ 1 . Then In the case in which f = P X we deduce Paradan's formula [28,Theorem 5.2]

The partition function
We assume that C(X ) is a pointed cone. Let us now consider a big cell . Given a big cell , let τ 1 , . . . , τ k be all the topes contained in . Then: Now in order to prove the statement for big cells, we need to see what happens when we cross a wall between two adjacent topes by the previous formulas one has Theorem 22 On ( − B(X )) ∩ , the partition function P X agrees with a quasipolynomial P X ∈ DM(X ).
This theorem was proven [14] by Dahmen-Micchelli for topes, and by Szenes-Vergne [34] for cells. In many cases, the sets − B(X ) are the maximal domains of quasi-polynomiality for T X .
There is an important point of the theory of Dahmen-Micchelli that we point out.

Theorem 23 Let be a big cell contained in C(X ).
Then P X is the unique element f ∈ DM(X ) such that f (0) = 1 and f (a) = 0, ∀a ∈ δ(c | X ), a = 0.

Equivariant K -theory
We briefly review the notations for K -theory that we will use, for a systematic treatment see Atiyah [2] and Segal [32]. Let G be a compact Lie group acting on a locally compact space N .
• One has the notion of the equivariant topological K -theory group K 0 For N locally compact setṄ = N ∪ ∞ be the one point compactification. Representatives of the K -theory group K 0 G (N ) can be described in the following way.
Given two G-equivariant complex vector bundles E 0 , E 1 on N and a G-equivariant bundle map f : E 0 → E 1 , the support supp( f ) of f is the set of points where f x : E 0 x → E 1 x is not an isomorphism.
A G-equivariant bundle map f with compact support defines an element [ f ] of K 0 G (N ). All elements can be described this way. Let f : E 0 → E 1 and g : F 0 → F 1 be two G-equivariant bundle maps. Using G-invariant Hermitian metrics on the bundles E i , F i we can define: The support of f g is the intersection of the supports of f, g thus f g induces an element in K 0 G (N ) as soon as one of the two f, g has compact support. In particular this defines a product In general take the projection π : N → pt, given τ ∈ R(G) and σ ∈ K 0 G (N ), we have that [π * (τ ) σ ] ∈ K 0 G (N ) and this gives a R(G) module structure to K 0 G (N ).

Clifford action.
Let W be a Hermitian vector space and let E = W . For w ∈ W, consider the exterior multiplication m(w) : E → E and the Clifford action of W on W .
We have c(w) 2 = − w 2 , so that c(w) is an isomorphism, if w = 0. We shall use this as follows:

Bott symbol and Thom isomorphism.
If p : W → M is a G-equivariant complex vector bundle over a G-space M, the fiberwise Clifford action c(w x ) : even W x → odd W x defines a morphism c W : p * even W → p * odd W of vector bundles over W, called the Bott symbol.
Take a bundle map f : E → F of complex equivariant vector bundles on M which is an isomorphism outside a compact set, and denote still by f its pull back f : p * E → p * F. Then f c W is a bundle map of bundles over W, which is an isomorphism outside the support of f embedded in W via the zero section. We set Thom isomorphism The This class does not depend of the choice of χ . We still say that this class σ is the symbol of A.
LetĜ be the set of equivalence classes of finite dimensional irreducible representations of G, and let C[Ĝ] be the group of Z-valued functions onĜ. Let χ τ (g) = Tr(τ (g)) be the character of the representation τ ∈Ĝ of G. We associate to an element f ∈ C[Ĝ] a formal (virtual) character ( f ) = τ f (τ )χ τ , that is a formal combination of the characters χ τ with multiplicities f (τ ) ∈ Z. When f (τ ) satisfies certain moderate growth conditions, then the series ( f )(g) = τ f (τ )χ τ (g) converges, in the distributional sense, to a generalized function on G.
The It follows also from Atiyah-Singer [1] that the series τ m(τ, A)χ τ (g) defines a generalized function on G. Thus we may also associate to A the generalized function is defined to be the index of j * (σ ). The excision property of the index shows that this is independent of the choice of the open embedding j and thus allows us to define the index map also for manifolds which can be embedded as open sets of compact ones.
In particular, if V is a vector space with a linear action of a compact group G, then V is diffeomorphic to the sphere, minus a point. Thus we can define the index of any σ ∈ K 0 G (T * G V ). More generally, if U is an open G-invariant subset of a vector bundle on a compact manifold, we can define the index of σ ∈ K 0 G (T * G U ). The problem of computing the index can be reduced, at least theoretically, to the case in which G is a torus. For a given compact manifold M, one embeds M into a linear representation and then is reduced to perform the computations in the representation. which at the level of K 0 is compatible with the index.
On the other hand, the space G × H M identifies with S 1 × M via the map [g, m] → [χ(g), g · m]. So (27) gives us the isomorphism (28). Thus we can take k := i ! i G H . • The tangential Cauchy-Riemann operator.
Assume that the group G is an abelian compact Lie group. An irreducible representation a of G is a one dimensional complex vector space L a , where G acts via a character χ a : G → S 1 , so thatĜ is identified with the abelian group of characters, denoted by .
The space M X is a G-manifold and our goal is the determination of K 0 G (T * G M X ). The basic tool that we shall use is the space of functions DM (G) (X ) on . This is defined as DM(X ) (and often denoted also DM(X )) but in this more general case of not necessarily connected G. The extension to non connected abelian groups is necessary in order to perform induction. In this case has a torsion subgroup t and Theorem 13 gives the statement that DM (G) (X ) is a free module over the group ring of t or rank δ(X ).
With a given G-invariant Hermitian structure on M let S be the unit sphere of M. Let P(M) be the complex projective space of M. Consider on S the differential operator δ acting on the pull back of the Dolbeault complex on the associated projective space P(M) using ∂ + ∂ * : 0,2 p → 0,2 p+1 . Then δ is a G-transversally elliptic differential operator the tangential Cauchy-Riemann operator on S.
Indeed, using the Hermitian structure, identify T * S with its tangent bundle T S ⊂ T M, the subspace H p of T * p S orthogonal to the line RJ u p is then identified to the complex subspace of M, orthogonal under the Hermitian form to p. We call it the horizontal cotangent space. The symbol of δ is σ ( p, ξ) = c(ξ 1 ) where ξ 1 is the projection of ξ on H p , and c the Clifford action of H p on H p . This morphism is invertible if ξ 1 = 0. We have also H p ⊕ Rρ(u) p = T * p S, as the eigenvalues of −iu on M are all positive. Thus we see that σ ( p, ξ) restricted to T * G S is invertible outside the zero section.

Theorem 24
Let M be provided with the complex structure J u and let δ be the tangential Cauchy-Riemann operator, on the unit sphere of M. Then where a X = a∈X a.
We recall briefly the proof. Let S 1 be the circle group acting by homotheties on M. We decompose solution spaces with respect to characters t → t n of S 1  Let us show that Thus the support of At u (v, ξ ) restricted to T * G M is the unique point v = 0, ξ = 0 and At u determines an element of K 0 G (T * G M), which depends only of the open face F of u in the hyperplane arrangement dual to X . We denote it by At F . The index of At F is computed in [1] (Theorem 8.1). In more detail, in the Appendix of [9], it is constructed an explicit G-transversally elliptic pseudo-differential operator A on the product of the projective lines P(L a ⊕ C).
If j : M X → a∈X P(L a ⊕ C) is the natural open embedding, it is shown that j * (At F ) is homotopic to the symbol of A. By definition, the index of At F is that of A and one has the explicit formula:

The index for M X
This section contains the main results on the index for the space M X , that is Theorems 28 and 29.

K -theory
Let G be, as before, a compact abelian Lie group of dimension s and M X := ⊕ a∈X L a as in (30). We assume that X has rank s. Given a vector v ∈ M X , its support is the sublist of elements a ∈ X such that v has a non zero coordinate in the summand L a .
If Y is the support of v, an element t of G stabilizes v if and only if t a = 1 for all a ∈ Y . If Y spans a rational subspace of dimension k, the G-orbit of v has dimension k.
For any rational subspace r , we may consider the subspace M r := ⊕ a∈X ∩r L a of M X . We set Definition 11.2 Given a rational subspace r , we denote by G r the subgroup of G joint kernel of the elements in ∩ r . The group G r is a torus and acts trivially on M r := ⊕ a∈X ∩r L a inducing an action of G/G r .
We define the set F(r ) to be the open set of M r where G/G r acts with finite stabilizers.
In other words, the connected component of the stabilizer of an element of F(r ) is exactly the group G r . Lemma 11.14 For each s ≥ i ≥ 0, the index multiplicity map ind m sends So Theorem 32 is crucial in establishing a cohomological formula valid for any transversally elliptic operator.

Equivariant cohomology and infdex
The purpose of this section is to establish cohomological formulas for the index of transversally elliptic operators (cf. [22] and [21]). We do this through a new invariant, the infinitesimal index.
This construction is motivated by taking the Fourier transform of the formula of Berline-Vergne for the equivariant index of a transversally elliptic operator [8,9,27] where one can also find the various notations and definitions. As we shall see the combinatorial spaces DM(X ) and F(X ) appearing in K -theory and index Theory will be replaced by spaces of distributions completely analogues to the ones appearing in the Theory of splines and giving rise to precise analogues.

Equivariant de Rham cohomology
Let M be a C ∞ manifold with a C ∞ action of a compact Lie group G, we are going to define its equivariant cohomology with compact support following Cartan (see [24]).
We define the space of compactly supported equivariant forms as with the grading given setting g * in degree 2. Here A c (M) is the algebra of differential forms on M with compact support.
Each element x ∈ g of the Lie algebra of G induces a vector field v x on M, the infinitesimal generator of the action: here the sign convention is that v x = d d exp(− x)·m in order that the map x → v x be a Lie algebra homomorphism. A vector field V on M induces a derivation ι V on forms, such that ι V (d f ) = V ( f ) and for simplicity we denote by ι x = ι v x . One defines the differential as follows. Given α ∈ A G,c (M), we think of α as an equivariant polynomial map on g with values in A c (M), thus for any x ∈ g we set where d is the usual de Rham differential. It is easy to see that D increases the degree by one and that D 2 = 0. Thus we can take cohomology and we get the G-equivariant cohomology of M with compact support. Notice that A G,c (U ) is an ideal in A G,c (M) so A G,c (Z , M) is a differential graded algebra and H * G,c (Z ) is a graded algebra (without 1 if Z is not compact). In this model, a representative of a class in H * G,c (Z ) is an equivariant form α(x) with compact support on M. The form α is not necessary equivariantly closed on M, but there exists a neighborhood of Z such that the restriction of α(x) to this neighborhood is equivariantly closed.

Action form and the moment map
Let G be a Lie group and M a G-manifold.

Definition 13.4 An action form is a G-invariant real one form σ on M.
The prime examples of this setting are when M is even dimensional and dσ is non degenerate. In this case dσ defines a symplectic structure on M.
Example 13.5 For every manifold N , we may take its cotangent bundle M := T * N with projection π : T * N → N . The canonical action form σ on a tangent vector v at a point (n, φ), n ∈ N , φ ∈ T * n N is given by In this setting, dσ is a canonical symplectic structure on T * N and, if r = dim(N ), the form dσ r r ! determines an orientation and a measure, the Liouville measure on T * N . If a group G acts on N , then it acts also on T * N preserving the canonical action form and hence the symplectic structure and the Liouville measure.

Remark 13.6
If M is a manifold with a G-invariant Riemannian structure, we can consider an invariant vector field instead of a 1-form.
Let v x be the vector field on M associated to x ∈ g and i x the derivation on forms induced by contraction with v x . Definition 13.7 Given an action form σ we define the moment map μ σ : M → g * associated to σ by: for m ∈ M, x ∈ g.
The moment map is related to the equivariant differential of σ as defined in the Cartan model (see Formula (38)).

Infinitesimal index
We fix a translation invariant Lebesgue measure dξ on g * . We choose a square root i of −1 and define the Fourier transform: We normalize dx on g so that the inverse Fourier transform is The measure dxdξ is independent of the choice of dξ . If f (ξ ) is a C ∞ function on g * with compact support in a ball B R of radius R in g * (for a choice of Euclidean structure on g * ) its Fourier transformf (x) is a rapidly decreasing function on g.
We formulate now the general index theorem. We denote by P(X ) ⊂ G the points of the arrangement.
Theorem 40 Let X be a sequence of elements in and let M := M X . Let Let be a G-invariant transversally elliptic symbol on M and ind m ( ) ∈ C Z [ ] be its multiplicity index.
For any g ∈ P(X ), let infdex −μ G (ch g ( )) be the distribution on g * associated to the cohomology class ch g ( ) ∈ H ∞,m G,c (T * G M g ) by the infinitesimal index. Then is a piecewise polynomial measure on g * . • Let c be an alcove having 0 in its closure. We have Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.