{'fraction_non_alphanumeric': 0.055704225352112675, 'fraction_numerical': 0.007652582159624413, 'mean_word_length': 3.7439866369710466, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 14, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Geometric realizations and duality for Dahmen-Micchelli modules and De Concini-Procesi-Vergne modules

We give an algebraic description of several modules and algebras related to the vector partition function, and we prove that they can be realized as the equivariant K-theory of some manifolds that have a nice combinatorial description. We also propose a more natural and general notion of duality between these modules, which corresponds to a Poincar\'e duality-type correspondence for equivariant K-theory.


Introduction
In recent years, the multivariate spline and the vector partition function have been studied by several authors. While the former is an essential tool in Approximation Theory, the latter has been studied in Combinatorics at least since Euler. Although they may seem quite different in nature, they can be viewed as the volume, and the number of integer points respectively, of a variable polytope; thus the partition function is the discretization of the spline. In their book [15], De Concini and Procesi brought these functions to the attention of geometrists, showing their relation with hyperplane arrangements and toric arrangements.
These functions are piecewise polynomial/quasi-polynomial respectively, meaning that their support can be divided in regions called big cells, such that on every big cell Ω, the spline agrees with a polynomial p Ω , and the partition function agrees with a quasi-polynomial q Ω .
The polynomials p Ω , together with their derivatives, form a vector space D(X). In a more algebraic language, D(X) is generated by the elements p Ω as module over the ring of polynomials, acting as derivations. On the other hand, the elements q Ω generate DM (X), as a module over the Laurent polynomials, acting as difference operators. The Dahmen-Micchelli modules (DM modules for short) D(X) and DM (X) are defined by a system of differential/difference equations respectively, having a simple combinatorial description in terms of the cocircuits of the associated matroid ( [12,13,14]).
For every big cell Ω, it is also natural to consider D Ω (X), the cyclic submodule of D(X) generated by the single polynomial p Ω , and DM Ω (X), the cyclic submodule of DM (X) generated by the quasi-polynomial q Ω . We call them the local DM modules.
Furthermore, studying the partition function and the spline led to the definition of two flags of modulesF i (X) andG i (X), of which DM (X) and D(X) are the smallest elements ( [16,17]). We call the modulesF i (X) and G i (X) the De Concini-Procesi-Vergne modules (or DPV modules for short).
The DM modules and the local DM modules naturally come together with their dual modules D * (X), D * Ω (X), DM * (X), DM * Ω (X), all described as quotients of the ring of polynomials by suitable ideals, and hence endowed by a structure of algebras. As we show in Theorem 3.7, this duality can be viewed in a more natural way via an Ext functor. This approach also allows us to define the dual DPV modules F * i (X), G * i (X) (see Definitions 3.9 and 3.11).
Surprisingly, all the modules and algebras above appear as invariants of geometric objects. In particular, the modules D(X) and D Ω (X) can be "geometrically realized" as the equivariant cohomology of two differentiable manifolds, while their discrete counterparts DM (X) and DM Ω (X) can be "geometrically realized" as the equivariant K-theory of the same manifolds.
The construction goes as follows. Let X be the list of vectors in Z d that defines the spline and the partition function. Each element of X defines a 1-dimensional representation of the torus G = (S 1 ) d , hence there is a representation M X which is the direct sum of all them. Let M f in Here the duality between D(X) and D * (X) is realized by the Poincaré duality between the ordinary cohomology H and compact support cohomology H c of the toric orbifold M f in X /G C (see Section 2.7). The first aim of this paper is to provide an analogue of this result for discrete DM modules; this is done by considering equivariant K-theory. However, this is intrinsically a compact support cohomology theory, thus lacking of a natural non-compact-support counterpart. Instead of looking for a different definition of equivariant K-theory non involving compact support, our approach will be based on compactifying the manifold M f in X . In fact we intersect it with the equivariant unit sphere and then remove open tubular neighborhoods of every resulting hypersurface, thus obtaining a compact manifold with corners S f in X . Then we have: c) K * G (M f in X ) ∼ = DM (X); d) K * G (S f in X ) ∼ = DM * (X). Facts a), b) and c) are consequences of statements proved by De Concini, Procesi and Vergne for DPV modules, that we will recall in Theorems 5.1, 5.2, 5.3. In particular the proof of a), given in [17], is based on index theory of transversally elliptic operators, and answers to a question that Atiyah raised about four decades ago ( [4]). The proof of c), given in [18], required the development of a further tool, the infinitesimal index. In this paper we prove Theorem 5.4 for dual DPV modules, that implies fact d).
The second aim of this paper is to provide an analogue of statement c) for the local DM modules: this is Theorem 6.4.
which has a nice combinatorial description; the algebraic property that the R(G)−modules DM Ω (X) generate DM (X) is reflected in the geometrical fact that the pushforward . We actually prove a more general statement (Theorem 6.2), by using Atiyah's index.
As we stated in Conjecture 6.3, we believe that the same path may be followed for the differentiable counterpart, by using the infinitesimal index; this will provide geometric realizations for the local modules D Ω (X).
We think that also other objects, such as the modules arising from the power ideals studied by Ardila and Postnikov [3], the semi-internal zonotopal modules introduced by Holtz, Ron and Xu in [25], and their generalizations proposed by Lenz [27], may admit a similar geometric realization, that we hope to study in future papers. = Hom(G, S 1 ) be its character group. We will assume for simplicity that G is connected, i.e., it is a torus. Let g be the Lie algebra of G: this can be identified to the tangent space at 1 of G, so that the differential dλ of every character λ is an element of the dual space g * ; by a slight abuse of notation we will sometimes identify dλ with λ. We denote by S[g * ] the symmetric algebra of polynomial functions on g, and by R(G) the character ring of G, that is, the group algebra of Γ.
For the sake of concreteness, let us say that g * (as well as g) is a real vector space V , and if we denote its dimension by d, G is isomorphic to (S 1 ) d , while Γ is isomorphic to Z d . S[g * ] is isomorphic to the ring of polynomials All the results in this paper may be extended to the case of G not being connected. In this case G is isomorphic to the product of a compact torus (S 1 ) d and a finite group G f , and Γ is isomorphic to Z d × G f . Then we have a projection Γ → V which forgets the torsion part of Γ.
Let C[Γ] be the space of Z-valued functions on Γ. On this space every element a ∈ Γ acts as the translation τ a ; this extends to an action of R(G). We define the difference operator ∇ a = 1 − τ a , i.e.: Let X = [a 1 , . . . , a n ] be a finite list of elements of Γ. We will always assume that rk(X) = rk(Γ), and that none of the elements of X is zero in V (otherwise, it is simple to reduce to this case).
Following [16], we say that a linear subspace s ⊆ V is rational if it is spanned by elements of X; we denote by R X the set of all rational subspaces.
With a little abuse of notation, we will indicate by s both the linear subspace of V and the sublattice of Γ generated by the same elements.
2.2. DM modules. We recall that A ⊂ X is a cocircuit if A = X \ s for some rational hyperplane s (i.e. for some s ∈ R X such that codim(s) = 1). Let us define the set For every A ⊆ X, we consider the difference operator ∇ A = a∈A ∇ a acting on C[Γ]. The discrete DM module is defined as This can be seen as the "discretization" of the differentiable DM module where the differential operator ∂ A = a∈A ∂ a is just the product of directional derivatives. Of course, as in many of the definitions that will follow, it is enough to check the equations above for the minimal elements of L(X), that is, the cocircuits. The space D(X) is naturally a module over S[g * ] ≃ R[x 1 , . . . , x d ], the action being given by derivation: , the action being given by difference operators: = ∇ x i f . The vector space underlying to D(X) is the space D(X) studied in [24]. We say that the list X is unimodular (or totally unimodular ) if every k ×k minor of X is equal to 1, −1 or 0, for every 1 ≤ k ≤ d; in other words, if every bases taken out from X generates the whole lattice Γ. In this case it is easy to see that D(X) = DM (X) ⊗ Z R.
Remark 2.1. One can see from the definition that D(X) essentially depends on the linear algebra of the vectors in X, while DM (X) also depends on the arithmetic of the vectors. The need to encode this arithmetic information in a combinatorial object led to the introduction of arithmetic matroids ( [10,8]) and matroids over Z ( [21]).
As proved in [11] and [12] respectively, the dimension of D(X) (as a real vector space) is equal to the number of bases which can be extracted from X, while the rank of DM (X) (as a free Z-module) is equal to the volume of the zonotope These spaces were introduced in order to study two important functions, that we are going to describe in the next subsection.

2.3.
Vector partion function and multivariate spline. For every λ ∈ Γ, we define P X (λ) as the number of ways we can write Since we want this number to be finite, we assume that all the elements a i of the list X lie on the same side of a hyperplane in V . We can always do that, eventually replacing some vectors by their opposites. We call P X (λ) the vector partition function, or simply the partition function. The equation above is indeed a system of diophantine equations, one for every coordinate of λ. We can rewrite this system as Xx = λ, where X is the matrix whose columns are the vectors a i ∈ X, and x is the vector whose entries are the variables x i .
This defines a subspace of R n . The intersection of this subspace with the positive orthant is a variable polytope P X (λ) = x ∈ (R ≥0 ) n | Xx = λ and the partition function is the number of its integer points: Then P X (λ) is related with another function: M X (λ) . = vol (P X (λ)). Indeed the number of integer points of a polytope the "discrete analogue" of its volume. The function M X (λ), which is well defined for every λ ∈ R n , is known as the multivariate spline (or simply the spline). Splines are used in Numerical Analysis to approximate functions. The word "spline" means that M X is piecewise polynomial and "as smooth as possible"; we will now make more precise this statement.
First of all, notice that both the functions M X , P X are supported on the cone For every cocircuit A, we consider the cone C(X \ A) spanned by X \ A. We define a big cell as a connected component of Then we have: Theorem 2.2 (de Boor-Hollig). For every big cell Ω, there is a polynomial p Ω ∈ D(X) such that M X and p Ω coincide on Ω. Moreover, all the polynomials p Ω have degree n−d and M X is continuous of class C n−d−1 (V ).
Since P X is the discretization of M X , it is natural to wish a discrete analogue of the theorem above. Such an analogue exists, but with two important differences.
The first can be understood by looking at the 1-dimensional example given by the list [2, 1] ∈ Z. Here P X (λ) = λ/2 + 1 when λ is even, and P X (λ) = λ/2 + 1/2 when λ is odd. Then, in general, we recall the following definition: a function q : Γ → Z is a quasi-polynomial if there exist a finite index subgroup of Γ such that on every coset, q coincides with a polynomial. As we will see, the partition function is piecewise quasi-polynomial.
The second issue is what a "discrete analogue" of smoothness should be. The natural idea is to require that the regions of quasi-polynomiality overlap a bit, that is, given two neighboring big cells, there is a stripe on which the corresponding quasi-polynomials agree.
More precisely we have: . For every big cell, there is a quasipolynomial q Ω ∈ DM (X) such that P X and q Ω coincide on Ω. Moreover, they coincide on a larger region, the Minkowski sum of Ω and −Z(X).
The two theorems above motivate the interest for the modules D(X), DM (X). In fact these modules contain the "local pieces" p Ω , q Ω respectively; more precisely, D(X) is the S[g * ]−module generated by the polynomials p Ω , and DM (X) is the R(G)−module generated by the quasipolynomials q Ω , where Ω ranges over all the big cells.

2.4.
Local DM modules and their generalizations. For every big cell Ω, it is also natural to consider D Ω (X), the cyclic submodule of D(X) generated by the polynomial p Ω , and DM Ω (X), the cyclic submodule of DM (X) generated by the quasi-polynomial q Ω . These modules admit a simple combinatorial description. Let us define Notice that L Ω (X) contains all the cocircuits and is closed under taking supsets. We have: Proof. If T contains all the cocircuits, then DM T (X) ⊆ DM (X), because it is defined by the same difference equations, plus further ones. Now let us assume that T does not contain a cocircuit A, and let s be a rational hyperplane such that A = X \ s. Then we will show that any function that is constant on s and on all its translates belongs to DM T (X). These functions are annihilated by all the operators ∇ a with a ∈ s, and then by all the ∇ B for all B such that B A; now, notice that all elements of T satisfy this condition, so that all these functions belong to DM T (X). Now, being s a proper subspace, its cosets in Γ are infinitely many, so such functions are a subset of DM T (X) of infinite rank over Z (in fact, a basis of it consists of uncountably many elements).
The same proof holds for D T (X).

DPV modules.
Altough DM (X) contains all the local pieces q Ω of the partition function P X , it does not contain P X itself. In fact all the elements of DM (X) are genuine quasi-polynomials, while the partition function is piecewise quasi-polynomial. It is then desirable to have a space that contains both DM (X) and P X . This is the Z-module By "supported on s" we mean that the support of f (i.e., the subset of the domain on which f takes nonzero values) is contained in s. This space comes with a natural filtration Clearly, these Z-modules are not invariant for the action by translations of R(G) (because after a translation ∇ X\s f is going to be supported on a translate of s). In fact they generate the discrete De Concini-Procesi-Vergne modules (discrete DPV modules for short) | ∇ X\s f = 0 for every rational proper subspace s of dimension < i ∇ X\s f is supported on a finite number of translates of s otherwise .
The space F(X) is a free Z-module of rank equal to the number of integer points of the zonotope Z(X). On the other hand,F(X) andF i (X) have clearly infinite rank over Z. All these spaces have been introduced and studied in [16,17].
In the same way, we can define a real vector space G(X) containing both the multivariate spline M X and the space D(X), with a filtration G i (X). The definitions are exactly the same, except for ∇ that is replaced by ∂. Again, these vector spaces generate the differentiable DPV S[g * ]-modules G(X),G i (X) respectively (see [19]). These modules have a natural grading, which is given by the degree of the homogeneous polynomials (or piecewise polynomials).
Remark 2.5. Interestingly, the external zonotopal space D + (X) studied in [24] has the same dimension as G(X) over R. When the list X is unimodular, this dimension is equal to the number of integer points of the zonotope Z(X). However, D + (X) and G(X) are different spaces: the former contains polynomials, the latter distributions. It is not surprising, then, that only the first space is closed under derivations, i.e., is an S[g * ]-module. It would be interesting, nevertheless, to establish some canonical correspondence between these two spaces.
The same considerations hold for the semi-external zonotopal spaces D + (X, I i ) studied in [25], when I i is the family of linearly independent sublists of X of rank at least i (0 ≤ i ≤ d). In fact, these spaces have the same dimension as the spaces G i (X).
2.6. Some representations. For every rational subspace s ∈ R X , let G s be the subgroup of G of the elements that are annihilated by all the characters in Γ ∩ s. Notice that dim(G s ) = codim(s).
To every element a of Γ correspond a 1-dimensional representation M a of G, on which every g ∈ G acts as the multiplication by the scalar a(g). Then for every list X of elements of G we consider the representation which has dimension n over C.
Given a G-invariant Hermitian product on M X , let S X be the unit sphere. This is a G-manifold of dimension 2n − 1 over R.
For every A ⊂ X we can consider the coordinate subspace M A ⊂ M X given by Notice that the stabilizer of M s . = M X∩s is G s . Given any subset T ⊆ 2 X closed under taking supsets, we can consider In particular when T = L(X), we get Notice that in the formula above it is sufficient to take the first union over the minimal elements (i.e. cocircuits) and the second union over the maximal elements (i.e. rational hyperplanes).
In the same way, we can stratify M X by dimension of orbits, and for every i ≤ d call M i X the subset of points having i-dimensional orbit. Then the subset M ≥i X of points whose orbit is i-dimensional or more is Then we have: 2.7. Toric varieties and orbifolds. For every big cell Ω, we have defined a subset M Ω X of M f in X . We now look at its quotient by the action of G C ≃ (C * ) d , the algebraic torus which is the complexification of the compact torus G.
Proposition 2.6. M Ω X /G C is a projective toric variety. Furthermore, M Ω X /G C is an orbifold, and it is smooth whenever X is unimodular.
(see for example [17,Section 8] or [19,Remark 3.8]). We recall that an orbifold is a space that is locally homeomorphic to the quotient of a vector space by a linear action of a finite group, with the usual compatibility relations. In particular, this means that an orbifold has just "mild" singularities.
When Ω ranges over the big cells, the sets M Ω is still a smooth (or orbifold) toric variety, but in general it is not separated, and hence not projective, as the example in Section 2.8 shows.
These toric varieties provide a new insight on our geometric realizations; in fact, instead of looking at equivariant cohomology, we can take the cohomology of the quotient: and the same for cohomology with compact support: . Thus, as we will see in Section 5, the algebraic duality between D(X) and D * (X) is realized by the Poincaré duality between ordinary cohomology and compact support cohomology (see for example [7,Chapter I.5]).
Remark 2.8. Unfortunately, the analogous relations of (2) do not hold for K-theory and equivariant K-theory, unless the list X is unimodular. However, this problem can be addressed by replacing equivariant K-theory by the orbifold K-theory studied in [2]. See in particular [1, Proposition 6.5].
2.8. An example. Let us take the list where k is a positive integer. Then we have two big cells; let us call Ω the one whose extremal rays are spanned by the vectors (1, 0) and (k, k) and Ω ′ the other one.
The torus G C = (C * ) 2 acts on M X = C 3 by (t, s).(z 1 , z 2 , z 3 ) = (tz 1 , sz 2 , t k s k z 3 ). Since the cocircuits are the three couples of vectors, we have that and DM (X) have a basis given by the three functions x, y and 1, over R and Z respectively. A basis of the local modules corresponding to Ω is given by y and 1, while a basis of the local modules corresponding to Ω ′ is given by x and 1.
The subset M Ω X is given by the condition z 1 z 2 = 0 or z 1 z 3 = 0, that is M Ω X = {(z 1 , z 2 , z 3 ) ∈ M X | z 1 = 0 and (z 2 , z 3 ) = (0, 0)}. Therefore M Ω X /G C ≃ P 1 (C), and similarly for Ω ′ . These projective lines intersect in an affine line, hence their union M f in X /G, which can be seen as a "projective line with double point at infinity", is not separated.
If k > 1, D(X) is unchanged while DM (X) is the free Z−module of rank 2k + 1 = vol Z(X) that is spanned by x, y and by all the functions that are constant in one of the two variables and k−periodic in the other. Furthermore M Ω X /G C and M Ω ′ X /G C are isomorphic to the weighted projective space P 1,k (C). The module DM Ω (X) ≃ K * G (M Ω X ) has rank k + 1 over Z.

Duality of modules
In this section we describe some duality relations between DM and DPV modules, that will be reflected in duality statements in equivariant K-theory. In the same way we can define an ideal of R(G) as Notice that these ideals are annihilators of D T (X) as S[g * ]-module and of DM T (X) as R(G)-module respectively. Then we define: By definition these are a S[g * ]-module and a R(G)-module respectively; but, unlike D T (X) and DM T (X), they also have a multiplicative structure, i.e they are algebras.
In particular when T = L(X) we get two algebras that we will denote by D * (X), DM * (X), and call the dual DM modules. While when T = L Ω (X) for a big cell Ω, we will denote the two corresponding algebras by D * Ω (X), DM * Ω (X).
Remark 3.1. The ideal J ∂ L(X) has been studied in [24], where is denoted by J (X). The vector space underlying to D * (X) is the space therein denoted by P(X), which is defined using a power ideal associated to X. Proof. Clear by definition: the homomorphism D * T (X) → R are the homomorphisms S[g * ] → R that are zero on J ∂ T , which correspond precisely to the polynomial functions on V = g * that satisfy the defining conditions for D T (X). Proof. Since Γ is a set of generators of R(G) as Z-module, we have an isomorphism of R(G)-modules Then, we can see by definition DM T (X) inside C[Γ] as the vanishing locus of the endomorphisms given by ∇ A for A ∈ T ; and we can see naturally as well Hom Z (DM * T (X), Z) sitting inside Hom Z (R(G), Z) as vanishing locus of the same elements of R(G), because DM * T (X) is the quotient of R(G) by these elements.

3.2.
Ext functor and duality. From an algebraic point of view, the duality described above is not completely satisfactory. In fact, since we are dealing with S[g * ]-modules and R(G)-modules, it would be more natural to have a duality involving these rings. Furthermore, in the case of DPV modules the attempt to build duals via the functors Hom R (·, R) and Hom Z (·, Z) does not give good results, because these modules have infinite rank (over R and over Z respectively).
On the other hand, defining a duality via the functors Hom S[g * ] (·, S[g * ]) and Hom R(G) (·, R(G)) would not yield desiderable results neither: in fact, we have Hom R(G) (DM * T (X), R(G)) = 0, as one can easily see by the fact that S[g * ] and R(G) are domains. Hence we propose to take a more abstract perspective, realizing the duality via the functors Ext * S[g * ] (·, S[g * ]) and Ext * R(G) (·, R(G)). The advantages of this choice will be multiple: it gives rise to a genuine duality in the category of finitely generated S[g * ]-modules (or R(G)-modules respectively), which corresponds to the Verdier duality in the derived categories. Furthermore, this algebraic duality is reflected in (and inspired by) the geometric duality the we will describe in the next Sections. Finally, this notion allows to build duals also for the DPV modules.
We recall that Ext * is the collection of the Ext i , which are the derived functors of Hom. In particular, they can be nonzero only for 0 ≤ i ≤ d (0 ≤ i ≤ d + 1 in the discrete case) and Ext 0 is Hom itself.
First of all, we check that we are actually extending the duality defined in the previous Subsection. In fact we have: Proposition 3.4. Let V be an S[g * ]-module, that is finite dimensional as a vector space. Then we have the isomorphisms of S[g * ]-modules: Proof. We are going to prove the two statements together. We will indicate by R the ring S[g * ] when dealing with the first statement and the ring R(G) when dealing with the second, denoting by x 1 , . . . , x d the variables in both rings. In the same way, F will stand for R in the first case and for Z in the second. Let {e 1 , . . . , e s } be a basis of M as a free F-module, and let us denote by M R the module R s , with a basis denoted by {p 1 , . . . , p s }, and the surjective R−linear map M R δ 0 − → M sending p i to e i . Note that the action Ψ of R on M is determined by specifying the d matrices Ψ(x i ); these have to be s × s matrices with coefficients in F, and commuting one with each other. In the second (i.e. discrete) case, we also need them to be invertible in GL s (Z).
These matrices also act on M R ; in particular, in M R the operators x i and Ψ(x i ) are different, one acting as a constant, one as linear transformation. Furthermore, in the module Hom F (M, F), denoting by {e * 1 , . . . , e * s } the Fdual basis, the action of R is given by the transpose matrices t Ψ(x i ). Starting from δ 0 , we will create a free resolution of M as R-module in which we need to describe the maps δ i : we are going to describe these maps block by block, using the decomposition The reason of this choice is, at first, to have the image of δ 1 in M R generated by all the vectors of the form x b j v − Ψ(x b j )v, that indeed is the kernel of δ 0 . As any other kind of de Rham complex, proving exactness is only a matter of symbol chasing; crucial point to prove it is that matrices Ψ(x i ) commute each other. With this free resolution, we can explicitly evaluate the Ext i functors, as cohomology of the following complex where the maps are called t δ i because the matrices giving them are exactly the transposes of the one given in the previous complex. Now, as we already noticed, Hom R (M, R) is 0; moreover, we have that and up to sign changes the complex is precisely the same as before, except for using t Ψ(x i ) in the maps t δ i . Therefore, in the end, we get that the sequence is exact, besides at the last spot, in which the cokernel (indeed, Ext d R (M, R)) is a module t M in which R acts in a transpose way (by transpose matrices), that as we have seen before is exactly Hom F (M, F). Remark 3.6. As a complex in the dual category of S[g * ]-modules (resp. R(G)-modules), the (derived) dual of any finite dimensional R-vector space (resp. free Z-module) has cohomology only supported in degree d; for the same reason, in some categories the definition of the dual includes also a shift in the derived category. This happens for instance for the category of coherent algebraic D-modules over an algebraic variety Z. It is interesting to remark that in this setting we have an analogue of Propositions 3.4 and 3.5: if M is an integrable connections (that is, a coherent D-module that is also a coherent sheaf), than its dual as D-module (obtained as Ext dim(Z) (M, D Z ) because of the shift) is isomorphic to its dual as coherent sheaf, that is, Hom(M, O Z ); for all details, see [26,Chapter 2.6].
We can now focus on the modules DM T (X) and D T (X), which are free and (under the assumption that T contains all the cocircuits) have finite rank over Z and R respectively. We have: Proof. By Lemma 2.4 D T (X) has finite dimension over R. Then by Lemma 3.2 also D * T (X) has finite dimension over R. Therefore statements i) and ii) follow from Proposition 3.4, since for every finite dimensional vector space In the same way, by Lemmas 2.4 and 3.3, DM T (X) and DM * T (X) have finite rank over Z. Then claims iii) and iv) follow from Proposition 3.5, since Hom Z ((Hom Z (M, Z), Z) ∼ = M for every free module of finite rank over Z.

3.3.
Duality for DPV modules. Since DPV modules have not finite rank, we can not invoke Lemmas 3.4 and 3.5. In order to study duality of these objects, we need first to better examine their structure.
We will treat only the discrete case, since everything is the same in the differentiable setting.
From [16], we get that a canonical isomorphism Here DM (X ∩ s) is a submodule of C[Γ ∩ s], the module of Z-valued functions on the lattice Γ ∩ s, which can be identified to functions in C[Γ] that are supported in Γ ∩ s.
By definition (see Section 2.6) the corresponding group is G/G s , so that the dual space DM * (X ∩ s) is a quotient of R(G/G s ).
Furthermore, the quotient actually splits in a non-canonical way, meaning that we have isomorphisms (depending on a choice of some bases) By taking on both sides the R(G)-modules generated, we get the isomorphisms Given this decomposition, we prove the following Lemma.
Lemma 3.8. We have the following isomorphisms: , R(G)) = 0 otherwise. Proof. Given the splitting in Formula (3), we can work on each component separately; in this way, we just need to check that Ext k R(G) (R(G)DM (X ∩ s), R(G)) = R(G) ⊗ R(G/Gs) DM * (X ∩ s) when k = dim(s), and 0 otherwise. By applying Theorem 3.7, everything follows from the sequence of isomorphisms Thus for DPV modules several Ext i s can be nonzero and must be taken into account. We give the following definition. .
We believe that this is a good definition for many reasons. First, the ideal J ∇ i is the annihilator of the R(G) module F i (X); this can be shown using the decomposition (3) above. Therefore F * i (X) is defined is the same spirit as the dual DM module DM * (X).
Second, this algebraic notion of duality corresponds to a geometric duality: that is, the dual DPV modules appear as the equivariant K-theory of some spaces related with F i (X), as we will see in the following sections.
Third, these modules actually keep track of all information contained in the whole Ext * . In fact, they admit filtration in which successive quotients are all the components of Ext * ( F i (X), R(G)). In particular, we have the following proposition.
։ 0 and we take { F * i,j (X)} as sequence of successive kernels. In this way we get isomorphisms where the last isomorphism comes from Lemma 3.8.
In the same way, in the differentiable case we can give the following definition. The same considerations and statements apply.

Recalls on equivariant K-theory
We will briefly recall some notions about equivariant K-theory for a compact group G. The reader is suggested to refer to [5] and [28] for details and proofs.

4.1.
Definition in the compact case. Given a compact topological space M with a continuous action of a compact Lie group G, one can consider equivariant complex vector bundles on M , that is, complex vector bundles E → M with a G-action on the total space E, respecting the action on M and acting linearly on fibers. The equivariant K-theory K * G (M ) of M is the group of integer linear combination of isomorphism classes of such objects, with sum operation given by direct sum of vector bundles.
This group is naturally a ring endowed with tensor product, having zero element given by the 0-dimensional vector bundle and identity given by the bundle C × M with the action of G only on the second coordinate. Furthermore, we have a class of trivial bundles V × M where V is a representation of G, so that we get an homomorphism R(G) → K G (M ) and hence giving K G (M ) the structure of a R(G)-algebra with identity.

4.2.
Definition in the locally compact case. To define equivariant Ktheory in the locally compact case, we will need a different definition. Following [28], in a compactly supported fashion, equivariant K-theory is defined in the following way. We are not going to explicitly describe the meaning of the word "classifies"; the only things to keep in mind are that there are (quite obvious) notions of isomorphism and homotopy between such objects, and that objects like {E id − → E} are set to be equivalent to 0. This definition agrees with the one given for compact spaces; in particular, the correspondence is obtained sending the object {E φ − → F } to the formal difference [E]−[F ]; note that in the compact case the map φ may be nowhere an isomorphism, and part of the proof of the equivalence is showing that we can move φ in an homotopic way to make it the zero function.
We have again a ring structure, in which the sum is given by the direct sum piece by piece, and the product is given by the formula where φ * is the dual map, defined by Hermitian metrics on E, E ′ , F and F ′ .
We have again an action of R(G) by the tensoring both E and F with the trivial elements V × M . So K G (M ) is again an R(G)-algebra, but not necessarily with identity: in fact, we do not have anymore a ring homomorphism R(G) → K G (M ).
where R i is given a trivial action of G, that again carries an R(G)-algebra structure (for i = 0 we get the same object we just defined). As we will see in Section 4.4, all the information is actually contained in K 0 G (M ) and K −1 G (M ).

4.3.
Examples. Proof of (4) is a slight modification of the classic proof of (2).

Thom isomorphism.
If E π − → M is a G-vector bundle, then we have an isomorphism as R(G)-modules (but not as rings) If M is compact, we can give an explicit description of the image of the identity element of K G (M ), so that we can see it as a generator of K G (E) as K G (M )-module.
We can consider E as an equivariant vector bundle on E itself (pulling back from the projection E → M ), and then consider all its exterior powers ∧ i E still as equivariant vector bundles on E. We will call ∧ odd E and ∧ even E the direct sum of all odd (resp. even) exterior powers of E; between this two vector bundles on E we have a special map c(E) (coming from the wedge product), called the Clifford map (see [17], pag. 795). So now, the data of

Geometric realization of dual DM and DPV modules
While the theorems in this Section can be stated without any Index Theory, their proof will make much use of it. The interested reader will find some basic notions on this topic in the Appendix.
Let M ≥i X (0 ≤ i ≤ d) be the spaces introduced in Section 2.6. We denote by T M X and T * M X the tangent and cotangent bundle of M X respectively. Let T * G M X be the subspace of the covectors in T * M X that are normal to the vectors in T M X that are tangent to orbits of the G-action.
Notice that orbits in M f in X = M ≥d X = M d X are all of the same dimension, so T * G M f in X is actually a vector bundle over M f in X (of rank n − d), while for i < d this is not true.
In [17], by using the index theorem, the following result is proved: The extremal cases are particularly interesting: the fact that is the answer to a question that Atiyah raised in [4]. On the other hand, when i = d, by the Thom isomorphism we get . In [19] there are analogues of these facts for equivariant cohomology with compact support: where, by retracting the total space T * G M f in X on the base space M f in X , the grading on the compact support cohomology is shifted by the (real) rank of the vector bundle, that is 2n − d.
The theorem above is proved in [18] by introducing an analogue of the index of transversally elliptic operators, the infinitesimal index; the correspondence is naturally with cohomology with compact support because of the "compact support nature" of equivariant K-theory.
In the same paper, there is also a calculation of the standard (meaning not with compact support) cohomology of the spaces M ≥i X , which does not use any index theory and turn out to yield the dual modules described in Section 3: = D * (X) since of course cohomology (with its natural grading) is preserved by the deformation retract.
We will now provide an analogue of Theorem 5.3 for equivariant K-theory. Instead of looking for a different definition of equivariant K-theory non involving compact support, we will compactify the spaces M ≥i X (actually, a deformation retract of them) to find spaces to perform the same inductive process on.
The geometric idea behind this kind of compactification is the following: instead of removing a closed subspace from a compact space (then, losing compactness), one removes a tubular neighborhood. In this way, the resulting space is still compact, but another property is lost: smoothness. More precisely, to perform differential geometry one has to enlarge the class of spaces to manifold with corners (for instance, see [22] and [23]); anyway, this is not an issue, because we are not requiring any smoothness or boundarylessness condition (we are not either going to use the tangent space).
We now describe in more detail our construction. First, let us restrict to the unitary sphere S X ⊂ M X , to have a compact space to start with. By Formula (1) S X ∩ M f in X , and more generally S X ∩ M ≥i X , are obtained by the compact space by removing some closed subsets S X ∩ M s . Now, for any such subset of S X , we consider a small G-invariant tubular neighbourhood where ε is a sufficiently small number (for the calculations we are going to do, we can assume ε being any number smaller than 1) and finally we are ready to define (in fact, we can take the union only on the rational subspaces of dimension i − 1) and to state one of the the main results of this paper.
G (S f in X ) = DM * (X) and K 1 G (S f in X ) = 0 Our proof will be based on a multiple induction. This requires to generalize it to a broader family of spaces; in order to do that, we define an admissible set of nonzero rational subspaces Q as a set closed under inclusions (if s ⊂ t and t ∈ Q, then also s ∈ Q), and let Of course, the set of all proper rational subspaces, and more in general the sets {dim(s) < i} are admissible.
Then Theorem 5.4 is a corollary of the following: Theorem 5.5. If Q is an admissible set of rational subspaces, then K 0 G (S Q X ) = R(G) /(∇ X\s ) s∈Q and K 1 G (S Q X ) = 0. The beginning step of our induction is given by the following lemma.
Let us consider the Mayer-Vietoris exact sequence The bottom left and center elements of this sequence are 0 (because of Thom isomorphism), so we have to look at kernel and cokernel of K 0 G (M X ) → K 0 G ({0}); now, both this two modules are isomorphic to R(G) (the latter being isomorphic to R(G) as ring too), and the map is the multiplication by the Clifford class [ odd M X ] − [ even M X ] that by a straightforward calculation appears to be exactly ∇ X ; so, K 0 G (M X \ {0}) = 0 and 5.1. Proof of Theorem 5.5. Let us proceed by a double induction on the cardinality of Q and on the dimension of the group G; we start with Q = ∅ and we will add one rational subspace at the time; this step will be based on the statement of the theorem for a lower dimensional group. When Q = ∅, lemma 5.6 gives us the statement of the theorem. Now, suppose we want to add a rational subspace s to Q, such that all rational subspaces t of s already belong to Q. On the geometric side, we have to remove from S Q X the open set S Q X ∩ U s , to get S So, we are ready to consider the exact sequence for the inclusions S Q where we already know that the lower middle and right modules are 0, and the upper left maps is the inclusion that is exactly what we needed to prove.
Remark 5.7. In this way we found a geometric realization for F * i (X) only for i ≥ 1, because these are the only ones that can be got from admissible set; for F * (X), the extremal case, we haven't either defined a compact space to match with it; this is because we should consider also points that are fixed for the G action, so the whole M X , that doesn't retract onto the unit sphere S X . But if we retract M X onto the origin, we immediately get K 0 G ({pt}) ∼ = F * (X) ∼ = R(G) and K 1 G ({pt}) = 0 How this theorem may lead to a more general statement about duality in equivariant K-theory is still a subject of investigation.

Geometric realization of local DM modules
As stated in Theorems 5.2 and 5.1, the modules D(X) and DM (X) have a geometric realization in terms of the manifold M f in X . We also know that these modules have interesting submodules, the local modules D Ω (X) and DM Ω (X); it is then natural to wonder if these submodules have geometric realizations in terms of submanifolds of M f in X . More generally, following definitions given in Section 2.4, we may ask whether we have K * G (T * G M T X ) = DM T (X) and H * G (T * G M T X ) = D T (X). Of course, this is not true in general: for instance, for T = ∅ we have Proof. Well-definedness of ∇ A comes right from the definitions, because A ∪ {a} ∈ T for every a / ∈ A. Let us prove exactness. In DM T ′ (X) this is clear; furthermore, DM T ′ (X) is defined in DM T (X) as the vanishing locus of ∇ A , so we have exactness also in DM T (X).
To prove surjectivity of ∇ A , let's recall the fact that DM T ′ (X) contains a quasi-polynomial q Ω agreeing with the partition function P X on the big cell Ω; this means, by definition of L Ω (X), that the cone C(X \ A) contains Ω as well; let now q ′ Ω be the quasi polynomial agreeing in Ω with the partition function P X\A ; note that by properties of the partition function, we have ∇ A q Ω = q ′ Ω . Let now X \ A = B ∪ C, where B is a Q-basis of Γ such that C(B) ⊇ Ω; we have now ∇ C P X\A = P B , that on Ω agrees with a quasi polynomial with value 1 on the lattice points in ZB and 0 otherwise, that is ∇ C q ′ Ω . Now, let's take an element f of DM S (X \ A); this function is constant modulo the sublattice Z(X \ A) (that contains the lattice ZB); this means that f can be obtained by ∇ C q ′ Ω by an element τ of R(G). We have then , thus f is in the image of ∇ A , because τ ∇ C q Ω is an element of the R(G)module DM T (X). Now that we have two exact sequences, we are going to glue them in a commutative diagram, with vertical arrows given by the index map, and all vertical arrows but one being an isomorphism; by the five lemma, this will lead us to the conclusion.
First, let's verify that such a diagram exist; the only check to do is proving the following lemma, that will be done in the Appendix.
Lemma 6.8. The index map sends K 0 G (T * G M T X ) into DM T (X). Note that the two other vertical arrows are well defined because of lemma 6.6 and by the inductive step on T ′ . We know have the diagram The only thing to be checked now is the commutativity of the two center squares: this will be done in the Appendix. Now, by the five lemma, we have that the index map K 0 G (T * G M T X ) → DM T (X) is an isomorphism, and completing the Mayer-Vietoris sequence we get K 0 G (T * G M T X ). This proves the theorem.