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Let M be a spin manifold with a circular action. Given an elliptic curve E, we introduce, as in Grojnowski, elliptic bouquets of germs of holomorphic equivariant cohomology classes on M. Following Bott-Taubes and Rosu, we show that integration of an elliptic bouquet is well defined. In particular, this imply Witten's rigidity theorem. We emphasize the similarity between integration in K-theory and in elliptic cohomology., Minor misprints corrected. Introduction expanded

We give an exposition of the Horn inequalities and their triple role characterizing tensor product invariants, eigenvalues of sums of Hermitian matrices, and intersections of Schubert varieties. We follow Belkale's geometric method, but assume only basic representation theory and algebraic geometry, aiming for self-contained, concrete proofs. In particular, we do not assume the Littlewood-Richardson rule nor an a priori relation between intersections of Schubert cells and tensor product invariants. Our motivation is largely pedagogical, but the desire for concrete approaches is also motivated by current research in computational complexity theory and effective algorithms., 51 pages

On a polarized compact symplectic manifold endowed with an action of a compact Lie group, in analogy with geometric invariant theory, one can define the space of invariant functions of degree k. A central statement in symplectic geometry, the quantization commutes with reduction hypothesis, is equivalent to saying that the dimension of these invariant functions depends polynomially on k. This statement was proved by Meinrenken and Sjamaar under positivity conditions. In this paper, we give a new proof of this polynomiality property. The proof is based on a study of the Atiyah-Bott fixed point formula from the point of view of the theory of partition functions, and a technique for localizing positivity.

We give a general description of the moment cone associated with an arbitrary finite-dimensional unitary representation of a compact, connected Lie group in terms of finitely many linear inequalities. Our method is based on combining differential-geometric arguments with a variant of Ressayre's notion of a dominant pair. As applications, we obtain generalizations of Horn's inequalities to arbitrary representations, new inequalities for the one-body quantum marginal problem in physics, which concerns the asymptotic support of the Kronecker coefficients of the symmetric group, and a geometric interpretation of the Howe-Lee-Tan-Willenbring invariants for the tensor product algebra., Comment: 42 pages, to appear in Journal of Symplectic Geometry

Let G be a connected compact Lie group acting on a manifold M and let D be a transversally elliptic operator on M. The multiplicity of the index of D is a function on the set of irreducible representations of G. Let T be a maximal torus of G with Lie algebra Lie(T). We construct a finite number of piecewise polynomial functions on the dual vector space Lie(T)*, and give a formula for the multiplicity in term of these functions. The main new concept is the formal equivariant $\hat A$ class., Comment: An example and a reference are added. Some misprints have been corrected

Motivated by applications to multiplicity formulas in index theory, we study a family of distributions $\Theta(m;k)$ associated to a piecewise quasi-polynomial function $m$. The family is indexed by an integer $k \in \mathbb{Z}_{>0}$, and admits an asymptotic expansion as $k \rightarrow \infty$, which generalizes the expansion obtained in the Euler-Maclaurin formula. When $m$ is the multiplicity function arising from the quantization of a symplectic manifold, the leading term of the asymptotic expansion is the Duistermaat-Heckman measure. Our main result is that $m$ is uniquely determined by a collection of such asymptotic expansions. We also show that the construction is compatible with pushforwards. As an application, we describe a simpler proof that formal quantization is functorial with respect to restrictions to a subgroup., Comment: 42 pages, minor corrections to section 2

We describe the admissible coadjoint orbits of a compact connected Lie group and their spin-c quantization., Comment: The authors have decided to divide the preprint "Multiplicities of equivariant Spin-c Dirac operators" into two separate publications. The present paper is one of them, the other part is entitled "Equivariant Dirac operators and differentiable geometric invariant theory" (arXiv:1411.7772, version 3)

The computation of Kronecker coefficients is a challenging problem with a variety of applications. In this paper we present an approach based on methods from symplectic geometry and residue calculus. We outline a general algorithm for the problem and then illustrate its effectiveness in several interesting examples. Significantly, our algorithm does not only compute individual Kronecker coefficients, but also symbolic formulas that are valid on an entire polyhedral chamber. As a byproduct, we are able to compute several Hilbert series., This article draws heavily from arXiv:1506.02472. It is an updated version of arXiv:1601.04325. We have clarified the structure and layout of the paper and we have isolated the algorithm into Appendix A, giving a concise exposition of it in pseudo-code and a detailed explanation of all steps, cross-referenced with the main text

In this paper, we give a geometric expression for the multiplicities of the equivariant index of a spin-c Dirac operator., Comment: The authors have decided to divide the preprint "Multiplicities of equivariant Spin-c Dirac operators" into two separate publications. The present paper is the first of them. The other part is the preprint arXiv:1512.02367 entitled "Admissible orbits for compact Lie group"

The 2016 Wigner Medal has been awarded to Bertram Kostant of the Massachusetts Institute of Technology (USA) for his fundamental contributions to the representation theory of Lie algebraic systems. Many of his results have led to new developments both in Mathematics and, as emphasized here, in Theoretical Physics.

In this article, we start to recall the inversion formula for the convolution with the Box spline. The equivariant cohomology and the equivariant K-theory with respect to a compact torus G of various spaces associated to a linear action of G in a vector space M can be both described using some vector spaces of distributions, on the dual of the group G or on the dual of its Lie algebra. The morphism from K-theory to cohomology is analyzed and the multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semidiscrete convolution with a box spline. Finally, the multiplicities of the index of a G-transversally elliptic operator on M are determined using the infinitesimal index of the symbol., Comment: 44 pages

In this note, we study an invariant associated to the zeros of the moment map generated by an action form, the infinitesimal index. This construction will be used to study the compactly supported equivariant cohomology of the zeros of the moment map and to give formulas for the multiplicity index map of a transversally elliptic operator., In this version, some misprints are corrected. An example have been changed of place

We present a formula for the degree of the discriminant of a smooth projective toric variety associated to a lattice polytope P, in terms of the number of integral points in the interior of dilates of faces of dimension greater or equal than $\lceil \frac {\dim P} 2 \rceil$., Introduction, last section and bibliography are revised

Using multiple Bernoulli series, we give a formula in the spirit of Euler MacLaurin formula. We also give a wall crossing formula and a decomposition formula. The study of these series is motivated by formulae of E.Witten for volumes of moduli spaces of flat bundles over a surface., 37 pages, 7 figures

Let G be a connected reductive real Lie group, and H a compact connected subgroup. Let M be a coadjoint admissible orbit of G and let Π be one of the unitary irreducible representations of G attached to M by Harish-Chandra. Using the character formula for Π , we give a geometric formula for the multiplicities of the restriction of Π to H , when the restriction map p : M → h ⁎ is proper. In particular, this gives an alternate proof of a result of Paradan.

The purpose of the workshop was to bring together mathematicians interested in ”quantization of manifolds” in a broad sense: given classical data, such as a Lie group G acting on a symplectic manifold M , construct a quantum version, that is a representation of G in a vector space Q(M) reflecting the classical properties of M . Mathematics Subject Classification (2000): 53Dxx, 58Jxx, 19Kxx. Introduction by the Organisers The workshop, Geometric Quantization in the Non-compact Setting, organized by Lisa Jeffrey (Toronto), Xiaonan Ma (Paris) and Michele Vergne (Paris) was held February 13th February 19th, 2011. The meeting was attended by 48 participants, representing researchers from many European countries, and Australia, Canada, China, Japan, USA. Unfortunately, Lisa Jeffrey could not be present because a minor injury before the meeting made it impossible for her to travel. The meeting was devoted to the following theme (and adjacent themes). Let G be a Lie group acting on a manifold M . Assume that G preserves some data (D), such as a symplectic structure, or a differential operator on M , or a fibration, . . ., then the aim of Geometric Quantization is to associate to these data a representation of G in a vector space Q(M,D), and to analyze the relations of the quantum space Q(M,D) with the classical data (M,D). There are diverse constructions of the quantum space Q(M,D). They should all have some functorial properties, summarized in the maxim (only a hope or guiding principle rather 426 Oberwolfach Report 09/2011 than an established fact in this very general setting): quantization commutes with reduction. To report on the recent progress overcoming a certain number of difficulties, arising in the case of a noncompact manifold, or a noncompact group G, was an important goal for this meeting. However, important results on the quantization in the case of compact manifolds were also reported in this meeting. Thus the topic of our meeting included deformation quantization of functions on a symplectic manifold via Toeplitz operators, branching rules for unitary representations of real Lie groups, equivariant index of transversally elliptic operators, quantization of Hamiltonian manifolds with proper moment maps, group valued moment maps, Lagrangian fibrations. It was not clear to the organizers that our choice of participants working on these many diverse topics and with many different techniques (topological K-theory, analytic estimates, C∗-algebras, representation theory) could lead to anything other than a series of talks with disjoint attendance. However, we think that the meeting was very successful in making bridges between the many different approaches towards a general common goal. This is certainly due to the very unique atmosphere of the Oberwolfach setting. There were 22 talks of approximately 50 minutes, 7 talks of 30 minutes, and a session of short talks by young postdocs. All the speakers presented interesting new results, and they were concerned with clearly communicating the results of their research to an audience, possibly not familiar with techniques used, although interested in same themes. Thus our meeting was successful due to the efforts of the speakers. Certainly, this meeting will produce new ideas in the future in the participants’ research, generated by attending a live presentation of new points of view. Let us give some details on the topics of the conference: • Quantization Q(M,L) of a noncompact symplectic manifold M provided with a line bundle L. Methods via C∗-algebras, or transversally elliptic operators or cutting methods were presented. Hamiltonian manifolds such as the cotangent bundle of a manifold are a classical topic in mechanics. The list of other interesting Hamiltonian manifolds include the coadjoint orbits of real reductive Lie groups, representation spaces of fundamental groups of a surface of genus g with value in a compact Lie group (moduli spaces of flat bundles) or in a complex lie group (Hitchin moduli spaces). Results on the quantisation of those manifolds were discussed. • Toeplitz algebras: this leads to quantisation of the algebra of functions on a symplectic manifold by studying asymptotic k estimates of the quantisation Q(M,L) when the line bundle L is raised to its k-th power. • The equivariant index of elliptic operators or of transversally elliptic operators. Methods via C∗-algebras, the heat kernel or topological K-theory were presented. Geometric Quantization in the Non-compact Setting 427 • Quantisation of integrable systems. This includes the theory of Lagrangian fibrations (possibly singular) and Bohr-Sommerfeld orbits. • Spectrum of the Laplacian or the hypoelliptic Laplacian. Zeta functions of the Laplacian, analytic torsion. Detailed information on the topics presented are given in the abstracts. We had asked several young researchers to prepare a poster on their research before coming to Oberwolfach. Wednesday evening was then devoted to a special session of short talks and posters. Talks given by the younger researchers were dynamic and very well prepared. Furthermore, although the talks were necessarily very short due to the lack of time, we had a poster session just after the introductory talks, and scientific informal discussions. This was the “must-see event” of the workshop, and it went very well. On behalf of all participants, we would like to thank the staff for their concern in providing the best material conditions for our stay. The setting of Oberwolfach is beautiful, the food of excellent quality, the library full of resources, and the staff extremely helpful. Thanks to Oberwolfach grants, four young researchers: Solha (Barcelona), Deltour (Montpellier), Hochs (Utrecht), Szilagyi (Geneve) could participate to our workshop. Finally, as organizers, we would like to thank the director and his staff for their great help in the scientific organization. In particular, the director explained the general policy of the Oberwolfach meeting to us, which was very helpful. 1. Program of the conference Monday, 14/02/2011 9h00-10h00 G. Marinescu Toeplitz operators and geometric quantization 10h10–11h10 P. Ramacher Singular equivariant heat asymptotics and Lefchetz formulas 11h20 -12h20 M. Duflo Kirillov’s formula and Box splines 14h30–15h30 P-E. Paradan Spin quantization in the compact and non-compact setting 16h00-17h00 T. Kobayashi Geometric quantization, limits and restrictions–some examples for elliptic and minimal orbits 17h10-18h10 B. Orsted 428 Oberwolfach Report 09/2011 Deformation of Fourier transformation

This paper settles the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We conclude the article with extensions to other polytopes, discussion of other available methods and experimental results., Tables added with new experimental results. References added

Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron \(\mathfrak{p} \subseteq \mathbb{R}^{d}\), sampled at the points of the lattice \(\mathbb{Z}^{d}/t\). We give an asymptotic expansion when \(t \rightarrow +\infty\), writing each coefficient of this expansion as a sum indexed by the faces \(\mathfrak{f}\) of the polyhedron, where the \(\mathfrak{f}\) term is the integral over \(\mathfrak{f}\) of a differential operator applied to the function h(x). In particular, if a Euclidean scalar product is chosen, we prove that the differential operator for the face \(\mathfrak{f}\) can be chosen (in a unique way) to involve only normal derivatives to \(\mathfrak{f}\).

Building on our earlier work on toric residues and reduction, we give a proof for the mixed toric residue conejecture of Batyrev and Materov. We simplify and streamline our technique of tropical degenerations, which allows one to interpolate between two localization principles: one appearing in the intersection theory toric quotients and the other in the calculus of toric residues. This quickly leads to the proof of the conjecture, which gives a closed form for the summation of a generating series whose coefficients are certain variants of the "numbers" of rational curves on toric complete intersection Calabi-Yau manifolds., misprints corrected, an example added; 35 pages

Let V be an n-dimensional real vector space endowed with a rank-n lattice Γ . The dual lattice Γ ∗ = Hom(Γ.Z) is naturally a subset of the dual vector space V ∗. Let Φ = [β1, β2, . . . , βN ] be a sequence of not necessarily distinct elements of Γ ∗, which span V ∗ and lie entirely in an open halfspace of V ∗. In what follows, the order of elements in the sequence will not matter. The closed cone C(Φ) generated by the elements of Φ is an acute convex cone, divided into open conic chambers by the (n− 1)-dimensional cones generated by linearly independent (n−1)-tuples of elements of Φ . Denote by ZΦ the sublattice of Γ ∗ generated by Φ . Pick a vector a ∈ V ∗ in the cone C(Φ), and denote by ΠΦ(a) ⊂ R+ the convex polytope consisting of all solutions x = (x1, x2, . . . , xN) of the equation ∑Nk=1 xkβk = a in nonnegative real numbers xk . This is a closed convex polytope called the partition polytope associated to Φ and a. Conversely, any closed convex polytope can be realized as a partition polytope. If λ ∈ Γ ∗, then the vertices of the partition polytope ΠΦ(λ) have rational coordinates. We denote by ιΦ(λ) the number of points with integral coordinates in ΠΦ(λ). Thus ιΦ(λ) is the number of solutions of the equation ∑N k=1 xkβk = λ in nonnegative integers xk . The function λ → ιΦ(λ) is called the vector partition function associated to Φ . Obviously, ιΦ(λ) vanishes if λ does not belong to C(Φ) ∩ZΦ .

We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449-1466]. By well-known decompositions, it is sufficient to consider the case of affine cones s+c, where s is an arbitrary real vertex and c is a rational polyhedral cone. For a given rational subspace L, we integrate a given polynomial function h over all lattice slices of the affine cone s + c parallel to the subspace L and sum up the integrals. We study these intermediate sums by means of the intermediate generating functions $S^L(s+c)(\xi)$, and expose the bidegree structure in parameters s and $\xi$, which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra, Found. Comput. Math. 12 (2012), 435-469] and [Intermediate sums on polyhedra: Computation and real Ehrhart theory, Mathematika 59 (2013), 1-22]. The bidegree structure is key to a new proof for the Baldoni--Berline--Vergne approximation theorem for discrete generating functions [Local Euler--Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes, Contemp. Math. 452 (2008), 15-33], using the Fourier analysis with respect to the parameter s and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes., Comment: 35 pages, 6 figures; v2 changes terminology regarding degrees, for consistency with arXiv:1410.8632

Consider the space RΔ of rational functions of several variables with poles on a fixed arrangement Δ of hyperplanes. We obtain a decomposition of RΔ as a module over the ring of differential operators with constant coefficients. We generalize the notions of principal part and of residue to the space RΔ, and we describe their relations to Laplace transforms of locally polynomial functions. This explains algebraic aspects of the work by L. Jeffrey and F. Kirwan about integrals of equivariant cohomology classes on Hamiltonian manifolds. As another application, we will construct multidimensional versions of Eisenstein series in a subsequent article, and we will obtain another proof of a residue formula of A. Szenes for Witten zeta functions.

Resume Si g est une algebre de Lie, nous definissons des fonctions F ( x , y ) et G ( x , y ) sur g⊕g a valeurs dans g telles que x + y − log (e x e y ) = (e ad x ) −1 F ( x , y +(1−e −ad y G ( x , y ). Si g est une algebre de Lie quadratique, nous prouvons une identite pour la trace de la matrice (ad x )∂ x F + (ad y )∂ y G . Cette identite est conjecturee dans [4] pour toute algebre de Lie, et demontree si g est resoluble. Elle implique (voir [4]) le prolongement naturel de l'isomorphisme de Duflo [2] aux algebres de convolution de distributions invariantes sur le groupe G et sur lalgebre de Lie g.

Let C be a complex algebraic cone, provided with an action of a compact Lie group K. The symplectic form of the ambient complex Hermitian space induces on the regular part of C a symplectic form. Let k be the Lie algebra of K. Let f : C → k∗ be the Mumford moment map, that is f(v)(X) = i(v,Xv), for X ∈ k and v ∈ C. The space R(C) of regular functions on C is a semi-simple representation of K. In this article, with the help of the moment map, we give some quantitative informations on the decomposition of R(C) in irreducible representations of K. For λ a dominant weight, let m(λ) be the multiplicity of the representation of highest weight λ in R(C). Then, if the moment map f : C → k∗ is proper, multiplicities m(λ) are finite and with polynomial growth in λ. Furthermore, the study of the pushforward by f of the Liouville measure gives us an asymptotic information on the function m(λ) . For example, in the case of a faithful torus action, the pushforward of the Liouville measure by the moment map is a locally polynomial homogeneous function `(λ) on the polyhedral cone f(C) ⊂ t∗, while the multiplicity function m(λ) for large values of λ is given by the restriction to the lattice of weights of a quasipolynomial function, with highest degree term equal to `(λ). If O is a nilpotent orbit of the coadjoint representation of a complex Lie group G, we show that the pushforward on k∗ of the G-invariant measure on O is the same that the pushforward of the Liouville measure on O associated to the symplectic form of the ambient complex vector space. Thus, this establishes for the case of complex reductive groups the relation, conjectured by D. Vogan, between the Fourier transform of the orbit O and multiplicities of the ring of regular functions on O.

Using Szenes formula for multiple Bernoulli series we explain how to compute Witten series associated to classical Lie algebras. Particular instances of these series compute volumes of moduli spaces of flat bundles over surfaces, and also certain multiple zeta values., 51 pages, 3 figures; formula in Proposition 3.1 for the Lie group of type G_2 is corrected; new references added

Let $P(b)\subset R^d$ be a semi-rational parametric polytope, where $b=(b_j)\in R^N$ is a real multi-parameter. We study intermediate sums of polynomial functions $h(x)$ on $P(b)$, $$ S^L (P(b),h)=\sum_{y}\int_{P(b)\cap (y+L)} h(x) \mathrm dx, $$ where we integrate over the intersections of $P(b)$ with the subspaces parallel to a fixed rational subspace $L$ through all lattice points, and sum the integrals. The purely discrete sum is of course a particular case ($L=0$), so $S^0(P(b), 1)$ counts the integer points in the parametric polytopes. The chambers are the open conical subsets of $R^N$ such that the shape of $P(b)$ does not change when $b$ runs over a chamber. We first prove that on every chamber of $R^N$, $S^L (P(b),h)$ is given by a quasi-polynomial function of $b\in R^N$. A key point of our paper is an analysis of the interplay between two notions of degree on quasi-polynomials: the usual polynomial degree and a filtration, called the local degree. Then, for a fixed $k\leq d$, we consider a particular linear combination of such intermediate weighted sums, which was introduced by Barvinok in order to compute efficiently the $k+1$ highest coefficients of the Ehrhart quasi-polynomial which gives the number of points of a dilated rational polytope. Thus, for each chamber, we obtain a quasi-polynomial function of $b$, which we call Barvinok's patched quasi-polynomial (at codimension level $k$). Finally, for each chamber, we introduce a new quasi-polynomial function of $b$, the cone-by-cone patched quasi-polynomial (at codimension level $k$), defined in a refined way by linear combinations of intermediate generating functions for the cones at vertices of $P(b)$. We prove that both patched quasi-polynomials agree with the discrete weighted sum $b\mapsto S^0(P(b),h)$ in the terms corresponding to the $k+1$ highest polynomial degrees., Comment: 41 pages, 13 figures; v2: changes to introduction, new graphics; v3: add more detailed references, move example to introduction; v4: fix references

In this note, we give a geometric expression for the multiplicities of the equivariant index of a Dirac operator twisted by a line bundle., Comment: 8 pages

in terms of fP(h) q(x)dx where the polytope P(h) is obtained from P by independent parallel motions of all facets. This extends to simple lattice polytopes the EulerMaclaurin summation formula of Khovanskii and Pukhlikov [8] (valid for lattice polytopes such that the primitive vectors on edges through each vertex of P form a basis of the lattice). As a corollary, we recover results of Pommersheim [9] and Kantor-Khovanskii [6] on the coefficients of the Ehrhart polynomial of P. Our proof is elementary. In a subsequent article, we will show how to adapt it to compute the equivariant Todd class of any complete toric variety with quotient singularities. The Euler-Maclaurin summation formula for simple lattice polytopes has been obtained independently by Ginzburg-Guillemin-Karshon [4]. They used the dictionary between convex polytopes and projective toric varieties with an ample divisor class, in combination with the Riemann-Roch-Kawasaki formula ([1], [7]) for complex manifolds with quotient singularities. A counting formula for lattice points in lattice simplices has been announced by Cappell and Shaneson [2], as a consequence of their computation of the Todd class of toric varieties with quotient singularities.

We obtain residue formulae for certain functions of several vari- ables. As an application, we obtain closed formulae for vector partition func- tions and for their continuous analogs. They imply an Euler-MacLaurin sum- mation formula for vector partition functions, and for rational convex poly- topes as well: we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope. Institut Fourier, B.P. 74, 38402 Saint-Martin d’Heres Cedex, France E-mail address: mbrion@fourier.ujf-grenoble.fr Ecole Normale Superieure, 45 rue d’Ulm, 75005 Paris Cedex 05, France E-mail address: vergne@dmi.ens.fr

Quantum mechanics associate to some symplectic manifolds M a quantum model Q ( M ), which is a Hilbert space. The space Q ( M ) is the quantum mechanical analogue of the classical phase space M . We discuss here relations between the volume of M and the dimension of the vector space Q ( M ). Analogues for convex polyhedra are considered.