Michele Vergne

• Institut de Mathématiques de Jussieu-Paris Rive Gauche, French National Centre for Scientific Research

Consider the convex cone of holomorphic orbits in the Lie algebra of \(U(p,q)\). As in the classical case of Hermitian matrices, the set of elliptic orbits contained in the sum of two holomorphic orbits can be described by “Horn inequalities.” Using representations of quivers, we give another proof of these Horn recursion inequalities obtained by P...

As shown by P-E Paradan, the set of orbits contained in the sum of two holomorphic orbits in the Lie algebra of U(p,q) is determined by a set of inequalities similar to the Horn inequalities for the sum of conjugacy classes of two Hermitian matrices. We give another proof of these inequalities using representations of quivers. We also discuss the i...

Motivated by applications to multiplicity formulas in index theory, we study a family of distributions Θ(m;k) associated to a piecewise quasi-polynomial function m. The family is indexed by an integer k∈Z>0, and admits an asymptotic expansion as k→∞, which generalizes the expansion obtained in the Euler–Maclaurin formula. When m is the multiplicity...

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...

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 $k>0$, and admits an asymptotic expansion as $k \rightarrow \infty$, which generalizes the expansion obtained in the Euler-Maclaurin formula...

We give inductive conditions that characterize the Schubert positions of subrepresentations of a general quiver representation. Our results generalize Belkale's criterion for the intersection of Schubert varieties in Grassmannians and refine Schofield's characterization of the dimension vectors of general subrepresentations. This implies Horn type...

We describe the admissible coadjoint orbits of a compact connected Lie group and their spin-c quantization.

Let G be a complex reductive group acting on a finite-dimensional complex vector space H. Let B be a Borel subgroup of G and let T be the associated torus. The Mumford cone is the polyhedral cone generated by the T-weights of the polynomial functions on H which are semi-invariant under the Borel subgroup. In this article, we determine the inequalit...

Consider a spin manifold M, equipped with a line bundle L and an action of a compact Lie group G. We can attach to this data a family Theta(k) of distributions on the dual of the Lie algebra of G. The aim of this paper is to study the asymptotic behaviour of Theta(k) when k is large, and M possibly non compact, and to explore a functorial consequen...

In this paper we study asymptotic distributions associated to piecewise quasi-polynomials. The main result obtained here is used in another paper of the authors "The equivariant index of twisted Dirac operators and semi-classical limits".

In this paper, we give a geometric expression for the multiplicities of the equivariant index of a spinc Dirac operator.

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...

Let G be a torus and M a G-Hamiltonian manifold with Kostant line bundle L and proper moment map. Let P be the weight lattice of G. We consider a parameter k and the multiplicity $m(\lambda,k)$ of the quantized representation associated to M and the k-th power of L . We prove that the weighted sum $\sum m(\lambda,k) f(\lambda/k)$ of the value of a...

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 pa...

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 pa...

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 Ĝ of irreducible representations of G. Let T be a maximal torus of G with Lie algebra t. We construct a finite number of piecewise polynomial functions on t∗, and give a formu...

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...

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 doe...

These notes are an expanded version of a talk given by the second author. Our main interest is focused on the challenging problem of computing Kronecker coefficients. We decided, at the beginning, to take a very general approach to the problem of studying multiplicity functions, and we survey the various aspects of the theory that comes into play,...

The purpose of the present paper is two-fold. First, we obtain a non-abelian localization theorem when M is any even dimensional compact manifold : following an idea of E. Witten, we deform an elliptic symbol associated to a Clifford bundle on M with a vector field associated to a moment map. Second, we use this general approach to reprove the [Q,R...

Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron in R^d, sampled at the points of the lattice Z^d/t. We give an asymptotic expansion when t goes to infinity, writing each coefficient of this expansion as a sum indexed by the faces f of the polyhedron, where the f-term is the integral over f of a differential op...

In this paper, we give a geometric expression for the multiplicities of the equivariant index of a spin^c Dirac operator.

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 su...

We give a general description of the moment cone of an arbitrary finite-dimensional unitary representation of a compact, connected Lie group. Our method is based on straightforward differential-geometric arguments combined with a variant of Ressayre's notion of a dominant pair. As applications, we obtain new inequalities for the one-body quantum ma...

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

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 ar...

For a given sequence $\mathbf{\alpha} = [\alpha_1,\alpha_2,\dots,,\alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\mathbf{\alpha})(t)$ that counts the nonnegative integer solutions of the equation $\alpha_1x_1+\alpha_2 x_2+\cdots+\alpha_{N} x_{N}+\alpha_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying...

We generalize Dahmen-Micchelli deconvolution formula for Box splines with parameters. Our proof is based on identities for Poisson summation of rational functions with poles on hyperplanes.

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.

For a given sequence α=[α 1 ,α 2 ,⋯,α N ,α N+1 ] of N+1 positive integers, we consider the combinatorial function E(α)(t) that counts the nonnegative integer solutions of the equation α 1 x 1 +α 2 x 2 +⋯+α N x N +α N+1 x N+1 =t, where the right-hand side t is a varying nonnegative integer. It is well-known that E(α)(t) is a quasipolynomial function...

International audience For a given sequence $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\alpha)(t)$ that counts the nonnegative integer solutions of the equation $\alpha_1x_1+\alpha_2 x_2+ \ldots+ \alpha_Nx_N+ \alpha_{N+1}x_{N+1}=t$, where the right-hand side $...

We study intermediate sums, 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]. For a given semi-rational polytope P and a rational subspace L, we integrate a given polynomial function h over all lattice slices of t...

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 s...

We define a set theoretic "analytic continuation" of a polytope defined by inequalities. For the regular values of the parameter, our construction coincides with the parallel transport of polytopes in a mirage introduced by Varchenko. We determine the set-theoretic variation when crossing a wall in the parameter space, and we relate this variation...

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 $\lceil \frac {\dim P} 2 \rceil$.

Let G be a connected reductive real Lie group, and H a compact connected subgroup. Harish-Chandra associates to a regular coadjoint admissible orbit M of G some unitary representations of G. Using the character formula for these representations, we show that the multiplicities of the restriction to H can be computed, under a suitable properness ass...

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 .

Using the Guichardet construction, we compute the cohomology groups of a complex of free Lie algebras introduced by Alekseev and Torossian.

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 grou...

Let G be a torus acting linearly on a complex vector space M and let X be the list of weights of G in M. We determine the topological equivariant K-theory of the open subset M f of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the character lattice [^(G)] \widehat{G} , satisfying the cocircuit di...

This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when th...

This document is a companion for the Maple program : Discrete series and K-types for U(p,q) available on:http://www.math.jussieu.fr/~vergne We explain an algorithm to compute the multiplicities of an irreducible representation of U(p)x U(q) in a discrete series of U(p,q). It is based on Blattner's formula. We recall the general mathematical backgro...

We study multiple Bernoulli series associated to a sequence ofvectors generating a lattice in a vector space. The associated multiple Bernoulliseries is a periodic and locally polynomial function, and we give an explicit formula(called wall crossing formula) comparing the polynomial densities in two adjacentdomains of polynomiality separated by a h...

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 invari...

In this note several computations of equivariant cohomology groups are performed. For the compactly supported equivariant cohomology, the notion of infinitesimal index developed in arXiv:1003.3525, allows to describe these groups in terms of certain spaces of distributions arising in the theory of splines. The new version contains a large number of...

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.

The semi-discrete convolution with the Box Spline is an important tool in approximation theory. We give a formula for the difference between semi-discrete convolution and convolution with the Box Spline. This formula involves multiple Bernoulli polynomials.

This document is a companion for the Maple program \textbf{Summing a polynomial function over integral points of a polygon}. It contains two parts. First, we see what this programs does. In the second part, we briefly recall the mathematical background.

In the 70’s, the notion of analytic index has been extended by Atiyah and Singer to the class of transversally elliptic operators. They did not, however, give a general cohomological formula for the index. This was accomplished many years later by Berline and Vergne. The Berline-Vergne formula is an integral of a non-compactly supported equivariant...

We describe a method for computing the highest degree coefficients of a weighted Ehrhart quasi-polynomial for a rational simple polytope. Comment: This paper has been withdrawn by the authors. This paper will be replaced soon by a more complete article with more authors

In the 80's, Quillen constructed a de Rham relative cohomology class associated to a smooth morphism between vector bundles, that we call the relative Quillen Chern character. In the first part of this paper we prove the multiplicativ property of the relative Quillen Chern character. Then we obtain a Riemann-Roch formula between the relative Chern...

This paper discusses volumes and Ehrhart polynomials in the context of flow polytopes. The general approach that studies these functions via rational functions with poles on arrangement of hyperplanes and the total residue of such functions allows us, via a unified approach, to reobtain many interesting calculations existing in the literature. In p...

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...

This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this...

We give an elementary algebraic proof of Paradan's wall crossing formulae for partition functions. We also express such jumps in volume and partition functions by one dimensional residue formulae. Subsequently we reprove the relation between them as given by the application of a generalized Khovanskii-Pukhlikov differential operator.

These notes form the next episode in a series of articles dedicated to a detailed proof of a cohomological index formula for transversally elliptic pseudo-differential operators and applications. The first two chapters are already available as math.DG/0702575 and arXiv:0711.3898. In this episode, we construct the relative equivariant Chern characte...

These notes are the first chapter of a monograph, dedicated to a detailed proof of the equivariant index theorem for transversally elliptic operators. In this preliminary chapter, we prove a certain number of natural relations in equivariant cohomology. These relations include the Thom isomorphism in equivariant cohomology, the multiplicativity of...

We extend to Barvinok's valuations the Euler-Maclaurin expansion formula which we obtained previously for the sum of values of a polynomial over the integral points of a rational polytope. This leads to an improvement of Barvinok's polynomial type algorithm for computing the highest coefficients of the corresponding Ehrhart quasi-polynomial.

We write the equivariant Todd class of a general complete toric variety as an explicit combination of the orbit closures, the coefficients being analytic functions on the Lie algebra of the torus which satisfy Danilov's requirement.

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These notes are very informal notes on the Langlands program. I had some pleasure in daring to ask colleagues to explain to me the importance of some of the recent results on Langlands program, so I thought I will record (to the best of my understanding) these conversations, and then share them with other mathematicians. These notes are intended fo...

We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic quotients and to indices of t...

We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic quotients and to indices of t...

This paper presents an algorithm to compute the value of the inverse Laplace transforms of rational functions with poles on arrangements of hyperplanes. As an application, we present an efficient computation of the partition function for classical root systems.

We give a local Euler-Maclaurin formula for rational convex polytopes in a rational euclidean space . For every affine rational polyhedral cone C in a rational euclidean space W, we construct a differential operator of infinite order D(C) on W with constant rational coefficients, which is unchanged when C is translated by an integral vector. Then f...

This paper presents an algorithm to compute the value of the inverse Laplace transforms of rational functions with poles on arrangements of hyperplanes. As an application, we present an efficient computation of the partition function for classical root systems.

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...

This paper discusses analytic algorithms and software for the enumeration of all integer flows inside a network. Concrete applications abound in graph theory, representation theory, and statistics. Our methods are based on the study of rational functions with poles on arrangements of hyperplanes; they surpass traditional exhaustive enumeration and...

The meeting brought together people with different mathematical background, who all use cohomological methods to study symmetries of manifolds. The main aim was the exchange of ideas, recent results, and the discussion of open problems and questions from diverse viewpoints. Altogether there were 27 talks (including an evening talk on computer progr...

The purpose of this paper is to find explicit formulae for the total residue of some interesting rational functions with poles on hyperplanes determined by roots of type A r = {(e i −e j )|1 ≤ i, j ≤ (r+1), i ≠ j}. As pointed out by Zeilberger [Z], these calculations are mere reformulations of Morris identities [M], where the total residue function...

This volume is devoted to the theme of Noncommutative Harmonic Analysis and consists of articles in honor of Jacques Carmona, whose scientific interests range through all aspects of Lie group representations. The topics encompass the theory of representations of reductive Lie groups, and especially the determination of the unitary dual, the problem...

We present a new integration formula for intersection numbers on toric quotients, extending the results of Witten, Jeffrey and Kirwan on localization. Our work was motivated by the Toric Residue Mirror Conjecture of Batyrev and Materov; as an application of our integration formula, we obtain a proof of this conjecture in a generalized setting. Comm...

This paper discusses new analytic algorithms and software for the enumeration of all integer flows inside a network. Concrete applications abound in graph theory \cite{Jaeger}, representation theory \cite{kirillov}, and statistics \cite{persi}. Our methods clearly surpass traditional exhaustive enumeration and other algorithms and can even yield fo...

We discuss some of the properties of the Bernoulli series\[{{B(p,t)}\over{p!}} = - \sum\limits_{n \ne 0}{{{e^{2i\pi nt}}\over{(2i\pi n)^p}}}\] and higher dimensional analogues, the Witten series. Similarly, we discuss trigonometric series\[V(q,k) = \sum\limits_{n = 1}^{k - 1} {{1\over{4^q (\sin (\pi n/k))^{2q}}}\]and higher dimensional analogues, t...

We first discuss here some classical results on the number of points with integral coordinates in convex rational polytopes P ⊂ Rn starting with Ehrhart's theorem. Then, following Baldoni-Vergne and Szenes-Vergne, we present some recent results giving number of points with integral coordinates in P in terms of multidimensional residues. In particul...

Given a finite set of vectors spanning a lattice and lying in a halfspace of a real vector space, to each vector $a$ in this vector space one can associate a polytope consisting of nonnegative linear combinations of the vectors in the set which sum up to $a$. This polytope is called the partition polytope of $a$. If $a$ is integral, this polytope c...