Admissible coadjoint orbits for compact Lie groups
We describe the admissible coadjoint orbits of a compact connected Lie group and their spin-c quantization.
math.RT↗
arXiv subjects
Paul-Emile Paradan
Publications and source records attributed to Paul-Emile Paradan.
Equivariant Dirac operators and differentiable geometric invariant theory
In this paper, we give a geometric expression for the multiplicities of the equivariant index of a spin-c Dirac operator.
math.DG↗
Stability property of multiplicities of group representations
This paper is dedicated to the study of the stability of multiplicities of group representations.
math.RT↗
The multiplicities of the equivariant index of twisted Dirac operators
In this note, we give a geometric expression for the multiplicities of the equivariant index of a Dirac operator twisted by a line bundle.
math.SG↗
On the structure of K_G(T_G M)
In this expository paper, we revisit the results of Atiyah-Singer and de Concini-Procesi-Vergne concerning the structure of the K-theory groups K_G(T_G M).
math.KT↗
Quantization commutes with reduction in the non-compact setting: the case of holomorphic discrete series
In this paper we show that the multiplicities of holomorphic discrete series representations relatively to reductive subgroups satisfy the credo "Quantization commutes with reduction".
math.SG↗
Spin-quantization commutes with reduction
In this paper, we prove that the "quantization commutes with reduction" phenomenon of Guillemin-Sternberg applies in the context of the metaplectic correction.
math.SG↗
Formal geometric quantization II
In this paper we pursue the study of formal geometric quantization of non-compact Hamiltonian manifolds. Our main result is the proof that two quantization process coincide. This fact was obtained by Ma and Zhang in the preprint arXiv:0812.3989 by completely different means.
math.SG↗
Multiplicities of the discrete series
The purpose of this paper is to show that the multiplicities of a discrete series representation relatively to a compact subgroup can be "computed" geometrically, in the way predicted by the "qantization commutes with reduction" principle of Guillemin-Sternberg.
math.RT↗
Quillen's relative Chern character is multiplicative
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 character of the Bott morphism and the relative Thom form.
math.DG↗
Index of transversally elliptic operators
In 1996, Berline and Vergne gave a cohomological formula for the index of a transversally elliptic operator. In this paper we propose a new point of view where the cohomological formulae make use of equivariant Chern characters with generalized coefficients and with compact suppport. This kind of Chern characters was studied by the authors in a previous paper (see arXiv:0801.2822).
math.DG↗
Equivariant Chern characters with generalized coefficients
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 character of a morphism of vector bundles, localized by a one form, and we prove a multiplicativity property of this generalized Chern character
math.DG↗
Equivariant relative Thom forms and Chern characters
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 the relative Chern characters, and the Riemann-Roch relation between the relative Chern character of the Bott symbol and of the relative Thom class.
math.DG↗
Formal Geometric Quantization
The aim of this article is to study the functorial properties of the ``formal geometric quantization'' procedure which is defined for non-compact Hamiltonian manifolds (when the moment map is proper). For this purpose, we introduce a technique of symplectic cutting that uses the wonderfull compactification of Concini-Procesi.
math.SG↗
Jump formulas in Hamiltonian Geometry
This paper is concerned with the Hamiltonian actions of a torus on a symplectic manifold. We are interested here in two global invariants: the Duistermaat-Heckman measure DH(M), and the Riemann-Roch chatacters RR(M,L^k),k>0, which are defined when the symplectic manifold is prequantized by a Kostant-Souriau line bundle L. We can associate to each connected component C of regular values of the moment map the following local invariants: the polynomial DH_c which coincides with DH(M) on C, and the periodic polynomial m_c which computes the multiplicity of RR(M,L^k), k>0, in the cone generated by C. The purpose of this paper is to compute the differences DH_c - DH_c' and m_c - m_c' when C and C' are two adjacent connected components of regular values of the moment map.
math.SG↗
Cohomologie equivariante et quantification geometrique
This is my Habilitation a Diriger des Recherches (French thing). In this paper I summarize the work I have done to realize the program of Witten called -non abelian localization-. This work deals first with problems of localization in equivariant cohomology. The second part of this paper concerns the Guillemin-Sternberg problem -Quantization commutes with reduction- in the geometric quantization procedure.
math.DG↗
Spin_c quantization and the K-multiplicities of the discrete series
In this paper we show that the `quantization commutes with reduction' principle of Guillemin-Sternberg holds for the coadjoint orbits that parametrize the discrete series of a real connected semi-simple Lie group.
math.DG↗
Localization of the Riemann-Roch character
We present a K-theoritic approach to the Guillemin-Sternberg conjecture, about the commutativity of geometric quantization and symplectic reduction, which was proved by Meinrenken and Tian-Zhang. Besides providing a new proof of this conjecture for the full non-abelian group action case, our methods lead to a generalisation for compact Lie group actions on manifolds that are not symplectic. Instead, these manifolds carry an invariant almost complex structure and an abstract moment map.
math.DG↗