arXiv subjects
Publications and source records attributed to Paul-Emile Paradan.
This is a monograph devoted to the study of Kirwan's real polytopes, i.e., in the context where there are involutions on the group and on the symplectic manifold. This work brings together and completes the two preprints I had already written on this subject (arXiv:2012.08837, arXiv:2111.13399).
The aim of this paper is twofold. First, we study HKKN stratifications, both algebraically and analytically, for a Cartesian product between a vector space and a compact K{ä}hler manifold. We then use these stratifications to prove convexity properties of the moment map for non-compact analytic subsets invariant under a Borel subgroup.
We withdraw this note because our calculation of the A(3,3) example, which initially contradicted one of the results of a 2005 paper by Fomin-Fulton-Li-Poon, was incorrect. In the second version of the prepublication arXiv:2303.11653, we explain how the description of the cone A(p,q) obtained by Fomin-Fulton-Li-Poon refines that obtained using the O'Shea-Sjamaar theorem.
The main focus of this work is the study of several cones relating the eigenvalues or singular values of a matrix to those of its off-diagonal blocks.
Let $G$ be a compact connected Lie group and let H be a subgroup fixed by an involution. A classical result assures that the action of the complex reductive group $H_C$ on the flag variety $F$ of $G$ admits a finite number of orbits. In this article we propose a formula for the branching coefficients of the symmetric pair $(G,H)$ that is parametrized by $H_C\backslash F$.
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 $k \in \mathbb{Z}_{>0}$, 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.
In this note, we show that the Horn cone associated with symplectic eigenvalues admits the same inequalities as the classical Horn cone, except that the equality corresponding to Tr(C) = Tr(A)+Tr(B) is replaced by the inequality corresponding to Tr(C) $\ge$ Tr(A)+Tr(B).
In this work, we study some convex cones associated to isotropic representations of symmetric spaces. We explain the inequalities that describe them by means of cohomological conditions. In particular, we study the singular Horn cone which is the counterpart of the classical Horn cone, where the eigenvalues of Hermitian square matrices are replaced by the singular values of rectangular matrices.
Let Z be the real part of a K{ä}hler Hamiltonian manifold M. The O'Shea-Sjamaar's Theorem tells us that the moment polytope Delta(Z) corresponds to the anti-invariant part of the Kirwan polytope Delta(M). The purpose of the present paper is to explain how to parameterize the equations of the facets of Delta(Z) in terms of real Ressayre's pairs of Z.
In this paper, we prove some convexity results associated to orbit projection of non-compact real reductive Lie groups.
The aim of this article is to explain how to parameterize the equations of the facets of the Kirwan polyhedron using the notion of Ressayre's pairs.
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 consequence of this formula for reduced spaces.
Let V be an Harish-Chandra discrete series representation of a real semi-simple Lie group G' and let G be a semi-simple subgroup of G'. In this paper, we give a geometric expression of the G-multiplicities in V when the representation V is supposed to be G-admissible.
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 prove a functorial aspect of the formal geometric quantization procedure of non-compact spin-c manifolds.
Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encompasses the fractional index formula of projective elliptic operator by Mathai-Melrose-Singer.
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] = 0 theorem of Meinrenken-Sjamaar in the Hamiltonian case, and we obtain mild generalizations to almost complex manifolds. This non-abelian localization theorem can be used to obtain a geometric description of the multiplicities of the index of general spin^c Dirac operators (see preprint arXiv:1411.7772).