The First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces

We give a formula for the first eigenvalue of the Dirac operator acting on spinor fields of a spin compact irreducible symmetric space $G/K$

A construction of symplectic connections through reduction

We give an elementary construction of symplectic connections through reduction. This provides an elegant description of a class of symmetric spaces and gives examples of symplectic connections with Ricci type curvature, which are not locally symmetric; the existence of such symplectic connections was unknown.Comment: 16 pages, Plain TeX fil

Special Symplectic Connections

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-K\"ahler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case.Comment: 35 pages, no figures. Exposition improved, some minor errors corrected. Version to be published by Jour.Diff.Geo

Primary decomposition of the ideal of polynomials whose fixed divisor is divisible by a prime power

We characterize the fixed divisor of a polynomial $f(X)$ in $\mathbb{Z}[X]$ by looking at the contraction of the powers of the maximal ideals of the overring ${\rm Int}(\mathbb{Z})$ containing $f(X)$. Given a prime $p$ and a positive integer $n$, we also obtain a complete description of the ideal of polynomials in $\mathbb{Z}[X]$ whose fixed divisor is divisible by $p^n$ in terms of its primary components.Comment: Fixed typos in (9) and (12

A remark on Berezin's quantization and cut locus

The consequences for Berezin's quantization on symmetric spaces of the identity of the set of coherent vectors orthogonal to a fixed one with the cut locus are stated precisely. It is shown that functions expressing the coherent states, the covariant symbols of operators, the diastasis function, the characteristic and two-point functions are defined when one variable does not belong to the cut locus of the other one.Comment: 8 pages, Latex2e, ams fonts, to appear in "Quantizations, Deformations and Coherent States", Edited by S. Twareque Ali, A. Odzijewicz and A. Strasburger, Proceedings of the XV Workshop on Geometric Methods in Physics, Bia\l owie\.{z}a, Poland, 1-7 july 1996, Rep. Math. Phys