Deformations of the Lie-Poisson sphere of a compact semisimple Lie algebra

A compact semisimple Lie algebra $\mathfrak{g}$ induces a Poisson structure $\pi$ on the unit sphere $S$ in $\mathfrak{g}^*$. We compute the moduli space of Poisson structures on $S$ around $\pi$. This is the first explicit computation of a Poisson moduli space in dimension greater or equal than three around a degenerate (i.e. not symplectic) Poisson structure.Comment: 10 pages, v3: published versio

On the domain of singular traces

The question whether an operator belongs to the domain of some singular trace is addressed, together with the dual question whether an operator does not belong to the domain of some singular trace. We show that the answers are positive in general, namely for any (compact, infinite rank) positive operator A we exhibit two singular traces, the first being zero and the second being infinite on A. However, if we assume that the singular traces are generated by a "regular" operator, the answers change, namely such traces always vanish on trace-class, non singularly traceable operators and are always infinite on non trace-class, non singularly traceable operators. These results are achieved on a general semifinite factor, and make use of a new characterization of singular traceability (cf. math.OA/0202108).Comment: 7 pages, LaTeX. Minor corrections, to appear on the International Journal of Mathematic

Extensions of positive definite functions on amenable groups

Let $S$ be a subset of a amenable group $G$ such that $e\in S$ and $S^{-1}=S$. The main result of the paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite operator-valued function on $S$ can be extended to a positive definite function on $G$. Several known extension results are obtained as a corollary. New applications are also presented

Self-adjointness and boundedness in quadratic quantization

We construct a counter example showing, for the quadratic quantization, the identity $(\Gamma(T))^*= \Gamma(T^*)$ is not necessarily true. We characterize all operators on the one-particle algebra whose quadratic quantization are self-adjoint operators on the quadratic Fock space. Finally, we discuss the boundedness of the quadratic quantization.Comment: 14 page

Thin buildings

Let X be a building of uniform thickness q+1. L^2-Betti numbers of X are reinterpreted as von-Neumann dimensions of weighted L^2-cohomology of the underlying Coxeter group. The dimension is measured with the help of the Hecke algebra. The weight depends on the thickness q. The weighted cohomology makes sense for all real positive values of q, and is computed for small q. If the Davis complex of the Coxeter group is a manifold, a version of Poincare duality allows to deduce that the L^2-cohomology of a building with large thickness is concentrated in the top dimension.Comment: This is the version published by Geometry & Topology on 24 May 200

Elementary invariants for centralizers of nilpotent matrices

We construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the Lie algebra gl_N(C). In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova (math.RT/0610049).Comment: 12 page

On the homomorphisms between scalar generalized Verma modules

We study the homomorphisms between scalar generalized Verma modules. We conjecture that any homomorphism between is composition of elementary homomorphisms. The purpose of this article is to show the conjecture is affirmative for many parabolic subalgebras under the assumption that the infinitesimal characters are regular.Comment: 46 pages, A reference is adde

Algebraic methods in the theory of generalized Harish-Chandra modules

This paper is a review of results on generalized Harish-Chandra modules in the framework of cohomological induction. The main results, obtained during the last 10 years, concern the structure of the fundamental series of $(\mathfrak{g},\mathfrak{k})-$modules, where $\mathfrak{g}$ is a semisimple Lie algebra and $\mathfrak{k}$ is an arbitrary algebraic reductive in $\mathfrak{g}$ subalgebra. These results lead to a classification of simple $(\mathfrak{g},\mathfrak{k})-$modules of finite type with generic minimal $\mathfrak{k}-$types, which we state. We establish a new result about the Fernando-Kac subalgebra of a fundamental series module. In addition, we pay special attention to the case when $\mathfrak{k}$ is an eligible $r-$subalgebra (see the definition in section 4) in which we prove stronger versions of our main results. If $\mathfrak{k}$ is eligible, the fundamental series of $(\mathfrak{g},\mathfrak{k})-$modules yields a natural algebraic generalization of Harish-Chandra's discrete series modules.Comment: Keywords : generalized Harish-Chandra module, (g,k)-module of finite type, minimal k-type, Fernando-Kac subalgebra, eligible subalgebra; Pages no. : 13; Bibliography : 21 item

On the continuity of spectra for families of magnetic pseudodifferential operators

For families of magnetic pseudodifferential operators defined by symbols and magnetic fields depending continuously on a real parameter $\epsilon$, we show that the corresponding family of spectra also varies continuously with $\epsilon$.Comment: 22 page