On some modules of covariants for a reflection group

Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak h$ and Weyl group $W$. We build up a graded map $(\mathcal H\otimes \bigwedge\mathfrak h\otimes \mathfrak h)^W\to (\bigwedge \mathfrak g\otimes \mathfrak g)^\mathfrak g$ of $(\bigwedge \mathfrak g)^\mathfrak g\cong S(\mathfrak h)^W$-modules, where $\mathcal H$ is the space of $W$-harmonics. In this way we prove an enhanced form of a conjecture of Reeder for the adjoint representation. New version with different title. Various improvements. New section 7.Comment: 18 Page

The quotient of a complete symmetric variety

We study the quotient of a completion of a symmetric variety G/H under the action of H. We prove that this is isomorphic to the closure of the image of an isotropic torus under the action of the restricted Weyl group. In the case the completion is smooth and toroidal we describe the set of semistable points.Comment: Dedicated to Ernest Vinberg on occasion of his 70th birthda

The adjoint representation inside the exterior algebra of a simple Lie algebra

For a simple complex Lie algebra $\mathfrak g$ we study the space of invariants $A=\left( \bigwedge \mathfrak g^*\otimes\mathfrak g^*\right)^{\mathfrak g}$, (which describes the isotypic component of type $\mathfrak g$ in $\bigwedge \mathfrak g^*$) as a module over the algebra of invariants $\left(\bigwedge \mathfrak g^*\right)^{\mathfrak g}$. As main result we prove that $A$ is a free module, of rank twice the rank of $\mathfrak g$, over the exterior algebra generated by all primitive invariants in $(\bigwedge \mathfrak g^*)^{\mathfrak g}$, with the exception of the one of highest degree.Comment: Final version. More misprints corrected. To appear in Advances in Mathematic

Infinitesimal index: cohomology computations

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 improvements

Projective wonderful models for toric arrangements

In this paper we illustrate an algorithmic procedure which allows to build projective wonderful models for the complement of a toric arrangement in a n-dimensional algebraic torus T. The main step of the construction, inspired by [9], is a combinatorial algorithm that produces a toric variety by subdividing in a suitable way a given smooth fan