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For any double Poisson algebra, we produce a double Poisson vertex algebra using the jet algebra construction. We show that this construction is compatible with the representation functor which associates to any double Poisson (vertex) algebra and any positive integer a Poisson (vertex) algebra. We also consider related constructions, such as Poisson reductions and Hamiltonian reductions, with the aim of comparing the different corresponding categories. This allows us to provide various interesting examples of double Poisson vertex algebras, in particular from double quivers., 69 pages, 7 figures. Comments are welcome

We apply results from the geometry of nilpotent orbits and nilpotent Slodowy slices, together with modularity and asymptotic analysis of characters, to prove many new isomorphisms between affine W-algebras and affine Kac-Moody vertex algebras and their finite extensions at specific admissible levels. In particular we identify many new collapsing levels for W-algebras., 92 pages. Minor corrections to text and to data tables. arXiv admin note: substantial text overlap with arXiv:math/0510205 by other authors

We give a simple description of the closure of the nilpotent orbits appearing as associated varieties of admissible affine vertex algebras in terms of primitive ideals., Proc. Lond. Math Soc. (published online)

There are several interesting filtrations on the Cartan subalgebra of a complex simple Lie algebra coming from very different contexts: one is the principal filtration coming from the Langlands dual, one is coming from the Clifford algebra associated with a non-degenerate invariant bilinear form, one is coming from the symmetric algebra and the Chevalley projection, and two other ones are coming from the enveloping algebra and Harish-Chandra projections. It is now known that all these filtrations coincide. This results from a combination of works of several authors (Rohr, Joseph, Alekseev and the second named author), and was essentially conjectured by Kostant. In this paper, we establish a direct correspondence between the enveloping filtration and the symmetric filtration for a simple Lie algebra of type A or C. Our proof is very different from Rohr and Joseph approaches. The idea is to use an explicit description of the symmetric and enveloping invariants in term of combinatorial objects, called weighted paths, in the crystal graph of the standard representation., 54 pages in English, 16 figures

In this note, we show that the free generators of the Mishchenko-Fomenko subalgebra of a complex reductive Lie algebra, constructed by the argument shift method at a regular element, form a regular sequence. This result was proven by Serge Ovsienko in the type A at a regular and semisimple element. Our approach is very different, and is strongly based on geometric properties of the nilpotent bicone., The proof of the main result has been shortened. To appear in Selecta Math

Let $G$ be a complex connected reductive algebraic group. Given a spherical subgroup $H \subset G$ and a subset $I$ of the set of spherical roots of $G/H$, we define, up to conjugation, a spherical subgroup $H_I \subset G$ of the same dimension of $H$, called a satellite. We investigate various interpretations of the satellites. We also show a close relation between the Poincar\'{e} polynomials of the two spherical homogeneous spaces $G/H$ and $G/H_I$., Comment: 27 pages in English. Several minor changes and significantly improved exhibition. Main results unchanged. Accepted for publication in Algebraic Geometry journal

Let g be a finite-dimensional simple Lie algebra of rank r over an algebraically closed field of characteristic zero, and let e be a nilpotent element of g. Denote by g^e the centralizer of e in g and by S(g^e)^{g^e} the algebra of symmetric invariants of g^e. We say that e is good if the nullvariety of some r homogeneous elements of S(g^e)^{g^e} in the dual of g^{e} has codimension r. If e is good then S(g^e)^{g^e} is polynomial. The main result of this paper stipulates that if for some homogeneous generators of S(g^e)^{g^e}, the initial homogeneous component of their restrictions to e+g^f are algebraically independent, with (e,h,f) an sl2-triple of g, then e is good. As applications, we obtain new examples of nilpotent elements that verify the above polynomiality condition, in in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type D_7., Comment: 59 pages. Final version, to appear in Math. Zeitschrift

For arbitrary connected reductive group G we consider the motivic integral over the arc space of an arbitrary Q-Gorenstein horospherical G-variety associated with a colored fan and prove a formula for the stringy E-function of a horospherical variety X which generalizes the one for toric varieties. We remark that in contrast to toric varieties the stringy E-function of a Gorenstein horospherical variety X may be not a polynomial if some cones in the fan of X have nonempty sets of colors. Using the stringy E-function, we can formulate and prove a new smoothness criterion for locally factorial horospherical varieties. We expect that this smoothness criterion holds for arbitrary spherical varieties., Comment: 27 pages, the proofs of Proposition 2.4 and Theorem 3.1 have been substantially modified, a general argument was added in the proof of Lemma 5.4., to appear in Compositio Math

Let g be a complex simple Lie algebra, and h be a Cartan subalgebra. In the end of 1990s, B. Kostant defined two filtrations on h, one using the Clifford algebras and the odd analogue of the Harish-Chandra projection $hc: Cl(g) \to Cl(h)$, and the other one using the canonical isomorphism $\check{h} = h^*$ (here $\check{h}$ is the Cartan subalgebra in the simple Lie algebra corresponding to the dual root system) and the adjoint action of the principal sl2-triple. Kostant conjectured that the two filtrations coincide. The two filtrations arise in very different contexts, and comparing them proved to be a difficult task. Y. Bazlov settled the conjecture for g of type A using explicit expressions for primitive invariants in the exterior algebra of g. Up to now this approach did not lead to a proof for all simple Lie algebras. Recently, A. Joseph proved that the second Kostant filtration coincides with the filtration on h induced by the generalized Harish-Chandra projection $(Ug \otimes g)^g \to Sh \otimes h$ and the evaluation at $\rho \in h^*$. In this note, we prove that Joseph's result is equivalent to the Kostant Conjecture. We also show that the standard Harish-Chandra projection $Ug \to Sh$ composed with evaluation at $\rho$ induces the same filtration on h., Comment: 10 pages

A connected algebraic group Q defined over a field of characteristic zero is quasi-reductive if there is an element of its dual of reductive type, that is such that the quotient of its stabiliser by the centre of Q is a reductive subgroup of GL(q), where q=Lie(Q). Due to results of M. Duflo, coadjoint representation of a quasi-reductive Q possesses a so called maximal reductive stabiliser and knowing this subgroup, defined up to a conjugation in Q, one can describe all coadjoint orbits of reductive type. In this paper, we consider quasi-reductive parabolic subalgebras of simple complex Lie algebras as well as all seaweed subalgebras of gl(n) and describe the classes of their maximal reductive stabilisers., Comment: 35 pages, 5 figures; International Mathematics Research Notices (2011) 45 pages

This note is a corrigendum to the previous version arXiv:0711.2735v3 published in J. Lie Theory. As it has been recently pointed out to me by Alexander Premet, Remark 3 of arXiv:0711.2735v3 is incorrect. We verify in this note thanks to recent results of Premet and Topley (see arXiv:1301.4653) that Theorem 25 of arXiv:0711.2735v3 remains correct in spite of this error., Comment: arXiv admin note: text overlap with arXiv:1301.4653 by other authors

I prefer taking off this paper for the moment because of a mistake in the lemma 2.1 of the secund version. Precisely, in the proof of this lemma, it is not clear that the morphism $r\_j$ is flat, that I claim it., This paper has been withdrawn

In \cite{indice}, we show the following result, conjectured by D. Panyushev \cite{Panyushev}, for $\g$ a semisimple Lie algebra: {\rm ind} \n(\g^{e}) = {\rm rk} \g-\dim \z(\g^{e}, where $\n(\g^{e})$ and $\z(\g^{e})$ are, respectively, the normaliser and the centre of the centraliser $\g^{e}$ of a nilpotent element $e$. This result is proved in \cite{indice} when $\g$ is a classical simple Lie algebra and when $e$ satisfies a certain property $(P)$. We present in this paper the computations, made using GAP4, which prove that distinguished, non-regular, nilpotent orbits in $E\_6$, $E\_7$, $E\_8$ and $F\_4$ satisfy the property $(P)$. This work completes the proof, presented in \cite{indice}, of the equality (\ref{princ}). The complete proof of this result was already presented in \cite{indice\_arxiv}., 34 pages en fran\c{c}ais