Very nilpotent basis and n-tuples in Borel subalgebras

A (vector space) basis B of a Lie algebra is said to be very nilpotent if all the iterated brackets of elements of B are nilpotent. In this note, we prove a refinement of Engel's Theorem. We show that a Lie algebra has a very nilpotent basis if and only if it is a nilpotent Lie algebra. When g is a semisimple Lie algebra, this allows us to define an ideal of S((g^n)^*)^G whose associated algebraic set in g^n is the set of n-tuples lying in a same Borel subalgebra.Comment: short note, 4 page

On the normality of the null-fiber of the moment map for θ\theta- and tori representations

Let (G, V) be a representation with either G a torus or (G, V) a locally free stable $\theta$-representation. We study the fiber at 0 of the associated moment map, which is a commuting variety in the latter case. We characterize the cases where this fiber is normal. The quotient (i.e. the symplectic reduction) turns out to be a specific orbifold when the representation is polar. In the torus case, this confirms a conjecture stated by C. Lehn, M. Lehn, R. Terpereau and the author in a former article. In the $\theta$-case, the conjecture was already known but the present approach yield another proof.Comment: 21 pages, comments welcom

Irregular locus of the commuting variety of reductive symmetric Lie algebras and rigid pairs

The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of Popov. In [Po], the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. [Po, Remark 1.13] suggests that the arguments and methods of [Po] are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain some results on the irregular locus of the component of maximal dimension of the "symmetric commuting variety". As a by-product, we study some pairs of commuting elements specific to the symmetric case, that we call rigid pairs. These pairs allow us to find all symmetric Lie algebras whose commuting variety is reducible.Comment: 29 pages, 3 table

Parabolic Conjugation and Commuting Varieties

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of the general linear group on the variety of nilpotent matrices in its Lie algebra. Lie-theoretically, it is natural to wonder about the number of orbits of this action. We translate the setup to a representation-theoretic one and obtain a finiteness criterion which classifies all actions with only a finite number of orbits over an arbitrary infinite field. These results are applied to commuting varieties and nested punctual Hilbert schemes.Comment: Comments welcom

Sheets, slice induction and G2(2) case

In this paper, we study sheets of symmetric Lie algebras through their Slodowy slices. In particular, we introduce a notion of slice induction of nilpotent orbits which coincides with the parabolic induction in the Lie algebra case. We also study in more details the sheets of the non-trivial symmetric Lie algebra of type G2. We characterize their singular loci and provide a nice desingularisation lying in so7.Comment: 22 pages. In this new version, computations of section 4 are pared down. Important modifications of the exposition of Section 3 on slice inductio

Sheets of Symmetric Lie Algebras and Slodowy Slices

38 pagesInternational audienceLet T be an involution of the finite dimensional complex reductive Lie algebra g and g=k+p be the associated Cartan decomposition. Denote by K the adjoint group of k. The K-module p is the union of the subsets p^{(m)}={x | dim K.x =m}, indexed by integers m, and the K-sheets of (g,T) are the irreducible components of the p^{(m)}. The sheets can be, in turn, written as a union of so-called Jordan K-classes. We introduce conditions in order to describe the sheets and Jordan K-classes in terms of Slodowy slices. When g is of classical type, the K-sheets are shown to be smooth; if g=gl_N a complete description of sheets and Jordan K-classes is then obtained

Composantes irr\'eductibles de la vari\'et\'e commutante nilpotente d'une alg\`ebre de Lie sym\'etrique semi-simple

Let \theta be an involution of the semisimple Lie algebra g and g=k+p be the associated Cartan decomposition. The nilpotent commuting variety of (g,\theta) consists in pairs of nilpotent elements (x,y) of p such that [x,y]=0. It is conjectured that this variety is equidimensional and that its irreducible components are indexed by the orbits of p-distinguished elements. This conjecture was established by A. Premet in the case (g \times g, \theta) where \theta(x,y)=(y,x). In this work we prove the conjecture in a significant number of other cases.Comment: 38 pages, in french, minor changes, Journal-ref adde

The closure of a sheet is not always a union of sheets, a short note

3 pagesAppendix of G. Carnovale's paper, Lusztig's partition and sheets, Mathematical Research Letters, 22 no. 3 (2015), 645--664.In this note we answer to a frequently asked question. If G is an algebraic group acting on a variety V, a G-sheet of V is an irreducible component of V^(m), the set of elements of V whose G-orbit has dimension m. We focus on the case of the adjoint action of a semisimple group on its Lie algebra. We give two families of examples of sheets whose closure is not a union of sheets in this setting

Dynamics of dikes versus cone sheets in volcanic systems

International audienceIgneous sheet intrusions of various shapes, such as dikes and cone sheets, coexist as parts of complex volcanic plumbing systems likely fed by common sources. How they form is fundamental regarding volcanic hazards, yet no dynamic model simulates and predicts satisfactorily their diversity. Here we present scaled laboratory experiments that reproduced dikes and cone sheets under controlled conditions. Our models show that their formation is governed by a dimensionless ratio (Π1), which describes the geometry of the magma source, and a dynamic dimensionless ratio (Π2), which compares the viscous stresses in the flowing magma to the host rock strength. Plotting our experiments against these two numbers results in a phase diagram evidencing a dike and a cone sheet field, separated by a sharp transition that fits a power law. This result shows that dikes and cone sheets correspond to distinct physical regimes of magma emplacement in the crust. For a given host rock strength, cone sheets preferentially form when the source is shallow, relative to its lateral extent, orwhen the magma influx velocity (or viscosity) is high. Conversely, dikes form when the source is deep compared to its size, or when magma influx rate (or viscosity) is low. Both dikes and cone sheets may form fromthe same source, the shift fromone regime to the other being then controlled by magma dynamics, i.e., different values of Π2. The extrapolated empirical dike-to-cone sheet transition is in good agreement with the occurrence of dikes and cone sheets in various natural volcanic settings