On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

We consider the finite $W$-algebra U(\g,e) associated to a nilpotent element e \in \g in a simple complex Lie algebra \g of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem, we verify a conjecture of Premet, that U(\g,e) always has a 1-dimensional representation, when \g is of type $G_2$, $F_4$, $E_6$ or $E_7$. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal in U(\g) whose associated variety is the coadjoint orbit corresponding to $e$.Comment: 14 pages, minor changes

A propos de quelques citations de Caecilius dans le de Senectute de Cicéron

En étudiant les restes des auteurs dramatiques latins, on rencontre parfois des fragments dont la source est double, par exemple avec une citation chez Cicéron et une citation parallèle chez un grammairien. Or, il arrive que le texte chez Cicéron ne soit pas le même que chez le grammairien. Il semble que ce soit souvent l ignorance de la métrique et de la prosodie anciennes qui provoque ces di!érences, puisque Cicéron – ou sa source – ne pouvait notamment plus lire un vers qui présentait un abrégement iambique et qu il le récrivait donc en corrigeant ce qui lui apparaissait comme une impossibilité métrique. Les situations sont certes très diverses, et les réponses doivent aussi varier selon qu on est un éditeur de Cicéron ou qu on veut retrouver le texte de Caecilius ou de Naevius.As we study the remains of Latin playwrights, we sometimes come across fragments whose source is twofold, as is the case with a quotation by Cicero and a parallel quotation by a grammarian. Yet it occurs at times that the text by Cicero is not the same as the grammarian's. It seems that it is often the ignorance of ancient metrics and prosody which causes these differences, since Cicero – or his source – was notably no more able to read a line of verse that displayed an iambic shortening, and would thus rewrite it, correcting what appeared as a metric impossibility. Admittedly, we encounter very diverse situations, and the answers must also vary depending upon whether we aim to edit Cicero's work or whether we want to get back to the original text by Caecilius or Naevius

L'instrumentation sur mesure à usage unique : le futur de la prothèse totale de genou ?

En termes d'évolution, le nombre d'arthroplasties totales de genou que l'on pratique aujourd'hui devrait être multiplié par 7 d'ici vingt ans. Jusqu'à récemment, cette intervention nécessitait un matériel très important. Il est aujourd'hui possible de rationaliser et d'optimiser cette procédure à l'aide d'une planification préopératoire tridimensionnelle, de guides de coupe personnalisés et d'un kit d'instrumentation à usage unique. Le but est de permettre des gains en termes de précision, de temps et de coûts ainsi que de participer à la réduction du risque infectieux. Cette technologie ouvre d'importantes perspectives sur un futur implant conçu entièrement sur mesure pour chaque patient

Classic and mirabolic Robinson-Schensted-Knuth correspondence for partial flags

In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line.Comment: 27 pages, slightly rewritten to combine two papers into one and clarify some section

Deconstructibility and the Hill lemma in Grothendieck categories

A full subcategory of a Grothendieck category is called deconstructible if it consists of all transfinite extensions of some set of objects. This concept provides a handy framework for structure theory and construction of approximations for subcategories of Grothendieck categories. It also allows to construct model structures and t-structures on categories of complexes over a Grothendieck category. In this paper we aim to establish fundamental results on deconstructible classes and outline how to apply these in the areas mentioned above. This is related to recent work of Gillespie, Enochs, Estrada, Guil Asensio, Murfet, Neeman, Prest, Trlifaj and others.Comment: 20 pages; version 2: minor changes, misprints corrected, references update

Corrigendum to `Orbit closures in the enhanced nilpotent cone', published in Adv. Math. 219 (2008)

In this note, we point out an error in the proof of Theorem 4.7 of [P. Achar and A.~Henderson, `Orbit closures in the enhanced nilpotent cone', Adv. Math. 219 (2008), 27-62], a statement about the existence of affine pavings for fibres of a certain resolution of singularities of an enhanced nilpotent orbit closure. We also give independent proofs of later results that depend on that statement, so all other results of that paper remain valid.Comment: 4 pages. The original paper, in a version almost the same as the published version, is arXiv:0712.107

Linear Koszul Duality II - Coherent sheaves on perfect sheaves

In this paper we continue the study (initiated in a previous article) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general setting, and prove its compatibility with morphisms of vector bundles and base change.Comment: Final version, to appear in JLMS. The numbering differs from the published version, and is the one used in our papers [MR2] and [MR3] from the bibliograph

Dynkin varieties

Let G be a linear algebraic group. The Dynkin variety ßx of an element x of G is the fixed point set of x on the variety ß of all Borel subgroups of G. We show that all irreducible components of this variety have the same dimension, and that ßx is connected if x is unipotent. Suppose now that G is reductive (but not necessarily connected) and that x is unipotent. We generalize an inequality linking dim ßxand dim Zꓖ (x) and some results on the action of A₀(x) on the set S(x) of all irreducible components of ßx where A₀(x) is the group of components of ZGo(x). We consider also regular and sub-regular elements in non-connected reductive groups. For classical groups we get a combinatorial description for S(x) and the action of A₀(x) on S(x) and a formula for dim ßx We generalize to non-connected reductive groups a theorem of Richardson which associates to each conjugacy class of parabolic subgroups of G a unipotent class of G and for classical groups we get a combinatorial description of this map. There is also some material on unipotent classes in arbitrary reductive groups

Orbit closures in the enhanced nilpotent cone

We study the orbits of $G=\mathrm{GL}(V)$ in the enhanced nilpotent cone $V\times\mathcal{N}$, where $\mathcal{N}$ is the variety of nilpotent endomorphisms of $V$. These orbits are parametrized by bipartitions of $n=\dim V$, and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition analogues of Kostka polynomials, defined by Shoji. Finally, we make a connection with Kato's exotic nilpotent cone in type C, proving that the closure ordering is the same, and conjecturing that the intersection cohomology is the same but with degrees doubled.Comment: 32 pages. Update (August 2010): There is an error in the proof of Theorem 4.7, in this version and the almost-identical published version. See the corrigendum arXiv:1008.1117 for independent proofs of later results that depend on that statemen

Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic

The classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite fiel