# Séminaire Groupes, Représentations et Géométrie

### salle 1016, 1er étage, Bâtiment Sophie Germain, 8 place Aurélie Nemours, 75013 Paris

Organisateurs : , , , , , , , et .

Page répertoriée sur l'AdM à partir de l'IMJ-PRG

Vendredi 10 mai 2019 à 14h00

Syu KATO (Kyoto University), Semi-infinite flag manifolds and quantum K-groups of flag manifolds.
[Semi-infinite flag manifolds, that is a variant of an affine flag manifolds, essentially appears in the considerations of Lusztig and Drinfeld in the late 1970s to early 1980s. It encodes representation theory of affine Lie algebras at the critical level as exhibited by Feigin and Frenkel. As typical in infinite-dimensional objects, it has several disguises. One of its incarnation (that we call the ind-model), realization as the space of quasi-maps, was pursued in detail by Braverman, Finkelberg, Mirkovic and their collaborators.
There is another incarnation (that we call the proj-model), that directly consider it as an ind-scheme of infinite type, existed from the beginning. However, it was not extensively studied as it denied ad hoc approaches to implement geometric objects on that.
In the first part of this talk, we begin by recalling representation theory of loop algebras and show that it captures the structure of the proj-models of semi-infinite flag manifolds (and some of the basic theorems analogous to those in the usual flag manifolds).
In the second part of my talk, we briefly explain how to make sense of equivariant $K$-groups of them. Using this, we construct an isomorphism between the equivariant $K$-groups of semi-infinite flag manifolds and the (equivariant small) quantum $K$-groups of the (finite-dimensional) flag manifold that respects natural bases offered by Schubert subvarieties.]

Vendredi 17 mai 2019 à 14h00

Mohamed BARAKAT (Universität Siegen), Chevalley’s Theorem on constructible images made constructive.
[Chevalley proved that the image of an algebraic morphism between algebraic varieties is a constructible set. Examples are orbits of algebraic group actions. A constructible set in a topological space is a finite union of locally closed sets and a locally closed set is the difference of two closed subsets. Simple examples show that even if the source and target of the morphism are affine varieties the image may neither be affine nor quasi-affine. In this talk I will present an Gröbner-basis-based algorithm which computes the constructible image of a morphism of affine spaces, along with applications from several fields.]

Vendredi 24 mai 2019 à 14h00

Ivan CHEREDNIK (University of North Carolina), Motivic and DAHA superpolynomials in any ranks for plane curve singularities.
[There was not much progress with the extension of theory of (moduli spaces of) vector bundles from smooth to singular curves. Surprisingly, this can be managed in a rather elementary way for plane curve singularities. The corresponding moduli spaces will be described; they generalize compactified Jacobians (the case of rank one). This theory is closely related to nil-elliptic affine Springer fibers (in type A'') with non-reduced spectral curves, their germs to be exact.
The corresponding motivic superpolynomials are expected to coincide with colored (by columns) DAHA superpolynomials, which is checked sufficiently well.
The former can be defined for any curve singularities but the connection to the latter is so far only in the planar case. A possible implication is to Hall-Ringel convolution algebras for curve singularities. This can potentially provide a variant of elliptic Hall algebras by Burban-Schiffmann-Vasserot for any (arithmetic) genus upon the restriction to curve singularities. The talk will require almost no knowledge of rings and modules; the DAHA superpolynomials will be defined.]

Vendredi 7 juin 2019 à 14h00

Anthony JOSEPH (Weizmann Institute of Science).

Vendredi 14 juin 2019 à 14h00

Linyuan LIU (IMJ-PRG), Cohomologie des fibrés en droites sur $G/B$ en caractéristique positive.
[Soit $G$ un groupe algébrique semi-simple sur un corps $k$ algébriquement clos de caractéristique positive et soit $B$ un sous-groupe de Borel. La cohomologie des fibrés en droites $G$-équivariants sur $G/B$ induits par des caractères de $B$ sont des objets importants dans la théorie des représentations de $G$. Dans cet exposé, je vais commencer par rappeler des résultats à leur sujet, dus à Kempf, Griffith, Andersen, Jantzen, Kuhne-Hausmann, Irving, Doty, Sullivan, Donkin, etc. Ensuite, je vais présenter les nouveaux résultats pour $G = SL_3$ obtenus dans ma thèse. Plus précisément, j’ai montré l’existence de deux filtrations de $H^i(G/B,\mu)$. La première existe pour $i = 1, 2$ et $\mu$ dans la région de Griffith ; la deuxième, qui généralise la $p$-filtration introduite par Jantzen, existe pour tout $i$ et pour tout $\mu$.]

Vendredi 21 juin 2019 à 14h00

Leonardo PATIMO (Albert-Ludwigs-Universität Freiburg).

Vendredi 28 juin 2019 à 14h00

Reda CHANEB (IMJ-PRG).

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 Groupes, représentations et géométrie. Dernière modification : le 07/05/2019

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