I will give an overview of a programme investigating projective embeddings of (exceptional) geometries which Hendrik Van Maldeghem and I started in 2010.
Tag - Algebraic groups
Previous work constructed an analogue of the Springer resolution for the universal cover of the principal nilpotent orbit. In joint work with Precup and Russell, we showed that in type A this generalized Springer resolution is closely connected with Lusztig's generalized Springer correspondence. In this talk we discuss the geometry of the fibres of the generalized Springer resolution, and in particular, show that the fibres have an analogue of an affine paving.
Buildings have been introduced by Tits in order to study semisimple algebraic groups from a geometrical point of view. One of the most important results in the theory of buildings is the classification of thick irreducible spherical buildings of rank at least 3. In particular, any such building comes from an RGD-system. The decisive tool in this classification is the Extension theorem for spherical buildings, i.e. a local isometry extends to the whole building.
Twin buildings were introduced by Ronan and Tits in the late 1980s. Their definition was motivated by the theory of Kac-Moody groups over fields. Each such group acts naturally on a pair of buildings and the action preserves an opposition relation between the chambers of the two buildings. This opposition relation shares many important properties with the opposition relation on the chambers of a spherical building. Thus, twin buildings appear to be natural generalizations of spherical buildings with infinite Weyl group. Since the notion of RGD-systems exists not only in the spherical case, one can ask whether any twin building (satisfying some further conditions) comes from an RGD-system. In 1992 Tits proves several results that are inspired by his strategy in the spherical case and he discusses several obstacles for obtaining a similar Extension theorem for twin buildings. In this talk I will speak about the history and developments of the Extension theorem for twin buildings.
We provide a unified combinatorial framework to study orbits in affine flag varieties via the associated Bruhat-Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine Weyl group. Finally, for Bruhat-Tits buildings, we relate these retractions and their effect on certain minimal galleries to double coset intersections in the corresponding affine flag variety.
Previous work constructed an analogue of the Springer resolution for the universal cover of the principal nilpotent orbit. In joint work with Precup and Russell, we showed that in type A this generalized Springer resolution is closely connected with Lusztig's generalized Springer correspondence. In this talk we discuss the geometry of the fibres of the generalized Springer resolution, and in particular, show that the fibres have an analogue of an affine paving.
We will survey a series of recent developments in the area of bounded generation and first-order descriptions of groups. The goal is to illuminate the known results relevant to logical characterizations of Chevalley and Kac-Moody groups. If time permits I will discuss related questions originated from universal algebraic geometry.
Compact hyperbolic manifolds are very interesting geometric objects. Maybe surprisingly, they are also interesting from an algebraic point of view: They are completely determined by their fundamental groups (this is Mostow's Theorem), which is naturally a subgroup of the rational valued invertible matrices in some dimension, GLn(ℚ). When the fundamental group essentially consists of the integer points of some algebraic subgroup of GLn we say that the manifold is arithmetic. A question arises: is there a simple geometric criterion for arithmeticity of hyperbolic manifolds? Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen. In a recent work with Fisher, Miller and Stover we proved this conjecture. Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which Alex Furman and I have developed in recent years. In my talk I will survey the subject and focus on the relation between the geometric, algebraic and arithmetic concepts
The purpose of my talk is to discuss the following results recently obtained in collaboration with A.Masuoka (Tsukuba University, Japan). First, we prove that a certain category of Harish-Chandra pairs is equivalent to the category of (not necessary affine) locally algebraic group superschemes. Using this fundamental equivalence we superize the famous Barsotti-Chevalley theorem and prove that the sheaf quotient of an algebraic group superscheme over its group super-subscheme is again a superscheme of finite type. I will also formulate some open problems whose solving would bring significant progress in the supergroup theory.
Kazhdan and Lusztig proved the Deligne-Langlands conjecture, a bijection between irreducible representations of unipotent principal block representations of a p-adic group with certain unipotent Langlands parameters in the Langlands dual group (plus the data of certain representations). We lift this bijection to a statement on the level of categories. Namely, we define a stack of unipotent Langlands parameters and a coherent sheaf on it, which we call the coherent Springer sheaf, which generates a subcategory of the derived category equivalent to modules for the affine Hecke algebra (or specializing at q, unipotent principal block representations of a p-adic group). Our approach involves categorical traces, Hochschild homology, and Bezrukavnikov's Langlands dual realizations of the affine Hecke category.
This talk will introduce spherical elements in a finite Coxeter system. These spherical elements are a generalization of Coxeter elements, that conjecturally, for Weyl groups, index Schubert varieties in the flag variety G/B that are spherical for the action of a Levi subgroup. We will see that this conjecture extends and unifies previous sphericality results for Schubert varieties in G/B due to P. Karuppuchamy, J. Stembridge, P. Magyar–J. Weyman-A. Zelevinsky. In type A, the combinatorics of Demazure modules and their key polynomials, multiplicity freeness, and split-symmetry in algebraic combinatorics are employed to prove this conjecture for several classes of Schubert varieties.
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