Academic literature on the topic 'Parabolic subgroups, projective homogeneous varieties'

We study the following question: Given a vector bundle on a projective variety [Formula: see text] such that the restriction of [Formula: see text] to every closed curve [Formula: see text] is ample, under what conditions [Formula: see text] is ample? We first consider the case of an abelian variety [Formula: see text]. If [Formula: see text] is a line bundle on [Formula: see text], then we answer the question in the affirmative. When [Formula: see text] is of higher rank, we show that the answer is affirmative under some conditions on [Formula: see text]. We then study the case of [Formula: see text], where [Formula: see text] is a reductive complex affine algebraic group, and [Formula: see text] is a parabolic subgroup of [Formula: see text]. In this case, we show that the answer to our question is affirmative if [Formula: see text] is [Formula: see text]-equivariant, where [Formula: see text] is a fixed maximal torus. Finally, we compute the Seshadri constant for such vector bundles defined on [Formula: see text].

We provide some general conditions which ensure that a system of inequalities involving homogeneous polynomials with coefficients in a \(S\)-adic field has nontrivial \(S\)-integral solutions. The proofs are based on the strong approximation property for Zariski-dense subgroups and adelic geometry of numbers. We give some examples of applications for systems involving quadratic and linear forms.

AbstractWe study K-orbits in G/P where G is a complex connected reductive group, P ⊆ G is a parabolic subgroup, and K ⊆ G is the fixed point subgroup of an involutive automorphism θ. Generalizing work of Springer, we parametrize the (finite) orbit set K \ G/P and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of θ-stable (resp. θ-split) parabolic subgroups. We also describe the decomposition of any (K, P)-double coset in G into (K, B)-double cosets, where B ⊆ P is a Borel subgroup. Finally, for certain K-orbit closures X ⊆ G/B, and for any homogeneous line bundle on G/B having nonzero global sections, we show that the restriction map resX : H0(G/B, ) → H0(X, ) is surjective and that Hi(X, ) = 0 for i ≥ 1. Moreover, we describe the K-module H0(X, ). This gives information on the restriction to K of the simple G-module H0(G/B, ). Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal K-types.

AbstractWe study the homogeneous ind-spaces $$\textrm{GL}(\textbf{s})/\textbf{P}$$ GL ( s ) / P where $$\textrm{GL}(\textbf{s})$$ GL ( s ) is a strict diagonal ind-group defined by a supernatural number $$\textbf{s}$$ s and $$\textbf{P}$$ P is a parabolic ind-subgroup of $$\textrm{GL}(\textbf{s})$$ GL ( s ) . We construct an explicit exhaustion of $$\textrm{GL}(\textbf{s})/\textbf{P}$$ GL ( s ) / P by finite-dimensional partial flag varieties. As an application, we characterize all locally projective $$\textrm{GL}(\infty )$$ GL ( ∞ ) -homogeneous spaces, and some direct products of such spaces, which are $$\textrm{GL}(\textbf{s})$$ GL ( s ) -homogeneous for a fixed $$\textbf{s}$$ s . The very possibility for a $$\textrm{GL}(\infty )$$ GL ( ∞ ) -homogeneous space to be $$\textrm{GL}(\textbf{s})$$ GL ( s ) -homogeneous for a strict diagonal ind-group $$\textrm{GL}(\textbf{s})$$ GL ( s ) arises from the fact that the automorphism group of a $$\textrm{GL}(\infty )$$ GL ( ∞ ) -homogeneous space is much larger than $$\textrm{GL}(\infty )$$ GL ( ∞ ) .

AbstractComplex projective algebraic varieties with $${{\mathbb {C}}}^*$$ C ∗ -actions can be thought of as geometric counterparts of birational transformations. In this paper we describe geometrically the birational transformations associated to rational homogeneous varieties endowed with a $${{\mathbb {C}}}^*$$ C ∗ -action with no proper isotropy subgroups.

Abstract Let 𝐺 be a real semisimple Lie group with finite centre and without compact factors, Q < G Q<G a parabolic subgroup and 𝑋 a homogeneous space of 𝐺 admitting an equivariant projection on the flag variety G / Q G/Q with fibres given by copies of lattice quotients of a semisimple factor of 𝑄. Given a probability measure 𝜇, Zariski-dense in a copy of H = SL 2 ⁡ ( R ) H=\operatorname{SL}_{2}(\mathbb{R}) in 𝐺, we give a description of 𝜇-stationary probability measures on 𝑋 and prove corresponding equidistribution results. Contrary to the results of Benoist–Quint corresponding to the case G = Q G=Q , the type of stationary measures that 𝜇 admits depends strongly on the position of 𝐻 relative to 𝑄. We describe possible cases and treat all but one of them, among others using ideas from the works of Eskin–Mirzakhani and Eskin–Lindenstrauss.

Cette thèse achève la classification des sous-schémas en groupes paraboliques des groupes algébriques semi-simples sur un corps algébriquement clos, en particulier de caractéristique deux et trois. Dans un premier temps, nous présentons la classification en supposant que la partie réduite de ces sous-groupes soit maximale, avant de passer au cas général. Nous parvenons à une description quasiment uniforme : à l'exception d'un groupe de type G₂ en caractéristique deux, chaque sous-schémas en groupes parabolique est obtenu en multipliant des paraboliques réduits par des noyaux d'isogénies purement inséparables, puis en prenant l'intersection. En conclusion, nous discutons quelques implications géométriques de cette classification
This thesis brings to an end the classification of parabolic subgroup schemes of semisimple groups over an algebraically closed field, focusing on characteristic two and three. First, we present the classification under the assumption that the reduced part of these subgroups is maximal; then we proceed to the general case. We arrive at an almost uniform description: with the exception of a group of type G₂ in characteristic two, any parabolic subgroup scheme is obtained by multiplying reduced parabolic subgroups by kernels of purely inseparable isogenies, then taking the intersection. In conclusion, we discuss some geometric implications of this classification