A Fulton–Hansen theorem for almost homogeneous spaces

Let $G$ be a simple algebraic group. A closed subgroup $H$ of $G$ is said to be spherical if it has a dense orbit on the flag variety $G/B$ of $G$. Reductive spherical subgroups of simple Lie groups were classified by Krämer in 1979. In 1997, Brundan showed that each example from Krämer’s list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, up to now there has been no classification of all such instances in positive characteristic. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic 2 which has no counterpart in Krämer’s classification. As one of our key tools, we prove a general deformation result for subgroup schemes that allows us to deduce the sphericality of subgroups in positive characteristic from the same property for subgroups in characteristic zero.

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A note on the mean value theorem for special homogeneous spaces

Nagoya Mathematical Journal ◽

10.1017/s0027763000005948 ◽

1996 ◽

Vol 143 ◽

pp. 111-117 ◽

Cited By ~ 1

Author(s):

Masanori Morishita ◽

Takao Watanabe

Keyword(s):

Algebraic Group ◽

Algebraic Variety ◽

Homogeneous Spaces ◽

Rational Numbers ◽

Rational Points ◽

Linear Algebraic Group ◽

Mean Value ◽

Mean Value Theorem ◽

The Mean

Let G be a connected linear algebraic group and X an algebraic variety, both defined over Q, the field of rational numbers. Suppose that G acts on X transitively and the action is defined over Q. Suppose that the set of rational points X(Q) is non-empty. Choosing x ∈ X(Q) allows us to identify G/Gx and X as varieties over Q, there Gx is the stabilizer of x.

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Dimension estimates for the set of points with non-dense orbit in homogeneous spaces

Mathematische Zeitschrift ◽

10.1007/s00209-019-02386-7 ◽

2019 ◽

Vol 295 (3-4) ◽

pp. 1355-1383

Author(s):

Dmitry Kleinbock ◽

Shahriar Mirzadeh

Keyword(s):

Homogeneous Spaces ◽

Dense Orbit ◽

Set Of Points

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Effectivity of uniqueness of the maximal entropy measure on -adic homogeneous spaces

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.148 ◽

2015 ◽

Vol 36 (6) ◽

pp. 1972-1988 ◽

Cited By ~ 2

Author(s):

RENE RÜHR

Keyword(s):

Dynamical System ◽

Probability Measure ◽

Algebraic Group ◽

Homogeneous Spaces ◽

Regular Representation ◽

Invariant Probability Measure ◽

Linear Algebraic Group ◽

Maximal Entropy ◽

Entropy Measure ◽

Effective Version

We consider the dynamical system given by an $\text{Ad}$-diagonalizable element $a$ of the $\mathbb{Q}_{p}$-points $G$ of a unimodular linear algebraic group acting by translation on a finite volume quotient $X$. Assuming that this action is exponentially mixing (e.g. if $G$ is simple) we give an effective version (in terms of $K$-finite vectors of the regular representation) of the following statement: If ${\it\mu}$ is an $a$-invariant probability measure with measure-theoretical entropy close to the topological entropy of $a$, then ${\it\mu}$ is close to the unique $G$-invariant probability measure of $X$.

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Translates of homogeneous measures associated with observable subgroups on some homogeneous spaces

Compositio Mathematica ◽

10.1112/s0010437x21007624 ◽

2021 ◽

Vol 157 (12) ◽

pp. 2657-2698

Author(s):

Runlin Zhang

Keyword(s):

Invariant Measure ◽

Algebraic Group ◽

Present Article ◽

Homogeneous Spaces ◽

Natural Extension ◽

Complete Classification ◽

Linear Algebraic Group ◽

Arbitrary Sequence ◽

Limiting Behavior ◽

Arithmetic Lattice

In the present article, we study the following problem. Let $\boldsymbol {G}$ be a linear algebraic group over $\mathbb {Q}$ , let $\Gamma$ be an arithmetic lattice, and let $\boldsymbol {H}$ be an observable $\mathbb {Q}$ -subgroup. There is a $H$ -invariant measure $\mu _H$ supported on the closed submanifold $H\Gamma /\Gamma$ . Given a sequence $(g_n)$ in $G$ , we study the limiting behavior of $(g_n)_*\mu _H$ under the weak- $*$ topology. In the non-divergent case, we give a rather complete classification. We further supplement this by giving a criterion of non-divergence and prove non-divergence for arbitrary sequence $(g_n)$ for certain large $\boldsymbol {H}$ . We also discuss some examples and applications of our result. This work can be viewed as a natural extension of the work of Eskin–Mozes–Shah and Shapira–Zheng.

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Divergent trajectories in arithmetic homogeneous spaces of rational rank two

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.96 ◽

2020 ◽

pp. 1-26

Author(s):

NATTALIE TAMAM

Keyword(s):

Algebraic Group ◽

Compact Subset ◽

Homogeneous Spaces ◽

Finite Collection ◽

Sufficient Condition ◽

Arithmetic Subgroup ◽

Rational Rank ◽

Closed Cone ◽

Full Classification ◽

Split Torus

Abstract Let G be a semisimple real algebraic group defined over ${\mathbb {Q}}$ , $\Gamma $ be an arithmetic subgroup of G, and T be a maximal ${\mathbb {R}}$ -split torus. A trajectory in $G/\Gamma $ is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which accounts for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in T, we study the existence of non-obvious divergent trajectories under its action in $G\kern-1pt{/}\kern-1pt\Gamma $ . We get a sufficient condition for the existence of a non-obvious divergence trajectory in the general case, and a full classification under the assumption that $\mathrm {rank}_{{\mathbb {Q}}}G=\mathrm {rank}_{{\mathbb {R}}}G=2$ .

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Application of Morse theory to some homogeneous spaces

Tohoku Mathematical Journal ◽

10.2748/tmj/1178242947 ◽

1969 ◽

Vol 21 (3) ◽

pp. 343-353 ◽

Cited By ~ 3

Author(s):

S. Ramanujan

Keyword(s):

Morse Theory ◽

Homogeneous Spaces

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Additive Groups in a Non-Symbolic ‘Artificial Algebra’

10.31234/osf.io/k935n ◽

2020 ◽

Author(s):

Randolph C Grace ◽

Nicola J. Morton ◽

Matt Grice ◽

Kate Stuart ◽

Simon Kemp

Keyword(s):

Algebraic Group ◽

Explicit Instruction ◽

Strong Support ◽

Line Length

Grace et al. (2018) developed an ‘artificial algebra’ task in which participants learn to make an analogue response based on a combination of non-symbolic magnitudes by feedback and without explicit instruction. Here we tested if participants could learn to add stimulus magnitudes in this task in accord with the properties of an algebraic group. Three pairs of experiments tested the group properties of commutativity (Experiments 1a-b), identity and inverse existence (Experiments 2a-b) and associativity (Experiments 3a-b), with both line length and brightness modalities. Transfer designs were used in which participants responded on trials with feedback based on sums of magnitudes and later were tested with novel stimulus configurations. In all experiments, correlations of average responses with magnitude sums were high on trials with feedback, r = .97 and .96 for Experiments 1a-b, r = .97 and .96 for Experiments 2a-b, and ranged between r = .97 and .99 for Experiment 3a and between r = .82 and .95 for Experiment 3b. Responding on transfer trials was accurate and provided strong support for commutativity, identity and inverse existence, and associativity with line length, and for commutativity and identity and inverse existence with brightness. Deviations from associativity in Experiment 3b suggested that participants were averaging rather than adding brightness magnitudes. Our results confirm that the artificial algebra task can be used to study implicit computation and suggest that representations of magnitudes may have a structure similar to an algebraic group.

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