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

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.

scholarly journals A note on the mean value theorem for special homogeneous spaces

1996 ◽  
Vol 143 ◽  
pp. 111-117 ◽  
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.

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

2015 ◽  
Vol 36 (6) ◽  
pp. 1972-1988 ◽  
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 $X$.

scholarly journals MULTIPLE FLAG IND-VARIETIES WITH FINITELY MANY ORBITS

2021 ◽  
Author(s):  
LUCAS FRESSE ◽  
IVAN PENKOV
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Interesting Case ◽  
Analogous Problem ◽  
Restrictive Condition ◽  
Dimensional Case ◽  
Essential Difference ◽  
Parabolic Subgroups ◽  
Finite Dimensional ◽  
Finite Dimensional Case

AbstractLet G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ.

Tangent bundle of hypersurfaces in G/P

2008 ◽  
Vol 4 (1) ◽  
pp. 91-100
Author(s):  
Indranil Biswas ◽  
Georg Schumacher
Keyword(s):  
Line Bundle ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Tangent Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Smooth Hypersurface ◽  
Canonical Line Bundle

AbstractLet G be a simple linear algebraic group defined over an algebraically closed field k of characteristic p ≥ 0, and let P be a maximal proper parabolic subgroup of G. If p > 0, then we will assume that dimG/P ≤ p. Let ι : H ↪ G/P be a reduced smooth hypersurface in G/P of degree d. We will assume that the pullback homomorphism is an isomorphism (this assumption is automatically satisfied when dimH ≥ 3). We prove that the tangent bundle of H is stable if the two conditions τ(G/P) ≠ d and hold; here n = dimH, and τ(G/P) ∈ is the index of G/P which is defined by the identity = where L is the ample generator of Pic(G/P) and is the anti–canonical line bundle of G/P. If d = τ(G/P), then the tangent bundle TH is proved to be semistable. If p > 0, and then TH is strongly stable. If p > 0, and d = τ(G/P), then TH is strongly semistable.

Restriction of the Tangent Bundle of G/P to a Hypersurface

2010 ◽  
Vol 53 (2) ◽  
pp. 218-222
Author(s):  
Indranil Biswas
Keyword(s):  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Tangent Bundle ◽  
Linear Algebraic Group ◽  
Smooth Hypersurface

AbstractLet P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over ℂ, such that n := dimℂG/P ≥ 4. Let ι : Z ↪ G/P be a reduced smooth hypersurface of degree at least (n – 1) · degree(T(G/P))/n. We prove that the restriction of the tangent bundle ι*TG/P is semistable.

scholarly journals Integral geometry under cut loci in compact symmetric spaces

1995 ◽  
Vol 137 ◽  
pp. 33-53 ◽  
Author(s):  
Hiroyuki Tasaki
Keyword(s):  
Invariant Measure ◽  
Lie Group ◽  
Integral Geometry ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Integral Invariant ◽  
Identity Component ◽  
Integral Invariants ◽  
Compact Symmetric Spaces ◽  
Group Of Isometries

The theory of integral geometry has mainly treated identities between integral invariants of submanifolds in Riemannian homogeneous spaces like as dμg(g) where M and N are submanifolds in a Riemannian homogeneous spaces of a Lie group G and I(M ∩ gN) is an integral invariant of M ∩ gN. For example Poincaré’s formula is one of typical identities in integral geometry, which is as follows. We denote by M(R2) the identity component of the group of isometries of the plane R2 with a suitable invariant measure μM(R2).

scholarly journals On the Steinberg Character of a Finite Simple Group of Lie Type

1971 ◽  
Vol 12 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Bhama Srinivasan
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Reductive Algebraic Group ◽  
Simple Algebraic Group ◽  
Simply Connected ◽  
Groups Of Lie Type ◽  
Steinberg Character

Let K be an algebraically closed field of characteristic ρ >0. If G is a connected, simple connected, semisimple linear algebraic group defined over K and σ an endomorphism of G onto G such that the subgroup Gσ of fixed points of σ is finite, Steinberg ([6] [7]) has shown that there is a complex irreducible character χ of Gσ with the following properties. χ vanishes at all elements of Gσ which are not semi- simple, and, if x ∈ G is semisimple, χ(x) = ±n(x) where n(x)is the order of a Sylow p-subgroup of (ZG(x))σ (ZG(x) is the centraliser of x in G). If G is simple he has, in [6], identified the possible groups Gσ they are the Chevalley groups and their twisted analogues over finite fields, that is, the ‘simply connected’ versions of finite simple groups of Lie type. In this paper we show, under certain restrictions on the type of the simple algebraic group G an on the characteristic of K, that χ can be expressed as a linear combination with integral coefficients of characters induced from linear characters of certain naturally defined subgroups of Gσ. This expression for χ gives an explanation for the occurence of n(x) in the formula for χ (x), and also gives an interpretation for the ± 1 occuring in the formula in terms of invariants of the reductive algebraic group ZG(x).

scholarly journals On flag varieties, hyperplane complements and Springer representations of Weyl groups

1990 ◽  
Vol 49 (3) ◽  
pp. 449-485 ◽  
Author(s):  
G. I. Lehrer ◽  
T. Shoji
Keyword(s):  
Algebraic Group ◽  
Weyl Group ◽  
Nilpotent Element ◽  
Parabolic Type ◽  
Linear Algebraic Group ◽  
Weyl Groups ◽  
Complex Numbers ◽  
Flag Varieties ◽  
Group Representations ◽  
Definition Of

AbstractLet G be a connected reductive linear algebraic group over the complex numbers. For any element A of the Lie algebra of G, there is an action of the Weyl group W on the cohomology Hi(BA) of the subvariety BA (see below for the definition) of the flag variety of G. We study this action and prove an inequality for the multiplicity of the Weyl group representations which occur ((4.8) below). This involves geometric data. This inequality is applied to determine the multiplicity of the reflection representation of W when A is a nilpotent element of “parabolic type”. In particular this multiplicity is related to the geometry of the corresponding hyperplane complement.

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