Subgroups of a spinor group containing a maximal split torus. I

Abstract Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results for translations of $U$-slices by elements of $A_U^+$ on a finite volume homogeneous space $G/\Gamma $ where $G$ is a Lie group containing $H$. More precisely, we prove quantitative nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/\Gamma $ has a spectral gap, we prove effective multiple equidistribution and pointwise equidistribution with an error rate. In this paper, we formulate the notion of the expanding cone and prove the dynamical results above in the more general setting where $H$ is a semisimple Lie group without compact factors. In the appendix, joint with Rene Rühr, we prove a multiple ergodic theorem with an error rate.

On a Certain Residual Spectrum of Sp8

Canadian Journal of Mathematics ◽

10.4153/cjm-2004-008-0 ◽

2004 ◽

Vol 56 (1) ◽

pp. 168-193

Author(s):

James Todd Pogge

Keyword(s):

Eisenstein Series ◽

Fundamental Problem ◽

Regular Representation ◽

Automorphic Representation ◽

Levi Subgroup ◽

Parabolic Subgroups ◽

Residual Spectrum ◽

The Right ◽

Maximal Split ◽

Split Torus

AbstractLet G = Sp2n be the symplectic group defined over a number field F. Let 𝔸 be the ring of adeles. A fundamental problem in the theory of automorphic forms is to decompose the right regular representation of G(𝔸) acting on the Hilbert space L2 (G(F) \ G(𝔸)). Main contributions have been made by Langlands. He described, using his theory of Eisenstein series, an orthogonal decomposition of this space of the form: , where (M, π) is a Levi subgroup with a cuspidal automorphic representation π taken modulo conjugacy. (Here we normalize π so that the action of the maximal split torus in the center of G at the archimedean places is trivial.) and is a space of residues of Eisenstein series associated to (M, π). In this paper, we will completely determine the space , when M ≃ GL2 × GL2. This is the first result on the residual spectrum for non-maximal, non-Borel parabolic subgroups, other than GLn.

Local Bounds for Torsion Points on Abelian Varieties

Canadian Journal of Mathematics ◽

10.4153/cjm-2008-026-x ◽

2008 ◽

Vol 60 (3) ◽

pp. 532-555 ◽

Cited By ~ 6

Author(s):

Pete L. Clark ◽

Xavier Xarles

Keyword(s):

Abelian Varieties ◽

Residue Field ◽

Rational Numbers ◽

Abelian Surface ◽

Torsion Subgroup ◽

Good Reduction ◽

Torsion Points ◽

Ramification Index ◽

Numerical Invariants ◽

Split Torus

AbstractWe say that an abelian variety over a p-adic field K has anisotropic reduction (AR) if the special fiber of its Néronminimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the K-rational torsion subgroup of a g-dimensional AR variety depending only on g and the numerical invariants of K (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of g, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an AR abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

Assessment of the Maximal Split-Half Coefficient to Estimate Reliability

Educational and Psychological Measurement ◽

10.1177/0013164409355688 ◽

2010 ◽

Vol 70 (2) ◽

pp. 232-251 ◽

Cited By ~ 8

Author(s):

Barry L. Thompson ◽

Samuel B. Green ◽

Yanyun Yang

Keyword(s):

Estimate Reliability ◽

Maximal Split

Subgroups of the Spinor Group that Contain a Split Maximal Torus. II

Journal of Mathematical Sciences ◽

10.1023/b:joth.0000042305.51231.e8 ◽

2004 ◽

Vol 124 (1) ◽

pp. 4698-4707

Author(s):

N. A. Vavilov

Keyword(s):

Maximal Torus ◽

Spinor Group ◽

Split Maximal Torus

Minimal parabolic k-subgroups acting on symmetric k-varieties corresponding to k-split groups

Journal of Algebra and Its Applications ◽

10.1142/s0219498821501991 ◽

2020 ◽

pp. 2150199

Author(s):

Mark Hunnell

Keyword(s):

Fixed Point ◽

Symmetric Spaces ◽

Point Group ◽

Natural Generalization ◽

Closed Fields ◽

Algebraically Closed Fields ◽

Split Torus

Symmetric [Formula: see text]-varieties are a natural generalization of symmetric spaces to general fields [Formula: see text]. We study the action of minimal parabolic [Formula: see text]-subgroups on symmetric [Formula: see text]-varieties and define a map that embeds these orbits within the orbits corresponding to algebraically closed fields. We develop a condition for the surjectivity of this map in the case of [Formula: see text]-split groups that depends only on the dimension of a maximal [Formula: see text]-split torus contained within the fixed point group of the involution defining the symmetric [Formula: see text]-variety.

Lifting of Outer Actions of Groups II

Algebra Colloquium ◽

10.1142/s1005386713000084 ◽

2013 ◽

Vol 20 (01) ◽

pp. 89-94

Author(s):

Fang Zhou ◽

Heguo Liu

Keyword(s):

Split Extensions ◽

Maximal Split

In this paper we continue to study the lifting of outer actions of groups and introduce the concept of equivalence of two lifting homomorphisms, maximal lifting homomorphisms and maximal split extensions. Some criteria are obtained.

On the action of the unitary group on the projective plane over a local field

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700001075 ◽

1997 ◽

Vol 62 (3) ◽

pp. 371-397 ◽

Cited By ~ 1

Author(s):

Harm Voskuil

Keyword(s):

Projective Plane ◽

Local Field ◽

Unitary Group ◽

Residue Field ◽

Equivariant Map ◽

Rank One ◽

Characteristic 2 ◽

Set Of Points ◽

Stable Points ◽

Split Torus

AbstractLet G be a unitary group of rank one over a non-archimedean local field K (whose residue field has a characteristic ≠ 2). We consider the action of G on the projective plane. A G(K) equivariant map from the set of points in the projective plane that are semistable for every maximal K split torus in G to the set of convex subsets of the building of G(K) is constructed. This map gives rise to an equivariant map from the set of points that are stable for every maximal K split torus to the building. Using these maps one describes a G(K) invariant pure affinoid covering of the set of stable points. The reduction of the affinoid covering is given.