Cross sections and an explicit formula for the Plancherel measure on a nilpotent Lie group

1988 ◽  
Vol 281 (2) ◽  
pp. 193-204
Author(s):  
Alan L. T. Paterson
Keyword(s):  
Explicit Formula ◽  
Cross Sections ◽  
Lie Group ◽  
Plancherel Measure ◽  
Nilpotent Lie Group

On Flag Curvature of Homogeneous Randers Spaces

2013 ◽  
Vol 65 (1) ◽  
pp. 66-81 ◽  
Author(s):  
Shaoqiang Deng ◽  
Zhiguang Hu
Keyword(s):  
Explicit Formula ◽  
Lie Group ◽  
Flag Curvature ◽  
Nilpotent Lie Group ◽  
Finsler Spaces ◽  
Rigidity Result ◽  
Randers Space ◽  
Negatively Curved

AbstractIn this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randersmetric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian.

PROPER ACTIONS ON SOME EXPONENTIAL SOLVABLE HOMOGENEOUS SPACES

2005 ◽  
Vol 16 (09) ◽  
pp. 941-955 ◽  
Author(s):  
ALI BAKLOUTI ◽  
FATMA KHLIF
Keyword(s):  
Maximal Subgroup ◽  
Homogeneous Space ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
Intersection Property ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Simply Connected

Let G be a connected, simply connected nilpotent Lie group, H and K be connected subgroups of G. We show in this paper that the action of K on X = G/H is proper if and only if the triple (G,H,K) has the compact intersection property in both cases where G is at most three-step and where G is special, extending then earlier cases. The result is also proved for exponential homogeneous space on which acts a maximal subgroup.

On Kähler nilmanifolds with top homology in codimension two

2006 ◽  
Vol 74 (1) ◽  
pp. 85-90
Author(s):  
Bruce Gilligan
Keyword(s):  
Lie Group ◽  
Homology Group ◽  
Discrete Subgroup ◽  
Nilpotent Lie Group ◽  
Codimension Two

SupposeGis a connected, complex, nilpotent Lie group and Γ is a discrete subgroup ofGsuch thatG/Γ is Kähler and the top nonvanishing homology group ofG/Γ (with coefficients in ℤ2) is in codimension two or less. We show thatGis then Abelian. We also note that an example from [12] shows that this fails if the top nonvanishing homology is in codimension three.

scholarly journals The spherical maximal function on the free two-step nilpotent Lie group

2006 ◽  
Vol 99 (1) ◽  
pp. 99 ◽  
Author(s):  
Véronique Fischer
Keyword(s):  
Lie Group ◽  
Maximal Function ◽  
Unit Sphere ◽  
Nilpotent Lie Group ◽  
Homogeneous Unit

We consider here the free two step nilpotent Lie group, provided with the homogeneous Korányi norm; we prove the $L^p$-boundedness of the maximal function corresponding to the homogeneous unit sphere, for some $p$.

CRITERION OF PROPER ACTIONS FOR 3-STEP NILPOTENT LIE GROUPS

2007 ◽  
Vol 18 (07) ◽  
pp. 783-795 ◽  
Author(s):  
TARO YOSHINO
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Lie Group ◽  
Closed Subgroup ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Affirmative Solution

For a nilpotent Lie group G and its closed subgroup L, Lipsman [13] conjectured that the L-action on some homogeneous space of G is proper in the sense of Palais if and only if the action is free. Nasrin [14] proved this conjecture assuming that G is a 2-step nilpotent Lie group. However, Lipsman's conjecture fails for the 4-step nilpotent case. This paper gives an affirmative solution to Lipsman's conjecture for the 3-step nilpotent case.

On Limit Multiplicities for Spaces of Automorphic Forms

1999 ◽  
Vol 51 (5) ◽  
pp. 952-976 ◽  
Author(s):  
Anton Deitmar ◽  
Werner Hoffmann
Keyword(s):  
Lie Group ◽  
Automorphic Forms ◽  
Plancherel Measure ◽  
Semisimple Lie Group ◽  
Selberg Trace Formula ◽  
Arithmetic Subgroup ◽  
Trivial Group ◽  
Rank One ◽  
Discrete Part ◽  
Cocompact Lattices

AbstractLet Γ be a rank-one arithmetic subgroup of a semisimple Lie group G. For fixed K-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of G, whose discrete part encodes the dimensions of the spaces of square-integrable Γ-automorphic forms. It is shown that this distribution converges to the Plancherel measure of G when Γ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices Γ follows from results of DeGeorge-Wallach and Delorme.

scholarly journals Harmonic analysis on the quotient spaces of Heisenberg groups

1991 ◽  
Vol 123 ◽  
pp. 103-117 ◽  
Author(s):  
Jae-Hyun Yang
Keyword(s):  
Quantum Mechanics ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Unitary Representation ◽  
Lie Group ◽  
Theta Series ◽  
Foundations Of Quantum Mechanics ◽  
Nilpotent Lie Group ◽  
Heisenberg Groups ◽  
Quotient Spaces

A certain nilpotent Lie group plays an important role in the study of the foundations of quantum mechanics ([Wey]) and of the theory of theta series (see [C], [I] and [Wei]). This work shows how theta series are applied to decompose the natural unitary representation of a Heisenberg group.

scholarly journals Ricci-flat and Einstein pseudoriemannian nilmanifolds

2019 ◽  
Vol 6 (1) ◽  
pp. 170-193 ◽  
Author(s):  
Diego Conti ◽  
Federico A. Rossi
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Ricci Flat ◽  
Open Questions ◽  
Expository Paper ◽  
New Criterion

AbstractThis is partly an expository paper, where the authors’ work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-˛at metrics on nilpotent Lie groups of dimension [eight.tf] are obtained. Some related open questions are presented.

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