Nilpotent Lie Groups Research Articles

On the complete integrability of the geodesic flow of pseudo-H-type Lie groups

Pseudo-H-type groups $$G_{r,s}$$ form a class of step-two nilpotent Lie groups with a natural pseudo-Riemannian metric. In this paper the question of complete integrability in the sense of Liouville is studied for the corresponding (pseudo-)Riemannian geodesic flow. Via the isometry group of $$G_{r,s}$$ families of first integrals are constructed. A modification of these functions gives a set of $$\dim G_{r,s}$$ functionally independent smooth first integrals in involution. The existence of a lattice L in $$G_{r,s}$$ is guaranteed by recent work of K. Furutani and I. Markina. The complete integrability of the pseudo-Riemannian geodesic flow of the compact nilmanifold $$L \backslash G_{r,s}$$ is proved under additional assumptions on the group $$G_{r,s}$$ .

Quantizations on Nilpotent Lie Groups and Algebras Having Flat Coadjoint Orbits

For a connected simply connected nilpotent Lie group $$\textsf {G}$$ with Lie algebra $${{\mathfrak {g}}}$$ and unitary dual $$\widehat{{\textsf {G}}}$$ one has (a) a global quantization of operator-valued symbols defined on $$\textsf {G}\times \widehat{{\textsf {G}}}$$ , involving the representation theory of the group, (b) a quantization of scalar-valued symbols defined on $$\textsf {G}\times {{\mathfrak {g}}}^*$$ , taking the group structure into account and (c) Weyl-type quantizations of all the coadjoint orbits $$\big \{\Omega _\xi \mid \xi \in \widehat{{\textsf {G}}}\big \}$$ . We show how these quantizations are connected, in the case when flat coadjoint orbits exist. This is done by a careful new analysis of the composition of two different types of Fourier transformations, interesting in itself. We also describe the concrete form of the operator-valued symbol quantization, by using Kirillov theory and the Euclidean version of the unitary dual and Plancherel measure. In the case of the Heisenberg group, this corresponds to the known picture, presenting the representation theoretical pseudo-differential operators in terms of families of Weyl operators depending on a parameter. For illustration, we work out a couple of examples and put into evidence some specific features of the case of Lie algebras with one-dimensional center. When $$\textsf {G}$$ is also graded, we make a short presentation of the symbol classes $$S^m_{\rho ,\delta }$$ , transferred from $$\textsf {G}\times \widehat{{\textsf {G}}}$$ to $$\textsf {G}\times {{\mathfrak {g}}}^*$$ by means of the connection mentioned above.

Transference for Banach space representations of nilpotent Lie groups. Part 1. Irreducible representations

We establish a general CCR (liminarity) property for uniformly bounded irreducible representations of nilpotent Lie groups on reflexive Banach spaces, extending the well-known property of unitary irreducible representations of these groups on Hilbert spaces. We also prove that this conclusion fails for many representations on non-reflexive Banach spaces. Our approach to these results blends the method of transference from abstract harmonic analysis and a systematic use of spaces of smooth vectors with respect to Lie group representations.

HARDY’S THEOREM FOR GABOR TRANSFORM

Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form$\mathbb{R}^{n}\times K$, where$K$is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group$G$which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form$G\times D$, where$D$is a discrete group.

English

We study a problem of isometric compact 2-step nilmanifolds ${M}/\Gamma $ using some information on their geodesic flows, where $M$ is a simply connected 2-step nilpotent Lie group with a left invariant metric and $\Gamma $ is a cocompact discrete subgroup of isometries of $M$. Among various works concerning this problem, we consider the algebraic aspect of it. In fact, isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, namely by normalizers. So, suitable factorization of normalizers and expression of a vector bundle as an associated fiber bundle to a principal bundle, lead us to a general framework, namely groupoids. In this way, drawing upon advanced ingredients of Lie groupoids, normal subgroupoid systems and other notions, not only an answer in some sense to our rigidity problem has been given, but also the dependence between normalizers, automorphisms and especially almost inner automorphisms, has been clarified.

The local limit theorem on nilpotent Lie groups

A local limit theorem is proven on connected, simply connected nilpotent Lie groups, for a class of generating measures satisfying a moment condition and a condition on the characteristic function of the abelianization. The result extends an earlier local limit theorems of Alexopoulos which treated absolutely continuous measures with a continuous density of compact support, and also extends local limit theorems of Breuillard and Diaconis-Hough which treated general measures on the Heisenberg group.

A Note on Hardy Inequalities on Homogeneous Groups

We provide the necessary and sufficient characterizations on a pair of positive radial functions so that the two-weight Hardy inequalities hold true on homogeneous groups, one of most general subclasses of nilpotent Lie groups. We also present a simple condition on a couple of positive radial weight functions for the Lp −Hardy type inequalities to be valid.

Isometry classes of simply connected nilmanifolds

We classify the isometry equivalence classes and determine the isometry groups of connected and simply connected Riemannian nilmanifolds on filiform Lie groups of arbitrary dimension and on five dimensional nilpotent Lie groups of nilpotency class > 2 . To achieve this classification we prove that up to one exceptional class the five dimensional non two-step nilmanifolds and the filiform nilmanifolds have isometry groups of the same (minimal) dimension as the nilmanifold. We give a detailed description of the exceptional case.

Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type

Let X X be a space of homogeneous type and let L \mathfrak {L} be a nonnegative self-adjoint operator on L 2 ( X ) L^2(X) enjoying Gaussian estimates. The main aim of this paper is twofold. Firstly, we prove (local) nontangential and radial maximal function characterizations for the local Hardy spaces associated to L \mathfrak {L} . This gives the maximal function characterization for local Hardy spaces in the sense of Coifman and Weiss provided that L \mathfrak {L} satisfies certain extra conditions. Secondly we introduce local Hardy spaces associated with a critical function ρ \rho which are motivated by the theory of Hardy spaces related to Schrödinger operators and of which include the local Hardy spaces of Coifman and Weiss as a special case. We then prove that these local Hardy spaces can be characterized by (local) nontangential and radial maximal functions related to L \mathfrak {L} and ρ \rho , and by global maximal functions associated to ‘perturbations’ of L \mathfrak {L} . We apply our theory to obtain a number of new results on maximal characterizations for the local Hardy type spaces in various settings ranging from Schrödinger operators on manifolds to Schrödinger operators on connected and simply connected nilpotent Lie groups.

Qualitative uncertainty principle for the Gabor transform on certain locally compact groups

Abstract Several classes of locally compact groups have been shown to possess a qualitative uncertainty principle for the Gabor transform. These include Moore groups, the Heisenberg group {\mathbb{H}_{n}} , the group {\mathbb{H}_{n}\times D} (where D is a discrete group) and other low-dimensional nilpotent Lie groups.

Orthonormal bases in the orbit of square-integrable representations of nilpotent Lie groups

Let G be a connected, simply connected nilpotent group and π be an irreducible unitary representation of G that is square-integrable modulo its center Z(G) on L2(Rd). We prove that under reasonably weak conditions on G and π there exist a discrete subset Γ of G/Z(G) and some (relatively) compact set F⊆Rd such that{|F|−1/2π(γ)1F|γ∈Γ} forms an orthonormal basis of L2(Rd). This construction generalizes the well-known example of Gabor orthonormal bases in time-frequency analysis.The main theorem covers graded Lie groups with one-dimensional center. In the presence of a rational structure, the set Γ can be chosen to be a uniform subgroup of G/Z.

Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

Consider the restriction of an irreducible unitary representation $\pi$ of a Lie group $G$ to its subgroup $H$. Kirillov's revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $\nu$ occurring in the restriction $\pi|_H$ could be read from the coadjoint action of $H$ on $O^G \cap pr^{-1}(O^H)$ provided $\pi$ and $\nu$ are "geometric quantizations" of a $G$-coadjoint orbit $O^G$ and an $H$-coadjoint orbit $O^H$,respectively, where $pr: \sqrt{-1} g^{\ast} \to \sqrt{-1} h^{\ast}$ is the projection dual to the inclusion $h \subset g$ of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits $O^G$ of a semisimple Lie group $G$ corresponding to highest weight modules of scalar type. We prove that the Corwin--Greenleaf number $\sharp(O^G \cap pr^{-1}(O^H))/H$ is either zero or one for any $H$-coadjoint orbit $O^H$, whenever $(G,H)$ is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits $O^H$ with nonzero Corwin-Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as "classical limits" of the multiplicity-free branching laws of holomorphic discrete series representations (T.Kobayashi [Progr.Math.2007]).

Lower bound of Riesz transform kernels and commutator theorems on stratified nilpotent Lie groups

We provide a study of the Riesz transforms on stratified nilpotent Lie groups, and obtain a certain version of the pointwise lower bound of the Riesz transform kernel. Then we establish the characterisation of the BMO space on stratified nilpotent Lie groups via the boundedness of the commutator of the Riesz transforms and the BMO function. This extends the well-known Coifman, Rochberg, Weiss theorem from Euclidean space to the setting of stratified nilpotent Lie groups. In particular, these results apply to the well-known example of the Heisenberg group. As an application, we also study the curl operator on the Heisenberg group and stratified nilpotent Lie groups, and establish the div–curl lemma with respect to the Hardy space on stratified nilpotent Lie groups.

Multilinear and multiparameter spectral multipliers on stratified groups

As far as we know, the study of multilinear spectral multipliers on nilpotent Lie groups is a very new research work. There is even no study of Hörmander‐type multiplier theorem for multilinear and multiparameter spectral multipliers on nilpotent Lie groups. In this paper, on product spaces of stratified groups G = G1 × ⋯ × GM, we prove Hörmander‐type multiplier theorems for multilinear and multiparameter spectral multipliers from to Lr(G) with , from to with , and from to Lr(·)(G) with or for all ℓ = 1,…,N.

The distribution of symmetry of a naturally reductive nilpotent Lie group

We show that the distribution of symmetry of a naturally reductive nilpotent Lie group coincides with the invariant distribution induced by the set of fixed vectors of the isotropy. This extends a known result on compact naturally reductive spaces. We also address the study of the quotient by the foliation of symmetry.

Einstein nilpotent Lie groups

We study the Ricci tensor of left-invariant pseudoriemannian metrics on Lie groups. For an appropriate class of Lie groups that contains nilpotent Lie groups, we introduce a variety with a natural GL(n,R) action, whose orbits parametrize Lie groups with a left-invariant metric; we show that the Ricci operator can be identified with the moment map relative to a natural symplectic structure. From this description we deduce that the Ricci operator is the derivative of the scalar curvature s under gauge transformations of the metric, and show that Lie algebra derivations with nonzero trace obstruct the existence of Einstein metrics with s≠0.Using the notion of nice Lie algebra, we give the first example of a left-invariant Einstein metric with s≠0 on a nilpotent Lie group. We show that nilpotent Lie groups of dimension ≤6 do not admit such a metric, and a similar result holds in dimension 7 with the extra assumption that the Lie algebra is nice.

Rigidity of equality of Lyapunov exponents for geodesic flows

We study the relationship between the Lyapunov exponents of the geodesic flow of a closed negatively curved manifold and the geometry of the manifold. We show that if each periodic orbit of the geodesic flow has exactly one Lyapunov exponent on the unstable bundle then the manifold has constant negative curvature. We also show under a curvature pinching condition that equality of all Lyapunov exponents with respect to volume on the unstable bundle also implies that the manifold has constant negative curvature. We then study the degree to which one can emulate these rigidity theorems for the hyperbolic spaces of nonconstant negative curvature when the Lyapunov exponents with respect to volume match those of the appropriate symmetric space and obtain rigidity results under additional technical assumptions. The proofs use new results from hyperbolic dynamics including the nonlinear invariance principle of Avila and Viana and the approximation of Lyapunov exponents of invariant measures by Lyapunov exponents associated to periodic orbits which was developed by Kalinin in his proof of the Livsic theorem for matrix cocycles. We also employ rigidity results of Capogna and Pansu on quasiconformal mappings of certain nilpotent Lie groups.

Laplacian solitons on nilpotent Lie groups

We investigate the existence of closed $G_2$-structures which are solitons for the Laplacian flow on nilpotent Lie groups. We obtain that seven of the twelve Lie algebras admitting a closed $G_2$-structure do admit a Laplacian soliton. Moreover, one of them admits a continuous family of Laplacian solitons which are pairwise non-homothetic and the Laplacian flow evolution on four of the Lie groups is not diagonal.

Graphs and metric 2-step nilpotent Lie algebras

Abstract Dani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra 𝔫 G from a simple directed graph G in 2005. There is a natural inner product on 𝔫 G arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group N with Lie algebra 𝔫 g . We classify singularity properties of the Lie algebra 𝔫 g in terms of the graph G. A comprehensive description is given of graphs G which give rise to Heisenberg-like Lie algebras. Conditions are given on the graph G and on a lattice Γ ⊆ N for which the quotient Γ \ N, a compact nilmanifold, has a dense set of smoothly closed geodesics. This paper provides the first investigation connecting graph theory, 2-step nilpotent Lie algebras, and the density of closed geodesics property.

Diophantine approximation on matrices and Lie groups

We study the general problem of extremality for metric diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over {mathbb{Q}}, we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest. In particular we prove that the diophantine exponent of rational nilpotent Lie groups exists and is a rational number, which we determine explicitly in terms of representation theoretic data.