Connected Nilpotent Lie Group Research Articles

Strong relative property (T) and spectral gap of random walks

We consider strong relative property (T) for pairs (Γ, G) where Γ acts on G. If N is a connected nilpotent Lie group and Γ is a group of automorphisms of N, we choose a finite index subgroup Γ 0 of Γ and obtain that (Γ , [Γ 0, N]) has strong relative property (T) provided Zariski-closure of Γ has no compact factor of positive dimension. We apply this to obtain the following: Let G be a connected Lie group with solvable radical R and a semisimple Levi subgroup S. If Snc denotes the product of noncompact simple factors of S and ST denotes the product of simple factors in S that have property (T), then we show that (Γ , R) or \({(\Gamma S_{T}, \overline{S_{T}R})}\) has strong relative property (T) for a ’Zariski-dense’ closed subgroup Γ of Snc if and only if R = [Snc, R]. We also provide some applications to the spectral gap of π (μ) = ∫ π (g) dμ (g) where π is a certain unitary representation and μ is a probability measure.

The non-existence of a non-abelian nilpotent Lie group with one discrete uniform subgroup up to automorphism

Let G be a connected, simply connected nilpotent Lie group and \({\fancyscript{S}(G)}\) the space of the discrete uniform subgroups of G. In this paper, we give some necessary and sufficient conditions for \({\fancyscript{S}(G)}\) under which the group G is abelian. In particular, we prove the non-existence of non-abelian nilpotent Lie groups with one discrete uniform subgroup up to automorphism.

Proper Actions of Certain Nilpotent Affine Groups with Codimension One Orbits

Criterion for proper actions has been established for a homogeneous space of reductive type by Kobayashi (Math. Ann. 1989, 1996). On the other hand, an analogous criterion to Kobayashi’s equivalent conditions was proposed by Lipsman (1995) for a nilpotent Lie group . Lipsman's Conjecture: Let be a simply connected nilpotent Lie group. Then the following two conditions on connected subgroups and are equivalent: (i) the action of on is proper; (ii) is compact for any The condition (i) is important in the study of discontinuous groups for the homogeneous space , while the second condition (ii) can easily be checked. The implication (i) (ii) is obvious, and the opposite implication (ii) (i) was known only in some lower dimensional cases. In this paper we prove the equivalence (i) ? (ii) for certain affine nilpotent Lie groups . Keywords: Affine nilpotent groups; Homogeneous manifolds; Proper actions; Properly discontinuous actions; Simply connected nilpotent Lie groups; Compact isotropy property (CI); Eigenvalues. © 2012 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v4i2.7889 J. Sci. Res. 4 (2), 315-326 (2012)

On the entropy of actions of nilpotent Lie groups and their lattice subgroups

AbstractWe consider a natural class $\mathcal {ULG}$ of connected, simply connected nilpotent Lie groups which contains ℝn, the group $\mathcal {UT}_n(\mathbb {R})$ of all triangular unipotent matrices over ℝ and many of its subgroups, and is closed under direct products. If $G \in \mathcal {ULG}$, then $\Gamma _1 = G\cap \mathcal {UT}_n(\mathbb {Z})$ is a lattice subgroup of G. We prove that if $G \in \mathcal {ULG}$ and Γ is a lattice subgroup of G, then a free ergodic measure-preserving action T of G on a probability space (X,ℬ,μ) has completely positive entropy (CPE) if and only if the restriction TΓ of T to Γ has CPE. We can deduce from this the following version of a well-known conjecture in this case: the action T has CPE if and only if T is uniformly mixing. Moreover, such T has a Lebesgue spectrum with infinite multiplicity. We further consider an ergodic free action T with positive entropy and suppose TΓ is ergodic for any lattice subgroup Γ of G. This holds, in particular, if the spectrum of T does not contain a discrete component. Then we show the Pinsker algebra Π(T) of T exists and coincides with the Pinsker algebras Π(TΓ) of TΓ for any lattice subgroup Γ of G. In this case, T always has Lebesgue spectrum with infinite multiplicity on the space ℒ20(X,μ)⊖ℒ20(Π(T)) , where ℒ20(Π(T)) contains all Π(T) -measurable functions from ℒ20(X,μ) . To prove these results, we use the following formula: h(T)=∣G(Γ)∣−1hK (TΓ) , where h(T) is the Ornstein–Weiss entropy of T, hK (TΓ) is a Kolmogorov–Sinai entropy of TΓ, and the number ∣G(TΓ)∣ is the Haar measure of the compact subset G(Γ) of G. In particular, h(T)=hK (TΓ1) , and hK (TΓ1)=∣G(Γ)∣−1hK (TΓ) . The last relation is an analogue of the Abramov formula for flows.

On the spectral theory of groups of affine transformations of compact nilmanifolds

Let N be a connected and simply connected nilpotent Lie group, � a lattice in N, and �\N the corresponding nilmanifold. Let Aff(�\N) be the group of affine transformations of �\N. We characterize the countable subgroups H of Aff(�\N) for which the action of H on �\N has a spectral gap, that is, such that the associated unitary representation U 0 of H on the space of functions from L 2 (�\N) with zero mean does not weakly contain the trivial representation. Denote by T the maximal torus factor associated to �\N. We show that the action of H on �\N has a spectral gap if and only if there exists no proper H-invariant subtorus S of T such that the projection of H on Aut(T/S) has an abelian subgroup of finite index. We first establish the result in the case where �\N is a torus. In the case of a general nilmanifold, we study the asymptotic behaviour of matrix coefficients ofU 0 using decay properties of metaplectic representations of symplectic groups. The result shows that the existence of a spectral gap for subgroups of Aff(�\N) is equivalent to strong ergodicity in the sense of K. Schmidt. Moreover, we show that the action of H on �\N is ergodic (or strongly mixing) if and only if the corresponding action of H on T is ergodic (or strongly mixing).

Mehler semigroups, Ornstein-Uhlenbeck processes and background driving Lévy processes on locally compact groups and on hypergroups

For finite dimensional vector spaces it is well-known that there exists a 1-1-correspondence between distributions of Ornstein-Uhlenbeck type processes (w.r.t. a fixed group of automorphisms) and (background driving) Lévy processes, hence between M- or skew convolution semigroups on the one hand and continuous convolution semigroups on the other. An analogous result could be proved for simply connected nilpotent Lie groups. Here we extend this correspondence to a class of commutative hypergroups.

Transitivity of Heisenberg group extensions of hyperbolic systems

AbstractWe show that amongCrextensions (r>0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group ℋnof dimension 2n+1, those that avoid an obvious obstruction to topological transitivity are generically topologically transitive. Moreover, if one considers extensions with fiber a connected nilpotent Lie group with a compact commutator subgroup (for example ℋn/ℤ), among those that avoid the obvious obstruction, topological transitivity is open and dense.

An Upper Bound for the Poisson Kernel on Higher Rank NA Groups

Let S be a semi direct product \(S=N\rtimes A\) where N is a connected and simply connected nilpotent Lie group and A is isomorphic with ℝk, k > 1. We obtain an upper bound for the Poisson kernel for the class of second order left-invariant differential operators on S.

On topological conjugacy of left invariant flows on semisimple and affine Lie groups

In this paper, we study the flows of nonzero left invariant vector fields on Lie groups with respect to topological conjugacy. Using the fundamental domain method, we are able to show that on a simply connected nilpotent Lie group any such flows are topologically conjugate. Combining this result with the Iwasawa decomposition, we find that on a noncompact semisimple Lie group the flows of two nilpotent or abelian fields are topologically conjugate. Finally, for affine groups G = HV, V ≅ n, we show that the conjugacy class of a left invariant vector field does not depend on its Euclidean component.

A smooth family of intertwining operators

Let $N$ be a connected, simply connected nilpotent Lie group with Lie algebra $\mathfrak{n}$ and let $\mathscr{W}$ be a submanifold of $\mathfrak{n}^*$ such that the dimension of all polarizations associated to elements of $\mathscr{W}$ is fixed. We choose $(\mathfrak{p}(w))_{w \in \mathscr{W}}$ and $(\mathfrak{p}'(w))_{w \in \mathscr{W}}$ two smooth families of polarizations in $\mathfrak{n}$. Let $\pi_w = \mathsf{ind}_{P(w)}^N \chi_w$ and $\pi'_w = \mathsf{ind}_{P'(w)}^N \chi_w$ be the corresponding induced representations, which are unitary and irreducible. It is well known that $\pi_w$ and $\pi'_w$ are unitary equivalent. In this paper, we prove the existence of a smooth family of intertwining operator $(T_w)_w$ for theses representations, where $w$ runs through an appropriate non-empty relatively open subset of $\mathscr{W}$. The intertwining operators are given by an explicit formula.

An extension of Minkowski’s theorem to simply connected 2-step nilpotent groups

This note extends the classical theorem of Minkowski on lattice points and convex bodies in ℝ^n to 2-step simply connected nilpotent Lie groups with a ℚ -structure. This includes all groups of Heisenberg type. More generally (and more naturally), it works for any simply connected nilpotent Lie group with a ℚ -structure whose Lie algebra admits a grading of length 2. Here a new invariant associated with the grading occurs which we call the degree . It explains why some directions are more equal than others.

Estimate of the Lp -Fourier transform norm for connected nilpotent Lie groups

Let G be a connected and simply connected nilpotent Lie group, and m the dimension of the generic coadjoint orbits of G . Then it is proved in [Baklouti, Smaoui, Ludwig, J. Funct. Anal. 199: 508–520, 2003] that the operator value Fourier transform norm satisfies , where and q being the conjugate of p . When the assumption on G to be simply connected is removed, we prove an analogue estimate of type , where H designates the maximal compact central subgroup of G and m the dimension of generic coadjoint orbits of the universal covering of G . The case of non-simply connected Heisenberg groups is treated as an example.

Remarks on a conjecture of Chabauty

The Chabauty conjecture for connected nilpotent Lie groups has been proved by S. P. Wang. We show that one reasoning flaw has infiltrated the proof. We therefore give a new proof of the validity of Chabauty's conjecture in this setup. More generally, we shall prove that the Chabauty conjecture is true for rigid lattices and Zariski dense lattices of connected solvable Lie groups. In particular, the Chabauty conjecture holds for solvable Lie groups of (R)-type.

Localized Hardy spaces 𝐻¹ related to admissible functions on RD-spaces and applications to Schrödinger operators

Let X {\mathcal X} be an RD-space, which means that X {\mathcal X} is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X {\mathcal X} . In this paper, the authors first introduce the notion of admissible functions ρ \rho and then develop a theory of localized Hardy spaces H ρ 1 ( X ) H^1_\rho ({\mathcal X}) associated with ρ \rho , which includes several maximal function characterizations of H ρ 1 ( X ) H^1_\rho ({\mathcal X}) , the relations between H ρ 1 ( X ) H^1_\rho ({\mathcal X}) and the classical Hardy space H 1 ( X ) H^1({\mathcal X}) via constructing a kernel function related to ρ \rho , the atomic decomposition characterization of H ρ 1 ( X ) H^1_\rho ({\mathcal X}) , and the boundedness of certain localized singular integrals on H ρ 1 ( X ) H^1_\rho ({\mathcal X}) via a finite atomic decomposition characterization of some dense subspace of H ρ 1 ( X ) H^1_\rho ({\mathcal X}) . This theory has a wide range of applications. Even when this theory is applied, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on R n \mathbb {R}^n , or to the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, some new results are also obtained. The Schrödinger operators considered here are associated with nonnegative potentials satisfying the reverse Hölder inequality.

On the multiplicity formula of compact nilmanifolds with flat orbits

Let G be a connected, simply connected nilpotent Lie group and Γ a lattice subgroup of G. The quasi regular representation decomposes as a direct sum of unitary irreducible representations of G. The nilmanifold G/Γ is called nilmanifold with flat orbits if the coadjoint orbit corresponding to any element of the spectrum of ℛΓ is flat. In this note, we give a new multiplicity formula related to the decomposition into irreducibles of ℛΓ in the case when G/Γ is a nilmanifold with flat orbits. As an application, we first prove that every nilmanifold with flat orbits satisfies the Moore formula. We also give partial answers to some related questions proposed by Brezin and by Corwin and Greenleaf.

Conformal actions of nilpotent groups on pseudo-Riemannian manifolds

We study conformal actions of connected nilpotent Lie groups on compact pseudo-Riemannian manifolds. We prove that if a type-(p,q) compact manifold M supports a conformal action of a connected nilpotent group H, then the degree of nilpotence of H is at most 2p+1, assuming p <= q; further, if this maximal degree is attained, then M is conformally equivalent to the universal type-(p,q), compact, conformally flat space, up to finite covers. The proofs make use of the canonical Cartan geometry associated to a pseudo-Riemannian conformal structure.

Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators

AbstractLet be a space of homogeneous type in the sense of Coifman and Weiss, and let be a collection of balls in . The authors introduce the localized atomic Hardy space the localized Morrey-Campanato space and the localized Morrey-Campanato-BLO (bounded lower oscillation) space with α ∊ ℝ and p ∊ (0, ∞) , and they establish their basic properties, including and several equivalent characterizations for In particular, the authors prove that when α &gt; 0 and p ∊ [1, ∞), then and when p ∈(0,1], then the dual space of is Let ρ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces and , respectively, by and when is determined by ρ. The authors then obtain the boundedness from of the radial and the Poisson semigroup maximal functions and the Littlewood-Paley g-function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝd, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators

AbstractLetbe a space of homogeneous type in the sense of Coifman and Weiss, and letbe a collection of balls in. The authors introduce the localized atomic Hardy spacethe localized Morrey-Campanato spaceand the localized Morrey-Campanato-BLO (bounded lower oscillation) spacewithα∊ ℝ andp∊ (0, ∞) , and they establish their basic properties, includingand several equivalent characterizations forIn particular, the authors prove that when α &amp;gt; 0 andp∊ [1, ∞), thenand whenp∈(0,1], then the dual space ofisLetρbe an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spacesand, respectively, byandwhenis determined byρ. The authors then obtain the boundedness fromof the radial and the Poisson semigroup maximal functions and the Littlewood-Paleyg-function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝd, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

Maximal function characterizations of Hardy spaces associated with Schrödinger operators on nilpotent Lie groups

Let G be a connected and simply connected nilpotent Lie group and L≡−Δ+W be the Schrodinger operator on L2(G), where \(0\le W\in L^{1}_{\mathrm{loc}}(G)\). In this paper, the authors establish some equivalent characterizations of the Hardy space \(H^{p}_{L}(G)\) for p∈(0,1] in terms of the radial maximal functions and non-tangential maximal functions associated with \(\{e^{-t^{2}L}\}_{t>0}\) and \(\{e^{-t\sqrt{L}}\}_{t>0}\), respectively. The boundedness of the Riesz transform \(\nabla L^{-\frac{1}{2}}\) from \(H^{p}_{L}(G)\) to Lp(G) with p∈(0,1] and from \(H^{p}_{L}(G)\) to Hp(G) with p∈(D/(D+1),1] are also obtained, where D is the dimension at infinity of G.

On closed manifolds which admit codimension one locally free actions of nilpotent Lie groups

We show that if a connected closed orientable manifold $M$ admits a codimension one locally free smooth action $\phi$ of a connected nilpotent Lie group such that any orbit of $\phi$ is non-compact, then $M$ is homeomorphic to a nilmanifold. And as an example of such an action, we study also a homogeneous action.