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For certain nilpotent real Lie groups constructed as semidirect products, algebras of invariant differential operators on some coadjoint orbits are used in the study of boundedness properties of the Weyl–Pedersen calculus of their corresponding unitary irreducible representations. Our main result is applicable to all unitary irreducible representations of arbitrary 3-step nilpotent Lie groups. [ABSTRACT FROM AUTHOR]

We examine a class of Grushin type operators P k where k ∈ N 0 defined in (1.1). The operators P k are non-elliptic and degenerate on a sub-manifold of R N + ℓ . Geometrically they arise via a submersion from a sub-Laplace operator on a nilpotent Lie group of step k + 1 . We explain the geometric framework and prove some analytic properties such as essential self-adjointness. The main purpose of the paper is to give an explicit expression of the fundamental solution of P k . Our methods rely on an appropriate change of coordinates and involve the theory of Bessel and modified Bessel functions together with Weber's second exponential integral. [ABSTRACT FROM AUTHOR]

Let G be a connected, simply connected nilpotent Lie group of step 1, 2 or 3, which contains a discrete cocompact subgroup Γ . Let τ = ind Γ G 1 be the quasi-regular representa-tion of G . Our main goals in this paper are twofold. The first is to give an orbital description of the decomposition of τ into irreducibles. The second main goal is to give an explicit intertwining operator between τ and its disintegration. As a straight application, we give a new multiplicity formula which is a partial answer to a question proposed by J. Brezin. [ABSTRACT FROM AUTHOR]

A locally compact group G is said to be approximated by discrete subgroups (in the sense of Tôyama) if there is a sequence of discrete subgroups of G that converges to G in the Chabauty topology (or equivalently, in the Vietoris topology). The notion of approximation of Lie groups by discrete subgroups was introduced by Tôyama in Kodai Math. Sem. Rep. 1 () 36-37 and investigated in detail by Kuranishi in Nagoya Math. J. 2 () 63-71. It is known as a theorem of Tôyama that any connected Lie group approximated by discrete subgroups is nilpotent. The converse, in general, does not hold. For example, a connected simply connected nilpotent Lie group is approximated by discrete subgroups if and only if G has a rational structure. On the other hand, if Γ is a discrete uniform subgroup of a connected, simply connected nilpotent Lie group G then G is approximated by discrete subgroups Γ containing Γ. The proof of the above result is by induction on the dimension of G, and gives an algorithm for inductively determining Γ. The purpose of this paper is to give another proof in which we present an explicit formula for the sequence (Γ) in terms of Γ. Several applications are given. [ABSTRACT FROM AUTHOR]

Let G = exp(g) be a connected, simply connected, nilpotent Lie group and let ω be a continuous symmetric weight on G with polynomial growth. We determine the structure of all the two-sided closed ideals of the weighted group algebra Lω1(G) which are attached to a flat co-adjoint orbit. [ABSTRACT FROM AUTHOR]

The primary purpose of this paper is to obtain explicit, coordinate-based descriptions of Ricci flow solutions-especially those corresponding to Ricci solitons-on two classes of nilpotent Lie groups. On the odd-dimensional classical Heisenberg groups, we determine the asymptotics of Ricci flow starting at any metric, and use Lott's blowdown method to demonstrate convergence to soliton metrics. On the groups of real unitriangular matrices, which are more complicated, we describe the solitons and corresponding solutions using a suitable ansatz. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

nilpotent Lie algebra is called an Einstein nilradical if the corresponding Lie group admits a left-invariant Ricci soliton metric. While these metrics are of independent interest, their existence is intimately related to the existence of Einstein metrics on solvable Lie groups. In this note we are concerned with the following question: How are the Einstein and non-Einstein nilradicals distributed among nilpotent Lie algebras? A full answer to this question is not known and we restrict to the class of 2-step nilpotent Lie groups. Within this class, it is known that a generic group admits a Ricci soliton metric. Using techniques from Geometric Invariant Theory, we study the set of non-generic algebras to learn more about the distribution of non-Einstein nilradicals. Many new (continuous) families of non-isomorphic, non-Einstein nilradicals are constructed. Moreover, the dimension of these families can be arbitrarily large (depending on the dimension of the underlying Lie group). To show such large classes of Lie groups are pairwise non-isomorphic, a new technique is developed to distinguish between Lie algebras. [ABSTRACT FROM AUTHOR]

We investigate continuity properties of operators obtained as values of the Weyl correspondence constructed by Pedersen (Invent. Math. 118:1-36, ) for arbitrary irreducible representations of nilpotent Lie groups. To this end we introduce modulation spaces for such representations and establish some of their basic properties. The situation of square-integrable representations is particularly important and in the special case of the Schrödinger representation of the Heisenberg group we recover the classical modulation spaces used in the time-frequency analysis. [ABSTRACT FROM AUTHOR]

Abstract: We study Anosov actions of nilpotent Lie groups on closed manifolds. Our main result is a generalization to the nilpotent case of a classical theorem by J.F. Plante in the 70''s. More precisely, we prove that, for what we call a good Anosov action of a nilpotent Lie group on a closed manifold, if the non-wandering set is the entire manifold, then the closure of stable strong leaves coincide with the closure of the strong unstable leaves. This implies the existence of an equivariant fibration of the manifold onto a homogeneous space of the Lie group, having as fibers the closures of the leaves of the strong foliation. [Copyright &y& Elsevier]

Let G be a connected and simply connected nilpotent Lie group and L≡−Δ+ W be the Schrödinger operator on L( 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 L( G) with p∈(0,1] and from $H^{p}_{L}(G)$ to H( G) with p∈( D/( D+1),1] are also obtained, where D is the dimension at infinity of G. [ABSTRACT FROM AUTHOR]

In this paper, we define an analog of the L- L Morgan's uncertainty principle for any exponential solvable Lie group G ( p, q ∈ [1,+∞]). When G is nilpotent and has a noncompact center, the proof of such an analog is given for p, q ∈ [2,+∞], extending the earlier settings ([2], [4] and [5]). Such a result is only known for some particular restrictive cases so far. We also prove the result for general exponential Lie groups with nontrivial center. [ABSTRACT FROM AUTHOR]

Let G = exp $${\mathfrak{g}}$$ be a connected, simply connected, nilpotent Lie group and let ω be a continuous symmetric weight on G with polynomial growth. In the weighted group algebra $${L^{1}_{\omega}(G)}$$ we determine the minimal ideal of given hull $${\{\pi_{l'} \in \hat{G} | l' \in l + \mathfrak{n}^{\perp}\}}$$, where $${\mathfrak{n}}$$ is an ideal contained in $${\mathfrak{g}(l)}$$, and we characterize all the L∞( G/ N)-invariant ideals (where $${N = {\rm exp}\, \mathfrak{n}}$$) of the same hull. They are parameterized by a set of G-invariant, translation invariant spaces of complex polynomials on N dominated by ω and are realized as kernels of specially built induced representations. The result is particularly simple if the co-adjoint orbit of l is flat. [ABSTRACT FROM AUTHOR]

We formulate and prove two versions of Miyachi's theorem for connected, simply connected nilpotent Lie groups. This allows us to prove the sharpness of the constant 1/4 in the theorems of Hardy and of Cowling and Price for any nilpotent Lie group. These theorems are proved using a variant of Miyachi's theorem for the group Fourier transform. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a connected simply connected nilpotent Lie group. In [A. Baklouti, N. Ben Salah, The version of Hardy''s Theorem on nilpotent Lie groups, Forum Math. 18 (2006) 245–262], we proved for the version of Hardy''s Theorem known as the Cowling–Price Theorem. In the setup where , the problem is still unsolved and the upshot is known only for few cases. We prove in this paper such a result in the context of nilpotent Lie groups. A proof of the analogue of Beurling''s Theorem is also provided in the same context. [Copyright &y& Elsevier]

Abstract: In this paper, we define the Littlewood–Paley and Lusin functions associated to the sub-Laplacian operator on nilpotent Lie groups. Then we prove the () boundedness of Littlewood–Paley and Lusin functions. [Copyright &y& Elsevier]

In this paper, we develop algebraic approaches to the construction of Anosov diffeomorphisms on compact manifolds. Two mutually dual constructions are described, which provide numerous new examples of Anosov diffeomorphisms on nilmanifolds. The basis of the constructions is the operation of tensor multiplication of Lie algebras by appropriate finite-dimensional associative-commutative algebras. Several examples illustrating the general method are given. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We describe the direct integral decomposition of a quasi regular representation of a connected and simply connected nilpotent Lie group G, which is induced from a discrete subgroup Γ. The solution is given explicitly in terms of orbital parameters. That is, the spectrum, multiplicity and spectral measure that constitute the decomposition are described completely in terms of natural objects associated to the co-adjoint orbits of G. We conclude with a study of the multiplicity function in certain cases. [ABSTRACT FROM AUTHOR]

On a Lie group S = NA , that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A , a probability measure μ is considered and treated as a distribution according to which transformations s ∈ S acting on N = S / A are sampled. Under natural conditions, formulated some over thirty years ago, there is a μ-invariant measure m on N . Properties of m have been intensively studied by a number of authors. The present article deals with the situation when μ( A ) = ℙ( s t ∈ A ), where ℝ + ∋ t → s t ∈ S is the diffusion on S generated by a second order subelliptic, hypoelliptic, left-invariant operator on S . In this article the most general operators of this kind are considered. Precise asymptotic for m at infinity and for the Green function of the operator are given. To achieve this goal a pseudodifferential calculus for operators with coefficients of finite smoothness is formulated and applied. [ABSTRACT FROM AUTHOR]

We consider the affine action of a nilpotent Lie group on ℝn. Lipsman (1995) conjectured that such an action is proper in the sense of Palais if and only if the action is (CI) in the sense of Kobayashi. The present paper gives a counterexample to Lipsman's conjecture for n ≥ 5. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a connected simply connected nilpotent Lie group, K an analytic subgroup of G and π an irreducible unitary representation of G. Let be the algebra of differential operators keeping invariant the space of vectors of π and commuting with the action of K on that space. In this paper, we assume that the restriction of π to K has finite multiplicities and we show that is isomorphic to a subalgebra of the field of rational K-invariant functions on the co-adjoint orbit associated to π, and for some particular cases, that is even isomorphic to the algebra of polynomial K-invariant functions on . We prove also the Frobenius reciprocity for some restricted classes of nilpotent Lie groups, especially in the cases where K is normal or abelian. [Copyright &y& Elsevier]

Let 1 and q be such that 1p+1q=1. It is well known that the norm of the Lp-Fourier transform of the additive group Rn is ||Fp(Rn)||=Apn, where Ap=p1pq1q12. For a nilpotent Lie group G, we obtain the estimate ||Fp(G)||⩽Ap2dimG−m2, where m is the maximal dimension of the coadjoint orbits. Such a result was known only for some particular cases. [Copyright &y& Elsevier]

In the first part of this article, we prove an explicit lower bound on the distance to the cut point of an arbitrary geodesic in a simply connected two-step nilpotent Lie group G with a lieft invariant metric. As a result, we obtaine a lower bound on the injectivity radius of a simply connected two-step nilpotent Lie group with a left invariant metric. We use this lower bound to determine the form of certain length minimizing geodesics from the identity to elements in the center of G. We also give an example of a two-step nilpotent Lie group G such that along most geodesics in this group, the cut point and the first conjugate point do not coincide. In the second part of this article, we examine the relation between the Laplace spectrum and the length spectrum on nilmanifolds by showing that a method developed by Gordon and Wilson for constructing families of isospectral two-step nilmanifolds necessarily yields manifolds with the same length spectrum. As a consequence, all known methods for constructing families of isospectral two-step nilmanifolds necessarily yield manifolds with the same length spectrum. [ABSTRACT FROM AUTHOR]

Let G be a step two nilpotent Lie group. In this paper, we give necessary and sufficient conditions on the operator valued symbols σ such that the associated pseudo-differential operators T σ on G are in the class of Hilbert-Schmidt operators. As a key step to prove this, we define (μ , ν) -Weyl transform on G and derive a trace formula for (μ , ν) -Weyl transform with symbols in L 2 (R 2 n). We show that Hilbert-Schmidt pseudo-differential operators on L 2 (G) are same as Hilbert-Schmidt (μ , ν) -Weyl transform with symbol in L 2 (R 2 n + r + k × R 2 n + r + k). Further, we present a characterization of the trace class pseudo-differential operators on G and provide a trace formula for these trace class operators. [ABSTRACT FROM AUTHOR]

Let G be a connected, simply connected one-parameter metabelian nilpotent Lie group, that means, the corresponding Lie algebra has a one-codimensional abelian subalgebra. In this article we show that G contains a discrete cocompact subgroup. Given a discrete cocompact subgroup Γ of G, we define the quasi-regular representation τ=ind_Γ^G 1 of G. The basic problem considered in this paper concerns the decomposition of τ into irreducibles. We give an orbital description of the spectrum, the multiplicity function and we construct an explicit intertwining operator between τ and its desintegration without considering multiplicities. Finally, unlike the Moore inductive algorithm for multiplicities on nilmanifolds, we carry out here a direct computation to get the multiplicity formula.

Let G be a connected and simply connected two-step nilpotent Lie group and Γ a lattice subgroup of G. In this note, we give a new multiplicity formula, according to the sense of Moore, of irreducible unitary representations involved in the decomposition of the quasi-regular representation Ind_Γ^G(1). Extending then the Abelian case.

The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if G = N × A is a connected, simply connected, nilpotent Lie group with an Abelian factor A, then every uniform subgroup of G is the direct product of a uniform subgroup of N and Z^r where r = dim A.

Let G be a connected simply connected nilpotent Lie group and let Γ be a discrete subgroup of G. We characterize the irreducible unitary representations of G that remain irreducible when restricted to Γ. [ABSTRACT FROM AUTHOR]