On Hardy’s uncertainty principle for connected nilpotent Lie groups

In (Kaniuth and Kumar in Math. Proc. Camb. Phil. Soc. 131, 487–494, 2001) Hardy’s uncertainty principle for was generalized to connected and simply connected nilpotent Lie groups. In this paper, we extend it further to connected nilpotent Lie groups with non-compact centre. Concerning the converse, we show that Hardy’s theorem fails for a connected nilpotent Lie group G which admits a square integrable irreducible representation and that this condition is necessary if the simply connected covering group of G satisfies the flat orbit condition.     

A restriction theorem for Métivier groups

In the spirit of an earlier result of D. Müller on the Heisenberg group we prove a restriction theorem on a certain class of two step nilpotent Lie groups. Our result extends that of Müller also in the framework of the Heisenberg group.     

Closed geodesics in compact nilmanifolds

We study the density of closed geodesics property on 2-step nilmanifolds Γ\N, where N is a simply connected 2-step nilpotent Lie group with a left invariant Riemannian metric and Lie algebra ?, and Γ is a lattice in N. We show the density of closedgeodesics property holds for quotients of singular, simply connected, 2-step nilpotent Lie groups N which are constructed using irreducible representations of the compact Lie group SU(2). Received: 8 November 2000 / Revised version: 9 April 2001     

On Anosov automorphisms of nilmanifolds

The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds, and the existence of such automorphisms is a really strong condition on the rational nilpotent Lie algebra determined by the lattice, so called an Anosov Lie algebra. We prove that n⊕?⊕n (s times, s≥2) has an Anosov rational form for any graded real nilpotent Lie algebra n having a rational form. We also obtain some obstructions for the types of nilpotent Lie algebras allowed, and use the fact that the eigenvalues of the automorphism are algebraic integers (even units) to show that the types (5,3) and (3,3,2) are not possible for Anosov Lie algebras.     

Riesz transforms on a solvable Lie group of polynomial growth

We study Riesz transforms associated with a sublaplacian H on a solvable Lie group G, where G has polynomial volume growth. It is known that the standard second order Riesz transforms corresponding to H are generally unbounded in L^p(G). In this paper, we establish boundedness in L^p for modified second order Riesz transforms, which are defined using derivatives on a nilpotent group G_N associated with G. Our method utilizes a new algebraic approach which associates a distinguished choice of Cartan subalgebra with the sublaplacian H. We also obtain estimates for higher derivatives of the heat kernel of H, and give a new proof (without the use of homogenization theory) of the boundedness of first order Riesz transforms. Our results can be generalized to an arbitrary (possibly non-solvable) Lie group of polynomial growth.     

Homogeneous nilmanifolds attached to representations of compact Lie groups

For each compact Lie algebra ? and each real representation V of ? we consider a two-step nilpotent Lie group N(?,V), endowed with a natural left-invariant riemannian metric. The homogeneous nilmanifolds so obtained are precisely those which are naturally reductive. We study some geometric aspects of these manifolds, finding many parallels with H-type groups. We also obtain, within the class of manifolds N(?,V), the first examples of non-weakly symmetric, naturally reductive spaces and new examples of non-commutative naturally reductive spaces. Received: 16 September 1998 / Revised version: 24 February 1999     

Distances on Lie groups of polynomial volume growth

For any simply connected solvable Lie group Q of polynomial volume growth, we introduce the notion of nil-shadow of Q. We shall give an explicit formula for the distance to the origin of an element q ∈ Q in terms of its exponential coordinates of the second kind taken in an appropriate basis. This result extends a previous result for nilpotent Lie groups [6, Theorem DN] and [7, Theorem 1].     

On theorems of Beurling and Cowling-Price for certain nilpotent Lie groups

Let G be a connected simply connected nilpotent Lie group. In [A. Baklouti, N. Ben Salah, The Lp−Lq version of Hardy's Theorem on nilpotent Lie groups, Forum Math. 18 (2006) 245-262], we proved for 2?p,q?+∞ the Lp−Lq version of Hardy's Theorem known as the Cowling-Price Theorem. In the setup where 1?p,q?+∞, 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 2-NPC nilpotent Lie groups. A proof of the analogue of Beurling's Theorem is also provided in the same context.     

Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds

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. In memory of Robert Brooks     

Analysis of restrictions of unitary representations of a nilpotent Lie group

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 DπK(G) be the algebra of differential operators keeping invariant the space of C^∞ 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 DπK(G) is isomorphic to a subalgebra of the field of rational K-invariant functions on the co-adjoint orbit Ω(π) associated to π, and for some particular cases, that DπK(G) is even isomorphic to the algebra of polynomial K-invariant functions on Ω(π). We prove also the Frobenius reciprocity for some restricted classes of nilpotent Lie groups, especially in the cases where K is normal or abelian.     

Symplectic or contact structures on Lie groups

We study left invariant contact forms and left invariant symplectic forms on Lie groups. In the case of filiform Lie groups we give a necessary and sufficient condition for the existence of a left invariant contact form and we prove the uniqueness of this contact form up to a nonzero scalar multiple. As an application we classify all symplectic structures on nilpotent Lie algebras of dimension ?6.     

Multiplicity-free actions and the geometry of nilpotent orbits

Let G be a reductive Lie group. Take a maximal compact subgroup K of G and denote their Lie algebras by and respectively. We get a Cartan decomposition . Let be the complexification of , and the complexified decomposition. The adjoint action restricted to K preserves the space , hence acts on , where denotes the complexification of K. In this paper, we consider a series of small nilpotent -orbits in which are obtained from the dual pair ([R. Howe, Transcending classical invariant theory. J. Amer. Math. Soc. 2 (1989), no. 3, 535–552]). We explain astonishing simple structures of these nilpotent orbits using generalized null cones. For example, these orbits have a linear ordering with respect to the closure relation, and acts on them in multiplicity-free manner. We clarify the -module structure of the regular function ring of the closure of these nilpotent orbits in detail, and prove the normality. All these results naturally comes from the analysis on the null cone in a matrix spaceW , and the double fibration of nilpotent orbits in and . The classical invariant theory assures that the regular functions on our nilpotent orbits are coming from harmonic polynomials on W with repspect to or . We also provide many interesting examples of multiplicity-free actions on conic algebraic varieties. Received November 1, 1999 / Published online October 30, 2000     

Distorsion des distances dans les groupes de Lie nilpotents

Let G be a connected nilpotent Lie group and H a connected subgroup of G. We give an explicit formula for the distance to the origin with the exponential coordinates of the second kind of g∈G. Using this fact, we prove that the distance to the origin of any element in H is bounded by a polynomial function of the distance to the origin in the group G. The degree of the polynomial is the nilpotency rank of the group G.     

Representations and the Jacobian Conjecture

In this paper we apply the representation theory of the Lie algebra sl₂(C) to the problem of describing Hessian nilpotent polynomials, which are important in the theory of the Jacobian Conjecture. In the two variable case we describe them as the maximal and minimal weight vectors of the irreducible representations of sl₂(C). For the first time this gives a characterization of the Hessian nilpotent polynomials in terms of linear differential operators.     

Littlewood-Paley and Lusin functions on nilpotent Lie groups

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 Lp (1<p<∞) boundedness of Littlewood-Paley and Lusin functions.     

The orbit method and Gelfand pairs, associated with nilpotent Lie groups

Let K be a compact Lie group acting by automorphisms on a nilpotent Lie group N. One calls (K, N) a Gelfand pair when the integrable K-invariant functions on N form a commutative algebra under convolution. We prove that in this case the coadjoint orbits for G:= K × N which meet the annihilator of the Lie algebra of K do so in single K-orbits. This generalizes a result of the authors and R. Lipsman concerning Gelfand pairs associated with Heisenberg groups.     

Curvatures of homogeneous Randers spaces

We study curvatures of homogeneous Randers spaces. After deducing the coordinate-free formulas of the flag curvature and Ricci scalar of homogeneous Randers spaces, we give several applications. We first present a direct proof of the fact that a homogeneous Randers space is Ricci quadratic if and only if it is a Berwald space. We then prove that any left invariant Randers metric on a non-commutative nilpotent Lie group must have three flags whose flag curvature is positive, negative and zero, respectively. This generalizes a result of J.A. Wolf on Riemannian metrics. We prove a conjecture of J. Milnor on the characterization of central elements of a real Lie algebra, in a more generalized sense. Finally, we study homogeneous Finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected Lie group admitting a left invariant Finsler metric with positive flag curvature is SU(2)$\text{SU}\left(2\right)$.