Connected Nilpotent Lie Group Research Articles

CR embeddings of nilpotent Lie groups

In this note we show that a connected, simply connected nilpotent Lie group with an integrable left-invariant complex structure on a generating and suitably complemented subbundle of the tangent bundle admits a Cauchy-Riemann (CR) embedding in complex space defined by polynomials. We also show that a similar conclusion holds on suitable quotients of nilpotent Lie groups. Our results extend the CR embeddings constructed by Naruki [Publ. Res. Inst. Math. Sci. 6 (1970), pp. 113–187] in 1970. In particular, our generalisation to quotients allows us to see a class of Levi degenerate CR manifolds as quotients of nilpotent Lie groups.

An uncertainty principle for the windowed Bochner–Fourier transform with the complex-valued window function

Abstract The generalization of Hardy uncertainty principle for the windowed Bochner–Fourier transform on the Heisenberg group is proved. We consider a Bochner measurable function Ψ : G → X {\Psi:G\to X} , where X is a completely separable Hilbert space X and Let G be a completely separable, unimodular, connected nilpotent Lie group. We establish that if ϕ ∈ C C ⁢ ( G ) {\phi\in C_{C}(G)} is a non-trivial window function and Ψ ∈ L 2 ⁢ ( G ) {\Psi\in L^{2}(G)} satisfies ∥ V ϕ ⁢ ( Ψ ) ⁢ ( g , χ ) ∥ HB ≤ c 2 ⁢ ( g ) ⁢ exp ⁡ ( - β ⁢ ∥ χ ∥ 2 ) \|V_{\phi}(\Psi)(g,\chi)\|_{\rm HB}\leq c_{2}(g)\exp(-\beta\|\chi\|^{2}) , β > 0 {\beta>0} , then Ψ = 0 {\Psi=0} almost everywhere.

Orthonormal bases arising from nilpotent actions

In this paper, we prove the existence of an orthonormal basis in at least one orbit of every generic irreducible representation of a simply connected and connected nilpotent Lie group. Our result has a wide-ranging impact, encompassing all irreducible representations of a nilpotent Lie group that are square-integrable modulo its center. This resolves a fundamental open problem in time-frequency analysis and frame theory, originally posed by Karlheinz Gröchenig. The implications of our findings are significant and far-reaching.

Poincaré profiles of Lie groups and a coarse geometric dichotomy

Poincaré profiles are analytically defined invariants, which provide obstructions to the existence of coarse embeddings between metric spaces. We calculate them for all connected unimodular Lie groups, Baumslag–Solitar groups and Thurston geometries, demonstrating two substantially different types of behaviour. For Lie groups, our dichotomy extends both the rank one versus higher rank dichotomy for semisimple Lie groups and the polynomial versus exponential growth dichotomy for solvable unimodular Lie groups. We provide equivalent algebraic, quasi-isometric and coarse geometric formulations of this dichotomy. As a consequence, we deduce that for groups of the form Ntimes S, where N is a connected nilpotent Lie group, and S is a rank one simple Lie group, both the growth exponent of N, and the conformal dimension of S are non-decreasing under coarse embeddings. These results are new even for quasi-isometric embeddings and give obstructions which in many cases improve those previously obtained by Buyalo–Schroeder.

Nilpotent Groups and Bi-Lipschitz Embeddings Into L1

Abstract We prove that if a simply connected nilpotent Lie group quasi-isometrically embeds into an $L^1$ space, then it is abelian. We reach this conclusion by proving that every Carnot group that bi-Lipschitz embeds into $L^1$ is abelian. Our proof follows the work of Cheeger and Kleiner, by considering the pull-back distance of a Lipschitz map into $L^1$ and representing it using a cut measure. We show that such cut measures, and the induced distances, can be blown up and the blown-up cut measure is supported on “generic” tangents of the original sets. By repeating such a blow-up procedure, one obtains a cut measure supported on half-spaces. This differentiation result then is used to prove that bi-Lipschitz embeddings can not exist in the non-abelian settings.

A proof of the polynomial conjecture for restrictions of nilpotent lie groups representations

Let G G be a connected and simply connected nilpotent Lie group, K K an analytic subgroup of G G and π \pi an irreducible unitary representation of G G whose coadjoint orbit of G G is denoted by Ω ( π ) \Omega (\pi ) . Let U ( g ) \mathscr U(\mathfrak g) be the enveloping algebra of g C {\mathfrak g}_{\mathbb C} , g \mathfrak g designating the Lie algebra of G G . We consider the algebra D π ( G ) K ≃ ( U ( g ) / ker ⁡ ( π ) ) K D_{\pi }(G)^K \simeq \left (\mathscr U(\mathfrak g)/\operatorname {ker}(\pi )\right )^K of the K K -invariant elements of U ( g ) / ker ⁡ ( π ) \mathscr U(\mathfrak g)/\operatorname {ker}(\pi ) . It turns out that this algebra is commutative if and only if the restriction π | K \pi |_K of π \pi to K K has finite multiplicities (cf. Baklouti and Fujiwara [J. Math. Pures Appl. (9) 83 (2004), pp. 137-161]). In this article we suppose this eventuality and we provide a proof of the polynomial conjecture asserting that D π ( G ) K D_{\pi }(G)^K is isomorphic to the algebra C [ Ω ( π ) ] K \mathbb C[\Omega (\pi )]^K of K K -invariant polynomial functions on Ω ( π ) \Omega (\pi ) . The conjecture was partially solved in our previous works (Baklouti, Fujiwara, and Ludwig [Bull. Sci. Math. 129 (2005), pp. 187-209]; J. Lie Theory 29 (2019), pp. 311-341).

Polynomial and horizontally polynomial functions on Lie groups

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset S of the algebra {mathfrak {g}} of left-invariant vector fields on a Lie group {mathbb {G}} and we assume that S Lie generates {mathfrak {g}}. We say that a function f:{mathbb {G}}rightarrow {mathbb {R}} (or more generally a distribution on {mathbb {G}}) is S-polynomial if for all Xin S there exists kin {mathbb {N}} such that the iterated derivative X^k f is zero in the sense of distributions. First, we show that all S-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent k in the previous definition is independent on Xin S, they form a finite-dimensional vector space. Second, if {mathbb {G}} is connected and nilpotent, we show that S-polynomial functions are polynomial functions in the sense of Leibman. The same result may not be true for non-nilpotent groups. Finally, we show that in connected nilpotent Lie groups, being polynomial in the sense of Leibman, being a polynomial in exponential chart, and the vanishing of mixed derivatives of some fixed degree along directions of {mathfrak {g}} are equivalent notions.

O-minimal flows on nilmanifolds

Let G be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of UT(n,R), and let Γ be a lattice in G, with π:G→G∕Γ the quotient map. For a semialgebraic X⊆G, and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of π(X) in the compact nilmanifold G∕Γ. Our theorem describes cl(π(X)) in terms of finitely many families of cosets of real algebraic subgroups of G. The underlying families are extracted from X, independently of Γ. We also prove an equidistribution result in the case of curves.

THE ASYMPTOTIC SEMIGROUP OF A SUBSEMIGROUP OF A NILPOTENT LIE GROUP

Let S be a subsemigroup of a simply connected nilpotent Lie group G. We construct an asymptotic semigroup S0 in the associated graded Lie group G0 of G. We can compute the image of S0 in the abelianization {G}_0^{mathrm{ab}}={G}^{mathrm{ab}}. This gives useful information about S. As an application, we obtain a transparent proof of the following result of E. B. Vinberg and the author: either there is an epimorphism f : G → ℝ such that f (s) ≥ 0 for every s in S or the closure overline{S} of S is a subgroup of G and G/overline{S} is compact.

A PARAMETRIZATION OF δ-SPHERICAL FUNCTIONS ON COMMUTATIVE TRIPLES ASSOCIATED WITH NILPOTENT LIE GROUPS

Let $N$ be a connected and simply connected nilpotent Lie group, $K$ be a compact subgroup of $Aut(N)$, the group of automorphisms of $N$ and $\delta$ be a class of unitary irreducible representations of $K$. The triple $(N,K,\delta)$ is a commutative triple if the convolution algebra $\mathfrak{U}_{\delta}^{1}(N)$ of $\delta$-radial integrable functions is commutative. In this paper, we obtain first a parametrization of $\delta$ spherical functions by means of the unitary dual $\widehat{N}$ and then an inversion formula for the spherical transform of $F\in \mathfrak{U}_{\delta}^{1}(N)$.

Generalized Ricci flow on nilpotent Lie groups

Abstract We define solitons for the generalized Ricci flow on an exact Courant algebroid. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.

Randers Ricci soliton homogeneous nilmanifolds

Let $F$ be a left-invariant Randers metric on a simply connected nilpotent Lie group $N$, induced by a left-invariant Riemannian metric $\hat{\boldsymbol{a}}$ and a vector field $X$ which is $I_{\hat{\boldsymbol{a}}}(M)$-invariant. We show that if the Ricci flow equation has a unique solution then, $(N,F)$ is a Ricci soliton if and only if $(N,F)$ is a semialgebraic Ricci soliton.

Global Sampled-Data Regulation of a Class of Fully Actuated Invariant Systems on Simply Connected Nilpotent Matrix Lie Groups

In this article, we examine a regulator problem for a class of fully actuated continuous-time invariant systems on Lie groups, using a discrete-time controller with constant sampling period. We present a smooth discrete-time control law that achieves global regulation on simply connected nilpotent Lie groups. We first solve the problem when both the plant state and exosystem state are available for feedback, then in the case where the plant state and plant output are available for feedback. The class of plant outputs considered includes, for example, the quantity to be regulated. This class of outputs allows us to use the classical Luenberger observer to estimate the exosystem states. In the full-information case, the regulation quantity on the Lie algebra is shown to decay exponentially to zero, which implies that it tends asymptotically to the identity on the Lie group.

Functional calculus on non-homogeneous operators on nilpotent groups

We study the functional calculus associated with a hypoelliptic left-invariant differential operator mathcal {L} on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator mathcal {L}_0 on the ‘local’ contraction G_0 of G, as well as of the corresponding Rockland operator mathcal {L}_infty on the ‘global’ contraction G_infty of G. We provide asymptotic estimates of the Riesz potentials associated with mathcal {L} at 0 and at infty , as well as of the kernels associated with functions of mathcal {L} satisfying Mihlin conditions of every order. We also prove some Mihlin–Hörmander multiplier theorems for mathcal {L} which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the ‘Plancherel measure’ associated with mathcal {L} from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.

Balian–Low Type Theorems on Homogeneous Groups

We prove strict necessary density conditions for coherent frames and Riesz sequences on homogeneous groups. Let N be a connected, simply connected nilpotent Lie group with a dilation structure (a homogeneous group) and let (π,Hπ) be an irreducible, square-integrable representation modulo the center Z(N) of N on a Hilbert space Hπ of formal dimension dπ. If g ∈ Hπ is an integrable vector and the set {π(λ)g : λ ∈ Λ} for a discrete subset Λ ⊆ N/Z(N) forms a frame for Hπ, then its density satisfies the strict inequality D − (Λ) > dπ, where D − (Λ) is the lower Beurling density. An analogous density condition D+(Λ) < dπ holds for a Riesz sequence in Hπ contained in the orbit of (π,Hπ). The proof is based on a deformation theorem for coherent systems, a universality result for p-frames and p-Riesz sequences, some results from Banach space theory, and tools from the analysis on homogeneous groups.

DISTAL ACTIONS OF AUTOMORPHISMS OF NILPOTENT GROUPS G ON SUBG AND APPLICATIONS TO LATTICES IN LIE GROUPS

AbstractFor a locally compact group G, we study the distality of the action of automorphisms T of G on SubG, the compact space of closed subgroups of G endowed with the Chabauty topology. For a certain class of discrete groups G, we show that T acts distally on SubG if and only if Tn is the identity map for some $n\in\mathbb N$ . As an application, we get that for a T-invariant lattice Γ in a simply connected nilpotent Lie group G, T acts distally on SubG if and only if it acts distally on SubΓ. This also holds for any closed T-invariant co-compact subgroup Γ in G. For a lattice Γ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on SubΓ. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group and that in a nilpotent Lie group. We also characterise automorphisms of a lattice Γ in a connected semisimple Lie group which act distally on SubΓ. For torsion-free compactly generated nilpotent (metrisable) groups G, we obtain the following characterisation: T acts distally on SubG if and only if T is contained in a compact subgroup of Aut(G). Using these results, we characterise the class of such groups G which act distally on SubG. We also show that any compactly generated distal group G is Lie projective.

On the Positivity of Kirillov’s Character Formula

We give a direct proof for the positivity of Kirillov’s character on the convolution algebra of smooth, compactly supported functions on a connected, simply connected nilpotent Lie group G. Then we use this positivity result to construct a representation of G × G and establish a G × G-equivariant isometric isomorphism between our representation and the Hilbert–Schmidt operators on the underlying representation of G. In fact, we provide a framework in which we establish the positivity of Kirillov’s character for coadjoint orbits of groups such as $\text {SL}(2, \mathbb {R})$ under additional hypotheses that are automatically satisfied in the nilpotent case. These hypotheses include the existence of a real polarization and the Pukanzsky condition.

On the irreducibility of some restrictions in nilpotent Lie groups

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.

On a type of maximal abelian torsion free subgroups of connected Lie groups

For an arbitrary real connected Lie group G we define $$\mathrm {p}(G)$$ as the maximal integer p such that $$\mathbb {Z}^p$$ is isomorphic to a discrete subgroup of G and $$\mathrm {q}(G)$$ is the maximal integer q such that $$\mathbb {R}^q$$ is isomorphic to a closed subgroup of G. The aim of this paper is to investigate properties of these two invariants. We will show that if G is a noncompact connected Lie group, then $$1\le \mathrm {q}(G)\le \mathrm {p}(G)\le \dim (G/K)$$ where K is a maximal compact subgroup of G. In the cases when G is an exponential Lie group or G is a connected nilpotent Lie group, we give explicit relationships between these two invariants and a well known Lie algebra invariant $$\mathcal M(\mathfrak {g})$$ , i.e. the maximum among the dimensions of abelian subalgebras of the Lie algebra $$\mathfrak {g}:=\mathrm {Lie}(G)$$ .

A classification of certain codimension one locally free actions of nilpotent Lie groups up to a differentiable orbital conjugacy

Let G be a simply connected nilpotent Lie group, let M be a compact connected manifold with dim(M) = dim(G)+1 and let Φ be a C∞ locally free action of G on M. If G admits no lattice and if the centralizer in G of its derived group G′ is the center of G′, then Φ is differentiably orbitally conjugated to a homogeneous action of the same kind.