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We prove an L p -spectral multiplier theorem for sub-Laplacians on Heisenberg type groups under the sharp regularity condition s > d 1 / p - 1 / 2 , where d is the topological dimension of the underlying group. Our approach relies on restriction type estimates where the multiplier is additionally truncated along the spectrum of the Laplacian on the center of the group. [ABSTRACT FROM AUTHOR]

We study the phase retrieval property for orbits of general irreducible representations of nilpotent groups, for the classes of simply connected Lie groups, and for finite groups. We prove by induction that in the Lie group case, all irreducible representations do phase retrieval. For the finite group case, we mostly focus on p-groups. Here our main result states that every irreducible representation of an arbitrary p-group with exponent p and size ≤ p 2 + p / 2 does phase retrieval. Despite the fundamental differences between the two settings, our inductive proof methods are very similar. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate which one of the non-isometric left-invariant Lorentz metrics g on four-dimensional nilpotent Lie groups H 3 × R and G 4 satisfies the Ricci soliton equation 2 Ric [ g ] + L X g + α g = 0 , here X is a vector field and α is a constant. Among the left-invariant Lorentzian metrics on H 3 × R , g λ - is a shrinking while g λ + and g μ are expanding and also g 0 1 , g 0 2 , g 0 3 have Ricci solitons. We exhibit among the non-isometric left-invariant Lorentz metric on the group G 4 only g 1 λ , g 2 λ have Lorentz Ricci solitons and g 2 λ is a shrinking. [ABSTRACT FROM AUTHOR]

In this paper, we study the conjecture of Bansal, Kumar and Sharma, which is an analog of Hardy’s theorem for Gabor transform in the setup of connected nilpotent Lie groups. To approach this conjecture, we use the orbit method and the Plancherel theory. When the Lie group G is simply connected, we show that the conjecture is true. [ABSTRACT FROM AUTHOR]

In this paper, we define and prove an analog of the Heisenberg uncertainty inequality for Gabor transform in the setup of connected, simply connected nilpotent Lie groups. When G is connected nilpotent and has a non-compact center, a proof of such an analog is given for functions in the Schwartz space of G. The representation theory and a localized Plancherel formula are fundamental tools in the proof of our results. [ABSTRACT FROM AUTHOR]

We survey the work of Elias M. Stein in the field of analysis on the Heisenberg group and other nilpotent Lie groups, together with its applications to complex analysis in several variables and partial differential equations. [ABSTRACT FROM AUTHOR]

Let N be a nilpotent Lie group and K a compact subgroup of the automorphism group Aut(N) of N. It is well-known that if (K ⋉ N , K) is a Gelfand pair then N is at most 2-step nilpotent Lie group. The notion of Gelfand pair was generalized when K is a non-compact group. In this work, we give an example of a 3-step nilpotent Lie group and a non-compact subgroup K of Aut(N) such that (K ⋉ N , N) is a generalized Gelfand pair. [ABSTRACT FROM AUTHOR]

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 SL (2 , 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. [ABSTRACT FROM AUTHOR]

We study local control of a mechanism with the growth vector (4,7). We study controllability and extremal trajectories of the nilpotent approximation as an example of the control theory on a Lie group. We provide solutions to the system and show examples of local extremal trajectories. [ABSTRACT FROM AUTHOR]

We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvature-dimension condition RCD(0, N) with N ∈ ℝ and N > 1. In fact, we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD(K, N), where K, N ∈ ℝ and N > 1. Along the way to the proofs, we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Carath´eodory spaces which may have independent interests. [ABSTRACT FROM AUTHOR]

We propose a general scheme for the search of a fundamental solution to the hypoelliptic diffusion equation in a "sufficiently good" sub-Riemannian manifold and the small-time asymptotics for the solution, which includes the generalized Fourier transform and the orbit method closely related to it, as well as an application of the perturbative method to the nilpotent approximation, and Trotter's formula. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a local uncertainty inequality for arbitrary connected, simply connected nilpotent Lie groups. This allows us to prove a couple of global uncertainty inequalities. In the nilpotent case, this type of result is only obtained for the Heisenberg group. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We establish a Fourier inversion theorem for general connected, simply connected nilpotent Lie groups G=exp(g) by showing that operator fields defined on suitable sub-manifolds of g∗ are images of Schwartz functions under the Fourier transform. As an application of this result, we provide a complete characterisation of a large class of invariant prime closed two-sided ideals of L1(G) as kernels of sets of irreducible representations of G. [ABSTRACT FROM AUTHOR]

Using the extension method, we study the one-parameter symmetry groups of the heat equation ∂tp = Δp, where Δ = X 1 2 + X 2 2 is the sub-Laplacian constructed by a Goursat distribution span({X1, X2}) in ℝn, where the vector fields X1 and X2 satisfy the commutation relations [X1, Xj] = Xj+1 (where Xn+1 = 0) and [Xj, Xk] = 0 for j ≥ 1 and k ≥ 1. We show that there are no such groups for n ≥ 4 (with exception of the linear transformations of solutions which are admitted by every linear equation). [ABSTRACT FROM AUTHOR]

Let $${\mathcal {N}}_m$$ be the group of $$m\times m$$ upper triangular real matrices with all the diagonal entries 1. Then it is an $$(m-1)$$ -step nilpotent Lie group, diffeomorphic to $${\mathbb {R}}^{\frac{1}{2} m(m-1)}$$ . It contains all the integer matrices as a lattice $$\Gamma _m$$ . The automorphism group of $${\mathcal {N}}_m \ (m\ge 4)$$ turns out to be extremely small. In fact, $$\mathrm {Aut}({\mathcal {N}})=\mathcal {I} \rtimes \mathrm {Out}({\mathcal {N}})$$ , where $$\mathcal {I}$$ is a connected, simply connected nilpotent Lie group, and $$\mathrm {Out}({\mathcal {N}})={{\tilde{K}}}={(\mathbb {R}^*)^{m-1}\rtimes \mathbb {Z}_2}$$ . With a nice left-invariant Riemannian metric on $${\mathcal {N}}$$ , the isometry group is $$\mathrm {Isom}({\mathcal {N}})= {\mathcal {N}} \rtimes K$$ , where $$K={(\mathbb {Z}_2)^{m-1}\rtimes \mathbb {Z}_2}\subset {{\tilde{K}}}$$ is a maximal compact subgroup of $$\mathrm {Aut}({\mathcal {N}})$$ . We prove that, for odd $$m\ge 4$$ , there is no infra-nilmanifold which is essentially covered by the nilmanifold $$\Gamma _m\backslash {\mathcal {N}}_m$$ . For $$m=2n\ge 4$$ (even), there is a unique infra-nilmanifold which is essentially (and doubly) covered by the nilmanifold $$\Gamma _m\backslash {\mathcal {N}}_m$$ . [ABSTRACT FROM AUTHOR]

We determine the space of primary ideals in the group algebra $$L^{1}(G) $$ of a connected nilpotent Lie group by identifying for every $$\pi \in \widehat{G} $$ the family $$\mathcal I^\pi $$ of primary ideals with hull $$\{\pi \} $$ with the family of invariant subspaces of a certain finite dimensional sub-space $$\mathcal P_Q^\pi $$ of the space of polynomials $$\mathcal P(G) $$ on G. [ABSTRACT FROM AUTHOR]

We investigate locally compact topological groups for which a generalized analog of the Heisenberg uncertainty inequality hold. In particular, it is shown that this inequality holds for $\mathbb{R}^{n} \times K$ (where K is a separable unimodular locally compact group of type I), Euclidean motion group and several general classes of nilpotent Lie groups which include thread-like nilpotent Lie groups, 2-NPC nilpotent Lie groups and several low-dimensional nilpotent Lie groups. [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]

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]

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]

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]

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]

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]

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]