# Questions tagged [heisenberg-groups]

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### Semidirect product of metaplectic group and Heisenberg group

I know that Symplectic group has an action on Heisenberg group. I am wondering how to extend this to non-trivial two fold metaplectic covering? Thanks in advance!
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### Carnot-Carathéodory metric

The metric in sub-Riemannian geometry is often called the Carnot-Carathéodory metric. Question 1. What is the origin of this name? Who was the first to introduce it? I believe that the "...
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### Heisenberg groups, Carnot groups and contact forms

The horizontal distribution in the Heisenberg group is the kernel of the standard contact form: $$\alpha = dt + 2 \sum_{j=1}^n (x_j \, dy_j - y_j \, dx_j).$$ Question. Can one describe ...
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### Explicit Quasisymmetric embedding into Euclidean space

It is known that every doubling metric space admits quasisymmetric map into Euclidean space. My question is, is there a known explicit (closed-form) quasisymmetry from the Heisenberg group into a ...
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### The extension class of a finite Heisenberg group

Let $\mathbb{K}$ be a field of characteristic $\neq 2$ and let $(V, \omega)$ be a symplectic vector space. Then the Heisenberg group $\mathsf{Heis}(V, \, \omega)$ is the central extension of the ...
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### Contact geometry: approximation of Legendrian mappings

Let $\alpha$ be a standard contact form on $\mathbb{R}^{2n+1}$. We say that a map $f:\mathbb{R}^k\to\mathbb{R}^{2n+1}$ contact if $f^*\alpha=0$. Question 1. Is it true that a $C^1$-contact ...
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### About the purpose of introducing '“groups of Heisenberg type”

I would like to know, can we say that the "groups of Heisenberg type" where introduced by A. Kaplan in "Kaplan, A. (1980). Fundamental solutions for a class of hypoelliptic PDE generated by ...
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### Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two

What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...
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### Non-isomorphic Heisenberg groups over rings

Suppose $R_1,R_2$ are finite unital commutative rings. Consider Heisenberg groups $H_3(R_1)$ and $H_3(R_2)$ (upper unitriangular marticies $3 \times 3$). Proposition. If $R_1 \not\cong R_2$ (as ...
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### Why the sub-Laplacian $\Delta_{sub}$ on the Heisenberg group $H^3$ is sub-elliptic but not elliptic?

I want to know why the sub-Laplacian $\Delta_{sub}= X^2 + Y^2$ on the Heisenberg group $H^3 = \mathbb C \times \mathbb R$ is sub-elliptic but not elliptic, where $X$ and $Y$ are the left-invariant ...
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I am trying to understand how the Heisenberg group is defined because I would like to understand the (irreducible) representations. Following this article given a symplectic bilinear form $\langle, \... 1answer 1k views ### Difference between the Laplacian and the sub-Laplacian of a Lie group Given a Lie group$G$, what is the difference between the Laplacian$\Delta$and the sub-Laplacian$\Delta_{sub}$of$G$. And what are the properties that we lose when going from sub-Laplace to ... 0answers 208 views ### How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic? The Heisenberg group$H^3$is the set$\mathbb C\times \mathbb R$endowed with the group law $$(z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right).$$ For$z=x+ i y \in \mathbb C$... 3answers 416 views ### Show that the Laplacian operator on the Heisenberg group is negative The Heisenberg group$H^3$is the set$\mathbb C\times \mathbb R$endowed with the group law $$(z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right).$$ For$z=x+ i y \in \mathbb C$... 1answer 173 views ### dirichlet problem in the heisenberg group Good morning everybody. I was looking just for a quick reference to know whether the Dirichlet problem has a solution in the Heisenberg group, that is$\mathbb R^3$endowed with coordinates$(x,y,z)$... 1answer 238 views ### Horizontal Sobolev space on Carnot group This question is connected with my previous: Heisenberg group: function without vertical derivative. Here I am trying to look from another side: what is a difference between Sobolev space and ... 0answers 363 views ### Heisenberg group: function without vertical derivative Let$\mathbb H$be Heisenberg group with vector fields $$X=\partial_x - \frac12y\partial_t,\quad Y=\partial_y + \frac12x\partial_t,\quad T=\partial_t$$ and$U\subset\mathbb H$is an open set. I am ... 1answer 895 views ### Irreducible representation of Heisenberg group with characteristic 2? As we all know that the irreducible representation for Heisenberg group can be classified easily when the group is over a finite field$\mathbb{F}_q$, where$q=p^n$and$p$is a prime greater than$2$.... 1answer 211 views ### totally geodesic submanifold of Heisenberg group Let$G= \left\{ \begin{pmatrix} 1&a&c\\0&1&b\\0&0&0 \end{pmatrix} \mid a,b,c\in \mathbb{R} \right\}$be the Heisenberg group. Is there a compact codimension one submanifold ... 0answers 160 views ### Joint representation of the semi-direct product of the metaplectic group and Heisenberg group Given a symplectic space$W$over a local field$F$and a additive character$\psi$of$F$, we can construct the Weil representation$\omega_\psi$, which can be viewed as a representation of the semi-... 2answers 315 views ### Hesse pencil and Schrodinger representation of Heisenberg group Let$E$be a smooth elliptic curve over an algebraically closed field of characteristic zero. Let$\mathcal{L}$be a line bundle of degree$3$. Heisenberg group$H_3$acts on global sections of$\...
Let $E$ be a smooth elliptic curve over algebraically closed field $k$ of characteristic zero, $\mathcal{L}$ is a line bundle over $E$, $\operatorname{deg}(\mathcal{L})=n \geq 1$. Then I define the ...
It is well known that: Theorem 1. For $f\in L^{2}(\mathbb H_{n}=\text{The Heisenberg group of dimesion } 2n+1)$ we have the expansion f(z, s)= (2\pi)^{-n} \sum_{k=0}^{\infty} \int_{0}^{\infty} f \...