Questions tagged [heisenberg-groups]

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Finite dimensional unitary representations of the discrete Heisenberg group

Let $H(\mathbb{Z})$ be the discrete Heisenberg group. What are the finite dimensional irreducible unitary representations of $H(\mathbb{Z})$? Do they all arise from the coordinate-wise quotient map to ...
0 votes
0 answers
57 views

Faithfull compact representation of the discrete Heisenberg group

Let $H$ be the discrete Heisenberg group. Is there an injective homomorphism $\varphi \colon H \to U(n)$ for some $n$? What about injective homomorphism $\varphi \colon H \to O(n)$ for some $n$?
1 vote
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Classification of the normal subgroups of the discrete Heisenberg group

Let $H$ be the discrete Heisenberg group, i.e., the set of matrices of the form $\begin{bmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{bmatrix}$ where $x,y,z \in \mathbb{Z}$...
1 vote
0 answers
382 views

Heat kernel on quaternion Heisenberg group

For the n Heisenberg($\Bbb C^n\times\Bbb R$) it is known that the heat kernel $q_s(z,t)=c_n\int_{\Bbb R} e^{-i\lambda t}\Big( \frac{\lambda}{\sinh(\lambda s)}\Big)^n e^{-\frac{\lambda|z|^2\coth(\...
3 votes
0 answers
51 views

Two pairings on the group $K(L)$ associated with a non-degenerate line bundle $L$ on an abelian variety

Let $A$ be an abelian variety over a field and let $L$ be a non-degenerate line bundle on $A$. Then $L$ gives rise to a morphism $\lambda:A\to A^*$ from $A$ to its dual. As usual, let $K(L):=\ker(\...
4 votes
0 answers
140 views

What is the analogue of the Jacobi theta function in the Weyl representation?

It is known (see https://en.wikipedia.org/wiki/Theta_representation) that the Jacobi theta function $$\vartheta(z; \tau) = \sum_{n\in\mathbb{Z}} \exp(\pi in^2\tau + 2\pi inz)$$ arises from a certain ...
1 vote
0 answers
38 views

Spectral decomposition of Heisenberg quotient

Let $H$ be the Heisenberg group over $\mathbb R$ of dimension $2n+1$ and let $\Gamma$ be a lattice in $H$. Then, I think, the spectral decomposition of $L^2(H/\Gamma)$ as unitary $H$-representation is ...
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1 vote
1 answer
142 views

Ratio limit results for restricted partition functions

This concerns difference/limit ratio results for special restricted partitions. Let $r,a, b$ be nonnegative integers; define $p(r,a,b)$ to be the number of partitions of the integer $r$ using at most $...
1 vote
0 answers
262 views

Canonical commutation relations-bounded vs. unbounded picture

Suppose that $Q,P$ are self-adjoint operators which satisfy the relation $$(1) \ \ \ \ \ [Q,P]=iI$$ One can easily show that in this case $P,Q$ cannot be bounded. However one can find unbounded ...
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2 votes
1 answer
159 views

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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20 votes
1 answer
881 views

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 "...
4 votes
1 answer
281 views

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 ...
1 vote
0 answers
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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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8 votes
1 answer
387 views

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 ...
5 votes
0 answers
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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 ...
4 votes
1 answer
186 views

Legendre's symbol in Schrödinger model for the Weil representation

I have a question concerning the Schrödinger model for the Weil representation over a finite field $\mathbb{F}_q$. The way to present the action of the Weil representation $\omega$ of $Sp(2n,\...
11 votes
3 answers
626 views

Non embedding of the Heisenberg group

It is well known that Heisenberg groups cannot be bi-Lipschitz embedded into Euclidean spaces. A standard proof uses the fact that a Lipschitz mapping from a Heisenberg group into a Euclidean space is ...
10 votes
0 answers
355 views

Lipschitz homotopy groups

There is an extensive literature on Lipschitz homotopies of Lipschitz maps. But I haven't seen anything about Lipschitz homotopy groups. We have introduced this notion in an article that you can find ...
5 votes
0 answers
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Modules of algebras with idempotents and the Stone-von Neumann theorem

The Stone-von Neumann theorem tells us that all unitary irreducible representations of the integrated/exponentiated/Weyl form of the canonical commutation relations (CCR) algebra in finite dimensions ...
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1 vote
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Factoring in discrete Heisenberg group $H_3(\mathbb{Z})$

Let $H_3(\mathbb{Z})$ be the discrete Heisenberg group generated by $x=\begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix},\ \ y=\begin{pmatrix} 1 & 0 &...
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3 votes
0 answers
123 views

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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1 vote
0 answers
174 views

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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10 votes
1 answer
530 views

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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1 vote
0 answers
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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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-1 votes
1 answer
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Representations of the $3\times 3$ Heisenberg group [closed]

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, \...
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6 votes
1 answer
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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 ...
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6 votes
0 answers
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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 $z=x+ i y \in \mathbb C$ ...
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5 votes
3 answers
497 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 $z=x+ i y \in \mathbb C$ ...
4 votes
1 answer
194 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 $(x,y,z)$ ...
4 votes
1 answer
267 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 ...
8 votes
0 answers
381 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 ...
4 votes
1 answer
1k 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 $2$....
2 votes
1 answer
230 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 ...
6 votes
0 answers
163 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-...
3 votes
2 answers
389 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 $\...
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2 votes
0 answers
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Finite Heisenberg groups action on cohomology of line bundles

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 ...
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3 votes
0 answers
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Harmonic analysis on the Heisenberg group

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 \...
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