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# 数学代写|表示论代写Representation Theory代考|Rational Structures and Uniform Subgroups

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## 数学代写|表示论代写Representation Theory代考|Rational Structures and Uniform Subgroups

In this subsection we present some results on discrete uniform subgroups of a nilpotent Lie group. Let $G$ be a nilpotent, connected and simply connected real Lie group and let $\mathfrak{g}$ be its Lie algebra. We say that $\mathfrak{g}$ (or $G$ ) has a rational structure if there is a Lie algebra $\mathfrak{g}{\mathbb{Q}}$ over $\mathbb{Q}$ such that $\mathfrak{g} \cong \mathfrak{g}{\mathbb{Q}} \otimes \mathbb{R}$. It is clear that $\mathfrak{g}$ has a rational structure if and only if $\mathfrak{g}$ has an $\mathbb{R}$-basis $\left{X_1, \ldots, X_n\right}$ with rational structure constants.

Let $\mathfrak{g}$ have a fixed rational structure given by $\mathfrak{g}{\mathbb{Q}}$ and let $\mathfrak{h}$ be an $\mathbb{R}$-subspace of $\mathfrak{g}$. Define $\mathfrak{h}{\mathbb{Q}}=\mathfrak{h} \cap \mathfrak{g}{\mathbb{Q}}$. We say that $\mathfrak{h}$ is rational if $\mathfrak{h}=\mathbb{R}$-span $\left{\mathfrak{h}{\mathbb{Q}}\right}$. A connected, closed subgroup $H$ of $G$ is rational if its Lie algebra $\mathfrak{h}$ is rational. The elements of $\mathfrak{g}{\mathbb{Q}}\left(\right.$ or $G{\mathbb{Q}}=\exp \mathfrak{g}_{\mathbb{Q}}$ ) are called rational elements (or rational points) of $\mathfrak{g}$ (or $G$ ).
A discrete subgroup $\Gamma$ is called uniform in $G$ if the quotient space $G / \Gamma$ is compact. The homogeneous space $G / \Gamma$ is called a compact nilmanifold. A criterion of Malcev states that $G$ admits a uniform subgroup $\Gamma$ if and only if $\mathfrak{g}$ admits a rational structure. Furthermore, if $G$ has a uniform subgroup $\Gamma$, then $\mathfrak{g}$ (hence $G$ ) has a rational structure such that $\mathfrak{g}{\mathbb{Q}}=\mathbb{Q}$-span ${\log (\Gamma)}$. Conversely, if $\mathfrak{g}$ has a rational structure given by some $\mathbb{Q}$-algebra $\mathfrak{g}{\mathbb{Q}} \subset \mathfrak{g}$, then $G$ has a uniform subgroup $\Gamma$ such that $\log (\Gamma) \subset \mathfrak{g}{\mathbb{Q}}$ (see [39] and [126]). If $G$ is endowed with the rational structure induced by a uniform subgroup $\Gamma$ and if $H$ is a Lie subgroup of $G$, then $H$ is rational if and only if $H \cap \Gamma$ is a uniform subgroup of $H$. Note that the notion of rationality depends on $\Gamma$. A real linear functional $f \in \mathfrak{g}^$ is called rational $\left(f \in \mathfrak{g}{\mathbb{Q}}^, \mathfrak{g}{\mathbb{Q}}=\mathbb{Q}\right.$-span $\left.(\log (\Gamma))\right)$ if $\left\langle f, \mathfrak{g}{\mathbb{Q}}\right\rangle \subset \mathbb{Q}$, or equivalently $\langle f, \log (\Gamma)\rangle \subset \mathbb{Q}$.
Let $\Gamma$ be a uniform subgroup of $G$. A strong Malcev basis $\left{X_1, \ldots, X_n\right}$ for $\mathfrak{g}$ is said to be strongly based on $\Gamma$ if
$$\Gamma=\exp \mathbb{Z} X_1 \cdots \exp \mathbb{Z} X_n$$

## 数学代写|表示论代写Representation Theory代考|Fundamental Domains for Uniform Subgroups

Let $\Gamma$ be a discrete uniform subgroup of a connected, simply connected nilpotent Lie group $G$, and let $\mathscr{B}=\left(X_1, \ldots, X_n\right)$ be a strong Malcev basis for the Lie algebra $\mathfrak{g}$ of $G$ strongly based on $\Gamma$. Define the mapping $\mathcal{E}{\mathscr{B}}: \mathbb{R}^n \longrightarrow G$ by $$\mathcal{E}{\mathscr{B}}(T)=\exp t_n X_n \cdots \exp t_1 X_1$$
where $T=\left(t_1, \ldots, t_n\right) \in \mathbb{R}^n$. It is well known that $\mathcal{E}{\mathscr{B}}$ is a diffeomorphism ([39]). Let $$\mathbb{I}=[0,1)={t \in \mathbb{R}: 0 \leq t<1}$$ and let $$\Omega=\mathcal{E}{\mathscr{B}}\left(\mathbb{I}^n\right)$$
Then $\Omega$ is a fundamental domain for $\Gamma$ in $G$ ([35, Lemma 3.6]), and the mapping $\mathcal{E}{\mathscr{B}}$ maps the Lebesgue measure $d t$ on $\mathbb{I}^n$ to the $G$-invariant probability measure $v$ on $G / \Gamma$, that is, for $\varphi$ in $\mathscr{C}(G / \Gamma)$, one has $$\int{G / \Gamma} \varphi(\dot{g}) d v(\dot{g})=\int_{\mathbb{I}^n} \varphi\left(\mathcal{E}{\mathscr{B}}(t)\right) d t$$ Furthermore, for $\phi \in \mathscr{C}_c(G)$ one has $$\int_G \phi(g) d \nu(g)=\sum{s \in \mathbb{Z}^n} \int_{\mathbb{I}^n} \phi\left(\mathcal{E}{\mathscr{B}}(t) \mathcal{E}{\mathscr{B}}(s)\right) d t$$

## 数学代写|表示论代写Representation Theory代考|Rational Structures and Uniform Subgroups

$$\Gamma=\exp \mathbb{Z} X_1 \cdots \exp \mathbb{Z} X_n$$

## 数学代写|表示论代写Representation Theory代考|Fundamental Domains for Uniform Subgroups

$$\mathcal{E} \mathscr{B}(T)=\exp t_n X_n \cdots \exp t_1 X_1$$

$$\mathbb{I}=[0,1)=t \in \mathbb{R}: 0 \leq t<1$$

$$\Omega=\mathcal{E} \mathscr{B}\left(\mathbb{I}^n\right)$$

$$\int G / \Gamma \varphi(\dot{g}) d v(\dot{g})=\int_{\mathbb{I}^n} \varphi(\mathcal{E} \mathscr{B}(t)) d t$$

## MATLAB代写

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