### 数学代写|表示论代写Representation theory代考|Branching Laws and the Multiplicity Function

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|表示论代写Representation theory代考|Generalities and Notations

We begin this section by reviewing some facts about induced and restricted representations of a solvable Lie group. One says that $G$ is an exponential solvable Lie group if the exponential mapping exp : $\mathfrak{g} \longrightarrow G$ is a diffeomorphism, where $\mathfrak{g}$ designates the Lie algebra of $G$. Let $\mathfrak{g}^{}$ be the dual space of $\mathfrak{g}$. The Lie algebra $\mathfrak{g}$ acts on $\mathfrak{g}$ by the adjoint representation $\mathrm{ad}{\mathfrak{g}}$, that is $$\operatorname{ad}{\mathfrak{g}}(X)(Y)=\operatorname{ad}(X)(Y)=[X, Y], X, Y \in \mathfrak{g}$$
The group $G$ acts on $g$ by the adjoint representation Ad $_{\mathfrak{g}}$, that is
$$\operatorname{Ad}{g}(g) Y=\operatorname{Ad}(g) Y=e^{\operatorname{ad}(X)} Y, g \in G, Y \in \mathfrak{g}$$ and on $\mathrm{g}^{}$ by the coadjoint representation $\mathrm{Ad}{\mathrm{g}}^{}$, i.e. $$\operatorname{Ad}_{\mathfrak{g}}^{}(g) l(X)=g \cdot l(X)=l\left(\operatorname{Ad}\left(g^{-1}\right) X\right), \quad g \in G, l \in \mathfrak{g}^{} X \in \mathfrak{g}$$ The coadjoint orbit $G \cdot l$ (or $G l$ for short) of $l$ is the set $$G \cdot l={g \cdot l, g \in G}$$ The space of coadjoint orbits is denoted by $\mathrm{g}^{} / G$.

## 数学代写|表示论代写Representation theory代考|Modular Functions and Quotient Measures

Let $G$ be an exponential solvable Lie group with Lie algebra $g$. Let $d g$ be a left Haar measure of $G$ and $\Delta_{G}$ the modular function of $G$, which is defined by the relation
$$\int_{G} F\left(g x^{-1}\right) d g=\Delta_{G}(x) \int_{G} F(g) d g(x \in G)$$
for any $F$ in the set $C_{c}(G)$ of continuous functions on $G$ with compact support. It is well known that
$$\Delta_{G}(x)=|\operatorname{det} \operatorname{Ad} x|^{-1}(x \in G)$$
Define for a subgroup $K$ in $G$ the function $\Delta_{K, G}(k)$ on $K$ by
$$\Delta_{K, G}(k)=\frac{\Delta_{K}(k)}{\Delta_{G}(k)},(k \in K) .$$
Let $\mathscr{E}(G, K)$ be the space of continuous functions $\xi$ on $G$ with compact support modulo $K$ which satisfy
$$\xi(g k)=\Delta_{K, G}(k) \xi(g) \quad(g \in G, k \in K) .$$
Then $G$ acts on $\mathscr{E}(G, K)$ by left translations. It is known [24] that, up to a positive scalar factor, there exists a unique $G$-invariant positive linear form on $E(G, K)$. We denote it by $\mu_{G, K}$ and write it as
$$\mu_{G, K}(\xi)=\oint_{G / K} \xi(g) \mu_{G, K}(g)$$
If $\Delta_{K}=\Delta_{G}$ on $K$ (this is for example the case if $K$ is a normal subgroup of $G$ ), then $\mu_{G, K}$ turns out to be a $G$-invariant measure on the homogeneous space $G / K$.

## 数学代写|表示论代写Representation theory代考|Branching Laws: Induced Representations

Let $p_{\mathfrak{h}}: \mathfrak{g}^{} \rightarrow \mathfrak{h}^{}$ be the canonical projection of $\mathfrak{h}$, and $\Omega_{\sigma}^{H}$ the coadjoint orbit associated to the representation $\sigma$ by the Kirillov-Bernat map $\Theta_{H}: \mathbf{h}^{} \rightarrow \hat{H}$, (where $\hat{H}$ is the unitary dual of $H$ ). The natural measure on $p_{h}^{-1}\left(\Omega_{\sigma}^{H}\right)$ is the fiber measure that has $H$-invariant measure on the base $\Omega_{\sigma}^{H}$ and Lebesgue measure on the affine fiber $\mathfrak{h}^{\perp}$. The representation $\tau(\sigma)$ obeys the orbital spectrum formula (cf. $[63])$ $$\tau(\sigma) \simeq \int_{G \cdot p_{h}^{-1}\left(\Omega_{\sigma}^{H}\right) / G}^{\oplus} n_{\phi}^{\sigma} \pi_{\phi} d v_{G, H}^{\sigma}(\phi)$$ where $\nu_{G, H}^{\sigma}$ is the push-forward of the natural measure on $p_{\mathrm{b}}^{-1}\left(\Omega_{\sigma}^{H}\right) \subset \mathrm{g}^{}$ under the mapping $p_{\mathrm{h}}^{-1}\left(\Omega_{\sigma}^{H}\right) \rightarrow G \cdot p_{\mathrm{h}}^{-1}\left(\Omega_{\sigma}^{H}\right) / G$, and the value of the multiplicity function $n_{\phi}^{\sigma}$ is the number of $H$-orbits in $p_{\mathrm{h}}^{-1}\left(\Omega_{\sigma}^{H}\right) \cap G \cdot \phi$. We denote this number by $#\left[p_{h}^{-1}\left(\Omega_{\sigma}^{H}\right) \cap G \cdot \phi / H\right]$ (the symbol #E denotes the cardinality of the set $E$ ).

If $G$ is a completely solvable Lie group , the multiplicity function $\phi \longmapsto n_{\phi}^{\sigma}$ is either uniformly infinite or uniformly finite and bounded. In particular, the multiplicities are finite if and only if
$$\operatorname{dim} G \cdot \phi-2 \operatorname{dim} H \cdot \phi+\operatorname{dim} \Omega_{\sigma}^{H}=0$$
for a generic $\phi \in p_{h}^{-1}\left(\Omega_{\sigma}^{H}\right)(\operatorname{see}[104])$

## 数学代写|表示论代写Representation theory代考|Generalities and Notations

G⋅l=G⋅l,G∈G共伴轨道的空间表示为G/G.

## 数学代写|表示论代写Representation theory代考|Modular Functions and Quotient Measures

∫GF(GX−1)dG=ΔG(X)∫GF(G)dG(X∈G)

ΔG(X)=|这⁡广告⁡X|−1(X∈G)

Δķ,G(ķ)=Δķ(ķ)ΔG(ķ),(ķ∈ķ).

X(Gķ)=Δķ,G(ķ)X(G)(G∈G,ķ∈ķ).

μG,ķ(X)=∮G/ķX(G)μG,ķ(G)

## 数学代写|表示论代写Representation theory代考|Branching Laws: Induced Representations

τ(σ)≃∫G⋅pH−1(ΩσH)/G⊕nφσ圆周率φd在G,Hσ(φ)在哪里νG,Hσ是自然措施的推进pb−1(ΩσH)⊂G在映射下pH−1(ΩσH)→G⋅pH−1(ΩσH)/G，以及多重性函数的值nφσ是数量H- 轨道pH−1(ΩσH)∩G⋅φ. 我们用这个数字表示#\left[p_{h}^{-1}\left(\Omega_{\sigma}^{H}\right) \cap G \cdot \phi / H\right]#\left[p_{h}^{-1}\left(\Omega_{\sigma}^{H}\right) \cap G \cdot \phi / H\right]（符号#E 表示集合的基数和 ).

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