### 数学代写|表示论代写Representation theory代考|MATH7333

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• 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|A Base Space of the Disintegration of Induced Representations

We fix in the whole section an exponential solvable Lie group $G=\exp g$ with Lie algebra $\mathrm{g}$. Let $f$ be an element of $\mathrm{g}^{*}$ and $H=\exp \mathrm{h}$ a normal subgroup of $G$. Recall the monomial representation $\tau=\operatorname{ind}{H}^{G} \chi{f}$, which is realized by left translations in the Hilbert space $\mathscr{H}_{2}$ of continuous functions $\xi$ on $G$ that satisfy the covariance relation (3.3.1) for all $g$ in $G$ and $h$ in $H$ and are square-integrable on $G / H$ for the $G$-invariant measure. A result on the disintegration of $\tau$ was obtained earlier (cf. Theorem 1.4.2). We first recall the precise disintegration formula:

Theorem 3.3.1 Let $G=\exp g$ be an exponential solvable group and $H=\exp$ h $a$ normal subgroup of $G$. Then
$$\tau \simeq \int_{f+\mathbf{b}^{\perp} / H}^{\oplus} \pi_{l} d \mu(I)$$
where $\mu$ is the image under the Kirillov-Bernat map of a finite positive measure on $\Gamma_{f} \subset \mathrm{g}^{\star}$ equivalent to the Lebesgue measure. On the other hand, the multiplicities involved in this decomposition are identically 1 or $+\infty$, depending on whether
$$\operatorname{dim}(H \cdot l)=\operatorname{dim}\left(G \cdot l \cap \Gamma_{f}\right)$$
or not, for l generic in $\Gamma_{f}$. Equivalently, we might have
$$2 \operatorname{dim}(H \cdot l)=\operatorname{dim}(G \cdot l)$$
or not. In either case the multiplicity of $\pi_{l}$ in $\tau$ is the number of $H$-orbits in $G \cdot l \cap \Gamma_{f} \cdot$.

## 数学代写|表示论代写Representation theory代考|Construction of the Intertwining Operator

Let $\mathfrak{s}=\left(a_{j}\right){j=0}^{n}$ be a good sequence of subspaces of $\mathfrak{g}$ adapted to $h$, and extract a coexponential basis $B=\left{X{1}, \ldots, X_{r}\right}$ to $\mathfrak{h}$ in $\mathfrak{g}$. Consider also the disintegration space $V$ endowed with the Lebesgue measure $d \lambda$ as in Sect.3.3.1 and formula (3.3.5). The basis $B$ defines an invariant measure on $G / H$, which allows to fix the norm
$$|\xi|_{L^{2}(G / H, f)}=\left(\int_{G / H}|\xi(g)|^{2} d_{G, H}(g)\right)^{1 / 2}$$
on $\mathscr{H}{r}=L^{2}(G / H, f)$ of $\tau$. We now build a Zariski open set $V{0}$ of $V$ and for $\phi \in V_{0}$ the Vergne polarization $\mathfrak{b}(\phi)$ at $\phi$ relatively to the good sequence $\mathfrak{s}$ as in Theorem 1.2.4. Since this good sequence is adapted to $\mathfrak{h}$, we must have $\mathfrak{h} \subset \mathfrak{b}(\phi)$ for all $\phi \in V_{0}$. We next construct, for $\phi \in V_{0}$, a coexponential basis $X(\phi)$ to $b(\phi)$ in $\mathfrak{g}$ and a coexponential basis $Y(\phi)$ to $\mathfrak{h}$. All these bases vary continuously on $V_{0}$. For $\phi \in V^{\prime}$ and $j=1, \ldots, n$, we set
$$J_{j}(\phi)=\left{k \in{1, \ldots, j}: \mathfrak{a}{j}\left(\phi{j}\right)+\mathfrak{a}{k-1} \varsubsetneqq \mathfrak{a}{j}\left(\phi_{j}\right)+\mathfrak{a}{k}\right} .$$ The set of indices $J{j}(\phi)$ is typically not constant for $\phi \in V^{\prime}$, but its cardinality is constant and equal to $d_{j}$ for all $j=1, \ldots, n$. Note then
$$J_{j}(\phi)=\left{i_{1}(\phi)<\cdots<i_{d_{j}}(\phi)\right},$$
We endow the set $\left{J_{j}(\phi), \phi \in V^{\prime}\right}$ with the lexicographic order defined by
$$\left{i_{1}(\phi)<\cdots<i_{d_{j}}(\phi)<i_{1}\left(\phi^{\prime}\right)<\cdots<i_{d_{j}}\left(\phi^{\prime}\right)\right},$$
if there exists $\sigma \in\left{1, \ldots, d_{j}\right}$ such that $i_{1}(\phi)=i_{1}\left(\phi^{\prime}\right), \ldots, i_{\sigma-1}\left(\phi^{\prime}\right), i_{\sigma}(\phi)<$ $i_{\sigma}\left(\phi^{\prime}\right)$. Using this order, let
$$J_{j}=\min {\phi \in V^{\prime}} J{j}(\phi)=\left{i_{1}<\cdots<i_{d_{j}}\right}$$

## 数学代写|表示论代写Representation theory代考|The Inverse Operator

We now prove that formulas (3.3.13) and (3.3.14), established in the proof of the theorem on $C_{c}^{\infty}(G / H, f)$, actually hold on $L^{2}(G / H, f)$. We will resume the cases studied in the previous theorem.
In the first case, it is clear that $L^{2}(G / H, f)=L^{2}\left(G_{0} / H, f_{0}\right)$ and that
$$\int_{V} L^{2}(G / B(\phi), \phi) d \lambda(\phi)=\int_{V_{0}} L^{2}\left(\mathbb{R}, L^{2}\left(G_{0} / B\left(\phi_{0}\right), \phi_{0}\right)\right) d \lambda^{0}\left(\phi_{0}\right)$$
Hence $U=\tilde{U}{0} \circ W$, where $W: L^{2}(G / H, f) \rightarrow L^{2}\left(\mathbb{R}, L^{2}\left(G{0} / H, f\right)\right)$ is the operator field defined by
\begin{aligned} &W(\xi)(t)\left(g_{0}\right)=\xi\left(\exp (t X) \cdot g_{0}\right)=\xi_{t}\left(g_{0}\right), g_{0} \in G_{0} \ &\text { and } \tilde{U}{0}(\xi)(t)\left(g{0}\right)=U_{0}\left(\xi_{t}\right)\left(g_{0}\right) \text {. } \end{aligned}
We move to the second case, so let $\phi \in V_{0}$ and $\phi_{0}=\phi_{\mid g_{0}}$. Then $\phi=\phi_{s}=$ $\phi_{0}+s X^{\star}$ for some $s \in \mathbb{R}$. For $\eta \in C_{c}^{\infty}\left(G / B\left(\phi_{0}\right)\right.$, $\left.\phi_{0}\right)$, let $\eta^{s}$ be the function defined on $G$ by
$$\eta^{s}(g)=\int_{\mathbb{R}} \eta\left(g \exp \left(t B_{n}(\phi)\right)\right) e^{-i t s} \Delta_{B(\phi), G}^{-1 / 2}\left(\exp \left(t B_{n}(\phi)\right)\right) e^{-i t \phi_{0}}\left(Z_{0}(\phi)\right) d t, g \in G$$

## 数学代写|表示论代写Representation theory代考|A Base Space of the Disintegration of Induced Representations

τ≃∫F+b⊥/H⊕圆周率ldμ(我)

2暗淡⁡(H⋅l)=暗淡⁡(G⋅l)

## 数学代写|表示论代写Representation theory代考|Construction of the Intertwining Operator

|X|大号2(G/H,F)=(∫G/H|X(G)|2dG,H(G))1/2

J_{j}(\phi)=\left{k \in{1, \ldots, j}: \mathfrak{a}{j}\left(\phi{j}\right)+\mathfrak{a}{ k-1} \varsubsetneqq \mathfrak{a}{j}\left(\phi_{j}\right)+\mathfrak{a}{k}\right} 。J_{j}(\phi)=\left{k \in{1, \ldots, j}: \mathfrak{a}{j}\left(\phi{j}\right)+\mathfrak{a}{ k-1} \varsubsetneqq \mathfrak{a}{j}\left(\phi_{j}\right)+\mathfrak{a}{k}\right} 。索引集Ĵj(φ)通常不是恒定的φ∈在′, 但它的基数是恒定的并且等于dj对所有人j=1,…,n. 然后注意

J_{j}(\phi)=\left{i_{1}(\phi)<\cdots<i_{d_{j}}(\phi)\right}，J_{j}(\phi)=\left{i_{1}(\phi)<\cdots<i_{d_{j}}(\phi)\right}，

\left{i_{1}(\phi)<\cdots<i_{d_{j}}(\phi)<i_{1}\left(\phi^{\prime}\right)<\cdots<i_{ d_{j}}\left(\phi^{\prime}\right)\right},\left{i_{1}(\phi)<\cdots<i_{d_{j}}(\phi)<i_{1}\left(\phi^{\prime}\right)<\cdots<i_{ d_{j}}\left(\phi^{\prime}\right)\right},

J_{j}=\min {\phi \in V^{\prime}} J{j}(\phi)=\left{i_{1}<\cdots<i_{d_{j}}\right}J_{j}=\min {\phi \in V^{\prime}} J{j}(\phi)=\left{i_{1}<\cdots<i_{d_{j}}\right}

## 数学代写|表示论代写Representation theory代考|The Inverse Operator

∫在大号2(G/乙(φ),φ)dλ(φ)=∫在0大号2(R,大号2(G0/乙(φ0),φ0))dλ0(φ0)

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