### 数学代写|表示论代写Representation theory代考|MAST90017

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• Statistical Inference 统计推断
• 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|Disintegration of Monomial Representations

Let $B=\exp b$ be a connected closed subgroup of $G$ and let $\left{Y_{1}, Y_{2}, \ldots, Y_{k}\right}$ be a Jordan-Hölder basis of $\mathbf{b}$. Then the map
$$\psi: \mathbb{R}^{k} \longrightarrow B,\left(t_{1}, \ldots, t_{k}\right) \longrightarrow \exp t_{1} Y_{1} \cdots \exp t_{k} Y_{k}$$

is a diffeomorphism. We recall from Sect. 1.2.2 how to choose normalized measures $d b$ on $B$ and $G / B$ : for any $C^{\infty}$-function with compact support $\eta$ in $B$,
$$\int_{B} \eta(b) d(b)=\int_{\mathbb{R}^{d}} \eta\left(\exp \left(t_{1} Y_{1}\right) \ldots \exp \left(t_{k} Y_{k}\right)\right) d t_{1} \ldots d t_{k} .$$
This measure is left-invariant and therefore a Haar measure. On the other hand, if we choose a Malcev basis $\left{X_{1}, \ldots, X_{d}\right}$ of $\mathfrak{g}$ relative to $b$, ie
$$\mathfrak{g}=\mathbb{R} X_{1} \oplus \ldots \oplus \mathbb{R} X_{d} \oplus \mathfrak{b} \text { and } \sum_{k=j}^{d} \mathbb{R} X_{k}+\mathfrak{b}$$
is a subalgebra of $\mathfrak{g}$ for all $j$, then the measure $d_{G, B}$ on $G / B$ is defined so that
$$\int_{G / B} \tilde{a}(g) d_{G, B}(g)=\int_{\mathbb{R}^{d}} \tilde{a}\left(\exp \left(t_{1} X_{1}\right) \cdots \exp \left(t_{d} X_{d}\right)\right) d t_{1} \cdots d t_{d}$$
is left-invariant for any continuous function with compact support $\tilde{a}$ on $G / B$. By normalizing one of the vectors $X_{j}$, we have that
$$\int_{G} q(g) d g=\int_{G / B}\left(\int_{B} q(x b) d b\right) d_{G, B}(x)$$
for any continuous function with compact support $q$ on $G$. We always choose the invariant measures $d_{G, B}$ on the quotient spaces $G / B$ in such a way that this identity holds.

Let $\mathfrak{b}{1}, \mathfrak{b}{2}$ be two polarizations at the point $\phi \in \mathfrak{g}^{\star}$, and $B_{1}, B_{2}$ the two associated subgroups. Notice
$$S\left(G / B_{i}, \phi\right)=\mathscr{H}{\mathrm{ind}{B_{i}}^{G} \chi_{\phi}}^{\infty}, i \in{1,2},$$
the space of $C^{\infty}$-vectors of the representation spaces ind ${ }{B{i}}^{G} \chi_{\phi}, i \in{1,2}$, which are $\pi_{\phi, B}$ denote the representation ind $G_{B}^{G} \chi_{\phi}$. If $d_{B_{2}, B_{2} \cap B_{1}}$ denotes the $B_{2}$-left-invariant measure on $B_{2} / B_{2} \cap B_{1}$, for any function $\bar{k}$ of $S\left(G / B_{1}, \phi\right)$ the integral
$$T_{B_{2}, B_{1}} \tilde{k}(g)=\int_{B_{2} / B_{2} \cap B_{1}} \tilde{k}(g b) \chi \phi(b) d_{B_{2}, B_{2} \cap B_{1}}(b)$$
defined for every $g \in G$ is absolutely convergent and defines an isomorphism of $S\left(G / B_{1}, \phi\right)$ on $S\left(G / B_{2}, \phi\right)$ which extends by continuity into an intertwining operator between $\pi_{\phi, B_{1}}$ and $\pi_{\phi, B_{2}}$. Furthermore, if the measures on the homogeneous spaces $G / B_{1}$ and $G / B_{2}$ are suitably normalized, $T_{B_{2}, B_{1}}$ is an isometry.

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

We now specify a flag $\mathscr{A}$ of ideals of $\mathfrak{g}$. The peculiarity comes from the fact that if $\mathscr{C}=\left{Z_{1}, \ldots, Z_{n}\right}$ is the Jordan-Hölder basis of $\mathfrak{g}$ extracted from $\mathscr{A}$, then $\mathscr{C}$ contains a Jordan-Hölder basis
$$\mathscr{D}=\left{V_{1}=Z_{l_{1}}, \ldots, V_{n-r}=Z_{l_{n-r}}\right}$$
of $\mathfrak{h}$. We also extract from the latter the Malcev basis $\mathscr{B}=\left{B_{1}, \ldots, B_{r}\right}$ of $\mathfrak{g}$ relative to $h$ as above. The basis $\mathscr{C}$ gives us the index sets $I^{H}$ and $L^{H}$ and allows to choose a family $R=\left{R_{1}, \ldots, R_{r}\right}$ of real affine functions on $\mathbb{R}^{k}$ having the properties defined above for $L^{H}$. Moreover, by what we saw earlier the basis $\mathscr{B}$ gives us a $G$-invariant measure $d_{G, H}$ on $G / H$, which allows to fix the norm
$$|\xi|_{L^{2}(G / H, f)}=\left(\int_{G / H}|\xi(g)|^{2} d_{G, H}(g)\right)^{\frac{1}{2}},$$
for $\xi \in \mathscr{H}{\tau}=L^{2}(G / H, f)$. Let $\mathscr{V}=\mathscr{V} R, \mathscr{B}$. We will now construct a Zariski open set $\mathscr{V}{0}$ of $\mathscr{V}$ on which all of the following objects will be well defined. For $\phi \in \mathscr{V} 0$, we will construct a polarization $\mathfrak{b}(\phi)$ at $\phi$, a Malcev basis $\mathscr{X}^{\prime}(\phi)=\left{X_{1}(\phi), \ldots, X_{l}(\phi)\right}$ of $\mathfrak{g}$ relative to $\mathfrak{b}(\phi)$, a Jordan-Hölder basis $\mathscr{D}(\phi)=$ $\left{V_{1}(\phi), \ldots, V_{q}(\phi)\right}$ of $\mathfrak{h} \cap \mathfrak{b}(\phi)$, a Malcev basis $\mathscr{Y}(\phi)=\left{Y_{1}(\phi), \ldots, Y_{m}(\phi)\right}$ of $\mathfrak{b}(\phi)$ relative to $\mathfrak{h} \cap \mathfrak{b}(\phi)$ and finally a basis $\mathscr{U}(\phi)=\left{U_{1}(\phi), \ldots, U_{p}(\phi)\right}$ of $\mathbf{h}$ relative to $\mathfrak{h} \cap \mathfrak{b}(\phi)$. Here the numbers $l, m, p$ do not depend upon $\phi \in \mathscr{V}{0}$. In addition, all vectors $X{j}(\phi), V_{j}(\phi), Y_{j}(\phi)$ and $U_{j}(\phi)$ vary rationally and smoothly on $\phi \in \mathscr{V}_{0}$.

The vectors $\left(X_{j}(\phi)\right)$ ) will define a $G$-invariant measure on $G / B(\phi)$, and hence the norm of the space $\pi_{\phi}$. Likewise, the vectors $\left(Y_{j}(\phi)\right){j}$ determine the $B(\phi)$-invariant measure on $B(\phi) / H \cap B(\phi)$, hence the infinitesimal intertwining operator $T{B(\phi), H}$ and the vectors $\left(U_{j}(\phi)\right)_{j}$ will determine the measure on $H / H \cap B(\phi)$. All objects mentioned above are going to be constructed inductively, step by step. Step $s=0$ consists in defining the bases using certain subalgebras, set of indices and Zariski open sets of $\mathscr{V}$. We will also introduce the tools at this stage, while the objects will be given at the intermediate step $s \in{0, \ldots, \operatorname{dim}(\mathfrak{g})}$. In (3.2.3) we will explain in detail the techniques for passing from $s$ to $s+1$ and exhibit the newly constructed objects. At the very end we will show that the procedure actually stops, at some $s_{0} \in{0, \ldots, \operatorname{dim}(\mathrm{g})}$, and that the outcome bases are convenient. Note that our constructions depend only upon $8, f$ and $h$.

## 数学代写|表示论代写Representation theory代考|The Case of Exponential Solvable Groups

We study in this section exponential solvable Lie groups $G=\exp g$ for which the inducing subgroup $H=\exp \mathrm{h}$ is normal. We still consider a monomial representation $\tau=\operatorname{ind}{H}^{G} \chi{f}$, where $\chi_{f}$ denotes a character of $H$. Starting from a good sequence of subalgebras $\mathfrak{s}=\left(\mathfrak{a}{i}\right){i=0}^{n}$ passing through $\mathfrak{h}$, we determine an affine subspace $V$ of $\Gamma_{f}$ and a measure $d \lambda$ on $V$ such that
$$\tau \simeq \int_{V}^{\oplus} \pi_{\phi} d \lambda(\phi),$$
where $\pi_{\phi}$ are the irreducible representations associated to $\phi$. We next construct an explicit unitary intertwining operator and we find its inverse. The construction of such an operator $U$ goes through the following steps. We start from the good sequence 5 , we construct a coexponential basis $B$ of $\mathrm{h}$ in $\mathrm{g}$, we obtain our disintegration space $V$ with Lebesgue measure $d \lambda$. The basis $B$ defines an invariant measure on $G / H$, hence the norm on the space $\mathscr{H}{\tau}$ of $\tau$. We build next a Zariski open set $V{0}$ of $V$, and for each $\phi \in V_{0}$ the Vergne polarization $\mathfrak{b}(\phi)$ at $\phi$ relative to the good sequence $\mathfrak{5}$. These polarizations obviously contain $\mathfrak{h}$. We determine then for all $\phi \in V_{0}$ a coexponential basis $X(\phi)$ of $\mathfrak{b}(\phi)$ in $\mathfrak{g}$, which fixes a $G$ invariant positive form $v_{G, B(\phi)}$ on the space $K(G, B(\phi)):=\mathscr{E}(G, B(\phi))$ as in Sect. 1.2.2. The latter is the space of continuous numerical functions on $G$, with compact support modulo $B(\phi)=\exp (\mathfrak{b}(\phi))$ and satisfying:
$$F(g b)=\Delta_{B(\phi), G}(b) F(g)(g \in G, b \in B(\phi)),$$
Then we have a norm on the space $\mathscr{H}{\phi}$ of the irreducible representation $\pi{\phi}=$ ind $_{B(\phi)}^{G} \chi_{\phi}$. We also construct a coexponential basis $Y(\phi)$ to $\mathfrak{h}$ in $\mathfrak{b}(\phi)$ then an invariant measure $d_{B(\phi), H}$ on $B(\phi) / H$. All these bases vary continuously with $\phi \in V_{0}$ and allow to define the set
$$\mathscr{H}=\int_{V}^{\oplus} \mathscr{H}_{\phi} d \lambda(\phi)$$ of the disintegration of $\tau$. Now we associate, to each smooth function $\xi$ on $G$ with compact support modulo $H$ satisfying the generalized covariance relation (1.2.2):
$$\xi(g h)=\chi_{f}\left(h^{-1}\right) \Delta_{H, G}^{1 / 2}(h) \xi(g),(g \in G, h \in H)$$
and to each $\phi \in V_{0}$, the $C^{\infty}$-vector
$$T_{b(\phi), h} \xi(g)=\int_{B(\phi) / H} \xi(g b) \chi_{\phi}(b) \Delta_{B(\phi), G}^{-1 / 2}(b) d_{B(\phi), H}(b), \quad g \in G$$
of $\mathscr{H}_{\phi}$. We prove next that this operator is invertible. We will also examine some examples and describe a smooth disintegration of $L^{2}(G)$ for an exponential solvable Lie group $G$.

## 数学代写|表示论代写Representation theory代考|Disintegration of Monomial Representations

ψ:Rķ⟶乙,(吨1,…,吨ķ)⟶经验⁡吨1是1⋯经验⁡吨ķ是ķ

∫乙这(b)d(b)=∫Rd这(经验⁡(吨1是1)…经验⁡(吨ķ是ķ))d吨1…d吨ķ.

G=RX1⊕…⊕RXd⊕b 和 ∑ķ=jdRXķ+b

∫G/乙一个~(G)dG,乙(G)=∫Rd一个~(经验⁡(吨1X1)⋯经验⁡(吨dXd))d吨1⋯d吨d

∫Gq(G)dG=∫G/乙(∫乙q(Xb)db)dG,乙(X)

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

\mathscr{D}=\left{V_{1}=Z_{l_{1}}, \ldots, V_{nr}=Z_{l_{nr}}\right}\mathscr{D}=\left{V_{1}=Z_{l_{1}}, \ldots, V_{nr}=Z_{l_{nr}}\right}

|X|大号2(G/H,F)=(∫G/H|X(G)|2dG,H(G))12,

## 数学代写|表示论代写Representation theory代考|The Case of Exponential Solvable Groups

τ≃∫在⊕圆周率φdλ(φ),

F(Gb)=Δ乙(φ),G(b)F(G)(G∈G,b∈乙(φ)),

H=∫在⊕Hφdλ(φ)的解体τ. 现在我们关联到每个平滑函数X上G带紧凑支撑模H满足广义协方差关系（1.2.2）：

X(GH)=χF(H−1)ΔH,G1/2(H)X(G),(G∈G,H∈H)

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