### 数学代写|表示论代写Representation theory代考|MTH4107

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## 数学代写|表示论代写Representation theory代考|The Key Point Is the Convergence

We keep all previous notation. In this last section we shall see that the key point is the convergence of the integral
$$\left(I_{h_{2} h_{1}}^{G} \varphi\right)(g):=\left(I_{h_{2} h_{1}} \varphi\right)(g)=\oint_{H_{2} /\left(H_{1} \cap H_{2}\right)} \varphi(g h) \chi_{f}(h) \Delta_{H_{2}, G}^{-1 / 2}(h) d v(h)(g \in G)$$
At least formally, it is clear that the function $I_{\mathfrak{h}{2} \mathfrak{h}{1}} \varphi$ satisfies the necessary covariance condition for belonging in the space $\mathscr{H}{\pi{2}}$, and also that $I_{\mathfrak{h}{2} \mathfrak{h}{1}}$ commutes with left translations. We can therefore assert that the convergence of the integral is one major issue to deal with. We call elements of $I(f, \mathfrak{g})$ Pukanszky polarizations.
In this section we look at all pairs $\left(\mathfrak{h}{1}, \mathfrak{h}{2}\right)$ of Pukanszky polarizations at $f$ which meet the following convergence property:
$(C P):$ The integral
$$\oint_{H_{2} /\left(H_{1} \cap H_{2}\right)}|\varphi(g h)| \Delta_{H_{2}, G}^{-\frac{1}{2}}(h) d v(h), g \in G$$
converges on a $G$-invariant and $\mathscr{U}(\mathrm{g})$-invariant dense subspace $\mathscr{H}$ of $\mathscr{H}{\pi{1}}^{\infty}$.
Proposition 2.8.1 Suppose that the pair of Pukanszky polarizations $\left(\mathfrak{h}{1}, \mathbf{h}{2}\right)$ satisfies property $(C P)$. Then for any function $\varphi \in \mathscr{K}\left(f, \mathfrak{h}{1}, G\right)$ we have that $$\oint{H_{2} /\left(H_{1} \cap H_{2}\right)}|\varphi(g h)| \Delta_{H_{2}, G}^{-\frac{1}{2}}(h) d v(h)<\infty, g \in G$$
and the function $g \mapsto\left(I_{b_{2} h_{1}} \varphi\right)(g)$ is continuous.

## 数学代写|表示论代写Representation theory代考|Intertwining Operators of Induced Representations

Let $G$ be a connected and simply connected nilpotent Lie group with Lie algebra g. Consider, as in (1.6.1), the monomial representation $\tau=$ ind $_{H}^{G} \chi$ induced by the unitary character $\chi$ of an analytic subgroup $H$ of $G$. Recall that $h$ is the Lie subalgebra corresponding to $H$, and $\chi$ is written in the form $\chi(\exp X)=$ $e^{i\langle f, X\rangle}\left(X \in\right.$ h) with $f \in \mathfrak{g}^{\star}$. Let $\Gamma_{f}=f+\mathfrak{h}^{\perp}$ be as in Eq. (1.4.8). In parallel to decomposition formula (1.6.2), the disintegration of $\tau$ into irreducibles reads:
$$\tau \simeq \int_{\Gamma_{f} / H}^{\oplus} \pi_{l} d l$$

where $d l$ is some natural measure on the space $\Gamma_{f} / H$ of $H$-orbits (cf. Theorem 1.4.2).

We construct in this section an intertwining operator of decomposition (3.2.1) for an arbitrary subgroup $H$. The idea is to make formula (3.2.1) explicit through the construction of certain affine subspaces
$$\mathscr{V}^{R, \mathfrak{B}}=\left{\sum_{i=1}^{r} R_{i}(T) f_{i} ; T \in \mathbb{R}^{k}\right}$$
of $\Gamma_{f}$, and a measure $d \lambda=d \lambda^{R, \mathfrak{B}}$ on $\mathscr{V} R, \mathfrak{B}$, so that
$$\tau \simeq \int_{\mathscr{V} R, \mathfrak{g}}^{\oplus} \pi_{\phi} d \lambda^{R, \mathfrak{B}}(\phi) .$$
Here $R$ denotes a family $\left{R_{1}, \ldots, R_{r}\right}$ of affine functions defined on $\mathbb{R}^{k}$, for some non-negative integer $k \in \mathbb{N}$, and $\mathfrak{B}^{\star}=\left{f_{r}, \ldots, f_{1}\right}$ denotes the dual of a Malcev basis $\mathfrak{B}$ relative to $\mathfrak{h}$. Such a basis defines an invariant measure on $G / H$ and so the norm on the space $\mathscr{H}{\mathrm{r}}$ of $\tau$. We then determine a Zariski open set $\mathscr{V}{0}$ of $\mathscr{V}^{R, \mathfrak{B}}$, a polarization $\mathfrak{b}(\phi)$ in $\phi$ for each $\phi \in \mathscr{V} 0$, a Malcev basis $\mathscr{X}(\phi)$ of $\mathfrak{b}(\phi)$, a Malcev basis $\mathscr{Y}(\phi)$ of $\mathfrak{b}(\phi)$ relative to $\mathfrak{h} \cap \mathfrak{b}(\phi)$ and a Malcev basis $\mathscr{U}$ of $\mathfrak{h}$ relative to $\mathfrak{h} \cap \mathfrak{b}(\phi)$. All these bases vary continuously in $\phi \in \mathcal{V}{0}$, which allows to fix the invariant measure $d{G, B(\phi)}$ on $G / B(\phi)$, where $B(\phi)=\exp (\mathfrak{b}(\phi)$ ) (and hence the norm on the space $\mathscr{H}{\phi}$ of the irreducible representation $\pi{\phi}=$ ind $\left.{B(\phi)}^{G} \chi{\phi}\right), d_{B(\phi), B(\phi) \cap H}$ on $B(\phi) / B(\phi) \cap H$, and $d_{H, H \cap B(\phi)}$ on $H / B(\phi) \cap H$. This permits to define the disintegration space
$$\mathscr{H}=\int_{\mathscr{y},, \mathfrak{B}}^{\oplus} \mathscr{H}{\phi} d \lambda(\phi)$$ of $\tau$. We now associate, to any sufficiently regular function $\xi$ of $\mathscr{H}{\tau}$ and any $\phi \in \mathscr{V}{0}$, the $C^{\infty}$-vector of $\mathscr{H}{\phi}$ :
$$T_{B(\phi), H} \xi(g)=\int_{B(\phi) / B(\phi) \cap H} \xi(g b) \chi_{\phi}(b) d_{B(\phi), B(\phi) \cap H(b), g \in G}$$
We will show that
$$\int_{\mathscr{Y}{0}}\left|T{B(\phi), H} \xi\right|_{\mathscr{H}{\phi}}^{2} d \phi=|\xi|{\mathscr{H}_{\mathrm{T}}}^{2}$$
therefore obtaining an isometric intertwining operator of (3.2.2). A (smooth) disintegration of $L^{2}(G)$ is also generated.

## 数学代写|表示论代写Representation theory代考|Notation and Backgrounds

Let $G$ be a connected and simply connected nilpotent Lie group with Lie algebra g. Let exp denote, as earlier, the exponential map, so that $G=\exp g$. Let $V, W$ be real vector spaces of finite dimension with $W \subset V$. We denote by $V^{}$ the dual vector space of $V$ and by $W^{\perp}$ the orthogonal to $W$ in $V^{}$. If $u_{1}, \ldots, u_{p},(p \in \mathbb{N})$ indicate linearly independent vectors in $V$, we denote by $\mathbb{R}-\operatorname{span}\left(u_{1}, \ldots, u_{p}\right)$ the vector subspace of $V$ they span, and we say the basis $\left{u_{1}, \ldots, u_{p}\right}$ generates this space. Given $l \in \mathrm{g}^{*}$ and $X \in \mathfrak{g}$, we denote by $\langle l, X\rangle$ the image of $X$ under $l$. Recall that the kernel of the bilinear form $B_{l}$, defined in Sect. 1.2.4 by $B_{l}(X, Y)=$ $\langle l,[X, Y]\rangle$, is denoted by $\mathrm{g}(l)=\left{X \in \mathfrak{g} ; B_{l}(X, Y)=0\right.$ for all $\left.Y \in \mathfrak{g}\right}$. The largest ideal contained in $\mathrm{g}(I)$ will be denoted by a $(l)$. Clearly
$$\mathrm{a}(l)=\bigcap_{\phi \in G-l} \mathrm{~g}(\phi)$$
Let $\mathfrak{h} \in S(l, \mathfrak{g})$ and let $\chi_{l}$ be the unitary character of the analytic subgroup $H=$ exp $h$ associated to $l$ by
$$\chi_{l}(\exp X)=e^{-i\langle l, X\rangle}$$
for $X \in 6$
Let $V$ be a real vector space of finite dimension and $\rho: G \longrightarrow \operatorname{End}(V)$ a unipotent action of $G$ on $V$. We designate by
$$(0)=V_{n} \subset V_{n-1} \subset \ldots \subset V_{1} \subset V_{0}=V$$
a Jordan-Hölder sequence for $G$ and call $\mathscr{L}=\left{v_{1}, \ldots, v_{n}\right}$ an associated JordanHölder basis $\left(v_{j} \in V_{j-1} \backslash V_{j}\right.$ ). If $v \in V$, we write $\rho(x) v=x \cdot v$ for all $x \in G$ and
$$X \cdot v=\left.\frac{d}{d t}{\rho(\exp (t X)) \cdot v}\right|{t=0},(X \in \mathfrak{g})$$ The set of indices $e^{\mathscr{L}}(v)=\left{i{1}<\ldots<i_{d}\right}$ of an element $v$ in $V$ relative to $\mathscr{L}$ is the following subset of ${1, \ldots, n}$ :
$$i \in e^{\mathscr{L}}(v) \Leftrightarrow \exists X \in \mathfrak{g}: X \cdot v \in V_{i-1} \backslash V_{i}$$
The cardinality $d$ of $e^{\mathscr{L}}(v)$ is the dimension of the $G$-orbit $\Omega$ of $v$ and does not vary with $v$ in $\Omega$.

## 数学代写|表示论代写Representation theory代考|The Key Point Is the Convergence

(我H2H1G披)(G):=(我H2H1披)(G)=∮H2/(H1∩H2)披(GH)χF(H)ΔH2,G−1/2(H)d在(H)(G∈G)

(C磷):积分

∮H2/(H1∩H2)|披(GH)|ΔH2,G−12(H)d在(H),G∈G

∮H2/(H1∩H2)|披(GH)|ΔH2,G−12(H)d在(H)<∞,G∈G

## 数学代写|表示论代写Representation theory代考|Intertwining Operators of Induced Representations

τ≃∫ΓF/H⊕圆周率ldl

\mathscr{V}^{R, \mathfrak{B}}=\left{\sum_{i=1}^{r} R_{i}(T) f_{i} ; T \in \mathbb{R}^{k}\right}\mathscr{V}^{R, \mathfrak{B}}=\left{\sum_{i=1}^{r} R_{i}(T) f_{i} ; T \in \mathbb{R}^{k}\right}

τ≃∫在R,G⊕圆周率φdλR,乙(φ).

H=∫是,,乙⊕Hφdλ(φ)的τ. 我们现在将任何足够规则的函数联系起来X的Hτ和任何φ∈在0， 这C∞-向量Hφ :

∫是0|吨乙(φ),HX|Hφ2dφ=|X|H吨2

## 数学代写|表示论代写Representation theory代考|Notation and Backgrounds

χl(经验⁡X)=和−一世⟨l,X⟩

(0)=在n⊂在n−1⊂…⊂在1⊂在0=在
Jordan-Hölder 序列为G并打电话\mathscr{L}=\left{v_{1}, \ldots, v_{n}\right}\mathscr{L}=\left{v_{1}, \ldots, v_{n}\right}相关的 JordanHölder 基(在j∈在j−1∖在j）。如果在∈在， 我们写ρ(X)在=X⋅在对所有人X∈G和

X⋅在=dd吨ρ(经验⁡(吨X))⋅在|吨=0,(X∈G)索引集e^{\mathscr{L}}(v)=\left{i{1}<\ldots<i_{d}\right}e^{\mathscr{L}}(v)=\left{i{1}<\ldots<i_{d}\right}一个元素的在在在关系到大号是以下子集1,…,n :

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