### 数学代写|表示论代写Representation theory代考|MATHS735

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## 数学代写|表示论代写Representation theory代考|Intertwining Operators of the Restriction

We return to the setting of a simply connected nilpotent Lie group $G=\exp g$ and a closed connected subgroup $H=\exp \mathrm{h}$. Given an irreducible unitary representation $\pi$ of $G$, we present an explicit disintegration of the restriction $\pi_{\mid H}$ of $\pi$ to $H$, which is based on a precise description of the space of double cosets $H \backslash G / B$, where $B$ is any closed connected subgroup of $G$, and the well-known smooth disintegration of monomial representations of nilpotent Lie groups. The aim is still to write down a smooth intertwining operator for the decomposition of $\pi_{\mid H}$ into irreducibles. As an application we produce a concrete disintegration of tensor products of irreducible representations of $G$ and a criterium for the irreducibility of these representations.
One should first study the general problem of describing a concrete disintegration of the restriction of an irreducible unitary representation of $G$ to a closed connected subgroup $H=\exp$ b. Since by Kirillov’s theory every $\pi \in \hat{G}$ is of the form $\pi=\pi_{l, \mathfrak{b}}=$ ind $_{B}^{G} \chi_{l}$, where $l \in \mathrm{g}^{}$ and $\mathfrak{b} \subset \mathfrak{g}$ is a polarization at $l$, it is known from Mackey [124] that the restriction of $\pi$ to $H$ disintegrates over the set of double cosets $H \backslash G / B$ and that the integrands are of the form ind ${ }{B(x)}^{H} \chi{I(x)}$, where $B(x)=H \cap \psi(x) B \psi(x)^{-1}, x \in H \backslash G / B$ and $l(x)=\operatorname{Ad}^{}(\psi(x)) l_{\mid \mathfrak{h}}$, $x \in H \backslash G / B$ and $\psi: H \backslash G / B \rightarrow G$ is a section for the double cosets. The idea is to describe an open dense subset of $H \backslash G / B$ and a section $\psi$ which give us an explicit description of $\pi_{\mid H}$ in term of an integral over $H \backslash G / B$ of the representations ind ${ }{B(x)}^{H} \chi{l(x)}$ (see Proposition 3.5.27). The results concerning the explicit disintegration of monomial representations are used to obtain a concrete disintegration of the restriction. This is somehow needed to get an “abstract” disintegration of the restriction into irreducibles to connected closed subgroups of simply connected nilpotent Lie groups. ‘Abstract’ here means that the measure class in $\hat{G}$ for the disintegration of the restriction and the multiplicities of the irreducibles appearing in the disintegration are given. These constructions will be applied to the disintegration of the tensor product of two irreducible representations $\pi$ and $\pi^{\prime}$.

## 数学代写|表示论代写Representation theory代考|Double-Coset Space

Let $\mathfrak{g}$ be a nilpotent Lie algebra, $\mathfrak{b}$ any subalgebra, $B \subset G$ their simply connected Lie groups. Recall that the exponential mapping exp : $\mathfrak{g} \rightarrow G$ is a diffeomorphism. Given a sequence of ideals
$$\mathfrak{g}{n+1}:={0} \varsubsetneqq \ldots \varsubsetneqq \mathfrak{g}{i} \varsubsetneqq \ldots \varsubsetneqq \mathfrak{g}{1}=\mathfrak{g}, \operatorname{dim}\left(\mathfrak{g}{i} / \mathfrak{g}_{i+1}\right)=1,$$ denote for every $i=1, \ldots, n, G_{i}:=\exp \mathfrak{g}{i}$ and choose a vector $Z{i} \in \mathfrak{g}{i} \backslash \mathfrak{g}{i+1}$, so that $\mathfrak{g}{i}=\mathbb{R}$-span $\left(Z{i}, \ldots, Z_{n}\right)$. One obtains in this way a Jordan-Hölder basis $\mathscr{Z}:=\left(Z_{1}, \ldots, Z_{n}\right)$ of $\mathfrak{g}$. To simplify the notations, let
$$V_{1} \cdot V_{2} \cdots V_{k}:=\exp \left(V_{1}\right) \cdot \exp \left(V_{2}\right) \cdots \exp \left(V_{k}\right) \in G$$
for given vectors $V_{1}, \ldots, V_{k} \in \mathfrak{g}$. Denote as before $d g$ the Haar measure on $G$. Using the basis $\mathscr{Z}$, one can express $d g$ in the following way.
$$\int_{G} f(g) d g=\int_{\mathbb{R}^{n}} f\left(z_{1} Z_{1} \cdots z_{n} Z_{n}\right) d z,\left(f \in L^{1}(G)\right) .$$
Since $G$ is nilpotent, the quotient space $G / B$ has a $G$-invariant measure which is unique up to a positive scalar multiple. This measure (denoted by $d \dot{g}$ ) is described in Chap. 1, Sect. 1.2.2. Let us recall such construction. Let
$$\mathscr{I}^{8 / b}=\left{k \in{1, \ldots, n}, Z_{k} \notin \mathfrak{b}+\mathfrak{g}{k+1}\right}=:\left{k{1}<\ldots<k_{p}\right} .$$
One obtains the sequence of subalgebras
$$\mathfrak{b}^{p+1}:=\mathfrak{b} \varsubsetneqq \ldots \varsubsetneqq \mathfrak{b}^{j}:=\mathbb{R} Z_{k_{j}} \oplus \mathfrak{b}^{j-1} \varsubsetneqq \ldots \ldots \mathfrak{b}^{1}=\mathfrak{g}$$
and the Malcev basis $\mathscr{M}:=\left(Z_{k_{1}}, \ldots, Z_{k_{p}}\right)$ of $\mathfrak{g}$ relative to $\mathfrak{b}$. The invariant measure $d \dot{g}$ is then given for $\varphi \in \mathscr{C}{c}(G / B)$ by: $$\mu{\mathcal{M}}(\varphi)=\mu_{\mathrm{g} / \mathrm{b}}(\varphi)=\int_{G / B} \varphi(\dot{g}) d \dot{g}:=\int_{\mathbb{R}^{p}} \varphi\left(w_{1} Z_{k_{1}} \cdots w_{p} Z_{k_{p}} \cdot B\right) d w$$
where $\mathscr{C}{c}(G / B)$ denotes the space of complex-valued continuous functions with compact support on $G / B$. This is a consequence of the fact that the mapping \begin{aligned} E{\mathscr{M}}^{}: \mathbb{R}^{p} & \longrightarrow \ w=\left(w_{1}, \ldots, w_{p}\right) & \longmapsto w_{1} Z_{k_{1}} \cdots w_{p} Z_{k_{p}} \cdot B=: E_{\mathscr{A}}^{}(w) \end{aligned}
is a diffeomorphism. If $\mathfrak{h}$ is another subalgebra of $\mathfrak{g}$, then denote by $\mathbb{I}^{\mathfrak{h}} \subset{1, \ldots, n}$ the index set
$$\mathbb{I}^{\mathfrak{h}}:=\left{i \in{1, \ldots, n} ; \mathfrak{h}+\mathfrak{g}{i+1}=\mathfrak{h}+\mathfrak{g}{i}\right}={1, \ldots, n} \backslash \mathbb{I}^{\mathfrak{g} / \mathfrak{h}}$$
One can then assume that the vectors $Z_{i}, i \in \mathbb{I}^{\mathfrak{h}}$, lie in $\mathfrak{h}$ so that $\mathfrak{h}=\mathbb{R}$-span $\left{Z_{i}, i \in\right.$ $\left.\mathbb{I}^{\mathrm{h}}\right}$

## 数学代写|表示论代写Representation theory代考|The Set of Double Cosets

For $g \in G$, denote by $\bar{g}$ its double coset $H \cdot g \cdot B={h g b,(h, b) \in H \times B}$. The aim is to find an open dense subset of $H \backslash G / B$ which will support the measure $d \gamma(\bar{g})$ and which is diffeomorphic to a Zariski open subset of $\mathbb{R}^{d}$ for some $d \in \mathbb{N}^{*}$. The following example illustrates this fact:

Example 3.5.1 Let $\mathfrak{g}$ be the 7-dimensional Lie algebra spanned by the JordanHölder basis $\mathscr{Z}=\left(Z_{1}, \ldots, Z_{7}\right)$, equipped with the brackets
$$\left[Z_{1}, Z_{4}\right]=Z_{6}, \quad\left[Z_{1}, Z_{5}\right]=Z_{7}, \quad\left[Z_{2}, Z_{3}\right]=Z_{7}$$
Consider its Abelian subalgebras $h=\mathbb{R}-\operatorname{span}\left(Z_{4}, Z_{5}, Z_{7}\right)$ and $\mathfrak{b}=\mathbb{R}-\operatorname{span}\left(Z_{3}, Z_{4}\right.$, $Z_{7}$. Since many products commute, the element $g=: z_{1} Z_{1} \cdots z_{7} Z_{7} \in G$, $\left(z_{1}, \ldots, z_{7}\right) \in \mathbb{R}^{7}$, can be described in the following way:
\begin{aligned} g &=\left(z_{1} Z_{1} \cdot z_{5} Z_{5}\right) \cdot z_{2} Z_{2} \cdot z_{6} Z_{6} \cdot\left(z_{3} Z_{3} \cdot z_{4} Z_{4} \cdot z_{7} Z_{7}\right) \ &=\left(z_{1} z_{5} Z_{7} \cdot z_{5} Z_{5} \cdot z_{1} Z_{1}\right) \cdot z_{2} Z_{2} \cdot z_{6} Z_{6} \cdot\left(z_{3} Z_{3} \cdot z_{4} Z_{4} \cdot z_{7} Z_{7}\right) \ & \in H \cdot z_{1} Z_{1} \cdot z_{2} Z_{2} \cdot z_{6} Z_{6} \cdot B \end{aligned}
This implies that $\bar{g}=H \cdot g \cdot B=H \cdot z_{1} Z_{1} \cdot z_{2} Z_{2} \cdot z_{6} Z_{6} \cdot B$. On the other hand if $z_{1} \neq 0$, the element $z_{1} Z_{1} \cdot z_{2} Z_{2} \cdot z_{6} Z_{6}$ can also be written as $z_{1} Z_{1} \cdot z_{2} Z_{2}$ conjugated by $\exp \left(-\frac{26}{z_{1}}\right) Z_{4}$, which is contained in $H \cap B$. Hence if $z 1 \neq 0$, then $\bar{g}=H \cdot z_{1} Z_{1} \cdot z_{2} Z_{2} \cdot B$. As a conclusion,
$H \backslash G / B=\left{H \cdot z_{1} Z_{1} \cdot z_{2} Z_{2} \cdot B,\left(z_{1}, z_{2}\right) \in \mathscr{V}\right} \dot{\cup}\left{H \cdot z_{2} Z_{2} \cdot z_{6} Z_{6} \cdot B,\left(z_{2}, z_{6}\right) \in \mathbb{R}^{2}\right}$
$=\quad \mathbf{p}\left(G \backslash G_{2}\right) \quad \dot{u} \quad \mathbf{p}\left(G_{2}\right)$
where $\mathbf{p}: g \longmapsto \tilde{g}$, is the canonical projection of $G$ on $H \backslash G / B$, and $\mathscr{V}:=$ $\mathbb{R}^{\star} \times \mathbb{R}$. Hence the space $H \backslash G / B$ is the disjoint union of two subsets, the first is the projection of a Zariski open subset of $G$ and the second of a Zariski closed subset. The measure $d \gamma(\bar{g})$ is shown to be supported on the first set.

## 数学代写|表示论代写Representation theory代考|Double-Coset Space

Gn+1:=0⫋…⫋G一世⫋…⫋G1=G,暗淡⁡(G一世/G一世+1)=1,表示每个一世=1,…,n,G一世:=经验⁡G一世并选择一个向量从一世∈G一世∖G一世+1， 以便G一世=R-跨度(从一世,…,从n). 以这种方式获得 Jordan-Hölder 基从:=(从1,…,从n)的G. 为了简化符号，让

∫GF(G)dG=∫RnF(和1从1⋯和n从n)d和,(F∈大号1(G)).

\mathscr{I}^{8 / b}=\left{k \in{1, \ldots, n}, Z_{k} \notin \mathfrak{b}+\mathfrak{g}{k+1}\右}=:\left{k{1}<\ldots<k_{p}\right} 。\mathscr{I}^{8 / b}=\left{k \in{1, \ldots, n}, Z_{k} \notin \mathfrak{b}+\mathfrak{g}{k+1}\右}=:\left{k{1}<\ldots<k_{p}\right} 。

bp+1:=b⫋…⫋bj:=R从ķj⊕bj−1⫋……b1=G

μ米(披)=μG/b(披)=∫G/乙披(G˙)dG˙:=∫Rp披(在1从ķ1⋯在p从ķp⋅乙)d在

\mathbb{I}^{\mathfrak{h}}:=\left{i \in{1, \ldots, n} ; \mathfrak{h}+\mathfrak{g}{i+1}=\mathfrak{h}+\mathfrak{g}{i}\right}={1, \ldots, n} \反斜杠 \mathbb{I} ^{\mathfrak{g} / \mathfrak{h}}\mathbb{I}^{\mathfrak{h}}:=\left{i \in{1, \ldots, n} ; \mathfrak{h}+\mathfrak{g}{i+1}=\mathfrak{h}+\mathfrak{g}{i}\right}={1, \ldots, n} \反斜杠 \mathbb{I} ^{\mathfrak{g} / \mathfrak{h}}

## 数学代写|表示论代写Representation theory代考|The Set of Double Cosets

[从1,从4]=从6,[从1,从5]=从7,[从2,从3]=从7

G=(和1从1⋅和5从5)⋅和2从2⋅和6从6⋅(和3从3⋅和4从4⋅和7从7) =(和1和5从7⋅和5从5⋅和1从1)⋅和2从2⋅和6从6⋅(和3从3⋅和4从4⋅和7从7) ∈H⋅和1从1⋅和2从2⋅和6从6⋅乙

H \反斜杠 G / B=\left{H \cdot z_{1} Z_{1} \cdot z_{2} Z_{2} \cdot B,\left(z_{1}, z_{2}\right) \in \mathscr{V}\right} \dot{\cup}\left{H \cdot z_{2} Z_{2} \cdot z_{6} Z_{6} \cdot B,\left(z_{2 }, z_{6}\right) \in \mathbb{R}^{2}\right}H \反斜杠 G / B=\left{H \cdot z_{1} Z_{1} \cdot z_{2} Z_{2} \cdot B,\left(z_{1}, z_{2}\right) \in \mathscr{V}\right} \dot{\cup}\left{H \cdot z_{2} Z_{2} \cdot z_{6} Z_{6} \cdot B,\left(z_{2 }, z_{6}\right) \in \mathbb{R}^{2}\right}
=p(G∖G2)在˙p(G2)

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