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# 数学代写|表示论代写Representation Theory代考|A Measure on H \backslash G / B

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## 数学代写|表示论代写Representation Theory代考|A Measure on H \backslash G / B

The aim now is to disintegrate the restriction of the irreducible representation $\pi_{l, \mathfrak{b}}$. The first step is to realize the Hilbert space $L^2\left(G / B, \chi_l\right)$ as a direct integral. Therefore one needs a measure $d \gamma$ on the space $H \backslash G / B$ such that
$$L^2\left(G / B, \chi_l\right) \simeq \int_{H \backslash G / B}^{\oplus} L^2\left(H / B(x), \chi_{l(x)}\right) d \gamma(x)$$
where the linear functional
$$l(x)=\mathrm{Ad}^*(\psi(x)) l, x \in H \backslash G / B$$

defines a character of the subalgebra $\mathfrak{b}(x):=\mathfrak{h} \cap \operatorname{Ad}(\psi(x)) \mathfrak{b}$ and where $B(x):=$ $H \cap \psi(x) B \psi(x)^{-1}$

Below we recall the construction of measures on homogeneous spaces as explained in Chap. 1, Sect. 1.2.2. Some notations will differ.

Definition 3.5.13 Let $H=\operatorname{exph}$ be a closed subgroup of $G$. Let $\mathscr{Y}=$ $\left{Y_1, \cdots, Y_m\right}$ be a Jordan-Hölder basis of $\mathfrak{h}$. As was said in Sect. 3.5.1, the mapping
$$E_{\mathscr{Y}}: \mathbb{R}^m \rightarrow H, \quad\left(y_1, \cdots, y_m\right) \mapsto \exp \left(y_1 Y_1\right) \cdots \exp \left(y_m Y_m\right)$$
is a diffeomorphism and the measure $d \mathscr{y}$ defined on $H$ by
$$\int_H f(h) d \mathscr{Y}(h):=\int_{\mathbb{R}^m} f \circ E_{\mathscr{Y}}(y) d y, \quad f \in C_c(H),$$
is a Haar measure of $H$. Furthermore, one knows that for any subalgebra $\mathfrak{k} \subset \mathfrak{h}$, if
$$\mathbb{I}^{\mathfrak{h} / \mathfrak{k}}:=\left{i \in{1, \cdots, m}, Y_i \notin \mathfrak{k}+\mathfrak{h}{i+1}\right}=\left{j_1<\cdots{\mathscr{N}}: \mathbb{R}^r \rightarrow H, \quad\left(s_1, \cdots, s_r\right) \mapsto \exp \left(s_1 Y_{j_1}\right) \cdots \exp \left(s_r Y_{j_r}\right)$$
composed with the mapping : $H \ni h \mapsto H \mapsto H=h \cdot K$, gives the diffeomorphism
$$E_{\mathscr{N}}^{\cdot}: \mathbb{R}^r \rightarrow H / K, \quad\left(s_1, \cdots, s_r\right) \mapsto \exp \left(s_1 Y_{j_1}\right) \cdots \exp \left(s_r Y_{j_r}\right) \cdot K$$

## 数学代写|表示论代写Representation Theory代考|A Concrete Intertwining Operator

We have seen in the previous section that the disintegration of the restriction of the irreducible representation $\pi_{l, \mathfrak{b}}$ to $H$ reduces now to the disintegration of the monomial representations $\sigma(x)=\tau_{\chi_l(x), B(x)}$ of $H$ into irreducibles. Let us use the techniques developed in Sect.3.2. Let $\mathscr{N}=\left{Y_{j_1}, \cdots, Y_{j_r}\right}$ be the Malcev basis defined in (3.5.16) and let us use the following notations:
\begin{aligned} & \mathfrak{b}(t):=\mathfrak{b}(\tilde{\phi}(t))=\mathfrak{b}{r+1} \subset \ldots \subset \mathfrak{b}_c(t)=\mathbb{R}-\operatorname{span}\left(Y{j_c}, \ldots, Y_{j_r}, \mathfrak{b}(t)>\subset \ldots \subset \mathfrak{b}1(t)=\mathfrak{h}\right. \ & B_c(t):=\exp \mathfrak{b}_c(t), \quad l(t):=\operatorname{Ad}^(\phi(t)) l{\mid \mathfrak{h}}, \quad t \in \mathscr{V}^{\prime}, c=1, \ldots, r . \ & \end{aligned}
One has for every $t \in \mathscr{V}^{\prime}$ a Jordan-Hölder basis $p_{\mathrm{a}}\left(G \cdot \Gamma_f\right)(t)=\left{X_{r+1}(t), \ldots\right.$, $\left.X_m(t)\right}$ of $\mathfrak{b}(t)$, such that the vectors $X_i(t), i=r+1, \ldots, m$, vary rationally without singularities in $t \in \mathscr{V}^{\prime}$. Let
$$\mathscr{Y}(t)=\left{X_1(t):=Y_{j_1}, \ldots, X_r(t):=Y_{j_r}, X_{r+1}(t), \ldots, X_m(t)\right}, \quad t \in \mathscr{V}^{\prime},$$
be the corresponding Malcev basis of $\mathfrak{h}$ and let
$$\mathscr{Y}(t)^=\left{X_1(t)^, \ldots, X_r(t)^, X_{r+1}(t)^, \ldots, X_m(t)^\right}, \quad t \in \mathscr{V}^{\prime}$$
be its dual basis in $\mathfrak{h}^$. For $t \in \mathscr{V}^{\prime}$, let $\Gamma(t)=l(t)+\mathfrak{b}(t)^{\perp}$ and for any $c \in{1, \ldots, r}$ let $$d_c(t):=\max \left(\operatorname{dim} \operatorname{Ad}^\left(B_c(t)\right)(f)\right) ; f \in \Gamma(t), t \in \mathscr{V}^{\prime}$$
and let $d_{r+1}(t)=0$. Let also
$$I(t):=\left{c_1<\ldots<c_{q(t)}\right}:=\left{c \in{1, \ldots, r}, \quad d_c(t)=d_{c+1}(t)\right}$$
and
$$\mathscr{R}(t):=\left{l(t)+\sum_{c_i \in I(t)} v_i X_{c_i}(t)^*, \quad\left(v_1, \ldots, v_{q(t)}\right) \in \mathbb{R}^{q(t)}\right}, t \in \mathscr{V}^{\prime}$$

## 数学代写|表示论代写Representation Theory代考|A Measure on H \backslash G /B

$$L^2(G / B, \chi l) \simeq \int_{H \backslash G / B}^{\oplus} L^2\left(H / B(x), \chi_{l(x)}\right) d \gamma(x)$$

$$l(x)=\operatorname{Ad}^*(\psi(x)) l, x \in H \backslash G / B$$

b. 正如在雇派中所兑的那样。3.5.1、映射
$$E_{\mathscr{y}}: \mathbb{R}^m \rightarrow H, \quad\left(y_1, \cdots, y_m\right) \mapsto \exp \left(y_1 Y_1\right) \cdots \exp \left(y_m Y_m\right)$$

$$\int_H f(h) d \mathscr{Y}(h):=\int_{\mathbb{R}^m} f \circ E_{\mathscr{Y}}(y) d y, \quad f \in C_c(H),$$

$\$ \$$\backslash \operatorname{mathbb}{1} \wedge{\backslash \operatorname{mathfrak}{\mathrm{h}} / \backslash \operatorname{mathfrak}{\mathrm{k}}}:=\backslash \operatorname{left}\left{i \backslash \operatorname{in}{1, \backslash\right. cdots, m}, Y_{-} i \backslash notin \backslash \operatorname{mathfrak}{\mathrm{k}}+\backslash \operatorname{mathfrak}{\mathrm{h}} {i+1} \backslash right }=\backslash left \left{j_{-} k \backslash\right. cdots {\backslash mathscr {\mathrm{N}}}: \backslash mathbb {\mathrm{R}} \wedge r \backslash rightarrow H, \quad \backslash left(s_l, lcolots, s_{-} r \backslash right) Imapsto \backslash \exp \backslash left(s_1Y_{j_l} 1 right) \mid cdots \backslash \exp \backslash left(s_r \left.Y_{-}\left{j_{-} r\right} \backslash r i g h t\right) composedwiththemapping : \ H \ni h \mapsto H \mapsto H=h \cdot K \$$, givesthediffeomorphism
$E_{-}{\backslash \operatorname{mathscr}{N}} \wedge{\backslash$ cdot $}: \backslash \operatorname{mathbb}{\mathrm{R}} \wedge r \backslash$ rightarrow $H / K$, \quad $\backslash$ left(s_1, lcdots, s_r $\backslash$ right) $\backslash$ mapsto lexp $\mid$ left(s_1 $1 Y_{-}\left{j_{-} 1\right} \backslash$ right) $\backslash$ cdots $\backslash \exp \backslash$ left(s_r $Y_{-}\left{j_{-} r\right} \backslash$ right $) \backslash c$ cot $K$
$\$ \$$## 数学代写|表示论代写Representation Theory代考|A Concrete Intertwining Operator 我们在上一节中已经看到，不可约表示的限制解体 \pi_{l, b} 到 H 现在减少到单项式表示的解体 \sigma(x)=\tau_{\chi l(x), B(x)} 的 H 成不可约的。让 我们使用第 3.2 节中开发的技术。让\left 缺少或无法识别的分隔符 是 (3.5.16) 中定义的 Malcev 基础，让我 们使用以下符号: 每个都有 t \in \mathscr{V}^{\prime} Jordan-Hölder 荃础 left 缺少或无法识别的分隔符 的 b(t), 这样向量 X_i(t), i=r+1, \ldots, m, 有理地变化而没有奇点 t \in \mathscr{V}^{\prime}. 让 〈left 缺少或无法识别的分隔符 是相应的 Malcev 基础然后让 〈left 缺少或无法识别的分隔符 成为它的双重基础缺少上标或下标参数 . 为了 t \in \mathscr{V}^{\prime} ，让 \Gamma(t)=l(t)+\mathfrak{b}(t)^{\perp} 对于任何 c \in 1, \ldots, r 让$$
d_c(t):=\max \left(\operatorname{dim} \operatorname{Ad}^{\left(B_c(t)\right)}(f)\right) ; f \in \Gamma(t), t \in \mathscr{V}^{\prime}


〈left 缺少或无法识别的分隔符

\left 缺少或无法识别的分隔符

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