### 数学代写|表示论代写Representation theory代考|Pseudo-Algebraic Geometry

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## 数学代写|表示论代写Representation theory代考|Pseudo-Algebraic Sets

Definition 1.3.1 Let $P$ be a polynomial function with real coefficients on $\mathbb{R}^{m+k}$ and $\lambda_{i}: \mathbb{R}^{m} \rightarrow \mathbb{R}, 1 \leq i \leq k$ real linear functionals. The function
$$F_{P}=F: \mathbb{R}^{m} \longrightarrow \mathbb{R}, x \longmapsto P\left(x, e^{\lambda_{1}(x)}, \ldots, e^{\lambda_{k}(x)}\right)$$
is called a pseudo-algebraic function associated to the polynomial $P$.

Definition 1.3.2 A subset $V$ of $\mathbb{R}^{m}$ is called pseudo-algebraic if it admits some representation of the form
$$V=\left{x \in \mathbb{R}^{m}: F_{1}(x)=\ldots=F_{r}(x)=0\right}$$
where $F_{i}, 1 \leq i \leq r$ are pseudo-algebraic functions on $\mathbb{R}^{m}$.
Definition 1.3.3 A subset $A$ of $\mathbb{R}^{m}$ is called semi-pseudo-algebraic if it admits some representation of the form
$$A=\left{x \in \mathbb{R}^{m}: F_{1}(x)=\ldots=F_{r}(x)=0, G_{1}(x)>0, \ldots, G_{l}(x)>0\right}$$
where $F_{i}=F_{P_{i}}, 1 \leq i \leq r$ and $G_{j}=G_{Q_{j}}, 1 \leq j \leq l$ are pseudo-algebraic functions on $\mathbb{R}^{m}$ associated to the polynomials $P_{i}$ and $Q_{j}$ respectively.

Then it is clear that any semi-pseudo-algebraic set of $\mathbb{R}^{m}$ is the intersection between the closed set of common zeroes of a finite number of pseudo-algebraic functions on $\mathbb{R}^{m}$ and an open set of $\mathbb{R}^{m}$.

Remark 1.3.4 When the linear forms $\lambda_{i}, i=1, \ldots, k$ are null, the notion of pseudoalgebraic set coincides with the familiar notion of algebraic set. Let us recall the definition of semi-algebraic sets.

Definition 1.3.5 A subset of $\mathbb{R}^{n}$ is said to be a semi-algebraic set if it admits a representation of the form
$$\left{x \in \mathbb{R}^{n}: h_{1}(x)=\cdots=h_{p}(x)=0, g_{1}(x)>0, \ldots, g_{q}(x)>0\right}$$
where $h_{1}, \ldots, h_{p}, g_{1}, \ldots, g_{q}$ are polynomial functions on $\mathbb{R}^{n}$.
The following theorem plays an important role in the sequel, since the analysis of the multiplicity function of mixed representations naturally involves pseudoalgebraic geometry (see the next section for details).

## 数学代写|表示论代写Representation theory代考|Structure of Coadjoint Orbits

The following theorem describes the structure of the space of coadjoint orbits of an exponential solvable Lie group, and is proved in [45].

Theorem 1.3.9 Let $G$ be a connected and simply connected exponential solvable Lie group with Lie algebra $\mathfrak{g}$. There exists a finite partition $\wp$ of $\mathfrak{g}^{}$ such that: (I) each $U \in \wp$ is $G$-invariant, (2) for a given $U \in \wp$, the dimension of coadjoint orbits in $U$ is constant, (3) there is a total ordering $U_{1}}$.
Given $U \in \wp$, there exist a subspace $V$ of $\mathrm{g}^{}$ and index sets $l$ and $\varphi$ such that for each $j \in I$ we have a complex-valued rational map $p_{j}$ on $\mathrm{g}^{}$ and for each $j \in \varphi$ a complex-valued rational map $q_{j}$ on $\mathrm{g}^{*}$ satisfying the following:
(4) the set $\Sigma=\left{l \in V \cap U: p_{j}(l)=0, j \in l,\left|q_{j}(l)\right|^{2}=1, j \in \varphi\right}$ is $a$ cross-section of the coadjoint orbits in $U$.

Moreover, there exists a G-invariant analytic function $P: U \rightarrow U$ such that $P(U)=\Sigma$

If $G$ is a completely solvable Lie group, the set $\varphi$ is empty and the $p_{j}$ are realvalued for each $j \in l$ (see [47]).

Corollary 1.3.10 Let $G$ be a connected and simply connected exponential solvable Lie group with Lie algebra g. Every coadjoint orbit $\Omega$ (lying in some $U$ of the above partition) is closed in $U$ and is a semi-analytic set in $\mathrm{g}^{}$. In particular $\Omega$ is locally closed in $\mathrm{g}^{}$.Proof Let $U$ be a layer in $\wp$ such that $\Omega \subset U$ and let $P: U \rightarrow U$ be the analytic $G$ invariant function such that $P(U)=\Sigma$, where $\Sigma$ is a cross-section of the coadjoint orbits in $U$, as in Theorem 1.3.9. Then $\Omega$ meets $\Sigma$ in a single point. Let ${l}=\Sigma \cap \Omega$. The subset $P^{-1}({l})$ is $G$-invariant and every orbit $\Omega^{\prime} \subset P^{-1}({l})$ intersects $\Omega$, so $\Omega=P^{-1}({l})$. This shows that $\Omega$ is closed in $U$, and semi-analytic as $P$ is analytic. Finally, as $U$ is a semi-algebraic set in $\mathfrak{g}^{}, \Omega$ is a semi-analytic set in $\mathrm{g}^{}$.

## 数学代写|表示论代写Representation theory代考|Up-Down Representations of Exponential Solvable Lie Groups

Let $G$ be a real Lie group, $A$ and $H$ closed connected subgroups of $G$ and $\sigma$ a unitary representation of $H$. The representation
$$\rho(G, H, A, \sigma)=\text { ind }\left.{H}^{G} \sigma\right|{A}$$ of $A$ is called an up-down representation.
When the unitary representation $\sigma$ of $H$ is replaced by a unitary character $\chi_{f}$, the representation $\rho(G, H, A, \sigma)$ will be simply denoted by $\rho(G, H, A, f)$. It is clear that $\rho(G, H, G, \sigma)=\tau(\sigma)$ and $\rho(G, G, A, \sigma)=\sigma_{\mid A}$. On the other hand, if the product $A H$ is a closed subgroup of $G$, then Mackey’s subgroup theorem allows us to say that
$$\rho(A H, H, A, \sigma) \simeq \rho\left(A, A \cap H, A, \sigma_{\mid A \cap H}\right) \simeq \operatorname{ind}{A \cap H}^{A}\left(\sigma{A \cap H}\right) .$$
If $G$ is exponential solvable and $\sigma$ is a unitary irreducible representation, then $\sigma$ is a monomial representation, which means that it is induced by a unitary character. We write $\sigma \simeq \operatorname{ind}{B}^{H} \chi{l}$ where $l \in \mathfrak{g}^{*}, B=\exp (\mathfrak{b})$ and $\mathfrak{b} \in I\left(l_{\mathfrak{h}}, \mathfrak{h}\right)$. Then it is immediate that
$$\rho(G, H, A, \sigma) \simeq \rho(G, B, A, l)$$
Thus, the study of the mixed representation (1.4.1) when $\sigma \in \widehat{H}$ can be reduced to mixed representations of type $\rho(G, H, A, f)$. This is exactly what many authors usually do. Some of our results will be proved in full generality for mixedtype representations (1.4.1). Relationship (1.4.2) is sometimes very useful for transferring known results on mixed representations of type $\rho(G, H, A, f)$ to general mixed representations. When $A=H$, the representation $\rho(G, H, H, \sigma)$ will be simply denoted by $\rho(G, H, \sigma)$.

Remark 1.4.1 Mackey studied the disintegration of such representations when $A$ and $H$ are regularly related in the context of arbitrary locally compact groups. At the same time, Lemma $4.2$ in [115] proves that if $H$ and $A$ are normal subgroups of a locally compact group $G$ and $A \subset H$, the support of the mixed representation (1.4.1) is of the form $\left(\sigma_{\mid A}\right)^{g}, g \in G$ where $\left(\sigma_{\mid A}\right)^{g}(a)=\sigma\left(g \cdot a \cdot g^{-1}\right), \in A$.

## 数学代写|表示论代写Representation theory代考|Pseudo-Algebraic Sets

F磷=F:R米⟶R,X⟼磷(X,和λ1(X),…,和λķ(X))

V=\left{x \in \mathbb{R}^{m}: F_{1}(x)=\ldots=F_{r}(x)=0\right}V=\left{x \in \mathbb{R}^{m}: F_{1}(x)=\ldots=F_{r}(x)=0\right}

A=\left{x \in \mathbb{R}^{m}: F_{1}(x)=\ldots=F_{r}(x)=0, G_{1}(x)>0, \ ldots, G_{l}(x)>0\right}A=\left{x \in \mathbb{R}^{m}: F_{1}(x)=\ldots=F_{r}(x)=0, G_{1}(x)>0, \ ldots, G_{l}(x)>0\right}

\left{x \in \mathbb{R}^{n}: h_{1}(x)=\cdots=h_{p}(x)=0, g_{1}(x)>0, \ldots, g_{q}(x)>0\right}\left{x \in \mathbb{R}^{n}: h_{1}(x)=\cdots=h_{p}(x)=0, g_{1}(x)>0, \ldots, g_{q}(x)>0\right}

## 数学代写|表示论代写Representation theory代考|Structure of Coadjoint Orbits

（4）集合\Sigma=\left{l \in V \cap U: p_{j}(l)=0, j \in l,\left|q_{j}(l)\right|^{2}=1, j \in \varphi\right}\Sigma=\left{l \in V \cap U: p_{j}(l)=0, j \in l,\left|q_{j}(l)\right|^{2}=1, j \in \varphi\right}是一个共伴轨道的横截面在.

## 数学代写|表示论代写Representation theory代考|Up-Down Representations of Exponential Solvable Lie Groups

ρ(G,H,一个,σ)= 工业 HGσ|一个的一个称为上下表示。

ρ(一个H,H,一个,σ)≃ρ(一个,一个∩H,一个,σ∣一个∩H)≃工业⁡一个∩H一个(σ一个∩H).

ρ(G,H,一个,σ)≃ρ(G,乙,一个,l)

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