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Harmonic Analysis on Exponential Solvable Lie Groups

verfasst von: Hidenori Fujiwara, Jean Ludwig

Verlag: Springer Japan

Buchreihe : Springer Monographs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is the first one that brings together recent results on the harmonic analysis of exponential solvable Lie groups. There still are many interesting open problems, and the book contributes to the future progress of this research field. As well, various related topics are presented to motivate young researchers.

The orbit method invented by Kirillov is applied to study basic problems in the analysis on exponential solvable Lie groups. This method tells us that the unitary dual of these groups is realized as the space of their coadjoint orbits. This fact is established using the Mackey theory for induced representations, and that mechanism is explained first. One of the fundamental problems in the representation theory is the irreducible decomposition of induced or restricted representations. Therefore, these decompositions are studied in detail before proceeding to various related problems: the multiplicity formula, Plancherel formulas, intertwining operators, Frobenius reciprocity, and associated algebras of invariant differential operators.

The main reasoning in the proof of the assertions made here is induction, and for this there are not many tools available. Thus a detailed analysis of the objects listed above is difficult even for exponential solvable Lie groups, and it is often assumed that G is nilpotent. To make the situation clearer and future development possible, many concrete examples are provided. Various topics presented in the nilpotent case still have to be studied for solvable Lie groups that are not nilpotent. They all present interesting and important but difficult problems, however, which should be addressed in the near future. Beyond the exponential case, holomorphically induced representations introduced by Auslander and Kostant are needed, and for that reason they are included in this book.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries: Lie Groups and Lie Algebras

Abstract

There are many texts on Lie groups and Lie algebras. Here following Sugiura [74] we gather general preliminaries. As other references we list [45, 62, 70, 71].

Hidenori Fujiwara, Jean Ludwig

Chapter 2. Haar Measure and Group Algebra

Hidenori Fujiwara, Jean Ludwig

Chapter 3. Induced Representations

Abstract

Let G be a locally compact group and H a closed subgroup of G. We are going to show that there exists a G-invariant Borel measure on the quotient space G∕H, if and only if for the modular functions Δ _H and Δ _G we have that

Hidenori Fujiwara, Jean Ludwig

Chapter 4. Four Exponential Solvable Lie Groups

Abstract

The group $(\mathbb{R}^{n},+),n \in \mathbb{N}^{{\ast}}$, is the only connected and simply connected abelian Lie group of dimension n. Its irreducible unitary representations are one-dimensional by Schur’s lemma (2.3.7), and define in this way unitary characters $\chi: \mathbb{R}^{n} \rightarrow \mathbb{C}$ of $\mathbb{R}^{n}$. Such a character χ is of the form χ _ℓ for some real-valued linear functional ℓ on $\mathbb{R}^{n}$, i.e.

$$\displaystyle{ \chi (x) = \chi _{\ell}(x) = e^{-2\pi i\ell(x)},\ x \in \mathbb{R}^{n}. }$$

Hidenori Fujiwara, Jean Ludwig

Chapter 5. Orbit Method

Abstract

Auslander and Kostant [3] extended the orbit method to solvable Lie groups of type I. To explain their theory, we first prepare the ingredients. Let G be a solvable Lie group with Lie algebra $\mathfrak{g}$ and $\mathfrak{g}^{{\ast}}$ the dual vector space of $\mathfrak{g}$. G acts on $\mathfrak{g}$ by the adjoint action and on $\mathfrak{g}^{{\ast}}$ by its contragradient representation:

$$\displaystyle{(g\cdot f)(X) = \left (\text{Ad}^{{\ast}}(g)\cdot f\right )(X) = f(\text{Ad}(g^{-1})X)\ (g \in G,f \in \mathfrak{g}^{{\ast}},X \in \mathfrak{g}).}$$

The action of G obtained in this way is called the coadjoint representation of G. Let G(f) be the stabilizer of $f \in \mathfrak{g}^{{\ast}}$ in G. Hence the Lie algebra of G(f) is

$$\displaystyle{\mathfrak{g}(f) =\{ X \in \mathfrak{g};f([X,Y ]) = 0,\forall \,Y \in \mathfrak{g}\}.}$$

Hidenori Fujiwara, Jean Ludwig

Chapter 6. Kirillov Theory for Nilpotent Lie Groups

Abstract

In this chapter, we examine in detail the Kirillov theory for nilpotent Lie groups, which are always assumed to be connected and simply connected.

Hidenori Fujiwara, Jean Ludwig

Chapter 7. Holomorphically Induced Representations ρ ( f , ?? , G $$\rho (f,\mathfrak{h},G$$ ) for Exponential Solvable Lie Groups

Para>Let us consider holomorphically induced representations for an exponential solvable Lie group $G = \text{exp}\ \mathfrak{g}$. Since the stabilizer G(f) in G of any $f \in \mathfrak{g}^{{\ast}}$ is connected (Theorem 5.3.2), there uniquely exists the homomorphism $\eta _{f}: G(f) \rightarrow \mathbb{T}$ such that $d\eta _{f} = if\vert _{\mathfrak{g}(f)}$. So, we write the holomorphically induced representation $\rho (f,\eta _{f},\mathfrak{h},G)$ constructed from a polarization $\mathfrak{h} \in P(f,G)$ satisfying the Pukanszky condition and its representation space $\mathcal{H}(f,\eta _{f},\mathfrak{h},G)$ simply as $\rho (f,\mathfrak{h},G)$ and $\mathcal{H}(f,\mathfrak{h},G)$. We show the following theorem.

Hidenori Fujiwara, Jean Ludwig

Chapter 8. Irreducible Decomposition

Abstract

>As usual let $G = \text{exp}\ \mathfrak{g}$ be an exponential solvable Lie group with Lie algebra $\mathfrak{g}$. It is well known that there is a strong duality between the induction and the restriction of representations. In this chapter, we study the irreducible decomposition of the representation induced from a subgroup or the representation restricted to a subgroup.

Hidenori Fujiwara, Jean Ludwig

Chapter 9. e $$\boldsymbol{e}$$ -Central Elements

Abstract

In order to practise a more detailed analysis of monomial representations, we suppose in this chapter that $G = \text{exp}\ \mathfrak{g}$ is a connected and simply connected nilpotent Lie group with Lie algebra $\mathfrak{g}$. Let us introduce e-central elements due to Corwin and Greenleaf [17]. Let

$$\displaystyle{ \{0\} = \mathfrak{g}_{0} \subset \mathfrak{g}_{1} \subset \cdots \subset \mathfrak{g}_{n-1} \subset \mathfrak{g}_{n} = \mathfrak{g},\ \mbox{ dim}(\mathfrak{g}_{k}) = k\ (0 \leq k \leq n) }$$

(9.1.1)

be a composition series of ideals of $\mathfrak{g}$. Let {X _j}_{1 ≤ j ≤ n} be a Malcev basis of $\mathfrak{g}$ according to this composition series, i.e. $X_{j} \in \mathfrak{g}_{j}\setminus \mathfrak{g}_{j-1}\ (1 \leq j \leq n)$ and $\{X_{j}^{{\ast}}\}_{1\leq j\leq n}$ its dual basis in $\mathfrak{g}^{{\ast}}$. We denote the coordinates of $\ell\in \mathfrak{g}^{{\ast}}$ by $(\ell_{1},\ldots,\ell_{n}),\ell_{j} =\ell (X_{j})$. Then $\mathfrak{g}_{j}^{\perp } = \langle X_{j+1}^{{\ast}},\ldots,X_{n}^{{\ast}}\rangle _{\mathbb{R}} \subset \mathfrak{g}^{{\ast}},\mathfrak{g}_{j}^{{\ast}}\mathop{\cong}\mathfrak{g}^{{\ast}}/\mathfrak{g}_{j}^{\perp }$ and the projection $p_{j}: \mathfrak{g}^{{\ast}}\rightarrow \mathfrak{g}_{j}^{{\ast}}$ intertwines the actions of G on $\mathfrak{g}^{{\ast}}$ and $\mathfrak{g}_{j}^{{\ast}}$. For $\ell\in \mathfrak{g}^{{\ast}}$, we define $e_{j}(\ell) = \mbox{ dim}\big(G\cdot p_{j}(\ell)\big),e(\ell) = (e_{1}(\ell),\ldots,e_{n}(\ell))$ and set $\mathcal{E} =\{ e(\ell);\ell\in \mathfrak{g}^{{\ast}}\}$. We also recognize e _j(ℓ) = dim(G _j⋅ ℓ) with $G_{j} = \text{exp}(\mathfrak{g}_{j})$. In fact,

$$\displaystyle\begin{array}{rcl} & & \mbox{ dim}\big(G\cdot p_{j}(\ell)\big) = \mbox{ dim}\big(\mathfrak{g}/\mathfrak{g}_{j}^{\ell}\big) = \mbox{ dim}(\mathfrak{g}/\mathfrak{g}(\ell)) -\mbox{ dim}\big(\mathfrak{g}_{ j}^{\ell}/\mathfrak{g}(\ell)\big) {}\\ & & = \mbox{ dim}(\mathfrak{g}/\mathfrak{g}(\ell)) -\big (\mbox{ dim}(\mathfrak{g}/\mathfrak{g}(\ell)) -\mbox{ dim}\big(\mathfrak{g}_{j}/\mathfrak{g}_{j}(\ell)\big)\big) = \mbox{ dim}(G_{j}\cdot \ell). {}\\ \end{array}$$

For $e \in \mathcal{E}$ we define the G-invariant layer $U_{e} =\{\ell\in \mathfrak{g}^{{\ast}};e(\ell) = e\}$. With e ₀ = 0 we define the set of jump indices $S(e) =\{ 1 \leq j \leq n;e_{j} = e_{j-1} + 1\}$ and that of non-jump indices $T(e) =\{ 1 \leq j \leq n;e_{j} = e_{j-1}\}$. $\mathcal{U}(\mathfrak{g})$ being the enveloping algebra of $\mathfrak{g}_{\mathbb{C}}$, $A \in \mathcal{U}(\mathfrak{g})$ is called an e-central element if, with $\pi _{\ell} =\hat{\rho } _{G}(\ell)$, π _ℓ(A) = d π _ℓ(A) is a scalar operator for any ℓ ∈ U _e.

Hidenori Fujiwara, Jean Ludwig

Chapter 10. Frobenius Reciprocity

Abstract

Let G be a σ-compact Lie group with Lie algebra $\mathfrak{g}$, and we only consider unitary representations π whose Hilbert space $\mathcal{H}_{\pi }$ are separable. First we remember the C ^∞-vectors. Let $v \in \mathcal{H}_{\pi }$. When the function $G \ni g\mapsto \pi (g)v \in \mathcal{H}_{\pi }$ is C ^∞, v is called a C ^∞-vector. We denote by $\mathcal{H}_{\pi }^{\infty }$ the space of the C ^∞-vectors of π. {ψ _n}_n = 1 ^∞ being the approximate identity of L ¹(G) introduced in Proposition 2.2.8 and chosen in $\mathcal{D}(G)$, we see that $\|\pi (\psi _{n})w - w\| \rightarrow 0\ (n \rightarrow \infty )$ for any $w \in \mathcal{H}_{\pi }$. As $\pi (\psi _{n})w \in \mathcal{H}_{\pi }^{\infty }$, $\mathcal{H}_{\pi }^{\infty }$ is a dense subspace of $\mathcal{H}_{\pi }$ and $\mathfrak{g}$ acts there by the differential representation d π of π:

$$\displaystyle{d\pi (X)v = \frac{d} {dt}\pi (\text{exp}(tX))v\vert _{t=0}\ (X \in \mathfrak{g},v \in \mathcal{H}_{\pi }^{\infty }).}$$

The differential representation d π is uniquely extended to a representation of $\mathcal{U}(\mathfrak{g})$. $\{X_{1},\ldots,X_{n}\}$ being a basis of $\mathfrak{g}$, $\mathcal{H}_{\pi }^{\infty }$ becomes a Fréchet space with semi-norms

$$\displaystyle{\rho _{d}(v) =\sum _{1\leq i_{k}\leq n}\|d\pi (X_{i_{1}}\cdots X_{i_{d}})v\|\ (d \in \mathbb{N}).}$$

We designate by $\mathcal{H}_{\pi }^{-\infty }$ the anti-dual space of $\mathcal{H}_{\pi }^{\infty }$, i.e. the space of the continuous anti-linear forms on $\mathcal{H}_{\pi }^{\infty }$ with values in $\mathbb{C}$. We call elements of $\mathcal{H}_{\pi }^{-\infty }$ generalized vectors of π. We equip $\mathcal{H}_{\pi }^{-\infty }$ with the strong dual topology of $\mathcal{H}_{\pi }^{\infty }$. Then, the anti-dual space of $\mathcal{H}_{\pi }^{-\infty }$ is identified with $\mathcal{H}_{\pi }^{\infty }$. For $a \in \mathcal{H}_{\pi }^{\pm \infty }$ and $b \in \mathcal{H}_{\pi }^{\mp \infty }$, we write $\langle a,b\rangle$ for the image of b by a. Hence $\langle a,b\rangle = \overline{\langle b,a\rangle }$. The actions of G and $\mathfrak{g}$ are continuously extended on $\mathcal{H}_{\pi }^{-\infty }$ by duality. Notice that

$$\displaystyle{\pi (\varphi )\left (\mathcal{H}_{\pi }^{-\infty }\right ) \subset \mathcal{H}_{\pi }^{\infty }}$$

for $\varphi \in \mathcal{D}(G)$. When a closed subgroup K and its character $\chi: K \rightarrow \mathbb{C}^{{\ast}}$ are given, set

$$\displaystyle{\left (\mathcal{H}_{\pi }^{-\infty }\right )^{K,\chi } =\{ a \in \mathcal{H}_{\pi }^{-\infty };\pi (k)a =\chi (k)a,\ \forall \,k \in K\}.}$$

Hidenori Fujiwara, Jean Ludwig

Chapter 11. Plancherel Formula

Abstract

As before, let $G = \text{exp}\ \mathfrak{g}$ be an exponential solvable Lie group with Lie algebra $\mathfrak{g}$, $f \in \mathfrak{g}^{{\ast}},\mathfrak{h} \in S(f,\mathfrak{g}),H = \text{exp}\ \mathfrak{h},\chi _{f}(\text{exp}\ X) = e^{if(X)}\ (X \in \mathfrak{h})$ and consider $\tau =\hat{\rho } (f,\mathfrak{h},G) = \text{ind}_{H}^{G}\chi _{f}$. We denote by e the unit element of G. Since every $\phi \in \mathcal{H}_{\tau }^{\infty }$ is an C ^∞-function [63] on G, $\delta _{\tau } \in \left (\mathcal{H}_{\tau }^{-\infty }\right )^{H,\chi _{f}\varDelta _{H,G}^{1/2} }$ is defined by $\delta _{\tau }(\phi ) = \langle \delta _{\tau },\phi \rangle = \overline{\phi (e)}$. In this chapter, we shall be interested in the abstract Plancherel formula due to Penney [59] and Bonnet [12] applied to the cyclic representation (τ, δ _τ).

Hidenori Fujiwara, Jean Ludwig

Chapter 12. Commutativity Conjecture: Induction Case

Abstract

Let G be a connected and simply connected nilpotent Lie group with Lie algebra $\mathfrak{g}$ and $H = \text{exp}\ \mathfrak{h}$ an analytic subgroup of G with Lie algebra $\mathfrak{h}$. Given a unitary character χ of H, we construct the monomial representation $\tau = \text{ind}_{H}^{G}\chi$ and examine the algebra D _τ(G∕H) of the G-invariant differential operators on the line bundle $G\times _{H}\mathbb{C}$ associated with χ. Our target is the commutativity conjecture due to Duflo [22] and Corwin and Greenleaf [17]. The latter proved one direction of implication: if τ has finite multiplicities, then D _τ(G∕H) is commutative. Therefore, we are interested in the inverse direction of implication.

Hidenori Fujiwara, Jean Ludwig

Chapter 13. Commutativity Conjecture: Restriction Case

Abstract

As Frobenius reciprocity suggests, there is a kind of duality between the induction and the restriction of representations. Under this guiding principle, let us formulate and prove for the restriction of representations the counterpart of the commutativity conjecture. These results were obtained by [6].

Hidenori Fujiwara, Jean Ludwig

Backmatter

Titel: Harmonic Analysis on Exponential Solvable Lie Groups
verfasst von: Hidenori Fujiwara
Jean Ludwig
Verlag: Springer Japan
Electronic ISBN: 978-4-431-55288-8
Print ISBN: 978-4-431-55287-1
DOI: https://doi.org/10.1007/978-4-431-55288-8

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries: Lie Groups and Lie Algebras

Chapter 2. Haar Measure and Group Algebra

Chapter 3. Induced Representations

Chapter 4. Four Exponential Solvable Lie Groups

Chapter 5. Orbit Method

Chapter 6. Kirillov Theory for Nilpotent Lie Groups

Chapter 7. Holomorphically Induced Representations ρ ( f , ?? , G $$\rho (f,\mathfrak{h},G$$ ) for Exponential Solvable Lie Groups

Chapter 8. Irreducible Decomposition

Chapter 9. e $$\boldsymbol{e}$$ -Central Elements

Chapter 10. Frobenius Reciprocity

Chapter 11. Plancherel Formula

Chapter 12. Commutativity Conjecture: Induction Case

Chapter 13. Commutativity Conjecture: Restriction Case

Backmatter

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