Représentations monomiales des groupes de Lie nilpotents

Fujiwara, H.

doi:10.2140/pjm.1987.127.329

Cited by 45 publications

(24 citation statements)

References 7 publications

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“…We shall achieve that by invoking Fujiwara's recent derivation of an explicit (that is distribution-theoretic) Plancherel formula for Indj^ 1 (see [7] [8]). We readjust our notation slightly to conform to that of Fujiwara [7], Set: …”

Section: Claim ψ θ =\mentioning

confidence: 99%

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Harmonic analysis on exponential solvable homogeneous spaces: the algebraic or symmetric cases

Lipsman¹

1989

Pacific J. Math.

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One of the main results of this paper is a complete description of the spectral decomposition of the quasi-regular representation of an arbitrary exponential solvable symmetric space. Benoist had shown previously that such a representation is multiplicity-free, but he was unable to compute the precise spectrum and spectral measure. More generally, the quasi-regular representation is considered for any exponential solvable homogeneous space. In previous work of the author and Messrs. Corwin, Greenleaf and Grelaud, the analysis of these representations was carried out in the nilpotent case. The spectral decomposition arrived at was in terms of the Kirillov orbital parameters. Corresponding results are obtained here for algebraic exponential solvable homogeneous spaces in case the stability subgroup is either: a Levi component, or its nilradical is multiplicity-free in the nilradical of the homogeneous group. The description of the spectral decomposition in the Mackey parameters is also obtained for these representations.

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Section: Claim ψ θ =\mentioning

confidence: 99%

“…(see [7]). Now if we translate h by an arbitrary group element and throw that term to the other side, we obtain that the infinitesimal components of the distributions τ(Λ)α are γ θ (h)a θ .…”

Section: Claim ψ θ =\mentioning

confidence: 99%

Harmonic analysis on exponential solvable homogeneous spaces: the algebraic or symmetric cases

Lipsman¹

1989

Pacific J. Math.

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“…The material dealt with here is quite standard, we refer the reader to the references [1,2,4,5,6,7,8,12,16] for more complete details. Throughout, g will be a n-dimensional real exponential Lie algebra, G will be the associated connected and simply connected exponential Lie group.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Intertwining operators of irreducible representations for exponential solvable Lie groups

Baklouti

Fujiwara

Ludwig³

2013

Forum Mathematicum

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Abstract. Let G be a real solvable exponential Lie group with Lie algebra g and let f ∈ g * . We take two polarizations p j , j = 1, 2, at f which meet Pukanszky's condition. Let P j := exp(p j ), j = 1, 2, be the associated subgroups in G. The linear functional f defines unitary characters χ j (exp(X)) := e −i f,X , X ∈ p j , of P j . Let τ j := ind G Pj χ j , j = 1, 2, be the corresponding induced representations, which are unitary and irreducible. It is well known that τ 1 and τ 2 are unitary equivalent. The description of the intertwining operator of such an equivalence is given via an abstract integral I p2,p1 over the homogeneous space P 2 /P 1 ∩ P 2 and one of the main problems is the convergence of such an integral. In this paper, we show that the product P 1 P 2 is closed in G. This then implies that our integral converges at least on a dense subspace of elements of the space of τ 1 and we can prove that this formal integral gives us a concrete intertwining operator. We can in this way avoid the use of a third Pukanszky polarization, which was necessary in the approach made in [1]. Finally, given three Pukanszky polarizations p i , i = 1, 2, 3 at f , we accurately determine the composition formula of I p3,p2 • I p2,p1 • I p1,p3 using Maslov's index.

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“…As usual we set (v,a) = a(v), v € Jf™, a e J^' 00 , a sesquilinear form. n(G) acts on both J^°°a nd Jtf-°°, and it is a well-known fact that 7 r (^( G ) )^-°° c ^°°, if S>(G) is the test space of compactly supported, infinitely differentiable functions on G. Given the direct integral decomposition (1.1), it is known from [6] and [14] that [4] The Plancherel formula for the horocycle spaces and generalizations, II 197 We refine the latter of these as follows. Set {J^-°°) We make use of the structure established in [12,Section 2].…”

Section: Nuclear Operators and The Plancherel Formulamentioning

confidence: 99%

“…Then we show how to derive explicitly the Plancherel theory for both cases by parallel techniques. This program has been carried out for G nilpotent in Fujiwara's two papers [4,5]; for Strichartz spaces with trivial stabilizer in [12]; and for general Strichartz spaces in [13]. In [13, item (1.3)], I proposed pursuing this program for these categories of homogeneous spaces:…”

Section: Introductionmentioning

confidence: 99%

The Plancherel formula for the horocycle spaces and generalizations, II

Lipsman

1998

J Aust Math Soc A

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The Plancherel formula for various semisimple homogeneous spaces with non-reductive stability group is derived within the framework of the Bonnet Plancherel formula for the direct integral decomposition of a quasi-regular representation. These formulas represent a continuation of the author's program to establish a new paradigm for concrete Plancherel analysis on homogeneous spaces wherein the distinction between finite and infinite multiplicity is de-emphasized. One interesting feature of the paper is the computation of the Bonnet nuclear operators corresponding to certain exponential representations (roughly those induced from infinite-dimensional representations of a subgroup). Another feature is a natural realization of the direct integral decomposition over a canonical set of concrete irreducible representations, rather than over the unitary dual.1991 Mathematics subject classification (Amer. Math. Soc): primary 22E46; secondary 22E45.

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Représentations monomiales des groupes de Lie nilpotents

Cited by 45 publications

References 7 publications

Harmonic analysis on exponential solvable homogeneous spaces: the algebraic or symmetric cases

Harmonic analysis on exponential solvable homogeneous spaces: the algebraic or symmetric cases

Intertwining operators of irreducible representations for exponential solvable Lie groups

The Plancherel formula for the horocycle spaces and generalizations, II

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