Paquets d'ondes dans l'espace de Schwartz d'un espace symétrique réductif

Ban, E.P. van den; Carmona, Jacques; Delorme, Patrick

doi:10.1006/jfan.1996.0084

Cited by 19 publications

(16 citation statements)

References 7 publications

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“…This in turn shows that the constant term (1.3) is regular for imaginary ν. By a result from [7] this implies that the Eisenstein integral E • (P : ν) is regular for imaginary ν, see Theorem 18.8. Because of the uniform tempered estimates formulated in Corollary 18.12, it becomes possible to define a spherical Fourier transform F P in the next section by the formula…”

Section: P0qmentioning

confidence: 92%

“…Proposition 19.6 asserts that F P is a continuous linear map into the Euclidean Schwartz space S(ia * P q ) ⊗ A 2,P , if P is of residue type. In Section 20 it is shown, using a result from [7], that the adjoint wave packet transform, given by the formula J P ϕ(x) = ia * P q E…”

Section: P0qmentioning

confidence: 99%

“…Let Λ ∈ L P (b, τ ) and fix ψ ∈ A 2,P (Λ) and η ∈ V * τ . Let ε 0 > 0; then the family F : (λ, x) → ηE • (P : λ : x)ψ belongs to the space II mer (Λ, ε 0 ) defined in [7], Def. 3.…”

Section: Proofmentioning

confidence: 99%

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The Plancherel decomposition for a reductive symmetric space. I. Spherical functions

2005

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We prove the Plancherel formula for spherical Schwartz functions on a reductive symmetric space. Our starting point is an inversion formula for spherical smooth compactly supported functions. The latter formula was earlier obtained from the most continuous part of the Plancherel formula by means of a residue calculus. In the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized Eisenstein integrals and of the Maass-Selberg relations satisfied by the associated C-functions.

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Section: P0qmentioning

confidence: 92%

Section: P0qmentioning

confidence: 99%

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The Plancherel decomposition for a reductive symmetric space. I. Spherical functions

2005

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“…One of the major advantages of Rosenberg's proof of the Plancherel formula, compared with Harish-Chandra's original proof, is that some rather intricate estimates of [23] are avoided. For the present generalization to G/H we have not been able to avoid the use of such Schwartz estimates (see Theorem 16.4); they are given in [9]. For the present generalization to G/H we have not been able to avoid the use of such Schwartz estimates (see Theorem 16.4); they are given in [9].…”

Section: And Amentioning

confidence: 93%

The Most Continuous Part of the Plancherel Decomposition for a Reductive Symmetric Space

Ban¹,

Schlichtkrull²

1997

The Annals of Mathematics

101

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“…The next corollary is an important classical application of the unitarity of the intertwiners (see [47,Théorème 2]). Proof .…”

Section: (I) In the 'Compact Realization' The Intertwining Mapmentioning

confidence: 99%

On the Spectral Decomposition of Affine Hecke Algebras

Opdam¹

2004

J. Inst. Math. Jussieu

343

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An affine Hecke algebra H contains a large abelian subalgebra A spanned by the Bernstein-Zelevinski-Lusztig basis elements θx, where x runs over (an extension of) the root lattice. The centre Z of H is the subalgebra of Weyl group invariant elements in A. The natural trace ('evaluation at the identity') of the affine Hecke algebra can be written as integral of a certain rational n-form (with values in the linear dual of H) over a cycle in the algebraic torus T = Spec(A). This cycle is homologous to a union of 'local cycles'. We show that this gives rise to a decomposition of the trace as an integral of positive local traces against an explicit probability measure on the spectrum W 0 \ T of Z. From this result we derive the Plancherel formula of the affine Hecke algebra. ContentsOn the spectral decomposition of affine Hecke algebras(2.13) The Iwahori-Hecke algebra as a Hilbert algebraMany of the results of this subsection are well known (see [31]). Let R be a root datum, and let q be a real number with q > 1. We assume that for all s ∈ S aff we are given a real number f s . Throughout this paper we use the convention that the labels as discussed in the previous subsection are defined by Convention 2.1. The labels are of the form(2.14)We write q := (q(s)) s∈S aff for the corresponding label function on S aff . The following theorem is well known.Theorem 2.2. There exists a unique complex associative algebra H = H(R, q) with C-basis (T w ) w∈W which satisfy the following relations.(a) If l(ww ) = l(w) + l(w ), then T w T w = T ww .(b) If s ∈ S aff , then (T s + 1)(T s − q(s)) = 0.Proof . This is elementary and well known. Use the Frobenius-Schur theorem for (iii), Dixmier's version of Schur's lemma for (iv) and Proposition 2.10. Corollary 2.12 (see also [31]). Restriction to H induces an injection of the setĈ into the spaceĤ of finite-dimensional irreducible * -representations of H.Consequently, the C * -algebra C is of finite type I. Proof .Because H ⊂ C is dense, it is clear that a representation of C is determined by its restriction to H and that (topological) irreducibility is preserved. Hence, by the previous corollary, all irreducible representations of C have finite dimension.We equip T and W 0 \ T with the analytic topology. Given π ∈Ĉ, we denote by W 0 t π ∈ W 0 \ T the character of Z such that χ π (z) = dim(π)z(t π ) (note that π(Z) can not vanish identically since 1 ∈ Z).By Proposition 2.9, the * -operator on Z is such that z * (t) = z(t −1 ). When π ∈Ĉ, we have χ π (x * ) = χ π (x). It follows that t −1 π ∈ W 0 t π for all π ∈Ĉ. Let us denote byProposition 2.13. The map p z :Ĉ → W 0 \ T defined by p z (π) = W 0 t π is continuous and finite. Its image S = p z (Ĉ) ⊂ W 0 \ T Herm is the spectrumẐ of the closureZ of Z in C. The map p z :Ĉ → S is closed.Proof . It is clear that the image is in W 0 \ T Herm and that the map is finite (by Proposition 2.10). Since W 0 \ T is Hausdorff andĈ is compact, the map p z is closed if it is continuous.

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Paquets d'ondes dans l'espace de Schwartz d'un espace symétrique réductif

Cited by 19 publications

References 7 publications

The Plancherel decomposition for a reductive symmetric space. I. Spherical functions

The Plancherel decomposition for a reductive symmetric space. I. Spherical functions

The Most Continuous Part of the Plancherel Decomposition for a Reductive Symmetric Space

On the Spectral Decomposition of Affine Hecke Algebras

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