Solutions analytiques des équations invariantes sur un groupe compact ou complexe réductif

Cérézo, André

doi:10.5802/aif.565

Cited by 2 publications

(7 citation statements)

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“…C , of what we refer to as the holomorphic Fourier series may be found already in Proposition 12 of [3]. See also Remark 5.5 below.…”

Section: Introductionmentioning

confidence: 87%

Kirillov’s character formula, the holomorphic Peter–Weyl theorem, and the Blattner–Kostant–Sternberg pairing

Huebschmann¹

2008

Journal of Geometry and Physics

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Abstract. Let K be a compact Lie group, endowed with a bi-invariant Riemannian metric, which we denote by κ. The complexification K C of K inherits a Kähler structure having twice the kinetic energy of the metric as its potential; let ε denote the symplectic volume form. Left and right translation turn the Hilbert space HL 2 (K C , e −κ/t ηε) of square-integrable holomorphic functions on K C relative to a suitable measure written as e −κ/t ηε into a unitary (K × K)-representation; here η is an additional term coming from the metaplectic correction, and t > 0 is a real parameter. In the physical interpretation, this parameter amounts to Planck 's constant .We establish the statement of the Peter-Weyl theorem for the Hilbert space HL 2 (K C , e −κ/t ηε) to the effect that (i) HL 2 (K C , e −κ/t ηε) contains the vector space of representative functions on K C as a dense subspace and that (ii) the assignment to a holomorphic function of its Fourier coefficients yields an isomorphism of Hilbert algebras from the convolution algebra HL 2 (K C , e −κ/t ηε) onto an algebra of the kind b ⊕End(V ). Here V ranges over the irreducible rational representations of K C and b ⊕ refers to a suitable completion of the direct sum algebra ⊕End(V ).Consequences are: (i) the existence of a uniquely determined unitary isomorphism between L 2 (K, dx) (where dx refers to Haar measure on K) and the Hilbert space HL 2 (K C , e −κ/t ηε), and (ii) a proof that this isomorphism coincides with the Blattner-Kostant-Sternberg pairing map from L 2 (K, dx) to HL 2 (K C , e −κ/t ηε), multiplied by (4πt) − dim(K)/4 . Among our crucial tools is Kirillov's character formula. Our methods are geometric, rely on the orbit method, and are independent of heat kernel harmonic analysis, which is used by B. C. Hall to obtain many of these results

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“…C , of what we refer to as the holomorphic Fourier series may be found already in Proposition 12 of [3]. See also Remark 5.5 below.…”

Section: Introductionmentioning

confidence: 87%

Kirillov’s character formula, the holomorphic Peter–Weyl theorem, and the Blattner–Kostant–Sternberg pairing

Huebschmann¹

2008

Journal of Geometry and Physics

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“…On sait alors que X est le plus haut des éléments de la famille { (ùy, 1 ^ j ^ d^ ] dans l'ordre induit par fa 4 '. C'est-à-dire, …”

Section: He^) Jgunclassified

“…Le lecteur vérifiera que la démons-tration qu'on y donne s'applique ici sans difficulté. [On utilise la propriété suivante de q ui résulte facilement de la caractérisation donnée à la remarque 1 : si A désigne la chambre de Weyl duale de û 4 …”

Section: Enveloppe (Pholomorphie Convexitéunclassified

“…Il vient (ii)^(i) : Pour tout compact K tel que 0 e K la fonction d'appui h^ est à valeurs ^ 0. La condition (ii) implique donc une condition de décroissance rapide sur ^, dont on sait (voir par exemple [4], p. 251) qu'elle est caractéristique des éléments de C°° (X). Par le (b) de la preuve du théorème 3, la série de Fourier-Laurent de t converge normalement sur tout compact de ^3-La somme est une fonction holomorphe dans ^ dont la restriction à X coïncide avec t.…”

Section: La Série De Fourier-laurent De T Converge Alors Normalement unclassified

“…On désigne par i û^ la chambre de Weyl de ; a' associée à S 4 ', a' + est un cône ouvert convexe de a' défini par a' 4 ' = {Hea' : Y(iH) > 0, VyeE'-}.…”

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Séries de Laurent des fonctions holomorphes dans la complexification d'un espace symétrique compact

Lassalle¹

1978

Ann. Sci. École Norm. Sup.

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Solutions analytiques des équations invariantes sur un groupe compact ou complexe réductif

Cited by 2 publications

References 1 publication

Kirillov’s character formula, the holomorphic Peter–Weyl theorem, and the Blattner–Kostant–Sternberg pairing

Kirillov’s character formula, the holomorphic Peter–Weyl theorem, and the Blattner–Kostant–Sternberg pairing

Séries de Laurent des fonctions holomorphes dans la complexification d'un espace symétrique compact

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