Abstract
Rankin–Cohen brackets and representations of conformal Lie groups
Highlights
Résumé Ce texte est une version étendue d’un cours donné par l’auteur lors de l’école d’été Formes quasimodulaires et applications qui s’est tenue à Besse en juin 2010
Before discussing our method let us mention a series of papers by Eholzer, Ibukayama and Ban [8, 1] where a construction of Rankin-Cohen brackets for Siegel modular forms was developed by means of the Howe θcorrespondence
Holomorphic induction from a maximal compact subgroup leads to a series of unitary representations of G, called holomorphic discrete series representations, that one usually realizes on holomorphic sections of holomorphic vector bundles over G/K
Summary
Let ( , ) denote the standard inner product on L2(S): This form is invariant with respect to the pairs (πμ−, π−−μ−2), and (πμ+, π−+μ−2). If μ = −1 + iσ, the representations πμ± are unitary, the inner product being ( , ). They are irreducible for any σ = 0 [18]. The map f → f (u, v)| u, v |−1+iσ, (σ ∈ R), is a unitary G-isomorphism between L2(G/H) and π−−1+iσ⊗ˆ 2 π−+1+iσ acting on L2(S × S) The latter space is provided with the usual inner product. The non-commutative product σ being defined on the whole space L2(G/H), it induces a ring structure on the set ⊕nEn+. It is noteworthy that up to a constant the coefficients cn(k1, k2, σ), that encode the associativity of the σ-product, were conjectured by Cohen, Manin and Zagier in [4]
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