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Function spaces on the Olśhanskiĭsemigroup and the Gel'fand-Gindikin program

Khalid KoufanyBent Ørsted — 1996

Annales de l'institut Fourier

For the scalar holomorphic discrete series representations of SU ( 2 , 2 ) and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside SU ( 2 , 2 ) . We construct a Cayley transform between the Ol’shanskiĭ semigroup having U ( 1 , 1 ) as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for SU ( 2 , 2 ) . This allows calculating the composition series in terms of harmonic analysis on U ( 1 , 1 ) . In particular we show that the Ol’shanskiĭ Hardy space for U ( 1 , 1 ) is different...

A Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a generalized Berezin transform

Bent ØrstedJorge Vargas — 2007

Annales de l’institut Fourier

In this note, we generalize the results in our previous paper on the Casimir operator and Berezin transform, by showing the ( L 2 , L 2 ) -continuity of a generalized Berezin transform associated with a branching problem for a class of unitary representations defined by invariant elliptic operators; we also show, that under suitable general conditions, this generalized Berezin transform is ( L p , L p ) -continuous for 1 p .

Conformally invariant trilinear forms on the sphere

Jean-Louis ClercBent Ørsted — 2011

Annales de l’institut Fourier

To each complex number λ is associated a representation π λ of the conformal group S O 0 ( 1 , n ) on 𝒞 ( S n - 1 ) (spherical principal series). For three values λ 1 , λ 2 , λ 3 , we construct a trilinear form on 𝒞 ( S n - 1 ) × 𝒞 ( S n - 1 ) × 𝒞 ( S n - 1 ) , which is invariant by π λ 1 π λ 2 π λ 3 . The trilinear form, first defined for ( λ 1 , λ 2 , λ 3 ) in an open set of 3 is extended meromorphically, with simple poles located in an explicit family of hyperplanes. For generic values of the parameters, we prove uniqueness of trilinear invariant forms.

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