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

Bent Ørsted, Jorge 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 .

A complete analogue of Hardy's theorem on semisimple Lie groups

Rudra P. Sarkar (2002)

Colloquium Mathematicae

A result by G. H. Hardy ([11]) says that if f and its Fourier transform f̂ are O ( | x | m e - α x ² ) and O ( | x | e - x ² / ( 4 α ) ) respectively for some m,n ≥ 0 and α > 0, then f and f̂ are P ( x ) e - α x ² and P ' ( x ) e - x ² / ( 4 α ) respectively for some polynomials P and P’. If in particular f is as above, but f̂ is o ( e - x ² / ( 4 α ) ) , then f = 0. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.

A remarkable contraction of semisimple Lie algebras

Dmitri I. Panyushev, Oksana S. Yakimova (2012)

Annales de l’institut Fourier

Recently, E.Feigin introduced a very interesting contraction 𝔮 of a semisimple Lie algebra 𝔤 (see arXiv:1007.0646 and arXiv:1101.1898). We prove that these non-reductive Lie algebras retain good invariant-theoretic properties of 𝔤 . For instance, the algebras of invariants of both adjoint and coadjoint representations of 𝔮 are free, and also the enveloping algebra of 𝔮 is a free module over its centre.

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