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Gelfand transforms of S O ( 3 ) -invariant Schwartz functions on the free group N 3 , 2

Véronique Fischer, Fulvio Ricci (2009)

Annales de l’institut Fourier

The spectrum of a Gelfand pair ( K N , K ) , where N is a nilpotent group, can be embedded in a Euclidean space. We prove that in general, the Schwartz functions on the spectrum are the Gelfand transforms of Schwartz K -invariant functions on N . We also show the converse in the case of the Gelfand pair ( S O ( 3 ) N 3 , 2 , S O ( 3 ) ) , where N 3 , 2 is the free two-step nilpotent Lie group with three generators. This extends recent results for the Heisenberg group.

General theory of Lie derivatives for Lorentz tensors

Lorenzo Fatibene, Mauro Francaviglia (2011)

Communications in Mathematics

We show how the ad hoc prescriptions appearing in 2001 for the Lie derivative of Lorentz tensors are a direct consequence of the Kosmann lift defined earlier, in a much more general setting encompassing older results of Y. Kosmann about Lie derivatives of spinors.

Generalized Verma module homomorphisms in singular character

Peter Franek (2006)

Archivum Mathematicum

In this paper we study invariant differential operators on manifolds with a given parabolic structure. The model for the parabolic geometry is the quotient of the orthogonal group by a maximal parabolic subgroup corresponding to crossing of the k -th simple root of the Dynkin diagram. In particular, invariant differential operators discussed in the paper correspond (in a flat model) to the Dirac operator in several variables.

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