Annales Mathematiques Africaines

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The quasi-spherical transformations

Université Marien NGOUABI BP : 69, Brazzaville, Congo

Université Félix Houphouët Boigny,
22 BP 582 Abidjan 22 Abidjan, Côte d'Ivoire

Mathematics Subject Classification: (MSC 2010) 22E25, 22E27, 22E60
Key words : Quasi-Gelfand pair, quasi-spherical function 


Let G be a locally compact group, K a compact subgroup of G. (G, K ) is a Gelfand pair if the convolution subalgebra L1(G ) of L1(G ) that are biinvariant under the action K, is commutative.

Let LK1(G) be the algebra of K-invariant L1-functions on G. L1(G) is isomorphic to LK1(G), then L1(G) is nilpotent is equivalent to the fact that LK1(G) is nilpotent.

Let Hn be the (2n + 1)-dimensional Heisenberg group. The action of the compact subgroup SO(n,ℝ) of the group of automorphisms Aut(Hn ) of Hn, is not of multiplicity free [1] then LSO(n,ℝ)1(Hn ) is not commutative and the couple (Hn,SO(n, ) is not a Gelfand pair.

In this work, we will interest in this case where the associated algebra is nilpotent but not commutative and we will construct and study the corresponding spherical transformation which will be called a quasi-spherical transformation.

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