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The quasi-spherical transformations
Cornelie MITCHA MALANDA
Université Marien NGOUABI BP : 69, Brazzaville, Congo
Kinvi KANGNI
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
Abstract:
Let G be a locally compact group, K a compact subgroup of G. is a Gelfand pair if the convolution subalgebra L1♮ of L1 that are biinvariant under the action K, is commutative.
Let LK1 be the algebra of K-invariant L1-functions on G. L1♮ is isomorphic to LK1(G), then L1♮ 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 of the group of automorphisms Aut of Hn, is not of multiplicity free [1] then LSO1 is not commutative and the couple (Hn,SO(n, ℝ) 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.