Analyse harmonique sur les espaces symétriques nilpotents (Q760631)

The author proves the Plancherel formula for a nilpotent symmetric space and that any invariant differential operator on such a space admits a fundamental solution. A nilpotent symmetric space is a homogeneous space G/H where G is a nilpotent Lie group and H is the set of the elements of G fixed by an involution. In order to prove the Plancherel formula the author determines first the irreducible unitary representations of G for which there exists a non zero H-invariant distribution vector. For such a representation the space of these vectors has dimension 1. Then a result of \textit{G. Schiffmann} [Bull. Soc. Math. Fr. 96, 347-355 (1968; Zbl 0167.438)] is generalized: There is a one to one correspondence between the H- invariant positive definite distributions on G, and the H-invariant positive definite distributions on the tangent space of G/H at the origin, considered as an Abelian group. Specializing the previous result one proves the Plancherel formula for the quasi regular representation of G in \(L^ 2(G/H)\), and, using a method due to \textit{M. Rais} [C.R. Acad. Sci., Paris, Ser. I, 273, 495-498 (1971; Zbl 0236.46047)] the author proves that any G-invariant differential operator on G/H admits a H- invariant fundamental solution which is tempered.