Bibliothèque de L'IMIST/CNRST

PH.D Université Mohammed V 2017

We look for an extension of the concept of "H-type groups" where introduced by A. Kaplan around 1980, which we call it " likewise Heisenberg group". Then, we focus on the construction of its Lie algebra, whose the main differential operator (Casimir-Laplacian) on this group defines as the sum of the squares of the basis of this Lie algebra. Moreover, we discuss in detail some concrete examples of such groups, for which we give the explicit expression of the corresponding Casimir-Laplacian. Next, we study the spectral analysis associated with the Casimir-Laplace operator on the quaternionic Heisenberg group. Namely, the heat kernel, the resolvent, the wave kernel and the Green function on this group have been investigated. From the resolvent, we have been able to explain the kernel of the spectral density associated with this Laplacian. In other hand, we give also an explicit expression of the Green kernel of the fractional power of the quaternionic Laplacian.