  Classes of multivariate and cone valued infinitely divisible Gamma
distributions are introduced. Particular emphasis is put on the cone-valued
case, due to the relevance of infinitely divisible distributions on the
positive semi-definite matrices in applications. The cone-valued class of
generalised Gamma convolutions is studied. In particular, a characterisation in
terms of an It\^o-Wiener integral with respect to an infinitely divisible
random measure associated to the jumps of a L\'evy process is established.
  A new example of an infinitely divisible positive definite Gamma random
matrix is introduced. It has properties which make it appealing for modelling
under an infinite divisibility framework. An interesting relation of the
moments of the L\'evy measure and the Wishart distribution is highlighted which
we suppose to be important when considering the limiting distribution of the
eigenvalues .
