Representations of semisimple Lie groups and special functions of a matrix argument

The method for investigation of irreducible representations of U compact and Gsub(O) non-compact real forms of the complex semisimple bound Lie G group is suggested. The method permits from the one hand to calculate zonal spherical functions of symmetrical spaces U/K and Gsub(O)/K where K is the maximum compact subgroup Gsub(O) and from the other hand - to determine and investigate new special functions of a matrix argument. These functions are turned out to be particular cases of more general functions, i.e. hypergeometric functions of a matrix arguments