Contents Introduction............................................................................................................................................................................... 5 Chapter I. Preliminaries and notations....................................................................................................................... 8 1. Manifolds—generalities................................................................................................. 8 2. Representations............................................................................................................. 8 3. Induced representations............................................................................................... 10 4. Elements of structure theory of semi-simple Lie groups........................................ 11 5. Homogoneous spaces of semi-simple Lie groups................................................ 12 6. Measures on G and its homogeneous spaces........................................................ 13 7. Invariant differential operators on homogeneous spaces...................................... 14 Chapter II. Spherical representations and spherical functions—General results..................................................... 15 Chapter III. Some representations in function spaces.................................................................................................... 19 Chapter IV. Conical representations and conical distributions...................................................................................... 27 Chapter V. Induced spherical representations................................................................................................................. 38 References............................................................................................................................................................................... 46