Complex analysis on symmetric spaces

I will discuss the role of complex analysis at harmonic analysis on symmetric spaces for both finite and infinite dimensional representations. It goes back to Weyl's unitary trick which was a way to avoid complex analysis and to work with compact groups. What kind advatages will we have if we'll develop complex analysis on complex semisimple groups?
In the case of real (semisimple) symmetric spaces we will consider the problem of the separation of series of representations which we'll interpret as a non commutative analogue of decopositions of function on the line as the sum of functions holomorphic at half-planes.