I have seen a lot of textbook recommendations on this site (e.g. mathoverflow.net/questions/2446/… or mathoverflow.net/questions/13/learning-about-lie-groups) so I thought it would be appropriate. Copied the question to math.stackexhange.

Thank you very much for your answer. I am not quite sure what about the direction of my studies mainly because I don't know much about geometric representation theory.

The question is not very well defined, because I don't quite understand what is understood by the resulting ``function'' in the case when, let's say, series is nowhere convergent? Your link is very interesting, maybe you should add it at least as a comment to the question? The problem for all "partially defined" functions seem to monstrous))