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My professor has told me to read "tube domain over symmetric cone". While saying so he said something related to complexificatin of real Lie algebra and representation theory. What is connection between all these concepts? What mathematical questions does it answer?

Given that I have taken courses on measure theory and topology and representation theory of finite groups only, on what books and references should I rely upon to write good M.Sc. thesis? Please narrate the references and their relevant concepts.

He has given me a book title "Analysis on the symmetric cones" by Jacques Faraut and Adam Korayani and told me to directly go to the chapter on "tube domain over symmetric cone".

I can not figure out as to how to proceed. Any help will be appreciated. Thanks.

Hanno
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    Shouldn't you talk (more) to your professor about this? – Zubin Mukerjee Jan 08 '15 at 07:02
  • One of the best places to read about symmetric cones is the book of Faraut and Koranyi; start at the beginning. Chapter III of the book of Stein and Weiss on Fourier analysis gives a different presentation of tube domains that is also helpful. For real lie algebras, there is a book of Onishchik, Lectures on Real Semisimple Lie Algebras and Their Representations, that explains the basics with careful attention to issues related to complexification (many books work only over the complex numbers). – Dan Fox Jan 08 '15 at 07:06
  • A helpful initial observation is that the simplest (symmetric) convex cone is the cone of positive real numbers. The tube domain over this cone is a half space. If one thinks of the positive reals as the positive imaginary axis, one can view the half space as hyperbolic space ... Almost any object attached to the positive real numbers or the half space, e.g. Laplace transform or hyperbolic metric - has some generalization to symmetric convex cones. – Dan Fox Jan 08 '15 at 07:10
  • @DanFox Please repost as an answer. –  Jun 19 '15 at 22:16

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