# Reference Request: Elementary introduction to holomorphic induction's role in index theory

I am currently working on an index theory senior project which involves Dirac operators, spin modules and holomorphic induction on the representation ring Lie groups. My primary reference is Sternberg's Lie algebra notes and *Spin Geometry*by Lawson and Michaelson. However I am reading these material very slowly and so far some of the material is still over my head(for example, I do not know how the holomorphic induction works explicitly since I even could not find its definition anywhere), so I am inquiring if there is any elementary article/notes/paper, etc on this topic. All I know is it is essential to under the index theorem at this stage. I am inquiring on here because somehow I lost contact with my advisor.

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It might help to show where this notion of holomorphic induction comes up in index theory. –  Eric O. Korman Jan 7 '12 at 14:47
@Eric: My advisor used this explicitly in his lectures to construct irreducible representations of semisimple Lie groups. But he told me "look up this online", that this was used by Atiyah and Bott in the 50s. I did looked up online but could not find much. –  Kerry Jan 8 '12 at 5:51
Maybe this is referring to the Borel-Weil theorem? –  Eric O. Korman Jan 10 '12 at 22:41
@Eric: This is a good guess, indeed Bott-Borel-Weil is related to my thesis, but so far I have not covered it myself so I do not know if this is holomorphic induction. Since my professor this was used mainly in 50s and re-picked up in the 90s, I assume it is not. I really appreciated your comment, though. Thank you. –  Kerry Jan 11 '12 at 16:29
I was trying to read Lurie's article, but it is obviously in a much higher level than I need (www-math.mit.edu/~lurie/papers/bwb.pdf), so I gave up in that time. I will try to read it again. –  Kerry Jan 11 '12 at 16:31