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Please Note: The main points of the question below are in bold in order to minimize the time required to read the question.

Let me begin by stating that I understand representation theory is a vast and deep area with many different subfields. Of course, any learning roadmap request for representation theory would necessarily have many different answers or at least one answer with many different suggestions. I would be more interested in "mainstream topics in representation theory"; one could define this as "the set of topics which every serious representation theorist should know" (although even this is subjective and varies from subfield to subfield). Of course, I am happy for people to suggest topics which they feel are not necessarily "mainstream representation theory"; I would be interested in as many suggestions as possible.

I am interested in representation theory both as a branch of mathematics in its own right and as a set of tools and ideas which one may use to study different (either related or a priori unrelated) areas of mathematics (please feel free to interpret this in a broad sense). My background in representation theory is almost all of (and will soon be exactly) the contents of the book entitled Lie Groups by Daniel Bump. The interdisciplinary nature of representation theory dictates that I have reasonable background in other branches of mathematics; I think that I have such a background but feel free to assume as prerequisites any branch of mathematics when giving suggestions.

I am interested in studying representation theory beyond that which is covered in Daniel Bump's Lie Groups. In other words, I am happy for suggestions for topics that a potential representation theorist should know after reading Bump's book (this is the key point). Of course, I am also interested in hearing suggestions for topics that a potential representation theorist should know even if they are virtually disjoint from Bump's book. I am certainly happy for suggestions to take either the form of a textbook, research monograph, research paper, or some other form that I have not thought about.

I am not really interested in suggestions for topics that are already subsumed in Bump's book; I certainly do not object to such suggestions but they would not really be in response to this request. (You can view/download free and legally the table of contents of Bump's book at the following website: http://www.springer.com/mathematics/algebra/book/978-0-387-21154-1.)

Thank you very much for all suggestions!

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  • $\begingroup$ It looks like Bump's book covers the representation theory of $\mathfrak{sl}_2$. From my perspective, understanding the representation theory of general semisimple Lie algebras (over $\mathbb{C}$, say) is a necessity for any serious representation theorist. Certainly it is prerequisite to many areas of current research $\endgroup$
    – Justin Campbell
    Commented Jul 6, 2012 at 16:09
  • $\begingroup$ I would recommend Serre's book Complex Semisimple Lie Algebras to this end. It is very terse (like most of Serre's writing) so for more details you might refer to Humphreys's Introduction to Lie Algebras and Representation Theory and Dixmier's Universal Enveloping Algebras. $\endgroup$
    – Justin Campbell
    Commented Jul 6, 2012 at 16:14
  • $\begingroup$ @JustinCampbell I think that the coverage of Bump's book is broader than simply the representation theory of $\mathfrak{sl}_{2}$. Of course, chapter 12 is entitled "Representations of $\mathfrak{sl}_{2}(\mathbb{C})$" but there are 50 chapters in Bump's book and many of them discuss a wide range of different topics. For example, general semisimple compact Lie groups are studied in chapter 23. Also, standard topics such as root systems, heighest-weight theory, the Iwasawa and Bruhat decompositions, symmetric spaces etc. are discussed in chapters 1 - 33 (i.e., the first two parts of the book). $\endgroup$
    – Amitesh Datta
    Commented Jul 6, 2012 at 16:16
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    $\begingroup$ Dear Amitesh, The theory of Harish-Chandra modules and its relationship to the theory of unitary representations of semisimple Lie groups is probably the natural next large topic following the classification of semisimple Lie groups and their finite-dimensional representations. There are some questions/answers here on MO and on Math.SE that give a quick overview, and there are various books; one that I like is Knapp's "Overview by examples". There is also the geometric perspective of Beilinson and Bernstein (a far-reaching sheaf-theoretic generalization of Borel--Weil--Bott), which ... $\endgroup$
    – Emerton
    Commented Jul 6, 2012 at 23:10
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    $\begingroup$ ... I think you would enjoy learning (based on my impression of your tastes), and which you would be well-positioned to learn after picking up a little background in the classical aspects of the theory. Regards, Matthew $\endgroup$
    – Emerton
    Commented Jul 6, 2012 at 23:11

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Meta-answer: There are short introductions to a variety of interesting topics in Representation Theory of Lie Groups, a conference proceedings containing lecture notes by Atiyah, Bott, & other luminaries.

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  • $\begingroup$ Thank you very much! I will take a look at this reference. $\endgroup$
    – Amitesh Datta
    Commented Jul 7, 2012 at 6:01

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