What are the most comprehensive textbooks on the structure of Lie groups and their infinite-dimensional representations if one is interested in their applications to number theory (so covering discrete subgroups and automorphic representations)?

  • $\begingroup$ It seems to me that (most?) number theorists are not interested by general discrete subgroups but some very special ones (arithmetic subgroups). $\endgroup$
    – YCor
    Aug 15 '20 at 11:14

I think the most comprehensive reference would be the following conference proceedings (Proceedings in Symposia in Pure Mathematics) :

  • Automorphic Forms, Representations, and L-functions, Parts 1&2, vol. 33
  • Motives, Parts 1&2, vol.51
  • Representation Theory and Automorphic Forms, vol. 61

However you might be also interested in the following books. An introduction to the Archimedean representation theory is given in "Representation Theory of Semisimple Groups" by Knapp. A slightly more advanced book is "Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups" by Borel and Wallach.

There is also a book "An Introduction to Automorphic Representations with a view toward Trace Formulae" by Getz and Hahn. Another recent introductory book is "Eisenstein Series and Automorphic Representations with Applications in String Theory" by Fleig, Gustafsson, Kleinschmidt, Persson.

  • 2
    $\begingroup$ Thank you! I was aware of the older texts and monographs, but not Getz and Hahn which looks wonderful in the freely available version, nor Fleig et al., again freely available on arXiv and full of concrete examples. $\endgroup$
    – user163784
    Aug 15 '20 at 5:23
  • 2
    $\begingroup$ PSPM 9, Algebraic groups and discontinuous subgroups, is another great in that series. $\endgroup$
    – LSpice
    Aug 15 '20 at 5:54

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