# What are limits of discrete series and which are cohomological?

Perhaps this is a bit too 'standard' for MO, but I'm struggling to dig it out of the literature (and maybe other people have had a similar experience), so it feels like a useful thing to ask and I hope people don't mind.

1) Harish-Chandra's theorem classifies all discrete series reps, which are defined to be the things occurring discretely in the spectrum of G. What is the correct definition of a 'limit of discrete series rep' and the corresponding classification result? Is it simply any non-discrete series tempered representation? (And if not, can I say anything intelligent about the other tempered representations?).

2-3) In his Ann Arbor paper, Harris mentions a theorem along the lines of: let G be a real Lie group admitting a Shimura datum. Then all representations contributing to the (appropriate) cohomology of the Shimura variety are 'nondegenerate' limits of discrete series.

2) What does 'nondegenerate' mean, and is the resulting theorem the strongest general result known in this direction?

3) Is any converse known/conjectured? Can we describe all cohomological representations at least for classical groups? (Where I guess it would be good to have answers interpreting 'cohomological' in both possible ways: coefficients in a local system or in coherent cohomology).

If clear references exist that treat things in full generality (not just GL2 or GLn) that would be helpful to know.

Thanks, Tom.

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Dear Tom, A limit of discrete series is not "any non-discrete series tempered representation". Rather, they are very particular such representations, for which the character formulas for discrete series continue to make sense (but the parameter that one plugs into this character formula, roughly speaking a regular highest weight, has been allowed to move into the walls of a Weyl chamber). Unfortunately, I'm not well-versed enough in the literature to give a more precise answer than this. (One thing to think about is "how deep" into the wall of the Weyl chamber one can go, e.g. can one ... –  Emerton Jun 25 '13 at 21:44
... get to highest weights that are in the intersection of two walls. I don't know if this is what "non-denerate" refers to, but I have a very vague suspicion that it's relevant.) Regards, Matthew –  Emerton Jun 25 '13 at 21:45
P.S. All cohomological unitary reps. (in the sense of local systems) are descibed in a paper of Vogan and Zuckerman (easily found once you know about it). I'm not sure about the coheren case, but would guess that a similar analysis to VZ is possible; I don't know if/where it's done, though. –  Emerton Jun 25 '13 at 21:47
Limits of discrete series are representations obtained from discrete series by Zucerman tensoring. They are precisely defined in Knapp (books.google.com/books?id=QCcW1h835pwC&pg=PA460), p. 460; "non degenerate" ones ibid., p. 615. –  Francois Ziegler Jun 25 '13 at 21:53
@FrancoisZiegler: Dear Tom and Francois, I wasn't able to access the google-books link, but this paper has a brief but reasonably detailed discussion in its first few pages. Note though that its discussion on associating Galois reps. to limits of discrete series automorphic forms is out-of-date: there is recent work on this problem by Wushi Goldring (Galois representations associated to holomorphic limits of discrete series I), available here. Regards, Matthew –  Emerton Jun 25 '13 at 23:19