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.

Galois representations associated to holomorphic limits of discrete series I), available here. Regards, Matthew – Emerton Jun 25 '13 at 23:19