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In "Automorphic representations of GSp(4)" (2004) (see http://www.math.toronto.edu/arthur/), James Arthur announces a classification of discrete automorphic representations of GSp(4). There are no proofs however, these are to appear in a monograph "Automorphic representations of classical groups" (in preparation), which still hasn't appeared. His 2012 article, "Classifying automorphic representations", makes no reference to the announced results for GSp(4). Arthur also mentions that the classification is conditional on the existence of a stabilized trace formula (and its twisted analogues) which require certain cases of the fundamental lemma. But of course there's been a lot of progress in this area since 2004.

My question is to what extent proofs of these claims can be found in the literature. Is it more or less known among experts how to prove them, maybe even unconditionally? Are there any interesting partial results, for instance, if you fix the archimedean component(s) to be a particular cohomological $(\mathfrak g, K)$-module? To what extent can the various cases in the classification be understood as functorial liftings from smaller groups?

Sorry about the open-ended question. Any information is appreciated.

I am aware that an equivalent of this classification for PGSp(4) is proven in Flicker's book: http://www.worldscientific.com/worldscibooks/10.1142/5883

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    $\begingroup$ Why don't you just ask Jim (Arthur)? $\endgroup$
    – David Handelman
    Commented Apr 3, 2014 at 19:56
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    $\begingroup$ Though I'm not a specialist in this area, I wonder if you've taken advantage of the citation trail in MathSciNet? This leads to many further papers. Aside from that, I certainly agree with David that it would be worth asking Jim Arthur himself for guidance. His expository article is reviewed here: MR2058604 (2005d:11074) Automorphic representations of GSp(4). Contributions to automorphic forms, geometry, and number theory, 65–81, Johns Hopkins Univ. Press, Baltimore, MD, 2004. $\endgroup$
    – Jim Humphreys
    Commented Apr 3, 2014 at 22:37

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This question is answered pretty definitively by the following recent paper:

Gee, Toby; Taïbi, Olivier, Arthur’s multiplicity formula for $\mathrm{GSp}_4$ and restriction to $\mathrm{Sp}_4$, Journal de l'École polytechnique — Mathématiques, Volume 6 (2019), p. 469-535.

Abstract: "We prove the classification of discrete automorphic representations of GSp4 explained in [Art04], as well as a compatibility between the local Langlands correspondences for GSp4 and Sp4."

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    $\begingroup$ Perhaps it might be worth adding to your answer the following quote from their paper? "We end this introduction with a small disclosure, and a comparison to other work.While we have said that the results of this paper are unconditional, they are only as unconditional as the results of [Art13] and [MW16a, MW16b]. In particular, they depend on cases of the twisted weighted fundamental lemma that were announced in [CL10], but whose proofs have not yet appeared in print, as well as on the references [A24], [A25], [A26] and [A27] in [Art13], which at the time of writing have not appeared publicly." $\endgroup$
    – user145307
    Commented Oct 30, 2019 at 1:32
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His monograph has appeared, albeit under a slightly different name:

James Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, Colloquium Publications 61 (2013) 590 pp, AMS bookstore.

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A student of Arthur, Bin Xu, has worked on the general problem of classifying automorphic representations for $GSp(2n)$. See his webpage here, or better:

Bin Xu, Endoscopic Classification of Representations of $GSp(2n)$ and $GSO(2n)$, PhD thesis, University of Toronto (2014) (http://hdl.handle.net/1807/68169).

and

Bin Xu, L-packets of quasisplit $GSp(2n)$ and $GO(2n)$, Math. Ann. 370 (2018) pp 71–189, doi:10.1007/s00208-016-1515-x, arXiv:1503.04897.

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  • $\begingroup$ URL no longer works. $\endgroup$
    – David Roberts ♦
    Commented Oct 29, 2019 at 20:45
  • $\begingroup$ @DavidRoberts I've done a little magic. How's it now? $\endgroup$
    – Robert Furber
    Commented Oct 29, 2019 at 20:57
  • $\begingroup$ @Robert thanks. I will link directly to the stable version of Bin Xu's thesis, then. $\endgroup$
    – David Roberts ♦
    Commented Oct 29, 2019 at 22:55

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