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>



What's the status of Arthur's announced classification for GSp(4)?
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 $\mathrm{Sp}\_4$**](http://dx.doi.org/10.5802/jep.99), 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."


