Arthur's long-awaited book project [is now published](http://bookstore.ams.org/COLL-61) (*The endoscopic classification of representations: orthogonal and symplectic groups*). However, in the book he takes some things for granted:


1. The stabilization of the twisted trace formula for GL($N$) and SO($2n$).
2. Orthogonality relations for elliptic tempered characters.
3. Weak spectral transfer of tempered $p$-adic characters
4. A twisted version of Shelstad's archimedean theory of endoscopy


Assumption 1, and implicitly 2 and 3, are in Arthur's Hypothesis 3.2.1. Assumption 4 is made in the proof of Arthur's Proposition 2.1.1 (see also Remark 5 after Theorem 2.2.1). (I think these are all of Arthur's working hypotheses, but let me know if I missed something.)



> 
> Q1: Recently, in a series of 10-11 preprints, Waldspurger and Moeglin
>  finished the stabilization of the twisted trace formula. This seems
>  to verify Assumption 1--does it also take care of implicit Assumptions 2
>  and 3? If yes, where are 2 and 3 proved?
> 
> 
> 


Arthur said that Assumption 4 should be taken care of by work in progress of Shelstad and Mezo.



> 
> Q2: Is Assumption 4 verified yet?
>  Mezo and Shelstad have some recent preprints, but I don't know if they prove exactly what's needed. If not, any explanation of what is still needed would be appreciated. 
> 
> 
> 


If not all of these assumptions are proven yet, are any of the specific main results of Arthur (beyond, say, SO(3)) now unconditional?


**Edit:** I was recently reminded that Arthur also refers to some of his preprints ([A24]-[A27] in the quasi-split case), which are still unpublished (unfinished?) as of mid 2018, for some proofs, but based on Arthur's earlier verbal confirmation to me, I will keep my answer as accepted unless other information comes to light.



What is the status of Arthur's book?
Now I think the answer is **yes, Arthur's work is now unconditional** for quasi-split special orthogonal and symplectic groups. Wee Teck Gan kindly directed me to the relevant papers of Waldspurger and Moeglin addressing the assumptions 2-4 above, though I was not able to verify with certainty that their stated results precisely cover what Arthur requires. Here are the relevant papers:


1. The stabilization is completed in Moeglin and Waldspurger's [Stabilisation X](http://arxiv.org/abs/1412.2981) paper.
2. The orthogonality for elliptic tempered characters appears to be completed in Waldspurger's [twisted local trace formulas](http://arxiv.org/abs/1205.1100) paper, generalizing Arthur's 1993 *Acta* paper. See Theorem 7.3.
3. I believe the $p$-adic weak spectral transfer that Arthur requires, specifically twisted analogues of Theorems 6.1 and 6.2 from his 1996 *Selecta* paper, is done by Moeglin in her [Représentations elliptiques...](http://webusers.imj-prg.fr/~jean-loup.waldspurger/stabilisationXIfinal.pdf) paper (which is unofficially called "Stabilisation XI").
4. Twisted archimedean endoscopy seems to be done in Waldspurger's [Stabilisation IV](http://arxiv.org/abs/1403.1454) paper, which uses Shelstad's 2012 *Annals* paper. This is independent from the treatment in Mezo's [Tempered spectral transfer...](http://people.math.carleton.ca/%7Emezo/spectrans3.pdf) paper, but I cannot tell if Mezo's results are unconditionally sufficient for Arthur's needs.


**Edit:** Arthur confirmed that the results in his book (except the last chapter on non-quasi-split forms) are now unconditional.


