I have not been able to post of late as I’ve been quite busy working on several projects.
I wanted to make a post though discussing a new draft with my collaborator A. Bertoloni Meli that I’m quite excited about. In it we discuss a method for characterizing the local Langlands conjecture for certain groups as in Scholze’s paper [Sch]. Namely we show that for certain classes of groups an equation like that in the Scholze–Shin conjecture (see [Conjecture 7.1, SS]) is enough to characterize the local Langlands conjecture (for supercuspidal parameters) at least if one is willing to assume that other expected properties of the local Langlands conjecture hold.
The main original idea of this paper is the realization that while the Langlands–Kottwitz–Scholze method only deals with Hecke operators at integral level (e.g. see the introduction to [Sch]) that one can circumvent the difficult questions this raises (e.g. see [Question 7.5,SS]) if one is willing to not only consider the local Langlands conjecture for in isolation, but also the local Langlands conjecture for certain groups closely related to (so-called elliptic hyperendoscopic groups). Another nice byproduct of this approach is that while the Scholze–Shin conjecture is stated as a set of equations for all endoscopic triples for our paper shows that one needs only consider the trivial endoscopic situation (for elliptic hyperendoscopic groups of ).
This paper is closely related to the paper mentioned in this previous post where me and A. Bertoloni Meli discuss the proof of the Scholze–Shin conjecture for unramified unitary groups in the trivial endoscopic triple setting.
[Sch] Scholze, Peter. The Local Langlands Correspondence for GL_n over -adic fields, Invent. Math. 192 (2013), no. 3, 663–715.
[SS] Scholze, P. and Shin, S., 2013. On the cohomology of compact unitary group Shimura varieties at ramified split places. Journal of the American Mathematical Society, 26(1), pp.261-294.