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I am not sure it makes sense to ask "what is the p-adic local Langlands conjecture for $\mathrm{GL}_1$". Nobody has succeeded in even formulating a reasonable candidate for a p-adic LLC for $\mathrm{G …

You are quite correct that the Langlands transfer map does not preserve cuspidality in general. E.g. if you take a modular form of CM type, coming from a Groessencharacter $\psi$ of some imaginary qua …

EDIT. A colleague wrote to me to point out that my original answer to this question was actually completely wrong: I had confused "isobaric" representations with "pure" representations (which are not …

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 $\mathrm{Sp}_4$, J …

It is indeed true that substantially more is known for holomorphic cusp forms than for general automorphic representations, as a consequence of modularity lifting theorems. The strongest result so f …

There is no way of reformulating local Langlands for $n > 1$ in terms of such a map. Local Langlands is a bijection between irreducible smooth representations of $\operatorname{GL}_n(K)$, and $n$-dime …

In Colmez's work on the p-adic local Langlands correspondence for ${\rm GL}_2(\mathbb{Q}_p)$, he works with ${\rm GL}_2(\mathbb{Q}_p)$-representations on $p$-adic Banach spaces which admit an invarian …

A fantastic place to start would be Toby Gee's notes from the 2013 Arizona Winter School. This gives a nice overview of the theory as it then existed -- things have of course moved on further since th …

You might like to read the introduction of Harris' 2013 Crelle paper "The Taylor-Wiles method for coherent cohomology" (see link). Here is an excerpt: In practice, all the higher-dimensional results, …

The simplest non-trivial case for this is for $G = SL_2$. If you take an unramified principal series representation of $\tilde G = GL_2$ and restrict it to $G$, then it will almost always be irreducib …

You seem to be expecting that mod $p$ local Langlands should satisfy the same compatibilities as "conventional" local Langlands (for smooth representations of $GL_2(\mathbf{Q}_p)$ and $WD(\mathbf{Q}_p …

This came up in a paper of mine not so long ago, and my coauthors and I were surprised that it wasn't made explicit in the standard references, so we wrote it out ourselves: Dembélé, Lassina; Loeffler …

There is a very interesting interaction between some aspects of the Langlands program and high-energy physics, involving elliptic polylogarithms (and other related generalisations of the classical pol …

You have asked a lot of questions at once, and it is impossible to give more than a hint at a small subset of these questions. I think the general theme here is: the existence of eigenvarieties doesn' …

If you're asking about admissible p-adic Banach space representations in the sense of Schneider--Teitelbaum, then I think virtually nothing is known in this setting about branching laws, even in the s …