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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 …

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 …

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 …

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 …

How about the following construction? Let $A$ be a principally-polarised abelian surface over $\mathbf{Q}$ which is "generic", i.e. $End_{\overline{\mathbf{Q}}}(A) = \mathbf{Z}$. Then the Galois acti …

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 …

The formula you quote from Harris defines a Galois representation, not a motive. We expect that there is a motive whose etale realisation is Harris' space, but that is not immediate. The problems are: …

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 …

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 …

(1) Borel's article in the Corvallis proceedings does this slightly differently: he chooses a specific Frobenius element $\sigma$, and then looks at the subset $\widehat{G} \times \{\sigma\}$ of ${}^L …

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' …

Let's look at the case of representations associated to modular forms. I'm going to switch the roles of $\ell$ and $p$, because I find $\ell$-adic Hodge theory disturbing; so I'm going to look at $\rh …

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 …

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 …