The cohomology of Shimura varieties and Drinfeld shtukas is conjectured to realize the representations sought for in the Langlands programme/conjectures, the cohomology of Deligne-Lusztig varieties realizes representations of the classical groups over finite fields: How did people find those varieties?
[Math] how to find the varieties whose cohomology realizes certain representations
ag.algebraic-geometryetale-cohomologynt.number-theory
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The question may be precised depending on what you call "show up". More precisely:
Concerning the cohomology of Shimura varieties there are two points of view: intersection cohomology or ordinary cohomology (this is of course the same for compact Shimura varieties).
Concerning the fact that the cohomology shows up in something we can add an option: it may show up potentially.
Then we can add another option: to proove it shows up in the Tannakian sub category of motives generated by the motives of Shimura varieties (even weaker (?): the class in the K_0 of motives of your variety is a virtual combination of the classes of motives showing up in Shimura varieties).
Let's first say we look at intersection cohomology. As stated your hope 4) is false for trivial reasons: the only Artin motives showing up in the intersection cohomology of Shimura varieties are the abelian one...In fact by purity they show up only in the H^0 that has been computed by Deligne and is abelian.
Of course if you put option (2) in my list this counterexample disappears.
Now you may say: yes but we can twist an Artin motive by a CM character and ask the same question. This is where I come to the following point: you're saying that because the twisting operation that is a particular case of Langlands functoriality is a known Langlands functoriality. Where I want to come is that in fact if you suppose Langlands functoriality known then the fact that your variety shows up in the Tannakian category generated by motives of Shimura varieties implies its L function is automorphic (tensor product functoriality).
If you suppose Langlands functoriality and your variety shows up potentially in the motive of a Shimura variety then its L-function is automorphic (existence of automorphic induction which implies for example Artin conjecture).
About the intersection cohomology of Shimura varieties: it is now pretty well understood and I think there is no reason why any variety would show up potentially in it. More precisely the Langlands parameters of automorphic representations showing up in the intersection cohomology of Shimura varieties factor through some representation $r_\mu:\;^L G_E\rightarrow GL_n$ where $G$ is the group attached to the Shimura variety (well to be more serious I woud have to invoke cohomological Arthur's parameter but it would take 5 hours to write this in details). Thus I clearly think the class of varieties that show up potentially in the cohomology of Shimura varieties has some serious restrictions...
Now there is another thing I did not speak about: the cohomology of non-compact Shimura varieties that may not be pure. For this little is known and it may be possible some interesting Galois representations that do not show up in the intersection cohomology of Shimura varieties show up in the cohomology...I know some people are looking at this (I won't give any name, even if I'm tortured) but as I said up to now little is known.
Well, I will stop here since this is an endless story and you can speak about this during hours...
Several remarks before answering your questions: (1) Langlands-Tunnell is a result in the other direction: from Galois representation to automorphic forms; it is therefore not relevant. (2) One expects to be able to attach Galois representations only to certain types of Mass forms, those whose component at infinity in algebraic (in the automorphic representation settings) or equivalently, whose eigenvalue for the Laplacian is $1/4$. (3) this Galois representation is expected to take values in Gl${}_2(\mathbb{C})$, hence to have finite image.
So you ask: "how does one construct a galois representation from a Maass form?". The answer is: one still doesn't know how to. It's one of the most striking open problem in the Langlands program. There was 25 years ago an announcement that this problem has been essentially solved (with published articles), but it was soon after retracted: see the two references given by Chandan in comments.
And for your displayed question about the infinite family, stripped of the reference to Langlands-Tunell and of the "infinite image" condition, the answer is no, as far as I know,
Best Answer
Regarding Shimura varieties:
One has to first consider the case of modular curves, which has served throughout as an impetus and inspiration for the general theory.
The study of modular curves (in various guises) goes back to the 19th century, with the work of Jacobi and others on modular equations (which from a modern viewpoint are explicit equations for the modular curves $X_0(N)$). The fact that these curves are defined over $\mathbb Q$ (or even $\mathbb Z$) also goes back (in some form) to the 19th century, in so far as it was noticed that modular equations have rational or integral coefficients. There is also the (strongly related) fact that interesting modular functions/forms have rational or integral $q$-expansion coefficients. Finally, there are the facts related to Kronecker's Jugendtraum, that modular functions/forms with integral Fourier coefficients, when evaluated at quadratic imaginary points in the upper half-plane, give algebraic numbers lying in abelian extensions of quadratic imaginary fields. These all go back to the 19th century in various forms, although complete theories/interpretations/explanations weren't known until well into the 20th century.
The idea that the cohomology of modular curves would be Galois theoretically interesting is more recent. I think that it goes back to Eichler, with Igusa, Ihara, Shimura, Serre, and then Deligne all playing important roles. It seems to be non-trivial to trace the history, in part because the intuitive idea seems to predate the formal introduction of etale cohomology (which is necessary to make the idea completely precise and general). Thus Ihara's work considers zeta-functions of modular curves (or of the Kuga--Satake varieties over them) rather than cohomology. (The zeta-function is a way of incarnating the information carried in cohomology without talking directly about cohomology). Shimura worked just with weight two modular forms (related to cohomology with constant coefficients), and instead of talking directly about etale cohomology worked with the Jacobians of the modular curves. (He explained how the Hecke operators break up the Jacobian into a product of abelian varieties attached to Hecke eigenforms.) [Added: In fact, I should add that Shimura also had an argument, via congruences, which reduced the study of cohomology attached to higher weight forms to the case of weight two forms; this was elaborated on by Ohta. These kinds of arguments were then rediscovered and further developed by Hida, and have since been used by lots of people to relate modular forms of different weights to one another.]
The basic idea, which must have been understood in some form by all these people, is that a given Hecke eigenform $f$ contributes two dimensions to cohomology, represented by the two differential forms $f d\tau$ and $\overline{f}d\tau$. Thus Hecke eigenspaces in cohomology of modular curves are two-dimensional. Since the Hecke operators are defined over $\mathbb Q$, these eigenspaces are preserved by the Galois action on etale cohomology, and so we get two-dimensional Galois reps. attached to modular forms.
As far as I understand, Shimura's introduction of general Shimura varieties grew out of thinking about the theory of modular curves, and in particular, the way in which that theory interacted with the theory of complex multiplication elliptic curves. In particular, he and Taniyama developed the general theory of CM abelian varieties, and it was natural to try to embed that more general theory into a theory of moduli spaces generalizing the modular curves. A particular challenge was to try to give a sense to the idea that the resulting varieties (i.e. Shimura varieties in modern terminology) had canonical models over number fields. This could no longer be done by studying rationality of $q$-expansions (since they could be compact, say, and hence have no cusps around which to form Fourier expansions). Shimura introduced the Shimura reciprocity law, i.e. the description of the Galois action on the special points (the points corresponding to CM abelian varieties) as the basic tool for characterizing and studying rationality questions for Shimura varieties.
In particular, Shimura varieties were introduced prior to the development of the Langlands programme, and for reasons other than the construction of Galois representations. However, once one had these varieties, naturally defined over number fields, and having their origins in the theory of algebraic groups and automorphic forms, it was natural to try to calculate their zeta-functions, or more generally, to calculate the Galois action on their cohomology, and Langlands turned to this problem in the early 1970s. (Incidentally, my understanding is that it was he who introduced the terminology Shimura varieties.) The first question he tried to answer was: how many dimensions does a given Hecke eigenspace contribute to the cohomology. He realized that the answer to this --- at least typically --- was given by Harish-Chandra's theory of (what are now called) discrete series $L$-packets, as is explained in his letters to Lang; the relationship of the resulting Galois representations to the Langlands program is not obvious --- in particular, it is not obvious how the dual group intervenes --- and this (namely, the intervention of the dual group) is the main topic of the letters to Lang. These letters to Lang are just the beginning of the story, of course. (For example, the typical situation does not always occur; there is the phenomenon of endoscopy. And then there is the problem of actually proving that the Galois action on cohomology gives what one expects it to!)
Regarding Drinfeld and Deligne--Lusztig varieties:
I've studied these cases in much less detail, but I think that Drinfeld was inspired by the case of Shimura varieties, and (as Jim Humphreys has noted) Deligne--Lusztig drew insipration from Drinfeld.
What can one conclude:
These theoretically intricate objects grew out of a long and involved history, with multiple motivations driving their creation and the investigations of their properties.
If you want to find a unifying (not necessarily historical) theme, one could also note that Deligne--Lusztig varieties are built out of flag varieties in a certain sense, in fact as locally closed regions of flag varieties, and that Shimura varieties are also built out of (in the sense that they are quotients of) symmetric spaces, which are again open regions in (partial) flag varieties. This suggests a well-known conclusion, namely that the geometry of reductive groups and the various spaces associated to them seems to be very rich.