TITLE: how to find the varieties whose cohomology realizes certain representations?
QUESTION [29 upvotes]: 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?

REPLY [14 votes]: The construction of Deligne-Lusztig representations is a very natural analog of the construction of discrete series representations of real groups, and follows a very general philosophy (associated I think with Gelfand and Kazhdan and most probably known to Drinfeld at the time) that all representations of reductive groups over any field are "forms of principal series". 
Namely we learn from Harish Chandra and Gelfand--Graev--Piatetskii-Shapiro the idea that representations of a reductive group are "always" labeled by conjugacy classes of tori and characters of the latter (up to Weyl group symmetry).
How do we construct these representations? The untwisted case is principal series, ie the case of a split torus. Then we are supposed to perform parabolic induction: look at the big G-representation of functions (or some form of cohomology) on the total space of the natural torus bundle $G/N\to G/B$ over the flag variety and decompose according to the torus action along the fibers (which commutes with G).  The theory of (standard/principal series) intertwining operators realizes a Weyl group symmetry on these induced representations.
The philosophy (explained beautifully by Kazhdan in various places, such as his ICM address)  is that all other "series" are given as "forms" of the principal series -- ie over the algebraic closure one only has principal series. Put more poetically, the dual of G should have an algebraic structure, so that representations over various fields can be simply specified by descent data from representations over field extensions, so at the end of the day they come from principal series. Over your given field you prescribe a conjugacy class of tori by descent data, which is a conjugacy class of Weyl group representations of your Galois group. So eg over R we just need a conjugacy class of  involutions in the Weyl group, over a finite field we can take any conjugacy class in the Weyl group, etc. Now the idea (not yet fully realized in its optimal strength) is that we should twist principal series over the algebraic closure using the "Weyl group action by intertwiners" to define the desired series of representations over the given field.
In any case this idea has a very concrete geometric realization. Recall principal series are functions or cohomologies of a torus bundle over the flag variety. Now given a field k (for simplicity assume the Galois group is cyclic, eg real or finite fields) we can decompose the flag variety over the algebraic closure of k into a collection of G(k) invariant subsets, by looking at flags which are in a given relative position (an element of $W$) with their Galois translates. In the case of $SL_2(R)$ this gives us the decomposition of $CP^1$ into $RP^1$ and the upper+lower half planes. In the finite field case these are Deligne-Lusztig varieties. This is a twisted k-version of G/B, and has a natural version of G/N over it, which is a torsor for precisely the k-torus we specified before by the same Weyl group element. So we can look at functions/cohomologies twisted by characters of this torus, finding a series of representations just as predicted by Harish Chandra. The principal series corresponds to the flag variety over k. For $SL2R$ (an example of a real group with compact torus) we recover the discrete series (though note the representation constructed this way is not irreducible, since we're considering upper and lower half planes together).