To my knowledge, the strongest modularity lifting theorem known to date can be found in the preprint **Potential automorphy and change of weight** by Thomas Barnet-Lamb, David Geraghty, Toby Gee and Richard Taylor (accessible through their webpages). 


A key idea in this work seems to be the introduction of a new local condition to impose on the liftings one wants to consider: the so-called *potential diagonalizability*. Since its original introduction by Richard Taylor, the notion of *potential automorphy* has similarly proven to be extremely useful, but the introduction of the latter notion seems to me much more easy to motivate than the former. 


My question is: what motivated the introduction of the notion of *potential diagonalizability*? 



Potential diagonalizability: motivation?
Graff's answer is (naturally) accurate. Historically I think it went something like this:


* observation that the fact that "automorphy of a point on a component of a universal deformation ring implies automorphy of the whole component" (i.e. the general TWK method) could profitably be combined with the Harris tensor product trick to attack Sato--Tate, using the known structure of the FL and pot-BT def rings (this is the first BLGG paper).
* independent observation that FL components contain "simple" points (the ones that imply potential diagonalizability for FL reps).
* the observation that these two arguments could be combined with potential automorphy for ordinary representations (and thus crucially that one could move from ordinary to non-ordinary reps - this wasn't done in the original BLGG paper). Initially this was a rather complicated argument that was specific to the FL case (indeed Taylor's Ihara avoidance was used in the FL case rather than the ordinary setting). It was then realised that it was easier to do all level raising/lowering away from l in the ordinary case, and at this point it became clear that the argument in fact worked for potentially diagonalizable representations.


So I guess the answer is: there was an argument that was designed to work in the FL case, but once it was put into its final form, it was obvious that the correct condition was potential diagonalizability.


As far as I know, outside of the ordinary, FL and pot-BT (GL\_2) cases, there aren't any other general cases in which one can prove potential diagonalizability at present. Of course there are some other concrete cases where one can, which can be useful e.g. for the weight part of Serre's conjecture (see the most recent BLGG paper).


