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*?