If you are studying finite groups, and you have classified all finite simple groups, your job is not done yet since potentially there might be some interesting phenomena in how two finite simple groups can be extended by each other.

In the context of the Langlands program, it has been said by Davin Ben-Zvi once (which, in my opinion, is somewhat similar in spirit to the remark above):

Where things get very rich geometrically is when you try to go beyond classification of irreducibles -- in the real local Langlands story we have that luxury since the first step is already done. How do you go beyond? you might ask for character formulas, relate standard and simple modules, or more ambitiously try to describe the full (derived) category of representations. Adams Barbasch Vogan introduce an interesting space with the same orbit structure as the real Langlands parameters but a much more interesting geometry, and they describe the K-group of representations in terms of equivariant perverse sheaves on this variety, finding a proper geometric context for Vogan's character duality.

In fact one can go much further. Soergel conjectures a real local Langlands classification for the entire derived category of representations (Harish-Chandra modules) of a real group, which lifts Adams-Barbasch-Vogan's picture on K-groups. Roughly speaking this is a derived equivalence between equivariant perverse sheaves on group orbits on flag varieties for Langlands dual groups -- one side gets identified with reps via Beilinson-Bernstein, the other is the ABV Langlands parameters. One can specify this conjecture much more -- it is supposed to be equivariant for intertwining operators/ braid group actions on the two sides, and it has a very particular interaction with t-structures (Koszul duality).

(I would be very interested to learn to what extent one might expect p-adic analogues of any of these more refined versions of local Langlands -- yes, I know, first one might want to prove the original conjectures! - but still it's interesting to dream.)

Do we have any understanding of what should be the correct analogues of the Soergel or ABV conjecture in the non-Archimedean setting? How does it interact with the other aspects of the local Langlands program?

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    $\begingroup$ Please indicate the source for the quote. $\endgroup$ – Abdelmalek Abdesselam Jun 5 at 12:44

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