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Apr 11, 2012 at 2:04 vote accept Alex B.
Feb 10, 2012 at 12:55 answer added Frieder Ladisch timeline score: 12
Feb 9, 2012 at 23:34 comment added Colin Reid For $l$ odd and different to $p$, see this paper by A. Weir: jstor.org/pss/2033424. Notice that they closely resemble the Sylow subgroups of symmetric groups. I don't know much this helps with the representation theory, though.
Feb 9, 2012 at 22:38 answer added Derek Holt timeline score: 15
Feb 9, 2012 at 19:56 answer added Jim Humphreys timeline score: 3
Feb 9, 2012 at 19:34 comment added Geoff Robinson R.Carter and P.Fong wrote a paper about the structure of Sylow $q$-subgroups of finite groups of Lie type when $q$ is a prime other than the natural characteristic of the group. Whether it would answer a question as precise as your semi-conjecture is another matter.
Feb 9, 2012 at 17:57 history edited Alex B. CC BY-SA 3.0
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Feb 9, 2012 at 17:40 history asked Alex B. CC BY-SA 3.0