Many features of the group GL_n over a field with q elements, including
its Sylow theory, character values, and modular
representation theory, suggest that the group should be viewed as the
specialization for x=q of a mysterious object.
The same applies to other reductive groups. Moreover, while the Weyl
group of an actual reductive
group must be a reflection group over Q, other reflection groups over
the complex numbers apparently give rise to other
mysterious objects. We present some evidence for these speculations.

Keywords:

Hyperplane arrangements; Sylow theorems; group characters; Levi subgroup.

Lecture #10719

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