Hyperplane arrangement and semisimple orbits
by
Jason Fulman
Dartmouth College

We describe conjectures about counting semisimple orbits of the adjoint action of a finite group of Lie type on its Lie algebra. By a theorem of Steinberg the total number of such orbits is q to the rank of the group, but their description remains a difficult problem. Grouping these orbits by "genus" suggests a formula involving counting solutions of equations which arose in work of Sommers on affine flag manifolds. These expressions have a quite different flavor from work of Lehrer in type A. Grouping these orbits by the conjugacy class of the Weyl group which they map to suggests formula involving characteristic polynomials of sublattices of the root hyperplane intersection lattice. These ideas tie in with a natural notion of riffle shuffling for real hyperplane arrangements.

http://www.dartmouth.edu/~jfulman

Date received: May 27, 1999