(I) Fourier transform as a triangular matrix / (II) Strata in reductive groups

Algebraic Geometry and Number Theory Seminar

(I) Fourier transform as a linear map from L^2(R) to L^2(R) has been diagonalized by Hermite in the late 1800's using Hermite polynomials.
We are interested in the Fourier transform F on the C-vector space of functions on a symplectic vector space over the field with two elements. We show that the following substitute of Hermite's result
holds: there is a remarkable C-basis of this vector space in which F acts as a triangular matrix.

(II) Let k_p be an algebraically closed field of characteristic p and let G_p be a reductive connected group over k_p of type independent of p; let W be the Weyl group of G_p. We define a partition of G_p into finitely many strata. Each stratum is a union of conjugacy classes of fixed dimension of G_p. The set of strata is independent of p.
It can be viewed as an enlargement of the set of unipotent classes of G_p. It can be identified with the image of a certain map from the set of conjugacy classes in W to the set of irreducible representations of W.