Oct 17, 2024
(I) Fourier transform as a triangular matrix / (II) Strata in reductive groups
Algebraic Geometry and Number Theory Seminar
Date: October 17, 2024 |
1:00 pm –
3:00 pm
Speaker:
George Lusztig, MIT Mathematics
Location: Office Bldg West / Ground floor / Heinzel Seminar Room (I21.EG.101)
Language:
English
(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.
