Abstract : The space of class functions on a reductive group over a finite field (such as GL(n,q), Sp(2n,q), etc.) admits two particularly interesting bases:
- an algebraic basis, given by the characters of irreducible representations,
- a geometric basis, given by the characteristic functions of character sheaves.
In this series of lectures, we will explain how these two bases are related. It involves a transformation that generalizes the classical Fourier transform on finite abelian groups, which was discovered by Lusztig when classifying the irreducible characters.
We will provide a new approach to understand this transformation, using traces of braid group operators acting on Deligne–Lusztig varieties. This viewpoint leads to a natural SL2(Z)-action on the space of class functions, bringing together the Fourier transform, the Frobenius eigenvalues on the cohomology of Deligne–Lusztig varieties, and Shintani's twisting operator, which exchanges the Frobenius morphism with its inverse.
If time permits, we will also discuss applications to the theory of Spetses — conjectural generalizations of finite reductive groups whose combinatorics are governed by complex reflection groups rather than Weyl groups. In this setting, we propose natural candidates for Fourier matrices and unipotent character sheaves.
(This is a work in progress with Bonnafé-Broué-Malle-Michel-Rouquier.)
Lecture 1,
June 29, 8:00 pm-9:30 pm,
Zoom ID: 825 9880 3208
Passcode: 869756
Lecture 2,
June 30, 2:30 pm-4:00pm,
Zoom ID: 873 6285 0228
Passcode: 524286
Lecture 3,
July 1, 2:30 pm-4:00pm,
Zoom ID: 842 8183 8381
Passcode: 485372
