Fourier transforms and Frobenius eigenvalues for finite Coxeter groups

Lusztig's classification of the unipotent characters of a finite Chevalley or Steinberg group involves a certain non-abelian Fourier transformation. We construct analogous transformations for the Suzuki and Ree groups, based on a set of axioms derived from Lusztig's theory of character sheaves. We also determine Fourier matrices for the “spetses” (in the sense of Broue, Michel, and the second author) associated with twisted dihedral groups. This completes the determination of Fourier matrices for all “spetses” associated with finite Coxeter groups. We end by collecting common properties of these Fourier matrices and the eigenvalues of Frobenius of character sheaves and unipotent characters.