Fourier matrices for unipotent characters

[This is a work in progress with Bonnafé-Broué-Malle-Michel-Rouquier.] In this talk I will explain how to construct a non-trivial action of SL2(Z) on the vector space spanned by unipotent characters. The existence of such an action follows from Lusztig's classification of unipotent characters, but I will present a global construction relying on traces of automorphisms of Deligne--Lusztig varieties. The advantage of this approach is that it can be naturally generalized to Spetses, thus giving candidates for Fourier matrices and unipotent character sheaves for Spetses.

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