A parametrization of unipotent representations

We define a map from the unipotent representations of a split semisimple group over a finite field to (essentially) the set of pairs of left cell representations of the Weyl group in the same two-sided cell.We use this map to parametrize the unipotent representations. 0.1.Let G be a simple algebraic group defined and split over a finite field F q .Let U be the set of isomorphism classes of irreducible unipotent representations (over C) of the finite group G(F q ).Let W be the Weyl group of G and let Irr(W ) be the set of isomorphism classes of irreducible representations (over C) of W .In [2] a partition of Irr(W ) into families is described and in [4] a partition U = ⊔ c U c of U (with c running over the families of Irr(W )) is introduced.Moreover, in [4, §4] to any family c we have associated a finite group G c and a bijection (a)U c ↔ M (G c ).Here, for any finite group Γ, M (Γ) is the set of Γ-conjugacy classes of pairs (x, ρ) were x ∈ Γ and ρ is an irreducible representation (over C) of the centralizer Z Γ (x) of x in Γ; let C[M (Γ)] (resp.N[M (Γ)]) be the vector space of formal C-linear combinations of elements in M (Γ) and let A Γ : C[M (Γ)] → C[M (Γ)] be the non-abelian Fourier transform of [2] (a linear isomorphism with square 1).Let N[M (Γ)] (resp.R ≥0 [M (Γ)]) be the set of vectors of C[M (Γ)] which are linear combinations with coefficients in N