1 Rationality Properties of Unipotent Representations
نویسنده
چکیده
0.1. Let k be an algebraic closure of a finite field Fq with q elements. Let G be a connected simple algebraic group of adjoint type over k with a fixed Fq-rational structure; let F : G −→ G be the corresponding Frobenius map. The fixed point set G is a finite group. Let W be the Weyl group of G. For w ∈ W let Rw be the character of the virtual representation R(w) of G defined in [DL, 1.5]. (The definition of Rw is in terms of l-adic cohomology but in fact Rw has integer values and is independent of l, see [DL, 3.3].) An irreducible representation ρ of G over C is said to be unipotent if its character χρ : G F −→ C occurs with 6= 0 multiplicity in Rw for some w ∈ W (see [DL, 7.8]). Let U be the set of isomorphism classes of unipotent representations of G . Let ŨQ = {ρ ∈ U|χρ(g) ∈ Q ∀g ∈ G }. Let UQ be the set of all ρ ∈ U such that ρ is defined over Q (that is, it can be realized by a Q[G ]-module). We have UQ ⊂ ŨQ ⊂ U . Unless otherwise specified, we assume that G is split over Fq. The following is one of our results.
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