Kazhdan-laumon Representations of Chevalley Groups, Character Sheaves and Some Generalization of the Lefschetz-verdier Trace Formula

Let G be a reductive group over a finite field Fq and let G(Fq) be its group of Fq-rational points. In 1976 P. Deligne and G. Lusztig (following a suggestion of V. Drinfeld for G = SL(2)) associated to any maximal torus T in G defined over Fq certain remarkable series (virtual) representations of the group G(Fq). These representations were realized in é tale l-adic cohomology of certain algebraic varieties over Fq, (called the Deligne-Lusztig varieties). Using these representations G. Lusztig has later given a complete classification of representations of G(Fq). He has also discovered that characters of these representations might be computed from certain geometrically defined l-adic perverse sheaves on the group G, called character sheaves. Despite the fact that both Deligne-Lusztig representations and character sheaves are defined by some simple (and beautiful) algebro-geometric procedures, the proof of the relation between the two is rather complicated (apart from the case, when T is split, where it becomes just the usual formula for the character of an induced representation). In 1987 D. Kazhdan and G. Laumon proposed a different (conjectural) geometric way for constructing representations of G(Fq). Their idea was based on exploiting the generalized Fourier-Deligne on the basic affine space X = G/U of G (here U is a maximal unipotent subgroup of G). In the first part of this paper we give a rigorous definition of (almost all) Kazhdan-Laumon representations. Namely for every Fq-rational maximal torus T in G and every quasi-regular (non-singular in the terminology of [6]) character θ : T (Fq) → Ql ∗ we construct certain representation Vθ of G(Fq). We show that this representation is finite-dimensional and that it is irreducible if θ is regular (in a generic position in the terminology of [6]). Moreover, it follows from the proof of irreducibility in essentially tautological way, that the character of Vθ is given by the trace function of the corresponding Lusztig’s character sheaf. Roughly speaking, all the above is achieved by replacing the generalized Fourier-Deligne transformations on X by the suitable version of Radon transformations. It follows from the above computation of the character of Vθ, that Vθ is equivalent to the corresponding Deligne-Lusztig representation Rθ. The second part of this paper is devoted to the construction of an explicit isomorphism between the two. This is done by using certain generalization of the Lefschetz-Verdier trace formula for Radon transformations, which we cannot establish in the full generality (however, we believe that this is a technical difficulty, which is there only because of authors’ laziness).