| Let G be a connected unipotent group over a finite field Fq . In this dissertation we define and study L-packets of irreducible characters of the finite group G( Fq ) in terms of certain geometric objects associated to G.;One definition of L-packets can be stated as follows. Let us fix a prime ℓ ≠ char( Fq ) and write DG (G) for the category of bounded constructible complexes of ℓ-adic sheaves on G that are equivariant with respect to the conjugation action of G on itself. It is a braided monoidal category under convolution with compact supports, defined by M * N = Rmu!( M ⊠ N) for all M, N ∈ DG (G), where mu : G x G → G is the multiplication morphism. For each M ∈ DG (G), let tM : G( Fq ) → Qℓ be the class function associated to M via the sheaves-to-functions dictionary.;We say that two irreducible representations rho1, rho 2 of G( Fq ) over Qℓ are L-indistinguishable if for every object e ∈ DG (G) with e * e ≅ e, the corresponding function te acts on rho1 and rho2 by the same scalar. The L-packets of irreducible representations are the equivalence classes for this equivalence relation.;The main results of our work provide a different, much more concrete, description of L-packets in terms of the so-called "admissible pairs" for G. Roughly speaking, these are pairs ( H, chi) consisting of a connected subgroup H ⊂ G and a multiplicative ℓ-adic local system chi on H satisfying a certain nondegeneracy condition. |
