Character sheaves on unipotent groups in positive characteristic: foundations

By M. Boyarchenko, V. Drinfeld. Selecta Mathematica, 2008

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In this article we formulate and prove the main theorems of the theory of character sheaves on unipotent groups over an algebraically closed field of characteristic $$p>0$$p>0. In particular, we show that every admissible pair for such a group $$G$$G gives rise to an $$\mathbb{L }$$L-packet of character sheaves on $$G$$G and that conversely, every $$\mathbb{L }$$L-packet of character sheaves on $$G$$G arises from a (nonunique) admissible pair. In the Appendices we discuss two abstract category theory patterns related to the study of character sheaves. The first Appendix sketches a theory of duality for monoidal categories, which generalizes the notion of a rigid monoidal category and is close in spirit to the Grothendieck–Verdier duality theory. In the second one we use a topological field theory approach to define the canonical braided monoidal structure and twist on the equivariant derived category of constructible sheaves on an algebraic group; moreover, we show that this category carries an action of the surface operad. The third Appendix proves that the “naive” definition of the equivariant $$\ell $$ℓ-adic derived category with respect to a unipotent algebraic group is equivalent to the “correct” one.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/def584dee06e6fcac61be5d49a225f02541233c2

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A duality formalism in the spirit of Grothendieck and Verdier (2011, background): “row 1 −→e. Remark 2.5. In the situation of Example 0.10 with Gunipotent the categories D(G) and DG(G) have many closed idempotents, see [BD11, §1] (especially §1.11 and Theorems 1.41(a), 1.49(c) from [BD11]). If e∈Mis a closed idempotent, we set eMe:= X∈M X∼=e⊗Y⊗efor some Y ∈M . (2.1) The tensor pro”