# A First Construction of Autodual Objects

# A First Construction of Autodual Objects

This chapter presents the construction of autodual objects. It begins with a geometrically irreducible middle extension sheaf Ƒ on *G/k* which is ι-pure of weight zero, and which is not geometrically an Lᵪ. Thus Ƒ(1/2)[1] is a geometrically irreducible object in in G_{arith}. Its dual in in G_{arith} is [*x* ↦ 1/*x*]★ ℱ¯ (1/2)[1], for 𝓕 with hook and macron the linear dual middle extension sheaf. Via ι, Ƒ and 𝓕 with hook and macron have complex conjugate trace functions; this holds by ι-purity on the dense open set where Ƒ is lisse, and then on all of 𝓖 by a result of Gabber [Fuj-Indep, Thm. 3], cf. also [Ka-MMP, proof of 1.8.1 (i)].

*Keywords:*
number theory, autoduality, autodual objects

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