Semilinear representations of PGL

Abstract

Let L be the function field of a projective space $$\mathbb{P}^{n}_{k} $$ over an algebraically closed field k of characteristic zero, and H be the group of projective transformations. An H-sheaf $$\mathcal{V}$$ on $$\mathbb{P}^{n}_{k} $$ is a collection of isomorphisms $$\mathcal{V} \to g^{ * } \mathcal{V}$$ for each g ∈ H satisfying the chain rule. We construct, for any n > 1, a fully faithful functor from the category of finite-dimensional L-semilinear representations of H extendable to the semigroup End(L/k) to the category of coherent H-sheaves on $$\mathbb{P}^{n}_{k} .$$ The paper is motivated by a study of admissible representations of the automorphism group G of an algebraically closed extension of k of countable transcendence degree undertaken in [4]. The semigroup End(L/k) is considered as a subquotient of G, hence the condition on extendability. In the appendix it is shown that, if $${ \ifmmode\expandafter\tilde\else\expandafter\~\fi{H}}$$ is either H, or a bigger subgroup in the Cremona group (generated by H and a certain pair of involutions), then any semilinear $$ \ifmmode\expandafter\tilde\else\expandafter\~\fi{H}{\text{ - representation}}$$ of degree one is an integral L-tensor power of $$\hbox{det}_{L} \Omega ^{1}_{{L/k}} .$$ It is also shown that this bigger subgroup has no non-trivial representations of finite degree if n > 1.