5 Admissible Semi - Linear Representations

The category of admissible (in the appropriately modified sense of representation theory of totally disconnected groups) semi-linear representations of the automorphism group of an algebraically closed extension of infinite transcendence degree of the field of algebraic complex numbers is described. Let k be a field of characteristic zero containing all ℓ-primary roots of unity for a prime ℓ, F be a universal domain over k, i.e., an algebraically closed extension of k of countable transcendence degree, and G F/k be the field automorphism group of F over k. We consider G F/k as a topological group with the base of open subgroups generated by {G F/k(x) | x ∈ F }.