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.

Admissible semi-linear representations

In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite field) are studied. Many results here are well-known to the experts, at least in the case of {\sl linear representations} of symmetric group. The presented results suggest, in particular, that an analogue of Hilbert's Theorem 90 should hold: in the case of faithful action of the group on the base field the irreducible smooth semilinear representations are one-dimensional (and trivial in appropriate sense).

On semilinear representations of the infinite symmetric group

Let $K$ be a field and $G$ be a group of its automorphisms. If $G$ is precompact then $K$ is a generator of the category of smooth (i.e. with open stabilizers) $K$-semilinear representations of $G$. There are non-semisimple smooth semilinear representations of $G$ over $K$ if $G$ is not precompact. 
In this note the smooth semilinear representations of the group $G$ of all permutations of an infinite set $S$ are studied. Let $k$ be a field and $k(S)$ be the field freely generated over $k$ by the set $S$ (endowed with the natural $G$-action). One of principal results describes the Gabriel spectrum of the category of smooth $k(S)$-semilinear representations of $G$. 
It is also shown, in particular, that (i) for any smooth $G$-field $K$ any smooth finitely generated $K$-semilinear representation of $G$ is noetherian, (ii) for any $G$-invariant subfield $K$ in the field $k(S)$, the object $k(S)$ is an injective cogenerator of the category of smooth $K$-semilinear representations of $G$, (iii) if $K\subset k(S)$ is the subfield of rational homogeneous functions of degree 0 then there is a one-dimensional $K$-semilinear representation of $G$, whose integral tensor powers form a system of injective cogenerators of the category of smooth $K$-semilinear representations of $G$, (iv) if $K\subset k(S)$ is the subfield generated over $k$ by $x-y$ for all $x,y\in S$ then there is a unique isomorphism class of indecomposable smooth $K$-semilinear representations of $G$ of each given finite length.

An analogue of Hilbert's Theorem 90 for infinite symmetric groups

This is a common introduction to math.RT/0101170, math.RT/0306333, math.RT/0506043, math.RT/0601028. Compared to these references there are new results including (i) a description of a separable closure of an extension of transcendence degree one of an algebraically closed field of positive characteristic; (ii) a "K\"unneth formula" for the products with curves; (iii) the semi-simplicity of the module of regular forms of top degree $\Omega^n_{F/k}$.

Representations of field automorphism groups

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 ${\mathcal V} \longrightarrow g^{\ast}{\mathcal V}$ for each $g\in H$ satisfying the chain rule. 
We construct, for any $n>1$, a fully faithful functor from the category of finite-dimensional $L$-semi-linear representations of $H$ extendable to the semi-group ${\rm End}(L/k)$ 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 \cite{rep}. The semi-group ${\rm End}(L/k)$ is considered as a subquotient of $G$, hence the condition on extendability. 
In the appendix it is shown that, if $\tilde{H}$ is either $H$, or a bigger subgroup in the Cremona group (generated by $H$ and a standard involution), then any semi-linear $\tilde{H}$-representation of degree one is an integral $L$-tensor power of $\det_L\Omega^1_{L/k}$. It is shown also that this bigger subgroup has no non-trivial representations of finite degree if $n>1$.

Semi-linear representations of PGL

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Semi-linear representations of PGL