Groups of finite Morley rank with a pseudoreflection action

In this work, we give two characterisations of the general linear group over an algebraically closed field as a group G of finite Morley rank acting on an abelian connected group V of finite Morley rank definably (in the sense that G⋉V is a group of finite Morley rank in which G and V are definable), faithfully and irreducibly. We prove that if the pseudoreflection rank of G is equal to the Morley rank of V, then V has a definable vector space structure over an algebraically closed field, G≅GL(V) and the action is the natural action. The same result holds also under the assumption of Prüfer 2-rank of G being equal to the Morley rank of V.