Geometric rigidity of simple modules for algebraic groups

Let k be a field, let G be an affine algebraic k-group and V a finite-dimensional G-module. We say V is rigid if the socle series and radical series coincide for the action of G on each indecomposable summand of V; say V is geometrically rigid (resp. absolutely rigid) if V is rigid after base change of G and V to k (resp. any field extension of k). We show that all simple G-modules are geometrically rigid, though not in general absolutely rigid. More precisely, we show that if V is a simple G-module, then there is a finite purely inseparable extension kV /k naturally attached to V such that V is absolutely rigid as a G-module after base change to kV. The proof turns on an investigation of algebras of the form K otimes E where K and E are field extensions of k; we give an example of such an algebra which is not rigid as a module over itself. We establish the existence of the purely inseparable field extension kV /k through an analogous version for artinian algebras.
In the second half of the paper we apply recent results on the structure and representation theory of pseudo-reductive groups to give a concrete description of kV when G is smooth and connected. Namely, we combine the main structure theorem of the Conrad-Prasad classification of pseudo-reductive G together with our previous high weight theory. For V a simple G-module, we calculate the minimal field of definition of the geometric Jacobson radical of EndG(V) in terms of the high weight of V and the Conrad-Prasad classification data; this gives a concrete construction of the field kV as a subextension of the minimal field of definition of the geometric unipotent radical of G. We also observe that the Conrad-Prasad classification can be used to hone the dimension formula for V we had previously established; we also use it to give a description of EndG(V) which includes a dimension formula.