Geometric rigidity of simple modules for algebraic groups
Michael Bate, David I. Stewart
TL;DR
This work proves that every simple G-module is geometrically rigid, by identifying a finite purely inseparable field extension k_V/k after which the module becomes absolutely rigid; it then provides a concrete description of k_V using endomorphism algebras, Jacobson radicals, and base-change techniques. The first half develops a general framework for rigidity via algebras End_G(V), Morita theory, and minimal fields of definition, including a non-rigid tensor-product example that shows limitations. The second half applies high-weight theory and the Conrad–Prasad classification of pseudo-reductive groups to give explicit constructions of k_V and End_G(V), and to derive a dimension formula and structural understanding of End_G(V) in terms of purely inseparable field data. Overall, the paper connects rigidity under base change to precise arithmetic data from pseudo-reductive group theory, providing explicit invariants and descriptions for simple modules across base fields.
Abstract
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.