Geometric rigidity of simple modules for algebraic groups

TL;DR

This paper investigates the geometric rigidity of simple modules for algebraic groups, proving their rigidity after a specific inseparable field extension and providing explicit descriptions of their endomorphism algebras.

Contribution

It introduces the concept of geometric and absolute rigidity for modules, proves all simple modules are geometrically rigid, and describes the associated field extensions and endomorphism structures.

Findings

All simple G-modules are geometrically rigid.

Existence of a finite purely inseparable extension for absolute rigidity.

Concrete description of endomorphism algebras and their dimensions.

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.