Simple modules for SL_2 over the algebraic closure of a finite field in cross characteristic — where the representation is over a field whose characteristic differs from the group's defining characteristic — form a rich but classifiable family. The classification depends on which modules admit T-stable lines, where T is the diagonal torus of SL_2.
A T-stable line is a one-dimensional subspace preserved by the torus action. Its existence constrains the module's weight structure: the module must have a weight space of dimension at least 1 at a specific weight determined by the T-action. In the cross-characteristic setting, the weight lattice has a different structure than in the defining characteristic, and the existence of T-stable lines becomes a subtle condition.
The classification determines exactly which simple modules admit T-stable lines, in terms of the highest weight and the relationship between the two characteristics. The answer depends on the residue of the highest weight modulo the characteristic of the representation field — a discrete condition that partitions the simple modules into finitely many families.
The result completes a piece of the modular representation theory of SL_2 that connects the defining-characteristic theory (where Steinberg's tensor product theorem applies) to the cross-characteristic theory (where the weight structure is governed by different combinatorics).
A classification problem in the simplest nonabelian algebraic group, made nontrivial by the interaction of two different prime characteristics. The T-stable line is the diagnostic: its presence or absence sorts the simple modules.