Small infinite dimensional modules for algebraic groups

A infinite dimensional module for an algebraic group is called small provided every proper submodule is finite dimensional. Small infinite dimensional modules exist provided that the characteristic is zero and the group has a non-trivial unipotent radical. The unipotent radical is shown to act through an abelian quotient, which allows a description, up to finite dimensional quotients, of the SID modules with trivial module socle via equivariant commutative algebra. In the case that the group is in fact unipotent, this description is used to calculate the Hilbert function of the ascending socle series of the module.