Varieties of $G_r$-summands in Rational $G$-modules

Let $G$ be a simple simply connected algebraic group over an algebraically closed field $k$ of characteristic $p$, with $r$-th Frobenius kernel $G_r$. Let $M$ be a $G_r$-module and $V$ a rational $G$-module. We put a variety structure on the set of all $G_r$-summands of $V$ that are isomorphic to $M$, and study basic properties of these varieties. We give a few applications of this work to the representation theory of $G$, primarily in providing some sufficient conditions for when a $G_r$-module decomposition of $V$ can be extended to a $G$-module decomposition. In particular we are interested in connections to Donkin's tilting module conjecture, and more generally to the problem of finding a $G$-structure for the projective indecomposable $G_r$-modules. To that end, we show that Donkin's conjecture is equivalent to determining the linearizability or non-linearizability of $G$-actions on certain affine spaces.