Given a reductive algebraic group $G$ and an invariant Hilbert function $h$, Alexeev and Brion have defined a moduli scheme $M$ which parametrizes affine $G$-schemes $X$ with the property that the coordinate ring of $X$ decomposes, as $G$-module, according to the function $h$. The talk will be about joint work with Bart Van Steirteghem (New York) which studies the moduli scheme $M$ under some additional assumptions.