Gregor Kemper
Universitaet Heidelberg

"A Characterization of Linearly Reductive Groups by their Invariants",
Transformation Groups 5 (2000), 85-92. 

Abstract:

  The theorem of Hochster and Roberts says that for every module $V$
  of a linearly reductive group $G$ over a field $K$ the invariant
  ring $K[V]^G$ is Cohen-Macaulay. We prove the following converse: if
  $G$ is a reductive group and $K[V]^G$ is Cohen-Macaulay for every
  module $V$, then $G$ is linearly reductive.