  We first propose a generalization of the image conjecture [Z3] for the
commuting differential operators related with classical orthogonal polynomials.
We then show that the non-trivial case of this generalized image conjecture is
equivalent to a variation of the Mathieu conjecture [Ma] from integrals of
$G$-finite functions over reductive Lie groups $G$ to integrals of polynomials
over open subsets of $\mathbb R^n$ with any positive measures. Via this
equivalence, the generalized image conjecture can also be viewed as a natural
variation of Duistermaat and van der Kallen's theorem [DK] on Laurent
polynomials with no constant terms. To put all the conjectures above in a
common setting, we introduce what we call the Mathieu subspaces of associative
algebras. We also discuss some examples of Mathieu subspaces from other sources
and derive some general results on this newly-introduced notion.
