象猜想和Mathieu猜想的推广
Generalizations of the Image Conjecture and the Mathieu Conjecture
摘要：
本文首先对与经典正交多项式有关的交换微分算子的象猜想[Z3]进行了推广。然后，我们表明，非平凡的情况下，这种广义的图像猜想是等价的Mathieu猜想[马]的变化，从积分的$G$ -有限功能约化李群$G$积分多项式开子集的$\mathbb R^n$与任何积极的措施。通过这种等价性，广义象猜想也可以看作是Duistermaat和van der Kallen关于无常数项的Laurent多项式定理[DK]的自然变形.为了把上面的所有结构放在一个公共的环境中，我们引入了我们称之为结合代数的Mathieu子空间。我们还讨论了一些例子Mathieu子空间从其他来源，并得出一些一般性的结果，这个新引入的概念。
英文摘要：
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.
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