有理Cherednik代数的不可约非完整模
Irreducible non-holonomic modules for rational Cherednik algebras
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中文总结 AI 辅助
该研究针对特征零代数闭域上的Weyl代数不变微分算子环，构造出特定维数的不可约非完整模，利用Morita等价推导有理Cherednik代数的对应结果，还证明Weyl代数在指定维数区间内存在所有维数的不可约模。
中文摘要 AI 辅助
设𝕂为特征零的代数闭域，Aₙ(𝕂)为第n个Weyl代数。我们证明：对每个复反射群G及每个n≥2，不变微分算子环Aₙ(𝕂)^G具有Gelfand-Kirillov维数为2n−1的不可约非完整模，并明确构造出这类模。借助Aₙ(𝕂)^G与整参数下的有理Cherednik代数H_𝔠的Morita等价，我们推得对n≥2，H_𝔠具有Gelfand-Kirillov维数为2n−1的不可约非完整模。附录中，我们遵循O. Mathieu提供的证明，得出Aₙ(𝕂)在区间[n,2n−1]内的每个Gelfand-Kirillov维数都存在不可约模。
英文摘要
Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and let $A_n(\mathbb{K})$ be the $n$-th Weyl algebra. We prove that for every complex reflection group $G$ and every $n \geq 2$ the ring of invariant differential operators $A_n(\mathbb{K})^G$ has irreducible non-holonomic modules of Gelfand-Kirillov dimension $2n-1$, and we exhibit such modules explicitly. Through the Morita equivalence between $A_n(\mathbb{K})^G$ and the rational Cherednik algebra $H_{\mathfrak c}$ at an integral parameter, we deduce that $H_{\mathfrak c}$ has irreducible non-holonomic modules of Gelfand-Kirillov dimension $2n-1$ for $n \geq 2$. In an appendix, following a proof kindly shared by O. Mathieu, we show that $A_n(\mathbb{K})$ has irreducible modules of every Gelfand-Kirillov dimension in the interval $[n, 2n-1]$.
