复叠向量丛上不变微分算子的可交换性
Commutativity of invariant differential operators on vector bundles on Hermitian symmetric spaces
AI总结：
本文研究了Hermitian对称空间上由$K$表示定义的向量丛上$G$不变微分算子环的可交换性，并构造了本征函数并研究了Weyl群对本征值的影响。
摘要：
设$G/K$为Hermitian对称空间，$V_τ$为$K$的不可约表示。我们研究由$K$的有限维表示$(V_τ, τ)$定义的向量丛$G×_{(K, τ)} V_τ$上$G$不变微分算子环$\mathcal D^G(G/K, V_τ)$。我们分类使得$\mathcal D^G(G/K, V_τ)$可交换的不可约表示$(V_τ, τ)$。我们构造微分算子的本征函数，并研究在$G$的限制实根系下Weyl群对本征值的不变性。
英文摘要：
Let $G/K$ be a Hermitian symmetric space and $V_τ$ an irreducible representation of $K$. We study the ring $\mathcal D^G(G/K, V_τ)$ of $G$-invariant differential operators on sections of vector bundles $G\times_{(K, τ)} V_τ$ over $G/K$ defined by a finite-dimensional representation $(V_τ, τ)$ of $K$. We classify irreducible representations $(V_τ, τ)$ such that $\mathcal D^G(G/K, V_τ)$ is commutative. We construct eigenfunctions for the differential operators and study the invariance property of the eigenvalues under the Weyl group for the restricted real root system of $G$.
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