Classification of multiplicity free symplectic representations
Friedrich Knop
备注：
25 pages; v2: minor changes; v3: typos fixed, final version; v3: As reported by Wen Chan, the table entries (1.8) and (11.2) contained mistakes and were corrected. The classification itself is not affected
设 G 为作用在有限维向量空间 V 上的连通约化群。假设 V 配备了一个 G-不变辛形式，那么多项式函数环 C[V] 成为 Poisson 代数。不变量环 C[V]^G 是子 Poisson 代数。若对所有不变量 f 和 g 均有 {f,g}=0，即 C[V]^G 是 Poisson 交换的，我们称 V 为无重数的。等价地，G 也作用于 Weyl 代数 W(V)，且 V 是无重数的当且仅当不变量子代数 W(V)^G 是交换的。本文分类了所有无重数辛表示。
英文摘要：
Let G be a connected reductive group acting on a finite dimensional vector space V. Assume that V is equipped with a G-invariant symplectic form. Then the ring C[V] of polynomial functions becomes a Poisson algebra. The ring C[V]^G of invariants is a sub-Poisson algebra. We call V multiplicity free if C[V]^G is Poisson commutative, i.e., if {f,g}=0 for all invariants f and g. Alternatively, G also acts on the Weyl algebra W(V) and V is multiplicity free if and only if the subalgebra W(V)^G of invariants is commutative. In this paper we classify all multiplicity free symplectic representations.
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