Study of global structure and invariants of symplectic quotients

首先，我们证明了对于任意紧李群，Weyl模中不变子空间的渐近维度与多重张量积中不变子空间的渐近维度成正比。其次，我们得到了具有可能负权的向量配分函数的渐近行为的一个公式，它推广了一个已知的无权向量配分函数的公式。第三，在一定的假设条件下，我们引入了多重重量簇的概念，并得到了它的体积公式。

First, we showed that for any compact Lie group, the asymptotic dimension of the invariant subspace in the Weyl module is proportional to that of the invariant subspace in the multiple tensor product. Second, we obtained a formula for the asymptotic behavior of a vector partition function with possibly negative weights, which generalizes a known formula for a vector partition function without weights. Third, we introduced the notion of a multiple weight variety, and obtained a formula for its volume under certain assumptions.