Limiting Distributions of Scaled Eigensections in a GIT-Setting

Let $\mathrm{\mathbf{L}\rightarrow \mathbf{X}}$ be a base point free $\mathrm{\mathbb{T}=T^{\mathbb{C}}}$-linearized hermitian line bundle over a compact variety $\mathrm{\mathbf{X}}$ where $\mathrm{T=\left(S^{1}\right)^{m}}$ is a real torus. The main focus of this paper is to describe the asymptotic behavior of a certain class of sequences $\mathrm{\left(s_{n}\right)_{n}}$ of $\mathrm{\mathbb{T}}$-eigensections $\mathrm{s_{n}\in H^{0}\left(\mathbf{X},\mathbf{L}^{n}\right)}$ as $\mathrm{n\rightarrow \infty}$, introduced by Shiffman, Tate and Zelditch, and its connection to the geometry of the Hilbert quotient $\mathrm{π\!:\!\bf{X}^{ss}_ξ\rightarrow \mathbf{X}^{ss}_ξ/\!\!/\mathbb{T}}$ where $\mathrm{ξ\in \mathfrak{t}^{*}}$.

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