On symplectic resolutions and factoriality of Hamiltonian reductions

Recently, Herbig–Schwarz–Seaton have shown that 3-large representations of a reductive group G give rise to a large class of symplectic singularities via Hamiltonian reduction. We show that these singularities are always terminal. We show that they are Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Q}}$$\end{document}-factorial if and only if G has finite abelianization. When G is connected and semi-simple, we show they are actually locally factorial. As a consequence, the symplectic singularities do not admit symplectic resolutions when G is semi-simple. We end with some open questions.