In radial quantization, the ground states of a gauge theory on ADE singularities ℝ4/Γ are characterized by flat connections that are maps from Γ to the gauge group. We study Class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} theory of type a1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{a}}_1 $$\\end{document} = su2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{su}(2) $$\\end{document} on a Riemann surface of genus g > 1, without punctures. The fundamental building block of Class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} theory is the trifundamental Trinion theory — a low energy limit of two M5 branes compactified on the three-punctured Riemann sphere. We show, through the superconformal index, that the supersymmetric Casimir energy of the trifundamental theory imposes a constraint on the set of allowed flat connections, which agrees with the prediction of a duality relating the ground state Hilbert space of Class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} on ADE singularities to the Hilbert space of a certain dual Chern-Simons theory whose gauge group is given by the McKay correspondence. The conjecture is shown to hold for Γ = ℤk, agreeing with the previous results of Benini et al. and Alday et al. A non-abelian generalization of this duality is analyzed by considering the example of the dicyclic group Γ = Dic2, corresponding to Chern-Simons gauge group SO(8).