Finite dimensional irreducible representations of finite W-algebras associated to even multiplicity nilpotent orbits in classical Lie algebras

We consider finite W-algebras \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${U(\mathfrak{g},e)}$$\end{document} associated to even multiplicity nilpotent elements in classical Lie algebras. We give a classification of finite dimensional irreducible \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${U(\mathfrak{g},e)}$$\end{document}-modules with integral central character in terms of the highest weight theory from Brundan et al. (Int. Math. Res. Notices 15, art. ID rnn051, 2008). As a corollary, we obtain a parametrization of primitive ideals of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${U(\mathfrak{g})}$$\end{document} with associated variety the closure of the adjoint orbit of e and integral central character.