Singular localization of \(\mathfrak{g}\)-modules and applications to representation theory (Q894192)

This paper develops a singular version of Beilinson-Bernstein localization theory for a complex semisimple Lie algebra. The main result is a singular version of the localization theorem which connects Lie algebra modules having a prescribed (singular) central character with certain twisted differential operators. As an application, various known results are reproved. In particular, a theorem of Bernstein and S.~Gelfand which connects category \(\mathcal{O}\) to Harish-Chandra bimodules; and a theorem of Miličić and Soergel which connects category \(\mathcal{O}\) to a certain category of Whittaker modules.