2020/02/04 by Romanov, Anna
#14F10 #17B10 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2002.01540
Let \mathfrakg be a complex semisimple Lie algebra. The Beilinson-Bernstein localization theorem establishes an equivalence of the category of \mathfrakg-modules of a fixed infinitesimal character and a category of modules over a twisted sheaf of differential operators on the flag variety of \mathfrakg. In this expository paper, we give four detailed examples of this theorem when \mathfrakg=\mathfraksl(2,ℂ). Specifically, we describe the D-modules associated to finite-dimensional irreducible \mathfrakg-modules, Verma modules, Whittaker modules, discrete series representations of SL(2,ℝ), and principal series representations of SL(2,ℝ).