Nonzero Infinitesimal Blocks of Category $\boldsymbol{\mathcal{O}}_{\textbf{\emph{S}}}$

Abstract

Category \(\mathcal{O}\) is a nice category of modules for a finite dimensional semisimple Lie algebra \(\mathfrak{g}\) It was first introduced by Bernstein–Gelfand–Gelfand in 1976. In the early 1980’s, Rocha–Caridi introduced parabolic category \(\mathcal{O}_S\), where S is a subset of simple roots. Category \(\mathcal{O}_S\) is a generalization of ordinary category O. These are highest weight categories that decompose into certain subcategories, called infinitesimal blocks. An infinitesimal block contains at most finitely many simple modules, and some contain only the zero module. The representation type of the infinitesimal blocks of category \(\mathcal{O}_S\) has been studied by Futorny–Nakano–Pollack, Brustle–Konig–Mazorchuk, and Boe–Nakano. The representation type of the singular blocks of category \(\mathcal{O}_S\) is still generally unknown, though Boe–Nakano classified certain of these. Understanding when a singular block is zero is an important step to understanding the singular blocks in general. In this work, we will answer this question. It is given in terms of nilpotent orbits of \(\mathfrak{g}\).