On the Structure of the Fundamental Series of Generalized Harish-Chandra Modules

Abstract

We continue the study of the fundamental series of generalized Harish-Chandra modules initiated in Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type. Generalized Harish-Chandra modules are $(\mathfrak{g}, \mathfrak{k})$-modules of finite type where $\mathfrak{g}$ is a semisimple Lie algebra and $\mathfrak{k} \subset \mathfrak{g}$ is a reductive in $\mathfrak{g}$ subalgebra. A first result of the present paper is that a fundamental series module is a $\mathfrak{g}$-module of finite length. We then define the notions of strongly and weakly reconstructible simple $(\mathfrak{g}, \mathfrak{k})$-modules $M$ which reflect to what extent $M$ can be determined via its appearance in the socle of a fundamental series module. In the second part of the paper we concentrate on the case $\mathfrak{k} \simeq sl(2)$ and prove a sufficient condition for strong reconstructibility. This strengthens our main result from Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type for the case $\mathfrak{k} = sl(2)$. We also compute the $sl(2)$-characters of all simple strongly reconstructible (and some weakly reconstructible) $(\mathfrak{g}, sl(2))$-modules. We conclude the paper by discussing a functor between a generalization of the category $\mathcal{O}$ and a category of $(\mathfrak{g}, sl(2))$-modules, and we conjecture that this functor is an equivalence of categories.