Asian Journal of Mathematics

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

Ivan Penkov and Gregg Zuckerman

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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.

Article information

Source
Asian J. Math., Volume 16, Number 3 (2012), 489-514.

Dates
First available in Project Euclid: 23 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.ajm/1353696019

Mathematical Reviews number (MathSciNet)
MR2989232

Zentralblatt MATH identifier
1281.17011

Subjects
Primary: 17B10: Representations, algebraic theory (weights) 17B55: Homological methods in Lie (super)algebras

Keywords
Generalized Harish-Chandra module fundamental series minimal $sl(2)$-type

Citation

Penkov, Ivan; Zuckerman, Gregg. On the Structure of the Fundamental Series of Generalized Harish-Chandra Modules. Asian J. Math. 16 (2012), no. 3, 489--514. https://projecteuclid.org/euclid.ajm/1353696019


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