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- W2963288672 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." @default.
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- W2963288672 date "2012-01-01" @default.
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- W2963288672 title "On the structure of the fundamental series of generalized Harish-Chandra modules" @default.
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- W2963288672 doi "https://doi.org/10.4310/ajm.2012.v16.n3.a8" @default.
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