
Journal of Lie Theory 20 (2010), No. 3, 581615 Copyright Heldermann Verlag 2010 Bounded Simple (g, sl(2))Modules for rk g = 2 Ivan Penkov Jacobs University Bremen, School of Engineering and Science, Campus Ring 1, 28759 Bremen, Germany i.penkov@jacobsuniversity.de Vera Serganova Dept. of Mathematics, University of California, Berkeley, CA 94720, U.S.A. serganov@math.berkeley.edu [Abstractpdf] \def\g{{\frak g}} \def\k{{\frak k}} \def\sL{\mathop{\rm sl}\nolimits} \def\sp{\mathop{\rm sp}\nolimits} This paper is a continuation of our work {\it On bounded generalized HarishChandra modules}, preprint (2009), math.jacobsuniversity.de/penkov, in which we prove some general results about simple $(\g, \k)$modules with bounded $\k$multiplicities (or bounded simple $(\g, \k)$modules). In the absence of a classification of bounded simple $(\g, \k)$modules in general, it is important to understand some special cases as best as possible. Here we consider the case $\k=\sL(2)$. It turns out that in order for an infinitedimensional bounded simple $(\g, \sL(2))$module to exist, $\g$ must have rank 2, and, up to conjugation, there are five possible embeddings $\sL(2)\rightarrow \g$ which yield infinitedimensional bounded simple $(\g, \sL(2))$modules. \par Our main result is a detailed description of the bounded simple $(\g, \sL(2))$modules in all five cases. When $\g \simeq \sL(2)\oplus \sL(2)$ we reproduce in modern terms some classical results from the 1940's. When $\g \simeq \sL(3)$ and $\sL(2)$ is a principal subalgebra, bounded simple $(\sL(3), \sL(2))$modules are HarishChandra modules and our result singles out all HarishChandra modules with bounded $\sL(2)$multiplicities. A case where the result is entirely new is the case of a principal $\sL(2)$subalgebra of $\g=\sp(4)$. Keywords: HarishChandra modules, bounded sl(2)multiplicities, sl(2)characters. MSC: 17B10; 22E46 [ Fulltextpdf (338 KB)] for subscribers only. 