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Journal of Lie Theory 20 (2010), No. 3, 581--615
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

Vera Serganova
Dept. of Mathematics, University of California, Berkeley, CA 94720, U.S.A.


\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 Harish-Chandra modules}, preprint (2009),, 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 infinite-dimensional 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 infinite-dimensional 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 Harish-Chandra modules and our result singles out all Harish-Chandra 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: Harish-Chandra modules, bounded sl(2)-multiplicities, sl(2)-characters.

MSC: 17B10; 22E46

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