Bounded Simple (g, sl(2))-Modules for rk g = 2

This paper is a continuation of our work [PS2] 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) g which yield infinite-dimensional bounded simple (g, sl(2)) -modules.Our main result is a detailed description of the bounded simple (g, sl(2))modules in all five cases.When g sl(2) sl(2) we reproduce in modern terms some classical results from the 1940's.When g 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) .