Bounded generalized Harish-Chandra modules

Let $\gg$ be a complex reductive Lie algebra and $\text{kk}\subset \gg$ be any reductive in $\gg$ subalgebra. We call a $(\gg ,\text{kk})$-module $M$ bounded if the $\text{kk}$-multiplicities of $M$ are uniformly bounded. In this paper we initiate a general study of simple bounded $(\gg ,\text{kk})$-modules. We prove a strong necessary condition for a subalgebra $\text{kk}$ to be bounded (Corollary \ref{cor1.6}), i.e. to admit an infinite-dimensional simple bounded $(\gg ,\text{kk})$-module, and then establish a sufficient condition for a subalgebra $\text{kk}$ to be bounded (Theorem \ref{thGroups2}). As a result we are able to classify all maximal bounded reductive subalgebras of $\gg =\text{sl}(n)$. In the second half of the paper we describe in detail simple bounded infinite-dimensional $(\gg ,\text{sl}(2))$-modules, and in particular compute their characters and minimal $\text{sl}(2)$-types. We show that if $\text{sl}(2)$ is a bounded subalgebra of $\gg$ which is not contained in a proper ideal of $\gg$, then $\gg \simeq \text{sl}(2)\oplus \text{sl}(2),\text{sl}(3),\text{sp}(4)$; alltogether, up to conjugation there are five possible embeddings of $\text{sl}(2)$ as a bounded subalgebra into $\gg$ as above. In two of these cases $\text{sl}(2)$ is a symmetric subalgebra, and many results about simple bounded $(\gg ,\text{sl}(2))$-modules are known. A case where our results are entirely new is the case of a principal $\text{sl}(2)$-subalgebra in $\text{sp}(4)$.