Bounded generalized Harish-Chandra modules
| dc.creator | Penkov, Ivan | |
| dc.creator | Serganova, Vera | |
| dc.date | 2007-10-03 | |
| dc.date.accessioned | 2026-07-07T08:34:04Z | |
| dc.date.available | 2026-07-07T08:34:04Z | |
| dc.description | Let $\gg$ be a complex reductive Lie algebra and $\kk\subset\gg$ be any reductive in $\gg$ subalgebra. We call a $(\gg,\kk)$-module $M$ bounded if the $\kk$-multiplicities of $M$ are uniformly bounded. In this paper we initiate a general study of simple bounded $(\gg,\kk)$-modules. We prove a strong necessary condition for a subalgebra $\kk$ to be bounded (Corollary \ref{cor1.6}), i.e. to admit an infinite-dimensional simple bounded $(\gg,\kk)$-module, and then establish a sufficient condition for a subalgebra $\kk$ to be bounded (Theorem \ref{thGroups2}). As a result we are able to classify all maximal bounded reductive subalgebras of $\gg=\sl(n)$. In the second half of the paper we describe in detail simple bounded infinite-dimensional $(\gg,\sl(2))$-modules, and in particular compute their characters and minimal $\sl(2)$-types. We show that if $\sl(2)$ is a bounded subalgebra of $\gg$ which is not contained in a proper ideal of $\gg$, then $\gg\simeq \sl(2)\oplus \sl(2), \sl(3),\sp(4)$; alltogether, up to conjugation there are five possible embeddings of $\sl(2)$ as a bounded subalgebra into $\gg$ as above. In two of these cases $\sl(2)$ is a symmetric subalgebra, and many results about simple bounded $(\gg,\sl(2))$-modules are known. A case where our results are entirely new is the case of a principal $\sl(2)$-subalgebra in $\sp(4)$. | |
| dc.identifier | https://arxiv.org/abs/0710.0906 | |
| dc.identifier | http://arxiv.org/abs/0710.0906 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/139306 | |
| dc.subject | Representation Theory | |
| dc.subject | 17B10; 22E46 | |
| dc.title | Bounded generalized Harish-Chandra modules | |
| dc.type | text |