Reductive Lie group

Summary

A reductive Lie group is a Lie group whose Lie algebra is a reductive Lie algebra. Such a Lie algebra is the direct sum of its centre and a semisimple Lie algebra. The class contains the semisimple groups, the tori, and the general linear groups $GL_n(\R)$ and $GL_n(\C)$, as well as every compact Lie group. It is the natural setting for much of representation theory and harmonic analysis. One reason is that Levi factors of parabolic subgroups of reductive groups are again reductive, while those of semisimple groups need not be semisimple. Authors differ on the extra conditions they impose, such as finitely many components or conditions on the adjoint action.