Editing Lie algebra representation (section)

== (g,K)-module ==
{{main|(g,K)-module|Harish-Chandra module}}

One of the most important applications of Lie algebra representations is to the representation theory of real reductive Lie groups. The application is based on the idea that if <math>\pi</math> is a Hilbert-space representation of, say, a connected real semisimple linear Lie group ''G'', then it has two natural actions: the complexification <math>\mathfrak{g}</math> and the connected [[maximal compact subgroup]] ''K''. The <math>\mathfrak{g}</math>-module structure of <math>\pi</math> allows algebraic especially homological methods to be applied and <math>K</math>-module structure allows harmonic analysis to be carried out in a way similar to that on connected compact semisimple Lie groups.