Harish-Chandra module

In mathematics, specifically in the representation theory of Lie groups, a Harish-Chandra module, named after the Indian mathematician and physicist Harish-Chandra, is a representation of a real Lie group, associated to a general representation, with regularity and finiteness conditions. When the associated representation is a $$(\mathfrak{g},K)$$-module, then its Harish-Chandra module is a representation with desirable factorization properties.

Definition
Let G be a Lie group and K a compact subgroup of G. If $$(\pi,V)$$ is a representation of G, then the Harish-Chandra module of $$\pi$$ is the subspace X of V consisting of the K-finite smooth vectors in V. This means that X includes exactly those vectors v such that the map $$\varphi_v : G \longrightarrow V$$ via


 * $$\varphi_v(g) = \pi(g)v$$

is smooth, and the subspace


 * $$\text{span}\{\pi(k)v : k\in K\}$$

is finite-dimensional.