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 ${\displaystyle ({\mathfrak {g}},K)}$-module, then its Harish-Chandra module is a representation with desirable factorization properties.

Let G be a Lie group and K a compact subgroup of G. If ${\displaystyle (\pi ,V)}$ is a representation of G, then the Harish-Chandra module of ${\displaystyle \pi }$ 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 ${\displaystyle \varphi _{v}:G\longrightarrow V}$ via

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