Harish-Chandra module

Summary

A Harish-Chandra module is the algebraic shadow of a representation of a real reductive group $G$. It keeps the vectors that transform in a finite way under a maximal compact subgroup $K$. On them it keeps the actions of $K$ and of the complexified Lie algebra $\mathfrak g$, and it forgets the topology. Harish-Chandra showed that an admissible representation is determined, up to a natural equivalence, by its Harish-Chandra module. This turns questions about infinite-dimensional representations into questions of algebra.