---
tag: 2MKHH636C3
kind: concept
name: Harish-Chandra module
msc: ["22E47", "22E46"]
sources:
  - url: https://en.wikipedia.org/wiki/(g,K)-module
same_as:
  wikidata: Q4544947
  wikipedia: (g,K)-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|SZV8Y806Y7]] is determined, up to a natural equivalence, by its Harish-Chandra module.
This turns questions about infinite-dimensional representations into questions of algebra.

## Explanations

### L1 human

A representation of a group like $\mathrm{SL}_2(\R)$ acts on a huge space of functions, where analysis is hard.
Inside it sit the vectors that only move in a finite-dimensional way under rotations, such as trigonometric polynomials on a circle.
On these vectors the group no longer acts, but its infinitesimal version, the Lie algebra, still does, together with the rotations.
The resulting algebraic object is the Harish-Chandra module.
For $\mathrm{SL}_2(\R)$ it can be drawn as a row of points, one per frequency $n$, with arrows that raise or lower $n$ by $2$.