# 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 -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 is a representation of *G*, then the *Harish-Chandra module* of 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 via

is smooth, and the subspace

is finite-dimensional.

## Notes

In 1973, Lepowsky showed that any irreducible -module *X* is isomorphic to the Harish-Chandra module of an irreducible representation of *G* on a Hilbert space. Such representations are *admissible*, meaning that they decompose in a manner analogous to the prime factorization of integers. (Of course, the decomposition may have infinitely many distinct factors!) Further, a result of Harish-Chandra indicates that if *G* is a *reductive* Lie group with maximal compact subgroup *K*, and *X* is an irreducible
-module with a positive definite Hermitian form satisfying

and

for all and , then *X* is the Harish-Chandra module of a unique irreducible unitary representation of *G*.

## References

- Vogan, Jr., David A. (1987),
*Unitary Representations of Reductive Lie Groups*, Annals of Mathematics Studies,**118**, Princeton University Press, ISBN 978-0-691-08482-4