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