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