In mathematics, specifically in the representation theory of Lie groups, a Harish-Chandra

Harish-Chandra

Harish-Chandra was an Indian mathematician, who did fundamental work in representation theory, especially Harmonic analysis on semisimple Lie groups. -Life:...

 module is a representation of a real Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

, 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 vector space X of K-finite

K-finite

In mathematics, a K-finite function is a type of generalized trigonometric polynomial. Here K is some compact group, and the generalization is from the circle group T....

 smooth vectors in V. That is, for every , the map via


is smooth, and the subspace


is finite-dimensional.

See also

• (g,K)-module
  (g,K)-module
  In mathematics, more specifically in the representation theory of reductive Lie groups, a -module is an algebraic object, first introduced by Harish-Chandra, used to deal with continuous infinite-dimensional representations using algebraic techniques...
  
  
• Admissible representation
  Admissible representation
  In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra....
  
  
• Unitary representation
  Unitary representation
  In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...