Harish-Chandra module

Summary

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.

DefinitionEdit

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.

NotesEdit

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.

ReferencesEdit

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

See alsoEdit