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In mathematics, specifically in the representation theory of Lie groups, a Harish-Chandra module is a representation of a real Lie group, associated to a general representation, with regularity and finiteness conditions. When the associated representation is a (\mathfrak{g},K)-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 (\pi,V) is a representation of G, then the Harish-Chandra module of \pi 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 \varphi_v : G \longrightarrow V via

\( \varphi_v(g) = \pi(g)v \)

is smooth, and the subspace

\( \text{span}\{\pi(k)v : k\in K\} \)

is finite-dimensional.
Notes

In 1973, Lepowsky showed that any irreducible (\mathfrak{g},K)-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 (\mathfrak{g},K)-module with a positive definite Hermitian form satisfying

\( \langle k\cdot v, w \rangle = \langle v, k^{-1}\cdot w \rangle \)

and

\( \langle Y\cdot v, w \rangle = -\langle v, Y\cdot w \rangle \)

for all \( Y\in \mathfrak{g} \) and \( k\in K \) , 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 978-0691084824