Harish-Chandra module
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 [math]\displaystyle{ (\mathfrak{g},K) }[/math]-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 [math]\displaystyle{ (\pi,V) }[/math] is a representation of G, then the Harish-Chandra module of [math]\displaystyle{ \pi }[/math] 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 [math]\displaystyle{ \varphi_v : G \longrightarrow V }[/math] via
- [math]\displaystyle{ \varphi_v(g) = \pi(g)v }[/math]
is smooth, and the subspace
- [math]\displaystyle{ \text{span}\{\pi(k)v : k\in K\} }[/math]
is finite-dimensional.
Notes
In 1973, Lepowsky showed that any irreducible [math]\displaystyle{ (\mathfrak{g},K) }[/math]-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 [math]\displaystyle{ (\mathfrak{g},K) }[/math]-module with a positive definite Hermitian form satisfying
- [math]\displaystyle{ \langle k\cdot v, w \rangle = \langle v, k^{-1}\cdot w \rangle }[/math]
and
- [math]\displaystyle{ \langle Y\cdot v, w \rangle = -\langle v, Y\cdot w \rangle }[/math]
for all [math]\displaystyle{ Y\in \mathfrak{g} }[/math] and [math]\displaystyle{ k\in K }[/math], 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 978-0-691-08482-4
See also
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