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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>(\mathfrak{g},K)</math>-module, then its Harish-Chandra module is a representation with desirable factorization properties.
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==Definition==
Let ''G'' be a Lie group  and ''K'' a compact [[subgroup]] of ''G''. If <math>(\pi,V)</math> is a representation of ''G'', then the ''Harish-Chandra module'' of <math>\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>\varphi_v : G \longrightarrow V</math> via
 
:<math>\varphi_v(g) = \pi(g)v</math>
 
is smooth, and the subspace
 
:<math>\text{span}\{\pi(k)v : k\in K\}</math>
 
is finite-dimensional.
 
==Notes==
In 1973, Lepowsky showed that any irreducible <math>(\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>(\mathfrak{g},K)</math>-module with a positive definite Hermitian form satisfying
:<math>
\langle k\cdot v, w \rangle = \langle v, k^{-1}\cdot w \rangle
</math>
and
:<math>
\langle Y\cdot v, w \rangle = -\langle v, Y\cdot w \rangle
</math>
for all <math>Y\in \mathfrak{g}</math> and <math>k\in K</math>, then ''X'' is the Harish-Chandra module of a unique irreducible unitary representation of&nbsp;''G''.
==References==
*{{Citation
| last=Vogan, Jr.
| first=David A.
| title=Unitary Representations of Reductive Lie Groups
| year=1987
| publisher=Princeton University Press
| series=Annals of Mathematics Studies
| volume=118
| isbn=978-0-691-08482-4
}}
 
==See also==
*[[(g,K)-module]]
*[[Admissible representation]]
*[[Unitary representation]]
 
{{DEFAULTSORT:Harish-Chandra Module}}
[[Category:Representation theory of Lie groups]]

Latest revision as of 18:12, 6 December 2014

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