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 $(\mathfrak,K)$-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\$

is finite-dimensional.

Notes

In 1973, Lepowsky showed that any irreducible $(\mathfrak,K)$-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
$(\mathfrak,K)$-module with a positive definite Hermitian form satisfying
:$\langle k\cdot v, w \rangle = \langle v, k^\cdot w \rangle$
and
:$\langle Y\cdot v, w \rangle = -\langle v, Y\cdot w \rangle$
for all $Y\in \mathfrak$ and $k\in K$, then ''X'' is the Harish-Chandra module of a unique irreducible unitary representation of ''G''.

References

*

See also

*(g,K)-module
*Admissible representation
*Unitary representation

{{DEFAULTSORT:Harish-Chandra Module
Representation theory of Lie groups