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 ${\displaystyle ({\mathfrak {g}},K)}$-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 ${\displaystyle (\pi ,V)}$ is a representation of G, then the Harish-Chandra module of ${\displaystyle \pi }$ 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 ${\displaystyle \varphi _{v}:G\longrightarrow V}$ via

${\displaystyle \varphi _{v}(g)=\pi (g)v}$

is smooth, and the subspace

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

is finite-dimensional.

Notes

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

${\displaystyle \langle k\cdot v,w\rangle =\langle v,k^{-1}\cdot w\rangle }$

and

${\displaystyle \langle Y\cdot v,w\rangle =-\langle v,Y\cdot w\rangle }$

for all ${\displaystyle Y\in {\mathfrak {g}}}$ and ${\displaystyle k\in K}$, 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 978-0-691-08482-4

See also

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