Harish-Chandra module

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

• {{Citation