Harish–Chandra module of K-finite vectors - Imported Knowlpedia Content

Core idea

Let $G$ be a real reductive Lie group, $K$ a maximal compact subgroup, and $(\pi,V)$ an admissible finite-length continuous representation in a class for which the $K$ -finite vectors are smooth. Its Harish–Chandra module of $K$ -finite vectors is

V_K=\{v\in V:\dim\operatorname{span}_{\mathbb C}\pi(K)v<\infty\}.

The restricted $K$ -action and the differentiated action of $\mathfrak g_{\mathbb C}$ make $V_K$ a Harish–Chandra module: it is admissible and finitely generated over $U(\mathfrak g_{\mathbb C})$ . The construction retains the algebraic infinitesimal representation while discarding the ambient Banach, Hilbert, or Fréchet topology.

Construction of the actions

The $K$ -action is simply $\pi|_K$ . Smoothness lets one differentiate:

d\pi(X)v=\left.\frac{d}{dt}\right|_{t=0} \pi(\exp tX)v, \qquad X\in\mathfrak g,\ v\in V_K.

The two actions obey the covariance relation required of a $(\mathfrak g_{\mathbb C},K)$ -module, and the differentiated action extends to the universal enveloping algebra. Admissibility gives finite $K$ -multiplicities; finite length supplies the standard finite-generation hypothesis Wallach, Chapter 4.

What the construction preserves

The passage to $V_K$ preserves $K$ -types, their multiplicities, invariant infinitesimal submodules, and the action of the center of the universal enveloping algebra. For irreducible admissible representations, the resulting Harish–Chandra module is irreducible. Conversely, globalization theorems recover canonical smooth representations from Harish–Chandra modules, but a choice of Hilbert globalization is not part of this construction Knapp, Chapter VIII.

Example and scope

For an irreducible unitary representation of $G$ , Harish–Chandra admissibility makes its $K$ -finite subspace a Harish–Chandra module. An arbitrary continuous representation is a near-miss: without admissibility or finite length, its $K$ -finite part can have infinite $K$ -multiplicities or fail finite generation.

References

Nolan R. Wallach, Real Reductive Groups I, Academic Press, 1988. Publisher record. Relevant: Chapter 4 on admissible representations and their $K$ -finite modules.
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. DOI record. Relevant: Chapter VIII on admissibility and underlying $(\mathfrak g,K)$ -modules.