Harish–Chandra admissibility theorem

Statement

Let $G$ be a real reductive group in the Harish–Chandra class, and let $K$ be a maximal compact subgroup. The Harish–Chandra admissibility theorem states that every irreducible unitary representation $(\pi,\mathcal H)$ of $G$ is admissible: for each irreducible finite-dimensional representation $\tau$ of $K$ ,

\dim\operatorname{Hom}_K(E_\tau,\mathcal H)<\infty.

Equivalently, every $K$ -type occurs in $\pi|_K$ with finite multiplicity. The theorem bounds each multiplicity separately; it does not say that only finitely many $K$ -types occur.

Why the conclusion is substantial

Compact-group theory decomposes the restriction $\pi|_K$ into $K$ -isotypic Hilbert subspaces, but compactness alone does not force their multiplicities to be finite. For example, an infinite Hilbert direct sum of the trivial representation of $K$ has infinite trivial-type multiplicity. The theorem uses irreducibility for the noncompact group $G$ , together with the structure of real reductive groups, to rule out this behavior.

Algebraic consequence

The $K$ -finite vectors $\mathcal H_K$ are dense in $\mathcal H$ and carry compatible actions of the complexified Lie algebra $\mathfrak g$ and $K$ . Admissibility supplies finite $K$ -multiplicities, while irreducibility supplies finite generation over $U(\mathfrak g)$ ; consequently $\mathcal H_K$ is an irreducible Harish–Chandra module. This passage replaces the Hilbert-space representation by a tractable algebraic core without discarding its infinitesimal and compact-subgroup data.

Hypotheses and scope

References

Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter VIII, especially Theorem 8.1.
Nolan R. Wallach, Real Reductive Groups I, Academic Press, 1988. Publisher record. Relevant: Chapter 3, §§3.3–3.5.