Admissible representation of a real reductive group

Definition

Let $G$ be a real reductive group and $K\subseteq G$ a maximal compact subgroup. A representation $V$ in the algebraic or continuous real-reductive category is admissible when every irreducible representation $\tau$ of $K$ occurs in its $K$ -finite part $V_K$ with finite multiplicity:

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

If $V$ is already an algebraic $K$ -module, take $V_K=V$ . For a continuous unitary Hilbert representation, this says that every $K$ -type isotypic component is finite-dimensional. The definition allows infinitely many distinct $K$ -types and applies equally to nonunitary globalizations and $(\mathfrak g,K)$ -modules.

Algebraic formulation

For a $(\mathfrak g,K)$ -module the core condition is imposed directly on its algebraic $K$ -decomposition. This is the form used in Harish–Chandra theory. If the module is also finitely generated over the universal enveloping algebra $U(\mathfrak g)$ , it is a Harish–Chandra module. The distinction matters: admissibility alone does not assert finite generation or finite length.

Harish–Chandra's finiteness theorem

For real reductive groups in the standard Harish–Chandra class, irreducible unitary representations are admissible. This deep theorem converts an infinite-dimensional Hilbert representation into a discrete family of finite-dimensional $K$ -isotypic pieces; it is developed in Wallach, §3.4 and Knapp, Chapter VIII. Admissibility of a reducible representation is not automatic: an infinite Hilbert direct sum of the trivial representation has infinite multiplicity for the trivial $K$ -type.

Conventions and scope

Some sources define admissibility for continuous representations on complete locally convex spaces rather than only Hilbert spaces. The invariant content is finite $K$ -multiplicity, together with enough regularity for the restriction to $K$ to decompose. Since maximal compact subgroups are conjugate, admissibility does not depend on the chosen $K$ , up to the corresponding transported types.

References

Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. DOI record. Relevant: Chapter VIII, “Admissible Representations.”
Nolan R. Wallach, Real Reductive Groups I, Academic Press, 1988. Publisher record. Relevant: Chapter 3, §§3.3–3.4.