Harish–Chandra module - Imported Knowlpedia Content

Definition

Let $G$ be a real reductive group, $K$ a maximal compact subgroup, and $\mathfrak g=\operatorname{Lie}(G)\otimes_{\mathbb R}\mathbb C$ . A Harish–Chandra module is a $(\mathfrak g,K)$ -module $V$ that is admissible and finitely generated as a module over the universal enveloping algebra $U(\mathfrak g)$ . Explicitly, every irreducible $K$ -representation occurs in $V$ with finite multiplicity, and finitely many vectors generate $V$ under $U(\mathfrak g)$ . These are independent finiteness requirements: neither is omitted from the standard definition used in real-reductive representation theory.

From group representations

If $\pi$ is an irreducible admissible representation of $G$ , its $K$ -finite vectors carry the differentiated $\mathfrak g$ -action and form a Harish–Chandra module. For an irreducible unitary representation, admissibility follows from Harish–Chandra’s finiteness theorem, and the resulting module is irreducible. This passage preserves the $K$ -type multiplicities and infinitesimal action while forgetting the Hilbert topology Wallach, §§3.3–3.5.

Structural consequences

Harish–Chandra modules have finite length. Their irreducible subquotients are again Harish–Chandra modules, and the center of $U(\mathfrak g)$ acts locally finitely. These finiteness properties make algebraic tools—central characters, composition series, homological functors, and induction—available for studying representations of noncompact groups; see Knapp–Vogan, Chapters I–II.

Conventions and scope

Some authors reserve “Harish–Chandra module” for a module obtained as the $K$ -finite vectors of a particular globalization; others use the intrinsic two-condition definition above. A bare $(\mathfrak g,K)$ -module need not be Harish–Chandra. Likewise, an admissible module with no finite set of $U(\mathfrak g)$ -generators is a decisive near-miss.

References

Nolan R. Wallach, Real Reductive Groups I, Academic Press, 1988. Publisher record. Relevant: Chapter 3, §§3.3–3.5.
Anthony W. Knapp and David A. Vogan Jr., Cohomological Induction and Unitary Representations, Princeton University Press, 1995. Publisher record. Relevant: Chapters I–II, especially the category $C(\mathfrak g,K)$ .