(g,K)-module - Imported Knowlpedia Content

Definition

Let $G$ be a real Lie group, $K\subseteq G$ a compact subgroup with Lie algebra $\mathfrak k_0$ , and $\mathfrak g=\operatorname{Lie}(G)\otimes_{\mathbb R}\mathbb C$ . A $(\mathfrak g,K)$ -module is a complex vector space $V$ with a $\mathfrak g$ -module structure and a $K$ -action such that every vector is $K$ -finite, the differential of the $K$ -action equals the restricted $\mathfrak k_0$ -action, and

k\cdot(X\cdot v)=(\operatorname{Ad}(k)X)\cdot(k\cdot v)

for $k\in K$ , $X\in\mathfrak g$ , and $v\in V$ . These conditions make the two actions infinitesimally and globally compatible.

Equivalent algebraic viewpoint

The $\mathfrak g$ -action extends uniquely to an action of the universal enveloping algebra $U(\mathfrak g)$ . The displayed compatibility then becomes

k\cdot(u\cdot v)=(\operatorname{Ad}(k)u)\cdot(k\cdot v) \qquad (u\in U(\mathfrak g)).

Thus a $(\mathfrak g,K)$ -module retains both differential operators coming from $G$ and the discrete decomposition into irreducible compact-group types. Wallach gives this definition and its basic algebraic consequences in §3.3.

Relation to group representations

For a suitable smooth representation of $G$ , its $K$ -finite smooth vectors inherit a $(\mathfrak g,K)$ -module structure by differentiation. Passing to this subspace discards the ambient topology but preserves much of the representation’s algebraic information. The reverse passage—from a $(\mathfrak g,K)$ -module to a topological representation of $G$ —is a globalization problem and is not part of the definition.

Conventions and scope

In the standard real-reductive setting, $K$ is a maximal compact subgroup and $\mathfrak g$ is the complexified Lie algebra. Some sources formulate the $K$ -action as an algebraic representation of a complex reductive group or impose connectedness assumptions. The local-finiteness and compatibility axioms above are the compact-group convention used in analytic representation theory.

References

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