(g,K)-module

In mathematics, more specifically in the representation theory of reductive Lie groups, a $$(\mathfrak{g},K)$$-module is an algebraic object, first introduced by Harish-Chandra, used to deal with continuous infinite-dimensional representations using algebraic techniques. Harish-Chandra showed that the study of irreducible unitary representations of a real reductive Lie group, G, could be reduced to the study of irreducible $$(\mathfrak{g},K)$$-modules, where $$\mathfrak{g}$$ is the Lie algebra of G and K is a maximal compact subgroup of G.

Definition
Let G be a real Lie group. Let $$\mathfrak{g}$$ be its Lie algebra, and K a maximal compact subgroup with Lie algebra $$\mathfrak{k}$$. A $$(\mathfrak{g},K)$$-module is defined as follows: it is a vector space V that is both a Lie algebra representation of $$\mathfrak{g}$$ and a group representation of K (without regard to the topology of K) satisfying the following three conditions
 * 1. for any v ∈ V, k ∈ K, and X ∈ $$\mathfrak{g}$$
 * $$k\cdot (X\cdot v)=(\operatorname{Ad}(k)X)\cdot (k\cdot v)$$
 * 2. for any v ∈ V, Kv spans a finite-dimensional subspace of V on which the action of K is continuous
 * 3. for any v ∈ V and Y ∈ $$\mathfrak{k}$$
 * $$\left.\left(\frac{d}{dt}\exp(tY)\cdot v\right)\right|_{t=0}=Y\cdot v.$$

In the above, the dot, $$\cdot$$, denotes both the action of $$\mathfrak{g}$$ on V and that of K. The notation Ad(k) denotes the adjoint action of G on $$\mathfrak{g}$$, and Kv is the set of vectors $$k\cdot v$$ as k varies over all of K.

The first condition can be understood as follows: if G is the general linear group GL(n, R), then $$\mathfrak{g}$$ is the algebra of all n by n matrices, and the adjoint action of k on X is kXk−1; condition 1 can then be read as
 * $$kXv=kXk^{-1}kv=\left(kXk^{-1}\right)kv.$$

In other words, it is a compatibility requirement among the actions of K on V, $$\mathfrak{g}$$ on V, and K on $$\mathfrak{g}$$. The third condition is also a compatibility condition, this time between the action of $$\mathfrak{k}$$ on V viewed as a sub-Lie algebra of $$\mathfrak{g}$$ and its action viewed as the differential of the action of K on V.