(g,K)-module

In mathematics, more specifically in the representation theory of reductive Lie groups, a $(𝔤,K)$-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 $(𝔤,K)$-modules, where $𝔤$ is the Lie algebra of G and K is a maximal compact subgroup of G.

Definition

Let G be a real Lie group. Let $𝔤$ be its Lie algebra, and K a maximal compact subgroup with Lie algebra $𝔨$. A $(𝔤,K)$-module is defined as follows: it is a vector space V that is both a Lie algebra representation of $𝔤$ 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 ∈ $𝔤$

$k\cdot (X\cdot v)=(\mathrm{Ad}(k)X)\cdot (k\cdot v)$

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 ∈ $𝔨$

${(\frac{d}{dt}\exp (tY)\cdot v)|}_{t=0}=Y\cdot v.$

In the above, the dot, $\cdot$, denotes both the action of $𝔤$ on V and that of K. The notation Ad(k) denotes the adjoint action of G on $𝔤$, 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 $𝔤$ 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

$kXv=kX{k}^{-1}kv=\left(kX{k}^{-1}\right)kv.$

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

References

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