(g,K)-module


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

Definition

Let G be a real Lie group. Let be its Lie algebra, and K a maximal compact subgroup with Lie algebra. A -module is defined as follows: it is a vector space V that is both a Lie algebra representation of and a group representation of K satisfying the following three conditions
In the above, the dot,, denotes both the action of on V and that of K. The notation Ad denotes the adjoint action of G on, and Kv is the set of vectors as k varies over all of K.
The first condition can be understood as follows: if G is the general linear group GL, then 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
In other words, it is a compatibility requirement among the actions of K on V, on V, and K on. The third condition is also a compatibility condition, this time between the action of on V viewed as a sub-Lie algebra of and its action viewed as the differential of the action of K on V.