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
*G* and *K* is a maximal compact subgroup of *G*.

## Definition

Let *G* be a real Lie group. Let
*K* a maximal compact subgroup with Lie algebra
*V* that is both a Lie algebra representation of
*K* (without regard to the topology of *K*) satisfying the following three conditions

*v*∈

*V*,

*k*∈

*K*, and

*X*∈

*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*∈

In the above, the dot,
*V* and that of *K*. The notation Ad(*k*) denotes the adjoint action of *G* on
*Kv* is the set of vectors
*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
*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*,
*V*, and *K* on
*V* viewed as a sub-Lie algebra of
*K* on *V*.