# (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,^{[1]} 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*.^{[2]}

## 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:^{[3]} it is a vector space *V* that is both a Lie algebra representation of 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*∈ - 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*∈

In the above, the dot, , denotes both the action of on *V* and that of *K*. The notation Ad(*k*) 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(*n*, **R**), 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*.

## Notes

- ↑ Page 73 of Template:Harvnb
- ↑ Page 12 of Template:Harvnb
- ↑ This is James Lepowsky's more general definition, as given in section 3.3.1 of Template:Harvnb

## References

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