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In [[mathematics]], more specifically in the [[Representation theory of Lie groups|representation theory]] of [[reductive Lie group]]s, a <math>(\mathfrak{g},K)</math>'''-module''' is an algebraic object, first introduced by [[Harish-Chandra]],<ref>Page 73 of {{harvnb|Wallach|1988}}</ref> used to deal with continuous infinite-dimensional representations using algebraic techniques. Harish-Chandra showed that the study of [[Unitary representation|irreducible unitary representations]] of a real reductive [[Lie group]], ''G'', could be reduced to the study of irreducible <math>(\mathfrak{g},K)</math>-modules, where <math>\mathfrak{g}</math> is the [[Lie algebra]] of ''G'' and ''K'' is a [[maximal compact subgroup]] of ''G''.<ref>Page 12 of {{harvnb|Doran|Varadarajan|2000}}</ref>

==Definition==
Let ''G'' be a real Lie group. Let <math>\mathfrak{g}</math> be its Lie algebra, and ''K'' a maximal compact subgroup with Lie algebra <math>\mathfrak{k}</math>. A <math>(\mathfrak{g},K)</math>-module is defined as follows:<ref>This is James Lepowsky's more general definition, as given in section 3.3.1 of {{harvnb|Wallach|1988}}</ref> it is a [[vector space]] ''V'' that is both a [[Lie algebra representation]] of <math>\mathfrak{g}</math> and a [[group representation]] of ''K'' (without regard to the [[topological space|topology]] of ''K'') satisfying the following three conditions
:1. for any ''v'' ∈ ''V'', ''k'' ∈ ''K'', and ''X'' ∈ <math>\mathfrak{g}</math>
::<math>k\cdot (X\cdot v)=(\operatorname{Ad}(k)X)\cdot (k\cdot v)</math>
: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'' ∈ <math>\mathfrak{k}</math>
::<math>\left.\left(\frac{d}{dt}\exp(tY)\cdot v\right)\right|_{t=0}=Y\cdot v.</math>
In the above, the dot, <math>\cdot</math>, denotes both the action of <math>\mathfrak{g}</math> on ''V'' and that of ''K''. The notation Ad(''k'') denotes the [[adjoint representation of a Lie group|adjoint action]] of ''G'' on <math>\mathfrak{g}</math>, and ''Kv'' is the set of vectors <math>k\cdot v</math> 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 <math>\mathfrak{g}</math> is the algebra of all ''n'' by ''n'' matrices, and the adjoint action of ''k'' on ''X'' is ''kXk''<sup>−1</sup>; condition 1 can then be read as
:<math>kXv=kXk^{-1}kv=\left(kXk^{-1}\right)kv.</math>
In other words, it is a compatibility requirement among the actions of ''K'' on ''V'', <math>\mathfrak{g}</math> on ''V'', and ''K'' on <math>\mathfrak{g}</math>. The third condition is also a compatibility condition, this time between the action of <math>\mathfrak{k}</math> on ''V'' viewed as a sub-Lie algebra of <math>\mathfrak{g}</math> and its action viewed as the differential of the action of ''K'' on ''V''.

==Notes==
{{Reflist}}

==References==
*{{Citation
| editor1-last=Doran
| editor1-first=Robert S.
| editor2-last=Varadarajan
| editor2-first=V. S.
| title=The mathematical legacy of Harish-Chandra
| publisher=[[American Mathematical Society|AMS]]
| series=Proceedings of Symposia in Pure Mathematics
| volume=68
| year=2000
| mr=1767886
| isbn=978-0-8218-1197-9
}}
*{{Citation
| last=Wallach
| first=Nolan R.
| title=Real reductive groups I
| year=1988
| publisher=Academic Press
| series=Pure and Applied Mathematics
| volume=132
| mr=0929683
| isbn=978-0-12-732960-4
| url-access=registration
| url=https://archive.org/details/realreductivegro0000wall
}}

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[[Category:Representation theory of Lie groups]]