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.
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
⋅
(
X
⋅
v
)
=
(
Ad
(
k
)
X
)
⋅
(
k
⋅
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
(
d
d
t
exp
(
t
Y
)
⋅
v
)
|
t
=
0
=
Y
⋅
v
.
In the above, the dot,
⋅
, 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
⋅
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
k
X
v
=
k
X
k
−
1
k
v
=
(
k
X
k
−
1
)
k
v
.
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.
