Let
G be a real Lie group. Let \mathfrak{g} be its Lie algebra, and
K a maximal compact subgroup with Lie algebra \mathfrak{k}. A (\mathfrak{g},K)-module is defined as follows: it is a
vector space V that is both a
Lie algebra representation of \mathfrak{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 ∈ \mathfrak{g} ::k\cdot (X\cdot v)=(\operatorname{Ad}(k)X)\cdot (k\cdot 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 ∈ \mathfrak{k} ::\left.\left(\frac{d}{dt}\exp(tY)\cdot v\right)\right|_{t=0}=Y\cdot v. In the above, the dot, \cdot, denotes both the action of \mathfrak{g} on
V and that of
K. The notation Ad(
k) denotes the
adjoint action of
G on \mathfrak{g}, and
Kv is the set of vectors k\cdot 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 \mathfrak{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 :kXv=kXk^{-1}kv=\left(kXk^{-1}\right)kv. In other words, it is a compatibility requirement among the actions of
K on
V, \mathfrak{g} on
V, and
K on \mathfrak{g}. The third condition is also a compatibility condition, this time between the action of \mathfrak{k} on
V viewed as a sub-Lie algebra of \mathfrak{g} and its action viewed as the differential of the action of
K on
V. ==Notes==