(g,K)-module explained
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 ∈
k ⋅ (X ⋅ v)=(\operatorname{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 ∈
\exp(tY) ⋅ v\right)\right|t=0=Y ⋅ v.
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
kXv=kXk-1kv=\left(kXk-1\right)kv.
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 and References
- Page 73 of
- Page 12 of
- This is James Lepowsky's more general definition, as given in section 3.3.1 of
