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 continuous3. 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.
