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
is the Lie algebra of G and K is a maximal compact subgroup of G.[2] Let G be a real Lie group.
be its Lie algebra, and K a maximal compact subgroup with Lie algebra
-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 In the above, the dot,
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 In other words, it is a compatibility requirement among the actions of K on V,
on V viewed as a sub-Lie algebra of