(g,K)-module

In mathematics, more specifically in the representation theory of reductive Lie groups, a (\mathfrak{g},K)-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 (\mathfrak{g},K)-modules, where \mathfrak{g} 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 \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:[3] 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

  1. ↑ Page 73 of Wallach 1988
  2. ↑ Page 12 of Doran & Varadarajan 2000
  3. ↑ This is James Lepowsky's more general definition, as given in section 3.3.1 of Wallach 1988

References